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2020JCAP...06..044D

Primordial black holes confront LIGO/Virgo data: current situation

2020-01-01

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The LIGO and Virgo Interferometers have so far provided 11 gravitational-wave (GW) observations of black-hole binaries. Similar detections are bound to become very frequent in the near future. With the current and upcoming wealth of data, it is possible to confront specific formation models with observations. We investigate here whether current data are compatible with the hypothesis that LIGO/Virgo black holes are of primordial origin. We compute in detail the mass and spin distributions of primordial black holes (PBHs), their merger rates, the stochastic background of unresolved coalescences, and confront them with current data from the first two observational runs, also including the recently discovered GW190412. We compute the best-fit values for the parameters of the PBH mass distribution at formation that are compatible with current GW data. In all cases, the maximum fraction of PBHs in dark matter is constrained by these observations to be f<SUB> PBH</SUB>≈ few× 10<SUP>-3</SUP>. We discuss the predictions of the PBH scenario that can be directly tested as new data become available. In the most likely formation scenarios where PBHs are born with negligible spin, the fact that at least one of the components of GW190412 is moderately spinning is incompatible with a primordial origin for this event, unless accretion or hierarchical mergers are significant. In the absence of accretion, current non-GW constraints already exclude that LIGO/Virgo events are all of primordial origin, whereas in the presence of accretion the GW bounds on the PBH abundance are the most stringent ones in the relevant mass range. A strong phase of accretion during the cosmic history would favour mass ratios close to unity, and a redshift-dependent correlation between high masses, high spins and nearly-equal mass binaries, with the secondary component spinning faster than the primary. Finally, we highlight that accretion can play an important role to relax current constraints on the PBH abundance, which calls for a better modelling of the mass and angular momentum accretion rates at redshift 0zlesssim3.

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https://arxiv.org/pdf/2005.05641.pdf

{'V. De Luca, a G. Franciolini, a,b P. Pani, c,d A. Riotto a,d': "- a Department of Theoretical Physics and Center for Astroparticle Physics (CAP) \n24 quai E. Ansermet, CH-1211 Geneva 4, Switzerland \n- b Instituto de F'ısica Te'orica UAM-CSIC, Universidad Aut'onoma de Madrid, Cantoblanco, Madrid, 28049 Spain\n- c Dipartimento di Fisica, 'Sapienza' Universit'a di Roma, Piazzale Aldo Moro 5, 00185, Roma, Italy \nd INFN, Sezione di Roma, Piazzale Aldo Moro 2, 00185, Roma, Italy \nE-mail: [email protected], [email protected], [email protected], [email protected] \nAbstract. The LIGO and Virgo Interferometers have so far provided 11 gravitational-wave (GW) observations of black-hole binaries. Similar detections are bound to become very frequent in the near future. With the current and upcoming wealth of data, it is possible to confront specific formation models with observations. We investigate here whether current data are compatible with the hypothesis that LIGO/Virgo black holes are of primordial origin. We compute in detail the mass and spin distributions of primordial black holes (PBHs), their merger rates, the stochastic background of unresolved coalescences, and confront them with current data from the first two observational runs, also including the recently discovered GW190412. We compute the best-fit values for the parameters of the PBH mass distribution at formation that are compatible with current GW data. In all cases, the maximum fraction of PBHs in dark matter is constrained by these observations to be f PBH ≈ few × 10 -3 . We discuss the predictions of the PBH scenario that can be directly tested as new data become available. In the most likely formation scenarios where PBHs are born with negligible spin, the fact that at least one of the components of GW190412 is moderately spinning is incompatible with a primordial origin for this event, unless accretion or hierarchical mergers are significant. In the absence of accretion, current non-GW constraints already exclude that LIGO/Virgo events are all of primordial origin, whereas in the presence of accretion the GW bounds on the PBH abundance are the most stringent ones in the relevant mass range. A strong phase of accretion during the cosmic history would favour mass ratios close to unity, and a redshift-dependent correlation between high masses, high spins and nearlyequal mass binaries, with the secondary component spinning faster than the primary. Finally, we highlight that accretion can play an important role to relax current constraints on the PBH abundance, which calls for a better modelling of the mass and angular momentum accretion rates at redshift z glyph[lessorsimilar] 30.", 'Contents': '| 1 Introduction | 1 Introduction | 1 Introduction | 1 |\n|------------------|-----------------------------------------------------------------|----------------------------------------------------------------------------------------------|----------------------------------------------------------------------------------------------|\n| 2 | The masses and spins of PBHs | The masses and spins of PBHs | 3 |\n| | 2.1 | Mass distribution at formation | 3 |\n| | 2.2 | Spin distribution at formation 3 | Spin distribution at formation 3 |\n| 2.3 | | The role of accretion | 4 |\n| | 2.3.1 | Accretion onto isolated PBHs | 4 |\n| | 2.3.2 | Accretion onto binary PBHs | 5 |\n| | 2.3.3 | Effects on the mass function and PBH abundance | 8 |\n| | 2.3.4 | Effects on the PBH spins | 9 |\n| 2.4 | Summary: theoretical distributions of the PBH binary parameters | Summary: theoretical distributions of the PBH binary parameters | 11 |\n| | | 2.4.1 Mass evolution | 11 |\n| | 2.4.2 | Spin evolution | 13 |\n| 2.5 | Limitations of the accretion model | Limitations of the accretion model | 14 |\n| 3 | Binary evolution | Binary evolution | 16 |\n| | 3.1 | GW-driven evolution | 17 |\n| | 3.2 | Accretion-driven evolution | 18 |\n| | PBH merger rates and phenomenology | PBH merger rates and phenomenology | |\n| 4 | | | 19 |\n| 4.1 | | Merger rates without accretion | 20 |\n| 4.2 | Merger rates with accretion | Merger rates with accretion | 20 |\n| | 4.3.1 | Likelihood analysis for GW observations without accretion | |\n| | 4.3 Phenomenology of PBH mergers without accretion | | 22 |\n| | 4.3.2 | Number of events and stochastic GW background without accretion | 24 |\n| 4.4 | Phenomenology of PBH mergers with accretion | Phenomenology of PBH mergers with accretion | 24 |\n| | 4.4.1 | Likelihood analysis for GWs observations with accretion | 24 |\n| | 4.4.2 | Number of events and GWs abundance with accretion | 24 |\n| 5 | Confrontation with LIGO/Virgo O1, O2, and GW190412 | Confrontation with LIGO/Virgo O1, O2, and GW190412 | 25 |\n| | 5.1 | Best-fit parameters for the PBH mass function 25 | Best-fit parameters for the PBH mass function 25 |\n| | 5.2 | Updated constraints on PBH abundance | 27 |\n| | 5.3 | Confrontation of the predicted distributions of the binary parameters with obser- vations 28 | Confrontation of the predicted distributions of the binary parameters with obser- vations 28 |', '1 Introduction': "Thanks to the current available measurements of the Gravitational Waves (GWs) from Black Holes (BHs) mergers in the observational runs O1 and O2 [1] and the more recent ongoing third phase [2] by the LIGO/Virgo collaborations, we have entered the era of GW astronomy. One of the most fundamental questions such observations raise is the nature of the BHs [3]. One fascinating hypothesis is that the merging BHs are of primordial origin, that is, they have formed early in the evolution of the universe (see Ref. [4] for a recent review). This possibility is also \ninteresting as Primordial Black Holes (PBHs) may comprise the totality or a fraction of the Dark Matter (DM) in the universe [5, 6]. \nIn order to assess if the merger events may be ascribed to PBHs one needs to go through various steps: \n- 1. First, one starts from a given mass function distribution (which might be theoretically justified by a given PBH formation mechanism). Such mass function is determined by a set of parameters (typically two of them) parametrising the characteristic PBH mass scale and the width of the distribution. Their central values may be estimated from observations by requiring that all GW events detected so far (or a fraction thereof) are explained by PBHs. This requires fitting the key observables, namely the merger rate, the BH masses, and the redshift at which the event is produced, resulting in the best-fit value for the PBH abundance f PBH in units of the DM one.\n- 2. Secondly, one has to check if the resulting model parameters and f PBH are compatible with the current constraints from other observations (e.g. lensing and CMB distortion) [7]. This second step is necessary to check which fraction of the PBHs may form the DM and, above all, to see if the observed events are compatible with the PBH scenario, or if their primordial origin is already excluded by other observations.\n- 3. Thirdly, one can confront the theoretical predictions obtained through the PBH hypothesis of some key quantities, e.g. the binary chirp masses, mass ratios, spins, with the observed values, thus assessing whether the predictions are compatible or in tension with observations. \nThe goal of this paper is to explore if the first two observational LIGO/Virgo runs - also including the event GW190412 recently discovered in O3 - are compatible with the hypothesis that the BHs are of primordial origin. In particular, we calculate the mass and spin distributions of PBHs, their merger rates, the stochastic background of unresolved coalescences, and confront them with current data. \nOne particularly relevant phenomenon to take into account when performing such analysis is PBH mass accretion. Indeed, PBHs may accrete efficiently during the cosmic history, see for instance [8-10]. First of all accretion changes the PBH masses, their mass functions, and their abundances. This is relevant when analysing the constraints at the present epoch for PBHs with masses larger than a few solar masses [11]. Furthermore, accretion may strongly influence the PBH merger rates, their final spins [12], as well as their mass ratios in the binaries. This is particularly important not only because the binary masses and effective spin parameter χ eff can be measured, but also because the recent GW190412 [2] and future events will provide fundamental information on the individual BH spin and on mass ratio distributions with a large hierarchy between the BH masses. \nThe paper is organised as follows. In Section 2 we discuss the masses and the spins of the PBHs including the accretion phenomenon. In Section 3 we provide details about the binary evolution, while Section 4 is devoted to PBH merger rates. Section 5 represents the main bulk of our paper as it contains the comparison of the recent LIGO/Virgo data with the theoretical predictions. Finally, we conclude in Section 6 by providing a list of our main findings that can be directly tested with current and future GW observations. \nA final note about notation. We are going to use the label 'i' for the quantities at formation time, to distinguish from those at the time of coalescence. Final values, e.g. PBH masses, will \ncarry no label. This distinction is relevant when accretion takes place. We use G = c = 1 units throughout.", '2 The masses and spins of PBHs': 'In this section we discuss the theoretical predictions for the masses and spins of binary BH components in the case the latter are of primordial origin. We consider two situations: a) accretion is negligible, hence the masses and spins of the binary components are those at formation; b) baryonic mass accretion is modelled through the cosmic history, hence the masses and spins of isolated and binary PBHs are different from those at formation. In the latter case also the mass distribution and the PBH abundance are affected by accretion. \nThe reader interested in the final results can skip the details of the modelling and jump directly to Sec. 2.4, where a summary of the theoretical predictions is presented. The limitations of the accretion model are discussed in Sec. 2.5.', '2.1 Mass distribution at formation': 'One can consider several initial shapes for the mass function at high redshift, depending on the details of the PBH formation mechanism. In Ref. [12] we considered a critical, spiky, lognormal, and power-law mass function. Since the first two cases are unrealistic, we restrict here to the latter two distributions. A distribution, motivated by the collapse of scale invariant perturbations, is given by a power-law mass function [13-16] \nψ ( M,z i ) = 1 2 ( M -1 2 min -M -1 2 max ) -1 M -3 2 . (2.1) \nThis is described by two free parameters, M min and M max . An alternative, and maybe more popular, mass function is the lognormal one \nψ ( M,z i ) = 1 √ 2 πσM exp ( -log 2 ( M/M c ) 2 σ 2 ) (2.2) \nexpressed again by two parameters, the width σ and the peak reference mass M c , first introduced in [17]. It represents a frequent parametrisation for the cases of a PBH population arising from a symmetric peak in the primordial power spectrum, see for example Ref. [18].', '2.2 Spin distribution at formation': 'The requirement that the cosmological abundance of PBHs is less than the DM abundance sets a bound on the PBHs mass fraction, which in turn requires the collapse of density perturbations generating a PBH to be a rare event. Applying the formalism of peak theory [19] in standard formation scenarios [20-24], one finds that high (and rare) peaks in the density contrast, which eventually collapse to form PBHs, are primarily spherical. However, at first order in perturbation theory, the presence of small departures from spherical symmetry introduces torques induced by the surrounding matter perturbations. This leads to the generation of a small angular momentum before collapse. Due to the small time scales which characterise the overdensity collapse, the action of the torque moments is indeed limited in time. \nThe estimated PBH spin at formation is [25] \nχ i = Ω m π σ δ √ 1 -γ 2 ∼ 10 -2 √ 1 -γ 2 , (2.3) \nwhere Ω m ∼ 0 . 3 is the current DM abundance, σ δ is the variance of the density perturbations at the horizon crossing time, and γ parametrises the shape of the power spectrum of the density perturbations in terms of its variances (being γ ∼ 1 for very narrow power spectra). The suppression factor due to γ arises because, as γ approaches unity, the velocity shear tends to be more strongly aligned with the inertia tensor. Thus, the initial spin of PBHs is expected to be below the percent level (see also Ref. [26]). Non-standard scenarios for the PBH formation, for instance during an early matter-dominated epoch [27] following inflation or from the collapse of Q-balls [28], may lead to higher values of the initial spin.', '2.3 The role of accretion': 'In this section we discuss the manifold roles of accretion onto PBHs during the cosmic history, reviewing and extending the analysis presented in previous work. Accretion can affect both the mass and spin of isolated and binary PBHs [12]. 1 For the latter, it can also affect the binary evolution before GW emission becomes dominant (see Sec. 3.2 below). Furthermore, accretion modifies the mass distribution of PBHs and the fraction of PBHs in DM in a redshift-dependent fashion [11].', '2.3.1 Accretion onto isolated PBHs': 'We model gas accretion onto an isolated PBH with mass M , moving with a relative velocity v rel with respect to the surrounding gas, through the Bondi-Hoyle mass accretion rate [8, 9, 32] \n˙ M B = 4 πλm H n gas v eff r 2 B (2.4) \nwhere v eff = √ v 2 rel + c 2 s is the effective velocity, c s is the speed of sound, and the gas number density is n gas glyph[similarequal] 200(1 + z/ 1000) 3 cm -3 . The Bondi-Hoyle radius reads \nr B ≡ M v 2 eff glyph[similarequal] 1 . 3 × 10 -4 ( M M glyph[circledot] ) ( v eff 5 . 7 kms -1 ) -2 pc . (2.5) \nFor a gas in equilibrium at the temperature of the intergalactic medium, \nc s glyph[similarequal] 5 . 7 ( 1 + z 1000 ) 1 / 2 [ ( 1 + z dec 1 + z ) β +1 ] -1 / 2 β kms -1 , (2.6) \nwith β = 1 . 72, and z dec glyph[similarequal] 130 being the redshift at which the baryonic matter decouples from the radiation fluid. The accretion parameter λ appearing in Eq. (2.4) keeps into account the effects of the Hubble expansion, the coupling of the CMB radiation to the gas through Compton scattering, and the gas viscosity [8]. The main formulas to compute λ are summarized in Appendix B of Ref. [12]. \nCurrent observational constraints imply that PBHs with masses larger than O ( M glyph[circledot] ) can comprise only a fraction of the DM [7]. Thus, accretion onto PBHs should include the presence of an additional DM halo. While direct DM accretion onto the PBH is negligible [8, 10], the halo acts as a catalyst enhancing the gas accretion rate. The DM halo has a typical spherical density profile ρ ∝ r -α (with approximately α glyph[similarequal] 2 . 25 [33, 34]), truncated at a radius r h glyph[similarequal] 0 . 019 pc( M/M glyph[circledot] ) 1 / 3 (1 + z/ 1000) -1 and with total mass \nM h ( z ) = 3 M ( 1 + z 1000 ) -1 . (2.7) \nFigure 1 . Schematic illustration of the relevant scales involved in the accretion process for a PBH binary system. The Bondi radius of the binary is much bigger than the orbital separation. \n<!-- image --> \nThe mass M h grows with time as long as the PBHs are isolated and eventually stops when all the available DM has been accreted, i.e. approximately when 3 f PBH (1 + z/ 1000) -1 = 1. In the presence of a DM halo, the mass entering in the Bondi-Hoyle formula (2.4) is M h , and this enhances the accretion rate. The halo extends much more than the Bondi radius ( r h glyph[greatermuch] r B ) and this effect is taken into account by the accretion parameter λ [8]. \nIt is customary to define the dimensionless accretion rate normalised to the Eddington one \n˙ m = ˙ M B ˙ M Edd with ˙ M Edd = 1 . 44 × 10 17 ( M M glyph[circledot] ) g s -1 , (2.8) \nwhose behaviour as a function of the redshift and PBH mass can be found in Refs. [8, 12]. It is noteworthy that ˙ m can be larger than unity (i.e., accretion can be super-Eddington) for z ∼ O (30) and for the masses of interest for this work. \nThe typical accretion time scale is given in terms of the Salpeter time by \nτ acc ≡ τ Salp ˙ m = σ T 4 πm p 1 ˙ m = 4 . 5 × 10 8 yr ˙ m , (2.9) \nwhere σ T is the Thompson cross section and m p is the proton mass. For z glyph[lessorsimilar] 100, τ acc is smaller than the typical age of the universe. Accretion can therefore play an important role in the mass evolution of PBHs [9, 12]. \nThe accretion rate (2.4) depends significantly on the relative velocity between the PBHs and the baryonic matter, and it is therefore sensitive to several physical processes that might increase the PBH characteristic velocities or the speed of sound in the gas, in turn reducing the accretion rate. We discuss this point in Sec. 2.5 below. Given the large uncertainties in the modelling of the accretion rate at relatively small redshift, here we shall adopt an agnostic view and consider several cut-off values z cut-off = (15 , 10 , 7), below which accretion is negligible.', '2.3.2 Accretion onto binary PBHs': 'In order to study the accretion onto binary PBHs, one has to take into account both global accretion processes (i.e., of the binary as a whole) and local accretion processes (i.e., onto the \nindividual components of the binary). Here we extend the analysis of Ref. [12] to account for generic mass ratios and eccentric orbits. We consider a binary of total mass M tot = M 1 + M 2 , reduced mass µ = M 1 M 2 / ( M 1 + M 2 ), mass ratio q = M 2 /M 1 ≤ 1, semi-major axis a , and eccentricity e . The situation is schematically illustrated in Fig. 1 and described below. \nTwo distinctive regimes occur depending on how the binary separation compares with the Bondi radius of the binary, \nr bin B = M tot v 2 eff , (2.10) \nwhere the effective velocity v eff = √ c 2 s + v 2 rel depends on the velocity v rel of the center of mass of the binary relative to the surrounding gas with mean cosmic density n gas . Indeed, if a glyph[greatermuch] r bin B , accretion occurs onto the two individual PBHs independently (each one moving at a characteristic velocity that also depends on the orbital one), as discussed in the previous section. However, as the binary hardens, the orbital separation becomes much smaller than the Bondi radius of the binary ( a glyph[lessmuch] r bin B ) and, as we shall discuss, this occurs much before GW emission becomes the dominant driving mechanism of the inspiral. In this case, accretion occurs on the binary as a whole at rate given by \n˙ M bin = 4 πλm H n gas v -3 eff M 2 tot . (2.11) \nWenow evaluate how the two binary components accrete in this configuration. The PBH positions and velocities with respect to the center of mass are given by [35] \nr 1 = q 1 + q r, v 1 = q 1 + q v ; r 2 = 1 1 + q r, v 2 = 1 1 + q v (2.12) \nin terms of their relative distance and velocity \nr = a (1 -e cos u ) , v = √ M tot ( 2 r -1 a ) , (2.13) \nboth expressed as a function of the semi-major axis a , eccentricity e , and angle u . The time evolution of the latter is implicitly given by √ a 3 /M tot ( u ( t ) -e sin u ( t )) = t -T , where T is an integration constant [35]. The PBH effective velocities with respect to the gas are given by \nv eff,1 = √ v 2 eff + v 2 1 , v eff,2 = √ v 2 eff + v 2 2 . (2.14) \nSince the Bondi radius of the binary is much bigger than the typical semi-axis of the binary (see Fig. 1), the total infalling flow of baryons towards the binary is constant, i.e. \n4 πm H n gas ( R ) v ff ( R ) R 2 = const = ˙ M bin (2.15) \nwhere the free fall velocity of the gas, v ff , is computed by assuming that at large distances, R ∼ r bin B , it reduces to the usual effective velocity v eff , i.e. \nv ff ( R ) = √ v 2 eff + 2 M tot R -2 M tot r bin B , (2.16) \nwhile n gas ( R ) identifies the density profile at a distance R from the center of mass of the binary, \nn gas ( R ) = ˙ M bin 4 πm H v ff ( R ) R 2 . (2.17) \nIn other words, being the infalling flow of baryons constant, their density near the binary increases relative to its mean cosmic value at the Bondi radius of the binary. \nThe accretion rates for the single components of the binary are then given in terms of the effective Bondi radii 2 of the binary components r B,i = M i /v 2 eff,i as \n˙ M 1 = 4 πm H n gas ( r B,1 ) v -3 eff,1 M 2 1 , ˙ M 2 = 4 πm H n gas ( r B,2 ) v -3 eff,2 M 2 2 . (2.18) \nNote that for the single accretion rates we considered two naked PBHs, for which the parameter λ ≈ 1 at low redshift, while we have described the binary with a dark halo clothing with parameter λ which takes into account the ratio of the Bondi radius of the binary with respect to the dark halo radius [9], as discussed above. \nPutting together the above formulas, the accretion rates (2.18) can be written in terms of the orbital parameters in a cumbersome (albeit analytical) form, \n˙ M 1 = ˙ M bin √ 1 + ζ +(1 -ζ ) γ 2 2(1 + ζ )(1 + q ) + (1 -ζ )(1 + 2 q ) γ 2 , (2.19) \n˙ M 2 = ˙ M bin √ (1 + ζ ) q +(1 -ζ ) q 3 γ 2 2(1 + ζ )(1 + q ) + (1 -ζ ) q 2 (2 + q ) γ 2 , (2.20) \nwhere we defined ζ = e cos u and γ 2 = av 2 eff /µq . The q → 0 limit is particularly simple and does not depend on e nor u , \n˙ M 1 = ˙ M bin + O ( q ) , ˙ M 2 = √ q 2 ˙ M bin + O ( q 3 / 2 ) . (2.21) \nIn the general case, since the orbital period is much smaller than the accretion time scale τ acc , we can average over the angle u and eliminate the explicit time dependence in Eqs. (2.19) and (2.20). After this averaging procedure the dependence on the eccentricity is negligible. Therefore, we can finally write \n˙ M 1 = ˙ M bin √ M 1 q 2 + a (1 + q ) v 2 eff (1 + q )[2 M 1 q 2 + a (1 + 2 q ) v 2 eff ] , ˙ M 2 = ˙ M bin √ q [ M 1 + a (1 + q ) v 2 eff ] (1 + q )[2 M 1 + a (2 + q ) v 2 eff ] . (2.22) \nThe above rates can be even further simplified when \nM 1 q 2 glyph[greatermuch] av 2 eff , (2.23) \nwhich is satisfied for M 1 ∼ O ( M glyph[circledot] ) and a ∼ O (10 6 ) M 1 for any q > 10 -2 . In this case, the previous expressions reduce to \n˙ M 1 = ˙ M bin 1 √ 2(1 + q ) , ˙ M 2 = ˙ M bin √ q √ 2(1 + q ) . (2.24) \nNote that the expected behaviour ˙ M 1 = ˙ M 2 = ˙ M bin / 2 is recovered in the limit q → 1. \nAs can be checked a posteriori, we will always be interested in a regime in which Eq. (2.24) is an excellent approximation. In the following we shall therefore use these simplified formulas, which have the great advantage to be independent of the orbital parameters. In other words, knowing ˙ M bin and the initial mass ratio, Eq. (2.24) provides the time evolution of the masses of the binary components, regardless of the orbital evolution. \nIn terms of the Eddington normalised rates, ˙ m i = τ Salp ˙ M i /M i , one gets \n˙ m 1 = ˙ m bin √ 1 + q 2 , ˙ m 2 = ˙ m bin √ 1 + q 2 q . (2.25) \nThe evolution equation for the mass ratio is given by \n˙ q = q ( ˙ M 2 M 2 -˙ M 1 M 1 ) = q τ Salp ( ˙ m 2 -˙ m 1 ) . (2.26) \nThe above equation shows an important point that will be relevant in the following: if ˙ m 2 > ˙ m 1 , the growth rate of the mass ratio is positive, i.e. the mass ratio grows until it reaches a stationary point when q = 1 and ˙ m 1 = ˙ m 2 . From Eq. (2.25), it is clear that ˙ m 2 > ˙ m 1 in any case. This shows that accretion onto a binary PBH implies that the binary masses tend to balance each other on secular time scales.', '2.3.3 Effects on the mass function and PBH abundance': 'In addition to changing the masses and spins of PBHs, accretion also affects their mass distribution function, as well as their mass fraction relative to that of the DM, in a redshift-dependent fashion [11]. \nLet us define the mass function ψ ( M,z ) as the fraction of PBHs with mass in the interval ( M,M + d M ) at redshift z . For an initial ψ ( M i , z i ) at formation redshift z i , its evolution is governed by [11, 12] \nψ ( M ( M i , z ) , z )d M = ψ ( M i , z i )d M i (2.27) \nwhere M ( M i , z ) is the final mass at redshift z for a PBH with mass M i at redshift z i . We stress that - since the evolution of the mass function is needed to re-weight existing constraints on the PBH abundance and since the latter are mainly due to isolated PBHs - for the evolution of the mass function we have considered the evolution of the mass of a single BH; in this case we have to follow Sec. 2.3.1. The main effect of accretion on the mass distribution is to make the latter broader at high masses, producing a high-mass tail that can be orders of magnitude above its corresponding value at formation [11]. A representative example is shown in Fig. 2, where we compare the final mass function (after the accretion phase) with that at formation for some choices of the initial mass distribution. \nFinally, also the value of f PBH is affected by accretion. Assuming for simplicity a nonrelativistic dominant DM component (whose energy density scales as the inverse of the volume), it is easy to show that [11] \nf PBH ( z ) = 〈 M ( z ) 〉 〈 M ( z i ) 〉 ( f -1 PBH ( z i ) -1) + 〈 M ( z ) 〉 , (2.28) \nwhere we defined the average mass \n〈 M ( z ) 〉 = ∫ d MMψ ( M,z ) . (2.29) \nDue to the presence of accretion, f PBH ( z ) can be significantly larger than f PBH ( z i ), see an example in Fig. 3. \n<!-- image --> \nFigure 2 . Example of evolution of the power-law (left) and lognormal (right) mass functions. \n<!-- image --> \nFigure 3 . Example of evolution of the PBH abundance for both power-law and lognormal mass functions. \n<!-- image -->', '2.3.4 Effects on the PBH spins': "The physics of accretion is very complex, since the accretion rate and the geometry of the accretion flow are intertwined, and they are both crucial in determining the evolution of the PBH mass. In addition, the infalling accreting gas onto a PBH can carry angular momentum which crucially determines the geometry of the accreting flow, and the evolution of the PBH spin [12] (see also [36]). 3 \nConditions for a thin-disk formation in isolated and binary PBHs. For accretion onto an isolated BH, angular momentum transfer is relevant if the typical gas velocity (given by the baryon velocity variance [9]) is larger than the Keplerian velocity close to the PBH. This implies that the minimum PBH mass for which the accreting gas flow is non-spherical is [9, 12] \nM ∼ > 6 × 10 2 M glyph[circledot] D 1 . 17 ξ 4 . 33 ( z ) (1 + z/ 1000) 3 . 35 [ 1 + 0 . 031 (1 + z/ 1000) -1 . 72 ] 0 . 68 , (2.30) \nwhere ξ ( z ) = Max[1 , 〈 v eff 〉 /c s ] describes the effect of a (relatively small) PBH proper motion in reducing the Bondi radius, and the constant D ∼ O (1 ÷ 10) takes into account relativistic corrections. \nBesides the necessary condition (2.30), the geometry of the disk also depends on the accretion rate. If ˙ m < 1 and accretion is non-spherical, an advection-dominated accretion flow (ADAF) may form [41]. When ˙ m glyph[greaterorsimilar] 1, the non-spherical accretion can give rise to a geometrically thin accretion disk [42]. For ˙ m glyph[greatermuch] 1 the accretion luminosity might be strong enough that the disk 'puffs up' and becomes thicker. For simplicity, here we follow Ref. [9] and assume that a thin disk forms when Eq. (2.30) is satisfied and \n˙ m ∼ > 1 . (2.31) \nIn the regimes we are interested in, the latter condition is always more stringent than condition (2.30). Therefore, ˙ m ∼ > 1 can be considered as the sufficient condition for the formation of a thin disk around an isolated PBH [12]. \nIf instead accretion occurs onto a PBH binary, angular momentum transfer on each PBH is much more efficient [12]. In this case each binary component has a velocity of the order of the orbital velocity v . Although the Bondi radius of the individual PBHs is much smaller than the Bondi radius of the binary, it is still parametrically larger than the radius of the innermost stable circular orbit (ISCO, see Eq. (2.34) below). Therefore, the accretion onto the binary components is never spherical and a disk can form. Compared to the aforementioned case of an isolated PBH, in this case condition (2.30) is absent. \nTo summarize, for both isolated and binary PBHs we can assume that a thin accretion disk forms whenever ˙ m glyph[greaterorsimilar] 1 along the cosmic history. Furthermore, since ˙ m never exceeds unit significantly [12], the thin-disk approximation should be reliable in the super-Eddington regime of PBHs. \nEvolution of the spin. When the conditions for formation of a thin accretion disk are satisfied, i.e. Eq. (2.31), mass accretion is accompanied by an increase of the PBH spin. A thin accretion disk is located along the equatorial plane [42, 43] and therefore the PBH spin is aligned perpendicularly to the disk plane. In such a configuration, one can use a geodesic model to describe the angular-momentum accretion [44]. \nFor circular disk motion the rate of change of the magnitude J ≡ | glyph[vector] J | ≡ χM 2 of the PBH angular momentum is related to the mass accretion rate (see also Refs. [44-47]) \n˙ J = L ( M,J ) E ( M,J ) ˙ M, (2.32) \nwhere \nE ( M,J ) = √ 1 -2 M 3 r ISCO and L ( M,J ) = 2 M 3 √ 3 ( 1 + 2 √ 3 r ISCO M -2 ) , (2.33) \nand the ISCO radius reads \nr ISCO ( M,J ) = M [ 3 + Z 2 -√ (3 -Z 1 ) (3 + Z 1 +2 Z 2 ) ] , (2.34) \nwith Z 1 = 1 + ( 1 -χ 2 ) 1 / 3 [ (1 + χ ) 1 / 3 +(1 -χ ) 1 / 3 ] and Z 2 = √ 3 χ 2 + Z 2 1 . \nFinally, Eq. (2.32) can be re-arranged to describe the time evolution of the dimensionless Kerr parameter \n˙ χ = ( F ( χ ) -2 χ ) ˙ M M , (2.35) \nwhere we have defined the combination F ( χ ) ≡ L ( M,J ) /ME ( M,J ), which is only a function of χ . The above spin evolution equation predicts that the spin grows over a typical accretion time \n<!-- image --> \nFigure 4 . Left: Evolution of the mass of the primary for fixed values of the initial mass ratio. Right: the region in the ( q i -M i 1 ) plane (shaded areas) that allows the total final mass of the binary to be below M tot = 10 4 M glyph[circledot] . In both panels we considered three different choices of z cut-off . \n<!-- image --> \nscale until it reaches extremality, χ = 1. However, radiation effects limit the actual maximum value of the spin to χ max = 0 . 998 [45]. Magnetohydrodynamic simulations of accretion disks around Kerr BHs suggest that the maximum spin might be smaller, χ max glyph[similarequal] 0 . 9 [48]. However, this limit may not apply to geometrically thin disks. The geometrically thin-disk approximation is expected to be valid for each PBH when ˙ m i glyph[greaterorsimilar] 1. For larger values of the accretion rates, the disk might be geometrically thicker and angular momentum accretion might be less efficient. However, the spin evolution time scale does not change significantly in more realistic accretion models [48]. \nTo summarize, another key prediction of the PBH scenario is the fact that the spin of sufficiently massive PBHs (those that go through epochs of super-Eddington accretion during the cosmic history) should be highly spinning. Note that the condition (2.31) is more easily fulfilled by binary PBHs than by the isolated ones, since in the former case accretion is enhanced by the larger total mass of the binary.", '2.4 Summary: theoretical distributions of the PBH binary parameters': 'In this section we summarize the results for the theoretical distributions of the binary parameters obtained as previously discussed. For ease of notation, we shall denote by M j and χ j the final mass and spin of the j -th binary component; likewise q will be the final mass ratio of the binary. The initial mass and spin of the j -th binary component are denoted by M i j and χ i j whereas the initial mass ratio is denoted by q i . 4 Clearly, in the absence of accretion or hierarchical mergers, the initial and final quantities coincide. In this case the mass and spin distributions are those at formation, see Secs. 2.1 and 2.2.', '2.4.1 Mass evolution': 'As a representative example, in the left panel of Fig. 4 we show the evolution of the primary component mass of the binary for various choices of the initial mass ratio q i , and for three choices of the cut-off redshift. One can notice that the effect of accretion becomes important above initial masses ∼ 10 M glyph[circledot] and is stronger for larger masses and for (initially) nearly-equal mass binaries. Overall, the masses can increase by one or two orders of magnitude due to accretion. Since the \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFigure 5 . From top to bottom: final masses M 1 , M 2 , and final mass ratio q as a function of the initial masses ( M i 1 , M i 2 ) . We consider three accretion scenarios parametrised by z cut-off = 15 (left panels), z cut-off = 10 (middle panels), and z cut-off = 7 (right panels). \n<!-- image --> \nmerger frequency of binaries with total mass M tot glyph[greaterorsimilar] 10 4 M glyph[circledot] is certainly below the frequency band of current ground-based detectors, we can safely neglect all the binaries that are pushed above that value by accretion. In the right panel of Fig. 4 we show the region in the q i -M i 1 plane that allows the total final mass of the binary to be below M tot = 10 4 M glyph[circledot] . The parameter space outside the shaded areas would give rise to binaries with M tot > 10 4 M glyph[circledot] , which are irrelevant for our study. \nNonetheless, it is intriguing to note that, due to a strong accretion phase at z ≈ (10 ÷ 30), PBHs formed with M i glyph[greaterorsimilar] 20 -100 M glyph[circledot] , can have much larger masses when they are detected at small redshift. These objects would be natural candidates for intermediate-mass BHs, which are sources for third-generation ground-based detectors and especially for LISA. \nSince the initial binary parameters are unmeasurable, it is more relevant to analyse the dependence of the final masses M 1 , M 2 and their mass ratio q in terms of the initial masses M i 1 , M i 2 . This is done in Fig. 5 for an accretion evolution until the cut-off redshift z cut-off = (15 , 10 , 7) (from left to right columns). \nAs shown in Sec. 2.3.2, accretion onto a PBH binary system is such that the secondary body experiences a stronger (specific) accretion rate compared to the primary body. Since the large majority of PBH binaries that merge in the LIGO/Virgo band are formed before accretion is relevant [4], it is expected that strongly-accreting binaries tend to have mass ratios close to unity. This is shown in Fig. 6, in which we present the distribution of the final mass ratio for an \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFigure 6 . Examples of evolution of the q distribution within the accretion scenario with z cut-off = 10 . The distributions are found imposing a cut on the total mass equal to M max tot = 10 4 M glyph[circledot] . Top and bottom panels correspond to power-law and lognormal mass functions, respectively. \n<!-- image --> \ninitial power-law and lognormal mass function for several choices of ( M min , M max ) and ( σ , M c ), respectively. The distributions are constructed by drawing the values of ( M i 1 , M i 2 ) from a given initial mass function, subject to the constraint that M max tot < 10 4 M glyph[circledot] . Note, however, that this plot does not represent the distribution of q expected in actual events, since it does not take into account neither how the merger rates depend on the parameters nor the sensitivity curve of LIGO/Virgo, which is optimal for frequencies corresponding to the merger of binary BHs with M max tot < 10 2 M glyph[circledot] . A direct confrontation with current GW events will be presented in Sec. 5.', '2.4.2 Spin evolution': 'In the absence of accretion, or if mass accretion is not efficient enough, the spin of PBHs is natal. As discussed in Sec. 2.2 the dimensionless spin parameter χ i in the most likely formation scenarios is of the order of the percent or smaller, although larger values are predicted in less standard scenarios, see e.g. [27, 28]. \nThe situation changes drastically in the case of efficient accretion. In Fig. 7 we show the final spins of the PBH binary components as a function of their final masses for three different choices of z cut-off . Besides the quantitative difference between different choices of z cut-off , the qualitative trend is the same. Namely, the final mass and final spin of the PBHs are correlated: low-mass PBHs are slowly spinning or non-spinning, whereas high-mass PBHs are rapidly spinning. The mass scale at which this continuous transition occurs depends on the cut-off redshift and it is always around (10 ÷ 40) M glyph[circledot] . In particular, smaller cut-offs favour a lower-mass transition, as expected by the fact that in this case the accreting phase lasts longer during the cosmic evolution. \nFigure 7 . Final spins χ 1 (top panels) and χ 2 (bottom panels) as functions of the final masses ( M 1 , M 2 ) . As in Fig. 5, from left to right we consider three cut-off redshifts below which accretion is suppressed. \n<!-- image --> \nIn Fig. 8 we show the distribution of the effective spin parameter defined as \nχ eff = glyph[vector] J 1 /M 1 + glyph[vector] J 2 /M 2 M 1 + M 2 · ˆ L, (2.36) \nwhere M 1 and M 2 are the individual BH masses, glyph[vector] J 1 and glyph[vector] J 2 are the corresponding angularmomentum vectors, and ˆ L is the direction of the orbital angular momentum. Following Ref. [12] we have averaged over the angles between the total angular momentum and the individual PBH spins. We have plotted χ eff as a function of the final PBH mass M 1 for different fixed values of the mass ratio parameter q . When accretion is present, such distributions reflect the transition from initially vanishing values of χ eff to large ones. In the top two rows of Fig. 8 (for q = 1 and q = 1 / 2), we have shown only the current data which are compatible (within their reported errors) with the corresponding chosen value of q . In the third row ( q = 1 / 4) we report only GW190412 as a reference.', '2.5 Limitations of the accretion model': "Accretion onto compact objects through the cosmic history is a complex phenomenon and the accretion rate relies on a set of assumptions, above all through the dependence on the velocity. While from the discussion of the previous sections it seems clear that accretion should play a relevant role, it is also important to spell out the main uncertainties in the accretion modelling [8, 9, 52]. \n- a) Local feedback : our analysis neglects the effect of feedback on the accretion flow. The effect of local heating for the PBH masses of interest for LIGO/Virgo can be safely neglected [8, 52].\n- b) Global feedback & X-ray pre-heating : this effect was estimated in [8], including the X-ray heating of the gas and the extra contribution due to PBH accretion, but neglecting other \nFigure 8 . The distribution of χ eff as a function of the PBH mass M 1 for selected values of the parameter q . Blue data points refer to the events listed in Ref. [1], whereas green and red data points refer to the events discovered in Refs. [49, 50]; the red data points refer to GW151216 and GW170403, for which the measured value of χ eff is significantly affected by the prior on the spin angles [51]. The cyan data point refers to GW190412 recently reported in Ref. [2]. \n<!-- image --> \npossible sources of X-ray heating [53]. However, since then, the analysis of the cosmic ionisation provided in Ref. [8] has been strongly revisited by later work, in particular after GW150914 and the suggestion that those BHs might be of primordial origin [5]. Indeed, the detailed analysis of Ref. [52] shows that global feedback is much less important for LIGO/Virgo BHs. In particular, taking all relevant effects into account, Ref. [52] found an accretion rate which is consistent with that estimated by Ref. [8] without the effect of global heating. Modelling the temperature of the intergalactic medium at redshift 10 glyph[lessorsimilar] z glyph[lessorsimilar] 30 is particularly relevant, since an increase in the temperature is followed by an increase in the sound speed and, in turn, by a reduction of the accretion rate. \n- c) DM halo : the accretion rate computed in our model is consistent with that of Ref. [52], with the important inclusion of a DM halo, which seems inevitable if PBHs form a small fraction of the DM, at least if the latter is of particle origin.\n- d) Structure formation : part of the population of PBHs starts falling in the gravitational potential well of large-scale structures after redshift around z glyph[similarequal] 10, experiencing an increase of the relative velocity up to one order of magnitude, see for example Ref. [54] for a recent analysis. This will result in a consequent suppression of the accretion rate [9, 55, 56]. While this motivates our choice z cut-off = 10, the precise dynamics depends on the complex modelling of the global thermal feedback and the change in the relative velocity due to the structure formation. Furthermore, it is hard to estimate the fraction of PBHs that stop accreting efficiently enough due to this effect. For example, the captured PBHs might settle at the center of the halo within a Hubble time, due e.g. to dynamical friction effects, and might keep accreting efficiently.\n- e) Spherical accretion & disk geometry : most of the semi-analytical studies on accretion onto compact objects necessarily assume a quasi-spherical flow. However, this approximation might break down, e.g. in the case of outflows [57]. The efficiency of the latter in reducing the accretion rate depends on the geometry of the accretion flow and on the relative direction of the outflow.\n- f) Angular momentum transfer : As discussed above, when ˙ m ∼ 1 the disk is geometrically thin and can be described with a geodesic model. When ˙ m glyph[lessmuch] 1 or ˙ m glyph[greatermuch] 1, the geometry of the disk is different, and this impacts on the accretion luminosity and feedback. In the super-Eddington regime the disk is expected to 'puff up', becoming geometrically thicker. Angular momentum accretion in this case is more complex, although numerical simulations suggest that the spin evolution time scale does not change significantly [48]. \nWhile the above details require complex and model-dependent simulations, the uncertainties in certain aspects of the accretion model can be parametrised in an agnostic way through a cut-off redshift z cut-off . For instance, as far as the X-ray pre-heating effect [53] on the accretion rate is concerned, the suppression factor in ˙ m with respect to the case of no X-ray pre-heating may be at most an order of magnitude or smaller. Most importantly, this effect can be caught in our model by increasing the value of z cut-off without including the X-ray pre-heating 5 . \nBased on the above discussion, we consider z cut-off = 10 as the most reasonable choice for the cut-off redshift; the value z cut-off = 7 advocated in Ref. [9] might be considered as an optimistic choice that assumes a non-negligible fraction of PBHs which keeps accreting significantly even after structure formation, whereas larger values of the cut-off, e.g. z cut-off = 15 would suppress the effect of accretion and correspond to a scenario in which the temperature of the intergalactic medium is high enough even before reionisation.", '3 Binary evolution': 'Having discussed the masses and spins of isolated and binaries PBHs, we now turn our attention to the evolution of PBH binaries. First we consider the case in which accretion is absent or \ninefficient, in which case the binary evolves only through GW radiation-reaction, which we review following Sec. 4.1 of Ref. [60]. Then, we consider the case in which baryonic mass accretion is modelled as discussed in the previous sections, extending the results of Ref. [61] to the case of accretion-driven inspiral for eccentric binaries with generic mass ratio.', '3.1 GW-driven evolution': 'In the absence of GW radiation-reaction, the eccentricity e and semi-major axis a are constants of motion and can be expressed in terms of the angular momentum and energy as \ne 2 = 1 + 2 EL 2 M 2 tot µ 3 , a = GM tot µ 2 | E | . (3.1) \nTo the leading order in the weak-field/slow-motion approximation, one can use the quadrupole formula to evaluate the energy and angular momentum losses through the GW emission. The energy and angular momentum of the binary evolves as [62, 63] \nd E d t = -32 5 µ 2 M 3 tot a 5 1 (1 -e 2 ) 7 / 2 ( 1 + 73 24 e 2 + 37 96 e 4 ) , d L d t = -32 5 µ 2 M 5 / 2 tot a 7 / 2 1 (1 -e 2 ) 2 ( 1 + 7 8 e 2 ) . (3.2) \nIn terms of the adiabatic evolution of e and a , one can recast the system of equations in the form \nd a d t = -64 5 µM 2 tot a 3 1 (1 -e 2 ) 7 / 2 ( 1 + 73 24 e 2 + 37 96 e 4 ) , \nd e d t = -304 15 µM 2 tot a 4 e (1 -e 2 ) 5 / 2 ( 1 + 121 304 e 2 ) . (3.3) \nThis system can be solved to find an estimate for the merging time t c , here defined as a ( t c ) = 0. For an initial orbit with e ( t i ) = e i and a ( t i ) = a i one finds \nt c ( a i , e i ) = t c ( a i ) 48 19 1 g 4 ( e i ) ∫ e i 0 d e g 4 ( e )(1 -e 2 ) 5 / 2 e (1 + 121 e 2 / 304) , (3.4) \nwhere we defined the function \ng ( e ) = e 12 / 19 1 -e 2 ( 1 + 121 304 e 2 ) 870 / 2299 . (3.5) \nFor a circular orbit, Eq. (3.4) reduces to \nt c ( a i , e i = 0) ≡ t c ( a i ) = 5 256 a 4 i M 2 tot µ . (3.6) \nConversely, in the limit e i → 1, one finds that \nt c ( a i , e i → 1) glyph[similarequal] t c ( a i ) 768 429 (1 -e 2 i ) 7 / 2 . (3.7) \nThe parameters leading to a coalescence time equal to the age of the universe, t 0 = 13 . 7 Gyr, are shown in Fig. 9. As one can notice, the initial value of a i diverges as e i tends towards unity, since the coalescence time tends to shrink rapidly in that limit. \nThe above relation is used by [4, 6] to compute an estimate of the merger rate, as we discuss in Sec. 4. \n<!-- image --> \n<!-- image --> \nFigure 9 . The contour lines indicate the combination of the parameters e i and a i giving a coalescence time equal to the age of the universe t c ( a i , e i ) = t 0 . The label indicates M 1 and the mass ratio considered is q = 1 , 1 / 2 and 1 / 8 respectively. \n<!-- image -->', '3.2 Accretion-driven evolution': 'Accretion introduces a further secular change to the orbital parameters, in addition to GW radiation-reaction [61, 64, 65]. As we are going to discuss, due to the time scales involved in the problem, we can study the accretion-driven phase and the GW-driven phase separately. Indeed, since for z < z cut-off accretion is drastically suppressed, the evolution of the binary from z cut-off to the redshift of detection (which we shall assume to be z ≈ 0 having in mind the current horizons of LIGO and Virgo) is governed by GW emission only. From Eq. (3.4), a merger occurring at z ≈ 0 corresponds to a binary with orbital separation a = O (10 11 m) at z cut-off = 10. In this configuration the time scale of variation of the semi-major axis due to GW emission is \nT GW ∼ a ˙ a ∣ ∣ ∣ GW = 4 × 10 17 ( a 1 . 4 × 10 11 m ) 4 ( M 30 M glyph[circledot] ) -3 ( 1 -e 2 0 . 1 ) 7 / 2 s (3.8) \nwhere the normalisation a = 1 . 4 × 10 11 m has been chosen as the one giving a merger in a time equal to the age of the universe for binary components of equal mass M = 30 M glyph[circledot] and e = 0 . 95. As can be directly checked, owing to the strong dependence T GW ∝ a 4 , for z > z cut-off the typical accretion time scale (2.9) is much smaller than the one governing GW radiation-reaction. In other words, we can assume 6 that the inspiral is driven solely by accretion when z > z cut-off and solely by GW radiation-reaction when z < z cut-off . \nIt is also worth noting that, for the range of parameters we are interested in, the mass accretion time scale (2.9) is always much larger than the characteristic orbital time scale, \nT orbital ∼ ( M tot a 3 ) -1 / 2 ∼ 8 × 10 5 ( M 30 M glyph[circledot] ) -1 / 2 ( a 1 . 4 · 10 11 m ) 3 / 2 s . (3.9) \nTherefore one can treat the mass accretion as an adiabatic process keeping constant the adiabatic invariants of the elliptical motion, as we now discuss. \nIf the masses vary adiabatically, one can compute the action variables I k = ∫ p d k/ (2 π ) for the elliptic motion (where k = r, φ are the polar coordinates) and then employ the fact that they are adiabatic invariants. Specifically, the adiabatic invariants for the Keplerian two-body problem are [66] \nI φ = 1 2 π ∫ 2 π 0 p φ d φ = L z , (3.10) \nI r = 1 2 π ∫ r max r min p r d r = -L z + √ M tot µ 2 a, (3.11) \nwhere we recall the definition of energy and angular momentum of the binary \nE = -µM tot 2 a , L z = √ M 2 tot µ 3 ( e 2 -1) 2 E = √ 1 -e 2 √ M tot µ 2 a. (3.12) \nThe invariance of I φ and I r implies that \nd I φ d t = ∂L z ∂e ∂e ∂t + ∂L z ∂a ∂a ∂t + ∂L z ∂µ ∂µ ∂t + ∂L z ∂M tot ∂M tot ∂t = 0 , d I r d t = ∂I r ∂e ∂e ∂t + ∂I r ∂a ∂a ∂t + ∂I r ∂µ ∂µ ∂t + ∂I r ∂M tot ∂M tot ∂t = 0 . (3.13) \nIt is easy to prove that the system of equations can be recast as \n∂a ∂t = -2 a µ ∂µ ∂t -a M tot ∂M tot ∂t , ∂e ∂t = 0 , (3.14) \nwhich shows that the eccentricity is a constant of motion for an accretion-driven inspiral. This does not come as a surprise as the eccentricity can be expressed in terms of the adiabatic invariants as [66] \ne = √ 1 -( I φ I φ + I r ) 2 . (3.15) \nThe only non-trivial dynamical equation can be written as \n˙ a a +2 ˙ µ µ + ˙ M tot M tot = 0 . (3.16) \nThis results recovers the one found in Ref. [61] in the absence of GW emission and for circular binaries, and extends it for a generic eccentricity, which remains a constant of motion. \nFinally, it is convenient to write Eq. (3.16) in terms of the mass accretion rates of the isolated objects \n˙ a a + M 2 ( M 1 +2 M 2 ) ˙ M 1 + M 1 (2 M 1 + M 2 ) ˙ M 2 M 1 M 2 ( M 1 + M 2 ) = 0 . (3.17) \nIn general, this equation is coupled to the evolution equations for M 1 and M 2 , namely Eqs. (2.22). However, within the aforementioned approximations, one can evolve M i using Eq. (2.24) and finally evolve the semi-axis major using Eq. (3.17). In the limit q → 1, i.e. M 1 = M 2 , Eq. (3.17) simplifies to the more familiar form \n˙ a a +3 ˙ M 1 M 1 = 0 . (3.18)', '4 PBH merger rates and phenomenology': 'After the discussion about the evolution of the PBH binary, we move to the computation of the PBH merger rate both without and including accretion, and we will then discuss its impact on the PBH phenomenology. 7', '4.1 Merger rates without accretion': 'Following the notation of Ref. [73], one can write down the differential merger rate at the time of coalescence t in the form \nd R = 1 . 6 × 10 6 Gpc 3 yr f 53 37 PBH ( z i ) ( t t 0 ) -34 37 η -34 37 i ( M i tot M glyph[circledot] ) -32 37 S ( M i tot , f PBH ( z i )) ψ ( M i 1 , z i ) ψ ( M i 2 , z i )d M i 1 d M i 2 , (4.1) \nwhere f PBH ( z i ) is the initial fraction of DM in the form of PBHs, η i = µ i /M i tot is the symmetric mass ratio, defined in terms of the reduced mass µ i = M i 1 M i 2 /M i tot and total mass M i tot = M i 1 + M i 2 of the binary components at the formation time, and ψ ( M i , z i ) identifies the PBH mass function at the formation time z i , normalised to unity. \nThe suppression factor S is introduced in order to keep into account the effect of the matter density perturbations and possible modifications due to the size of the empty region around the binary. Its expression is given by [73] \nS ( M i tot , f PBH ( z i )) = e -¯ N ( y ) Γ(21 / 37) ∫ d vv -16 37 exp [ -¯ N ( y ) 〈 m 〉 ∫ d m m ψ ( m,z i ) F ( m 〈 m 〉 v ¯ N ( y ) ) -3 σ 2 M v 2 10 f 2 PBH ( z i ) ] (4.2) \nin terms of the generalised hypergeometric function \nF ( z ) = F 1 2 ( -1 2 ; 3 4 ; 5 4 ; -9 z 2 16 ) -1 , (4.3) \nthe rescaled variance of matter density perturbations σ 2 M = (Ω M / Ω DM ) 2 〈 δ 2 M 〉 glyph[similarequal] 3 . 6 · 10 -5 at the time at which the binary is formed, and \n〈 m 〉 = (∫ 1 m ψ ( m,z i )d m ) -1 . (4.4) \nThe number ¯ N ( y ) of PBHs in a spherical volume of radius y is chosen such that the binaries do not get destroyed by other PBHs, i.e. \n¯ N ( y ) = M i tot 〈 m 〉 f PBH ( z i ) f PBH ( z i ) + σ M . (4.5) \nThe integral over the masses in the suppression factor gives an estimate of the typical mass of the perturber PBHs responsible for the torque, which prevents two PBHs to collide directly and is responsible for forming the binary itself. \nIn Fig. 10 we show the behaviour of the suppression factor for the cases of a lognormal and power-law mass function.', '4.2 Merger rates with accretion': 'In this subsection we will describe the main impact of accretion on the PBH merger rate. \nAs already discussed in Sec. 3, due to the time scales involved in the problem, one can study the accretion-driven and the GW-driven binary evolution separately. Indeed, accretion dominates the evolution of the binary up to redshift z cut-off , while GW radiation-reaction is dominant from that redshift up to the detection time. \nTo understand the effect of accretion on the merger rate we remind that, using the formalism of Ref. [73], the latter is defined by integrating the probability distribution of the orbital parameters \nd R ( t ) = ∫ d a d e d P d e δ ( t -t c ( µ, M tot , a, e )) (4.6) \n<!-- image --> \nFigure 10 . Left: Suppression factor for a power-law mass function with M max = 10 M min and various M i tot . Right: Suppression factor for a lognormal mass function with σ = 0 . 5 . \n<!-- image --> \nat the coalescence time of the binary, which has been computed in Sec. 3 without accretion and which we report here for convenience in a more compact form \nt c = 3 85 a 4 i (1 -e 2 i ) 7 / 2 η ( z i ) M 3 tot ( z i ) , (4.7) \nwhere all the quantities are set at the formation time z i . \nIn the presence of accretion the masses change in time. Even though the ellipticity is a constant of motion, the semi-major axis a does evolve with time due to the masses evolution, as one can see from Eq. (3.16). This will have an impact on the coalescence time of the binary. Since the accretion-driven phase occurs earlier and independently of GW emission, once accretion is included the coalescence time becomes \nt acc c = 3 85 N 4 a 4 i (1 -e 2 i ) 7 / 2 η ( z cut-off ) M 3 tot ( z cut-off ) ≡ N 4 S t c ( M i j ) , (4.8) \nwhere we have defined the factor \nS = η ( z cut-off ) M 3 tot ( z cut-off ) η ( z i ) M 3 tot ( z i ) (4.9) \nwhich keeps into account the masses evolution from initial time z i to the cut-off redshift z cut-off , and the shrinking factor of the orbit \nN ≡ a ( z cut-off ) a i = exp [ -∫ t cut-off t i d t ( ˙ M tot M tot +2 ˙ µ µ )] , (4.10) \nwhich properly considers the semi-major axis evolution. As we discussed, for z < z cut-off accretion is negligible, so the binary proceeds in the standard GW radiation-reaction scenario, but with different masses with respect to the no accretion case. \nImplementing the fact that the suppression factor does not depend on time and using Eq. (2.27) for the mass function evolution, one can rescale the coalescence time to compute the final differential merger rate as \nd R acc ( t, M j , f PBH ( z i )) = S N 4 d R ( t S / N 4 , M i j , f PBH ( z i ) ) = N -12 / 37 S 3 / 37 d R ( t, M i j , f PBH ( z i ) ) (4.11) \nwhere with M i j and M j we identify respectively the couple ( M 1 , M 2 ) at the formation and final time. In the last line the merger rate in terms of the initial quantities is explicitly given by Eq. (4.1). As such, the bounds coming from merger rates do not depend upon the evolution of the mass functions discussed in subsection 2.3.3.', '4.3 Phenomenology of PBH mergers without accretion': 'In this subsection we will discuss the implications of the physics of the PBH mergers on their phenomenology, by focusing on the constraints on the PBH abundance both from the total number of observed binary BH merger events by the LIGO/Virgo collaboration, and from the absence of the stochastic GW background produced by unresolved sources. We will follow the procedure outlined in Ref. [73], without including the effects of accretion. The latter will be discussed in Sec. 4.4.', '4.3.1 Likelihood analysis for GW observations without accretion': "One can start by performing a maximum-likelihood analysis, considering all the BH merger events observed by the LIGO/Virgo collaboration to date [1, 2] and assuming that they all have a primordial origin, to find out the best-fit values for the parameters of the PBH mass function. For concreteness, we shall assume either a lognormal or a power-law distribution. \nThe log-likelihood function is given by \nL = ∑ j ln ∫ d R ( M 1 , M 2 , z ) / (1 + z )d V c ( z ) p j ( M j, 1 | M 1 ) p j ( M j, 2 | M 2 ) p j ( z j | z )Θ( ρ ( M 1 , M 2 , z ) -ρ c ) ∫ d R ( M 1 , M 2 , z ) / (1 + z )d V c ( z )Θ( ρ ( M 1 , M 2 , z ) -ρ c ) , (4.12) \nwhere the experimental uncertainties of the detected events are assumed to be described by Gaussian probabilities p j ( M j | M ) to observe a BH mass M j given that the BH has mass M , and by Gaussian probabilities p j ( z j | z ) to observe a merger at redshift z j given that it happens at redshift z . \nThe integral is performed over the masses and redshift, in terms of the PBH merger rate d R ( M 1 , M 2 , z ) shown in Eq. (4.1), and of the comoving volume per unit redshift \nd V c ( z ) d z = 4 π H 0 D 2 c ( z ) E ( z ) = 4 π H 2 0 1 E ( z ) (∫ z 0 d z ' E ( z ' ) ) 2 , (4.13) \nwhere the comoving distance D c is \nD c ( z ) = 1 H 0 ∫ z 0 d z ' E ( z ' ) (4.14) \nwith \nE ( z ' ) = √ Ω r (1 + z ' ) 4 +Ω m (1 + z ' ) 3 +Ω K (1 + z ' ) 2 +Ω Λ , (4.15) \nand Ω K = 0 . 0007, Ω r = 5 . 38 × 10 -5 , Ω Λ = 0 . 685, Ω m = 0 . 315, h = 0 . 674, H 0 = 1 . 0227 × 10 -10 h yr -1 . The additional factor of redshift 1 / (1+ z ) is introduced to account for the difference in the clock rates at the time of merger and detection. The Heaviside function Θ is introduced in Eq. (4.12) to implement a detectability threshold based on the signal-to-noise ratio (SNR) of the GW events, ρ c = 8. The optimal signal-to-noise ratio ρ opt of individual GW events for a source with masses M 1 , M 2 at a redshift z is given by [74] \nρ 2 opt ( M 1 , M 2 , z ) ≡ ∫ ∞ 0 4 | ˜ h ( ν ) | 2 S n ( ν ) d ν (4.16) \nwhere the strain noise for the O2 run has been taken from Ref. [75] and its analytical fit in the frequency range ν ∈ [10 , 5000] Hz is given by \nS 1 / 2 n, O2 ( ν ) = exp [ 6 ∑ i =0 c i log i ( ν ) ] , (4.17) \nwith \nc 0 = 33 . 3329 , c 1 = -75 . 7393 , c 2 = 27 . 1742 , c 3 = -5 . 10534 , c 4 = 0 . 524229 , c 5 = -0 . 0273956 , c 6 = 0 . 000557901 . (4.18) \nThe GW strain signal ˜ h is given in Fourier space by [74] \n˜ h ( ν ) = √ 5 8 1 D L π 1 ν ( d E GW ( ν ) d ν ) 1 / 2 e iφ ( ν ) (4.19) \nwhere φ ( ν ) is the phase of the waveform, not relevant for the SNR, and D L identifies the luminosity distance from the source in terms of the comoving distance D c as \nD L ( z ) = (1 + z ) D c ( z ) = (1 + z ) 1 H 0 ∫ z 0 d z ' E ( z ' ) . (4.20) \nFor the GW energy spectrum d E GW with frequency between ( ν, ν +d ν ) we use a phenomenological expression which, in the non-spinning limit, is given by [74, 76] 8 \nd E GW ( ν ) d ν = π 2 / 3 3 M 5 / 3 tot η × ν -1 / 3 for ν < ν 1 , ν ν 1 ν -1 / 3 for ν 1 ≤ ν < ν 2 , ν 2 ν 1 ν 4 / 3 2 σ 4 (4( ν -ν 2 ) 2 + σ 2 ) 2 for ν 2 ≤ ν < ν 3 , (4.21) \nwhere \nπM tot ν 1 = (1 -4 . 455 + 3 . 521) + 0 . 6437 η -0 . 05822 η 2 -7 . 092 η 3 , πM tot ν 2 = (1 -0 . 63) / 2 + 0 . 1469 η -0 . 0249 η 2 +2 . 325 η 3 , πM tot σ = (1 -0 . 63) / 4 -0 . 4098 η +1 . 829 η 2 -2 . 87 η 3 , πM tot ν 3 = 0 . 3236 -0 . 1331 η -0 . 2714 η 2 +4 . 922 η 3 . (4.22) \nThe final SNR can be then obtained by performing an average over the isotropic sky locations and orientations, finding that [78, 79] \nρ 2 ( M 1 , M 2 , z ) = 1 5 ρ 2 opt ( M 1 , M 2 , z ) , (4.23) \nwhich should be compared with the detectability threshold assumed to be ρ c = 8. \nFrom all these ingredients one can perform a maximum-likelihood analysis and find the best-fit values for the PBH mass function parameters, which can then be used as benchmark values to show the constraints on the PBH abundance from GWs events.", '4.3.2 Number of events and stochastic GW background without accretion': 'The predicted number of PBH merger detections in a time interval ∆ t is given by [73, 80, 81] \nN = ∆ t ∫ d z d M 1 d M 2 1 1 + z d V c ( z ) d z d R ( M 1 , M 2 , z ) d M 1 d M 2 Θ( ρ ( M 1 , M 2 , z ) -ρ c ) (4.24) \nin terms of the PBH merger rate, including a redshift factor 1 / (1+ z ) to account for the difference in the clock rates at the time of merger and detection. The errors N -N min and N max -N on N can be estimated using a Poisson statistics by computing the number of events with the PBH mass function parameters and f PBH at which the likelihood is maximum, and the reference masses at the 2 σ confidence level. One can use the range N min < N < N max to constrain the fraction of DM as PBHs assuming that all the observed BH merger events are primordial, and setting an upper bound on f PBH . We will show the results of this procedure in Sec. 5. \nPBHs mergers which are not individually resolved (i.e. ρ < ρ c = 8) contribute to a stochastic GW background, which in turn can be used to constrain the PBHs abundance, see Ref. [73, 8285]. From the differential merger rate d R ( z ) at redshift z , one can compute the spectrum of the stochastic GW background of frequency ν as \nΩ GW ( ν ) = ν ρ 0 ∫ ν 3 ν -1 0 d z d M 1 d M 2 1 (1 + z ) H ( z ) d R ( M 1 , M 2 , z ) d M 1 d M 2 d E GW ( ν s ) d ν s Θ( ρ c -ρ ( M 1 , M 2 , z )) , (4.25) \nwhere ρ 0 = 3 H 2 0 / 8 π , ν s = ν (1 + z ) is the redshifted source frequency, and now the Heaviside function is introduced to subtract the contribution from events which can be observed individually. \nBy calculating the stochastic GW background arising from the coalescences of PBH binaries and comparing its strength to the sensitivity of LIGO [86, 87], one can constrain the fraction of DM in PBHs. The result of this procedure will be shown in Sec. 5.', '4.4 Phenomenology of PBH mergers with accretion': 'In this subsection we include the effect of accretion on the likelihood analysis as well as on the estimates of the number of BH merger events and of the stochastic GW background from unresolved sources.', '4.4.1 Likelihood analysis for GWs observations with accretion': 'Following the same procedure of the previous subsection, one can start by performing a maximumlikelihood analysis to find out the preferred values for the parameters of the PBH initial mass function which best-fit the data. The log-likelihood function is given by \nL acc = ∑ j ln ∫ d R acc ( M 1 , M 2 , z ) / (1 + z )d V c ( z ) p j ( M j, 1 | M 1 ) p j ( M j, 2 | M 2 ) p j ( z j | z )Θ( ρ ( M 1 , M 2 , z ) -ρ c ) ∫ d R acc ( M 1 , M 2 , z ) / (1 + z )d V c ( z )Θ( ρ ( M 1 , M 2 , z ) -ρ c ) \n(4.26) \nin terms of the merger rate including the effect of accretion d R acc ( z ) given in Eq. (4.11). We stress that, once accretion is included, the final masses enter both in the Gaussian probabilities for the experimental uncertainties of the detected events as well as in the SNR.', '4.4.2 Number of events and GWs abundance with accretion': 'Including the effect of accretion, the predicted number of PBH mergers detected in a time ∆ t is given by \nN acc = ∆ t ∫ d z d M 1 d M 2 1 1 + z d V c ( z ) d z d R acc ( M 1 , M 2 , z ) d M 1 d M 2 Θ( ρ ( M 1 , M 2 , z ) -ρ c ) (4.27) \nin terms of the accretion-included merger rate d R acc ( M 1 , M 2 , z ) given by Eq. (4.11). \nLikewise, also the GW background gets modified by accretion as \nΩ acc GW ( ν ) = ν ρ 0 ∫ ν 3 ν -1 0 d z d M 1 d M 2 1 (1 + z ) H ( z ) d R acc ( M 1 , M 2 , z ) d M 1 d M 2 d E GW ( ν s ) d ν s Θ( ρ c -ρ ( M 1 , M 2 , z )) . (4.28) \nThe results for the constraints with the inclusion of accretion will be shown in the next section.', '5 Confrontation with LIGO/Virgo O1, O2, and GW190412': 'In this section we present the main results of this work. We confront the theoretical predictions discussed above with current observations, assuming that the BHs involved in the merger events are of primordial origin. We do so by assuming two mass functions (power-law and lognormal) and various scenarios, namely no accretion and accretion with three different cut-off redshifts ( z cut-off = 15 , 10 , 7). As for the data, we include the LIGO/Virgo observation runs O1 and O2 [1] and the recent GW190412 [2], which overall comprise N obs = 11 events. 9 \nWe have proceeded by assuming that all the events seen by LIGO/Virgo are due to a firstgeneration merger of PBHs (hierarchical mergers [29] will be briefly discussed later on). This is admittedly a strong assumption as of course a fraction of these events (if not all) might be due to BHs of astrophysical origin. Our goal is therefore to analyse whether motivated PBH mass functions (power-law and lognormal) are compatible with current GW data.', '5.1 Best-fit parameters for the PBH mass function': 'In Fig. 11 we present the likelihood on the parameter space for a power-law (top panels) and lognormal (bottom panels) mass function. A few comments are in order: first, the best-fit values in all cases end up providing a similar value of the PBH fraction in DM, namely f PBH ≈ few × 10 -3 ; this upper bound becomes less stringent in the case of strong accretion (small z cut-off ); secondly, increasing the accretion effect (i.e., decreasing z cut-off ) makes the 2 σ and 3 σ contours to shrink. This is due to the fact that the best-fit values correspond to a narrower initial distributions and smaller initial central masses; finally, we note that, for the case of no accretion, our best-fit values for the lognormal distribution agree with those recently computed in Ref. [90]. The latter reported M c = 17 M glyph[circledot] and σ = 0 . 75, which agree with the values shown in the bottom left panel of Fig. 11 within 1 σ . This shows the robustness of these values, given the fact that our analysis and that of Ref. [90] are different; for example Ref. [90] did not include the suppression factor for the merger rate and fitted only the chirp masses of the events. \nIn Fig. 12 we present the comparison between the initial and the final mass functions for the best-fit values obtained from the previous likelihood analysis. To highlight the differences at large masses, for the lognormal case we also show the same results in a log-linear scale. The effect of accretion is to shift the tail of the mass function to larger PBH masses, this shift being more pronounced when the accretion is stronger, with a consequent decrease of the amplitude of the peak. In other words, accretion tends to make the mass distribution broader [11]. Note, however, that the effect on the mass functions for the specific values of the parameters selected by the likelihood is much less pronounced than in the example shown in Fig. 2. This happens because the best-fit distributions peak at relatively low mass. We also notice that this evolution is for isolated PBHs, which is the relevant case for the constraints inferred from the observations \nFigure 11 . Likelihood on the parameter space for the power-law (top) and lognormal (bottom) mass functions requiring N obs = 11 . The red dashed (solid) contours corresponds to 2 σ ( 3 σ ) respectively. The leftmost panels correspond to the case in which accretion is negligible, whereas the second to fourth columns correspond to accretion suppressed at z cut-off = (15 , 10 , 7) , respectively. \n<!-- image --> \nFigure 12 . Primordial mass function and its evolution if accretion is present. In this plot we use the best-fit values for the parameters of the PBH mass distributions at formation, as obtained from the previous likelihood analysis. The organization of the panels is the same as in Fig. 11. \n<!-- image --> \nother than those from GW observations. However, one should take into account that for PBHs in binaries the effect of accretion is larger, thus allowing PBH masses larger than those indicated in Fig. 12.', '5.2 Updated constraints on PBH abundance': 'In Fig. 13 we present the constraints on the PBH abundance as a function of the mass, again for the best-fit values obtained from the previous likelihood analysis. In the range of masses of interest for LIGO/Virgo, the most important constraints come from lensing, dynamical processes, formation of structures, and accretion related phenomena. The lensing bounds include those from supernovae [91], the MACHO and EROS experiments [92, 93], ICARUS [94] and radio observations, such as [95] and [96] (Ogle). They all consider lensing sources at low redshift z glyph[lessmuch] z cut-off . Dynamical constraints involve disruption of wide binaries [97], and survival of star clusters in Eridanus II [98] and Segue I [99] at small redshifts. Bounds also arise by observations of the Lymanα forest at redshift before z ≈ 4 [100]. Other constraints involve bounds from Planck data on the CMB anisotropies induced by X-rays emitted by spherical or disk (Planck S and Planck D, respectively) [52, 101] accretion at high redshifts or bounds on the observed number of X-ray (XRay) [102, 103] and X-ray binaries (XRayB) at low redshifts [104]. \nIn Fig. 13 we show a selection of the above constraints, identified by the nickname in parenthesis in the list above, and computed as discussed in Ref. [11]. In addition, we show the bounds coming from the absence of stochastic GW background in LIGO/Virgo data (black line) and those from the merger rate (red lines), computed as discussed in the previous section. 10 \nThe red dashed and continuous lines correspond to the 2 σ values for the expected number of events. In other words, the red continuous line corresponds to the upper bound from the observed merger rate, since larger values of f PBH ( z = 0) would yield a merger rate higher than observed at a given mass. On the other hand, the red dashed line corresponds to a lower bound on f PBH ( z = 0), assuming all the observed events are of primordial origin, since smaller values of f PBH ( z = 0) would yield a merger rate lower than observed at a given mass. Clearly, this lower bound can be made less stringent by assuming that only a fraction of events is of primordial origin. The vertical dashed red lines indicate the 2 σ interval around the best-fit value of the mass parameter, as obtained by the likelihood analysis. \nWhen accretion gets stronger (for masses around ≈ 10 M glyph[circledot] ), we observe two main effects: \n- i) In the leftmost part of the red curve (i.e. on the left of the minimum) the GW bounds on f PBH become more stringent , since accretion enhances the merger rate in that region. The opposite is true on the right of the minimum, because in that region accretion pushes the masses to large values, outside the optimal sensitivity range of the detectors.\n- ii) the non-GW bounds become weaker , due to the broadening of the mass function and the evolution of f PBH ( z ). The interested reader can find a more detailed discussion of this phenomenon in Ref. [11]. \nThe net result is that, if accretion is negligible, in the range of the mass-function parameters selected by the likelihood analysis, non-GW constraints (in particular Planck D) would already marginally exclude the possibility that all BH merger events detected so far are of primordial origin. However, when accretion is significant the opposite is true: LIGO/Virgo constraints are the most stringent ones in the relevant mass range. Overall, the upper bound on the PBH abundance coming from LIGO/Virgo rates is f PBH ( z ) glyph[lessorsimilar] few × 10 -3 in the relevant mass range, with the upper bound becoming more stringent in the case of strong accretion. Moreover, in the relevant mass range existing constraints seem to exclude the possibility to detect the GW \nFig. 15 is analogous to Fig. 14 but for the observable distribution of the mass ratio parameter q . Again, the histograms (blue lines) inferred from the data have been obtained by summing up, in each q bin, the data weighted by the corresponding posterior probability. \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFigure 13 . Constraints on f PBH from experiments not related to GWs (see legend and main text), absence of stochastic GW background in LIGO O2 (black-solid), and upper (red-solid indicating the 2 σ exclusion region) and lower (red-dashed, obtained assuming that all BH merger events detected so far are primordial) bounds coming from the observed merger rates. As in Figs. 11 and 12, the leftmost panels correspond to the case in which accretion is negligible, whereas the second to fourth columns correspond to accretion suppressed at z cut-off = (15 , 10 , 7) , respectively. \n<!-- image --> \nstochastic background from PBH mergers, because detecting the latter would require a PBH abundance which is already excluded.', '5.3 Confrontation of the predicted distributions of the binary parameters with observations': 'In Fig. 14 we show the observable distribution of the chirp mass M≡ µ 3 / 5 M 2 / 5 tot obtained using the the best-fit values calculated from the previous likelihood analysis and compared to LIGO/Virgo data. We have plotted the differential R det , i.e. the number of events in the chirp mass bin normalised with respect to the total number of events (red lines). The blue histograms indicate the distribution inferred from the data. They have been obtained by plotting, in each chirp mass bin, the integral of the posterior probability summed over all measurements in that bin. On the top of the figures we have shown the individual posterior probability for each measured event. One can appreciate that the larger the accretion, the wider the distribution becomes because the high-mass tail gets more spread. For the power-law mass function, accretion shifts the peak of the distribution to smaller masses because the likelihood prefers smaller M min . For the same reason, for very strong accretion the predicted distribution has also support for small chirp masses. For the extreme case z cut-off = 7 and a power-law distribution, the predicted merger rate at small chirp mass seems in tension with current observations. \nHere we notice that the distributions, when accretion is included, move towards q = 1. This is predicted by the the discussion in Sec. 2.4, where we showed that q = 1 is a fixed-point of the binary evolution. \nCurrent observed rates have a peak at about q ≈ 0 . 7. However, it is important to note that current errors on the mass ratio (especially for LIGO/Virgo O1 and O2 events [1]) are quite \nFigure 14 . Observable distribution of chirp mass M (red histogram) compared to the data available (blue histogram). The organization of the panels is the same as in Fig. 13. \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFigure 15 . Same as Fig. 14, but for the observable distribution of mass ratio q compared to the data available. \n<!-- image --> \nlarge. In fact, all O1 and O2 events are compatible with q ≈ 1; GW190412 is the only BH merger detected by LIGO/Virgo to date which has a mass ratio significantly different from unity, as also shown in the posterior distributions on the small top panels of Fig. 15. Overall, one might expect that the distributions of q will change significantly as new events in O3 become available. Nonetheless, a general qualitative result can be drawn: extreme accretion scenarios (like z cut-off glyph[lessorsimilar] 7) are in tension with GW190412, since they would predict that the vast majority of events should have q ∼ 1. Furthermore, the cases of a power-law and a lognormal mass functions with z cut-off glyph[similarequal] 10, see Fig. 14, which seemed to be in good agreement as far as the chirp mass distribution is concerned, seem to be in tension with the corresponding mass-ratio distributions. This is of course only a preliminary result which should wait for confirmation or disproval when the new flow of data will be available. \nFinally, let us now turn our attention to the distributions of the binary spins. In Fig. 16 we plot the observable distributions of the individual spins as a function of q with the same color code chosen to reflect the 1-, 2- and 3σ regions of Fig. 8. For each bin in q we have simulated a number of binaries and built a distribution weighting the individual contributions by the relative detection rates. Notice that for the less massive PBH component of the binary, the corresponding spin χ 2 is always larger than the spin χ 1 of the more massive component. This is predicted by the spin evolution described in Sec. 2.4.2 and is due to the fact that ˙ m 2 is increased by a factor q -1 / 2 with respect to ˙ m 1 . Also, for large values of q , the magnitude of the spin χ 1 increases when accretion becomes stronger (smaller z cut-off ), showing a strong correlation between large values of q and the spins. \nIt is interesting to note that at least one of the components of GW190412 is moderately spinning 11 . Indeed, assuming priors that are uniform in the spin magnitudes and isotropic in the directions, the LIGO/Virgo collaboration estimated χ 1 = 0 . 43 +0 . 16 -0 . 26 , while χ 2 is essentially unconstrained. On the other hand, using astrophysically-motivated priors, Ref. [107] has imposed a non-spinning primary component 12 and a uniform prior on glyph[vector] χ 2 · ˆ L , inferring glyph[vector] χ 2 · ˆ L = 0 . 88 +0 . 11 -0 . 24 . The latter can also be used as an estimate on χ 2 , assuming the orbit is not significantly tilted by the natal kick during the supernovae that produced the secondary. As shown in Fig. 16, both cases inferred by the analyses of Refs. [2, 107] are incompatible with a primordial origin for GW190412 unless: (i) accretion is significant during the cosmic history of PBHs; (ii) PBHs can be formed with non-negligible natal spin (as in some scenarios [27, 28]), in contrast with the most likely formation scenarios [25], in which the spin is at the percent level; (iii) GW190412 is actually a higher-generation merger [29], in which the spinning binary underwent a previous merger in the past. This possibility has been recently explored in Refs. [108-110], under the hypothesis that GW190412 originates from first-generation or hierarchical mergers of astrophysical origin in the context of both the field and cluster formation scenarios. However, in the case of PBHs the possibility of hierarchical merger is less likely, since the merger rates for higher-generation mergers are much smaller than those of first-generation mergers [12, 30, 31]. The possibility that GW190412 was a higher-generation merger (of either astrophysical or primordial origin) requires a better assessment of its spin and a comparison with multiple-generation scenarios. A robust assessment will probably require to wait for more events like GW190412 in O3 or future observational runs 13 . \nFinally, in Fig. 17 we plot the observable distribution of the effective spin as a function of q compared to the data available. These plots show, when accretion is important, a strong correlation between χ eff and q . Obviously, in the case of no accretion, the effective spin parameter is vanishing, owing to the initial conditions of the individual spins.', '6 Conclusions: Key predictions of the PBH scenario for GW astronomy': 'With the current and upcoming wealth of data on BH binaries from the LIGO/Virgo observatories, it becomes possible to perform model selection and rule out or corroborate specific formation \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFigure 16 . Observable distribution of the individual spins as a function of q for the power-law (first two blocks) and the lognormal (last two blocks) mass functions. The cyan data point refers to the measurement of χ 1 for GW190412 provided in Ref. [2], whereas the purple data point to the measurement of χ 2 for the same system as provided in Ref. [107] (where a prior χ 1 = 0 was imposed). The grey bands indicate those regions where the PBH model does not provide a number of observable events with a sufficiently high statistical significance. \n<!-- image --> \nscenarios for BHs in the LIGO/Virgo band. We have investigated whether current data - including O1, O2 and the recently discovered GW190412 - are compatible with the hypothesis that LIGO/Virgo BHs are of primordial origin. \nWe conclude by summarising our main findings and listing the main predictions of the PBH scenario, which can be directly tested with current and future GW observations: \n- 1. For a given PBH mass distribution at formation, current merger rates set an upper bound on the PBH abundance at the level of f PBH ∼ < (10 -2 ÷ 10 -3 ), depending on the mass function. The best-fit values selected by the likelihood provide mass distributions which are optimally compatible with current events. For the case of a lognormal distribution and no accretion, \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFigure 17 . Observable distribution of the effective spin as a function of q compared to the data available for the power-law (first block) and the lognormal (last block) mass functions. The structure of the panels and the grey bands have the same meaning as in Fig. 16. \n<!-- image --> \nour best-fit parameters are in agreement with the recent analysis in Ref. [90] within 1 σ . In addition, we found that accretion can alter quantitatively the distributions, but not the qualitative aspects of this analysis. \n- 2. Although not directly relevant for the LIGO/Virgo frequency range, it is intriguing that accretion can be very efficient for PBHs with initial mass M i glyph[greaterorsimilar] 10 M glyph[circledot] , increasing the latter by some orders of magnitude during the cosmic history. Thus, it might occur that PBHs formed with masses ∼ (10 ÷ 100) M glyph[circledot] can have much larger masses when detected at smaller redshift. These objects would be natural candidates for intermediate-mass BHs, which are sources for ET and LISA. If these objects were born in the stellar-mass range in the primordial universe, they should also have nearly-extremal spin and mass ratio close to unity.\n- 3. The fact that at least one of the components of GW190412 is moderately spinning is incompatible with a primordial origin for this event, unless: (i) PBHs are born with non-negligible spin, in contradiction with the most likely formation scenarios; (ii) at least the spinning component of GW190412 is a second-generation BH (however, this would require a better assessment of its spin and a comparison with multiple-generation merger scenarios, which are unlikely for PBHs [12]); (iii) accretion is significant during the cosmic history of PBHs.\n- 4. Despite the uncertainties in the accretion modelling, the role of accretion onto PBHs is manifold. Due to the effect of super-Eddington accretion at z ≈ 30, and to the fact that accretion onto the smaller binary component is stronger than on the primary, the mass ratio of PBH binaries with total mass above a transition value, M tot ∼ > 10 M glyph[circledot] tends to be close to unity. The precise value of the transition depends on the cut-off redshift at which accretion ceases to be efficient. This produces a peculiar distribution of the binary chirp mass (Fig. 14) and mass ratio (Fig. 15), which is absent in the stellar-origin scenario. However, for the best-fit values of the lognormal distribution, the effect of accretion on the expected distribution of q for the detected events is small. \n- 5. Overall, in the case of efficient accretion the spins of the binary components at detection are non-negligible and can be close to extremality for individual PBH with final masses M i ∼ > (10 ÷ 30) M glyph[circledot] . The transition value depends on the initial mass ratio and on the cut-off redshift.\n- 6. For the same reason as above, assuming PBHs are formed with negligible angular momentum, the spin of the smaller binary component at detection is always higher than the one of the primary, unless hierarchical mergers occur (which are, however, unlikely [12]).\n- 7. In all cases, the final spin of the secondary is close to extremality if the mass of the primary M 1 ∼ > 40 M glyph[circledot] .\n- 8. It is worth noting that - if accretion is inefficient - current non-GW constraints (in particular from the absence of extra CMB distortions [52, 101]) already exclude that LIGO/Virgo events are all of primordial origin, whereas in the presence of accretion the GW bounds on the PBH abundance are the most stringent ones in the relevant mass range.\n- 9. Overall, a strong phase of accretion during the cosmic history would favour mass ratios close to unity, and a redshift-dependent correlation between high PBH masses, high spins, and q ≈ 1. These correlations can be used to distinguish the accreting PBH scenario from that of astrophysical-origin BH binaries.\n- 10. Extreme accretion scenarios (in our study represented by z cut-off glyph[lessorsimilar] 7) predict that most of the events should be clustered around q ∼ 1. This is in tension with the recent GW190412 data.\n- 11. The individual spin evolution results in a broad distribution of the effective spin parameter of the binary, which is compatible with the observed distribution of the GW events detected so far [1, 49-51, 112], including GW190412 [2]. In particular, in the accreting PBH scenario the dispersion of χ eff around zero grows with the mass (as GW data suggest [113]), with a dispersion ≈ 0 for low binary masses and O (1) at larger masses.\n- 12. The above properties produce a peculiar distribution of χ i and χ eff as a function of the mass ratio (Figs. 16 and 17), which is absent in the stellar-origin scenario.\n- 13. According to our theoretical predictions, low mass PBHs, M ∼ < O (10) M glyph[circledot] , should have tiny spins. This property might help to distinguish PBH binaries from those composed by neutron stars [114], in the case the spin of the latter is non-negligible. \nIt is worth mention that - in confronting our theoretical predictions with GW observations we have relied on the measurements obtained by the LIGO/Virgo collaboration. The latter are based on agnostic priors on the masses and spins. A different choice of the priors - possibly motivated by a sound PBH scenario - might affect the posterior distributions of the observed parameters, especially for those which are poorly measured (see Ref. [107] for an example in the context of astrophysical priors for the spin of GW190412). We plan to investigate this interesting problem in a future work [115]. \nFurthermore, given the important role that accretion on PBHs can play in the phenomenology of GW coalescence events and in relaxing current constraints on the PBH abundance, we advocate the urgent need of a better modelling of the accretion rate at redshift z glyph[lessorsimilar] 30. \nFinally, the question whether the PBHs may be responsible or not of the LIGO/Virgo data has the following answer: while in the absence of accretion the LIGO/Virgo events are \nincompatible with the primordial nature of BHs, the situation - in the presence of accretion - is at the moment not conclusive, even though the theoretical predictions of the PBH scenario are rather sharp. The next forthcoming data from the O3 campaign and from the Advanced LIGO might provide a more definite answer.', 'Acknowledgments': "We are indebted to Christopher Berry and Luigi Stella for interesting correspondence, and to Christian Byrnes for spotting a typo in the value of σ M given in Ref. [73] which mildly affected the suppression factor used in the previous version of this manuscript. We also thank Andreas Finke, Francesco Lucarelli and Michele Maggiore for useful discussions. Some computations were performed at University of Geneva on the Baobab cluster. V.DL., G.F. and A.R. are supported by the Swiss National Science Foundation (SNSF), project The Non-Gaussian Universe and Cosmological Symmetries , project number: 200020-178787. G.F. would like to thank the Instituto de Fisica Teorica (IFT UAM-CSIC) in Madrid for its support via the Centro de Excelencia Severo Ochoa Program under Grant SEV-2012-0249. 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2018MNRAS.481.5445S

Black Hole Mergers From Globular Clusters Observable by LISA I: Eccentric Sources Originating From Relativistic N-body Dynamics

2018-01-01

['gravitation', 'gravitational waves', 'stars kinematics and dynamics', 'clusters globular', '-']

[]

We show that nearly half of all binary black hole (BBH) mergers dynamically assembled in globular clusters have measurable eccentricities (e &gt; 0.01) in the LISA band (10<SUP>-2</SUP> Hz), when General Relativistic corrections are properly included in the N-body evolution. If only Newtonian gravity is included, the derived fraction of eccentric LISA sources is significantly lower, which explains why recent studies all have greatly underestimated this fraction. Our findings have major implications for how to observationally distinguish between BBH formation channels using eccentricity with LISA, which is one of the key science goals of the mission. We illustrate that the relatively large population of eccentric LISA sources reported here originates from BBHs that merge between hardening binary-single interactions inside their globular cluster. These results indicate a bright future for using LISA to probe the origin of BBH mergers.

[]

https://arxiv.org/pdf/1804.06519.pdf

{'No Header': ', 1-6 (2018)', 'Black Hole Mergers From Globular Clusters Observable by LISA I: Eccentric Sources Originating From Relativistic N -body Dynamics': "Johan Samsing 1 ,/star , Daniel J. D'Orazio 2 \n1 Department of Astrophysical Sciences, Princeton University, Peyton Hall, 4 Ivy Lane, Princeton, NJ 08544, USA 2 Department of Astronomy, Harvard University, 60 Garden Street Cambridge, MA 01238, USA \nAccepted XXX. Received YYY; in original form ZZZ", 'ABSTRACT': 'We show that nearly half of all binary black hole (BBH) mergers dynamically assembled in globular clusters have measurable eccentricities ( e > 0 . 01) in the LISA band (10 -2 Hz), when General Relativistic corrections are properly included in the N -body evolution. If only Newtonian gravity is included, the derived fraction of eccentric LISA sources is significantly lower, which explains why recent studies all have greatly underestimated this fraction. Our findings have major implications for how to observationally distinguish between BBH formation channels using eccentricity with LISA, which is one of the key science goals of the mission. We illustrate that the relatively large population of eccentric LISA sources reported here originates from BBHs that merge between hardening binary-single interactions inside their globular cluster. These results indicate a bright future for using LISA to probe the origin of BBH mergers. \nKey words: gravitation - gravitational waves - stars: black holes - stars: kinematics and dynamics - globular clusters: general', '1 INTRODUCTION': "Gravitational waves (GWs) from merging binary black holes (BBHs) have been observed (Abbott et al. 2016b,c,a, 2017a,b); however with the sparse sample collected to far, it is not clear where and how these BBHs formed in our Universe. From a theoretical perspective, several formation channels have been suggested including isolated field binaries (Dominik et al. 2012, 2013, 2015; Belczynski et al. 2016b,a), dense stellar clusters (Portegies Zwart & McMillan 2000; Banerjee et al. 2010; Tanikawa 2013; Bae et al. 2014; Rodriguez et al. 2015, 2016a,b,b; Askar et al. 2017; Park et al. 2017), single-single GW captures of primordial BHs (Bird et al. 2016; Cholis et al. 2016; Sasaki et al. 2016; Carr et al. 2016), active galactic nuclei discs (Bartos et al. 2017; Stone et al. 2017; McKernan et al. 2017), galactic nuclei (O'Leary et al. 2009; Hong & Lee 2015; VanLandingham et al. 2016; Antonini & Rasio 2016; Hoang et al. 2017), and very massive stellar mergers (Loeb 2016; Woosley 2016; Janiuk et al. 2017; D'Orazio & Loeb 2017). Although these proposed pathways seem to give rise to similar merger rates and observables, recent work interestingly suggests that careful measurements of the BBH orbital eccentricity and relative spins might be the key to disentangling them. For example, BBH mergers forming as a result of field binary evolution are likely to have correlated spin orientations, except if a third object is bound and the three objects form a hierarchical triple \n( e.g. , Liu & Lai 2017; Antonini et al. 2017), whereas BBH mergers forming in clusters are expected to have randomized orientations due to frequent exchanges ( e.g. , Rodriguez et al. 2016c). Regarding eccentricity, it was recently shown by Samsing (2017) that ≈ 5% of all BBH mergers forming in globular clusters (GCs) are likely to have a notable eccentricity ( > 0 . 1) when entering the observable range of the 'Laser Interferometer Gravitational-Wave Observatory' (LIGO). As argued by Samsing (2017), this population originates from GW capture mergers forming in chaotic three-body interactions ( e.g. , Gültekin et al. 2006; Samsing et al. 2014) during classical hardening, which explains why all recent Newtonian N -body studies have failed in resolving the correct fraction. In fact, it was analytically derived in Samsing (2017) that a Newtonian N -body code will always result in a rate of eccentric mergers that is ≈ 100 times lower compared to the (correct) General Relativistic (GR) prediction. These results were recently confirmed by Rodriguez et al. (2017) and Samsing et al. (2018c), using data from simulated GCs. As isolated BBH mergers forming in the field are expected to be circular when entering the LIGO band, these studies show that eccentricity could play a key role in distinguishing formation channels from each other. \nIn this paper we study how BBH mergers that form dynamically in GCs distribute and evolve in the sensitivity band of the proposed 'Laser Interferometer Space Antenna' mission (LISA; Amaro-Seoane et al. 2017), when GR effects are included in the dynamical modeling. Dissipative effects, such as GW emission, which usually are modeled using the post-Newtonian (PN) formalism ( e.g. , \nSamsing, D'Orazio \nBlanchet 2014), have previously been shown to play a crucial role in resolving eccentric LIGO populations ( e.g. , Samsing et al. 2018b); however, the possible effects related to LISA have not yet been properly studied. Our motivation is to explore what can be learned about where and how BBH mergers form in our Universe from a LISA mission; we identify possible observable differences between different BBH populations formed in GCs compared to those formed in the field. Motivated by previous studies, we focus in this paper on the eccentricity distribution. We note that the recent work by Breivik et al. (2016) did indeed look into this; however, the data used for that study did not include GR effects, which we in this paper show are extremely important. \nUsing a semi-analytical approach, we find that ≈ 4 times more BBH mergers will appear eccentric ( > 0 . 01) in the LISA band (10 -2 Hz) compared to the results reported by Breivik et al. (2016), when GR effects are included. This leads to the exciting conclusion that about 40% of all GC BBH mergers are expected to have a measurable eccentricity in the LISA band, whereas a field BBH population in comparison will have ≈ 0%. As we describe, the merger population that leads to this increase in eccentric LISA sources, originates from BBHs that merge between their hardening binarysingle interactions inside their GC ( e.g. , Rodriguez et al. 2017). This population was not included in the recent study by Samsing et al. (2018a), which focused solely on the BBH mergers forming during the binary-single interactions. These BBH mergers where shown to elude the LISA band, and joint observations with LIGO are therefore necessary to tell their GC origin. The fact that BBH mergers can be jointly observed by LISA and LIGO was recently pointed out by Sesana (2016) and Seto (2016). Discussions on BBH merger channels and eccentricity distributions relevant for LISA were presented in Nishizawa et al. (2016) and (Nishizawa et al. 2017). However, we note again that all previous studies have greatly underestimated the fraction of eccentric LISA sources from GCs, mainly due to the omitted GR effects in the data set derived in Rodriguez et al. (2016a). It would be interesting to see how the results presented in this paper affect those previous studies. \nThe paper is organized as follows. In Section 2 we describe the approach we use for modeling the dynamical evolution of BBHs inside GCs and their path towards merger, when GR effects are included in the problem. Our main results are discussed in Section 3, where we show for the first time that, with the inclusion of GR effects, nearly half of all BBH mergers forming in GCs are expected be eccentric in LISA. We conclude our study in Section 4.", '2 BLACK HOLE DYNAMICS IN GLOBULAR CLUSTERS': "In this section we describe the new approach we use in this paper for estimating the distribution of GW frequency, f GW , and eccentricity, e , of BBH mergers forming in globular clusters (GCs). Using this, we explore the possible observable differences between different BBH populations forming in GCs and those forming in the field for an instrument similar to LISA, and the role of GR in that modeling. As described in the Introduction, in the recent work by Breivik et al. (2016) it was claimed that ≈ 10% of the GC mergers will have an eccentricity > 0 . 01 at 10 -2 Hz, compared to ≈ 0% for the field population. However, the simulations used is Breivik et al. (2016) did not treat the relativistic evolution of BBHs inside the GC correctly, which essentially prevented BBHs to merge inside their GCs (see Rodriguez et al. (2017) for a description). To improve on their study we combine in the sections below a simple Monte Carlo (MC) method with the analytical framework from Samsing (2017) \nFigure 1. The graphics in the three columns above illustrate the three different dynamical pathways for merging BBHs to form, each of which result in a different type of GW merger. The horizontal steps from top to bottom illustrate the stepwise decrease in the BBH's SMA due to hardening binary-single interactions, which progresses as δ 0 a HB, δ 1 a HB, ..., until a merger or an ejection takes place. The illustration complements the description of our model from Section 2. In short, our model assumes the BBH in question starts with an SMA = a HB, after which it hardens through binary-single interactions, each of which leads to a decrease in its SMA from a to δ a . This hardening continues until the SMA reaches a ej, below which the BBH will be ejected from the GC through three-body recoil. If the BBH merges outside the GC within a Hubble time, we label it an 'ejected merger' (left column). The ejected merger progenitors form via interactions involving Newtonian gravity alone; however, when GR effects are included, the BBH can also merge inside the cluster, before ejection takes place (e.g. Samsing 2017; Rodriguez et al. 2017). This can happen either between or during its hardening interactions, outcomes we refer to as a '2-body merger' (middle column) and a '3-body merger' (right column), respectively. All previous studies on the eccentricity distribution of LISA sources have only considered the 'ejected mergers'; however, as we show in this paper, the '2-body mergers' clearly dominate the eccentric population observable by LISA ( e > 0 . 01 at 10 -2 Hz). In comparison, the '3-body mergers' dominate the eccentric population observable by LIGO ( e > 0 . 1 at 10 Hz). \n<!-- image --> \nto estimate what the actual BBH eccentricity distribution is expected to be in the LISA band, taking into account that BBHs can form both during and between hardening binary-single interactions ( e.g. , Rodriguez et al. 2017). Although our approach is highly simplified, we do clearly find that GR effects play a central role in such a study. Figure 1 schematically illustrates our dynamical model described below.", '2.1 Binary Black Holes Interacting in Clusters': "We assume that the dynamical history of a BBH in a GC from its formation to final merger follows the idealized model described in Samsing (2017), in which it first forms dynamically at the hardbinary (HB) limit ( e.g. , Heggie 1975; Aarseth & Heggie 1976; \nHut & Bahcall 1983), after which it hardens through equal mass three-body interactions. Each interaction leads to a fixed decrease in its semi-major axis (SMA) from a to δ a , where the average value of δ is 7 / 9 using the distributions from Heggie (1975), as shown by Samsing (2017). For simplicity we will use this value of δ for our modeling. The BBH will harden in this way until it either merges inside the GC, or its three-body recoil velocity exceeds the escape velocity of the GC, v esc, after which it escapes. In this model, such an 'ejection' can only happen if the SMA of the BBH is below the following characteristic value (Samsing 2017), \na ej ≈ 1 6 ( 1 δ -1 ) Gm v 2 esc , (1) \nwhere m is the mass of one of the three interacting (assumed equal mass) BHs. The mergers that are normally considered, using Newtonian prescriptions, are the BBHs that will merge outside the GC, i.e the subset of the ejected BBHs that has a GW lifetime that is less than the Hubble time, t H . However, when GR effects are included, a BBH can also merge inside the GC in at least two different ways ( e.g. , Samsing 2017; Rodriguez et al. 2017): The first way is between its hardening binary-single interactions - a merger type we will refer to in short as a '2-body merger' (2b). A BBH will undergo such a merger if its GW lifetime is shorter than the time it takes for the next interaction to occur. The second way is during its hardening binary-single interactions - a merger type we will refer to in short as a '3-body merger' (3b). Such a merger occurs if two of the three interacting BHs undergo a two-body GW capture merger during the chaotic evolution of the three-body system (Gültekin et al. 2006; Samsing et al. 2014). \nThese three different types of mergers (ejected merger, 2-body merger, and 3-body merger) arise, as described, from different mechanisms that each have their own characteristic time scale (Hubble time, binary-single encounter time, three-body orbital time), which explains why they give rise to different distributions in GW frequency and eccentricity, as will be shown in Section 3. Below we describe how we construct these distributions from our simple model.", '2.2 Deriving Eccentricity and GW Frequency Distributions': "Westart by considering two BHs each with mass m , in a binary with SMA equal to their HB value given by ( e.g. , Hut & Bahcall 1983), \na HB ≈ 3 2 Gm v 2 dis , (2) \nwhere v dis is the velocity dispersion of the interacting BHs. As described in Section 2.1 and shown in Figure 1, we assume that the dynamical evolution of this BBH is governed by isolated binarysingle interactions that lead to a stepwise decrease in its SMA as follows, δ 0 a HB , δ 1 a HB , δ 2 a HB , ..., δ n a HB , ..., until δ N ej a HB ≈ a ej , where n is the n 'th binary-single interaction, and N ej is the number of interactions it takes to bring the BBH to its ejection value. \nFor deriving the BBH merger fractions, GW frequencies, and eccentricity distributions, we perform the following calculations at each interaction step n starting from n = 0, until the BBH either undergoes a merger or escapes the GC: We first estimate if the BBH will undergo a 2-body merger, i.e. merge before the next encounter. For this estimation, we start by calculating the time between successive binary-single interactions, t bs , which can be approximated by ≈ 1 /( n s σ bs v dis ) , where n s is the number density of single BHs, and σ bs is the cross section for a binary-single interaction at step n \n( e.g. , Samsing et al. 2018b). We then derive the GW-inspiral lifetime of the BBH assuming its eccentricity is = 0, denoted by t c, using the prescriptions from Peters (1964). From these two derived time scales, we can then calculate what the minimum eccentricity of the BBH must be for it to undergo a GW merger before its next encounter, denoted by e 2b , which is the solution to the following relation t bs = t c ( 1 -e 2 2b ) 7 / 2 , assuming e 2b /greatermuch 0 (Peters 1964). From this follows, \ne 2b ≈ √ ( 1 - ( t bs / t c ) 2 / 7 ) . (3) \nTo now determine if the BBH will actually undergo a 2-body merger inside the GC at this interaction step n , we draw a value for the eccentricity of the BBH, e , assuming a thermal distribution P ( e ) = 2 e (Heggie 1975). If the drawn eccentricity is ≥ e 2b , the BBH will undergo a 2-body merger, and we record its orbital elements. If the BBHdoes not merge, i.e. if the drawn eccentricity is < e 2b , we then move on to estimate if the BBH instead undergoes a merger during its next binary-single interaction. \nFor estimating the probability of a 3-body merger we use the framework first presented in Samsing et al. (2014), in which the binary-single interaction is pictured as a series of states composed of a binary, referred to as an intermediate state (IMS) binary, and a bound single. As described in Samsing (2017), on average about N IMS ≈ 20 IMS binaries will form per binary-single interaction, each with a SMA that is about the initial SMA of the target binary and an eccentricity that is drawn from the thermal distribution P ( e ) = 2 e . The probability for a 3-body merger to form during the three-body interaction is equal to the probability for an IMS binary to undergo a GW merger within the orbital time of the bound single. To calculate this probability, we first estimate the characteristic pericentre distance an IMS binary must have for it to undergo a GW capture merger during the interaction, a distance we denote by r 3b . Although this distance changes between each IMS in the three-body interaction (Samsing et al. 2014), one finds that on average r 3b is about the distance for which the energy loss through GW emission integrated over one IMS binary orbit is similar to the initial total energy of the three-body system. From this it follows that, r 3b ≈ R m ×( a / R m ) 2 / 7 , where R m here denotes the Schwarzschild radius of a BH with mass m , and a is the SMA of the target binary, which in our step wise hardening series equals δ n a HB for step n (see Samsing 2017). The minimum eccentricity of an IMS BBH needed to undergo a GW capture merger during the interaction is then given by, \ne 3b ≈ 1 -r 3b / a . (4) \nAs for the 2-body mergers, we then draw a value for the IMS BBH eccentricity from the thermal distribution 2 e . We do this up to N IMS = 20 times for each interaction. If one of the drawn eccentricities is ≥ e 3b a 3-body merger has formed and we save its orbital elements. \nIf neither a 2-body nor a 3-body merger has formed at the considered step n , we move on to the next SMA step in the hardening series, which after the last binary-single interaction is now δ n + 1 a HB , and redo the above calculations. If no merger has formed when the BBH SMA falls below a ej , we assume the BBH escapes the cluster with a SMA = a ej . For this BBH we then calculate what its minimum eccentricity must be for it to merge within in a Hubble time, e Ht . Weagain draw from a thermal distribution in eccentricity, and if the value is > e Ht we label the BBH as an ejected BBH merger. \nFor this paper we follow 10 6 such BBHs starting at their a HB from which we then derive BBH merger fractions, frequency and ec-", 'GW freq ency f GW at formation': 'Figure 2. Distribution of GW peak frequency f GW at formation for merging BBHsforming dynamically in GCs. The distributions here are derived using our simple BBH hardening model described in Section 2.1, which combines an MCapproach with the analytical framework presented in (Samsing 2017). Blue: Distribution of BBHs that are ejected from the GC and merge within a Hubble time (ejected mergers). Green: Distribution of BBHs that merge inside the GC between their hardening binary-single interactions (2-body mergers). Red: Distribution of BBHs that merge inside the GC during their hardening binary-single interactions (3-body mergers). The relative contribution from each population depends on the masses of the interacting BHs, the density of single BHs in the GC core, and the escape velocity of the GC; however, all reasonable values lead to about half of all merging BBHs merging inside the cluster (green/red), where about 5% of merging BBHs form in 3-body mergers. We emphasize that the 2-body and 3-body merger populations only can be resolved with GR included in the N -body modeling. \n<!-- image --> \ncentricity distributions, from the above procedure by going through each of the hardening steps. This allows us to investigate the role of GR effects in what effectively corresponds to > 10 7 2.5 PN binary-single scatterings in just a few seconds. Our results relevant for LISA are described below.', '3 ECCENTRIC BLACK HOLE MERGERS IN LISA': 'The following results are derived using the method described above, applied to the scenario for which the interacting BHs are identical, with a mass 30 M /circledot , and for which the population of GCs all have an escape velocity of 50 kms -1 ( e.g. , Harris 1996). We further assume that the number density of single BHs, n s, in each GC core is 10 5 pc -3 . This number is highly uncertain; however, one finds that the relative number of 2-body mergers only scales weakly with density as n -2 / 7 s , which follows from Samsing (2017). Finally, we note that our chosen values robustly result in that ≈ 50% of all BBH mergers are in the form of 2-body mergers, which is in agreement with the recent PN simulations presented in Rodriguez et al. (2017). \nThe distributions of peak GW frequency, f GW , at the time of formation of the BBHs that are merging through the three different pathways considered in this work (ejected merger, 2-body merger, 3-body merger) are shown in Figure 2. For this we used the approximation f GW = π -1 √ 2 Gm / r 3 p , where r p is the pericentre \nFigure3. Distribution of BBH orbital eccentricity e at 10 -2 Hzderivedusing all the BBH mergers from the set presented in Figure 2 that have an initial f GW < 10 -2 Hz. As the 3-body mergers peak at much higher frequencies, the considered set is completely dominated by the ejected (blue) and 2-body (green) mergers. As seen, the 2-body mergers dominate the fraction that will have a resolvable eccentricity ( > 0 . 01) in the LISA band (10 -2 Hz). This population will therefore play a key role in determining the origin of BBH mergers using a LISA-like instrument, as, e.g. , field BBHs are expected to be circular to a much higher degree in LISA. \n<!-- image --> \ndistance at the time of formation of the merging BBH ( e.g. , Wen 2003; Samsing 2017). The ejected mergers (blue) initially distribute at relatively low f GW with a peak between 10 -5 -10 -4 Hz, and will therefore drift through both LISA and LIGO. The possibility for joint observations have been suggested for such a population ( e.g. , Sesana 2016; Seto 2016). The 3-body mergers (red) all have a much higher initial f GW with a peak between 10 0 -10 1 Hz, and will therefore elude the LISA band and form directly in the proposed DECIGO (Kawamura et al. 2011; Isoyama et al. 2018)/Tian Qin (Luo et al. 2016) band before entering the LIGO band ( e.g. , Chen & Amaro-Seoane 2017; Samsing et al. 2018a). We note here that these two distributions are in full agreement with those found in Samsing et al. (2018a), in which the distributions were resolved using full numerical 2.5 PN scatterings using data from the MOCCA code (Giersz et al. 2013; Askar et al. 2017). This validates at least this part of our framework. The 2-body mergers (green) interestingly distribute between the ejected and the 3-body mergers, with a peak only slightly below the maximum sensitivity region of LISA. Aproper understanding and modeling of this population is required for using LISA to determine the origin of BBH mergers. As stated in Section 2, we note that this population has not been studied in this context before. In Paper II of this series we investigate in detail the GW signatures of these three dynamically formed populations in the LISA band. \nThe eccentricity distribution of the BBH mergers evaluated at 10 -2 Hz, near the peak of the LISA sensitivity, is shown in Figure 3. To derive this distribution, we use the evolution equations from Peters (1964) to propagate the initial BBH eccentricity distribution, with initial peak GW frequency f GW < 10 -2 Hz, to the value at f GW = 10 -2 Hz. As seen, the 2-body mergers, i.e. the BBHs that merge between encounters inside the GC, completely domin- \nFigure 4. Cumulative distribution of the eccentricity distributions shown in Figure 3. When the 2-body mergers are included (black), ≈ 40% of merging BBHs will have an eccentricity > 0 . 01 at 10 -2 Hz, near the peak sensitivity of the LISA band. We note that this fraction is about 4 times higher than recently reported by (Breivik et al. 2016), who effectively only considered the ejected population. A substantial fraction of eccentric BBH mergers are therefore expected in LISA if the dynamical GC channel contributes to the BBHmergerrate. This finding should be taken into account when optimizing science cases and instrumental designs. \n<!-- image --> \nate the fraction of mergers that will have an eccentricity resolvable by LISA ( e > 0 . 01). To clarify this statement, Figure 4 shows the corresponding cumulative distribution. As seen, if only the ejected mergers are considered (as was effectively done in Breivik et al. (2016)), then only ≈ 10% will have an eccentricity > 0 . 01 at 10 -2 Hz (blue); however, when the 2-body mergers are included ≈ 40% of all the mergers will have an eccentricity > 0 . 01 (black). This is an important correction, as some recent studies have argued that eccentric populations would hint for BBH mergers forming near massive BHs ( e.g. , Nishizawa et al. 2017). Our results show that GCs can produce eccentric mergers in LISA as well, greatly motivating further and more detailed studies on systems. \nFrom this we conclude that BBH mergers forming in GCs are expected to lead to a notable distribution of eccentric sources ( > 0 . 01) in the LISA band (10 -2 Hz), with a relative fraction that is significantly higher than recently reported by Breivik et al. (2016). This not only shows the importance of a proper inclusion of GR terms in current N -body studies, but also the bright prospects of observationally distinguishing where and how BBH mergers form in our Universe with LISA.', '4 CONCLUSIONS': 'In Paper I of this series, we have explored the role of GR effects in the dynamical evolution of BBHs inside GCs, and found that the population that merges through GW emission between their hardening binary-single interactions, referred to as 2-body mergers, all appear with a notable eccentricity ( > 0 . 01) in the LISA band (10 -2 Hz). Using a simple MC approach together with the analytical framework presented in Samsing (2017), we find that with the inclusion of these 2-body mergers, ≈ 40% of all BBH mergers from GCs will be eccentric in LISA, which is ≈ 4 times more than \nrecently stated by Breivik et al. (2016), in which only Newtonian gravity was included. \nThat GCs are expected to have much richer distributions in eccentricity across the LISA band than previously thought, has important implications for how to observationally distinguish BBH merger channels from each other using LISA, as well as LIGO ( e.g. , Nishizawa et al. 2016; Breivik et al. 2016; Nishizawa et al. 2017; Samsing et al. 2018a). The reason is that different channels will have different eccentricity distributions, e.g. isolated field binaries are believed to have almost circularized once entering LISA, whereas BBHmergers assembled near massive black holes have been shown to have a notable eccentricity in LISA ( e.g. , Nishizawa et al. 2017). \nOur results likewise indicate that the background of unresolved sources observable by LISA, is likely to have a significant fraction of eccentric sources. Including such a population will lead to changes in the expected background spectrum, which often is assumed to be dominated by circular BBHs partly due to the Newtonian results derived in Rodriguez et al. (2016a), that we argue greatly underestimates the true fraction of eccentric sources. In Paper II of this series, we explore the tracks of individually resolvable, eccentric BBHs through the LISA band as well as the effect of unresolvable eccentric systems on the gravitational wave background detectable by LISA, each a result of the GR effects discussed in this paper.', 'ACKNOWLEDGEMENTS': 'It is a pleasure to thank M. Giersz, A. Askar, E. Kovetz, and M. Kamionkowski for helpful discussions. J.S. acknowledges support from the Lyman Spitzer Fellowship. D.J.D. acknowledges financial support from NASA through Einstein Postdoctoral Fellowship award number PF6-170151. D.J.D. also thanks Adrian PriceWhelan and Lauren Glattly for their hospitality during the conception of this work.', 'References': "Aarseth S. J., Heggie D. C., 1976, A&A, 53, 259 \nAbbott B. P., et al., 2016a, Physical Review X, 6, 041015 \nAbbott B. P., et al., 2016b, Physical Review Letters, 116, 061102 \nAbbott B. P., et al., 2016c, Physical Review Letters, 116, 241103 \nAbbott B. 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2021PhRvD.103f3538M

Threshold for primordial black holes. II. A simple analytic prescription

2021-01-01

['-', '-', '-']

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Primordial black holes could have been formed in the early universe from nonlinear cosmological perturbations reentering the cosmological horizon when the Universe was still radiation dominated. Starting from the shape of the power spectrum on superhorizon scales, we provide a simple prescription, based on the results of numerical simulations, to compute the threshold δ<SUB>c</SUB> for primordial black hole formation. Our procedure takes into account both the nonlinearities between the Gaussian curvature perturbation and the density contrast and, for the first time in the literature, the nonlinear effects arising at horizon crossing, which increase the value of the threshold by about a factor two with respect to the one computed on superhorizon scales.

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https://arxiv.org/pdf/2011.03014.pdf

{'The Threshold for Primordial Black Hole Formation: a Simple Analytic Prescription': "Ilia Musco, 1, 2, ∗ Valerio De Luca, 1, † Gabriele Franciolini, 1, ‡ and Antonio Riotto 1, 3, § \n1 D'epartement de Physique Th'eorique and CAP, Universit'e de Gen'eve, 24 quai E. Ansermet, CH-1211 Geneva, Switzerland 2 Instituto Galego de F'ısica de Altas Enerx'ıas, Universidade de Santiago de Compostela, E-15782 Santiago de Compostela, Spain 3 INFN, Sezione di Roma, Piazzale Aldo Moro 2, 00185, Roma, Italy \nPrimordial black holes could have been formed in the early universe from non linear cosmological perturbations re-entering the cosmological horizon when the Universe was still radiation dominated. Starting from the shape of the power spectrum on superhorizon scales, we provide a simple prescription, based on the results of numerical simulations, to compute the threshold δ c for primordial black hole formation. Our procedure takes into account both the non linearities between the Gaussian curvature perturbation and the density contrast and, for the first time in the literature, the non linear effects arising at horizon crossing, which increase the value of the threshold by about a factor two with respect to the one computed on superhorizon scales.", 'I. INTRODUCTION AND SUMMARY': 'It has been suggested that Primordial Black Holes (PBHs) might form in the radiation dominated era of the early Universe by gravitational collapse of sufficiently large-amplitude cosmological perturbations [1-3] (see Refs. [4, 5] for recent reviews), and that they can comprise a significant fraction of the dark matter in the universe, see Ref. [6] for a review of the current experimental constraints on the PBH abundance. This idea has recently received renewed attention given the possibility that PBHs might have given rise to gravitational waves detected during the O1/O2 and O3 observational runs [710] by the LIGO/Virgo Collaboration. This has motivated several studies on the primordial origin of these events [11-26]. In particular, the GWTC-2 catalog is found to be compatible with the primordial scenario [27]. Furthermore, a possible detection of a stochastic gravitational wave background by the NANOGrav collaboration [28] could be ascribed to PBHs [29-34]. \nDespite some pioneering numerical studies [35-37], it has only recently become possible to fully understand the mechanism of PBH formation with detailed spherically symmetric numerical simulations [38-41], showing that a cosmological perturbation collapses to a PBH if it has an amplitude δ greater than a certain threshold value δ c . This quantity has been estimated initially using a simplified Jeans length argument in Newtonian gravity [42], obtaining δ c ∼ c 2 s , where c 2 s = 1 / 3 is the sound speed of the cosmological radiation fluid measured in units of the speed of light. More recently, this value has been refined generalising the Jeans length argument with the theory of General Relativity, obtaining a value of δ c glyph[similarequal] 0 . 4 for a radiation dominated Universe [43]. This analytical computation gives just a lower bound for the value of the threshold because it is not able to account for the non \nlinear effects of pressure gradients, which require full numerical relativistic simulations. A recent detailed study has shown that there is a clear relation between the value of the threshold δ c and the initial curvature (or energy density) profile, with 0 . 4 ≤ δ c ≤ 2 / 3, where the shape is identified by a single parameter [44, 45]. \nA consistent way to measure the amplitude of a perturbation is by using the relative mass excess inside the length scale of the perturbation, that for a consistent comparison between different shapes, should be measured at horizon crossing, when the length scale of the perturbation is equal to the cosmological horizon [44]. \nNumerical simulations have also shown that the mechanism of critical collapse discovered by Choptuik [46] is arising during the formation of PBHs, characterising the mass spectrum [47]. A crucial aspect to fully describe this mechanism was the implementation of an Adaptive Mesh Refinement (AMR), which allows study of the critical behavior down to very small values of ( δ δ c ) [48, 49]. \nNumerical simulations modelling PBH formation start from initial conditions specified on superhorizon scales, when the curvature perturbations describing adiabatic perturbations are time independent [50]. This allows expression of the initial conditions of the numerical simulations, such as the energy density and velocity field, only in terms of a time independent curvature profile [51, 52], which can be derived, in the Gaussian approximation using peak theory [53], from the shape of the inflationary power spectrum of cosmological perturbations measured on superhorizon scales [54, 55]. \n- \nThe relation between the shape of the peak of the curvature power spectrum and the initial conditions used in simulations for PBH formation has recently been investigated in both the Gaussian approximation, with the aim of obtaining a proper estimate of the cosmological abundance of PBHs [54-57], and including also corrections coming from non linearities [58-62] and nonGaussianities [63-67]. On the other hand, numerical simulations have been used to reconstruct the shape of the peak of the inflationary power spectrum, understanding to which extent this is consistent with the observational constraint for PBH formation on different scales [68]. \n- 1. The power spectrum of the curvature perturbation : take the primordial power spectrum P ζ of the Gaussian curvature perturbation and compute, on superhorizon scales, its convolution with the transfer function T ( k, η ) \nP ζ ( k, η ) = 2 π 2 k 3 P ζ ( k ) T 2 ( k, η ) . \n- 2. The comoving length scale ˆ r m of the perturbation is related to the characteristic scale k ∗ of the power spectrum P ζ . Compute the value of k ∗ ˆ r m by solving the following integral equation \n∫ dkk 2 [ ( k 2 ˆ r 2 m -1) sin( k ˆ r m ) k ˆ r m +cos( k ˆ r m ) ] P ζ ( k, η ) = 0 . \n- 3. The shape parameter: compute the corresponding shape parameter α of the collapsing perturbation, including the correction from the non linear effects, by solving the following equation \nF ( α ) [1 + F ( α )] α = -1 2 [ 1+ˆ r m ∫ dkk 4 cos ( k ˆ r m ) P ζ ( k, η ) ∫ dkk 3 sin ( k ˆ r m ) P ζ ( k, η ) ] \nF ( α ) = √ 1 -2 5 e -1 / α α 1 -5 / 2 α Γ ( 5 2 α ) -Γ ( 5 2 α , 1 α ) . \n- 4. The threshold δ c : compute the threshold as function of α , fitting the numerical simulations.\n- · At superhorizon scales making a linear extrapolation at horizon crossing ( aHr m = 1). \nδ c glyph[similarequal] α 0 . 047 -0 . 50 0 . 1 glyph[lessorsimilar] α glyph[lessorsimilar] 7 α 0 . 035 -0 . 475 7 glyph[lessorsimilar] α glyph[lessorsimilar] 13 α 0 . 026 -0 . 45 13 glyph[lessorsimilar] α glyph[lessorsimilar] 30 \n- · At horizon crossing taking into account also the non linear effects. \nδ c glyph[similarequal] α 0 . 125 -0 . 05 0 . 1 glyph[lessorsimilar] α glyph[lessorsimilar] 3 α 0 . 06 +0 . 025 3 glyph[lessorsimilar] α glyph[lessorsimilar] 8 1 . 15 α glyph[greaterorsimilar] 8 \nThe difference between these two values of the threshold δ c is discussed later in section V. \nThe aim of the present paper is to enable the interested reader to calculate the threshold for PBH formation, when the Universe is still radiation dominated, without the need for running numerical simulations. Although non linear cosmological density perturbations are \ndescribed by a non Gaussian random field, we provide a simple prescription to compute the threshold δ c from the shape of the Gaussian inflationary power spectrum. The algorithm, divided into a few simple steps, accounts for both the non linearities associated with the relation between the Gaussian curvature perturbation and the density contrast as well as for those, so far neglected in the present literature, arising at horizon crossing. While a more refined description of the various steps will be found in the rest of the paper, we here provide the reader in the adjacent column with an overview. \nFollowing the present Introduction, Section II reviews the mathematical formulation of the problem. In Section III we discuss the relation between the threshold δ c and the shape of cosmological perturbation. In Section IV we show how to compute the typical value of the threshold δ c as a function of the shape of the power spectrum, analysing in detail some explicit examples. In Section V we compute, using numerical simulations, the amplitude of the threshold δ c as a function of the shape parameter, discussing the difference when this is computed at horizon crossing or on superhorizon scales. Finally in Section VI conclusions are presented, making a summary of the results. Throughout we use c = G = 1.', 'A. Gradient expansion': "PBHs form from the collapse of non-linear cosmological perturbations after they re-enter the cosmological horizon. Following the standard result for extreme peaks we assume spherical symmetry on superhorizon scales [53]. The local region of the Universe characterised by such perturbations is described by an asymptotic form of the metric, usually written as \nd s 2 = -d t 2 + a 2 ( t ) [ d r 2 1 -K ( r ) r 2 + r 2 dΩ 2 ] = -d t 2 + a 2 ( t ) e 2 ζ (ˆ r ) [ dˆ r 2 + ˆ r 2 dΩ 2 ] , (1) \nwhere a ( t ) is the scale factor, while K ( r ) and ζ (ˆ r ) are the conserved comoving curvature perturbations defined on a super-Hubble scale, converging to zero at infinity where the Universe is taken to be unperturbed and spatially flat. The equivalence between the radial and the angular parts gives \n r = ˆ re ζ (ˆ r ) , d r √ 1 -K ( r ) r 2 = e ζ (ˆ r ) dˆ r , (2) \nand the difference between the two Lagrangian coordinates r and ˆ r is related to the 'spatial gauge' of the comoving coordinate, which is fixed by the form chosen to specify the curvature perturbation put into the \nmetric, i.e. K ( r ) or ζ (ˆ r ). From a geometrical point of view the coordinate ˆ r considers the perturbed region as a local FRW separated universe, with the curvature perturbation ζ (ˆ r ) modifying the local expansion, while the curvature profile K ( r ) is defined with respect to the background FRW solution ( K = 0). Combining the two expressions in (2) one gets \nK ( r ) r 2 = -ˆ rζ ' (ˆ r ) [2 + ˆ rζ ' (ˆ r )] , (3) \nshowing that K ( r ) is more directly related to the spatial geometry of the spacetime, obtained as a quadratic correction in terms of ˆ rζ ' (ˆ r ). \nOn the superhorizon scales, where the curvature profile is time independent, we use the gradient expansion approach [39, 51, 69, 70], based on expanding the time dependent variables such as energy density and velocity profile, as power series of a small parameter glyph[epsilon1] glyph[lessmuch] 1 up to the first non zero order, where glyph[epsilon1] is conveniently identified with the ratio between the Hubble radius and the length scale of the perturbation. This approach reproduces the time evolution of linear perturbation theory but also allows having non linear curvature perturbations if the spacetime is sufficiently smooth on the scale of the perturbation (see [50]). This is equivalent to saying that pressure gradients are small when glyph[epsilon1] glyph[lessmuch] 1 and are not then playing an important role in the evolution of the perturbation. \nIn this approximation, the energy density profile can be written as [44, 52] \nδρ ρ b ≡ ρ ( r, t ) -ρ b ( t ) ρ b ( t ) = 1 a 2 H 2 3(1 + w ) 5 + 3 w [ K ( r ) r 3 ] ' 3 r 2 = -1 a 2 H 2 4(1 + w ) 5 + 3 w e -5 ζ (ˆ r ) / 2 ∇ 2 e ζ (ˆ r ) / 2 , (4) \nwhere H ( t ) = ˙ a ( t ) /a ( t ) is the Hubble parameter, ρ b is the mean background energy density and K ' ( r ) denotes differentiation with respect to r while ζ ' (ˆ r ) and ∇ 2 ζ (ˆ r ) denote differentiation with respect to ˆ r . The parameter w is the coefficient of the equation of state p = wρ relating the total (isotropic) pressure p to the total energy density ρ . From now on we are going to consider just the standard scenario for PBH formation assuming a radiation dominated Universe with w = 1 / 3.", 'B. The compaction function': "The criterion to distinguish whether a cosmological perturbation is able to form a PBH depends on the amplitude measured at the peak of the compaction function [39, 44] defined as \nC ≡ 2 δM ( r, t ) R ( r, t ) , (5) \nwhere R ( r, t ) is the areal radius and δM ( r, t ) is the difference between the Misner-Sharp mass within a \nsphere of radius R ( r, t ), and the background mass M b ( r, t ) = 4 πρ b ( r, t ) R 3 ( r, t ) / 3 within the same areal radius but calculated with respect to a spatially flat FRW metric. In the superhorizon regime (i.e. glyph[epsilon1] glyph[lessmuch] 1) the compaction function is time independent, and is simply related to the curvature profile by \nC = 2 3 K ( r ) r 2 = -2 3 ˆ rζ ' (ˆ r ) [2 + ˆ rζ ' (ˆ r )] . (6) \nAs shown in [44], the comoving length scale of the perturbation is the distance from r = r m , where the compaction function reaches its maximum (i.e. C ' ( r m ) = 0), which gives \nK ( r m ) + r m 2 K ' ( r m ) = 0 , (7) \nor \nζ ' (ˆ r m ) + ˆ r m ζ '' (ˆ r m ) = 0 . (8) \nGiven the curvature profile, the parameter glyph[epsilon1] of the gradient expansion is defined as \nglyph[epsilon1] ≡ R H ( t ) R b ( r m , t ) = 1 aHr m = 1 aH ˆ r m e ζ (ˆ r m ) , (9) \nwhere R H = 1 /H is the cosmological horizon and R b ( r, t ) = a ( t ) r is the background component of the areal radius. With these definitions, the expression written in Eq. (4) is valid for glyph[epsilon1] glyph[lessmuch] 1.", 'C. The perturbation amplitude and the threshold': "We are now able to define consistently the perturbation amplitude as being the mass excess of the energy density within the scale r m , measured at the cosmological horizon crossing time t H , defined when glyph[epsilon1] = 1 ( aHr m = 1). Although in this regime the gradient expansion approximation is not very accurate, and the horizon crossing defined in this way is only a linear extrapolation, this provides a well defined criterion to measure consistently the amplitude of different perturbations, understanding how the threshold is varying because of the different initial curvature profiles (see [44] for more details). Later in Section V we are going to extend the present discussion to include the non linear effect on the threshold when the cosmological horizon crossing is fully computed with numerical simulations. \nThe amplitude of the perturbation measured at t H , which we refer to as δ m ≡ δ ( r m , t H ), is given by the excess of mass averaged over a spherical volume of radius R m , defined as \nδ m = 4 π V R m ∫ R m 0 δρ ρ b R 2 d R = 3 r 3 m ∫ r m 0 δρ ρ b r 2 d r , (10) \nwhere V R m = 4 πR 3 m / 3. The second equality is obtained by neglecting the higher order terms in glyph[epsilon1] , approximating \nR m glyph[similarequal] a ( t ) r m , which allows to simply integrate over the comoving volume of radius r m . Inserting the expression for δρ/ρ b given by (4) into (10), one obtains δ m = C ( r m ) and a simple calculation seen in [44] gives the fundamental relation \nδ m = 3 δρ ρ b ( r m , t H ) . (11) \nPBHs form when the perturbation amplitude δ m > δ c , where the value of the threshold δ c depends on the shape of the energy density profile, with 2 / 5 ≤ δ c ≤ 2 / 3, as shown in [44]. Defining the quantity Φ ≡ -ˆ rζ ' (ˆ r ) we can write δ m as \nδ m = 4 3 Φ m ( 1 -1 2 Φ m ) (12) \nwhere Φ m = Φ(ˆ r m ), and the corresponding threshold for Φ is such that 0 . 37 glyph[lessorsimilar] Φ c 1. \nThis shows that there are two different values of Φ m corresponding to the same value of δ m , with a maximum value of δ m = 2 / 3 for Φ m = 1. This degeneracy in the amplitude of the perturbation measured with δ m is related to the difference between cosmological perturbations of Type I and Type II that have been carefully analysed in [71]. Here we review this analysis in the context of PBHs formation. \n≤ \nThe quantity Φ m measures the perturbation amplitude in terms of the local curvature, uniquely defined, while the quantity δ m is related to the global geometry, related to the compactness of the region or radius r m , which has a degeneracy: there are two possible geometrical configurations of the spacetime with the same compactness as shown in Figure 3 of [71]. When Φ > 1 the spatial geometry of the spacetime starts to close on itself, up to Φ = 2 corresponding to the Separate Universe limit. \nComputing the first and second derivatives of C in terms of Φ gives \nC ' (ˆ r ) = 4 3 Φ ' (ˆ r ) (1 -Φ(ˆ r )) , (13) \nC '' (ˆ r ) = 4 3 [ Φ '' (ˆ r ) (1 -Φ(ˆ r )) -(Φ ' (ˆ r )) 2 ] . (14) \nFor a positive peak of the density contrast δρ/ρ b we have C ' (ˆ r m ) = 0 and C '' (ˆ r m ) < 0, and one can distinguish between PBHs of Type I and Type II from the sign of Φ '' m . \n- · PBHs of Type I: δ c <δ m ≤ 2 / 3 and Φ c < Φ m ≤ 1. In this case δ m is increasing for larger values of Φ m and C ' (ˆ r m ) = 0 implies that Φ ' m = 0, corresponding to the condition for ˆ r m given in (8). When Φ m ≤ 1 we have C '' (ˆ r m ) < 0 corresponding to Φ '' m < 0. In the limiting case of Φ m = 1 we have that both Φ '' m and C '' (ˆ r m ) are converging towards -∞ (see after (23) for more explanations).\n- · PBHs of Type II: 2 / 3 > δ m ≥ 0 and 1 < Φ m ≤ 2. In this case δ m is decreasing for larger values of Φ m , and as before C ' (ˆ r m ) = 0 implies that Φ ' m = 0 \n(see Eq.(8)). For Φ m > 1 we have C '' (ˆ r m ) < 0 while Φ '' m > 0, changing sign with respect to Type I solutions. \nAll of the possible values of the threshold are within the regime of PBHs of Type I, where the mass spectrum of PBHs has a behavior described by the scaling law of critical collapse [44] \nM PBH = K ( δ m -δ c ) γ M H , (15) \nwith γ glyph[similarequal] 0 . 36 for a radiation dominated fluid, where M H indicates the mass of the cosmological horizon measured at time t H and K is a coefficient depending on the particular profile of δρ/ρ b . Numerical simulations have shown that 1 glyph[lessorsimilar] K glyph[lessorsimilar] 10, and that (15) is valid with γ constant when δ m -δ c glyph[lessorsimilar] 10 -2 .", 'III. THE SHAPE PARAMETER': "As seen in [44, 45], the threshold for PBHs depends on the shape of the cosmological perturbation, characterised by the width of the peak of the compaction function, measured by a dimensionless parameter defined as \nα = -C '' ( r m ) r 2 m 4 C ( r m ) , (16) \nwhere the family of curvature profiles K ( r ) given by \nK ( r ) = A exp [ -1 α ( r r m ) 2 α ] (17) \nidentifies a basis of profiles which describes the main features of all of the possible shapes. In Figure 1 - taken from [44] - one can see the energy density profile δρ/ρ b plotted against r/r m , obtained by inserting (17) into (4) for different values of α , normalised at horizon crossing ( aHr m = 1). The shape of the energy density contrast becomes peaked for α < 1 (red lines) corresponding to a broad profile of the compaction function, where the dashed line describes the typical Mexican-hat profile ( α = 1). On the contrary the shape of the compaction function C is more peaked for values of α > 1 (blue lines), corresponding to broad profiles of the density contrast. \nIt is important to appreciate that when replacing (17) or any other K -profile into (16), the value of α is independent of the amplitude δ m = C ( r m ) of the perturbation, related to the peak A of the curvature profile: this value is just an overall factor which cancels out in the ratio between the second derivatives and the value of the peak of the compaction function. The parameter α is therefore distinguishing between different shapes of the perturbation, independently of their amplitude. \nAs shown in [45], the average value of C ( r ) integrated over a volume of comoving radius r m , defined as \n¯ C ( r m ) = 3 r 3 m ∫ r m 0 C ( r ) r 2 d r , (18) \nFIG. 1. This figure, taken from [44], shows the behavior of δρ/ρ b given by (4) plotted against r/r m when aHr m = 1, for α = 0 . 5 , 0 . 75 , 1 , 2 , 3 , 5 , 10. The profiles with α ≤ 1 are plotted with a red line (a dashed line for α = 1) while blue lines are used for profiles with α > 1. \n<!-- image --> \nhas a nearly constant value when computed at the threshold for PBH formation, which is ¯ C c glyph[similarequal] 2 / 5. This allows derivation of an analytic expression to compute the threshold δ c as a function of the shape parameter α , up to a few percent precision [45] \nδ c glyph[similarequal] 4 15 e -1 / α α 1 -5 / 2 α Γ ( 5 2 α ) -Γ ( 5 2 α , 1 α ) , (19) \nwhere Γ identifies the special Gamma-functions. This is consistent with the analysis made in [44] where it was shown that the effects of additional parameters modifying the simple basis given by (17) are negligible. \nThe corresponding peak amplitude δρ 0 /ρ b , corresponding to the overdensity amplitude evaluated at the centre of symmetry, is related to the value of δ m by δρ 0 /ρ b = e 1 / α δ m , which combined with (19) gives \n( δρ 0 ρ b ) c glyph[similarequal] 4 15 α 1 -5 / 2 α Γ ( 5 2 α ) -Γ ( 5 2 α , 1 α ) . (20) \nThe shape parameter α describes the main features of the profile in the region 0 < r glyph[lessorsimilar] r m where PBHs form, while any other additional parameters describe only secondary modification of the tail, r glyph[greaterorsimilar] r m , giving only a few percent deviation of the value of δ c with respect to the one obtained with (17). \nThe shape is not correlated with the amplitude of the perturbation when the shape is measured in the r -gauge of the comoving coordinate, while a correlation arises \nwhen measured in the ˆ r -gauge. Using the coordinate transformation of (2) one obtains that \nC '' ( r m ) = 1 e 2 ζ (ˆ r m ) [1 + ˆ r m ζ (ˆ r m )] 2 C '' (ˆ r m ) , (21) \nwhere the additional term proportional to C ' (ˆ r ) is equal to zero when calculated at ˆ r m because of (8). As stated in the introduction, the prime denotes spatial derivative with respect to the variable written explicitly in the argument of the function. The shape parameter can therefore be written as \nα = -C '' (ˆ r m )ˆ r 2 m 4 C (ˆ r m ) [ 1 -3 2 C (ˆ r m ) ] , (22) \nshowing that the peak of the compaction function does not cancel out with the peak of the second derivative, when computed with respect to ˆ r instead of r . \nUsing (12), this can be written as \nα = -Φ '' m ˆ r 2 m 4Φ m ( 1 -1 2 Φ m ) (1 -Φ m ) , (23) \nshowing that in general, when varying the amplitude of the perturbation, the values of Φ '' m and Φ m are not independent, but correlated, changing according to the given value of α . It is interesting to note that both Type I (Φ '' m ≤ 0 , Φ m ≤ 1) and Type II (Φ '' m > 0 , Φ m > 1) perturbations have α > 0, consistently with (16). \nIn general there is a correlation between the shape of Φ and the amplitude of the peak Φ m . In the upper limit of the Type I solution, when Φ m → 1, one finds α →∞ which implies from (14) that Φ '' m →-∞ because C '' ( r m ) < 0 for any positive peak of the compaction function. From the geometrical point of view the shape of the compaction function is forced to be a Dirac delta (a top hat in the energy contrast) when Φ m = 1, corresponding to the threshold for PBH formation when α . \nTo give an explicit example of the correlation between the amplitude and the shape of ζ (ˆ r ) we can consider the profile used in [59] \n→∞ \nζ (ˆ r ) = B exp [ -( ˆ r ˆ r m ) 2 β ] , (24) \nthat inserted into the (23) gives \nα = β 2 (1 -βζ (ˆ r m ))(1 -2 βζ (ˆ r m )) . (25) \nIn the linear approximation B glyph[lessmuch] 1 ⇒ βζ (ˆ r m ) glyph[lessmuch] 1, which gives α glyph[similarequal] β 2 , showing that for a given value of α , the corresponding value of β is fixed and there is no correlation between the shape and the amplitude, while when we are considering a perturbation amplitude of the order of the threshold δ c , one has B ∼ 1 (corresponding to A r 2 m ∼ 1) and the correlation is not negligible. For example, when α = 1 one has a typical Mexican-hat shape \nand a value of the threshold δ c glyph[similarequal] 0 . 5, while for a value of β = 1 corresponding to a Mexican-hat shape in the linear approximation, the value of the threshold is δ c glyph[similarequal] 0 . 55, as seen in [59].", 'The non linear component of the shape': "If ζ is a Gaussian random variable, also Φ m , and Φ '' m ˆ r 2 m obey Gaussian statistics. In such case we can write the shape parameter given by (22) as \nα = α G ( 1 -1 2 Φ m ) (1 -Φ m ) , (26) \nwhere \nα G = -Φ '' m ˆ r 2 m 4Φ m (27) \nis the Gaussian shape parameter obtained in the linear approximation (Φ m glyph[lessmuch] 1), independent of the amplitude of Φ m since Φ '' m ∝ Φ m as, for instance, one can understand by computing the average of Φ '' m given a realisation of Φ m using conditional probability. \nThe value of Φ m introduces a correction, which is negligible in the linear regime when Φ m glyph[lessmuch] 1. On the other hand, when the value of Φ m is non linear, the term (1 -Φ m )(1 -Φ m / 2) gives a non negligible modification of the value of α with respect α G . In general α depends on the statistics of Φ '' m and the amplitude Φ m . \nConsidering Type I solutions one can write Φ m as a function of δ c using (12), which gives \nΦ m = 1 -√ 1 -3 2 δ c , (28) \nand then inserting this equation combined with (19) into (26) one obtains \nF ( α ) [1 + F ( α )] α = 2 α G , (29) \nwhere \nF ( α ) = √ 1 -2 5 e -1 / α α 1 -5 / 2 α Γ ( 5 2 α ) -Γ ( 5 2 α , 1 α ) . (30) \nThe numerical solution of equation (29) gives a value of α as a function of α G . By inserting this into (19), one can compute the value of δ c as a function of α G , which is plotted in the left panel of Figure 2 using a solid line. This is compared with the analytic behavior of δ c G = δ c ( α G ) plotted with the dashed line. \nThe right panel of Figure 2 shows the ratio of these two quantities as function of α G , and one can appreciate the correction of δ c due to the modification of the shape with respect to the one obtained in the Gaussian approximation, because of the non linear effects. Because at the boundaries F (0) → 1 ( F ( ∞ ) = 1), there is no modification with respect to the Gaussian case and δ c = δ c G in the limits α → 0 ( α →∞ ).", 'IV. THE AVERAGE VALUE OF δ c': "The aim of this section is to describe how to calculate the average value of the shape parameter α , identifying which is the typical perturbation shape associated with a given cosmological power spectrum, which gives the corresponding averaged value of the threshold δ c . \nAssuming Gaussian statistics for the comoving curvature perturbation ζ , the first step is to compute the value of α G from the power spectrum P ζ ( k, η ) defined as \nP ζ ( k, η ) = 2 π 2 k 3 P ζ ( k ) T 2 ( k, η ) , (31) \ncomputed at the proper time η when ˆ r m glyph[greatermuch] r H , where r H = 1 /aH is the comoving Hubble radius. P ζ ( k ) is the dimensionless form of the power spectrum, and the linear transfer function T ( k, η ), given by \nT ( k, η ) = 3 sin( kη/ √ 3) -( kη/ √ 3) cos( kη/ √ 3) ( kη/ √ 3) 3 , (32) \nhas the effect of smoothing out the subhorizon modes, playing the role of the pressure gradients during the collapse. This smoothing should be done when ˆ r m glyph[similarequal] 10 r H or larger, according to the gradient expansion approach used to specify the initial condition of the numerical simulations. This ensures that modes collapsing within the scale r H does not affect the collapse on the larger scale ˆ r m . The details of how to apply the smoothing have been extensively discussed in [68], showing that using just the transfer function on superhorizon scales avoids the need of introducing a window function on the scale ˆ r m of the perturbation, which introduces corrections in the calculation of the threshold that, however, are reduced when computing the PBH abundance if the same window function is adopted for evaluating the variance [72]. \nThe radius ˆ r m is obtained from condition (8), which can be expressed in terms of the power spectrum using Gaussian peak theory to write ζ (ˆ r ) \nζ (ˆ r ) = ζ 0 ∫ dkk 2 sin( k ˆ r ) k ˆ r P ζ ( k, η ) , (33) \nand, applying Φ ' (ˆ r m ) = 0, one finally gets \n∫ dkk 2 [ ( k 2 ˆ r 2 m -1) sin( k ˆ r m ) k ˆ r m +cos( k ˆ r m ) ] P ζ ( k, η ) = 0 , (34) \nwhere this integral equation, in general, has to be solved numerically given the expression of P ζ . \nThe Gaussian shape parameter can be computed from the average profile of ζ (ˆ r ) shown in (33), which allows α G to be written as \nα G = 1 2 -ˆ r 2 m 4 ζ ''' (ˆ r m ) ζ ' (ˆ r m ) , (35) \nwhere we have used the constraint relation Φ ' (ˆ r m ) = 0, which gives \nˆ r 2 m Φ '' m = ˆ r m [2 ζ ' (ˆ r m ) -ˆ r 2 m ζ ''' (ˆ r m )] . (36) \n<!-- image --> \nFIG. 2. The left panel of this figure shows the behavior of δ c and δ c G = δ c ( α G ) plotted as a function of the Gaussian shape parameter α G . The right panel shows the ratio of these two quantities: the difference is due to the non linear effects coming from the solution of (29). \n<!-- image --> \nInserting (33) into the expression for α G , combined with (34) one obtains \nα G = -1 4 [ 1 + ˆ r m ∫ dkk 4 cos ( k ˆ r m ) P ζ ( k, η ) ∫ dkk 3 sin ( k ˆ r m ) P ζ ( k, η ) ] , (37) \nshowing that α G , and the corresponding value of α computed using (29), are varying with the shape of the cosmological power spectrum. The same holds for the value of ˆ r m given by the solution of (34). The values of α G and α can then be used in (19) so as to calculate the corresponding values of δ c G and δ c , obtaining a direct relation between the threshold and the particular shape of the cosmological power spectrum P ζ . \nIn the following we are going to apply this prescription to study the extent to which, given a particular form of the power spectrum, the amplitude of the threshold δ c is varying.", 'Peaked Power Spectrum': 'The simplest cosmological power spectrum of the comoving curvature perturbation that can be considered is monochromatic, behaving like a Dirac-delta distribution, typically written as \nP ζ ( k ) = P 0 k ∗ δ D ( k -k ∗ ) . (38) \nInserting this into (34) we get k ∗ ˆ r m glyph[similarequal] 2 . 74, which gives δ c, G glyph[similarequal] 0 . 51, a value of the threshold in the Gaussian approximation consistent with the one obtained in [54]. Solving equation (29), we can see the corresponding modification of δ c due to the non linear effects, giving α glyph[similarequal] 6 . 33 corresponding to δ c glyph[similarequal] 0 . 59. \nIt is interesting to note that this value of the threshold is consistent with the one found if the average profile of ζ ( r ) for a peaked power spectrum, characterised by the sync function, is inserted into (4) to specify the initial conditions for the numerical simulations [67]. This is consistent with the fact that using peak theory in ζ or in the density contrast δρ/ρ b is equivalent when the power spectrum is very peaked, behaving like a Dirac delta [60].', 'Broad Power Spectrum': 'Aclass of models with a broad and flat power spectrum of the curvature perturbations of the form [73, 74] \nP ζ ( k ) = P 0 Θ( k -k min )Θ( k max -k ) , k max glyph[greatermuch] k min (39) \nis another simple toy model, corresponding to the top hat shape of the primordial power spectrum, which is considered in [54]. In this case, from (34) we have k max ˆ r m 4 . 49, which gives α G glyph[similarequal] 0 . 9 and δ c, G 0 . 48. \nglyph[similarequal] \nglyph[similarequal] The values of k max ˆ r m and δ c, G obtained here are different from the values k max ˆ r m glyph[similarequal] 3 . 5 and δ c, G glyph[similarequal] 0 . 51 found in [54], because in that analysis peak theory was applied directly to the linearised density contrast δρ/ρ b while here, instead, we are using peak theory to compute the average curvature perturbation ζ and account for the non-linear relation with the compaction function. For this reason the integrals in peak theory for finding ˆ r m and the shape profile ζ (ˆ r ) are characterised by a higher power in k . \nglyph[similarequal] \nSolving equation (29) to include the non linear effects gives α glyph[similarequal] 3 . 14 corresponding to δ c glyph[similarequal] 0 . 56. \n<!-- image --> \nFIG. 3. Left: Peak position of the compaction function ˆ r m for both the case of a gaussian and lognormal shape of the power spectrum. Right: Average threshold for collapse for both the δ c G ( α G ) and its corresponding non-linearly corrected δ c in both cases. In the limit of σ → 0, both spectra converge to the monochromatic case and the result is compatible with the Dirac delta example. \n<!-- image -->', 'Gaussian Power Spectrum': 'The gaussian shape of the curvature power spectrum given by \nP ζ ( k ) = P 0 exp [ -( k -k ∗ ) 2 / 2 σ 2 ] , (40) \nis characterised by the central reference scale k ∗ and width σ . Solving (34), the relation between the length scale ˆ r m of the perturbation and the scale k ∗ is shown in the left panel of Fig. 3. As one can appreciate, in the limit of the narrow case σ → 0 the result converges to the one obtained for a monochromatic shape of the curvature power spectrum (studied previously in the peaked case), while for broader shapes the expected length scale of the overdensity multiplied by k ∗ is decreasing. This is a result of the fact that, for broader shapes, more modes are contributing to the collapse, resulting in a narrower curvature profile. \nThe behavior of the shape parameter α , which decreases as σ increases, reflects the fact that when multiple modes are participating in the collapse, the compaction function becomes flatter. As a consequence, the pressure gradients are reduced, facilitating the collapse, and the corresponding threshold for PBHs decreases for larger values of σ , as one can appreciate in the right panel of the same figure. As discussed in the previous section and shown in Fig. 2, as non-linearities are taken into account, the critical threshold δ c reaches larger values than tat for δ c G computed in the Gaussian approximation.', 'Lognormal Power Spectrum': 'The lognormal power spectrum is expressed as \nP ζ ( k ) = P 0 exp [ -ln 2 ( k/k ∗ ) / 2 σ 2 ] , (41) \ncharacterised by a width σ and a central scale k ∗ . The relation between the length scale of the overdensity and the scale k ∗ is plotted in the left panel of Fig. 3, while the right panel is showing the average threshold for PBHs, showing the same qualitative behaviors found for the Gaussian power spectrum. \nBecause σ in this case identifies the width of the power spectrum in logarithmic space, larger values of σ allow for more modes to be part of the collapse. As a consequence, if compared to the Gaussian case, the trends for the relative change of k ∗ ˆ r m and δ c are amplified.', 'Cut-Power-Law Power Spectrum': 'The cut-power-law curvature power spectrum is given by \nP ζ ( k ) = P 0 ( k k ∗ ) n ∗ s exp [ -( k/k ∗ ) 2 ] , (42) \nexpressed in terms of a tilt n ∗ s and with an exponential cut-off at the momentum scale k ∗ . The relation between the length scale of the overdensity and the scale k ∗ is shown in the left panel of Fig. 4, while the right panel of this figure is showing the behavior of the average threshold for PBHs. \nAs n ∗ s increases, the spectrum becomes narrower, with a shift towards a higher value of the power spectrum \n<!-- image --> \nFIG. 4. The same as in Fig. 3, but for the cut-power-law power spectrum. \n<!-- image --> \npeak which is identified by the maximum of the combined product of k n ∗ s and the exponential cut-off. In agreement with the behavior seen in the previous examples, as the spectral tilt decreases, a larger number of modes participate in the collapse, resulting in a lower value of the threshold δ c .', 'Summary': 'The analysis of this section of different power spectra shows that, when the shape is broader, the value of the threshold δ c is lower because more modes are involved in the collapse. The maximum value we have found is δ c glyph[similarequal] 0 . 59 when the power spectrum behave like a Dirac Delta (corresponding to a single mode). The behaviour for the lognormal power spectrum that one can extrapolate looking at the right panel of Figure 3 indicates the possibility of getting closer to the lower boundary of 0 . 4 for very large values of σ . In conclusion the shapes of the power spectra we have considered here allows 0 . 4 glyph[lessorsimilar] δ c glyph[lessorsimilar] 0 . 6, when the threshold is computed on superhorizon scales.', 'V. THE NON LINEAR HORIZON CROSSING': "In this section we study the effects on the threshold when the cosmological horizon crossing is computed during the numerical evolution, measuring the amplitude δ m of the perturbation when the length scale R m is equal to the cosmological horizon radius R H defined with respect to the perturbed medium. The numerical code used for the simulations is the same as used in previous works (see [44] and references therein for more details). \nThe threshold for PBHs has so far been computed at \ncosmological horizon crossing by making a linear extrapolation from the superhorizon regime, where the curvature is time independent, imposing aHr m = 1 in equation (4), where the cosmological horizon R H = 1 /H is defined with respect to the background. In this way one is extending the validity of the gradient expansion approximation up to glyph[epsilon1] = 1, which is not very accurate. Although this represents a well defined criterion for measuring the perturbation amplitude, and has been widely used in the literature to compute the threshold δ c for PBHs, it does not give the correct amplitude of the perturbation at the 'real' cosmological horizon crossing, because it is neglecting the non linear effects of the higher orders in the gradient expansion approach, which are taking into account that the curvature profile ζ starts to vary with time when glyph[epsilon1] 1. \nIn general the cosmological horizon is a marginally trapped surface within an expanding region, which in spherical symmetry is simply defined by the condition R ( r, t ) = 2 M ( r, t ), where R ( r, t ) is the areal radius and M ( r, t ) is the mass within a given sphere of radius R ( r, t ), called the Misner-Sharp mass. This relation for a trapped surface is very general, assuming only spherical symmetry, and allows computation of the location of any apparent horizon: if we have an expanding medium, this is a cosmological horizon, while if the medium is collapsing then it is a black hole apparent horizon [75, 76]. \n∼ \nIn simulations of PBH formation, because we are in a locally closed Universe, the rate of expansion of the cosmological horizon is less than that of the spatially flat background, and this gives rise to an additional growth of the amplitude of the perturbation before reaching the horizon crossing. \nThe left plot of Figure 5 shows the critical energy density profile obtained with the curvature perturbation given by (17), with α = 1 corresponding to a Mexican-hat \n<!-- image --> \nFIG. 5. The left panel shows the critical Mexican-hat profile of the energy density, obtained from (17) with α = 1, computed at the horizon crossing, linearly extrapolated ( glyph[epsilon1] = 1) with a blue line, and computed numerically at the non linear horizon crossing (red line), corresponding in this case to glyph[epsilon1] glyph[similarequal] 1 . 46. Both profiles are plotted against R/R H , where R H is the radius of the cosmological horizon computed at the corresponding time. The right panel shows how the non linear horizon crossing, measured in terms of glyph[epsilon1] , varies when plotted against the shape parameter α , compared to glyph[epsilon1] = 1 at the linear horizon crossing. The dashed line is a polynomial fit of numerical data given by the dots. \n<!-- image --> \nshape, computed at the horizon crossing linearly extrapolated (blue line) and at the non linear horizon crossing obtained from the numerical simulations (red line). The second profile shows an additional growth of the amplitude, which is not negligible when the value of the energy density obtained with the linear extrapolation is nonlinear. Part of this extra growth is due to the longer time necessary to reach the non linear horizon crossing which can be seen explicitly in the right plot where glyph[epsilon1] ( t H ) is plotted against the shape parameter α , with the dashed line fitting the numerical results given by the dots. We can appreciate that the value of glyph[epsilon1] ( t H ) at the non linear horizon crossing, 1 . 3 glyph[lessorsimilar] glyph[epsilon1] ( t H ) glyph[lessorsimilar] 1 . 5, is larger than one given by the linear horizon crossing ( aHr m = 1). In particular, for α = 1 the non linear horizon crossing is obtained at glyph[epsilon1] ( t H ) glyph[similarequal] 1 . 46, corresponding to an amplitude of the central peak calculated with (4) equal to δρ 0 /ρ b glyph[similarequal] 1 . 98, as compared with the value of δρ 0 /ρ b glyph[similarequal] 1 . 35 computed at the linear horizon crossing (blue line). The additional growth of the profile, with a peak value of the density contrast δρ 0 /ρ b glyph[similarequal] 3 . 34 obtained numerically at the non linear horizon crossing (red line), is explained by the higher orders in the gradient expansion which need to be taken into account when glyph[epsilon1] ∼ 1. This effect is genuinely non linear. \nThe delay of the horizon crossing due to the non linear effects gives an increase of the final mass of PBHs because of the corresponding increase of the cosmolog- \nical horizon mass M H : during the radiation dominated universe M H ∼ glyph[epsilon1] 2 and from Figure 5 one can appreciate the change of the horizon scale introducing a correction value about equal to 2 (i.e. 1 . 7 ÷ 2 . 2) in the cosmological horizon mass. \nIn the left panel of Figure 6 we are comparing the critical energy density profiles obtained from (17) for α = 0 . 15, which gives a very sharp profile, almost like a Dirac-delta, while in the right panel we plot the critical profiles computed for α = 30, which gives a very broad profile, very similar to a top-hat. As with the Mexican-hat shape, the profile in the right frame computed at the non linear horizon crossing is characterised by an extra growth of the peak: the numerical evolution gives δρ 0 /ρ b glyph[similarequal] 1 . 46 at glyph[epsilon1] ( t H ) glyph[similarequal] 1 . 31 as compared with δρ 0 /ρ b glyph[similarequal] 0 . 66 obtained at glyph[epsilon1] = 1 with the linear extrapolation. As in Figure 5 for α = 1, this difference is a result of the combination of the extra linear growth due to the larger value of glyph[epsilon1] and the non linear effects. \nFor the very sharp profile plotted in the left panel ( α = 0 . 15) we can observe instead that the value of the peak amplitude is significantly reduced at the non-linear horizon crossing with respect the one computed with a linear extrapolation at glyph[epsilon1] = 1. This is because for α = 0 . 15 the profile is not smooth in the center, and there is a significant effect of the local pressure gradients, which are smoothing the profile during the evolution, giving at the non linear horizon crossing time a smooth profile with a \n<!-- image --> \nFIG. 6. The two plots of this figure show the critical energy density profiles obtained from (17) with α = 0 . 15 (left panel) and α = 30 (right panel), plotted against R/R H , computed at the horizon crossing linearly extrapolated (blue line) and at the non linear one (red line). \n<!-- image --> \nmuch lower amplitude of the peak: δρ 0 /ρ b glyph[similarequal] 7 as compared with δρ 0 /ρ b glyph[similarequal] 338 linearly extrapolated at glyph[epsilon1] = 1. A similar effect happens in the under dense region for the top-hat like profile ( α = 30). \nIn Figures 5 and 6 we have analysed three sample cases of the energy density profiles, seeing how the shape is modified at the non linear horizon crossing with respect to the one imposed at initial conditions on super horizon scales, discovering the following general behavior: if the profile is initially smooth, the peak amplitude computed at the non linear horizon crossing is higher than the one extrapolated linearly due to non linear effects which give an extra growth factor, while when the profile is sharp the behavior is the opposite, due to the non linear effects of the pressure gradients smoothing the profile. In general very large values of the peak amplitude at horizon crossing are strongly suppressed because of the smoothing induced by the pressure gradients. \nIn general the critical amplitude of the peak δρ c /ρ b depends on the shape, and in the left panel of Figure 7 we can see how this quantity is varying with respect to α , for all of the range of shape described by 0 . 15 ≤ α ≤ 30. The linearly extrapolated values of the critical amplitude of the peak, given by (20), are plotted with a blue line, while the values computed at the non linear horizon crossing are plotted with a red line. \nThe linearly extrapolated critical peak values can be computed analytically from (20) while, as shown in the plot, the critical values computed at the non-linear horizon crossing are given with a good approximation by a \nsimple fit, divided in two regimes. \nδρ c ρ b glyph[similarequal] { 10 0 . 53 -0 . 17 ln α α glyph[lessorsimilar] 8 1 . 52 α glyph[greaterorsimilar] 8 (43) \nIn the right panel of Figure 7 we show the ratio between the critical amplitude computed at the non linear horizon crossing and the one linearly extrapolated. This shows clearly the two different regimes: the first one, for α glyph[lessorsimilar] 8, with the critical amplitude varying with α , and the second one for α glyph[greaterorsimilar] 8 which is almost independent of α , with the peak amplitude converging towards an almost constant value. \nThe linearly extrapolated value is equal to the one computed numerically for α glyph[similarequal] 0 . 45, because the energy density profiles obtained from (17) are not smooth if α ≤ 0 . 5, with a non vanishing first derivative in the center. On the contrary, for α > 0 . 5 the energy density profiles are smooth in the center and the perturbation is free to grow without any relevant smoothing of the shape produced by the pressure gradients, reaching a larger value of the critical peak amplitude at the non linear horizon crossing with respect to the one linearly extrapolated. \nIn Figure 8 the same analysis is made for the threshold δ c , with the left plot showing the threshold δ c ( t i ) linearly extrapolated (blue line) and the threshold δ c ( t H ) computed at the non linear horizon crossing (red line). The linearly extrapolated threshold, described with a very good approximation by the analytic expression of equation (19), can be divided into three different regimes, each \n<!-- image --> \nFIG. 7. The left panel of this figure shows the two behaviors of the critical amplitude of the peak δρ c /ρ b , in one case extrapolated linearly at horizon crossing (blue line) and in the other one computed at the non linear horizon crossing (red line), plotted as function of the shape parameter α . The right panel of this figure shows the corresponding ratio of these two quantities. \n<!-- image --> \none described by a simple fit. \nδ c ( t i ) glyph[similarequal] α 0 . 047 -0 . 50 0 . 1 glyph[lessorsimilar] α glyph[lessorsimilar] 7 α 0 . 035 -0 . 475 7 glyph[lessorsimilar] α glyph[lessorsimilar] 13 α 0 . 026 -0 . 45 13 glyph[lessorsimilar] α glyph[lessorsimilar] 30 (44) \nwhere the first range 0 . 1 glyph[lessorsimilar] α glyph[lessorsimilar] 7 is corresponding with good approximation to all of the shapes of the power spectrum analysed in Section IV, suggesting that the other two ranges are suppressed by the smoothing. They describes energy density profiles which are very sharp around ˆ r m where the threshold is computed, and therefore such profiles are smoothed by the pressure gradients, as we have seen in the right panel of Figure 6, suppressing the values δ c glyph[greaterorsimilar] 0 . 6. \nThis interpretation is enforced when the threshold is computed at the non linear horizon crossing, which is well described by another fit, again divided into three different regimes. \nδ c ( t H ) glyph[similarequal] α 0 . 125 -0 . 05 0 . 1 glyph[lessorsimilar] α glyph[lessorsimilar] 3 α 0 . 06 +0 . 025 3 glyph[lessorsimilar] α glyph[lessorsimilar] 8 1 . 15 α glyph[greaterorsimilar] 8 (45) \nHere the first regime of (44) is basically splitting into two different behaviors of the threshold computed at the non linear horizon crossing time, while the second and the third regimes of (44), corresponding to δ c glyph[greaterorsimilar] 0 . 6 computed at superhorizon scales, saturate to an almost constant value of the threshold when is computed at t H . \nThe right panel of Figure 8 shows that the ratio between δ c ( t H ) and δ c ( t i ), where one can distinguish two different regimes: the first one, when α glyph[lessorsimilar] 3, is corresponding to the increasing behavior of this ratio, and explains the first regime of (45). The second regime, when α glyph[greaterorsimilar] 3, has a decreasing behavior of the ratio between the two thresholds, corresponding to the second and third regime of (45), which can be distinguished in the right panel of Figure 7. \nThe lower ( α glyph[greaterorsimilar] 0 . 1) and the upper ( α glyph[lessorsimilar] 30) boundaries of validity of the fit are given by the numerical simulations that are not able to handle very extreme shapes beyond these values. We are however neglecting only a range of α which is not significant as we are already close enough to the limits of δ c . \nFinally we can observe that the difference between the threshold computed at the non linear horizon crossing and the linearly extrapolated one is an almost constant numerical coefficient, varying between 1 . 7 and 2. This underlines the fact that the threshold δ c is a much more stable quantity than the local critical amplitude of the peak, and has to be preferred for distinguishing between cosmological perturbations forming PBHs and the ones that are bouncing back into the expanding medium.", 'VI. CONCLUSIONS': 'PBHs could have formed in the early universe from the collapse of cosmological perturbations at the horizon \n<!-- image --> \nFIG. 8. The left panel of this figure shows the two behaviors of the threshold δ c in one case extrapolated linearly at horizon crossing (blue line) and in the other one computed at the non linear horizon crossing (red line), plotted as a function of the shape parameter α . The right panel of this figure shows the corresponding ratio of these two quantities. \n<!-- image --> \nre-entry, provided that their amplitude is larger than a certain critical threshold. In this paper we have provided a simple analytical prescription, summarised in Fig. 9, to compute the threshold of collapse for PBHs, embedding results coming from numerical simulations. \nFrom Gaussian curvature perturbations, one can compute the mean profile on superhorizon scales using peak theory and find the characteristic comoving scale of the perturbations from the given shape of the curvature power spectrum. From the computation of the profile shape parameter on superhorizon scales, one can determine the value of the threshold, also taking into account the effects of non-linearities arising at the cosmological horizon crossing fitted from numerical simulations. In particular we stress that the thresholds calculated at horizon crossing differs by a factor of order two from the values traditionally adopted in the literature. \nBy analysing different explicit examples of the curvature power spectrum, we have seen that in general the value of the threshold δ c is larger for a monochromatic power spectrum, modelled by a Dirac delta, than for a broader shape which allows more modes to contribute to the collapse. The latter gives a broader and flatter profile of the compaction function describing a cosmological perturbation collapsing to form a PBH, corresponding to a lower value of δ c . This allows, using (44), to compute the threshold of PBHs measured on super-horizon scales by correctly identifying the shape parameter for a given curvature power spectrum, obtaining 0 . 4 glyph[lessorsimilar] δ c glyph[lessorsimilar] 0 . 6. \nHowever, if the threshold is computed at the non linear horizon crossing time (i.e. around the time when they \nFIG. 9. This diagram summarises our prescription for computing the threshold δ c starting from the power spectrum of cosmological curvature perturbations P ζ . \n<!-- image --> \nare really formed), the physical range of the threshold δ c obtained from (45) for all of the possible shapes of the power spectrum is 0 . 7 glyph[lessorsimilar] δ c ( t H ) glyph[lessorsimilar] 1 . 15 . This might introduce a sizeable contribution in the calculation of the corresponding abundance of PBHs which is exponentially \n<latexit sha1\\_base64="b05Ut0nFaftOumo1ZSFWAqSXXd0=">AAAB+HicbVDLSsNAFJ3UV62PRl26GSxC3YTEB7osiuCygn1AG8pkOm2HTiZx5kasoV/ixoUibv0Ud/6N0zYLbT1w4XDOvdx7TxALrsF1v63c0vLK6lp+vbCxubVdtHd26zpKFGU1GolINQOimeCS1YCDYM1YMRIGgjWC4dXEbzwwpXkk72AUMz8kfcl7nBIwUscutoE9Qnp97+Bx+cQ96tgl13GnwIvEy0gJZah27K92N6JJyCRQQbRueW4MfkoUcCrYuNBONIsJHZI+axkqSci0n04PH+NDo3RxL1KmJOCp+nsiJaHWozAwnSGBgZ73JuJ/XiuB3oWfchknwCSdLeolAkOEJyngLleMghgZQqji5lZMB0QRCiarggnBm395kdSPHe/McW9PS5XLLI482kcHqIw8dI4q6AZVUQ1RlKBn9IrerCfrxXq3PmatOSub2UN/YH3+ACOIkhY=</latexit> Eq. ( \n<latexit sha1\\_base64="XLN2J23Ey27Hv4NdKCKMccpfNq4=">AAAB+HicbVBNS8NAEN34WetHox69LBahXkpiFT0WRfBYwX5AG8pmu2mXbjZxdyLW0F/ixYMiXv0p3vw3btsctPXBwOO9GWbm+bHgGhzn21paXlldW89t5De3tncK9u5eQ0eJoqxOIxGplk80E1yyOnAQrBUrRkJfsKY/vJr4zQemNI/kHYxi5oWkL3nAKQEjde1CB9gjpNf3ZTwuVSrHXbvolJ0p8CJxM1JEGWpd+6vTi2gSMglUEK3brhODlxIFnAo2zncSzWJCh6TP2oZKEjLtpdPDx/jIKD0cRMqUBDxVf0+kJNR6FPqmMyQw0PPeRPzPaycQXHgpl3ECTNLZoiARGCI8SQH3uGIUxMgQQhU3t2I6IIpQMFnlTQju/MuLpHFSds/Kzu1psXqZxZFDB+gQlZCLzlEV3aAaqiOKEvSMXtGb9WS9WO/Wx6x1ycpm9tEfWJ8/KBeSGQ==</latexit> Eq. ( \n<latexit sha1\\_base64="Oug9sDdsper/wY9YoYDqoAPT+wk=">AAAB/XicbVDJTgJBEO3BDXEbl5uXjsQELjiDEjkSjYlHTGRJgJCepgc69Cx21xhxQvwVLx40xqv/4c2/sYE5KPiSSl7eq0pVPScUXIFlfRuppeWV1bX0emZjc2t7x9zdq6sgkpTVaCAC2XSIYoL7rAYcBGuGkhHPEazhDC8nfuOeScUD/xZGIet4pO9zl1MCWuqaB21gDxBf3RXwOFcs509yp6V818xaBWsKvEjshGRRgmrX/Gr3Ahp5zAcqiFIt2wqhExMJnAo2zrQjxUJCh6TPWpr6xGOqE0+vH+NjrfSwG0hdPuCp+nsiJp5SI8/RnR6BgZr3JuJ/XisCt9yJuR9GwHw6W+RGAkOAJ1HgHpeMghhpQqjk+lZMB0QSCjqwjA7Bnn95kdSLBbtUsG7OspWLJI40OkRHKIdsdI4q6BpVUQ1R9Iie0St6M56MF+Pd+Ji1poxkZh/9gfH5A2Gekzc=</latexit> Eq. ( \n<latexit sha1\\_base64="kix+dZetSKJN9CQFdHxlOsHqNGs=">AAAB+HicbVBNS8NAEN34WetHox69LBahXkpSKnosiuCxgv2ANpTNdtMu3Wzi7kSsob/EiwdFvPpTvPlv3LY5aOuDgcd7M8zM82PBNTjOt7Wyura+sZnbym/v7O4V7P2Dpo4SRVmDRiJSbZ9oJrhkDeAgWDtWjIS+YC1/dDX1Ww9MaR7JOxjHzAvJQPKAUwJG6tmFLrBHSK/vy3hSqlZOe3bRKTsz4GXiZqSIMtR79le3H9EkZBKoIFp3XCcGLyUKOBVsku8mmsWEjsiAdQyVJGTaS2eHT/CJUfo4iJQpCXim/p5ISaj1OPRNZ0hgqBe9qfif10kguPBSLuMEmKTzRUEiMER4mgLuc8UoiLEhhCpubsV0SBShYLLKmxDcxZeXSbNSds/Kzm21WLvM4sihI3SMSshF56iGblAdNRBFCXpGr+jNerJerHfrY966YmUzh+gPrM8fKBiSGQ==</latexit> Eq. ( \nsensitive to the squared value of the threshold ν c ≡ δ c /σ , where σ is the variance of the density field of cosmological perturbations. So far in the literature those have been computed on super horizon scales, which gives only the leading order computation of the abundance. Non linear effects, becoming important close to the horizon crossing, can give rise to corrections to the probability of collapse estimated on super horizon scale. 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2019MNRAS.488...47F

Black hole and neutron star mergers in galactic nuclei

2019-01-01

['stars kinematics and dynamics', '-', 'stars black holes', 'galaxy center', 'galaxy kinematics and dynamics', '-', '-', '-']

[]

Nuclear star clusters surrounding supermassive black holes (SMBHs) in galactic nuclei contain large numbers of stars, black holes (BHs), and neutron stars (NSs), a fraction of which are likely to form binaries. These binaries were suggested to form a triple system with the SMBH, which acts as a perturber and may enhance BH and NS mergers via the Lidov-Kozai mechanism. We follow-up previous studies, but for the first time perform an extensive statistical study of BH-BH, NS-NS, and BH-NS binary mergers by means of direct high-precision regularized N-body simulations, including post-Newtonian (PN) terms up to order PN2.5. We consider different SMBH masses, slopes for the BH mass function, binary semimajor axis and eccentricity distributions, and different spatial distributions for the binaries. We find that the merger rates are a decreasing function of the SMBH mass and are in the ranges ∼0.17-0.52, ∼0.06-0.10, and ∼0.04-0.16 Gpc<SUP>-3</SUP> yr<SUP>-1</SUP> for BH-BH, BH-NS, and NS-NS binaries, respectively. However, the rate estimate from this channel remains highly uncertain and depends on the specific assumptions regarding the star formation history in galactic nuclei and the supply rate of compact objects (COs). We find that {∼ } 10-20{{ per cent}} of the mergers enter the LIGO band with eccentricities ≳0.1. We also compare our results to the secular approximation, and show that N-body simulations generally predict a larger number of mergers. Finally, these events can also be observable via their electromagnetic counterparts, thus making these CO mergers especially valuable for cosmological and astrophysical purposes.

[]

https://arxiv.org/pdf/1811.10627.pdf

{'Black Hole and Neutron Star Mergers in Galactic Nuclei': "Giacomo Fragione 1 glyph[star] , Evgeni Grishin 2 , Nathan W. C. Leigh 3 , 4 , 5 , Hagai B. Perets 2 , Rosalba Perna 3 , 6 \n1 Racah Institute for Physics, The Hebrew University, Jerusalem 91904, Israel \n2 Physics Department, Technion - Israel institute of Technology, Haifa 3200002, Israel \n- 3 Departamento de Astronom'ıa, Facultad de Ciencias F'ısicas y Matem'aticas, Universidad de Concepci'on, Concepci'on, Chile\n- 4 Department of Astrophysics, American Museum of Natural History, New York, NY 10024, USA\n- 5 Department of Physics and Astronomy, Stony Brook University, Stony Brook, NY 11794-3800, USA \n6 Center for Computational Astrophysics, Flatiron Institute, New York, NY 10010, USA \n13 June 2019", 'ABSTRACT': 'Nuclear star clusters surrounding supermassive black holes (SMBHs) in galactic nuclei contain large numbers of stars, black holes (BHs) and neutron stars (NSs), a fraction of which are likely to form binaries. These binaries were suggested to form a triple system with the SMBH, which acts as a perturber and may enhance BH and NS mergers via the Lidov-Kozai mechanism. We follow-up previous studies, but for the first time perform an extensive statistical study of BH-BH, NS-NS and BH-NS binary mergers by means of direct high-precision regularized N -body simulations, including Post-Newtonian (PN) terms up to order PN2.5. We consider different SMBH masses, slopes for the BH mass function, binary semi-major axis and eccentricity distributions, and different spatial distributions for the binaries. We find that the merger rates are a decreasing function of the SMBH mass and are in the ranges ∼ 0 . 17-0 . 52 Gpc -3 yr -1 , ∼ 0 . 06-0 . 10 Gpc -3 yr -1 and ∼ 0 . 04-0 . 16 Gpc -3 yr -1 for BH-BH, BH-NS and NS-NS binaries, respectively. However, the rate estimate from this channel remains highly uncertain and depends on the specific assumptions regarding the star-formation history in galactic nuclei and the supply rate of compact objects. We find that ∼ 10%-20% of the mergers enter the LIGO band with eccentricities glyph[greaterorsimilar] 0 . 1. We also compare our results to the secular approximation, and show that N -body simulations generally predict a larger number of mergers. Finally, these events can also be observable via their electromagnetic counterparts, thus making these compact object mergers especially valuable for cosmological and astrophysical purposes. \nKey words: Galaxy: centre - Galaxy: kinematics and dynamics - stars: black holes - stars: kinematics and dynamics - galaxies: star clusters: general - stars: neutron', '1 INTRODUCTION': "Recently, the LIGO-Virgo 1 collaboration detected six sources of gravitational waves (GWs), five from merging BHBH binaries (Abbott et al. 2016b,a, 2017a,b,c) and one from merging NS-NS binaries (Abbott et al. 2017d). With ongoing improvements to LIGO and upcoming instruments such as LISA 2 and the Einstein Telescope 3 , hundreds of BH-BH, BH-NS and NS-NS binary sources may be detected within a few years. Thus the modeling of the formation and evolution \n- glyph[star] E-mail: [email protected]\n- 1 https://www.ligo.caltech.edu/\n- 2 http://www.et-gw.eu \nof BH and NS binaries is crucial for interpreting the signals from all the GW sources we expect to observe. \nThe origins of BH and NS mergers are actively debated. Several scenarios have been proposed, such as isolated binary evolution in the galactic field (Belczynski et al. 2016), gas-assisted mergers (Bartos et al. 2017; Stone et al. 2017a; Tagawa et al. 2018), triple systems in the field (Antonini et al. 2017; Silsbee & Tremaine 2017; Liu & Lai 2018) and dynamically formed in dense clusters (Wen 2003; Antonini et al. 2014), mergers of binaries in galactic nuclei (Miller & Lauburg 2009; O'Leary et al. 2009; Antonini & Perets 2012; Prodan et al. 2015; Antonini & Rasio 2016; VanLandingham et al. 2016; Chen et al. 2017; Petrovich & Antonini 2017; Chen & Han 2018; Fern'andez & Kobayashi 2018; Hamers et al. 2018; Hoang et al. 2018; Randall & Xianyu 2018) \nand other dense stellar systems (Askar et al. 2017; Banerjee 2018; Choksi et al. 2018; Fragione & Kocsis 2018; Rodriguez et al. 2018; Rodriguez & Loeb 2018; Samsing et al. 2018a). Each model predicts different rates (generally of the order of ∼ few Gpc -3 yr -1 ) and can in principle be distinguished from other channels using the observed mass, eccentricity, spin and redshift distributions (see e.g. O'Leary et al. 2016). For instance, dynamically assembled mergers are expected to have a non-negligible probability of appearing eccentric when observed (Antonini & Perets 2012; Samsing et al. 2018b; Zevin et al. 2018). \nMost of the literature pertaining to dynamicallyinduced mergers focuses on BH and NS binaries forming in globular clusters, while only a few studies have paid attention to the formation of compact-object binaries in the vicinity of a super massive black hole (SMBH) and in nuclear star clusters (e.g. Hoang et al. 2018; Leigh et al. 2016, 2018). The pioneering work by Antonini et al. (2010) and Antonini & Perets (2012) showed that SMBHs can induce Lidov-Kozai (LK) oscillations on BH-BH and NS-NS binaries orbiting in its vicinity (Lidov 1962; Kozai 1962), thus enhancing the probability of merging compact binaries. In this scenario, the eccentricity of the BH/NS binary reaches large values (for a review on LK mechanisms see Naoz 2016), then GW emission drives the binary to merge. While Antonini & Perets (2012) adopted a secular treatment for the equations of motion at the quadruple level of approximation, Hoang et al. (2018) considered soft binaries and the importance of expansion up to the octuple order. These calculations adopt the secular approximation to study triples, which must satisfy hierarchical conditions (Naoz 2016). In some cases, the inner binary may undergo rapid oscillations in the angular momentum and eccentricity, thus the secular theory is not anymore an adequate description of the three-body equations of motion (Antonini & Perets 2012; Antognini, Shappee, Thompson & Amaro-Seoane 2014; Antonini, Murray & Mikkola 2014; Luo, Katz & Dong 2016; Liu & Lai 2018; Grishin, Perets & Fragione 2018). For these cases, direct precise N -body simulations, including regularization schemes and Post-Newtonian (PN) terms, are required to follow accurately the orbits of the objects up to the final merger (Grishin et al. 2018; Fragione & Leigh 2018a,b). Recently, VanLandingham et al. (2016) used N -body simulations of small ( ∼ 300-4000 stars) clusters surrounding 10 3 -10 4 M glyph[circledot] black holes to study the effect of LK oscillations in dense environments, while Arca-Sedda & Gualandris (2018) used few-body simulations to check the merger rates of BHBH binaries delivered by infalling star clusters at typical distances of ∼ few pc. \nIn this paper, we revisit the SMBH-induced mergers of compact binaries orbiting in its vicinity. We consider a threebody system consisting of an inner binary comprised of a BH-BH/NS-NS/BH-NS binary, and an outer binary comprised of the SMBH and the centre of mass of the inner binary. Figure 1 depicts the system we study in the present paper. We denote the mass of the SMBH as M SMBH and the mass of the objects in the inner binary as m 1 and m 2 , while the semimajor axis and eccentricity of the inner orbit are a in and e in , respectively, and for the outer orbit, these are a out and e out , respectively. While previous papers mainly adopted a secular approximation for the equations of motion, with a few direct N-body integrations in comparison \nFigure 1. The three-body system studied in the present work. We denote the mass of the SMBH as M SMBH , and the masses of the binary components as m 1 and m 2 . The semimajor axis and eccentricity of the outer orbit are a out and e out , respectively, while for the inner orbit they are a in and e in , respectively. \n<!-- image --> \nto secular evolution (e.g. Antonini & Perets 2012; Hoang et al. 2018), here, we make the first systematic and statistical study of BH-BH, NS-NS and BH-NS mergers in the proximity of an SMBH by means of direct high-precision N -body simulations, including Post-Newtonian (PN) terms up to order PN2.5. Moreover, we consider how different masses of the SMBH affect the mergers of compact binaries, and adopt a mass spectrum for the BHs, while also studying different spatial distributions for the merging binaries. Finally, we discuss observational diagnostics that can help discriminate this compact object merger channel from other ones. \nThe paper is organized as follows. In Section 2, we discuss the properties and dynamics of BHs and NSs in galactic nuclei. In Section 3, we discuss the state-of-the-art secular approximations currently being used in the literature. In Section 4, we present our numerical methods to determine the rate of BH-BH, NS-NS and BH-NS mergers in galactic nuclei, for which we discuss the results. In Section 5, we discuss the predicted rate of compact object mergers in galactic nuclei and compare our results to secular approximations, while, in Section 6, we discuss the observational signatures of these events. Finally, in Section 7, we discuss the implications of our findings and draw our conclusions.", '2 BLACK HOLE AND NEUTRON STAR BINARIES IN GALACTIC NUCLEI': 'The evidence in favour of the presence of close binaries composed of compact objects (COs) such as white dwarfs (WDs), and especially NSs and BHs, in dense stellar environments is rapidly growing. Recently, Hailey & Mori (2017) reported observations of a dozen quiescent X-ray binaries that form a central density cusp within ∼ 1 parsec of Sagittarius A ∗ . The authors argue that the emission spectra they observe are inconsistent with a population of accreting WDs, suggesting that the X-ray binaries must contain mostly NSs and BHs. However, the relative numbers of these two types \nof COs is an open question. Six other X-ray transients are known to be present in the inner parsec of the Galactic Centre, which are also strongly indicative of binaries containing COs (e.g. Muno et al. 2005; Hailey & Mori 2017). Given the very dense stellar environments in galactic nuclei combined with the presence of a central SMBH, these CO binaries can undergo several fates. They can be hardened to shorter orbital periods, either by direct scattering interactions with single stars (Perets 2009a; Leigh et al. 2018), or by LK oscillations due to the SMBH combined with gravitational wave (GW) emission acting at pericentre (e.g. Antonini & Perets 2012; Prodan et al. 2015). \nIn their Table 1, Generozov et al. (2018) combine the reported statistics from the literature to provide estimates for the numbers of BHs and NSs in the Galactic Centre. In short, the number of NS X-ray Binaries (XRBs) per stellar mass in the Galactic Centre is roughly three orders of magnitude higher than in the field, and comparable to the number expected to be in globular clusters. Similarly, the number of BH XRBs per stellar mass is roughly three orders of magnitude higher than in the field, and roughly an order of magnitude higher than in any known globular cluster (e.g. Strader et al. 2012; Leigh et al. 2014, 2016). \nThese large numbers of NS and BH binaries must come from somewhere. Yet, little is known about binaries in our Galactic Centre. A likely explanation is that they are the remnants of massive O/B stars, since hundreds of these are known to be present in the inner ∼ 1 pc of Sagittarius A*. In the Solar neighborhood, the massive-star binary fraction is very high ( glyph[greaterorsimilar] 70%) and the most massive binaries have semimajor axes of up to a few AU (Sana et al. 2012). This alone is suggestive of a high rate of NS and BH formation in nuclear star clusters (e.g. Levin & Beloborodov 2003; Genzel et al. 2008). Interestingly, the discovery of even a single magnetar within glyph[lessorsimilar] 0 . 1 pc of Sgr A* would also argue in favour of a high rate of NS formation, given their short active lifetimes (Mori et al. 2013). \nThis population of CO binaries is continuously depleted through dynamical interactions with other stars and COs (evaporation) and GW mergers. A few mechanisms may replenish the CO population. The binaries may come from outside the central region near the SMBH, which thus serves as a continuous source term (Hopman 2009; Alexander 2017). In this scenario, CO binaries form far from the innermost region around the SMBH and gradually migrate on a 2-body relaxation timescale, \nT 2 b = 1 . 6 × 10 10 yr ( σ 300 kms -1 ) 3 ( m M glyph[circledot] ) -1 × × ( ρ 2 . 1 × 10 6 M glyph[circledot] pc -3 ) -1 ( ln Λ 15 ) -1 , (1) \ntowards smaller distances, where they become active in the Lidov-Kozai regime. Here, ρ and σ are the 1-D density and velocity dispersion in the Galactic Centre, respectively, ln Λ is the Coulomb logarithm and m is the average stellar mass. On the other hand, our Galactic Centre contains a large population of young massive O-type stars, many of which have been observed to reside in a stellar disk. Likely, most of them were born in-situ as a consequence of the fragmentation of a gaseous disk formed from an infalling gaseous clump (Genzel et al. 2010). An important question is which \nprocess can make their orbit, which are not observed closer than ∼ 0 . 05 pc, approach the SMBH, where efficient LidovKozai oscillations take place. Both planet-like migration in the gaseous disc (Baruteau et al. 2011) and disc instability (Madigan et al. 2009) have been proposed to make the in situ binaries migrate, but to which innermost distance with respect to the SMBH is not known exactly. Other mechanisms include triple/quadruple disruptions, where a triple/quadruple is disrupted and the inner binary is left orbiting the SMBH (Perets 2009b; Ginsburg & Perets 2011; Fragione & Gualandris 2018; Fragione 2018), and infalls of star clusters (Antonini & Merritt 2012; Fragione, CapuzzoDolcetta & Kroupa 2017). \nOutside our own Milky Way, Secunda et al. (2018) recently showed that CO binaries can form efficiently in Active Galactic Nucleus (AGN) disks. The COs migrate in the disk due to differential torques exerted by the gas, moving toward migration traps, where the torques actually cancel (e.g. Bellovary et al. 2016). The COs drift toward the trap and get stuck there, waiting for the next CO to migrate toward it. Once close enough, the two COs can undergo a strong interaction and end up forming a bound binary due to the dissipative effects of the gas. COs can also accrete from the disk gas in this scenario, possibly growing substantially and in some cases even forming an IMBH (McKernan et al. 2012, 2014).', '2.1 Outer semi-major axis and eccentricity': "The numbers and spatial profiles of BHs and NSs are poorly known in galactic nuclei. In general, stars tend to form a power-law density cusp around an SMBH. The classical result by Bahcall & Wolf (1976) shows that a population of equal-mass objects forms a power-law density cusp around an SMBH, n ( r ) ∝ r -α , where α = 7 / 4. For multi-mass distributions, lighter and heavier objects develop shallower and steeper cusps, respectively (Hopman & Alexander 2006a; Freitag et al. 2006; Alexander & Hopman 2009a; Preto & Amaro-Seoane 2010; Aharon & Perets 2016; Baumgardt et al. 2018), while source terms may make the cusp steeper as well (Aharon & Perets 2015; Fragione & Sari 2018). Recent observations of the Milky Way's centre showed that the slope of the cusp appears to be shallower ( α ∼ 5 / 4; GallegoCano et al. 2018; Schodel et al. 2018). As BHs and NSs are heavier than average stars, they are expected to relax into steeper cusps, with BHs relaxing into steeper cusps than NSs (Freitag et al. 2006; Hopman & Alexander 2006b). If not all BHs have the same mass, but follow a mass distribution, only the more massive ones would have steeper slopes, while low-mass BHs should follow shallower slopes (Aharon & Perets 2016). \nIn the present study, we assume that the BH and NS number densities follow a cusp with α = 2. We also study the effects of the cusp slope, by considering a steeper cusp ( α = 3) and a uniform density profile ( α = 0) for BHs, and a shallower cusp ( α = 1 . 5) for NSs. For the maximum outer semi-major axis, we take a M out = 0 . 1 pc following Hoang et al. (2018), which approximately corresponds to the value at which the eccentric Lidov-Kozai timescale is equal to the timescale over which accumulated fly-bys from single stars tend to unbind the binary (see Eqs. 6-11). However, the ratio of these two timescales generally depends on the binary \nTable 1. Models: name, SMBH mass ( M SMBH ), binary type, slope of the BH mass function ( β ), slope of the outer semi-major axis distribution ( α ), a in distribution, e in distribution, merger fraction ( f merge ). \n| Name | M SMBH (M glyph[circledot] ) | Binary Type | β | α | f ( a in ) | f ( e in ) | f merge |\n|--------|--------------------------------|---------------|-----|-------|--------------------------|--------------|-----------|\n| MW | 4 × 10 6 | BH-BH | 1 | 2 | Hoang et al. (2018) | uniform | 0 . 045 |\n| MW | 4 × 10 6 | BH-BH | 1 | 2 | Antonini & Perets (2012) | thermal | 0 . 261 |\n| MW | 4 × 10 6 | BH-BH | 2 | 2 | Hoang et al. (2018) | uniform | 0 . 05 |\n| MW | 4 × 10 6 | BH-BH | 3 | 2 | Hoang et al. (2018) | uniform | 0 . 036 |\n| MW | 4 × 10 6 | BH-BH | 4 | 2 | Hoang et al. (2018) | uniform | 0 . 041 |\n| MW | 4 × 10 6 | BH-BH | 1 | 0 | Hoang et al. (2018) | uniform | 0 . 035 |\n| MW | 4 × 10 6 | BH-BH | 1 | 3 | Hoang et al. (2018) | uniform | 0 . 051 |\n| MW | 4 × 10 6 | BH-NS | 1 | 2 | Hoang et al. (2018) | uniform | 0 . 026 |\n| MW | 4 × 10 6 | BH-NS | 2 | 2 | Hoang et al. (2018) | uniform | 0 . 029 |\n| MW | 4 × 10 6 | BH-NS | 3 | 2 | Hoang et al. (2018) | uniform | 0 . 028 |\n| MW | 4 × 10 6 | NS-NS | - | 1 . 5 | Hoang et al. (2018) | uniform | 0 . 032 |\n| MW | 4 × 10 6 | NS-NS | - | 2 | Hoang et al. (2018) | uniform | 0 . 028 |\n| MW | 4 × 10 6 | NS-NS | - | 2 | Antonini & Perets (2012) | thermal | 0 . 079 |\n| GN | 1 × 10 8 | BH-BH | 1 | 2 | Hoang et al. (2018) | uniform | 0 . 072 |\n| GN | 1 × 10 8 | BH-BH | 1 | 2 | Antonini & Perets (2012) | thermal | 0 . 258 |\n| GN | 1 × 10 8 | BH-NS | 1 | 2 | Hoang et al. (2018) | uniform | 0 . 054 |\n| GN | 1 × 10 8 | NS-NS | - | 2 | Hoang et al. (2018) | uniform | 0 . 044 |\n| GN | 1 × 10 8 | NS-NS | - | 2 | Antonini & Perets (2012) | thermal | 0 . 133 |\n| GN2 | 1 × 10 9 | BH-BH | 1 | 2 | Hoang et al. (2018) | uniform | 0 . 087 |\n| GN2 | 1 × 10 9 | BH-BH | 1 | 2 | Antonini & Perets (2012) | thermal | 0 . 3 |\n| GN2 | 1 × 10 9 | BH-NS | 1 | 2 | Hoang et al. (2018) | uniform | 0 . 061 |\n| GN2 | 1 × 10 9 | NS-NS | - | 2 | Hoang et al. (2018) | uniform | 0 . 048 |\n| GN2 | 1 × 10 9 | NS-NS | - | 2 | Antonini & Perets (2012) | thermal | 0 . 055 | \nand cusp properties since wide binaries could be affected by the LK mechanism at larger distances. We also consider one model where we take a M out = 0 . 5 pc, to assess how our results depend on this parameter. As discussed, in-situ formation occurs in our Galactic Centre at larger distances and some mechanism that delivers such CO binaries closer to the SMBH has to be invoked. Finally, we sample the outer orbital eccentricity from a thermal distribution (Jeans 1919).", '2.2 Inner semi-major axis and eccentricity': 'The inner binary (BH-BH/NS-NS/BH-NS) semi-major axis and eccentricity are not well known. Different models predict different distributions. Moreover, the dense environments characteristic of galactic nuclei should cause both distributions to diffuse over time, thus changing the relative distributions. Hopman (2009) made the only attempt to model binaries very close to the SMBH, even though this pioneering study accounted only for the 2-body relaxation process. Other relaxation processes, such as resonant relaxation (Rauch & Tremaine 1996), may affect the distribution as well (Hamers et al. 2018). For what concerns eccentricity, though it mostly depends on the natal-kick and the commonenvelope phase, it also depends on the scattering of the CO binaries by local COs and stars. As a consequence, in-situ formation would probably favour circular binaries, while the migration scenario would most likely prefer a thermal distribution, as a consequence of the energy exchange through many dynamical encounters of the CO binaries with other stars and COs in galactic nuclei. \nAntonini & Perets (2012, see Fig. 1) used the results of Belczynski et al. (2004) for the initial distribution of BH-BH \nbinaries orbits, while, for NSs, they used the observed pulsar binary population as found in the ATNF pulsar catalog 4 (Manchester et al. 2005). We note that their distributions referred to isolated binaries and used a simplified approach for to account for the softening and binary-destruction due to the crowded environments (following the approach in (Perets et al. 2007)). Inner eccentricities were sampled from a uniform distribution. Recently, Hoang et al. (2018) drew the inner semi-major axes from a log-uniform distribution in the range 0 . 1-50 AU, somewhat consistent with the observed distribution from Sana et al. (2012), which favors short period binaries, and the inner eccentricities from a uniform distribution (Raghavan et al. 2010). Note the caveat that this distribution corresponds to massive MS binaries, and not to CO binaries which already evolved and change their configurations.', '2.3 Masses': "The mass distribution of BHs is unknown, even in isolation. Moreover, in galactic nuclei, irrespective of the original mass function, mass-segregation can make the effective BH mass function even steeper in the inner-most regions (Aharon & Perets 2016). This would in turn also affect the outer orbit distribution, since more massive BHs would have steeper slopes (Aharon & Perets 2016). \nIn our models, we sample the masses of the BHs from \ndN dm ∝ M -β , (2) \nin the mass range 5 M glyph[circledot] -100 M glyph[circledot] 5 . To check how the results depend on the slope of the BH mass function, we run models with β = 1, 2, 3, 4 for both the BH-BH and BH-NS binaries (O'Leary et al. 2016). For NSs, we fix the mass to 1 . 3 M glyph[circledot] (e.g. Fragione, Pavl'ık & Banerjee 2018).", '2.4 Inclinations and relevant angles': 'We draw the initial mutual inclination i 0 between the inner and outer orbit from an isotropic distribution (i.e. uniform in cos i ). The other relevant angles, such as the arguments of pericentre, nodes and mean anomalies, are drawn randomly.', '2.5 Timescales in galactic nuclei': 'In the dense stellar environment of a galactic nucleus, several dynamical processes other than 2-body relaxation (Eq. 1) can take place and affect the evolution of the stellar and compact object populations. On smaller timescales than T 2 b , resonant relaxation (Rauch & Tremaine 1996; Kocsis & Tremaine 2015) randomizes the direction and magnitude (hence eccentricity) of the outer orbit on a typical timescale \nT RR = 9 . 2 × 10 8 yr ( M SMBH 4 × 10 6 M glyph[circledot] ) 1 / 2 ( a out 0 . 1 pc ) 3 / 2 ( m M glyph[circledot] ) -1 (3) \nOn even shorter timescales, vector resonant relaxation changes the direction (hence the relative inclination) of the outer orbit angular momentum on a typical timescale \nT VRR = 7 . 6 × 10 6 yr ( M SMBH 4 × 10 6 M glyph[circledot] ) 1 / 2 × × ( a out 0 . 1 pc ) 3 / 2 ( m M glyph[circledot] ) -1 ( N 6000 ) -1 / 2 , (4) \nwhere N is the number of stars within a out . In the context of Kozai-Lidov oscillations, vector resonant relaxation plays a role, since it may affect the initial inclination of the inner and outer orbit of the CO binary on timescales comparable to or even shorter than the Kozai-Lidov timescale (Hamers et al. 2018). \nFinally, binaries may evaporate due to dynamical interactions with field stars in the dense environment of a galactic nucleus when \nE b ( m 1 + m 2 ) σ 2 glyph[lessorsimilar] 1 , (5) \nwhere E b is the binary internal orbital energy and σ is the velocity dispersion. This happens on an evaporation timescale (Binney & Tremaine 1987) \nT EV = 3 . 2 × 10 7 yr ( m 1 + m 2 2 M glyph[circledot] ) ( σ 300 kms -1 ) ( m M glyph[circledot] ) -1 × × ( a in 1 AU ) -1 ( ρ 2 . 1 × 10 6 M glyph[circledot] pc -3 ) -1 ( ln Λ 15 ) -1 . (6) \n.', '3 SECULAR AVERAGING TECHNIQUES': "The merger time of an isolated binary of component masses m 1 , m 2 , semimajor axis a and eccentricity e emitting GWs is (Peters 1964) \nT GW ( a, e ) = 5 256 c 5 a 4 G 3 m 1 m 2 ( m 1 + m 2 ) (1 -e 2 ) 7 / 2 . (7) \nFor a triple system made up of an inner binary that is orbited by an outer companion, the inner eccentricity can be pumped by the tidal potential of a distant body via the Lidov-Kozai (LK) mechanism (Lidov 1962; Kozai 1962). The LK oscillations occur on a secular timescale (Antognini 2015) \nt sec = 8 15 π m tot m out P 2 out P in (1 -e 2 out ) 3 / 2 , (8) \nwhere m out = M SMBH and m tot = M SMBH + m bin ≈ M SMBH , P in and P out are the orbital periods of the inner and outer binary, respectively. The large values attained by the inner eccentricity make the overall merger time of the inner binary shorter since it efficiently dissipates energy when e ∼ e max (e.g., see Antonini & Perets 2012). The LK-induced merger time is (Antonini & Perets 2012; Randall & Xianyu 2018; Liu & Lai 2018) \nT LK GW ( a, e max ) ≈ T GW ( a, e max ) / √ 1 -e 2 max . (9) \nThe maximal eccentricity is a function mostly of the initial mutual inclination, i 0 , and is usually evaluated by the secular approximation, which relies on double-averaging of both the inner and the outer orbits (Randall & Xianyu 2018). In the leading, quadrupole order, the system is integrable and has been widely studied (see recent review by Naoz 2016, and references therein), where coupled oscillations between the eccentricity and inclination of the inner binary are excited for sufficiently large initial mutual inclinations. The inner binary eccentricity approaches almost unity as i 0 approaches ∼ 90 deg. \nWhen the outer orbit is eccentric and the inner binary has an extreme mass ratio, octupole-level perturbations turn the system from integrable to chaotic (Li et al. 2014), and can potentially induce extreme orbital eccentricities, orbital flips and even direct collisions (Katz et al. 2011; Lithwick & Naoz 2011). The strength of the octupole perturbation is encapsulated in the octupole parameter as \nglyph[epsilon1] oct ≡ m 1 -m 2 m 1 + m 2 a in a out e out 1 -e 2 out . (10) \nGenerally, increasing glyph[epsilon1] oct will increase the parameter space corresponding to orbital flips and very large eccentricities. The typical timescale for an orbital flip is (Antognini 2015) \nt flip = 8 π √ 10 glyph[epsilon1] oct t sec . (11) \nThe secular approximation assumes that the triple system is hierarchical, namely that P in /P out ∝ ( a in /a out ) 2 / 3 glyph[lessmuch] 1, thus ignoring short-term variations ( t ∼ P out ) of the osculating elements, whose typical strength can be parametrized by the so-called 'single-averaging parameter' (Luo et al. 2016) \nglyph[epsilon1] SA ≡ ( a in a out (1 -e 2 out ) ) 3 / 2 ( M SMBH m bin ) 1 / 2 = P out 2 πτ sec . (12) \nAlso, Luo et al. (2016) point out that in addition to the fluctuating terms, additional secular evolution can take place. Consequently, the resulting fate of the system could be different, since additional extra apsidal and nodal precession changes the structure of the LK resonance. Grishin et al. (2017) showed that extra apsidal precession shifts the critical inclination for the LK resonance and affects the Hill stability limit of irregular satellites. \nRecently, Grishin et al. (2018) used Luo et al. (2016)'s result to find an analytic formula for the maximal eccentricity that can be reached due to LK oscillations \ne max = ¯ e max + δe , ¯ e max = √ 1 -5 3 cos 2 i 0 1 + 9 8 glyph[epsilon1] SA cos i 0 1 -9 8 glyph[epsilon1] SA cos i 0 , δe = 135 128 ¯ e SA max glyph[epsilon1] SA [ 16 9 √ 3 5 √ 1 -¯ e 2 max + glyph[epsilon1] SA -2 glyph[epsilon1] SA ¯ e 2 max ] . (13) \nIf the inner binary is too compact or the tertiary is too far away, the maximal eccentricity could be quenched, e.g. by GR (general relativistic) precession, and the average maximal eccentricity is given by solving (Grishin et al. 2018) \n¯ A (1 -¯ e 2 max ) = 8 glyph[epsilon1] GR ¯ e 2 max √ 1 -¯ e 2 max +15 ¯ j 2 z ( 1 + 9 8 glyph[epsilon1] SA ¯ j z ) ¯ A ( ¯ j z , ¯ e max ) ≡ 9 -glyph[epsilon1] SA 81 8 ¯ j z +8 glyph[epsilon1] GR ¯ e 2 max , (14) \nwhere ¯ j z = √ (1 -e 2 0 ) cos i 0 is the initial normalized angular momentum of the inner binary and \nglyph[epsilon1] GR ≡ 3 m bin (1 -e 2 out ) 3 / 2 m out ( a out a ) 3 Gm bin ac 2 (15) \nmeasures the ratio between the apsidal precession rates induced by Lidov-Kozai and GR perturbations. The above formulae are valid in the limit glyph[epsilon1] oct = 0. \nIn some configurations the inner binary may undergo rapid oscillations in the angular momentum and eccentricity, thus the secular theory is no longer an adequate description of the three-body equations of motion (Antonini & Perets 2012; Antognini et al. 2014). For instance, this can happen when the typical time-scale for the angular momentum of the inner orbit to change by of order itself becomes comparable to (or even shorter than) the outer or inner orbital periods (Antonini et al. 2014). Thus, in this case, the secular approximation can fail to predict both the correct maximum eccentricity and merger time. Although computationally expensive (in particular in the case of a third very massive companion as in this paper), direct N -body simulations including Post-Newtonian (PN) terms represent the most reliable option for accurately studying the effects of the tertiary companion in reducing the GW merger time of the inner binary.", '4 N-BODY SIMULATIONS: BLACK HOLE AND NEUTRON STAR MERGERS IN GALACTIC NUCLEI': 'In this section, we use N -body simulations to study the fate of BH-BH, NS-NS and BH-NS binaries in galactic nuclei that \n<!-- image --> \nFigure 2. Inclination as a function of the initial semi-major axis a in and the total mass in merged BH-BH binaries for all models with f ( a in ) from Hoang et al. (2018). Most BH binaries that merge have initial inclinations ∼ 90 · , where the enhancement in the maximum eccentricity is expected to be larger due to LidovKozai oscillations. \n<!-- image --> \nhost an SMBH. We consider three different SMBH masses, i.e. M SMBH = 4 × 10 6 M glyph[circledot] for a Milky-Way-like nucleus (Models MW), M SMBH = 10 8 M glyph[circledot] for a M31-like nucleus (Models GN) and M SMBH = 10 9 M glyph[circledot] for more massive host galaxies (Models GN2). For the inner semi-major axis and eccentricities, we follow the prescriptions given by Hoang et al. (2018), but also run some models with the sampling suggested by Antonini & Perets (2012) to check how the initial conditions affect the final rates. Given the set of initial parameters as described in Sect. 2, we draw the main parameters of the three-body system and require that the inner binary does not cross the Roche limit of the SMBH at its orbital pericentre distance \na out a in > η 1 + e in (1 -e out ) ( 3 M SMBH m 1 + m 2 ) 1 / 3 . (16) \nFollowing Antonini & Perets (2012), we set η = 4 since at shorter distances the inner binary is unstable. We then integrate the triple SMBH-CO-CO differential equations of motion \nr i = -G ∑ j glyph[negationslash] = i m j ( r i -r j ) | r i -r j | 3 , (17) \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFigure 3. Cumulative distributions of the inner semi-major axis a in (left) and of the outer semi-major axis a out (right) for BH-BH (top), BH-NS (centre), NS-NS (bottom) binaries that merge in all models with f ( a in ) from Hoang et al. (2018). \n<!-- image --> \nwith i = 1,2,3. The integrations are performed using the archain code (Mikkola & Merritt 2006, 2008). This code is fully regularized and is able to model the evolution of objects of arbitrary mass ratios and eccentricities with extreme accuracy, even over long periods of time. We include PN corrections up to order PN2.5. \nFor each set of parameters in Tab. 1, we run ∼ 1500 simulations up to a maximum integration time of T = 1 Myr, for a total of ∼ 35000 simulations. From the computational point of view, this limit represents a good compromise between the numerical effort (large mass ratios and \nGW effects slow down the code) and the size of the statistical sample we want to take into account. From the physical point of view, we note that our total integration time is smaller than the typical timescale for vector resonant relaxation to operate ( ∼ few Myr, see Eq. 4; Rauch & Tremaine 1996; Kocsis & Tremaine 2015), which reorients the binary centre-of-mass orbital plane with respect to the SMBH, thus affecting the relative inclination of the inner and outer orbits and the relative LK dynamics, rendering the 3-body approximation insufficient (Hamers et al. 2018). Furthermore, in-plane precession induced by the nuclear cluster potential \n<!-- image --> \nFigure 4. Distribution of the total ( m 1 + m 2 ) BH-BH mass (left) and primary BH mass ( m 1 ) in BH-NS binaries (right) for all models with f ( a in ) from Hoang et al. (2018). As expected, the main parameter that affects the mass distribution is the slope β of the BH mass function. \n<!-- image --> \nand departure from spherical symmetry of the galactic nucleus would make the CO center of mass orbit precess even faster than vector resonant relaxation alone in a MW-like nucleus (Petrovich & Antonini 2017). Finally, we also note that our total integration time is smaller than the typical evaporation time of the CO binaries in galactic nuclei (see Eq. 6). Taken all together, these considerations justify our choice of maximum integration time. Nevertheless, we also take into account in our rate calculations the binaries that are not affected by the LK cycles within 1 Myr and merge by emission of GWs alone on longer timescales, without the assistance of LK oscillations 6 . Thus, our estimations correspond to a lower limit. We consider in total 23 different models, which take into account different COs in the inner binary (BH-BH, BH-NS and NS-NS), different masses of the SMBHs, different slopes of the BH/NS mass functions, different spatial distributions of the CO binaries, and different inner semi-major axis and eccentricity distributions. Table 1 summarizes all the models considered in this work. \nIn our simulations the CO binary has three possible fates: (i) the CO binary can survive on an orbit perturbed with respect to the initial one; (ii) the CO binary can be tidally broken apart by differential forces exerted by the SMBH, and its components will either be captured by the SMBH or ejected from the galactic nucleus; (iii) the CO binary merges producing an GW merger event. We distinguish among these possible outcomes by computing the mechanical energy of the CO binary. If the relative energy remains negative, we consider the binary survived (case (i)), otherwise we consider the binary unbound (case (ii)). Finally, if the CO binary merges, which occurs if the relative radii of the two COs overlap directly, we have a GW merger event (case (iii)). \nTable 2. BH masses already detected via GW emission. \n| Name | m 1 (M glyph[circledot] ) | m 2 (M glyph[circledot] ) |\n|-----------|-----------------------------|-----------------------------|\n| GW150914 | 36 . 0 +5 . 0 - 4 . 0 | 29 . 0 +4 . 0 - 4 . 0 |\n| GW151226 | 14 . 2 +8 . 3 - 3 . 7 | 7 . 5 +2 . 3 - 2 . 3 |\n| GW170104 | 31 . 2 +8 . 4 - 6 . 0 | 19 . 4 +5 . 3 - 5 . 9 |\n| GW170608 | 12 . 0 +7 . 0 - 2 . 0 | 7 . 0 +2 . 0 - 2 . 0 |\n| GW170814 | 30 . 5 +5 . 7 - 3 . 0 | 25 . 3 +2 . 8 - 4 . 2 |\n| LVT151012 | 23 . 0 +18 . 0 - 6 . 0 | 13 . 0 +4 . 0 - 5 . 0 |', '4.1 Inclination distribution': 'We illustrate in Fig. 2 the inclination as a function of semimajor axis a in and total mass in merged BH-BH binaries in our simulations for all the models with f ( a in ) from Hoang et al. (2018). Most of the BH-BH binaries that merge have initial inclinations ∼ 90 · , where the enhancement of the maximum eccentricity is expected to be larger due to LK oscillations. As discussed in Sect. 3, the exact LK window angle depends on the physical quantities of the system (Grishin et al. 2017, 2018), and the final distribution of surviving systems lacks highly inclined binaries (Fragione & Leigh 2018b). The lack of highly inclined systems shows the importance of the LK mechanism, since BH-BH binaries that successfully undergo a merger event originally orbit in a plane highly inclined with respect to the outer orbital plane (Fragione & Leigh 2018b). In these binaries, the LK mechanism influences the dynamics of the system and induces oscillations both in eccentricity and inclination, whenever not suppressed by GR precession. Figure 2 also shows that some systems that merge have inclinations far from ∼ 90 · , in particular when the total binary mass is large and the inner semi-major axis is small. Also, these systems typically have relatively high initial inner eccentricities. \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFigure 5. Density maps of the masses of the BH-BH binaries that merge in our models (with f ( a in ) from Hoang et al. (2018)), along with a comparison to the observed BH masses detected via GW emission (see Tab. 2; Abbott et al. 2016b,a, 2017a,b,c). \n<!-- image -->', '4.2 Inner and outer orbital parameter distributions': "We present in Fig. 3 the cumulative distribution of a in (left panel) for BH-BH (top), BH-NS (centre), NS-NS (bottom) binaries for all models with f ( a in ) from Hoang et al. (2018). Both the SMBH mass and the slope α of the CO binary spatial distribution around the SMBH significantly affect the inner and outer semi-major axes of merging binaries, as a consequence of their Hill stability (Grishin et al. 2017). \nLarger SMBH masses imply that, on average, smaller values of a in are needed to avoid tidal disruption of the CO binaries after only a few orbits about the SMBH. For the same reason, large SMBH masses typically produce mergers at larger distances from the SMBH. Obviously, steep binary distributions (i.e., large α 's) imply that the simulated CO binaries will, on average, be closer to the SMBH when they merge. Hence, smaller inner semi-major axes are needed to avoid tidal dissociation by the SMBH. Finally, the total mass of the CO binary plays some role in shaping the final outer or- \nital semi-major axis distribution. CO binaries with smaller total masses ( m 1 + m 2 ) typically merge at larger distances from the SMBH, since their binding energy is easily overcome by the gravitational pull of the SMBH, which tends to break the binaries at smaller distances. For BHs, this translates into steeper mass functions typically producing mergers at farther distances from the SMBH.", '4.3 Mass distribution': 'Figure 4 shows the distribution of the total ( m 1 + m 2 ) BHBH mass (left) and the primary BH mass ( m 1 ) in BHNS binaries (right) that merge in all models with f ( a in ) from Hoang et al. (2018). The resulting mass distribution is barely affected by the slope of the binary spatial distribution around the SMBH, with a roughly constant shape in the range ∼ 25 M glyph[circledot] -125 M glyph[circledot] and a tail extending up to ∼ 180 M glyph[circledot] for BH-BH binaries. Also, the mass of the central SMBH does not significantly affect the mass distribution. As expected, the parameter that governs the resulting shape of the mass distribution is the slope β of the BH mass function: the shallower the BH mass function, the larger the typical total mass of merging BH-BH binaries. In the case β = 1, we find that ∼ 95% of the mergers have m 1 + m 2 glyph[lessorsimilar] 150 M glyph[circledot] , while roughly all the mergers have total masses glyph[lessorsimilar] 100 M glyph[circledot] , glyph[lessorsimilar] 50 M glyph[circledot] , and glyph[lessorsimilar] 25 M glyph[circledot] for β = 2, β = 3, and β = 4, respectively. Similar results also hold for BH-NS binaries. \nThe slope of the BH mass function is unknown. We can use the results of our simulations along with the BHBH merger events observed by LIGO (see Tab. 2; Abbott et al. 2016b,a, 2017a,b,c) to constrain the BH mass function, assuming these mergers took place in a galactic nucleus. We show in Fig. 5 density maps for the masses of the two merging BHs ( m 1 > m 2 ), along with data from the LIGO-observed BH merger events. It is clear that a steep mass function ( β > 1) seems to be disfavored by the current data, which suggest a shallower BH mass function. However, we note that BH mass measurements via GW observations are biased towards more massive BHs, since these are more easily observed by LIGO. We also note that, although the mass distribution is only slightly affected by the SMBH mass, the data seem to prefer more massive nuclei than the Milky-Way. Note, however, that mass-segregation processes, that can give rise to much steeper effective massfunctions of BHs in galactic nuclei (Aharon & Perets 2016), only operate in small-SMBH nuclei where relaxation (and mass-segregation) times are short.', '4.4 Eccentricity': "For the systems that merge in our simulations, we compute a proxy for the GW frequency of the merging binaries. This is taken to be the frequency corresponding to the harmonic that gives the maximal emission of GWs (Wen 2003) \nf GW = √ G ( m 1 + m 2 ) π (1 + e in ) 1 . 1954 [ a in (1 -e 2 in )] 1 . 5 . (18) \nFigure 6 reports the distribution of eccentricities at the moment the binaries enter the LIGO frequency band (10 Hz) for BH-BH mergers in a Milky Way-like nucleus, for different values of β and α . The distributions have a double peak \nFigure 6. Distribution of eccentricities at the moment the binaries enter the LIGO frequency band (10 Hz) for BH-BH mergers in a Milky Way-like nucleus, for different values of β and α . The vertical line shows the minimum e 10Hz = 0 . 081 where LIGO/VIRGO/KAGRA network may distinguish eccentric sources from circular sources (Gond'an & Kocsis 2019). A significant fraction of binaries have a significant eccentricity in the LIGO band. \n<!-- image --> \nat e 10Hz ∼ 10 -2 and e 10Hz ∼ 1. In our model where we take a M out = 0 . 5 pc, we find a similar distribution of eccentricities. Binaries merging in galactic nuclei typically have larger eccentricities than those formed through most other channels, particularly mergers in isolated binary evolution and in SBHBs ejected from star clusters. However, mergers that follow from the GW capture scenario in clusters (Zevin et al. 2018) and galactic nuclei (Gond'an et al. 2018; Rasskazov & Kocsis 2019), from resonant binary-single scattering in clusters (Samsing et al. 2018a), from hierarchical triples (Antonini et al. 2017; Fragione & Kocsis 2019; Fragione et al. 2019; Fragione & Loeb 2019), and from BH binaries orbiting intermediate-mass black holes in star clusters (Fragione & Bromberg 2019) also present a similar peak at high eccentricities. We find that typically ∼ 20-30% of binaries have e glyph[greaterorsimilar] 0 . 1 in the LIGO band (Gond'an & Kocsis 2019). In our runs, we also find that some of the CO binaries that merge do not merge due to LK oscillations, but instead merge by emission of GWs on timescales longer than 1 Myr and with eccentricities at 10 Hz much smaller than the typical eccentricity reported in Figure 6. Their relative fraction is typically ∼ 20%-50% of the total mergers we find in our simulations. When their contribution is taken into account, the fraction of binaries that enter the LIGO band with e glyph[greaterorsimilar] 0 . 1 decreases to ∼ 10%-20%. This fraction is still larger than previously estimated values ( ∼ 1%) found in the literature (Antonini & Perets 2012; VanLandingham et al. 2016; Randall & Xianyu 2018), probably due to the different integration schemes adopted. In a secular approach, the averaged equations of motion could smear out the peak at high eccentricities and lower the number of binaries entering the LIGO band with very high eccentricities. \nFinally, the high eccentricities we find in those binaries that merge may imply that a fraction of these binaries could emit their maximum power at higher frequencies, possibly in the range of LISA.", '5 MERGER TIME DISTRIBUTIONS AND RATES': 'As discussed, the SMBH plays a fundamental role in reducing the merger timescale from the nominal value in Eq. (7) (Antonini & Perets 2012; Hoang et al. 2018; Fragione & Leigh 2018a). In Fig. 7, we present for all models the cumulative distribution of merger times ( t merge ; left panel) for BH-BH (top), BH-NS (centre) and NS-NS (bottom) binaries. The merger time distribution is nearly independent of our assumptions for the BH mass function slope β and the CO binary spatial distribution slope α . It depends only on the SMBH mass. Larger SMBH masses imply shorter merger times due to more intense perturbations from the SMBH. In the right panel of Fig. 7, we show t merge as a function of the nominal (Peters 1964) GW merger time-scale T GW , for all binaries that merge in our simulations. Due to oscillations in the orbital elements, the CO binaries merge much faster than predicted by Eq. 7, by several orders of magnitude.', '5.1 Comparing secular techniques and N-body simulations': "In order to compare the distribution of merger times with different prescriptions, we take the initial conditions and calculate the merger time from Eq. (9). We find the maximal eccentricity by solving Eq. (14) (Grishin et al. 2018). For the secular case, we use glyph[epsilon1] SA = 0, which also implies δe = 0 from Eq. (13), while for the corrected case, glyph[epsilon1] SA > 0 is set from the system's initial conditions. If the merger time is shorter than the secular LK timescale that is required to reach the maximal eccentricity, we use the secular LK time. We then filter out the merger times longer than 1 Myr and compare to the simulated merger times. \nWe use the initial conditions of the MW NS-NS case with α = 2 to calculate T LK GW as prescribed above. In addition, in order to compare with archain , we draw N = 10 4 initial conditions from the same distribution and use a secular code to evolve them up to T = 1 Myr and record their merger times. For the secular code, we use SecuLab 7 , a publicly available code that solves the secular equations of motion up to octupole order with additional secular 2.5PN terms. \nOverall, the number of events from both simulations exceeds their expected analytic estimate. For the secular case, the ratio between the merger rate obtained numerically from SecuLab and analytically from Eq. (9) is 0 . 004 / 0 . 0028 = 1 . 43. Similarly, for the N-body case, this ratio is 0 . 028 / 0 . 01 = 2 . 8. The ratio of merger rates for both \nTable 3. Fraction of merger for the MW NS-NS case. First and second row are the fractions from SecuLab and archain , respectively. Third and forth row are the expected mergers from evaluating Eq. 9 with different maximal eccentricity evaluations (see text).Table 4. D and p values for double-sided Kolmogorov-Smirnov (KS) tests of the cumulative distributions of the merger times. \n| | SecuLab | ARCHAIN | T LK GW sec | T LK GW | GPF18 |\n|---------|-----------|-----------|---------------|-----------|---------|\n| f merge | 0 . 004 | 0 . 028 | 0 . 0028 | 0 . | 01 | \nTable 3 shows the expected merger fractions from the direct N-body and the secular codes, respectively, together with different methods of evaluating the merger time. T LK GW is evaluated once using the secular approximation (sec), and the corrected eccentricity (Grishin et al. 2018; GPF18). The merger rate is given by the fraction of initial conditions that result in T LK GW glyph[lessorequalslant] 1 Myr. We see that T LK GW slightly overestimates the merger rate in the secular regime, but underpredicts it in the N-body regime. \n| ( D,p ) values | SecuLab | ARCHAIN |\n|------------------|-----------------------|------------------|\n| LK GW sec | 0 . 29 , 0 . 58 | T 0 . 4 , 0 . 23 |\n| T LK GW GPF18 | 0 . 2 , 0 . 52 0 . 19 | , 0 . 5 | \nsimulations is about 0 . 028 / 0 . 004 = 7. The possible origins of these discrepancies are discussed below. \nFigure 8 shows the cumulative distribution of the merger times for the MW NS-NS case, where both secular and direct N-body simulations were used. Overall, the total number of mergers predicted by the corrected GPF18 model is larger, since larger eccentricities are involved, which is compatible with the numerical results. The secular model is less accurate with decreasing merger timescales; this is because the eccentricities involved are extreme, therefore deviations from the secular regime are more severe. \nOn longer merger timescales, t merge glyph[greaterorsimilar] 10 4 yr, if the typical LK timescales are short enough the maximal eccentricity attained, e max , is usually larger than the predicted one from 1PN theory alone (see their Fig. 5 of Grishin et al. 2018). Therefore, the mergers occur of faster than expected and the CDF is underestimated. On the other hand, for short merger times, the maximal eccentricity of the GPF18 model unbound, and the merger occurs on the secular timescale. If, however, t LK √ 1 -e 2 max glyph[lessorsimilar] P in , the GPF18 model also breaks down and cannot describe the system (Antonini et al. 2017), while the value of e max is stochastic. This is because the fraction of time the inner orbit spends near e max (i.e. ∼ t LK √ 1 -e 2 max ) is too short for the inner orbit to complete one revolution and reach pericentre. Thus it takes longer time (at least a few secular times) to merge and the CDF is overestimated. \nIn order to get a better sense of the corrected prescription, we preform two sided Kolmogorov-Smirnov (KS) tests comparing the cumulative distribution function (CDF) from the simulated results versus the secular merger time distributions. Table 4 shows the resulting D values and p values upon comparing the N -body and secular simulations with the secular and the corrected GPF18 model. \nFor the KS statistics, the D values are better for the GPF18 model, since the distance between the simulated and GPF18 CDFs is smaller. The p values are comparable for all models. A possible statistical artifact could be the low number of predicted mergers for the secular case. This suggesting \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFigure 7. Left panel: cumulative merger time ( t merge ) distribution for BH-BH (top), BH-NS (centre) and NS-NS (bottom) binaries, for all models with f ( a in ) from Hoang et al. (2018). Right panel: t merge as a function of the nominal Peters (1964) GW merger time-scale T GW , for the same models as shown in the left panel. \n<!-- image --> \nthat neither of the models is comparable with the simulated distributions. \nTo summarize, the simulated distribution cannot be fully described by the GPF18 model in a statistical sense, but the overall trend of larger merger fractions on shorter merger timescales is consistent with the simulations. Thus, further improvements in the analytic understanding both in the GPF18 model and in the Peters (1964) formulae are highly desired and deserve future work.", '5.2 Merger Rates': "Although we explore only a limited number of SMBH masses, we note that the range is relatively representative of what is expected for galactic nuclei. With the results of our simulations in hand, we can derive the expected merger rates of BH-BH, BH-NS and NS-NS binaries. With this, we can infer the dependence of the rate on the distribution of SMBH masses in the nearby Universe. \nFollowing Hamers et al. (2018), we calculate the merger \nTable 5. Rates of CO binary mergers (in Gpc -3 yr -1 ) as a function of the SMBH mass. \n| M SMBH ( M glyph[circledot] ) | Γ BH - BH | Γ BH - NS | Γ NS - NS | Γ SF NS - NS |\n|---------------------------------|-------------|-------------|-----------------|----------------|\n| 4 × 10 6 | 0 . 52 | 0 . 10 | 1 . 71 × 10 - 3 | 0 . 16 |\n| 10 8 | 0 . 24 | 0 . 08 | 1 . 27 × 10 - 3 | 0 . 11 |\n| 10 9 | 0 . 17 | 0 . 06 | 0 . 41 × 10 - 3 | 0 . 04 | \nFigure 8. Cumulative distribution of merger times for merged orbits. Red: archain N-body simulation. The analytic merger estimate for the archain initial conditions is from Eq. 9 is shown in blue (secular approximation) and green (GPF18 correction, see text). Cyan: SecuLab secular simulation. The distribution of initial conditions is the same but the sample is larger ( N = 10 4 ). \n<!-- image --> \nrate for CO binaries as \nΓ( M SMBH ) = n gal f SMBH Γ sup CO f bin f merge , (19) \nwhere n gal is the galaxy density, f SMBH ≈ 0 . 5 is the fraction of galaxies containing an SMBH (Antonini et al. 2015a,b), Γ sup CO is the compact object supply rate, f bin is the fraction of stars forming compact object binaries, and f merge is the fraction of mergers we find in our simulations. For galaxies, we assume that the SMBH number density scales as Φ( M SMBH ) ∝ 1 /M SMBH (Aller & Richstone 2002), hence the integrated number density of galaxies scales as \nn gal ∝ ∫ Φ( M SMBH ) dM SMBH ∝ log( M SMBH ) . (20) \nAs in Hamers et al. (2018), we neglect the weak dependence on the SMBH mass and fix n gal = 0 . 02 Mpc -3 (Conselice, Blackburne & Papovich 2005). \nThe fraction of CO binaries in the GC strongly depends on the assumptions regarding their origins. Several possibilities have been discussed in Antonini & Perets (2012); here we focus on two: ex-situ and in-situ origins. In the ex-situ scenario, stars form outside the nuclear cluster and then diffuse inwards. In the in-situ formation scenario, stars are formed in-situ close to the SMBH. We use simplified assumptions to estimate the supply rate in both cases. For a relaxed nuclear cluster, Fokker-Planck, Monte-Carlo and Nbody simulations suggest that the fractions of BHs and NSs \nin the central 0.1 pc are of the order of γ CO = 0 . 06 , 0 . 01 for BHs and NSs, respectively (the higher BH fractions are due to mass-segregation), assuming the background stellar population has a continuous star-formation rate (Hopman & Alexander 2006a). Following Antonini & Perets (2012) we take initial binary fractions of f bin = 0 . 1 , 0 . 07 for BHs and NSs, respectively. For the compact object formation rate, we assume that the compact objects are supplied to the galactic nucleus by 2-body relaxation and mass segregation \nΓ sup CO = γ CO N ∗ (0 . 1 pc) t seg (0 . 1 pc) ∝ M SMBH (3 -β ) /β , (21) \nwhere γ CO is the fractional number of compact objects; t seg = T 2b ( m bin / M glyph[circledot] ) is the timescale for mass segregation for binaries with mass m bin ; and we assume M SMBH ∝ σ 4 (Merritt & Ferrarese 2001). \nNormalizing the rates to the Milky Way's Galactic Centre \nΓ sup BH = 2 . 5 × 10 -6 ( 4 × 10 6 M glyph[circledot] M SMBH ) 1 / 4 yr -1 . (22) \nΓ sup NS = 2 . 3 × 10 -8 ( 4 × 10 6 M glyph[circledot] M SMBH ) 1 / 4 yr -1 . (23) \nThe final expression for our rate becomes \nΓ BH ( M SMBH ) = 3 . 5 f merge Gpc -3 yr -1 × ( 4 × 10 6 M glyph[circledot] M SMBH ) 1 / 4 , (24) \nΓ NS ( M SMBH ) = 3 . 2 × 10 -2 f merge Gpc -3 yr -1 × ( 4 × 10 6 M glyph[circledot] M SMBH ) 1 / 4 , (25) \nwhich is weakly dependent on the SMBH mass, and where the merger fraction f merge , which is typically a few up to a few tens of percents for the various models we considered, can be found in Tab. 1. We note that in f merge we have also included the CO binaries that would merge by emission of GWs in timescales > 1 Myr, without the assistance of LK oscillations. These are typically a few percent of the total mergers in the case of BH-NS and NS-NS binaries, while ∼ 20-50% in the case of BH-BH binaries. \nNote that the evaporation time of NS binaries (see Eq. 6) could become comparable to the segregation time, and therefore the rates of NS-NS mergers could be even lower, if supplied from outside the central region of the nuclear cluster. If more massive stellar-BHs exist, the most massive ones will dominate the inner regions due to strong mass-segregation (Alexander & Hopman 2009b; Aharon & Perets 2016), and can be resupplied into the inner regions faster, enhancing the rates by up to a factor of a few. The larger numbers and faster supply will therefore bias the mass-function of merging BHs through this process to higher \nmasses. Also, the relaxation and mass-segregation times in non-cuspy nuclear clusters, or when no nuclear cluster exists (e.g. for SMBHs more massive than ∼ 10 8 M glyph[circledot] ), could be so long that the stellar density around the SMBHs is low. As a consequence, the resupply of stars close to the SMBH cannot be efficiently attained through 2-body relaxation processes, but is more likely to depend on the gas-inflow and in-situ star formation close to the SMBH (Antonini 2013, 2014). \nThe star-formation rate close to non-resolved regions around SMBHs is difficult to estimate theoretically. Here, we try to use an empirical estimate based on our own resolved Galactic Centre (see e.g. Bartko et al. 2009). Approximately ∼ 200 O-stars (likely to later form stellar black holes) are observed and inferred to have formed over the last 10 Myrs in the young stellar disk close ( ∼ 0 . 05 -0 . 5 pc) to the SMBH. The number of lower-mass B-stars in the same environment suggests that similar continuous star-formation has not occurred over the last 100 Myr. Based on these observations we may consider an in-situ formation rate of BHs of ∼ 200 / 10 8 = 2 × 10 -6 yr -1 , i.e. comparable to the estimated supply rate of 2 . 5 × 10 -6 from mass-segregation of BHs from outside the central regions. The comparable formation rate of NSs, however, would increase their resupply rates to the same level as BHs, i.e. much higher than the resupply from NSs migrating in from the outside ( ∼ 2 × 10 -6 yr -1 ) and thereby \nΓ SF NS ( M SMBH ) = 2 . 8 f merge Gpc -3 yr -1 × ( 4 × 10 6 M glyph[circledot] M SMBH ) 1 / 4 . (26) \nTable 5 reports the resulting rates as a function of the SMBH mass. Upon using the semi-major axis and eccentricity distributions following the prescriptions of Antonini & Perets (2012), we typically get a merger fraction ∼ 2 -5 times larger than in the case of adopting the initial conditions from Hoang et al. (2018). This is probably related to the fact that the semi-major axes of CO binaries are typically smaller in the former case. BH-NS binaries should have mass segregation times similar to BH-BH binaries, hence we use Eq. 24. For all SMBH masses considered in this study, the rates are in the range ∼ 0 . 17-0 . 52 Gpc -3 yr -1 , ∼ 0 . 060 . 10 Gpc -3 yr -1 and ∼ 0 . 41-1 . 71 × 10 -3 Gpc -3 yr -1 for BH-BH, BH-NS and NS-NS binaries, respectively. In the star-formation channel, the NS-NS rate may be as high as ∼ 0 . 04-0 . 16 Gpc -3 yr -1 . We note that the merger rate is a decreasing function of the SMBH mass, even though the relative fraction of merger events is typically larger for more massive SMBHs (see Tab. 1). On the other hand, more massive SMBHs imply longer relaxation times, that contribute to a reduction in the merger fraction and make the relative rates smaller. Finally, we also run one model where we take a M out = 0 . 5 pc, to check how the results depend on this parameter; wide binaries can be affected by LK cycles at distances larger than ∼ 0 . 1 pc. In this case, we find that typically our merger fraction f merge is reduced by a factor of ∼ 2-3 8 .", '5.3 Comparison of merger rates to previous studies': 'We find rates comparable with though somewhat different from the ones of Antonini & Perets (2012) and those of Petrovich & Antonini (2017) for spherical clusters (the latter finds ten times higher rates for the case of a non-spherical nuclear cluster - not modelled, or compared with in our work), but lower with respect to other works that explored the role of the SMBH in reducing the merger timescale of binaries due to Lidov-Kozai oscillations (Hoang et al. 2018). Other merger channels typically predict larger rates. Typical values for globular clusters are ∼ 2 -10 Gpc -3 yr -1 (Askar et al. 2017; Fragione & Kocsis 2018; Rodriguez et al. 2018) and for nuclear star clusters are ∼ 1 -15 Gpc -3 yr -1 (Antonini & Rasio 2016). For reference, the BH-BH merger rate inferred by LIGO is ∼ 12 -213 Gpc -3 (Abbott et al. 2017a). \nBefore directly comparing different estimates, we emphasize that any rate estimate is highly uncertain and should be considered as an order-of-magnitude estimate, since it depends on the specific assumptions regarding the starformation history in galactic nuclei and the supply rate of compact objects at various distances from the SMBH, which remains poorly constrained in the literature. \nWe note that previous studies that used a similar approach made use of much larger supply rates, ∼ 10-40 times higher than considered here. The differences arise for several reasons, in particular from the different assumptions on the star-formation rate in the Galactic Centre and nuclear star cluster. Petrovich & Antonini (2017), Hamers et al. (2018) and Hoang et al. (2018) considered a resupply rate of BHs of 10 -5 -10 -4 yr -1 from star-formation, where the latter rate is derived assuming a top-heavy initial mass function from (Maness et al. 2007). However, this gives rise to several difficulties: (1) This rate is based on the formation rates derived by Lockmann & Baumgardt (2009) and Lockmann et al. (2009) for both NSs and BHs lumped together, while the formation rate for NSs is actually ∼ 8 times higher than that of BHs for regular (e.g. Kroupa or Miller-Scalo) IMFs, and becomes comparable only for top-heavy IMFs; one can therefore not discuss the number of BHs taking these numbers at face value, but consider the division between BHs and NSs (this issue was accounted for by Petrovich & Antonini (2017), Hamers et al. (2018), but not in Hoang et al. (2018)), and its dependence on the assumed IMF. (2) The higher rate estimates are based on a top-heavy IMF, which in-turn is based on results from Maness et al. (2007). However, this top-heavy IMF was only derived from observations of old low-mass stars and the results are therefore highly problematic for the use in this context, as they are based on a large extrapolation from the low-mass regime up to that of NS and BH progenitors not probed at all by Maness et al. (2007). Also note that low-mass stars could already dynamically evolve through mass-segregation and their observed distribution in the GC does not necessarily reflect the actual IMF (e.g. see Aharon & Perets 2015). Moreover, direct observations of massive stars in the GC today, though suggestive of a somewhat top-heavy IMF (Lu et al. 2013), find a much steeper power-law of -1 . 7 ± 0 . 2 compared with ∼ -0 . 8 in the Maness et al. study of low-mass stars (which also supersedes the shallow power-law results from from Bartko et \nal. 2010 regarding massive stars), compared with -2 . 3 for Salpeter or Kroupa IMFs in the relevant mass-range. (3) Even more important, all of these estimates considered star formation throughout the nuclear cluster, rather than the innermost regions, where the induced mergers actually take place (especially in the cases considered by Hoang et al. (2018)), i.e. the COs would have to migrate inwards over long timescales, and one should then refer to the ex-situ resupply rate discussed above. Indeed, a recent paper by Zhang et al. (2019) exploring the long-term evolution of binaries in the cluster shows that the long-term softening and disruption of binaries due to stars and the SMBH effectively quench the contribution of secular evolution induced mergers, consistent with the points raised above in the case of ex-situ supply. The overall numbers of COs derived when assuming in-situ formation in these other papers are therefore at least 10 times higher than actually expected in the central parts, and, in fact, are at least 10 times higher than what one can infer from the observed young massive stars in the GC (Bartko et al. 2009, 2010), or those inferred from X-ray sources (Muno et al. 2005). \nWe also briefly note that the binary fraction of BHs and NSs in nuclear clusters is not well known, and the different studies make somewhat different assumptions and definitions in their calculations; for example, we calculate the fraction of BHs/NSs given the fraction of massive progenitors in the population and then multiply by the binary fraction, making use of past theoretical and observational studies of BH/NSs, while Petrovich & Antonini (2017) and Hamers et al. (2018) calculated the total binary fraction, including the dependence on the mass-function derived from stellar evolution. The differences on this point, however, are of order a factor of 1-2, and do not amount to a significant difference. Finally, our models originally considered binaries up to 0.1 pc where most of the current star-formation is observed (the young stellar disk). As mentioned above the merger fractions we find at larger separations up to 0.5 pc are smaller, and therefore, in our models these can contribute at most a comparable number of sources. This potential factor of two in the rate could then also help reconcile some of the differences in comparisons with the previous models which effectively considered star-formation throughout the central pc. \nIn summary, we believe the overall higher rate estimates near SMBHs by a factor of a few up to ∼ 10 s in some of the previous studies mostly arise from the different assumptions on the star-formation rate in the Galactic Centre and nuclear star cluster. If one assumes that it formed only through in-situ star formation and that star formation occurred as observed today (at glyph[lessorsimilar] 0 . 1 pc) throughout the last ∼ 10 Gyr, then the resupply rate considered in previous studies should be valid. For the Milky Way, however, these assumptions seem not to be consistent both with the star-formation history over the last 100 Myrs given the observations of OB stars and X-ray sources (as discussed above), and with the inferred long-term star-formation history over the lifetime of the Galaxy (Nogueras-Lara et al., in prep; R. Schodel, private communication), suggesting most of the stars in the Galactic Centre formed very early ( glyph[greaterorsimilar] 8 Gyrs), and at most a few percents formed in the last Gyr. Nevertheless, the overall star-formation history in our Galaxy remains poorly constrained, and one can not exclude that a more vigorous \nand continuous star-formation could occur in the nuclei of other galaxies, in which case a higher rate estimates may apply, thus a higher merger rate of COs.', '6 ELECTROMAGNETIC COUNTERPARTS AND OBSERVATIONAL SIGNATURES OF SMBH-INDUCED CO MERGERS': "As we noted previously (cfr Sec. 4) the very high (close to unity) eccentricity, with which the GW signal enters the LIGO band in the scenario explored potentially proviude an important observational diagnostic of CO mergers induced by LK oscillations. In the following, we discuss further observational diagnostics of this merger channel in relation to possible electromagnetic (EM) counterparts to the mergers. In particular, mergers of compact object binaries are expected to be associated with a strong release of electromagnetic radiation, if the right conditions arise to power an energetic outflow. \nIn the case of a NS-NS merger, tidal disruption during the inspiral phase leaves behind an accretion torus surrounding the merged object (either a NS or a BH), unless the two NSs have identical masses (Shibata et al. 2006; Rezzolla et al. 2010; Giacomazzo et al. 2013; Hotokezaka et al. 2013; Kiuchi et al. 2014; Ruiz et al. 2016; Radice et al. 2016). An energetic engine can be driven by rapid accretion onto to the remnant object and/or by dipole radiation losses if the remnant is an hypermassive or stable NS (Giacomazzo & Perna 2013; Ciolfi et al. 2017). Growth and collimation of magnetic fields during the merger, as well as neutrino losses, are then believed to power a relativistic outflow. Dissipation within the expanding flow, and later interaction of the flow with the interstellar medium, gives rise to radiation that spans a wide window in the electromagnetic spectrum, from high-energy γ -rays down to the radio. This basic scenario has been observationally confirmed with the recent event GW170817/GRB170817A (Abbott et al. 2017). \nMergers of BH-NS binaries (always resulting in a BH as the resulting compact remnant) are expected to be accompanied by the formation of an hyperaccreting disk only if the mass ratio between the BH and NS does not exceed the value ∼ 3 -5, with the precise value depending on the equation of state of the NS and the BH spin (Pannarale et al. 2011; Foucart 2012; Foucart et al. 2018). For larger mass ratios, the tidal disruption radius of the NS is smaller than the radius of the innermost stable circular orbit, and no disk will form, resulting in a direct plunge into the BH (see e.g. Bartos et al. 2013 for a review). If a rapidly accreting disk forms, then the resulting EM phenomenology is expected to be similar to that of the NS-NS case, at least in so far as the bulk properties are concerned. For the initial conditions explored in this work, ∼ 10-20% of mergers is expected to have mass ratios glyph[lessorsimilar] 5, and hence possibly giving rise to an accretion-powered EM counterpart. \nIn the case of a BH-BH binary merger, there is no tidally disrupted material which can readily supply the accretion power for a relativistic outflow 9 . However, following the tentative detection of a γ -ray counterpart by the Fermi satellite \nto the event GW150914 (Connaughton et al. 2016), several ideas were proposed for providing the merged BH with a baryonic remannt to accrete from (Perna et al. 2016; Loeb 2016; Woosley 2016; Murase et al. 2016; Stone et al. 2017b; Bartos et al. 2017; Kimura et al. 2017; Janiuk et al. 2017; de Mink & King 2017). Within the context of this study, the scenario proposed by Bartos et al. (2017) is of particular relevance; they note that BH-BH binaries merging within an AGN disk can accrete a significant amount of gas from the disk, well above the Eddington rate, and possibly give rise to high-energy EM emission. \nElectromagnetic counterparts to binary mergers provide crucial information on the production mechanism of the binaries, since they can potentially allow a much better localization compared to GWs alone. A distinctive signature of binary mergers enhanced by LK oscillations in the vicinity of SMBHs is their relatively short merger timescale compared to that of other formation channels. For example, the classical channel of isolated binary evolution predicts merger times ∼ 100 Myr-15 Gyr (Belczynski et al. 2006). The short lifetimes of the binaries, coupled with their production in the galactic centers, lead to correspondingly short distances traveled prior to mergers. We find that these distances are typically glyph[lessorsimilar] 0 . 1 pc, which makes these merger events practically occurring within the close nuclear region. This constitutes a major difference with respect to the standard isolated binary evolution scenario (Perna & Belczynski 2002; Belczynski et al. 2006; O'Shaughnessy et al. 2017; Perna et al. 2018): whether it is a small or a large galaxy, the bulk of the merger events occurs at projected distances (from the galaxy center) glyph[greaterorsimilar] 100 pc 10 . Localization via EM counterparts hence becomes an especially useful discriminant. \nShort GRBs associated with NS-NS mergers, and BHNS mergers with a small enough mass ratio to allow tidal disruption, are expected to be followed by broadband radiation called afterglow, resulting from the dissipation of a relativistic shock propagating in the interstellar medium (Sari, Piran & Narayan 1998). The maximum flux intensity (at any wavelength) is given by \nF ν, max = 110 n 1 1 / 2 ξ B 1 / 2 E 52 D -2 28 (1 + z ) mJy , (27) \nwhere E 52 is the explosion energy in units of 10 52 erg, D 28 the luminosity distance in units of 10 28 cm, z is the redshift, n 1 the number density of the interstellar medium in cm -3 , and it is assumed that the magnetic field energy density in the shock rest frame is a fraction ξ B of the equipartition value. \nThe broadband spectrum evolves with time, and we compute it numerically using the formalism of Sari et al. (1998). For an energy E = 10 50 erg as typical of short GRBs, standard assumptions for the shock parameters and medium ambient density ∼ a few cm -3 (more typical of the inner \ncretion to produce energy (Zhang 2016; Liebling & Palenzuela 2016). \n10 Note that, even if the isolated binary evolution scenario does predict a fraction of tight binaries with sub-Myr lifetimes (Belczynski et al. 2006), and even ultra-short merger times (Michaely & Perets 2018), but the merger sites are still dominated by large scales since isolated binaries are born throughout the galactic disk. \nregions of a galaxy), the afterglow luminosity in some representative bands (X-rays and radio) at some typical observation times is found to be L [2 -10]kev ∼ 5 × 10 45 erg s -1 at t obs = 1 hr and L 5 GHz ∼ 6 × 10 30 erg s -1 Hz -1 at t obs = 7 days. In the X-rays, a representative flux threshold is the Swift /XRT flux sensitivity of F lim = 2 . 5 × 10 -13 erg s -1 cm -2 , while in the radio, a 1hr integration with the VLA leads to a flux threshold for detection of F lim ≈ 50 µ Jy. In both these bands, detection would be possible up to a redshift of ∼ 2, considerably larger than the LIGO horizon 11 . The detection distances are larger than in the isolated binary evolution scenario, for which the large traveled distances lead to a sizable fraction of mergers to occur in low-density environments, where the afterglow luminosity is considerably dimmer. \nAdditionally note that, independently of the postmerger EM signal, a fraction on the order of a few × 10 -3 of the GW sources is expected to be accompanied by an SN-type precursor (Michaely & Perets 2018). This is due to the fact that the distribution of the delay time between the last SN explosion and the binary merger has a non-negligible tail of ultra-short times, on the order of 1-100 yr (see also (Dominik et al. 2012)). \nDetection of an EM counterpart to a GW event generally allows a redshift measurement. The redshift distribution of the channel studied here would be one that follows the star-formation rate, since the merger times are shorter or at most comparable to the lifetimes of the most massive stars (as a reference, the lifetime of a ∼ 100 M glyph[circledot] star is about 1 Myr). This would hence constitute another observational diagnostics. \nThe relatively easier prospects for detecting EM counterparts from CO mergers in galactic nuclei makes this channel especially useful for extraction of astrophysical and cosmological information from combined GW/EM detections. This includes, among other, measurements of the Hubble constant, new tests of the Lorentz invariance, constraints on the speed of GWs, probes of the physics of mergers and jet formation, constraints on the equation of state of neutron stars (see Abbott et al. 2017, and references therein).", '7 DISCUSSION AND SUMMARY': "In this paper, we have revisited the SMBH-induced mergers of compact binaries orbiting within its sphere of influence. While previous studies in the literature adopted the secular approximation for the equations of motion (Antonini & Perets 2012; Hamers et al. 2018; Hoang et al. 2018), here we have performed an extensive statistical study of BH-BH, NS-NS and BH-NS binary mergers by means of ∼ 35000 direct high-precision regularized N -body simulations, including Post-Newtonian (PN) terms up to order PN2.5. \nWe have shown that the secular approach breaks down for systems with mild and extreme hierarchies. We used \n11 It should however be noted that in the X-rays, when the shock is still moving at relativistic speeds, relativistic beaming of the emission will lead to a reduced luminosity for jets which are not observed on-axis. The fraction of on-axis jets is expected to be on the order of 1 / 20 by taking the jet size of ∼ 16 deg inferred for short GRBs (Fong et al. 2015). \nthe recent corrections to the maximal eccentricity e max and merger times in the quasi-secular regime (Grishin et al. 2018, GPF18) and tested it against N-body population synthesis integrations. The total number of mergers is under-predicted by a factor of ∼ 6 -10 in the secular approach, and by a factor of ∼ 2 -3 by the corrected GPF18 model. The CDF of merger times fail to fit either of the distributions, although the D-value distance between the simulated CDF and the GPF18 model is closer than for the secular approach. The difference can be attributed to the original underestimate of e max in the latter model, which leads to slower and less frequent mergers. \nIn our numerical simulations, we have considered different SMBH masses, different slopes for the BH mass function and the binary spatial distributions, and different CO binary semi-major axis and eccentricity distributions. We find that the majority of binary mergers happen when the mutual inclination of the binary orbit and its center of mass orbit around the SMBH is i 0 ∼ 90 · , as a consequence of the LidovKozai mechanism. We have also shown that the distributions of the inner and outer semi-major axes of the merging binaries depend mainly on the mass of the SMBH and on the slope α of the binary spatial distribution around the SMBH. On the other hand, the shape of the resulting CO mass distributions depend on the slope β of the BH mass function. BH mergers observed by LIGO seem to favour β ∼ 1, if those mergers were to happen around SMBHs. We have also discussed that the fraction of binaries that enter the LIGO band with e glyph[greaterorsimilar] 0 . 1 is ∼ 10%-20%, larger than previous values found in the literature (Antonini & Perets 2012; VanLandingham et al. 2016; Randall & Xianyu 2018), due to the different integration schemes adopted. \nWe have also calculated the resulting rates as a function of the SMBH mass. We find that the merger rates are a decreasing function of the SMBH mass and are in the ranges ∼ 0 . 17-0 . 52 Gpc -3 yr -1 , ∼ 0 . 06-0 . 10 Gpc -3 yr -1 and ∼ 0 . 41-1 . 71 × 10 -3 Gpc -3 yr -1 for BH-BH, BH-NS and NSNS binaries, respectively. In the star-formation channel, the NS-NS rate may be as high as ∼ 0 . 04-0 . 16 Gpc -3 yr -1 . We find rates consistent with Antonini & Perets (2012), but lower with respect to previous works (Petrovich & Antonini 2017; Hamers et al. 2018; Hoang et al. 2018), which may have overestimated the amount of CO binaries supplied to galactic nuclei through star-formation. \nWe have also discussed the possible EM counterparts of these events. Due to their locations, these mergers may have higher probabilities of being detected also via their EMcounterparts, hence making these CO mergers especially valuable for cosmological and astrophysical purposes. \nFinally, we note that we have adopted 1 Myr for the maximum integration time in our simulations, since this limit sets a good compromise between the computational effort and the size of the statistical sample we generate. This choice is further justified by noting that the typical timescale for vector resonant relaxation to operate is ∼ 1-10 Myr, over which the mutual orbital inclination is reoriented by interactions with other background objects, and renders the 3-body approximation insufficient (Hamers et al. 2018). Also, we have neglected the possible precession of the CO binaries' motion induced by continual weak interactions with other stars and COs in the stellar cusp surrounding the SMBH (Alexander 2017). A comprehensive N -body study over long \nintegration timescales that includes both the SMBH-induced Lidov-Kozai oscillations and the detailed effects of the background stars surrounding the CO binaries deserves consideration in future work.", 'ACKNOWLEDGEMENTS': 'GFis supported by the Foreign Postdoctoral Fellowship Program of the Israel Academy of Sciences and Humanities. GF also acknowledges support from an Arskin postdoctoral fellowship at the Hebrew University of Jerusalem. EG acknowledges support from the Technion Irwin and Joan Jacobs Excellence Fellowship for outstanding graduate students. EG and HBP acknowledge support by Israel Science Foundation I-CORE grant 1829/12. NL and RP acknowledge support by NSF award AST-1616157. GF thanks Seppo Mikkola for helpful discussions on the use of the code archain . Simulations were run on the Astric cluster at the Hebrew University of Jerusalem. The Center for Computational Astrophysics at the Flatiron Institute is supported by the Simons Foundation.', 'REFERENCES': "Abbott B. P., Abbott R., Abbott T. D., Acernese F., Ackley K., Adams C., Adams T., Addesso P., Adhikari R. X., Adya V. B., et al. 2017, ApJL, 848, L13 \nAbbott B. P., et al., 2016a, Physical Review Letters, 116, 241103 Abbott B. P., et al., 2016b, Physical Review Letters, 116, 061102 Abbott B. P., et al., 2017a, Physical Review Letters, 118, 221101 Abbott B. P., et al., 2017b, ApJ Lett, 851 Abbott B. 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2000ApJ...536..668K

General Relativistic Simulations of Early Jet Formation in a Rapidly Rotating Black Hole Magnetosphere

2000-01-01

['accretion', 'accretion disks', 'black hole physics', 'galaxies jets', 'stars magnetic fields', 'methods numerical', 'mhd', 'relativity', 'astrophysics', '-']

[]

To investigate the formation mechanism of relativistic jets in active galactic nuclei and microquasars, we have developed a new general relativistic magnetohydrodynamic code in Kerr geometry. Here we report on the first numerical simulations of jet formation in a rapidly rotating (a=0.95) Kerr black hole magnetosphere. We study cases in which the Keplerian accretion disk is both corotating and counter-rotating with respect to the black hole rotation, and investigate the first ~50 light-crossing times. In the corotating disk case, our results are almost the same as those in Schwarzschild black hole cases: a gas pressure-driven jet is formed by a shock in the disk, and a weaker magnetically driven jet is also generated outside the gas pressure-driven jet. On the other hand, in the counter-rotating disk case, a new powerful magnetically driven jet is formed inside the gas pressure-driven jet. The newly found magnetically driven jet in the latter case is accelerated by a strong magnetic field created by frame dragging in the ergosphere. Through this process, the magnetic field extracts the energy of the black hole rotation.

[]

https://arxiv.org/pdf/astro-ph/9907435.pdf

{'GENERAL RELATIVISTIC SIMULATIONS OF JET FORMATION IN A RAPIDLY ROTATING BLACK HOLE MAGNETOSPHERE': 'Shinji Koide \nFaculty of Engineering, Toyama University, Gofuku 3190, Toyama 930-8555, Japan \nDavid L. Meier \nJet Propulsion Laboratory, 4800 Oak Grove Dr. Pasadena, CA 91109, USA. \nKazunari Shibata \nKwasan and Hida Observatories, Kyoto University, Yamashina, Kyoto, 607-8471, Japan. \nKwasan and Hida Observatories, Kyoto University, Yamashina, Kyoto, 607-8471, Japan. \nTakahiro Kudoh \nNational Astronomical Observatory, Mitaka, Tokyo 181-8588, Japan \nDraft version October 2, 2018', 'ABSTRACT': 'To investigate the formation mechanism of relativistic jets in active galactic nuclei and micro-quasars, we have developed a new general relativistic magnetohydrodynamic code in Kerr geometry. Here we report on the first numerical simulation of jet formation in a rapidly-rotating ( a = 0 . 95) Kerr black hole magnetosphere. We study cases in which the Keplerian accretion disk is both co-rotating and counterrotating with respect to the black hole rotation. In the co-rotating disk case, our results are almost the same as those in Schwarzschild black hole cases: a gas pressure-driven jet is formed by a shock in the disk, and a weaker magnetically-driven jet is also generated outside the gas pressure-driven jet. On the other hand, in the counter-rotating disk case, a new powerful magnetically-driven jet is formed inside the gas pressure-driven jet. The newly found magnetically-driven jet in the latter case is accelerated by a strong magnetic field created by frame dragging in the ergosphere. Through this process, the magnetic field extracts the energy of the black hole rotation. \nSubject headings: accretion, accretion disks - black hole physics - galaxies: jets - magnetic field - methods: numerical - MHD - relativity', '1. INTRODUCTION': "Radio jets ejected from radio loud active galactic nuclei (AGNs) sometimes show proper motion with apparent velocity exceeding the speed of light c (Pearson et al. 1981, Hughes 1991). The widely-accepted explanation for this phenomenon, called superluminal motion, is relativistic jet flow in a direction along the observer's line-of-sight with a Lorentz factor greater than 2 (Rees 1966). Such relativistic motion is thought to originate from a region very close to the putative supermassive black hole which is thought to power each AGN (Linden-Bell 1969, Rees 1984). On the other hand, the great majority of AGNs are radio quiet and do not produce powerful relativistic radio jets (Rees 1984). These two classes of active objects (radio loud and quiet) are also found in the black hole candidates (BHCs) in our own Galaxy. Objects with superluminal jets, such as GRS 1915+105 and GRO J165540, belong to the radio loud class (Mirabel & Rodriguez 1994, Tingay et al. 1995). Other objects such as Cyg X1 and GS 1124-68 are relatively radio quiet and produce little or no jet. \nWhat causes the difference between the two classes? Recent observations of the BHCs in our Galaxy suggest that the Galactic superluminal sources contain very rapidly rotating black holes (normalized angular momentum, a ≡ J/ [ GM 2 BH /c ] = 0 . 9 -0 . 95, where G and M BH are the gravitational constant and black hole mass, respectively), while the black holes in Cyg X-1 and GS \n1124-68 are spinning much less rapidly ( a = 0 . 3 -0 . 5) (Cui, Zhang, & Chen 1998). A similar rapidly rotating black hole is also suggested in the AGN of the Seyfert 1 galaxy MCG-6-30-15 by the X-ray satellite ASCA (Iwasawa et al. 1996). According to recent (nonrelativistic) studies of magnetically-driven jets from accretion disks by Kudoh & Shibata (1995, 1997a), the terminal velocity of the formed jet is comparable to the rotational velocity of the disk at the foot of the jet. Further nonrelativistic simulations of jet formation confirm these results (Kudo & Shibata 1997b, Ouyed, Pudritz, & Stone 1997), except for the extremely large magnetic field/high jet-power case (Meier et al. 1997, Meier 1999) in which very fast jets can be produced. The rotation velocity at the innermost stable orbit of the Schwarzschild black hole ( r = 3 r S ) is 0 . 5 c , where r S = 2 GM BH /c 2 is the Schwarzschild radius. In addition, it appears that the poloidal magnetic field strength in disks around non-rotating black holes may be not extremely strong if the magnetic field energy density is comparable with that of the radiation (Begelman, Blandford, & Rees 1984, Rees 1984). Therefore, a jet produced by MHD acceleration from an accretion disk around a non-rotating black hole should be sub-relativistic and very weak. In fact, numerical simulations of jet formation in a Schwarzschild metric show only sub-relativistic jet flow (Koide, Shibata, & Kudoh 1999a), except for the case when the initial black hole corona is in hydrostatic equilibrium rather than free fall (Koide, Shibata, & Ku- \ndoh 1998). \nSeveral mechanisms for relativistic jet formation from rotating black holes have been proposed (Blandford & Znajek 1977, Takahashi et al. 1990). However, up until now no one has performed a self-consistent numerical simulation of the dynamic process of jet formation in a rotating black hole magnetosphere. To this end, we have developed a Kerr general relativistic magnetohydrodynamic (KGRMHD) code. In this paper we report briefly on what we believe are some of the first calculations of their kind simulation of jet formation in a rotating black hole magnetosphere.", '2. NUMERICAL METHOD': 'We use a 3 + 1 formalism of the general relativistic conservation laws of particle number, momentum, and energy and Maxwell equations with infinite electric conductivity (Thorne, Price, & Macdonald 1986). The Kerr metric, which describes the spacetime around a rotating black hole, is used in the calculation. When we use BoyerLindquist coordinates, x 0 = ct , x 1 = r , x 2 = θ , and x 3 = φ , the Kerr metric g µν is written as follows, \nds 2 = g µν dx µ dx ν = -h 2 0 ( cdt ) 2 + 3 ∑ i =1 h 2 i ( dx i ) 2 -2 h 3 Ω 3 cdtdx 3 . (1) \nBy modifying the lapse function in our Schwarzschild black hole code ( α = √ 1 -r S /r ) to be α = √ h 2 0 +Ω 2 3 , and adding some terms of Ω 3 to the time evolution equations, we were able to develop a KGRMHD code relatively easily. (See Appendix C in Koide, Shibata, & Kudoh 1999a for more details on this procedure and the meaning of symbols used.) \nWe use the Zero Angular Momentum Observer (ZAMO) system for the 3-vector quantities, such as velocity v , magnetic field B , and so on. For scalars, we use the frame comoving with the fluid flow. The simulation is performed in the region 0 . 75 r S ≤ r ≤ 20 r S , 0 ≤ θ ≤ π/ 2 with 210 × 70 mesh points, assuming axisymmetry with respect to the z -axis and mirror symmetry with respect to the plane z = 0. A free boundary condition is employed at r = 0 . 75 r S and r = 20 r S . In the simulations, we use simplified tortoise coordinates, x = log( r/r H -1), where r H is the radius of the black hole horizon. To avoid numerical oscillations, we use a simplified TVD method (Davis 1984, Koide, Nishikawa & Mutel 1996, Koide 1997, Koide, Shibata, & Kudoh 1999a). We checked the KGRMHD code by computing Kepler motion around a rotating black hole and comparing with analytic results (Shapiro & Teukolsky 1983).', '3. RESULTS': "The simulations were performed for two cases in which the disk co-rotates and counter-rotates with respect to the black hole rotation. Figures 1a-c illustrate the time evolution of the counter-rotating disk case and Fig. 1d the final state of the co-rotating case. These figures show the rest mass density (color), velocity (vectors), and magnetic field (solid lines) in 0 ≤ R ≡ r sin θ ≤ 7 r S , 0 ≤ z ≡ r cos θ ≤ 7 r S . The black region at the origin shows the inside of the black hole horizon. The angular \nmomentum parameter of the black hole is a = 0 . 95 and the radius is r H = 0 . 656 r S . The initial state in the simulation consists of a hot corona and a cold accretion disk around the black hole (Fig. 1a). In the corona, plasma is assumed to be in nearly stationary infall, with the specific enthalpy h/ρc 2 = 1+Γ p/ [(Γ -1) ρc 2 ] = 1 . 3, where ρ is the rest mass density, p is the pressure, and Γ is specific heat ratio and set Γ = 5 / 3. Far from the hole, it becomes the stationary transonic solution exactly. The accretion disk is located at | cot θ | ≤ 0 . 125, r ≥ r D = 3 r S and the initial velocity of the disk is assumed to be the velocity of a circular orbit around the Kerr black hole. The co-rotating disk is stable, but the counter-rotating disk is unstable in the region R ≤ 4 . 4 r S . Except for the disk rotation direction, we use the same initial conditions in both cases. The mass density of the disk is 100 times that of the corona at the inner edge of the disk. The mass density profile is given by that of a hydrostatic equilibrium corona with a scale height of r c ∼ 3 r S . The disk is in pressure balance with the corona, and the magnetic field lines are perpendicular to the accretion disk. We use the azimuthal component of the vector potential A φ of the Wald solution to set the magnetic field, which provides a uniform magnetic field far from the Kerr black hole (Wald 1974). Here the magnetic field strength far from the black hole is 0 . 3 √ ρ 0 c 2 , where ρ 0 is the initial corona density at r = 3 r S . However, we do not use the time component of the vector potential A t from Wald solution; instead, we use the ideal MHD condition E + v × B = 0 to determine the electric field E . Here the Alfv'en velocity and plasma beta value at the disk ( r = 3 . 5 r S ) are v A = 0 . 03 c and β ∼ 3 . 4, respectively. \nFigure 1b shows the state at t = 30 τ S , where τ S is defined as τ S ≡ r S /c . By this time the inner edge of the disk has rotated 0 . 75 cycles, if we assume the edge is at R = 3 r S . 1 Actually, the edge falls toward the black hole and rotates faster at R = 2 r S . The rapid infall produces a shock at R = 3 . 1 r S , and the high pressure behind it begins to produce the jet. This is the same pressure-driven jet formation process seen previously in the Schwarzschild case (Koide, Shibata, & Kudoh 1999a). \nFigure 1c shows the final state of the counter-rotating disk case at t = 47 τ S when the inner edge of the disk rotated 1.2 cycles. The accretion disk continues to fall rapidly toward the black hole, with the disk plasma entering the ergosphere and then crossing the horizon, as shown by the crowded magnetic field lines near r = 0 . 75 r S . The magnetic field lines become radial due to dragging by the disk infall near the black hole. The jet is ejected almost along the magnetic field lines. Its maximum total and poloidal velocities are the same, v = v p = 0 . 44 c at R = 3 . 2 r S , z = 1 . 6 r S . The mass density plot (color) shows that the jet consists of two layers. One is an inner, low density, fast, magnetically-driven jet and the other is an outer, high density, slow, gas pressure-driven jet. The latter comes from the disk near the shock at R = 3 . 1 r S and is, therefore, similar to the gas pressure-driven jet of Koide, Shibata, & Kudoh (1998). The former is new and has never been seen in the Schwarzschild black hole case. It comes from the disk near the ergosphere, and is accelerated as follows. As there is no stable orbit at R ≤ 4 . 4 r S , \nthe disk falls rapidly into the ergosphere. Inside the static limit , the velocity of frame dragging exceeds the speed of light ( c Ω 3 /α > c ), causing the disk to rotate in the same direction of the black hole rotation (relative to the fixed Boyer-Lindquist frame), even though it was initially counter-rotating. The rapid, differential frame dragging greatly enhances the azimuthal magnetic field, which then accelerates the flow upward and pinches it into a powerful collimated jet. \nFigure 1d shows a snapshot of the co-rotating disk case at t = 47 τ S . The disk stops its infall near R = 3 r S due to the centrifugal barrier with a shock at r = 3 . 4 r S . The high pressure behind the shock causes a gas pressure-driven jet with total and poloidal velocities of v = v p = 0 . 30 c at R = 3 . 4 r S , z = 2 . 4 r S . A detailed analysis shows that a weak magnetically-driven jet is formed outside the gas pressuredriven jet with maximum total and poloidal velocities of v = 0 . 42 c and v p = 0 . 13 c , respectively. This two-layered shell structure is similar to that of Schwarzschild black hole case (Koide, Shibata, & Kudoh 1998). The centrifugal barrier makes the disk take much long time to reach the ergosphere, which causes the difference between the co-rotating and counter-rotating disk cases. \nTo more fully illustrate the physics of the jet formation mechanism, in figure 2 we show the plasma beta, β ≡ p/ ( B 2 / 2) (color) and the toroidal component of the magnetic field, B φ (contour) in the counter-rotating and co-rotating disk cases at t = 47 τ S . The blue color shows the region where magnetic field dominates the gas pressure; light red-yellow shows where gas pressure is dominant; and solid contour line shows negative azimuthal magnetic field ( B φ < 0), while the broken line the positive value ( B φ > 0). The toroidal component of the magnetic field B φ is negative and its absolute value is very large above the black hole in both cases. The field increases to more than 10 times the initial magnetic field. This amplification is caused by the shear of the plasma flow in the Boyer-Lindquist frame due to the frame dragging effect of the rotating black hole (Yokosawa, Ishizuka, & Yabuki 1991, Yokosawa 1993, Meier 1999). Under the simplifying assumption that the plasma is at rest in the ZAMO frame, the general relativistic Faraday law of induction and ideal MHD condition yield, \n∂B φ ∂t = f 1 B r + f 2 B θ , (2) \nwhere f 1 = c ( h 3 /h 1 ) ∂ (Ω 3 /h 3 ) /∂r , and f 2 = c ( h 3 /h 2 ) ∂ (Ω 3 /h 3 ) /∂θ . This expression is almost identical to that of ω -dynamo effect from the field of the terrestrial magnetism. Noted that f 2 is one order smaller than f 1 when a ∼ 1. \nWhere does the magnetic field amplification energy comes from? It does not come from the gravitational energy or thermal energy of the disk, because frame dragging effect occurs even when the plasmas of the disk and corona are rest and cool. The only other possible energy source is the rotation of the black hole itself. Indeed, the increase in the azimuthal magnetic field component (eq. (2)) depends on the shear of the rotational variable Ω 3 . We conclude that the amplification energy of the magnetic field is supplied by extraction of the rotational energy of the black hole. \nThe distribution of the plasma beta ( β ) and the azimuthal component of the magnetic field ( B φ ) of the counter-rotating and co-rotating disk cases are quite different. In the co-rotating disk case, they are similar to those of the Schwarzschild black hole case. In the counterrotating disk case, the outer part has a positive azimuthal component of the magnetic field ( B φ > 0), which is caused by the counter-rotating disk, and the outer part has the high plasma beta. The inner part has a negative azimuthal magnetic field ( B φ < 0) and low plasma beta. Note that the very high plasma beta region (yellow region) is outside of the jet; at this point it has almost stopped and eventually will fall into the black hole. The negative azimuthal magnetic field is caused by the disk around the ergosphere, where the disk rotates in the same direction as the black hole in the Boyer-Lindquist frame. \nTo confirm the jet acceleration mechanism, we estimate the power from the electromagnetic field, W EM = v · ( E + J × B ) and the gas pressure, W gp = -v · ∇ p along the line, z = 1 . 1 r S which crosses the jet foot (Fig. 3). At t = 47 τ S , the gas pressure is dominant in the corotating disk case (Fig. 3b). However, in the counterrotating disk case, the electromagnetic power is dominant near the black hole even through the gas pressure power is the same as that of the co-rotating disk case (Fig. 3a). The magnetically-driven jet in this latter case is accelerated by the magnetic field anchored to the ergospheric disk. The frame dragging effect rapidly rotates the disk in the same direction as the black hole rotation, increasing the azimuthal component of the magnetic field and the magnetic tension which, in turn, accelerates the plasma by the magnetic pressure and centrifugal force, respectively. (A detailed analysis shows that both component of the magnetic forces are comparable.) This mechanism of jet production, therefore, is a kind of Penrose process that uses the magnetic field to extract rotational energy of the black hole and eject a collimated outflow from very near the horizon.", '4. DISCUSSION': 'We have presented general relativistic simulations of jet formation from both counter-rotating and co-rotating disks in a Kerr black hole magnetosphere. We have found that jets are formed in both cases. At the time when the simulations were stopped ( t = 47 τ S (53 τ S ), after the inner edge of the disk had rotated 1.2 (1.4) cycles in the counter-rotating (co-rotating) disk case) the poloidal velocities of the jets were v ∼ 0 . 4 c (counter-rotating), ∼ 0 . 3 c (co-rotating), both sub-relativistic. In the corotating disk case, the jet has a two-layered structure: inner, gas pressure-driven jet and outer, magnetically-driven jet. On the other hand, in the counter-rotating case, a new magnetically-driven jet has been found inside the gas pressure-driven jet. The new jet is accelerated by the magnetic field induced by the frame dragging effect in the ergosphere. In this case, existence of a magnetically-driven jet is not clear outside the gas pressure-driven jet. A longer term simulation may show a three-layered structure including the outer, magnetically-driven jet. \nUnfortunately, the counter-rotating (co-rotating) disk case could not be continued beyond t = 47 τ S ( t = 53 τ S ) because of numerical problems. We have performed one other case previously - the infall of a magnetized non- \ntating disk into a rapidly-rotating black hole (Koide, Meier, Shibata, & Kudoh 1999b). The disk falls toward the black hole more rapidly than the counter-rotating case. At later times (after almost two inner disk turns) it developed a relativistic jet with a velocity of v ∼ 0 . 9 c (Lorentz factor ∼ 2). We believe that, if we had been able to perform longer-term simulations here, in at least the counterrotating disk case the magnetically-driven jet also would have been accelerated to relativistic velocities (and possibly the co-rotating case as well). Despite its low speed, the magnetically-driven jet in the counter-rotating disk case is nevertheless noteworthy because it extracts rotational energy from the black hole. While the process is similar to the Blandford-Znajek mechanism (Blandford & Znajek 1977), it appears more closely related to the model of Takahashi et al. (1990). In our case, the electromagnetic field energy is transformed immediately into kinetic energy in the jet. A more detailed analysis and further calculations will be reported in our next paper. \nRecently, Wardle et al. (1998) detected circularly polarized radio emission from the jets of the archetypal quasar 3C279. They concluded that electron-positron pairs are \nimportant components of the jet plasma. Similar detections in three other radio sources have been made (Homan et al. 1999), which suggests that, in general, extragalactic radio jets may be composed mainly of an electron-positron pair plasma. The electron-positron plasma is probably produced very near the black hole. The mechanism we have investigated here, magnetically-driven jet powered by extraction of rotational energy of a black hole, is a strong candidate for explaining the acceleration of such electronpositron jets. \nS. K. thanks M. Inda-Koide for discussions and important comments for this study. We thank K.-I. Nishikawa, M. Takahashi, A. Tomimatsu, P. Hardee, and J.-I. Sakai for discussions and encouragement. We appreciate the support of the National Institute for Fusion Science and the National Astronomical Observatory in the use of their super-computers. Part of this research was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administrations.', 'REFERENCES': 'Begelman, .M.C., Blandford, R.D., Rees, M.J. 1984, Review of Modern Physics, 56 , 255. \nBelloni, T., van der Klis, M., Lewin, W. H. G., van Paradijs, J., Dotani, T., Mitsuda, K., & Miyamoto, S. 1997, A & A 322 , 857. Blandford, R. D. & Payne, D. G. 1982, MNRAS 199 , 883. 179 \n- Cui, W., Zhang, S. N., Focke, W., & Swank, J. H. 1997, MNRAS 290 , L65.\n- Blandford, R. D. & Znajek, R. 1977, MNRAS , 433. Cui, W., Zhang, S. N., & Chen, W. 1998, ApJ 484 , 383.\n- Davis, S. F. 1984, NASA Contractor Rep. 172373, ICASE Rep. No. 84-20.\n- Homan, D. C., Wardle, J. F. C., Roberts, D. H., and Ojha, R. 1999, preprint.\n- Hughes, P. A. 1991, eds., Beams and Jets in Astrophysics (Cambridge University Press, New York).\n- Iwasawa, K., Fabian, A. C., Reynolds, C. S., Nandra, K., Otani, C. Inoue, H. Hayashida, K. Brandt, W. N., Dotani, T., Kunieda, H., Matsuoka, M., Tanaka, Y. 1996, MNRAS, 282 , 1038. Koide, S. 1997, ApJ 478 , 66.\n- Koide, S., Nishikawa, K.-I., & Mutel, R. L. 1996, ApJ 463 , L71.\n- Koide, S., Shibata, K., & Kudoh, T., 1999a, ApJ 522 , 175.\n- Koide, S., Shibata, K., & Kudoh, T. 1998, ApJ 495 , L63.\n- T. Montmerle, & J. Paul (World Scientific Press, Paris) in press. \nKudoh, T. & Shibata, K. 1995, ApJ 452 , L41. Kudoh, T. & Shibata, K. 1997a, ApJ 474 , 362. \n- Kudoh, T. & Shibata, K. 1997b, ApJ 476 , 632. \nLinden-Bell, D. 1969, Nature 223 , 690. Meier, D. L. 1999, ApJ 522 , in press. \n- Meier, D. L., Edgingon, S., Godon, P., Payne, D. G., & Lind, K. R. 1997, Nature 388 , 350.\n- Mirabel, I. F. & Rodriguez, L. F. 1994, Nature 371 , 46. \nMorgan, E. H., Remillard, R. A., & Greiner, J. 1977, ApJ 482 , 993. Ouyed, R., Pudritz, R. E., & Stone, J. M. 1997, Nature 385 , 409. Pearson, J. J. 1981, et al. Nature 290 , 365. Rees, M. J. 1966, Nature 211 , 468. \n- Rees, M. J. 1984, Ann. Rev. Astron. Ap. 22 , 471.\n- Rees, M. J. 1984, Ann. Rev. Astron. Ap. 22 , 471.\n- Remillard, R. A, Morgan, E. H, McClintock, J. E, Bailyn, C. D., Oroszek, A., & Greiner, J. 1999, in Proc. 18th Texas Symp. on Relativistic Astrophysics, ed. A. Olinto, J. Frieman, & D. Schramm (World Scientific Press, Singapore) in press.\n- Shapiro, S. L. & Teukolsky, S. A. 1983, Black Holes, White Dwarfs, and Neutron Stars , (John Wiley & Sons Inc., New York) Shibata, K. & Uchida, Y. 1986, PASJ 38 , 631.\n- Takahashi, M., Nitta, S., Tatematsu, Y., & Tomimatsu, A. 1990, ApJ 363 , 206.\n- Thorne, K. S., Price, R. H., & Macdonald, D. A. 1986, Membrane Paradigm (Yale University Press, New Haven and London). \nTingay, S. J. 1995, et al. Nature 374 , 141. Uchida, Y. & Shibata, K. 1985, PASJ 37 , 515. \n- Wald, R. M. 1974, Phys. Rev. D 10 , 1680.\n- Wardle, J. F. C., Homan, D. C., Ojha, R., & Roberts, D. H. 1998, Nature 395 , 457.\n- Yokosawa, M. 1993, PASJ 45 , 207.\n- Yokosawa, M., Ishizuka, T., & Yabuki, Y. 1991, PASJ 43 , 427.', 'Figure Captions': '- Fig. 1. Time evolution of jet formation in the counter-rotating disk case and the final state of the co-rotating disk case. Color shows the logarithm of the proper mass density; vectors indicate velocity; solid lines show the poloidal magnetic field. The black fan-shaped region at the origin shows the horizon of the Kerr black hole ( a = 0 . 95). The dashed line near the horizon is the inner boundary of the calculation region.\n- At t = 0 and t = 30 τ S the state of the co-rotating and counter-rotating disk cases are almost identical. However, at t = 47 τ S , while the infall of the disk in the co-rotating disk stops (due to a centrifugal barrier), the unstable orbits of the counter-rotating disk plasma continue to spiral rapidly toward the black hole horizon. This difference causes the magnetohydrodynamic jet formation mechanisms in the two cases to differ drastically, resulting in a powerful jet emanating from deep within the ergosphere.\n- Fig. 2. Plasma beta (color) and azimuthal component of the magnetic field B φ (contour) in the counter-rotating and co-rotating disk cases. Solid lines show negative value of B φ and the dashed lines positive values. The two cases differ significantly in structure, with the jet in the counter-rotating case originating much closer to the black hole.\n- Fig. 3. Power contribution to jet acceleration along the line, z = 1 . 1 r S due to the gas pressure ( W gp ) and the electromagnetic force ( W EM ) for both the counter-rotating and co-rotating disk cases. The jet in the counter-rotating disk case is accelerated mainly by electromagnetic forces, while that in the co-rotating disk is accelerated mainly by gas pressure. Note that, while the power in the gas jet component is comparable in the two cases, the power in the MHD jet component is nearly two orders of magnitude greater in the counter-rotating case than the co-rotating case. \nt=47 counter-rotating disk (c) \n<!-- image --> \nFig. 1.- \n<!-- image --> \nc \n(b) \ncounter-rotating disk \nt=30 \n(d) t=47 co-rotating disk \n<!-- image --> \n<!-- image -->', 'counter-rotating disk (a)': 't=47 \nco-rotating disk \nt=47 \nFig. 2.- \n<!-- image --> \n(b) \n(a) \ncounter-rotating disk \nt=47, z=1.1 \nFig. 3.- \n<!-- image --> \n(b) \nco-rotating disk \nt=47, z=1.1 \n<!-- image -->'}

2003CQGra..20L.277M

LETTER TO THE EDITOR: The black holes of topologically massive gravity

2003-01-01

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We show that an analytical continuation of the Vuorio solution to three-dimensional topologically massive gravity leads to a two-parameter family of black-hole solutions, which are geodesically complete and causally regular within a certain parameter range. No observers can remain static in these spacetimes. We discuss their global structure, and evaluate their mass, angular momentum and entropy, which satisfy a slightly modified form of the first law of thermodynamics.

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https://arxiv.org/pdf/gr-qc/0303042.pdf

{'The black holes of topologically massive gravity': "Karim Ait Moussa a,b ∗ , G'erard Cl'ement a † and C'edric Leygnac a ‡ \na \nLaboratoire de Physique Th'eorique LAPTH (CNRS), \nB.P.110, F-74941 Annecy-le-Vieux cedex, France \nb Laboratoire de Physique Math'ematique et Physique Subatomique, D'epartement de Physique, Facult'e des Sciences, Universit'e Mentouri, Constantine 25000, Algeria \nOctober 29, 2018", 'Abstract': "We show that an analytical continuation of the Vuorio solution to three-dimensional topologically massive gravity leads to a two-parameter family of black hole solutions, which are geodesically complete and causally regular within a certain parameter range. No observers can remain static in these spacetimes. We discuss their global structure, and evaluate their mass, angular momentum, and entropy, which satisfy a slightly modified form of the first law of thermodynamics. \nIt is well known that Einstein gravity in 2+1 dimensions is dynamically trivial, without propagating degrees of freedom. The addition to the Einstein-Hilbert action of a gravitational Chern-Simons term leads to topologically massive gravity [1] (TMG), with massive spin 2 excitations. A long standing question [2] is that of the existence of Schwarzschild- or Kerrlike black hole solutions to TMG. The equations of TMG with a negative cosmological constant are trivially solved [3] by the BTZ black-hole metric [4]. Other black hole solutions to cosmological TMG were written down in [5, 6], however it is possible to show that these solutions are not bona fide black holes for the range of parameters considered in [5, 6]. A number of exact solutions to TMG with vanishing cosmological constant are known [7, 8, 9, 10], but these do not include black hole solutions. The purpose of this Letter is to show that a simple extension of the Vuorio solution [7] leads to regular TMG black holes, and to present an introductory exploration of their properties. \nThe field equations of TMG are \nG µ ν + 1 µ C µ ν = 0 , (1) \nwhere G µ ν ≡ R µ ν -1 2 Rδ µ ν is the Einstein tensor, \nC µ ν ≡ ε µαβ D α ( R βν -1 4 g βν R ) (2) \nis the Cotton tensor (the antisymmetrical tensor is ε µαβ = | g | -1 / 2 η µαβ , with η 012 = +1), and µ is the topological mass constant. Vuorio searched for stationary rotationally symmetric solutions to these equations, and noticed that they could be solved exactly by assuming a constant g tt , which he normalized to the Minkowski value 1 g tt = -1. Vuorio's solution is, in units such that µ = +3, \nds 2 = -[ d ˜ t -(2 cosh σ + ˜ ω ) d ˜ ϕ ] 2 + dσ 2 +sinh 2 σ d ˜ ϕ 2 , (3) \nwhere ˜ ϕ is assumed to be periodic with period 2 π , and we have reintroduced an integration constant ˜ ω which Vuorio set to -2 for regularity. This spacetime is homogeneous, with constant curvature scalars, and admits four Killing vectors generating the Lie algebra of SL (2 , R ) × U (1) [7, 11]. However, it has one undesirable property. From (3) we obtain \ng ˜ ϕ ˜ ϕ = -3cosh 2 σ +4˜ ω cosh σ -˜ ω 2 -1, which is negative definite if ˜ ω 2 < 3, and negative outside a critical radius σ c if ˜ ω 2 > 3, leading to closed timelike circles 2 for all σ > σ c . \nTo overcome this indesirable violation of causality, let us analytically continue the solution (3) by the combined imaginary coordinate transformation (which does not change the overall Lorentzian signature) ˜ t = i √ 3 t , ˜ ϕ = i ( ρ 0 / √ 3) ϕ , with ρ 0 an arbitrary positive constant, leading to \nds 2 = 3 [ dt -( 2 ρ 3 + ω ) dϕ ] 2 + dρ 2 ρ 2 -ρ 2 0 -ρ 2 -ρ 2 0 3 dϕ 2 , (4) \nwhere ω = ρ 0 ˜ ω/ 3, the new radial coordinate is ρ = ρ 0 cosh σ , and ϕ is again assumed to be an angular variable with period 2 π . This new solution of the field equations can also be derived directly 'a la Vuorio by making the unconventional ansatz g tt = +3 (instead of -1). The fact that the Killing vector field ∂ t is spacelike, not timelike, means that there can be no static observers in such a geometry. Furthermore it is easily seen that no linear combination of the Killing vectors ∂ t and ∂ ϕ can remain timelike for ρ →∞ . This situation is quite similar to that inside the ergosphere of the Kerr metric, except that here the ergosphere extends to spacelike infinity. As in the case of the Kerr ergosphere, the solution is however locally stationary. At a given radius ρ , observers moving with uniform angular velocity Ω remain timelike, ( ∂ t +Ω ∂ ϕ ) 2 < 0, provided \n2 ρ +3 ω -√ ρ 2 -ρ 2 0 r 2 < Ω < 2 ρ +3 ω + √ ρ 2 -ρ 2 0 r 2 . (5) \nThe divergence of the metric component g ρρ at ρ = ± ρ 0 suggests that (4) is a black hole solution, which is made obvious by rearranging the metric as \nwith \nds 2 = -ρ 2 -ρ 2 0 r 2 dt 2 + dρ 2 ρ 2 -ρ 2 0 + r 2 ( dϕ -2 ρ +3 ω r 2 dt ) 2 , (6) \nr 2 = ρ 2 +4 ωρ +3 ω 2 + ρ 2 0 / 3 . (7) \nWhile this metric is not asymptotically Minkowskian, the squared lapse N 2 = ( ρ 2 -ρ 2 0 ) /r 2 and the shift N ϕ = -(2 ρ +3 ω ) /r 2 respectively go to 1 \nand 0 at spacelike infinity ρ → ±∞ . For ρ 2 0 > 0, there are two horizons located at ρ h ± = ± ρ 0 , of perimeter and angular velocity \nA h ± ≡ 2 πr h ± = 2 π | 2 ρ 0 ± 3 ω | / √ 3 , Ω h ± = 3 / ( ± 2 ρ 0 +3 ω ) . (8) \nIf ω /negationslash = ∓ 2 ρ 0 / 3, the metric may be extended through these horizons by the usual Kruskal method. Just as in the case of the BTZ black holes, the metric (4) is regular for all ρ /negationslash = ± ρ 0 , so that the maximally extended spacetime is geodesically complete, with a Penrose diagram similar to that of the Kerr black hole (except of course for the ring singularity). However, singularities in the causal structure do occur for a range of values of ω . If ω 2 < ρ 2 0 / 3, the Killing vector ∂ ϕ is everywhere spacelike. On the contrary, if ω 2 > ρ 2 0 / 3, ∂ ϕ becomes timelike in the range ρ c -< ρ < ρ c + , with \nρ c ± = -2 ω ± √ ω 2 -ρ 2 0 / 3 . (9) \nIt is easily seen that these two zeros of ( ∂ ϕ ) 2 are timelike lines belonging to the same stationary Kruskal patch, so that the acausal regions r 2 < 0 are safely hidden behind the horizon for an observer at ρ = + ∞ if ω > 0, which we shall assume henceforth. The Penrose diagram for the maximally extended spacetime with the acausal regions cut out is the same as for Reissner-Nordstrom black holes. The limiting case ρ 0 = 0 leads to extreme black holes, with a double horizon at ρ = 0. In this case, there is an acausal region ( ∂ ϕ ) 2 < 0 behind the horizon for all positive values of ω . The resulting Penrose diagram (again with the acausal regions cut out) is identical to that of extreme Reissner-Nordstrom black holes. \nThe case ω = 2 ρ 0 / 3 deserves special consideration. In this case the metric (6) reduces to \nds 2 = -ρ -ρ 0 ρ +5 ρ 0 / 3 dt 2 + dρ 2 ρ 2 -ρ 2 0 +( ρ +5 ρ 0 / 3)( ρ + ρ 0 ) ( dϕ -2 dt ρ +5 ρ 0 / 3 ) 2 . (10) \nThis has only one horizon at ρ = + ρ 0 , where Kruskal extension can be carried out as usual. Near the causal singularity ρ = -ρ 0 , the metric (10) can be approximated by \nds 2 /similarequal 3 dt 2 + dσ 2 , dσ 2 = -dρ 2 2 ρ 0 ( ρ + ρ 0 ) +(2 ρ 0 / 3)( ρ + ρ 0 ) d ˆ ϕ 2 , (11) \nwith d ˆ ϕ = dϕ -3 dt . Clearly the two-dimensional metric dσ 2 becomes null for ρ →-ρ 0 , and is non-extendible because of the periodicity condition on ϕ , so \nthat ρ = -ρ 0 is a spacelike singularity of the metric (10). The corresponding Penrose diagram is thus identical to that of the Schwarzschild black hole. \nThe even more special case ω = ρ 0 = 0 lies at the intersection of the preceding case and of the extreme black hole case ρ 0 = 0. In this case the metric (6) reduces to \nds 2 = -dt 2 + dρ 2 ρ 2 + ρ 2 ( dϕ -2 ρ dt ) 2 , (12) \nwhich is devoid of horizons, and thus qualifies as the ground state or 'vacuum' of our two-parameter family of black-hole solutions. This metric does not appear to be extendible beyond the manifest singularity ρ = 0, which can be shown to be at finite affine distance. Moreover, the angular velocity of stationary observers approaching this singularity increases without bound, so that the very concept of a Penrose diagram breaks down. \nThe black hole metric (6) depends on two parameters ρ 0 and ω , which should somehow be related to the physical parameters, mass and angular momentum. The standard approach for computing these quantities in the case of non-asymptotically flat spacetimes 3 uses the idea of quasilocal energy [13]. From the action functional for a self-gravitating system with boundary conditions on a given hypersurface, one derives canonically a Hamiltonian, given by the sum of a bulk integral, which vanishes on shell, and of a surface term. The quasilocal energy is the (substracted) on-shell value of the Hamiltonian in the limit where the spatial boundary is taken to infinity. A canonical formulation of topologically massive gravity was given in [14]. However integrations by part were freely performed in [14], so that at present we do not know what is the correct surface term. Instead we shall bypass the standard quasilocal approach by suitably extending the recently proposed super angular momentum approach to the computation of conserved quantities in 2+1 gravity [15]. This approach relies on the observation that the dimensional reduction of a self-gravitating system with two Killing vectors ∂ t and ∂ ϕ leads to a mechanical system with the SL (2 , R ) ∼ SO (2 , 1) invariance. This mechanical system has a conserved super angular momentum, two components of which may be identified as the mass and angular momentum of the (2+1)-dimensional gravitating configuration. These identifications have been shown in [15] to correspond to a well-defined 'finite part' prescription for computing the (unsubstracted) quasilocal conserved \nquantities, and to lead to consistent results for Einstein-scalar and (up to some gauge ambiguity) for Einstein-Maxwell black holes. \nThe conserved super angular momentum for TMG has been given in [10]. The general stationary rotationally symmetric metric may be written in the 2+1 form \nds 2 = λ ab ( ρ ) dx a dx b + ζ -2 ( ρ ) R -2 ( ρ ) dρ 2 , (13) \nwhere λ is the 2 × 2 matrix \nλ = ( T + X Y Y T -X ) , (14) \nR 2 = X 2 = -T 2 + X 2 + Y 2 is the Minkowski pseudo-norm of the vector X ( ρ ) = ( T, X, Y ), and the function ζ ( ρ ) allows for arbitrary reparametrizations of the radial coordinate ρ . In the gauge ζ = 1, the conserved generalized angular momentum for TMG is \nJ = 1 2 κ ( X ∧ X ' + 1 2 µ [ X ' ∧ ( X ∧ X ' ) -2 X ∧ ( X ∧ X '' ) ]) , (15) \nwhere κ = 8 πG is the Einstein gravitational constant, the prime is the derivative d/dρ , and the wedge product is defined by ( X ∧ Y ) A = η AB /epsilon1 BCD X C Y D (with η AB the inverse Minkowski metric, and /epsilon1 012 = +1). Assuming that the identifications of Einstein-scalar or Einstein-Maxwell black hole conserved quantities proposed in [15] can be extended to TMG, the (2+1)-dimensional mass and angular momentum are given by \nM = 2 π J Y , (16) \nJ = 2 π ( J T -J X ) . (17) \n- \nWe first test these formulas on the example of the BTZ solution, for which the quasilocal mass and angular momentum have recently been computed in the wider framework of a Poincar'e gauge theory [16] (the action for this theory reduces to that of TMG when a constraint for vanishing torsion is added). The BTZ metric is given by \nds 2 = ( -2 l -2 ρ +M / 2) dt 2 -J dt dϕ +(2 ρ +M l 2 / 2) dϕ 2 +[4 l -2 ρ 2 -(M 2 l 2 -J 2 ) / 4] -1 dρ 2 , (18) \nwith Λ = -l -2 the cosmological constant. The parametrization (14) of this metric corresponds to [15] \nX = ∣ ∣ ∣ ∣ ∣ ∣ (1 -l -2 ) ρ +(1 + l -2 )M l 2 / 4 -(1 + l -2 ) ρ -(1 -l -2 )M l 2 / 4 -J / 2 , (19) \n∣ \nwith ζ = 1. The computation of the superangular momentum (15) is straightforward and gives the TMG mass and angular momentum of the BTZ black holes in terms of the Einstein conserved quantities M and J, \nM = π κ ( M -J µl 2 ) , J = π κ ( J -M µ ) . (20) \nThese values (which reduce to the Einstein values in the limit µ → ∞ ) coincide with the values (22) and (23) obtained in [16] in the special case of a vanishing torsion T = 0 (the identification is 2 /lscriptθ L = -1 /µ , Λ eff = -Λ = l -2 , /lscript = κ = π , χ = 1). \nStrengthened by this agreement, we proceed with the computation of the conserved quantities M and J for the Λ = 0 TMG black holes. The parametrization (14) of the metric (6) corresponds to \nX = ∣ ∣ ∣ ∣ ∣ ρ 2 / 2 + 2 ωρ +3( ω 2 +1) / 2 + ρ 2 0 / 6 -ρ 2 / 2 -2 ωρ -3( ω 2 -1) / 2 -ρ 2 0 / 6 -2 ρ -3 ω , (21) \n∣ \n∣ with ζ = 1. The computation of the super angular momentum (15) leads to \nM = π κ ω, J = π κ ( ω 2 -5 ρ 2 0 / 9) . (22) \nSurprisingly, the mass depends only on the parameter ω , and not on the ostensible horizon radius ρ 0 . These relations can be inverted to yield \nω = κ π M, ρ 2 0 = 9 κ 2 5 π 2 ( M 2 -πJ/κ ) . (23) \nExtreme black holes thus have J = κM 2 /π , Schwarzschild-like black holes have J = -κM 2 / 4 π , while the mass and angular momentum vanish for the vacuum solution (12), as expected. \nWhat are the thermodynamical properties of these black holes? The Hawking temperature depends only on the metric, not on the particular theory of gravity considered, and is given by the inverse of the period in imaginary time, \nWe obtain here the temperature \nT H ≡ 1 2 π n ρ ∂ ρ N | ( ρ = ρ h ) = ζRR ' 2 π √ V ∣ ∣ ∣ ∣ ( ρ = ρ h ) . (24) \nT H = √ 3 2 π ρ 0 2 ρ 0 +3 ω , (25) \nwhich as usual vanishes for extreme black holes ( ρ 0 = 0). On the other hand, the black hole entropy should depend on the specific theory under consideration, and we have no reason to expect that it is given by the Einstein value S E = (2 π/κ ) A h . To determine the black hole entropy, we use the first law of thermodynamics, according to which the entropy variation is given by \nThis can be integrated to yield \n∂S ∂M ∣ ∣ ∣ ∣ J = T -1 H . (26) \nS = 2 π 2 3 √ 3 κ (5 ρ 0 +6 ω ) , (27) \nup to an arbitrary additive function of J . Assuming that this function vanishes, we obtain from (27), (25), (22) and (8) a modified form of the first law of black hole thermodynamics: \ndM = T H dS + 1 2 Ω h dJ . (28) \nThe anomalous factor 1 / 2 in front of the angular velocity is another surprising effect of TMG. Finally, a simple quadratic combination of the same (undifferentiated) quantities yields the Smarr-like formula \nM = T H S +Ω h J , (29) \nto be compared with the Smarr-like formula for 2+1 Einstein-scalar black holes [15] M = T H S/ 2 + Ω h J . \nA problem with the value (22) for the black hole mass is that, contrary to the case of four-dimensional Einstein gravity, the gravitational constant κ must be negative in TMG to avoid the occurrence of ghosts [1]. This means that, for ω 2 > ρ 2 0 / 3, causally regular black holes, with ω > 0, have a negative mass (as well as a negative entropy). For ω 2 < ρ 2 0 / 3, all black holes are regular, but only those with ω < 0 have a positive mass (but a negative entropy). We do not know how to solve this problem, but point out that a similar problem arises with the BTZ black holes viewed as solutions of TMG with negative gravitational constant. It follows from (20) that, for κ < 0, regular BTZ black holes, with M > 0 and J 2 ≤ M 2 l 2 , necessarily have a negative TMG mass M if µ 2 l 2 > 1, and do not necessarily have a positive TMG mass if µ 2 l 2 1. \nWe have shown that an analytical extension of the Vuorio solution to TMG leads to a two-parameter family of black hole solutions, which are \n≤ \ncausally regular within a certain parameter range. While their metric is not asymptotically Minkowskian, the AdM lapse and shift functions go to the Minkowski values 1 and 0 at spacelike infinity, so that their Penrose diagrams are similar to those of Kerr or Reissner-Nordstrom black holes. In this respect, these black hole spacetimes are closer to four-dimensional black holes than the asymptotically AdS BTZ black holes of [4]. No observers can remain static in these spacetimes, however stationary observers are allowed, which is all that is needed to discusss physical experiments such as wave scattering. We have evaluated the mass, angular momentum, and entropy of these black holes, which satisfy a slightly modified form of the first law of thermodynamics. At present these evaluations remain tentative. Our formulas (16) and (17) for computing the black hole mass and angular momentum generalize formulas previously derived and tested in the case of Einstein-scalar or Einstein-Maxwell black holes, and have been tested here in the specific case of the BTZ solution to cosmological TMG. However a full computation of the quasilocal energy and angular momentum in TMG should be carried out in order to ascertain the validity of our evaluations (the computations of [16] cannot be adapted for this purpose because the field equations of the Poincar'e gauge theory considered there are stronger than those of TMG, and do not admit our black holes as solutions). A direct computation of the black hole entropy is also desirable. We intend to address these questions, and to further elucidate the properties of our black holes, in a future publication.", 'References': "- [1] S. Deser, R. Jackiw and S. Templeton, Phys. Rev. Lett. 48 (1982) 975; Ann. Phys., NY 140 (1982) 372.\n- [2] S. Deser, private communication.\n- [3] N. Kaloper, Phys. Rev. D 48 (1993) 2598.\n- [4] M. Ba˜nados, C. Teitelboim and J. Zanelli, Phys. Rev. Lett. 69 (1992) 1849; M. Ba˜nados, M. Henneaux, C. Teitelboim and J. Zanelli, Phys. Rev. D 48 (1993) 1506.\n- [5] Y. Nutku, Class. Quant. Grav. 10 (1993) 2657.\n- [6] M. Gurses, Class. Quant. Grav. 11 (1994) 2585. \n- [7] I. Vuorio, Phys. Lett. 163B (1985) 91; R. Percacci, P. Sodano and I. Vuorio, Ann. Phys., NY 176 (1987) 344.\n- [8] G.S. Hall, T. Morgan and Z. Perj'es, Gen. Rel. Grav. 19 (1987) 1137.\n- [9] Y. Nutku and P. Baekler, Ann. Phys., NY 195 (1989) 16.\n- [10] G. Cl'ement, Class. Quantum Grav. 11 (1994) L115.\n- [11] M.E. Ortiz, Ann. Phys, NY 200 (1990) 345.\n- [12] S. Deser and B. Tekin, Class. Quantum Grav. 20 (2003) L259.\n- [13] J.D. Brown and J.W. York, Phys. Rev. D 47 (1993) 1407.\n- [14] S. Deser and X. Xiang, Phys. Lett. B 263 (1991) 39.\n- [15] G. Cl'ement, Phys. Rev. D 68 (2003) 024032.\n- [16] A.A. Garc'ıa, F.W. Hehl, C. Heinicke and A. Mac'ıas, Phys. Rev. D 67 (2003) 124016."}

2008ApJ...673..703M

Comparing and Calibrating Black Hole Mass Estimators for Distant Active Galactic Nuclei

2008-01-01

['black hole physics', 'galaxies active', 'galaxies evolution', 'galaxies quasars', 'astrophysics']

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Black hole mass (M<SUB>BH</SUB>) is a fundamental property of active galactic nuclei (AGNs). In the distant universe, M<SUB>BH</SUB> is commonly estimated using the Mg II, Hβ, or Hα emission line widths and the optical/UV continuum or line luminosities as proxies for the characteristic velocity and size of the broad-line region. Although they all have a common calibration in the local universe, a number of different recipes are currently used in the literature. It is important to verify the relative accuracy and consistency of the recipes, as systematic changes could mimic evolutionary trends when comparing various samples. At z = 0.36, all three lines can be observed at optical wavelengths, providing a unique opportunity to compare different empirical recipes. We use spectra from the Keck Telescope and the Sloan Digital Sky Survey to compare M<SUB>BH</SUB> estimators for a sample of 19 AGNs at this redshift. We compare popular recipes available from the literature, finding that M<SUB>BH</SUB> estimates can differ up to 0.38 +/- 0.05 dex in the mean (or 0.13 +/- 0.05 dex, if the same virial coefficient is adopted). Finally, we provide a set of 30 internally self-consistent recipes for determining M<SUB>BH</SUB> from a variety of observables. The intrinsic scatter between cross-calibrated recipes is in the range 0.1-0.3 dex. This should be considered as a lower limit to the uncertainty of the M<SUB>BH</SUB> estimators.

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https://arxiv.org/pdf/0710.1839.pdf

{'COMPARING AND CALIBRATING BLACK HOLE MASS ESTIMATORS FOR DISTANT ACTIVE GALACTIC NUCLEI': 'Kathryn L. McGill 1 , Jong-Hak Woo 1,2 , Tommaso Treu 1,3 , Matthew A. Malkan 4 (Received June 15 2007; Accepted October 9 2007) Draft version October 31, 2018', 'ABSTRACT': 'Black hole mass (M BH ) is a fundamental property of active galactic nuclei (AGNs). In the distant universe, M BH is commonly estimated using the MgII, H β , or H α emission line widths and the optical/UV continuum or line luminosities, as proxies for the characteristic velocity and size of the broad-line region. Although they all have a common calibration in the local universe, a number of different recipes are currently used in the literature. It is important to verify the relative accuracy and consistency of the recipes, as systematic changes could mimic evolutionary trends when comparing various samples. At z = 0 . 36, all three lines can be observed at optical wavelengths, providing a unique opportunity to compare different empirical recipes. We use spectra from the Keck Telescope and the Sloan Digital Sky Survey to compare M BH estimators for a sample of nineteen AGNs at this redshift. We compare popular recipes available from the literature, finding that M BH estimates can differ up to 0 . 38 ± 0 . 05 dex in the mean (or 0 . 13 ± 0 . 05 dex, if the same virial coefficient is adopted). Finally, we provide a set of 30 internally self consistent recipes for determining M BH from a variety of observables. The intrinsic scatter between cross-calibrated recipes is in the range 0 . 1 -0 . 3 dex. This should be considered as a lower limit to the uncertainty of the M BH estimators. \nSubject headings: black hole physics: accretion - galaxies: active - galaxies: evolution - quasars: general', '1. INTRODUCTION': "Understanding the growth of supermassive black holes along with their host galaxies is one of the fundamental questions in current astrophysics (e.g. Di Matteo et al. 2005; Croton et al. 2006). Black hole mass (M BH ) is a key parameter in revealing the nature of black holegalaxy coevolution as well as the physics of active galactic nuclei (AGNs). However, direct mass measurements using the motions of gas and stars in the sphere of influence of a central black hole is limited to very nearby galaxies (e.g. Kormendy & Gebhardt 2001; Ferrarese & Ford 2005). \nBeyond the very local universe, the so-called 'virial' or 'empirically calibrated photo-ionization' method based on the reverberation sample is popularly used for active galaxies (e.g. Wandel et al. 1999; Kaspi et al. 2000, 2005; Bentz et al. 2006). This method utilizes broad line widths as velocity indicators and monochromatic continuum or line luminosities as indicators of broadline region size, hence estimating virial M BH . A combination of the MgII, H β , or H α broad emission line widths and the 3000 ˚ A, 5100 ˚ A, H β , or H α luminosities is typically used, depending on the redshift of the source and the observational setup. Several equations have been presented in the literature to estimate M BH using various combinations of these indicators (e.g., Woo & Urry 2002a,b; McLure & Jarvis 2002; Treu et al. 2004; Kollmeier et al. 2006; Greene & Ho \n1 Department of Physics, University of California, Santa Barbara, CA 93106-9530; [email protected], [email protected], [email protected] \n- 2 Corresponding author\n- 3 Alfred P. Sloan Research Fellow\n- 4 Department of Physics and Astronomy, University of California at Los Angeles, CA 90095-1547, [email protected] \n2005; Vestergaard & Peterson 2006; Woo et al. 2006; Salviander et al. 2007; Netzer & Trakhtenbrot 2007; Treu et al. 2007). \nAlthough all three emission lines have a common calibration based on the reverberation sample in the local universe, it is important to verify that different recipes give consistent results; any systematic changes could mimic evolutionary trends given that different recipes are often used in various studies. \nAt z = 0 . 36, all three lines can be observed at optical wavelengths, providing a unique opportunity to crosscalibrate the different methods of M BH estimation. Using data from the Keck Telescope and the Sloan Digital Sky Survey for a sample of nineteen AGNs at z = 0 . 36, we compare the different methods of estimating M BH , and derive a set of self-consistent equations for M BH estimates using every combination of velocity scale (FWHM and line dispersion σ line of MgII, H β , or H α ) and luminosity (3000 ˚ A, 5100 ˚ A- nuclear and total - H β , or H α ). \nThe paper is organized as follows. In § 2 we describe the sample selection, observations, and data reduction. In § 3 we describe our line fitting process, based on expansion in Gauss-Hermite series, and the resulting luminosity and width measurements. In § 4 we review the various formulae adopted in the literature, and compare the various M BH estimators. In § 5 we present our selfconsistent recipes. Section 6 summarizes our results. \nThroughout this paper magnitudes are given in the AB scale. We assume a concordance cosmology with matter and dark energy density Ω m = 0 . 3, Ω Λ = 0 . 7, and Hubble constant H 0 =70 kms -1 Mpc -1 . \nFig. 1.Flux-calibrated spectra. The SDSS spectra are shown in black, and the Keck spectra are shown in blue and red. The MgII line can be seen on the far left of each wavelength range, while H β is located in the center and H α to the far right. \n<!-- image --> \nThe AGN sample was initially selected for stellar velocity dispersion ( σ ∗ ) measurements to measure the M BH -σ ∗ relation at z = 0 . 36 (Treu et al. 2004; Woo et al. 2006). Readers are referred to the papers by Woo et al. (2006) and Treu et al. (2007) - where Keck red spectra (5900 ˚ A7500 ˚ A) and Hubble images of the sample were presented - for more details. The relevant properties of the observed objects are listed in Table 1. \nHigh signal-to-noise ratio spectra of nineteen targets were obtained with the Low Resolution Imaging Spectrometer (Oke et al. 1995, hereafter LRIS) at the KeckI telescope in five runs between March 2003 and July 2005, as detailed by Woo et al. (2006). The red setup is described by Woo et al. (2006). In the blue, the 600 lines mm -1 grism was used, yielding a pixel scale of 0.63 ˚ A × 0 . '' 135 and a resolution of ∼ 145 km s -1 . Note that objects S16, S31, and S99 included in the papers by Woo et al. (2006) or Treu et al. (2007) lack Keck and/or SDSS spectra and are therefore not considered here. \nThe reduction of the blue spectra was very similar to that of the red spectra described by Woo et al. (2006), except that arc lamp emission lines for Hg and Cd were used for wavelength calibration due to the paucity of sky lines. The flux was calibrated using spectrophotometric stars or A0V type Hipparcos stars. Galactic extinction correction was applied to all data based on the average extinction law derived in Cardelli et al. (1989). The fi- \nnal step in the reduction process involved normalizing all spectra to the proper AB magnitude values from the Sloan photometric database. This was achieved by calculating synthetic g ' and r ' magnitudes from the spectra taking into account the SDSS bandpasses and finding the constant multiplicative factor appropriate to match the SDSS photometry. The final flux-calibrated spectra of all nineteen observed AGNs are shown in Figures 1 and 2.", '3. MEASUREMENTS': 'The relevant quantities for our purpose are line widths and line and continuum fluxes. In this paper, line widths are measured as FWHM or line dispersion, i.e. the square root of the second central moment: \nσ 2 line = ∫ (F λ -C)( λ -λ 0 ) 2 dλ (F λ -C)d λ (1) \nwhere F λ is the flux density, C is the continuum, λ is the wavelength and λ 0 is the central wavelength of the line. As described in this section, we derive line widths and fluxes by fitting a Gauss-Hermite series to the data after subtraction of the continuum, Fe emission, and when necessary a narrow component of the line. While the Gauss-Hermite fit is not necessary for the high S/N ratio Keck data - where the quantities can be measured directly from the data (Treu et al. 2004; Woo et al. 2006) - the fitting procedure is needed to obtain robust mea- \n∫ \nFig. 2.As in Figure 1 for objects S11 to S29. \n<!-- image --> \nFig. 3.Example of nuclear Fe subtraction. The top panel shows a typical observed spectrum (black histogram), together with the nuclear iron emission template matched in intensity and resolution (red line). The bottom line shows the residuals after Fe subtraction. Note that the bump redwards of MgII emission has disappeared in the residuals. \n<!-- image --> \nsurements for the noisier SDSS data. As we demonstrate below by comparing the H β measurements from Keck and SDSS data, this fitting procedure gives consistent \nresults between the two datasets, and therefore indicates that the quality of the SDSS data is adequate to measure the width and flux of H β and, especially, H α , since the latter is considerably stronger.', '3.1. Gauss-Hermite Fitting': "The emission lines, especially for H β , are often asymmetrical, thus making a symmetrical Gaussian approximation of the line profiles undesirable. To account for the asymmetries in the emission lines, we fit a truncated Gauss-Hermite series to the profiles (van der Marel & Franx 1993). The main advantage of the Gauss-Hermite expansion is that it provides an orthonormal basis set and that the coefficients of the Hermite polynomials (commonly referred to as h 3 , h 4 , etc.) can be derived by straightforward linear minimization, leaving only two non-linear parameters (the center and the width of the Gaussian). Furthermore, the coefficients can be interpreted in terms of the kinematics of the tracing population (e.g. Gerhard 1993). The best fit profiles are then used to measure the luminosity, FWHM, and σ line of the emission lines as parameters for the M BH formulae. \nWe begin the fitting process with continuum subtraction. We identify the continuum level on each side of the three emission lines, using a window of 60 ˚ A. In the case of MgII, a narrower 40 ˚ A window is used because the blue continuum of MgII is close to the end of the \nFig. 4.Gauss-Hermite broad line fits. From left to right, the boxes show the fits for the Keck MgII line, Keck H β line, SDSS H β line, and SDSS H α line for each object. The continuum-subtracted line is shown in black, the broad component is shown in blue, and the broad-line fit is shown in red. \n<!-- image --> \nFig. 5.As in Figure 4 for objects S07 to S12. \n<!-- image --> \nobserved spectral range. For MgII, the ranges typically used are 2660-2700 ˚ A for the blue continuum level and 2930-2970 ˚ A for the red continuum level. Similarly for H β , the ranges are 4670-4730 ˚ A and 5080-5140 ˚ A for blue and red, respectively, and for H α the ranges are 6290-6350 ˚ A and 6700-6760 ˚ A. We then independently subtract the continuum by linear interpolation for each line. \nFig. 6.As in Figure 4 for objects S21 to S29. \n<!-- image --> \nTogether with the featureless continuum, we also remove broad nuclear Fe emission underneath MgII and H β . For this task we use template spectra of I Zw 1 kindly provided by Todd Boroson and Ross McLure. The procedure is similar to that described by Woo et al. (2006). A typical example of Fe subtraction in the wavelength region around MgII is shown in Figure 3. Removing nuclear Fe emission changes the measured width of H β by a negligible amount (the FWHM is unchanged and σ line is reduced by 0.013 dex), but it has a significant effect on MgII (the FWHM is reduced by 0.027 dex, while σ line is reduced by 0.106 dex). \nBefore fitting the broad lines, we subtract out the narrow lines, extending the procedure described by Woo et al. (2006). For H β this involves subtraction of the [OIII] λ 4959 and λ 5007 narrow lines, as well as the narrow H β line. We subtract [OIII] λ 5007 directly, and subtract [OIII] λ 4959 by dividing [OIII] λ 5007 by 3 and blueshifting. The narrow component of H β is subtracted by rescaling and blueshifting [OIII] λ 5007. The line ratio H β narrow / [O III ] λ 5007 was allowed to range between 1/20 and the maximum value consistent with the absence of 'dips' in the broad component (typically 1/10-1/7; e.g. Marziani et al. 2003). The adopted value of the scale factor is listed in Table 2. The narrow component of H α is subtracted by multiplying the determined H β narrow component by 3.1 (Malkan 1983; Osterbrock 1989) and redshifting. \nNo attempt is made to remove narrow NII emission lines around H α , since they are effectively rejected by the Gauss-Hermite fitting procedure as noise spikes, nor the narrow component (if present) of MgII. \nThe final step in approximating the line profiles involves fitting the Gauss-Hermite series to the broad lines. The fitting procedure finds the minimum χ 2 by increasing the order of the Hermite polynomials as required by the data (i.e. only if the goodness of fit, measured by \nthe reduced χ 2 , improves). Most objects required up to order 6 polynomials (i.e. h 6 ) to plateau in reduced χ 2 . Figures 4 to 6 show our best fits to the MgII, H β , and H α lines for the nineteen objects. The derived measurements of σ line and FWHM, after removal of the instrumental resolution, are listed in Table 2.", '3.2. Luminosities': 'The formulae for M BH estimates require line luminosities or monochromatic continuum luminosities at given wavelengths. For consistency with Greene & Ho (2005) the line luminosities are calculated from the total flux for the combined broad and narrow components of the H β and H α emission lines, where the broad component is taken as the best Gauss-Hermite fit, and the narrow component is taken as the appropriately scaled and shifted [OIII] λ 5007 narrow line (see § 3.1). We note that the narrow components contribute only a small fraction of the total flux of the Balmer lines (e.g., ∼ 5% for H β - see Table 2, and similarly to H α ) and therefore they make a contribution of about 0.01 dex to the broad-line size estimates. The H α to H β flux ratio ranges between 3 and 7 as expected for Seyfert 1s (e.g. Lacy et al. 1982). \nThe total continuum luminosity at 3000 (5100) ˚ A is calculated from the average flux in the 2950-3050 (50505150) ˚ A rest frame. In this paper, we use the term total continuum luminosity to indicate the total luminosity as measured within the spectroscopic aperture, i.e. without removing the host galaxy contamination. The fraction of host galaxy contamination depends on the properties of the individual object as well as on the instrumental setup, and it is hence a source of scatter. Nevertheless, the total luminosity is often the only measurement available and therefore it is important to investigate estimators based on this quantity. \nThe total continuum luminosities at 5100 ˚ A measured from the SDSS spectra agree to within a few per cent of those inferred from the Keck spectra and those listed in the paper by Woo et al. (2006). By comparing the values listed here with respect to those given by Woo et al. (2006), we infer 0.014 dex as the error on the continuum (L 3000 and L 5100 ). For line luminosities, we compare measurements on the fit with measurements on the data, and we take the r.m.s. scatter as the average error. This error is 0.011 and 0.062 dex on L H β and L H α , respectively. Nuclear luminosities are taken from the HST measurements presented by Treu et al. (2007), except for three objects (S11, S28, and S29), where the nuclear luminosity has been estimated from scaling the total luminosity by the average nuclear fraction for the sample, 0.31. In addition to measurement errors, AGN variability effectively limits the accuracy of the calibration of luminosity-based estimators, to the typical level of variability of 5-10% (e.g., Webb & Malkan 2000; Woo et al. 2007).', '3.3. Line Widths': 'The M BH estimators depend either on σ line or on the FWHM as a measurement of line width, and hence of the broad line kinematics. Both of these quantities were measured from the Gauss-Hermite fit to the broad lines. Errors on the Keck line width measurements are obtained by comparison with those reported by Woo et al. (2006), \nFig. 7.Distribution of FWHM to σ line ratios for MgII, H β , and H α . For comparison, the expected value for a Gaussian is 2.35. The typical errors on the measurements are 0.03 dex for MgII and H β and 0.06 dex for H α . \n<!-- image --> \nwhich were measured independently and did not rely on Gauss-Hermite expansion. The average error is 0.017 dex for the MgII and H β FWHM and σ line . The errors on the SDSS H α line width measurements are obtained by comparing the results for H β with the Keck fit and assuming that the error is the same for H α and H β . The average error is 0.051 dex for σ line and 0.040 dex for the FWHM. \nThe distribution of the FWHM to σ line ratios is shown in Figure 7. The average ratio for H β and H α is close to the Gaussian value (2.35), although with large scatter, consistent with the sample of Peterson et al. (2004). For MgII the average is considerably smaller indicating significant departure from Gaussianity. A large range of FWHM/ σ line ratios indicates a large scatter between M BH based on FWHM and M BH based on σ line for the sample since velocity is simply derived either from FWHM or σ line by multiplying by a constant as shown in 4 (see detailed discussion by Collin et al. (2006)). \nFigure 8 compares the FWHM of MgII with that of H β and H α . The FWHM are correlated albeit with substantial scatter. Comparing all the velocity scales, the average ratios and r.m.s. scatters (in parenthesis) are: 〈 log(FWHM MgII / FWHM H β ) 〉 = 0 . 02 ± 0 . 03(0 . 13), 〈 log(FWHM H β / FWHM H α ) 〉 = 0 . 09 ± 0 . 02(0 . 07), 〈 log( σ MgII /σ H β ) 〉 = 0 . 13 ± 0 . 02(0 . 10), 〈 log( σ H β /σ H α ) = 0 . 07 ± 0 . 02(0 . 10). \n〉 \n§ \n〈 ± Summarizing these relations, the width of H α is generally narrower by ∼ 20% than that of H β as expected from other studies (e.g. Shuder 1984; Greene & Ho 2005), while MgII and H β are similar in FWHM but not in σ line . These differences are expected given that different lines trace different species and that their shapes reflect the ionization and excitation variations throughout the region. This finding implies that each line has to be calibrated independently as a velocity estimator and that one cannot go from FWHM to σ line using the simple scaling for a Gaussian distribution. Therefore, we conclude that in general Mg II and Balmer line width cannot be used interchangeably, although in the case of \nFig. 8.Comparison of the width of MgII with that of H β and that of H α . The average errors are presented in the bottom right conner. The two objects furthest away from the ratio=1 line (dashed line) are labeled for easy comparison with the line profiles shown in Figures 5 and 6. \n<!-- image -->', '4. REVIEW OF OPTICAL-UV M BH ESTIMATORS': 'the FWHM of Mg and H β the ratio is close to unity (see McLure & Jarvis 2002; Salviander et al. 2007, for further discussion). \nAs far as M BH estimators are concerned, the scatter is of order 0.1 dex (i.e. significantly larger than the measurement errors), which sets a lower limit of ∼ 0 . 1 -0 . 2 dex on the relative uncertainty of the cross calibration of simple velocity estimators based on line widths. \nWe begin this section with a list of the 12 formulae for M BH estimation considered in this paper ( § 4.1). In these formulae, we adopt a notation where L 5100 ,t =total luminosity λL λ at λ = 5100 ˚ A, L 5100 ,n =nuclear luminosity λL λ at λ = 5100 ˚ A, and L 3000 = λL λ at λ = 3000 ˚ A. All formulae are given in the original notation, without applying any correction for different assumptions on the virial coefficient. \nIn § 4.2 we will compare the various estimators to infer how much they differ when applied to the same sample of objects before presenting our cross-calibrated recipes in § 5.', '4.1. Summary of existing recipes': 'McLure & Jarvis (2002), Kollmeier et al. (2006) and Salviander et al. (2007) give equations based on the width of the MgII line and the optical/UV continuum luminosity: \nM M = 3 . 37 ( L 3000 10 37 W ) 0 . 47 ( FWHM MgII km s -1 ) 2 M /circledot , (2) \nM K = 2 . 04 ( L 3000 10 44 erg s -1 ) 0 . 88 ( FWHM MgII km s -1 ) 2 M /circledot , (3) \nM Sa = 10 7 . 69 ( L 5100 ,t 10 44 erg s -1 ) 0 . 5 ( FWHM MgII 3000 km s -1 ) 2 M /circledot . (4) \nGreene & Ho (2005) and Vestergaard & Peterson (2006) present equations based on the width and luminosities of the H β and H α broad lines: \nM G β = 3 . 6 × 10 6 ( L H β 10 42 erg s -1 ) 0 . 56 ( FWHM H β 1000 km s -1 ) 2 M /circledot , (5) \nM G α = 2 . 0 × 10 6 ( L H α 10 42 erg s -1 ) 0 . 55 ( FWHM H α 1000 km s -1 ) 2 . 06 M /circledot , (6) \nM V β = 10 6 . 67 ( L H β 10 42 erg s -1 ) 0 . 63 ( FWHM H β 1000 km s -1 ) 2 M /circledot . (7) \nShields et al. (2003), Greene & Ho (2005), Vestergaard & Peterson (2006), Woo et al. (2006), Netzer & Trakhtenbrot (2007), and Treu et al. (2007) adopt the following formulae based on L 5100 and the width of the H β broad line: \nM Sh = 10 7 . 69 ( L 5100 ,t 10 44 erg s -1 ) 0 . 5 ( FWHM H β 3000 km s -1 ) 2 M /circledot , (8) \nM G51 = 4 . 4 × 10 6 ( L 5100 ,n 10 44 erg s -1 ) 0 . 64 ( FWHM H β 1000 km s -1 ) 2 M /circledot , (9) \nM V = 10 6 . 91 ( L 5100 ,t 10 44 erg s -1 ) 0 . 5 ( FWHM H β 1000 km s -1 ) 2 M /circledot , (10) \nM W = 2 . 15 × 10 8 ( L 5100 , t 10 44 erg s -1 ) 0 . 69 ( σ H β 3000 km s -1 ) 2 M /circledot , (11) \nM N = 1 . 05 × 10 8 ( L 5100 ,t 10 46 erg s -1 ) 0 . 65 ( FWHM H β 1000 km s -1 ) 2 M /circledot , (12) \nM T = 10 8 . 58 ( L 5100 ,n 10 44 erg s -1 ) 0 . 518 ( σ H β 3000 km s -1 ) 2 M /circledot . (13) \nNote that the formulae listed above adopt different estimators of broad-line region velocity ( σ line or FWHM) and size (continuum luminosity or line luminosity).', '4.2. Comparison of M BH estimators': 'Fig. 9.Comparison of the consistency of various estimators adopted in the literature, as summarized in Section 4. The average difference (log M BH -log M T ) and r.m.s. scatter in dex are listed in the bottom right corner. The formulae are taken directly from the literature with no adjustments for the difference in the choice of virial coefficients. \n<!-- image --> \nIn this section we assess the relative consistency of the M BH estimators taken from the literature by comparison with our fiducial black hole mass M T (as listed in the paper by Treu et al. 2007). The choice of the fiducial estimator is based on reverberation mapping studies of local AGNs, which are mostly based on H β and L 5100 . These studies show that σ line is the most robust velocity estimator (Peterson et al. 2004; Collin et al. 2006). They determine the slope of the size-luminosity relation, taking into account the host galaxy contamination (Bentz et al. 2006), and they set the virial coefficient by requiring that the M BH -σ ∗ relation be the same for active and quiescent galaxies, modulo selection effects (Onken et al. 2004; Lauer et al. 2007). \nIt is important to notice that the formulae M G51 , M G β , M G α , M Sh , and M Sa have been calibrated on the isotropic spherical virial coefficient (f=3/4 in the notation of Netzer, 1990), M M and M K adopt f=1, and M T , M W , M V , M V β , and M N are based on the recalibration of the virial coefficient given by Onken et al. (2004) 5 . Therefore, we expect the first set of M BH estimators to give lower values than M T by log 1 . 8 = 0 . 255 dex, and M M and M K to give lower values by log(1 . 8 × 3 / 4) = 0 . 130 dex. In the case of M Sh and M Sa , an isotropic spherical virial coeffi- \nt was used. However, their equations were based on a different size-luminosity relation, hence making these formulae approximately equivalent to having the same virial coefficient as M T , M W , and M N (Salviander et al. 2007). As noted by Treu et al. (2007), M W , M N , M Sh , and M Sa agree on average to within a few per cent of M T . The more discrepant black hole masses are those with different virial coefficients, which can differ on average by as much as 0 . 38 ± 0 . 05 dex (i.e. more than a factor of two). Even renormalizing these formulae to the same virial coefficient would still leave discrepancies of order 0.1 dex: after renormalization, M V and M G β are approximately 0.1 dex larger than M T while M G51 , M M , and M K are still 0.1 dex smaller than M T . This is approximately twice the expected error on the mean given the size of the sample. \nThese results show that systematic errors as large as 0 . 38 ± 0 . 05 dex can be introduced when comparing M BH estimates based on different diagnostics or when comparing AGN M BH estimates to local samples with direct M BH measurements from stellar or gaseous kinematics. \nWe conclude by discussing the absolute calibration of our fiducial mass estimator. A study by Collin et al. (2006) finds that for the local sample with reverberation based M BH , the widths of the Balmer lines measured on mean spectra are systematically larger than those measured on the r.m.s. spectra for variable objects, suggest- \ning that a smaller virial coefficient than that advocated by Onken et al. (2004) and used for our fiducial mass estimator, should be adopted when measuring widths from mean spectra. This could possibly indicate that our fiducial M BH are overestimated by approximately 0.15 dex. To investigate this effect, in Section 5 we apply our calibrated recipes - based on our fiducial estimator - to the local sample with reverberation based M BH , using published line widths and fluxes measured from single epoch and mean spectra. As discussed in the next section, we find an excellent agreement with the mass inferred from reverberation mapping based on line widths from the r.m.s. spectra, indicating that no such correction is necessary. Data for a larger number of objects with reverberation-based masses are needed to determine the zero point of the virial scalings more accurately, as discussed in the next Section.', '5. A SET OF SELF-CONSISTENT RECIPES': 'In this Section we examine all possible combinations of velocity and flux estimators to produce a set of cross calibrated recipes. By comparing with our fiducial mass estimator M T , we compute the r.m.s. scatter of the residuals to infer a lower limit on the intrinsic uncertainty of each recipe. \nIn practice, we adopt the following relation: \nlog M BH = α +2logv 1000 + β log L , (14) \nwhere v 1000 is a velocity estimator in units of 1000 km s -1 , L is a luminosity estimator in units of 10 44 erg s -1 or 10 42 erg s -1 , respectively for continuum or line luminosity, and we find the α that best matches the M T fiducial estimates. Our range in luminosities is too small to fit for β as well, and therefore, we assert the following fiducial values: β = 0 . 47 for L 3000 , 0.518 for L 5100 ,n , 0.67 for L 5100 ,t , 0.56 for H β , and 0.55 for H α . These choices are based on the most current calibration of the size-luminosity relation for each wavelength/line. In particular, following the results of the study by Bentz et al. (2006) we adopt 0.518 as the slope of the broad-line region size vs nuclear luminosity relation, while for the size vs total luminosity (i.e. including host galaxy contamination within the spectroscopic aperture) relation we adopt the slope given by Kaspi et al. (2005). Although the former slope is to be preferred when the nuclear luminosity is available, we also provide results for the second slope, which appears to be the best estimate whenever the light from the nucleus and from the host galaxy cannot be disentangled. In general, extrapolations well outside the range considered here are to be done with caution, since most of the local calibrators for the size-luminosity relation are Seyferts and PG quasars in 10 42 < L 5100 (ergs -1 ) < 10 46 . \nWe emphasize that this procedure produces self consistent mass estimates, but these all share a common uncertainty in the zero point. In practice, all α can be shifted by a constant if it turns out that a different value of the virial coefficient for the local sample of reverberation mapped AGNs is to be preferred. As a sanity check, we applied our calibrated recipes to the local sample with reverberation M BH (Peterson et al. (2004) and recent updates by Bentz et al. (2006), Denney et al. (2006), and Bentz et al. (2007)). Based on the single epoch mea- \nurements given by Vestergaard & Peterson (2006) we compared our two estimators based on the FWHM of H β , and on L 5100 ,t and H β line flux, with the reverberation masses. The agreement is excellent, with average ∆logM BH , rev -∆logM BH equal to 0.009 dex and 0.023, respectively, with r.m.s. scatter of 0.46 dex and 0.49 dex. Based on the measurements from mean spectra given by Collin et al. (2006) we compared our two estimators based on the FWHM and line dispersion of H β , and on L 5100 ,t , with the reverberation masses. The agreement is again excellent, with average ∆ log M BH , rev -∆logM BH equal to 0.004 dex and 0.018, respectively, with r.m.s. scatter of 0.35 and 0.29 dex. The scatter of the difference is combination of effects from uncertainties in time-lag, line width, and luminosity measurements, and scatter in the size-luminosity relation. \nThe best fit values of α together with the r.m.s. scatter are listed in Table 3. The best fit relations are shown in Figure 10. We note that we included in the recipes the combination of H β σ line and L 5100 ,n that was used to compute M T by Treu et al. (2007). The goal of this exercise is to estimate the measurement errors associated with the relation, given that the input parameters were measured independently for this paper, based on the Gauss-Hermite polynomial expansion fit. Since measurement errors such as narrow line and continuum subtraction dominate over pixel noise, given the high S/N of the Keck data, the r.m.s. of these residuals divided by √ 2 is effectively the total measurement error, i.e. 0 . 03 dex. Thus we can conclude that, for all practical purposes, the r.m.s. scatter that we observe for the other recipes is intrinsic scatter in the relation and not measurement error. This is also the rationale for not showing measurement error bars in the plots. \nLooking at Table 3, it appears that the smallest relative scatter is obtained when comparing M T to black hole masses based on the same velocity scale σ H β . This is not surprising, as the velocity scale enters with the square in the M BH estimate and we have seen that velocity scales have typical relative scatter of 0.1 dex. However, the scatter means that adopting optical/UV continuum luminosity as a proxy of the size of the broad-line region introduces an uncertainty of ∼ 0.10-0.15 dex in the M BH estimate, with the larger r.m.s. for the line luminosities. The velocity scale that best matches the σ line of H β is the line width of MgII, which gives an r.m.s. scatter of 0.17 dex. This is better than the σ line of H α (0.20 dex) and the FWHM of H β (0.23 dex) - as expected from the large distribution of FWHM/ σ line ratios - which are in turn slightly better than the other indicators. The worst match is obtained for line luminosities with FWHM, with an r.m.s. scatter of ∼ 0 . 24 -0 . 33 dex. From this study, we infer a lower limit to the relative accuracy of the various indicators of order 0.1-0.2 dex, depending on the choice of estimators. If the relationships presented here were to be extended beyond the range of M BH considered, it is likely that the scatter will increase, as suggested by the slope of the points in the corresponding panels. Otherwise, the slope β of the size luminosity relation will have to be fitted independently.', '6. SUMMARY': "In this paper we have used Keck and SDSS spectra of nineteen Seyferts at z = 0 . 36, to perform a compre- \nFig. 10.Comparison of M BH estimates according to the new cross-calibrated formulae as discussed in Section 5. A relation of the form log M BH = α + 2log v 1000 + β log L is assumed, where v 1000 is a velocity estimator in units of 1000 km s -1 , and L is a luminosity estimator in units of 10 44 erg s -1 or 10 42 erg s -1 , respectively for continuum or line luminosity. The slope β is fixed at 0.47 for L 3000 , 0.518 for L 5100 ,n , 0.69 for L 5100 ,t , 0.56 for L H β , and 0.55 for L H α as taken from the literature. The best fit coefficients α are given in Table 3 together with the r.m.s. scatter of the comparison. The typical measurement error bar is 0.05 dex. \n<!-- image --> \nhensive study of 'virial' black hole mass estimators for broad line AGNs. The main results can be summarized as follows: \n- 1. We have fit Gauss-Hermite series to the data in order to measure the FWHM and σ line of MgII, H β , and H α , as well as H α and H β luminosities and continuum luminosities at 3000 ˚ A and 5100 ˚ A. Measurement errors are approximately 0.02 dex on the MgII and H β line widths, 0.04-0.05 on the H α line widths, 0.01 dex on the continuum luminosity, 0.01 dex on the H β luminosity, and 0.06 dex on the H α luminosity.\n- 2. We have compared twelve formulae taken from the literature, showing that M BH estimates can differ systematically by as much as 0 . 38 ± 0 . 05 dex (or 0 . 13 ± 0 . 05 dex, if the same virial coefficient is \nadopted). Such differences should be taken into account when comparing data obtained with different methods. \n- 3. We have cross-calibrated a set of 30 empirical recipes based on all combinations of the velocity and luminosity indicators corresponding to the Mg II , H β , and H α broad lines. Taking the masses measured by Treu et al. (2007) as our fiducial black hole masses, we find that: the absolute scale of the different indicators is calibrated to within ∼ 0.05 dex; the best agreement is found when using the line dispersion of H β as a velocity estimator, with the residual 0.1 dex r.m.s. scatter resulting from the various continuum luminosity estimators; adopting the line dispersion of Mg II raises the scatter to 0.2 dex; for the other estimators the intrinsic scatter is in the range 0.2-0.38 dex. This implies \na lower limit of 0.1-0.2 dex on the validity of each estimator for each individual case. \nThe newly calibrated recipes should be useful to reduce the sources of systematic uncertainties when comparing different studies. \nThis work is based on data obtained with the Hubble Space Telescope - obtained 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 program 10216 -, and with the 10m W.M. Keck Telescope, which is operated as a scientific partnership among the California Institute of Technology, the University of California and the National Aeronautics and Space Administration. The Observatory was made \npossible by the generous financial support of the W.M. Keck Foundation. The authors wish to recognize and acknowledge the very significant cultural role and reverence that the summit of Mauna Kea has always had within the indigenous Hawaiian community. We are most fortunate to have the opportunity to conduct observations from this mountain. We acknowledge financial support from NASA through HST grant GO-10216 and AR10986. TT acknowledges support from the NSF through CAREERawardNSF-0642621, and from the Sloan Foundation through a Sloan Research Fellowship. We thank Todd Boroson, Ross McLure, and Marianne Vestergaard for providing nuclear Fe emission templates, and Brad Peterson for providing Table 1 in the paper by Collin et al. (2006) in electronic format. We thank the referee for a careful report which improved the manuscript.", 'REFERENCES': "- Bentz, M. C., Peterson, B. M., Pogge, R. W., Vestergaard, M., & Onken, C. A. 2006, ApJ, 644, 133 \nBentz, M. C., et al. 2006, ApJ, 651, 775 \n- Bentz, M. C., et al. 2007, ApJ, 662, 205\n- Cardelli, J. 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M. 2002a, ApJ, 579, 530\n- -. 2002b, ApJ, 581, L5 \nTABLE 1 Sample properties \n| Name (1) | RA (J2000) (2) | DEC (J2000) (3) | z (4) | i' (5) |\n|------------|------------------|-------------------|---------|----------|\n| S01 | 15 39 16.23 | +03 23 22.06 | 0.3592 | 18.74 |\n| S02 | 16 11 11.67 | +51 31 31.12 | 0.3544 | 18.94 |\n| S03 | 17 32 03.11 | +61 17 51.96 | 0.3588 | 18.2 |\n| S04 | 21 02 11.51 | -06 46 45.03 | 0.3578 | 18.41 |\n| S05 | 21 04 51.85 | -07 12 09.45 | 0.353 | 18.35 |\n| S06 | 21 20 34.19 | -06 41 22.24 | 0.3684 | 18.41 |\n| S07 | 23 09 46.14 | +00 00 48.91 | 0.3518 | 18.11 |\n| S08 | 23 59 53.44 | -09 36 55.53 | 0.3585 | 18.43 |\n| S09 | 00 59 16.11 | +15 38 16.08 | 0.3542 | 18.16 |\n| S10 | 01 01 12.07 | -09 45 00.76 | 0.3506 | 17.92 |\n| S11 | 01 07 15.97 | -08 34 29.40 | 0.3557 | 18.34 |\n| S12 | 02 13 40.60 | +13 47 56.06 | 0.3575 | 18.12 |\n| S21 | 11 05 56.18 | +03 12 43.26 | 0.3534 | 17.21 |\n| S23 | 14 00 16.66 | -01 08 22.19 | 0.351 | 18.08 |\n| S24 | 14 00 34.71 | +00 47 33.48 | 0.3615 | 18.21 |\n| S26 | 15 29 22.26 | +59 28 54.56 | 0.3691 | 18.88 |\n| S27 | 15 36 51.28 | +54 14 42.71 | 0.3667 | 18.8 |\n| S28 | 16 11 56.30 | +45 16 11.04 | 0.366 | 18.59 |\n| S29 | 21 58 41.93 | -01 15 00.33 | 0.3576 | 18.77 | \nNote . - Col. (1): Target ID. Col. (2): RA. Col. (3): DEC. Col. (4): Redshift from SDSS DR4. Col. (5): Extinction corrected i ' AB magnitude from SDSS photometry. \nTABLE 2 Measured Properties \n| Name (1) | σ MgII (2) | σ H β (3) | σ H α (4) | FWHM MgII (5) | FWHM H β (6) | FWHM H α (7) | L 3000 (8) | L 5100 ,n (9) | L 5100 ,t (10) | L H β (11) | L H α (12) | f H β, [ OIII ] (13) | f H β,nt (14) |\n|------------|--------------|-------------|-------------|-----------------|----------------|----------------|--------------|-----------------|------------------|--------------|--------------|------------------------|-----------------|\n| S01 | 2260 | 2133 | 1847 | 4418 | 4755 | 3420 | 1.91 | 0.74 | 1.66 | 2.29 | 7.24 | 0.1 | 0.05 |\n| S02 | 2914 | 1928 | 2113 | 4000 | 5188 | 3442 | 2.14 | 0.36 | 1.51 | 3.09 | 21.38 | 0.17 | 0.1 |\n| S03 | 2147 | 1745 | 1698 | 3345 | 2945 | 2634 | 3.98 | 1.69 | 2.82 | 3.8 | 14.45 | 0.13 | 0.04 |\n| S04 | 2253 | 2392 | 983 | 3636 | 3100 | 2617 | 1.86 | 1.42 | 2.24 | 1.35 | 6.92 | 0.08 | 0.06 |\n| S05 | 3784 | 3297 | 2686 | 6315 | 5220 | 3751 | 3.39 | 2.04 | 2.75 | 4.37 | 16.6 | 0.11 | 0.05 |\n| S06 | 2353 | 1664 | 1382 | 4023 | 4625 | 4306 | 2.75 | 0.54 | 2.69 | 2 | 7.94 | 0.1 | 0.13 |\n| S07 | 3297 | 2500 | 2531 | 5561 | 4815 | 4326 | 3.72 | 2.26 | 3.02 | 4.9 | 15.14 | 0.1 | 0.06 |\n| S08 | 2365 | 1538 | 1015 | 3380 | 3372 | 3017 | 2.29 | 1.25 | 2.57 | 1.12 | 4.47 | 0.1 | 0.08 |\n| S09 | 2542 | 2013 | 1462 | 3824 | 2865 | 2710 | 3.16 | 0.78 | 3.09 | 3.8 | 15.49 | 0.11 | 0.04 |\n| S10 | 2897 | 1690 | 2056 | 4532 | 4410 | 3498 | 7.08 | 1.11 | 3.8 | 4.79 | 17.78 | 0.11 | 0.02 |\n| S11 | 2858 | 1590 | 1435 | 4291 | 2733 | 2569 | 3.24 | 0.88 | 2.45 | 2.75 | 10.72 | 0.08 | 0.01 |\n| S12 | 2672 | 3213 | 3232 | 4132 | 9005 | 7163 | 4.37 | 1.05 | 3.24 | 5.01 | 23.99 | 0.06 | 0.02 |\n| S21 | 3450 | 3172 | 3208 | 6582 | 7681 | 7493 | 2.69 | 2.3 | 7.24 | 9.12 | 63.1 | 0.07 | 0.03 |\n| S23 | 3941 | 3196 | 2688 | 7378 | 9700 | 6870 | 3.09 | 1.2 | 3.55 | 3.47 | 14.13 | 0.1 | 0.05 |\n| S24 | 3480 | 2886 | 2667 | 6628 | 7864 | 4468 | 2.82 | 0.44 | 2.95 | 3.47 | 12.59 | 0.1 | 0.04 |\n| S26 | 3370 | 1862 | 1657 | 6265 | 5451 | 4440 | 1.78 | 0.5 | 1.58 | 2.69 | 6.46 | 0.07 | 0.12 |\n| S27 | 2699 | 1609 | 1157 | 3766 | 2567 | 1832 | 2.19 | 0.92 | 1.82 | 2.45 | 8.32 | 0.17 | 0.05 |\n| S28 | 3926 | 2313 | 2221 | 8436 | 5116 | 5412 | 2.45 | 0.76 | 2.29 | 1.91 | 6.61 | 0.09 | 0.03 |\n| S29 | 2556 | 1744 | 1520 | 4003 | 3190 | 2216 | 2.09 | 0.59 | 1.74 | 2.04 | 9.12 | 0.1 | 0.05 | \nNote . - Col. (1): Target ID. Col. (2): σ line of MgII (km s -1 ) measured on the line model fit to Keck data. The average error is 0.017 dex. Col. (3): σ line of H β (km s -1 ) measured on the line model fit to Keck data. The average error is 0.017 dex. Col. (4): σ line of H α (km s -1 ) measured on the line model fit to SDSS data. The average error is 0.051 dex. Col. (5): FWHM of MgII (km s -1 ) measured on the line model fit to Keck data. The average error is 0.017 dex. Col. (6): FWHM of H β (km s -1 ) measured on the line model fit to Keck data. The average error is 0.017 dex. Col. (7): FWHM of H α (km s -1 ) measured on the line model fit to SDSS data. The average error is 0.040 dex. Col. (8): Rest frame luminosity at 3000 ˚ A in 10 44 erg s -1 . The average error is 0.014 dex. The actual error will be dominated by variability of order 10%, as for all luminosities listed in this Table. Col. (9): Rest frame nuclear luminosity at 5100 ˚ A in 10 44 erg s -1 , from Treu et al. (2007). The average error is 0.08 dex. Col. (10): Rest frame total luminosity at 5100 ˚ A in 10 44 erg s -1 . The average error is 0.014 dex. Col. (11): Rest frame H β line luminosity in 10 42 erg s -1 . The average error is 0.011 dex. Col. (12): Rest frame H α line luminosity in 10 42 erg s -1 . The average error is 0.062 dex. Col. (13): Narrow component of H β to [OIII] λ 5007 flux ratio. Col. (14): Fraction of flux of the H β line in the narrow component. \nTABLE 3 M BH Estimator Factors \n| | L 3000 (0.47) | L 5100 ,n (0.518) | L 5100 ,t (0.69) | L H β (0.56) | L H α (0.55) |\n|-----------|----------------------|----------------------|----------------------|----------------------|----------------------|\n| σ MgII | 7.207 ± 0.052, 0.219 | 7.429 ± 0.039, 0.166 | 7.133 ± 0.045, 0.191 | 7.150 ± 0.053, 0.226 | 6.824 ± 0.051, 0.218 |\n| σ H β | 7.458 ± 0.027, 0.112 | 7.680 ± 0.021, 0.090 | 7.383 ± 0.028, 0.119 | 7.401 ± 0.038, 0.163 | 7.074 ± 0.042, 0.178 |\n| σ H α | 7.588 ± 0.061, 0.260 | 7.810 ± 0.048, 0.205 | 7.514 ± 0.060, 0.253 | 7.532 ± 0.074, 0.313 | 7.205 ± 0.075, 0.319 |\n| FWHM MgII | 6.767 ± 0.055, 0.233 | 6.990 ± 0.045, 0.191 | 6.693 ± 0.053, 0.224 | 6.711 ± 0.059, 0.251 | 6.384 ± 0.056, 0.236 |\n| FWHM | ± 0.069, 0.292 | ± | ± | 6.747 ± 0.077, 0.327 | |\n| H β | 6.803 | 7.026 0.056, 0.235 | 6.729 0.071, 0.300 | | 6.420 ± 0.079, 0.335 |\n| FWHM H α | 6.986 ± 0.064, 0.270 | 7.209 ± 0.055, 0.233 | 6.912 ± 0.069, 0.293 | 6.930 ± 0.071, 0.303 | 6.603 ± 0.073, 0.311 | \nNote . - Normalization constants for M BH . Entries are α ± error, r.m.s. scatter of log M BH vs. log M T (Fig. 10). The α coefficients are determined using the general formula, log M BH = α +2logv 1000 + β log L, where v 1000 is the velocity estimator in units of 1000 km s -1 , and L is the luminosity estimator, which is divided by 10 44 erg s -1 for continuum luminosity measurements and by 10 42 erg s -1 for line luminosity measurements. The values used for β are listed below the luminosities."}

2012ApJ...756...93H

Rapidly Accreting Supergiant Protostars: Embryos of Supermassive Black Holes?

2012-01-01

['accretion', 'accretion disks', 'cosmology theory', 'cosmology early universe', 'galaxies formation', 'stars fundamental parameters', '-', '-']

[]

Direct collapse of supermassive stars (SMSs) is a possible pathway for generating supermassive black holes in the early universe. It is expected that an SMS could form via very rapid mass accretion with \dot{M}_*\sim 0.1{--}1 \,M_\odot \,yr^{-1} during the gravitational collapse of an atomic-cooling primordial gas cloud. In this paper, we study how stars would evolve under such extreme rapid mass accretion, focusing on the early evolution until the stellar mass reaches 10<SUP>3</SUP> M <SUB>⊙</SUB>. To this end, we numerically calculate the detailed interior structure of accreting stars with primordial element abundances. Our results show that for accretion rates higher than 10<SUP>-2</SUP> M <SUB>⊙</SUB> yr<SUP>-1</SUP>, stellar evolution is qualitatively different from that expected at lower rates. While accreting at these high rates, the star always has a radius exceeding 100 R <SUB>⊙</SUB>, which increases monotonically with the stellar mass. The mass-radius relation for stellar masses exceeding ~100 M <SUB>⊙</SUB> follows the same track with R <SUB>*</SUB>vpropM <SUP>1/2</SUP> <SUB>*</SUB> in all cases with accretion rates &gt;~ 10<SUP>-2</SUP> M <SUB>⊙</SUB> yr<SUP>-1</SUP> at a stellar mass of 10<SUP>3</SUP> M <SUB>⊙</SUB>, the radius is ~= 7000 R <SUB>⊙</SUB> (sime 30 AU). With higher accretion rates, the onset of hydrogen burning is shifted toward higher stellar masses. In particular, for accretion rates exceeding \dot{M}_*\gtrsim 0.1 \,M_\odot \,yr^{-1}, there is no significant hydrogen burning even after 10<SUP>3</SUP> M <SUB>⊙</SUB> have accreted onto the protostar. Such "supergiant" protostars have effective temperatures as low as T <SUB>eff</SUB> ~= 5000 K throughout their evolution and because they hardly emit ionizing photons, they do not create an H II region or significantly heat their immediate surroundings. Thus, radiative feedback is unable to hinder the growth of rapidly accreting stars to masses in excess of 10<SUP>3</SUP> M <SUB>⊙</SUB> as long as material is accreted at rates \dot{M}_*\gtrsim 10^{-2} \,M_\odot \,yr^{-1}.

[]

https://arxiv.org/pdf/1203.2613.pdf

{'RAPIDLY ACCRETING SUPERGIANT PROTOSTARS: EMBRYOS OF SUPERMASSIVE BLACK HOLES?': 'Takashi Hosokawa 1,2 , Kazuyuki Omukai 2 , Harold W. Yorke 1 Draft version May 26, 2018', 'ABSTRACT': "Direct collapse of supermassive stars (SMSs) is a possible pathway for generating supermassive black holes in the early universe. It is expected that an SMS could form via very rapid mass accretion with ˙ M ∗ ∼ 0 . 1 -1 M /circledot yr -1 during the gravitational collapse of an atomic-cooling primordial gas cloud. In this paper we study how stars would evolve under such extreme rapid mass accretion, focusing on the early evolution until the stellar mass reaches 10 3 M /circledot . To this end we numerically calculate the detailed interior structure of accreting stars with primordial element abundances. Our results show that for accretion rates higher than 10 -2 M /circledot yr -1 , stellar evolution is qualitatively different from that expected at lower rates. While accreting at these high rates the star always has a radius exceeding 100 R /circledot , which increases monotonically with the stellar mass. The mass-radius relation for stellar masses exceeding ∼ 100 M /circledot follows the same track with R ∗ ∝ M 1 / 2 ∗ in all cases with accretion rates /greaterorsimilar 10 -2 M /circledot yr -1 ; at a stellar mass of 10 3 M /circledot the radius is /similarequal 7000 R /circledot ( /similarequal 30 AU). With higher accretion rates the onset of hydrogen burning is shifted towards higher stellar masses. In particular, for accretion rates exceeding ˙ M ∗ /greaterorsimilar 0 . 1 M /circledot yr -1 , there is no significant hydrogen burning even after 10 3 M /circledot have accreted onto the protostar. Such 'supergiant' protostars have effective temperatures as low as T eff /similarequal 5000 K throughout their evolution and because they hardly emit ionizing photons, they do not create an HII region or significantly heat their immediate surroundings. Thus, radiative feedback is unable to hinder the growth of rapidly accreting stars to masses in excess of 10 3 M /circledot , as long as material is accreted at rates ˙ M ∗ /greaterorsimilar 10 -2 M /circledot yr -1 . \nSubject headings: cosmology: theory - early universe - galaxies: formation - stars: formation accretion", '1. INTRODUCTION': "Recent observations reveal that supermassive black holes (SMBHs) exceeding 10 9 M /circledot already existed in the universe less than 1 Gyr after the Big Bang (e.g., Fan 2006; Mortlock et al. 2011; Treister et al. 2011). The origins of such SMBHs must be intimately related to structure formation in the early universe. Some scenarios on the birth and growth of SMBHs postulate the existence of remnant BHs from Population III (Pop III) stars as their seeds (e.g., Madau & Rees 2001; Schneider et al. 2002). For several decades theoretical studies have predicted that the majority of Pop III stars were very massive, exceeding 100 M /circledot (e.g., Bromm & Larson 2004). Pop III stars more massive than 300 M /circledot end their lives by directly collapsing to form BHs (e.g., Heger & Woosley 2002). If such a ∼ 100 M /circledot BH grows via continuous mass accretion at the Eddington limited rate, its mass barely attains 10 9 M /circledot in 1 Gyr. \nThis scenario, however, has recently been challenged. First, it is suspected that most Pop III stars were much less massive than previously thought. A circumstellar disk forming after the cloud's collapse easily fragments due to gravitational instability and could produce multiple protostars (e.g., Machida et al. 2008; Stacy et al. 2010; Clark et al. 2011). The final stellar masses would \nbe reduced as the accreting gas is shared by multiple stars. Moreover, strong stellar UV light creates an HII region around the protostar when the stellar mass exceeds a few × 10 M /circledot . The resulting feedback terminates the growth of Pop III protostars via mass accretion at a few × 10 M /circledot (e.g., McKee & Tan 2008; Hosokawa et al. 2011b; Stacy et al. 2012). A large amount of gas would be expelled from the dark halo due to the expansion of HII regions and the onset of core-collapse supernovae (e.g., Whalen et al. 2004; Kitayama et al. 2004; Kitayama & Yoshida 2005), which quenches the supply of gas to any remnant BH. Even if a BH gets some gas supply, radiative feedback from the BH accretion disk could regulate mass accretion onto the BH-disk system (e.g., Alvarez et al. 2009; Milosavljevi'c et al. 2009; Jeon et al. 2011). \nAnother pathway for generating SMBHs is BH binary mergers. However, this process could be also limited due to the strong recoil resulting from gravitational wave emission (e.g., Campanelli et al. 2007; Herrmann et al. 2007). It is not straightforward that a seed BH /lessorsimilar 100 M /circledot can grow to a ∼ 10 9 M /circledot SMBH within 1 Gyr of its birth. \nAn alternative possibility is that massive BHs exceeding 10 5 M /circledot form directly in some rare occasions in the primordial gas (e.g., Bromm & Loeb 2003). The primary cooling process in the primordial gas is line emission of molecular hydrogen. However, the thermal evolution of a gravitationally collapsing cloud can change significantly, if this cooling process is suppressed, for example, due to photodissociation of molecules by strong background radiation (Omukai 2001; Oh & Haiman 2002; Shang et al. \n2010; Inayoshi & Omukai 2011) or collisional dissociation in dense shocks (Inayoshi & Omukai 2012). If the dark halo is sufficiently massive ( /greaterorsimilar 10 8 M /circledot ), the baryonic gas can contract with atomic hydrogen cooling even without molecular hydrogen. The collapse proceeds nearly isothermally at /similarequal 8000 K. Without efficient molecular cooling fragmentation is suppressed and single or binary protostars form within dense cloud cores of /greaterorsimilar 10 5 M /circledot (Bromm & Loeb 2003; Regan & Haehnelt 2009). The protostar's mass is initially ∼ 10 -2 M /circledot but quickly increases via mass accretion. The expected accretion rates are 0 . 1 -1 M /circledot yr -1 , more than 100 times higher than the standard value /similarequal 10 -3 M /circledot yr -1 expected for Pop III star formation. The stellar mass could reach 10 5 -6 M /circledot in ∼ 1 Myr with this very rapid mass accretion. General relativity predicts that such a supermassive star (SMS) becomes unstable (e.g. Chandrasekhar 1964) and collapses to form a BH, which subsequently swallows most of the surrounding stellar material (e.g., Shibata & Shapiro 2002). Some authors are exploring a different picture, whereby only a central part of the SMS collapses to form a ∼ 100 M /circledot BH and heat input from the accreting BH inflates the outer envelope of the SMS ('quasi-star', Begelman et al. 2006, 2008; Begelman 2010; Ball et al. 2011; Dotan et al. 2011). \nHowever, we only have limited knowledge on how stars evolve under such extreme conditions of rapid mass accretion. Begelman (2010) predicts that, based on simple analytic arguments, such stars have a very different structure from their main-sequence counterparts. Stellar evolution at lower accretion rates ˙ M ∗ /lessorsimilar 10 -2 M /circledot yr -1 has been studied in detail by numerically solving the stellar interior structure (e.g., Stahler et al. 1986; Omukai & Palla 2001, 2003; Ohkubo et al. 2009; Hosokawa & Omukai 2009; Hosokawa et al. 2010). Omukai & Palla (2001, 2003) showed that rapid mass accretion with ˙ M ∗ > 4 × 10 -3 M /circledot yr -1 causes the protostar's abrupt expansion before its arrival to the Zero-Age Main Sequence (ZAMS). Further comprehensive studies on stellar evolution with rapid mass accretion are indispensable for considering their radiative feedback and observational signatures (e.g., Johnson et al. 2011). \nWe present here our first results of this sort, whereby as a first step, we study the early evolution up to a stellar mass of 10 3 M /circledot . Our results show that rapid accretion with ˙ M ∗ /greaterorsimilar 10 -2 M /circledot yr -1 causes the star to bloats up like a red giant. The stellar radius increases monotonically with stellar mass and reaches /similarequal 7000 R /circledot ( /similarequal 30 AU) at a mass of 10 3 M /circledot . Unlike the cases with lower accretion rates previously studied, the mass-radius relation in this phase is almost independent of the assumed accretion rate. Such massive 'super-giant' protostars could be the progenitors that eventually evolve to the observed SMBHs in the early universe. \nThe organization of this paper is as follows. First, we briefly review our numerical method and summarize the calculated cases in Section 2. We describe our results in Section 3; we first focus on the fiducial case with ˙ M = 0 . 1 M /circledot yr -1 and then examine effects of varying accretion rates and boundary conditions. Finally, summary and discussions are described in Section 4.", '2.1. Method': "We calculate stellar evolution with mass accretion using the numerical codes developed in our previous work (see Omukai & Palla 2003; Hosokawa & Omukai 2009; Hosokawa et al. 2010, for details). The four stellar structure equations, i.e., equations of continuity, hydrostatic equilibrium, energy conservation, and energy transfer, including effects of mass accretion are solved assuming spherical symmetry. We focus on the early evolution until slightly after the ignition of hydrogen fusion in this paper. To this end, the appropriate nuclear network for the thermo-nuclear burning of deuterium, hydrogen, and helium is considered. \nThe codes are designed to handle two different outer boundary conditions for stellar models: shock and photospheric boundaries. The shock boundary condition presupposes spherically symmetric accretion onto a protostar, whereby the inflow directly hits the stellar surface and forms a shock front (e.g., Stahler et al. 1980; Hosokawa & Omukai 2009). We solve for the structure of both the stellar interior and outer accretion flow. With this boundary condition the photosphere is located outside the stellar surface where the accretion flow is optically thick to the stellar radiation. The photospheric boundary condition, on the other hand, presupposes a limiting case of mass accretion via a circumstellar disk, whereby accretion columns connecting the star and disk are geometrically compact and most of the stellar surface radiates freely (e.g., Palla & Stahler 1992; Hosokawa et al. 2010). In this case we only solve the stellar interior structure without considering details of the accretion flow; the location of photosphere always coincides with the stellar surface. \nThe different outer boundary conditions correspond to the two extremes of accretion flow geometries, or more specifically to different thermal efficiencies of mass accretion, which determine the specific entropy of accreting materials (e.g., Hosokawa et al. 2011a). The accretion thermal efficiency controls the entropy content of the star, which determines the stellar structure. With the shock boundary condition, the accreting gas obtains a fraction of the entropy generated behind the shock front at the stellar surface. The resulting thermal efficiency is relatively high ('hot' accretion). \nFor the photospheric boundary condition, on the other hand, the accreting gas is assumed to have the same entropy as in the stellar atmosphere. The underlying idea is that, when the accreting gas slowly approaches the star via angular momentum transport in the disk, its entropy should be regulated to the atmospheric value. This is a limiting case of thermally inefficient accretion ('cold' accretion). In general, with even a small amount of angular momentum, mass accretion onto the protostar would be via a circumstellar disk, perhaps with geometrically narrow accretion columns connecting the disk with the star. For extremely rapid mass accretion, however, the innermost part of the disk becomes hot and entropy generated within the disk is advected into the stellar interior (e.g., Popham et al. 1993). Thus, cold accretion as envisioned for the photospheric boundary condition is not appropriate for the case of rapid mass accretion (also see discussions in Hartmann et al. 1997; Smith et al. 2011). \nTABLE 1 Cases Considered \n| Case | ˙ M ∗ ( M /circledot yr - 1 ) | M ∗ , 0 ( M /circledot ) | R ∗ , 0 ( R /circledot ) | Notes | References |\n|------------|---------------------------------|----------------------------|----------------------------|----------------------------------------------------------|---------------|\n| MD1e0 | 1 . 0 | 2.5 | 298.4 | | Sec. 3.2 |\n| MD3e1 | 0 . 3 | 2.0 | 238.3 | | Sec. 3.2, 3.3 |\n| MD3e1-HC10 | 0 . 3 | 2.0 (10) | 238.3 (437.1) | shock → photo. BC at M ∗ = 10 M /circledot | Sec. 3.3 |\n| MD3e1-HC50 | 0 . 3 | 2.0 (50) | 238.3 (826.8) | shock → photo. BC at M ∗ = 50 M /circledot fiducial case | Sec. 3.3 |\n| MD1e1 | 0 . 1 | 2.0 | 177.8 | shock → photo. BC at M ∗ = 50 M /circledot fiducial case | Sec. 3.1, 3.2 |\n| MD6e2 | 0 . 06 | 1.0 | 118.6 | | Sec. 3.2 |\n| MD3e2 | 0 . 03 | 1.0 | 95.2 | | Sec. 3.2 |\n| MD6e3 | 0 . 006 | 1.0 | 52.3 | also see Omukai & Palla (2003) | Sec. 3.2 |\n| MD1e3 | 0 . 001 | 0.05 | 12.5 | also see Omukai & Palla (2003) | Sec. 3.1, 3.2 | \nNote . - Col. 2: mass accretion rate, Col. 3: initial stellar mass, Col. 4: initial radius. For cases MD3e1-HC10 and MD3e1-HD50 the values when the boundary condition is switched is given in parentheses. \nWe therefore expect that the shock boundary condition is a good approximation for the extremely high accretion rates considered here and mostly focus on stellar evolution with the shock boundary condition. We also consider a few cases with the photospheric boundary condition for comparison to test potential effects of reducing the accretion thermal efficiency (also see Sec. 2.2 below).", '2.2. Cases Considered': 'The cases considered are summarized in Table 1. In this paper, we only consider the evolution with constant accretion rates for simplicity. The adopted accretion rates range from 10 -3 M /circledot yr -1 to 1 M /circledot yr -1 . Stellar evolution with accretion rates less than 10 -2 M /circledot yr -1 (cases MD1e3 and MD6e3) has been studied in detail in our previous work (e.g., Omukai & Palla 2003; Hosokawa & Omukai 2009). As described in Section 2.1 above, we calculate the protostellar evolution assuming shock outer boundary conditions for most of the cases. Cases MD3e1-HC10 and MD3e1-HC50 are the only exceptions, whereby we switch to the photospheric boundary condition after the stellar mass reaches 10 M /circledot and 50 M /circledot , respectively, at a constant accretion rate 0 . 3 M /circledot yr -1 . The underlying idea for switching the boundary at some moment is that the specific angular momentum of the inflow and thus the circumstellar disk grows with time, which reduces entropy of the accreted matter. The higher mass at switching point corresponds to the higher angular momentum of the parental core. As in our previous work, we start the calculations with initial stellar models constructed assuming that the stellar interior is in radiative equilibrium (e.g., Hosokawa & Omukai 2009). We adopt a slightly higher initial stellar mass of ∼ 1 M /circledot for stability reasons. The calculated initial stellar radii are also summarized in Table 1.', '3.1. Evolution in the Fiducial Case ( ˙ M ∗ = 0 . 1 M /circledot yr -1 )': "We first consider the fiducial case (MD1e1), whereby the stellar mass increases with the constant accretion rate of ˙ M ∗ = 0 . 1 M /circledot yr -1 . The calculated evolution of the stellar interior structure is presented in Figure 1. We see that the stellar radius is very large and increases monotonically with the stellar mass. The stellar radius exceeds 10 3 R /circledot when the stellar mass is M ∗ /similarequal 45 M /circledot and reaches \nFig. 1.Evolution of the stellar interior structure for the fiducial case, whereby the stellar mass increases at a rate of 0 . 1 M /circledot yr -1 (case MD1e1). The thick solid line depicts the stellar surface, which is the position of the accretion shock front. The dotted lines show the radial positions of the mass coordinates of M = 3, 10, 30, 100, and 300 M /circledot . The dot-solid line indicates the radial position within which 70 % of the stellar mass is enclosed. The gray-shaded areas represent the convective layers. The hatched areas indicate layers of active nuclear burning, where the energy production rate exceeds 10% of the steady rate 0 . 1 L D , st /M ∗ for deuterium burning (see eq. 10), and 0 . 1 L ∗ /M ∗ for hydrogen burning. The blue solid line shows the evolution of the radius of a star accreting material at 10 -3 M /circledot yr -1 for comparison. \n<!-- image --> \n/similarequal 6500 R /circledot at M ∗ /similarequal 10 3 M /circledot . This evolution differs qualitatively from that calculated assuming a lower accretion rate 10 -3 M /circledot yr -1 as depicted in Figure 1 by the blue line (see also e.g., Omukai & Palla 2003). At the lower accretion rate the stellar radius initially increases with stellar mass but begins to decrease at M ∗ /greaterorsimilar 6 M /circledot . \nA key quantity for understanding the contrast between the two cases is the balance between the two characteristic timescales: the accretion timescale \nt acc ≡ M ∗ ˙ M ∗ (1) \nand Kelvin-Helmholtz (KH) timescale \nt KH ≡ GM 2 ∗ R ∗ L ∗ , (2) \nwhere R ∗ and L ∗ are the stellar radius and luminosity, and G is the gravitational constant (e.g., Stahler et al. \nFig. 2.Evolution of the stellar surface luminosity L ∗ (dashed line) and maximum luminosity within the star L max (solid line) for the cases with 10 -3 M /circledot yr -1 (MD1e3; upper panel ) and 0 . 1 M /circledot yr -1 (fiducial case MD1e1; lower panel ). The massluminosity relations given by equations (3), (4) and (5) are shown with the thin green, blue, and red lines, respectively. In each panel the vertical dot-dashed line (magenta) indicates the epoch when the accretion time is equal to the KH time. \n<!-- image --> \n1986; Omukai & Palla 2003; Hosokawa & Omukai 2009). In the early stage, during which the stellar radius increases with mass, the timescale balance is t acc /lessmuch t KH and radiative loss of the stellar energy is negligible ( adiabatic accretion stage ). However, the radiative energy loss becomes more efficient as the stellar mass increases. This is because opacity in the stellar interior, which is due to the free-free absorption following Kramers' law κ ∝ ρT -3 . 5 , decreases and the stellar luminosity L ∗ increases with the stellar interior temperature (and thus with its mass M ∗ ). The upper panel of Figure 2 indeed shows that the maximum luminosity within the star L max increases as a power-law function of M ∗ . \nThis increase of L ∗ is consistent with the analytic scaling relation for radiative stars with Kramers' opacity, L ∝ M 11 / 2 ∗ R -1 / 2 ∗ (e.g., Hayashi et al. 1962). Our numerical results are well fitted by the analytic relation \nL max /similarequal 0 . 6 L /circledot ( M ∗ M /circledot ) 11 / 2 ( R ∗ R /circledot ) -1 / 2 . (3) \nThe increase of L ∗ causes an inversion of the timescale balance to t acc > t KH at low accretion rates. The star contracts by losing its energy ( KH contraction stage ), which is seen for M ∗ /greaterorsimilar 6 M /circledot . The opacity in the stellar interior has fallen down to the constant value of electron scattering. Figure 2 shows that, in this stage, luminosity takes its maximum value at the stellar surface and \nFig. 3.Evolution of several physical quantities for the fiducial case with 0 . 1 M /circledot yr -1 (MD1e1). Top panel: Comparison between the accretion timescale t acc (dashed line) and KH timescale t KH (dotted line). The vertical magenta dot-dashed line indicates the epoch when t KH is equal to t acc . The thin solid line represents 100 times of the stellar free-fall timescale t ff ≡ √ 3 π/ 32 G ¯ ρ , where ¯ ρ is the average mass density of the star. The fact that t ff is much shorter than t KH and t acc verifies the hydrostatic balance assumption implicit in the stellar structure equations. Middle panel: Evolution of the accretion luminosity L acc (dashed line), stellar luminosity L ∗ (dotted line), and total luminosity L tot ≡ L acc + L ∗ (solid line). The red line indicates the Eddington luminosity at each stellar mass. Bottom panel: Evolution of the radial positions of the photosphere R ph ( R /circledot ) (dashed line) and stellar surface R ∗ ( R /circledot ) (black solid line). The red solid and dashed lines denote the analytic formulae for these radii by Stahler et al. (1986) (eqs 6 and 7). The evolution of the stellar effective temperature T eff ( K ) is also shown with the blue line. \n<!-- image --> \nincreases as L ∗ = L max ∝ M 3 ∗ , which is valid for the constant opacity cases (e.g., Hayashi et al. 1962). The relation \nL max /similarequal 10 L /circledot ( M ∗ M /circledot ) 3 (4) \nroughly agrees with our results. Temperature at the stellar center increases during the KH contraction stage. Hydrogen burning finally begins and the stellar radius begins to increase following the mass-radius relation of ZAMS stars for M ∗ /greaterorsimilar 50 M /circledot . Figure 2 shows that the stellar luminosity gradually approaches to the Eddington \nlimit \nL Edd ( M ∗ ) /similarequal 3 . 8 × 10 6 L /circledot ( M ∗ 100 M /circledot ) . (5) \nBy contrast, there is no contraction stage for the case with a much higher accretion rate ˙ M ∗ = 0 . 1 M /circledot yr -1 (MD1e1). Nevertheless, the evolution of the timescales still follows the picture described above (Fig. 3 a). We see that the timescale balance changes from t acc < t KH to t acc > t KH at M ∗ /similarequal 40 M /circledot . The protostar is in the adiabatic accretion stage for M ∗ /lessorsimilar 40 M /circledot . The accretion luminosity L acc ≡ GM ∗ ˙ M ∗ /R ∗ at this stage is much higher than the stellar luminosity L ∗ , since the luminosity ratio L acc /L ∗ is equal to the timescale ratio t KH /t acc by definition. Stahler et al. (1986) derived the approximate analytic formulae describing radial positions of the stellar surface R ∗ and photosphere R ph (located within the accretion flow) during the adiabatic stage: \nR ∗ /similarequal 26 R /circledot ( M ∗ M /circledot ) 0 . 27 ( ˙ M ∗ 10 -3 M /circledot yr -1 ) 0 . 41 , (6) \nR ph /similarequal 1 . 4 R ∗ . (7) \nThe bottom panel of Figure 3 shows that these formulae still agree with our numerical results with 0 . 1 M /circledot yr -1 . Equation (6) shows that the stellar radius is larger for the higher accretion rate at the same mass. The larger radius is due to the higher specific entropy of accreting material and the resulting higher entropy content of the star (e.g., Hosokawa & Omukai 2009). Comparing two stars of the same mass, the one with a larger radius has a lower interior temperature, which then implies a higher opacity due to the strong T -dependence of Kramers' law κ ∝ ρT -3 . 5 . For this reason the adiabatic accretion stage is prolonged up to higher stellar masses for higher accretion rates. The stellar luminosity L ∗ thus increases with stellar mass. Figure 2 (b) shows that the evolution of the stellar maximum luminosity L max still obeys equations (3) - (5). Unlike in the case for 10 -3 M /circledot yr -1 , however, it is only after L max approaches the relation of L max ∝ M 3 ∗ (eq. 4) that t KH becomes equal to t acc . The rapid heat input via mass accretion prevents the star from losing internal energy until the stellar luminosity becomes sufficiently high. \nThe fact that t KH is shorter than t acc for M ∗ /greaterorsimilar 40 M /circledot indicates that most of the stellar interior is contracting, as shown by the trajectories of the mass coordinates (dashed lines in Figure 1). The figure also shows that the bloated surface layer occupies only a small fraction of the total stellar mass. When the stellar mass is /similarequal 300 M /circledot , for example, the layer which has 30 % of the total mass measured from the surface covers more than 98 % of the radial extent. The star has a radiative core surrounded by an outer convective layer. Although the convective layer covers a large fraction of the stellar radius, even the central radiative core is much larger than a ZAMS star with the same mass (compare with the blue curve for M ∗ /greaterorsimilar 40 M /circledot in Fig. 1). Figure 4 shows the radial distributions of physical quantities, i.e., specific entropy, luminosity, temperature, and density, in the stellar interior. We see that the specific entropy is at its maximum value near the boundary between the radiative core and \nconvective layer. The stellar entropy distribution is controlled by the energy equation, \nT ( ∂s ∂t ) M = /epsilon1 -( ∂L ∂M ) t , (8) \nwhere s is the specific entropy and /epsilon1 is the energy production rate by nuclear fusion. For M ∗ /greaterorsimilar 40 M /circledot most of the stellar luminosity comes from the release of gravitational energy. In fact, as seen in Figure 4 (b), ( ∂L/∂M ) t > 0 in the radiative core, which means that the internal energy of the gas is decreasing. The local luminosity in the radiative core is close to the Eddington value given by equation (5), using the mass coordinate M rather than stellar mass M ∗ (Fig. 4 b). Note that the opacity in the radiative core is only slightly higher than that expected from electron-scattering alone. \nIn the outer parts of the star, where temperature and density are lower, however, opacity is higher than in the core because of bound-free absorption of H, He atoms and the H -ion. Energy transport via radiation is inefficient there, and a part of the energy coming from the core is carried outward via convection. Figure 4 shows that the surface convective layer lies in the temperature and density range where T /lessorsimilar 10 5 K and ρ /lessorsimilar 10 -8 cm -3 , which is almost independent of the stellar mass. We also see that the specific entropy is not constant over the convective layer, decreasing toward the stellar surface (a so-called super-adiabatic layer). This is because convective heat transport is inefficient near the stellar surface. A part of the outflowing energy is absorbed there, as indicated by the fact that the surface layer has a negative luminosity gradient ( ∂L/∂M ) t < 0. This explains the high specific entropy in the outer convective layer. \nThe outermost part of the star has a density inversion, i.e. the density increases toward the stellar surface. Here, opacity assumes very high values because of H -absorption. Radiation pressure is so strong that the hydrostatic balance is not achieved only with gravity; the additional inward force by the negative gas pressure gradient helps maintain the hydrostatic structure. Note, however, that this density inversion could be unstable in realistic multidimensions (see e.g., Begelman et al. 2008). \nAlthough deuterium burning is ignited when the stellar mass is /similarequal 50 M /circledot , its influence on the subsequent evolution is negligible (Fig. 1). The total energy production rate by deuterium burning is approximately \nL D , st ≡ ˙ M ∗ δ D (9) \n=1 . 5 × 10 5 L /circledot ( ˙ M ∗ 0 . 1 M /circledot yr -1 ) ( [D / H] 2 . 5 × 10 -5 ) , \nwhere δ D is the energy available from deuterium burning per unit gas mass. Since this is much lower than the Eddington luminosity L Edd in the mass range considered, energy production by deuterium burning contributes only slightly to the luminosity in the stellar interior. Hosokawa & Omukai (2009) showed that deuterium burning influences the stellar evolution only when the accretion rate is low ˙ M ∗ /lessorsimilar 10 -4 M /circledot yr -1 . \nFigure 4 (c) shows that temperature in the stellar interior increases with total mass. The central temperature reaches 10 8 K when the stellar mass is /similarequal 600 M /circledot . Soon \nFig. 4.Radial profiles of the specific entropy ( panel a ), luminosity ( b ), temperature ( c ) and gas mass density ( d ) in the stellar interior for the fiducial case with 0 . 1 M /circledot yr -1 (MD1e1). The profiles when the stellar mass is 200 M /circledot , 600 M /circledot , and 10 3 M /circledot are shown in each panel. The magenta parts indicate the layers are convective. The thin red line in panel (b) represents the Eddington-limit luminosity as a function of the mass coordinate M . \n<!-- image --> \nafter that, hydrogen burning begins and a central convective core develops (Fig. 1). This convective core can also be seen in the radial profiles for the 10 3 M /circledot model in Figure 4 (indicated by magenta). The luminosity profile tells that most of the energy produced by hydrogen burning is absorbed within the convective core. The star still shines largely by releasing its gravitational energy even after hydrogen ignition. \nWhen the stellar radius is sufficiently large, the accretion flow reaches the stellar surface before becoming opaque to the outgoing stellar light. In fact, soon after the end of the adiabatic accretion stage, the accreting en- \nFig. 5.Evolution of the protostellar radius for various accretion rates. Upper panel: the different curves represent the cases with ˙ M ∗ = 10 -3 M /circledot yr -1 (case MD1e3, black), 6 × 10 -3 M /circledot yr -1 (MD6e3, blue), 3 × 10 -2 M /circledot yr -1 (MD3e2, red), and 6 × 10 -2 M /circledot yr -1 (MD6e2, magenta). The open and filled circles on each curve denote the epoch when t KH = t acc and when the central hydrogen burning begins, respectively. Lower panel: same as the upper panel but for higher accretion rates of 6 × 10 -2 M /circledot yr -1 (MD6e2, magenta), 0 . 1 M /circledot yr -1 (MD1e1, red), 0 . 3 M /circledot yr -1 (MD3e1, blue), and 1 M /circledot yr -1 (MD1e0, black). The case MDe2 is illustrated in both panels as a reference. For the cases with 0 . 3 M /circledot yr -1 and 1 M /circledot yr -1 (MD3e1 and MD1e0) hydrogen fusion has not ignited by the time the stellar mass reaches 10 3 M /circledot . In the both panels the thin green line represents the mass-radius relation given by equation (12). \n<!-- image --> \nvelope remains optically thin throughout (Fig. 3 c). We also see that the stellar effective temperature is almost constant at T eff /similarequal 5000 K during this period. In general, the stellar effective temperature never assumes a lower value due to the strong temperature-dependence of H -absorption opacity (e.g., Hayashi 1961). Stars that have a compact core and bloated envelope (e.g., red giants) commonly have an almost constant effective temperature, regardless of their stellar masses.", '3.2. Cases with Different Accretion Rates': 'We now investigate how stellar evolution changes with the accretion rate. Figure 5 shows the evolution of the stellar radius for several cases, including ˙ M ∗ = 10 -3 M /circledot yr -1 (case MD1e3) and 0 . 1 M /circledot yr -1 (case MD1e1) explained in Section 3.1. We see that, with accretion rates higher than 6 × 10 -2 M /circledot yr -1 , the evolution becomes similar to that of the fiducial case for 0 . 1 M /circledot yr -1 (MD1e1); the stellar radius monotonically increases with mass. The protostars undergo adiabatic accretion in the early stage. As equation (6) shows, the stellar radius is larger for higher accretion rates at a given stellar mass, say, at M ∗ = 10 M /circledot . \nFig. 6.Upper panel: Evolution of radial positions of the stellar surface (solid) and photosphere (dotted). The black, blue, and magenta curves denote cases with ˙ M ∗ = 10 -3 M /circledot yr -1 (case MD1e3), 6 × 10 -3 M /circledot yr -1 (MD6e3), and 1 M /circledot yr -1 (MD1e0), respectively. The mass-radius relation given by equation (12) is plotted with the thin green line. Lower panel: The stellar effective temperature for the same cases as in the upper panel. \n<!-- image --> \nFig. 7.Evolutionary tracks in the HR diagram. The different colors denote the evolution for different accretion rates, 10 -3 M /circledot yr -1 (case MD1e3, black), 6 × 10 -3 M /circledot yr -1 (MD6e3, blue), 0 . 1 M /circledot yr -1 (MD1e1, red), 1 M /circledot yr -1 (MD1e0, magenta). For each case the values at the stellar photosphere ( L tot and T eff ) and at the stellar surface ( L ∗ and T eff , ∗ ≡ L ∗ / 4 πσR 2 ∗ ) are plotted with solid and dashed lines, respectively. The thick dashed line represents the loci of non-accreting ZAMS stars taken from Marigo et al. (2001) ( M ∗ ≤ 100 M /circledot ) and Bromm et al. (2001) ( M ∗ ≥ 100 M /circledot ). The filled circles and open squares on the lines mark the positions for M ∗ = 30 M /circledot , 100 M /circledot , 300 M /circledot , and 10 3 M /circledot in ascending order. \n<!-- image --> \nFig. 8.Comparisons of the interior structure of 10 3 M /circledot stars produced by different accretion rates. Radial profiles of mass ( panel a ), specific entropy ( b ), and luminosity ( c ) are shown. In each panel the solid and dashed curves represent the cases for 0 . 3 M /circledot yr -1 (MD3e1), 0 . 1 M /circledot yr -1 (MD1e1), 3 × 10 -2 M /circledot yr -1 (MD3e2), and 10 -3 M /circledot yr -1 (MD1e3). The magenta lines mark convective layers in the stellar interior. The thin red line in panel (c) represents the Eddington luminosity as a function of mass M . \n<!-- image --> \nAdiabatic accretion occurs up to a higher stellar mass when the accretion rate is higher. We derive an analytic expression describing this dependence in the following. As long as the KH timescale t KH is longer than the accretion timescale t acc , we have adiabatic accretion. (e.g., Hosokawa & Omukai 2009). Figure 2 shows that L max , whose evolution is well described by simple analytic expressions (eqs. 3 and 4), converges to L ∗ at the end of the adiabatic accretion stage. Thus, the stellar mass when the adiabatic accretion terminates M ∗ , teq can be estimated with the relation t acc /similarequal t KH = GM 2 ∗ /R ∗ L max . Eliminating L max and R ∗ with equations (4) and (6), we obtain \nM ∗ , teq /similarequal 14 . 9 M /circledot ( ˙ M ∗ 10 -2 M /circledot yr -1 ) 0 . 26 . (10) \nWe have confirmed that the epoch of the timescale equality in our numerical calculation is well described by this equation. \nEven after t KH becomes shorter than t acc , the stellar radius continues to increase for accretion rates /greaterorsimilar \n10 -2 M /circledot yr -1 . The variations of radii among cases with different accretion rates gradually disappear. The stellar radii finally converge to a unique mass-radius relation with R ∗ ∝ M 1 / 2 ∗ in all the cases. We can derive the approximate mass-radius relation from the following simple argument. First, the stellar luminosity is generally written as \nL ∗ = 4 πR 2 ∗ σT 4 eff , (11) \nwhere σ is the Stefan-Boltzman constant. As Figure 3 indicates, the stellar luminosity approaches the Eddington value L Edd ( M ∗ ) for M ∗ /greaterorsimilar 100 M /circledot . As explained in Section 3.1 the stellar effective temperature stays at the constant value T eff /similarequal 5000 K after the inversion of the timescales (also see Figs. 6 and 7). Substituting these relations into equation (11), we obtain \nR ∗ /similarequal 2 . 6 × 10 3 R /circledot ( M ∗ 100 M /circledot ) 1 / 2 . (12) \nFigure 5 shows that our numerical results approximately follow this relation. \nBegelman (2010) also considered stellar evolution with very rapid mass accretion using simple analytic arguments. His model predicts that the stellar radius is proportional to the mass accretion rate (his eq. 24), which does not agree with our numerical results. Begelman (2010), however, does not take into account the detailed structure of the outermost layer of the star, where H -opacity is important. The fact that the strong T -dependence of H -opacity keeps the stellar effective temperature almost constant is essential for our results. \nCases MD3e2 and MD6e3 model stellar evolution at intermediate accretion rates (3 × 10 -2 and 6 × 10 -3 M /circledot yr -1 , respectively) and exhibit a different behavior than the higher accretion-rate cases described above. For an accretion rate 3 × 10 -2 M /circledot yr -1 (MD3e2) the protostar initially contracts after the adiabatic accretion stage. At M ∗ /similarequal 70 M /circledot , however, the stellar radius sharply increases and ultimately converges to the mass-radius relation given by equation (12). The case with 6 × 10 -3 M /circledot yr -1 exhibits an oscillatory behavior of the stellar radius for M ∗ /greaterorsimilar 70 M /circledot . In this case the accreting envelope remains optically thick after the onset of KH contraction (Figs. 6 and 7). Its photospheric radius still follows the mass-radius relation R ph ∝ M 1 / 2 ∗ . The effective temperature also assumes the constant value T eff /similarequal 6000 K . These evolutionary features for ˙ M ∗ < 10 -2 M /circledot yr -1 have also been found in previous studies (e.g., Omukai & Palla 2001, 2003). \nWe have seen that the stellar radius at M ∗ /similarequal 10 3 M /circledot is almost independent of accretion rate as long as ˙ M ∗ /greaterorsimilar 3 × 10 -2 M /circledot yr -1 . However, the stellar interior structure at this moment is not identical among these cases (Fig. 8). Although each of these stars has a radiative core and a convective envelope, the mass is more strongly centrally concentrated for the lower accretion rates; the less massive envelopes have a higher entropy and inflate even more to achieve the same stellar radius. \nThe evolution of the stellar maximum temperature is helpful for understanding the variation of the stellar interior structure (Fig. 9 a) with accretion rate. As equation (10) shows, the central part of the star begins to contract \nFig. 9.Evolution of the maximum temperature in the stellar interior ( upper panel ) and Eddington ratio L tot /L Edd ( lower panel ) with increasing stellar mass. The solid and dashed curves alternately represent the cases with different accretion rates, 10 -3 M /circledot yr -1 (case MD1e3), 6 × 10 -3 M /circledot yr -1 (MD6e3), 3 × 10 -2 M /circledot yr -1 (MD3e2), 0 . 1 M /circledot yr -1 (MD1e1), and 0 . 3 M /circledot yr -1 (MD3e1). \n<!-- image --> \nand release gravitational energy at a lower stellar mass for the lower accretion rates. The central temperature quickly increases with stellar mass once the KH time becomes shorter than the accretion time. The maximum temperature T max reaches 10 8 K at a lower stellar mass for the lower accretion rate. After that T max assumes an almost constant value due to the strong T -dependence of the energy production rate of hydrogen burning. For the case with 3 × 10 -2 M /circledot yr -1 (MD3e2) hydrogen burning begins at M ∗ /similarequal 200 M /circledot ; the resulting central convective core is seen in the profiles in Figure 8. This feature is not seen for the case with 0 . 3 M /circledot yr -1 , because hydrogen has not yet ignited by the time M ∗ = 10 3 M /circledot for ˙ M ∗ /greaterorsimilar 0 . 3 M /circledot yr -1 . \nFigure 5 shows that for ˙ M ∗ /greaterorsimilar 6 × 10 -3 M /circledot yr -1 the protostar cannot reach the ZAMS stage by KH contraction. Omukai & Palla (2003) pointed out that this is because the total luminosity L tot ≡ L ∗ + L acc becomes close to the Eddington limit during the contraction to the ZAMS. For the cases with 6 × 10 -3 and 3 × 10 -2 M /circledot yr -1 (cases MD6e3 and MD3e2), for example, the abrupt expansion terminates the KH contraction when the total luminosity is nearly at the Eddington limit (Fig. 9 b). Omukai & Palla (2003) analytically derived the maximum accretion rate /similarequal 4 × 10 -3 M /circledot yr -1 with which the protostar can reach the ZAMS following KH contraction. Figure 5 indicates that there is another critical accretion rate /similarequal 6 × 10 -2 M /circledot yr -1 , above which the stellar evolution changes qualitatively; the KH contraction stage disappears entirely at higher rates. This critical \nrate can also be derived from a similar argument as the one above. \nNote that the increase of stellar mass during the KH contraction stage is smaller for the case with 3 × 10 -2 M /circledot yr -1 than with 6 × 10 -3 M /circledot yr -1 . Extending this fact for our critical case, the total luminosity would nearly reach the Eddington limit just at the end of the adiabatic accretion stage, i.e., when t KH /similarequal t acc . Since the opacity in the surface layer is higher than from Thomson scattering during this epoch, having the total luminosity only slightly lower than the Eddington value causes the star to expand. Thus, the condition for the critical case is \n2 L max /similarequal C Edd L Edd , (13) \nwhere C Edd is a factor less than the unity and we have used the fact that the total luminosity is written as L tot /similarequal 2 L max when t KH /similarequal t acc . Using C Edd = 0 . 25 as a fiducial value (Fig. 9 b) and equation (4) for L max , the stellar mass which satisfies the condition (13) is \nM ∗ , Edd , teq /similarequal 21 . 7 M /circledot ( C Edd 0 . 25 ) 0 . 5 . (14) \nOn the other hand, equation (10) also gives the stellar mass when t KH /similarequal t acc for a given accretion rate. Equating M ∗ , teq and M ∗ . Edd , teq with equations (10) and (14), we obtain the critical mass accretion rate \n˙ M cr /similarequal 4 . 7 × 10 -2 M /circledot yr -1 ( C Edd 0 . 25 ) 1 . 9 , (15) \nwhich agrees with our numerical results.', '3.3. Effects of Lower-Entropy Accretion': "We have used the shock outer boundary condition for the stellar models presented and discussed above. As discussed in Section 2.1, the shock boundary condition, which implies that the accreting gas joins the star with relatively high entropy, would be valid for cases with the very rapid mass accretion considered in this paper. If the accred gas had lower entropy, however, the stellar radius would be reduced because of the resulting lower entropy thoughout the stellar interior. Here, we examine potential effects of the colder mass accretion by adopting the photospheric boundary conditions (e.g., Hosokawa et al. 2010, 2011a). Figure 10 (a) shows the evolution of the stellar radius for three cases with 0 . 3 M /circledot yr -1 , whereby the shock boundary condition is used throughout in one case (MD3e1), whereas the outer boundary condition is changed to the photospheric one for > 10 M /circledot (MD3e1HCm10) and > 50 M /circledot (MD3e1-HCm50), respectively. The stars are still in the adiabatic accretion stage when the boundary condition is changed in both cases. The different outer boundary conditions do affect the stellar evolution. For the case where the photospheric boundary condition is adopted at M ∗ = 10 M /circledot (MD3e1-HCm10), for example, the star initially contracts after the boundary condition is switched at M ∗ = 10 M /circledot , and then abruptly inflates at M ∗ /similarequal 45 M /circledot . The stellar radius exceeds 10 3 R /circledot and gradually increases with the stellar mass thereafter. In spite of the different behaviors in the early stages, the subsequent evolution for M ∗ /greaterorsimilar 100 M /circledot \nFig. 10.Effect of reducing the thermal efficiency of mass accretion ( upper panel: stellar radius, lower panel: maximum temperature within the star). The same accretion rate of 0 . 3 M /circledot yr -1 is adopted for all three cases presented. The solid line represents the evolution with the shock boundary condition, i.e, thermally efficient or 'hot' accretion, throughout (MD3e1). The dashed and dot-dashed lines show the evolution in cases MD3e1-HC10 and MD3e1-HC50, where the photospheric boundary condition (i.e., thermally inefficient or 'cold' accretion) is adopted after the stellar mass exceeds 10 M /circledot and 50 M /circledot , respectively. In the lower panel the dot-dashed line is indistinguishable from the solid line. \n<!-- image --> \nis quite similar to that in the case with the shock boundary condition throughout (MD3e1). The evolution when the boundary condition switching occurs at M ∗ = 50 M /circledot (MD3e1-HCm50) is much closer to that in the shockboundary case. The uniqueness of the mass-radius relation for M ∗ /greaterorsimilar 100 M /circledot can be explained by the fact that the argument leading to the analytic expression, equation (12), does not assume a specific boundary condition. When the boundary condition is switched at M ∗ = 10 M /circledot (MD3e1-HCm10), the stellar interior temperature is higher and thus the opacity in the stellar interior ( ∝ T -3 . 5 according to Kramars's law) is lower than for the shock-boundary case (MD3e1) at the same stellar mass. As a result the star begins to release its internal energy earlier than for the shock-boundary case. Indeed, the timescale equality between t KH and t acc occurs at M ∗ /similarequal 40 M /circledot , earlier than for the shock-boundary case, which occurs at the time of abrupt expansion of the stellar radius.", '4. SUMMARY AND DISCUSSIONS': "We have studied the evolution of stars growing via very rapid mass accretion with 10 -2 M /circledot yr -1 /lessorsimilar ˙ M ∗ /lessorsimilar 1 M /circledot yr -1 , which potentially leads to formation of SMBHs in the early universe. In contrast to previous attempts to address this problem, we study the stars' evolution by numerically solving the stellar structure equations including mass accretion. Our calculations show that stellar evolution in such cases is qualitatively different from that expected for normal Pop III star formation, which proceeds at much lower accretion rates \n∼ 10 -3 -10 -2 M /circledot yr -1 . Rapid mass accretion causes the star to inflate; the stellar radius further increases monotonically with stellar mass at least up to M ∗ /similarequal 10 3 M /circledot . For masses exceeding ∼ 100 M /circledot , the star consists of a contracting radiative core and a bloated surface convective layer. The surface layer, which contains only a small fraction of the total stellar mass, fills out most of the stellar radius. The evolution of the stellar radius in this stage follows a unique mass-radius relation R ∗ ∝ M 1 / 2 ∗ , which reaches /similarequal 7000 R /circledot ( /similarequal 30 AU) at M ∗ = 10 3 M /circledot , in all the cases with /greaterorsimilar 10 -2 M /circledot yr -1 . Hydrogen burning begins only after the star becomes very massive ( M ∗ /greaterorsimilar 100 M /circledot ); its onset is shifted toward higher masses for higher accretion rates. With very high accretion rates ˙ M ∗ /greaterorsimilar 0 . 1 M /circledot yr -1 , hydrogen is ignited after the stellar mass exceeds 10 3 M /circledot . The stellar radius continues to grow as R ∗ ∝ M 1 / 2 ∗ even after hydrogen ignition. \nIn this paper we have focused on the early evolution until the stellar mass reaches 10 3 M /circledot . The subsequent evolution remains unexplored because of convergence difficulties with the current numerical codes. If the star continues to expand following the same mass-radius relation (12) also for M ∗ > 10 3 M /circledot , the stellar radius at 10 5 M /circledot would be /similarequal 400 AU. Since the stellar effective temperature remains /similarequal 5000 K, the star hardly emits ionizing photons during accretion. Therefore, it is unlikely that stellar growth is limited by the radiative feedback via formation of an HII region as discussed by Hosokawa et al. (2011b). Johnson et al. (2011) also reached an analogous conclusion that UV feedback does not hinder SMS formation. In their argument, however, the star is assumed to reach the ZAMS and to emit a copious amount of ionizing photons, but the expansion of the HII region is squelched by rapid spherical inflow. They also expected that, as a result of confinement of the HII region, strong emission lines reprocessed from the ionizing photons (e.g., Ly α and He II) would escape from the accretion envelope to be an observational signature of these objects. By contrast, our calculations show that the stellar UV luminosity and thus the luminosities in those lines should be much weaker than supposed. Note that the argument by Johnson et al. (2011) assumes perfect spherical symmetry, which allows the HII region to be confined within the accretion envelope. Given that mass accretion will likely occur through a circumstellar disk, the HII region should grow toward the polar region \nwhere the gas density is much lower than the spherical accretion flow (e.g., Hosokawa et al. 2011b). This should be the case with the high stellar UV luminosity assumed in Johnson et al. (2011). \nEven without stellar radiative feedback, stellar growth via mass accretion might be hindered by some other process, e.g., rapid mass loss. Indeed, evolved massive stars in the Galaxy ( M ∗ ∼ 10 -100 M /circledot ), which have large radii ( R ∗ /greaterorsimilar 100 R /circledot ) and high luminosities close to the Eddington limit ( L ∗ /similarequal 10 6 L /circledot ), generally have strong stellar winds with mass losses ∼ 10 -4 M /circledot yr -1 (e.g., Humphreys & Davidson 1994). Although the line-driven winds of primordial stars are predicted to be weak or non-existent (Krtiˇcka & Kub'at 2006), pulsational instability of massive stars has also been found to drive mass loss (Baraffe et al. 2001; Sonoi & Umeda 2011). Further work is necessary to address how massive SMSs could form via mass accretion in spite of such disruptive effects. \nStellar evolution under conditions of very rapid mass accretion as presented and discussed here is mostly relevant to the formation of stars in the atomic-cooling halos. However, our results could be also important for normal Pop III star formation where H 2 molecular cooling operates. The typical mass accretion rate for this case is around 10 -3 M /circledot yr -1 , but in some exceptional situations, e.g., when a progenitor cloud core is extremely slow rotating, higher accretion rates ˙ M ∗ ∼ 10 -2 M /circledot yr -1 can be realized (e.g., Hosokawa et al. 2011b). Since the stellar effective temperature is low at /similarequal 5000 K with rapid mass accretion, formation of the HII region would be postponed until the mass accretion rate falls below 10 -2 M /circledot yr -1 . 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2011PhRvL.106l1301C

Universal Area Product Formulas for Rotating and Charged Black Holes in Four and Higher Dimensions

2011-01-01

['-', '-', '-', '-', '-', '-']

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We present explicit results for the product of all horizon areas for general rotating multicharge black holes, both in asymptotically flat and asymptotically anti-de Sitter spacetimes in four and higher dimensions. The expressions are universal, and depend only on the quantized charges, quantized angular momenta and the cosmological constant. If the latter is also quantized these universal results may provide a “looking glass” for probing the microscopics of general black holes.

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https://arxiv.org/pdf/1011.0008.pdf

{'M.Cvetiˇc, 1 , 4 G.W. Gibbons, 2 C.N. Pope 2 , 3': "1 Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA 19104, USA 2 DAMTP, Centre for Mathematical Sciences, Cambridge University, Wilberforce Road, Cambridge, CB3 0WA, UK 3 George P. & Cynthia W. Mitchell Institute for Fundamental Physics and Astronomy, Texas A&M University, College Station, TX 77843-4242, USA \n- 4 Center for Applied Mathematics and Theoretical Physics, University of Maribor, Maribor, Slovenia \nWe present explicit results for the product of all horizon areas for general rotating multi-charge black holes, both in asymptotically flat and asymptotically anti-de Sitter spacetimes in four and higher dimensions. The expressions are universal, and depend only on the quantized charges, quantized angular momenta and the cosmological constant. If the latter is also quantized these universal results may provide a 'looking glass' for probing the microscopics of general black holes. \nExplaining the origin of the black hole entropy S = 1 4 A at the microscopic level, where A is the area of the outer event horizon, is an outstanding problem for quantum theories of gravity. Significant insights have been achieved for supersymmetric, asymptotically flat, multicharged black holes in four and five dimensions [1], where the microscopic degrees of freedom can be explained in terms of a two-dimensional conformal field theory. More recent work has focused on the microscopic entropy of extreme rotating solutions [2]. By contrast, the detailed microscopic origin of the entropy of non-extremal rotating charged black holes remains an open problem, although recently there has been some promising progress [3]. \nGreybody factors (i.e. absorption coefficients) and radiation spectra provide another approach to probing the black hole structure. An intriguing property of multicharged rotating black holes (in maximally supersymmetric supergravity theories) is that their wave equations are separable. The radial equation has poles at the locations of the horizons, where the radial component of the metric degenerates, with residues proportional to the inverse squares of the surface gravities, and so the Green functions are sensitive to the geometry near all the black hole horizons, and not just the outermost one. The thermodynamic properties, including the surface gravity and area at each horizon, can therefore be expected to play a role in understanding the entropy at the microscopic level. \nSome of these ideas have been explored for asymptotically flat, rotating, multi-charged black holes in four and five spacetime dimensions. (Explicit solutions were given in [4, 5], as generating solutions of maximally supersymmetric N = 4 (or N = 8) supergravities, obtained as toroidal compactifications of the heterotic string (or of Type IIA string or M-theory).) In addition to their mass M , in four dimensions these solutions are specified by four charges Q i ( i = 1 , · · · , 4) and one angular momentum J , and in five dimensions by three charges Q i ( i = 1 , 2 , 3) and two angular momenta J 1 , 2 . These black holes have just two horizons, and the area of the outer \nhorizon has the tantalizing form [4] \nS + = 2 π ( √ N L + √ N R ) . (1) \nwhere the integers N L and N R may be viewed as the excitation numbers of the left and right moving modes of a weakly-coupled two-dimensional conformal field theory. N L and N R depend explicitly on all the black hole parameters. It was pointed out, first in the static case [6] and later for the general rotating black holes [7, 8], that the entropy of the inner horizon, S -= 1 4 A -, is \nS -= 2 π ( √ N L -√ N R ) . (2) \nFrom this and (1), it follows that the product of the inner and outer horizon entropies satisfies S + S -= 4 π 2 ( N L -N R ), which in terms of the underlying conformal field theory would be interpreted in terms of a level-matching condition. S + S -should therefore also be an integer [68]. (This point was recently re-emphasized in [9].) It was found that S + S -is indeed quantized, and intriguingly, it is expressed solely in terms of the quantized charges and quantized angular momenta. In particular, it is modulusindependent, taking the forms \nS + S -= 4 π 2 ( 4 ∏ i =1 Q i + J 2 ) (3) \nS + S -=4 π 2 ( 3 ∏ i =1 Q i + J 2 R -J 2 L )=4 π 2 ( 3 ∏ i =1 Q i + J a J b ) (4) \nin four and five dimensions respectively. (These results were implicit in [7, 8], though not explicitly evaluated.) The solutions considered here can be viewed as 'seed solutions' from which the complete families can be generated. The expressions for S + S -would be expressed in terms of S-, T- and U-duality invariants built from the charges in the general case. \nIn a parallel development, Ansorg and collaborators [12-18] studied general axisymmetric stationary solutions of Einstein-Maxwell theory in four dimensions, with \nsources external to the horizons. They obtained striking 'universal' formulae expressing the areas A ± of the outer and inner Killing horizons in terms of the total angular momentum J and total charge Q . In particular, for KerrNewman black holes, they found (in the normalisation conventions we use in the remainder of this paper) \nA 2 + ≤ A + A -= (8 πJ ) 2 +(4 πQ 2 ) 2 , (5) \nin agreement (after conversion to our conventions) with the result given above in the special case that the four charges are set equal. Note the inequality (5) may be interpreted as a general criterion for extremality, and has been used to prove a No-Go theorem for the possibility of force balance between two rotating black holes [20]. \nIt is natural to enquire whether analogous properties hold for more general classes of black holes; and especially, for those where the radial metric function has more than two zeroes. Examples include charged or rotating black holes in four or five dimensional gauged supergravity, and in more than five dimensions with or without gauging. The wave equations in these backgrounds will have dominant contributions associated with poles at each of these zeroes. One can therefore again expect that the thermodynamics associated with each pole will play a role in governing the properties of the black hole at the microscopic level. At event horizons or Cauchy horizons, the metric at fixed radius has signature (0 , + , + , · · · , +); that is, it describes a null hypersurface. However, it may happen that the induced metric has signature (0 , -, + , + , · · · , +); in other words that the hypersurface is time-like, and the area of this 'pseudo-horizon' [10] is pure imaginary. The metric radial function may also have zeroes for complex values of the radial variable; these occur in conjugate pairs. In what follows, we shall just refer to zeroes of the radial function as horizons, regardless of whether the areas are real, imaginary or complex. \nIf it is indeed the case that geometries near all the horizons are involved in governing the microscopic behaviour of the black hole, one might expect that the formulae (3) and (4) should generalise, for the more general black hole examples, to expressions involving the products of all the horizon entropies or areas. This would suggest the possibility of an explanation for the microscopic behaviour of such black holes in terms of a field theory in more than two dimensions. \nWe shall present results for the products of horizon areas in examples that include certain rotating black hole solutions in gauged supergravities in dimensions 4, 5, 6 and 7, and also Kerr-anti de Sitter rotating black holes in arbitrary spacetime dimensions. For the sake of brevity, we shall not present the details of our calculations in all cases, and instead, we have selected one example, namely the rotating black hole in five-dimensional minimal gauged supergravity, for which we present the calculation of the area-product formula in more detail. \nThe formulae that we obtain for the area products are universal; they depend only on quantized charges, quan- \ntized angular momenta and the cosmological, or gaugecoupling, constant. In the case that the latter is also quantized (such as arises in compactifications of string theory, as discussed, for example, in [11]), these results are indeed suggestive of some underlying microscopics. For example, one may speculate that asymptotically antide Sitter black holes in four and five dimensions, for which there are three horizons, may have a microscopic origin in three-dimensional Chern-Simons theory. \nWe shall use normalisation conventions where the Lagrangian density for gravity and Maxwell field(s) is of the form \nL = 1 16 πG ( R -∑ i Φ i ( φ ) F i µν F i µν +( D -1)( D -2) g 2 ) , (6) \nwhere the functions of scalar fields (if present) are such that Φ i ( φ ) tends to unity at infinity for the black-hole solutions. We define charge(s) and angular momenta by \nQ i = 1 4 π ∫ Φ i ( φ ) ∗ F i , J i = 1 16 π ∫ ∗ dK i , (7) \nwhere K i = K i µ dx µ and K i µ ∂ µ = ∂/∂ψ i , where ψ i is the azimuthal coordinate, with period 2 π , in the 2-plane associated with the angular momentum J i . \nOur results for the products of the horizon areas for rotating black holes in gauged supergravities in dimensions 4, 5, 6, and 7 are as follows: \nD = 4 ungauged 4-charge [4]: \nA + A -= (8 πJ a )(8 πJ b ) + 256 π 2 ∏ 4 i =1 Q i ,", 'D = 4 gauged pairwise equal charges [21]:': "∏ 4 α =1 A α = (4 π ) 2 g -4 (8 πJ ) 2 +4 g -4 (4 πQ 1 ) 2 (4 πQ 2 ) 2 . \nD = 5 ungauged 3-charge [5]: \nA + A -= (8 πJ a )(8 πJ b ) + 256 π ∏ 3 i =1 Q i , \nD = 5 minimal gauged [22]: \n∏ 2 α =0 A α = -2i π 2 g -3 (8 πJ a )(8 πJ b ) -i g -3 ( 8 πQ √ 3 ) 3 , \nD = 5 gauged Q 1 = Q 2 = Q 3 [23]: \n∏ 3 α =0 A α = -2i π 2 g 3 (8 πJ a )(8 πJ b ) -i g 3 (8 πQ 1 ) 2 (8 πQ 3 ). \n/negationslash \nD = 6 gauged [24]: \n∏ 6 α =1 A α = g -8 ( 8 π 2 3 ) 2 (8 πJ a ) 2 (8 πJ b ) 2 + g -6 ( 8 πQ 3 ) 6 . \nD = 7 gauged [25]: \n∏ 4 α =1 A α = π 3 g -5 ∏ 3 i =1 (8 πJ i ) -g -4 (2 πQ ) 4 . \nNote that we have included the cases of the 4-charge D = 4, and the 3-charge D = 5, solutions in ungauged supergravities, which were already presented as entropyproduct formulae in the introduction. This is done for the \nsake of uniformity, using the normalisation conventions that we follow in the rest of the body of the paper. The citation in each heading above refers to the paper where the black hole solution was constructed. \nTo illustrate how these calculations may be performed, we shall present the example of the rotating black hole in five-dimensionsal minimal gauged supergravity. The horizons are located at the roots of the radial function \n∆( r ) = (1+ g 2 r 2 )( r 2 + a 2 )( r 2 + b 2 )+ q 2 +2 abq -2 mr 2 (8) \nthat appears in the metric found in [22]. This is a cubic polynomial in r 2 , and so there are six roots in total, occurring in pairs for which r 2 takes the same value. We may view x = r 2 as the radial variable, and thus just consider 3 roots. We may write ∆ as \n∆( r ) = g 2 2 ∏ α =0 ( r 2 -r 2 α ) . (9) \nThe horizon areas are \nA α = 2 π 2 [( r 2 α + a 2 )( r 2 α + b 2 ) + abq ] Ξ a Ξ b r α . (10) \nUsing (8) and ∆( r α ) = 0, we can write this as \nA α = -2 π 2 (2 m + abqg 2 ) Ξ a Ξ b (1 + g 2 r 2 α ) r α [ q ( q + ab ) 2 m + abqg 2 -r 2 α ] . (11) \nNoting from (8) and (9) that we may write ∏ α ( c 2 -r 2 α ) as g -2 ∆( c ), for any c , it is then straightforward to evaluate the product of the A α . With the angular momenta and the charge given in terms of the rotation parameters a and b , the mass parameter m , and the charge parameter q by [22] \nJ a = π [2 am + qb (1 + g 2 a 2 )] 4Ξ 2 a Ξ b , (12) J b = π [2 bm + qa (1 + g 2 b 2 )] 4Ξ 2 b Ξ a , Q = √ 3 π q 4Ξ a Ξ b , \nwhere Ξ a = 1 -a 2 g 2 and Ξ b = 1 -b 2 g 2 , a straightforward calculation then gives the result we listed above. The calculations for the other examples can be performed in a similar manner. \nFor the Kerr-AdS metrics in arbitrary dimensions [26, 27], it is necessary to separate the cases of even dimensions, D = 2 N +2, and odd dimensions, D = 2 N +1. In each case there are 2 N + 2 horizons and N angular momenta J i . When D = 2 N +1, the radial metric function is a function of r 2 , and the product over all horizons is equivalently expressible as the square of the product over just N +1 horizons corresponding to a single choice of square root for each r 2 α . Our results for the horizon area products in D -dimensional Kerr-AdS are \nD = 2 N +2 : 2 N +2 ∏ α =1 A α = g -4 N ( A D -2 ) 2 N ∏ i =1 (8 πJ i ) 2 , D = 2 N +1 : N ∏ α =0 A α = g -2 N +1 c N A D -2 ∏ i (8 πJ i ) , \nwhere c N = ( -1) ( N +1) / 2 , and A D -2 = 2 π ( D -1) / 2 / Γ[( D -1) / 2] is the volume of the unit ( D -2)-sphere. \n-The results presented above for black holes in gauged supergravities, and for Kerr-AdS black holes in pure gravity with a cosmological constant, admit straightforward limits to the ungauged, or zero cosmological constant, case. The radial functions in the metrics have a universal feature, as can be seen in (8) for the example of five-dimensional gauged supergravity, that the degree of the polynomial in r is reduced by 2 when the gauge coupling g is set to zero. In this limit, the locations of these two 'lost horizons' approach r = ± i g -1 , and the areas of the lost horizons in the cases of even and odd dimensional black holes are \nD = 2 N +2 : A lost = ( -1) N g -2 N A D -2 , (13) D = 2 N +1 : A lost = ∓ i ( -1) N g -2 N +1 A D -2 . \nIf these areas are factored out from our previous expressions for the horizon area products, and then g is sent to zero, we can obtain the analogous formulae for the corresponding ungauged supergravities, and for asymptotically-flat rotatating black holes in arbitrary dimensions. For the black holes in four and five dimensional supergravities, the limits yield expressions encompassed by those given above for the ungauged cases. For the black holes in gauged six and seven dimensional supergravities, it is interesting to note that the electric charge terms scale to zero in the ungauged limit. The resulting expressions are then just the D = 6 and D = 7 specialisations of the limiting forms for asymptotically-flat black holes in arbitrary dimensions, which we find to be \nD = 2 N +2 : 2 N ∏ α =1 A α = N ∏ i =1 (8 πJ i ) 2 , D = 2 N +1 : N ∏ α =1 A α = ∏ i (8 πJ i ) . (14) \nWe have also worked out the area product formulae for a general class of charged rotating black holes in D > 5 ungauged supergravities [28], and we find the same phenomenon as in the D = 6 and D = 7 ungauged limits described above. 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B658 , 64 (2007), arXiv:0709.0559 [hep-th].\n- [24] D.D.K. Chow, Charged rotating black holes in six dimensional gauged supergravity , Class. Quant. Grav. 27 , 065004 (2010), arXiv:0808.2728 [hep-th].\n- [25] D.D.K. Chow, Equal charge black holes and seven dimensional gauged supergravity , Class. Quant. Grav. 25 , 175010 (2008), arXiv:0711.1975 [hep-th].\n- [26] G.W. Gibbons, H. Lu, D.N. Page and C.N. Pope, The general Kerr-de Sitter metrics in all dimensions, J. Geom. Phys. 53 , 49 (2005), hep-th/0404008.\n- [27] G.W. Gibbons, H. Lu, D.N. Page and C.N. Pope, Rotating black holes in higher dimensions with a cosmological constant, Phys. Rev. Lett. 93 , 171102 (2004), hep-th/0409155.\n- [28] M. Cvetiˇc and D. Youm, Near-BPS-saturated rotating electrically charged black holes as string states , Nucl. Phys. B477 , 449 (1996), hep-th/9605051.'}

1999ApJ...520..650F

Merging White Dwarf/Black Hole Binaries and Gamma-Ray Bursts

1999-01-01

['accretion', 'accretion disks', 'black hole physics', 'gamma rays', 'hydrodynamics', 'stars white dwarfs', 'accretion', 'accretion disks', 'black hole physics', 'gamma rays', 'hydrodynamics', 'stars white dwarfs', 'astrophysics']

[]

The merger of compact binaries, especially black holes and neutron stars, is frequently invoked to explain gamma-ray bursts (GRBs). In this paper, we present three-dimensional hydrodynamical simulations of the relatively neglected mergers of white dwarfs and black holes. During the merger, the white dwarf is tidally disrupted and sheared into an accretion disk. Nuclear reactions are followed, and the energy release is negligible. Peak accretion rates are ~0.05 M<SUB>solar</SUB> s<SUP>-1</SUP> (less for lower mass white dwarfs) and last for approximately a minute. Many of the disk parameters can be explained by a simple analytic model that we derive and compare to our simulations. This model can be used to predict accretion rates for white dwarf and black hole (or neutron star) masses that are not simulated here. Although the mergers studied here create disks with larger radii and longer accretion times than those from the merger of double neutron stars, a larger fraction of the white dwarf's mass becomes part of the disk. Thus the merger of a white dwarf and a black hole could produce a long-duration GRB. The event rate of these mergers may be as high as 10<SUP>-6</SUP> yr<SUP>-1</SUP> per galaxy.

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https://arxiv.org/pdf/astro-ph/9808094.pdf

{'Merging White Dwarf/Black Hole Binaries and Gamma-Ray Bursts': 'Chris L. Fryer, S. E. Woosley Lick Observatory, University of California Observatories, Santa Cruz, CA 95064 [email protected] \nMarc Herant \nWashington University School of Medicine, Box 8107, 660 S. Euclid St. Louis, MO 63110 \nMelvyn B. Davies Cambridge Institute for Astronomy, Madingley Road, Cambridge CH3 0HA 1', 'ABSTRACT': "The merger of compact binaries, especially black holes and neutron stars, is frequently invoked to explain gamma-ray bursts (GRB's). In this paper, we present three dimensional hydrodynamical simulations of the relatively neglected mergers of white dwarfs and black holes. During the merger, the white dwarf is tidally disrupted and sheared into an accretion disk. Nuclear reactions are followed and the energy release is negligible. Peak accretion rates are ∼ 0.05 M /circledot s -1 (less for lower mass white dwarfs) lasting for approximately a minute. Many of the disk parameters can be explained by a simple analytic model which we derive and compare to our simulations. This model can be used to predict accretion rates for white dwarf and black hole (or neutron star) masses which are not simulated here. Although the mergers studied here create disks with larger radii, and longer accretion times than those from the merger of double \nneutron stars, a larger fraction of the white dwarf's mass becomes part of the disk. Thus the merger of a white dwarf and a black hole could produce a long duration GRB. The event rate of these mergers may be as high as 10 -6 yr -1 per galaxy. \nSubject headings: Gamma-Rays: Bursts, Black Hole Physics, Accretion: Accretion Disks, Stars: White Dwarfs", '1. Introduction': "As evidence supporting the extra-galactic nature of gamma-ray bursts (GRB's) mounts (Metzger et al. 1997; Frail et al. 1997), the class of models based on hyper-accreting black holes has become the favorite mechanism for driving these explosions (e.g., Popham, Woosley, & Fryer 1998; Eberl, Ruffert, & Janka 1998). Calculations show that a fraction of the gravitational potential energy released as the material in the disk ( M disk ≈ 0 . 01 -2M /circledot ) accretes into a small black hole ( M BH ≈ 3 -10M /circledot ) can be converted into a 'fireball' which produces the observed gamma-rays (Meszaros & Rees 1992). Such systems form in collapsars or hypernovae (Woosley 1993, 1996; Paczynski 1997) and in the merger of compact binaries consisting of: two neutron stars or a neutron star and a black hole (Paczy'nski 1991; Narayan, Paczy'nski, & Piran 1992); a helium star and a black hole (Fryer & Woosley 1998), and, the topic of this paper, a white dwarf and a black hole. \nSeveral mechanisms have been proposed to facilitate the conversion of potential energy into GRB explosion energy. Due to the high densities involved during the merging process, the potential energy may be emitted in the form of neutrinos, and the subsequent annihilation of these neutrinos could power a GRB (Goodman, Dar, & Nussinov 1987). Meszaros & Rees (1992) first pointed out the advantages of disk geometry for enhancing neutrino annihilation. However, unless the accretion rate is very high, over a few hundredths of a solar mass per second, Popham et al. (1998) have shown that neutrino emission is inefficent as the energy released in the disk is advected into the hole. Alternatively, and especially for lower accretion rates and lower disk viscosity, the amplification of magnetic field in the disk can tap either the black hole rotational energy or the potential energy of the accreting material to drive to drive a relativistic jet and create a GRB (Blandford & Znajek 1977; MacDonald et al. 1986; Paczynski 1991,1997; Woosley 1996; Meszaros & Rees 1997, Katz 1997). However, these magnetically based models are currently not sufficiently accurate to make quantitative predictions (Livio, Ogilvie, & Pringle 1998). \nOne scenario to form binaries consisting of a black hole and a white dwarf (WD/BH binaries) begins with main-sequence stellar systems having extreme mass ratios ( M primary > ∼ 30M /circledot , M secondary ≈ 1 -8M /circledot ). The formation scenario for white dwarf binaries with neutron star companions is similar, only with a primary star mass between ≈ 8 and ≈ 30M /circledot . As the massive star evolves off the main sequence, a common envelope phase may occur that ejects the primary's hydrogen envelope. Beyond some critical mass (roughly 30M /circledot ), massive stars are thought to form black holes, either by a failed explosion, or through significant fallback (Woosley & Weaver 1995, Fryer 1998). The more massive primary eventually collapses into a 3 -15M /circledot black hole and forms a binary consisting of a black hole and a main-sequence star. As the secondary expands and the orbit shrinks, a second mass transfer phase commences. This phase is observed in the closest of these systems as a low-mass X-ray binary (e.g. J0422+32-Nova Per, 2023+338 Nova Cyg). Roughly 20 binary systems with a black hole and a low-mass companion have been observed and many, as yet undetected systems may exist (see Tanaka & Shibazaki, 1996, for a review). These systems generally involve main-sequence secondaries that lose a significant amount of their mass while still burning hydrogen and do not evolve into massive ( > 0 . 5M /circledot ) white dwarfs. However, slightly wider binaries do not undergo mass transfer until after the secondary has evolved off the main sequence and produce the systems which we model in this paper. Unfortunately, the formation rate of the observed low mass X-ray binaries is difficult to determine, and estimating the number of wider systems from observations is impossible. It is these more massive companions that dominate the merging WD/BH binaries and hence, the typical merging system consists of a black hole and massive ( ∼ > 0 . 9M /circledot ) white dwarf. \nAn alternative evolution scenario for WD/BH binaries begins with less extreme mass ratios in a system where the primary initially forms a neutron star in a supernova explosion. As the secondary expands off the main-sequence and a common envelope phase ensues, the neutron star accretes rapidly via neutrino emission (allowing the accretion rate to greatly exceed the photon Eddington limit) and eventually collapses to a low mass black hole (see Fryer, Benz, & Herant 1996, Bethe & Brown 1998). Because this common envelope phase happens quickly, it is unlikely that systems evolving through this scenario will be observed, and no merger rate can be predicted from the observations. \nUnfortunately, uncertainties in black hole formation and binary evolution also make it difficult to make any firm predictions using population synthesis studies, but the merger rate is likely to lie in the range from 10 -9 -10 -6 yr -1 per Milky-Way like galaxy (Fryer, Woosley, & Hartmann 1999). The large uncertainties in the merger rate are primarily due to uncertainties in the critical mass beyond which massive progenitor stars collapse to black holes and in the kick imparted to black holes. \nBlack holes and white dwarfs can also merge through collisions in dense star regions such as in galactic centers and globular clusters. Sigurdsson & Rees (1997) predict a neutron star/white dwarf merger rate of ∼ 10 -7 yr -1 per galaxy. Low mass black holes will merge with white dwarfs roughly at the same rate (or an order of magnitude less) depending upon the black hole formation rate. This rate is comparable to the merger rate predicted by Quinlan & Shapiro (1987, 1989, 1990). Depending on the beaming fraction, this rate is certainly sufficient to give the observed GRB statistics (Wijers et al. 1998), provided, of course, that the merger produces a GRB. \nIn this paper, we model the merger of black holes and white dwarfs on the computer using a three-dimensional hydrodynamics code based on the Smooth Particle Hydrodynamics (SPH) method (Benz 1990). We follow the merger from the initial Roche-lobe overflow through the complete disruption of the white dwarf into a disk. Roche-lobe overflow for compact objects differs from that for giant stars in several ways: (b) due to the degeneracy of the compact object, its radius increases as it loses mass, and (c) the orbital angular momentum is far from conserved ( ≡ the mass transfer is not 'conservative'). We discuss this physics, applicable to most compact object mergers, in § 2. A description of the code along with a presentation of the simulations, including a comparison to the analytic estimates of § 2, is given in § 3. We conclude with a discussion of the accretion disks formed in these mergers and their suitability as GRB models.", '2. Accretion Disk Formation': 'Whether or not the accretion disks formed in WD/BH mergers produce the necessary GRB explosion energies is determined by the accretion rate and the angular momentum of those disks, which in turn, depends upon the size and mass of the accretion disk. It is important to know, then, how quickly, and at what radius, the white dwarf is torn up by the gravitational potential of the black hole and transformed into an accretion disk which might fuel a GRB. One might naively assume that, since the white dwarf is less massive than the black hole ( M BH ∼ 3 -10M /circledot , M WD ∼ 0 . 5 -1 . 3M /circledot ), that stable accretion will occur and the white dwarf will slowly accrete onto the black hole over many orbital periods. However, as we shall discuss in this section, several aspects of physics conspire to destabilize this mass transfer, leading to the rapid transformation of most of the white dwarf into an accretion disk. This was seen in the merger of double white dwarf binaries by Davies, Benz & Hills (1991).', '2.1. Gravitational Radiation': 'Gravitational radiation plays an important role in the merging of double neutron star or black hole/neutron star systems. For these systems, the emission of gravitational waves tightens the binaries on a timescale comparable to those of the hydrodynamical evolution. However, white dwarfs fill their Roche lobes at much wider separations where the gravitational wave merging timescale is 1-100 yr (depending upon the white dwarf and black hole masses). Although gravitational radiation does cause the orbit to tighten sufficiently to drive the white dwarf to fill its Roche lobe in the first place, once Roche-lobe overflow occurs, the mass transfer rate from the white dwarf onto a disk around the black hole is determined by the transfer of angular momentum and the white dwarf mass-radius relationship which drive unstable mass transfer on much shorter timescales ( ∼ minutes).', '2.2. Effects of Degeneracy': 'One such destabilizing effect is the inverse relationship of the radii of degenerate objects (neutron star, white dwarf) with respect to mass. For a Γ = 5 / 3 polytrope approximation of a white dwarf equation of state, this relationship is (Nauenberg 1972): \nR WD ≈ 10 4 ( M WD 0 . 7M /circledot ) -1 / 3 [ 1 -( M WD M CH ) 4 / 3 ] 1 / 2 ( µ e 2 ) -5 / 3 km (1) \nwhere R WD , M WD , and µ e are the radius, mass, and mean molecular weight per electron of the white dwarf and M CH ≈ 1 . 4 is the Chandrasekhar mass. Our simulated white dwarfs differ slightly from this simple relation due to deviations from a simple Γ = 5 / 3 polytrope (Fig. 1). The radius of the white dwarf and the masses of the white dwarf and black hole determine the orbital separation at which Roche-lobe overflow commences (Eggleton 1983): \nA 0 = R WD 0 . 6 q 2 / 3 +ln(1 + q 1 / 3 ) 0 . 49 q 2 / 3 (2) \nwhere q = M WD /M BH is the mass ratio (Fig. 1). Stable mass transfer would require that as the white dwarf loses mass, its orbit widens to place it just at this critical Roche lobe separation. If the white dwarf binary instead remains at a constant orbital separation, the accretion will quickly become unstable as the white dwarf itself expands. This effect is important for all merging systems involving a compact secondary. \nand \nM BH+disk = β ( M 0 WD -M WD ) + M 0 BH , (7) \nand where superscript 0 denotes pre-mass transfer phase values. \nIn Roche-lobe overflow onto compact objects (neutron stars or black holes), much of the angular momentum is placed into a disk around that compact object. For the merger of binaries consisting of a black hole and a neutron star, roughly half of the orbital angular', '2.3. Non-Conservative Mass Transfer': "The orbital separation need not remain constant. In conservative mass transfer, when an object accretes onto a more massive companion, orbital angular momentum conservation requires that the orbit expands. However, some fraction of the material can be lost from the system and carry away angular momentum. Hence, although the total angular momentum is conserved, the orbital angular momentum of the binary system decreases, and the orbital separation may actually decrease during mass transfer. This 'non-conservative' mass-transfer can be parameterized and solved (see Podsiadlowski, Joss, & Hsu 1992 and references therein). In addition, for mergers with black holes or neutron stars, some of the orbital angular momentum is converted to angular momentum of the accretion disk or to spin angular momentum of the black hole. The change of orbital angular momentum ( δJ orbit ) of the binary is then given by: \nδJ orbit = [ j ejecta (1 -β ) + j disk β ] δM WD 2 πA 2 P (3) \nwhere β is the fraction of mass lost by the white dwarf that is accreted by the black hole (or becomes part of the black hole's accretion disk), j ejecta and j disk are the specific angular momenta (in the rest frame of the black hole) of the ejected material and the material which is either accreted onto the black hole or becomes part of the accretion disk. A and P are the orbital separation and period of the binary system. Following the procedure of Podsiadlowski, Joss, & Hsu (1992), we derive the orbital separation ( A ) of the binary during mass transfer including the loss of angular momentum to the accretion disk: \nA A 0 = M WD + M BH+disk M 0 WD + M 0 BH ( M WD M 0 WD ) C 1 ( M BH M 0 BH ) C 2 (4) \nwhere the values of the constants differ only slightly from those derived by Podsiadlowski, Joss, & Hsu (1992): \nC 1 ≡ 2 j ejecta (1 -β ) -2 + 2 j disk β (5) \nC 2 ≡ -2 j ejecta β (1 -β ) -2 -2 j disk (6) \nmomentum is fed directly into spinning up the black hole (Eberl, Ruffert, & Janka 1998). For wider Roche-lobe overflow systems (e.g. white dwarf mergers) much of the orbital angular momentum is converted into disk angular momentum. (see Papaloizou & Lin 1995 for a review). In systems where the mass transfer is stable and the primary, disk and secondary coexist for many orbital periods, the angular momentum of the disk can be transferred back to the orbital angular momentum of the binary. However, for the runaway accretion caused by the expansion of the white dwarf, the white dwarf is disrupted quickly (2-3 orbital periods) and the torques between the disk and the binary stars are unable convert the disk angular momentum back to that of the orbit before the disruption of the white dwarf. \nFigure 2 shows the orbital evolution for 0 . 7 and 1 . 1M /circledot white dwarfs merging with a 3M /circledot black hole for a range of values of j disk (in terms of the white dwarf specific angular momentum ≡ j WD ) and assuming no mass is ejected from the system ( β = 1). The critical separation for Roche lobe overflow, shown in Figure 2, marks the dividing line between stable and unstable mass accretion. If the orbit widens faster than the white dwarf expands, the accretion rate onto the black hole is limited to the gravitational wave timescale (1-100 yr) and the merger occurs on these timescales. However, if j disk > 0 . 3 , 0 . 1 j WD for 0 . 7 , 1 . 1M /circledot white dwarfs respectively, the mass transfer is unstable. If the accreting material transports all of its angular momentum to the accretion disk, then the angular momentum of the disk is roughly the angular momentum of the material at the Lagrange point. Assuming tidal locking, for the binary systems we model, this angular momentum is roughly: j disk ≈ ( A -R WD ) 2 /A 2 j WD ∼ 0 . 5 j WD where A is the orbital separation, R WD is the white dwarf radius, and j WD is the specific angular momentum of the white dwarf in the rest frame of the black hole ( ∼ 10 18 cm 2 s -1 ). For these high values of j disk , unstable mass transfer is inevitable, and we expect the white dwarf to be tidally disrupted rapidly. But to accurately calculate the mass transfer, and ultimately, the mass accretion rate onto the black hole, we must resort to numerical simulations.", '3. Simulations': 'For our simulations, we use a three dimensional SPH code (Davies, Benz, & Hills 1991) with 6000-16000 particles. We employ the equation of state developed by Lattimer & Swesty (1991) for densities above 10 11 g cm -3 and, for low densities, the equation of state by Blinnikov, Dunina-Barkovskaya & Nadyozhin (1996). We include a nuclear burning network for temperatures above 4 × 10 8 K (Woosley 1986), though we find that burning is not important, except well within the accretion disk. As we are concerned with the tidal \ndisruption of the white dwarf and not the accretion of matter in the disk formed from this disruption, we model the black hole (or neutron star) as a point mass and remove particles that fall within 2 -3 × 10 8 cm of the black hole, well before general relativistic effects are important. Similarly, since we are not following the evolution of the accretion disk, the the numerically determined artificial viscosity should not impact our results. However, as a check, we have varied the artificial viscosity by an order of magnitude and find it does not effect the radius at which the white dwarf is disrupted or the initial structure of the accretion disk formed by this disruption. On the other hand, the true physical viscosity does affect the rate at which material is accreted onto the black hole, and hence the pair fireball energy, which we discuss in § 4. \nWith this code, we modeled the tidal disruption of 4 binary systems consisting of a white dwarf (with masses of 0.7, 1.1 M /circledot ) and a black hole (with masses of 3, 10 M /circledot ) and one system consisting of a 1 . 1 M /circledot white dwarf and a 1 . 4 M /circledot neutron star. We followed the evolution from the initial Roche-lobe overflow through the destruction of the white dwarf and the formation of an accretion disk (Figs. 3, 4). We assume that gravitational radiation has brought the white dwarf close enough to its black hole companion to overfill its Roche-lobe and transfer mass onto the black hole. We estimate this critical separation using eq. (2). By increasing the separation by 20-30%, we see that no mass transfer takes place (Fig. 3), assuring that our initial separation is within 30% of the actual Roche-lobe overflow separation. We will come back to this error estimate in our discussion of the accretion disk properties at the end of this section. \nBefore we discuss the disk properties, let us first validate our physical picture of the tidal disruption process. From § 2, we expect the specific angular momentum of the disk (in the rest frame of the black hole) to be initially ∼ 0 . 5 j WD and then rise as more of the white dwarf it disrupted. The angular momentum of the simulated disk is ∼ 0 . 6 j WD and then increases to 1 . 0 j WD as the white dwarf is disrupted (Fig. 5). Physically, this means that as the white dwarf transfers mass onto a disk around the black hole, the angular momentum of matter at the Lagrange point is first added to the disk. When the white dwarf is finally disrupted, nearly all of its angular momentum is immediately put into the disk, and the average disk angular momentum equals the initial white dwarf angular momentum. The disk must then shed this angular momentum before this material can accrete onto the black hole (see § 4). \nBecause much of the orbital angular momentum is converted into disk angular momentum, the orbital separation does not expand as one might expect in conservative mass-transfer, and the white dwarf is quickly disrupted by tidal forces. As the black hole accretes mass, the orbital separation of the white dwarf/black hole binary is described \nby equation (4). Using equation (4) and assuming no mass is ejected from the system (very little mass is ejected in our simulations, see Figs. 3, 4), we can plot data from the simulations along with the derived separations for a range j disk values (Fig. 6). The remarkable agreement of the best fit of j disk using equation (4) and the actual j disk values from Figure 5 suggests that we have indeed found the relevant physics, and that the orbital separation can be estimated by our simple mass-transfer model. \nIn these simulations, the mass transfer from the white dwarf becomes increasingly unstable as more of the white dwarf expands beyond its Roche radius and accretes onto a disk around the black hole. Our simulations show that after losing ∼ 0 . 2M /circledot , the transfer rate becomes so great that the white dwarf is disrupted. This occurs rapidly (in an orbit time), dumping the remains of the white dwarf into an accretion disk around the black hole. This critical mass loss after which the accretion runs away is the one parameter not determined by our analytic model. Using our simulations to constrain this parameter, we are able to describe both the angular momentum and the mass growth rate of the disk from the tidal disruption of the white dwarf. \nThe specific angular momentum of the disk is given by \nj ≈ √ GA ( M BH + M disruption WD ) (8) \nwhere G is the gravitational constant, M BH is the black hole mass, and M disruption WD ≈ 0 . 5 , 0 . 8 M /circledot (for initial white dwarf masses of 0.7, 1.1 M /circledot respectively) is the white dwarf mass at the time of disruption taken from our simulations. The orbital separation ( A ) can be derived from equation (4). The mass transfer rate of the white dwarf onto the disk is roughly \n˙ M = M disruption WD /T orbit (9) \nwhere T orbit is the orbital timescale for the binary system after the white dwarf has lost 0 . 2M /circledot . These results are summarized in Table 1 and the mass-transfer rate can be compared to the simulated rates shown in Figure 7. Note that mass-transfer rates derived from analytical estimates agree within a factor of 2 with those obtained from our simulations. The actual accretion rate onto the black hole is not likely to exceed this mass-transfer rate. \nMany of these results rely upon our knowing the exact separation where Roche lobe overflow commences. As we have already mentioned, by increasing the separation by 30%, we find no accretion occurs over many orbits, which suggests that the error in the initial separation is less than 30%. If the errors in the initial orbital separation are less than 30%, our maximum mass-transfer rates are accurate to ∼ < 30% and the disk angular momenta are accurate to ∼ < 15%. Even changing the initial separation by a factor of 2 only results in a \nfactor of 3 change in the maximum accretion rate and a change in the angular momenta by less than 40%.', '4. Accretion Disk Powered Gamma-Ray Bursts': "With these results, we can now address the viability of WD/BH mergers as a GRB model. The mass transfer rate of the white dwarf onto the black hole accretion disk should, in a steady state, balance the accretion rate into the black hole. The actual accretion rate is determined by the efficiency at which the angular momentum is removed from the disk 2 (Popham, Woosley,& Fryer 1998): \n˙ M acc ≈ 0 . 37 αM disk M 1 / 2 BH r -3 / 2 disk , 9 M /circledot s -1 , (10) \nwhere α is the standard accretion disk parameter, M disk and M BH are, respectively, the mass of the disk and the black hole in M /circledot , and r disk , 9 is the outer disk radius in 10 9 cm. Figure 8 shows the mass of the disk as a function of radius for our two M BH = 3M /circledot models, from which, given a value of α , we can determine the accretion rate onto the black hole. For values of α < 0 . 5, the accretion rate is limited by the disk accretion and not the mass-transfer rate. Using equation (10), we estimate the effective disk viscosity ( α ) from accretion rate onto the black hole of our hydrodynamical simulations to be ∼ 0 . 1. \nThe energy from neutrino annihilation can be estimated by integrating the following approximate fit to the pair luminosity results of Popham, Woosley, & Fryer (1998): \nlog L ν, ¯ ν (erg s -1 ) ≈ 43 . 6 + 4 . 89 log ( ˙ M 0 . 01M /circledot s -1 ) +3 . 4 a (11) \nwhere a ≡ J BH c/GM 2 BH is the spin parameter. This fit is reasonably accurate for accretion rates between 0 . 01 and 0 . 1M /circledot s -1 . Table 2 gives the maximum energies for each of our simulations. In the optimistic situation where α > 0 . 5 and the disk accretion rate equals the mass-transfer rate from the white dwarf into a black hole accretion disk, the disruption of a white dwarf around a black hole cannot explain the most energetic gamma-ray bursts without requiring that the mechanism produce strongly beamed jets. Indeed, with isotropic energy requirements as high as 3 × 10 53 erg (Kulkarni et al. 1998), the beaming must be extremely high (the burst must be constrained to 0.1% of the sky, that is, the beaming \nfactor > 1000). However, a wide range of GRB energies may exist, and WD/BH mergers may only constitute a subset of the observations. If α = 0 . 1, the accretion rate drops by about a factor of 5. For the most optimistic mergers of a 1 . 1M /circledot white dwarf with a black hole, this lowers the accretion rate on the black hole to 0 . 01 -0 . 02M /circledot s -1 and increases the accretion time, causing a net decrease in the total energy produced by neutrino annihilation of roughly 1-2 orders of magnitude. With beaming factors of ∼ 100, WD/BH mergers could still explain bursts with inferred isotropic energies between 10 48 -10 51 erg. \nAlternatively, and perhaps more likely for the low-mass accretion rates derived here, the GRB can be powered by the magnetic fields of the disk, which become stretched and amplified as the material accretes. These magnetic fields then extract the rotational energy of the black hole (Blandford & Znajek 1977; MacDonald et al. 1986; Paczynski 1991,1997; Woosley 1993; Katz 1994, 1997; Hartmann & Woosley 1995; Thompson 1996; Meszaros & Rees 1997; Popham et al. 1998). Very roughly, using Blandford-Znajek for example, \nL rot = 10 50 ( jc GM BH ) 2 ( M BH 3M /circledot ) 2 ( B 10 15 Gauss ) 2 erg s -1 (12) \nwhere j is the specific angular momentum of the black hole and B is the magnetic field strength in the disk. Table 2 lists the total energy that an initially non-rotating black hole would emit over its accretion timescale assuming the magnetic field energy is 10% of the equipartition energy, or 0 . 1 ρv 2 . These high magnetic fields are reasonable if the disk viscosity depends upon the magnetic field strength. In this case, the viscosity is initially ∼ 0 allowing the disk to continue winding the magnetic field until a sufficiently strong equipartion field is generated, thereby increasing the viscosity and allowing the disk to accrete. \nA successful GRB explosion must also avoid excessive baryonic contamination. The disruption of the white dwarf forms a hot thick disk around the black hole (Figure 9), with some of the matter along the angular momentum axis above the black hole (Table 2). The explosion will force its way along this polar region, sweeping up this material (and possibly pushing some aside). Assuming all of the material is swept along with the burst, we can estimate a lower limit for the Lorentz factors (Table 2). Beaming factors of at least 100 are required to achieve the high Lorentz factors needed to power a gamma-ray burst. Even assuming that beaming factors of 100 do occur, low mass white dwarfs do not produce enough energy (or high enough Lorentz factors) to power a gamma-ray burst. Thus, there is some critical white dwarf mass (between 0 . 7 -1 . 1M /circledot depending upon beaming) below which no visible GRB will form. Because of the strong dependence of the GRB luminosity on the Lorentz factor, the transition from observed gamma-ray burst to non-detectable explosion is sharp. Those explosions that do not achieve the high Lorentz factors will only \nbe observable in our own Galaxy, and, given the low event rate, will not be detected. We reiterate, however, that most of the merging white dwarfs will be massive (Fryer, Woosley, & Hartmann 1999) and a large fraction of the merging systems may become GRBs. \nThe merger of a black hole and a massive white dwarf can produce the energies (10 48 -10 51 erg) and the high Lorentz factors to explain the long duration GRBs if the bursts themselves are highly beamed (beaming factors > 100). Assuming the GRB rate for isotropic bursts is 10 -7 yr -1 per galaxy of roughly the Milky Way's size (Wijers et al. 1998), the merger rate of massive white dwarfs and black holes with beaming factors > 10 must be ∼ > 10 -6 yr -1 per galaxy, within the uncertainties of the predicted rates (see § 1). \nFrom these results, one might conclude that mergers of white dwarfs and black holes are not likely to play a major role in the production of gamma-ray bursts. However, our estimates of the energy released via magnetic fields are very uncertain. Some magnetic field mechanisms may convert a large fraction of the potential energy of the accreting material into burst energy. If a magnetic field mechanism can be constructed which converts 10% of the potential energy into burst energy, WD/BH mergers would have energies in excess of 10 52 erg. With beaming into 10gamma-ray bursts. \nThe merger of a neutron star and a white dwarf is a different story. At these accretion rates, Popham, Woosley, & Fryer (1998) found that much of the energy is advected into the black hole. The hard surface of the neutron star acts as a plug, stopping up this accretion. Unless the neutron star mass quickly exceeds the upper neutron star mass limit, causing it to collapse to a black hole and removing this plug, the accreting material will flow around the neutron star, building up a spherically symmetric atmosphere. Any explosion from the surface will be baryon rich with velocities much less than the speed of light (Fryer, Benz, & Herant 1994). These outbursts will be too dim to observe beyond our Galaxy, and are too rare to observe within our Galaxy. \nThis research has been supported by NASA (NAG5-2843 and MIT SC A292701), and the NSF (AST-97-31569). We would like to thank Bob Popham, Thomas Janka, and William Lee for many useful corrections and comments. 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WD/BH Mergers \n| Disk Parameters | M WD = 0 . 7 M BH = 10 | M WD = 0 . 7 M BH = 3 | M WD = 1 . 1 M BH = 10 | M WD = 1 . 1 M BH = 3 | M WD = 1 . 1 M NS = 1 . 4 |\n|--------------------------------------|--------------------------|-------------------------|--------------------------|-------------------------|-----------------------------|\n| A (10 9 cm) | 4.98 | 3.1 | 2.37 | 1.49 | 1.27 |\n| T orbit (s) | 58.4 | 48.1 | 18.9 | 15.4 | 12.1 |\n| j disk (10 18 cm 2 s - 1 ) | 2.67 | 1.24 | 1.87 | 0.903 | 0.651 |\n| ˙ M (M /circledot s - 1 ) | 0.00856 | 0.01 | 0.0477 | 0.0584 | 0.074 |\n| A sim (10 9 cm) a | 5 | 3.2 | 2.5 | 1.6 | 1.2 |\n| M disr WD (M /circledot ) b | 0.5 | 0.5 | 0.8 | 0.8 | 0.8 |\n| ˙ M sim peak (M /circledot s - 1 ) c | 0.012 | 0.008 | 0.063 | 0.079 | 0.075 | \na The last three rows of data come directly from the simulations, the other rows are derived from the equations in § 2,3. \nb This is the simulated white dwarf mass at the onset of the white dwarf disruption. For our derivations, we assume M disr WD ≈ 0 . 5 , 0 . 8M /circledot for initial white dwarf masses of 0.7,1.1 M /circledot respectively. \nc The peak mass-transfer rate from Figure 5. \nTable 2. Powering a GRB \n| Observables | M WD = 0 . 7 M BH = 10 | M WD = 0 . 7 M BH = 3 | M WD = 1 . 1 M BH = 10 | M WD = 1 . 1 M BH = 3 |\n|----------------------------------------------|--------------------------|-------------------------|--------------------------|-------------------------|\n| a ≡ jc/GM BH E max ν, ¯ ν (10 49 Erg) a 49 b | 0.21 | 0.54 | 0.31 | 0.69 |\n| a ≡ jc/GM BH E max ν, ¯ ν (10 49 Erg) a 49 b | 0.001 | 0.003 | 3 | 50 |\n| E rot (10 erg) M axis (M /circledot ) c | ∼ 0 . 1 | ∼ 1 | ∼ 1 | ∼ 4 |\n| | 10 - 5 ( < 10 - 5 ) | 10 - 4 ( < 10 - 5 ) | 10 - 3 ( < 10 - 5 ) | 10 - 3 ( < 10 - 5 ) |\n| ( E ν, ¯ ν /M axis c 2 ) d | > 5 × 10 - 4 | > 1 . 5 × 10 - 3 | > 2 . 5 | > 25 | \nb We use the magnetic field estimate of Popham, Woosley, & Fryer (1998) which assumes that the magnetic field energy is 10% of the equipartition energy, or 0 . 1 ρv 2 . \nc This corresponds to the mass from the white dwarf which lies along the accretion disk axis and is likely to be swept up in the explosion for a beaming factor of 10(100). \nd E ν, ¯ ν /M axis c 2 ≈ γ when E ν, ¯ ν /M axis c 2 /greatermuch 1. We assume beaming factors of 100. \nFig. 1.- White dwarf radii and separations at which Roche-lobe overflow occurs vs. white dwarf mass. The dotted line shows the white dwarf radius using eq. (1) in comparison to our simulated radii. The orbital separations are given for two black hole masses (3 , 10 M /circledot ) and demarkate the limit within which the white dwarf overfills its Roche lobe. \n<!-- image --> \n<!-- image --> \nFig. 2.- Evolutionary paths of the binary separation as a white dwarf accretes onto a 3 M /circledot black hole for a range of j disk (fraction of specific angular momentum of the the white dwarf) values. The critical Roche-lobe separation is plotted for comparison. If the separation remains above this critical separation, stable accretion occurs. Otherwise, the accretion is unstable and the white dwarf quickly accretes onto the black hole. \n<!-- image --> \nFig. 3.- Time evolution of a M WD = 0 . 7M /circledot , M BH = 3 . 0M /circledot simulation. Here we show slices about the z-axis from -10000-10000 km. Note that very little accretion occurs for over 600s and then, very rapidly, the white dwarf is torn apart. However, for orbital separations just 20% further out (lower right panel), there is no mass transfer. This suggests our initial conditions are roughly accurate. \n<!-- image --> \nFig. 4.- Time evolution of a M WD = 1 . 1M /circledot , M BH = 3 . 0M /circledot simulation. Here we show slices about the z-axis from -10000-10000 km. Note that very little accretion occurs for over 70s and then, very rapidly, the white dwarf is torn apart. In the last slide (T=110s), nearly all traces of the white dwarf have been removed and half of the white dwarf mass has been accreted onto the black hole. \n<!-- image --> \nFig. 5.- The specific angular momentum ( j disk ) of the disk (in units of the white dwarf specific angular momentum) as a function of time (in orbital time). The orbital times of the systems are given in Table 1. As the white dwarf is disrupted, its entire angular momentum is put in the disk. \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFig. 6.- Orbital separation as a function of white dwarf mass. The circles are the data from the simulations and the solid lines denote the predicted separations from equation (4). The dashed line is the separation at which the white dwarf overfills its Roche radius. Note that after losing ∼ 0 . 2M /circledot , the orbital separation evolution remains constant. This occurs as the white dwarf is torn apart by tidal forces. \n<!-- image --> \nFig. 7.- Mass Transfer Rate vs. time for the 4 white dwarf/black hole mergers. The more massive white dwarfs merge more quickly and have correspondingly higher mass-transfer rates. This rate gives a maximum for the disk accretion rate onto the black hole. \n<!-- image --> \nFig. 8.- Enclosed Mass vs. radius just after white dwarf disruption. The accretion rate of this material onto the disk is roughly given by: ˙ M acc ≈ α Ω M disk . \n<!-- image --> \nFig. 9.- Distribution of the white dwarf material along the rotation axis for our two M BH = 3M /circledot simulations. Any GRB explosion must plow through this material (and possibly sweep it up) as it expands. If the explosion is beamed, the total swept up mass is very small (see Table 2) and will not effect the GRB. \n<!-- image -->"}

2004PhRvD..70l4006B

Quasinormal modes and classical wave propagation in analogue black holes

2004-01-01

['-', '-', '-', '-', '-', 'astrophysics', '-', '-', '-']

[]

Many properties of black holes can be studied using acoustic analogues in the laboratory through the propagation of sound waves. We investigate in detail sound wave propagation in a rotating acoustic (2+1)-dimensional black hole, which corresponds to the “draining bathtub” fluid flow. We compute the quasinormal mode frequencies of this system and discuss late-time power-law tails. Because of the presence of an ergoregion, waves in a rotating acoustic black hole can be superradiantly amplified. We also compute superradiant reflection coefficients and instability time scales for the acoustic black hole bomb, the equivalent of the Press-Teukolsky black hole bomb. Finally we discuss quasinormal modes and late-time tails in a nonrotating canonical acoustic black hole, corresponding to an incompressible, spherically symmetric (3+1)-dimensional fluid flow.

[]

https://arxiv.org/pdf/gr-qc/0408099.pdf

{'Emanuele Berti ∗': 'McDonnell Center for the Space Sciences, Department of Physics, Washington University, St. Louis, Missouri 63130, USA', 'Vitor Cardoso †': "Centro de F'ısica Computacional, Universidade de Coimbra, P-3004-516 Coimbra, Portugal \nJos'e P. S. Lemos ‡ \nCentro Multidisciplinar de Astrof'ısica - CENTRA, Departamento de F'ısica, Instituto Superior T'ecnico, Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal (Dated: August 20, 2018) \nMany properties of black holes can be studied using acoustic analogues in the laboratory through the propagation of sound waves. We investigate in detail sound wave propagation in a rotating acoustic (2 + 1)-dimensional black hole, which corresponds to the 'draining bathtub' fluid flow. We compute the quasinormal mode frequencies of this system and discuss late-time power-law tails. Due to the presence of an ergoregion, waves in a rotating acoustic black hole can be superradiantly amplified. We compute superradiant reflection coefficients and instability timescales for the acoustic black hole bomb, the equivalent of the Press-Teukolsky black hole bomb. Finally we discuss quasinormal modes and late-time tails in a non-rotating canonical acoustic black hole, corresponding to an incompressible, spherically symmetric (3 + 1)-dimensional fluid flow. \nPACS numbers: 04.70.-s, 43.20.+g, 04.80.Cc", 'I. INTRODUCTION': "Black holes are one of the most fascinating predictions of general relativity. Being a solution of the classical Einstein field equations in vacuum, they are the simplest objects which can be built out of spacetime itself. Hawking [1] showed that when quantum effects are taken into account, black holes are not really black: they slowly evaporate by emitting an almost thermal radiation. Hawking's prediction has been tested time and again in very different ways. It is now clear that the appearance of Hawking radiation does not depend on the dynamics of the Einstein equations, but only on their kinematical structure, and more specifically on the existence of an apparent horizon [2, 3]. The discovery of Hawking radiation uncovered a number of fundamental questions: among them the information puzzle, the issue of the black hole final state, and so on. For these reasons an experimental verification of the Hawking effect would be of the utmost importance. Unfortunately astrophysical black holes, having a temperature much smaller than the temperature of the cosmic microwave background, accrete matter more efficiently than they evaporate. An alternative could be provided by the existence of large extra dimensions: if gravity is effective at the TeV scale, black holes could be produced in particle accelerators [4]. However, even if these mini-black holes are actually produced in colliders, it is unlikely that they will yield any conclusive evidence of the existence of Hawking radiation. \nProspects for detecting Hawking radiation changed when Unruh [5] realized that the basic ingredients of Hawking radiation can experimentally be reproduced in the laboratory. Since Hawking radiation crucially depends on the existence of an apparent horizon, the experimental setup should display the essential features that define apparent horizons in general relativity. Unruh considered precisely such a system: a fluid moving with a space-dependent velocity, for example water flowing through a nozzle. Where the fluid velocity exceeds the sound velocity we get the equivalent of an apparent horizon for sound waves. This is the acoustic analogue of a black hole, or a dumb hole . Following on Unruh's dumb hole proposal many different kinds of analogue black holes have been devised, based on condensed matter physics, slow light etcetera [6, 7]. At present the Hawking temperatures associated to these analogues are too low to be detectable, but the situation is likely to change in the near future [8, 9]. \nIn order to detect Hawking radiation, a full understanding of the classical physics of analogue black holes is \nnecessary. Not only must one control what happens in the experimental situation, but the understanding of classical phenomena may bring clues on how to favor the probabilities to detect Hawking radiation. It is also worth stressing that some purely classical phenomena shed light on quantum aspects of (analogue and general-relativistic) black hole physics. For example, positive and negative norm mixing at the horizon leads to non-trivial Bogoliubov coefficients in the calculations of Hawking radiation [10]; superradiant instabilities of the Kerr metric are related to the quantum process of Schwinger pair production [11]; and more speculatively (classical) highly damped black hole oscillations could be related to area quantization [12]. \nIn this paper we carry out a comprehensive study of wave propagation in analogue black holes. We consider in detail two acoustic black hole metrics: the 'draining bathtub' model for a (2+1)-dimensional fluid flow and the 'canonical' metric for a non-rotating, spherically symmetric (3+1)-dimensional acoustic black hole [7]. For each acoustic metric we compute the characteristic oscillation frequencies using a WKB approach. These complex characteristic frequencies, called quasinormal modes (QNMs), play a very important role in the classical physics of black holes. They govern the late time behavior of waves propagating outside the black hole: any perturbation of the black hole, after an initial transient, will damp exponentially in the so-called 'ringdown phase'. The frequencies and damping times of this ringdown signal depend only on the black hole parameters, such as the mass, charge and angular momentum [13], and can therefore be used to estimate black hole parameters from observational data [14]. Therefore, in cases where a 'formal' definition of (say) the mass and angular momentum of acoustic black holes is missing, quasinormal (QN) frequencies may be used to define these quantities operationally . According to recent speculations, highly damped QNMs may yield some information on quantum properties of a black hole. In particular, it has been conjectured that highly damped QN frequencies could be linked to black hole area quantization [12]. Motivated by these conjectures we also study highly damped QNMs of acoustic black holes. We find that for a (2+1)-dimensional acoustic black hole there are no asymptotic QN frequencies, whereas for the (3 + 1)-dimensional canonical black hole QN frequencies are asymptotically given by 4 πω = log [(3 -√ 5) / 2] -i(2 n +1) π . This result does not seem to support any of the recent conjectures, but perhaps this is no surprise, since Hod's argument relies heavily on black hole thermodynamics. Even the very formulation of the laws of black hole thermodynamics for analogue black holes is a non-trivial matter [2]. \nAfter the exponential decay characteristic of the ringdown phase, black hole perturbations decay with a power-law tail [15] due to backscattering off the background curvature. Here we compute the late-time tails of wave propagation in the draining bathtub and canonical acoustic black hole metrics. We show that for the (2+1)-dimensional fluid flow the field falloff at very late times is of the form Ψ ∼ t -(2 m +1) , where m is the angular quantum number. This time exponent is characteristic of any (2 + 1)-dimensional flow, and not just of a black hole. For the (3 + 1)-dimensional canonical acoustic geometry, the field falloff is of the form Ψ t -(2 l +6) . \nThe rotating draining bathtub metric, possessing an ergoregion, can display the phenomenon of superradiance [16, 17, 18]. We compute reflection coefficients for this superradiant scattering by numerical integration of the relevant equations. Enclosing the acoustic black hole by a reflecting mirror we can exploit superradiance to destabilize the system, making an initial perturbation grow exponentially with time: we have an 'acoustic black hole bomb' [19, 20, 21]. We compute analytically and numerically the frequencies and growing timescales for this instability. An interesting feature of acoustic geometries is that the acoustic black hole spin can be varied independently of the black hole mass . Therefore, at variance with the Kerr metric, the spin can be made (at least in principle) very large, and rotational superradiance in acoustic black holes can be very efficient. It should not be difficult to set up an experimental apparatus to observe an acoustic black hole bomb in the lab. A concrete example of an experimental setup in which this idea can be realized with moderate experimental effort has been fully worked out by Schutzhold and Unruh [16], and is based on the study of gravity waves in shallow water. \n∼ \nThe paper is organized as follows. In Section II we discuss the (2 + 1)-dimensional draining bathtub metric. We first introduce the formalism describing sound propagation in this acoustic metric, and shortly describe the possible experimental setup (gravity waves in a shallow basin). Then we compute QNMs, discuss late-time tails, introduce the phenomenon of superradiant amplification and quantify the timescales for the acoustic black hole bomb instability. In Section III we repeat the analysis for the canonical (3 + 1)-dimensional acoustic black hole metric (of course, the absence of an ergoregion means that we have no superradiance in this case). The conclusions follow in Section IV.", 'II. DRAINING BATHTUB: A ROTATING ACOUSTIC BLACK HOLE': "In this Section we consider a simple 'draining bathtub' model, first introduced in [7], for a rotating acoustic black hole. We write down the acoustic metric and the wave equation describing sound propagation in this model, and describe a concrete example for a possible experimental setup. We compute QNMs and discuss the late-time tail behavior of the system. Due to the presence of an ergoregion, sound waves in this acoustic black hole can be superradiantly amplified. We quantify this amplification and discuss the possibility to build an acoustic black hole bomb in the lab. \nP 1 ( r ) \n= \nA 2 + r 2 c 2 +2i A ( Bm -r 2 ω ) r ( r 2 c 2 A 2 ) , \n-Q 1 ( r ) = -2i ABm -B 2 m 2 + c 2 m 2 r 2 +2 Bmωr 2 -r 4 ω 2 r 2 ( r 2 c 2 -A 2 ) . (6) \nNotice that if we take the incompressible fluid limit c →∞ we get well known equations from fluid dynamics [23, 24]. We now introduce a tortoise coordinate r ∗ defined by the condition \ndr ∗ dr = ∆ , (7) \nwhere ∆ ≡ (1 -A 2 /c 2 r 2 ) -1 . Explicitly, \nSetting R = ZH we get: \nr ∗ = r + A 2 c log ∣ ∣ ∣ ∣ cr -A cr + A ∣ ∣ ∣ ∣ . (8) \nZ ∆ 2 H ,r ∗ r ∗ +[∆(2 Z ,r + P 1 Z ) + ∆ ' Z ] H ,r ∗ +( Z ,rr + P 1 Z ,r + Q 1 Z ) H = 0 . (9) \nTo obtain a Schrodinger-like equation we impose the coefficient of H ,r ∗ to be zero: \nZ ,r + 1 2 [ ( c 2 r 2 -A 2 ) + 2i A ( Bm -r 2 ω ) r ( c 2 r 2 -A 2 ) ] Z = 0 . (10)", 'A. Formalism and basic equations': "It is simple to show that the acoustic metric corresponding to the most general spherically symmetric flow of an incompressible fluid is given by [7] \nds 2 = -c 2 ( 1 -r 4 0 r 4 ) dt 2 + ( 1 -r 4 0 r 4 ) -1 dr 2 + r 2 ( dθ 2 +sin θ 2 dφ 2 ) . (49) \nThis metric does not correspond to any of the geometries typically considered in general relativity, but it describes (in the sense specified above) a 'canonical' acoustic black hole. The propagation of small disturbances (sound waves) is again described by the massless Klein-Gordon equation ∇ µ ∇ µ Ψ = 0 in this background. We can separate variables by the substitution \nΨ( t, r, φ ) = Φ( ω, r ) r e -i ωt Y lm ( θ ) , (50) \nwhere Y lm ( θ ) are the usual spherical harmonics. This yields the wave equation \nΦ ,r ∗ r ∗ + ( ω 2 c 2 -V ) Φ = 0 , (51) \nwhere \nV = ( 1 -r 4 0 r 4 )[ l ( l +1) r 2 + 4 r 4 0 r 6 ] , (52) \nand the tortoise coordinate r ∗ is defined, as usual, by dr/dr ∗ = (1 -r 4 0 /r 4 ). Since this is a spherical symmetric problem, the azimuthal number m does not play any role, only the angular momentum l is important here. In the \nTABLE III: The fundamental ( n = 0) QN frequencies for the non-rotating canonical acoustic black hole, using three WKB computational schemes. ω (1) QN is the result for the QN frequency using only the lowest approximation [25], ω (3) QN is the value obtained using 3rd order improvements [26], and finally ω (6) QN was computed using 6th order corrections [28]. \n| l ω (1) QN | ω (3) QN ω (6) QN |\n|-------------------------|------------------------------------|\n| | 0 1.13-0.73i 0.19-1.11i 0.06+0.87i |\n| 1 1.37-0.68i 0.53-0.71i | 1.09-0.39i |\n| 2 1.82-0.64i 1.41-0.61i | 1.41-0.70i |\n| 3 2.36-0.62i 2.10-0.62i | 2.12-0.62i |\n| 4 2.94-0.62i 2.75-0.62i | 2.75-0.62i | \nnext Section we summarize our results for QNMs and wave tails of the canonical acoustic black hole. When presenting our numerical results we will choose units such that c = r 0 = 1. This is of course equivalent to a simple rescaling of the radial variable (ˆ r = r/r 0 ) and of the frequency (ˆ ω = ω/c ), but we will omit hats in the following.", 'B. A possible experimental setup': "The acoustic metric (1) can be realized in a simple experimental setup, that was described in detail by Schutzhold and Unruh [16]. In this setup it no longer describes a sonic analogue but rather a shallow basin gravity wave analogue of a black hole. The idea is to use gravity waves in a viscosity free, incompressible liquid with irrotational flow: under appropriate circumstances, one can envisage the use of common fluids like water or mercury. Schutzhold and Unruh assumed a shallow water, long wavelength approximation: the gravity wave amplitude δh , their wavelength λ and the depth of the basin h B are such that δh /lessmuch h B /lessmuch λ . Relaxing the assumptions that the bottom of the tank and the background flow surfaces are flat and parallel, they showed that the most general rotationally symmetric and locally irrotational background flow profile can be described precisely by the draining bathtub metric (1) when the radial slope of the bottom of the tank - that in cylindrical coordinates ( z, r, φ ) will be described by some function f ( r ) -is \nf ' ( r ) /lessmuch 1. In this gravity wave black hole analogue the constants A and B are proportional to the radial and tangential components of the background flow velocity: \nv φ = B r 2 , v r = -Ah ∞ rh 1 + f ' ( r ) 2 , (18) \nwhere h ∞ is the height of the tank far from the black hole, and the slope of the tank satisfies the relation \n√ \nf ( r ) = -( A 2 + B 2 ) gr 2 . (19) \nIn the previous equations g is the gravitational acceleration, related to the constant c of the acoustic black hole metric (1) by the relation \nc = √ gh ∞ . (20) \n√ One of the main advantages of this acoustic black hole is apparent from this equation: the speed of the gravity waves can simply be tuned to one's needs by adjusting the height of the basin h ∞ . Another advantage, that we will not exploit here, is that the inclusion of surface tension and viscosity can be used to manipulate the waves' dispersion relation. To be concrete we will sometimes consider the following plausible choice of physical parameters, as suggested in [16]: gravity waves of amplitude δh ∼ 1 mm and wavelength λ ∼ 10 cm; a tank of height h ∞ ∼ 1 cm (so that c ∼ 0 . 31 m s -1 ); and a typical characteristic size of the acoustic black hole horizon r H ∼ 1 m, corresponding to A = cr H ∼ 0 . 31 m 2 s -1 .", 'C. Quasinormal modes': "Numerical and analytical studies of a fairly general class of initial data show that the evolution of the perturbations of a black hole spacetime can roughly be divided in three parts: (i) The first part is the prompt response at very early times. In this phase, which is the obvious counterpart of light cone propagation, the form of the signal depends strongly on the initial conditions. (ii) At intermediate times the signal is dominated by an exponential decay, whose frequencies and damping times are determined by the black hole QNMs. In this 'ringdown' phase the signal depends entirely on the black hole parameters (typically mass, charge and angular momentum). (iii) Due to backscattering off the spacetime curvature, at late times the propagating wave leaves a 'tail' behind, usually a power law falloff of the field. This power law seems to be highly independent of the initial data, and persists even if there is no black hole horizon. In fact it only depends on the asymptotic far region. In the following we study the QNM ringing phase of our acoustic black hole metric; then we will consider its late-time behavior. \nThe characteristic QNMs of the rotating acoustic black hole can be defined in the usual way, imposing appropriate boundary conditions and solving the corresponding eigenvalue problem. Close to the event horizon the solutions of equation (16) behave as \nH ∼ e ± i( ω -Bm ) r ∗ . (21) \nClassically, only ingoing waves - that is, waves falling into the black hole - should be present at the horizon. This means (according to our conventions on the time dependence of the perturbations) that we must choose the minus sign in the exponential. At spatial infinity the solutions of (16) behave as \nH ∼ e ± i ωr ∗ . (22) \nIn this case we require that only outgoing waves (waves leaving the domain under study) should be present, and correspondingly choose the plus sign in the exponential. This boundary condition at infinity may be cause for objections. Indeed, no actual physical apparatus will be accurately described by these boundary conditions: a real acoustic black hole experiment will certainly not extend out to infinity. However, we may imagine using some absorbing device to simulate the 'purely outgoing' wave conditions at infinity (for another example in which an absorbing device modeling spatial infinity could be required, cf. Section XI of [16] - in particular their Fig. 5). In any event, later on we shall consider the alternative possibility of 'boxed' boundary conditions, describing a closed system. Boxed boundary conditions were also considered in [20]. \nFor assigned values of the rotational parameter B and of the angular index m there is a discrete (and infinite) set of QN frequencies, ω QN , satisfying the wave equation (16) with boundary conditions specified by Eqs. (21) and (22). The QN frequencies are in general complex numbers, the imaginary part describing the decay or growth of \nTABLE I: The fundamental ( n = 0) QN frequencies for the non-rotating acoustic black hole, using three WKB computational schemes. ω (1) QN is the result for the QN frequency using only the lowest approximation [25], ω (3) QN is the value obtained using 3rd order improvements [26], and finally ω (6) QN was computed using 6th order corrections [28]. Notice how for m > 1 the three schemes yield very similar answers. \n| m ω (1) | ω (3) QN | ω (6) QN |\n|------------------------------------------|------------|------------|\n| QN | | |\n| 1 0.696-0.353i 0.321-0.389i 0.427-0.330i | | |\n| 2 1.105-0.349i 0.940-0.353i 0.945-0.344i | | |\n| 3 1.571-0.351i 1.465-0.353i 1.468-0.352i | | |\n| 4 2.054-0.352i 1.975-0.353i 1.976-0.353i | | | \nthe perturbation, because the time dependence is given by e -i ωt . We expect the black hole to be stable against small perturbations, and therefore ω QN is expected to have a negative imaginary part, so that the perturbation decays exponentially as time goes by. We could not prove stability of the draining bathtub metric in general, but we managed to derive upper bounds on the frequencies of unstable modes (if they exist at all). We give the derivation of these upper bounds in Appendix A. As usual, we will order the QN frequencies ω QN according to the absolute value of their imaginary part: the fundamental mode (labeled by an integer n = 0) will have the smallest imaginary part (in modulus), and so on.", '1. Slowly decaying modes of non-rotating black holes': "The lowest QNMs control the ringing behavior of any classical perturbation outside the black hole. Higher overtones, having a larger imaginary part, are damped more quickly and play a negligible role. In particular, it is known [13] that the fundamental ( n = 0) QNM effectively determines the response of the black hole to exterior perturbations. For these slowly damped QNMs, a WKB approximation - as developed by Schutz, Will and others [25, 26, 27] - is accurate enough. As a convergence check, we will sometimes extend the WKB treatment to sixth order [28]. The WKB method demands that the generalized potential Q defined by Eq. (17) has a single maximum outside the horizon. For non-rotating black holes ( B = 0) this is certainly true provided m is not zero, and one can easily extract the lowest QN frequencies in this case. We show them in Table I, where we also show the convergence of the WKB scheme as the order of approximation increases. For m = 0 the situation is not so simple, and we haven't been able to extract the QN frequencies using this method, because the potential possesses two extrema. Due to the symmetry of the potential (17), QN frequencies of non-rotating black holes are independent of the sign of m (to any positive QN frequency for positive m corresponds a negative QN frequency for negative m ). \nFrom Table I we see that the imaginary part is nearly constant as a function of m , whereas the real part approximately scales with m . Indeed, in the limit of large m one can show directly from the WKB formula [25] that ω QN behaves as \nω ∼ m 2 -i 2 n +1 2 √ 2 , m →∞ , (23) \nand indeed already for m = 4 this formula yields very good agreement with the results shown in Table I. This very same result can also be obtained using a Poschl-Teller fitting potential [29]. Just to give an idea of the orders of magnitude involved consider an m = 2 mode: if we take the 'typical' gravity wave analogue experimental parameters of Section II B we get (for the fundamental QNM with m = 2) a frequency ω R = 0 . 945 × c 2 /A = 0 . 293 Hz and a damping timescale τ = 1 / | ω I | = 1 / 0 . 344 × A/c 2 = 9 . 37 s.", '2. Slowly decaying modes of rotating black holes': "For rotating black holes, the generalized potential has more than one extremum. This immediately raises problems for the applicability of the WKB technique. Previous work on the Kerr geometry [26, 27] showed that the WKB method can still be used, yielding good results, as long as the rotation parameter is small. The way to handle the several extrema of the potential is the following: start with B = 0 and compute the roots of dQ/dr ∗ = 0 (to find the extrema; as we said before, for B = 0 the generalized potential Q has a single maximum outside the horizon as long as m = 0). Compute ω QN in the non-rotating case. Then add some small rotation B to the black hole, and compute \n/negationslash \n/.notdef \n<!-- image --> \n/.notdef \nFIG. 1: In the left panel we show the real part of the fundamental QN frequency ω R (and in the right panel we show the imaginary part ω I ) as a function of the rotation parameter B , for selected values of m . For m> 0 ( m< 0) ω R and | ω I | increase (decrease) with rotation. \n<!-- image --> \nthe roots of dQ/dr ∗ = 0. Now the roots will depend on ω , but only one root yields the correct non-rotating limit as B → 0. It is this root that corresponds to the QNMs of the rotating black hole. Our results, which can be trusted for B /lessorsimilar 0 . 2, are shown in Fig. 1. For m > 0 ( m < 0) ω R and | ω I | increase (decrease) with rotation, at least in the range in which the WKB method can be applied. The QN frequency change with rotation is not dramatic, but it can probably be used to apply a 'fingerprint analysis' of the acoustic black hole parameters ' a la Echeverria [14]: that is, once we measure at least two QNM frequencies we may infer the acoustic black hole parameters A and B . Carrying out such experiments in the lab can shed some light on the applicability of similar ideas to test the no-hair theorem in the astrophysical context [30]. A more complete analysis of the QNMs is still needed to probe the high rotation regime: the results shown in Fig. 1 seem to indicate that instability may set in for large Bm . A continued fraction analysis could be used to test this hypothesis, but it is beyond the scopes of this paper [31].", '3. Highly damped modes': "QNMs with a large imaginary part, i.e. with a very large overtone number n , have recently become a subject of intense scrutiny for black holes in general relativity and similar theories. The interest in these modes comes from Hod's proposal [12] that they could be related to black hole area quantization, and from Dreyer's suggestion that a similar argument could be used to fix the Barbero-Immirzi parameter in Loop Quantum Gravity [32]. In this context, an analytical calculation of highly damped QNMs was first carried out by Motl for the Schwarzschild black hole [33]. Subsequently Motl and Neitzke [34] used a complex-integration technique to compute highly-damped QNMs of the Schwarzschild and Reissner-Nordstrom black holes. Their analytical results are in striking agreement with numerical data [35, 36] and alternative analytical calculations [37]. The complex integration method has also been generalized with success to other black hole geometries [38, 39], yielding again predictions in excellent agreement with the numerical results [40]. In view of this, we shall now build on the results and techniques of [34] to compute highly damped QNMs of a rotating acoustic black hole. \nLet us first consider a non-rotating black hole. In the limit B → 0 the wave equation (16) reduces to \nwhere the potential \nd 2 H dr 2 ∗ + [ ω 2 -V ( r ) ] H = 0 , (24) \nV ( r ) ≡ ( 1 -1 r 2 )( m 2 -1 / 4 r 2 + 5 4 r 4 ) . (25) \nFollowing ([34]), we can determine the highly damped QNMs looking at the behaviour of the potential near the \nsingular point r = 0. In our case the leading term as r → 0 is \nV ∼ -5 4 r 6 , (26) \nand in the same limit we have r ∗ ∼ -r 3 / 3. Thus \nV ∼ -5 36 r 2 ∗ = j 2 -1 4 r 2 ∗ , (27) \nfor j = 2 / 3 (this is precisely the same power-law behavior found in the case of the Schwarzschild black hole, except for the different value of j ). The result in [34] carries over directly: \ne 4 πω = -(1 + 2 cos πj ) . (28) \nHowever now (1 + 2 cos πj ) = 0. This means that there are no asymptotic QN frequencies for this black hole. A similar situation occurs for electromagnetic perturbations of a five-dimensional black hole [40]. If we add rotation to the black hole, i.e., if B is non-zero, a similar analysis [41] implies that the asymptotic QN frequencies behave as \ne 4 π ( ω -mB ) = -(1 + 2 cos πj ) , (29) \nwhere again j = 2 / 3. Thus also in this case there will be no asymptotic QN frequencies. \nThis result is quite puzzling. It could mean either that the real part grows without bound, or that it just doesn't converge to any finite value. In any case, an application of Hod's conjecture to these black holes seems impossible: the very definition of the classical laws of black hole thermodynamics is non-trivial in the analogue case. In fact, as argued by Visser [2], the laws of black hole thermodynamics arise solely from the Einstein equations for the metric; from a different perspective, it has been shown that the Einstein equations themselves can be derived from purely thermodynamical arguments [42]. A well-known 'weakness' of analogue models is the fact that they can be used to reproduce the kinematical aspects of the Einstein equations, but not their dynamical aspects (see eg. [43] for the implications of this distinction on the causal structure of acoustic spacetimes). Hod's arguments [12] assume a thermodynamic relation between black hole surface area and entropy: even if Hod's conjecture is true, the absence of any such relation for analogues could explain the missing link between the QNM spectrum and area quantization.", 'D. Late-time tails': "After the exponential QNM decay characteristic of the ringdown phase black hole perturbations usually decay with a power-law tail [15], due to the backscattering of waves off the background curvature. The existence of late-time tails in black hole spacetimes is by now well established. A strong body of evidence comes from analytical and numerical calculations using linear perturbation theory and even non-linear evolutions, for massless or massive fields [15, 44, 45, 46, 47]. In a very complete analysis, Ching, Leung, Suen and Young [45, 46] considered the late-time tails appearing when one deals with evolution equations of the form (16), and the potential V is of the form \nV ( r ∗ ) ∼ ν ( ν +1) r 2 ∗ + c 1 log r ∗ + c 2 r α ∗ , r ∗ →∞ . (30) \nBy a careful study of the branch cut contribution to the associated Green's function they concluded that in general the late-time behavior is dictated by a power-law or by a power-law times a logarithm. The exponents of the power-law depend on the leading term at very large spatial distances. The case of interest for us here is when c 1 = 0. Their conclusions, which we will therefore restrict to the c 1 = 0 case, are (see Table 1 in [45] or [46]): \n(i) if ν is an integer the term ν ( ν + 1) /r 2 ∗ does not contribute to the late-time tail. Since this term represents just the pure centrifugal barrier, characteristic of flat space, one can expect that indeed it does not contribute, at least in four-dimensional spacetimes. Therefore, for integer ν , it is the c 2 /r α ∗ term that contributes to the late-time tail. In this case the authors of [45, 46] find that the tail is given by a power-law, \nΨ ∼ t -µ , µ > 2 ν + α , α odd integer < 2 ν +3 . (31) \nwhere the exponent µ = 2 ν +2 α -2. For all other real α , the tail is \nΨ ∼ t -(2 ν + α ) , all other real α. (32) \n(ii) if ν is not an integer, then the main contribution to the late-time tail comes from the ν ( ν +1) /r 2 ∗ term. In this case the tail is \nΨ ∼ t -(2 ν +2) , noninteger ν. (33) \nAs an example of non-integer ν , Cardoso et al. [47] have shown that the late-time tails of wave propagation appearing in odd dimensional spacetimes do not depend on the presence of the black hole at all. For odd dimensions the powerlaw is determined not by the presence of the black hole, but by the very fact that the spacetime is odd dimensional. In this case the field decays as \nΨ ∼ t -(2 l + D -2) , (34) \nwhere l is the angular index determining the angular dependence of the field, and D the number of spacetime dimensions. One can show directly from the flat space Green's function that such a power-law is indeed expected in flat, odd dimensional spacetimes. \nFrom the aforementioned arguments we should expect late-time tails in our draining bathtub acoustic black hole to be directly related to the dimensionality of the underlying flow, and not to the presence of a black hole, since we are dealing with a (2 + 1)-dimensional flow. Indeed, at leading order the potential behaves as \nV ∼ m 2 -1 / 4 r 2 ∗ = ν ( ν +1) r 2 ∗ , (35) \nwhere ν = m -1 / 2. We are thus in case (ii) above, because ν is not an integer. So any perturbation in the vicinities of this black hole will die out as a late-time tail of the form \nH ∼ t -(2 m +1) . (36) \nThis power-law is just the one given by the general expression (34), if one substitutes D = 3. \nStrictly speaking, the asymptotic form (35) for the potential is valid only for non-rotating acoustic black holes. For the general rotating case the leading term in the potential is ω -dependent: \nV ∼ m 2 -1 / 4 r 2 ∗ = ν ( ν +1) + 2 ωBm r 2 ∗ . (37) \nSince late-times are associated with small frequencies, and the term ν ( ν +1) gives a contribution to the late-time tail, then it follows that the late-time tails of rotating acoustic black holes are the same as those for non-rotating black holes, expression (36).", 'E. Superradiance': "Superradiance is a general phenomenon in physics. Inertial motion superradiance has long been known [48], and refers to the possibility that a (possibly electrically neutral) object endowed with internal structure, moving uniformly through a medium, may emit photons even when it starts off in its ground state. Some examples of inertial motion superradiance include the Cherenkov effect, the Landau criterion for disappearance of superfluidity, and Mach shocks for solid objects travelling through a fluid (cf. [49] for a discussion). Non-inertial rotational motion also produces superradiance. This was discovered by Zel'dovich [50], who pointed out that a cylinder made of absorbing material and rotating around its axis with frequency Ω can amplify modes of scalar or electromagnetic radiation of frequency ω , provided the condition \nω < m Ω (38) \n(where m is the azimuthal quantum number with respect to the axis of rotation) is satisfied. Zel'dovich realized that, accounting for quantum effects, the rotating object should emit spontaneously in this superradiant regime. He then suggested that a Kerr black hole whose angular velocity at the horizon is Ω will show both amplification and spontaneous emission when the condition (38) for superradiance is satisfied. This suggestion was put on firmer ground by a substantial body of work [51]. In particular, it became clear that (even at the purely classical level) superradiance is required to satisfy Hawking's area theorem [19, 52]. \nSuperradiance is essentially related to the presence of an ergosphere, allowing the extraction of rotational energy from a black hole through a wave equivalent of the Penrose process [53]. Under certain conditions, superradiance can \nbe used to induce instabilities in Kerr black holes [11]. Indeed, all spacetimes admitting an ergosphere and no horizon are unstable due to rotational superradiance. This was shown rigorously in [54], but the growth rate of the instability is too slow to observe it in an astrophysical context [55]. Kerr black holes are stable, but if enclosed by a reflecting mirror they can become unstable due to superradiance [20, 56]; we will discuss this 'black hole bomb' instability as applied to the analogue acoustic black hole. \nThe possibility to observe rotational superradiance in analogue black holes was considered by Schutzhold and Unruh [16], and more extensively by Basak and Majumdar [17, 18], who computed analytically the reflection coefficients in the low frequency limit ωA/c 2 /lessmuch 1. In particular, the authors of [16] showed that the ergoregion instability in gravity wave analogues is related to the existence of an 'energy function' [their Eq. (68)] that is not positive definite inside the ergosphere. In the context of analogues, inertial superradiance based on superfluid 3 He has been studied by Jacobson and Volovik [57]. \nHere we present a quantitative calculation of the efficiency of superradiant amplification for the draining bathtub metric. We work in the frequency representation, and consider an incident plane wave of unit amplitude at infinity, frequency ω and azimuthal index m . Part of this wave will be reflected back by the medium, the reflection coefficient being some (complex) number R ωm . In terms of the wave equation (16), this determines the following boundary condition at infinity: \nH ∼ R ωm e i ωr ∗ + e -i ωr ∗ , r →∞ . (39) \nAt the sonic horizon ( r → 1, r ∗ →-∞ ) the solution behaves like \nH ∼ T ωm e -i( ω -mB ) r ∗ , r → 1 . (40) \nwhere T ωm is the transmission coefficient. An easy way to prove the existence of superresonance is to compute the Wronskian of a solution of equation (16) and of its adjoint at the sonic horizon and at infinity. From the constancy of the Wronskian as a function of the radial coordinate, using the boundary conditions (39) and (40) we find the following 'energy conservation' condition: \n1 -| R ωm | 2 = ( 1 -mB ω ) | T ωm | 2 . (41) \nTherefore, if ω < mB the reflection coefficient R ωm | 2 > 1, i.e. we have superradiance. \n| \n| To compute numerically with improved accuracy the reflection coefficient R ωm , we have used a refined condition at the horizon: \nH ∼ T ωm e -i( ω -mB ) r ∗ [1 + y 1 ( r -1) + . . . ] , (42) \nwhere the leading-order coefficient is \ny 1 = (1 + 2 B 2 ) m 2 -2 ωBm +1 2[i( mB -ω ) + 1] . (43) \nWe also keep higher-order terms to extract the reflection coefficient at infinity. Namely, for the outgoing wave we use an expansion of the form: \nwhere \nH ∼ e i ωr ∗ [ 1 + z 1 r + z 2 r 2 + . . . ] , (44) \nz 1 = i B +(4 m 2 -1) 8 ω , z 2 = i z 1 [ Bm 2 + (4 m 2 -9) 16 ω ] , \nand for the ingoing wave we use the complex conjugate of Eq. (44). As a check of our code we have reproduced results by Andersson et al. [58] for the superradiant amplification of a scalar field in the vicinities of a Kerr black hole. Our results are in perfect agreement with Fig. 1 of [58]. In particular, we find a maximum amplification coefficient given by 1 - | R ωm | 2 /similarequal 0 . 2 % for l = m = 2 scalar perturbations of a near-extremal extremal Kerr black hole (in early numerical work [19, 56] the maximum amplification was found to be 1 R ωm | 2 0 . 3 %). \n-| \n/similarequal \n| Results of the numerical integrations for the draining bathtub metric are shown in Fig. 2. Panels on the left show the reflection coefficient | R ω 1 | 2 for m = 1, and panels on the right show | R ω 2 | 2 for m = 2, for selected values of the black hole rotation B . Panels on top show that, as expected, in the superradiant regime 0 < ω < mB the reflection \n/.notdef \n<!-- image --> \n/.notdef \n<!-- image --> \n/.notdef \n<!-- image --> \n/.notdef \n/.notdef \n<!-- image --> \n/.notdef \n/.notdef \n<!-- image --> \n/.notdef \nFIG. 2: Reflection coefficient | R ωm | 2 as a function of ω for m = 1 (left panels) and m = 2 (right panels). Each curve corresponds to a different value of B , as indicated. The top panels show that the reflection coefficient decays exponentially at the critical frequency for superradiance, ω SR = mB . The middle panels show a close-up view in the superradiant regime for B < 1: at B = 1 the maximum amplification is 21.2 % ( m = 1) and 4.7 % ( m = 2). The bottom panels show that superradiant amplification can become much more efficient for values of the rotation parameter B > 1. \n<!-- image --> \ncoefficient | R ωm | 2 ≥ 1. Furthermore, as one increases B the reflection coefficient increases, and for fixed B , the reflection coefficient | R ωm | 2 attains a maximum at ω ∼ mB , after which it decays exponentially as a function of ω outside the superradiant interval. This is very similar to what happens when one deals with massless fields in the vicinities of rotating Kerr black holes [19]. In particular, from the close-up view in the middle panels we see that, for B = 1, the maximum amplification is 21.2 % ( m = 1) and 4.7 % ( m = 2). \nAs a final remark, and as we have anticipated, an important difference between the acoustic black hole metric and the Kerr metric is that in the present case there is no mathematical upper limit on the black hole's rotational velocity B . In the bottom panels we show that, considering values of B > 1, we can indeed have larger amplification factors for acoustic black holes. \nSummarizing: if we are clever enough to build in the lab an acoustic black hole that spins very rapidly, rotational superradiance can be particularly efficient in analogues. This is an important result, considering that the detection of rotational superradiance in the lab is by no means an easy task, as originally predicted by Zel'dovich [50] and confirmed by recent reconsiderations of the problem [49]. Of course, in any real-world experiment the maximum rotational parameter will be limited. At the mathematical level, the equations describing sound propagation (which are written assuming the hydrodynamic approximation) will eventually break down. Physically, if the angular component of the velocity v θ becomes very large the dispersion relation for the fluid will change, invalidating the assumptions under which we have derived our acoustic metric [16]. To be more concrete, let us consider again the gravity wave analogue described in Section II B. Then we can use a very simple argument to limit the acoustic black hole spin. The derivation of the rotating acoustic black hole metric is based on the assumption that the bottom of the tank should not be too steep, that is, f ' ( r ) /lessmuch 1. This condition translates into a condition for B : \nB /lessmuch ( gr 3 2 -A 2 ) 1 / 2 , (45) \nwhich must be satisfied for all values of r , and in particular at the acoustic black hole horizon r = r H (notice that here, and here only, we have switched back to physical units). Setting r = r H in the previous inequality and using the physical parameters quoted in Section II B we get the very stringent condition that B /lessmuch 2 . 2 m 2 s -1 , or (in the dimensionless units we use throughout this paper) ˆ B /lessmuch 7 . 0 . In other words, we can only get values of the rotation parameter larger than ˆ B ∼ 1 when the slope of the tank is so large that the assumptions underlying the derivation of the acoustic metric are not valid any more. Of course, this example does not mean that such a constraint applies to every possible experimental realization of the draining bathtub metric. However, it serves as an illustration of the kind of experimental difficulties we may expect to encounter in practice. \nThe superradiant phenomena we have described are purely classical in nature. However, an interesting suggestion to observe quantum effects in acoustic superradiance was put forward in [18]. To write down our acoustic metric we required the flow to be irrotational and nonviscous. As a natural choice, we could use a fluid which is well known to possess precisely these properties: superfluid HeII. In this case the presence of vortices with quantized angular momenta may lead to a quantized energy flux. The heuristic argument presented in [18] goes as follows. Let us imagine that our black hole is a vortex with a sink at the centre. In the quantum theory of HeII the wavefunction is of the form Ψ = exp [ i ∑ j φ ( /vectorr j )Φ ground ] , where /vectorr j is the position of the j -th particle of HeII. The velocity at any point is given by the gradient of the phase at that point, /vectorv = ∇ φ , so that (roughly speaking) the velocity potential (3) can be identified with the phase of the wavefunction. This phase will be singular at the sink r = 0. Continuity of the phase around a circle surrounding the sink requires that the change of the wavefunction satisfies ∆ φ = 2 πB . For the wavefunction to be single valued, B (that is, the black hole's angular velocity at the horizon) must be the integer multiple of some minimum value ∆ B , i.e., B = n ∆ B . Then the angular momentum of the acoustic black hole would be forced to change in integer multiples of ∆ B . Correspondingly, the spectrum of the reflection coefficients may be given by equally-spaced peaks with different strengths. This discrete amplification could enhance chances of observing superradiance in acoustic black holes, and rule out (or provide empirical support to) some of the many competing heuristic approaches to black hole quantization.", 'F. Superradiant instabilities: the acoustic black hole bomb': "The very existence of an ergoregion in the acoustic black hole metric allows immediately for the possibility to make the system unstable. Suppose we enclose the system by a reflecting mirror at constant radius r 0 . Now, throw in a wave having frequency ω = ω R + i ω I such that ω R < m Ω: the wave will be amplified at the expense of the black hole's rotational energy and travel back to the mirror. There it will be reflected and move again towards the black hole, this time with increased amplitude. Through repeated reflections, the waves' amplitude will grow exponentially \n/.notdef \n<!-- image --> \nFIG. 3: Real part (left) and imaginary part (right) of the fundamental BQN frequency and the first two overtones as a function of mirror location r 0 . Both panels refer to m = 1 and B = 1 (but the B -dependence of the real part of the BQN frequency is very weak). With excellent accuracy ω I crosses zero (and the instability abruptly shuts off) at the critical radius predicted by Eq. (48), as can be verified taking a glance at the first row of Table II. \n<!-- image --> \nwith time. This 'black hole bomb' was first proposed by Press and Teukolsky [56], and recently it has been studied in detail by some of us [20] in the context of the Kerr metric. The detailed analysis performed in [20] showed that the black hole bomb can be characterized by a set of complex resonant frequencies, the Boxed QuasiNormal Modes (BQNMs). The real part of a BQNM is, not surprisingly, proportional to 1 /r 0 , where r 0 is the mirror radius: this is essentially the condition for the existence of standing waves in the region enclosed by the mirror. The imaginary part of the BQNMs is proportional to ( ω R -m Ω): the system can only become unstable if it is in the superradiant regime. Combined with the standing wave condition this implies that the mirror should be placed at some radius r 0 /greaterorsimilar 1 /m Ω in order for the system to be unstable. As rotational energy is extracted from the system the black hole will spin down. For any given r 0 the instability will eventually shut off, as the condition r 0 /greaterorsimilar 1 /m Ω will no longer be satisfied. \nAn additional reason to study the system 'acoustic black hole+mirror' is related to the problem of boundary conditions at infinity for QNMs (cf. Section II C). Whatever analogue model we consider, no physical apparatus will ever extend out to infinity. We are left with two possibilities: we can either use some absorbing device to simulate spatial infinity, or we can impose alternative boundary conditions on the system. A natural choice is to have a reflecting mirror surrounding the apparatus, so that we are effectively building an acoustic black hole bomb. \nIn the following we will assume that we are in the presence of a perfectly reflecting mirror. Correspondingly, we shall impose the boundary condition H = 0 at r = r 0 . At the horizon we demand, as usual, the presence of purely ingoing waves (waves headed towards the horizon), i.e. H ∼ e -i( ω -Bm ) r ∗ . We are again in presence of an eigenvalue problem. However, since the boundary conditions have changed we shall not refer to the characteristic frequencies as QN frequencies, but rather as Boxed QuasiNormal frequencies (BQN frequencies). The reader is referred to [20], where this terminology was first introduced. Using a direct integration of the wave equation (see Section II E for details) it is quite easy to compute the BQN frequencies. We simply integrate Eq. (16) outwards from the horizon, where we impose the condition (42), to the mirror location r 0 . The (complex) BQNM frequencies are those frequencies for which \nH ( ω BQNM , r 0 ) = 0 . (46) \nSome results for the fundamental BQNM and the first two overtones are shown in Fig. 3. There we fixed the black hole rotation parameter B = 1, and m = 1, as a typical case. \nFrom the left panel we see that the real part of the BQNM frequency for the n -th overtone (where we use the convention that n = 0 corresponds to the fundamental mode) scales as 1 /r 0 , as anticipated. The proportionality constant can be obtained by analytical arguments similar to those in [20]. The result is \nω R = j m,n r 0 , (47) \nwhere the j m,n 's are zeros of the Bessel function of integer order m (recall that for a Kerr black hole bomb one gets ω R = j l +1 / 2 ,n /r 0 instead, cf. [20]). The numerical values of the j m,n 's can be found, e.g., in [59]. For reference, we list \nTABLE II: Zeros of the Bessel functions j m,n for the first few values of m . The real part of the BQNM frequency is well approximated by Eq. (47). \n| m j m, 0 j m, 1 j m, 2 |\n|-----------------------------|\n| 1 3.83171 7.01559 10.17347 |\n| 2 5.13562 8.41724 11.61984 |\n| 3 6.38016 9.76102 13.01520 |\n| 4 7.58834 11.06471 14.37254 |\n| 5 8.77148 12.33860 15.70017 | \n<!-- image --> \nFIG. 4: Growth timescales for BQNMs, as a function of mirror location r 0 , for different values of B . The left panel refers to m = 1, the right panel to m = 2. Notice the different scales for the ω I axis. \n<!-- image --> \nthe first few values in Table II. Our numerical calculations are in excellent agreement with the predictions of formula (47). This is particularly true for small values of B , but the B -dependence of the real part of the BQN frequencies is very weak anyway, as it is for the Kerr metric [20]. \nFrom the right panel we see the behavior predicted by the qualitative arguments at the beginning of this Section: for any BQNM, the instability is more and more efficient as the mirror radius r 0 becomes smaller, until eventually the mirror radius becomes small enough that the instability shuts off. \nA few remarks are in order: (i) The real part of the BQN frequency (slowly) increases with overtone number n . (ii) Higher overtones become stable at larger distances, but they also attain a smaller maximum growing rate. (iii) With excellent accuracy, the instability switches off at the critical radius predicted by the analytical formula (47) supplemented by the superradiance condition (38), that is: \nr 0 ,c /similarequal j m,n mB . (48) \nSince j m,n grows linearly with m (in particular, this is asymptotically true for high m 's [59]) the critical radius is almost m -independent. However, the instability is not so efficient for higher m as it is for small m , at least when B /lessorsimilar 1. This is apparent in Fig. 4, where we compare growth timescales with m = 1 (left) and m = 2 (right) for acoustic black holes rotating at different rates. For example, the maximum growth rate for B = 0 . 3 is ∼ 10 -4 for m = 1, and ∼ 10 -5 for m = 2. The growth rate for m = 2 becomes roughly comparable to the growth rate for m = 1 when B > 1: look for example at the curves corresponding to B = 5 . 0. However, other physical considerations may forbid the construction of acoustic black holes rotating at such rates (see the arguments at the end of Section II E). \nTo get an idea of the orders of magnitude involved, let us consider again the 'typical' parameters for the gravity wave analogue we introduced in Section II B. Let us pick m = 1 and an acoustic black hole rotation parameter ˆ B = 1, which - according to the arguments in Section II E - is close to the maximum rotation rate we may hope to achieve (recall hats distinguish dimensionless quantities from quantities in physical units). To take advantage of the process the mirror should be located close to the maximum of ω I , but not quite at the maximum. For example, if we place the mirror at ˆ r 0 /similarequal 10 (cf. the right panel of Fig. 3) we get ˆ ω I /similarequal 4 · 10 -3 (in this case the maximum growth rate \nwould be ˆ ω I = 8 . 5 · 10 -3 ). This corresponds to a growth time τ /similarequal 800 s /similarequal 13 minutes (at the maximum we would get τ /similarequal 6 minutes). Roughly speaking, this means that the amplitude will double every 13 minutes or so, and will be amplified by orders of magnitude on timescales of the order of a few hours. To reduce the typical timescale to seconds, one just has to ajust the horizon radius and wave velocity. For example, working with r H ∼ 0 . 1 m and c ∼ 10 m s -1 , we would get a typical timescale of about 2 seconds. This looks like a perfectly reasonable timescale to observe an acoustic black hole bomb in the lab. \nWhen the instability sets in the acoustic black hole loses energy and angular momentum (∆ E ∼ B ∆ J ). The critical radius r 0 ,c increases with decreasing B . To estimate the efficiency we can use a simple argument. We can take advantage of the mechanism by picking some large B and r 0 /similarequal 2 r 0 ,max , where r 0 ,max is the radius of maximum growth timescale at the given B . As the hole emits angular momentum B will decrease. For any given mirror location r 0 , the bomb eventually switches off when condition (48) is satisfied. From the difference in initial and final angular momentum ∆ B we can infer the extracted energy ∆ E . For more details we refer to [20]. \nAs a final remark we want to stress that, when we talk about surrounding the acoustic black hole by a reflecting mirror, what we have in mind is a generic type of mirror. For instance, in the shallow basin gravity wave analogue, the mirror could be a circular rubber band of radius r 0 reflecting gravity waves. A possible alternative to implement a mirror could involve, for example, variations in the dispersion relation for sound waves. In particular, Schutzhold and Unruh [16] suggested that a simple change of the height h ∞ of the basin would imply a change of the gravity wave speed, hence of the effective refractive index. The changed refractive index may be used to trap the waves, which can then be amplified via superradiant scattering until non-linear effects dominate.", 'III. THE CANONICAL NON-ROTATING ACOUSTIC BLACK HOLE': "In the previous Section we considered the metric describing a rotating (2 + 1)-dimensional acoustic black hole. Rotation implies the presence of an ergosphere, allowing us to study in the lab some of the most interesting phenomena concerning black hole physics (eg. superradiance and the related rotational instabilities). However, the non-rotating limit of the acoustic metric we considered is not the 'natural' metric describing a non-rotating acoustic black hole. In this Section we are going to consider another class of non-rotating, (3 + 1)-dimensional acoustic metrics. These metrics are associated to the most general spherically symmetric flow of an incompressible fluid: in this sense they represent 'canonical' metrics for a non-rotating acoustic black hole.", 'B. Quasinormal modes': "A 6th order WKB analysis [28] reveals an unstable QNM for l = 0. However, it is possible to prove stability of the canonical black hole since the potential is positive-definite. We therefore computed the QN frequencies using first the lowest approximation [25], then 3rd order corrections [26] and finally 6th order corrections. The results are presented in Table III. The QN frequencies for l = 0 and l = 1 seem to be the problem here. For any other l the value quickly converges as we increase the correction order of the WKB method. Notice for example that the l = 0 mode suffers a variation of almost an order of magnitude as we go from the lowest approximation to the 3rd order correction scheme, and gets unstable for the scheme using 6th order corrections. The reason for this failure is most likely related to the breakdown of the basic WKB assumptions, namely that the ratio of the derivatives of the potential to the potential itself is small. Indeed a close inspection shows that as l increases these ratios tend to decrease. This means that the WKB method is more reliable for higher l : this was first observed in the early works on the subject [25, 26]. For l = 0 and l = 1 the lowest WKB approximation gives the most reliable results. This is confirmed by recent numerical work [36]. \nIn the limit of large l one finds \nω ∼ l √ 2 3 3 / 4 -i √ 2(1 + 2 n ) 3 3 / 4 , (53) \na result which agrees very well with our WKB data already for l = 3. \nThe calculation of highly damped QNMs proceeds along the same lines sketched for the (2+1)-dimensional acoustic black hole. In this case the index j = 3 / 5, which implies that \n4 πω = log 3 -√ 5 2 -i(2 n +1) π . (54) \nOnce again asymptotic QN frequencies are not given by the logarithm of an integer, as required in Hod's construction [12]. As we remarked earlier this does not necessarily imply that the conjecture is wrong, since a thermodynamical interpretation of the black hole area is possible only when the dynamics of the system are described by the Einstein equations.", 'C. Late-time tails': 'The analysis of late-time tails proceeds as in the case of the (2 + 1)-dimensional acoustic black hole. We can easily show that asymptotically the potential behaves as \nV ∼ l ( l +1) r 2 ∗ + 12 -5 l ( l +1) 3 r 6 ∗ , r →∞ . (55) \nFollowing the previous analysis we now find α = 6, and since l is an integer the power-law falloff is of the form \nΦ ∼ t -(2 l +6) . (56) \nThus any perturbation eventually dies off as t -(2 l +6) , much more quickly (for m = l ) than in the (2 + 1)-dimensional acoustic black hole background.', 'IV. CONCLUSIONS': "In this paper we have considered two acoustic black hole metrics: the (2+1)-dimensional 'draining bathtub' metric, and the (3 + 1)-dimensional canonical non-rotating acoustic black hole. We have studied QNMs and late-time tails in both metrics, and superradiance in the 'draining bathtub' metric. \nMore specifically, we numerically computed slowly damped QNMs of the draining bathtub metric using a thirdorder WKB approach for all values of m /negationslash = 0. We analytically evaluated QN frequencies in the largem limit [Eq. (23)] and set upper limits on frequencies of unstable modes (Appendix A). We showed that highly damped modes do not tend to any simple limit. At late times, power-law tails decay as t -(2 m +1) . This behavior is typical of any odddimensional spacetime: independently of the presence of a black hole, if D is odd the power-law falloff is proportional to t -(2 l + D -2) [47] [the (2 + 1)-dimensional case, D = 3, is special: the azimuthal and angular numbers are the same, l = m , since there is only one angular coordinate]. The draining bathtub metric, possessing an ergoregion, can superradiantly amplify waves. We computed reflection coefficients for this superradiant scattering by numerical integration. When the (dimensionless) acoustic black hole rotation B = 1, the maximum amplification is 21.2 % for m = 1 and 4.7 % for m = 2. Enclosing the acoustic black hole by a reflecting mirror we can destabilize the system, making an initial perturbation grow exponentially with time: we have an acoustic black hole bomb [19], or 'dumb hole bomb'. We computed analytically and numerically the frequencies and growing timescales for this instability. An interesting feature of acoustic geometries is that the acoustic black hole spin can be varied independently of the black hole mass . Therefore, at variance with the Kerr metric, the spin can be made (at least in principle) arbitrarily large, and rotational superradiance in acoustic black holes can be very efficient. Gravity waves in shallow water [16] provide a concrete example of an experimental setup for studying classical physics in a draining bathtub metric [and of experimental difficulties in increasing arbitrarily the rotation rate: see the discussion leading to Eq. (45)]. In this case, and for a typical choice of parameters, growth times for the black hole bomb instability would be of the order of minutes: this appears to be well within the range of experimental possibilities. Finally one can speculate, following [18], that superfluid HeII could be used to observe some quantum effects, such as a discrete superradiant amplification. \nFor the canonical acoustic black hole we computed slowly-damped QNMs using first, third and sixth order WKB. For l < 2 the WKB method does not seem to converge, and even yields an unstable frequency at 6th order when l = 0. This is only due to bad convergence properties of the WKB technique, since the canonical acoustic black hole is stable. In the high-damping limit QNMs are given by 4 πω = log [(3 -√ 5) / 2] -i(2 n + 1) π , so they are not proportional to the logarithm of an integer, as required by recent conjectures. This is not surprising since there are no Einstein equations for acoustic metrics, i.e., the acoustic metric evolution is not governed by Einstein's equations. Therefore, acoustic black holes can teach us a lot about quantum gravity [7, 10], but cannot shed any light on its possible connection with highly damped quasinormal modes. \nThe late time falloff in the canonical acoustic black hole metric is proportional to t -(2 l +6) , to be compared with the t -(2 l +3) decay of 4-dimensional Schwarzschild black holes and with the t -(2 l +3 D -8) decay of even-dimensional spherically symmetric black holes with D > 4 [47].", 'Acknowledgements': "We are grateful to Carlos Barcel'o and Ted Jacobson for useful discussions. This work was partially funded by Funda¸c˜ao para a Ciˆencia e Tecnologia (FCT) - Portugal through project CERN/FNU/43797/2001. V.C. acknowledges financial support from FCT through grant SFRH/BPD/2003. This work was supported in part by the National Science Foundation under grant PHY 03-53180.", 'APPENDIX A: UPPER BOUNDS FOR QUASINORMAL FREQUENCIES OF UNSTABLE MODES': "To derive some bounds on the magnitude of QN frequencies of possible unstable QNMs we shall follow Detweiler and Ipser [60]. Let us begin with the Klein-Gordon equation for the evolution of the field, ∇ µ ∇ µ Ψ = 0. Using the \nmetric (14) and performing a mode decomposition Ψ( t, r, φ ) = R ( r ) e i( mφ -ωt ) , this can be written as \nr fc 2 ω 2 R + d dr [ rfR ' ] -2 Bmω c 2 fr R -( 1 r -B 2 c 2 r 3 f ) m 2 R = 0 , (A1) \nwhere we have defined f ∆ -1 = 1 -A 2 /c 2 r 2 . \n≡ \n-Multiply by R ∗ and integrate from the horizon to spatial infinity. The result is \n∫ ∞ r H dr [ r fc 2 ω 2 | R | 2 -rf | R ' | 2 -2 Bmω c 2 fr | R | 2 -( 1 r -B 2 c 2 r 3 f ) m 2 | R | 2 ] = 0 , (A2) \nwhere we have used an integration by parts and discarded the surface integrals. In fact, for unstable modes (and for these only) the boundary conditions guarantee that an unstable mode vanishes exponentially as r → r H or r →∞ . The imaginary part of this equation yields \n∫ ∞ r H dr ( r 2 ω R -Bm ) | R | 2 frc 2 = 0 , (A3) \nwhere we used ω = ω R +i ω I . Therefore ω R and Bm must have the same sign for unstable modes. Furthermore, since | R | 2 /frc 2 is always positive, the quantity ( r 2 ω R -Bm ) must be negative somewhere for the integral to vanish. Since r 2 ω R increases with r , if this quantity is somewhere negative, then it certainly is negative at the horizon. In this way we get an upper bound for ω R : \nω R < Bm r 2 H . (A4) \nTo get a similar bound on ω I consider now the real part of Eq. (A2): \n∫ ∞ r H dr [ ω 2 I -ω 2 R + 2 Bmω R r 2 + c 2 fm 2 r 2 -B 2 m 2 r 4 + f 2 c 2 | R ' | 2 | R | 2 ] r | R | 2 fc 2 = 0 . 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1996PhRvL..77..430H

Statistical Entropy of Nonextremal Four-Dimensional Black Holes and U Duality

1996-01-01

['-', '-']

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We identify the states in string theory which are responsible for the entropy of near-extremal rotating four-dimensional black holes in N = 8 supergravity. For black holes far from extremality (with no rotation), the Bekenstein-Hawking entropy is exactly matched by a mysterious duality invariant extension of the formulas derived for near-extremal black holes states.

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https://arxiv.org/pdf/hep-th/9603195.pdf

{'Statistical Entropy of Nonextremal Four-Dimensional Black Holes and U-Duality': 'Gary T. Horowitz † , David A. Lowe † and Juan M. Maldacena /natural \n- ‡ Department of Physics, University of California, Santa Barbara, CA 93106, USA\n- /natural Joseph Henry Laboratories, Princeton University, Princeton, NJ 08544, USA', 'Abstract': "We identify the states in string theory which are responsible for the entropy of nearextremal rotating four-dimensional black holes in N = 8 supergravity. For black holes far from extremality (with no rotation), the Bekenstein-Hawking entropy is exactly matched by a mysterious duality invariant extension of the formulas derived for near-extremal black holes states. \nMarch, 1996 \nRecent developments in string theory have led, for the first time, to an understanding of black hole entropy from a microscopic point of view. In [1] it was shown that the Bekenstein-Hawking entropy of an extremal, nonrotating, five-dimensional black hole precisely counts the number of BPS states in string theory with the given charges (in the limit of large charges). This agreement has since then been extended in a number of directions. Extreme 5D black holes with rotation [2], extreme 4D black holes [3], and slightly nonextreme 5D black holes [4,5,6] have all been shown to have a Bekenstein-Hawking entropy which agrees with the number of corresponding states in string theory. One goal of the present work is to show that this agreement continues to hold for slightly nonextremal 4D black holes. \nThe restriction to extreme or near-extreme black holes arises since we can only count states at weak coupling, while black holes only exist at strong coupling. For extremal configurations, one can argue that interactions are absent on the basis of supersymmetry and the extrapolation of the number of states from weak to strong coupling is justified. For near-extremal black holes there are situations in which the interactions are again suppressed. For black holes far from extremality, there appears to be no reason why a weakly coupled description is applicable. \nNevertheless, it was shown in [7] that there is a sense in which even black holes far from extremality can be thought of as composed of weakly interacting fundamental objects in string theory. The objects one needs are the same ones which yield the states of extremal black holes: extended solitons known as D-branes and fundamental strings. More precisely, ref. [7] considered a class of five-dimensional black holes labeled by the energy, three charges, and the asymptotic values of two scalars. The three charges are carried by onebranes, fivebranes and strings (or anti-branes, which are just branes with the opposite orientation and with the opposite sign of the charge). One can replace the original six parameters in the solution by the number of branes, anti-branes and strings ( N 1 , N ¯ 1 , N 5 , N ¯ 5 , n R , n L ) by matching the energy, three gauge charges and two scalar charges of these noninteracting objects with that of the black hole. In terms of these new variables the black hole entropy takes the suggestive form \nS = 2 π ( √ N 1 + √ N ¯ 1 )( √ N 5 + √ N ¯ 5 )( √ n L + √ n R ) . (1) \nThis expression applies to all black holes, even those which are far from extremality. The symmetry of this expression is consistent with U-duality which permutes the three types of fundamental objects. It was argued [7] that (1) arises naturally from counting states in string theory, in the sense that it correctly reproduces the number of string states in three different weak-coupling limits, and is the simplest duality invariant expression with this property. However, no derivation of the general formula directly from counting states in string theory is currently available. \nSince the significance of the above expression for the black hole entropy is not yet well understood, it is important to know whether it is special to five dimensions or if it applies more generally. In this paper we will show that the entropy of four-dimensional black holes can be expressed in a form directly analogous to (1). In [3] it was shown that the states of extremal four-dimensional black holes can be described in terms of D-twobranes, solitonic fivebranes, D-sixbranes, and open strings. We will consider the nonextremal version of \nthese solutions which is an eight parameter family of four-dimensional black holes. By comparing the mass, gauge charges, and scalar charges (which are pressures in the internal directions) of the black hole with those of a set of noninteracting branes and anti-branes, we will rewrite the Bekenstein-Hawking entropy in a form analogous to (1). We will show that in certain limits (corresponding to near-extremal black holes), the entropy formula we obtain indeed represents the number of states of this collection of branes at weak coupling. \nThe generalization of these black hole solutions to include rotation has recently been found [8]. We will show that in the limit of small rotation and near extremality the black hole entropy again agrees with the number of string states. \nWe will be considering Type II string theory compactified on T 6 = T 4 × S 1 × ˆ S 1 , which gives N = 8 supergravity in four dimensions. In ref. [9] general spherically symmetric black hole solutions of N = 4 supergravity in four dimensions were considered. Using these solutions it is straightforward to construct the general class of black holes in N = 8 supergravity. The starting point for this construction is a solution with four nonzero U (1) gauge fields (carrying two electric and two magnetic charges) and three nontrivial scalars [9]. The Einstein metric is \nds 2 = -f -1 / 2 ( r ) ( 1 -r 0 r ) dt 2 + f 1 / 2 ( r ) [ ( 1 -r 0 r ) -1 dr 2 + r 2 ( dθ 2 +sin 2 θdφ 2 ) ] , f ( r ) = ( 1 + r 0 sinh 2 α 2 r )( 1 + r 0 sinh 2 α 5 r )( 1 + r 0 sinh 2 α 6 r )( 1 + r 0 sinh 2 α p r ) . (2) \nThis metric is parameterized by the five independent quantities α 2 , α 5 , α 6 , α p and r 0 . The event horizon lies at r = r 0 . The special case α 2 = α 5 = α 6 = α p corresponds to the Reissner-Nordstrom metric. The overall solution contains three additional parameters which are related to the asymptotic values of the three scalars. From the ten-dimensional viewpoint, these are the volume of the 4-torus (2 π ) 4 V , and the radii of S 1 and ˆ S 1 , R 1 and R 2 . \nThe physical charges are expressed in terms of these quantities as \nQ 2 = r 0 V g sinh 2 α 2 , Q 5 = r 0 R 2 sinh 2 α 5 , Q 6 = r 0 g sinh 2 α 6 , n = r 0 V R 2 1 R 2 g 2 sinh 2 α p , (3) \nwhere g is the ten-dimensional string coupling and we have chosen conventions such that α ' = 1 and the four-dimensional Newton constant is G 4 = g 2 / (8 V R 1 R 2 ). Note that in these conventions the string coupling is such that g → 1 /g under S duality. \nThe ADM mass of the solution is \nM = r 0 V R 1 R 2 g 2 (cosh 2 α 2 +cosh2 α 5 +cosh2 α 6 +cosh2 α p ) (4) \nand the Bekenstein-Hawking entropy is \nS = A 4 G 4 = 8 πr 2 0 V R 1 R 2 g 2 cosh α 2 cosh α 5 cosh α 6 cosh α p . (5) \nThere are three nontrivial scalar fields present in the solution and associated with these scalar fields are three pressures (scalar charges) \nP 1 = r 0 V R 1 R 2 g 2 (cosh 2 α 2 +cosh2 α 6 -cosh 2 α 5 -cosh 2 α p ) , P 2 = r 0 V R 1 R 2 g 2 (cosh 2 α 2 -cosh 2 α 6 ) , P 3 = r 0 V R 1 R 2 g 2 (cosh 2 α 5 -cosh 2 α p ) . (6) \nIn the ten-dimensional theory, the four charges (3) are carried by twobranes, fivebranes, sixbranes and strings. The D-sixbranes wrap around T 4 × S 1 × ˆ S 1 , the solitonic fivebranes wrap around T 4 × S 1 and the D-twobranes wrap around S 1 × ˆ S 1 . The strings carry momentum along the S 1 direction. In the spirit of [7] we calculate the values for the mass and scalar charges of each type of brane or string. This can be calculated from the solution we have presented by taking the four extremal limits: r 0 → 0 , α i →±∞ with Q i and α j ( j = i ) fixed. We find that D-twobranes have mass and pressures \n/negationslash \nM = P 1 = P 2 = R 1 R 2 g , P 3 = 0 , (7) \nwhile for the sixbranes we have \nM = P 1 = -P 2 = V R 1 R 2 g , P 3 = 0 . (8) \nFor the solitonic fivebrane we have \nM = -P 1 = P 3 = V R 1 g 2 , P 2 = 0 , (9) \nand for the momentum we find \nM = -P 1 = -P 3 = 1 R 1 , P 2 = 0 . (10) \nUsing these relations plus the charges (3) we trade in the eight parameters of the solution for the eight quantities ( n R , n L , N 2 , N ¯ 2 , N 5 , N ¯ 5 , N 6 , N ¯ 6 ) which are the numbers of right(left)-moving momentum modes, twobranes, anti-twobranes, fivebranes, antifivebranes, sixbranes and anti-sixbranes. We do this by matching the mass (4), pressures \n(6), and gauge charges (3) with those of a collection of noninteracting branes. This leads to \nn R = r 0 V R 2 1 R 2 2 g 2 e 2 α p , N 2 = r 0 V 2 g e 2 α 2 , N 5 = r 0 R 2 2 e 2 α 5 , N 6 = r 0 2 g e 2 α 6 , n L = r 0 V R 2 1 R 2 2 g 2 e -2 α p , N ¯ 2 = r 0 V 2 g e -2 α 2 , N ¯ 5 = r 0 R 2 2 e -2 α 5 , N ¯ 6 = r 0 2 g e -2 α 6 . (11) \nIn terms of the brane numbers, the ADM mass is reexpressed as \nM = 1 R 1 ( n R + n L ) + R 1 R 2 g ( N 2 + N ¯ 2 ) + V R 1 g 2 ( N 5 + N ¯ 5 ) + V R 1 R 2 g ( N 6 + N ¯ 6 ) , (12) \nthe gauge charges are simply differences of the brane numbers, and the other parameters are \nV = √ N 2 N ¯ 2 N 6 N ¯ 6 , R 2 = √ N 5 N ¯ 5 g 2 N 6 N ¯ 6 , R 2 1 R 2 = √ g 2 n R n L N 2 N ¯ 2 . (13) \nThe entropy (5) then takes the surprisingly simple form \nS = 2 π ( √ n R + √ n L )( √ N 2 + √ N ¯ 2 )( √ N 5 + √ N ¯ 5 )( √ N 6 + √ N ¯ 6 ) . (14) \nThis is the analog of (1) for four-dimensional black holes. When one term in each factor vanishes, the black hole is extremal. In this case, (14) agrees with the number of bound states of these branes at weak coupling [3]. Although we cannot derive the general formula from counting string states, we can do so in certain limits corresponding to near-extremal black holes. Consider the case when N ¯ 2 = N ¯ 5 = N ¯ 6 = 0 and R 1 is large. We see from (12) that the lightest excitations will be the momentum modes. The extremal limit is obtained by also setting the number of left movers to zero n L = 0. In that case the entropy can be calculated [3] as the entropy of a one-dimensional gas of 4 N 2 N 5 N 6 bosonic particles plus an equal number of fermionic particles with total energy E = n R /R 1 , which gives S = 2 π √ N 2 N 5 N 6 n R . In the near-extremal limit we also include left movers, which will be noninteracting if R 1 is large. Hence the entropy will be the sum \nS = 2 π √ N 2 N 5 N 6 ( √ n R + √ n L ) (15) \nwhich clearly agrees with (14) when N ¯ 2 ∼ N ¯ 5 ∼ N ¯ 6 ∼ 0. Note that these antibrane excitations are very massive when R 1 is large, so one can see from (11), (12) that their number will be very small in the near-extremal limit and their contribution to the entropy will be negligible. We could do a similar calculation for the cases in which the lightest particles are the other branes. Since U-duality interchanges the different branes and strings, one expects a result similar to (15) with the indices permuted. Equation (14) is clearly the simplest duality invariant expression which agrees with these different nonextremal limits. \nWe have considered only four types of charges. Reducing Type II string theory to four dimensions on T 6 leads to a theory whose low energy limit is N = 8 supergravity. This contains 28 gauge fields and 70 scalars. The gauge fields can carry either electric or magnetic charges, so there are 56 possible charges. Each of these charges is carried by a different type of soliton in the ten-dimensional theory. From black hole uniqueness theorems [10] it is clear that the Bekenstein-Hawking entropy of the general solution depends on the energy and 56 'solution generating parameters' that add charge. However, these parameters are not the physically normalized charges, but also involve the asymptotic values of the scalars. From the special form of the coupling of scalars to gauge fields in N = 8 supergravity [11], one sees that a basis may be chosen for the scalars in which only 56 of them enter in the normalization of the gauge charges. One can view these parameters as 55 scalars and the total energy. The entropy can then be viewed as a function of 56+56 parameters which may be interpreted as the number of solitons and anti-solitons. \nSince the full theory should be E 7 invariant we should be able to write the general entropy formula in an invariant way. If we denote by V A 1 the 56-dimensional vector giving the number of solitons and by V A 2 the number of anti-solitons, the formula for the entropy may take the form \nS = 2 π ∑ i,j,k,l √ T ABCD V A i V B j V C k V D l , (16) \nwhere T ABCD is the quartic invariant considered in [12], where this formula was derived for the extremal case ( V A 2 = 0). \nWe now consider adding rotation to the black holes discussed above. Since the rotation dependent terms in the solution fall off faster at infinity than the charges, the definition of the brane numbers (11) is unchanged. If we again take nearly extremal black holes with N ¯ 2 ∼ N ¯ 5 ∼ N ¯ 6 ∼ 0, and R 1 large, the Bekenstein-Hawking entropy takes the form [8] 1 \nS = 2 π ( √ n R N 2 N 5 N 6 + √ n L N 2 N 5 N 6 -J 2 ) . (17) \nwhere J is the angular momentum of the black hole. This agrees precisely with the counting of string states as follows. With R 1 much larger than the other compact dimensions and with just twobranes and sixbranes present, the D-brane excitations of this system are described by a 1+1-dimensional field theory which turns out to be a (4 , 4) superconformal sigma model [2]. The fivebrane breaks the right-moving supersymmetry [13], leaving us with (0 , 4) superconformal symmetry. The N = 4 superconformal algebra gives rise to a left-moving SU (2) symmetry. Since fermionic states in the sigma model become spinors in spacetime, the action of O (3) spatial rotations has a natural action on this SU (2). The charge F L under one U (1) subgroup of this SU (2) will then be related to the fourdimensional angular momentum (along one of the three axes) carried by the left movers by J = F L / 2. Due to the presence of the fivebrane the right-moving SU (2) symmetry of the original (4 , 4) superconformal field theory is broken and the right movers cannot carry macroscopic angular momentum. The number of states with fixed n L , n R , F L /greatermuch 1 may \nbe computed as in [2,6] to yield the entropy \nS = 2 π √ c 6 ( √ n R + √ ˜ n L ) , (18) \nwhere ˜ n L = n L -6 J 2 /c is the effective number of left movers that one is free to change once one has demanded that we have a given macroscopic angular momentum. For our problem the central charge is c = 6 N 2 N 5 N 6 [3], thus the entropy (17) agrees with the D-brane formula (18). \nIt is interesting to take the extremal limit of these rotating black holes, when the mass takes the minimum value consistent with given angular momentum and charges. This happens when ˜ n L = 0, so the left movers are constrained to just carry the angular momentum and do not contribute to the entropy. When the angular momentum is nonzero, even the extremal black hole is not supersymmetric. Using (18) and writing the result in terms of the charge n = n R -n L we find \nS = 2 π √ J 2 + nQ 2 Q 5 Q 6 , (19) \nwhich indeed agrees with the entropy of an extremal charged rotating black hole [8]. Notice the surprising fact that although we derived this formula in the large R 1 regime (and J/M 2 /lessmuch 1), it continues to be valid for arbitrary values of the parameters. Since this is far from the BPS state, we had no reason to expect the weak-coupling counting to continue to agree with the black hole entropy. † \nTo summarize, we first considered four-dimensional nonrotating black holes. We argued that there is a sense in which one can view the general nonextremal black hole as composed of a collection of noninteracting branes and anti-branes. The number of branes of each type is determined by matching physical properties of the branes with those of the black hole. In terms of these numbers, the Bekenstein-Hawking entropy takes the simple form (14). We were able to show that in certain limits, this expression agrees with the number of states of this collection of branes and anti-branes at weak coupling. A complete derivation of these formula remains an outstanding challenge. We also showed that for nearly extremal rotating black holes, the entropy again agrees with the number of string states. Surprisingly, the extremal rotating black hole entropy was precisely matched by a D-brane counting argument, even far beyond the regime in which this counting was done. \nThere have been earlier indications [4] that the counting of string states at weak coupling agrees with the black hole entropy even in situations where one could not justify the extrapolation to strong coupling. We found another example of this in the case of extremal rotating black holes. The surprising success of these weak-coupling arguments indicates that understanding black hole entropy may be even simpler than it appears today.", 'Acknowledgements': 'We would like to thank M. Cvetic, R. Kallosh, B. Koll, A. Peet, A. Strominger, and L. Susskind for useful discussions. The research of G.H. is supported in part by NSF Grant PHY95-07065, D.L is supported in part by NSF Grant PHY91-16964, and J. M. is supported in part by DOE grant DE-FG02-91ER40671.', 'References': '- [1] A. Strominger and C. Vafa, hep-th/9601029.\n- [2] J. Breckenridge, R. Myers, A. Peet and C. Vafa, hep-th/9602065.\n- [3] J. Maldacena and A. Strominger, hep-th/9603060; C. Johnson, R. Khuri, R. Myers, hep-th/9603061.\n- [4] C. Callan and J. Maldacena, hep-th/9602043.\n- [5] G. Horowitz and A. Strominger, hep-th/9602051.\n- [6] J. Breckenridge, D. Lowe, R. Myers, A. Peet, A. Strominger and C. Vafa, hepth/9603078.\n- [7] G. Horowitz, J. Maldacena and A. Strominger, hep-th/9603109.\n- [8] M. Cvetic and D. Youm, hep-th/9603147.\n- [9] M. Cvetic and D. Youm, hep-th/9508058; hep-th/9512127.\n- [10] P. Breitenlohner, D. Maison, and G. Gibbons, Commun. Math. Phys. 120 (1988) 295.\n- [11] E. Cremmer and B. Julia, Nucl. Phys. B159 (1979) 141; B. De Wit and H. Nicolai, Nucl. Phys. B208 (1982) 323.\n- [12] R. Kallosh and B. Koll, hep-th/9602014; R. Dijkgraaf, E. Verlinde and H. Verlinde, hep-th/9603126.\n- [13] J.A. Harvey and A. Strominger, Nucl. Phys. B449 (1995) 535.'}

2013PASJ...65..118S

Rotation Curve and Mass Distribution in the Galactic Center - From Black Hole to Entire Galaxy

2013-01-01

['galaxies', 'galaxies kinematics and dynamics', 'galaxies structure', 'ism kinematics and dynamics', 'astronomy radio', '-']

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Analyzing high-resolution longitude-velocity (LV) diagrams of the Galactic Center observed with the Nobeyama 45-m telescope in the CO and CS line emissions, we obtained a central rotation curve of the Milky Way. We combined it with data for the outer disk, and constructed a logarithmic rotation curve of the entire Galaxy. The new rotation curve covers a wide range of radius from r ∼ 1 pc to several hundred kpc without a gap of data points. It links, for the first time, the kinematical characteristics of the Galaxy from the central black hole to the bulge, disk and dark halo. Using this grand rotation curve, we calculated the radial distribution of the surface mass density in the entire Galaxy, where the radius and derived mass densities vary over a dynamical range with several orders of magnitudes. We show that the galactic bulge is deconvolved into two components: the inner (core) and main bulges. Both of the two bulge components are represented by exponential density profiles, but the de Vaucouleurs law was found to fail to represent the mass profile of the galactic bulge.

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https://arxiv.org/pdf/1307.8241.pdf

{'Rotation Curve and Mass Distribution in the Galactic Center - From Black Hole to Entire Galaxy -': 'Yoshiaki Sofue 1 , 2 \n1. Institute of Astronomy, The University of Tokyo, Mitaka, 181-0015 Tokyo, 2. Department of Physics, Meisei University, Hinoshi-shi, 191-8506 Tokyo Email:[email protected] \n(Received 2013 0; accepted 2013 0)', 'Abstract': 'Analyzing high-resolution longitude-velocity (LV) diagrams of the Galactic Center observed with the Nobeyama 45-m telescope in the CO and CS line emissions, we obtain a central rotation curve of the Milky Way. We combine it with the data for the outer disk, and construct a logarithmic rotation curve of the entire Galaxy. The new rotation curve covers a wide range of radius from r ∼ 1 pc to several hundred kpc without a gap of data points. It links, for the first time, the kinematical characteristics of the Galaxy from the central black hole to the bulge, disk and dark halo. Using this grand rotation curve, we calculate the radial distribution of surface mass density in the entire Galaxy, where the radius and derived mass densities vary over a dynamical range with several orders of magnitudes. We show that the galactic bulge is deconvolved into two components: the inner (core) and main bulges. Both the two bulge components are represented by exponential density profiles, but the de Vaucouleurs law was found to fail in representing the mass profile of the galactic bulge. \nNote: Preprint with full figures is available from http://www.ioa.s.u-tokyo.ac.jp/ sofue/htdocs/2013rc/ \n∼ Key words: galaxies: Galactic Center - galaxies: mass - galaxies: the Galaxy - galaxies: rotation curve', '1. Introduction': "Determination of the mass distribution in the Galaxy is one of the most fundamental subjects in galactic astronomy, and is usually obtained by analyzing rotation curves (Sofue and Rubin 2001). The rotation curve from the inner disk to the dark halo and their mass distributions have been obtained with significant accuracy (Sofue et al 2009; Sofue 2012; Honma et al. 2012; and the literature therein). The innermost mass structure within a few pc around the central black hole has been extensively studied by analyzing stellar kinematics (Crawford et al. 1985; Genzel and Townes 1987; Rieke and Rieke 1988; Lindqvist et al. 1992; Genzel et al. 1994, 2010; Ghez et al. 2005, 2008; Gillessen et al. 200). \nThe mass structure between the central black hole and the disk, and therefore, the dynamical mass structure inside the bulge, is not thoroughly studied. We derive the rotation curve in the Galactic Center, which has remained as the last unresolved problem of the rotation curve study of the Galaxy. We derive a central rotation curve using longitude-velocity diagrams obtained by the highest resolution molecular line observations. The curve will be deconvolved into classical mass components: the black hole, bulge, disk, and dark halo. During the analysis, we show that the de Vaucouleurs ( e -( r/a ) 1 / 4 ) law cannot fit the bulge's mass structure, and that the bulge is composed of two concentric mass components with exponential density profile ( e -r/a ). The fitted parameters will become \nthe guideline to analyze perturbations often highlighted as non-circular motions and bar. \nThe dynamical parameters of the Galaxy to be determined from observations are summarized in table 2 in Appendix. In the present paper we try to fix the parameters (1) to (10) in the table for the most fundamental axisymmetric part. Non-circular motions have often been stressed in the discussion of central dynamics (Binney et al. 1991; Jenkins and Binney 1994; Athnasoula 1992; Burton and Liszt 1993). However, discussing non-axisymmetric dynamics first is akin to calculating epicyclic frequency without angular velocity. The present analysis is limited only to items (1) to (10) for the axisymmetric part, which describe the first approximation of the galactic structure. The second-order parameters (11) to (27) are beyond the scope of this paper. Although the accuracy of the obtained result may not be as good as that of the outer disk rotation curve, the present parameters will become the basis for analyses of the second-order parameters such as bars and arms. \nThe galactocentric distance and the circular velocity of the Sun are taken to be ( R 0 , V 0 )=(8.0 kpc, 200 km s -1 ). We also examine a case for the newest values of ( R 0 ,V 0 )=(8.0 kpc, 238 km s -1 ) obtained by recent VERAVLBI observations (Honma et al. 2012). \nFig. 1. Integrate intensity map of the 12 CO( J =1 -0) line (115.27 GHz) made from data by Oka et al. (1998). \n<!-- image --> \nFig. 2. Longitude-velocity (LV) diagram in the CO line averaged over latitudes from b = -30 ' to +30 ' (data from Oka et al. 1998). \n<!-- image -->", '2. Longitude-Velocity Diagrams': "Figure 1 shows the intensity distribution of molecular gas in the 12 CO( J = 1 -0) line (115.27 GHz) for the central ± 1 · region of the Galactic Center as produced from the survey data using the Nobeyama 45-m telescope by Oka et al. (1998). Although the telescope beam was 15 '' , the observations were obtained with 34 '' gridding, resulting in an effective resolution of 37 '' . \nFigure 2 shows a longitude-velocity (LV) diagram of the central molecular disk averaged from b = -20 ' to +20 ' . The LV diagram is characterized by two major structures. The tilted LV ridges are the most prominent features, which correspond to dense nuclear gas arms in the disk (Sofue 1995a). The tilted LV ridges shares most of the intensities, while the high-velocity components shares only a few percents of the total gas (Sofue 1995a,b). The tilted LV ridges correspond to the major disk at low latitudes making ring-like arms. The high-velocity arcs come from fainter structures extending perpendicular to the disk at higher latitudes. \nThe high-velocity arcs are visible as parts of an ellipse \ncorresponding either to parallelogram (Binney et al. 1991) or to an expanding molecular ring (Kaifu et al. 1972; Scoville 1972; Sofue 1995b). The LV ridges and the highvelocity arcs must not be physically related to each other, but are present at different locations, since their distributions and kinematics cannot be originating from the same gas cloud, unless different gas streams can cross each other. In the present analysis, we focus on the LV ridges corresponding to the major disk. \nIn order to examine the kinematics of the molecular disk in more details, and to see if the major gas disk is characterized by the tilted-ridge structures, we present LV diagrams at different latitudes by slicing the disk. Figures 3 and 4 show the LV diagrams in the 12 CO( J = 1 -0) (115.27 GHz; Oka et al. 1998) and C 32 S( J =1 -0) (48.99 GHz; Tsuboi et al. 1999) line emissions. The CS data cube had an effective resolution of 46 '' arising from the telescope beam of 35 '' of the 45-m telescope and the grid spacing of 30 '' during the observations. \nThe figures show that most of the molecular gas is distributed on the tilted ridges, representing the central molecular zone (CMZ), or the main Galactic Center disk. The major LV ridge features are indicated in the figures such as the ridge of dense molecular gas running from ( l,v ) = ( 0 . 6 , -150) to (0 . 2 , +130) (in degree and km s -1 ). \n- \n-The tilted ridges make the fundamental structure in the LV diagram, whereas the parallelogram/expanding shell is much fainter. From comparison of figures 3 and 4, we learn that the denser gas represented by CS line is more strongly concentrated in the tilted ridges. Also, the highvelocity arcs are hardly seen in the CS line emission. We consider that the dynamics of the Galactic Center disk is better represented by these dense molecular features than by the high-velocity arcs. In this paper, we thus analyze the tilted LV molecular gas ridges. \nFigure 5 shows an LV diagram in the 12 CO( J =1 -0) line emission observed using the NRO 45-m telescope at a 15 '' with 7 '' . 5 Nyquist-sampling gridding. In this highest resolution LV diagram, we can also recognize a tilted LV ridge running from ( l, v ) = ( -1 . 5 ' , -80) to (1 . 2 ' , 100).", '3.1. Terminal Velocities by LV Ridge Tracing': "Tilted ridges in the LV diagrams are naturally interpreted as due to arms and rings rotating around the GC, as illustrated in figure 6. In fact, the major LV ridges have been shown to represent ring/arm like structures in the main disk (Sofue 1995a, b). We now trace the LV ridges on the observed LV diagrams shown in figures 3, 4 and 5. By applying the terminal velocity method (Rubin and Sofue 2001) to each LV ridge, we determine rotational velocities on individual LV ridges. Here, we adopted the terminal velocity as the velocity showing the steepest gradient. \nSince the original LV diagrams in the Galactic Center are superposed by various kinematical features such as molecular clouds, star forming regions (Sgr B, C, etc..), high-velocity wings (e.g. Oka et al. 1998), and fore- \nNo. ] \nFig. 3. Longitude-velocity diagrams at different latitudes in the 12 CO( J = 1 -0) line (115.27 GHz; Oka et al. 1998) from b = -11 ' . 9 (top-left) to b = +6 ' . 23 (bottom-right) at 2 ' . 26 latitude interval. Overlaid are traced LV ridges and corresponding terminal velocity tracing. \n<!-- image --> \nFig. 4. Same as figure 3 but in C 32 S( J =1 -0) line (48.99 GHz) data from Tsuboi et al. 1999), showing denser gas kinematics, with latitude interval 1 ' . 553 from b = -9 ' . 033 (top left) to +4 ' . 187 (bottom right). \n<!-- image --> \n<!-- image --> \nFig. 5. Top: High-resolution LV diagrams of the Galactic Center in the 12 CO ( J =1 -0) line observed with the NRO 45-m telescope at resolution of 15 '' at latitude interval 19 '' . 56 from b = 3 ' 14 '' (top left) to 0 ' 37 '' (bottom right). Bottom: Same for the most clear LV ridge at b = -2 ' 08 '' . \n<!-- image --> \nFig. 6. Circular rotation (full lines) vs streaming motion and expanding ring (dashed lines) projected on the galactic plane and LV plane. \n<!-- image --> \nground absorption, it was not practical to write a program to determine the terminal velocity automatically from machine-readable data. So, we read the velocities by eye, judging individually the LV behaviors in velocity and longitudinal extents. The thus measured velocities and positions included errors of the order of ∼ ± 10 km s -1 in velocity and about twice the effective angular resolutions. \nThe obtained terminal velocities are shown in figure 7. The velocities are scattered locally by 20-30 km s -1 . The east-west asymmetry in velocity distribution is greater than the local scatter, and amounts to almost 30-40 km s -1 . Open circles in figure 7 shows a rotation curve produced from the thus measured terminal velocities. The filled circles in the bottom panel in figure 7 show runningaveraged values every 1.3 times the neighboring radius with a Gaussian weighting width of 0.3 times the radius. \nStandard deviations among the neighboring data points during the running-averaging process are indicated by the horizontal (position) and vertical (velocity) error bars. Although each velocity includes small error of order ∼ 10 km s -1 , the running-averaged values have errors of ∼ ± 20 -30%. This large scatter (error) cannot be removed, because it arises from the dynamical property of molecular gas in the Galactic Center.", '3.2. Grand rotation curve from the black hole to halo': 'Figure 8 shows the obtained running-averaged rotation curve for the central region combined with the rotation curve of the whole galactic disk. The plotted data beyond the bulge were taken from our earlier papers (Sofue et al 2009; Sofue 2009). The bulge component with its peak at R ∼ 0 . 3 kpc seems to decline toward the center faster than \nthat expected for the de Vaucouleurs law as calculated by Sofue (2012). Figure 7 is the same within 0.5 kpc, which shows that the velocity is followed by a flat part at R ∼ 0 . 1 to 0.01 kpc. The rotation velocity within the bulge at R ≤∼ 0 . 5 kpc seems to be composed of two separate components, one peaking at R ∼ 0 . 3 kpc, and the other a flat part at R 0 . 01 -0 . 1 kpc. \n∼ \n-Figure 9 is a logarithmic plot of the measured rotation velocities combined with the grand rotation curve of the Galaxy covering the dark halo (Sofue 2012). The logarithmic representation is essential to analyze the central part, as it enlarges the radial scale toward the center according to the variation of dynamical scale. In the figure, the disk to bulge rotation data have been adopted from the existing HI and molecular line observations (Burton and Gordon 1978; Clemens 1985; and the literature in Sofue (2009)). The inner straight dashed line represents the central massive black hole of mass 3 . 6 × 10 6 M /circledot (Genzel et al. 2000; Ghez et al. 2005; Gillesen et al. 2009). \nHere, in figure 9, the observed rotation velocities have been running averaged by Gaussian convolution around each representative radius at every 1 + /epsilon1 times the neighboring inner point with a Gaussian width of ± η times the radius. Here we take /epsilon1 = η =0 . 1 for radius 3 <R< 15 kpc where data points are dense, and otherwise 0.3. For an initial radius a , the j -th radius is given by \nr =(1+ /epsilon1 ) ( j -1) a, (1) \nand the Gaussian width for running average (Gaussian convolution) is taken as \n∆ r = ηr. (2) \nThe mean value of an observable f ( r ), which is either the radius r or the velocity V , at r is calculated as \nf ( r ) = Σ f i w i Σ w i , (3) \nwhere w i is the weight given by \nw i =exp [ -( r i -r ∆ r ) 2 ] . (4) \nThe statistical error of the observable is calculated by \nσ i = [ { Σ( f i -f ) 2 w i } Σ w i -1 ] 1 / 2 . (5) \nThe curve is drawn to connect the central rotation curve smoothly to the Keplerian law by the central massive black hole. This figure demonstrates, for the first time, continuous variation of the rotational velocity from the central black hole to the dark halo. Table 3 lists the obtained rotation velocities.', '4.1. Rotation Curve around the Galactic Center': "The logarithmic rotation curve shows the central disk behavior more appropriately than the linear rotation curves. The main bulge component has a velocity peak \n<!-- image --> \nFig. 7. Top: Rotation velocities (grey dots) in the Galactic Center obtained by LV ridge terminal method. The error bars represent effective spatial resolutions taken to be twice the effective angular resolution in the data (15 '' for our innermost CO data by 45-m telescope; 37 '' for CO data by Oka et al. (1989); 48 '' for CS data by Tsuboi et al. (1991)), and eye-estimated terminal velocity errors of ± 10 km s -1 . Open circles show running-averaged values every 3 points using the neighboring 5 data points. The error bars denote standard deviation in the averaged data used in each plotted point. Bottom: Same up to 0.6 kpc, but Gaussian-running averaged velocities combined with the data by Sofue et al. (2009). Deconvolved components (inner and main bulges, disk and dark halo) are indicated by the full lines. \n<!-- image --> \nat r /similarequal 400 pc with V = 250 km s -1 . It declines toward the center steeply, followed by a plateau-like hump at r ∼ 30 -3 pc. The plateau-like hump is then merged by the Keplerian rotation curve corresponding to the central black hole at r < ∼ 2 pc. Here we use a black hole mass of M BH = 4 × 10 6 M /circledot , taking the mean of the recent values converted to the case for R 0 = 8 . 0 kpc, i.e., 2 . 6 -4 . 4 × 10 6 M /circledot (Genzel et al. 2000, 2010), 4 . 1 -4 . 3 × 10 6 M /circledot (Ghez et al. 2005), and 3 . 95 × 10 6 M /circledot (Gillessen et al. 2009).", '4.2. Broad velocity maximum by de Vaucouleurs law': "We first compare the observations with the best-fit de Vaucouleurs law for the galactic bulge as obtained in our previous works (Sofue et al. 2009), which is shown by the dashed lines in figure 11. It is obvious that the model fit is not sufficient in the inner several hundred pc. It is valuable to revisit de Vaucouleurs rotation curve, which \n<!-- image --> \nFig. 8. Top: Rotation curve of the Galaxy from Sofue et al. (2009) combined with the running-averaged rotation velocities in the Galactic Center. Bottom: Same, but smoothed curve by Gaussian-running mean. Deconvolved components are indicated by the full lines. The Upper curve with open circles is a smoothed rotation curve for the newest values of ( R 0 =8 . 0 kpc, V 0 =238 km s -1 ). \n<!-- image --> \nFig. 9. Logarithmic rotation curve of the entire Galaxy by combining the present data with the outer curve from Sofue et al. (2009). The thin line represents the central black hole of mass 3 . 6 × 10 6 M /circledot . The innermost three dots are interpolated values using the data and a Keplerian curve for the black hole. \n<!-- image --> \nis represented by Σ ∝ e -( r/a ) 1 / 4 with Σ and a being the surface mass density and scale radius. By definition scale radius a used here is equal to R b / 3460 . , where R b is the half-surface mass radius used in usual de Vaucouleurs law expression (e.g. Sofue et al. 2009). \nSince Σ is nearly constant at r /lessmuch a , the volume density varies as ∝ 1 /r and the enclosed mass ∝ r 2 . This leads to circular velocity V = √ GM/r ∝ r 1 / 2 near the center. Thus, the rotation velocity rises very steeply with infinite gradient at the center. It should be compared with the mildly rising velocity as V ∝ r at the center in the other models. \nAt large r > a , the de Vaucouleurs law has slower density decrease due to the weaker dependence on r ( r 1 / 4 effect) than the other models. This leads to more gentle decrease after the maximum. Figure 10 shows normalized behaviors of rotation velocity for de Vaucouleurs and other models. As the result of steeper rise near the center and slower decrease at large radii, the de Vaucouleurs rotation curve shows a much broader maximum in logarithmic plot compared to the other models. \nWe here define the half-maximum logarithmic velocity width by ∆ log =log r 2 -log r 1 , where r 2 and r 1 ( r 2 >r 1 ) are the radii at which the rotational velocity becomes equal to a half of the maximum velocity. From the calculated curves in figure 10, we obtain ∆ log = 3 . 0 for de Vaucouleurs , while ∆ log =1 . 5 for the other models. Thus the de Vaucouleurs 's logarithmic curve width is twice the others, and the curve's shape is much milder. Note that the logarithmic curve shape keeps the similarity against changed parameters such as the mass and scale radius. \nIn figure 11 (top) we show LogRC calculated for the de Vaucouleurs model by dashed lines and compare with the observations. It is obvious that the de Vaucouleurs law cannot reproduce the observations inside ∼ 200 pc. Note that the shape of the curve is scaling free in the logarithmic plot. The de Vaucouleurs curve can be shifted in both directions by changing the total mass and scale radius, but the shape is kept same.", '4.3. Exponential spheroid model': "Since the de Vaucouleurs law was found to fail to fit the observed LogRC, we now try to represent the inner rotation curve inside ∼ 100 pc by different models. We propose a new functional form for the central spheroidal component, which we call the exponential sphere model. In this model, the volume mass density ρ is represented by an exponential function of radius r with a scale radius a as \nρ ( r ) = ρ c e -r/a . (6) \nThe mass involved within radius r is given by \nM ( R ) = M 0 F ( x ) , \n(7) \nwhere x = r/a and \nF ( x ) = 1 -e -x (1 + x + x 2 / 2) . (8) \nThe total mass is given by \n- (8) to the data by trial and error by changing the parameter values. \nFig. 10. Comparison of normalized rotation curves for the exponential spheroid, de Vaucouleurs spheroid, and other typical models, for a fixed total mass. The exponential spheroid model is almost identical to that for Plummer law model. \n<!-- image --> \nM 0 = ∫ ∞ 0 4 πr 2 ρdr =8 πa 3 ρ c . (9) \nThe circular rotation velocity is then calculated by \nV ( r ) = √ GM/r = √ GM 0 a F ( r a ) (10) \n√ where G is the gravitational constant. \nThis model is simpler than the canonical bulge models such as the de Vaucouleurs profiles. Since the density decreases faster, the rotational velocity has narrower peak near the characteristic radius in logarithmic plot as shown in figure 10. The exponential-sphere model is nearly identical to that for the Plummer's law, and the rotation curves have almost identical profiles. Hence, the results in the present paper may not be much changed, even if we adopt the Plummer potential.", '5.1. Mass Component Fitting': 'In order to fit the observed rotation curve by models, we assume the following components, each of which should be determined of the parameters as listed in table 2 : \n· \n- The central black hole with mass M BH =4 × 10 6 M /circledot .\n- × · An innermost spheroidal component with the exponential-sphere density profile, or a central massive core.\n- · A spheroidal bulge with the exponential-sphere density profile.\n- An exponential flat disk.\n- · A dark halo with NFW profile. \n· \nThe approximate parameters for the disk and dark halo are adopted from the current study such as by Sofue (2012), and were adjusted here in order to better fit the data. The inner two spheroidal components were fitted to the data by trial and error by changing the parameter values. \nFig. 11. Logarithmic rotation curve of the Galaxy compared with model curves and deconvolution into mass components. Solid lines represent the best-fit curve with two exponential-spherical bulges, exponential flat disk, and NFW dark halo. The classical de Vaucouleurs bulge is shown by dashed line, which is significantly displaced from the observation. Open circles are a new rotation curve adopting the recently determined circular velocity of the Sun, V 0 =238 km s -1 (Honma et al. 2012). \n<!-- image --> \nAfter a number of trials, we obtained the best-fit parameters as listed in table 1. Figure 11 shows the calculated rotation curve for these parameters. The result satisfactorily represents the entire rotation curve from the central black hole to the outer dark halo. \nWe find that the fitting is fairly good in the Galactic Center, and the inner two peaks of rotation curve at r ∼ 0 . 01 kpc and ∼ 0 . 5 kpc are well reproduced by the two exponential spheroids. The figure also demonstrates that the present model is better than the de Vaucouleurs model shown by the dashed line. Figure 8 shows the usual presentation of the rotation curve up to 15 kpc in linear scales. The bottom panel enlarges the central several hundred pc region. \nTable 1 lists the fitted parameters for individual components. The disk and halo parameters are about the same as those determined in our earlier paper (Sofue 2012). The classical bulge is composed of two superposed components. The inner bulge, or a massive core, has a mass of 5 × 10 7 M /circledot , scale radius of 3 . 8 pc, and the central density of 4 × 10 4 M /circledot pc -3 . The main bulge has a mass 10 10 M /circledot , scale radius 120 pc, and central density 200 M /circledot pc -3 . The central volume densities are consistent with the surface mass density (SMD) of the order of ∼ 10 5 M /circledot pc -2 at r ∼ 3 -10 pc directly calculated from the rotation curve (Paper I).', '5.2. Volume density': 'Figure 12 shows the resulting volume density profiles in the entire Galaxy for the total and individual mass components as functions of radius in logarithmic presentation. The bottom panel shows the same but for the Galactic Center in semi-logarithmic scaling. \nThe calculated dark matter distribution shows a steep cusp near the nucleus because of the 1 /x factor in the NFW profile. However, since the functional form was derived from numerical simulations with much broader res- \nolution (Navarro et al. 1996), the exact behavior in the immediate vicinity of the nucleus may not be taken so serious, but it may include significant uncertainties.', '5.3. Surface mass density': 'Using the best-fit model rotation curve, we calculated the surface-mass density (SMD) as a function of radius. Figure 13 shows the calculated results, both for the spherical and flat-disk assumptions by applying the method developed by Takamiya and Sofue (2000). In the figure, we also show the SMD distributions directly calculated using the observed rotation curve. The observed SMD is thus reproduced by the present model within errors of a factor of ∼ 1 . 5 -2 throughout the Galaxy. Here, the errors were estimated by eyes from the plots in the figure.', '5.4. Direct Calculation of Surface Mass Distribution': "Using the obtained rotation curve, we can also calculate the distribution of surface mass density (SMD) more directly (Takamiya and Sofue 2000). For a spherically symmetric model, the mass M ( r ) inside the radius r is calculated by using the rotation curve as: \nM ( r ) = rV ( r ) 2 G , (11) \nwhere V ( r ) is the rotation velocity at r . Then the SMD Σ S ( R ) at R is calculated by, \nΣ S ( R ) = 2 ∞ ∫ 0 ρ ( r ) dz, (12) \n= 1 2 π ∞ ∫ R 1 r √ r 2 -R 2 dM ( r ) dr dr. (13) \nHere, R , r and z are related by r = √ R 2 + z 2 . The SMD for a thin flat-disk, Σ D ( R ), is derived by solving \n<!-- image --> \nFig. 12. Left: Logarithmic plots of volume density of the exponential spheroids, exponential disk, and NFW halo calculated for the fitted parameters. Right: Same but by semi-logarithmic plot for the innermost region. \n<!-- image --> \n<!-- image --> \nFig. 13. Left: Surface-mass density profiles. Solid line shows SMD calculated by using the model rotation curve in spherical assumption. Thin dashed lines show individual components. Thick dashed line is SMD in flat-disk assumption. Grey dots and open circles show SMD calculated by using the observed rotation curve in spherical and flat-disk assumptions, respectively. Right: Same, but direct mass alone in semi-logarithmic presentation with the exponential disk by dashed line. \n<!-- image --> \nTable 1. Parameters for exponential sphere model of the bulge. \n| Mass component | Total mass ( M /circledot ) | Scale radius (kpc) | Center density ( M /circledot pc - 3 ) |\n|--------------------|-------------------------------|----------------------|------------------------------------------|\n| Black hole | 4E6 | - | - |\n| Inner bulge (core) | 5.0E7 | 0.0038 | 3.6E4 |\n| Main Bulge | 8.4E9 | 0.12 | 1.9E2 |\n| Disk | 4.4E10 | 3.0 | 15 |\n| Dark halo | 5E10 ( r ≤ h ) | h =12 . 0 | ρ 0 =0 . 011 | \nthe Poisson's equation: \nΣ D ( R ) = 1 π 2 G × (14) \n 1 R R ∫ 0 ( dV 2 dr ) x K ( x R ) dx + ∞ ∫ R ( dV 2 dr ) x K ( R x ) dx x , \n/similarequal \nwhere K is the complete elliptic integral and becomes very large when x R (Binney & Tremaine 2008). \nFigure 13 shows the calculated SMD both for sphericalsymmetric mass model and for flat-disk model from R =1 pc to 1 Mpc. Both results agree with each other within a scatter of a factor of two. The radial profiles of SMD for the two models are similar to each other. The dashed lines represent the de Vaucouleurs law and exponential law approximately representing the bulge and disk components, respectively. The bottom panel of this figure shows the same but in semi-logarithm scaling, so that the exponential disk is represented by a straight line. It is also clearly shown that the outer SMD profile beyond R ∼ 10 kpc is significantly displaced from the disk's profile due to the dark matter halo. Table 4 lists the calculated SMD values. \nThe present SMD plot has sufficient resolution to reveal the connection between the central black hole and bulge in the central R ∼ 1 pc to 1 kpc region. The bulge and black hole appears to be connected by a dense core component, which fills the gap of rotation velocity between black hole and bulge. Figure 13 also demonstrates smooth variation of SMD from the central black hole to the outer dark halo. The bulge at R ∼ 0 . 5 kpc and exponential disk at R ∼ 3 kpc clearly show up as the two bumps, and are followed by the dark halo extending to ∼ 400 kpc. The mass distribution based on the grand rotation curve beyond 0.5 kpc has been extensively studied by Sofue (2012).", '6. Discussion': 'In contrast to the extensive research of the disk and outer rotation curve of the Galaxy (e.g., Sofue and Rubin 2001; Sofue et al. 2009), the Galactic Center kinematics, particularly rotation curve and mass distribution, has not been thoroughly highlighted. This is mainly due to the too much emphasized complexity of kinematics due to the supposed bar and non-circular stream motions such as items (11) to (27) in Appendix 1. \nIn this paper, we abstracted simpler structures in the Galactic Center molecular line data represented by the straight LV ridges observed at high-resolution in the \n12 CO( J = 1 -0) and C 32 S( J = 1 -0) lines. We have argued that the LV ridges can represent approximate circular rotation of the dense gas disk, and obtained a central rotation curve inside 1 ∼ 100 pc. The central rotation curve was connected to the inner curve corresponding to the nuclear black hole, and also to the outer curve of the bulge, disk and dark halo. Thus, a grand rotation curve covering the entire Milky Way, from the central black hole to dark matter halo, was constructed for the first time.', '6.1. Main bulge: Exponential profile and failure in the de Vaucouleurs law': 'The classical de Vaucouleurs law was found to fail to fit the observations (figure 11). This fact was recognized for the first time by using the logarithmic rotation curve. As argued in section 2, the de Vaucouleurs profile for the surface mass density requires a central cusp, yielding steeply rising circular velocity as V ∝ r 1 / 2 at the center. Beyond the velocity maximum at r > a , it declines more slowly due to the extended outskirt. Thus, the logarithmic halfmaximum velocity width is about twice that for the exponential spheroid or the Plummer law as shown in figure 10. This profile was found to be inappropriate to reproduce the observations as indicated in figure 11. \nIn order to reproduce the observations, we proposed a new bulge model, in which the volume density is represented by a simple exponential function as ρ = ρ 0 e -r/a . The main bulge was found to be represented well by the exponential-spheroid model with mass 8 × 10 9 M /circledot and scale radius 120 pc as shown in figure 11.', '6.2. Inner bulge (core): Dynamical link to the black hole': 'Inside the main bulge, a significant excess of rotation velocity was observed over those due to the black hole and main bulge (figure 11). This component was well explained by an additional inner spheroidal bulge of a mass of ∼ 5 × 10 7 M /circledot with the same exponential density profile as the main bulge with scale radius 3.8 pc. \nConsidering the relatively large scatter and error of data at r ∼ 3 -20 pc, the density profile may not be strictly conclusive. However, the velocity excess should be taken as the evidence for existence of an additional mass component filling the space between the black hole and main bulge, which we called the inner bulge. As an alternative mass model to explain the plateau-like velocity excess, an isothermal sphere with flat rotation might be a candidate. However, it yields constant velocity from the center to halo, so that some artificial cut off of the sphere is required at some radius. Such a sphere with an artificial \nboundary may not be a good model for the Galaxy. \nThe Keplerian velocity by the central black hole of mass 4 × 10 6 M /circledot declines to 100 km s -1 at r = 1 . 5 pc, where the observed velocities are about the same. This implies that the mass of the inner bulge enclosed in this radius is negligible compared to the black hole mass. In fact, the present model indicates that the mass inside r = 1 pc is only ∼ 1 . 2 × 10 5 M /circledot , an order of magnitude smaller than the black hole mass. \nThe central ∼ 1 pc region is, therefore, controlled by the strong gravity of the massive black hole. Stars there can no longer remain as a gravitationally bound system, but are orbiting around the black hole individually by Keplerian law. As an ensemble of the stars orbiting the black hole may show velocity dispersion on the order of v σ ( r ) ∼ 125( r/ 1pc) -1 / 2 km s -1 .', '6.3. Comparison with the previous works': 'It is worthwhile to examine if the present result is consistent with the previous works by other authors. For this purpose, we compare our result with the measurements and compilation of enclosed mass data by Genzel et al. (1994), which have been obtained using various kinds of objects such as giant stars, He I stars, HI and CO gases, circumnuclear disk, and mini spirals (See the literature therein for details). \nFigure 14 shows the enclosed mass as a function of radius calculated by the presently fitted model. In the figure we overlaid the results by Genzel et al. (1994), where their data have been converted to the case of R 0 =8 . 0 kpc from 8.5 kpc adopted in their paper, multiplying the radius scale by 8.0/8.5=0.94. The mass scale was also multiplied by the same amount, as the mass is proportional to ∝ rv 2 , while radial velocities v toward the Galactic Center are hardly affected by the galacto-centric distance. \nThe enclosed mass for the black hole is trivially constant. The inner and main bulges have constant density near the center, which yields enclosed mass approximately proportional to ∝ r 3 after volume integration. The disk model has constant surface density near the center, yielding enclosed mass ∝ r 2 for surface integration. However, it would be much less because of the finite thickness for the real galactic disk. The NFW model predicts a high density cusp near the center as ∝ r -1 , yielding enclosed mass proportional to ∝ r 2 . However, the dark matter density may not be taken so serious because of the unknown accuracy of the model in the vicinity of the nucleus. \nThe figure shows that the present result is in good agreement with the previous observations. We stress that the wavy variation in our profile due to the two-component bulge structure is also observed in the stellar kinematics results.', '6.4. Effect of a bar and the limitation of the present analysis': "First of all, the rotation curve analysis cannot treat the non-axisymmetric part of the Galaxy (11) to (27) as listed in table 2. It is true that the galactic disk is superposed by non-circular streaming motions such as due to bars, arms \nFig. 14. Comparison of enclosed mass calculated for the present rotation curve with the current results compiled by Genzel et al. (1994: see the literature therein for the plotted data). Horizontal line indicates the central black hole, thin lines show the inner bulge, main bulge and disk. The dashed line shows dark matter cusp. \n<!-- image --> \nand expanding rings. However, it is not easy to derive non-axisymmetric mass distribution from the existing observations. Simulations based on given parameters of bar potential can produce LV diagrams, and may be compared with the observations (Binney et al. 1991; Jenkins and Binney 1994; Athnasoula 1992; Burton and Liszt 1993). The present analysis would be a practical way to approach the dynamical mass structure of the central Galaxy. \nWe comment on the accuracy of the present analysis. The non-circular motions observed in the LV diagrams are as large as ∼ ± 20 -30% of the circular velocity. In the present analysis, these motions yield systematic errors of ∼ 40 -60% of mass estimation, and the accuracy of obtained mass is about ± 60%, or within a factor of ∼ 1 . 6. The accuracy is obviously not as good as those for the outer disk and halo galactic parameters. However, it may be sufficient for examining the fundamental mass structure for the first approximation in view of the large dynamical range of order of six in the logarithmic plots of SMD as in figures 12, 13 and 14. \nThe agreement of the present analysis with those from the stellar dynamics by Genzel et al. (1994) as shown in figure 14 may indicate that the bar's effect will not be so significant in the central several tens of parsecs. Stellar bar dynamics and stability analysis in the close vicinity of the massive black hole would be a subject for the future. It is an interesting subject to examine if such a strong gravity by the central mass structures may allow for a long-lived bar.", '6.5. Correction for the solar rotation velocity': 'Throughout the paper, we adopted the galactic parameters, ( R 0 , V 0 ) = (8 . 0 , 200) (kpc, km s -1 ). In their recent trigonometric measurements of positions and velocities using VERA, Honma et al. (2012) obtained a faster circular velocity of the Sun of V 0 =238 km s -1 . If we adopt this new value, the general rotation velocities also increases by several to ∼ 20% in the outer disk. We have corrected the observed rotational velocities for the difference between the new and current circular velocities of the Sun, ∆ V =238 -200=38 km s -1 , using the following equation. \nV c ( r ) = V ( r ) + ∆ V ( r R 0 ) . (15) \nFor globular clusters and satellite galaxies, the rotation velocities were obtained by multiplying √ 2 to their radial velocities to yield expected Virial velocities. We have not applied of the above correction to these cases, because their radial velocities are influenced only statistically by the change of solar velocity, and their mean values do not significantly change by different V 0 . \nIn figure 11 we plot the newly determined corrected rotation curve by open circles. The curve is not significantly changed in the central region as the above equation indicates. The outer most halo rotation curve using satellite galaxies also remains almost unchanged. A large difference is observed at r ∼ 6 to 20 kpc, where the rotation velocity is no longer flat. The rotation velocity increases up to ∼ 20 kpc, attaining a maximum at V /similarequal 270 km s -1 . Beyond r ∼ 20 kpc, the new rotation curve V c declines more steeply than that for the NFW profile, but rather consistent with Keplerian curve. This indicates that the dark matter is empty beyond ∼ 20 kpc. At this moment it is not clear if such a cut off of dark halo is indeed present, or if the simple extrapolation of the solar velocity to the outer part is allowed. \nAcknowledgement: The author thanks Dr. Fumi Egusa for data reduction of CO-line observations using the Nobeyama 45-m telescope, and for allowing him to use the LV diagram prior to publication of the result.', 'References': "Athanassoula, E. 1992 MNRAS 259, 345. \nBally, J., Stark, A. A., Wilson, R. W., & Henkel, C. 1987, ApJS, 65, 13 \nBinney, J. and Tremain, S. 2008, in 'Galactic Dynamics', 2nd ed., Chap. 2. \nBinney, J., Gerhard, O. E., Stark, A. A., Bally, J., & Uchida, K. I. 1991, MNRAS, 252, 210 Burton W B, Gordon M A. 1978. AA 63: Burton, W. B. and Liszt H. S. 1993 AA 274, 765. Clemens, D. P. 1985. Ap. J. 295:422 \nCrawford, M. K., Genzel, R., Harris, A. I., et al. 1985, Nature, 315, 467 \nGenzel, R., & Townes, C. H. 1987, ARA&A, 25, 377 \nGenzel, R., Hollenbach, D., & Townes, C. H. 1994, Reports on Progress in Physics, 57, 417 \nGenzel, R., Pichon, C., Eckart, A., Gerhard, O. E., & Ott, T. 2000, MNRAS, 317, 348 \n- Genzel, R., Eisenhauer, F., & Gillessen, S. 2010, Reviews of Modern Physics, 82, 3121\n- Ghez, A. M., Salim, S., Hornstein, S. D., et al. 2005, ApJ, 620, 744\n- Ghez, A. M., Salim, S., Weinberg, N. N., et al. 2008, ApJ, 689, 1044 \nGillessen, S., Eisenhauer, F., Trippe, S., et al. 2009, ApJ, 692, 1075 \n- Honma, M., Nagayama, T., Ando, K., et al. 2012, PASJ, 64, 136 \nJenkins, A., & Binney, J. 1994, MNRAS, 270, 703 Kaifu, N. Kato, T., Iguchi, T. 1972 Nature, 238, 105. \n- Lindqvist, M., Habing, H. J., & Winnberg, A. 1992, A&A, 259, 118\n- Navarro, J. F., Frenk, C. S., White, S. D. M., 1996, ApJ, 462, 563\n- Oh, C. S., Kobayashi, H., Honma, M., et al. 2010, PASJ, 62, 101\n- Oka, T., Hasegawa, T., Sato, F., Tsuboi, M., & Miyazaki, A. 1998, ApJS, 118, 455 \nRieke, G. H., & Rieke, M. J. 1988, ApJL, 330, L33 \n- Sakai, N., Honma, M., Nakanishi, H., et al. 2012, PASJ, 64, 108 \nScoville, N. 1972, ApJ 175, L127. Sofue, Y. 2012, PASJ, 64, 75 Sofue, Y., Honma, M., & Omodaka, T. 2009, PASJ, 61, 227 Sofue, Y., Rubin, V.C. 2001 ARAA 39, 137 Sofue, Y. 1995, PASJ, 47, 551 Sofue, Y. 1995, PASJ, 47, 527 Sofue, Y. 1996, ApJ, 458, 120 Sofue, Y. 2009, PASJ, 61, 153 Sofue, Y. 2012, PASJ, 64, 75 Sofue, Y., Honma, M., & Omodaka, T. 2009, PASJ, 61, 227 Takamiya, T., & Sofue, Y. 2000, ApJ, 534, 670 Tsuboi, M., Handa, T., & Ukita, N. 1999, ApJS, 120, 1 Xue, X. X., Rix, H. W., Zhao, G., et al. 2008, ApJ, 684, 1143", 'Appendix 1. List of dynamical parameters of the Galaxy': 'We list in table 2 the dynamical parameters of the Galaxy to be determined by the rotation curve analysis. Also listed are possible parameters for the second-order perturbations causing various non-circular motions.', 'Appendix 2. Tables of rotation curve and surface mass density of the Galaxy': 'In this appendix, we present the observed rotation curve and surface mass density (SMD) of the entire Galaxy in tables. Table 3 lists the radius r , Gaussian width of the running mean radius δr , observed rotation velocity V and its statistical error δV . Table 4 lists SMD, Σ s and Σ f , calculated using the rotation curve in table 3 for spherical symmetry and thin-disk assumptions, respectively. Digitized data are available from http://www.ioa.s.utokyo.ac.jp/ sofue/htdocs/2013rc . \nTable 2. Dynamical parameters for Galaxy mass study. \n| Subject | No. | Component | Parameter | Method |\n|--------------------------------|-----------|----------------|--------------------------------------|----------------------------|\n| I. Axisymmetric | (1) | Black hole | Mass | Stellar kinematics |\n| structure | (2) | Bulge(s) | Mass | Rotation curve |\n| structure | (3) | | Radius | |\n| structure | (4) | | Profile (function) | |\n| structure | (5) | Disk | Mass | |\n| structure | (6) | | Radius | |\n| structure | (7) | | Profile (function) | |\n| structure | (8) | Dark halo | Mass | |\n| structure | (9) | | Scale radius | |\n| structure | (10) | | Profile (function) | |\n| II. Non-axisymmetric structure | (11) | Bar(s) | Mass | LV, κ analysis, simulation |\n| II. Non-axisymmetric structure | (12) | | Major axial length | |\n| II. Non-axisymmetric structure | (13) | | Minor axial length | |\n| II. Non-axisymmetric structure | (14) | | z -directional axial length | |\n| II. Non-axisymmetric structure | (15) | | Major axis profile | |\n| II. Non-axisymmetric structure | (16) | | Minor axis profile | |\n| II. Non-axisymmetric structure | (17) (18) | | z-directional profile Position angle | |\n| II. Non-axisymmetric structure | (19) | | Pattern speed Ω p | |\n| II. Non-axisymmetric structure | (20) | Arms | Density amplitude | LV, κ analysis |\n| II. Non-axisymmetric structure | (21) | | Velocity amplitude | |\n| II. Non-axisymmetric structure | (22) | | Pitch angle | |\n| II. Non-axisymmetric structure | (23) | | Position angle | |\n| II. Non-axisymmetric structure | (24) | | Pattern speed Ω p | |\n| III. Radial flow | (25) | Expanding ring | Mass | LV |\n| III. Radial flow | (26) | | Velocity | |\n| III. Radial flow | (27) | | Radius | | \n(Note) LV stands for longitude-velocity diagram; κ and Ω p are the epicyclic frequency and pattern speed. The bulge and bar may be multiple, increasing the number of parameters. \nTable 3. Rotation curve of the Galaxy. \n| r (kpc) | δr (kpc) | V (km s - 1 ) | δV (km s - 1 ) |\n|-----------|------------|-----------------|------------------|\n| 0.00112 | 0.000255 | 121 | 18.2 |\n| 0.00146 | 0.000359 | 108 | 17.2 |\n| 0.0019 | 0.000463 | 97.5 | 16 |\n| 0.00247 | 0.00102 | 93.5 | 20.1 |\n| 0.00418 | 0.000546 | 97.9 | 30 |\n| 0.00543 | 0.00122 | 98.2 | 30.8 |\n| 0.0119 | 0.00264 | 109 | 42.5 |\n| 0.0155 | 0.00266 | 112 | 41.8 |\n| 0.0202 | 0.00549 | 107 | 37.1 |\n| 0.0262 | 0.00609 | 99.6 | 21.6 |\n| 0.0341 | 0.00649 | 97.1 | 13.8 |\n| 0.0443 | 0.0128 | 104 | 24.4 |\n| 0.0576 | 0.0206 | 128 | 33.5 |\n| 0.0748 | 0.0167 | 137 | 25.8 |\n| 0.0973 | 0.0171 | 140 | 33.7 |\n| 0.126 | 0.0269 | 145 | 46.9 |\n| 0.164 | 0.0546 | 156 | 60.6 |\n| 0.214 | 0.074 | 214 | 67.7 |\n| 0.278 | 0.0733 | 243 | 37.6 |\n| 0.361 | 0.101 | 249 | 19.9 |\n| 0.47 | 0.108 | 247 | 11.7 |\n| 0.61 | 0.135 | 240 | 12.1 |\n| 0.794 | 0.234 | 227 | 13.3 |\n| 1.02 | 0.075 | 216 | 2.98 |\n| 1.12 | 0.083 | 215 | 2.79 |\n| 1.23 | 0.0976 | 213 | 6.83 |\n| 1.36 | 0.117 | 207 | 7.85 |\n| 1.49 | 0.11 | 204 | 6 |\n| 1.64 | 0.116 | 201 | 4.78 |\n| 1.81 | 0.163 | 200 | 3.49 |\n| 1.99 | 0.149 | 199 | 5.95 |\n| 2.19 | 0.163 | 196 | 7.22 |\n| 2.4 | 0.166 | 193 | 6.01 |\n| 2.65 | 0.193 | 192 | 4.33 | \n| r (kpc) | δr (kpc) | V (km s - 1 ) | δV (km s - 1 |\n|---------------------|------------|-----------------|----------------|\n| 0.291E+01 | 0.209 | 193 | 5.29 |\n| 0.320E+01 | 0.232 | 197 | 7.79 |\n| 0.352E+01 | 0.251 | 201 | 7.97 |\n| 0.387E+01 | 0.282 | 207 | 10.1 |\n| 0.426E+01 | 0.306 | 211 | 8.43 |\n| 0.469E+01 | 0.349 | 209 | 6.87 |\n| 0.515E+01 | 0.355 | 208 | 8.38 |\n| 0.567E+01 | 0.421 | 208 | 12 |\n| 0.624E+01 | 0.426 | 209 | 16.6 |\n| 0.686E+01 | 0.521 | 207 | 13 |\n| 0.755E+01 | 0.464 | 205 | 9.65 |\n| 0.830E+01 | 0.587 | 200 | 15.8 |\n| 0.913E+01 | 0.691 | 186 | 24.2 |\n| 0.100E+02 | 0.719 | 182 | 35.2 |\n| 0.110E+02 | 0.825 | 190 | 49.6 |\n| 0.122E+02 | 0.83 | 192 | 49.9 |\n| 0.134E+02 | 1.01 | 184 | 54.5 |\n| 0.147E+02 | 1.27 | 189 | 64.7 |\n| | 5.93 | 189 | 59.9 |\n| 0.185E+02 0.240E+02 | 8.61 | 182 | 70.6 |\n| 0.312E+02 | 11.3 | 172 | 75.1 |\n| 0.406E+02 | 14.4 | 157 | 64.5 |\n| 0.528E+02 | 15.4 | 136 | 56.3 |\n| 0.686E+02 | 18.5 | 143 | 49.6 |\n| 0.892E+02 | 19.1 | 135 | 58.2 |\n| 0.116E+03 | 29.8 | 104 | 64.1 |\n| 0.151E+03 | 55.5 | 77.9 | 64.2 |\n| 0.196E+03 | 81.2 | 90.2 | 110 |\n| 0.431E+03 | 133 | 61.3 | 80.9 |\n| 0.560E+03 | 173 | 76.6 | 84.8 |\n| 0.728E+03 | 122 | 104 | 85.2 |\n| 0.946E+03 | 197 | 107 | 87.4 |\n| 0.123E+04 | 274 | 143 | 101 |\n| 0.160E+04 | 373 | 168 | 96.1 | \nTable 4. Surface mass density (SMD) of the Galaxy calculated for the rotation curve in table 3. \n| r (kpc) | Σ s ( M /circledot pc - 2 ) | Σ f ( M /circledot pc - 2 |\n|-----------|-------------------------------|-----------------------------|\n| 0.00112 | 287000 | 395000 |\n| 0.00146 | 288000 | 225000 |\n| 0.0019 | 302000 | 177000 |\n| 0.00247 | 289000 | 154000 |\n| 0.00418 | 183000 | 109000 |\n| 0.00543 | 163000 | 89200 |\n| 0.0119 | 82400 | 49900 |\n| 0.0155 | 57700 | 38600 |\n| 0.0202 | 49900 | 30300 |\n| 0.0262 | 49600 | 26600 |\n| 0.0341 | 50200 | 25200 |\n| 0.0443 | 50600 | 24500 |\n| 0.0576 | 38000 | 21900 |\n| 0.0748 | 31000 | 18100 |\n| 0.0973 | 28500 | 15600 |\n| 0.126 | 27000 | 14200 |\n| 0.164 | 31100 | 13900 |\n| 0.214 | 20000 | 12300 |\n| 0.278 | 11500 | 8670 |\n| 0.361 | 7420 | 5740 |\n| 0.47 | 4810 | 3820 |\n| 0.61 | 3080 | 2530 |\n| 0.794 | 2390 | 1730 |\n| 1.02 | 2030 | 1300 |\n| 1.12 | 1690 | 1170 |\n| 1.23 | 1440 | 1030 |\n| 1.36 | 1420 | 927 |\n| 1.49 | 1340 | 851 |\n| 1.64 | 1270 | 785 |\n| 1.81 | 1150 | 721 |\n| 1.99 | 999 | 654 |\n| 2.19 | 939 | 594 |\n| 2.4 | 933 | 553 | \n| r (kpc) | Σ s ( M /circledot pc - 2 ) | Σ f ( M /circledot pc - 2 ) |\n|-----------|-------------------------------|-------------------------------|\n| 2.65 | 909 | 523 |\n| 2.91 | 877 | 496 |\n| 3.2 | 786 | 464 |\n| 3.52 | 721 | 428 |\n| 3.87 | 577 | 385 |\n| 4.26 | 431 | 330 |\n| 4.69 | 402 | 285 |\n| 5.15 | 366 | 252 |\n| 5.67 | 313 | 223 |\n| 6.24 | 239 | 191 |\n| 6.86 | 200 | 161 |\n| 7.55 | 148 | 135 |\n| 8.3 | 101 | 110 |\n| 9.13 | 173 | 99.9 |\n| 10 | 203 | 102 |\n| 11 | 142 | 95.9 |\n| 12.2 | 89.5 | 80.3 |\n| 13.4 | 125 | 71.4 |\n| 14.7 | 90.8 | 65.7 |\n| 18.5 | 50.8 | 44.9 |\n| 24 | 28.4 | 27.3 |\n| 31.2 | 13.7 | 16.1 |\n| 40.6 | 6.46 | 9.15 |\n| 52.8 | 15.1 | 7.04 |\n| 68.6 | 4.12 | 5.45 |\n| 116 | 0.661 | 0.995 |\n| 151 | 4.28 | 1.56 |\n| 196 | 1.74 | 1.76 |\n| 431 | 2.42 | 1.33 |\n| 560 | 2.41 | 1.43 |\n| 728 | 1.54 | 1.3 |\n| 946 | 1.89 | 1.19 |\n| 1230 | 1.14 | 1.08 |'}

2006JHEP...09..021R

Noncommutative inspired black holes in extra dimensions

2006-01-01

['-', '-', '-']

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In a recent string theory motivated paper, Nicolini, Smailagic and Spallucci (NSS) presented an interesting model for a noncommutative inspired, Schwarzschild-like black hole solution in 4-dimensions. The essential effect of having noncommutative co-ordinates in this approach is to smear out matter distributions on a scale associated with the turn-on of noncommutativity which was taken to be near the 4-d Planck mass. In particular, NSS took this smearing to be essentially Gaussian. This energy scale is sufficiently large that in 4-d such effects may remain invisible indefinitely. Extra dimensional models which attempt to address the gauge hierarchy problem, however, allow for the possibility that the effective fundamental scale may not be far from ~ 1 TeV, an energy regime that will soon be probed by experiments at both the LHC and ILC. In this paper we generalize the NSS model to the case where flat, toroidally compactified extra dimensions are accessible at the Terascale and examine the resulting modifications in black hole properties due to the existence of noncommutativity. We show that while many of the noncommutativity-induced black hole features found in 4-d by NSS persist, in some cases there can be significant modifications due the presence of extra dimensions. We also demonstrate that the essential features of this approach are not particularly sensitive to the Gaussian nature of the smearing employed by NSS.

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https://arxiv.org/pdf/hep-ph/0606051.pdf

{'Noncommutative Inspired Black Holes in Extra Dimensions ∗ †': 'Thomas G. Rizzo a \nStanford Linear Accelerator Center, 2575 Sand Hill Rd., Menlo Park, CA, 94025', 'Abstract': 'In a recent string theory motivated paper, Nicolini, Smailagic and Spallucci (NSS) presented an interesting model for a noncommutative inspired, Schwarzschild-like black hole solution in 4-dimensions. The essential effect of having noncommutative co-ordinates in this approach is to smear out matter distributions on a scale associated with the turn-on of noncommutativity which was taken to be near the 4-d Planck mass. In particular, NSS took this smearing to be essentially Gaussian. This energy scale is sufficiently large that in 4-d such effects may remain invisible indefinitely. Extra dimensional models which attempt to address the gauge hierarchy problem, however, allow for the possibility that the effective fundamental scale may not be far from ∼ 1 TeV, an energy regime that will soon be probed by experiments at both the LHC and ILC. In this paper we generalize the NSS model to the case where flat, toroidally compactified extra dimensions are accessible at the Terascale and examine the resulting modifications in black hole properties due to the existence of noncommutativity. We show that while many of the noncommutativity-induced black hole features found in 4-d by NSS persist, in some cases there can be significant modifications due the presence of extra dimensions. We also demonstrate that the essential features of this approach are not particularly sensitive to the Gaussian nature of the smearing employed by NSS. \n† e-mail: \na [email protected]', '1 Introduction and Background': "The theoretical effort that has gone into understanding the full details of string/M theory has inspired a number of ideas which, on their own, have had a significant impact on particle physics model building and phenomenology. One of the more recent developments of this kind has been the resurgence[1] of interest in noncommutative (NC) quantum field theories[2] and, in particular, the question of how a NC version of the Standard Model (SM) may be constructed[3] and probed experimentally[4]. \nThe essential idea behind NC constructions is that the commutator of two spacetime coordinates, now thought of as operators, is no longer zero. In its simplest form, for a space with an arbitrary number of dimensions, D , this is can be written explicitly as \n[ x A , x B ] = iθ AB = i c AB Λ 2 NC , (1) \nwhere Λ NC is the mass scale associated with NC and c AB is normally taken to be a frameindependent , dimensionless anti-symmetric matrix with constant, real, typically O (1) entries; it is not a tensor. Here we assume that vastly different NC scales do not exist depending upon the values of A,B . ‡ In a general string theory context one might imagine that Λ NC would naturally not be far from the 4-d Planck scale, M Pl , and that the c AB are generated due to the presence of background 'electric' or 'magnetic' type fields. Most of the phenomenological studies of NC models[4] have assumed that we live in 4-d and that Λ NC ∼ 1-10 TeV so that we have access to this scale at, e.g. , the LHC or ILC. However, if the NC scale Λ NC is indeed of order M Pl , then probing NC physics directly may prove difficult in the near term. \nOne way of possibly observing NC is its effects on the properties of black holes (BH). In order to analyze this problem at a truly fundamental level one would need to successfully construct the NC equivalent of General Relativity. Attempts along these lines have been made \nin the literature[5] but no complete and fully compelling theory of this type yet exists. Recently, Nicolini, Smailagic and Spallucci (NSS) [6] have considered a physically motivated and tractable model of the possible NC modifications to Schwarzschild BH solutions. The essential ideas of this picture are: ( i ) General Relativity in its usual commutative form as described by the EinsteinHilbert action remains applicable. This seems justifiable, at least to a good approximation, if NC effects can be treated perturbatively. The authors in Ref.[5] have indeed shown that the leading NC corrections to the form of the Einstein-Hilbert action are at least second order in the θ AB parameters. ( ii ) NC leads to a 'smearing' of matter distributions on length scales of order ∼ Λ -1 NC . Thus the usual ' δ -function' matter source of the conventional Schwarzschild solution is replaced by a centrally peaked, spherically symmetric (and time-independent) mass distribution which has a size of order ∼ Λ -1 NC . This, too, seems justifiable based on the results presented in Ref.[5], which note that matter actions from which the stress-energy tensors are derived are modified at leading order in the θ AB parameters. Based on earlier work[7], NSS took this smeared distribution to be in the form of a spherical 3-d Gaussian in 4-d whose size, due to the spherical symmetry, was set by a single parameter, θ , indicative of the NC scale. Though such a picture leads to many interesting properties for the resulting BH (to be elaborated on below), since M Pl ∼ Λ NC was assumed, as would be natural in 4-d, such BH are not immediately accessible to experiment or to direct observation. \nAnother interesting prediction of string theory is that several extra dimensions must exist. However, extra spatial dimensions, in models with an effective fundamental scale M ∗ now in the TeV range, have been discussed as possible solutions to the hierarchy problem[8, 9]. In the case of 'flat' extra dimensions, e.g. , in the model of Arkani-Hamed, Dimopoulos and Dvali (ADD)[8], the 4-d Planck and fundamental scales are related by the volume of the compactified extra dimensions: \nM 2 Pl = V n M n +2 ∗ , (2) \nwhere V n is the volume of the compactified manifold. Assuming for simplicity that these extra \ndimensions form an n -dimensional torus, if all compactification radii ( R c ) are the same, then V n = (2 πR c ) n . In such a scenario gravity becomes strong at M ∗ and not at M Pl which is viewed as an artifact of our inability to probe gravity at scales smaller than R c . This scenario has gotten a lot of attention over the last few years and the collider phenomenology of these types of models has been shown to be particularly rich[10]. In such a scheme it would be natural that the NC scale, Λ NC , would now also be of order M ∗ ∼ TeV allowing it to be accessible to colliders. Furthermore, the copious production of TeV-scale BH at colliders also becomes possible[11] and the nature of such BH could then be examined experimentally in some detail. Thus it is reasonable to ask if the properties of such TeV-scale BH may be influenced by NC effects, which originate at a similar scale, and if these effects are large enough to be observable in collider data. \nThe goal of the present paper is to begin to address these issues. In particular we will examine how NC BH in D dimensions differ from those in 4-d as well as from the more conventional D -dimensional commutative Schwarzschild BH traditionally analyzed at colliders. Furthermore, we will demonstrate that the essential features of this scenario are not particularly sensitive to the detailed nature of the NC smearing.", '2 Analysis and Results': 'We begin our analysis by reminding the reader that we will assume that D = 4 + n -dimensional gravity, and BH in particular, can still be described by the conventional Einstein-Hilbert (EH) action, i.e. , \nS = M n +2 ∗ 2 ∫ d 4+ n x √ -g R, (3) \nwith R being the Ricci scalar and M ∗ being the (reduced) fundamental scale as appearing in the ADD relationship above. It is important to recall that for the ADD scenario with n ≥ 2, M ∗ is only weakly constrained by current collider experiments, i.e. , one finds that M ∗ ≥ 0 . 38 -0 . 60 TeV, depending on the value of n , when the bound on the more commonly used GRW[10] parameter', 'M D > 1 . 5 TeV is employed.': "In the present and NSS approaches the basic effect of NC is proposed to be the smearing out of conventional mass distributions. Thus, following NSS[6], we will take, instead of the point mass, M , described by a δ -function distribution, a static, spherically symmetric, Gaussian-smeared matter source whose NC 'size' is determined by the parameter √ θ ∼ Λ -1 NC : \nρ = M (4 πθ ) ( n +3) / 2 e -r 2 / 4 θ . (4) \nHere will we explicitly assume that both the horizon size of our BH and the NC parameter √ θ are far smaller than the compactification scale R c so that the BH physics is not sensitive to the finite size of the compactified dimensions. For ADD-type models this can be easily verified from the numerical results we obtain below as the BH horizon size will typically be of order ∼ 1 /M ∗ while R c is generally many orders of magnitude larger as long as the value of D is not too large[12] as is certainly the case when D ≤ 11. Note that the value of √ θ is directly correlated with the NC scale and is certainly proportional to it; however, within this treatment the exact nature of this relationship is unspecified and would require a more detailed model to explicitly determine. It is sufficient to remember only that θ ≈ 1 / Λ NC . It is important to realize that many such parameterizations of this peaked smeared mass distribution are possible which should lead to qualitatively similar physics results. However, the various predictions arising from these may differ only at the O(1) level or less, as long as the detailed structure of the peaked mass distribution is not probed. We will discuss this issue further below. \nThe metric of our D -dimensional space is assumed to be given by the usual D -dimensional Schwarzschild form \nds 2 = e ν dx 2 0 -e µ dr 2 -r 2 d Ω 2 D -2 . (5) \nHere we will be searching for Schwarzschild-like, spherically symmetric and static solutions with ν and µ being functions only of the co-ordinate r and we will further demand that e ν,µ → 1 as \nr → ∞ ; this will require that ν = -µ in the solutions of Einstein's equations as in the usual commutative scenario since the EH action and resulting field equations remains applicable. Note that the surface of the sphere, Ω D -2 , can be simply described by a set of D -2 = n +2 angles, φ i , where i = 1 , .., n +2. \nGiven the assumed form of the matter density, ρ , above and the expectation that ν = -µ as usual, two components of the diagonal stress-energy tensor, T AB , are already determined, i.e. , T r r = T 0 0 = ρ . As noted by NSS, the remaining components, T i i (no summation), all n +2 of which are identical due to the spherical symmetry, can be obtained from the requirement that T AB have a vanishing divergence, T AB ; B = 0, where the semicolon denotes covariant differentiation. Given the nature of our metric it is easily seen that both T 00 ; 0 = 0 and T ii ; i = 0, for all i , automatically. The remaining equation T rr ; r = 0 then yields the explicit result \n0 = ∂ r T r r + 1 2 g 00 ( T r r -T 0 0 ) ∂ r g 00 + 1 2 ∑ i g ii ( T r r -T i i ) ∂ r g ii . (6) \nSince by construction T r r = T 0 0 and, noting that g ii ∂ r g ii = 2 /r , for all (unsummed) i , this yields \nT i i = ρ + r n +2 ∂ r ρ, (7) \nfor all i (without summation). This reproduces the 4-d NSS result in the limit when n → 0. \nWith our metric the non-zero components of the Ricci tensor are given by (with the index i not summed) \nR 0 0 = R r r = -e ν 2 [ ν '' +( ν ' ) 2 +( n +2) ν ' r ] R i i = -1 r 2 [ e ν (1 + n + rν ' ) -( n +1) ] , (8) \nwhere now a prime denotes partial differentiation with respect to r . The Einstein equations resulting from the EH action augmented with our matter distribution can be conveniently written in the form \nR B A = 1 M n +2 ∗ ( T B A -δ B A T n +2 ) , (9) \nwhere T is the trace of the stress-energy tensor, T = T A A . Note that given our assumptions there are only two distinct Einstein equations. Writing g 00 = e ν = 1 -A ( r ), the R i i Einstein equation leads to the following first order differential equation for A ( r ): \nA ' + n +1 r A = 1 M n +2 ∗ 2 rρ n +2 , (10) \nfrom which we obtain the following solution, after substituting the above expression for ρ and demanding that A ( r ) → 0 as r →∞ : \nA ( r ) = 1 M n +2 ∗ M ( n +2) π ( n +3) / 2 1 r n +1 ∫ r 2 / 4 θ 0 dt e -t t ( n +1) / 2 . (11) \nThis result is seen to reduce to that of NSS when n → 0 as well as to the usual D -dimensional Schwarzschild solution when θ → 0. The remaining R 0 0 Einstein equation just returns to us the continuity equation for the stress-energy tensor, hence, nothing new. Note that given our assumptions these results allow us to calculate A ( r ) for any chosen form of the mass distribution ρ ( r ). \nThe horizon radius, R H , occurs at values of r where g 00 = 0, i.e. , where A ( r ) = 1. Defining for convenience the dimensionless quantities m = M/M ∗ , x = M ∗ R H and y = M ∗ √ θ , along with the constant \nc n = ( n +2) π ( n +3) / 2 Γ( n +3 2 ) , (12) \nthe horizon radius can be obtained by solving the equation \nx n +1 = m c n F n ( z ) , (13) \nwhere z = x/y and the functions F n ( z ) are given by the integrals \nF n ( z ) = 1 Γ( n +3 2 ) ∫ z 2 / 4 0 dt e -t t ( n +1) / 2 . (14) \nThese integrals be performed analytically when n is odd, e.g. , F 1 = 1 -e -q (1+ q ), F 3 = 1 -e -q (1+ q + q 2 / 2) and F 5 = 1 -e -q (1 + q + q 2 / 2 + q 3 / 6), etc , with q = z 2 / 4. (For n even these functions can be expressed in terms of combinations of error functions.) Given their definition it is clear that the F n vanish as ∼ z n +3 when z → 0 and they are seen to monotonically increase with increasing z ; as z →∞ our normalization is such that F n → 1. This implies that the BH mass, m , diverges as either x → 0 , ∞ for fixed y and that a minimum value of m must exist for some value of x . Since x appears on both sides of the above equation determining the horizon radius, a trivial relationship between m and x no longer occurs as it does for the ordinary D -dimensional Schwarzschild BH solution. In the θ, y → 0 limit, corresponding to the usual commutative result, the upper limit of the integral defining the functions F n becomes infinite and we arrive at the well-known standard result[11] as then F n → 1: \nm = c n x n +1 . (15) \nNote that if we had chosen a different form for the matter distribution representing the smeared point mass source only the set of functions F n ( z ) would be changed but their general properties would be identical to those above. For example, if we had taken a smearing of the modified Lorentzian form, ρ ∼ ( r 2 + θ ' 2 ) -( n +4) / 2 , (with the parameter θ ' not necessarily being the same as θ ) then the corresponding functions, which we'll call G n , would also vanish as z → 0 in a power law manner and go → 1 as z → ∞ in a monotonic fashion. For example, one obtains G 0 ( z ) = 2 π (tan -1 z -z 1+ z 2 ) which we observe has these same limiting properties and is quite similar to the F n qualitatively. More generally we find that the G n are given by the integrals \nG n ( z ) = 2 π ( n +2)!! ( n +1)!! ∫ z 0 dt t n +2 (1 + t 2 ) ( n/ 2+2) . (16) \nThese basic properties of the F n (and G n ) capture the essential aspects of the NC physics. Other possible smearings, such as a straightforward exponential, would also lead to quite similar results. § We will have more to say about this below but note that many of the specific expressions that we will obtain remain applicable if we take a different form for the smeared mass distribution. \nIt is interesting to inquire how the NC value of x , for the moment explicitly denoted as x NC , compares with the usual D -dimensional result. Clearly for any fixed value of m the ratio x NC /x can be expressed solely via the functions F n and is thus only dependent upon the ratio z = x/y and n ; the result of this calculation is shown in Fig. 1. In examining these results, as well as those in the following figures, it is important to remember than when n = 0, M ∗ = M Pl ; for all larger values of n , M ∗ ∼ 1 TeV. In this figure we see that for large values of z > ∼ 3 we recover the commutative result for all n since the NC scale is far smaller than the horizon size in this case. However when the two scales are comparable or when the horizon shrinks inside the NC scale the value of x is greatly reduced for fixed m . Of course, we really should not trust the details of our modeling of the NC effects when z is extremely small, i.e. , when x is very much less than y . It is also in this very region where most of the differences between, e.g. , the Gaussian and Lorentzian forms of the smeared mass distribution would be expected to begin to appear. \nMore generally, we can now calculate m as a function of x for fixed y . Instead of a monotonically increasing function ∼ x n +1 , we find that m ( x ) is now a function with a single minimum and which grows large as either x → 0 or → ∞ . For small x we find the scaling m ∼ y n +3 /x 2 while the usual commutative behavior is obtained at large x , ∼ x n +1 . Fig. 2 gives a first rough indication of this behavior. The existence of a minima has several implications: ( i ) a minimum value of m implies that there is a physical mass threshold below which BH will not form. ( ii ) The inverse function, x ( m ), is double valued indicating the existence of two possible horizons for a fixed value of the BH mass; this is a potentiality first pointed out by NSS and something we will return \nFigure 1: Horizon size in the NC scenario compared to the commutative result as a function of z = x/y for fixed values of m . On the left-hand side of the figure, from bottom to top, the curves correspond to n = 1 to 7. \n<!-- image --> \nto below. In Fig. 2 we see several additional features: first, as y increases for fixed n so does m except where x is large and we are thus residing in the commutative limit. Also for large y we see that m increases with n as it usually does in the commuting case. This is not too surprising given the small x scaling behavior of m above. Secondly, we see that for large y the values m are always large and the position of the mass minimum moves out to ever larger values of x as y increases. For example, if y = n = 1 then m > ∼ 400 and lighter BH do not form. Such large mass values are far beyond the range accessible to the LHC if we assume M ∗ ∼ 1 TeV, i.e. , the interesting range roughly being 1 < ∼ m < ∼ 10 or so. ¶ \nSince we are interested here in BH that can be created at colliders we will restrict our attention to smaller y values. In fact a short calculation shows that we need 0 . 05 < ∼ y < ∼ 0 . 2 in order to get into the LHC accessible mass region 1 < ∼ m < ∼ 10. To demonstrate this we must find the minimum value of the BH mass, m min as a function of y for various values of n . This can be \ndone in a two-step procedure: first we find where in x the mass minimum occurs for fixed values of y , i.e. , where ∂m/∂x = 0. Calculating this derivative we see that the minimum can be obtained by solving the equation (using for convenience the variable q = x 2 / 4 y 2 introduced above): \nF n ( q ) -2 q ( n +3) / 2 e -q ( n +1)Γ( n +3 2 ) = 0 , (17) \nwhich has a single, non-zero root q 0 ( n ). For any y this tells us the value of the horizon radius where the minimum mass occurs, x min = 2 y √ q 0 ( n ), which we can now use to obtain m min employing the equations above. The result of this calculation is shown in Fig. 3. Here we see that for n in the range 1 to 7 the relevant values of y are rather narrow, not differing from y /similarequal 0.1 by more than about a factor of 2. \nIt is important to recall that for the ordinary commutative D -dimensional BH solution both m min and x min are algebraically zero. However, since we believe that m min > ∼ 1 is required to produce a BH, stronger conditions are usually imposed. In our case the results shown in Fig. 3 represent the algebraic lower bound on m which certainly → 0 as θ → 0. Physically, we might crudely imagine that m min /similarequal Max (1 , m a min ) where m a min is the result shown in Fig. 3. We note that having a finite m min implies a BH production threshold while a finite x min implies a minimum BH production cross section, σ BH /similarequal πx 2 min /M 2 ∗ , at colliders such as the LHC as is shown in Fig. 4. Here we see that far above threshold the BH production cross section scales like ∼ m 2 / ( n +1) as would be expected in the commutative theory. However, for lighter BH this cross section falls significantly below this simple scaling rule and becomes quite small in the neighborhood of m min , almost but not quite vanishing. \nWe can now focus on this region of small y ; for simplicity in what follows we will generally concentrate our results on the case y = 0 . 1. Fig. 5 shows m ( x ) for y = 0 . 1 -0 . 2 for n in the range 0 to 7. Note that generally larger n leads to smaller horizon radii for fixed y and already at y = 0 . 2 we see explicitly that BH are too massive to be produced at the LHC as expected from the above \nFigure 2: BH mass as a function of the horizon size for n = 0(solid), n = 1(dashed) and n = 2(dotted). The upper(middle, lower) set of curves correspond to y = 10(1 , 0 . 1). \n<!-- image --> \nFigure 3: Minimum BH mass as a function of y showing the (very) approximate mass range accessible to the LHC between the dashed lines. On the left-hand side from top to bottom the curves correspond to n = 1 , 3 , 5 and 7, respectively. The allowed parameter range is above and to left of each curve. \n<!-- image --> \nFigure 4: NC BH production subprocess cross section as a function of m for y = 0 . 1. From right to left at the bottom of the figure the curves correspond to n =1 to 7. \n<!-- image --> \nanalysis if M ∗ ∼ 1 TeV. One also sees that at x ∼ 1 the asymptotic behavior, m ∼ x n +1 , has already begun to set in for all n . \nAs an example of the insensitivity of these results to our Gaussian parameterization of the smearing due to NC effects, let us briefly considered the modified Lorentzian form mentioned above. Setting y ' = M ∗ √ θ ' = 0 . 1 (which is not necessarily the same as y = 0 . 1), we again evaluate m as a function of x . The result of this calculation is shown in Fig. 6 which we should compare with the top panel of Fig. 5 that yields essentially the same result in the asymptotic large x > ∼ 0 . 5(as it should since this is the commutative limit). In both cases a minimum mass occurs at relatively small x with somewhat similar values of m . m min decreases in both cases as n is increased and the corresponding value of x min also decreases as n is increased. The m min values are seen to be quite comparable in the two cases. The greatest difference in the two results is seen to occur in the region of x below x min where there is the most sensitivity to NC effects and the detailed shape of the BH mass distribution. This is just what we would have expected; in the region where NC \nFigure 5: Same as in Fig. 2 but now for y = 0 . 10(top), 0.15(middle) and 0.20(bottom) with the dotted curve being for n = 0. From top to bottom on the left-hand side of the figure the solid curves are for n =1 to 7. \n<!-- image --> \neffects just begin to be felt the detailed nature of the peaked mass distribution is not actually being probed. What is really being probed in this parameter range is the fact that there is a peaked mass distribution instead of a δ -function source, i.e. , the BH has a form factor due its finite size, and not the details of its shape. Only at smaller values of the radii (relative to the values of y or y ' ) do the differences in the details of the mass distribution become important. In fact, we can if we wish tune our chosen value of y ' to make these two sets of curves even more alike. Since the majority of the effects we will discuss are sensitive only to physics with x ≥ x min , this short analysis shows that the general features of the results that we obtain below are not particularly sensitive to how the NC smearing of the mass distribution is performed. \nFigure 6: BH mass as a function of the horizon size x for y ' = 0 . 1 using Lorentzian smearing. From top to bottom in the middle of the figure the curves correspond to n = 0, 2, 4 and 6, respectively. \n<!-- image --> \nIn order to understand the possibility of the formation of two (or no) horizons, we follow NSS and consider the metric tensor component g 00 as a function of the dimensionless radius co-ordinate M ∗ r as shown in Fig. 7. Here we will assume that m = 5, a typical value which is kinematically accessible at LHC, for demonstration purposes. Recall that horizons occur when g 00 = 0. With y = 0 . 1 all of the curves pass through g 00 = 0 twice corresponding to two horizons, one on either \nside of x min . This explains why x ( m ) is double valued in Fig. 5, i.e. , the two solutions correspond to the two radii where g 00 vanishes. For n = 0, the case studied by NSS, these two horizons are rather close in radius but this separation grows significantly as n increases. When y = 0 . 15, we see that for n = 0 -4 no horizon form as g 00 > ∼ 0 . 13. This corresponds to the result observed in Fig. 5 for y = 0 . 15 where we see that for this range of n the value m = 5 is not allowed. However, for n ≥ 5, we again obtain two horizons; clearly a tuning of parameters will allow the two horizons to converge to the case of a single degenerate horizon at x min as found by NSS. For both values of y we note that as M ∗ r → 0 the metric is no longer singular as in the pure Schwarzschild case as was noted by NSS in 4-d (as we are inside a well-behaved mass distribution), independently of the existence of any horizons. This is further confirmed by constructing the Ricci curvature invariant, R , as can be easily done from the Einstein equations above; in fact, we find that as x → 0, R ∼ y -( n +3) . Furthermore, and more explicitly, apart from an overall numerical factor, R ∼ my -( n +3) [ n +4 -( M ∗ r ) 2 2 y 2 ] e -( M ∗ r ) 2 / 4 y 2 so that R is seen to vanish as M ∗ r →∞ as expected and undergoes a change of sign in the region, e.g. , M ∗ r ∼ 0 . 2 -0 . 5 for y = 0 . 1 independently of m and only weakly dependent on the value of n . As we will now see only the 'outer' horizon, i.e. , the one with x ≥ x min is actually relevant to us. \nOur next step is to determine the thermodynamic behavior of these NC BH; to do this we first must calculate the Hawking temperature of the BH. This can be done in the usual manner by remembering that \nT H = 1 4 π de ν dr | r = x/M ∗ , (18) \ni.e. , the temperature is essentially the r (radial) co-ordinate derivative of the metric evaluated at the horizon radius. Defining for convenience the dimensionless temperature, T = T H /M ∗ , we obtain from the above form of the metric \nT = n +1 4 πx [ 1 -2 q ( n +3) / 2 e -q F n ( q )( n +1)Γ( n +3 2 ) ] . (19) \nFigure 7: g 00 as a function of M ∗ r assuming m = 5 for either y = 0 . 10(top) or 0.15(bottom). The dotted curve corresponds to n = 0 while from top to bottom on the left-hand side of the figure the solid curves are for n =1 to 7. The dashed line corresponds to g 00 = 0. \n<!-- image --> \nNote that as expected T returns to the usual commutative result in the q → ∞ limit, i.e. , the quantity in the large square bracket above → 1 in this limit. It is instructive to compare the temperature we obtain in both the NC and commutative cases for various values of the parameters; the ratio of these quantities is shown in Fig. 8. This ratio is seen to be near unity for large z as one would expect but it decreases rapidly as z approaches the ∼ 2-3 range from above. The temperature is also seen to vanish at the same z value where the BH mass is minimized. For smaller z , T becomes negative (which is where the second horizon occurs) and thus we enter a region that might usually be considered unphysical. If we had instead chosen the Lorentzian smearing these results would be quite similar qualitatively. \nFigure 8: The ratio of the BH NC temperature to that obtained in the commutative limit where there is only a point matter source. The dotted curve corresponds to n = 0 whereas, from bottom to top on the left-hand side of the figure, the solid lines correspond to n =1 to 7. \n<!-- image --> \nFig. 9 shows the actual NC temperature as a function of x for y = 0 . 1 and different n values. Here we see that T is quite close to its commutative value for x near unity, goes through a maximum as x decreases and then falls to zero at the BH mass minimum point. From these results and Fig. 5 above we can now trace the history of the entire semiclassical BH evaporation \nprocess. ‖ Consider a BH formed in a suitable parameter space region with moderately large values of m ∼ 8 -10 (and, hence, with large x ). For such BH their Schwarzschild radii are too large to feel the effects of the NC scale in the formation process since x >> y . As in the usual commuting picture, when the BH emits Hawking radiation it loses mass and gets hotter and thus radiates even more quickly. As the BH shrinks it begins to feel the NC effects and the temperature reaches a maximum in the mass region m /similarequal 1 -7, the specific number depending on the value of n . As the BH continues to lose mass its temperature now decreases so that it radiates ever more slowly. Finally, as m approaches m min the semiclassical radiation emission processes ceases since T → 0 has been reached leaving a classically stable remnant. Though sounding somewhat unusual such a possibility has been discussed in the literature for a number of alternative BH scenarios which go beyond the basic picture presented by General Relativity based on just the EH action[11, 13]. ∗∗ Whether quantum effects destabilize such a relic is not known. \nIt is easy to convince oneself that a classically stable remnant is the natural outcome of this scenario. In the usual treatment of BH decays, the mass loss rate (assuming a perfect radiator) is given by \ndm dt = -Ξ d x d +2 T d +4 , (20) \nwith Ξ d being a positive numerical constant. For decays dominantly to bulk(brane) fields we have d = n (0). In either case, clearly the lifetime of the BH is then given by \nt BH = -Ξ -1 d ∫ m min m initial dm x d +2 ( m ) T d +4 ( m ) , (21) \nwith m initial being the original BH mass. We recall from above that while x ( m ) is double valued it is well-behaved and never vanishes. However, on the otherhand, we also know that T ( m ) → 0 as m → m min . Thus for all n (and any d ), t BH will be driven to infinity due to the presence of a \nsingular denominator in the integrand implying a stable relic. (Of course, an initial very massive BH will radiate down to a mass very close to m min quite quickly.) This must happen in any model that predicts T → 0 at finite x due to a corresponding singularity. For the commuting case, since T ∼ 1 /x , the integrand is never singular so that the BH lifetime remains finite. \nThe lower panel in Fig. 9 summarizes this discussion where we see that as m decreases from a large value the temperature increases, reaches a maximum value and then falls to zero at m min leaving a classically stable remnant. \nThis unusual temperature behavior in the NC case can also be studied more fully by examining the BH heat capacity/specific heat, C . In the commutative case, C is always finite (away from x = 0) and negative since the BH gets hotter as it loses mass. Let we define \nC = ∂m ∂T = ∂m ∂x ( ∂T ∂x ) -1 , (22) \nwhich we can explicitly write as \nC = -4 πc n x n +2 F n ( q ) -1 [ 1 -2 H n ( q ) ( n +1) 1 -2 H n ( q ) ( n +1) + 4 qH n ( q ) ( n +1) ( ( n +3) 2 q -H n ( q ) q -1 ) ] , (23) \nwhere for convenience we have defined the set of auxiliary functions \nH n ( q ) = q ( n +3) / 2 e -q F n ( q )Γ( ( n +3) 2 ) . (24) \nUsing this expression, Fig. 10 shows the results for the NC BH heat capacity as a function of both x and m for our standard choice of y = 0 . 1. We first see that C remains negative at large x and asymptotes to its commutative value, C = -4 πc n x n +2 , as it should. Further we note that at x = x min (or at m = m min ) we find that C → 0 as was expected. BH with mass m min are no longer capable of mass loss since they have zero temperature. Between these two regimes the \nFigure 9: The NC BH temperature as a function of x (top panel) and m (lower panel) for fixed y = 0 . 1. The dotted curve corresponds to case of n = 0 whereas, from bottom to top on the right-hand side of the figure, the solid lines correspond to n =1 to 7. \n<!-- image --> \nbehavior of C is quite interesting. Consider C as a function of large and decreasing x . C at first decreases in magnitude as it does in the commutative case. However as we know from above, for some n -dependent x value, T reaches a maximum and then decreases. This implies that the magnitude of C then increases and becomes singular for this particular x value. For lower x the sign of C changes as now ∂T ∂x > 0 and then approaches zero at m min . This is even more obvious when we consider C as a function of m . For large m the BH radiates as in the commutative case as C decreases in magnitude as m is reduced. However, at some point the NC effects turn on and -C begins to increase, becoming singular where ∂T ∂m = 0, the location of temperature the maximum. For smaller masses, as the BH radiates and gets lighter the temperature decreases so that we are now in a region of positive heat capacity. As m decreases further to m min , C becomes zero for the remnant. \nOur next step in examining the thermodynamics of NC BH is to consider the value of the entropy which is defined via \nS = ∫ dx T -1 ∂m ∂x = 4 πc n ∫ x x min dv v n +1 F n ( v 2 / 4 y 2 ) , (25) \nwhere here we have made the natural choice that the entropy vanish at x min where the BH mass m is minimized for fixed values of y and n . In the commutative limit where θ, y → 0 then x min → 0 and we recover the usual lower limit of integration. Fig. 11 shows the values of the entropy for various n as a functions of x or m for our canonical choice of y = 0 . 1. We again observe that the commutative power law behavior, S = 4 πc n x n +2 ( n +2) , is recovered in the large x > ∼ 1 limit as expected. It is more interesting to consider S as a function of m ; as we see from the Figure, while the entropy scales as ∼ m ( n +2) / ( n +1) for large m it rapidly falls below this scaling law to zero as m approaches m min from above. \nFinally, we also examine the free energy of the NC BH which is simply given by combining \nFigure 10: Negative of the NC BH heat capacity/specific heat as a function of x (top panel) and m (bottom panel) for fixed y = 0 . 1. The dotted curve corresponds to case of n = 0 whereas, from bottom to top in the middle of the figure, the solid lines correspond to n =1 to 7. In the bottom panel, the parameter region with negative temperatures has been removed. \n<!-- image --> \nFigure 11: The NC BH entropy as a function of x (top panel) and m (bottom panel) for fixed y = 0 . 1. From left to right (right to left) at the bottom of the top(bottom) panel the the curves are for n = 1, 3, 5 and 7, respectively. \n<!-- image --> \nthe above thermodynamic quantities: \nF = m -TS . (26) \nAs seen in Fig. 12, it too returns to its commutative value, F = m/ ( n +2), in the limit of large x > ∼ 1 or large m . However, for x = x min it is clear that F = m for all values of y and n since both T and (by definition) S vanish at this point. Immediately above x = x min (or m min ), F decreases slightly, until it matches onto the asymptotic ∼ x n +1 behavior. For some parameter values F can even become negative in this intermediate mass regime. Generally we see the unusual behavior that m = m min is not the location of a minimum in F as either a function of x or m . This minimum lies near the values of x and m where T is maximized.", '3 Discussion and Conclusions': 'Space-time noncommutativity and extra dimensions are both well-motivated ideas within the string theory context and it is natural for them to make their appearance felt as one approaches the fundamental scale. In more than four dimensions it is possible for this scale to be not far from ∼ TeV and thereby address the gauge hierarchy problem. If this mass parameter is indeed this low it is likely that black holes will be produced at the LHC in sufficient quantities that their properties will be well measured. The occurrence of NC also at a similar scale could lead to significant modifications in the anticipated properties of these BH. Since a complete NC theory of gravity does not yet exist it becomes necessary to model the NC effects within the commutative General Relativity framework. \nNicolini, Smailagic and Spallucci presented a physically motivated model of this kind in 4-d, where the essential aspects were the Gaussian smearing of matter distributions on the NC scale and the continued applicability of the EH action. They then went on to examine NC effects on BH physics. In this paper we extended this NC BH analysis in several ways: ( i ) we generalized the NSS study to the case of extra dimensions with a fundamental scale in the TeV range so that \nFigure 12: The NC BH free energy as a function of x (top panel) and m (bottom panel) for y = 0 . 1. From top to bottom on the left-hand side the curves correspond to n = 1, 3, 5 and 7, respectively \n<!-- image --> \nthe associated BH can be produced with large cross sections and studied in detail at the LHC. While much of the BH behavior observed in extra dimensions was similar to that obtained in 4-d, some significant modifications to the previously obtained 4-d results were observed. However, there appears to be an overall dominance of NC effects over those that arise due to the existence of the extra dimensions. ( ii ) We demonstrated that the essential physics induced by NC smearing is not particularly sensitive to the nature of the smearing function. In particular we explicitly showed that Gaussian and Lorentzian smearing lead to essentially the same behavior for the expected modifications of the BH mass-radius relationship due to NC effects. ( iii ) We extended the NSS analysis to include several other thermodynamic quantities which are of interest in the study of NC BH such as their entropy, heat capacity and free energy. \nPerhaps the most important qualitative influence of NC on BH physics was already observed in 4-d by NSS, i.e. , the existence of a classically stable remnant whose mass and radius are completely fixed by the NC scale and the number of dimensions. Within the framework of extra dimensions, if the fundamental scale is not too large then BH and their remnants will be copiously produced at the LHC and studied in detail. The observation of NC effects in the properties of these BH can open a new window on the fundamental theory of gravity and space-time.', 'Acknowledgments': 'The author would like to thank J.L. Hewett and B. Lillie for discussions related to this work.', 'References': '- [1] For very early work on NC, see H. S. Snyder, Phys. 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Grav. 20 , L205 (2003) [arXiv:hep-ph/0305223] and references therein. See also, S. Hossenfelder, Phys. Lett. B 598 , 92 (2004) [arXiv:hep-th/0404232]; M. Bojowald, R. Goswami, R. Maartens and P. Singh, gr-qc/0503041; T. G. Rizzo in Ref.[11]'}

2012ApJ...748...36B

Radio Monitoring of the Tidal Disruption Event Swift J164449.3+573451. I. Jet Energetics and the Pristine Parsec-scale Environment of a Supermassive Black Hole

2012-01-01

['galaxies nuclei', 'techniques interferometric', '-']

[]

We present continued radio observations of the tidal disruption event Swift J164449.3+573451 extending to δt ≈ 216 days after discovery. The data were obtained with the EVLA, AMI Large Array, CARMA, the SMA, and the VLBA+Effelsberg as part of a long-term program to monitor the expansion and energy scale of the relativistic outflow, and to trace the parsec-scale environment around a previously dormant supermassive black hole (SMBH). The new observations reveal a significant change in the radio evolution starting at δt ≈ 1 month, with a brightening at all frequencies that requires an increase in the energy by about an order of magnitude, and an overall density profile around the SMBH of ρvpropr <SUP>-3/2</SUP> (0.1-1.2 pc) with a significant flattening at r ≈ 0.4-0.6 pc. The increase in energy cannot be explained with continuous injection from an Lvpropt <SUP>-5/3</SUP> tail, which is observed in the X-rays. Instead, we conclude that the relativistic jet was launched with a wide range of Lorentz factors, obeying E(&gt; Γ<SUB> j </SUB>)vpropΓ<SUP>-2.5</SUP> <SUB> j </SUB>. The similar ratios of duration to dynamical timescale for Sw 1644+57 and gamma-ray bursts (GRBs) suggest that this result may be applicable to GRB jets as well. The radial density profile may be indicative of Bondi accretion, with the inferred flattening at r ~ 0.5 pc in good agreement with the Bondi radius for a ~few × 10<SUP>6</SUP> M <SUB>⊙</SUB> black hole. The density at ~0.5 pc is about a factor of 30 times lower than inferred for the Milky Way Galactic Center, potentially due to a smaller number of mass-shedding massive stars. From our latest observations (δt ≈ 216 days) we find that the jet energy is E <SUB> j, iso</SUB> ≈ 5 × 10<SUP>53</SUP> erg (E<SUB>j</SUB> ≈ 2.4 × 10<SUP>51</SUP> erg for θ<SUB> j </SUB> = 0.1), the radius is r ≈ 1.2 pc, the Lorentz factor is Γ<SUB> j </SUB> ≈ 2.2, the ambient density is n ≈ 0.2 cm<SUP>-3</SUP>, and the projected angular size is r <SUB>proj</SUB> ≈ 25 μas, below the resolution of the VLBA+Effelsberg. Assuming no future changes in the observed evolution and a final integrated total energy of E<SUB>j</SUB> ≈ 10<SUP>52</SUP> erg, we predict that the radio emission from Sw 1644+57 should be detectable with the EVLA for several decades and will be resolvable with very long baseline interferometry in a few years.

[]

https://arxiv.org/pdf/1112.1697.pdf

{'No Header': 'RADIO MONITORING OF THE TIDAL DISRUPTION EVENT SWIFT J164449.3+573451. I. JET ENERGETICS AND THE PRISTINE PARSEC-SCALE ENVIRONMENT OF A SUPERMASSIVE BLACK HOLE \nE. BERGER 1 , A. ZAUDERER 1 , G. G. POOLEY 2 , A. M. SODERBERG 1 , R. SARI 1,3 , A. BRUNTHALER 4,5 , AND M. F. BIETENHOLZ 6,7 Draft version November 9, 2018', 'ABSTRACT': 'We present continued radio observations of the tidal disruption event Swift J164449.3+573451 extending to δ t ≈ 216 days after discovery. The data were obtained with the EVLA, AMI Large Array, CARMA, the SMA, and the VLBA+Effelsberg as part of a long-term program to monitor the expansion and energy scale of the relativistic outflow, and to trace the parsec-scale environment around a previously-dormant supermassive black hole (SMBH). The new observations reveal a significant change in the radio evolution starting at δ t ≈ 1 month, with a brightening at all frequencies that requires an increase in the energy by about an order of magnitude, and an overall density profile around the SMBH of ρ ∝ r -3 / 2 (0 . 1 -1 . 2 pc) with a significant flattening at r ≈ 0 . 4 -0 . 6 pc. The increase in energy cannot be explained with continuous injection from an L ∝ t -5 / 3 tail, which is observed in the X-rays. Instead, we conclude that the relativistic jet was launched with a wide range of Lorentz factors, obeying E ( > Γ j ) ∝ Γ -2 . 5 j . The similar ratio of duration to dynamical timescale for Sw 1644+57 and GRBs suggests that this result may be applicable to GRB jets as well. The radial density profile may be indicative of Bondi accretion, with the inferred flattening at r ∼ 0 . 5 pc in good agreement with the Bondi radius for a ∼ few × 10 6 M /circledot black hole. The density at ∼ 0 . 5 pc is about a factor of 30 times lower than inferred for the Milky Way galactic center, potentially due to a smaller number of mass-shedding massive stars. From our latest observations ( δ t ≈ 216 d) we find that the jet energy is E j , iso ≈ 5 × 10 53 erg ( Ej ≈ 2 . 4 × 10 51 erg for θ j = 0 . 1), the radius is r ≈ 1 . 2 pc, the Lorentz factor is Γ j ≈ 2 . 2, the ambient density is n ≈ 0 . 2 cm -3 , and the projected angular size is r proj ≈ 25 µ as, below the resolution of the VLBA+Effelsberg. Assuming no future changes in the observed evolution and a final integrated total energy of Ej ≈ 10 52 erg, we predict that the radio emission from Sw 1644+57 should be detectable with the EVLA for several decades, and will be resolvable with VLBI in a few years. \nSubject headings:', '1. INTRODUCTION': 'The discovery of the unusual γ -ray/X-ray transient Swift J164449.3+573451 (hereafter, Sw 1644+57), which coincided with the nucleus of an inactive galaxy at z =0 . 354, has opened a new window into high-energy transient phenomena, with potential implications to our understanding of relativistic outflows in systems such as gamma-ray bursts (GRBs) and active galactic nuclei (AGN). The prevailing interpretation for this event is the tidal disruption of a star by a dormant supermassive black hole (SMBH) with a mass of M BH ∼ 10 6 -10 7 M /circledot (Bloom et al. 2011; Burrows et al. 2011; Levan et al. 2011; Zauderer et al. 2011; but see Krolik & Piran 2011; Ouyed et al. 2011; Quataert & Kasen 2011 for alternative explanations). The argument for a tidal disruption origin is based on: (i) a positional coincidence ( /lessorsimilar 0 . 2 kpc) with the host galaxy nucleus; (ii) rapid time variability in γ -rays and X-rays ( /lessorsimilar 10 2 s), which requires a compact source of /lessorsimilar 0 . 15 AU, a few times the Schwarzschild radius of a ∼ 10 6 M /circledot \n- 1 Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138\n- 2 Mullard Radio Observatory, Cavendish Laboratory, Cambridge, CB3 0HE UK\n- 3 Racah Institute of Physics, The Hebrew University, 91904 Jerusalem, Israel\n- 4 Max-Planck-Institut f ur Radioastronomie, Auf dem H u gel 69, 53121 Bonn, Germany \nblack hole; (iii) high γ -ray and X-ray luminosity of ∼ 10 47 erg s -1 , which exceeds the Eddington limit of a ∼ 10 6 M /circledot black hole by 2 -3 orders of magnitude; (iv) a lack of previous radio to γ -ray activity from this source to much deeper limits than the observed outburst, pointing to a rapid onset; and (v) long-term X-ray luminosity evolution following LX ∝ t -5 / 3 , as expected from the fallback of tidally disrupted material (e.g., Rees 1988; Strubbe & Quataert 2009). \nEqually important, Sw1644+57 was accompanied by bright radio synchrotron emission, with an initial peak in the millimeter band ( F ν ≈ 35 mJy) and a steep spectral slope at lower frequencies indicative of self-absorption (Zauderer et al. 2011; hereafter, ZBS11). The properties of the radio emission established the existence of a relativistic outflow with a Lorentz factor of Γ ∼ few (ZBS11, Bloom et al. 2011). The spectral energy distribution also demonstrated that the lack of detected optical variability required significant rest-frame extinction ( AV /greaterorsimilar 5 mag; ZBS11, Levan et al. 2011), and that the X-rays were produced by a distinct emission component, rather than inverse Compton scattering by the radio-emitting relativistic electrons (ZBS11). Finally, the evolution of the radio emission on a timescale of δ t ∼ 5 -22 d pointed to an ambient density with a radial profile of roughly ρ ∝ r -2 , as well as a mild increase in the energy of the outflow (ZBS11). \nThe formation of a relativistic jet with dominant X-ray and radio emission were not predicted in standard tidal disruption models (e.g., Rees 1988; Strubbe & Quataert 2009), which instead focused on the thermal optical/UV emission from the long-term accretion of the stellar debris. A signature of the \nlatter process is a mass accretion rate that evolves as ˙ M ∝ t -5 / 3 , presumably leading to emission with the same temporal dependence (e.g., Komossa & Greiner 1999; Gezari et al. 2008; van Velzen et al. 2011). Shortly before the discovery of Sw1644+57, Giannios & Metzger (2011) investigated the potential signature of a putative relativistic outflow, and concluded that the interaction of the outflow with the ambient medium will lead to radio emission on a timescale of δ t ∼ 1 yr (for typical off-axis observers). While the mechanism for the radio emission from Sw 1644+57 is interaction with an external medium, the actual light curves differ from the offaxis prediction. To address this issue, in a follow-up paper Metzger et al. (2011) (hereafter, MGM11) reconsidered the model for a relativistic jet interacting with an ambient medium. They draw on the inferences from the early radio emission described in ZBS11 to infer the properties of the environment and the jet kinetic energy, and use this information to predict the future evolution of the radio emission. \nThis long-term radio evolution is of great interest because it can provide several critical insights: \n- · The integrated energy release in the relativistic outflow, including the anticipated injection from on-going accretion.\n- · The density profile around a previously-dormant SMBH on ∼ 0 . 1 -10 pc scales, which cannot be otherwise probed in AGN.\n- · The potential to spatially resolve the outflow with very long baseline interferometry (VLBI), and hence to measure the dynamical evolution (expansion and potentially spreading) of a relativistic jet.\n- · Predictions for the radio emission from tidal disruption jets as viewed by off-axis observers on timescales of months to years to decades. \nThe energy scale and jet dynamics are of particular importance since the total energy input and the structure of the jet may also have implications for relativistic jets in GRBs and AGN. The ability to trace the environment on parsec scales provides a unique probe of gas inflow or outflow around an inactive SMBH on scales that cannot be probed outside of the Milky Way. Finally, the long-term radio emission from Sw1644+57 will inform future radio searches for tidal disruption events (TDEs) that can overcome the low detection rate in γ -rays/X-rays (due to beaming), and obscuration due to extinction in the optical/UV (as in the case of Sw 1644+57). \nTo extract these critical properties we are undertaking longterm monitoring of the radio emission from Sw 1644+57 using a wide range of centimeter- and millimeter-band facilities. Here we present radio observations of Sw 1644+57 that extend to δ t ≈ 216 d, and use these observations to determine the evolution of the total energy and ambient density. We find that the evolution of both quantities deviates from the behavior at δ t /lessorsimilar 1 month (presented in ZBS11), thereby providing crucial insight into the structure of the relativistic outflow and the ambient medium. This paper is the first in a series that will investigate the long-term radio evolution of Sw 1644+57 and the implications for relativistic jets and parsec-scale environments around supermassive black holes, including efforts to resolve the source with VLBI and to measure polarization. \nThe current paper is organized as follows. We describe the radio observations in §2, and summarize the radio evolution \nat δ t ≈ 5 -216 d in §3. In §4 we present our modeling of the radio emission, which utilizes the formulation of MGM11. The implications for the energy scale and ambient density are discussed in §4.1 and §4.2, respectively, and we finally consider the implications for relativistic jets and the parsec-scale environments of SMBHs in §5.', '2. RADIO OBSERVATIONS': 'Although Sw 1644+57 first triggered the Swift Burst Alert Telescope on 2011 March 28.55 UT, discernible γ -ray emission was detected starting on 2011 March 25 UT (Burrows et al. 2011). We therefore consider 2011 March 25.5 UT to be the actual initial time for the event and the associated relativistic outflow that powers the observed radio emission. Our radio observations of Sw 1644+57 commenced on 2011 March 29.36 UT (19 . 4 hr after the Swift trigger and about 3 . 9 d after the initial γ -ray detection). Observations extending to δ t ≈ 26 d were presented in ZBS11. Here we report observations extending to δ t ≈ 216 d. Throughout the paper we use the standard cosmological constants with H 0 = 70 km s -1 Mpc -1 , Ω m = 0 . 27 and Ω Λ = 0 . 73.', '2.1. Expanded Very Large Array': 'We observed Sw 1644+57 with the EVLA 8 using the new Wideband Interferometric Digital Architecture (WIDAR) correlator to obtain up to 2 GHz of bandwidth at several frequencies. At all frequencies, we used 3C286 for bandpass and flux calibration. At 1.4 GHz, we used J1634+6245 for phase calibration. For phase calibration at all other frequencies, we used J1638+5720, and also included a third calibrator, J1639+5357, at 5.8 GHz. The data were reduced and imaged with the Astronomical Image Processing System (AIPS) software package. The observations are summarized in Table 1. Minor changes to data values with respect to those reported in ZBS11 are due to additional flagging of the data. The errors reported in Table 1 are statistical uncertainties only; the overall systematic uncertainty in the flux calibration is /lessorsimilar 5%.', '2.2. AMI Large Array': 'We observed with the AMI Large Array at 15.4 GHz with a bandwidth of 3.75 GHz. The maximum baseline is about 110 m, with a resulting angular resolution of 25 arcsec. Observations ranged in duration from 45 min to 11 hr. Observations of the compact source J1638+5720 were interleaved at intervals of 10 min as a phase reference, and the flux density scale was established by regular observations of the calibrators 3C48 and 3C286. The telescope measures linearly-polarized signals (Stokes I+Q). The observations are summarized in Table 1.', '2.3. Combined Array for Research in Millimeter Astronomy': 'We observed Sw1644+57 with CARMA at frequencies of 87.3 and 93.6 GHz with a total bandwidth ranging between 6.8 and 7.8 GHz. We used Neptune as our primary flux calibrator, and J1824+568 and J1638+573 as bandpass and phase calibrators. The overall uncertainty in the absolute flux calibration is ≈ 20%. Data calibration and imaging were done with the MIRIAD and AIPS software packages. The observations are summarized in Table 1. We note some changes \n8 The EVLA is operated by the National Radio Astronomy Observatory, a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc. The observations were obtained as part of programs 10C-145, 11A-262, and 11A-266 \nwith respect to the flux densities reported in ZBS11, especially those epochs where the total integration time on source was less than ∼ 20 min ( δ t ≈ 10 -25 d).', '2.4. Submillimeter Array': 'We observed Sw 1644+57 with the SMA using at least seven of the eight antennas, in a wide range of weather conditions, with τ 225 ranging from 0.04 to 0.3. In each observation we combined the two sidebands, each with a bandwidth of 4 GHz separated by 10 GHz, to increase the signal-to-noise ratio. The data were calibrated using the MIR software package developed at Caltech and modified for the SMA, while for imaging and analysis we used MIRIAD. Gain calibration was performed using J1642+689, 3C345, and J1849+670. Absolute flux calibration was performed using real-time measurements of the system temperatures, with observations of Neptune to set the overall scale. Bandpass calibration was done using 3C454.3, J1924-292, and 3C279. The observations are summarized in Table 1.', '2.5. Very Long Baseline Interferometry': "We observed Sw 1644+57 with the NRAO Very Long Baseline Array (VLBA 9 ) and the 100-m Effelsberg telescope in six epochs 10 between 2011 April 2 and September 17 UT at 8.4 and 22 GHz. The observations were performed with eight frequency bands of 8 MHz bandwidth each in dual circular polarization, resulting in a total data rate of 512 Mbps. ICRFJ1638+5720 (Ma et al. 1998), located only 0.92 · from Sw1644+57 was used for phase-referencing at both frequencies. At 22 GHz, we switched between the target and calibrator every 40 s, while at 8.4 GHz we spent 40 s on the calibrator and 90 s on the target. This resulted in a total integration time on Sw 1644+57 of about 93 min at 8.4 GHz and 78 min at 22 GHz in epochs 1, 2, 5, and 6. Epochs 3 and 4 were 1 hour shorter, resulting in integration times on source of 75 min at 8.4 GHz and 60 min at 22 GHz. A second calibrator, J1657+5705 (Beasley et al. 2002), was observed for 6 min at each frequency to check the overall data calibration. At 22 GHz, we also performed ∼ 30-min blocks of geodetic observations to perform atmospheric calibration (for details see Brunthaler et al. 2005; Reid et al. 2009). \nThe results from epoch 1 were presented in ZBS11. Here, we only discuss the 22 GHz data; the full VLBI data set will be presented together with future VLBI observations in an upcoming paper. The data were correlated at the VLBA Array Operations Center in Socorro, New Mexico and calibrated using AIPS and ParselTongue (Kettenis et al. 2006). We applied the latest values of the Earth orientation parameters and performed zenith delay corrections based on the results of the geodetic block observations. Total electron content maps of the ionosphere were used to correct for ionospheric phase changes. Amplitude calibration used system temperature measurements and standard gain curves. We performed a 'manual phase-calibration' using the data from one scan of J1638+5720 to remove instrumental phase offsets among the frequency bands. We then fringe-fitted the data from J1638+5720. Since J1638+5720 has extended structure, we performed phase self-calibration, and later amplitude and phase self-calibration on J1638+5720 to construct robust models of J1638+5720 at both frequencies. These \n9 The data were collected as part of programs BS210 and BS212. \nmodels were then used to fringe-fit the data again. Finally, the calibration was transferred to Sw 1644+57 and J1657+5705. The data were imaged in AIPS using robust weighting (with ROBUST=0 ). \nAlinear fit to the positions of Sw 1644+57 from epochs 2-6 gives a proper motion of 81 ± 37 µ as yr -1 in right ascension and 46 ± 59 µ as yr -1 in declination, with an additional systematic uncertainty of about 40 µ as yr -1 . This is consistent with no detectable motion. The corresponding 3 σ upper limit on the motion at the redshift of Sw 1644+57 is /lessorsimilar 1 pc. Since the first epoch was correlated at a different position, it was not included in this fit to avoid second order systematic errors. \nTo obtain an accurate estimate of the angular size in our latest epoch (2011 September 17 UT), we fit Gaussian models directly to the visibility data. We first self-calibrated the data in phase using solution intervals of 20 min, and then fitted a circular Gaussian model using a weighted least-squares fit. The source is not resolved in our observation. For unresolved sources the fitted size can be strongly correlated with the antenna amplitude gains, which are only imprecisely known. Our uncertainty on the fitted FWHM size, and consequently our upper limit, was estimated from a Monte-Carlo simulation in which we randomly varied the amplitude gains of the antennas by conservative factors of ± 25% (for a more elaborate discussion of the uncertainties in estimating angular source sizes from a similar procedure see Bietenholz et al. 2010). The 3 σ upper limit on the FWHM source size is 0 . 22 mas, corresponding to a projected physical diameter of /lessorsimilar 1 . 1 pc. For spherical expansion this corresponds to an upper limit on the apparent expansion velocity of /lessorsimilar 3 . 8 c . For a collimated relativistic source with an opening angle θ j the projected diameter is ≈ 4 Γ 2 ct θ j , indicating that for θ j = 0 . 1 the limit on the size of Sw 1644+57 corresponds to Γ /lessorsimilar 5 at δ t ≈ 176 d. We compare this result with estimates of the source size from modeling of the radio emission in §4.1.", '3. OBSERVED EVOLUTION OF THE RADIO EMISSION': 'The light curves at observed frequencies of 4.9, 6.7, 15.4, 19, 24, 43, 90, and 230 GHz are presented in Figure 1. As described in ZBS11, the light curves at 4.9 to 24 GHz exhibit an initial rapid increase, with a flux density of F ν ∝ t 1 . 5 -t 2 , followed by a shallower increase, with F ν ∝ t 0 . 5 . In the millimeter band, the light curves peak on a timescale of a few days and then decline as F ν ∝ t -1 . This evolution is due to a synchrotron spectrum that is optically thin in the millimeter band and optically thick in the centimeter band, with a peak frequency and flux density that decline as a function of time as ν p ∝ t -1 . 3 and F ν , p ∝ t -0 . 8 (ZBS11). The properties of the spectral energy distribution, along with measurements of interstellar scintillation, established that the outflow is relativistic ( Γ ≈ 2 -5), that the density profile of the circumnuclear medium (CNM) is roughly ρ CNM ∝ r -2 , and that the post-shock energy increases as E ∝ t 0 . 5 (ZBS11). \nThe most striking result from our new observations is that the millimeter flux on a timescale of ∼ 100 d is significantly brighter than expected from an extrapolation of the early decline. Indeed, at 90 GHz the brightness is comparable to the initial peak at δ t ∼ 10 d. A similar effect is observed at lower frequencies, where the light curves exhibit an upturn starting at ≈ 30 d, with F ν ∝ t 0 . 7 . The 90 GHz light curve is already in a declining (i.e., optically thin) phase at δ t /greaterorsimilar 100 d, while the light curves at 4.9 to 24 GHz peak at δ t ≈ 100 d (19 and 24 GHz), δ t ≈ 125 d (15.4 GHz), δ t ≈ 170 d (8.4 GHz), and δ t ≈ 190 d (4.9 and 6.7 GHz), with peak flux densities of ≈ 31 \nmJy, ≈ 28 mJy, ≈ 23 mJy, and ≈ 20 mJy, respectively. These observations are at odds with the light curve evolution predicted by MGM11 based on our radio data at 5 -22 d, using a model that assumes a constant energy and a steady ρ ∝ r -2 CNM profile (Figure 1). \nThe observed change in the light curve evolution at δ t /greaterorsimilar 1 month requires an increase in the outflow kinetic energy and/or the CNM density. The evolution of the peak frequency, roughly ν p ∝ t -1 at δ t ∼ 30 -100 d, is slower than expected for the characteristic synchrotron frequency, ν m ∝ t -1 . 5 (see §4). Since ν m only depends on the kinetic energy, ν m ∝ E 0 . 5 j t -3 / 2 (Granot & Sari 2002), the observed evolution requires an increase in the energy, with Ej ∝ t . At the same time, the similar light curve evolution at 4.9 and 24 GHz, which straddle the self-absorption frequency (see §4), indicates that the density must also change. In particular, at 4.9 GHz we expect F ν ∝ E 1 . 5 j A -1 . 5 t 0 . 5 , while at 24 GHz we expect F ν ∝ E 5 / 6 j A 0 . 5 t -0 . 5 ; here A is a fiducial density that parametrizes the CNM density, with ρ = Ar -2 . For the observed light curve evolution of F ν ∝ t 0 . 7 at both frequencies, and using the inferred evolution of Ej , we find that A ∝ t 0 . 85 . Thus, both the energy and density normalization appear to increase at δ t /greaterorsimilar 30 d. In the next section we model the radio emission in detail to extract the temporal evolution of the energy and density.', '4. DETAILED MODELING OF THE RADIO EMISSION': 'To determine the temporal evolution of the relativistic outflow and the radial density profile we use the formulation of MGM11, which draws on the GRB afterglow model of Granot & Sari (2002) (hereafter, GS02). The MGM11 model is largely motivated by the basic properties of the outflow that were determined in our initial study of the radio emission (ZBS11), as well as by the observed evolution of the Xray emission (Burrows et al. 2011). The model assumes that the outflow is collimated, with an opening angle θ j , and has a Lorentz factor of Γ j ; in ZBS11 we demonstrated that Γ j ∼ few (see also Bloom et al. 2011). The kinetic luminosity of the outflow, L j , iso, is assumed to be constant for a timescale t j , followed by a decline as L j , iso ∝ t -5 / 3 at t /greaterorsimilar t j . This is expected for fallback accretion in a TDE, and is observationally motivated by the evolution of the X-ray light curve, which also indicates that t j ≈ 10 6 s (Burrows et al. 2011). As a result of the long-term injection of energy, the expectation is that the kinetic energy will gradually increase from an initial value of E j , iso = L j , iso t j to a final level of (5 / 2) L j , iso t j . From the initial X-ray luminosity, L X , iso ≈ 6 × 10 46 erg s -1 , the expected total energy is E j , iso ≈ 3 × 10 53 ( /epsilon1 X / 0 . 5) -1 erg, where /epsilon1 X is the efficiency of the jet in producing X-rays (MGM11), while the predicted beaming-corrected energy is Ej ≈ 1 . 5 × 10 51 erg. \nThe interaction of the outflow with the CNM leads to synchrotron emission due to the acceleration of electrons and amplification of magnetic fields. Based on the early radio observations, ZBS11 demonstrated that the CNM density roughly follows ρ CNM ∝ r -2 (hereafter, Wind medium), which is the profile adopted for the analysis in MGM11. The Lorentz factor of the fluid behind the forward shock is Γ sh , FS = Γ j (1 -Γ j √ 2 n CNM / 7 nj ), where n CNM is the circumnuclear density and nj = L j , iso / 4 π r 2 mpc 3 Γ 2 j is the density of the ejecta (see MGM11); this relation is appropriate for the case of a Newtonian reverse shock, i.e., Γ 2 j /lessmuch nj / n CNM (Sari & Piran 1995), which as we find below is appropriate for Sw 1644+57 (Table 2). A key difference between the MGM11 model and the \nGS02 GRB afterglow model is that the former includes a suppression of the flux density by a factor of ( Γ sh , FS θ j ) 2 / 2 due to the finite extent of the collimated outflow (i.e., Γ sh , FS /lessorsimilar 1 /θ j ). Thus, the synchrotron emission from the relativistic outflow is determined by L j , iso, n CNM, and the fractions of post-shock energy in the radiating electrons ( /epsilon1 e ) and the magnetic field ( /epsilon1 B ). \nWeuse this model (Equations 7-11 of MGM11), along with the smoothing formulation of GS02, to fit snap-shot broadband spectra of Sw 1644+57 on a timescale of δ t ≈ 5 -216 d. In each epoch we fix /epsilon1 e = /epsilon1 B = 0 . 1 and p = 2 . 5, and we then determine the best-fit values of L j , iso and n CNM; here p is the power law index of the electron Lorentz factor distribution, N ( γ ) ∝ γ -p . For our choice of p value the synchrotron frequencies (self-absorption: ν a ; peak: ν m ) and flux normalization [ F ν ( ν = ν a )] for t /greaterorsimilar t j are given by (MGM11, GS02): \nν a ( t ) = 4 . 2 × 10 9 /epsilon1 -1 e , -1 /epsilon1 0 . 2 B , -1 L -0 . 4 j , iso , 48 t -1 j , 6 n 1 . 2 18 ( t t j ) -0 . 6 Hz , (1) \nF ν [ ν a ( t )] = 345 /epsilon1 e , -1 L 1 . 5 j , iso , 48 t 2 j , 6 n -1 . 5 18 θ 2 j , -1 ν 2 a , 10 ( t t j ) 0 . 5 mJy , (3) \nν m ( t ) = 3 . 5 × 10 11 /epsilon1 2 e , -1 /epsilon1 0 . 5 B , -1 L 0 . 5 j , iso , 48 t -1 j , 6 ( t t j ) -1 . 5 Hz , (2) \nwhere we use the notation X ≡ 10 y Xy , and n 18 is the CNM density ( n CNM) at a fiducial radius of 10 18 cm. At t /lessorsimilar t j we use the same equations but with the time dependencies modified to ν a ∝ t -1 , ν m ∝ t -1 , and F ν ( ν = ν a ) ∝ t 2 (MGM11). The synchrotron spectrum is given by (c.f., Spectrum 1 of Granot & Sari 2002): \nF ν = F ν ( ν a ) [ ( ν ν a ) -s 1 β 1 + ( ν ν a ) -s 1 β 2 ] -1 / s 1 × [ 1 + ( ν ν m ) s 2( β 2 -β 3) ] -1 / s 2 , (4) \nwhere s 1 and s 2 are smoothing parameters (GS02), and β 1 = 2, β 2 = 1 / 3, and β 3 = (1 -p ) / 2 are the power law indices for each segment of the synchrotron spectrum. \nEquations 1 and 4 are appropriate when ν a <ν m , and have to be modified when ν m <ν a as follows (GS02, MGM11 11 ): \nν a ∝ t -3( p + 2) / 2( p + 4) , (5) \nF ν = F ν ( ν m ) [ ( ν ν m ) -s 1 β 1 + ( ν ν m ) -s 1 β 2 ] -1 / s 1 × [ 1 + ( ν ν a ) s 2( β 2 -β 3) ] -1 / s 2 , (6) \nwhere the spectrum is now normalized at ν = ν m , s 1 and s 2, take on different values (GS02) and β 1 = 2, β 2 = 5 / 2, and β 3 = (1 -p ) / 2. To smoothly connect the evolution in the two phases we use a weighted average of Equations 4 and 6, with the weighting determined by the time difference relative to the transition time, t am, defined as ν a ( t am) = ν m ( t am). We find \nthat t am ≈ 275 d, so this transition does not affect the data presented in this paper. \nInstead of imposing a specific temporal evolution on L j , iso and n CNM, we model each broad-band radio spectrum independently to extract the evolution of these quantities, and in turn the time evolution of the emission radius ( r ) and Γ sh (and hence Γ j ), as well as the radial density profile. The results of the individual fits are shown in Figure 2 and the relevant extracted parameters are plotted in Figure 3 and summarized in Table 2. The broad-band SEDs reveal a complex evolution. At δ t ≈ 5 -22 d both the peak frequency and peak flux density decrease with time (as noted by ZBS11), but starting at δ t ≈ 36 d the peak flux density begins to increase, reaching a maximum at δ t ≈ 100 d, and subsequently declining again. \nThe VLBI size limit of r proj /lessorsimilar 0 . 55 pc at δ t ≈ 175 d can be used to set an independent upper bound on the ratio L j , iso , 48 / n 18. In particular, on this timescale we expect r proj ≈ 2 . 9 t j , 6( L j , iso , 48 / n 18) 0 . 5 θ j pc, indicating that L j , iso , 48 / n 18 /lessorsimilar 3 . 6 (for θ j = 0 . 1). From the results of our radio modeling we find on a similar timescale that L j , iso , 48 ≈ 0 . 45 and n 18 ≈ 3 (Table 2), and hence L j , iso , 48 / n 18 ≈ 0 . 15 is in agreement with the VLBI size limit. On the other hand, if we assume /epsilon1 e = 0 . 01 the resulting value of L j , iso , 48 increases by about a factor of 15, while n 18 decreases by about a factor of 2, leading to L j , iso , 48 / n 18 ≈ 4, in violation of the VLBI size limit 12 We therefore conclude from the VLBI results that /epsilon1 e /greaterorsimilar 0 . 01.', '4.1. The Evolution of E j , iso': 'A detailed analysis of the evolution of ν m (shown in Figure 4) reveals a complex behavior. While it continuously declines as a function of time, the decline rate is always shallower than the expected t -3 / 2 for constant energy. Since ν m only depends on L j , iso and the equipartition fractions (which are not expected to change with time), the shallower than expected evolution directly implies that L j , iso continuously increases with time. In particular, at 5 -22 d and /greaterorsimilar 100 d, the observed evolution of ν m ∝ t -1 . 15 indicates that L j , iso ∝ t 0 . 7 , while in the intermediate phase (22 -100) d, ν m ∝ t -0 . 95 points to a more rapid increase in the energy scale, with L j , iso ∝ t 1 . 1 . \nThe time evolution of L j , iso is shown in Figure 5. We find that at δ t /lessorsimilar 20 d L j , iso ≈ 4 . 5 × 10 46 erg s -1 (or E j , iso ≈ 4 . 5 × 10 52 erg for t j = 10 6 s). Changing the values of the equipartition fractions from our assumed values /epsilon1 e = /epsilon1 B = 0 . 1, we find that L j , iso increases by about a factor of 3 if /epsilon1 B = 0 . 01 or by a factor of 15 if /epsilon1 e = 0 . 01 ( /epsilon1 e /lessorsimilar 0 . 01 is ruled out by the VLBI limits on the source size). These scalings hold for the overall time evolution of L j , iso. \nAlso shown in Figure 5 is the expected evolution for a model with L j , iso = const at t < t j and L j , iso ∝ t -5 / 3 at t > t j , as inferred from the X-ray light curve (Burrows et al. 2011, MGM11). In this model, we expect the integrated value of L j , iso (i.e., the total isotropic-equivalent energy) to approach (5 / 2) L j , iso as t →∞ , and to roughly double within δ t ∼ 5 t j ≈ 50 -60 d. This seems to be the case based on the radio data at δ t /lessorsimilar 30 d, but the subsequent rapid increase in energy by about an order of magnitude at 30 -216 d cannot be explained with continued injection from a t -5 / 3 tail. On the other hand, the X-ray light curve of Sw 1644+57 (Figure 5) agrees well with this simple luminosity evolution, with L X , iso , 0 ≈ 6 × 10 46 erg s -1 at t /lessorsimilar 13 d. A comparison to the inferred value of \nL j , iso , 0 indicates a high efficiency 13 of producing X-rays of /epsilon1 X ≈ 0 . 5. \nSince continuous energy injection from an L ∝ t -5 / 3 tail cannot explain the inferred rise in integrated kinetic energy, the radio data require a different energy injection mechanism. Onepossibility is that the outflow has a distribution of Lorentz factors, with increasing energy as a function of decreasing Lorentz factor, Ej ( > Γ j ) ∝ Γ 1 -s j (Sari & Mészáros 2000). In this scenario the energy will increase with time as material with a lower Lorentz factor catches up with decelerated ejecta that initially had a higher Lorentz factor. The continuous injection of energy will also lead to a more rapid increase in radius, with r ∝ t (1 + s ) / (7 + s -2 m ) (Sari & Mészáros 2000); here m is the power law index describing the CNM density profile ( ρ ∝ r -m ). In Figure 3 we plot the inferred radius as a function of time, and find that it follows r ∝ t 0 . 6 , faster than expected in a simple Wind model ( r ∝ t 0 . 5 ). We also find from the density profile that m ≈ 1 . 5 (see §4.2), thereby leading to s ≈ 3 . 5. Thus, we expect in this scenario a fairly steep profile for the energy as a function of Lorentz factor of E ( > Γ j ) ∝ Γ -2 . 5 j . \nWith this relation, and the inferred time evolution of the jet Lorentz factor (Figure 3) we can infer the expected increase in energy. We find Γ j ≈ 5 . 5 at δ t = 10 d and Γ j ≈ 2 . 2 at δ t = 216 d, indicating an expected increase in energy by about a factor of 10 for our inferred profile. This is in excellent agreement with the data. Thus, a jet with a distribution of Lorentz factors of E ( > Γ j ) ∝ Γ -2 . 5 j naturally explains the evolution of source size and the substantial increase in energy beyond the on-going input from a t -5 / 3 tail.', '4.2. The Radial Density Profile': 'In addition to the substantial increase in energy, we also find that the normalization of the density profile, n 18 ≡ n CNM r 2 18 , changes as a function of time (Figure 3). This indicates that the radial density structure is not a simple Wind profile, for which n 18 is by definition constant. In particular, we find that n 18 decreases at 5 -10 d, followed by a steady increase at 10 -216 d is about a factor of 8. The resulting radial density profile is shown in Figure 6, and clearly reflects the complex evolution of n 18. The density profile at r ≈ 0 . 1 -0 . 4 pc is ρ ∝ r -1 . 5 , followed by a uniform density at r ≈ 0 . 4 -0 . 6 pc, and a slow transition back to ρ ∝ r -1 . 5 by r ≈ 1 . 2 pc. The inferred density is n ≈ 0 . 2 -2 cm -3 at r ≈ 1 . 2 -0 . 1 pc, respectively. These values scale with the choice of equipartition fractions: for /epsilon1 B = 0 . 01 the density is about 3 times larger, while for /epsilon1 e =0 . 01 it is about 1.7 times lower. Thus, the inferred density is fairly robust to large changes in the equipartition fractions. \nThe flattening at r ≈ 0 . 4 -0 . 6 pc coincides with the rapid increase in energy at δ t ≈ 35 -100 d. This can be understood in the context of an outflow with a distribution of Lorentz factors since the relative increase in density compared to the previous r -1 . 5 profile leads to enhanced contribution from lower Lorentz factor ejecta. A similar effect has been observed in the radio emission from some core-collapse supernovae, which have a steep ejecta profile with E ∝ v -5 (e.g., Chevalier 1982; Matzner & McKee 1999; Berger et al. 2002; Soderberg et al. 2006). \nAlso shown in Figure 6 are the inferred densities in the central parsec of the Milky way galactic center from X-ray measurements (Baganoff et al. 2003). The Galactic Center density is similar that inferred for Sw 1644+57 at r ≈ 0 . 05 pc, but \nis about 30 times larger at r ≈ 0 . 4 pc. If the bulk of the gas in the central parsec is due to mass loss from massive stars (e.g., Melia 1992; Baganoff et al. 2003; Quataert 2004), the lower density inferred here may indicate a much smaller number of massive stars in the central parsec of the host galaxy of Sw1644+57. Although we cannot investigate star formation on scales smaller than ∼ 1 kpc in the host of Sw 1644+57, the overall star formation rate in this galaxy is indeed a factor of several times lower than in the Milky Way (Levan et al. 2011). Regardless of the reason for the difference in density, it is quite remarkable that the radio emission from Sw 1644+57 offers as detailed a view (or better) of the density profile in the inner parsec around an inactive SMBH at z = 0 . 354 as available for the Galactic Center at a distance of only 8.5 kpc.', '4.3. Optical and Near-Infrared Emission': 'Using the radio modeling we can also predict the optical and near-IR emission from the jet. In Figure 7 we plot the optical r -band upper limits from Levan et al. (2011), as well as their near-IR K -band measurements. The latter include the total flux from Sw 1644+57 and its host galaxy since transientfree templates are not presently available. We estimate the host contribution to be about 20 µ Jy ( K ≈ 20 . 6 AB mag) by requiring that the overall shape of the K -band light curve match our predicted light curve. We note that this only affects the light curve shape at δ t /greaterorsimilar 15 d since at earlier times the observed flux is dominated by the transient itself. A simple extrapolation of our model over-predicts the K -band flux density by about an order of magnitude, indicating the presence of an additional break in the spectrum between the radio and optical/near-IR band. Such a break is indicative of the synchrotron cooling frequency, ν c . To explain the K -band flux density requires ν c ∼ 10 13 Hz. Even if we include this break, the observed r -band limits are still a factor of about 70 times fainter than the model prediction. To explain this discrepancy requires host galaxy extinction of A V , host /greaterorsimilar 3 . 5 mag (see also Levan et al. 2011, ZBS11). With the addition of a cooling break and host galaxy extinction we find an excellent match between our model and the near-IR fluxes. Since these data were not used in the fitting, the agreement between our model and the data provides an independent confirmation of the model.', '5. IMPLICATIONS FOR RELATIVISTIC JETS AND THE ENVIRONMENTS OF SUPERMASSIVE BLACK HOLES': 'Our on-going radio observations have uncovered two unexpected and critical results regarding the nature of the relativistic outflow from Sw 1644+57 and the parsec-scale environment around a previously-dormant SMBH. We find that the relativistic outflow is not dominated by a single Lorentz factor, as typically assumed in GRB jet models. Instead, the outflow has a Lorentz factor distribution with Ej ( > Γ j ) ∝ Γ -2 . 5 j , at least over the range of Lorentz factors probed so far, Γ j ≈ 2 . 2 -5 . 5. This scenario is reminiscent of supernova ejecta, in which the energy is a strong function of velocity, E ∝ v -5 (Chevalier 1982; Matzner & McKee 1999). In the SN case, this coupling reflects shock acceleration through the steep density gradient in the outer envelope of the star. In the case of Sw 1644+57 we do not expect such a density gradient, suggesting instead that the ejecta structure may be an intrinsic property of relativistic jet launching, potentially through the Blandford-Znajek mechanism (Blandford & Znajek 1977). In this case, the same structure may also apply to the relativistic jets of GRBs, which are generally assumed to have a single \nLorentz factor. \nTo assess this possibility we compare the ratio of event duration to the accretion disk dynamical timescale, t dyn ∝ M BH( r / rg ) 3 / 2 , where rg = 2 GM BH / c 2 is the black hole Schwarzschild radius. For similar ratios of ( r / rg ), since M BH , Sw1644 + 57 ∼ 10 6 M /circledot ∼ 10 5 M BH , GRBs, the dynamical timescale in Sw 1644+57 is about 10 5 times longer than in GRBs. The ratio of durations is similar, ∼ 10 6 s for Sw1644+57 and ∼ 10 s for GRBs. Therefore, the ratio of duration to dynamical timescale is similar, and the jet launching mechanism may imprint a similar profile in the case of Sw1644+57 and GRBss. Of course, in long GRBs the jet still has to propagate through the stellar envelope, which may modify the jet structure. \nWenext address the potential implications of the radial density profile and the flattening at r ≈ 0 . 4 -0 . 6 pc. One possibility is that this scale represents the Bondi radius for spherical accretion from the CNM, rB = GM / c 2 s = ( c / cs ) 2 rg / 2; here cs is the sound speed in the CNM and rg = 2 GM / c 2 is the Schwarzschild radius of the SMBH. For a CNM gas temperature of ∼ 10 7 K (Baganoff et al. 2003; Narayan & Fabian 2011), we find rB ≈ 10 6 rs ≈ 0 . 1 -1 pc for the SMBH mass of 10 6 -10 7 M /circledot inferred for Sw 1644+57. Thus, the radius at which we infer a density enhancement is in reasonable agreement with the expected Bondi radius. In addition, we note that the expected density profile inside the Bondi radius, ρ ∝ r -3 / 2 , matches the inferred profile for Sw 1644+57 on a scale of ∼ 0 . 15 -0 . 4 pc. In the standard scenario, the medium outside the Bondi radius is assumed to have a uniform density, while here we infer a continued decline in the density at r ≈ 0 . 6 -1 . 2 pc. This may reflect the complex conditions in the inner few parsecs around the SMBH in which the interstellar density profile may be mainly influenced by mass loss from stars (as in the case of the Milky Way galactic center; e.g., Krabbe et al. 1991). Indeed, from the inferred density of n CNM ≈ 0 . 6 cm -3 at r ≈ 0 . 5 pc, the required mass loss rate is ˙ M ≈ 5 × 10 -5 M /circledot yr -1 for a wind velocity of 10 3 km s -1 . \nIn particular, in the case of the Galactic Center it has long been believed that the gas in the central parsec is supplied by mass loss from massive stars, with the bulk of the gas ( /greaterorsimilar 90%) being thermally expelled in a wind (Melia 1992; Baganoff et al. 2003; Quataert 2004, but see Sazonov et al. 2011 for an alternative interpretation of the X-ray emission). Using a spherically symmetric hydrodynamic simulation of the gas supplied by stellar winds, Quataert (2004) showed that beyond about 0.4 pc gas is mainly expelled with a resulting Wind profile ( ρ ∝ r -2 ) on larger scales, while on scales of /lessorsimilar 0 . 2 pc the profile is roughly as expected for Bondi accretion ( ρ ∝ r -3 / 2 ); see Figure 6. The profile we find here is somewhat different from this model, but this may be due to a different distribution of massive stars, or to the simplifying assumption of spherical symmetry in the model.', '6. CONCLUSIONS AND FUTURE DIRECTIONS': 'We presented radio observations of Sw 1644+57 extending to δ t ≈ 216 d and spanning a wide range of frequencies. The evolution of the radio emission changes dramatically at δ t /greaterorsimilar 1 month, requiring an increase in the total energy by about an order of magnitude, a density profile of ρ ∝ r -3 / 2 (0 . 1 -1 . 2 pc), and a flattening at r ≈ 0 . 4 -0 . 6 pc. A comparison of the model to optical limits and near-IR detections indicates a cooling break at ν c ∼ 10 13 Hz and host galaxy extinction of A V , host /greaterorsimilar 3 . 5 mag. \nThe increase in energy cannot be explained by injection from an L ∝ t -5 / 3 tail that is expected in tidal disruption events and which matches the evolution of the X-ray emission. We conclude that a natural explanation is a structured outflow with E ( > Γ j ) ∝ Γ -2 . 5 j . The inferred density profile and the radial scale of the density enhancement are in rough agreement with the expectation for Bondi accretion from a circumnuclear medium. The jet energetics and structure, as well as the detailed density profile on ∼ 0 . 1 -1 pc scale are a testament to the important insight that can be gained from continued radio observations of Sw 1644+57. In particular, the radial density profile is traced in greater detail than even the inner parsec of the Milky Way. Continued radio observations will probe the environment to a scale of ∼ 10 pc in the coming decade. \nUsing the results of our analysis we can predict the future evolution of the radio emission (modulo any future unpredictable changes in energy and/or density as we have found here). We use the evolution of L j , iso and n CNM as inferred from the data at δ t /lessorsimilar 216 d, and assume that the density will continue to evolve as ρ ∝ r -1 . 5 and that the energy will increase to a maximum beaming-corrected value of Ej = L j , iso t j [1 -cos( θ j )] with Ej = 10 52 , 3 × 10 52 , 10 53 erg. The resulting light curves at 6 and 22 GHz are shown in Figure 8. The long-term evolution is marked by a break when Ej achieves its maximum value, corresponding to about 5, 28, and 180 yr for our three choices of maximum energy. Using the 5 σ sensitivity of the EVLA in an observation of a few hours 14 , we find that the emission at 22 and 6 GHz should be detectable for at least ∼ 40 yr and ∼ 80 yr, respectively. Indeed, any significant upgrades to the EVLA or the construction of more sensitive radio facilities in the coming decades may extend the range of detectability to centuries 15 . The same is true if the total energy scale is ∼ 10 53 erg. \nAn equally important question is whether the jet will be resolvable with VLBI in the future. The projected radius is r proj ≈ r θ j , as long as the jet maintains its collimation. In Figure 9 we plot the predicted future evolution of r using the prescription described above. We find that for θ j = 0 . 1 and a best-case VLBI angular resolution of ≈ 0 . 2 mas (FWHM), the source should become resolvable at δ t ≈ 6 yr. On this timescale the 22 GHz flux density is expected to be only ≈ 2 mJy (Figure 8), still accessible with VLBI. While the flux density at 6 GHz is expected to be larger by about a factor of 2 . 6, the angular resolution at this frequency is poorer by \n14 At 22 GHz we use the sensitivity for the full 8 GHz bandwidth that will become available some time in 2012. \nabout a factor of 3.7, making it less competitive than 22 GHz. Thus, we conclude that the radio emission from Sw 1644+57 may be marginally resolved in a few years. On the other hand, if the jet undergoes significant spreading on the timescale at which it becomes non-relativistic (as expected for GRB jets: e.g., Livio & Waxman 2000) it is possible that it will become resolvable at δ t ∼ 1 -2 yr when the expected 22 GHz flux density is still ∼ 10 mJy. \nWe are undertaking continued multi-frequency radio monitoring of Sw 1644+57 to follow the long-term evolution of the relativistic outflow and the radial profile of the ambient medium. Even in the absence of any future dramatic changes relative to the current evolution, we expect that in the next few years we may be able to determine the total energy of the relativistic outflow, measure the spreading of the jet, and study the radial density profile to a scale of ∼ 10 pc. Future papers in this series will detail these results. \nWe thank Ramesh Narayan, Ryan Chornock, and Nicholas Stone for helpful discussions, and Glen Petitpas for assistance with the SMA data reduction. E.B. acknowledges support from Swift AO6 grant NNX10AI24G and from the National Science Foundation through Grant AST-1107973. A.B. is supported by a Marie Curie Outgoing International Fellowship (FP7) of the European Union (project number 275596). This work is partially based on observations with the 100m telescope of the MPIfR (Max-Planck-Institut f ur Radioastronomie) at Effelsberg. The Submillimeter Array is a joint project between the Smithsonian Astrophysical Observatory and the Academia Sinica Institute of Astronomy and Astrophysics, and is funded by the Smithsonian Institution and the Academia Sinica. Support for CARMA construction was derived from the Gordon and Betty Moore Foundation, the Kenneth T. and Eileen L. Norris Foundation, the James S. McDonnell Foundation, the Associates of the California Institute of Technology, the University of Chicago, the states of California, Illinois, and Maryland, and the National Science Foundation. Ongoing CARMA development and operations are supported by the National Science Foundation under a cooperative agreement, and by the CARMA partner universities. The AMI arrays are supported by the University of Cambridge and the STFC. 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A., et al. 2011, Nature, 476, 425; ZBS11 \nTABLE 1 RADIO OBSERVATIONS OF SW 1644+57 \n| δ t a (d) | Facility | Frequency (GHz) | Flux Density (mJy) | Flux Density (mJy) |\n|---------------|----------------------|-------------------|-----------------------------------|-----------------------------------|\n| | | | 0 . 21 ± 0 . 08 | 0 . 21 ± 0 . 08 |\n| 6.79 | EVLA | 1.4 | | |\n| 126.59 | | 1.4 | 1 . 10 ± 0 . 10 . 11 | 1 . 10 ± 0 . 10 . 11 |\n| 174.47 | EVLA EVLA | 1.4 | 1 . 60 ± 0 | 1 . 60 ± 0 |\n| 197.41 | EVLA | 1.4 | 1 . 80 ± | 1 . 80 ± |\n| 3.87 | EVLA | 4.9 | 0 . 10 0 . 25 ± 0 . 02 | 0 . 10 0 . 25 ± 0 . 02 |\n| 4.76 | EVLA | 4.9 | 0 . 34 ± 0 . 02 | 0 . 34 ± 0 . 02 |\n| 5.00 | EVLA | 4.9 | 0 . 34 ± 0 . 02 ± | 0 . 34 ± 0 . 02 ± |\n| 5.79 | | 4.9 | 0 . 61 0 . 02 | 0 . 61 0 . 02 |\n| 6.78 | EVLA EVLA | 4.9 | 0 . 82 ± 0 . 02 | 0 . 82 ± 0 . 02 |\n| 7.77 | EVLA | 4.9 | 1 . 48 ± 0 . 02 | 1 . 48 ± 0 . 02 |\n| 9.79 | EVLA | 4.9 | 1 . 47 ± 0 . 02 | 1 . 47 ± 0 . 02 |\n| 14.98 | EVLA | 4.9 | 1 . 80 ± 0 . 03 | 1 . 80 ± 0 . 03 |\n| | | 4.9 | 2 . 10 ± 0 . 01 | 2 . 10 ± 0 . 01 |\n| 22.78 | EVLA | | | |\n| 35.86 | EVLA | 4.9 | 4 . 62 ± 0 . 02 | 4 . 62 ± 0 . 02 |\n| 50.65 67.61 | EVLA EVLA | 4.9 | 4 . 84 ± 0 . 03 5 . 86 ± 0 . 03 | 4 . 84 ± 0 . 03 5 . 86 ± 0 . 03 |\n| | | 4.9 | 9 . 06 ± 0 . | 9 . 06 ± 0 . |\n| 94.64 | EVLA | 4.9 | 03 | 03 |\n| 111.62 126.51 | EVLA EVLA | 4.9 | 9 . 10 ± 0 . 03 9 . 10 ± 0 . 03 | 9 . 10 ± 0 . 03 9 . 10 ± 0 . 03 |\n| | | 4.9 | 11 . 71 ± 0 . 03 | 11 . 71 ± 0 . 03 |\n| 143.62 | EVLA | 4.9 | | |\n| 164.38 | EVLA | 4.9 | 12 . 93 ± 0 . 05 ± | 12 . 93 ± 0 . 05 ± |\n| 174.47 197.41 | EVLA EVLA | 4.9 4.9 | 12 . 83 0 . 06 13 . 29 ± 0 . 03 | 12 . 83 0 . 06 13 . 29 ± 0 . 03 |\n| 213.32 | EVLA | 4.9 | 12 . 43 ± 0 . 04 | 12 . 43 ± 0 . 04 |\n| | | 6.7 | 0 . 38 ± 0 . 02 | 0 . 38 ± 0 . 02 |\n| 3.87 4.76 | EVLA EVLA | | ± | ± |\n| 5.00 | | 6.7 6.7 | 0 . 63 0 . 02 ± | 0 . 63 0 . 02 ± |\n| | EVLA | | 0 . 64 0 . 02 | 0 . 64 0 . 02 |\n| 5.79 | EVLA | 6.7 | 1 . 16 ± 0 . 02 | 1 . 16 ± 0 . 02 |\n| 6.79 | EVLA | 6.7 | 1 . 47 ± 0 . 02 | 1 . 47 ± 0 . 02 |\n| 7.77 9.79 | EVLA EVLA | 6.7 6.7 | 1 . 50 ± 0 . 02 ± | 1 . 50 ± 0 . 02 ± |\n| 14.98 | EVLA | 6.7 | 2 . 15 0 . 02 | 2 . 15 0 . 02 |\n| | | | 3 . 79 ± 0 . 03 3 . 44 ± 0 . 01 | 3 . 79 ± 0 . 03 3 . 44 ± 0 . 01 |\n| 22.78 | EVLA | 6.7 | 6 . 39 ± 0 . 02 | 6 . 39 ± 0 . 02 |\n| 35.86 50.65 | EVLA | 6.7 | 5 . 70 ± 0 . 02 | 5 . 70 ± 0 . 02 |\n| | EVLA | 6.7 | ± | ± |\n| 67.61 | EVLA | | 8 . 94 0 . 03 | 8 . 94 0 . 03 |\n| 94.64 | EVLA | 6.7 6.7 | 13 . 43 ± 0 . 03 | 13 . 43 ± 0 . 03 |\n| 111.62 | EVLA | 6.7 | 13 . 66 ± 0 . 03 | 13 . 66 ± 0 . 03 |\n| 126.51 | | | . 16 ± 0 . 04 | . 16 ± 0 . 04 |\n| 143.62 | EVLA EVLA | 6.7 6.7 | 14 16 . 85 ± 0 . 04 | 14 16 . 85 ± 0 . 04 |\n| 164.38 174.47 | EVLA EVLA | 6.7 | 18 . 27 ± 0 . 06 19 . 59 ± 0 . 16 | 18 . 27 ± 0 . 06 19 . 59 ± 0 . 16 |\n| 197.41 | EVLA | 6.7 6.7 | | |\n| 213.32 | EVLA | 6.7 | 19 . 34 ± 0 . 03 | 19 . 34 ± 0 . 03 |\n| 14.97 | | 8.4 | 18 . 02 ± 0 . 05 ± | 18 . 02 ± 0 . 05 ± |\n| 127.69 | EVLA | 8.4 | 5 . 49 0 . 09 | 5 . 49 0 . 09 |\n| | EVLA | | 19 . 03 ± 0 . 14 22 . 15 ± 0 . 20 | 19 . 03 ± 0 . 14 22 . 15 ± 0 . 20 |\n| 159.77 174.47 | EVLA EVLA | 8.4 8.4 | 23 . 19 ± 0 . 38 | 23 . 19 ± 0 . 38 |\n| 177.50 | EVLA | 8.4 | 23 . 65 ± 0 . 16 | 23 . 65 ± 0 . 16 |\n| 197.41 | EVLA | 8.4 | 22 . 42 ± 0 . 10 | 22 . 42 ± 0 . 10 |\n| 213.32 | EVLA | 8.4 8.4 | 22 . 04 ± 0 . 13 21 ± | 22 . 04 ± 0 . 13 21 ± |\n| 219.22 | EVLA | | . 52 0 . 09 | . 52 0 . 09 |\n| 5.81 | AMI-LA | 15.4 | 2 . 69 ± 0 . 44 | 2 . 69 ± 0 . 44 |\n| 6.64 | AMI-LA | 15.4 | 3 . 62 ± 0 . 22 | 3 . 62 ± 0 . 22 |\n| 7.62 | AMI-LA | 15.4 | 4 . 32 ± 0 . 28 5 ± | 4 . 32 ± 0 . 28 5 ± |\n| 8.55 9.56 | AMI-LA | 15.4 | . 07 0 . 6 ± | . 07 0 . 6 ± |\n| | AMI-LA | 15.4 | 36 . 68 0 . 51 | 36 . 68 0 . 51 |\n| | | 15.4 | 6 . 74 ± 0 . 48 | 6 . 74 ± 0 . 48 |\n| 10.78 11.56 | AMI-LA AMI-LA | 15.4 | 7 . 50 ± 0 . ± | 7 . 50 ± 0 . ± |\n| 13.64 14.57 | AMI-LA | 15.4 | 44 . 02 0 . 71 . 43 ± 0 . 32 | 44 . 02 0 . 71 . 43 ± 0 . 32 |\n| | | 15.4 | 8 | 8 |\n| 16.47 | AMI-LA | 15.4 | 8 . 86 ± 0 . | 8 . 86 ± 0 . |\n| 18.65 | AMI-LA AMI-LA | 15.4 | 8 58 8 . 62 ± 0 . 40 ± 63 | 8 58 8 . 62 ± 0 . 40 ± 63 |\n| 19.73 21.78 | AMI-LA | 15.4 | 10 . 04 0 . . 91 ± 0 . 89 | 10 . 04 0 . . 91 ± 0 . 89 |\n| | AMI-LA AMI-LA | 15.4 | 10 . 01 ± 0 . 77 | 10 . 01 ± 0 . 77 |\n| 22.76 25.38 | AMI-LA | 15.4 15.4 | 11 . 28 ± 0 . | 11 . 28 ± 0 . |\n| 26.74 | AMI-LA | 15.4 | 10 58 11 . 36 ± 0 . 80 . 24 ± 0 . | 10 58 11 . 36 ± 0 . 80 . 24 ± 0 . |\n| 31.45 | | | 11 . 14 ± 0 . | 11 . 14 ± 0 . |\n| | | | . 89 ± 0 . | . 89 ± 0 . |\n| | AMI-LA AMI-LA AMI-LA | 15.4 15.4 | 41 16 25 . 39 ± 0 . | 41 16 25 . 39 ± 0 . |\n| 33.61 34.79 | | 15.4 | 12 | 12 |\n| | | | 11 . 63 ± 0 . ± | 49 45 |\n| 35.71 | | | | |\n| 37.49 | AMI-LA AMI-LA | 15.4 15.4 | 13 13 | 13 13 |\n| 38.60 | AMI-LA | 15.4 | 13 . 72 0 . 39 | 13 . 72 0 . 39 | \nTABLE 1 Continued \n| δ t a (d) | Facility | Frequency (GHz) | Flux Density (mJy) |\n|----------------------|----------------------|-------------------|-----------------------------------------------------|\n| 39.78 | AMI-LA | 15.4 | . 80 ± 0 . |\n| | AMI-LA | | 91 . 38 ± 0 . 40 |\n| 40.67 | | 15.4 15.4 | 10 13 13 . 64 ± 0 . 21 |\n| 41.69 43.63 | AMI-LA AMI-LA | 15.4 | . 21 ± 1 . 24 . 33 ± 0 . 37 |\n| 45.74 | AMI-LA | | 14 12 14 ± |\n| 48.69 | AMI-LA | 15.4 15.4 | . 06 0 . 41 13 . 94 ± 0 . 05 |\n| 49.64 50.54 | AMI-LA | 15.4 | . 39 ± 0 . 25 |\n| 53.39 | AMI-LA | 15.4 | 14 15 . 94 ± 0 . 59 |\n| | AMI-LA | 15.4 | ± |\n| 55.62 56.52 | AMI-LA | 15.4 | . 28 0 . 33 . 91 ± 0 . 07 |\n| | AMI-LA | | 15 17 |\n| 60.73 | | 15.4 | 15 . 00 ± 1 . 18 19 . ± |\n| 62.54 | AMI-LA AMI-LA | 15.4 15.4 | 23 0 . 62 |\n| 63.45 | AMI-LA | | . 47 ± 0 . 21 ± |\n| | AMI-LA | 15.4 | 16 18 . 77 0 . 37 |\n| 65.51 68.60 | AMI-LA | 15.4 15.4 | 19 . 06 ± 0 . 48 |\n| 70.69 | AMI-LA | 15.4 | 20 . 34 ± 0 . 38 |\n| | AMI-LA | | 19 . 36 ± 0 . 56 |\n| 71.38 | | 15.4 | |\n| 73.48 74.48 | AMI-LA AMI-LA | 15.4 15.4 | 21 . 27 ± 0 . 34 21 . 84 ± 0 . 59 |\n| 76.65 | | 15.4 | 21 . 40 ± 0 . 51 |\n| 77.47 | AMI-LA AMI-LA | 15.4 | 23 . 08 ± 0 . 26 |\n| 78.49 | AMI-LA | 15.4 | 23 . 26 ± 0 . 65 |\n| 79.49 | AMI-LA | 15.4 | 23 . 22 ± 0 . 24 22 ± |\n| 80.47 81.48 | AMI-LA | 15.4 | . 94 0 . 50 21 . 74 ± 0 . 45 |\n| | AMI-LA | 15.4 | |\n| 83.42 | AMI-LA | 15.4 | 23 . 99 ± 0 . 50 22 . |\n| 86.65 | AMI-LA | 15.4 | 94 ± 0 . 33 |\n| 88.37 | AMI-LA | 15.4 | 24 . 69 ± 0 . 26 |\n| 89.64 | AMI-LA | 15.4 | 25 . 90 ± 0 . 50 07 ± |\n| 91.58 | AMI-LA | 15.4 | 26 . 1 . 04 ± 0 . 55 |\n| 92.58 | AMI-LA | 15.4 | 25 . 40 34 ± 0 . 72 |\n| 95.39 | AMI-LA | 15.4 | 25 . 26 . 15 ± 0 . 25 0 . |\n| 98.42 100.61 | AMI-LA | 15.4 15.4 | ± 57 ± 0 . 91 |\n| 101.60 | AMI-LA | 15.4 | 27 . 83 25 . 96 |\n| 102.60 | | 15.4 | 26 . 60 ± 0 . 26 . 68 ± 0 . |\n| 105.59 | AMI-LA AMI-LA AMI-LA | 15.4 | 26 21 |\n| 107.38 | AMI-LA | 15.4 | . 89 ± 0 . 33 |\n| 108.32 | AMI-LA | 15.4 | 27 26 . 84 ± 0 . |\n| 110.34 112.50 | AMI-LA AMI-LA | 15.4 | 40 28 . 24 ± 0 . 22 27 |\n| | AMI-LA | 15.4 | . 10 ± 0 . 58 |\n| 113.50 | | 15.4 | 82 ± 0 . 31 39 ± 0 . 23 |\n| | | | 28 . . ± |\n| 115.49 118.31 119.48 | AMI-LA AMI-LA AMI-LA | 15.4 15.4 15.4 | 28 28 . 58 0 . 47 26 . 90 ± 0 . 80 27 . 88 ± 0 . 26 |\n| 120.55 | AMI-LA | 15.4 15.4 | 35 ± 0 . 73 |\n| 121.55 122.18 | AMI-LA AMI-LA | 15.4 | . 92 29 . 87 ± 0 . 61 |\n| 123.45 | AMI-LA AMI-LA | 15.4 | 0 . 41 0 . 97 |\n| | | | ± ± ± 0 . 56 |\n| 124.38 125.51 127.38 | AMI-LA | 15.4 15.4 | 28 . 86 . 46 . 40 |\n| | AMI-LA | 15.4 | 27 27 28 . 96 ± 0 . 28 . |\n| | AMI-LA | 15.4 | 39 ± 1 . 00 ± 64 |\n| 128.35 130.38 131.41 | AMI-LA | 15.4 15.4 | 69 27 . 87 0 . 28 . 94 ± 1 . 11 |\n| 132.49 133.35 | AMI-LA AMI-LA | 15.4 | 29 . 39 ± 0 . 96 30 . 81 ± 1 . 73 |\n| | AMI-LA AMI-LA | 15.4 15.4 | 03 29 . 0 . 52 |\n| 134.36 135.32 | | 15.4 | ± 31 . 25 ± 0 . 59 |\n| 136.46 | AMI-LA AMI-LA | 15.4 | 29 . 31 1 . 29 |\n| 137.51 | AMI-LA | 15.4 | ± 29 . ± 0 . 23 |\n| 138.51 139.51 | AMI-LA | 15.4 | 58 28 . 50 ± 0 . 69 28 . 96 ± 0 . 57 |\n| 140.50 | AMI-LA AMI-LA | 15.4 15.4 | 29 . 0 . 52 |\n| | AMI-LA AMI-LA | | 22 ± 29 . ± 0 . 45 |\n| 141.13 | | 15.4 | 03 26 . 60 ± 0 . |\n| 142.30 | AMI-LA AMI-LA | 15.4 | 31 28 . |\n| | | 15.4 | ± 0 . 61 |\n| 143.26 144.42 | | 15.4 15.4 | 96 28 . 36 ± 0 . 57 ± |\n| 146.48 | AMI-LA | | 92 0 . 64 ± 0 . |\n| 147.48 148.27 | | 15.4 | . 44 76 49 ± 0 . 97 |\n| | AMI-LA AMI-LA | 15.4 | 28 27 . . |\n| 149.27 150.47 | AMI-LA AMI-LA | 15.4 15.4 15.4 | 29 29 . 82 ± 0 . 43 . 83 ± 0 . . 31 ± 0 . |\n| 151.22 | | | 29 23 29 |\n| | | | ± |\n| | | | 28 . 10 1 . |\n| | AMI-LA AMI-LA | | |\n| 152.27 | | | |\n| | | 15.4 | 73 30 | \nTABLE 1 Continued \n| δ t a (d) | Facility | Frequency (GHz) | (mJy) | Flux Density |\n|---------------|---------------|-------------------|----------------------|----------------------------------|\n| 153.46 | AMI-LA | 15.4 | 26 . 71 ± | |\n| 155.45 | | | 27 . 80 | 0 . 38 |\n| | AMI-LA | 15.4 | | ± 0 . 45 ± 0 . 89 |\n| 156.35 | AMI-LA | 15.4 15.4 | 29 . 92 27 . 77 | 0 . 31 |\n| 157.45 158.29 | AMI-LA AMI-LA | 15.4 | 27 . 72 | ± ± 0 . 83 ± |\n| 159.28 | AMI-LA | 15.4 | 27 . 98 | 0 . 59 ± 0 . 44 |\n| 160.27 161.20 | AMI-LA | 15.4 | 28 . 93 | 0 . 84 |\n| | AMI-LA | 15.4 | 26 . 94 | ± ± 1 |\n| 162.21 | AMI-LA | 15.4 | 26 . 81 | . 21 68 |\n| 163.28 | AMI-LA | 15.4 | 28 . 26 . | 0 . 0 . 58 |\n| 166.00 | AMI-LA | 15.4 | 82 92 | ± |\n| 168.13 | AMI-LA | | 25 . 90 | ± ± 0 . 64 |\n| 169.13 | AMI-LA | 15.4 15.4 | 25 . 47 | ± 0 . 83 |\n| 172.25 | AMI-LA | 15.4 | 26 . 12 | 0 . 48 |\n| 174.33 | AMI-LA | 15.4 | 26 . 12 | ± ± 0 . 15 |\n| 177.32 | AMI-LA | 15.4 | | ± 0 . 37 |\n| | | | 26 . 31 24 . 37 | ± 0 . 85 ± |\n| 181.30 | AMI-LA AMI-LA | 15.4 | . 06 | 0 . 77 |\n| 184.03 187.19 | AMI-LA | 15.4 15.4 | 23 24 . 24 | 0 . 43 |\n| 187.91 | AMI-LA | | 24 . | ± ± 0 . 45 |\n| 189.37 | AMI-LA | 15.4 15.4 | 28 24 . 60 | ± 0 . 59 |\n| 193.19 | AMI-LA | 15.4 | 26 . 15 | ± 0 . 75 ± |\n| 194.01 | AMI-LA | 15.4 | 24 . | 0 . 34 |\n| 197.02 | AMI-LA | 15.4 | 53 26 . 72 | ± 0 . 52 ± |\n| 200.01 203.01 | AMI-LA | 15.4 | 25 . 90 | 0 . 15 0 . |\n| | AMI-LA | 15.4 | 24 . 12 | ± 41 ± |\n| 207.08 | AMI-LA | 15.4 | 25 . 06 | 0 . 45 |\n| 210.19 | AMI-LA | 15.4 | 24 . 24 | ± 0 . 31 ± |\n| 214.17 217.16 | AMI-LA | 15.4 15.4 | 22 . 64 | 0 . 38 0 . 25 |\n| 220.20 | AMI-LA | | 24 . 16 | ± ± 0 . 05 |\n| 221.22 | AMI-LA AMI-LA | 15.4 15.4 | 23 . 74 24 . 33 | ± 0 . 49 |\n| 225.91 | AMI-LA | 15.4 | 23 . | ± 0 . 14 ± 0 . 16 |\n| 232.87 | AMI-LA | | 80 | |\n| | | 15.4 | 23 . 72 | 0 . 16 |\n| 235.01 237.85 | AMI-LA | 15.4 | 21 . 64 | ± ± 0 . 75 |\n| 4.79 | AMI-LA EVLA | 15.4 19.1 | 20 . 86 2 . 12 | ± 0 . 02 36 ± 0 . 05 |\n| 6.75 7.77 | EVLA EVLA | 19.1 19.1 | 4 . | ± |\n| 8.87 9.78 | EVLA EVLA | 19.1 | 5 . 25 6 . 38 5 . 42 | 0 . 03 ± 0 . 06 ± |\n| | EVLA | 19.1 | | 0 . 03 01 ± 0 . 03 |\n| 21.89 31.74 | EVLA | 19.1 | 12 . | |\n| 35.86 | EVLA | 19.1 19.1 | 13 . 50 13 . 97 | ± 0 . 05 ± 0 . 05 |\n| | EVLA | | 17 . 11 | ± 0 . 06 |\n| 50.65 | | 19.1 | | |\n| 67.61 | EVLA | 19.1 | 23 . 03 | ± 0 . 06 |\n| 94.64 111.62 | EVLA EVLA | 19.1 19.1 | 31 . 36 30 . 21 ± | ± 0 . 07 0 . 10 |\n| 127.83 142.62 | EVLA EVLA | 19.1 19.1 | 29 . 29 . 57 | 75 ± 0 . 22 ± 0 . 13 10 ± 0 . 26 |\n| | | | . | ± 0 . 16 |\n| 159.77 | EVLA | 19.1 | 26 | 24 ± |\n| 177.50 | EVLA | 19.1 | 24 . | . 02 0 . 12 |\n| 198.22 | EVLA | 19.1 | 23 | . 15 ± 0 . 07 3 . 01 ± 0 . 03 |\n| 219.22 | EVLA | 19.1 | 23 | |\n| 4.79 6.75 | EVLA EVLA | 24.4 24.4 | 5 . 58 | ± 0 . 06 ± 0 . 03 ± |\n| 7.77 | EVLA | 24.4 24.4 | 6 . 70 | 0 . 12 |\n| 8.87 9.78 | EVLA | | 7 . 88 ± | |\n| | EVLA | 24.4 24.4 | 6 . 84 12 . 69 ± | 0 . 03 0 . 02 |\n| 21.89 31.74 | EVLA | | . 80 | 0 . 06 |\n| 35.86 | EVLA | 24.4 | 13 ± . 95 ± | 0 . 05 |\n| 50.65 | EVLA EVLA | 24.4 24.4 | 14 18 . 30 | ± 0 . 06 0 . 06 |\n| 67.61 94.64 | EVLA | 24.4 | . 62 . 67 | |\n| | EVLA | | 25 | ± ± 0 . 07 |\n| 111.62 | EVLA | 24.4 | 30 . 20 | ± 0 . 13 ± |\n| | EVLA | 24.4 24.4 | 28 . 29 . 73 | 0 . 33 |\n| 127.83 | | 24.4 | 28 24 ± . 83 | . 16 |\n| 142.62 | EVLA EVLA | | 23 20 . 40 | 0 . 40 |\n| 159.77 177.50 | EVLA | 24.4 24.4 | ± 19 . 88 | 0 0 . 19 |\n| 198.22 | EVLA | 24.4 | ± | ± 0 . 15 |\n| 219.22 5.75 | EVLA EVLA | 24.4 43.6 | 21 . 40 ± | 0 . 08 |\n| 6.75 | EVLA | 43.6 43.6 | 7 . 70 ± 9 62 ± | 8 . 11 ± 0 . 16 0 . 14 |\n| 8.87 | | | . | 0 . 14 |\n| | EVLA | | | | \nTABLE 1 Continued \n| δ t a (d) | Facility | Frequency (GHz) | Flux Density (mJy) |\n|-------------|------------|-------------------|----------------------|\n| 111.62 | EVLA | 43.6 | 22 . 09 ± 0 . 65 |\n| 127.83 | EVLA | 43.6 | 20 . 84 ± 0 . 48 |\n| 159.77 | EVLA | 43.6 | 15 . 43 ± 0 . 63 |\n| 177.5 | EVLA | 43.6 | 15 . 73 ± 0 . 38 |\n| 198.22 | EVLA | 43.6 | 11 . 70 ± 0 . 26 |\n| 219.22 | EVLA | 43.6 | 15 . 32 ± 0 . 15 |\n| 4.9 | CARMA | 87 | 15 . 66 ± 0 . 51 |\n| 8.19 | CARMA | 87 | 20 . 16 ± 0 . 96 |\n| 9.14 | CARMA | 87 | 21 . 67 ± 0 . 33 |\n| 10.23 | CARMA | 87 | 14 . 70 ± 1 . 28 |\n| 12.14 | CARMA | 87 | 17 . 67 ± 0 . 94 |\n| 17.66 | CARMA | 87 | 12 . 15 ± 1 . 32 |\n| 22.11 | CARMA | 87 | 11 . 70 ± 1 . 74 |\n| 25.12 | CARMA | 87 | 16 . 95 ± 3 . 48 |\n| 99.75 | CARMA | 87 | 18 . 79 ± 0 . 65 |\n| 131.52 | CARMA | 87 | 15 . 18 ± 0 . 35 |\n| 148.66 | CARMA | 87 | 11 . 88 ± 0 . 35 |\n| 175.56 | CARMA | 87 | 9 . 39 ± 0 . 31 |\n| 19.25 | SMA | 200 | 14 . 10 ± 1 . 50 |\n| 24.32 | SMA | 200 | 10 . 70 ± 1 . 00 |\n| 10.3 | SMA | 230 | 14 . 90 ± 1 . 50 |\n| 11.13 | SMA | 230 | 11 . 70 ± 1 . 40 |\n| 17.23 | SMA | 230 | 13 . 30 ± 1 . 50 |\n| 18.25 | SMA | 230 | 9 . 90 ± 1 . 40 |\n| 20.24 | SMA | 230 | 8 . 20 ± 1 . 40 |\n| 21.25 | SMA | 230 | 8 . 30 ± 2 . 20 |\n| 125.05 | SMA | 230 | 6 . 10 ± 0 . 65 |\n| 5.13 | SMA | 345 | 35 . 10 ± 0 . 80 | \nNOTE. a All values of δ t are relative to the initial γ -ray detection: 2011 March 25.5 UT. \nTABLE 2 RESULTS OF BROAD-BAND SPECTRAL ENERGY DISTRIBUTION FITS \n| δ t (d) | log( ν a ) (Hz) | log( ν m ) (Hz) | log( F ν a ) (mJy) | log( r 18) (cm) | log( Γ sh) | log( Γ j ) | log( L j , iso , 48) (erg s - 1 ) | log( n 18) (cm - 3 ) | log( n CNM) (cm - 3 ) | log( nj ) (cm - 3 ) |\n|-----------|-------------------|-------------------|----------------------|-------------------|--------------|--------------|-------------------------------------|------------------------|-------------------------|-----------------------|\n| 5 | 11 . 01 | 11 . 74 | 1 . 47 | - 0 . 79 | 0 . 65 | 0 . 78 | - 1 . 33 | 0 . 28 | 1 . 82 | 3 . 53 |\n| 10 | 10 . 11 | 11 . 41 | 0 . 96 | - 0 . 26 | 0 . 61 | 0 . 74 | - 1 . 39 | - 0 . 28 | 0 . 25 | 2 . 39 |\n| 15 | 10 . 02 | 11 . 21 | 0 . 98 | - 0 . 13 | 0 . 55 | 0 . 65 | - 1 . 33 | - 0 . 22 | 0 . 05 | 2 . 19 |\n| 22 | 9 . 96 | 10 . 99 | 0 . 97 | - 0 . 04 | 0 . 51 | 0 . 60 | - 1 . 25 | - 0 . 15 | - 0 . 07 | 2 . 08 |\n| 36 | 9 . 95 | 10 . 78 | 1 . 11 | 0 . 08 | 0 . 47 | 0 . 54 | - 1 . 05 | 0 . 01 | - 0 . 17 | 2 . 03 |\n| 51 | 9 . 96 | 10 . 62 | 1 . 21 | 0 . 16 | 0 . 43 | 0 . 50 | - 0 . 92 | 0 . 13 | - 0 . 18 | 2 . 01 |\n| 68 | 9 . 99 | 10 . 53 | 1 . 39 | 0 . 24 | 0 . 41 | 0 . 47 | - 0 . 72 | 0 . 28 | - 0 . 20 | 2 . 04 |\n| 97 | 9 . 95 | 10 . 39 | 1 . 52 | 0 . 36 | 0 . 40 | 0 . 45 | - 0 . 53 | 0 . 39 | - 0 . 34 | 1 . 99 |\n| 126 | 9 . 97 | 10 . 26 | 1 . 58 | 0 . 41 | 0 . 36 | 0 . 41 | - 0 . 46 | 0 . 49 | - 0 . 33 | 1 . 97 |\n| 161 | 9 . 82 | 10 . 13 | 1 . 51 | 0 . 52 | 0 . 36 | 0 . 40 | - 0 . 39 | 0 . 44 | - 0 . 60 | 1 . 82 |\n| 197 | 9 . 83 | 10 . 04 | 1 . 56 | 0 . 56 | 0 . 34 | 0 . 38 | - 0 . 32 | 0 . 52 | - 0 . 60 | 1 . 81 |\n| 216 | 9 . 90 | 9 . 99 | 1 . 63 | 0 . 55 | 0 . 32 | 0 . 35 | - 0 . 29 | 0 . 60 | - 0 . 50 | 1 . 85 | \nNOTE. - Inferred parameters of the relativistic outflow and environment of Sw 1644+57 from model fits of individual multi-frequency epochs. The model is described in §4. \nFIG. 1.- Radio light curves of Sw 1644+57 extending to δ t ≈ 216 d. The data at δ t ≈ 5 -22 d were previously presented in ZBS11. The solid lines are models based on independent fits of broad-band SEDs (Figure 2) using the model described in §4 (see also MGM11). The dashed lines are the predicted light curves from MGM11, which assumed a constant energy and a steady density profile of ρ ∝ r -2 . The secondary maximum in the millimeter band and the continued increase in brightness to a peak time of δ t /greaterorsimilar 100 -200 d in the centimeter bands require an increase in the energy and density relative to the initial evolution. \n<!-- image --> \nFIG. 2.- Radio spectral energy distributions of Sw 1644+57 at δ t ≈ 5 -216 d. The data at δ t ≈ 5 -22 d were previously presented in ZBS11. The solid lines are fits based on the model described in §4 (see also MGM11). In each epoch we fit for L j , iso and n 18 with fixed values of /epsilon1 e = /epsilon1 B = 0 . 1 and p = 2 . 5. The dashed gray lines mark the peak flux density and peak frequency at δ t = 10 d to help track the evolution of the spectrum as a function of time. The dashed red lines mark the expected SEDs based on the evolution at δ t ≈ 5 -22 d. \n<!-- image --> \nFIG. 3.- Model and extracted parameters for each broad-band SED shown in Figure 2. Shown are the time evolution of the synchrotron parameters ( ν a , ν m , and F ν a ), L j , iso , 48, n 18, r , Γ sh, and Γ j . The substantial increase in energy and density is clearly seen. In addition, we find r ∝ t 0 . 6 , a steeper increase than expected in a simple Wind model with constant energy ( r ∝ t 0 . 5 ). \n<!-- image --> \nFIG. 4.- Temporal evolution of the synchrotron frequencies ν m (top) and ν a (bottom) relative to the expected evolution in a simple model with a constant energy and a Wind profile. The shallower decline of ν m ∝ E 0 . 5 j , with a particularly shallow evolution at δ t ≈ 30 -100 d, is indicative of a continuous increase in energy. Similarly, the shallower decline of ν a , followed by a rapid increase, is indicative of a density profile of ρ ∝ r -1 . 5 and a flattening at δ t ≈ 30 -100 d. \n<!-- image --> \nFIG. 5.- Temporal evolution of the integrated luminosity (or alternatively E j , iso = L j , iso t j ; black circles) in comparison to the X-ray luminosity (gray dots) as parametrized with a simple luminosity evolution (gray line). The red curve is the integrated luminosity derived from the simple model. The observed X-ray luminosity indicates that the fraction of total energy emitted in X-rays is comparable to the energy in the relativistic outflow (i.e., /epsilon1 X ≈ 0 . 5). The large increase in energy inferred from the radio observations cannot be explained by injection from a L ∝ t -5 / 3 tail. \n<!-- image --> \nFIG. 6.- Radial density profile in the inner parsec around Sw 1644+57 as inferred from the radio observations (black circles). The overall profile follows ρ ∝ r -3 / 2 , with a significant flattening at r ≈ 0 . 4 -0 . 6 pc. Following the flattening, the profile appears to recover to r -3 / 2 by about 1 pc. Also shown is the density inferred from X-ray observations of the Galactic center (gray squares; Baganoff et al. 2003), which is about a factor of 30 times larger at ≈ 0 . 5 pc. The dashed line is a scaled-down model of the Galactic center density profile assuming gas feeding from massive stars in which the bulk of the gas is thermally expelled in a wind (Quataert 2004). In this model the inner profile ( /lessorsimilar 0 . 2 pc) is ∝ r -3 / 2 , while the outer profile ( /greaterorsimilar 0 . 4 pc) has a Wind ( ∝ r -2 ) profile. \n<!-- image --> \nFIG. 7.- Predicted optical ( r -band; blue) and near-infrared ( K -band; red) light curves using the results of the radio modeling. The upper limits in r -band and detections in K -band are from Levan et al. (2011). Since the K -band fluxes are the total for Sw 1644+57 and its host galaxy we have subtracted an estimated host contribution of about 20 µ Jy ( K ≈ 20 . 6 AB mag). The solid lines are models without a cooling break between the radio and optical/near-IR, which clearly over-estimate the K -band flux. The dashed lines include a cooling break at ν c ≈ 10 13 Hz, and the dotted lines add host galaxy extinction of AV ≈ 3 . 5 mag to account for the optical non-detections. The combination of a cooling break and extinction provides an excellent fit to the near-IR evolution. \n<!-- image --> \nFIG. 8.- Predicted evolution of the radio light curves at 6 GHz (blue) and 22 GHz (red) assuming a radial density profile of ρ ∝ r -1 . 5 at r /greaterorsimilar 1 pc and three values for the maximum integrated beaming-corrected energy (solid: Ej = 10 52 erg; dashed: Ej = 3 × 10 52 erg; dotted: Ej = 10 53 erg). The thin horizontal lines mark the 5 σ sensitivity of the EVLA, and indicate that the radio emission from Sw 1644+57 should be detectable for decades (and perhaps centuries) at centimeter wavelengths. \n<!-- image --> \nFIG. 9.- Predicted evolution of the jet radius assuming a radial density profile of ρ ∝ r -1 . 5 at r /greaterorsimilar 1 pc and three values for the maximum beaming energy (solid: Ej = 10 52 erg; dashed: Ej = 3 × 10 52 erg; dotted: Ej = 10 53 erg). The thin horizontal line marks the resolution of VLBI for a jet opening angle of θ j = 0 . 1. The source should become resolvable at δ t ∼ 6 yr, with an expected 22 GHz flux density of about 2 mJy (Figure 8). If the jet instead begins to undergo significant spreading it may become resolvable at ∼ 1 yr when the 22 GHz flux density is still ∼ 10 mJy. \n<!-- image -->'}

2015MNRAS.447..948C

The old nuclear star cluster in the Milky Way: dynamics, mass, statistical parallax, and black hole mass

2015-01-01

['galaxy center', 'galaxy kinematics and dynamics', '-']

[]

We derive new constraints on the mass, rotation, orbit structure, and statistical parallax of the Galactic old nuclear star cluster and the mass of the supermassive black hole. We combine star counts and kinematic data from Fritz et al., including 2500 line-of-sight velocities and 10 000 proper motions obtained with VLT instruments. We show that the difference between the proper motion dispersions σ<SUB>l</SUB> and σ<SUB>b</SUB> cannot be explained by rotation, but is a consequence of the flattening of the nuclear cluster. We fit the surface density distribution of stars in the central 1000 arcsec by a superposition of a spheroidal cluster with scale ∼100 arcsec and a much larger nuclear disc component. We compute the self-consistent two-integral distribution function f(E, L<SUB>z</SUB>) for this density model, and add rotation self-consistently. We find that (i) the orbit structure of the f(E, L<SUB>z</SUB>) gives an excellent match to the observed velocity dispersion profiles as well as the proper motion and line-of-sight velocity histograms, including the double-peak in the v<SUB>l</SUB>-histograms. (ii) This requires an axial ratio near q<SUB>1</SUB> = 0.7 consistent with our determination from star counts, q<SUB>1</SUB> = 0.73 ± 0.04 for r &lt; 70 arcsec. (iii) The nuclear star cluster is approximately described by an isotropic rotator model. (iv) Using the corresponding Jeans equations to fit the proper motion and line-of-sight velocity dispersions, we obtain best estimates for the nuclear star cluster mass, black hole mass, and distance M<SUB>*</SUB>(r &lt; 100 arcsec) = (8.94 ± 0.31|<SUB>stat</SUB> ± 0.9|<SUB>syst</SUB>) × 10<SUP>6</SUP> M<SUB>⊙</SUB>, M<SUB>•</SUB> = (3.86 ± 0.14|<SUB>stat</SUB> ± 0.4|<SUB>syst</SUB>) × 10<SUP>6</SUP> M<SUB>⊙</SUB>, and R<SUB>0</SUB> = 8.27 ± 0.09|<SUB>stat</SUB> ± 0.1|<SUB>syst</SUB> kpc, where the estimated systematic errors account for additional uncertainties in the dynamical modelling. (v) The combination of the cluster dynamics with the S-star orbits around Sgr A* strongly reduces the degeneracy between black hole mass and Galactic Centre distance present in previous S-star studies. A joint statistical analysis with the results of Gillessen et al., gives M<SUB>•</SUB> = (4.23 ± 0.14) × 10<SUP>6</SUP> M<SUB>⊙</SUB> and R<SUB>0</SUB> = 8.33 ± 0.11 kpc.

[]

https://arxiv.org/pdf/1403.5266.pdf

{'The old nuclear star cluster in the Milky Way: dynamics, mass, statistical parallax, and black hole mass': '- S. Chatzopoulos 1 /star , T. K. Fritz 2 , O. Gerhard 1 , S. Gillessen 1 , C. Wegg 1 , R. Genzel 1 , 3 ,', 'O. Pfuhl 1': '- 1 Max Planck Institut fur Extraterrestrische Physic, Postfach 1312, D-85741, Garching Germany\n- 2 Department of Astronomy, University of Virginia, 530 McCormick Road Charlottesville VA 22904-4325 USA\n- 3 Department of Physics, Le Conte Hall, University of California, 94720 Berkeley, USA \nSubmitted 2014 March 15', 'ABSTRACT': "We derive new constraints on the mass, rotation, orbit structure and statistical parallax of the Galactic old nuclear star cluster and the mass of the supermassive black hole. We combine star counts and kinematic data from Fritz et al. (2014), including 2'500 line-of-sight velocities and 10'000 proper motions obtained with VLT instruments. We show that the difference between the proper motion dispersions σ l and σ b cannot be explained by rotation, but is a consequence of the flattening of the nuclear cluster. We fit the surface density distribution of stars in the central 1000 '' by a superposition of a spheroidal cluster with scale ∼ 100 '' and a much larger nuclear disk component. We compute the self-consistent two-integral distribution function f ( E,L z ) for this density model, and add rotation self-consistently. We find that: (i) The orbit structure of the f ( E,L z ) gives an excellent match to the observed velocity dispersion profiles as well as the proper motion and line-of-sight velocity histograms, including the double-peak in the v l -histograms. (ii) This requires an axial ratio near q 1 = 0 . 7 consistent with our determination from star counts, q 1 = 0 . 73 ± 0 . 04 for r < 70 '' . (iii) The nuclear star cluster is approximately described by an isotropic rotator model. (iv) Using the corresponding Jeans equations to fit the proper motion and line-of-sight velocity dispersions, we obtain best estimates for the nuclear star cluster mass, black hole mass, and distance M ∗ ( r< 100 '' )=(8 . 94 ± 0 . 31 | stat ± 0 . 9 | syst ) × 10 6 M /circledot , M · =(3 . 86 ± 0 . 14 | stat ± 0 . 4 | syst ) × 10 6 M /circledot , and R 0 =8 . 27 ± 0 . 09 | stat ± 0 . 1 | syst kpc, where the estimated systematic errors account for additional uncertainties in the dynamical modeling. (v) The combination of the cluster dynamics with the S-star orbits around Sgr A ∗ strongly reduces the degeneracy between black hole mass and Galactic centre distance present in previous S-star studies. A joint statistical analysis with the results of Gillessen et al. (2009) gives M · =(4 . 23 ± 0 . 14) × 10 6 M /circledot and R 0 =8 . 33 ± 0 . 11 kpc. \nKey words: galaxy center, nuclear cluster, kinematics and dynamics.", '1 INTRODUCTION': 'Nuclear star clusters (NSC) are located at the centers of most spiral galaxies (Carollo et al. 1997; Boker et al. 2002). They are more luminous than globular clusters (Boker et al. 2004), have masses of order ∼ 10 6 -10 7 M /circledot (Walcher et al. 2005), have complex star formation histories (Rossa et al. 2006; Seth et al. 2006), and obey scaling-relations with host galaxy properties as do central supermassive black holes (SMBH; Ferrarese et al. 2006; Wehner & Harris 2006); see Boker (2010) for a review. Many host an AGN, i.e., a SMBH \n/star \nE-mail: [email protected], [email protected] \n(Seth et al. 2008), and the ratio of NSC to SMBH mass varies widely (Graham & Spitler 2009; Kormendy & Ho 2013). \nThe NSC of the Milky Way is of exceptional interest because of its proximity, about 8 kpc from Earth. It extends up to several hundred arcsecs from the center of the Milky Way (Sgr A*) and its mass within 1 pc is ∼ 10 6 M /circledot with ∼ 50% uncertainty (Schodel et al. 2009; Genzel et al. 2010). There is strong evidence that the center of the NSC hosts a SMBH of several million solar masses. Estimates from stellar orbits show that the SMBH mass is M · = (4 . 31 ± 0 . 36) × 10 6 M /circledot (Schodel et al. 2002; Ghez et al. 2008; Gillessen et al. 2009). Due to its proximity, individual stars can be resolved and \nnumber counts can be derived; however, due to the strong interstellar extinction the stars can only be observed in the infrared. A large number of proper motions and line-of-sight velocities have been measured, and analyzed with spherical models to attempt to constrain the NSC dynamics and mass (Haller et al. 1996; Genzel et al. 1996, 2000; Trippe et al. 2008; Schodel et al. 2009; Fritz et al. 2014). \nThe relaxation time of the NSC within 1 pc is t r ∼ 10 10 yr (Alexander 2005; Merritt 2013), indicating that the NSC is not fully relaxed and is likely to be evolving. One would expect from theoretical models that, if relaxed, the stellar density near the SMBH should be steeply-rising and form a Bahcall & Wolf (1976) cusp. In contrast, observations by Do et al. (2009); Buchholz et al. (2009); Bartko et al. (2010) show that the distribution of old stars near the SMBH appears to have a core. Understanding the nuclear star cluster dynamics may therefore give useful constraints on the mechanisms by which it formed and evolved (Merritt 2010). \nIn this work we construct axisymmetric Jeans and twointegral distribution function models based on stellar number counts, proper motions, and line-of-sight velocities. We describe the data briefly in Section 2; for more detail the reader is referred to the companion paper of Fritz et al. (2014). In Section 3 we carry out a preliminary study of the NSC dynamics using isotropic spherical models, in view of understanding the effect of rotation on the data. In Section 4 we describe our axisymmetric models and show that they describe the kinematic properties of the NSC exceptionally well. By applying a χ 2 minimization algorithm, we estimate the mass of the cluster, the SMBH mass, and the NSC distance. We discuss our results and summarize our conclusions in Section 5. The Appendix contains some details on our use of the Qian et al. (1995) algorithm to calculate the two-integral distribution function for the fitted density model.', '2 DATASET': "We first give a brief description of the data set used for our dynamical analysis. These data are taken from Fritz et al. (2014) and are thoroughly examined in that paper, which should be consulted for more details. The coordinate system used is a shifted Galactic coordinate system ( l ∗ , b ∗ ) where Sgr A* is at the center and ( l ∗ , b ∗ ) are parallel to Galactic coordinates ( l, b ). In the following we always refer to the shifted coordinates but will omit the asterisks for simplicity. The dataset consists of stellar number densities, proper motions and line-of-sight velocities. We use the stellar number density map rather than the surface brightness map because it is less sensitive to individual bright stars and non-uniform extinction. \nThe stellar number density distribution is constructed from NACO high-resolution images for R box < 20 '' , in a similar way as in Schodel et al. (2010), from HST WFC3/IR data for 20 '' < R box < 66 '' , and from VISTA-VVV data for 66 '' < R box < 1000 '' . \nThe kinematic data include proper motions for ∼ 10'000 stars obtained from AO assisted images. The proper motion stars are binned into 58 cells (Figure 1; Fritz et al. 2014) according to distance from Sgr A* and the Galactic plane. This binning assumes that the NSC is symmetric with re- \nect to the Galactic plane and with respect to the b -axis on the sky, consistent with axisymmetric dynamical modeling. The sizes of the bins are chosen such that all bins contain comparable numbers of stars, and the velocity dispersion gradients are resolved, i.e., vary by less than the error bars between adjacent bins. \nRelative to the large velocity dispersions at the Galactic center (100 km/s), measurement errors for individual stars are typically ∼ 10%, much smaller than in typical globular cluster proper motion data where they can be ∼ 50% (e.g., in Omega Cen; van de Ven et al. (2006)). Therefore corrections for these measurement errors are very small. \nWe also use ∼ 2'500 radial velocities obtained from SINFONI integral field spectroscopy. The binning of the radial velocities is shown in Fig. 2. There are 46 rectangular outer bins as shown in Fig. 2 plus 6 small rectangular rings around the center (not shown; see App. E of Fritz et al. 2014). Again the outer bins are chosen such that they contain similar numbers of stars and the velocity dispersion gradients are resolved. The distribution of radial velocity stars on the sky is different from the distribution of proper motion stars, and it is not symmetric with respect to l = 0. Because of this and the observed rotation, the binning is different, and extends to both positive and negative longitudes. Both the proper motion and radial velocity binning are also used in Fritz et al. (2014) and some tests are described in that paper. \nFinally, we compare our models with (but do not fit to) the kinematics derived from about 200 maser velocities at r > 100 '' (from Lindquist et al. 1992; Deguchi et al. 2004). As for the proper motion and radial velocity bins, we use the mean velocities and velocity dispersions as derived in Fritz et al. (2014). \nThe assumption that the NSC is symmetric with respect to the Galactic plane and the b = 0 axis is supported by the recent Spitzer/IRAC photometry (Schodel et al. 2014) and by the distribution of proper motions (Fritz et al. 2014). The radial velocity data at intermediate radii instead show an apparent misalignment with respect to the Galactic plane, by ∼ 10 · ; see Feldmeier et al. (2014) and Fritz et al. (2014). We show in Section 4.2 that, even if confirmed, such a misaligned structure would have minimal impact on the results obtained here with the symmetrised analysis.", '3 SPHERICAL MODELS OF THE NSC': 'In this section we study the NSC using the preliminary assumption that the NSC can be described by an isotropic distribution function (DF) depending only on energy. We use the DF to predict the kinematical data of the cluster. Later we add rotation self-consistently to the model. The advantages of using a distribution function instead of common Jeans modeling are that (i) we can always check if a DF is positive and therefore if the model is physical, and (ii) the DF provides us with all the moments of the system. For the rest of the paper we use ( r, θ, ϕ ) for spherical and ( R,ϕ,z ) for cylindrical coordinates, with θ = 0 corresponding to the z-axis normal to the equatorial plane of the NSC. \nFigure 1. Binning of the proper motion velocities. The stars are binned into cells according to their distance from Sgr A* and their smallest angle to the Galactic plane (Fritz et al. 2014). \n<!-- image --> \nFigure 2. Binning of the line-of-sight velocities. The stars are binned into 46 rectangular outer cells plus 6 rectangular rings at the center. The latter are located within the white area around l = b = 0 and are not shown in the plot; see App. E of Fritz et al. (2014). \n<!-- image --> \nFigure 3. A combination of two γ -models gives an accurate approximation to the spherically averaged number density of latetype stars versus radius on the sky (points with error bars). Blue line: inner component, purple line: outer component, brown line: both components. \n<!-- image --> \nFigure 4. Isotropic DF for the two-component spherical model of the NSC in the joint gravitational potential including also a central black hole. Parameters for the NSC are as given in (4), and M · / ( M 1 + M 2 ) = 1 . 4 × 10 -3 . \n<!-- image -->', '3.1 Mass model for the NSC': "The first step is to model the surface density. We use the well-known one-parameter family of spherical γ -models (Dehnen 1993): \nρ γ ( r ) = 3 -γ 4 π Ma r γ ( r + a ) 4 -γ , 0 /lessorequalslant γ < 3 (1) \nwhere a is the scaling radius and M the total mass.The model behaves as ρ ∼ r -γ for r → 0 and ρ ∼ r -4 for r →∞ . Dehnen γ models are equivalent to the η -models of Tremaine et al. (1994) under the transformation γ = 3 -η . Special cases are the Jaffe (1983) and Hernquist (1990) models for γ = 2 and γ = 1 respectively. For γ = 3 / 2 the model approximates de Vaucouleurs R 1 / 4 law. In order to improve the fitting of the surface density we use a combination of two γ -models, i.e. \nρ ( r ) = 2 ∑ i =1 3 -γ i 4 π M i a i r γ i ( r + a i ) 4 -γ i . (2) \nThe use of a two-component model will prove convenient later when we move to the axisymmetric case. The projected density is \nΣ( R s ) = 2 ∫ ∞ R s ρ ( r ) r/ ( r 2 -R 2 s ) 1 / 2 dr (3) \nand can be expressed in terms of elementary functions for integer γ , or in terms of elliptic integrals for half-integer γ . For arbitrary γ 1 and γ 2 the surface density can only be calculated numerically using equation (3). The surface density diverges for γ > 1 but is finite for γ < 1. \nThe projected number density profile of the NSC obtained from the data of Fritz et al. (2014) (see Section 2) is shown in Figure 3. The inflection point at R s ∼ 100 '' indicates that the NSC is embedded in a more extended, lower-density component. The surface density distribution can be approximated by a two-component model of the form of equation (2), where the six parameters ( γ 1 , M 1 , a 1 , γ 2 , M 2 , a 2 ) are fitted to the data subject to the following constraints: The slope of the inner component", 'S. Chatzopoulos et al.': "should be γ 1 > 0 . 5 because isotropic models with a black hole and γ 1 < 0 . 5 are unphysical (Tremaine et al. 1994), but it should be close to the limiting value of 0.5 to better approximate the observed core near the center (Buchholz et al. 2009). For the outer component γ 2 /lessmuch 0 . 5 so that it is negligible in the inner part of the density profile. In addition M 1 < M 2 and a 1 < a 2 . With these constraints we start with some initial values for the parameters and then iteratively minimize χ 2 . The reduced χ 2 resulting from this procedure is χ 2 /ν = 0 . 93 for ν = 55 d.o.f. and the corresponding bestfit parameter values are: \nγ 1 = 0 . 51 a 1 = 99 '' γ 2 = 0 . 07 a 2 = 2376 '' M 2 M 1 = 105 . 45 . (4) \nHere we provide only the ratio of masses instead of absolute values in model units since the shape of the model depends only on the ratio. The surface density of the final model is overplotted on the data in Figure 3.", '3.2 Spherical model': "With the assumption of constant mass-to-light ratio and the addition of the black hole the potential (Φ = -Ψ) will be (Dehnen 1993) \nΨ( r ) = 2 i =1 GM i a i 1 (2 -γ i ) ( 1 -( r r + a ) 2 -γ i ) + GM · r (5) \n∑ \nwhere M · is the mass of the black hole. Since we now know the potential and the density we can calculate the distribution function (DF) numerically using Eddington's formula, as a function of positive energy E = Ψ -1 2 υ 2 , \nf ( E ) = 1 √ 8 π 2 [∫ E 0 d Ψ √ E -Ψ d 2 ρ d Ψ 2 + 1 √ E ( dρ d Ψ ) Ψ=0 ] . (6) \nThe 2nd term of the equation vanishes for reasonable behavior of the potential and the double derivative inside the integral can be calculated easily by using the transformation \nd 2 ρ d Ψ 2 = [ -( d Ψ dr ) -3 d 2 Ψ dr 2 ] dρ dr + ( d Ψ dr ) -2 d 2 ρ dr 2 . (7) \nFigure 4 shows the DF of the two components in their joint potential plus that of a black hole with mass ratio M · / ( M 1 + M 2 ) = 1 . 4 × 10 -3 . The DF is positive for all energies. We can test the accuracy of the DF by retrieving the density using \nρ ( r ) = 4 π Ψ ∫ 0 dEf ( E ) √ Ψ -E (8) \nand comparing it with equation (2). Both agree to within 0 . 1%. The DF has the typical shape of models with a shallow cusp of γ < 3 2 . It decreases as a function of energy both in the neighborhood of the black hole and also for large energies. It has a maximum near the binding energy of the stellar potential well (Baes et al. 2005). \nFor a spherical isotropic model the velocity ellipsoid (Binney & Tremaine 2008) is a sphere of radius σ . The intrinsic dispersion σ can be calculated directly using \nσ 2 ( r ) = 4 π 3 ρ ( r ) ∫ ∞ 0 dυυ 4 f ( 1 2 υ 2 -Ψ) . (9) \nFigure 5. Line-of-sight velocity dispersion σ los of the twocomponent spherical model with black hole, compared to the observed line-of-sight dispersions (black) and the proper motion dispersions in l (red) and b (blue). The line-of-sight data includes the outer maser data, and for the proper motions a canonical distance of R 0 = 8 kpc is assumed. \n<!-- image --> \nThe projected dispersion is then given by: \nΣ( R s ) σ 2 P ( R s ) = 2 ∫ ∞ R s σ 2 ( r ) ρ ( r ) r √ r 2 -R 2 s dr. (10) \nIn Figure 5 we see how our two-component model compares with the kinematical data using the values R 0 = 8 kpc for the distance to the Galactic centre, M · = 4 × 10 6 M /circledot for the black hole mass, and M ∗ ( r < 100 '' ) = 5 × 10 6 M /circledot for the cluster mass inside 100'. The good match of the data up to 80 '' suggests that the assumption of constant mass-to-light ratio for the cluster is reasonable. Later-on we will see that a flattened model gives a much better match also for the maser data.", '3.3 Adding self-consistent rotation to the spherical model': "We describe here the effects of adding self-consistent rotation to the spherical model, but much of this also applies to the axisymmetric case which will be discussed in Section 4. We assume that the rotation axis of the NSC is aligned with the rotation axis of the Milky Way disk. We also use a cartesian coordinate system ( x, y, z ) where z is parallel to the axis of rotation as before, y is along the line of sight, and x is along the direction of negative longitude, with the center of the NSC located at the origin. The proper motion data are given in Galactic longitude l and Galactic latitude b angles, but because of the large distance to the center, we can assume that x ‖ l and z b . \n‖ Whether a spherical system can rotate has been answered in Lynden Bell (1960). Here we give a brief review. Rotation in a spherical or axisymmetric system can be added self-consistently by reversing the sense of rotation of some of its stars. Doing so, the system will remain in equilibrium. This is equivalent with adding to the DF a part that is odd with respect to L z . The addition of an odd part does not affect the density (or the mass) because the integral of the odd part over velocity space is zero. The most effective way to \n‖ \nFigure 6. Mean line-of-sight velocity data compared to the prediction of the two-component spherical model with added rotation for F = -0 . 90 and two κ values for illustration. Each data point corresponds to a cell from Figure 2. Velocities at negative l have been folded over with their signs reversed and are shown in red. The plot also includes the maser data at R s > 100 '' . The model prediction is computed for b = 20 '' . For comparison, cells with centers between b = 15 '' and b = 25 '' are highlighted with filled triangles. \n<!-- image --> \nadd rotation to a spherical system is by reversing the sense of rotation of all of its counterrotating stars. This corresponds to adding f -( E,L 2 , L z ) = sign( L z ) f ( E,L 2 ) (Maxwell's daemon, Lynden Bell 1960) to the initially non-rotating DF, and generates a system with the maximum allowable rotation. The general case of adding rotation to a spherical system can be written f ' ( E,L 2 , L z ) = (1 + g ( L z )) f ( E,L 2 ) where g ( L z ) is an odd function with max | g ( L z ) | < 1 to ensure positivity of the DF. We notice that the new distribution function is a three-integral DF. In this case the density of the system is still rotationally invariant but f -is not. \nIn Figure 5 we notice that the projected velocity dispersion in the l direction is larger than the dispersion in the b direction which was first found by Trippe et al. (2008). This is particularly apparent for distances larger than 10 '' . A heuristic attempt to explain this difference was made in Trippe et al. (2008) where they imposed a rotation of the form υ ϕ ( r, θ ) along with their Jeans modeling, as a proxy for axisymmetric modeling. Here we show that for a selfconsistent system the difference in the projected l and b dispersions cannot be explained by just adding rotation to the cluster. \nSpecifically, we show that adding an odd part to the distribution function does not change the proper motion dispersion σ x . The dispersion along the x axis is σ 2 x = υ 2 x -υ x 2 . Writing υ x in spherical velocity components (see the beginning of this section for the notation), \nυ x = υ R x R -υ ϕ y R = υ r sin θ x R + υ θ cos θ x R -υ ϕ y R (11) \nwe see that \nυ 2 x = ∫ dυ r ∫ dυ θ ∫ dυ ϕ υ 2 x (1 + g ( L z )) f + = = ∫ dυ r ∫ dυ θ ∫ dυ ϕ υ 2 x f + +0 . (12) \nThe second term vanishes because f + ( E,L 2 ) g ( L z ) is even in υ r , υ θ and odd in υ ϕ , so that the integrand for all terms of f + g υ 2 x is odd in at least one velocity variable. We also have \nυ x = ∫ dυ r ∫ dυ θ ∫ dυ ϕ υ x (1 + g ( L z )) f + = = 0 -dυ r ∫ dυ θ dυ ϕ υ ϕ y R f + g. (13) \n∫ \n∫ The first part is zero because υ x f + is odd. The second part is different from zero; however when projecting υ ϕ along the line-of-sight this term also vanishes because f + g is an even function of y when the integration is in a direction perpendicular to the L z angular momentum direction. Hence the projected mean velocity υ x is zero, and the velocity dispersion σ 2 x = υ 2 x is unchanged. \n∫ \nAn alternative way to see this is by making a particle realization of the initial DF (e.g. Aarseth et al. 1974). Then we can add rotation by reversing the sign of L z of a percentage of particles using some probability function which is equivalent to changing the signs of υ x and υ y of those particles. υ 2 x will not be affected by the sign change and the υ 2 x averaged over the line-of-sight will be zero because for each particle at the front of the system rotating in a specific direction there will be another particle at the rear of the system rotating in the opposite direction. In this work we do not use particle models to avoid fluctuations due to the limited number of particles near the center. \nFor the odd part of the DF we choose the two-parameter function from Qian et al. (1995). This is a modified version of Dejonghe (1986) which was based on maximum entropy arguments: \ng ( L z ) = G ( η ) = F tanh( κη/ 2) tanh( κ/ 2) (14) \nwhere η = L z /L m ( E ), L m ( E ) is the maximum allowable value of L z at a given energy, and -1 < F < 1 and κ > 0 are free parameters. The parameter F works as a global adjustment of rotation while the parameter κ determines the contributions of stars with different L z ratios. Specifically for small κ only stars with high L z will contribute while large κ implies that all stars irrespective of their L z contribute to rotation. For F=1 and κ /greatermuch 0, g ( L z ) = sign( L z ) which corresponds to maximum rotation. \nFrom the resulting distribution function f ( E,L z ) we can calculate υ ϕ ( R,z ) in cylindrical coordinates using the equation \nυ ϕ ( R,z ) = 4 π ρR 2 Ψ ∫ 0 dE R √ 2(Ψ -E ) ∫ 0 f -( E,L z ) L z dL z . (15) \nTo find the mean line-of-sight velocity versus Galactic longitude l we have to project equation (15) to the sky plane \nυ los ( x, z ) = 2 Σ ∫ ∞ x υ ϕ ( R,z ) x R ρ ( R,z ) RdR √ R 2 -x 2 . (16) \nFigure 6 shows the mean line-of-sight velocity data vs Galactic longitude l for F = -0 . 9 and two κ values for the parameters in equation (14). Later in the axisymmetric section we constrain these parameters by fitting. Each data point corresponds to a cell from Figure 2. The maser data ( r > 100 '' ) are also included. The signs of velocities for negative l are reversed because of the assumed symmetry. The line shows the prediction of the model with parameters determined with equation (16). Figure 2 shows that the line-of-sight velocity \nFigure 7. Axisymmetric two-component model for the surface density of the nuclear cluster. The points with error bars show the number density of late-type stars along the l and b directions (Fritz et al. 2014) in red and blue respectively. The blue lines show the model that gives the best fit to the surface density data with parameters as in 19. \n<!-- image --> \ncells extend from b=0 to up to b = 50 '' , but most of them lie between 0 and b = 20 '' . For this reason we compute the model prediction at an average value of b = 20 '' .", '4 AXISYMMETRIC MODELING OF THE NSC': "We have seen that spherical models cannot explain the difference between the velocity dispersions along the l and b directions. The number counts also show that the cluster is flattened; see Figure 7 and Fritz et al. (2014). Therefore we now continue with axisymmetric modeling of the nuclear cluster. The first step is to fit the surface density counts with an axisymmetric density model. The available surface density data extend up to 1000 '' in the l and b directions. For comparison, the proper motion data extend to ∼ 70 '' from the centre (Figure 1). We generalize our spherical twocomponent γ -model from equation (2) to a spheroidal model given by \nρ ( R,z ) = 2 ∑ i =1 3 -γ i 4 πq i M i a i m γ i i ( m i + a i ) 4 -γ i (17) \nwhere m 2 i = R 2 + z 2 /q 2 i is the spheroidal radius and the two new parameters q 1 , 2 are the axial ratios (prolate > 1, oblate < 1) of the inner and outer component, respectively. Note that the method can be generalized to N components. The mass of a single component is given by 4 πq i ∞ 0 m 2 i ρ ( m i ) dm i . \nFrom Figure 7 we expect that the inner component will be more spherical than the outer component, although when the density profile gets flatter near the center it becomes more difficult to determine the axial ratio. In Figure 7 one also sees that the stellar surface density along the l direction is larger than along the b direction. Thus we assume that the NSC is an oblate system. To fit the model we first need to project the density and express it as a function of l and b . \n∫ \nThe projected surface density as seen edge on is \nΣ( x, z ) = 2 ∞ ∫ x ρ ( R,z ) R √ R 2 -x 2 dR. (18) \nIn general, to fit equation (18) to the data we would need to determine the eight parameters γ 1 , 2 , M 1 , 2 , a 1 , 2 , q 1 , 2 . However, we decided to fix a value for q 2 because the second component is not very well confined in the 8-dimensional parameter space (i.e. there are several models each with different q 2 and similar χ 2 ). We choose q 2 = 0 . 28, close to the value found in Fritz et al. (2014). For similar reasons, we also fix the value of γ 2 to that used in the spherical case. The minimum value of γ 1 for a semi-isotropic axisymmetric model with a black hole cannot be smaller than 0 . 5 (Qian et al. 1995), as in the spherical case. For our current modeling we treat γ 1 as a free parameter. Thus six free parameters remain. To fit these parameters to the data in Fig. 7 we apply a Markov chain Monte Carlo algorithm. For comparing the model surface density (18) to the star counts we found it important to average over angle in the inner conical cells to prevent an underestimation of the q 1 parameter. The values obtained with the MCMC algorithm for the NSC parameters and their errors are: \nγ 1 = 0 . 71 ± 0 . 12 a 1 = 147 . 6 '' ± 27 '' q 1 = 0 . 73 ± 0 . 04 γ 2 = 0 . 07 a 2 = 4572 '' ± 360 '' q 2 = 0 . 28 M 2 /M 1 = 101 . 6 ± 18 \n(19) \nThe reduced χ 2 that corresponds to these parameter values is χ 2 /ν SD = 0 . 99 for ν SD = 110 d.o.f. Here we note that there is a strong correlation between the parameters a 2 and M 2 . The flattening of the inner component is very similar to the recent determination from Spitzer/IRAC photometry (0 . 71 ± 0 . 02, Schodel et al. 2014) but slightly more flattened than the best value given by Fritz et al. (2014), 0 . 80 ± 0 . 04. The second component is about 100 times more massive than the first, but also extends more than one order of magnitude further. \nAssuming constant mass-to-light ratio for the star cluster, we determine its potential using the relation from Qian et al. (1995), which is compatible with their contour integral method (i.e. it can be used for complex R 2 and z 2 ). The potential for a single component i is given by: \nΨ i ( R,z ) = Ψ 0 i -2 πGq i e i ∞ ∫ 0 ρ i ( U ) [ R 2 (1+ u ) 2 + z 2 ( q 2 i + u ) 2 ] × (arcsin e i -arcsin e 1 √ 1+ u ) du (20) \nwith e i = √ 1 -q 2 i , U = R 2 1+ u + z 2 q 2 i + u , and where Ψ 0 i is the central potential (for a review of the potential theory of ellipsoidal bodies consider Chandrasekhar (1969)). The total potential of the two-component model is \nΨ( R,z ) = 2 ∑ i =1 Ψ i ( R,z ) + GM · √ R 2 + z 2 . (21)", '4.1 Axisymmetric Jeans modeling': "Here we first continue with axisymmetric Jeans modeling. We will need a large number of models to determine the best values for the mass and distance of the NSC, and for \nthe mass of the embedded black hole. We will use DFs for the detailed modeling in Section 4.3, but this is computationally expensive, and so a large parameter study with the DF approach is not currently feasible. In Section 4.3 we will show that a two-integral (2I) distribution function of the form f ( E,L 2 z ) gives a very good representation to the histograms of proper motions and line-of-sight velocities for the nuclear star cluster in all bins. Therefore we can assume for our Jeans models that the system is semi-isotropic, i.e., isotropic in the meridional plane, υ 2 z = υ 2 R . From the tensor virial theorem (Binney & Tremaine 2008) we know that for 2I-models υ 2 Φ > υ 2 R in order to produce the flattening. In principle, for systems of the form f ( E,L z ) it is possible to find recursive expressions for any moment of the distribution function (Magorrian & Binney 1994) if we know the potential and the density of the system. However, here we will confine ourselves to the second moments, since later we will recover the distribution function. By integrating the Jeans equations we get relations for the independent dispersions (Nagai & Miyamoto 1976): \nυ 2 z ( R,z ) = υ 2 R ( R,z ) = -1 ρ ( R,z ) ∫ ∞ z dz ' ρ ( R,z ' ) ∂ Ψ ∂z ' υ 2 ϕ ( R,z ) = υ 2 R ( R,z ) + R ρ ( R,z ) ∂ ( ρυ 2 R ) ∂R -R ∂ Ψ ∂R (22) \nThe potential and density are already known from the previous section. Once υ 2 z is found it can be used to calculate υ 2 ϕ . The intrinsic dispersions in l and b direction are given by the equations: \nσ 2 b = υ 2 z σ 2 l = υ 2 x = υ 2 R sin 2 θ + υ 2 ϕ cos 2 θ υ 2 los = υ 2 y = υ 2 R cos 2 θ + υ 2 ϕ sin 2 θ (23) \nwhere sin 2 θ = x 2 /R 2 and cos 2 θ = 1 -x 2 /R 2 . Projecting the previous equations along the line of sight we have: \nΣ σ 2 l ( x, z ) = 2 ∫ ∞ x [ υ 2 R x 2 R 2 + υ 2 ϕ ( 1 -x 2 R 2 )] ρ ( R,z ) √ R 2 -x 2 dR, Σ σ 2 b ( x, z ) = 2 ∫ ∞ x υ 2 z ( R,z ) ρ ( R,z ) √ R 2 -x 2 dR, Σ υ 2 los ( x, z ) = 2 ∫ ∞ x [ υ 2 R ( 1 -x 2 R 2 ) + υ 2 ϕ x 2 R 2 ] ρ ( R,z ) √ R 2 -x 2 dR, (24) \nwhere we note that the last quantity in (23) and (24) is the 2 nd moment and not the line-of-sight velocity dispersion. \nIn order to define our model completely, we need to determine the distance R 0 and mass M ∗ of the cluster and the black hole mass M · . To do this we apply a χ 2 minimization technique matching all three velocity dispersions in both sets of cells, using the following procedure. First we note that the inclusion of self-consistent rotation to the model will not affect its mass. This means that for the fitting we can use υ 2 los 1 / 2 for each cell of Figure 2. Similarly, since our model is axisymmetric we should match to the υ 2 l,b 1 / 2 for each proper motion cell; the υ l,b terms should be and indeed are negligible. Another way to see this is that since the system is axially symmetric, the integration of υ l,b along the line-ofsight should be zero because the integration would cancel out for positive and negative y . \nWith this in mind we proceed as follows, using the clus- \nFigure 8. Velocity dispersions σ l and σ b compared to axisymmetric, semi-isotropic Jeans models. The measured dispersions σ l (red points with error bars) and σ b (blue points) for all cells are plotted as a function of their two-dimensional radius on the sky, with the Galactic centre at the origin. The black lines show the best model; the model velocity dispersions are averaged over azimuth on the sky. The dashed black lines show the same quantities for a model which has lower flattening ( q 1 = 0 . 85 vs q 1 = 0 . 73) and a smaller central density slope (0 . 5 vs 0 . 7). \n<!-- image --> \nFigure 9. Root mean square line-of-sight velocities compared with the best model, as a function of two-dimensional radius on the sky as in Fig. 8. In both plots the stellar mass of the NSC is 7 . 73 × 10 6 M /circledot within m< 100 '' , the black hole mass is 3 . 86 × 10 6 M /circledot , and the distance is 8 . 3 kpc (equation 25). All the maser data are included in the plot. \n<!-- image --> \nter's density parameters 1 as given in (19). First we partition the 3d space ( R 0 , M ∗ , M · ) into a grid with resolution 20 × 20 × 20. Then for each point of the grid we calculate the corresponding χ 2 using the velocity dispersions from all cells in Figs. 1 and 2, excluding the two cells at the largest radii (see Fig. 8). We compare the measured dispersions with the model values obtained from equations (24) for the centers \nFigure 10. All three projected velocity disperions compared. Red: σ l , Blue: σ b , Brown: σ los = υ 2 los 1 / 2 . Note that σ b is slightly lower than σ los . The difference between σ b and σ l comes from the flattening of both the inner and outer components of the model. \n<!-- image --> \nof these cells. Then we interpolate between the χ 2 values on the grid and find the minimum of the interpolated function, i.e., the best values for ( R 0 , M ∗ , M · ). To determine statistical errors on these quantities, we first calculate the Hessian matrix from the curvature of χ 2 surface at the minimum, ∂χ 2 /∂p i ∂p j . The statistical variances will be the diagonal elements of the inverted matrix. \nWith this procedure we obtain a minimum reduced χ 2 /ν Jeans = 1 . 07 with ν Jeans = 161 degrees of freedom, for the values \nR 0 = 8 . 27 kpc M ∗ ( m< 100 '' ) = 7 . 73 × 10 6 M /circledot M · = 3 . 86 × 10 6 M /circledot , (25) \nwhere \nM ∗ ( m ) ≡ ∫ m 0 4 πm 2 [ q 1 ρ 1 ( m ) + q 2 ρ 2 ( m )] dm, (26) \nand the value given for M ∗ in (25) is not the total cluster mass but the stellar mass within elliptical radius 100 '' . In Section 4.2 we will consider in more detail the determination of these parameters and their errors. The model with density parameters as in (19) and dynamical parameters as in (25) will be our best model. In Section 4.3 we will see that it also gives an excellent prediction to the velocity histograms. \nFirst, we now look at the comparison of this model with the velocity data. Figure 8 shows how the azimuthally averaged dispersions σ l and σ b compare with the measured proper motion dispersions. Figure 9 shows how this best model, similarly averaged, compares with the line-of-sight mean square velocity data. The maser data are also included in the plot. It is seen that the model fits the data very well, in accordance with its χ 2 /ν Jeans = 1 . 07 per cell. Figure 10 shows how all three projected dispersions of the model compare. σ b is slightly lower than σ los due to projection effects. The fact that all three velocity dispersion profiles in Figs. 8, 9 are fitted well by the model suggests that the assumed semi-isotropic dynamical structure is a reasonable approximation. \nThe model prediction in Fig. 8 is similar to Figure 11 \nof Trippe et al. (2008) but the interpretation is different. As shown in the previous section, the difference in projected dispersions cannot be explained by imposing rotation on the model. Here we demonstrated how the observational finding σ l > σ b can be quantitatively reproduced by flattened axisymmetric models of the NSC and the surrounding nuclear disk. \nMost of our velocity data are in the range 7'-70', i.e., where the inner NSC component dominates the potential. In order to understand the dynamical implications of these data on the flattening of this component, we have also constructed several density models in which we fixed q 1 to values different from the q 1 = 0 . 73 obtained from star counts. In each case we repeated the fitting of the dynamical parameters as in (25). We found that models with q 1 in a range from ∼ 0 . 69 to ∼ 0 . 74 gave comparable fits ( χ 2 /ν ) to the velocity dispersion data as our nominal best model but that a model with q 1 = 0 . 77 was noticeably worse. We present an illustrative model with flattening about half-way between the measured q 1 = 0 . 73 and the spherical case, for which we set q 1 = 0 . 85. This is also close to the value given by (Fritz et al. 2014), q 1 = 0 . 80 ± 0 . 04. We simultaneously explore a slightly different inner slope, γ 1 = 0 . 5. We then repeat the fitting of the starcount density profile in Fig. 7 (model not shown), keeping also γ 2 and q 2 fixed to the previous values, and varying the remaining parameters. Our rounder comparison model then has the following density parameters: \nγ 1 = 0 . 51 a 1 = 102 . 6 '' q 1 = 0 . 85 γ 2 = 0 . 07 a 2 = 4086 '' q 2 = 0 . 28 M 2 M 1 = 109 . 1 (27) \nThe best reduced χ 2 that we obtain for the velocity dispersion profiles with these parameters is χ 2 /ν Jeans = 1 . 16 and corresponds to the values \nR 0 = 8 . 20 kpc M ∗ ( m< 100 '' ) = 8 . 31 × 10 6 M /circledot M · = 3 . 50 × 10 6 M /circledot , (28) \nCompared to the best and more flattened model, the cluster mass has increased and the black hole mass has decreased. The sum of both masses has changed only by 2% and the distance only by 1%. In Figures 8, 9 we see how the projected velocity dispersions of this model compare with our best model. The main difference seen in σ l comes from the different flattening of the inner component, and the smaller slope of the dispersions near the center of the new model is because of its smaller central density slope.", '4.2 Distance to the Galactic Center, mass of the star cluster, and mass of the black hole': "Wenow consider the determination of these parameters from the NSC data in more detail. Fig 11 shows the marginalized χ 2 -plot for the NSC model as given in equation (19), for pairs of two parameters ( R 0 , M · ), ( M · , M ∗ ), ( R 0 , M ∗ ), as obtained from fitting the Jeans dynamical model to the velocity dispersion profiles. The figure shows contour plots for constant χ 2 /ν Jeans with 1 σ , 2 σ and 3 σ in the three planes for the two-dimensional distribution of the respective parameters. We notice that the distance R 0 has the smallest relative error. \nThe best-fitting values for ( R 0 , M ∗ , M · ) are given in equation (25); these values are our best estimates based on \n<!-- image --> \n<!-- image --> \nFigure 11. Contour plots for the marginalized χ 2 in the three parameter planes ( R 0 , M · ), ( M · , M ∗ ), ( R 0 , M ∗ ). Contours are plotted at confidence levels corresponding to 1 σ , 2 σ and 3 σ of the joint probability distribution. The minimum corresponds to the values R 0 = 8 . 27kpc, M ∗ ( m< 100 '' ) = 7 . 73 × 10 6 M /circledot , M · = 3 . 86 × 10 6 M /circledot , with errors discussed in Section 4.2. \n<!-- image --> \nthe NSC data alone. For the dynamical model with these parameters and the surface density parameters given in (19), the flattening of the inner component inferred from the surface density data is consistent with the dynamical flattening, which is largely determined by the ratio of σ l /σ b and the tensor virial theorem. \nStatistical errors are determined from the Hessian matrix for this model. Systematic errors can arise from uncertainties in the NSC density structure, from deviations from the assumed axisymmetric two-integral dynamical structure, from dust extinction within the cluster (see Section 5), and other sources. We have already illustrated the effect of varying the cluster flattening on ( R 0 , M · , M ∗ ) with our second, rounder model. We have also tested how variations of the cluster density structure ( a 2 , q 2 , M 2 ) beyond 500 '' impact the best-fit parameters, and found that these effects are smaller than those due to flattening variations. \nWe have additionally estimated the uncertainty introduced by the symmetrisation of the data if the misalignment found by Feldmeier et al. (2014); Fritz et al. (2014) were intrinsic to the cluster, as follows. We took all radial velocity stars and rotated each star by 10 · clockwise on the sky. Then we resorted the stars into our radial velocity grid (Fig. 2). Using the new values υ 2 los 1 / 2 obtained in the cells we fitted Jeans models as before. The values we found for R 0 , M ∗ , M · with these tilted data differed from those in equation (25) by ∆ R 0 = -0 . 02 kpc, ∆ M ∗ ( m < 100 '' ) = -0 . 15 × 10 6 M /circledot , and ∆ M · = +0 . 02 × 10 6 M /circledot , respectively, which are well within the statistical errors. \nPropagating the errors of the surface density parameters from the MCMC fit and taking into account the correlation of the parameters, we estimate the systematic uncertainties from the NSC density structure to be ∼ 0 . 1 kpc in R 0 , ∼ 6% in M · , and ∼ 8% M ∗ ( m < 100 '' ). We will see in Section 4.3 below that the DF for our illustrative rounder NSC model gives a clearly inferior representation of the velocity histograms than our best kinematic model, and also that the systematic differences between both models appear comparable to the residual differences between our preferred model and the observed histograms. Therefore we take the \ndifferences between these models, ∼ 10% in M ∗ , ∼ 10% in M · , and ∼ 0 . 1kpc in R 0 , as a more conservative estimate of the dynamical modeling uncertainties, so that finally \nR 0 = 8 . 27 ± 0 . 09 | stat ± 0 . 1 | syst kpc M ∗ ( m< 100 '' ) = (7 . 73 ± 0 . 31 | stat ± 0 . 8 | syst ) × 10 6 M /circledot M · = (3 . 86 ± 0 . 14 | stat ± 0 . 4 | syst ) × 10 6 M /circledot . (29) \nWe note several other systematic errors which are not easily quantifiable and so are not included in these estimates, such as inhomogeneous sampling of proper motions or lineof-sight velocities, extinction within the NSC, and the presence of an additional component of dark stellar remnants. \nBased on our best model, the mass of the star cluster within 100 '' converted into spherical coordinates is M ∗ ( r < 100 '' ) = (8 . 94 ± 0 . 32 | stat ± 0 . 9 | syst ) × 10 6 M /circledot . The model's mass within the innermost pc (25 '' ) is M ∗ ( m < 1pc) = 0 . 729 × 10 6 M /circledot in spheroidal radius, or M ∗ ( r < 1pc) = 0 . 89 × 10 6 M /circledot in spherical radius. The total mass of the inner NSC component is M 1 = 6 . 1 × 10 7 M /circledot . Because most of this mass is located beyond the radius where the inner component dominates the projected star counts, the precise division of the mass in the model between the NSC and the adjacent nuclear disk is dependent on the assumed slope of the outer density profile of NSC, and is therefore uncertain. \nThe distance and the black hole mass we found differ by 0 . 7% and 12%, respectively, from the values R 0 = 8 . 33 ± 0 . 17 | stat ± 0 . 31 | syst kpc and M · = 4 . 31 ± 0 . 36 × 10 6 M /circledot for R 0 = 8 . 33 kpc, as determined by Gillessen et al. (2009) from stellar orbits around Sgr A ∗ . Figure 12 shows the 1 σ to 3 σ contours of marginalized χ 2 for ( R 0 , M · ) jointly from stellar orbits (Gillessen et al. 2009), for the NSC model of this paper, and for the combined modeling of both data sets. The figure shows that both analyses are mutually consistent. When marginalized over M ∗ and the respective other parameter, the combined modeling gives, for each parameter alone, R 0 = 8 . 33 ± 0 . 11 kpc and M · = 4 . 23 ± 0 . 14 × 10 6 M /circledot . We note that these errors for R 0 and M · are both dominated by the distance error from the NSC modeling. Thus \nFigure 12. Blue: χ 2 contours in the ( R 0 , M · ) plane from stellar orbits of S-stars, as in Figure 15 of Gillessen et al. (2009), at confidence levels corresponding to 1 σ , 2 σ , 3 σ for the joint probability distribution. Brown: Corresponding χ 2 contours from this work. Black: Combined contours after adding the χ 2 values. \n<!-- image --> \nour estimated additional systematic error of 0 . 1 kpc for R 0 in the NSC modeling translates to a similar additional error in the combined R 0 measurement and, through the SMBH mass-distance relation given in Gillessen et al (2009), to an additional uncertainty /similarequal 0 . 1 × 10 6 M /circledot in M · . We see that the combination of the NSC and S-star orbit data is a powerful means for decreasing the degeneracy between the SMBH mass and Galactic center distance in the S-star analysis.", '4.3 Two-integral distribution function for the NSC.': "Now we have seen the success of fitting the semi-isotropic Jeans models to all three velocity dispersion profiles of the NSC, and determined its mass and distance parameters, we proceed to calculate two-integral (2I) distribution functions. We use the contour integral method of Hunter & Qian (1993, HQ) and Qian et al. (1995). A 2I DF is the logical, next-simplest generalization of isotropic spherical models. Finding a positive DF will ensure that our model is physical. Other possible methods to determine f ( E,L z ) include reconstructing the DF from moments (Magorrian 1995), using series expansions as in Dehnen & Gerhard (1994), or grid-based quadratic programming as in Kuijken (1995). We find the HQ method the most suitable since it is a straightforward generalization of Eddington's formula. The contour integral is given by: \nf + ( E,L z ) = 1 4 π 2 i √ 2 ∮ dξ ( ξ -E ) 1 / 2 ˜ ρ 11 ( ξ, L 2 z 2( ξ -E ) 1 / 2 ) (30) \nwhere ˜ ρ 11 (Ψ , R ) = ∂ 2 ∂ Ψ 2 ρ (Ψ , R ). Equation (30) is remarkably similar to Eddington's formula. Like in the spherical \ncase the DF is even in L z . The integration for each ( E,L z ) pair takes place on the complex plane of the potential ξ following a closed path (i.e. an ellipse) around the special value Ψ env . For more information on the implementation and for a minor improvement over the original method see Appendix A. We find that a resolution of (120 × 60) logarithmically placed cells in the ( E,L z ) space is adequate to give us relative errors of the order of 10 -3 when comparing with the zeroth moment, i.e., the density, already known analytically, and with the second moments, i.e., the velocity dispersions from Jeans modeling. \nThe gravitational potential is already known from equations (20) and (21). For the parameters (cluster mass, black hole mass, distance) we use the values given in (25). Figure 13 shows the DF in ( E,L z ) space. The shape resembles that of the spherical case (Fig. 4). The DF is a monotonically increasing function of η = L z /L z max ( E ) and declines for small and large energies. The DF contains information about all moments and therefore we can calculate the projected velocity profiles (i.e., velocity distributions, hereafter abbreviated VPs) in all directions. The normalized VP in the line-of-sight (los) direction y is \nV P ( υ los ; x, z ) = 1 Σ ∫∫∫ E> 0 f ( E,L z ) dυ x dυ z dy. (31) \nUsing polar coordinates in the velocity space ( υ x , υ z ) → ( υ ⊥ , ϕ ) where υ x = υ ⊥ cos ϕ and υ z = υ ⊥ sin ϕ we find \nV P ( υ los ; x, z ) = 1 2Σ y 2 ∫ y 1 dy 2Ψ -υ 2 los ∫ 0 dυ 2 ⊥ 2 π ∫ 0 dϕf ( E,L z ) (32) \nwhere \nE = Ψ( x, y, z ) -1 2 ( υ 2 los + υ 2 ⊥ ) , L z = xυ los -yυ ⊥ cos ϕ. (33) \nand y 1 , 2 are the solutions of Ψ( x, y, z ) -v 2 los / 2 = 0. Following a similar path we can easily find the corresponding integrals for the VPs in the l and b directions. \nThe typical shape of the VPs in the l and b directions within the area of interest ( r < 100 '' ) is shown in Figure 14. We notice the characteristic two-peak shape of the VP along l that is caused by the near-circular orbits of the flattened system. Because the front and the back of the axisymmetric cluster contribute equally, the two peaks are mirror-symmetric, and adding rotation would not change their shapes. \nThe middle panels of Figure 15 and Figures B1 and B2 in Appendix B show how our best model (with parameters as given in (19) and (27)) predicts the observed velocity histograms for various combinations of cells. The reduced χ 2 for each histogram is also provided. The prediction is very good both for the VPs in υ l and υ b . Specifically, for the l proper motions our flattened cluster model predicts the two-peak structure of the data pointed out by several authors (Trippe et al. 2008; Schodel et al. 2009; Fritz et al. 2014). In order to calculate the VP from the model for each cell we averaged over the VP functions for the center of each cell weighted by the number of stars in each cell and normalized by the total number of stars in all the combined cells. \nFigure 15 compares two selected υ l -VPs for our two main models with the data. The left column shows how the observed velocity histograms (VHs) for corresponding cells compare to the model VPs for the less flattened model with parameters given in (27) and (28), the middle column compares with the same VPs from our best model with parameters given in (19) and (25). Clearly, the more flattened model with q 1 = 0 . 73 fits the shape of the data much better than the more spherical model with q 1 = 0 . 85, justifying its use in Section 4.2. \n<!-- image --> \nFigure 13. We used the HQ algorithm to calculate the 2I-DF for our best Jeans model. The left plot shows the DF in E and η = L z /L z max ( E ) space. The DF is an increasing function of η . The right plot shows the projection of the DF on energy space for several values of η . The shape resembles that of the spherical case in Figure 4. \n<!-- image --> \nThis model is based on an even DF in L z and therefore does not yet have rotation. To include rotation, we will (in Section 4.4) add an odd part to the DF, but this will not change the even parts of the model's VPs. Therefore, we can already see whether the model is also a good match to the observed los velocities by comparing it to the even parts of the observed los VHs. This greatly simplifies the problem since we can think of rotation as independent, and can therefore adjust it to the data as a final step. Figure B3 shows how the even parts of the VHs from the los data compare with the VPs of the 2I model. Based on the reduced χ 2 , the model provides a very good match. Possible systematic deviations are within the errors. The los VHs are broader than those in the l direction because the los data contain information about rotation (the broader the even part of the symmetrized los VHs, the more rotation the system possesses, and in extreme cases they would show two peaks).", '4.4 Adding rotation to the axisymmetric model: is the NSC an isotropic rotator?': "As in the spherical case, to model the rotation we add an odd part in L z to the initial even part of the distribution function, so that the final DF takes the form f ( E,L z ) = (1 + g ( L z )) f ( E,L z ). We use again equation (14); this adds two additional parameters ( κ , F) to the DF. Equation (16) \nFigure 14. Typical velocity distributions for l and b -velocities within the area of interest ( r < 100 '' ). The red line shows the VPs in the b direction, the blue line in the l direction. The VPs along l show the characteristic two-peak-shape pointed out from the data by several authors (Schodel et al. 2007; Trippe et al. 2008; Fritz et al. 2014). \n<!-- image --> \ngives the mean los velocity vs Galactic longitude. In order to constrain the parameters ( κ , F) we fitted the mean los velocity from equation (16) to the los velocity data for all cells in Fig. 2. The best parameter values resulting from this 2D-fitting are κ = 2 . 8 ± 1 . 7, F = 0 . 85 ± 0 . 15 and χ 2 r = 1 . 25. Figure B4 shows that the VPs of this rotating model compare well with the observed los VHs. \nAn axisymmetric system with a DF of the form f ( E,L z ) is an isotropic rotator when all three eigenvalues of the dispersion tensor are equal (Binney & Tremaine 2008) and therefore \nυ 2 ϕ = υ 2 ϕ -υ 2 R . (34) \nIn order to calculate υ ϕ from equation (34) it is not neces- \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFigure 15. Predicted distributions of υ l velocity compared to the observed histograms. In each row, model VPs and observed VHs are shown averaged over the cells indicated in red in the right column, respectively. Left column: predictions for the less flattened model which we use as an illustration model, i.e., for parameters given in (26) and (27). Middle column: predicted VPs for our best model with parameters given in (18) and (24). This more flattened model with q 1 = 0 . 73 fits the data much better than the rounder cluster model with q 1 = 0 . 85. \n<!-- image --> \nry for the DF to be known since υ 2 ϕ and υ 2 R are already known from the Jeans equations (22). Figure 16 shows the fitted u los velocity from the DF against the isotropic rotator case calculated from equation (34), together with the mean los velocity data. The two curves agree well within ∼ 30 '' , and also out to ∼ 200 '' they differ only by ∼ 10 km/s. Therefore according to our best model the NSC is close to an isotropic rotator, with slightly lower rotation and some tangential anisotropy outwards of 30'.", '5 DISCUSSION': "In this work we presented a dynamical analysis of the Milky Way's nuclear star cluster (NSC), based on ∼ 10'000 proper motions, ∼ 2'700 radial velocities, and new star counts from the companion paper of Fritz et al. (2014). We showed that an excellent representation of the kinematic data can be obtained by assuming a constant mass-to-light ratio for the cluster, and modeling its dynamics with axisymmetric twointegral distribution functions (2I-DFs), f ( E,L z ). The DF modeling allows us to see whether the model is physical, i.e., whether the DF is positive, and to model the proper motion (PM) and line-of-sight (los) velocity histograms (VHs). One open question until now has been the nature of the double peaked VHs of the v l -velocities along Galactic longitude, and the bell-shaped VHs of v b along Galactic latitude, which cannot be fitted by Gaussians (Schodel et al. 2009). Our 2I DF approximation of the NSC gives an excellent prediction for the observed shapes of the v l -, v b , and v los -VHs. The models show that the double-peaked shape of the v l -VHs is a result of the flattening of the NSC, and suggest that the cluster's dynamical structure is close to an isotropic rotator. \nFigure 16. Best fitting model from the 2I DF compared to the isotropic rotator model. Each data point corresponds to a cell from Figure 2. Velocities at negative l have been folded over with their signs reversed and are shown in red. The plot also includes the maser data at R s > 100 '' . The predictions of both models are computed for b = 20 '' . For comparison, cells with centers between b = 15 '' and b = 25 '' are highlighted with full triangles. \n<!-- image --> \nBecause both PMs and los-velocities enter the dynamical models, we can use them also to constrain the distance to the GC, the mass of the NSC, and the mass of the Galactic centre black hole. To do this efficiently, we used the semiisotropic Jeans equations corresponding to 2I-DFs. In this section, we discuss these issues in more detail. \nFigure 17. Average differential extinction of nuclear cluster stars plotted as a function of v l proper motion. The differential extinction is inferred from the difference in the color of a star to the median color of its 16 nearest neighbours, using the extinction law of Fritz et al. (2011), and correcting also for the weak color variation with magnitude. For this plot we use all the proper motion stars in the central and extended fields of Fritz et al. (2014) and bins of 0.2 mas/yr. The differential extinction is larger for stars with negative l -proper motions which occur preferentially at the back of the cluster. \n<!-- image -->", '5.1 The dynamical structure of the NSC': "The star count map derived in Fritz et al. (2014) suggests two components in the NSC density profile, separated by an inflection point at about ∼ 200 '' ∼ 8 pc (see Fig. 7 above). To account for this we constructed a two-component dynamical model for the star counts in which the two components are described as independent γ -models. The inner, rounder component can be considered as the proper NSC, as in Fritz et al. (2014), while the outer, much more flattened component may represent the inner parts of the nuclear stellar disk (NSD) described in Launhardt et al. (2002). \nThe scale radius of the inner component is ∼ 100 '' , close to the radius of influence of the SMBH, r h ∼ 90 '' (Alexander 2005). The profile flattens inside ∼ 20 '' to a possible core (Buchholz et al. 2009; Fritz et al. 2014) but the slope of the three-dimensional density profile for the inner component is not well-constrained. \nThe flattening for the inner NSC component inferred from star counts is q 1 = 0 . 73 ± 0 . 04, very close to the value of q = 0 . 71 ± 0 . 02 found recently from Spitzer multi-band photometry (Schodel et al. 2014). It is important that these determinations agree with the dynamical flattening of our best Jeans dynamical models: the dynamical flattening is robust because it is largely determined by the ratio of σ b /σ l and the tensor virial theorem. Because star counts, photometric, and dynamical values for the inner NSC flattening agree, this parameter can now be considered securely determined. \nAssuming constant mass-to-light ratio for the NSC, we found that a 2I-DF model gives an excellent description of the proper motion and los velocity dispersions and VHs, in particular of the double-peaked distributions in the v l -velocities. This double-peaked structure is a direct consequence of the flattening of the star cluster; the detailed \nagreement of the model VPs with the observed histograms therefore confirms the value q 1 = 0 . 73 for the inner cluster component. For an axisymmetric model rotation cannot be seen directly in the proper motion VHs when observed edge-on, as is the case here, but is apparent only in the los velocities. When a suitable odd part of the DF is added to include rotation, the 2I-DF model also gives a very good representation of the skewed los VHs. From the amplitude of the required rotation we showed that the NSC can be approximately described as an isotropic rotator model, rotating slightly slower than that outside 30 '' . \nIndividual VHs are generally fitted by this model within the statistical errors, but on closer examination the combined v l VHs show a slightly lower peak at negative velocities, as already apparent in the global histograms of Trippe et al. (2008); Schodel et al. (2009). Fig. 17 suggests that differential extinction of order ∼ 0 . 2 mag within the cluster may be responsible for this small systematic effect, by causing some stars from the back of the cluster to fall out of the sample. The dependence of mean extinction on v l independently shows that the NSC must be rotating, which could otherwise only be inferred from the los velocities. In subsequent work, we will model the effect of extinction on the inferred dynamics of the NSC. This will then also allow us to estimate better how important deviations from the 2I-dynamical structure are, i.e., whether three-integral dynamical modeling (e.g., De Lorenzi et al. 2013) would be worthwhile. \n∼", '5.2 Mass of the NSC': "The dynamical model results in an estimate of the mass of the cluster from our dataset. Our fiducial mass value is M ∗ ( m< 100 '' ) = (7 . 73 ± 0 . 31 | stat ± 0 . 8 | syst ) × 10 6 M /circledot interior to a spheroidal major axis distance m =100 '' . This corresponds to an enclosed mass within 3-dimensional radius r = 100 '' of M ∗ ( r< 100 '' ) = (8 . 94 0 . 31 | stat 0 . 9 | syst ) 10 6 M /circledot . \n∗ \n∗ | | /circledot The fiducial mass M ∗ ( r< 100 '' ) for the best axisymmetric model is larger than that obtained with spherical models. The constant M/L spherical model with density parameters as in Section 3, for R 0 = 8 . 3 kpc and the same black hole mass has M ( r< 100 '' ) = 6 . 6 × 10 6 M . \n± \n± \n× \n× There are two reasons for this difference: (i) At ∼ 50 '' where the model is well-fixed by kinematic data the black hole still contributes more than half of the interior mass. In this region, flattening the cluster at constant mass leaves σ l and σ los approximately constant, but decreases σ b to adjust to the shape. To fit the same observed data, the NSC mass must be increased. (ii) Because of the increasing flattening with radius, the average density of the axisymmetric model decreases faster than that of the spherical density fit; thus for the same observed velocity dispersion profiles a larger binding mass for the NSC is required. \n/circledot \nFigure 18 shows the enclosed stellar mass within the spheroidal radius m as in equation (26), as well as the mass within the spherical radius r . E.g., the mass within 1 pc (25 '' ) is M ∗ ( r < 1pc ∼ 25 '' ) = 0 . 89 × 10 6 M /circledot . This is compatible with the spherical modeling of Schodel et al. (2009) who gave a range of 0 . 6 -1 . 7 × 10 6 M /circledot , rescaled to R 0 = 8 . 3 kpc, with the highest mass obtained for their isotropic, constant M/L model. According to Fig. 18, at r /similarequal 30 '' = 1 . 2 pc the NSC contributes already /similarequal 25% of the interior mass \n<!-- image --> \n<!-- image --> \nFigure 18. Upper panel: Enclosed mass of the NSC, as function of three-dimensional radius r and spheroidal radius m , and total enclosed mass including the black hole. Middle panel: Enclosed mass of the inner component of the NSC (inner component M 1 ), the NSD (outer component M 2 ), and total enclosed stellar mass, as function of three-dimensional radius r . Lower panel: Axis ratios of the stellar density and total potential as functions of the cylindrical radius R . \n<!-- image --> \n( /similarequal 45% at r /similarequal 50 '' = 2 pc), and beyond r /similarequal 100 '' = 4 pc it clearly dominates. \nAn important point to note is that the cluster mass does not depend on the net rotation of the cluster but only on its flattening. This is because to add rotation self-consistently to the model we need to add an odd part to the DF which does not affect the density or the proper motion dispersions σ l and σ b . \nOur NSC mass model can be described as a superposition of a moderately flattened nuclear cluster embedded in a highly flattened nuclear disk. The cumulative mass distributions of the two components are shown in the middle \npanel of Figure 18. The NSD starts to dominate at about 800 '' which is in good agreement with the value found by Launhardt et al. (2002). \nApproximate local axis ratios for the combined density and for the total potential including the central black hole are shown in the lower panel of Fig. 18. Here we approximate the axial ratio of the density at radius R by solving the equation ρ ( R, 0) = ρ (0 , z ) for z and writing q ρ = z/R , and similarly for q Ψ . The density axis ratio q ρ ( R ) shows a strong decrease between the regions dominated by the inner and outer model components. The equipotentials are everywhere less flattened. At the center, q Ψ = 1 because of the black hole; the minimum value is not yet reached at 1000 '' . Therefore, we can define the NSC proper as the inner component of this model, similar to Fritz et al. (2014). \nThe total mass of the inner component, M 1 = 6 . 1 × 10 7 M /circledot (Section 4.2), is well-determined within similar relative errors as M ∗ ( m < 100 '' ). However, identifying M 1 with the total mass of the Galactic NSC at the center of the nuclear disk has considerable uncertainties: because the outer NSD component dominates the surface density outside 100 '' -200 '' , the NSC density profile slope at large radii is uncertain, and therefore the part of the mass outside ∼ 200 '' ( ∼ 64% of the total) is also uncertain. A minimal estimate for the mass of the inner NSC component is its mass within 200 '' up to where it dominates the star counts. This gives M NSC > 2 10 7 M /circledot . \n× \n/circledot Finally, we use our inferred dynamical cluster mass to update the K-band mass-to-light ratio of the NSC. The bestdetermined mass is within 100 '' . Comparing our M ∗ ( r < 100 '' ) = (8 . 94 ± 0 . 31 | stat ± 0 . 9 | syst ) × 10 6 M /circledot with the K-band luminosity of the old stars derived in Fritz et al. (2014), L 100 '' = (12 . 12 ± 2 . 58) × 10 6 L /circledot , Ks , we obtain M/L Ks = (0 . 76 ± 0 . 18) M /circledot /L /circledot , Ks . The error is dominated by the uncertainty in the luminosity (21%, compared to a total 10% in mass from adding statistical and systematic errors in quadrature). The inferred range is consistent with values expected for mostly old, solar metallicity populations with normal IMF (e.g., Courteau et al. 2013; Fritz et al. 2014).", '5.3 Evolution of the NSC': "After ∼ 10 half mass relaxation times t rh a dense nuclear star cluster will eventually evolve to form a BahcallWolf cusp with slope γ = 7 / 4 (Merritt 2013); for rotating dense star clusters around black holes this was studied by Fiestas & Spurzem (2010). The minimum allowable inner slope for a spherical system with a black hole to have a positive DF is γ = 0 . 5. From the data it appears that the Galactic NSC instead has a core (Buchholz et al. 2009; Fritz et al. 2014), with the number density possibly even decreasing very close to the center ( r < 0 . 3 pc). This is far from the expected Bahcall-Wolf cusp, indicating that the NSC is not fully relaxed. It is consistent with the relaxation time of the NSC being of order 10 Gyr everywhere in the cluster (Merritt 2013). \nFrom Fig. 16 we see that the rotational properties of the Milky Way's NSC are close to those of an isotropic rotator. Fiestas et al. (2012) found that relaxation in rotating clusters causes a slow ( ∼ 3 t rh ) evolution of the rotation profile. Kim et al. (2008) found that it also drives the velocity dispersions towards isotropy; in their initially already \nnearly isotropic models this happens in ∼ 4 t rh . On a similar time-scale the cluster becomes rounder (Einsel & Spurzem 1999). Comparing with the NSC relaxation time suggests that these processes are too slow to greatly modify the dynamical structure of the NSC, and thus that its properties were probably largely set up at the time of its formation. \nThe rotation-supported structure of the NSC could be due to the rotation of the gas from which its stars formed, but it could also be explained if the NSC formed from merging of globular clusters. In the latter model, if the black hole is already present, the NSC density and rotation after completion of the merging phase reflects the distribution of disrupted material in the potential of the black-hole (e.g. Antonini et al. 2012). Subsequently, relaxation would lead to shrinking of the core by a factor of ∼ 2 in 10 Gyr towards a value similar to that observed (Merritt 2010). In the simulations of Antonini et al. (2012), the final relaxed model has an inner slope of γ = 0 . 45, not far from our models (note that in flattened semi-isotropic models the minimum allowed slope for the density is also 0.5 (Qian et al. 1995)). Their cluster also evolved towards a more spherical shape, however, starting from a configuration with much less rotation and flattening than we inferred here for the present Milky Way NSC. Similar models with a net rotation in the initial distribution of globular clusters could lead to a final dynamical structure more similar to the Milky Way NSC.", '5.4 Distance to the Galactic center': "From our large proper motion and los velocity datasets, we obtained a new estimate for the statistical parallax distance to the NSC using axisymmetric Jeans modeling based on the cluster's inferred dynamical structure. From matching our best dynamical model to the proper motion and los velocity dispersions within approximately | l | , | b | < 50 '' , we found R 0 = 8 . 27 ± 0 . 09 | stat ± 0 . 1 | syst kpc. The statistical error is very small, reflecting the large number of fitted dispersion points. The systematic modeling error was estimated from uncertainties in the density structure of the NSC, as discussed in Section 4.2. \nOur new distance determination is much more accurate than that of Do et al. (2013) based on anisotropic spherical Jeans models of the NSC, R 0 = 8 . 92 +0 . 58 -0 . 58 kpc, but is consistent within their large errors. We believe this is mostly due to the much larger radial range we modeled, which leaves less freedom in the dynamical structure of the model. \nThe new value for R 0 is in the range R 0 = 8 . 33 ± 0 . 35 kpc found by Gillessen et al. (2009) from analyzing stellar orbits around Sgr A ∗ . A joint statistical analysis of the NSC data with the orbit results of Gillessen et al. (2009) gives a new best value and error R 0 = 8 . 33 ± 0 . 11 kpc (Fig. 12, Section 4.2). Our estimated systematic error of 0 . 1 kpc for R 0 in the NSC modeling translates to a similar additional uncertainty in this combined R 0 measurement. \nMeasurements of R 0 prior to 2010 were reviewed by Genzel et al. (2010). Their weighted average of direct measurements is R 0 = 8 . 23 ± 0 . 20 ± 0 . 19 kpc, where the first error is the variance of the weighted mean and the second the unbiased weighted sample variance. Two recent measurements give R 0 = 8 . 33 ± 0 . 05 | stat ± 0 . 14 | syst kpc from RR Lyrae stars (Dekany et al. 2013) and R 0 = 8 . 34 ± 0 . 14 kpc from fitting axially symmetric disk models to trigonometric \nparallaxes of star forming regions (Reid et al. 2014). These measurements are consistent with each other and with our distance value from the statistical parallax of the NSC, with or without including the results from stellar orbits around Sgr A ∗ , and the total errors of all three measurements are similar, ∼ 2%.", '5.5 Mass of the Galactic supermassive black hole': "Given a dynamical model, it is possible to constrain the mass of the central black hole from 3D stellar kinematics of the NSC alone. With axisymmetric Jeans modeling we found M · = (3 . 86 ± 0 . 14 | stat ± 0 . 4 | syst ) × 10 6 M /circledot , where the systematic modeling error is estimated from the difference between models with different inner cluster flattening as discussed in Section 4.2. Within errors this result is in agreement with the black hole mass determined from stellar orbits around Sgr A ∗ (Gillessen et al. 2009). \nOur dataset for the NSC is the largest analyzed so far, and the axisymmetric dynamical model is the most accurate to date; it compares well with the various proper motion and line-of-sight velocity histograms. Nonetheless, future improvements may be possible if the uncertainties in the star density distribution and kinematics within 20' can be reduced, the effects of dust are incorporated, and possible deviations from the assumed 2I-axisymmetric dynamical structure are taken into account. \nSeveral similar analyses have been previously made using spherical isotropic or anisotropic modeling. Trippe et al. (2008) used isotropic spherical Jeans modeling for proper motions and radial velocities in 1 '' < R < 100 '' ; their best estimate is M · ∼ 1 . 2 × 10 6 M /circledot , much lower than the value found from stellar orbits. Schodel et al. (2009) constructed isotropic and anisotropic spherical broken power-law models, resulting in a black hole mass of M · = 3 . 6 +0 . 2 -0 . 4 × 10 6 M /circledot . However, Fritz et al. (2014) find M · ∼ 2 . 27 ± 0 . 25 × 10 6 M /circledot , also using a power-law tracer density. They argue that the main reason for the difference to Schodel et al. (2009) is because their velocity dispersion data for R > 15 '' are more accurate, and their sample is better cleaned for young stars in the central R < 2 . 5 '' . Assuming an isotropic spherical model with constant M/L, Fritz et al. (2014) find M · ∼ 4 . 35 ± 0 . 12 × 10 6 M /circledot . Do et al. (2013) used 3D stellar kinematics within only the central 0 . 5 pc of the NSC. Applying spherical Jeans modeling, they obtained M · = 5 . 76 +1 . 76 -1 . 26 × 10 6 M /circledot which is consistent with that derived from stellar orbits inside 1 '' , within the large errors. However, in their modeling they used a very small density slope for the NSC, of γ = 0 . 05, which does not correspond to a positive DF for their quasi-isotropic model. \nBased on this work and our own models in Section 4, the black hole mass inferred from NSC dynamics is larger for constant M/L models than for power law models, and it increases with the flattening of the cluster density distribution. \nThe conceptually best method to determine the black hole mass is from stellar orbits close to the black hole (Schodel et al. 2002; Ghez et al. 2008; Gillessen et al. 2009), as it requires only the assumption of Keplerian orbits and is therefore least susceptible to systematic errors. Gillessen et al. (2009) find that the largest uncertainty in the value obtained for M · is due to the uncertainty in R 0 , \n/circledot \nand that M · scales as M · ∝ R 2 . 19 0 . Therefore using our improved statistical parallax for the NSC also leads to a more accurate determination of the black hole mass. A joint statistical analysis of the axisymmetric NSC modeling together with the orbit modeling of Gillessen et al. (2009) gives a new best value and error for the black hole mass, M · = (4 . 26 ± 0 . 14) × 10 6 M /circledot (see Fig. 12, Section 4.2). An additional systematic error of 0.1 kpc for R 0 in the NSC modeling, through the BH mass-distance relation given in Gillessen et al (2009), translates to an additional uncertainty 0 . 1 × 10 6 M in M · . \n/similarequal \n/similarequal \n× · Combining this result with the mass modeling of the NSC, we can give a revised value for the black hole influence radius r infl , using a common definition of r infl as the radius where the interior mass M ( < r ) of the NSC equals twice the black hole mass (Merritt 2013). Comparing the interior mass profile in Fig. 18 as determined by the dynamical measurement with M · = 4 . 26 × 10 6 M /circledot , we obtain r infl 94 '' = 3 . 8 pc. \nThe Milky Way is one of some 10 galaxies for which both the masses of the black hole and of the NSC have been estimated (Kormendy & Ho 2013). From these it is known that the ratio of both masses varies widely. Based on the results above we estimate the Milky Way mass ratio M · /M NSC = 0 . 12 ± 0 . 04, with the error dominated by the uncertainty in the total NSC mass.", '6 CONCLUSIONS': "Our results can be summarized as follows: \n- · The density distribution of old stars in the central 1000 '' in the Galactic center can be well-approximated as the superposition of a spheroidal nuclear star cluster (NSC) with a scale length of ∼ 100 '' and a much larger nuclear disk (NSD) component.\n- · The difference between the proper motion dispersions σ l and σ b cannot be explained by rotation alone, but is a consequence of the flattening of the NSC. The dynamically inferred axial ratio for the inner component is consistent with the axial ratio inferred from the star counts which for our two-component model is q 1 = 0 . 73 ± 0 . 04.\n- ± · The orbit structure of an axisymmetric two-integral DF f ( E,L z ) gives an excellent match to the observed doublepeak in the v l -proper motion velocity histograms, as well as to the shapes of the vertical v b -proper motion histograms. Our model also compares well with the symmetrized (even) line-of-sight velocity histograms.\n- · The rotation seen in the line-of-sight velocities can be modelled by adding an odd part of the DF, and this shows that the dynamical structure of the NSC is close to an isotropic rotator model.\n- · Fitting proper motions and line-of-sight dispersions to the model determines the NSC mass within 100 '' , the mass of the SMBH, and the distance to the NSC. From the star cluster data alone, we find M ∗ ( r< 100 '' )=(8 . 94 ± 0 . 31 | stat ± 0 . 9 | syst ) × 10 6 M /circledot , M · =(3 . 86 ± 0 . 14 | stat ± 0 . 4 | syst ) × 10 6 M /circledot , and R 0 =8 . 27 ± 0 . 09 | stat ± 0 . 1 | syst kpc, where the estimated systematic errors account for additional uncertainties in the dynamical modeling. The fiducial mass of the NSC is larger than in previous spherical models. The total mass of the NSC is significantly more uncertain due to the surrounding \nnuclear disk; we estimate M NSC =(2 -6) × 10 7 M /circledot . The mass of the black hole determined with this approach is consistent with results from stellar orbits around Sgr A ∗ . The Galactic center distance agrees well with recent accurate determinations from RR Lyrae stars and masers in the Galactic disk, and has similarly small errors. \n- · Combining our modeling results with the stellar orbit analysis of Gillessen et al. 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E., 2006, ApJL, 644, L17 Wolfram Research, Inc., Mathematica, Version 8.0, Champaign, IL (2011)', 'APPENDIX A: TWO-INTEGRAL DISTRIBUTIONS FUNCTIONS': 'In this part we give implementation instructions for the 2IDF algorithm of Hunter & Qian (1993, HQ). We will try to focus on the important parts of the algorithm and also on the tests that one has to make to ensure that the implementation works correctly. Our implementation is based on Qian et al. (1995) and made with Wolfram Mathematica. For the theory the reader should consider the original HQ paper. \nWe will focus on the even part of the DF and for the case where the potential at infinity, Ψ ∞ , is finite and therefore can be set to zero. First one partitions the ( E,η ) space where η ≡ L z /L z max ( E ) takes values in (0 , 1). The goal of the HQ algorithm is to calculate the value of the DF on each of these points on a 2D grid and subsequently end up with a 3D grid where we can apply an interpolation to obtain the final smooth function f ( E, L z ). The energy values on the 2D grid are placed logarithmically within an interval of interest [ E min , E max ] (higher E max value is closer to the center) and the values of η are placed linearly between 0 and 1. Physically allowable E and L z correspond to bound orbits in the potential Ψ and therefore E > 0. In addition at each energy there is a maximum physically allowed L z corresponding to circular orbits with z = 0. This is given by the equations: \nE = Ψ( R 2 c , 0) + R 2 c d Ψ( R 2 c , 0) dR 2 | R = R c L 2 z = -2 R 4 c d Ψ( R 2 c , 0) dR 2 | R = R c (A1) \nwhere R c is the radius of the circular orbit and the value L z max ≡ L z ( R c ) is the maximum allowed value of L z at a specific E . The L z max ( E ) function can be found by solving the 1st equation for R c and substituting in the second one therefore making a map E → L z max . The value of the potential of a circular orbit with energy E is denoted by Ψ env ( E ) and can be found from Ψ env ( E ) = Ψ( R 2 c , 0) after solving the 1st of equation (A1) for R c . The value Ψ env ( E ) is important for evaluation of f ( E,L z ) and it is used in the contour of the complex integral. \nTo calculate the even part f + ( E,L z ) of the DF for each point of the grid we have to apply the following complex contour integral on the complex ξ -plane using a suitable path: \nf + ( E,L z ) = 1 4 π 2 i √ 2 dξ ( ξ -E ) 1 / 2 ˜ ρ 11 ( ξ, L 2 z 2( ξ -E ) 1 / 2 ) (A2) \n∮ \nwhere the subscripts denotes the second partial derivative with respect to the first argument. A possible path for the contour is shown in figure A1. The loop starts at the point 0 on the lower side of the real ξ axis, crosses the real ξ axis at the point Ψ env ( E ) and ends at the upper side of real ξ axis. The parametrization of the path in general could be that of an ellipse: \nξ = 1 2 Ψ env ( E )(1 + cos θ ) + ih sin θ, -π /lessorequalslant θ /lessorequalslant π (A3) \nwhere h is the highest point of the ellipse. The value of h should not be too high because we want to avoid other singularities but not too low either to maintain the accuracy.', '18 S. Chatzopoulos et al.': "We optimize our implementation by integrating along the upper part of the loop and multiply the real part of the result by 2 (this is because of the Schwarz reflection principle). \nIn order to calculate the integrand of the integral we need the following transformation: \n˜ ρ 11 ( ξ, R 2 ) = ρ 22 ( R 2 , z 2 ) [Ψ 2 ( R 2 , z 2 )] 2 -ρ 2 ( R 2 , z 2 )Ψ 22 ( R 2 , z 2 ) [Ψ 2 ( R 2 , z 2 )] 3 (A4) \nin which each subscript denotes a partial differentiation with respect to z 2 . This equation is analogous to equation (7) of the spherical case. In addition ˜ ρ is the density considered as a function of ξ and R 2 as opposed to R 2 and z 2 . The integrand of the contour integral A2 depends only on θ angle for a given ( E,L z ) pair. Therefore we need the maps R → ξ and z → ξ in order to find the value of the integrand for a specific θ . The first map is given by R 2 = 1 2 L 2 z / ( ξ -E ). The second is given by solving the equation ξ = Ψ [ L 2 z 2( ξ -E ) , z 2 ] for z . It is very important that the solution of the previous equation corresponds to the correct branch in which the integrand attains its physically achieved values. In order to achieve that for each pair ( E,L z max ) we start at the point ξ = Ψ env ( E )( θ = 0) which belongs to the physical domain and we look for the unique real positive solution. For the next point of the contour we use as initial guess the value of z from the previous step that we already know that belongs to the correct branch. Using this method we can calculate the integrand in several values of θ then make an interpolation of the integrand and calculate the value of the DF using numerical integration. \nFigure A2 shows the shape of the DF for η = 0 . 5 for the potential we use in the fourth section of the paper for one value of h , using the aforementioned procedure. We notice that for large energies fluctuations of the DF appear. In order to solve this we introduce a minor improvement of the procedure, by generalizing the h value of the contour to an energy-dependent function h = h ( E ). The h ( E ) could be a simple step function that takes four or five different values. For our model the h ( E ) function is a decreasing function of E . This means that the minor axis of the ellipse should decrease as the E increases to avoid such fluctuations. In general we can write h = h ( E,L z ) so that the contour depends both on E and L z . \nOnce we implement the algorithm it is necessary to test it. Our first test is to check that the lower half of the integration path in figure A1 is the complex conjugate of the upper half. Probably the next most straightforward test is against the spherical case. It is possible to use the HQ algorithm to calculate a DF for spherical system. This DF should be equal to that obtained from Eddington's formula for the same parameters. After calculating our 2I-DF we compare its low-order moments with those of Jeans modeling. The 0th and 2nd moments of the DF (the 1st is 0 for the even part) are given from the integrals. \nFigure A1. The contour used for the numerical evaluation of f ( E,L z ) for the case where Ψ ∞ = 0. We optimize our implementation by integrating only along the upper or lower part and then multiplying the result by 2. \n<!-- image --> \nρ ( R,z ) = 4 π R Ψ ∫ 0 dE R √ 2(Ψ -E ) ∫ 0 dL z f + ( E,L z ) ρ ( R,z ) υ 2 ϕ ( R,z ) = 4 π R Ψ ∫ 0 dE R √ 2(Ψ -E ) ∫ 0 dL z ( L z R ) 2 f + ( E,L z ) ρ ( R,z ) υ 2 z ( R,z ) = 2 π R Ψ ∫ 0 dE R √ 2(Ψ -E ) ∫ 0 dL z [ 2(Ψ -E ) -( L z R ) 2 ] f + ( E,L z ) (A5) \nComparison with the 0th moment (density) is straight forward since the density is analytically known from the start. The 1st moments should be 0 within the expected error. In our implementation the error between Jeans modeling and the DF is of the order of 10 -3 within the area of interest. An additional test would be to integrate the VPs over the velocity space. Since the VPs integrals are normalized with the surface density the integral of a VP over the whole velocity space should be 1 within the expected error. \nFigure A2. This shows our best DF for η = 0 . 5 (green line). Fluctuations (red lines) appear for large energies because we used a constant h for equation (A3). To resolve this we used a more general function h = h ( E ) or h = h ( E,L z ) even closer to the center. \n<!-- image -->", 'APPENDIX B: VELOCITY HISTOGRAMS FOR THE 2-I MODEL': "<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFigure B1. VHs and VPs in the l and b directions predicted by the 2I model in angular bins. The reduced χ 2 is also provided. The size of the bins is 0.6mas/yr ( ∼ 23 . 6 km/s) for the upper two plots and 0.5mas/yr ( ∼ 19 . 6 km/s) for the rest of the diagrams. The right column shows which cells have been used for the VHs and VPs. \n<!-- image --> \nFigure B2. VHs and VPs in the l and b directions predicted by the 2I model in radial bins. The reduced χ 2 is also provided. The size of the bins is 0.5mas/yr ( ∼ 19 . 6 km/s) for the 1st and 4th column and 0.6mas/yr ( ∼ 23 . 6 km/s) for the rest of the diagrams. The right column shows which cells have been used for the VHs and VPs. \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFigure B3. VHs for the symmetrized los data compared with the corresponding even VPs of the model. The reduced χ 2 is also provided. The size of the bins is 40km/s. For the upper left we use stars with 20 '' < | l | < 30 '' and | b | < 20 '' , for the upper right stars with 30 '' < | l | < 40 '' and | b | < 20 '' , for the bottom left 40 '' < | l | < 50 '' and | b | < 20 '' , and for the bottom right 50 '' < | l | < 70 '' and | b | < 20 '' . \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFigure B4. Los VHs compared with the corresponding VPs of the model including rotation. The reduced χ 2 is also provided. The size of the bins is 40km/s. For the upper left we use stars with 20 '' < | l | < 30 '' and | b | < 20 '' , for the upper right stars with 30 '' < | l | < 40 '' and | b | < 20 '' , for the bottom left 40 '' < | l | < 50 '' and | b | < 20 '' , and for the bottom right 60 '' < | l | < 80 '' and | b | < 20 '' . \n<!-- image -->"}

2012PhRvD..86b4028B

Instabilities of wormholes and regular black holes supported by a phantom scalar field

2012-01-01

['-', '-', '-', '-', '-', '-', '-']

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We test the stability of various wormholes and black holes supported by a scalar field with a negative kinetic term. The general axial perturbations and the monopole type of polar perturbations are considered in the linear approximation. Two classes of objects are considered: (i) wormholes with flat asymptotic behavior at one end and anti-de Sitter on the other (Minkowski-anti-de Sitter wormholes) and (ii) regular black holes with asymptotically de Sitter expansion far beyond the horizon (the so-called black universes). A difficulty in such stability studies is that the effective potential for perturbations forms an infinite wall at throats, if any. Its regularization is in general possible only by numerical methods, and such a method is suggested in a general form and used in the present paper. As a result, we have shown that all configurations under study are unstable under spherically symmetric perturbations, except for a special class of black universes where the event horizon coincides with the minimum of the area function. For this stable family, the frequencies of quasinormal modes of axial perturbations are calculated.

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https://arxiv.org/pdf/1205.2224.pdf

{'K. A. Bronnikov ∗': 'Center for Gravitation and Fundamental Metrology, VNIIMS, Ozyornaya 46, Moscow 119361, Russia; Institute of Gravitation and Cosmology, PFUR, ul. Miklukho-Maklaya 6, Moscow 117198, Russia', 'R. A. Konoplya †': "DAMTP, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK; Centro de Estudios Cient'ıficos (CECS), Casilla 1469, Valdivia, Chile.", 'A. Zhidenko ‡': "Centro de Matem'atica, Computa¸c˜ao e Cogni¸c˜ao, Universidade Federal do ABC (UFABC), Rua Santa Ad'elia, 166, 09210-170, Santo Andr'e, SP, Brazil \nWe test the stability of various wormholes and black holes supported by a scalar field with a negative kinetic term. The general axial perturbations and the monopole type of polar perturbations are considered in the linear approximation. Two classes of objects are considered: (i) wormholes with flat asymptotic behavior at one end and AdS on the other (M-AdS wormholes) and (ii) regular black holes with asymptotically de Sitter expansion far beyond the horizon (the so-called black universes). A difficulty in such stability studies is that the effective potential for perturbations forms an infinite wall at throats, if any. Its regularization is in general possible only by numerical methods, and such a method is suggested in a general form and used in the present paper. As a result, we have shown that all configurations under study are unstable under spherically symmetric perturbations, except for a special class of black universes where the event horizon coincides with the minimum of the area function. For this stable family, the frequencies of quasinormal modes of axial perturbations are calculated. \nPACS numbers: 04.30.Nk, 04.50.+h", 'I. INTRODUCTION': "Modern observations [1] indicate that the Universe is expanding with acceleration. The most favored explanation of this acceleration is nowadays that the Universe is dominated (to about 70 %) by some unknown form of energy density with large negative pressure, termed dark energy (DE), while the remaining 30 % consisting of baryonic and nonbaryonic visible and dark matter. It is often admitted that DE can be modeled by a self-interacting scalar field with a potential. Such a field acts as a negative pressure source; it is called quintessence if its pressure to density ratio p/ρ = w > -1 and a phantom field if w < -1 while w = -1 corresponds to a cosmological constant Λ. Since observations admit a range of w including w = -1, all sorts of models are under study. \nOne should note that values w < -1 seem to be not only admissible but even preferable for describing an increasing acceleration, as follows from the most recent estimates: w = -1 . 10 ± 0 . 14 (1 σ ) [2] (according to the 7-year WMAP data) and w = -1 . 069 +0 . 091 -0 . 092 [3] (mainly from data on type Ia supernovae from the SNLS3 sample). In this connection, cosmological models with phan- \nm scalar fields, i.e., those with a negative kinetic term, have gained considerable attention in the recent years [4]. \nIf such fields can play an important role in cosmology, it is natural to expect that they manifest themselves in local phenomena, for instance, in the existence and properties of black holes and wormholes [5]. Quite a number of scalar field configurations of this kind have been described in the literature, see, e.g., examples of black holes with scalar fields (the so-called scalar hair) in [6] and wormholes supported by scalar fields in [7-9] and references therein. Thus, in [9] it was shown that, in addition to wormholes, a phantom scalar can support a regular black hole where a possible explorer, after crossing the event horizon, gets into an expanding universe instead of a singularity. Thus such hypothetic configurations combine the properties of a wormhole (absence of a center, a regular minimum of the area function) and a black hole (a Killing horizon separating R and T regions). Moreover, the Kantowski-Sachs cosmology that occurs in the T region is asymptotically isotropic and approaches a de Sitter regime of expansion, which makes such models potentially viable as models of our accelerating Universe. Such configurations, termed black universes, were later shown to exist with other scalar field sources exhibiting a phantom behavior such as k-essence [10] and some brane world models [11] (in the latter case, even without a phantom field). \nBoth wormholes and black universes have been shown to exist as well in models where a scalar field exhibits \nphantom properties only in a strong-field region while in the weak-field region it obeys the canonical field equation (the so-called trapped-ghost models ) [12, 13] \nTo see whether or not such solutions can lead to viable models of black holes and wormholes, one needs to check their stability against various perturbations. Previously, gravitational stability as well as passage of radiation through wormholes supported by a phantom scalar field were considered, in particular, in [14-18], with a special emphasis on massless wormholes (see also references therein). \nIn the present paper, we consider the linear stability of various static, spherically symmetric solutions to the field equations of general relativity with minimally coupled scalar fields, describing compact objects of interest such as asymptotically flat and AdS wormholes (M-AdS wormholes, for short, where M stands for 'Minkowski') and black universes (in other words, M-dS regular black holes), and use as examples solutions obtained in [9]. We show that M-AdS wormholes are unstable in the whole range of the parameters while among black universes there is a stable subfamily which corresponds to the event horizon located precisely at the minimum of the area function. \nThe particular solutions whose stability is studied here are certainly not general. A more comprehensive study is prevented by the fact that a sufficiently general solution describing self-gravitating scalar fields with nonzero potentials is unknown, therefore it seems to be a natural decision to study the properties of known special solutions. On the other hand, in [9] all possible regular static, spherically symmetric solutions to the field equations were classified for phantom minimally coupled scalar fields with arbitrary potentials. One can see that the solution studied here, being certainly special, still represents a very simple but quite typical example reproducing the generic features of such configurations with flat, dS and AdS asymptotics; its another advantage is that it reproduces as a cpecial case the well-known Ellis wormhole, for which the stability results are already known [14-16]. \nThe paper is organized as follows. Sec. II presents the backgrounds to be considered. Sec. III develops a general formalism for axial gravitational and Maxwell field perturbations in a static, spherically symmetric background. Sec. IV is devoted to polar spherically symmetric perturbations. Sec. V discusses the stability of the black universes and wormholes under consideration and analyzes the quasinormal radiation spectrum for the cases where the background configuration is linearly stable. In addition, we there develop a numerical tool for reducing the wavelike equation with a singular potential to the one with a regular potential. In Sec. VI we summarize the results and mention some open problems.", 'II. STATIC BACKGROUND CONFIGURATIONS': "Let us consider Lagrangians of the form \nL = √ -g ( R + /epsilon1g αβ φ ; α φ ; β -2 V ( φ ) -F µν F µν ) , (1) \nwhich includes a scalar field, in general, with some potential V ( φ ), and an electromagnetic field F µν ; /epsilon1 = ± 1 distinguishes normal, canonical scalar fields ( /epsilon1 = +1) and phantom fields ( /epsilon1 = -1). In what follows, we present a perturbation analysis for static, spherically symmetric solutions for this general type of Lagrangian and then study the stability of some particular (electrically neutral) solutions. \nThe general static, spherically symmetric metric can be written in the form \nds 2 = A ( r ) dt 2 -A ( r ) -1 dr 2 -R ( r ) 2 d Ω 2 , (2) \nwhere d Ω 2 = dθ 2 + sin 2 θdϕ 2 is the linear element on a unit sphere. \nWe will consider the following static background [9]: \nR ( r ) = ( r 2 + b 2 ) 1 / 2 , b = const > 0 , A ( r ) = ( r 2 + b 2 ) × [ c b 2 + 1 b 2 + r 2 + 3 m b 3 ( br b 2 + r 2 +tan -1 r b )] , φ = √ 2tan -1 ( r/b ) . (3) \nIt is a solution to the Einstein-scalar equations that follow from (1) with F µν ≡ 0 and the potential \nV ( φ ) = -c b 2 [ 3 -2 cos 2 ( φ √ 2 )] -3 m b 3 [ 3 sin φ √ 2 cos φ √ 2 + φ √ 2 ( 3 -2 cos 2 φ √ 2 )] . (4) \nThe solution behavior is controlled by the scale b and two integration constants: c that moves the curve B ( r ) ≡ A/R 2 up and down, and m showing the position of the maximum of B ( r ). Both R ( r ) and B ( r ) are even functions if m = 0, otherwise B ( r ) loses this symmetry. Asymptotic flatness at r = + ∞ implies \n2 bc = -3 πm, (5) \nwhere m is the Schwarzschild mass defined in the usual way. \nUnder this asymptotic flatness assumption, for m = c = 0, we obtain the simplest symmetric configuration, the Ellis wormhole [7]: A ≡ 1, V ≡ 0. With m < 0, we obtain a wormhole with an AdS metric at the far end, corresponding to the cosmological constant V ( φ ) ∣ ∣ r →-∞ = V -< 0. Further on, such configurations will be referred to as M-AdS wormholes (where M stands for Minkowski, the flat asymptotic). Assuming m > 0, at large negative r we obtain negative A ( r ), such that \n| A ( r ) | ∼ R 2 ( r ), and a potential tending to V -> 0. Thus it is a regular black hole with a de Sitter asymptotic behavior far beyond the horizon, precisely corresponding to the above description of a black universe. \nIn black-universe solutions, the horizon radius depends on both parameters m and b (min R ( r ) = b ), which also plays the role of a scalar charge since φ/ √ 2 ≈ π/ 2 -b/r at large r . Since A (0) = 1 + c , the minimum of R ( r ), located at r = 0, occurs in the R region if c > -1, i.e., if 3 πm < 2 b (it is then a throat, like that in wormholes), right at the horizon if c = -1 (i.e., 3 πm = 2 b ) and in the T region beyond it if c < -1, that is, 3 πm > 2 b . It is then not a throat, since r is a time coordinate, but a bounce in one of the scale factors R ( r ) of the Kantowski-Sachs cosmology; the other scale factor is A ( r ). \nLet us mention that another important case of the system (1), the one with V ≡ 0, has already been studied in a number of papers. In this case we are dealing with Fisher's famous solution ([19], 1948) for a canonical massless scalar in general relativity ( /epsilon1 = +1) and three branches of its counterpart for /epsilon1 = -1, sometimes called the anti-Fisher solution, found for the first time by Bergmann and Leipnik [20] in 1957 and repeatedly rediscovered afterwards (as well as Fisher's solution). In the latter case the solution consists of three branches, one representing wormholes [7, 8], the other two also containing throats but with singularities or horizons of infinite area at the far end instead of another spatial infinity [8, 21]. An instability of Fisher's solution under spherically symmetric perturbations was established long ago [22], a similar instability of the wormhole branch was discovered in [14, 16] and the same for the other two branches in [23]. The case of zero mass in the anti-Fisher solution represents the Ellis wormhole and is common with the solution under study, (3), m = 0. \nIn what follows, we will assume m /negationslash = 0 and use the mass m (which has the dimension of length and is equal to half the Schwarzschild radius in the units employed) as a natural length scale, putting, for convenience, | 3 m | = 1. Then the constant c is used as a family parameter while b is found from the relation (5). \nWe should remark that in all particular examples to be tested here for stability the background electromagnetic field is zero, but the general perturbation formalism is developed in Sec III for axial perturbations of systems with non-zero F µν as well. Actually in Sec IV non-zero F µν is also allowed, simply because the monopole perturbations do not excite an electromagnetic field in a spherically symmetric background. Perturbations of the electromagnetic field appear in higher multipoles of polar modes. The general formalism developed for non-zero F µν is intended to be used in further studies of systems with both scalar and electromagnetic fields.", 'III. LINEAR AXIAL PERTURBATIONS: GENERAL ANALYSIS': "In our analysis of axial perturbations, we use Chandrasekhar's notations: x 0 = t , x 1 = φ , x 2 = r , x 3 = θ , so that the coordinates along which the background has Killing vectors are enumerated first. Following Chandrasekhar [24], we consider the metric (2) as a special case of the metric \nds 2 = e 2 ν dt 2 -e 2 ψ ( dφ -σdt -q 2 dr -q 3 dθ ) 2 -e 2 µ 2 dr 2 -e 2 µ 3 dθ 2 . (6) \nThus in (2) \ne 2 ν = A ( r ) , e 2 µ 2 = A -1 ( r ) , e 2 µ 3 = R ( r ) 2 , e 2 ψ = R ( r ) 2 sin 2 θ. (7) \nThe background electromagnetic field is taken in the form \nF 02 = -Q ∗ /R ( r ) 2 , (8) \nthat is, only a radial electric field, and Q ∗ is the (effective) charge. \nAll calculations and results can be easily rewritten for magnetic fields, with nonzero F 13 , owing to the Maxwell field duality. One can bear in mind that configurations like wormholes and black universes can possess electric or magnetic fields without any real electric charges or magnetic monopoles, due to their geometry, actually realizing Wheeler's concept of a 'charge without charge'. \nIt is easy to see that axial perturbations of a scalar field vanish. Then, σ , q 2 and q 3 are perturbed, while ψ , µ 2 , µ 3 and ν remain unperturbed. \nThe axial gravitational perturbations obey the equations \nδR 13 = 0 , δR 12 = 2 Q ∗ R -2 F 01 (9) \nLet us introduce new variables: \nQ ik = q i,k -q k,i , Q i 0 = q i, 0 -σ ,i , (10) \nwith i, k = 2 , 3, and \nE ≡ F 01 sin θ. (11) \nRecall that we use the numbers 0 , 1 , 2 , 3 for t , φ , r and θ coordinates, respectively. The Maxwell equations, subject to only first-order perturbations, have the form \n( e ψ + µ 2 F 12 ) , 3 +( e ψ + µ 3 F 31 ) , 2 = 0 , (12) \n( e ψ + ν F 01 ) , 2 +( e ψ + µ 2 F 12 ) , 0 = 0 , (13) \n( e ψ + ν F 01 ) , 3 +( e ψ + µ 3 F 13 ) , 0 = 0 , (14) \n( e µ 2 + µ 3 F 01 ) , 0 +( e ν + µ 3 F 12 ) , 2 +( e ν + µ 2 F 13 ) , 3 = e ψ + µ 3 F 02 Q 02 . (15) \nAfter some algebra the Maxwell equations can be written in the form \nRe -ν E , 0 , 0 -( e 2 ν ( Re ν E ) ,r ) ,r + e ν R sin θ ( E ,θ 1 sin θ ) ,θ = -Q ∗ ( σ , 2 , 0 -q 2 , 0 , 0 ) sin 2 θ (16) \nFrom δR 13 = 0 and δR 12 = 2 Q ∗ R -2 F 01 it follows \n( e 3 ψ + ν -µ 2 -µ 3 Q 23 ) , 2 -( e 3 ψ -ν + µ 2 -µ 3 Q 03 ) , 0 = 0 (17) \nand \n( e 3 ψ + ν -µ 2 -µ 3 Q 23 ) , 3 -( e 3 ψ -ν -µ 2 + µ 3 Q 02 ) , 0 = e 2 ψ + ν + µ 3 Q ∗ R -2 F 01 . (18) \nAfter introducing the new function \nQ ≡ R 2 AQ 23 sin 3 θ, (19) \nEqs. (17) and (18) can be written in the form \nA R 2 sin 3 θ ∂Q ∂r = σ , 3 , 0 -q 3 , 0 , 0 , (20) \nA R 4 sin 3 θ ∂Q ∂θ = -σ , 2 , 0 + q 2 , 0 , 0 + 4 Q ∗ e ν E R 2 sin 2 θ . (21) \nLet us differentiate Eq. (20) in r and Eq.(21) in θ and then add the results. After some algebra we have \nR 4 ∂ ∂r ( A R 2 ∂Q ∂r ) +sin 3 θ ∂ ∂θ ( 1 sin 3 θ ∂Q ∂θ ) -QR 2 e -2 ν = 4 Q ∗ e ν R sin 3 θ ∂ ∂θ ( E sin 2 θ ) . (22) \nNow let us return to the Maxwell field perturbation equation (16). Using (21), we can get rid of the term containing σ , 2 , 0 -q 2 , 0 , 0 in (10). After some algebra and using the relations E ∼ e iωt , Q ∼ e iωt , we obtain \n( e 2 ν ( Re ν E ) ,r ) ,r + E ( ω 2 Re -ν -4 Q 2 ∗ e ν R -3 ) -e ν R -1 sin θ ( E ,θ 1 sin θ ) ,θ = -Q ∗ R 4 sin θ ∂Q ∂θ . (23) \nThe angular variable can be separated by the following ansatz: \nQ ( r, θ ) = Q ( r ) C -3 / 2 /lscript +2 ( θ ) , (24) \nE ( r, θ ) = E ( r ) sin θ dC -3 / 2 /lscript +2 ( θ ) dθ = 3 E ( r ) C -1 / 2 /lscript +1 ( θ ) , (25) \nwhere C b a are Gegenbauer polynomials. Eqs. (22) and (23) then read \n∆ d dr ( ∆ R 4 dQ dr ) -µ 2 ∆ R 4 Q + ω 2 Q = -4 Q ∗ µ 2 ∆ e ν E R 3 , (26) ( e 2 ν ( Re ν E ) ,r ) ,r -( µ 2 +2) e ν R -1 E + E ( ω 2 Re -ν -4 Q 2 ∗ e ν R -3 ) = -Q ∗ QR -4 , (27) \nwhere ∆ = R 2 e 2 ν , µ 2 = ( /lscript -1)( /lscript + 2), and we use the tortoise coordinate r ∗ defined by d/dr ∗ = ∆ R -2 d/dr . \nAfter passing over from Q and E to the new functions H 1 and H 2 using the relations \nQ = RH 2 , Re ν E = -H 1 2 µ , (28) \nEqs. (26) and (27) can be reduced to \nΛ 2 H 2 = ( -R ,r ∗ r ∗ R + 2 R 2 ,r ∗ R 2 + µ 2 ∆ R 4 ) H 2 + 2 Q ∗ µ ∆ R 5 H 1 , (29) \nΛ 2 H 1 = ∆ R 4 ( ( µ 2 +2) + 4 Q 2 ∗ R 2 ) H 1 + 2 Q ∗ µ ∆ R 5 H 2 , (30) \nwhere we have introduced the operator \nΛ 2 = d 2 dr 2 ∗ + ω 2 . (31) \nFor an electrically neutral background Q ∗ = 0, H 1 = 0, and Eq.(29) reduces to the Schrodinger-like equation \nd 2 H 2 dr 2 ∗ +( ω 2 -V eff ( r )) H 2 = 0 (32) \nwith the effective potential \nV eff ( r ) = A ( r ) ( /lscript +2)( /lscript -1) R 2 + R ( R -1 ) ,r ∗ ,r ∗ , (33) \nAs a partial verification of the above relations, we can check that Eqs. (29) and (30) reproduce some known special cases. Thus, Eq.(33), obtained here in the Chandrasekhar approach [24], coincides with Eq. (4.13) of [25], obtained in the Regge-Wheeler approach. In addition, Eqs. (29), (30) coincide with Eqs. (144), (145), p. 230 of [24] in the Reissner-Nordstrom limit, and, as a result, the potential (33) coincides with the well-known ReggeWheeler potential for Schwarzschild black holes.", 'IV. POLAR PERTURBATIONS': "For polar perturbations let us consider the Einsteinscalar equations following from (1) with F µν = 0 \nR µν + /epsilon1∂ µ Φ ∂ ν Φ -δ µν V ( φ ) = 0 , /epsilon1 /square Φ+ V φ = 0 , V φ ≡ dV/dφ, (34) \nwhere /square = ∇ α ∇ α is the d'Alembert operator. In the polar case we can put σ = q 2 = q 3 = 0, while δν , δµ 2 , δµ 3 , and δψ do not vanish. In addition, we perturb the scalar field, \nφ ( r, t ) = φ ( r ) + δφ ( r, t ) . \nLet us restrict ourselves to the lowest frequency modes, corresponding to spherically symmetric (or radial) perturbations, /lscript = 0. This gives δψ = δµ 3 . Then we can use the gauge freedom and fix 1 \nδψ = δµ 3 ≡ 0 = ⇒ δR ( r, t ) = 0 . (35) \nTedious calculations allow us to reduce the perturbation equations to a single wave equation \ne -2 µ 2 -2 ν δ ¨ φ -δφ '' -δφ ' ( ν ' -2 µ ' 3 + µ ' 2 ) + Uδφ = 0 , (36) \nwhere \nU ≡ e -2 µ 2 ( /epsilon1 ( e 2 µ 3 -V ) ( φ ' ) 2 ( µ ' 3 ) 2 -2 φ ' µ ' 3 V ,φ + /epsilon1V ,φφ ) . (37) \nIntroducing the new function Ψ by putting \nδφ = Ψ e µ 3 + iωt , (38) \nwe bring the wave equation to the Schrodinger-like form \nd 2 Ψ dr 2 ∗ +( ω 2 -V eff ( r ∗ ))Ψ = 0 , (39) \nwith the effective potential \nV eff = U + 1 R d 2 R dr 2 ∗ . (40) \nU A = /epsilon1φ ' 2 R ' 2 ( R 2 V -1) + 2 φ ' RV ,φ R ' + /epsilon1V ,φ,φ . (41)", 'A. Methods': "Finite potentials. If the effective potential V eff is finite and positive-definite, the differential operator \nW = -d 2 dr 2 ∗ + V eff (42) \nis a positive self-adjoint operator in L 2 ( r ∗ , dr ∗ ), the space of functions satisfying proper boundary conditions. Then W has no negative eigenvalues, in other words, there are no normalizable solutions to the corresponding Schrodinger equation with well-behaved initial data (smooth data on a compact support) that grow with time, and the system under study is then stable under this particular form \nof perturbations. Therefore, in our stability studies, our main concern will be regions where the effective potentials are negative since possible instabilities are indicated by such regions. \nThe response of a stable black hole or wormhole to external perturbations is dominated at late times by a set of damped oscillations, called quasinormal modes (QNMs). Quasinormal frequencies do not depend on the way of their excitation but are completely determined by the parameters of the configuration itself. Thus quasinormal modes form a characteristic spectrum of proper oscillations of a black hole or a wormhole and could be called their 'fingerprints'. Apart from black hole physics, QNMs are studied in such areas as gauge/gravity correspondence, gravitational wave astronomy [26, 27] and cosmology [28]. \nThe quasinormal boundary conditions for black hole perturbations imply pure incoming waves at the horizon and pure outgoing waves at spatial infinity. For asymptotically flat solutions, the quasinormal boundary conditions are \nΨ ∝ e ± iωr ∗ , r ∗ →±∞ . (43) \nThe proper oscillation frequencies correspond in a sense to a 'momentary perturbation', that is, to the situation where one looks at a response of the system (say, a wormhole) to initial perturbation when the source of the perturbation stopped to act. This is the essence of the word 'proper'. Thus, in the wormhole case, no incoming waves are allowed coming from either of the infinities. The issue of the boundary conditions for quasinormal modes of wormholes is not completely new and was considered in [29]. \nTherefore, for wormholes the condition 'pure incoming waves at the horizon' is replaced with 'pure outgoing waves at the other spatial infinity' [29, 30]. For asymptotically anti-de Sitter black holes or wormholes, the AdS boundary creates an effective confining box [26], so that on the AdS boundary one usually requires the Dirichlet boundary conditions \nΨ → 0 , r ∗ →∞ . (44) \nThis choice is not only dictated by the asymptotic of the wave equation at infinity but is also consistent with the limit of purely AdS space-time: the QNMs of an AdS black hole approach the normal modes of empty AdS space-time in the limit of a vanishing black hole radius [31]. Thus quasinormal modes of a compact object (a black hole or wormhole) in AdS space-time look like normal modes of empty AdS space-time 'perturbed' by the compact object. \nIf the effective potential is negative in some region, growing quasinormal modes can appear in the spectrum, indicating an instability of the system under such perturbations. It turns out that some potentials with a negative gap still do not imply instability. If there are no growing quasinormal modes in the black-hole or wormhole spectrum, this object is stable against linear perturbations. \nSingular potentials and their regularization. It can be seen from (40) (provided that V R 2 < 1) that the effective potential V eff for radial perturbations forms an infinite wall located at a throat, with the generic behavior V eff ∼ ( r -r throat ) -2 , since we have there R ' ( r ) ∼ r -r throat . As a result, perturbations are independent at different sides of the throat, necessarily turn to zero at the throat itself, and we thus lose part of the information on their possible properties. To remove the divergence, one can use a method on the basis of S-deformations as described in [32, 33] for solutions with arbitrary potentials V ( φ ). For anti-Fisher wormholes ( V ( φ ) 0) it was applied in [14]. \nIf the scalar potential V ( φ ) is zero, the effective potential (40) takes the form \n≡ \nV eff ( r ) = A R 2 -A 2 R ' 2 R 2 -/epsilon1Aφ ' 2 R ' 2 . (45) \nIn this case the general static solution ( ω = 0) to Eq. (39) is \nΨ 0 ( r ) = C 1 R ( 1 -/epsilon1ARφ ' 2 A ' R ' ) + C 2 R [ φ A ' -/epsilon1ARφ ' 3 A ' 2 R ' ( φ φ ' -2 A A ' )] , (46) \nand its special cases with C 1 = 0 and C 2 = 0 were used in [14, 23] to remove the divergence in V eff . Indeed, we can introduce the new wave function \nΨ = S Ψ -d Ψ dr ∗ , where S = 1 Ψ 0 d Ψ 0 dr ∗ , (47) \nwhich satisfies the equation \nd 2 Ψ dr 2 ∗ + ( ω 2 -V eff ( r ∗ ) ) Ψ = 0 (48) \nwith the effective potential \nV eff = 2 S 2 -V eff , (49) \nand the new effective potential is everywhere regular if either C 1 = 0 or C 2 = 0. This transformation was used to prove the instability of anti-Fisher wormholes [14] and other anti-Fisher solutions [23], \nIt turns out that C 1 and C 2 can be both nonzero, leading to regularized effective potentials with the same quasinormal spectrum, though not preserving the symmetry φ ↔-φ . This means that we can fix the boundary conditions arbitrarily and, if we find a 'good' static solution, we can use it to remove the singularity of the effective potential. \nFor nonzero scalar potentials V ( φ ), analytical expressions for static perturbations are unknown. Therefore, such suitable solutions must be found numerically under proper boundary conditions. \nFirst we notice that since Ψ 0 is a static solution to Eq.(39), S satisfies the Riccati equation \ndS dr ∗ + S 2 -V eff = 0 . (50) \nSubstituting (40) into (50), we find an expansion for S near the throat as [23] \nS ( r ) = -1 + c r -4 c 2 3 mπ 2 + 4 c 2 (4 c 2 +(1 + c ) π 2 ) 9 m 2 (1 + c ) π 4 r + Kr 2 + . . . , (51) \nwhere K is an arbitrary constant. \nWith this expression we find the boundary condition for the function S ( r ) at some points close to the throat on both sides. Then we integrate (50) numerically and find S ( r ) between the throat and both asymptotical regions. Having S ( r ) at hand, we find the regularized effective potential (49); it is finite at the throat due to (51). \nWe assigned different values to the free constant K . As a rule, we were able to integrate Eq. (50) numerically in a sufficiently wide range of r near the throat. The regularized effective potentials obtained in this way always lead to the same growth rate of the perturbation function. However, for some particular values of K , the numerical integration with the Wolfram Mathematica R © built-in function encounters a growing numerical error. Apparently, in these cases some alternative methods of integration should be used. \nIn addition, we used the same method to find numerically the regularized potential for the Branch B antiFisher solution whose stability properties had been studied previously [23]. Although the numerically found potential differs from Eq. (68) of [23], the time-domain profile again shows the same growth rate. This confirms the correctness of the method suggested and used here.", 'B. M-AdS wormholes with negative mass': 'Polar perturbations. Now we are in a position to apply the above method to the special case of M-AdS wormholes (3) with m < 0, which are asymptotically AdS as r → -∞ . In Fig. 1 one can see that although the initial (singular) effective potentials V eff are positive definite, the regularized potentials V eff are negative in some range around the throat and at the AdS boundary. This behavior usually indicates an instability but does not guarantee it [34]. Therefore, to prove the instability of M-AdS wormholes, we have used the time-domain integration method proposed by Gundlach et al. [35] and later used by a number of authors (see, e.g., [36]). The method shows a convergence of the time-domain profile with diminishing the integration grid and increasing the precision of all computations. We imposed the Dirichlet boundary conditions at the AdS boundary, as described in [37]. Fig. 2 shows examples of time-domain profiles for the evolution of perturbations. The growth of the signal allows us to conclude that such wormholes are unstable against radial perturbations. At larger c we found the regularized potential with deeper negative wells and observed a quicker growth of the signal. Therefore we conclude that all such M-AdS wormholes are unstable. \nFIG. 1: Effective potentials for spherically symmetric perturbations of M-AdS wormholes with m = -1 / 3, c = 0 . 3 (red), c = 0 . 5 (green), c = 0 . 8 (blue) with divergences at the throat (left figure) and the corresponding regular ones found numerically (right figure). Larger values of c correspond to larger absolute values of V eff at the AdS boundary (they are positive for divergent potentials and negative for regular ones). \n<!-- image --> \nFIG. 2: Time-domain evolution of spherically symmetric perturbations of M-AdS wormholes with m = -1 / 3, c = 0 . 5 (green, bottom), and c = 0 . 8 (red, top). \n<!-- image --> \nAxial perturbations. The effective potentials V eff for axial perturbations of M-AdS wormholes V eff are plotted in Fig. 3. One can see that above some threshold value of c , V eff ( r ) has a negative gap. This threshold value of c is c ≈ 1 . 737 in units for which m = -1 / 3. A further increase of c makes the negative gap deeper, however, time-domain profiles for the evolution of axial perturbations show a decay of the signal without any indication of instability (see Fig. 4).', 'C. Black universes': "Polar perturbations. Black universes correspond to the metric (2), (3) with m > 0, c < 0. Black universes with c ≤ -1, or equivalently 3 πm ≥ 2 b , have no throat \nin the static region. From Fig.5 one can see that for c = -1 the effective potential has a negative gap, however, time-domain integration proves that in this case the black universe is stable. For smaller values of c , an additional negative gap appears between the peak and the horizon, leading to an instability even for c = -1 . 001. For large negative c the potential peak disappears, and the potential becomes negative everywhere (see Fig. 6), which inevitably creates an instability. \nAt c > -1 the throat r = 0 is in the static region. In this case we numerically find the regularized effective potentials which have large negative gaps leading to instabilities (see Figs. 7, 8). As c approaches zero, the growth rate of time-domain profile decreases, still remaining positive because the effective potential remains negative in a wide region near the throat (Fig. 7). \nAs | c | grows the negative gap becomes deeper and narrower, giving way to a small positive hill, which becomes broader (see Fig. 8) as c approaches -1. However, we do not observe a decrease in the growth rate as expected when approaching the parametric region of stability near c = -1. This can be an indication that the only parameter for which a black universe can be stable is c = -1. \nAxial perturbations and quasinormal oscillation frequencies. From Fig. 9 we can see that in the static region the effective potential is positive-definite. Therefore, if we perturb a black hole 'on the right' of the event horizon (Fig. 9), such perturbations are stable. Beyond the event horizon, in the cosmological region, the effective potential can take large negative values, but this has no effect on perturbations propagating outside the horizon. We therefore conclude that the black universes under consideration are stable against axial perturbations in the static region. \nAs we have shown, black universes at c = -1 ( b = 3 πm/ 2) are stable against polar monopole perturbations, \nFIG. 3: Effective potential as a function of the radial coordinate r (left figure) and of the tortoise coordinate r ∗ (right figure) for axial gravitational perturbations of M-AdS wormholes with m = -1 / 3 for /lscript = 2, c = 0 . 8 (no negative gap), 1 . 0, 1 . 8 (a negative gap), 2 . 2 (a deep negative gap). \n<!-- image --> \nFIG. 4: Time-domain profiles of the evolution of axial ( /lscript = 2) perturbations of M-AdS wormholes ( m = -1 / 3) with c = 1 (left) and c = 5 (right). \n<!-- image --> \nFIG. 5: Left panel: Effective potentials for radial perturbations of a black universe with /lscript = 0, m = 1 / 3, c = -1 (red, top), c = -1 . 001 (green), c = -1 . 01 (blue, bottom). The potentials vanish at the horizon. Right panel: time-domain perturbation evolution for c = -1 (red, stable), and c = -1 . 001 (green, unstable). \n<!-- image --> \nFIG. 6: Effective potential for radial perturbations of a black universe with /lscript = 0, m = 1 / 3, c = -1 . 5 (red, top), c = -2 (green), c = -3 (blue, bottom). The potentials vanish at the horizon. \n<!-- image --> \nTABLE I: Quasinormal modes of axial perturbations of black universes with c = -1 ( m = 1 / 3). \n| /lscript | WKB6 time-domain fit |\n|--------------|-----------------------------------------------------|\n| 2 | 0 . 712299 - 0 . 157848 i - |\n| 3 | 1 . 168844 - 0 . 176095 i 1 . 169144 0 . 176019 i |\n| 5 | 1 . 997792 - 0 . 185591 i 1 . 997801 - 0 . 185588 i |\n| /greatermuch | . 191226(2 /lscript +1 - i ) |\n| | 1 0 | \nand their response to external perturbations is dominated at late times by the quasinormal (QN) frequencies. Supposing Ψ ∝ e -iωt , quasinormal modes can be written in the form \nω = ω Re -iω Im , \nwhere a positive ω Im is proportional to the decay rate of a damped QN mode. The low-lying axial QN frequencies have the smallest decay rates in the spectrum and thus dominate in a signal at sufficiently late times. They can be calculated with the help of the WKB approach [38, 39]. \nIntroducing Q = ω 2 -V eff , the 6-th order WKB formula reads \niQ 0 2 Q '' 0 -6 ∑ k =2 Λ k = n + 1 2 , n = 0 , 1 , 2 . . . , (52) \nwhere the correction terms Λ k were obtained in [38, 39]. Here Q 0 and Q k 0 are the value and the k -th derivative of Q at its maximum with respect to the tortoise coordinate r ∗ , and n labels the overtones. The WKB formula (52) was effectively used in a lot of papers (see, e.g., [40] and references therein). \n√ \nThe WKB formula gives an accurate result for large multipole numbers (see Table I). Expanding the WKB formula it powers of /lscript , we find the following asymptotical expression for c = -1: \nωb = √ 2 π tan -1 ( 2 π )( /lscript + 1 2 -i ( n + 1 2 ) ) + O ( 1 /lscript ) . \nIn Table I, the asymptotic formula for the fundamental mode (the one that dominates at late times) is presented in units m = 1 / 3, so that b = π/ 2. \nThe WKB formula (52) used here is developed for effective potentials which have the form of a barrier with only one peak (see, e.g., Fig. 9 for black universes), so that there are two turning points given by the equation ω 2 -V eff = 0. Therefore, it cannot be used for effective potentials which have negative gaps, that is, for testing the stability of all questionable cases.", 'VI. CONCLUSIONS': 'We have developed a general formalism for analyzing axial gravitational perturbations of an arbitrary static, spherically symmetric solution to the Einstein-Maxwellscalar equations where the scalar field, which can be both normal and phantom, is minimally coupled to gravity and possesses an arbitrary potential. This can be used for studying the stability and QNM modes of diverse charged and neutral scalar field configurations in general relativity. As to polar perturbations, we have restricted ourselves to the monopole mode, i.e., to spherically symmetric (radial, for short) perturbations. \nWe have applied this formalism to some electrically neutral wormholes and black holes supported by a phantom scalar field. The main results which were obtained here are: \n- 1. M-AdS wormholes described by the solution (3) with negative mass are shown to be unstable under radial perturbations, although the initial effective potential with a singularity at the throat is positive everywhere. These features are similar to those of anti-Fisher wormholes [14] which are twice asymptotically flat.\n- 2. Black universes (i.e., regular black holes with de Sitter expansion far beyond the horizon), described by the solution (3) without throats in the static region, are shown to be stable only in the special case where the horizon coincides with the minimum of the area function R ( r ) (the parameters: c = -1 , b = 3 πm/ 2) and unstable for c = -1. \n/negationslash \n- 3. Quasinormal modes of axial perturbations have been calculated for stable black universes. \nQuite a lot of other problems of interest are yet to be studied. One can mention a full nonlinear analysis of perturbations in all relevant cases. Next, the formalism developed here for linear perturbations allows us to include the electromagnetic field into consideration and to study charged solutions with both normal and phantom scalar fields. Last but not least, using the well-known conformal mappings that relate Einstein and Jordan frames of scalar-tensor and curvature-nonlinear theories of gravity, one can extend the studies to solutions of these theories. \nFIG. 7: Regularized effective potentials (left panel, top to bottom) and time-domain profiles (right panel, bottom to top) for radial perturbations of a black universe with /lscript = 0, m = 1 / 3, c = -0 . 1 (red), -0 . 2 (green), -0 . 3 (blue). \n<!-- image --> \nFIG. 8: Regularized effective potentials (left panel) and time-domain profiles (right panel) for radial perturbations of a black universe with /lscript = 0, m = 1 / 3, c = -0 . 90 (blue), -0 . 95 (green), -0 . 99 (red). Larger negative values of c correspond to deeper negative gaps and later growing phase of the signal. \n<!-- image --> \nFIG. 9: The effective potential (left figure) and the time-domain profile (right figure) for axial gravitational perturbations of black universes for /lscript = 2, m = 1 / 3, c = -1. The potential vanishes at the horizon r = 0. \n<!-- image -->', 'Acknowledgments': "The work of K.B. was supported in part by RFBR grant 09-02-00677a, by NPK MU grant at PFUR, and by FTsP 'Nauchnye i nauchno-pedagogicheskie kadry innovatsionnoy Rossii' for the years 2009-2013. R. K. acknowledges the Centro de Estudios Cientifcos (Valdivia, Chile) where part of this work was done. At its initial stage the work was partially funded by the Conicyt grant ACT-91: 'Southern Theoretical Physics Lab- \n- [1] N. A. Bachall, J. P. Ostriker, S. Perlmutter and P. J. Steinhardt, Science 284 1481 (1999); S. J. Perlmutter et al, Astrophys. J. 517 565 (1999); V. Sahni and A. A. Starobinsky, Int. J. 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2009PhRvD..80l4010S

Simulations of binary black hole mergers using spectral methods

2009-01-01

['-', '-', '-', '-', 'methods numerical', 'methods numerical', 'methods numerical', 'techniques spectroscopic', 'methods numerical', '-']

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Several improvements in numerical methods and gauge choice are presented that make it possible now to perform simulations of the merger and ringdown phases of “generic” binary black hole evolutions using the pseudospectral evolution code SpEC. These improvements include the use of a new damped-wave gauge condition, a new grid structure with appropriate filtering that improves stability, and better adaptivity in conforming the grid structures to the shapes and sizes of the black holes. Simulations illustrating the success of these new methods are presented for a variety of binary black hole systems. These include fairly generic systems with unequal masses (up to 2∶1 mass ratios), and spins (with magnitudes up to 0.4M<SUP>2</SUP>) pointing in various directions.

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https://arxiv.org/pdf/0909.3557.pdf

{'Simulations of Binary Black Hole Mergers Using Spectral Methods': "B'ela Szil'agyi, Lee Lindblom, and Mark A. Scheel Theoretical Astrophysics 350-17, California Institute of Technology, Pasadena, CA 91125 (Dated: November 26, 2024) \nSeveral improvements in numerical methods and gauge choice are presented that make it possible now to perform simulations of the merger and ringdown phases of 'generic' binary black-hole evolutions using the pseudo-spectral evolution code SpEC. These improvements include the use of a new damped-wave gauge condition, a new grid structure with appropriate filtering that improves stability, and better adaptivity in conforming the grid structures to the shapes and sizes of the black holes. Simulations illustrating the success of these new methods are presented for a variety of binary black-hole systems. These include fairly 'generic' systems with unequal masses (up to 2:1 mass ratios), and spins (with magnitudes up to 0 . 4 M 2 ) pointing in various directions.", 'I. INTRODUCTION': "Black-hole science took a great stride forward in 2005 when Pretorius [1] performed the first successful full nonlinear dynamical numerical simulation of the inspiral, merger, and ringdown of an orbiting black-hole binary system; this initial success then stimulated other groups to match this achievement within months [2, 3]. These developments lead quickly to advances in our understanding of black-hole physics: investigations of the orbital mechanics of spinning binaries [4, 5, 6, 7, 8, 9], studies of the recoil from the merger of unequal mass binary systems [10, 11, 12, 13], the remarkable discovery of unexpectedly large recoil velocities from the merger of certain spinning binary systems [7, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], investigations into the mapping between the binary black-hole initial conditions (individual masses and spins) and the final state of the merged black hole [27, 28, 29, 30, 31, 32, 33, 34], and improvements in our understanding of the validity of approximate binary black-hole orbital calculations using postNewtonian methods [35, 36, 37, 38, 39, 40, 41, 42]. \nThese first results on binary black-hole systems were obtained by several different groups using different codes based on two different formulations of the Einstein equations (BSSN and generalized harmonic), using two different methods for treating the black-hole interiors (moving puncture and excision), and using rather different gauge conditions to fix the spacetime coordinates. All of these early results, however, were obtained with codes based on finite difference numerical methods and adaptive mesh refinement for computational efficiency. The Caltech/Cornell collaboration decided a number of years ago to follow a different path by developing an Einstein evolution code, called SpEC, based on spectral methods. The advantages of spectral methods are their superior accuracy and computational efficiency, and their extremely low numerical dissipation that is ideal for solving wave propagation problems with high precision. The disadvantages are the relative complexity of spectral codes, the lack of any appropriate pre-existing spectral code infrastructure (analagous to CACTUS [43] for example), and the extreme sensitivity of spectral algorithms to develop- \ning instabilities when any aspect of the solution method is mathematicaly ill-posed. Consequently it has taken our group somewhat longer to bring SpEC up to the level of the state-of-the-art codes in the field. \nWe, along with our Caltech/Cornell collaborators, have developed a large number of numerical and analytical tools over the years that make it possible for SpEC to evolve binary black-hole systems with greater precision than any other code (at the present time) [44]. These technical developments include the derivation and implementation of constraint preserving and physical (no incoming gravitational wave) boundary conditions [45, 46], dual frame evolution methods and feedback and control systems to lock the computational grids onto the location of the black holes [47], and special angular filtering methods needed to cure an instability that occurs when tensor fields are evolved [48]. Using these methods we and our collaborators have performed a number of high precision evolutions of the inspiral portions of binary black-hole systems, and have used the gravitational waveforms from these evolutions to calibrate the accuracy of the approximate post-Newtonian waveforms that are widely used in gravitational wave data analysis [49, 50, 51, 52]. \nDuring the past year our group has begun to have some success in performing the more dynamical and difficult (for spectral codes) merger and ringdown portions of binary black-hole evolutions [50, 52, 53, 54, 55, 56]. The techniques we used to achieve these first merger calculations turned out to be rather non-robust however, requiring a great deal of hands-on adjustment and fine tuning for each new case we attempted. In this paper we report on several new numerical and analytical developments that allow us now to perform stable and accurate binary merger and ringdown simulations of a fairly wide range of binary black-hole systems. These new methods appear to be quite robust, and no fine tuning is required: the same basic method works on each of the cases we have attempted. \nThe three new technical breakthroughs that allow us now to perform successful binary black-hole merger and ringdown simulations are described in Sec. II. These major advances include the development of a new gauge condition that works extremely well with the generalized harmonic formulation of the Einstein system used in our \ncode. This new gauge chooses spatial coordinates that are solutions of a damped wave equation, and chooses the time coordinate in a way that limits the growth of √ g/N , here g is the determinant of the spatial metric and N is the lapse function. This new gauge condition is described in some detail in Sec. II A. Another important development, reported in Sec. II B, is the construction of a new grid structure on which our binary black-hole evolutions are performed. This new non-overlapping grid structure, and a certain type of spectral filtering used with the new grid structure, removes a class of numerical instabilities that were a limiting factor in our ability to perform the highly dynamical portions of merger calculations, and allows us to increase resolution near the black holes in a more targeted and computationally efficient way. The third technical development is a new more efficient and robust method to conform the structure of the grid to the shape and size of the black holes in an automatic dynamical way. These developments are described in Sec. II C and the Appendix . \nUsing these new technical tools we have now performed successful merger and the succeeding ringdown simulations of a fairly wide range of binary black-hole systems. We describe in Sec. III merger simulations for six reasonably diverse cases. These include two systems in which the black holes are non-spinning: one has equal mass black holes, the other has holes with a 2:1 mass ratio. We also describe successful merger and ringdown calculations with two different equal-mass binary systems in which the holes have identical intrinsic spins of magnitude 0 . 4 M 2 : aligned to the orbital angular momentum in one system, and anti-aligned in the other. We have also performed successful merger and ringdown simulations on two fairly generic binary systems. These two systems have black holes with the same mass ratio M 2 /M 1 = 2, and the same 'randomly oriented' initial spins of magnitudes 0 . 2 M 2 1 and 0 . 4 M 2 2 respectively. But the initial separations of the holes are different in the two cases: one starts about 8.5 orbits before merger, and another (used to perform careful convergence studies) starts about 1.5 orbits before merger. The same methods were used to perform all of these merger and ringdown simulations, and no particular fine tuning was required.", 'II. TECHNICAL DEVELOPMENTS': 'This section describes the three new technical breakthroughs that allow us now to perform successful binary black-hole merger and ringdown simulations. These major advances include the development of a new gauge condition, described in Sec. II A, in which the spatial coordinates satisfy a damped wave equation and the time coordinate is chosen in a way that controls the growth of the spatial volume element. Another important development, described in Sec. II B, is a new non-overlapping grid structure for our binary black-hole simulations, and a type of spectral filtering for these new grid structures \nthat improves their accuracy and stability. The third new technical development, described in Sec. II C, is a more efficient and robust method of conforming the structure of the grid to the shape and size of the black holes in a dynamical and automatic way.', 'A. Damped-Wave Gauge': "Harmonic gauge is defined by the condition that each coordinate x a satisfies the co-variant scalar wave equation: \n∇ c ∇ c x a = H a = 0 . (1) \nHarmonic coordinates have proven to be extremely useful for analytical studies of the Einstein equations, but have found only limited success in numerical problems like simulations of complicated highly dynamical blackhole mergers. One reason for some of these difficulties is the wealth of 'interesting' dynamical solutions to the harmonic gauge condition itself, Eq. (1). Since all 'physical' dynamical fields are expressed in terms of the coordinates, an ideal gauge condition would limit coordinates to those that are simple, straightforward, dependable, and non-singular; having 'interesting' dynamics of their own is not a desirable feature for coordinates. The dynamical range available to harmonic coordinates can be reduced by adding a damping term to the equation: [57, 58] \n∇ c ∇ c x a = µ S t c ∂ c x a = µ S t a , (2) \nwhere t a is the future directed unit normal to the constantt hypersurfaces. Adding such a damping term to the equations for the spatial coordinates x i tends to remove extraneous gauge dynamics and drives the coordinates toward solutions of the co-variant spatial Laplace equation on the timescale 1 /µ . Choosing 1 /µ to be comparable to (or smaller than) the characteristic timescale of a particular problem should remove any extraneous coordinate dynamics on timescales shorter than the physical timescale. The addition of such a damping term in the time-coordinate equation is not appropriate however. Such a damped-wave time coordinate is driven toward a constant value, and therefore toward a state in which it fails to be a useful time coordinate at all. It makes sense then to use the damped-wave gauge condition only for the spatial coordinates: \n∇ c ∇ c x i = H i = µ S t i = -µ S N i /N, (3) \nwhere N i is the shift, and N is the lapse. The appropriate contra-variant version of this damped-wave gauge condition is therefore \nH a = -µ S g ai N i /N, (4) \nwhere g ab is the spatial metric of the constantt hypersurfaces. (Note that we use Latin letters from the beginning of the alphabet, a, b, c, . . . , to denote four-dimensional \nspacetime indices, and letters from the middle of the alphabet, i, j, k, . . . , for three-dimensional spatial indices.) \nWhile the damped-wave gauge is a poor choice for the time coordinate, the idea of imposing a gauge that adds dissipation to the gauge dynamics of the time coordinate is attractive. To find the appropriate expression for t a H a , the component of H a not fixed by Eq. (4), we note that the gauge constraint H a + Γ a = 0 implies that t a H a is given by \nt a H a = t a ∂ a log ( √ g N ) -N -1 ∂ k N k , (5) \nwhere g = det g ij is the spatial volume element. In our experience, a frequent symptom of the failure of simpler gauge conditions in binary black-hole simulations is an explosive growth of g in the spacetime region near the black-hole horizons. This suggests that a good use of the remaining gauge freedom would be to attempt to control the growth of the spatial volume element, g . Choosing the gauge condition, \nt a H a = -µ L log ( √ g N ) , (6) \ntogether with Eq. (5) implies the following evolution equation for √ g/N : \nt a ∂ a log ( √ g N ) + µ L log ( √ g N ) = N -1 ∂ k N k , (7) \nwhose solutions tend to suppress growth in √ g/N . The discussion of this gauge condition in Ref. [57] shows that it also implies that the lapse N satisfys a damped wave equation, with the damping factor µ L . So in this sense, the gauge condition on the time coordinate, Eq. (6), is the natural extension of the spatial-coordinate dampedwave gauge condition, Eq. (4). \nCombining this new lapse condition, Eq. (6), with the damped-wave spatial coordinate condition, Eq. (4), gives the gauge-source function for our full damped-wave gauge condition: \nH a = µ L log ( √ g N ) t a -µ S N -1 g ai N i . (8) \nThe damping factors µ L ≥ 0 and µ S ≥ 0 can be chosen quite arbitrarily as functions of spacetime coordinates x a , or even as functions of the spacetime metric ψ ab . The gauge source function H a depends only on coordinates and the spacetime metric ψ ab in this case, so these gauge conditions can be implemented directly in the generalized harmonic Einstein system without the need for a gauge driver. Previous studies of this condition (developed initially as a test of the first-order gauge-driver system [57]) showed it to be quite useful for evolving single black-hole spacetimes. In those tests, which included several different evolutions of maximal-slice Schwarzschild initial data with large non-spherical gauge perturbations, the black \nhole always evolved quickly toward a non-singular timeindependent equilibrium state. \nThe solutions to the lapse gauge condition, Eq. (7), can be thought of as equating log( √ g/N ) to a certain weighted time average of ∂ k N k /N . The timescale associated with this time averaging is set by µ L , which determines for example the rate at which √ g/N is driven toward an asymptotic equilibrium state. When µ L is constant, it is easy to show that √ g/N is driven exponentially toward this asymptotic state. In highly dynamical spacetimes, however, we find that √ g/N must be driven even faster than exponential in order to prevent the formation of singularities in g . This can be accomplished by making µ L larger whenever g becomes large. In practice we find that the choice µ L = µ 0 [log( √ g/N )] 2 , (where µ 0 is a constant or perhaps a function of time) is very effective at suppressing the growth of these singularities. We find that choosing the same damping factor µ S , \nµ S = µ L = µ 0 [ log ( √ g N )] 2 , (9) \nfor the spatial part of the gauge condition is also quite effective. The binary black-hole merger and ringdown simulations described in Sec. III use these µ L and µ S , with µ 0 taken to be an order-unity function of time (to accomodate starting up evolutions from initial data satisfing a different gauge condition).", 'B. Grid Structure and Filtering': "The pseudo-spectral numerical methods used by our code, the Spectral Einstein Code (SpEC) [45, 47, 59], represent dynamical fields on a spatial grid structure that is specially constructed for each problem. For binary blackhole simulations, our group has been using grid structures constructed from layers of spherical-shell subdomains centered on each black hole, surrounded and connected by cylindrical, cylindrical-shell, and/or rectangular-block subdomains, all surrounded by spherical-shell subdomains that extend to the outer boundary of the computational domain (located far from the holes). The intermediate-zone cylindrical-shell and/or rectangularblock subdomains must overlap both the inner and outer spherical-shell subdomains in these grid structures to cover the computational domain completely. These grid structures are quite efficient, and have allowed our group to perform long stable inspiral calculations for a variety of simple binary black-hole systems [47, 49, 52, 60], and also simulations of the merger and ringdown phases of a few of these simple cases [53, 56]. The overlap regions in these grid structures are well behaved in these successful cases. But in more generic inspiral and merger simulations, these overlap regions become significant sources of numerical error and instability. \nThe overlap-related instability discussed above is most likely caused by the method we use to exchange information between adjacent subdomains. We set the incoming \nFIG. 1: Illustrates a basic element of the grid structure used to fill the 3D volume between a sphere and a concentric cube. \n<!-- image --> \ncharacteristic fields, whose values are determined by interpolation from the adjacent subdomain, using a penalty method [61, 62, 63, 64]. This penalty method for imposing boundary conditions was derived explicitly for the case where adjacent subdomain boundaries touch but do not overlap. One possibility for resolving this instability would be to re-derive the appropriate penalty boundary terms for the case of overlapping subdomains. We have chosen the simpler option of constructing new grid structures without overlapping subdomains. We do this by changing the intermediate-zone grid structure from the cylindrical-shells and/or rectangular blocks used previously into a series of deformed rectangular blocks whose boundaries conform to the spherical shells used near the inner and outer boundaries. The new type of grid element needed for this construction is illustrated in Fig. 1, which shows how the 3D volume between a sphere and a concentric cube can be mapped into six deformed cubes. This construction is based on the 'cubed-sphere' coordinate representations of the sphere (i.e. the mapping of the six faces of the cube onto the sphere as illustrated in Fig. 1), that is widely used in atmospheric and geophysical modeling [65, 66, 67, 68]. Three-dimensional grid structures based on cubed spheres have also been used in other types of simulations, including a few black-hole simulations [69, 70]. \nOur new grid structure is illustrated in Figs. 2 and 3. It consists of a series of spherical-shell subdomains and several layers of cubed-sphere subdomains surrounding each black hole, as illustrated in Fig. 2. The two cubic blocks containing the black holes are surrounded by a series of rectangular-block subdomains to fill out the volume of a large cube. The center of this large cube is placed at the center-of-mass of the binary system, as illustrated in Fig. 3 for a system having a 2:1 mass ratio. This large cube is connected to the outer spherical-shell \nFIG. 2: Grid structure around each black hole. A series of spherical shells (two shells shown) is surrounded by a series of cubed-sphere layers (three shown) to fill up a cube. This figure illustrates one quarter of the cube structure surrounding one of the black holes. \n<!-- image --> \nFIG. 3: Illustrates the intermediate grid structure surrounding the two black holes, consisting of a set of rectangular blocks placed around the cubic block containing each hole. This collection of blocks filles up a cube which is surrounded by a series of cubed-sphere layers (two shown) all surrounded by a series of spherical shells (one shell shown) that are centered on the center of mass of the binary system. \n<!-- image --> \nsubdomains by several additional layers of cubed-sphere subdomains, also illustrated in Fig. 3. \nPrevious attempts to use this type of cubed-sphere grid structure in SpEC have always proven unsuccessful, due to numerical instabilities at the interdomain boundaries. These previous attempts used no spectral filtering, or a two-thirds anti-aliasing filter, in the cubed-sphere subdomains. It has been shown, however, that an appropriate filter is required for stability (and improved accuracy) of Chebyshev polynomial spectral expansions, such as those used in these cubed-sphere subdomains. Such a filter must satisfy two important conditions. The first criterion is that when represented in physical space, this filter must act like a dissipation term in the evolution equations that vanishes on the subdomain boundaries (to avoid conflicting with the boundary conditions). The second criterion is that it must set the highest spectral coefficient to zero. Adding this type of spectral filtering turns out to be one \nof the key elements in improving our numerical method enough to make possible the binary black-hole simulations described in Sec. III. The needed spectral filter is applied to each field u ( x, t ) that is expanded as a sum of Chebyshev polynomials: \nu ( x, t ) = ∑ k u k ( t ) T k ( x ) . (10) \nThese are used for the radial-coordinate expansions in the spherical-shell subdomains, and for all three spectralcoordinate expansions in the rectangular-block and the cubed-sphere subdomains. Each spectral coefficient u k in these expansions is filtered according to the expression, \nF ( u k ) = u k e -α ( k/k max ) 2 p , (11) \nafter each time step. This filter satisfies the first necessary filter criterion by adding a 2 p th -order dissipation term to the evolution equations which vanishes on each subdomain boundary [64]. For the binary black-hole simulations presented here, we use the filter parameters α = 36 and p = 32. The choice α = 36 guarantees that this filter satisfies the second necessary filter criterion by ensuring that the highest spectral coefficient is set to (double precision) zero, while the choice p = 32 insures that this is a very mild filter that only influences the largest few spectral coefficients. This new filter is applied wherever Chebyshev expansions, Eq. (10), are used. We use the same filter we have used in previous binary black-hole simulations in the angular directions in the spherical-shell subdomains: we set to zero the top four /lscript -coefficients in the tensor spherical-harmonic represention of each dynamical field [47, 48].", 'C. Adaptive Conforming Grid': "One of the important developments that allowed our group (several years ago) to begin performing successful binary black-hole inspiral simulations was the introduction of the dual-coordinate-frame evolution method [47]. This method uses two distinct coordinate frames: One is a non-rotating and asymptotically Cartesian coordinate system used to construct the tensor basis for the components of the various dynamical fields evolved by our code. The second is a coordinate system chosen to follow (approximately) the motions of the black holes. We fix the computational grid to this second coordinate frame, and solve the evolution equations for the 'inertial-frame' tensor field components as functions of these 'grid-frame' coordinates. The map M connecting these grid frame coordinates ¯ x i to the inertial frame x i , can be written as the composition of more elementary maps: \nM = M K · M S . (12) \nM K represents a 'kinematical' map that keeps the centers of the black holes located (approximately) at the \ncenters of the excised holes in our grid structures (cf. Sec. II B), and M S is a 'shape-control' map that makes the grid conform (approximately) to the shapes of the black holes. The third major technical development (which makes it possible now for us to perform robust merger simulations) consists of improvements in the choice of the shape-control map, M S , and improvements in the way this map is adapted to the dynamically changing shapes of the black holes. \nThe kinematical map M K can itself be decomposed into more elementary maps: \nM K = M T · M E · M R . (13) \nThe maps M T , M E , and M R each move the centers of the black holes, but do not significantly distort their shapes: The map M T translates the grid to account for the motion of the center of the system due to linear momentum being exchanged with the near field [54] and being emitted in gravitational radiation. The map M E does a conformal rescaling that keeps the coordinate distance between the centers of the two black holes fixed in the grid frame, as they inspiral in the inertial frame. And M R rotates the frame so that the centers of the two black holes remain on the ¯ x axis in the grid frame, as they move along their orbits in the inertial frame. These maps are also used during inspiral simulations in the same way we use them for our merger simulations. So here we focus on the new shape-control map M S , which is one of the critical new developments that made the merger simulations reported in Sec. III possible. \nThe interiors of the black holes are excised in our simulations at the spherical boundaries shown in Figs. 2 and 3. This is possible because, if this excision boundary is chosen wisely , the spacetime inside this boundary can not influence the spacetime region covered by our computational grid. For hyperbolic evolution systems, like the generalized harmonic form of the Einstein equations used in our code [45], boundary conditions must be placed on each incoming characteristic field at each boundary point. Apparent horizons are surfaces that are often used to study black holes, because they can be found numerically in a fairly straightforward way, and because (if they exist at all) they are always located within the true event horizons. If the excision bounaries for our computational domain were placed exactly on the apparent horizons, then all the characteristic fields of the Einstein system would be outflowing (with respect to the computational domain) since apparent horizons are spacelike (or null) hypersurfaces. Boundary conditions would not be needed on any field at such boundaries. Unfortunately we are not able to place excision boundaries precisely on the apparent horizons. So the best that can be done numerically is to place them slightly inside the apparent horizons (i.e., the apparent horizons must remain within the computational domain), if we are to avoid the need for boundary conditions on these excision surfaces. \nHowever, if an excision boundary is placed inside an apparent horizon, the outflow condition is no longer auto- \nmatic or simple: the condition depends on the shape and the location of the excision boundary, its motion with respect to the horizon, and the gauge. In our simulations, the excision boundaries are kept somewhat inside the apparent horizons, so the outflow condition can-and does-fail if we are not careful. One reason (and probably the principal reason) we need to control the shapes of the computational domains through the map M S is to keep pure outflow conditions on all the dynamical fields at the excision boundaries. We do this, as described in more detail below, by requiring that the excision boundaries closely track the shapes and sizes of the horizons. \nAnother (probably secondary) reason that shape and size control are needed in our numerical simulations is related to finite numerical resolution. If the excision boundaries had different shapes than the horizons, then some points on the boundaries would be located deeper into the black-hole inerior and hence closer to the spacetime singularity. Higher numerical resolution would be needed at these points, and for fixed resolution, the amount of constraint violation and errors in the solution would be largest there. In many situations we find that numerical instabilities at these points cause our simulations to fail. This mode of failure can be eliminated by keeping the shapes and sizes of the boundaries close to those of the horizons. \nWe have decomposed the map M , which connects the grid-frame coordinates ¯ x i with inertial-frame coordinates x i , into kinematical and shape-control parts: M = M K · M S . It will be convenient to have a name for the intermediate coordinate system whose existence is implied by this split. The map M K connects the inertialframe coordinates x i with coordinates ˜ x i in which the centers of the black holes are at rest (approximately). So it is natural to call these intermediate-frame coordinates, ˜ x i , the 'rest frame' coordinate system. The shape control map M S connects the grid-frame coordinates ¯ x i with these rest-frame coordinates ˜ x i . \nIt is useful to express the shape-control map M S as the composition of maps acting on each black hole individually: M S = M S 1 · M S 2 . Such individual black-hole shape-control maps can be written quite generally in the form \n˜ θ A = ¯ θ A , (14) \n˜ φ A = ¯ φ A , (15) \n˜ r A = ¯ r A -f A (¯ r A , ¯ θ A , ¯ φ A ) × /lscript max ∑ /lscript =0 /lscript ∑ m = -/lscript λ /lscriptm A ( t ) Y /lscriptm ( ¯ θ A , ¯ φ A ) , (16) \nwhere (¯ r A , ¯ θ A , ¯ φ A ) and (˜ r A , ˜ θ A , ˜ φ A ) are, respectively, the grid-frame and rest-frame spherical polar coordinates centered at the (fixed) grid-coordinate location of black hole A = { 1 , 2 } . The functions f A (¯ r A , ¯ θ A , ¯ φ A ) are fixed functions of space in the grid frame. We note that these maps become the identity whenever f A (¯ r A , ¯ θ A , ¯ φ A ) = 0, so making f A (¯ r A , ¯ θ A , ¯ φ A ) vanish as ¯ r A →∞ ensures that \nthe distortion is limited to the neighborhood of each black hole. The parameters λ /lscriptm A ( t ) specify the angular structure of the distortion map. They are determined by a feedback control system (discussed in Appendix ) that dynamically adjusts the shape and overall size of the grid frame coordinates relative to the rest-frame coordinates. \nThe Caltech/Cornell collaboration has used shapecontrol maps of this form, Eqs. (14)-(16), in all of our recent binary black-hole simulations [53, 54, 56]. In those cases the functions f A (¯ r A , ¯ θ A , ¯ φ A ) were taken to be smooth functions of ¯ r A alone: roughly constant near each black hole and falling off to zero rapidly enough that M S 1 and M S 2 approximately commute. These functions had large gradients which made it difficult to control the grid distortion near the horizons without introducing additional unwanted distortions elsewhere. As a result, we were never able to achieve very robust shape control using these maps. \nOne of our major breakthroughs, leading to the successful merger results reported here, is an improvement in our choice of the map functions f A (¯ r A , ¯ θ A , ¯ φ A ). We now use simpler functions f A (¯ r A , ¯ θ A , ¯ φ A ) that have much smaller gradients and that are exactly zero outside (nonintersecting) compact regions surrounding each black hole. This means the new maps M S 1 and M S 2 commute, exactly. These new maps can be defined everywhere between the black holes because of the new non-overlapping grid structure discussed in Sec. II B, but these improvements are achieved at the expense of smoothness: the new f A (¯ r A , ¯ θ A , ¯ φ A ) are smooth except at subdomain boundaries where they may only be continuous, not differentiable 1 . Fortunately, this lack of smoothness at subdomain boundaries is not a problem for our basic evolution method. In our multidomain code the equations are solved independently on each subdomain, and subdomains communicate only by equating the appropriate characteristic fields at their mutual boundaries. These boundary conditions require no differentiation, so the grid coordinates themselves need not be smooth across these boundaries. \nThe new f A (¯ r A , ¯ θ A , ¯ φ A ) are chosen to be unity inside a sphere of radius ¯ b A centered on black hole A = { 1 , 2 } , then decrease linearly with radius along a ray through the center of the black hole, and finally vanish on the surface of a centered cube of size 2¯ a A . A two-dimensional sketch of this grid-frame domain is shown in Fig. 4, which is merely an abstract version of the type of grid structures we use around each black hole as illustrated in Figs. 1 and 2. This function f A (¯ r A , ¯ θ A , ¯ φ A ) can be expressed \nFIG. 4: Two-dimensional illustration of the grid-frame coordinate domain on which the function f A (¯ r A , ¯ θ A , ¯ φ A ) is defined in Eq. (17). \n<!-- image --> \nanalytically as \nf A (¯ r A , ¯ θ A , ¯ φ A ) = 1 , ¯ r A ≤ ¯ b A , ¯ r A -¯ ρ A ¯ b A -¯ ρ A , ¯ b A ≤ ¯ r A ≤ ρ A , 0 , ¯ ρ A ≤ ¯ r A , (17) \nwhere \n¯ ρ A ( ¯ θ A , ¯ φ A ) = ¯ a A (18) × max ( | sin ¯ θ A cos ¯ φ A | , | sin ¯ θ A sin ¯ φ A | , | cos ¯ θ A | -1 \n[ \n)] \nWe note that the choice of f A (¯ r A , ¯ θ A , ¯ φ A ) in Eq. (17) implies that the shape-control map M S , and hence the map between grid-frame and rest-frame coordinates is not smooth. Since the kinematical map M K is smooth, this also imples that the full map M relating grid-frame to inertial-frame coordinates is not smooth either. This non-smoothness does not cause a problem for our basic evolution method, but it implies that calculations that require smoothness must not be performed in the grid frame. For example, the representation of a smooth apparent horizon in grid coordinates will not be smooth. So apparent horizon finding and other calculations that require smoothness are done in the smooth rest-frame or inertial-frame coordinates. \n( is the value of ¯ r A on the surface of the cube of size 2¯ a A centered on black hole A = { 1 , 2 } . The different cases of the maxima which appear in the denominator of Eq. (18) correspond to the different faces of this cube. \nThe angular structures of the shape-control maps defined in Eq. (14)-(16) are determined by the parameters λ /lscriptm A ( t ). These are chosen dynamically to ensure that the shapes of the excision boundaries match the evolving shapes of the apparent horizons. These maps also control the sizes of the excision boundaries by adjusting λ 00 A , which are chosn to ensure that the excision boundaries remain close to but safely inside the apparent horizons. Figure 5 illustrates the effect of this distortion map on the structure of the grid surrounding a black hole in one of \nFIG. 5: Illustrates the inertial frame representation of the grid structure around one black hole, just at the time of merger. This grid structure has been distorted in relation to the inertial frame by the shape-control maps M S described here. \n<!-- image --> \nthe generic merger simulations described in Sec. III. The choices of suitable target shapes and sizes for these shapecontrol maps, and the feedback control system that we use to implement these choices dynamically in our simulations, are also among the critical technical advances that allow us now to perform robust merger and ringdown simulations. The details of exactly how this is done are somewhat complicated, however, so we defer their discussion to the Appendix .", 'III. MERGER AND RINGDOWN SIMULATIONS': 'In this section we present simulations of the merger and ringdown of binary black-hole systems using our Spectral Einstein Code (SpEC). In Section III A we describe the binary black-hole initial data sets that we evolve. In Section III B we discuss our evolution algorithm, pointing out at which stage we employ the various improvements described in Section II and what goes wrong when these improvements are not used. Finally, in Section III C we describe six different binary black-hole merger and ringdown simulations peformed using these new methods. We present snapshots of the shapes and locations of the horizons at various times during the merger and ringdown, and a demonstration of convergence of the constraint violations.', 'A. Initial Data': 'Our initial data are in quasi-equilibrium [71, 72, 73] (see also [74, 75]), and are built using the conformal thin sandwich formalism [76, 77] with the simplifying \nTABLE I: Black hole binary configurations run through merger and ringdown using the methods presented here. For the initial spin parameters /vector S 1 /M 2 1 and /vector S 2 /M 2 2 , the ˆ z direction is parallel to the orbital angular momentum. \n| Case | M 2 /M 1 | /vector S 1 /M 2 1 | /vector S 2 /M 2 2 | N orbits |\n|--------|------------|-----------------------------------|-----------------------------|------------|\n| A | 1 | 0 | 0 | 16 |\n| B | 2 | 0 | 0 | 15 |\n| C | 1 | - 0 . 4ˆ z | - 0 . 4ˆ z | 11 |\n| D | 1 | 0 . 4ˆ z | 0 . 4ˆ z | 15 |\n| E | 2 | 0 . 2(ˆ z - ˆ x ) / √ 2 0 . 4(ˆ z | - + ˆ y ) / √ 2 | 8.5 |\n| F | 2 | 0 . 2(ˆ z - ˆ x ) / 2 | √ - 0 . 4(ˆ z + ˆ y ) / √ 2 | 1.5 | \nchoices of conformal flatness and maximal slicing. Quasiequilibrium boundary conditions are imposed on spherical excision boundaries for each black hole, with the lapse boundary condition given by Eq. (33a) of Ref. [73]. The spins of the black holes are determined by boundary conditions on the shift vector at each excision surface [72]. \nThis formalism for constructing initial data also requires the initial radial velocity v r of each black hole toward its partner and an initial orbital angular frequency Ω 0 (chosen to be about the z axis without loss of generality). These parameters determine the orbital eccentricity of the binary, and can be tuned by an iterative method [56, 60] to produce data with very low eccentricity. This has been done for most of the initial data sets described here (cases A through D in Table I), but is unnecessary for the purposes of the present paper, since our goal here is simply to document our improved methods for evolving binaries through merger. For cases E and F, v r and Ω 0 are chosen roughly by using post-Newtonian formulae that ignore spins. \nTable I shows the mass ratio and initial spins for the black hole binary configurations we evolve here. The initial data and first 15 orbits of inspiral for case A are identical to those presented in [49, 53]. The initial data and first 9 orbits of inspiral for case C are identical to those presented in [56]. Cases E and F are fully generic, with unequal masses and unaligned spins, and exhibit precession and radiation-reaction recoil.', 'B. Evolution procedure': "In this section we summarize our procedure for evolving black hole binary systems through merger and ringdown, concentrating on the improvements discussed in Section II. This procedure can be divided into several stages, beginning with the early inspiral and extending through ringdown: \n- 1. Evolve early inspiral using 'quasi-equilibrium' gauge.\n- 2. Transition smoothly to damped harmonic gauge.\n- 3. Eliminate overlapping subdomains. \n- 4. Turn on shape control.\n- 5. Replace remaining inner spherical shells with cubed spheres.\n- 6. Turn on size control just before holes merge.\n- 7. After merger, interpolate to a single-hole grid to run through ringdown. \nThe above ordering of these stages is not the only possible choice: we have exchanged the order of some of these stages without trouble. For example, for the runs described here we perform stage 3 at the instant we start stage 2. The transitions between these stages are relatively simple, require little or no fine tuning, and can be automated. We now discuss each of these stages in turn. For several of these stages, we will illustrate the effects of new improvements (Section II) for one particular evolution, case F of Table I.", '1. Early inspiral': "The gauge is chosen early in the evolution following the procedure of Ref. [49]: the initial data are constructed in quasi-equilibrium, so we choose the initial H a to make the time derivatives of the lapse and shift zero. We attempt to maintain this quasi-equilibrium condition by demanding that H a remains constant in time in the grid frame in the following sense: we require \n∂ ¯ t ˜ H ¯ a = 0 , (19) \nwhere ˜ H a is a tensor (note that H a is not a tensor) defined so that ˜ H a = H a in the inertial frame. The bars in Eq. (19) refer to the grid frame coordinates. We refer to this as 'quasi-equilibrium' gauge, although strictly speaking this gauge maintains quasi-equilibrium only when the spin directions are constant in the grid frame. \nFor cases A-D, the simulations follow many orbits using this quasi-equilibrium gauge. For cases E and F, in which the spins change direction in the grid frame, this gauge is not appropriate and causes difficulties, so for these cases we transition to harmonic gauge H a = 0 very early in the inspiral. \nDuring inspiral, the map M E uniformly contracts the grid so that the grid-coordinate distance between the centers of the two black holes remains constant. Because this contraction is uniform, the spherical-shell subdomains inside each apparent horizon shrink relative to the horizon. This means that the excision boundary (the inner boundary of the innermost spherical shell) inside each black hole moves further into the strong-field region in the interior, towards the singularity. If this motion continues unchecked, gradients on the innermost spherical shell increase until the solution is no longer resolved. However, the inner boundary of the next-to-innermost spherical shell is also shrinking, and it eventually shrinks \nFIG. 6: The minimum of the outgoing (into the hole) characteristic speeds on the nine innermost spherical grid boundaries (labeled A 0 . . . A 8 ) around the larger black hole, for part of run F. \n<!-- image --> \nenough that the characteristic fields on this boundary all become outflowing (into the black hole); when this happens, the innermost spherical shell no longer influences the exterior solution, so we simply drop that shell from the domain. This shell-dropping process occurs automatically as successive spherical shells shrink relative to the horizons, This process is illustrated in Figure 6, which shows the minimum of the (into the hole) characteristic speeds on spherical grid boundaries around one of the black holes. At any given time, the innermost of these boundaries is an excision boundary. All characteristic speeds at all points on the excision boundary (and hence the minimum) must be positive for the excision algorithm to be well-posed. The innermost shell is dropped whenever the characteristic speeds are positive on the inner boundary of the next shell. For example, surface A 3 in Figure 6 is the excision boundary at t = 24 M , but after the characteristic speeds on A 4 become positive (around t = 26 M ), the innermost shell is dropped and A 4 becomes the excision boundary. Similarly, A 5 becomes the excision boundary at t = 38, and so on. It is important that the characteristic speed on surface A n in Figure 6 does not become negative before the speed on surface A n +1 becomes positive; otherwise this shelldropping procedure will fail. Shell dropping works well during most of the evolution but it fails in the last stage before merger, which is discussed in Section III B 6. \nWhen describing mergers, it is useful to introduce a measure of distance between the black holes. One such measure of distance is indicated on the horizontal axis at the top of Figure 6. This is the spatial (in each time slice) proper separation between the apparent horizon surfaces, obtained by integrating along the rest-frame ˜ x axis (recall \nthat in the rest frame, the centers of the horizons always remain along this axis). This is not the true proper separation between the horizons because we do not minimize over all possible paths between all possible pairs of points on the two surfaces; however, we use this only as an approximate measure of how far the black holes are from merger. Note that the horizons touch each other when (or possibly before) this proper separation measure falls to zero. \nIt turns out that the improvements discussed in Section II are unnecessary during early inspiral for cases A-D and are not yet used at this stage. For instance, the domain decomposition uses overlapping cylinders and spherical shells, and not the cubed-sphere subdomains described in Sec. II B. We find no problems while the holes are far enough apart that they remain roughly in equilibrium. However, the binary eventually becomes extremely dynamical, and the black holes become significantly distorted. Unless the algorithm is modified, the simulation fails (typically because of large constraint violations) before the black holes merge. \nTypically, it is possible to extend the quasi-equilibrium inspiral stage until the proper separation decreases to about 7 M or 8 M , but the next stages, Secs. III B 3 and III B 2, can be done sooner if desired. (For example, cases E and F use a non-overlapping cubed-sphere grid from t = 0.) For the simulations presented here, the length of this quasi-equilibrium inspiral stage is variable. For instance, for case A in Table I this stage lasts for 15 orbits, but for case F we transition to damped harmonic gauge and eliminate overlapping subdomains starting at t = 0 because the black holes are initially very close together.", '2. Transition to damped harmonic gauge': "When the inspirling black holes reach a proper separation of about 8 M , we smoothly transition from quasiequilibrium gauge to damped harmonic gauge at t = t g ( g stands for 'gauge') by choosing \nH a ( t ) = ˜ H a ( t ) e -( t -t g ) 4 /σ 4 g + µ L log ( √ g N ) t a -µ S N -1 g ai N i , (20) \nwhere ˜ H a ( t ) is the value of H a ( t ) obtained from the quasiequilibrium gauge condition (19), and σ g is a constant. The last line in Eq. (20) is the damped harmonic gauge condition, Eq. (8). The values of µ L and µ S are set according to Eq. (9), using \nµ 0 = { 0 , t < t d 1 -e -( t -t d ) 2 /σ 2 d , t > t d , (21) \nwhere t d and σ d are constants. It is necessary to choose σ g and σ d sufficiently large or else the gauge becomes \nunnecessarily dynamical, possibly resulting in incoming characteristic fields on the excision boundary. \nFor all runs shown here, t d corresponds approximately to a proper separation of 8 M . We use the values σ g = 20 M,σ d = 50 M for the equal mass binaries (cases A, C, and D), σ g = 15 M,σ d = 100 M for case B, and σ g = 25 M,σ d = 50 M for case F. For these cases, t g = t d . For case E, the gauge is rolled off to harmonic early in the inspiral, so at t = t d we need only to roll on the damped harmonic gauge, and we use σ d = 40 M . The constants in Eqs. (20) and (21) can be chosen quite flexibly: the only constraint is that σ g and σ d must be large enough to avoid significant spurious gauge dynamics, but small enough so that the simulation is using damped harmonic gauge before the black holes approach each other too closely. The σ g and σ d can always be made longer by choosing earlier transition times t g and t d .", '3. Eliminate overlapping subdomains': 'Although the domain decomposition consisting of overlapping spherical shells, cylinders, and cylindrical shells is quite efficient for the early inspiral, it often suffers from numerical instabilities, particularly as the black holes approach merger. Therefore, we regrid onto the cubedsphere subdomains described in Sec. II B. Use of these non-overlapping subdomains eliminates these remaining instabilities, and effectively increases our resolution. For most of the runs described here, we happen to regrid at the same time t d that we begin the transition to damped harmonic gauge. However, for runs E and F, the entire simulation uses the non-overlapping cube-sphere subdomains. As far as we know, there is no reason that regridding cannot occur at times other than t d or t g .', '4. Shape control': "The next stage is to turn on shape control by introducing the map M S defined in Eqs. (14-18). The parameters λ /lscriptm A that appear in this map are determined (via a feedback control system) by Eqs. (A.3) and (A.6) for /lscript > 0, and λ 00 A is set to zero. This map deforms the grid so that the boundaries of the spherical shells inside the horizons, including the excision boundaries, are mapped to closely match the horizons' shapes. \nWe do not turn on shape control until time t g +2 . 5 σ g , which is when the coefficient multiplying the gauge term ˜ H a ( t ) in Eq. (20) becomes smaller than double-precision roundoff. This ensures that H a remains a smooth function of space in inertial coordinates. If shape control were turned on too early, H a would fail to be everywhere smooth because the quasi-equilibrium-gauge quantity ˜ H ¯ a is a smooth function of space in the grid-frame coordinates (and its numerical value is constant in time for each grid point), and the deformation is only C 0 across subdomain boundaries. \nFIG. 7: The minimum of the outgoing (into the hole) characteristic speeds on the remaining four spherical-shell boundaries (labeled A 5 . . . A 8 ) around the larger black hole, for a portion of run F. The dashed curves correspond to runs without shape control, and the solid curves correspond to runs with shape control turned on at t = 62 M . The run without shape control terminates at proper separation 5 . 2 M (time t = 106 M ) because the excision algorithm fails on surface A 7 . \n<!-- image --> \nShape control is necessary for the shell-dropping procedure described in Section III B 1 to remain successful as the holes approach each other. For a shell to be dropped, the entire inner boundary of that shell must remain an outflow surface until the entire inner boundary of the next shell becomes an outflow surface. If these boundaries do not have the same shape as the horizon, then typically some portion of either boundary moves too close or too far from the horizon, and violates this condition. An example is shown in Figure 7. Like Figure 6, Figure 7 shows the minimum characteristic speeds on the boundaries of inner spherical shells for a portion of case F, but Figure 7 shows two separate evolutions: one with shape control turned on at t = 62 M (solid curves) and one with no shape control (dashed curves). Without shape control, the minimum characteristic speed on surface A 7 stops growing around t = 100 M and becomes negative soon thereafter. Because A 7 is the excision surface at t ∼ 100 M , the excision algorithm becomes ill-posed and the simulation stops at t = 106 M , well before merger (which occurs at t ∼ 128 M ). In the simulation with active shape control, the same boundary A 7 shows no such problems-in fact it is dropped at t = 102 M when all speeds on the next shell, A 8 become positive. \nFIG. 8: Constraint norm ||C GH || as a function of time for a portion of case F. The dashed curve corresponds to a run with spherical shells around each hole, and the solid curve corresponds to an identical run with these inner spherical shells replaced by cubed-sphere shells at t = 80 M . The constraint norm grows larger and begins to grow earlier when spherical shells are used. \n<!-- image -->", '5. Eliminate inner spherical shells': 'Spherical-shell subdomains are extremely efficient when the solution is nearly spherically symmetric, because these subdomains use Y lm basis functions, which are well-suited for nearly spherical functions. However, once the holes become sufficiently distorted, it is difficult to resolve the region near each horizon using spherical shells. At this point in the evolution, we replace the inner spherical shells with cubed-sphere shells. Figure 8 shows the constraints as a function of time for two evolutions of case F: one with inner spherical shells (dashed curves), and another in which the inner spherical shells were replaced with cubed-sphere shells at t = 80 M (solid curves). The quantity plotted is ||C GH || , the L 2 norm over all the constraint fields of our first-order generalized harmonic system, normalized by the L 2 norm over the spatial gradients of all dynamical fields (see Eq. (71) of Ref.[45]). The L 2 norms are taken over the portion of the computational volume that lies outside the apparent horizons. For the spherical-shell case in Figure 8, the constraints in the spherical shells grow large enough that the simulation terminates at t = 129 . 5 M , when the proper separation has fallen to 0 . 5 M . Note that these simulations typically proceed through merger even if we do not replace spherical shells with cubed-sphere shells. So while this replacement step is not strictly necessary, eliminating spherical shells improves the accuracy of the simulation (as measured by the constraint quantity ||C GH || ) by almost an order of magnitude.', '6. Size control': 'Eventually, even using shape control, shell dropping typically fails before the horizons merge: the outflow condition on the inner boundary of the inner shell fails before the outflow condition on the inner boundary of the next shell becomes satisfied. This problem occurs because the map M E shrinks the entire grid-including regions inside each hole-faster and faster as the holes approach each other. As soon as the velocity of excision boundary towards the center of the hole becomes large enough, the boundary becomes timelike, and the excision algorithm fails. To remedy this, we turn off shell dropping and we turn on size control. Size control slows down the infall of the excision surface (by pulling the excision sphere towards the horizon), thus keeping the characteristic speeds from changing sign. Size control is implemented by changing the map parameters λ 00 A ( t ), which previously were set to zero, to values given by Eqs. (A.3) and (A.10). This causes the size of each excision boundary to be driven towards some fraction η of the size of the appropriate horizon. For the equal-mass runs A, C, and D we use η = 0 . 8 for both black holes. For the unequal-mass run B and the generic cases E and F we use η = 0 . 9 and η = 0 . 7 for the smaller and larger black holes, respectively. These values do not require finetuning; for instance, case E still runs through merger and ringdown if we change η to 0 . 9 for both black holes. The main criterion for choosing this parameter is that a grid sphere of radius ¯ b A (see Figure 4) should not be mapped outside the cube of side 2¯ a A . If this occurs, the map M S becomes singular and the run fails. Figure 9 shows the minimum characteristic speed at particular domain boundaries for two evolutions of case F: one with size control and one without. Without size control, the outflow condition on the excision boundary fails at proper separation 1 . 67 M and time t = 127 . 24 M . With size control, the evolution proceeds to proper separation 0 . 015 M , time t = 130 . 17, well past the formation of a common apparent horizon, which occurs at proper separation 1 . 4 M (time t = 128 . 23 M ).', '7. Ringdown': 'At some time t = t m shortly after a common apparent horizon forms, we define a new grid, composed only of spherical shells. This grid has only a single excision boundary inside the common horizon. We also define a new map M ringdown : ¯ x i → x i that can be written as a composition of simpler maps: \nM ringdown := M Tf · M Ef · M R · M S 3 . (22) \nThis is similar to the ringdown maps described in Refs. [53, 56] (which had no translation) and Ref.[54] (which had no rotation). The new translation map M Tf is tied to the center of the new common horizon, and \nFIG. 9: The minimum of the outgoing (into the hole) characteristic speeds on the two innermost cubed-sphere boundaries (labeled B 0 and B 1 ) around the smaller black hole, for a portion of run F. The dashed curves correspond to runs without size control, and the solid curves correspond to runs with size control turned on at t = 120 . 2 M . Without size control, the innermost cubed-sphere is dropped at t = 124 . 5 M and B 1 becomes the excision boundary. However, the characteristic speeds become negative on B 1 at t = 127 M and excision fails. With size control, the characteristic speeds on the excision boundary B 0 remain positive until proper separation 0 . 075 M , well after merger. \n<!-- image --> \nis not continuous with the old translation map M T except near the outer boundary where both translation maps are the identity. The new expansion map M Ef is chosen to be continuous with the old expansion map M E at the outer boundary, but it is the identity near the merged black hole. This new expansion map M Ef smoothly becomes constant in time shortly after merger. The rotation map M R is continuous at merger, and after merger it smoothly becomes constant in time. The map M S 3 has the same form as Eqs. (14-17) except the function ¯ ρ A ( ¯ θ A , ¯ φ A ) defined in Eq. (18) is replaced by ¯ ρ 3 = constant, where the constant value corresponds to a subdomain boundary. \nAs described in Ref. [53], at t = t m the parameters λ /lscriptm A are chosen so that the common apparent horizon is stationary and spherical in the grid frame. Similarly, the map M Tf is chosen at t = t m so that the center of the horizon is located at the grid-frame origin and is stationary in the grid frame. \nOnce we have defined the maps at t = t m , we then interpolate all variables from the old grid to the new grid. Note that the grid frame changes discontinuously in time at t = t m . Because of this, we take care to properly use both the new map M ringdown and the old map M in doing this interpolation, so that the inertial frame and \nFIG. 10: Three snapshots of the apparant horizons of the non-spinning equal-mass binary black hole merger (case A). Solid curves are the orbital plane cross section when a common horizon is first detected; dotted curves represent the time of transition between the binary merger and the single hole ringdown evolutions; dashed curve shows the final equilibrium horizon. \n<!-- image --> \nquantities defined in that frame remain smooth. In particular, the gauge source function H a is still determined by Eq. (20) during ringdown, and remains smooth at t = t m . \nFor t > t m , the parameters λ /lscriptm A ( t ) are determined by a feedback control system that keeps the common apparent horizon stationary in the grid frame. Likewise, the map M Tf keeps the common apparent horizon centered at the origin in the grid frame.', 'C. Results': 'In this section we present some results from the simulations listed in Table I. We first show snapshots of the inertial-coordinate shapes of the apparent horizons during merger and after ringdown. Figures 10, 11, 12, and 13 show cross-sections of the horizons in the orbital plane for cases A-D; in these cases, the orbital plane is well-defined and is constant in time. For all cases, we show the apparent horizons of the individual black holes and the common apparent horizon at the time when the common horizon is first detected (solid curves), and at the time t m when we transition to a new grid that has a single excision boundary (dotted curves). We also show the apparent horizon of the final remnant black hole after it has reached equilibrium (dashed curves). Figures 14 and 15 show cross-sections of the horizons for cases E and F, in which the orbital plane precesses. For these cases we show cross-sections in the coordinate plane (defined \nFIG. 13: Apparent horizon snapshots as in Figure 10, except for an aligned-spin equal-mass binary black hole merger (case D). \n<!-- image --> \nFIG. 11: Apparent horizon snapshots as in Figure 10, except for a 2:1 mass ratio non-spinning binary black hole merger (case B). \n<!-- image --> \n<!-- image --> \nFIG. 12: Apparent horizon snapshots as in Figure 10, except for an anti-aligned-spin equal-mass binary black hole merger (case C). \n<!-- image --> \nusing the flat metric) that is perpendicular to the instantaneous orbital angular velocity at the time the common horizon is first detected, and that passes through the coordinate center of the common horizon at this time. We show cross-sections at three different times: the time of horizon formation, the time t m , and a time 300 M after merger, all with respect to the same plane. The remnant black hole in cases E and F have nonzero linear momentum both because of radiation reaction and because of some nonzero linear momentum present in the initial data; this is why the centers of the horizons shown in Figures 14 and 15 change with time. \nFor case F, we also show the constraint norm as a func- \nFIG. 14: Three snapshots of the apparant horizons of a generic binary black-hole merger (case E). The plane of the cross-sections is a coordinate plane perpendicular to the instantenous orbital axis at the time the common horizon is first detected. The additional degree of freedom is fixed by having this plane go through the coordinate-center of the shape of the common horizon at its first detection. Solid curves are the cross section when a common horizon is first detected; dotted curves represent the time of transition between the binary merger and the single hole ringdown evolutions; dashed curve shows the horizon at a time well after merger. \ntion of time in Figure 16. We plot the same quantity ||C GH || as shown in Figure 8. This quantity is shown for four numerical resolutions, and is convergent at all times (although the convergence rate is smaller near merger when the solution is most dynamical). The constraints are largest at t = t m , when we transition to a grid with \nFIG. 15: Apparent horizon snapshots as in Figure 14, except for the generic binary black-hole merger case F in Table I. \n<!-- image --> \nFIG. 16: Constraint norm ||C GH || as a function of time for the generic (case F) binary black-hole merger and ringdown, computed with four different numerical resolutions. The small inset graph shows that numerical convergence is maintained (at a reduced rate) even during the most dynamical part of the merger. \n<!-- image --> \na single excision boundary. Just after t = t m the constraints decrease discontinuously by a small amount because part of the computational domain has been newly excised. The approximate number of grid points for the four resolutions is { 79 3 , 87 3 , 95 3 , 103 3 } just before merger and { 44 3 , 51 3 , 57 3 , 63 3 } during ringdown.', 'Acknowledgments': 'We thank Jan Hesthaven for helpful discussions. We are grateful to Luisa Buchman, Larry Kidder, and Harald Pfeiffer for the improved kinematical coordinate maps and associated control systems that handle inspirals of precessing and unequal-mass binaries, and we thank Harald Pfeiffer for help preparing the initial data. We also thank Luisa Buchman, Tony Chu, Geoffrey Lovelace, and Harald Pfeiffer for producing some of the BBH inspiral simulations that we here extend through merger and ringdown. We acknowledge use of the Spectral Einstein Code (SpEC). This work was supported in part by grants from the Sherman Fairchild Foundation, by NSF grants PHY-0601459, PHY-0652995, and by NASA grant NNX09AF97G.', 'APPENDIX: CONTROLLING THE MAPS': "The purpose of the shape-control maps M S A , with A = { 1 , 2 } , is to distort the grid structures around the black holes so the excision boundaries of the computational domain are mapped to surfaces lying just inside and having the same basic shapes as the apparent horizons. Choosing the right maps is equivalent to choosing the right target values for the parameters λ /lscriptm A ( t ) that define these maps, cf. Eq. (16). This Appendix describes in some detail how these parameters are chosen, and what target surfaces are used to fix these maps in the merger simulations described in Sec. III. \nThe grid structures used in our merger simulations, cf. Sec. II B, have excision boundaries that are grid-frame coordinate spheres. The target surfaces S T A to which we want to map these grid-frame spheres can be written in the form \n˜ r A = /lscript max ∑ /lscript =0 /lscript ∑ m = -/lscript Λ /lscriptm A ( t ) Y /lscriptm ( ˜ θ A , ˜ φ A ) . (A.1) \nwhere the expansion coefficients Λ /lscriptm A ( t ) define the target surface and (˜ r A , ˜ θ A , ˜ φ A ) are rest-frame coordinates. \nLet ¯ r target A denote the radii of the grid-frame coordinate spheres that are to be mapped onto the target surfaces S T A . The rest-frame representations of these coordinate spheres, using Eqs. (14-16), are: \n˜ r A = ¯ r target A -/lscript max ∑ /lscript =0 /lscript ∑ m = -/lscript λ /lscriptm A ( t ) Y /lscriptm ( ˜ θ A , ˜ φ A ) . (A.2) \nNote that the functions f A that appear in Eq. (17) are set equal to unity here because we always choose the target grid-spheres ¯ r target A to be smaller than the outer radii of the largest spherical subdomain layers: ¯ b a ≥ ¯ r target A . It follows that the target spheres will be mapped to the \nshapes of S T A when \nλ /lscriptm A ( t ) = -Λ 00 A ( t ) + ¯ r target A Y 00 , /lscript = 0 , -Λ /lscriptm A ( t ) , /lscript > 0 . (A.3) \nThese conditions can not be imposed directly for a number of reasons, but they can be enforced approximately using a feedback control system. The control system used in the merger calculations described in Sec. III is the same one used to perform our earlier binary black-hole simulations [47], so we will not describe it in detail here. \nTo complete the specification of the shape-control maps we must choose the target grid-spheres ¯ r target A , and the target surface parameters Λ /lscriptm A . From Eq. (A.3) it follows that the choice of ¯ r target A only effects the /lscript = 0 size control part of the distortion map. So until size control is activated late in our runs, there is no need to specify what ¯ r target A actually is. We think of it as being the radius of the suitably scaled apparent horizon, but this fact does not influence the shape control part of the map in any way. Once size control is activated late in the run, then we require ¯ r target A to be the radius of the excision boundary: \n¯ r target A = ¯ r ex A . (A.4) \nNext consider the choice of target surfaces S T A , starting with the /lscript > 0 contributions that control the shape but not the overall scaling of the map. The idea is to choose the target surfaces S T A to be similar in shape to the apparent horizons H A . The H A can be represented as smooth surfaces in the rest-frame coordinate system: \n˜ r A = /lscript max ∑ /lscript =0 /lscript ∑ m = -/lscript ˜ S /lscriptm A ( t ) Y /lscriptm ( ˜ θ, ˜ φ ) . (A.5) \nTo keep the shapes of the target surfaces similar to the shapes of the apparent horizons, the target surface parameters Λ /lscriptm A should be made proportional to S /lscriptm A . We find it is appropriate to scale these coefficients by a factor, G ( ˜ R A ), which depends on the average radius of the apparent horizon, ˜ R A = S 00 A Y 00 : \nΛ /lscriptm A ( t ) = G ( ˜ R A ) S /lscriptm A ( t ) . (A.6) \nThe larger the apparent horizon radius in relation to the desired excision radius, the smaller this scaling factor must be to maintain an appropriate shape for the excision boundry. In practice, we find the scale factor \nG ( ˜ R A ) = ¯ a A ˜ R A tanh p ( ˜ R A ¯ a A ) 1 /p , (A.7) \nworks quite well for p = 2. This scaling factor is near unity when ˜ R A < ¯ a A and decreases like 1 / ˜ R A for larger values of ˜ R A . The tanh x function is introduced here because it is linear for small values of | x | , and approaches \n1 for x /greatermuch 1. The p -dependence maintains these asymptotic forms, but allows the transition between them to occur more quickly than the p = 1 case. \nWe have found no need to introduce the /lscript = 0 size control parts these maps, until the final plunge phase just before the black holes merge. So during most of our merger simulations we simply set λ 00 A ( t ) = 0. This allows the apparent horizons to grow relative to the grid coordinates as the map M E contracts the rest-frame relative to the inertial coordinates. This growth in the apparent horizons in the grid frame allows us to drop a number of spherical subdomain layers as the evolution progresses, and this helps remove unwanted constraint violations from the computational domain. \nAt some point however, we may not be able to drop additional spherical subdomain layers: either because we run out of layers, or because the inner boundary experiences incoming characteristic fields when the apparent horizon expands too quickly during the final plunge. When either of these conditions occurs we turn on the /lscript = 0 part of the shape-control maps, to keep the radius of the excision boundary ¯ r ex A close to the size of the apparent horizon. It will be useful to define \n∆ r A = η ˜ R A -¯ r ex A . (A.8) \nWe choose η so that when ∆ r A < 0 there is no need to impose size control, and use η in the range 0 . 7 ≤ η ≤ 0 . 9 for the merger simulations reported in Sec. III. When ∆ r A becomes positive we want to turn on size control, but we want to do it in a fairly smooth and continuous way. This is done by defining the function: \nP (∆ r A ) = 0 , ∆ r A < 0 tanh ( 40∆ r A ˜ R A ) , ∆ r A ≥ 0 , (A.9) \n which vanishes for ∆ r A ≤ 0, and asymptotically approaches P (∆ r A ) → 1 for ∆ r A /greatermuch ˜ R A / 40. We find that an appropriate value for the overall scale of the target surfaces S T A , i.e. their /lscript = 0 components, to be \nΛ 00 A = ¯ r ex A +∆ r A P (∆ r A ) Y 00 . (A.10) \nOnce size control is activated, the target radius is given by Eq. (A.4): ¯ r target A = ¯ r ex A . 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2021PhRvL.126a1104B

Spin-Induced Black Hole Scalarization in Einstein-Scalar-Gauss-Bonnet Theory

2021-01-01

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We construct black hole solutions with spin-induced scalarization in a class of models where a scalar field is quadratically coupled to the topological Gauss-Bonnet term. Starting from the tachyonically unstable Kerr solutions, we obtain families of scalarized black holes such that the scalar field has either even or odd parity, and we investigate their domain of existence. The scalarized black holes can violate the Kerr rotation bound. We identify "critical" families of scalarized black hole solutions such that the expansion of the metric functions and of the scalar field at the horizon no longer allows for real coefficients. For the quadratic coupling considered here, solutions with spin-induced scalarization are entropically favored over Kerr solutions with the same mass and angular momentum.

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https://arxiv.org/pdf/2009.03905.pdf

{'Spin-induced black-hole scalarization in Einstein-scalar-Gauss-Bonnet theory': "Emanuele Berti, 1, ∗ Lucas G. Collodel, 2, † Burkhard Kleihaus, 3, ‡ and Jutta Kunz 3, § \n1 Department of Physics and Astronomy, Johns Hopkins University, 3400 N. Charles Street, Baltimore, Maryland 21218, US 2 Theoretical Astrophysics, Eberhard Karls University of Tübingen, D-72076 Tübingen, Germany 3 Institute of Physics, University of Oldenburg, D-26111 Oldenburg, Germany \n(Dated: January 12, 2021) \nWe construct black hole solutions with spin-induced scalarization in a class of models where a scalar field is quadratically coupled to the topological Gauss-Bonnet term. Starting from the tachyonically unstable Kerr solutions, we obtain families of scalarized black holes such that the scalar field has either even or odd parity, and we investigate their domain of existence. The scalarized black holes can violate the Kerr rotation bound. We identify 'critical' families of scalarized black hole solutions such that the expansion of the metric functions and of the scalar field at the horizon no longer allows for real coefficients. For the quadratic coupling considered here, solutions with spininduced scalarization are entropically favored over Kerr solutions with the same mass and angular momentum. \nIntroduction. Compact objects in gravity theories involving scalar degrees of freedom can undergo a phase transition induced by a tachyonic instability, known as 'spontaneous scalarization': the solutions of the general relativistic field equations become unstable in certain regions of parameter space, developing scalar 'hair.' This instability comes in different flavors. Matter-induced spontaneous scalarization was originally proposed for compact neutron stars in scalar-tensor theories [1], but more recently it was shown that spontaneous scalarization is possible also in the absence of matter. Curvatureinduced spontaneous scalarization of black holes (BHs) was first studied in Einstein-scalar-Gauss-Bonnet (EsGB) theories [2-4]. Charge-induced scalarization can also occur in Einstein-scalar-Maxwell theories [5]. \nIn this paper we consider EsGB theories with action \nS = 1 16 π ∫ d 4 x √ -g [ R -1 2 ( ∂ µ φ ) 2 + f ( φ ) R 2 GB ] , (1) \nwhere we use geometrical units ( G = c = 1 ), and φ is a real scalar field coupled to the Gauss-Bonnet (GB) invariant R 2 GB = R µνρσ R µνρσ 4 R µν R µν + R 2 . \nWe will focus on the simple quadratic coupling function f ( φ ) = 1 8 ηφ 2 . Early work showed that the GB invariant acts as a tachyonic mass term for the scalar ('curvatureinduced' BH scalarization) when η > 0 [2, 3], but here we focus on the case η < 0 . Recent work pointed out that the Kerr BH solutions of general relativity can still scalarize when η < 0 [6]: the Gauss-Bonnet scalar for the Kerr metric can become negative close to the horizon, producing 'spin-induced' scalarization when the dimensionless Kerr spin parameter j ≡ J/M 2 /greaterorsimilar 0 . 5 [6]. This conclusion was confirmed analytically [7] and numerically [8] by \n- \nstudying linear perturbations of Kerr BHs. These works concluded that the instability threshold depends on the Gauss-Bonnet (GB) coupling η/M 2 and on the (even or odd) symmetry of the scalar field under parity transformation. For small values of the GB coupling and associated large values of j , the thresholds for even and odd parity differ, whereas for large values of the GB coupling the two thresholds almost coincide. \nHere we show that stationary and axisymmetric BH solutions with spin-induced scalar hair do indeed exist in the nonlinear regime. We construct these BHs numerically, starting from the respective threshold solutions. We then vary the input parameters to map out the domain of existence of scalarized BHs for both even- and odd-parity scalar fields. The expansions of the metric functions and of the scalar field at the horizon yield an analytic criterion to identify critical solutions that form the second boundary of the domain of existence. \nWe investigate the thermodynamical stability of these BH solutions by computing their entropy. Solutions with curvature-induced scalarization are entropically disfavored with respect to Schwarzschild and Kerr BHs when f ( φ ) is quadratic [3, 9], but they become entropically favored when we add a quartic term [10] or for exponential coupling functions [2, 11]. Linear perturbation theory shows that the entropically favored (disfavored) fundamental scalarized solutions are mode stable (unstable) [10, 12, 13]. Here we find that BH solutions with spin-induced scalarization are entropically favored over Kerr solutions with the same mass and angular momentum, but their dynamical stability remains an open question. \nGeneral framework. The generalized Einstein and scalar field equations follow by varying the action (1) with respect to the metric g µν and the scalar field φ : \nG µν = T µν , ∇ 2 φ + df dφ R 2 GB = 0 , (2) \nwhere the effective stress-energy tensor \nT µν = -1 4 g µν ∂ ρ φ∂ ρ φ + 1 2 ∂ µ φ∂ ν φ -1 2 ( g ρµ g λν + g λµ g ρν ) η κλαβ ˜ R ργ αβ ∇ γ ∂ κ f ( φ ) , (3) \n- \nwith ˜ R ργ αβ = η ργστ R σταβ and η ργστ = /epsilon1 ργστ / √ g . \nTo construct stationary, axially symmetric spacetimes with two commuting Killing vector fields ( ξ = ∂ t and η = ∂ ϕ ) we employ a Lewis-Papapetrou-type ansatz [14, 15] \nds 2 = -be F 0 dt 2 + e F 1 ( dr 2 + r 2 dθ 2 ) + e F 2 r 2 sin 2 θ ( dϕ + ωdt ) 2 , (4) \nwhere r is a quasi-isotropic radial coordinate, r H is the isotropic horizon radius, b = (1 -r r H ) 2 . The metric functions F i ( i = 0 , 1 , 2 ) and ω depend on the coordinates r and θ , and they are even under parity. The scalar field φ = φ ( r, θ ) can be either even or odd with respect to parity transformation, i.e. φ ± ( r, π -θ ) = ± φ ± ( r, θ ) . Both parity-even and parity-odd scalar fields are consistent with the field equations, since the generalized Einstein equations are quadratic and the generalized KleinGordon equation is linear in φ (note that parity is a symmetry only when f ( φ ) is even in φ ). Scalarized BHs with an even scalar field and no radial nodes are the fundamental scalarized solutions, whereas those with an odd scalar field are angularly excited solutions. \nThe proper set of boundary conditions is obtained by considering symmetry, regularity and asymptotic flatness of the solutions. This implies F i ( ∞ ) = 0 ( i = 0 , 1 , 2) , ω ( ∞ ) = φ ( ∞ ) = 0 as r →∞ . For a massless scalar field one can construct an approximate solution of the field equations as a power series in 1 /r , with the dominant term being of monopole type in the even case and of dipole type in the odd case: φ + = Q/r + . . . and φ -= P cos θ/r 2 + . . . , where Q and P are interpreted as the scalar charge and the dipole moment of the scalar field, respectively. \nThe boundary conditions at the event horizon, located at a surface of constant r = r H , are obtained by considering a power-series expansion in terms of δ = ( r -r H ) /r H : ∂ r F 0 ( r H ) = 1 /r H , ∂ r F 1 ( r H ) = -2 /r H , ∂ r F 2 ( r H ) = -2 /r H , ω ( r H ) = ω H , ∂ r φ ( r H ) = 0 , where ω H is a constant. On the symmetry axis ( θ = 0 , π ), axial symmetry and regularity impose ∂ θ F i | θ =0 ,π = 0 ( i = 0 , 1 , 2) , ∂ θ ω | θ =0 ,π = ∂ θ φ | θ =0 ,π = 0 . Since all functions are either even or odd, it is sufficient to consider the range 0 ≤ θ ≤ π/ 2 for the angular variable θ in the numerical calculations. Consequently, we impose the following boundary conditions on the equatorial plane: ∂ θ F i | θ = π/ 2 = 0 ( i = 0 , 1 , 2) , ∂ θ ω | θ = π/ 2 = ∂ θ φ + θ = π/ 2 = φ -| θ = π/ 2 = 0 . \n| | From the horizon metric we obtain the Hawking temperature [14] \n| \nT H = 1 2 πr H e ( F 0 -F 1 ) / 2 . (5) \nIn fact, the equation G θ r = T θ r implies that F 0 /F 1 (and therefore the Hawking temperature) is constant. This \nFIG. 1. Dimensionless discriminant as a function of θ for ω H = 0 . 065 and selected values of η . When η < η cr /similarequal -179 the local maximum becomes positive, and scalarized BH solutions cease to exist. \n<!-- image --> \nobservation can be used to test the numerical accuracy of our solutions. \nThe horizon area is given by \nA H = 2 πr 2 H ∫ π 0 dθ sin θe ( F 1 + F 2 ) / 2 . (6) \nThe entropy of Kerr BHs is a quarter of the horizon area [14], but the entropy of EsGB BHs can be computed as an integral over the spatial cross section of the horizon [16] and it acquires an extra contribution: \nS = 1 4 ∫ Σ H d 2 x √ h (1 + 2 f ( φ ) ˜ R ) , (7) \nwhere h is the determinant of the induced metric on the horizon and ˜ R is the corresponding scalar curvature. \nThe mass M and the angular momentum J can be found from the asymptotic behavior of the metric functions: g tt = -1 + 2 M/r + . . . , g ϕt = -2 J sin 2 θ/r + . . . . \nNumerical Results. To obtain the EsGB BHs with spin-induced scalarization we need to solve for the functions ( F 0 , F 1 , F 2 , ω ; φ ) subject to the boundary conditions specified above, that guarantee regularity and asymptotic flatness. REDUCE files with the field equations and expansions are available from the authors upon request. \nWe provide three input parameters ( η , r H and ω H ) and we follow the numerical procedure of Refs. [9, 15]. We introduce the radial variable x = 1 -r H /r , mapping the interval [ r H , ∞ ) to the interval [0 , 1] , and discretize the equations on a nonequidistant grid in x and θ , covering the integration region 0 ≤ x ≤ 1 and 0 ≤ θ ≤ π/ 2 . We perform the integrations using the package FIDISOL/CADSOL [17, 18], based on a NewtonRaphson method, and we extract the physical properties of the BHs when the estimated truncation error is within \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFIG. 2. Top left: dimensionless charge Q/M of BHs with even scalar field (solid lines) and dimensionless dipole moment P/M 2 of BHs with odd scalar field (dashed lines). The other panels show the dimensionless angular momentum j (top right), dimensionless horizon area A H / 8 πM 2 (bottom left) and dimensionless Hawking temperature T H M (bottom right) of even-parity BHs (solid lines) and odd-parity BHs (dashed lines). All quantities are shown for selected values of ω H as functions of -η/ 4 M 2 . \n<!-- image --> \nthe required accuracy, i.e., when the maximal numerical error for the functions at any point is estimated to be of order 10 -3 or less. \nWe now address the second boundary of the domain of existence of scalarized BH solutions, given by the set of critical solutions. To this end, we consider higher-order terms of local solutions close to the horizon: F 0 = F 0 , H + δ + f 0 , 2 δ 2 / 2+ . . . , F i = F i, H -2 δ + f i, 2 δ 2 / 2+ . . . , ( i = 1 , 2) , ω = ω H + ω 2 δ 2 / 2 + ω 3 δ 3 / 6 + . . . , φ = φ H + φ 2 δ 2 / 2 + . . . . \nWe obtain equations for the coefficients of the higherorder terms f 0 , 2 , f 1 , 2 , f 2 , 2 , ω 3 , and φ 2 which allow us to express the higher-order coefficients in terms of F 0 , H , F 1 , H , F 2 , H , ω 2 , φ H , and their first and second derivatives with respect to θ . Solving these equations yields a quartic equation for φ 2 . The existence of real solutions of the quartic equation depends on the sign of the discriminant D = ( p/ 3) 3 +( q/ 2) 2 of the reduced cubic resolvent, v 3 + pv + q = 0 . Real solutions exist if D ( θ ) ≤ 0 for 0 ≤ θ ≤ π/ 2 . The numerical calculations show that D ( θ ) is always negative for ω H > 0 . 073 . However, for ω H ≤ 0 . 073 the \nfunction D ( θ ) developes a local maximum, which tends to zero when η is decreased to some critical value η cr . Solutions for η < η cr cease to exist. \nThis is demonstrated in Fig. 1, where we show the dimensionless discriminant ∆ ≡ 4 27 p 3 q 2 + 1 for ω H = 0 . 065 and decreasing values of η . Note however that for symmetric BHs with large angular momentum the condition for real solutions is violated at the equator of the horizon. \nWe are now ready to discuss the physical properties of these spin-induced spontaneously scalarized black holes. To map out their domain of existence, we have calculated numerous families of scalarized BH solutions with fixed horizon angular velocity ω H while varying the coupling constant η . \nIn Fig. 2 we show various BH properties as functions of the dimensionless coupling parameter η/ 4 M 2 for families of solutions with fixed values of ω H . The different panels show the dimensionless scalar charge Q/M (dipole moment P/M 2 ) for the fundamental even (odd) solutions (top left); the dimensionless angular momentum j (top \nright); the dimensionless horizon area A H / 8 πM 2 (bottom left); and the dimensionless Hawking temperature T H M (bottom right). \nFigure 2 provides important new insight into the domain of existence of BHs with spin-induced scalarization. Bifurcation from the Kerr solutions takes place at some threshold solutions ('Kerr-thr' in the legend) representing the first boundary of the domain of existence. These thresholds are rather close for even and odd solutions, especially for large values of | η/M 2 | . The second boundary is given by the critical solutions ('crit' in the legend) such that the discriminant D ( θ ) vanishes somewhere. For large values of | η/M 2 | the threshold lines and the critical lines approach each other, and the domain of existence becomes narrower. \nIf present, the third boundary of the domain of existence should correspond to extremal scalarized BHs, which are numerically difficult to explore. The bottomright panel of Fig. 2 shows that the Hawking temperature approaches zero in this limit. The previously studied case of rotating dilatonic GB BHs suggests that these extremal scalarized solutions might not be regular [1921]. Some of these solutions have angular momentum exceeding the Kerr bound j = 1 . In fact, the bound is already exceeded by nonextremal odd solutions when | η/M 2 | < 1 . 53 . For even solutions, the Kerr bound is exceeded only marginally when η/M 2 | < 0 . 55 . \n| \n| The violation of the Kerr bound is also clear from Fig. 3, where we plot the dimensionless entropy S/ 2 πM 2 as a function of j = J/M 2 for the same families of solutions. The inset of the figure shows the charge-to-mass ratio Q/M as a function of j . Interpolation yields a maximum value Q/M = 0 . 1225 at j = 0 . 9989 < 1 . Most importantly, Fig. 3 allows us to draw a crucial conclusion: for a given mass and angular momentum, BHs with spin-induced scalarization have larger entropy than Kerr BHs, and therefore they are entropically favored. Close to the Kerr bound, the area of even- and odd-parity BHs with spin-induced scalarization can exceed the area of their Kerr counterparts by about 30%. This could have interesting observational consequences, e.g. in terms of telling them apart from Kerr BHs with very-long baseline interferometry of their shadow [22]. \nConclusions. Starting from even- and odd-parity threshold solutions, we have mapped out the domain of existence of BHs with spin-induced scalarization in EsGB theories with a quadratic coupling function. The second boundary of the domain of existence corresponds to critical solutions beyond which the horizon expansion of the metric functions and of the scalar field no longer admit real coefficients. If present, a third boundary should correspond to extremal scalarized BHs, but this regime is hard to explore numerically. Scalarized BHs can violate the Kerr bound when | η/M 2 | < 1 . 53 ( | η/M 2 | < 0 . 55 ) for odd (even) solutions. This violation seems to occur only in the vicinity of the extremal solutions, and it is of the order of 5% (0.5%) for odd (even) solutions. \nScalarized BHs are entropically favored over Kerr BHs \nFIG. 3. Dimensionless entropy S/ (2 πM 2 ) as a function of j for odd- and even-parity BHs for selected values of ω H . Inset: Q/M as a function of j for even-parity BHs. \n<!-- image --> \nwith the same mass and angular momentum. If previous studies of curvature-induced scalarization are a useful analogy, this would suggest that BHs with spin-induced scalarization are (linearly) mode stable under perturbations. This may come as a surprise, since we have employed a simple quadratic coupling function; however we chose a negative coupling constant, in contrast with previous work on curvature-induced scalarization. The dynamical stability of BHs with spin-induced scalarization is an important open question that will require further work. Perturbations of rotating BHs in modified gravity are a notoriously difficult technical problem, because the equations are nonseparable (see e.g. Ref. [23] for recent progress on scalar perturbations in a slow-rotation expansion). Time evolutions may provide a practical way to find out if these solutions are dynamically stable. \nLast but not least, the problems of well-posedness, gravitational collapse, and gravitational waveforms from binary BH mergers in EsGB theories are very active research areas in analytical and numerical relativity [2431]. The new solutions discussed in this paper may have important implications in this context. \nAcknowledgments. We thank C. Herdeiro, E. Radu, H. O. Silva, T. P. Sotiriou and N. Yunes for sharing with us, while this paper was nearing completion, a manuscript in which they derive independently similar results for a different coupling function. The authors gratefully acknowledge support by the DFG Research Training Group 1620 Models of Gravity and the COST Actions CA15117 CANTATA and CA16104 GWverse . L.C. is thankful for the financial support obtained through the DFG Emmy Noether Research Group under grant no. DO 1771/1-1. E.B. is supported by NSF Grants No. PHY-1912550 and AST-2006538, NASA ATP Grants No. 17-ATP17-0225 and 19-ATP19-0051, and NSF-XSEDE Grant No. PHY-090003. This work has received funding \nfrom the European Union's Horizon 2020 research and in- \n- [1] T. Damour and G. Esposito-Farese, Phys. Rev. Lett. 70 , 2220 (1993).\n- [2] D. D. Doneva and S. S. Yazadjiev, Phys. Rev. Lett. 120 , 131103 (2018), arXiv:1711.01187 [gr-qc].\n- [3] H. O. Silva, J. Sakstein, L. Gualtieri, T. P. Sotiriou, and E. Berti, Phys. Rev. Lett. 120 , 131104 (2018), arXiv:1711.02080 [gr-qc].\n- [4] G. Antoniou, A. Bakopoulos, and P. Kanti, Phys. Rev. Lett. 120 , 131102 (2018), arXiv:1711.03390 [hep-th].\n- [5] C. A. R. Herdeiro, E. Radu, N. Sanchis-Gual, and J. A. Font, Phys. Rev. Lett. 121 , 101102 (2018), arXiv:1806.05190 [gr-qc].\n- [6] A. Dima, E. Barausse, N. Franchini, and T. P. Sotiriou, (2020), arXiv:2006.03095 [gr-qc].\n- [7] S. Hod, (2020), arXiv:2006.09399 [gr-qc].\n- [8] D. D. Doneva, L. G. Collodel, C. 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Lett. 86 , 3704 \nnovation programme under the Marie Skłodowska-Curie grant agreement No. 690904. \n- (2001), arXiv:gr-qc/0012081 [gr-qc].\n- [16] R. M. Wald, Phys. Rev. D48 , R3427 (1993), arXiv:gr-qc/9307038 [gr-qc].\n- [17] W. Schönauer and R. Weiß, Journal of Computational and Applied Mathematics 27 , 279 (1989), special Issue on Parallel Algorithms for Numerical Linear Algebra.\n- [18] M. Schauder, R. Weiß, and W. Schönauer, The CADSOL Program Package , Universität Karlsruhe, Interner Bericht Nr. 46/92 (1992).\n- [19] B. Kleihaus, J. Kunz, and E. Radu, Phys. Rev. Lett. 106 , 151104 (2011), arXiv:1101.2868 [gr-qc].\n- [20] B. Kleihaus, J. Kunz, S. Mojica, and E. Radu, Phys. Rev. D93 , 044047 (2016), arXiv:1511.05513 [gr-qc].\n- [21] P. V. P. Cunha, C. A. R. Herdeiro, B. Kleihaus, J. Kunz, and E. Radu, Phys. Lett. B768 , 373 (2017), arXiv:1701.00079 [gr-qc].\n- [22] K. Akiyama et al. (Event Horizon Telescope), Astrophys. J. Lett. 875 , L1 (2019), arXiv:1906.11238 [astro-ph.GA].\n- [23] P. A. Cano, K. Fransen, and T. Hertog, (2020), arXiv:2005.03671 [gr-qc].\n- [24] H. Witek, L. Gualtieri, P. Pani, and T. P. Sotiriou, Phys. Rev. D99 , 064035 (2019), arXiv:1810.05177 [gr-qc].\n- [25] J. L. Ripley and F. Pretorius, Phys. Rev. D99 , 084014 (2019), arXiv:1902.01468 [gr-qc].\n- [26] J. L. Ripley and F. Pretorius, Class. Quant. Grav. 36 , 134001 (2019), arXiv:1903.07543 [gr-qc].\n- [27] F.-L. Julié and E. Berti, Phys. Rev. D100 , 104061 (2019), arXiv:1909.05258 [gr-qc].\n- [28] J. L. Ripley and F. Pretorius, Class. Quant. Grav. 37 , 155003 (2020), arXiv:2005.05417 [gr-qc].\n- [29] M. Okounkova, (2020), arXiv:2001.03571 [gr-qc].\n- [30] F.-L. Julié and E. Berti, Phys. Rev. D101 , 124045 (2020), arXiv:2004.00003 [gr-qc].\n- [31] H. Witek, L. Gualtieri, and P. Pani, Phys. Rev. D101 , 124055 (2020), arXiv:2004.00009 [gr-qc]."}

2019ApJ...878...85S

Orbital Migration of Interacting Stellar Mass Black Holes in Disks around Supermassive Black Holes

2019-01-01

['accretion', 'accretion disks', 'black hole physics', 'galaxy nucleus', '-', '-']

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The merger rate of stellar-mass black hole binaries (sBHBs) inferred by the Advanced Laser Interferometer Gravitational-Wave Observatory (LIGO) suggests the need for an efficient source of sBHB formation. Active galactic nucleus (AGN) disks are a promising location for the formation of these sBHBs, as well as binaries of other compact objects, because of powerful torques exerted by the gas disk. These gas torques cause orbiting compact objects to migrate toward regions in the disk where inward and outward torques cancel, known as migration traps. We simulate the migration of stellar mass black holes in an example of a model AGN disk, using an augmented N-body code that includes analytic approximations to migration torques, stochastic gravitational forces exerted by turbulent density fluctuations in the disk, and inclination and eccentricity dampening produced by passages through the gas disk, in addition to the standard gravitational forces between objects. We find that sBHBs form rapidly in our model disk as stellar-mass black holes migrate toward the migration trap. These sBHBs are likely to subsequently merge on short timescales. The process continues, leading to the build-up of a population of over-massive stellar-mass black holes. The formation of sBHBs in AGN disks could contribute significantly to the sBHB merger rate inferred by LIGO.

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https://arxiv.org/pdf/1807.02859.pdf

{'No Header': "Draft version November 14, 2019 Typeset using L A T E X twocolumn style in AASTeX62 \nOrbital Migration of Interacting Stellar Mass Black Holes in Disks Around Supermassive Black Holes \nAmy Secunda, 1 Jillian Bellovary, 1, 2 Mordecai-Mark Mac Low, 1, 3 K.E. Saavik Ford, 1, 4, 5 Barry McKernan, 1, 4, 5 Nathan W. C. Leigh, 6, 7, 1 Wladimir Lyra, 8, 9 and Zsolt S'andor 10, 11 \n1 Department of Astrophysics, American Museum of Natural History, Central Park West at 79th Street, New York, NY 10024, USA 2 Department of Physics, Queensborough Community College, Bayside, NY 11364 \n3 Center for Computational Astrophysics, Flatiron Institute, 162 Fifth Avenue, New York, NY 10010 \n4 Department of Science, Borough of Manhattan Community College, City University of New York, New York, NY 10007 5 Physics Program, The Graduate Center, CUNY, New York, NY 10016 \n6 Departamento de Astronom'ıa, Facultad de Ciencias F'ısicas y Matem'aticas,Universidad de Concepci'on, Concepci'on, Chile 7 Department of Physics and Astronomy, Stony Brook University, Stony Brook, NY 11794-3800, USA \n8 Department of Physics and Astronomy, California State University Northridge, 18111 Nordhoff Street, Northridge, CA 91330, USA 9 Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, CA 91109, USA 10 Department of Astronomy, Eotvos Lor'and University, P'azm'any P'eter s'et'any 1/A, H-1117 Budapest, Hungary \n11 Konkoly Observatory, Hungarian Academy of Sciences, Konkoly-Thege Mikl'os 'ut 15-17, H-1121 Budapest, Hungary \n(Received 2018 July 8; Revised 2019 May 7; Accepted 2019 May 8)", 'ABSTRACT': 'The merger rate of stellar-mass black hole binaries (sBHBs) inferred by the Advanced Laser Interferometer Gravitational-Wave Observatory (LIGO) suggests the need for an efficient source of sBHB formation. Active galactic nucleus (AGN) disks are a promising location for the formation of these sBHBs, as well as binaries of other compact objects, because of powerful torques exerted by the gas disk. These gas torques cause orbiting compact objects to migrate towards regions in the disk where inward and outward torques cancel, known as migration traps. We simulate the migration of stellar mass black holes in an example of a model AGN disk, using an augmented N-body code that includes analytic approximations to migration torques, stochastic gravitational forces exerted by turbulent density fluctuations in the disk, and inclination and eccentricity dampening produced by passages through the gas disk, in addition to the standard gravitational forces between objects. We find that sBHBs form rapidly in our model disk as stellar-mass black holes migrate towards the migration trap. These sBHBs are likely to subsequently merge on short time-scales. The process continues, leading to the build-up of a population of over-massive stellar-mass black holes. The formation of sBHBs in AGN disks could contribute significantly to the sBHB merger rate inferred by LIGO. \nKeywords: black hole physics - accretion disks - galaxies:nuclei', '1. INTRODUCTION': "The Advanced Laser Interferometer GravitationalWave Observatory (LIGO) has detected the merger of stellar mass black holes (sBHs) more massive than those previously inferred from electromagnetic observations in our own Galaxy. Additionally, while isolated binary evolution could potentially account for the high sBH merger rate inferred from LIGO detections, 52.9 +55 . 6 -27 . 0 Gpc -3 yr -1 (Belczynski et al. 2016; The \nCorresponding author: Amy Secunda [email protected] \nLIGO Scientific Collaboration & The Virgo Collaboration 2018), an additional mechanism of sBH mergers in the Local Universe would ease several of the assumptions necessary in these models. \nIt has been suggested that over-massive sBHs are most likely to form in galactic nuclear star clusters (Hopman & Alexander 2006; O'Leary et al. 2009; Antonini & Rasio 2016; Rodriguez et al. 2016). The gas disks in active galactic nuclei (AGN) are particularly promising locations for the formation and merger of over-massive sBHs. As McKernan et al. (2014, 2018) point out, these gas disks will act to decrease the inclination of intersecting orbiters and harden existing binaries, already making \nthem interesting possible locations for LIGO detections of merging sBHs. The recent discovery of a possible black hole (BH) cusp in the core of our own Galaxy (Bahcall & Wolf 1976; Hailey et al. 2018) lends further weight to this possibility. \nOrbiters in a gas disk exchange angular momentum with the surrounding gas, leading to a change in semimajor axis known as migration. Migration of objects embedded within the disk provides opportunities for sBHs to form binaries if they encounter each other at small relative velocities; in particular at far smaller relative velocities than in gas-free star clusters (McKernan et al. 2012, 2018; Leigh et al. 2018). If a gas disk is locally isothermal, the gas torques cause all isolated orbiters to migrate inward (Goldreich & Tremaine 1979; Ward 1997; Tanaka et al. 2002). However in the more realistic case of a disk with an adiabatic midplane, for some values of the radial density and temperature gradients the torque from the disk can also lead to outward migration (Paardekooper & Mellema 2006). \nPaardekooper et al. (2010) used analytic arguments and numerical simulations to model the sign and strength of migration, and found that there are regions of gas disks where outward and inward torques cancel out; leading to a region of zero net torque where migration halts. Lyra et al. (2010) showed that such regions of zero net torque, or migration traps, are predicted by standard models of protoplanetary disks, and Horn et al. (2012) showed that the migration of protoplanets towards these migration traps can lead to the rapid collisional build-up of giant planet cores. \nWhile Paardekooper et al. (2010) considered only fully unsaturated torques (where the angular momentum of the corotational region is continuously replenished by viscous mixing, thus continuously driving migration), updated work showed migration rates including saturation Paardekooper et al. (2011). The basic change due to saturation is twofold. First, only larger orbiters with mass ratio q glyph[greaterorsimilar] 10 -5 will experience sustained outward migration. For lower masses, the width of the horseshoe region is small enough that diffusion saturates the torques; rapid inward migration occurs for planets outside of a narrow range in mass. Second, inclusion of saturation introduces a mass-dependency to both the location and existence of convergence zones (Hellary & Nelson 2012; Coleman & Nelson 2014; Dittkrist et al. 2014). The theory of planet migration continues to be refined, with a dynamic corotation torque found (Paardekooper 2014; Pierens 2015) dependent on the migration rate and viscosity, stemming from an asymmetry in the coorbital region as the planet moves. This torque can stall inward and boost outward migration, taking planets away from \nthe convergence zone and essentially enlarging the region of outward migration. Its action requires a ShakuraSunyaev (1973) viscosity parameter α glyph[lessorsimilar] 10 -2 for orbiters of mass ratio ≈ 10 -5 (see Appendix A). These torques were included in N-body calculations by Sasaki &Ebisuzaki (2016), who found that these torques helped form cores of giant planets. Finally, a heating torque was found by Ben'ıtez-Llambay et al. (2015), resulting from the protoplanets accretional luminosity, and found to counteract inward migration. The theory of this heating torque has been further developed by Masset (2017) and Eklund & Masset (2017), showing that it can lead to significant eccentricity and inclination pumping. A torque formula for inclusion in evolutionary simulations has been extracted by Jim'enez & Masset (2017). The state of the art in the application of these models for planet population synthesis calculations is discussed in Mordasini et al. (2017). \nIt is entirely plausible that migration models will undergo significant modifications in the future, driven by advances stemming from the unabated rate of exoplanet discoveries. Yet, some of the differences between AGN and protoplanetary disks cause pause, first and foremost the fact that the latter are relatively cold and thus poorly ionized, with large swaths not unstable to the MRI (Blaes & Balbus 1994; Gammie 1996; Wardle 1999; Bai & Stone 2010; Lesur et al. 2014; Lyra & Umurhan 2018). Application of planet migration theory to AGN disks should thus focus on results for high-viscosity and turbulent gas. In this respect, dynamical torques, requiring α glyph[lessorsimilar] 10 -2 , should probably not be too relevant (see Appendix A). The heating torque, on the other hand, should also exist for black hole orbiters in AGN disks: even though they do not have a surface to heat via accretional shocks, the accretion disks they develop are hot and luminous and should heat up the surrounding AGN gas. We defer exploring this sBH hole feedback effect to a future publication. \nIn this work, as in Horn et al. (2012), we prefer to work with the unsaturated torque because Nelson & Papaloizou (2004), Baruteau & Lin (2010), Uribe et al. (2011), and Baruteau et al. (2011) find that the co-rotational torques in turbulent disks are subject to stochastic turbulent fluctuations that keep the corotational torque unsaturated even in locally isothermal simulations. The result has been corroborated by more recent simulations (Guilet et al. 2013; Comins et al. 2016; Uribe et al. 2015); yet, because they could not resolve the width of the corotational region for smaller objects, saturation remains a possibility if the turbulent fluctuations are strong enough to wipe out their horseshoe turns. \nMcKernan et al. (2012) drew on the work of Lyra et al. (2010) and Horn et al. (2012) to develop a model describing a BH merger hierarchy in the AGN disk. McKernan et al. (2014) explored the consequences of this model and predicted that LIGO should detect gravitational waves from a previously unconsidered population of merging overweight sBH in AGN disks. Bellovary et al. (2016) explored this analogy applying the Paardekooper et al. (2010) migration torque model to two steady-state analytic supermassive black hole (SMBH) accretion disk models derived by Sirko & Goodman (2003) and Thompson et al. (2005). Bellovary et al. (2016) showed that migration traps do exist in both AGN disk models. \nHere we build on Bellovary et al. (2016), by using a modified version of the N-body code described by S'andor et al. (2011) and Horn et al. (2012) that implements several manifestations of the gravity of the gas disk around the SMBH in addition to the standard gravitational forces between particles. The additional effects include migration torques, a stochastic gravitational force exerted by turbulent density fluctuations in the disk, and inclination and eccentricity dampening produced by passages through the gas disk on inclined orbits ˙ In order to explore the dynamical behavior of multiple interacting sBHs approaching a migration trap, we take as an example the migration rates and other disk parameters derived from the analytic AGN disk model of Sirko & Goodman (2003). \nEmbedded sBHs will migrate towards the migration traps modeled in Bellovary et al. (2016), and due to this migration, sBHs on prograde orbits encounter each other at low relative velocities. These encounters provide favorable conditions for fast sBHB formation and evolution, resulting in frequent mergers detectable by LIGO. Future constraints from LIGO on this merger channel (e.g. from spins or rates) will allow us to constrain AGN disk physics better than present spectroscopic modeling efforts (see McKernan et al. (2018) for a discussion of which parameters can be best constrained by LIGO).", '2. METHODS': 'In this section we describe in detail our modified Nbody simulations. Our simulations neglect forces exerted by sBHs on the gas disk aside from those implicitly modeled by the migration torques, the effects of accretion onto either the central SMBH or orbiting sBHs, and general relativistic effects. We also only consider sBHs on prograde orbits and ignore sBHs on retrograde orbits around the central object. We defer detailed modeling of retrograde objects until the torques on them have been derived in work in progress. \n2.1. Disk Models \nFigure 1. SMBH accretion disk model used in our simulations (Sirko & Goodman 2003). From top to bottom are plotted the midplane temperature T , surface density Σ (in g cm -2 ), disk aspect ratio h ( H/r ), optical depth τ , and Toomre Q as a function of Schwarzschild radius R s . The top axis represents the translation from Schwarzschild radius to parsecs for a 10 8 M glyph[circledot] SMBH. \n<!-- image --> \nThe Sirko & Goodman (2003) model is a modification of the classic Keplerian viscous disk model (Shakura & Sunyaev 1973), with a constant high accretion rate fixed at Eddington ratio 0.5. The disk is assumed to be marginally stable to gravitational fragmentation; however the model does not directly take into account magnetic fields or general relativistic effects. The Sirko & Goodman (2003) model assumes some additional unspecified heating mechanism in the outer disk in order to maintain the stability of the disk and prevent fragmentation. \nSirko & Goodman (2003) use the opacity models from Iglesias & Rogers (1996) and Alexander & Ferguson (1994) for the high and low temperature regimes, respectively. The inner disk is optically thick due to a high rate of Thompson scattering from electrons produced by the ionization of hydrogen. The intermediate region of the disk has a lower electron density, and is therefore less optically thick and cooler. \nWe use a SMBH mass of M glyph[star] = 10 8 M glyph[circledot] . The total mass of the disk integrated out to 2 × 10 5 AU is 3 . 7 × 10 7 M glyph[circledot] . The midplane temperature, surface density, scale height, optical depth, and Toomre Q as a function of radius in this model are plotted in Figure 2.', '2.2. Torque Model': 'We model the disk torque on the sBHs using the analytical prescription of Paardekooper et al. (2010) which incorporates the effects of non-isothermal co-rotation torques. For the azimuthally isothermal case the normalized torque is \nΓ iso / Γ 0 = -0 . 85 -α -0 . 9 β, (1) \nwhile for the purely adiabatic case the normalized torque is \nγ Γ ad / Γ 0 = -0 . 85 -α -1 . 7 β +7 . 9 ξ/γ. (2) \nThe adiabatic index γ = 5 / 3, and the variables α , β and ξ represent the negative local gradients of density, temperature and entropy, respectively, and are defined as \nα = -∂ ln Σ ∂ ln r ; β = -∂ ln T ∂ ln r ; ξ = β -( γ -1) α. (3) \nThe torques are normalized by \nΓ 0 = ( q/h ) 2 Σ r 4 Ω 2 , (4) \nwhere q is the mass ratio of the migrator to the SMBH, h is the aspect ratio of the disk and Ω is the rotational velocity. \nThe effective torque is interpolated between the isothermal and adiabatic torque models using \nΓ = Γ ad Θ 2 +Γ iso (Θ + 1) 2 , (5) \nwhere Θ is the ratio of the radiative timescale to the dynamical timescale. Lyra et al. (2010) show that Θ depends on the local disk properties as \nΘ = c v ΣΩ τ eff 12 πσT 3 , (6) \nwhere c v is the thermodynamic constant at constant volume, σ is the Stefan-Boltzmann constant, and the effective optical depth taken at the midplane is (Hubeny', '1990; Kley & Crida 2008)': 'τ eff = 3 τ 8 + √ 3 4 + 1 4 τ . (7) \nThe true optical depth τ is given by \nτ = κ Σ 2 (8) \nwhere κ is the opacity used in the Sirko & Goodman (2003) models (see Section 2.1). \nEach component of the torque depends on the local disk gradients of density, temperature and entropy. These torques are implemented into our N-body code as forces on the particles with vector dependence \nF mig = Γ r ˆ θ (9)', '2.3. Turbulence': 'AGN disks are sufficiently ionized (certainly in the inner regions) that the magnetorotational instability (MRI) will drive turbulence. We use a model for turbulence developed by Laughlin & Bodenheimer (1994) and further modified by Ogihara et al. (2007) that gives the gravitational forces exerted by turbulent density fluctuations as \nF turb = -C ∇ Φ , (10) \nwhere C is a scaling factor relating the fraction of the force exerted on the gas by the potential Φ to the force that is exerted by the gas on a migrator embedded in the disk. This fraction is given as \nC = 64Σ r 2 π 2 M glyph[star] . (11) \nThe turbulent potential, Φ, is taken to be the sum of n = 200 independent, scaled oscillation modes \nΦ c , m = ψr 2 Ω 2 Λ c , m , (12) \nwhere ψ is a dimensionless measure of the strength of the turbulent force in comparison to the migration forces (see Section 2.2). It is related to the Shakura & Sunyaev (1973) viscosity parameter α by Baruteau & Lin (2010) as \nψ glyph[similarequal] 8 . 5 × 10 -2 hα 1 / 2 (13) \nwhere h is the aspect ratio of the disk and comes from the mode lifetime being set by the speed of sound. In our model h is not constant, but to fix the scaling in Equation (13) we set h = 0 . 05. MHD simulations of accretion disks suggest typical values for α of 10 -3 -0.1 (Davis et al. 2010). A value of α = 0 . 01 gives us ψ = 4 . 25 × 10 -4 . \nIn Equation (12), Λ c , m is one oscillation mode defined as \nΛ c , m = ξe -( r -r c ) 2 /σ 2 cos( mθ -φ c -Ω c ˜ t ) sin ( π ˜ t ∆ t ) . (14) \nEach oscillation mode is defined by m , an azimuthal wavenumber chosen from a log normal distribution between 1 and 64, and c denotes the initial center of the perturbation. The position c is given in cylindrical coordinates r c and φ c selected from uniform distributions from the inner boundary to the outer boundary of the disk and from 0 to 2 π , respectively. The z coordinate is assumed to be small enough to have a negligible effect. Ω c is the Keplerian angular velocity at r c . \nThe mode evolves as a function of ˜ t = t 0 + t , where t 0 is the time when the mode comes into existence. The lifetime of the perturbation is \n∆ t = 2 πr c mc s , (15) \nwhich represents the sound-crossing time for each mode. The radial scale of the perturbation is chosen from a Gaussian distribution and scales as σ = πr c / 4 m . \nAt the beginning of the simulation there are n = 200 modes. When one mode expires another mode is created so that there are always 200 modes. Ogihara et al. (2007) showed that all modes m > 6 can be left out of the summation to determine the total potential Φ. We use this simplification in our model and only include Φ perturbations where m< 7. Equation (10) is used in our model to calculate the turbulent force on a given migrator at position ( r, θ ) as a function of the local speed of sound, Keplerian angular velocity, surface density of the gas, and time. \nWe note that when the net vertical magnetic flux of the disk is not sufficiently large, spiral acoustic waves or even radiation stresses dominate angular momentum transport and accretion power instead of MRI turbulence (Jiang et al. 2017). While the perturbations generated through these mechanisms will not be identical to those produced by MRI turbulence, as modeled above, we anticipate they will have qualitatively the same effect on our simulations (see Section 4).', '2.4. Eccentricity and Inclination Dampening': "Tanaka & Ward (2004) have shown that the gas disk exerts a force on migrators that acts to dampen their orbital eccentricity, e , and inclination, i , leading to the co-planar circularization of orbiters. They give the timescale \nt damp = M 2 glyph[star] h 4 m Σ a 2 Ω , (16) \nwhere m is the mass of the migrator and a is the semimajor axis of the migrator. We follow the timescales given in Cresswell & Nelson (2008) for eccentricity and inclination, respectively: \nt e = t damp 0 . 780 (1 -0 . 14 glyph[epsilon1] 2 +0 . 06 glyph[epsilon1] 3 +0 . 18 glyph[epsilon1]l 2 ) (17) \nt i = t damp 0 . 544 (1 -0 . 30 l 2 +0 . 24 l 3 +0 . 14 lglyph[epsilon1] 2 ) (18) \nwhere glyph[epsilon1] = e/h and l = i/h . \nThe resulting forces acting on these timescales as a function of position and velocity of an orbiting body are \nF damp , r = -2 ( v · r ) r r 2 t e m ˆ r (19) \nF damp , z = -v z t i m ˆ z , (20) \nwhere ˆ r and ˆ z are unit vectors in the r and z directions, respectively. \n2.5. N-Body Code \nWe use the Bulirsch-Stoer N-body code described by S'andor et al. (2011) that was modified by Horn et al. (2012) to include the additional forces outlined above in Sections 2.2, 2.3, and 2.4. The total force acting on each sBH in our simulation is \nF total = F nbody + F mig + F damp + F turb . (21) \nThe forces acting from the gas disk, F mig , F damp and F turb , are calculated at the beginning of each BulirschStoer timestep and not recalculated during the modified midpoint method used to calculate F nbody . However, the Bulirsch-Stoer timestep is a small fraction of the dynamical timescales of the sBHs and is reduced during close encounters. Therefore holding these forces from the gas disk constant throughout each Bulirsch-Stoer timestep does not have a significant effect on the simulations. \nOur simulations consider two sBHs to have formed a new sBHB once two conditions have been met. First, they must approach each other within a mutual Hill radius, \nR mH = ( m i + m j 3 M glyph[star] ) 1 / 3 ( r i + r j 2 ) , (22) \nwhere m i and m j represent the masses of the two sBHs and r i and r j represent their distances from the SMBH. Second, the relative kinetic energy of the binary, \nK rel = 1 2 µv 2 rel , (23) \nwhere µ is the reduced mass of the binary, and v rel is the relative velocity between the two sBHs, must be less than the binding energy, \nU = Gm i m j 2 R mH . (24) \nDue to the complex and poorly understood interactions between sBHBs and the gas disk within the Hill sphere, for simplicity once a gravitationally bound sBHB forms, our model assumes that it is merged. Indeed it is likely given the conditions of our simulations that all sBHBs will merge within approximately 10-500 yr (Baruteau & Lin 2010), which is a short timescale compared to any dynamical timescales. However, escapes from within a mutual Hill sphere are of course possible. We discuss the merging of sBHBs in our simulations further in Section 5.", '3. INITIAL STELLAR MASS BH POPULATIONS': "In this section we describe the two models for the initial sBH populations used in our simulations, which are outlined in Table 1. We choose the number of sBHs in each model based on the lower limit of about 10 3 sBHs within 0.1 pc of a SMBH estimated by Antonini (2014) based on the distribution of S-Star orbits around Sgr A glyph[star] . This estimate is consistent with the population of O(10 4 ) sBHs within 1 pc of Sgr A ∗ inferred by Hailey et al. (2018). Assuming sBHs are uniformly distributed throughout the disk, we estimate that around 1% of sBHs in an AGN disk will be within the inner 1000 AU ( ≈ 0.005 pc). Both of our models therefore include ten sBHs within roughly 1000 AU. \nThe gravitational wave decay lifetime of a sBH a few hundred AU from a SMBH in a gas-free nucleus is (Peters 1964), \nT ( a 0 , e 0 ) ≈ 768 425 (1 -e 2 0 ) 7 / 2 a 4 0 4 β , (25) \nwhere β is, \nβ = 64 5 G 3 m 1 m 2 ( m 1 + m 2 ) c 5 . (26) \nUsing m 1 = 10 8 M glyph[circledot] , m 2 = 30 M glyph[circledot] , e 0 = 0.05, and a 0 =650 AU as fiducial values that are used in our runs (see below), gives a decay time of approximately 3 . 72 × 10 11 yr. Since this value is several orders of magnitude longer than the run time of our simulations, our models do not include the gravitational wave decay of the orbits of the sBHs around the SMBH. \nOur three fiducial models (labeled F1-F3 in Table 1) contain 10 sBHs of uniform masses. This uniform mass distribution is different for each fiducial model, and \nTable 1. Models. Column 1: Name of run; Column 2: initial masses (or range of masses) of bodies in M glyph[circledot] ; Column 3: the total combined mass of all bodies in the run in M glyph[circledot] ; Column 4: the time it takes for all bodies to reach the migration trap or resonant orbits in megayears; Column 5: the time for a sBHB of over 50 M glyph[circledot] to form in megayears; Column 6: the mass in M glyph[circledot] of the most massive sBH at the end of the run. \n| Run (1) | M sBH (2) | m tot (3) | T mig (4) | T form (5) | m max (6) |\n|-----------|-------------|-------------|-------------|--------------|-------------|\n| F1 | 10 | 100 | 0.14 | 0.129 | 70 |\n| F2 | 20 | 200 | 0.025 | 0.008 | 100 |\n| F3 | 30 | 300 | 0.014 | 0.002 | 240 |\n| LMA | 5-15 | 74 | 0.7 | N/A | 46 |\n| LMB | 5-15 | 100 | 2.8 | 1.5 | 65 |\n| HMA | 5-30 | 97 | 0.45 | 0.24 | 60 |\n| HMB | 5-30 | 95 | 0.8 | 0.56 | 59 | \nranges from 10 M glyph[circledot] in F1 to 30 M glyph[circledot] in F3. The innermost sBH has an initial semi-major axis of 500 AU. The semi-major axis of each successive sBH is separated by 30 R mH from the one before it (see Equation 22). These initial positions are chosen to create a distribution of sBHs around the migration trap found at roughly 667 AU by Bellovary et al. (2016). We note that this initial distribution is somewhat arbitrary, however, these fiducial runs are mainly used as a baseline example to show how sBHs of different masses and initial semi-major axes that are initially not under each other's gravitational influence can migrate to form sBHBs in an AGN disk. \nIn our second set of models the masses of the sBHs vary in a more physically realistic manner. We draw them from the initial mass function for massive stars given by Kroupa (2002), by drawing from a Pareto power law probability distribution of sBHs with a probability density \np ( x ) = am a 0 x a +1 , (27) \nwhere a = 1 . 35, m 0 is a scale factor of 5 M glyph[circledot] , and x is a mass that is drawn from the distribution. \nIn our two lower mass runs, denoted LMA and LMB, a randomly generated mass is rejected if it is greater than 15 M glyph[circledot] , so the masses of the sBHs range from 5-15 M glyph[circledot] . In our two higher mass runs, denoted HMA and HMB, the mass is allowed to range from 5-30 M glyph[circledot] . Despite being denoted higher and lower mass runs, the total mass of the higher mass runs does not always exceed that of the lower mass runs because of random variation. This is the case for LMB, for example, which has the highest total mass of 100 M glyph[circledot] . The initial semimajor axes for the sBHs in these models are chosen randomly from a uniform distribution ranging from 300-1000 AU. We do not use an initial-final mass relation for the sBHs \n(i.e. Fryer et al. 2012) which would require us to make assumptions of the metallicity and supernova explosion model of our simulations. However, our distribution of initial masses for our sBHs remains similar to what they would be if such a relation had been used. \nFor all models, the initial eccentricities and inclinations of the sBHs are selected randomly from a Gaussian distribution. The mean value for the initial eccentricity is 0.05, with a standard deviation of 0.02. Selections are made until the value is positive. The mean value for the inclination is 0 with a standard deviation of 0 . 05 · , and the absolute value of the randomly selected value is used. The initial mean anomaly and pericenter values are chosen randomly from a uniform distribution ranging from 0 to 2 π . \nFor the variable mass models the distance between sBHs is calculated based on the randomly generated positional coordinates. If any two sBHs are within 10 AU of each other, a new distribution is generated until no two sBHs are within 10 AU of each other. \nThe masses of the sBHs remain constant over the course of the simulations, i.e. the sBHs are not accreting gas. This is a realistic simplifying assumption based on the Eddington-limited accretion rates, which would give a mass doubling time of about 40 Myr. Since our simulations are only run for 10 Myr and most of the mergers take place within the first few megayears, the additional mass due to accretion is insignificant, to both the mass of the object, and the migration rate. Gas accretion onto the sBHs could have a significant effect on the gas disks around the sBHs (i.e. feedback). However, these back reactions have not been well quantified and so we defer the study of the effects of gas accretion to future work. \nThese models were run for 10 Myr which is within the range of estimated lifetimes for an AGN disk (Haehnelt & Rees 1993; King & Nixon 2015; Schawinski et al. 2015). However, the final orbits of all sBHs in all seven models are established in less than 3 Myr, and these orbits remain stable for the remainder of the run. Over longer periods of time we would expect more sBHs to migrate inwards towards the SMBH from the outer disk. These sBHs may perturb the stable resonant orbiters or sBHs in the migration trap. We defer investigation of this evolution to future work.", '4.1. Fiducial Model': "Figure 2 shows the migration history for runs F1, F2, and F3 in the top, middle, and bottom panels, respectively (see Table 1). In the top two panels of Figure 2 the figures on the left show the migration history from \nthe start of the run to shortly after the final merger. The orbits of the remaining sBHs stay the same until the end of the 10 Myr run. The figures on the right in the top two panels are zoomed in views of mergers for runs F1 and F2. \nThe bottom left panel of Figure 2 shows the main period of mergers for the F3 run. The orbits of the four remaining sBHs remain the same for over 5 Myr. However, a turbulent mode (see Section 2.3) opens up near the remaining orbiters at around 5.3 Myr causing the 60 M glyph[circledot] sBH to form a sBHB with the 180 M glyph[circledot] sBH. This merger is shown in the bottom right panel of Figure 2. The turbulent mode continues to cause the semimajor axes of the orbits of the sBHs' in the migration trap to oscillate. The oscillations are more distinctive in the semimajor axis of the 30 M glyph[circledot] sBH, because it is significantly less massive than the 240 M glyph[circledot] sBH. \nIn these fiducial runs it is clear that more massive bodies migrate faster towards the migration trap, as expected since the migration torque is proportional to the square of the mass of the orbiter and so the acceleration is linearly proportional to mass. Thus, more massive sBHs reach the migration trap more rapidly. For example, the sBHs in model F3 all reach the migration trap or nearby resonant orbits in roughly 14 kyr, whereas it takes the sBHs in model F1 around 140 kyr. In all cases the last sBHs to reach the migration trap region are the innermost sBHs. These innermost sBHs have the slowest migration rates because within 1000 AU of the SMBH the aspect ratio of the disk increases with proximity to the SMBH (see Figure 2). The higher aspect ratio of the inner disk also means that the innermost sBHs will remain on eccentric orbits longer than sBHs since the damping force, F damp , r is inversely proportional to h 4 (see Equations 16 - 19). \nFigure 3 shows the growth of sBHs through mergers over time. Massive bodies approaching or reaching the migration trap encounter each other at high rates. Since binaries form at greater rates as sBHs migrate towards the migration trap, the faster migration rate of the more massive bodies leads to faster sBHB formation in the more massive fiducial models. For example, F2 and F3 both have four sBHBs form within the first 10 kyr, whereas it takes nearly 50 kyr for a sBHB to form in F1. \nFigure 4 shows the eccentricity of all ten sBHs over the first 200 kyr for the F1 run. While the initial eccentricities of the sBHs' orbits are dampened by the gas within the first 10 kyr, these eccentricities can actually delay sBHB formation at earlier times in our simulations. sBHs that are on eccentric orbits may pass within a Hill radius of each other, but because their orbits have \nFigure 2. The migration of ten sBHs for all three fiducial runs. The initial masses of the sBHs are 10 M glyph[circledot] (top), 20 M glyph[circledot] (middle), and 30 M glyph[circledot] (bottom). Each colored line represents one sBH and is labeled by its final mass in M glyph[circledot] . Each vertical dashed black line represents a time at which a bound binary forms. The figures on the left show the main period during which binary formation occurs. In the top two panels the sBHs remain on the same orbits that they are on at the end of the figures for the remainder of the simulations. In the bottom panel turbulence knocks a sBH out of resonance after roughly 5 Myr (see bottom right panel). The figures on the right show zoomed in views of various episodes of binary formation. The 100 M glyph[circledot] and 40 M glyph[circledot] sBHs in the center left panel and the 240 M glyph[circledot] and 30 M glyph[circledot] in the bottom right panel end up on the trojan orbits discussed in Section 4.2 \n<!-- image --> \nFigures 5-8 show the migration histories for runs LMA, LMB, HMA, and HMB, respectively (see Table 1). The top panel of all four figures shows the migration histories of the simulation up until all sBHs have reached stable orbits. In each simulation the sBHs remain at these final radii for the remainder of the 10 Myr run. The bottom two panels of these figures show examples of sBH interactions. \n<!-- image --> \n<!-- image --> \nFigure 3. The masses of the sBHs over time for runs F1, F2, and F3 are shown in the top, middle, and bottom figures, respectively. Each colored line represents a sBH. The dashed black line represents the total mass of all sBHs in the model. \n<!-- image --> \ndifferent pericenter phases their relative velocities are great enough that the relative kinetic energy of the two sBHs remains greater than their binding energy (see Section 2.5). \nOscillations in the eccentricity of the orbits of the sBHs that occur later in the run are due to interactions between sBHs. As the sBHs migrate into closer proximity with each other they will be pulled towards each other. This feature can be seen in Figure 2 as little spikes in the semimajor axes of the orbiters. These spikes can be periodic if they occur when two orbiters with similar semimajor axes are in phase with each other. The change in semimajor axis drives the eccentricity of the sBHs. The gas disk will dampen these eccentricities, leading to a decrease in eccentricity until another close pass occurs. These interactions are what cause the oscillations in Figure 4. The eccentricity of the sBH orbits rarely increases to more than 10 -2 . This feature is common in our simulations and is discussed further in Section 4.2. \nWhen sBHs pass within just a couple of Hill radii of each other, whether on not their relative kinetic energy is low enough to form a sBHB, the effective semimajor axis of their orbits around the SMBH often spike dramatically as their orbits are strongly perturbed from Keplerian orbits, as can be seen in Figure 2. However, this should be interpreted as a dramatic change in velocity rather than position. \nIn our runs the most massive sBH consistently ends up closest to the migration trap. However, in some cases, such as in the F2 run, no sBH ends up precisely in the migration trap. Instead the most massive sBH ended up roughly 2.5 AU away from the migration trap. At these small distances, the migration torque is very minimal, and the dynamics due to the high density of sBHs in the region play a larger role in determining the orbits' positions. Less massive sBHs end up either on Trojan or resonant orbits that exchange angular momentum with the other sBHs. These final configurations tend to be stable on megayear time scales. However, MRI turbulence can lead to sBHB formation even after these stable orbits are established if it knocks a sBH out of resonance, as happened in the F3 run. Encounters with other objects either being ground down into the disk, or migrating inward from further out in the disk might also disturb the steady configurations over longer time scales.", '4.2. Varying Masses': "Figure 4. The eccentricities over the first 200 kyr of all ten sBHs in the F1 run. Initial eccentricities are quickly dampened by the gas in less than 10 kyr, however interactions between the sBHs in close proximity drives the eccentricity of the sBHs' orbits as they are pulled towards each other. This eccentricity is then dampened, until another close passage occurs. \n<!-- image --> \nAs the more massive sBHs migrate through the disk they overtake less massive sBHs and frequently form sBHBs. The time that elapses before the first binary capture of the simulation varies among the four runs from a few hundred years to roughly 20 kyr due to the randomly generated initial positions and eccentricities. Even if two sBHs have similar initial positions at the beginning of a simulation, if the orbits of the sBHs are too eccentric the relative kinetic energy of the two sBHs that approach each other within 1 R mH may be higher than their binding energy preventing them from forming a sBHB (see Section 4.1). \nFigure 9 shows the build up in mass of the sBHs due to mergers for runs LMA (top left), LMB (top right), HMA (bottom left), and HMB (bottom right). In our simulation two sBHs are considered merged as soon as they form a sBHB (i.e. approach each other within 1 \nR mH ; see Section 2.5). 6-8 mergers occur in each run. The most massive sBH at the end of each run ranges from 45-65 M glyph[circledot] , which represents 60-65% of the total mass of the run. \nThe time that elapses before all sBHs reach the migration trap also varies and depends on the random generation of positions and masses. The smaller the initial semimajor axis of a sBH the longer it will take to migrate towards the trap, especially if it has a smaller initial mass. \nDynamical effects can produce some exceptions. For example, in the LMB run (see Figure 6), there are three sBHs with very small initial semimajor axes ranging from 310 AU to 320 AU. Being in such close initial proximity causes the sBHs to interact with each other from the start, but they do not immediately form a sBHB. The least massive sBH only has a mass of 5 M glyph[circledot] and \n<!-- image --> \nFigure 5. The migration of ten sBHs of varying mass in model LMA. Each colored line represents one sBH and is labeled by its final mass in M glyph[circledot] . Each vertical dashed black line represents a time at which a collision occurs. The top figure shows the first 1.1 Myr which is the period during which binary formation occurs, and all sBHs migrate towards the migration trap to stable orbits where they remain for the rest of the 10 Myr run. The middle figure is a zoomed in view of the first binary capture (so early that it is barely visible in the top panel) and the bottom figure is a zoomed in view of a later period. \n<!-- image --> \nFigure 6. Migration in the LMB run, with the same notation as Figure 5. The top figure shows the first 4.5 Myr which is the period during which binary formation occurs, and all sBHs migrate towards the migration trap to stable orbits where they remain for the rest of the 10 Myr run. The middle figure is a zoomed in view of the first period of binary formation and the bottom figure is a zoomed in view of the interaction between the three innermost sBHs, two of which end up co-orbital. \n<!-- image --> \nFigure 8. Migration in the HMB run, with the same notation as Figure 5. The top figure shows the first 1.1 Myr, which is roughly the period during which binary formation occurs, and all sBHs migrate towards the migration trap to stable orbits where they remain for the rest of the 10 Myr run. The bottom two figures are zoomed in views of the first (middle panel) and last (bottom panel) periods of binary formation. In the bottom panel the 5 M glyph[circledot] sBH that is the last to reach the migration trap region merges with a 9 M glyph[circledot] sBH that is on a resonant orbit with the other sBHs. This event breaks the resonance of the sBHs orbiting near the migration trap. \n<!-- image --> \nFigure 7. Migration in the HMA run, with the same notation as Figure 5. The top figure shows the first 600 kyr which is roughly the period during which binary formation occurs, and all sBHs migrate towards the migration trap to stable orbits where they remain for the rest of the 10 Myr run. The middle figure shows a zoomed in view of binary formation that breaks apart two co-orbital sBHs and the bottom panel shows a zoomed in view of a later period of binary formation. In the top panel the 6 M glyph[circledot] sBH is the last to reach the region of the migration trap, because it has a small initial semimajor axis. When it reaches the trap it ends up on its own resonant orbit, instead of merging with other sBHs. \n<!-- image --> \nFigure 11 shows the semimajor axes (upper panels) and eccentricities (lower panels) of the sBHs in or near the migration trap for the LMA (top left), LMB (top right), HMA (bottom left), and HMB (bottom right) \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFigure 9. The growth of sBHs through mergers over time for the LMA (top left), LMB (top right), HMA (bottom left), and HMB (bottom right) runs starting at 400 yr and ending at 2 Myr after which no mergers take place. Each colored line represents a sBH. The dashed black line represents the total mass of all sBHs in the model. \n<!-- image --> \nafter the three-body interaction ends up on its own at approximately 300 AU. This low mass sBH left alone in a region with a very low migration rate takes nearly 3 Myr to finally make it to the migration trap. The two more massive sBHs (8 M glyph[circledot] and 14 M glyph[circledot] ) end up in a stable horseshoe co-orbit as modeled by Cresswell & Nelson (2006), who found that it was common for planets in a protoplanetary disk to become co-orbital, occupying either horseshoe or tadpole orbits that survived for the duration of their runs. Figure 10 shows the relative phase, semimajor axes, and ratio of the orbital period around the SMBH for these two co-orbital sBHs. Over a period of thousands of orbits the phase difference between the two sBHs oscillates between 180 o and 20 o . When the phase difference is at a minimum the two sBHs swap radial positions. Occasionally the migration rate of the more massive, 14 M glyph[circledot] , sBH is large enough compared to the migration rate of the less massive, 8 M glyph[circledot] , sBH that it overtakes it while the two are out of phase. However, the two sBHs still swap radial positions when they are closest to being in phase. As a result the 8 M glyph[circledot] sBH mi- \ngrates at the rate of the 14 M glyph[circledot] sBH, which means the 8 M glyph[circledot] sBH reaches the migration trap at nearly double the rate it would alone. \nIn the HMA run, the cyan and purple sBHs in the center panel of Figure 7 are also on a horseshoe co-orbit until the orbit is destabilized by the presence of a 26 M glyph[circledot] sBH, which the cyan sBH merges with. The coorbital tadpole (i.e. Trojan) orbits that were observed by Cresswell & Nelson (2006) are seen in runs F2 and F3 (see Figure 2). \nIn all cases, after several hundred kyr one sBH becomes massive enough to dominate the region closest to the migration trap and lock all other less massive sBHs in high-order resonant orbits. sBHs migrating towards the trap at later times will either merge with the sBHs already populating resonant orbits (Figure 8), or end up on their own resonant orbit (Figure 7). \n<!-- image --> \n<!-- image --> \n45000 46000 47000 48000 49000 50000 51000 52000 53000 Time (Years) \nFigure 10. Two sBHs from the LMB run in a stable horseshoe co-orbital configuration. The top panel shows the relative phase between the sBHs, the middle panel shows the semimajor axes of the two sBHs, which are labeled by their current masses, and the bottom panel shows the ratio of orbital periods around the SMBH. \n<!-- image --> \nruns. As in the F2 run (see Section 4.1), in the LMA, LMB, and HMB runs no sBH ends up exactly in the migration trap. Instead the most massive sBH ends up 1-2.5 AU from the migration trap, where it becomes \nlocked in a resonant orbit with the other sBHs. The semimajor axes of the sBHs around the SMBH spike periodically as the sBHs on resonant orbits exchange angular momentum with each other and the sBHs in the migration trap get pushed back into resonance. The sudden change in the orbit's semimajor axis causes a spike in eccentricity that is then dampened by the gas. Figure 12 shows one example from the HMB run of these interactions of two sBHs on a 27:28 resonance. When the phase difference between the sBH in the migration trap and the sBH on a resonant orbit is zero, the two are pulled towards each other by their mutual gravitational attraction. This temporarily drives an increase in the eccentricity of their orbits, before it is gradually dampened once again by the gas disk. \nThese orbits remain stable for 9 Myr to the end of runs LMA, HMA, and HMB, suggesting that trapping sBHs in resonant orbits around a migration trap could prevent more massive sBHs from building up. However in the LMB run, as in the F3 run (see Section 4.1), a perturbative force caused by disk turbulence pushes the 23 M glyph[circledot] sBH out of resonance so that it merges with the 42 M glyph[circledot] sBH in the migration trap. Therefore disk turbulence could provide a mechanism to break resonances, and create more massive sBHs. Horn et al. (2012) showed that increasing levels of disk turbulence makes this mechanism even more efficient. Batygin & Adams (2017) worked out an analytic solution for the breaking of resonances by turbulence for protoplanetary disks and found that the disruption of resonances by turbulence depends most strongly on the migrator-central mass ratio. For the migrator-central mass ratios and other relevant parameters in our simulations, their analytic solution agrees with our conclusion that turbulence could play a role in disrupting resonances. \nThe initial inclinations of the sBHs were very small, and all sBHs were quickly ground down into flat orbits in less than 50 yr. The initial eccentricities played a role in our models in preventing early sBHB formation, but were also a transient effect and were dampened by the gas in roughly 10 kyr. Larger initial values for inclination and eccentricity would likely delay sBHB formation because it would increase the relative kinetic energy of two sBHs. However, these larger inclinations and eccentricities will eventually be dampened by the gas disk, and as sBHs are ground down into the disk and their orbits are circularized, they would start to form sBHBs with other sBHs at later times.", '5. SUMMARY AND DISCUSSION': "We have simulated the migration of compact objects in a model AGN disk (Sirko & Goodman 2003), using an \nFigure 11. Clockwise from the top left are zoomed in views of the stable resonant orbits of runs LMA, LMB, HMA, and HMB. The top figure for each run shows the semimajor axis and the bottom figure shows the eccentricity. In each plot each line represents one sBH and is labeled by its final mass in M glyph[circledot] . \n<!-- image --> \nanalytic model developed from simulations of the migra- \ntion of protoplanets in protoplanetary disks. We have \nFigure 12. A zoomed in view of the interactions between two sBHs from the HMB run on resonant orbits. The top plot shows the semimajor axis of the two sBHs, the middle plot shows the phase difference between the two sBHs, and the bottom plot shows the eccentricity of the two sBHs. When the phase difference reaches zero (represented by the vertical dashed line) the two sBHs are pulled towards each other by gravity making their orbits more eccentric. This eccentricity is then dampened until they are pulled towards each other again. \n<!-- image --> \nFigure 13. Various timescales in years are plotted as a function of radial distance from the SMBH in AU. The blue, yellow, and green lines represent the approximate time for 10 M glyph[circledot] , 20 M glyph[circledot] and 50 M glyph[circledot] sBHs to migrate from their current location to the SMBH due to only migration torques. The dashed and dotted black lines represent the merger time for a sBHB that forms when two sBHs are within a mutual Hill radius (see Equation 22) for a prograde orbiting sBHB and a retrograde orbiting sBHB, respectively. The merger timescales are significantly shorter than the migration timescales, suggesting that the probability of the sBHB failing to merge due to an encounter with a tertiary body is low. \n<!-- image --> \nfound that migration due to gas torques in AGN disks can provide an efficient mechanism to create a population of hard compact object binaries remarkably quickly, replicating the results of Horn et al. (2012) for protoplanets in a protostellar disk, but for the case of sBHs in an AGN disk. \nMcKernan et al. (2018) parameterized the rate of sBHsBH mergers in AGN disks as, \nR = 12 Gpc -3 yr -1 N GN 0 . 006 Mpc -3 N BH 2 × 10 4 f AGN 0 . 1 X f d 0 . 1 f b 0 . 1 glyph[epsilon1] 1 ( τ AGN 10 Myr ) -1 , (28) \nwhere N BH is the number of sBHs in an AGN disk, N GN is the average number density of galactic nuclei in the Universe, f AGN is the fraction of galactic nuclei with AGN that last for time τ AGN , f d is the fraction of sBHs that end up in the AGN disk, f b is the fraction of \nsBHs that form binaries, and glyph[epsilon1] represents the fractional change in N BH over one full AGN duty cycle. Using our finding that within the inner 1000 AU of an AGN disk 60-80% of sBHs form sBHBs in the lifetime of our AGN disk, we can use 0.6-0.8 as an upper limit on f b , giving an upper limit on the merger rate of 72 Gpc -3 yr -1 . This value is an upper limit because, although our model assumes a uniform distribution of sBHs throughout the disk, sBHs in the outer disk, further from the migration trap, may merge less frequently. In addition, this upper limit assumes that sBHs orbiting in the retrograde direction would have similar sBHB formation rates, which is unlikely because migration torques on retrograde orbiters should be much weaker. We defer a more realistic prediction of retrograde orbiter merger rates and merger rates of sBHs in the outer disk to future work. \nUncertainties in AGN disk structure result in a wide variety of plausible theoretical models to describe these disks. However, migration traps should occur in any disk where there is a rapid change in the surface density gradient (Bellovary et al. 2016). Such rapid changes are likely to occur in most actual disks, since radiation pressure is expected to inflate the inner disk. This paper is intended to highlight the qualitative behavior of objects at the migration trap. Regardless of the location of these migration traps, whether they are at 331 R s as in the Sirko & Goodman (2003) model or about 225 R s as in the Thompson et al. (2005) model (Bellovary et al. 2016), most of the binary formation will take place in the immediate vicinity of the migration traps. We expect the qualitative behavior around the migration trap to be similar regardless of AGN disk model, although having a migration trap at a different distance from the SMBH as in Thompson et al. (2005) will affect how long it will take sBHs to migrate to the trap. Additionally, different disk models have different surface densities, which will affect migration rates. If these surface densities are lower than in Sirko & Goodman (2003), as they are in Thompson et al. (2005), the migration rates will be lower. We defer simulations of migration in alternative AGN disk models to future work. \nWe highlight that although we have taken the compact objects to be sBHs here, similar results apply to any objects embedded in the AGN disk, including neutron stars, white dwarfs, or main sequence or evolved stars, although their typically lower masses will result in slower migration rates. Our demonstration of how quickly binaries can form in AGN disks may help us to understand the behavior of other objects embedded in AGN disks. For example, Davies et al. (1998) attributed the observed lack of red giant stars in the galactic center to direct collisions during single-binary encounters. \nWe might suggest a simple alternative, albeit analogous mechanism motivated by our results in this paper: mainsequence turn-off stars efficiently form (or are exchanged into) compact binaries, such that they form common envelope binaries (or some other variation of the myriad of possible binary evolution pathways) when the turnoff star evolves up the giant branch, preventing it from evolving in to a normal red giant star. In short, a myriad of binary and stellar exotica could form in AGN disks. These additional compact objects could also contribute non-negligibly to subsequent binary mergers and interactions (Leigh et al. 2016), and even produce exotic populations that might contribute to the total light distribution in galactic nuclei non-negligibly, once the gas disk has dissipated and the SMBH is no longer actively accreting at high rates. \nOne assumption of our model is that sBHBs merge as soon as they form. These binaries actually harden due to gas torques on a timescale that depends on the distribution of gas in the Hill sphere of the binary, and which also involves the complicated effects of accretion onto the sBHB and the resulting feedback. We justify our assumption by comparing the migration timescale to the binary hardening time scale. Baruteau et al. (2011) modeled the hardening of binaries in a gas disk. Their models showed that it takes roughly 1000 orbits of binary stars around the binary's center of mass to reduce the semimajor axis of the binary by a factor of two if the binary is rotating in the prograde direction with respect to its orbit around the central mass, and only 200 orbits for retrograde rotation. \nWe assume that after the binary's semimajor axis has been halved 20 times, the sBHB separation is small enough that gravitational radiation will rapidly merge the sBHB to form a single sBH of mass m i + m j . The binary inspiral time due to gravitational wave emission alone (Peters 1964), neglecting any gas hardening effects, exceeds the binary hardening timescale of 4-200 × 10 3 orbits as long as the binary eccentricity e < 0 . 9995. Note that this estimate may be a significant underestimate of the actual time to merger, since gas hardening may become less efficient as the binary shrinks. However, we have also neglected the possibility of hardening encounters due to tertiary objects in the disk, which will accelerate the rate of binary hardening (Leigh et al. 2016, 2018). Both of these complications will require further study in future work. \nGiven our assumptions, Figure 13 shows the approximate radial dependence of the timescales of mergers for sBHBs rotating in prograde and retrograde directions, for sBHBs orbiting in the prograde direction through the disk. In our simulations these timescales will be equiva- \nlent for all mass sBHBs because sBHs are considered to form a sBHB when they approach each other within a mutual Hill radius, which is ∝ ( m i + m j ) 1 / 3 . For comparison, in Figure 13 we plot the time for 10 M glyph[circledot] , 20 M glyph[circledot] and 50 M glyph[circledot] sBHs to migrate from their current radial location to the SMBH due to migration torques. Recall that the migration torques vary as a function of radius and temperature, surface density and disk aspect ratio at each radius. Since the migration timescales of these objects are at least an order of magnitude larger than the time it would take for a sBHB of the same mass to merge, we can see that the likelihood of a tertiary encounter from another sBH is low. \nThis low likelihood is important because while a tertiary encounter could accelerate a binary merger (Leigh et al. 2018), the third sBH would be ejected in the process, and because the sBHB will already be merged in our simulation, it is not possible for a third body to gain energy from a three-body encounter. However, our simulations do permit binary formation to occur via three-body interactions in a limited set of realistic circumstances. That is, three initially isolated sBHs could end up in a sufficiently small volume that their mutual gravitational attraction dominates locally, and a chaotic three-body interaction ensues. If one star is ejected, the other two remaining sBHs could form a binary. \nThe dissipative effects of the gas actually enhance the probability of such three-body mediated binary formation occurring. The critical orbital separation of a sBHB for which the kinetic energy of a third isolated sBH is equal to the orbital energy of the sBHB is known as the hard-soft boundary. Third body encounters with hard binaries promote hardening, while with soft binaries they can promote ionization. In an AGN disk the hard-soft boundary for a sBHB in a circular orbit is (Leigh et al. 2018) \na HS , disk = (12) 1 / 3 R H ( µ b /M 3 ) 1 / 3 , (29) \nwhere R H is the Hill radius, µ b is the reduced mass M 1 M 2 / ( M 1 + M 2 ) of the binary, and M 3 is the mass of the third sBH. Since we consider sBHBs to be merged once they are within a Hill radius, as long as 12 µ b /M 3 > 1, the kinetic energy from a prograde tertiary sBH should not be enough to ionize a sBHB in our simulations. \nLooking at examples in our simulations of sBHBs that have a close encounter with a third sBH on timescales shorter than the merger timescales in Figure 13, we find only one instance where a third sBH is massive enough that a HS , disk < R H . However in this case the third sBH never approaches closer than 10 R H from the sBHB, making it too distant to ionize the sBHB. Therefore in \nour models the ionization of our binaries by a prograde third body interaction appears to be rare. \nFinally it is possible for one of two sBHs within a mutual Hill radius of each other to be ejected, even if the binding energy of the two sBHs is less than their relative kinetic energy and there is no tertiary interaction. Preliminary results from Secunda et al. (in prep.) suggest that this is rare. In Secunda et al. (in prep.) the merger boundary is reset to 0.65 R mH for the same runs as in this paper. This boundary was chosen to allow us to study some of the properties of the sBHBs we were forming without requiring unreasonably large computational resources, in the form of integration time. In 4 out of 32 cases, sBHs that would have merged under the criteria presented in this paper did not merge when the boundary for merger is 0.65 R mH . Instead these sBHs swapped orbits. This orbit swapping in place of sBHB formation is already seen in runs with the sBHB formation criterion set to 1 R mH (see the yellow and brown lines in the center panel of Figure 6). Additionally, those sBHs that failed to merge initially later were able to merge with other sBHs. Therefore, the merger histories of runs with a more stringent merger criterion were qualitatively identical to those presented above. \nFuture work that includes the relevant gas physics should evolve sBHBs to much smaller merger boundaries of 0.1 to 0.01 R mH to further probe the poorly understood evolution of sBHBs in a gas disk. We do not do so here for simplicity's sake, since whether a sBHB in a gaseous accretion disk will be able to merge is still an open question. For example, recent work by Moody et al. (2019) found that the circumbinary disk around BHBs can actually exert a net positive torque on the BHB, causing its semi-major axis to increase. The properties of the sBHBs formed in our simulations should serve as useful, physically motivated inputs for future hydrodynamic simulations of BHB evolution in gas disks. \nOur model is efficient at building up massive sBHBs on timescales far shorter than the lifetime of the AGN disks that host them. We argue that these sBHBs are likely to merge, producing gravitational wave events such as those observed by LIGO. However, future work using hydrodynamic simulations is needed to better describe the interactions between the gas disk and the sBHs, in particular examining the binary hardening timescale due to gas torques. More work is also needed to model the evolution of the sBH population as additional compact objects either drift inward or have their orbital inclination ground down into the inner region of the disk where they may be able to break resonances and form additional sBHBs and more massive sBHs. We have also completely \nthe migration time scale \nignored the role of retrograde orbiters in this paper, and the population of objects on retrograde orbits that can ionize binaries embedded in the disk. General relativistic effects are also not included in our model. Ultimately a full, three-dimensional, time-evolving AGN disk model should be used to provide the most accurate predictions for the merger rates of sBHs and the build-up of overmassive sBHs. In the meantime, constraints from the next few LIGO runs on mergers from this model channel should help put limits on models of AGN disks (in particular, the presence or absence of density gradients likely to produce migration traps), such as those used here. \nWe thank BridgeUP: STEM Brown Scholars Juliette Cornelis, Denelis Ferreira, Ariba Khan, Anna Li, Audrey Soo, and Anay Vicente for their test runs and \npreliminary figures, and the anonymous referee for a useful report that improved the clarity and accuracy of the paper. A.S. was supported by a fellowship from the Helen Gurley Brown Revocable Trust, and N.W.C.L was supported by a Kalbfleisch Fellowship, both at the American Museum of Natural History. M.M.M.L. was partly funded by NASA Astrophysical Theory Grant NNX14AP27G and NSF Grant AST18-15461. N.W.C.L. and M.-M.M.L. were partly funded by NSF Grant AST11-0395. B.M. and K.E.S.F. was partly supported by NSF PAARE AST11-53335 and NSF PHY1125915. J.M.B. acknowledges support from PSC-CUNY award 60303-00 48, and W.L. acknowledges support from Space Telescope Science Institute through grant HST-AR-14572 and the NASA Exoplanet Research Program through grant 16-XRP16 2-0065", 'A. DYNAMICAL TORQUE': "As we note in Section 1, the underlying physics of migration remains uncertain. For example, Paardekooper (2014) found that for low enough viscosity, if there is a radial gradient in vortensity the planets in a protoplanetary disk can experience dynamical torques in addition to the static torques implemented in our model. These dynamical torques could act to slow down inward migration and even lead to runaway outward migration. However, at least in the particular example we focus on in this paper, these effects do not necessarily act. Paardekooper (2014) emphasizes that dynamic migration only sets in when \nk ∼ m c τ ν /τ mig > 1 / 2 , (A1) \nwhere the coorbital gas mass in planet masses is \nm c = 4 q d ˜ x s /q, (A2) \nτ mig = π 2 h 2 q d q Ω , (A3) \nand the time for viscosity to adapt the co-orbital vortensity to the ambient value is \nτ ν = x 2 s /ν. (A4) \nThe half-width of the horseshoe region, in units of the planet's orbital radius r p , is ˜ x s glyph[similarequal] ( q/h ) 1 / 2 (Paardekooper & Papaloizou 2009). \nIn our example, with central SMBH mass M = 10 8 M glyph[circledot] , the mass ratio of the disk to the central SMBH q d = 0 . 37, the mass ratio of the orbiter to the SMBH, q = 1-3 × 10 -7 (Tab. 1), the angular velocity of the orbiter Ω = ( GM/r 3 ) 1 / 2 , the Shakura-Sunyaev (1973) viscosity parameter α = c s H/ν = 0 . 01 (Sirko & Goodman 2003), and, at the trap radius r = 3 . 2 × 10 -3 pc (Sect. 3), the disk aspect ratio h = H/r = 0 . 05 (see Fig. 1), and the sound speed c s glyph[similarequal] 10 7 cm s -1 (Fig. 2, Sirko & Goodman 2003, including both radiation and thermal pressure). We can derive \nk = 8 π ( GM r ) 1 / 2 q 3 / 2 q 2 d αc s h 9 / 2 . (A5) \nIn our case, we find k = 0 . 09-0.27, satisfying the condition (Eq. A1) that dynamical migration be ineffective. 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2020ApJ...892...78K

Black Hole Parameter Estimation from Its Shadow

2020-01-01

['-', 'galaxy center', 'black hole physics', 'gravitation', 'gravitational lensing', '-', '-', '-', '-', '-', '-']

[]

The Event Horizon Telescope (EHT), a global submillimeter wavelength very long baseline interferometry array, unveiled event-horizon-scale images of the supermassive black hole M87* as an asymmetric bright emission ring with a diameter of 42 ± 3 μas, and it is consistent with the shadow of a Kerr black hole of general relativity. A Kerr black hole is also a solution of some alternative theories of gravity, while several modified theories of gravity admit non-Kerr black holes. While earlier estimates for the M87* black hole mass, depending on the method used, fall in the range ≈ 3× 10<SUP>9</SUP> M<SUB>⊙</SUB> - 7×10<SUP>9</SUP> M<SUB>⊙</SUB>, the EHT data indicated a mass for the M87* black hole of (6.5 ± 0.7) × 10<SUP>9</SUP> M<SUB>⊙</SUB>. This offers another promising tool to estimate black hole parameters and to probe theories of gravity in its most extreme region near the event horizon. The important question arises: Is it possible by a simple technique to estimate black hole parameters from its shadow, for arbitrary models? In this paper, we present observables, expressed in terms of ordinary integrals, characterizing a haphazard shadow shape to estimate the parameters associated with black holes, and then illustrate its relevance to four different models: Kerr, Kerr-Newman, and two rotating regular models. Our method is robust, accurate, and consistent with the results obtained from existing formalism, and it is applicable to more general shadow shapes that may not be circular due to noisy data.

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https://arxiv.org/pdf/1811.01260.pdf

{'No Header': 'Draft version May 12, 2020 Typeset using L A T E X twocolumn style in AASTeX63', 'Black Hole Parameter Estimation from Its Shadow': 'Rahul Kumar 1 and Sushant G. Ghosh 1, 2 \n1 Centre for Theoretical Physics, Jamia Millia Islamia, New Delhi 110025, India \n2 Astrophysics and Cosmology Research Unit, School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Private Bag 54001, Durban 4000, South Africa', 'ABSTRACT': 'The Event Horizon Telescope (EHT), a global submillimeter wavelength very long baseline interferometry array, unveiled event-horizon-scale images of the supermassive black hole M87* as an asymmetric bright emission ring with a diameter of 42 ± 3 µ as, and it is consistent with the shadow of a Kerr black hole of general relativity. A Kerr black hole is also a solution of some alternative theories of gravity, while several modified theories of gravity admit non-Kerr black holes. While earlier estimates for the M87* black hole mass, depending on the method used, fall in the range ≈ 3 × 10 9 M /circledot -7 × 10 9 M /circledot , the EHT data indicated a mass for the M87* black hole of (6 . 5 ± 0 . 7) × 10 9 M /circledot . This offers another promising tool to estimate black hole parameters and to probe theories of gravity in its most extreme region near the event horizon. The important question arises: Is it possible by a simple technique to estimate black hole parameters from its shadow, for arbitrary models? In this paper, we present observables, expressed in terms of ordinary integrals, characterizing a haphazard shadow shape to estimate the parameters associated with black holes, and then illustrate its relevance to four different models: Kerr, Kerr -Newman, and two rotating regular models. Our method is robust, accurate, and consistent with the results obtained from existing formalism, and it is applicable to more general shadow shapes that may not be circular due to noisy data. \nKeywords: Astrophysical black holes (98); Galactic center (565); Black hole physics (159); Gravitation (661); Gravitational lensing (670)', '1. INTRODUCTION': "Black holes are one of the most remarkable predictions of Einstein's theory of general relativity, which also provides a means to probe them via unstable circular photon orbits (Bardeen 1973). A black hole, due to its defining property at the event horizon along with the surrounding photon region, casts a dark region over the observer's celestial sky, which is known as a shadow (Bardeen 1973; Falcke et al. 2000). Astronomical observations suggest that each galaxy hosts millions of stellar-mass black holes, and also a supermassive black hole at the nucleus of the galaxy (Melia & Falcke 2001; Shen et al. 2005). However, the majority of these black holes have very low accretion luminosity and thus are very faint. Due to relatively very large size and close proximity, the black hole candidates at the center of the Milky \nCorresponding author: Rahul Kumar \[email protected] \nWay and in the nearby galaxy Messier 87, respectively, Sgr A* and M87*, are prime candidates for black hole imaging (Broderick & Narayan 2006; Doeleman et al. 2008, 2012). Probing the immediate environment of black holes will not only provide images of these objects and the dynamics of nearby matter but will also enable the study of the strong gravity effects near the horizon. The Event Horizon Telescope (EHT) 1 , a global array of millimeter and submillimeter radio observatories, is using the technique of very long baseline interferometry (VLBI) by combining several synchronized radio telescopes around the world. This Earth-sized virtual telescope has achieved an angular resolution of 20 µ as, sufficient to obtain the horizon-scale image of supermassive black holes at a galaxy's center. The EHT has published the first direct image of the M87* black hole (Akiyama et al. 2019a,b,c,d). Further, fitting geometric models to the observational data and extracting feature parameters \nin the image domain indicates that we see emission from near the event horizon that is gravitationally lensed into a crescent shape around the photon ring (Akiyama et al. 2019c,d). \nIt turns out that photons may propagate along the unstable circular orbits due to the strong gravitational field of the black hole, and these orbits have a very important influence on quasinormal modes (Cardoso et al. 2009; Hod 2009; Konoplya & Stuchlik 2017), gravitational lensing (Stefanov et al. 2010), and the black hole shadow (Bardeen 1973). Synge (1966) and Luminet (1979), in pioneering works, calculated the shadow cast by a Schwarzschild black hole, and thereafter Bardeen (1973) studied the shadows of Kerr black holes over a bright background, which turn out to deviate from a perfect circle. The past decade saw more attention given to analytical investigations, observational studies, and numerical simulations of shadows (see Cunha & Herdeiro 2018). The shadows of modified theory black holes are smaller and more distorted when compared with the Kerr black hole shadow (Bambi & Freese 2009; Johannsen & Psaltis 2010; Falcke & Markoff 2013; Broderick et al. 2014; Younsi et al. 2016; Giddings & Psaltis 2018; Mizuno et al. 2018; Long et al. 2019; Konoplya & Zhidenko 2019; Held et al. 2019; Wang et al. 2019; Yan 2019; Kumar et al. 2019; Vagnozzi & Visinelli 2019; Breton et al. 2019). \nThe no-hair theorem states that the Kerr black hole is the unique stationary vacuum solution of Einstein's field equations, however, the exact nature of astrophysical black holes has not been confirmed (Johannsen & Psaltis 2011; Bambi 2018), and the possible existence of non-Kerr black holes cannot be completely ruled out (Johannsen 2013a, 2016). Indeed, the Kerr metric remains a solution in some modified theories of gravity (Psaltis et al. 2008). For rotating black holes, significant deviations from the Kerr solution are found in modified theories (Bambi & Modesto 2013; Berti et al. 2015). The Bardeen perspective of a shadow of a black hole in front of a planar-emitting source was applied to several black hole models, e.g., Kerr-Newman black hole (Young 1976; De Vries 2000), Chern-Simons modified gravity black hole (Amarilla et al. 2010), Kaluza-Klein rotating dilaton black hole (Amarilla & Eiroa 2013), rotating braneworld black hole (Amarilla & Eiroa 2012; Eiroa & Sendra 2018), regular black holes (Abdujabbarov et al. 2016; Amir & Ghosh 2016; Kumar et al. 2019), and black holes in higher dimensions (Papnoi et al. 2014; Abdujabbarov et al. 2015a; Amir et al. 2018; Singh & Ghosh 2018). The black hole shadow in asymptotically de-Sitter spacetime has also been analyzed (Grenzebach et al. 2014; Perlick et al. 2018; Eiroa & Sendra 2018). Black hole shadows have been investigated for a parameterized axisymmetric rotating black hole, which \ngeneralizes all stationary and axisymmetric black holes in any metric theory of gravity (Rezzolla & Zhidenko 2014; Younsi et al. 2016; Konoplya et al. 2016). \nHowever, developing a methodological way to estimate parameters from astrophysical observations of a black hole image is a promising avenue to advance our understanding of black holes. The observations commonly used for the estimation of the mass and size of a black hole are based on the motion of nearby stars and spectroscopy of the radiation emitted from the surrounding matter in the Keplerian orbits, i.e., stellar dynamical and gas dynamical methods (Gebhardt et al. 2000; Schodel et al. 2002; Shafee et al. 2006). The dynamical mass measurements from X-ray binaries only provide lower limits of the black hole's mass (Haring & Rix 2004; Narayan 2005; Casares & Jonker 2014). Unlike for the mass, effects of the black hole's spin and any possible deviation from standard Kerr geometry are manifest only at the small radii. The two most commonly used model-dependent techniques to estimate the spin are the analysis of the K α iron line (Fabian et al. 1989) and the continuum-fitting method (McClintock et al. 2014). Though black hole parameters have been inferred in a number of contexts through the gravitational impact on the dynamics of surrounding matter (Matt & Perola 1992; Narayan et al. 2008; Broderick et al. 2009; Steiner et al. 2009, 2011; McClintock et al. 2011; Narayan & McClintock 2012; Bambi 2013), the EHT observations can put stringent bounds on the parameters. Furthermore, it is found that the non-Kerr black hole shadows strongly depend on the deviation parameter apart from the spin (Atamurotov et al. 2013; Johannsen 2013b; Wang et al. 2017, 2018). Thus, shadow observations of astrophysical black holes can be regarded as a potential tool to probe their departure from an exact Kerr nature, and in turn, to determine the black hole parameters (Johannsen & Psaltis 2010). Hioki and Maeda (2009) discussed numerical estimations of Kerr black hole spin and inclination angle from the shadow observables, which was extended to an analytical estimation by Tsupko (2017). These observables, namely shadow radius and distortion parameter, were extensively used in the characterization of black holes shadows (De Vries 2000; Amarilla et al. 2010; Amarilla & Eiroa 2012, 2013; Papnoi et al. 2014; Abdujabbarov et al. 2015a, 2016; Amir & Ghosh 2016; Amir et al. 2018; Singh & Ghosh 2018; Eiroa & Sendra 2018). However, it was found that the distortion parameter is degenerate with respect to the spin and possible deviations from the Kerr solution; a method for discriminating the Kerr black hole from other rotating black holes using the shadow analysis is presented by Tsukamoto et al. (2014). An analytic description of distortion parameters of the shadow has also been discussed in a coordinate-independent manner (Abdujabbarov et al. \n2015b). Motivated by the above, we construct observables that can uniquely characterize shadows to estimate the black hole parameters. \nThe aim of this paper is to give simple shadow observables and show their applicability to determining the black hole parameters with emphasis on the characterization of various black hole shadows of more general shape and size. The proposed observables do not presume any symmetries in the shadow and completely depend upon the geometry of the shadow. The characterization of the shadow's size and shape is not restricted to a circle and is applicable to a large variety of shadows. The prescription is applied to four models of rotating black holes to get an estimation of black hole parameters, and when compared with the existing results (Tsukamoto et al. 2014), we find that our prescription gives an accurate estimation. Thus, this simple approach enables us to estimate the black hole parameters accurately, and the method is robust, as it is applicable to haphazard shadow shapes that may result from the noisy data. \nThe paper is organized as follows. In Sect. 2, we discuss the propagation of light in rotating black hole spacetime. Further, in Sect. 3, we present the observables for shadow characterization and use them to estimate the parameters associated with four black holes in Sect. 4. In Sect. 5, we summarize our main results. We use geometrized units G = 1, c = 1, unless units are specifically defined.", '2. BLACK HOLE SHADOW': "The metric of a general rotating, stationary, and axially symmetric black hole, in Boyer -Lindquist coordinates, reads (Bambi & Modesto 2013) \nds 2 = -( 1 -2 m ( r ) r Σ ) dt 2 -4 am ( r ) r Σ sin 2 θdt dφ + Σ ∆ dr 2 +Σ dθ 2 + [ r 2 + a 2 + 2 m ( r ) ra 2 Σ sin 2 θ ] sin 2 θdφ 2 , (1) \nand \nΣ = r 2 + a 2 cos 2 θ ; ∆ = r 2 + a 2 -2 m ( r ) r, (2) \nwhere m ( r ) is the mass function such that lim r →∞ m ( r ) = M and a is the spin parameter defined as a = J/M ; J and M are, respectively, the angular momentum and ADM mass of a rotating black hole. Obviously metric (1) reverts back to the Kerr (1963) and Kerr -Newman (Newman et al. 1965) spacetimes when m ( r ) = M and m ( r ) = M -Q 2 / 2 r , respectively. Photons moving in a general rotating spacetime (1) exhibit two conserved quantities, energy E and angular momentum L , associated with Killing vectors ∂ t and ∂ φ . To study the geodesics motion in spacetime (1), we adopt the Carter (1968) separability prescription of the Hamilton -Jacobi equation. The complete set of \nequations of motion in the first-order differential form read (Carter 1968; Chandrasekhar 1985) \nΣ dt dτ = r 2 + a 2 r 2 -2 m ( r ) r + a 2 ( E ( r 2 + a 2 ) -a L ) a ( a E sin 2 θ ) , (3) \n- \n-L \n√ Σ dθ dτ = ± Θ( θ ) , (5) \nE Σ dr dτ = ± √ R ( r ) , (4) \nΣ dφ dτ = a r 2 -2 m ( r ) r + a 2 ( E ( r 2 + a 2 ) -a L ) -( a E L sin 2 θ ) , (6) \n√ \nwith the expressions for R ( r ) and Θ( θ ), respectively, are given by \nR ( r ) = ( ( r 2 + a 2 ) E a L ) 2 -( r 2 -2 m ( r ) r + a 2 )( K +( a E - L ) 2 ) , (7) \nE - L Θ( θ ) = K-( L 2 sin 2 θ -a 2 E 2 ) cos 2 θ. (8) \nThe conserved quantity Q associated with the hidden symmetry of the conformal Killing tensor is related to the Carter integral of motion K through Q = K + ( a E - L ) 2 (Carter 1968). One can minimize the number of parameters by defining two dimensionless impact parameters η and ξ as follows (Chandrasekhar 1985) \nξ = L / E , η = K / E 2 . (9) \nDue to spacetime symmetries, geodesics along t and φ coordinates do not reveal nontrivial features of orbits, therefore the only concern will be mainly for Eqs. (4) and (5). Rewriting Eq. (5) in terms of µ = cos θ , we obtain \nΣ ∫ dµ √ Θ µ = ∫ dτ ; Θ µ = η -( ξ 2 + η -a 2 ) µ 2 -a 2 µ 4 . (10) \nObviously η ≥ 0 is required for possible θ motion, i.e., Θ µ ≥ 0 (see Figure 1). For the Schwarzschild black hole, due to spherical symmetry, all null circular orbits are planar, i.e., orbits with ˙ θ = 0. However, in the Kerr black hole, the frame-dragging effects may lead to nonplanar orbits as well. Indeed, planar and circular orbits around Kerr black hole are possible only in the equatorial plane ( θ = π/ 2) that leads to a vanishing Carter constant ( K = 0). Furthermore, generic bound orbits at a plane other than θ = π/ 2 are nonplanar ( ˙ θ /negationslash = 0) and cross the equatorial plane while oscillating symmetrically about it. These orbits are identified by K > 0 (or η > 0) and are commonly known as spherical orbits (Chandrasekhar 1985), and θ motion freezes \nFigure 1. Left panel: schematic of a photon region around a rotating black hole. Right panel: variation of Θ µ with µ for η = 1 and ξ = 1 . 2. Horizontal dashed lines correspond to the maximum and minimum values of Θ µ . \n<!-- image --> \nonly in the equatorial plane. Equation (10) reveals that the maximum latitude of a spherical orbit, θ max = cos -1 ( µ max ), depends upon the angular momentum of photons, i.e., the smaller the angular momentum of photons the larger the latitude of orbits; µ max correspond to the solution of Θ µ ( µ ) = 0. Only photons with zero angular momentum ( ξ = 0) can reach the polar plane of the black hole ( θ = 0 , µ = 1) and cover the entire span of θ coordinates (see Figure 1). \nDepending on the values of the impact parameters η and ξ , photon orbits can be classified into three categories, namely scattering orbits, unstable circular and spherical orbits, and plunging orbits. Indeed, the unstable orbits separate the plunging and scattering orbits, and their radii ( r p ) are given by (Chandrasekhar 1985) \nR| ( r = r p ) = ∂ R ∂r ∣ ∣ ∣ ( r = r p ) = 0 , ∂ 2 R ∂r 2 ∣ ∣ ∣ ∣ ( r = r p ) ≤ 0 . (11) \n∣ \n∣ Solving Equation (11) yields the critical locus ( η c , ξ c ) associated with the unstable photon orbits, that for nonrotating black holes are at a fixed radius, e.g., r p = 3 M for a Schwarzschild black hole, and construct a spherical photon sphere. In the rotating black hole spacetime, photons moving in unstable circular orbits at the equatorial plane can either corotate with the black hole or counterrotate, and their radii can be identified as the real positive solutions of η c = 0 for r , r -p and r + p , respectively. Photon orbit radii are an explicit function of black hole spin and lie in the range M ≤ r -p ≤ 3 M and 3 M ≤ r + p ≤ 4 M for the Kerr black hole, and r -p ≤ r + p due to the Lens -Thirring effect. Whereas, spherical photon orbits (orbits at θ /negationslash = π/ 2) are no longer affixed to a fixed plane and instead are three-dimensional orbits with radii in the interval [ r -p , r + p ], i.e., for η c > 0 orbits, radii lie in the range r -p < r p < r + p . Although rotating black holes generically have two distinct photon regions, viz., \ninside the Cauchy horizon ( r -) and outside the event horizon ( r + ), for a black hole shadow we will be only focusing on the latter, i.e., for r p > r + (Grenzebach et al. 2014). The critical impact parameter ξ c is a monotonically decreasing function of r p with ξ c ( r -p ) > 0 and ξ c ( r + p ) < 0, such that at r p = r 0 p ( r -p < r 0 p < r + p ) ξ c is vanishing. Even though for orbits at r 0 p the angular momentum of photons is zero, they still cross the equatorial plane with nonzero azimuthal velocity ˙ φ = 0 (Wilkins 1972; Chandrasekhar 1985). \nA black hole in a luminous background of stars or glowing accreting matter leads to the appearance of a dark spot on the celestial sky accounting for the photons which are unable to reach the observer, popularly known as a black hole shadow. Photons moving on unstable orbits construct the edges of the shadow. A far distant observer perceives the shadow as a projection of a locus of points η c and ξ c on the celestial sphere on to a two-dimensional plane. Let us introduce the celestial coordinates (Bardeen 1973) \n/negationslash \nα = lim r O →∞ ( -r 2 O sin θ O dφ dr ) , β = lim r O →∞ ( r 2 O dθ dr ) . (12) \nHere, we assume the observer is far away from the black hole ( r O → ∞ ) and θ O is the angle between the line of sight and the spin axes of black hole, namely, the inclination angle. Since the black hole spacetime is asymptotically flat, we can consider a static observer at an arbitrarily large distance, and this yields \nα = -ξ c sin θ O , β = ± √ η c + a 2 cos θ 2 O -ξ 2 c cot 2 θ O . \n(13) \nFor an observer in the equatorial plane θ O = π/ 2, Eq. (13) reduces to \nα = -ξ c , β = ± √ η c . (14) \nSolving Eq. (11) for rotating metric (1) and using Eq. (14), the celestial coordinates of the black hole shadow boundary take the following form \nα = -[ a 2 -3 r 2 p ] m ( r p ) + r p [ a 2 + r 2 p ][1 + m ' ( r p )] a [ m ( r p ) + r p [ -1 + m ' ( r p )]] , β = ± 1 a [ m ( r p ) + r p [ -1 + m ' ( r p )]] [ r 3 / 2 p [ -r 3 p (1 + m ' ( r p ) 2 ) + m ( r p )[4 a 2 +6 r 2 p -9 r p m ( r p )] -2 r p [2 a 2 + r 2 p -3 r p m ( r p )] m ' ( r p ) ] 1 / 2 ] , (15) \nand whereas for m ( r ) = M , Eq. (15) yields \nα = r 2 p ( r p -3 M ) + a 2 ( M + r p ) a ( r p -M ) , β = ± r 3 / 2 p (4 a 2 M -r p ( r p -3 M ) 2 ) 1 / 2 a ( r p -M ) , (16) \nand which is exactly the same as obtained for the Kerr black hole (Hioki & Maeda 2009). The contour of a nonrotating black hole shadow ( a = 0) can be delineated by \nα 2 + β 2 = 2 r 4 p +[ m ( r p ) + r p m ' ( r p )][ -6 r 2 p m ( r p ) + 2 r 3 p m ' ( r p )] [ m ( r p ) + r p [ -1 + m ' ( r p )]] 2 , \n(17) \nwhich implies that the shadow of a nonrotating black hole is indeed a perfect circle, and further returns to α 2 + β 2 = 27 M 2 for the Schwarzschild black hole m ( r ) = M . Though the shape of the shadow is determined by the properties of null geodesics, it is neither the Euclidean image of the black hole horizon nor that of its photon region, rather it is the gravitationally lensed image of the photon region. For instance, the horizon of Sgr A*, with M ≈ 4 . 3 × 10 6 M /circledot at a distance d ≈ 8 . 35 kpc, spans an angular size of 20 µ as, whereas its shadow has an expected angular size of ≈ 53 µ as. Whereas, EHT measured the angular size of the M87* gravitational radius as 3 . 8 ± 0 . 4 µ as, and its crescent-shaped emission region has an angular diameter of 42 ± 3 µ as, with a scaling factor in the range 10 . 7 -11 . 5 (Akiyama et al. 2019c,d).", '3. CHARACTERIZATION OF THE SHADOW VIA NEW OBSERVABLES': "A nonrotating black hole casts a perfectly circular shadow. However, for a rotating black hole, an observer placed at a position other than in the polar directions witnesses an off-center displacement of the shadow along the direction of black hole rotation. Furthermore, for sufficiently large values of the spin parameter, a distortion appears in the shadow because of the Lense-Thirring effect (Johannsen & Psaltis 2010). Hioki and Maeda (2009) characterized this distortion and shadow size by the two observables δ s and R s , respectively. The shadow is approximated to a circle passing through three points located at the top, bottom, and right edges of the shadow, such that R s is the radius of this circle and δ s is the deviation of the left edge of the shadow from the circle boundary (Hioki & Maeda 2009). It was found that the applicability of these observables was limited to a specific class of shadows, demanding some symmetries in their shapes, and they may not precisely work for black holes in some modified theories of gravity (Abdujabbarov et al. 2015b), which leads to the introduction of new observables (Schee & Stuchlik 2009; Johannsen 2013b; Tsukamoto et al. 2014; Cunha et al. 2015; Abdujabbarov et al. 2015b; Younsi et al. 2016; Tsupko 2017; Wang et al. 2018). EHT observations can constrain the key physical parameters of the black holes, including the black hole mass and other parameters. However, EHT observations do not give any estimation of angular momentum (Akiyama et al. 2019a,d). Their \nmeasurement of the black hole mass in M87* is consistent with the prior mass measurement using stellar dynamics, but is inconsistent with the gas dynamics measurement (Gebhardt et al. 2011; Walsh et al. 2013; Akiyama et al. 2019d). Here, we would propose new observables for the characterization of the black hole shadow, which unlike previous observables (Hioki & Maeda 2009), do not require the apparent shadow shape to be approximated as a circle. We consider a shadow of general shape and size to propose new observables, namely the area ( A ) enclosed by a black hole shadow, the circumference of the shadow ( C ), and the oblateness ( D ) of the shadow. The observables A and C , respectively, are defined by \nA = 2 ∫ β ( r p ) dα ( r p ) = 2 ∫ r + p r -p ( β ( r p ) dα ( r p ) dr p ) dr p , (18) \nand \nC =2 ∫ √ ( dβ ( r p ) 2 + dα ( r p ) 2 ) =2 ∫ r + p r -p √ √ √ √ ( ( dβ ( r p ) dr p ) 2 + ( dα ( r p ) dr p ) 2 ) dr p . (19) \nThe prefactor 2 is due to the black hole shadow's symmetry along the α -axis. A and C have dimensions of [ M ] 2 and [ M ], respectively. A shadow silhouette can be taken as a parametric curve between celestial coordinates as a function of r p for r -p ≤ r p ≤ r + p , i.e., a plot of β ( r p ) versus α ( r p ). We can also characterize the shadow of rotating black hole through its oblateness (Takahashi 2004; Grenzebach et al. 2015; Tsupko 2017), the measure of distortion (circular asymmetry) in a shadow, by defining the dimensionless parameter D as the ratio of horizontal and vertical diameters: \nD = α r -α l β t -β b . (20) \nThe subscripts r, l, t , and b stand for right, left, top, and bottom, respectively, of the shadow boundary. For a spherically symmetric black hole shadow, D = 1, while for a Kerr shadow √ 3 / 2 ≤ D < 1 (Tsupko 2017). Thus, D /negationslash = 1 indicates that the shadow has distortion and hence corresponds to a rotating black hole. In particular, the quasi-Kerr black hole metric may lead to a shadow with D > 1 or D < 1, depending on the sign of the quadrupole deviation parameters (Johannsen & Psaltis 2010). The definitions of these observables require neither any nontrivial symmetry in shadow shape nor any primary curve to approximate the shadow. It can be expected that an observer targeting the black hole shadow through astronomical observations can measure the area, the length of the shadow boundary, and also the horizontal and vertical diameters. In what follows, we show that these \nTable 1. Table representing the values of observables, solid angle, and angular diameter with varying spin parameter for the Sgr A* black hole shadow. \n| a/M | A (10 20 m 2 | C ) (10 10 m ) | D | Ω (10 - 3 µ as 2 | ϑ m ) ( µ as) |\n|-------|----------------|------------------|----------|--------------------|-----------------|\n| 0 | 34.079 | 20.6942 | 1 | 2.1818 | 52.7344 |\n| 0.1 | 34.06 | 20.6884 | 0.999443 | 2.18059 | 52.705 |\n| 0.2 | 34.0025 | 20.671 | 0.997748 | 2.1769 | 52.6156 |\n| 0.3 | 33.9046 | 20.6413 | 0.994847 | 2.17064 | 52.4626 |\n| 0.4 | 33.7629 | 20.5984 | 0.990607 | 2.16157 | 52.239 |\n| 0.5 | 33.572 | 20.5406 | 0.984808 | 2.14934 | 51.9332 |\n| 0.6 | 33.3227 | 20.4655 | 0.977083 | 2.13338 | 51.5259 |\n| 0.7 | 32.9998 | 20.3688 | 0.966783 | 2.11271 | 50.9827 |\n| 0.8 | 32.5742 | 20.2427 | 0.952608 | 2.08546 | 50.2352 |\n| 0.9 | 31.9754 | 20.0699 | 0.931145 | 2.04713 | 49.1033 |\n| 0.998 | 30.7793 | 19.776 | 0.876375 | 1.97055 | 46.2151 | \nobservables uniquely characterize the shadow and it is possible to estimate the black hole parameters from these observables. \nThe EHT observations indicated that the M87* black hole shadow is consistent with that of Kerr black hole, however, the exact nature of the Sgr A* black hole is still elusive. Astronomical observations have place constraints on their masses and distances from Earth as M = 4 . 3 × 10 6 M /circledot and d = 8 . 35 kpc for Sgr A* (Ghez et al. 2008; Gillessen et al. 2009; Falcke & Markoff 2013; Reid et al. 2014), and M = (6 . 5 ± 0 . 7) × 10 9 M /circledot and d = (16 . 8 ± 0 . 8)Mpc for M87* (Akiyama et al. 2019a). Presuming the exact Kerr nature of these black holes, we determine the area spanned by their shadows, the solid angle covered by them on the celestial sky, and also their angular sizes. In general, for rotating black holes, the vertical (or major ϑ M ) and horizontal (or minor ϑ m ) angular diameters are not the same and can be defined as \nϑ M = β t -β b d , ϑ m = α r -α l d , (21) \nand the solid angle is Ω = A/d 2 . Clearly, ϑ M is not dependent on black hole spin. \nObviously, for a = 0, ϑ M = ϑ m = 52 . 7344 µ as for Sgr A* and ϑ M = ϑ m = 39 . 6192 µ as for M87*. The shadow observables and angular diameters of the Sgr A* and M87* black hole shadows are calculated for various values of spin parameter a (see Table 1 and Table 2). Nevertheless, the shadow observables for a Schwarzschild black hole take the values A/M 2 = 84 . 823, C/M = 32 . 6484, and D = 1, whereas for a maximally rotating Kerr black hole A/M 2 = 76 . 6101, C/M = 31 . 1998, and D = 0 . 876375. \nTable 2. Table representing the values of observables, solid angle, and angular diameter with varying spin parameter for the M87* black hole shadow. \n| a/M | A (10 27 m 2 | C ) (10 14 m ) | D | Ω (10 - 4 µ as 2 | ϑ m ) ( µ as) |\n|-------|----------------|------------------|----------|--------------------|-----------------|\n| 0 | 7.78711 | 3.12819 | 1 | 1.23151 | 39.6192 |\n| 0.1 | 7.78278 | 3.12732 | 0.999443 | 1.23083 | 39.5971 |\n| 0.2 | 7.76963 | 3.12468 | 0.997748 | 1.22875 | 39.53 |\n| 0.3 | 7.74726 | 3.12019 | 0.994847 | 1.22521 | 39.4151 |\n| 0.4 | 7.7149 | 3.1137 | 0.990607 | 1.22009 | 39.2471 |\n| 0.5 | 7.67126 | 3.10498 | 0.984808 | 1.21319 | 39.0173 |\n| 0.6 | 7.6143 | 3.09362 | 0.977083 | 1.20418 | 38.7113 |\n| 0.7 | 7.54052 | 3.079 | 0.966783 | 1.19252 | 38.3032 |\n| 0.8 | 7.44327 | 3.05994 | 0.952608 | 1.17714 | 37.7416 |\n| 0.9 | 7.30644 | 3.03382 | 0.931145 | 1.1555 | 36.8912 |\n| 0.998 | 6.99973 | 2.65529 | 0.866025 | 1.10699 | 34.3112 |", '4. APPLICATION TO VARIOUS BLACK HOLE SPACETIMES': 'We examine several rotating black holes such as Kerr -Newman, Bardeen, and nonsingular black holes. In general these black holes are given by metric (1) with an appropriate choice of mass function m ( r ). We assume that the observer is in the equatorial plane, i.e., the inclination angle θ O = π/ 2 for the estimation. One can use either of the two observables A or C along with D to estimate the black hole parameters. For the sake of brevity, we shall use only A and D for our purpose, but shall calculate all three.', '4.1. Kerr -Newman black hole': 'We start with a Kerr -Newman black hole, which encompasses Kerr, Reissner -Nordstrom, and Schwarzschild black holes as special cases. One can analyze null geodesics to the shadow of a Kerr -Newman black hole (Young 1976; De Vries 2000). In the case of the Kerr -Newman black hole, the mass function m ( r ) has a form \nm ( r ) = M -Q 2 2 r . (22) \nIn Figure 2, we have shown the allowed range of parameters a and Q for the existence of a black hole horizon. The Kerr -Newman black hole shadows are distorted from a perfect circle and possess a dent on the left side of shadow (De Vries 2000). This distortion reduces as the observer moves from the equatorial plane to the axis of black hole symmetry, and eventually disappears completely for θ O = 0 , π \nFigure 2. Allowed parametric space ( a, Q ) for the existence of a Kerr -Newman black hole. The solid line corresponds for the extremal black hole with degenerate horizons and demarcates the black hole case from the no black hole case. \n<!-- image --> \n(De Vries 2000). It is straightforward to calculate the celestial coordinates α and β using the m ( r ) in Eq. (15). Though for these α and β the observables A , C , and D could not be obtained in exact analytic form, we have calculated them approximately in the Appendix A. \nIn Figures 3 and 4, respectively, charge Q and spin parameter a are plotted with varying observables A , C , and D . Interestingly, estimated values of black hole parameters decrease with independently increasing observables. For a far extremal black hole, parameters decrease rapidly with observables, whereas for a near extremal black hole, parameters decrease relatively slowly with increasing D . Therefore, one can conclude that the size of the shadow decreases with an increase in the electric charge, which is consistent with the earlier results (De Vries 2000). On the other hand, Figure 3 suggests that shadows of Kerr -Newman black holes get more distorted as the charge increase. Shadow observables for Kerr -Newman black holes are numerically compared with those for Kerr black holes in Figure 4, and it can be inferred that observables for Kerr -Newman black holes are smaller than those for Kerr black holes for fixed values of a . \nThe apparent shape and size of the Kerr -Newman black hole shadow depend on the a and Q (De Vries 2000). Next, we see the possibility of estimation of a and Q for the Kerr -Newman black hole by using the two observables A and D , expecting that mass M can be fixed through other astrophysical observations. We plot the contour map of the observables A and D in the ( a, Q ) plane (see Figure 5). Each point of the contour plot in Figure 5 has coordinates ( a, Q ) that can be described as a unique intersection of the lines of constant A and D . Hence, from Figure 5, it is clear \nthat intersection points give an exact estimation of parameters a and Q when one has the values of A and D for a Kerr -Newman black hole. In Table 3, we have presented the estimated values of a and Q for given shadow observables A and D for the Kerr -Newman black hole.', '4.1.1. Kerr Black Hole': 'When the electric charge is switched off ( Q = 0), the Kerr -Newman spacetime becomes Kerr with m ( r ) = M . We plot the spin parameter a (0 ≤ a ≤ 1) with varying observables A , C , and D in Figure 4. It is evident that with increasing observables A , C , and D the estimated Kerr spin parameter decreases. Figure 4 indicates that the black hole shadow gets smaller and more distorted for a rapidly rotating black hole, as shown in earlier studies as well (Bardeen 1973). \nKerr black holes have only two parameters associated with them, namely, mass M and spin a , however, presuming the knowledge of only mass through the stellar motion around the black hole, one has only one unknown parameter i.e., spin. The spin parameter for the Kerr black hole can be uniquely determined by knowing any one of the shadow observables defined above (see Figure 4).', '4.2. Rotating Bardeen Black Hole': 'The first regular black hole was proposed by Bardeen (1968) with horizons and no curvature singularity -a modification of the Reissner -Nordstrom black hole. The rotating Bardeen black hole (Bambi & Modesto 2013) belongs to the prototype non-Kerr family with the mass function m ( r ) given by \nm ( r ) = M ( r 2 r 2 + g 2 ) 3 / 2 . (23) \nThe Bardeen black hole is an exact solution of the Einstein field equations coupled with nonlinear electrodynamics associated with the magnetic monopole charge g (Ayon-Beato & Garcia 1999). The Kerr black hole can be recovered in the absence of the nonlinear electrodynamics ( g = 0). For the existence of a black hole, the allowed values of a and g are constrained and shown in Figure 6, and the extremal values of parameters correspond to those lying on the boundary line. The shadows of rotating Bardeen black holes get more distorted and their sizes decrease due to the magnetic charge g (Abdujabbarov et al. 2016). \nThe Bardeen black hole parameters g and a versus the observables A , C , and D are depicted in Figures 7 and 8, respectively. Within the allowed parameter space, they have a similar behavior to that of the Kerr -Newman black hole. The parameters decrease with increasing observables, however, for a near \n<!-- image --> \n<!-- image --> \nFigure 3. Charge parameter Q vs. observables A , C , and D for the Kerr -Newman black hole, for a/M = 0 (solid black curve), for a/M = 0 . 3 (solid green curve), for a/M = 0 . 5 (dashed blue curve), and for a/M = 0 . 8 (dotted red curve). \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFigure 4. Spin parameter a vs. observables A , C , and D for the Kerr -Newman black hole, for Kerr black hole Q/M = 0 . 0 (solid black curve), for Q/M = 0 . 4 (dashed blue curve), for Q/M = 0 . 6 (dotted dashed magenta curve), and for Q/M = 0 . 8 (dotted red curve). \n<!-- image --> \nTable 3. Estimated values of parameters for different black hole models from known shadow observables A and D . \n| | | Black Hole Parameters | Black Hole Parameters | Black Hole Parameters | Black Hole Parameters | Black Hole Parameters | Black Hole Parameters |\n|-------------------|-------------------|-------------------------|-------------------------|-------------------------|-------------------------|-------------------------|-------------------------|\n| Shadow Observable | Shadow Observable | Kerr - Newman | Kerr - Newman | Bardeen | Bardeen | Nonsingular | Nonsingular |\n| A/M 2 | D | a/M | Q/M | a/M | g/M | a/M | k/M |\n| 82.0 | 0.995 | 0.28284 | 0.29079 | 0.26954 | 0.28938 | 0.283082 | 0.042578 |\n| 82.0 | 0.98 | 0.55344 | 0.20523 | 0.5396 | 0.20487 | 0.553528 | 0.021183 |\n| 82.0 | 0.97 | 0.66713 | 0.11830 | 0.66130 | 0.11840 | 0.66714 | 0.0070 |\n| 80.0 | 0.995 | 0.27224 | 0.392001 | 0.24911 | 0.388453 | 0.272684 | 0.077867 |\n| 80.0 | 0.97 | 0.64252 | 0.29326 | 0.608567 | 0.292265 | 0.642787 | 0.043316 |\n| 80.0 | 0.95 | 0.80151 | 0.18123 | 0.783569 | 0.181885 | 0.801557 | 0.016468 |\n| 75.0 | 0.995 | 0.245457 | 0.56651 | 0.20174 | 0.5550 | 0.24706 | 0.165303 |\n| 75.0 | 0.95 | 0.724557 | 0.4614554 | 0.623425 | 0.45764 | 0.726799 | 0.108487 |\n| 75.0 | 0.87 | 0.9725687 | 0.231468 | N.A. | N.A. | N.A. | N.A. |\n| 70.0 | 0.995 | 0.2180449 | 0.692545 | 0.15845 | 0.669714 | 0.2218 | 0.251415 |\n| 70.0 | 0.95 | 0.64521 | 0.620045 | 0.479736 | 0.607512 | 0.653064 | 0.199267 |\n| 70.0 | 0.90 | 0.825472 | 0.536881 | N.A. | N.A. | 0.8321695 | 0.1476488 |\n| 67.0 | 0.995 | 0.20126 | 0.75515 | 0.1337 | 0.723888 | 0.20681 | 0.30231 |\n| 67.0 | 0.98 | 0.394672 | 0.735403 | 0.263005 | 0.707514 | 0.40475 | 0.2856 |\n| 67.0 | 0.90 | 0.76314 | 0.628296 | N.A. | N.A. | 0.776511 | 0.2042426 |\n| 55.0 | 0.995 | 0.13035 | 0.944071 | N.A. | N.A. | 0.1478 | 0.497943 |\n| 55.0 | 0.95 | 0.381534 | 0.91421 | N.A. | N.A. | 0.435251 | 0.460537 |\n| 55.0 | 0.92 | N.A. | N.A. | N.A. | N.A. | 0.518359 | 0.437666 | \nFigure 5. Contour plot of the observables A and D in the plane ( a, Q ) for a Kerr -Newman black hole. Each curve is labeled with the corresponding values of A and D . The solid red lines correspond to the area observable A , and the dashed blue lines correspond to the oblateness observable D . \n<!-- image --> \nFigure 6. The allowed parametric space of a and g for the existence of a rotating Bardeen black hole. The solid line corresponds to the extremal black hole with degenerate horizons. \n<!-- image --> \nextremal black hole, parameters decrease comparatively slowly with increasing D . Further, the observables of a rotating Bardeen black hole are smaller when compared with the Kerr black hole for a given a , i.e., A ( g /negationslash = 0) < A ( g = 0) and D ( g /negationslash = 0) < D ( g = 0) (see Figure 8). An interesting comparison between shadows of Bardeen and Kerr black holes shows that for some values of parameters, a \nBardeen black hole ( M = 1 , a/M = 0 . 5286 , g/M = 0 . 6) casts a similar shadow to that of Kerr black hole ( M = 0 . 9311 , a/M = 0 . 9189) (Tsukamoto et al. 2014). In this case, the observables for a Bardeen black hole are A = 69 . 1445 , C = 29 . 5269, and D = 0 . 925402, whereas for a Kerr black hole, they are A = 68 . 68015, C = 29 . 4213, and D = 0 . 925402. Thus, the A and C for the two black holes differ by 0 . 671% and 0 . 357%, respectively. The differences in their major and minor angular diameters are 0 . 0331% and 2 . 158%, respectively. Figure 9 shows the contour map of observables A and D for the rotating Bardeen black hole as a function of ( a, g ). In Table 3, we have shown the estimated values of Bardeen parameters a and g for given shadow observables A and D , and compare them with the estimated values of other black hole parameters. Thus, from Figure 9 and Table 3 it is clear that if A and D are known from the observations, this uniquely determines the a and g .', '4.3. Rotating Nonsingular Black Hole': "The Bardeen regular black holes have a de-Sitter region at the core. Next, we consider a class of rotating regular black holes with asymptotically Minkowski cores (Simpson & Visser 2020), which have an additional parameter k = q 2 / 2 M > 0 due to nonlinear electrodynamics that deviates from Kerr and asymptotically ( r >> k ) goes over to a Kerr -Newman black hole (Ghosh 2015). Whilst this rotating regular black hole shares many properties with Bardeen rotating regular black holes, there is also a significant contrast, and for definiteness, we name it a rotating nonsingular black hole. It also belongs to the non-Kerr family with mass function \nm ( r ) = Me -k/r . (24) \nFigure 10 shows the allowed values of parameters a and k for the black hole's existence. The effect of varying observables A , C , and D on the inferred rotating nonsingular black hole parameters k and a are depicted in Figures 11 and 12, respectively. The characteristic behavior is again similar to that for the Kerr -Newman, but the effect on k is visible for both nonrotating and rotating nonsingular black holes (see Figures 11 and 12). The estimated values of k show a similar sharp decreasing behavior with increasing A and C , whereas it slowly decreases with D for near extremal black holes. The observables for rotating nonsingular black holes are examined in contrast with those for Kerr black holes in Figure 12, and for a fixed value of a they turn out to be smaller. This indicates that shadows of rotating nonsingular black holes are smaller and more distorted than those of Kerr black holes (Amir & Ghosh 2016). Contour maps of A and D as a function of ( a, k ) are shown in Figure 13. We \n<!-- image --> \n<!-- image --> \nFigure 7. The magnetic charge parameter g vs. observables A , C , and D for the Bardeen black hole, for a nonrotating Bardeen black hole a/M = 0 . 0 (solid black curve), for a rotating Bardeen black hole with a/M = 0 . 3 (solid green curve), for a/M = 0 . 5 (dashed blue curve), and for a/M = 0 . 8 (dotted red curve). \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFigure 8. The spin parameter a vs. observables A , C , and D for the Bardeen black hole, for g/M = 0 . 0 (solid black curve), for g/M = 0 . 4 (dashed blue curve), for g/M = 0 . 6 (dotted dashed magenta curve), and for g/M = 0 . 7 (dotted red curve). \n<!-- image --> \nFigure 9. Contour plot of the observables A and D in the plane ( a, g ) for a Bardeen black hole. Each curve is labeled with the corresponding values of A (solid red curve) and D (dashed blue curve). \n<!-- image --> \ncan easily determine the specific points where curves of constant A and D intersect each other in the black hole parameter space, yielding the unique values of a and k . \nFigure 10. The allowed parametric space of a and k for the existence of rotating nonsingular black hole. The solid line corresponds to the extremal black hole with degenerate horizons. \n<!-- image -->", '4.4. Comparison of Estimated Black Hole Parameters': 'Applying the method described in section 3, the numerical values of the three considered rotating black holes parameters, for a given shadow area A and oblateness D , are summarized in the Table 3. Here, we compare the estimated black hole parameters for the three black holes. For a given shadow area A , we find that the spin parameter decreases with increasing \n<!-- image --> \n<!-- image --> \nFigure 11. Charge parameter k vs. observables A , C , and D for the nonsingular black hole, for a/M = 0 . 0 (solid black curve), for a/M = 0 . 2 (solid green curve), for a/M = 0 . 5 (dashed blue curve) and for a/M = 0 . 8 (dotted red curve). \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFigure 12. Spin parameter a vs. observables A , C , and D for the nonsingular black hole, for k/M = 0 . 0 (solid black curve), for k/M = 0 . 2 (dashed blue curve), for k/M = 0 . 4 (dotted dashed magenta curve), and for k/M = 0 . 6 (dotted red curve). \n<!-- image --> \noblateness D , and that for a fixed area A and oblateness D we obtain that the spin parameters are a NS > a KN > a Bardeen and the charge parameters are Q > g > k . For a fixed oblateness D , the charge parameters Q , g , and k increase and the spin parameter a decreases with a decrease in the area A . For small area A and oblateness D , e.g., A = 55 M 2 and D = 0 . 92, one could estimate parameters associated with only the rotating nonsingular black hole (see Table 3).', '5. CONCLUSION': 'The EHT has obtained the first image of the M87* black hole, and thus its shadow becomes an important probe of spacetime structure, parameter estimation, and testing gravity in the extreme region near the event horizon. Even though most of the available tests are consistent with general relativity, deviations from the Kerr black hole (or non-Kerr black hole) arising from modified theories of gravity are not ruled out (Johannsen & Psaltis 2011; Berti et al. 2015). These non-Kerr black holes, in Boyer -Lindquist coordinates, are defined by the metric (1) with mass function m ( r ), and Kerr black holes are included as special case when m ( r ) = M . In this paper, we have proposed observables, namely, shadow area ( A ), its circumference ( C ), and oblateness ( D ). The observables A and C characterize the size of the shadow, and D defines its shape asymmetry. These observables are calculated for Sgr A* and M87*, \nFigure 13. Contours of constant A and D as a function of ( a, k ) for a rotating nonsingular black hole. Each curve is labeled with the corresponding value of A (solid red curve) and D (dashed blue curve). \n<!-- image --> \nassuming their Kerr nature, and we find that their angular diameters are approximately 52 µ as and 39 µ as, respectively, and decrease for a rapidly rotating black \nhole. This is consistent with other predicted results (Falcke & Markoff 2013; Fish et al. 2014; Brinkerink et al. 2019; Akiyama et al. 2019a,b,d). We highlight several other results that are obtained by our analysis. \n- 1. The method can estimate, at most, two parameters by using either A or C along with D ; for example, the Kerr black hole parameters a and θ O can be estimated. In order to estimate a single parameter, we require any one of these observables.\n- 2. For given shadow observables, we have estimated parameters associated with Kerr -Newman ( a, Q ), rotating Bardeen ( a, g ), and rotating nonsingular ( a, k ) black holes. Here, our analysis assumes that the observer is in the equatorial plane, i.e., at a fixed inclination angle θ O = π/ 2.\n- 3. Our results for the considered black holes are consistent with existing results (Tsukamoto et al. 2014).\n- 4. We have interpolated the numerical values of observables from integrals in Equations (18) and (19) and used Equation (20) to approximate these observables as polynomials in terms of the black hole parameters.\n- 5. Our analysis is applicable to a large variety of shadow shapes and does not require approximating the shadow as a circle. \nThus, by comparing the theoretically calculated values of these observables with those obtained from the astrophysical observations, it is expected that one can completely determine information about a black hole. Our analysis is clearly different from other approaches but leads to the correct estimation of black hole parameters. Our framework can be extended to other classes of black holes. \n- A set of shadow observables can correspond to various black holes with different values of parameters involved (see Table 3). Indeed, we find that a strong correlation between the spin and the deviation parameters from the Kerr solution makes it difficult to discern two black hole models with given shadow observables. It will be interesting to find new observables characterizing the shadows in the presence of an accretion disk; this and related projects are being investigated.', '6. ACKNOWLEDGMENTS': 'S.G.G. would like to thank the DST INDO-SA bilateral project DST/INT/South Africa/P-06/2016 and also IUCAA, Pune for the hospitality while this work was being done. R.K. would like to thank UGC for providing SRF, and also Md Sabir Ali and Balendra Pratap Singh for fruitful discussions. The authors would like to thank the anonymous reviewer for providing insightful comments which immensely helped to improve the paper.', 'A. ANALYTIC FORM OF OBSERVABLES': 'The celestial coordinates α and β can be calculated via Equation (15) for a given mass function, and in turn, they help us to calculate observables A , C , and D numerically. Here, we present an approximate and analytic form of A , C , and D obtained from the best fit of the numerical data for the three discussed rotating black holes. For a Kerr -Newman black hole it yields \nA ( a, Q ) M 2 =84 . 823 -0 . 0241486 a M -3 . 92067 a 2 M 2 -7 . 64929 a 3 M 3 +31 . 6438 a 4 M 4 -70 . 2995 a 5 M 5 +73 . 3373 a 6 M 6 -30 . 7606 a 7 M 7 -2 . 21765 Q M -6 . 0038 aQ M 2 +52 . 4736 a 2 Q M 3 -210 . 27 a 3 Q M 4 +415 . 819 a 4 Q M 5 -509 . 279 a 5 Q M 6 +356 . 297 a 6 Q M 7 +11 . 9166 Q 2 M 2 +43 . 6266 aQ 2 M 3 -342 . 836 a 2 Q 2 M 4 +1144 . 53 a 3 Q 2 M 5 -1460 . 97 a 4 Q 2 M 6 +988 . 7 a 5 Q 2 M 7 -287 . 055 Q 3 M 3 -110 . 416 aQ 3 M 4 +786 . 043 a 2 Q 3 M 5 -2403 . 56 a 3 Q 3 M 6 +1916 . 18 a 4 Q 3 M 7 +1064 . 15 Q 4 M 4 +116 . 604 aQ 4 M 5 -757 . 586 a 2 Q 4 M 6 +2203 . 57 a 3 Q 4 M 7 -2257 . 51 Q 5 M 5 -44 . 1061 aQ 5 M 6 +253 . 501 a 2 Q 5 M 7 +2735 . 71 Q 6 M 6 -1769 . 27 Q 7 M 7 , C ( a, Q ) M =32 . 6484 -0 . 00333693 a M -0 . 801683 a 2 M 2 -1 . 00595 a 3 M 3 +4 . 08401 a 4 M 4 -9 . 22737 a 5 M 5 +9 . 63857 a 6 M 6 -4 . 08849 a 7 M 7 -0 . 252252 Q M -0 . 952349 aQ M 2 +7 . 90764 a 2 Q M 3 -30 . 7126 a 3 Q M 4 +55 . 564 a 4 Q M 5 -65 . 4834 a 5 Q M 6 +44 . 8428 a 6 Q M 7 -0 . 986894 Q 2 M 2 +7 . 56037 aQ 2 M 3 -56 . 8018 a 2 Q 2 M 4 +187 . 173 a 3 Q 2 M 5 -212 . 661 a 4 Q 2 M 6 +137 . 518 a 5 Q 2 M 7 -30 . 7211 Q 3 M 3 -20 . 9274 aQ 3 M 4 +141 . 551 a 2 Q 3 M 5 -438 . 524 a 3 Q 3 M 6 +299 . 757 a 4 Q 3 M 7 +108 . 328 Q 4 M 4 +24 . 2099 aQ 4 M 5 -149 . 282 a 2 Q 4 M 6 +450 . 497 a 3 Q 4 M 7 -219 . 805 Q 5 M 5 -10 . 0047 aQ 5 M 6 +54 . 27 a 2 Q 5 M 7 +250 . 578 Q 6 M 6 -151 . 118 Q 7 M 7 , D ( a, Q ) = 1 . -0 . 000544168 a M -0 . 0377214 a 2 M 2 -0 . 172164 a 3 M 3 +0 . 719728 a 4 M 4 -1 . 57686 a 5 M 5 +1 . 6425 a 6 M 6 -0 . 683473 a 7 M 7 -0 . 160047 aQ M 2 +1 . 38395 a 2 Q M 3 -5 . 40696 a 3 Q M 4 +10 . 432 a 4 Q M 5 -12 . 6347 a 5 Q M 6 +8 . 78616 a 6 Q M 7 +1 . 19921 aQ 2 M 3 -9 . 34042 a 2 Q 2 M 4 +30 . 3103 a 3 Q 2 M 5 -37 . 3007 a 4 Q 2 M 6 +24 . 7655 a 5 Q 2 M 7 -3 . 14047 aQ 3 M 4 +22 . 4071 a 2 Q 3 M 5 -65 . 711 a 3 Q 3 M 6 +49 . 9938 a 4 Q 3 M 7 +3 . 4426 aQ 4 M 5 -22 . 7217 a 2 Q 4 M 6 +62 . 2473 a 3 Q 4 M 7 -1 . 3531 aQ 5 M 6 +8 . 14017 a 2 Q 5 M 7 . (A1) \nClearly, A , C , and D are functions of spin a and charge Q . For the Bardeen black hole, they depend upon the magnetic charge g in addition to a , and are given by \n```\nA ( a, g ) M 2 =84 . 823 -0 . 024064 a M -3 . 92154 a 2 M 2 -7 . 64519 a 3 M 3 +31 . 6335 a 4 M 4 -70 . 2853 a 5 M 5 +73 . 327 a 6 M 6 -30 . 7577 a 7 M 7 +0 . 0715772 g M +10 . 1889 ag M 2 -88 . 754 a 2 g M 3 +310 . 385 a 3 g M 4 -492 . 683 a 4 g M 5 +323 . 536 a 5 g M 6 +9 . 27673 a 6 g M 7 -33 . 5425 g 2 M 2 -157 . 492 ag 2 M 3 +1350 . 8 a 2 g 2 M 4 -4604 . 46 a 3 g 2 M 5 +7243 . 43 a 4 g 2 M 6 -5690 . 2 a 5 g 2 M 7 +77 . 0126 g 3 M 3 +837 . 866 ag 3 M 4 -7045 . 26 a 2 g 3 M 5 +21471 . 2 a 3 g 3 M 6 -28065 . 2 a 4 g 3 M 7 -483 . 12 g 4 M 4 -1957 . 29 ag 4 M 5 +16337 . 0 a 2 g 4 M 6 -42607 . 5 a 3 g 4 M 7 +1519 . 55 g 5 M 5 +1913 . 41 ag 5 M 6 -17719 . 5 a 2 g 5 M 7 -2628 . 46 g 6 M 6 +289 . 67 ag 6 M 7 -2343 . 44 g 7 M 7 , C ( a, g ) M =32 . 6484 -0 . 00340862 a M -0 . 801464 a 2 M 2 -1 . 00523 a 3 M 3 +4 . 07939 a 4 M 4 -9 . 21854 a 5 M 5 +9 . 63115 a 6 M 6 -4 . 08615 a 7 M 7 +0 . 08745 g M +1 . 41767 ag M 2 -12 . 2238 a 2 g M 3 +41 . 4626 a 3 g M 4 -62 . 7011 a 4 g M 5 +38 . 1636 a 5 g M 6 +4 . 13287 a 6 g M 7 -8 . 15235 g 2 M 2 -23 . 2831 ag 2 M 3 +198 . 21 a 2 g 2 M 4 -660 . 393 a 3 g 2 M 5 +1000 . 05 a 4 g 2 M 6 -765 . 47 a 5 g 2 M 7 +30 . 7989 g 3 M 3 +128 . 559 ag 3 M 4 -1084 . 82 a 2 g 3 M 5 +3228 . 88 a 3 g 3 M 6 -4041 . 99 a 4 g 3 M 7 -173 . 259 g 4 M 4 -308 . 254 ag 4 M 5 +2625 . 19 a 2 g 4 M 6 -6718 . 79 a 3 g 4 M 7 +521 . 402 g 5 M 5 +302 . 965 ag 5 M 6 -883 . 676 g 6 M 6 -28 . 9381 ag 6 M 7 +782 . 985 g 7 M 7 , D ( a, g ) = 1 . -0 . 000544168 a M -0 . 0377214 a 2 M 2 -0 . 172164 a 3 M 3 +0 . 719728 a 4 M 4 -1 . 57686 a 5 M 5 +1 . 6425 a 6 M 6 -0 . 683473 a 7 M 7 +0 . 28401 ag M 2 -2 . 47242 a 2 g M 3 +8 . 78248 a 3 g M 4 -14 . 4606 a 4 g M 5 +10 . 5194 a 5 g M 6 -1 . 18703 a 6 g M 7 -4 . 30871 ag 2 M 3 +36 . 6622 a 2 g 2 M 4 -124 . 886 a 3 g 2 M 5 +197 . 753 a 4 g 2 M 6 -155 . 984 a 5 g 2 M 7 +22 . 8559 ag 3 M 4 -188 . 187 a 2 g 3 M 5 +573 . 07 a 3 g 3 M 6 -751 . 811 a 4 g 3 M 7 -54 . 0374 ag 4 M 5 +433 . 096 a 2 g 4 M 6 -1125 . 49 a 3 g 4 M 7 +55 . 5103 ag 5 M 6 -466 . 675 a 2 g 5 M 7 -13 . 6694 ag 6 M 7 . (A2)\n``` \nFor the rotating nonsingular black hole, they are functions of a and k and read as \nA ( a, k ) M 2 =84 . 823 -0 . 024298 a M -3 . 91914 a 2 M 2 -7 . 6566 a 3 M 3 +31 . 6622 a 4 M 4 -70 . 3248 a 5 M 5 +73 . 3553 a 6 M 6 -30 . 7658 a 7 M 7 -56 . 3846 k M +0 . 53105 ak M 2 -20 . 4783 a 2 k M 3 +139 . 145 a 3 k M 4 -410 . 288 a 4 k M 5 +467 . 355 a 5 k M 6 -221 . 434 a 6 k M 7 -5 . 25777 k 2 M 2 -2 . 43019 ak 2 M 3 +40 . 2944 a 2 k 2 M 4 -663 . 862 a 3 k 2 M 5 +1494 . 64 a 4 k 2 M 6 -735 . 119 a 5 k 2 M 7 +5 . 63201 k 3 M 3 -1 . 5995 ak 3 M 4 +37 . 8412 a 2 k 3 M 5 +901 . 603 a 3 k 3 M 6 -1720 . 55 a 4 k 3 M 7 -14 . 1295 k 4 M 4 +3 . 844 ak 4 M 5 -239 . 61 a 2 k 4 M 6 , C ( a, k ) M =32 . 6484 -0 . 00339904 a M -0 . 800873 a 2 M 2 -1 . 00981 a 3 M 3 +4 . 09283 a 4 M 4 -9 . 23782 a 5 M 5 +9 . 64471 a 6 M 6 -4 . 0899 a 7 M 7 -10 . 8185 k M +0 . 699341 ak M 2 -6 . 44615 a 2 k M 3 +22 . 6968 a 3 k M 4 -65 . 7711 a 4 k M 5 +70 . 7009 a 5 k M 6 -32 . 8498 a 6 k M 7 -3 . 24875 k 2 M 2 -7 . 91142 ak 2 M 3 +38 . 7444 a 2 k 2 M 4 -118 . 877 a 3 k 2 M 5 +268 . 241 a 4 k 2 M 6 -123 . 612 a 5 k 2 M 7 +1 . 99375 k 3 M 3 +27 . 4976 ak 3 M 4 -111 . 238 a 2 k 3 M 5 +169 . 666 a 3 k 3 M 6 -334 . 206 a 4 k 3 M 7 -5 . 67007 k 4 M 4 -29 . 498 ak 4 M 5 +82 . 1988 a 2 k 4 M 6 , D ( a, k ) = 1 . -0 . 000544168 a M -0 . 0377214 a 2 M 2 -0 . 172164 a 3 M 3 +0 . 719728 a 4 M 4 -1 . 57686 a 5 M 5 +1 . 6425 a 6 M 6 -0 . 683473 a 7 M 7 +0 . 014452 ak M 2 -0 . 490382 a 2 k M 3 +3 . 7850 a 3 k M 4 -10 . 804 a 4 k M 5 +11 . 709 a 5 k M 6 -5 . 37221 a 6 k M 7 -0 . 071081 ak 2 M 3 +1 . 20427 a 2 k 2 M 4 -19 . 6311 a 3 k 2 M 5 +43 . 3697 a 4 k 2 M 6 -19 . 823 a 5 k 2 M 7 +0 . 0486 ak 3 M 4 +1 . 1621 a 2 k 3 M 5 +28 . 0021 a 3 k 3 M 6 -52 . 7191 a 4 k 3 M 7 +0 . 1211 ak 4 M 5 -7 . 30097 a 2 k 4 M 6 . (A3) \nHere, we have presented the series up to O ( M -7 ). The nonrotating black hole ( a = 0) casts a perfect circular shadow (Synge 1966; Chandrasekhar 1985), which is also fully consistent with Equations (A1)-(A3), i.e., D (0 , Q ) = D (0 , g ) = D (0 , k ) = 1.', 'B. OBSERVABLES IN ASSOCIATION WITH NOISY DATA': "The observables A , C , and D are described in terms of the celestial coordinates ( α, β ), which are easy to calculate for a given black hole. Astronomical observations may not give a sharp shadow boundary demarcating the bright and dark regions; rather there will be intrinsic uncertainty in determining the shadow boundary because of noise in the observational data. In such observational data, we consider the set of visibility data points ( α i , β i ) along the hazy shadow boundary. The geometric center ( α G , β G ) of the apparent shadow reads \nα G = 1 N N ∑ i =1 α i ; β G = 1 N N ∑ i =1 β i , (B4) \nwhere N is the total number of data points. In the coordinate system centered at ( α G , β G ), the shadow boundary can be parameterized by ( α ' i , β ' i ) \nThus, we can calculate the shadow observables, A and C , respectively, as \nα ' i = α i -α G ; β ' i = β i -β G . (B5) \nA = N ∑ i =1 | β ' i -1 + β ' i | 2 | α ' i -α ' i -1 | , (B6) \nand \nwhere α ' 0 = 0 or α 0 = α G and data points are arranged such that | α ' i | ≥ | α ' i -1 | . In this case, the oblateness D becomes \n<!-- image --> \n<!-- image --> \nFigure 14. Probability density distribution of shadow observables for perturbed Kerr black hole shadows. \n<!-- image --> \nC = N ∑ i =1 ( ( α ' i -α ' i -1 ) 2 +( β ' i -β ' i -1 ) 2 ) 1 / 2 , (B7) \nD = α ' r -α ' l β ' t -β ' b , (B8) \nwhere ( α ' l , 0) and ( α ' r , 0) are, respectively, coordinates for the left and right edges of the shadow boundary, and ( α ' t , β ' t ) and ( α ' b , β ' b ) are for the top and bottom edges. 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2000NuPhB.568...93D

Self-gravitating fundamental strings and black holes

2000-01-01

['-', '-']

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The configuration of typical highly excited ( M≫ M<SUB>s</SUB>∼( α') <SUP>-1/2</SUP>) string states is considered as the string coupling g is adiabatically increased. The size distribution of very massive single string states is studied and the mass shift, due to long-range gravitational, dilatonic and axionic attraction, is estimated. By combining the two effects, in any number of spatial dimensions d, the most probable size of a string state becomes of order ℓ <SUB>s</SUB>= 2 α' when g<SUP>2</SUP>M/ M<SUB>s</SUB>∼1. Depending on the dimension d, the transition between a random-walk-size string state (for low g) and a compact (∼ℓ <SUB>s</SUB>) string state (when g<SUP>2</SUP>M/ M<SUB>s</SUB>∼1) can be very gradual ( d=3), fast but continuous ( d=4), or discontinuous ( d⩾5). Those compact string states look like nuggets of an ultradense state of string matter, with energy density ρ∼ g<SUP>-2</SUP>M<SUB>s</SUB><SUP>d+1 </SUP>. Our results extend and clarify previous work by Susskind, and by Horowitz and Polchinski, on the correspondence between self-gravitating string states and black holes.

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https://arxiv.org/pdf/hep-th/9907030.pdf

{'Self-gravitating fundamental strings and black holes': 'Thibault Damour \nInstitut des Hautes Etudes Scientifiques, 91440 Bures-sur-Yvette, France \nGabriele Veneziano Theory Division, CERN, CH-1211 Geneva 23, Switzerland', 'Abstract': "The configuration of typical highly excited ( M /greatermuch M s ∼ ( α ' ) -1 / 2 ) string states is considered as the string coupling g is adiabatically increased. The size distribution of very massive single string states is studied and the mass shift, due to long-range gravitational, dilatonic and axionic attraction, is estimated. By combining the two effects, in any number of spatial dimensions d , the most probable size of a string state becomes of order /lscript s = √ 2 α ' when g 2 M/M s ∼ 1. Depending on the dimension d , the transition between a random-walk-size string state (for low g ) and a compact ( ∼ /lscript s ) string state (when g 2 M/M s ∼ 1) can be very gradual ( d = 3), fast but continuous ( d = 4), or discontinuous ( d ≥ 5). Those compact string states look like nuggets of an ultradense state of string matter, with energy density ρ ∼ g -2 M d +1 s . Our results extend and clarify previous work by Susskind, and by Horowitz and Polchinski, on the correspondence between self-gravitating string states and black holes.", 'I. INTRODUCTION': "Almost exactly thirty years ago the study of the spectrum of string theory (known at the time as the dual resonance model) revealed [1] a huge degeneracy of states growing as an exponential of the mass. This led to the concept of a limiting (Hagedorn) temperature T Hag in string theory. Only slightly more recently Bekenstein [2] proposed that the entropy of a black hole should be proportional to the area of its horizon in Planck units, and Hawking [3] fixed the constant of proportionality after discovering that black holes, after all, do emit thermal radiation at a temperature T Haw R -1 BH . \nWhen string and black hole entropies are compared one immediately notices a striking difference: string entropy 1 is proportional to the first power of mass in any number of spatial dimensions d , while black hole entropy is proportional to a d -dependent power of the mass, always larger than 1. In formulae: \n∼ \nS s ∼ α ' M /lscript s ∼ M/M s , S BH ∼ Area G N ∼ R d -1 BH G N ∼ ( g 2 M/M s ) d -1 d -2 g 2 , (1.1) \nwhere, as usual, α ' is the inverse of the classical string tension, /lscript s ∼ √ α ' /planckover2pi1 is the quantum length associated with it 2 , M s ∼ √ /planckover2pi1 /α ' is the corresponding string mass scale, R BH is the Schwarzschild radius associated with M : \nR BH ∼ ( G N M ) 1 / ( d -2) , (1.2) \nand we have used that, at least at sufficiently small coupling, the Newton constant and α ' are related via the string coupling by G N ∼ g 2 ( α ' ) ( d -1) / 2 (more geometrically, /lscript d -1 P g 2 /lscript d -1 s ). \n∼ Given their different mass dependence, it is obvious that, for a given set of the fundamental constants G N , α ' , g 2 , S s > S BH at sufficiently small M , while the opposite is true at sufficiently large M . Obviously, there has to be a critical value of M , M c , at which S s = S BH . This observation led Bowick et al. [4] to conjecture that large black holes end up their Hawking-evaporation process when M = M c , and then transform into a higher-entropy string state without ever reaching the singular zero-mass limit. This reasoning is confirmed [5] by the observation that, in string theory, the fundamental string length /lscript s should set a minimal value for the Schwarzschild radius of any black hole (and thus a maximal value for its Hawking temperature). It was also noticed [4], [6], [7] that, precisely at M = M c , R BH = /lscript s and the Hawking temperature equals the Hagedorn temperature of string theory. For any d , the value of M c is given by: \n∼ \nM c ∼ M s g -2 . (1.3) \n2 Below, we shall use the precise definition /lscript s ≡ √ 2 α ' /planckover2pi1 , but, in this section, we neglect factors of order unity. \nSusskind and collaborators [6], [8] went a step further and proposed that the spectrum of black holes and the spectrum of single string states be 'identical', in the sense that there be a one to one correspondence between (uncharged) fundamental string states and (uncharged) black hole states. Such a 'correspondence principle' has been generalized by Horowitz and Polchinski [9] to a wide range of charged black hole states (in any dimension). Instead of keeping fixed the fundamental constants and letting M evolve by evaporation, as considered above, one can (equivalently) describe the physics of this conjectured correspondence by following a narrow band of states, on both sides of and through, the string /harpoonleftright black hole transition, by keeping fixed the entropy 3 S = S s = S BH , while adiabatically 4 varying the string coupling g , i.e. the ratio between /lscript P and /lscript s . The correspondence principle then means that if one increases g each (quantum) string state should turn into a (quantum) black hole state at sufficiently strong coupling, while, conversely, if g is decreased, each black hole state should 'decollapse' and transform into a string state at sufficiently weak coupling. For all the reasons mentioned above, it is very natural to expect that, when starting from a black hole state, the critical value of g at which a black hole should turn into a string is given, in clear relation to (1.3), by \ng 2 c M ∼ M s , (1.4) \nand is related to the common value of string and black-hole entropy via \ng 2 c ∼ 1 S BH = 1 S s . (1.5) \nNote that g 2 c /lessmuch 1 for the very massive states ( M /greatermuch M s ) that we consider. This justifies our use of the perturbative relation between G N and α ' . \nIn the case of extremal BPS, and nearly extremal, black holes the conjectured correspondence was dramatically confirmed through the work of Strominger and Vafa [10] and others [11] leading to a statistical mechanics interpretation of black-hole entropy in terms of the number of microscopic states sharing the same macroscopic quantum numbers. However, little is known about whether and how the correspondence works for non-extremal, non BPS black holes, such as the simplest Schwarzschild black hole 5 . By contrast to BPS states whose mass is protected by supersymmetry, we shall consider here the effect of varying g on the mass and size of non-BPS string states. \n5 For simplicity, we shall consider in this work only Schwarzschild black holes, in any number d ≡ D -1 of non-compact spatial dimensions. \nAlthough it is remarkable that black-hole and string entropy coincide when R BH = /lscript s , this is still not quite sufficient to claim that, when starting from a string state, a string becomes a black hole at g = g c . In fact, the process in which one starts from a string state in flat space and increases g poses a serious puzzle [6]. Indeed, the radius of a typical excited string state of mass M is generally thought of being of order \nR rw s ∼ /lscript s ( M/M s ) 1 / 2 , (1.6) \nas if a highly excited string state were a random walk made of M/M s = α ' M//lscript s segments of length /lscript s [12]. [The number of steps in this random walk is, as is natural, the string entropy (1.1).] The 'random walk' radius (1.6) is much larger than the Schwarzschild radius for all couplings g ≤ g c , or, equivalently, the ratio of self-gravitational binding energy to mass (in d spatial dimensions) \nG N M ( R rw s ) d -2 ∼ ( R BH ( M ) R rw s ) d -2 ∼ g 2 ( M M s ) 4 -d 2 (1.7) \nremains much smaller than one (when d > 2, to which we restrict ourselves) up to the transition point. In view of (1.7) it does not seem natural to expect that a string state will 'collapse' to a black hole when g reaches the value (1.4). One would expect a string state of mass M to turn into a black hole only when its typical size is of order of R BH ( M ) (which is of order /lscript s at the expected transition point (1.4)). According to Eq. (1.7), this seems to happen for a value of g much larger than g c . \nHorowitz and Polchinski [13] have addressed this puzzle by means of a 'thermal scalar' formalism [14]. Their results suggest a resolution of the puzzle when d = 3 (four-dimensional spacetime), but lead to a rather complicated behaviour when d ≥ 4. More specifically, they consider the effective field theory of a complex scalar field χ in d (spacetime) dimensions (with period β in Euclidean time τ ), with mass squared m 2 ( β ) = (4 π 2 α ' 2 ) -1 [ G ττ β 2 -β 2 H ], where G µν is the string metric and β -1 H the Hagedorn temperature. They took into account the effect of gravitational (and dilatonic) self-interactions in a mean field approximation. This leads to an approximate Hartree-like equation for χ ( x ), which admits a stable bound state, in some range g 0 < g < g c , when d = 3. They interpret the size of the bound state 'wave function' χ ( x ) as the 'size of the string', and find that (in d = 3) this size is of order \n/lscript χ ∼ 1 g 2 M . (1.8) \nThey describe their result by saying that 'the string contracts from its initial (large) size', when g ∼ g 0 ∼ ( M/M s ) -3 / 4 , down to the string scale when g ∼ g c ∼ ( M/M s ) -1 / 2 . This interpretation of the length scale /lscript χ , characterizing the thermal scalar bound state, as 'the size of the string' is unclear to us, because of the formal nature of the auxiliary field χ which has no direct physical meaning in Minkowski spacetime. Moreover, the analysis of Ref. [13] in higher dimensions is somewhat inconclusive, and suggests that a phenomenon of hysteresis takes place (when d ≥ 5): the critical value of g corresponding to the string /harpoonleftright black hole transition would be g 0 ∼ ( M/M s ) ( d -6) / 4 > g c for the direct process (string → black hole), and g c for the reverse one. Finally, they suggest that, in the reverse process, a black hole becomes an excited string in an atypical state. \nThe aim of the present work is to clarify the string /harpoonleftright black hole transition by a direct study, in real spacetime, of the size and mass of a typical excited string, within the microcanonical ensemble of self-gravitating strings. Our results lead to a rather simple picture of the transition, in any dimension. We find no hysteresis phenomenon in higher dimensions. The critical value for the transition is (1.4), or (1.5) in terms of the entropy S , for both directions of the string /harpoonleftright black hole transition. In three spatial dimensions, we find that the size (computed in real spacetime) of a typical self-gravitating string is given by the random walk value (1.6) when g 2 ≤ g 2 0 , with g 2 0 ∼ ( M/M s ) -3 / 2 ∼ S -3 / 2 , and by \nR typ ∼ 1 g 2 M , (1.9) \nwhen g 2 0 ≤ g 2 ≤ g 2 c . Note that R typ smoothly interpolates between R rw s and /lscript s . This result confirms the picture proposed by Ref. [13] when d = 3, but with the bonus that Eq. (1.9) (which agrees with Eq. (1.8)) refers to a radius which is estimated directly in physical space (see below), and which is the size of a typical member of the microcanonical ensemble of self-gravitating strings. In all higher dimensions 6 , we find that the size of a typical selfgravitating string remains fixed at the random walk value (1.6) when g ≤ g c . However, when g gets close to a value of order g c , the ensemble of self-gravitating strings becomes (smoothly in d = 4, but suddenly in d ≥ 5) dominated by very compact strings of size ∼ /lscript s (which are then expected to collapse with a slight further increase of g because the dominant size is only slightly larger than the Schwarzschild radius at g c ). \nOur results confirm and clarify the main idea of a correspondence between string states and black hole states [6], [8], [9], [13], and suggest that the transition between these states is rather smooth, with no apparent hysteresis, and with continuity in entropy, mass and typical size. It is, however, beyond the technical grasp of our analysis to compute any precise number at the transition (such as the famous factor 1 / 4 in the Bekenstein-Hawking entropy formula).", 'II. SIZE DISTRIBUTION OF FREE STRING STATES': "The aim of this section is to estimate the distribution function in size of the ensemble of free string states of mass M , i.e. to count how many massive string states have a given size R . This estimate will be done while neglecting the gravitational self-interaction. The effect of the latter will be taken into account in a later section. \nLet us first estimate the distribution in size by a rough, heuristic argument based on the random walk model [12] of a generic excited string state. In string units ( /lscript s ∼ M -1 s ∼ 1), the geometrical shape (in d -dimensional space) of a generic massive string state can be roughly identified with a random walk of M steps of unit length. We can constrain this random walk to stay of size < ∼ R by considering a diffusion process, starting from a point source at the origin, in presence of an absorbing sphere S R of radius R , centered on the origin. In the \ncontinuous approximation, the kernel K t ( x , 0 ) giving the conditional probability density of ending, at time t , at position x , after having started (at time 0) at the origin, without having ever gone farther from the origin than the distance R , satisfies: (i) the diffusion equation ∂ t K t = ∆ K t , (ii) the initial condition K 0 ( x , 0 ) = δ ( x ), and (iii) the 'absorbing' boundary condition K t = 0 on the sphere S R . The kernel K t can be decomposed in eigenmodes, \nK t ( x , 0 ) = ∑ n ψ n ( x ) ψ n ( 0 ) e -E n t , (2.1) \nwhere ψ n ( x ) is a normalized (real) L 2 basis ( ∫ d d xψ n ( x ) ψ m ( x ) = δ nm ; ∑ n ψ n ( x ) ψ n ( y ) = δ ( x -y )) satisfying \n∆ ψ n = -E n ψ, (2.2) \nin the interior, and vanishing on S R . The total conditional probability of having stayed within S R after the time t is the integral ∫ B R d d xK t ( x , 0 ) within the ball B R : | x | ≤ R . For large values of t , K t is dominated by the lowest eigenvalue E 0 , and the conditional probability goes like c 0 e -E 0 t , where c 0 is a numerical constant of order unity. The eigenvalue problem (2.2) is easily solved (in any dimension d ), and the s -wave ground state can be expressed in terms of a Bessel function: ψ 0 ( r ) = N J ν ( k 0 r ) / ( k 0 r ) ν with ν = ( d -2) / 2. Here, k 0 = √ E 0 is given by the first zero j ν of J ν ( z ) : k 0 = j ν /R . The important information for us is that the ground state energy E 0 scales with R like E 0 = c 1 /R 2 , where c 1 = O (1) is a numerical constant. This scaling is evident for the Dirichlet problem(2.2), whatever be the shape of the boundary. Finally, remembering that the number of time steps is given by the mass, t = M , we expect the looked for conditional probability, i.e. the fraction of all string states at mass level M which are of size < ∼ R , to be asymptotically of order \nf ( R ) ∼ e -c 1 M/R 2 . (2.3) \nThis estimate is expected to be valid when M/R 2 /greatermuch 1, i.e. for string states which are much smaller (in size) than a typical random walk R 2 rw ∼ M (but still larger than the string length, R > ∼ 1). In the opposite limit, R 2 /greatermuch M , the kernel K t ( x , 0 ) can be approximated by the free-space value K (0) t ( x , 0 ) = (4 π t ) -d/ 2 exp( -x 2 / (4 t )), with t = M , so that the fraction of string states of size > R /greatermuch R rw will be of order e -c 2 R 2 /M , with c 2 = O (1). \n/greatermuch O As the result (2.3) will be central to the considerations of this paper, we shall now go beyond the previous heuristic, random walk argument and derive the fraction of small string states by a direct counting of quantum string states. For simplicity, we shall deal with open bosonic strings ( /lscript s ≡ √ 2 α ' , 0 ≤ σ ≤ π ) \n∼ \nX µ ( τ, σ ) = X µ cm ( τ, σ ) + ˜ X µ ( τ, σ ) , (2.4) \nX µ cm ( τ, σ ) = x µ +2 α ' p µ τ , (2.5) \n˜ X µ ( τ, σ ) = i /lscript s ∑ n /negationslash =0 α µ n n e -inτ cos nσ. (2.6) \n∼ \nHere, we have explicitly separated the center of mass motion X µ cm (with [ x µ , p ν ] = i η µν ) from the oscillatory one ˜ X µ ([ α µ m , α ν n ] = mδ 0 m + n η µν ). The free spectrum is given by α ' M 2 = N -1 where ( α · β ≡ η µν α µ β ν ≡ -α 0 β 0 + α i β i ) \nHere N n ≡ a † n · a n is the occupation number of the n th oscillator ( α µ n = √ n a µ n ,[ a µ n , a ν † m ] = η µν δ nm , with n, m positive). \nN = ∞ ∑ n =1 α -n · α n = ∞ ∑ n =1 nN n . (2.7) \nThe decomposition (2.4)-(2.6) holds in any conformal gauge (( ∂ τ X µ ± ∂ σ X µ ) 2 = 0). One can further specify the choice of worldsheet coordinates by imposing \nn µ X µ ( τ, σ ) = 2 α ' ( n µ p µ ) τ , (2.8) \nwhere n µ is an arbitrary timelike or null vector ( n · n ≤ 0) [15]. Eq. (2.8) means that the n -projected oscillators n µ α µ m are set equal to zero. The usual 'light-cone' gauge is obtained by choosing a fixed, null vector n µ . The light-cone gauge introduces a preferred ('longitudinal') direction in space, which is an inconvenience for defining the (rms) size of a massive string state. As we shall be interested in quasi-classical, very massive string states ( N /greatermuch 1) it should be possible to work in the 'center of mass' gauge, where the vector n µ used in Eq. (2.8) to define the τ -slices of the world-sheet is taken to be the total momentum p µ of the string. This gauge is the most intrinsic way to describe a string in the classical limit. Using this intrinsic gauge, one can covariantly 7 define the proper rms size of a massive string state as \nR 2 ≡ 1 d 〈 ( X µ ⊥ ( τ, σ )) 2 〉 σ,τ , (2.9) \nwhere ˜ X µ ⊥ ≡ ˜ X µ -p µ ( p · ˜ X ) / ( p · p ) denotes the projection of ˜ X µ ≡ X µ -X µ cm ( τ ) orthogonally to p µ , and where the angular brackets denote the (simple) average with respect to σ and τ . The factor 1 /d in Eq. (2.9) is introduced to simplify later formulas. So defined R is the rms value of the projected size of the string along an arbitrary, but fixed spatial direction. [We shall find that this projected size is always larger than √ 3 α ' / 2; i.e. string states cannot be 'squeezed', along any axis, more than this.] \n˜ \nIn the center of mass gauge, p µ ˜ X µ vanishes by definition, and Eq. (2.9) yields simply \n˜ R 2 = 1 d /lscript 2 s R , (2.10) \nwith \nR≡ ∞ ∑ n =1 α -n · α n + α n · α -n 2 n 2 . (2.11) \nThe squared-size operator R , Eq. (2.11), contains the logarithmically infinite contribution ∑ 1 / (2 n ). Without arguing with the suggestion that this contribution may have a physical meaning (see, e.g., [16]), we note here that this contribution is state-independent. We are interested in this work in the relative sizes of various highly excited, quasi-classical states. A concept which should reduce to the well-defined, finite rms size of a classical Nambu string in the classical limit. We shall therefore discard this state-independent contribution, i.e. work with the normal-ordered operator \n: R : = ∞ ∑ n =1 a † n · a n n = ∞ ∑ n =1 N n n . (2.12) \nWe shall assume that we can work both in the center-of-mass (worldsheet) gauge ( p µ α µ m → 0) and in the center-of-mass (Lorentz) frame (( p µ ) = ( M, 0 )). This means that the scalar product in the level occupation number N n runs over the d spatial dimensions: \nN n = a † n · a n = d ∑ i =1 ( a i n ) † a i n . The 'wrong sign' time oscillators α 0 n are set equal to zero. \nThe problem we would like to solve is to count the number of physical states, in the Fock space of the center-of-mass oscillators α i n , having some fixed values of N and R (we henceforth work only with the normal-ordered operator (2.12) without adorning it with the : : notation). The Virasoro constraints make this problem technically quite difficult. However, we know from the exact counting of physical states (without size restriction) in the light-cone gauge that the essential physical effect of the Virasoro constraints is simply to reduce the number of independent oscillators at any level n from d = D -1 (in the center-of-mass gauge) to d -1 = D -2. If we (formally) consider d as a large parameter 8 , this change in the number of effective free oscillators should have only a small fractional effect on any other coarse-grained, counting problem. We shall assume that this is the case, and solve the much simpler counting problem where the d oscillators α i n are considered as independent 9 . To solve this problem we pass from a microcanonical problem (fixed values \nThe Virasoro constraints then imply, besides the mass formula α ' M 2 = N -1, the usual sequence of constraints on physical states, L n | φ 〉 = 0, with L n = 1 2 ∑ m d ∑ i =1 α i n -m α i m . These constraints mean that the d oscillators α i n at level n are not physically independent. \n9 We tried to work in the light-cone gauge, with d -1 independent oscillators. However, the necessary inclusion of the longitudinal term M -2 ( p · ˜ X ) 2 in (2.9), which is quadratic in the longitudinal oscillators α -n = ( p + ) -1 L transverse n , leads to a complicated, interacting theory of the d -1 transverse oscillators. \nof N and R ) to a grand canonical one (fixed values of some thermodynamical conjugates of N and R ). Let us introduce the formal 'partition function' \nZ d ( β, γ ) ≡ ∑ { N i n } exp( -β N [ N i n ] -γ R [ N i n ]) , (2.13) \nwhere the sum runs over all sequences (labelled by n ≥ 1 and i = 1 , . . . , d ) of independent occupation numbers N i n = ( a i n ) † a i n = 0 , 1 , 2 , . . . , and where N [ N i n ] and R [ N i n ] are defined by \nEqs. (2.7), and (2.12), with N n ≡ i =1 N i n . Note that (2.13) is not the usual thermodynamical \npartition function, and that β is not the usual inverse temperature. Indeed, β is a formal conjugate to N /similarequal α ' M 2 and not to the energy M . In particular, because the degeneracy grows exponentially with M (and not M 2 ) its Laplace transform (2.13) is defined for arbitrary values of β . We associate with the definition (2.13) that of a formal grand canonical ensemble of configurations, with the probability \nd ∑ \np [ { N i n } ] = Z -1 d ( β, γ ) exp( -β N [ N i n ] -γ R [ N i n ]) (2.14) \nof realization of the particular sequence N i n of occupation numbers. The mean values of N [ N i n ] and R [ N i n ] in this ensemble are \nN = -∂ ψ d ( β, γ ) ∂ β , R = -∂ ψ d ( β, γ ) ∂ γ , (2.15) \nwhere we denote \nψ d ( β, γ ) ≡ ln Z d ( β, γ ) . (2.16) \nThe second derivatives of the thermodynamical potential ψ d ( β, γ ) give the fluctuations of N and R in this grand canonical ensemble: \n(∆ N ) 2 = ∂ 2 ψ d ( β, γ ) ∂ β 2 , (∆ R ) 2 = ∂ 2 ψ d ( β, γ ) ∂ γ 2 . (2.17) \nLet us define as usual the entropy S ( β, γ ) as the logarithm of the number of string configurations having values of N and R equal to N and R , Eqs. (2.15), within the precision of the rms fluctuations (2.17) [17]. This definition means that, in the saddle-point approximation, Z d ( β, γ ) /similarequal exp [ S -β N -γ R ], i.e. \nψ d ( β, γ ) /similarequal S ( β, γ ) -β N -γ R , (2.18) \nor \nS /similarequal ψ d ( β, γ ) -β ∂ ψ d ( β, γ ) ∂ β -γ ∂ ψ d ( β, γ ) ∂ γ . (2.19) \nIn other words, the entropy S ( N, R ) is the Legendre transform of ψ d ( β, γ ). \nR Because of the (assumed) independence of the d oscillators in (2.13), one has \ni.e. \nwith \nψ 1 ( β, γ ) = -∞ ∑ n =1 ln [ 1 -exp ( -βn -γ n )] . (2.22) \nWe shall check a posteriori that we are interested in values of β and γ such that \nβ /lessmuch β γ /lessmuch 1 . (2.23) \nFor such values, one can approximate the discrete sum (2.22) by a continuous integral over x = β n . This yields \nand \n√ \nψ 1 ( β, γ ) = I ( δ ) β , where δ ≡ √ β γ , (2.24) \nI ( δ ) ≡ -∫ ∞ 0 dx ln [1 -e -( x + δ 2 /x ) ] . (2.25) \n∫ I ' (0) = lim δ → 0 [ -2 ∫ ∞ 0 du u δ e δ ( u +1 /u ) -1 ] = -2 ∫ ∞ 0 du u 2 +1 = -π . \nAs δ = √ β γ /lessmuch 1, we can try to expand I ( δ ) in powers of δ : I ( δ ) = I (0) + δ I ' (0) + o ( δ ). [Though the integral (2.25) is expressed in terms of δ 2 , its formal expansion in powers of δ 2 leads to divergent integrals.] The zeroth-order term is I (0) = -∫ ∞ 0 dx ln (1 -e -x ) = π 2 / 6, while \nHence, using (2.21), \nwith \nψ d ( β, γ ) = 1 β [ C -D √ β γ + o ( √ β γ )] (2.26) \nC = π 2 6 d , D = π d . (2.27) \n[The notation D in (2.27) should not be confused with the space-time dimension d +1.] The thermodynamic potential (2.26) corresponds to the mean values \nN /similarequal C -1 2 D √ β γ β 2 , R/similarequal D 2 δ , (2.28) \nand to the entropy \nZ d ( β, γ ) = ∞ ∏ n =1 [1 -e -( βn + γ/n ) ] -d , (2.20) \nψ d ( β, γ ) = d ψ 1 ( β, γ ) , (2.21) \ni.e. \nS /similarequal 2 C -Dδ β /similarequal 2 √ C N [ 1 -D 2 8 C R ] , (2.29) \nS /similarequal 2 π √ d 6 N [ 1 -3 d 4 1 R ] /similarequal 2 π ( d α ' 6 ) 1 2 M [ 1 -3 4 /lscript 2 s R 2 ] . (2.30) \nThe lowest-order term 2 π √ d N/ 6 is the usual (Hardy-Ramanujan) result for d independent oscillators, without size restriction. The factor in bracket, 1 -(3 / 4)( /lscript 2 s /R 2 ), with /lscript 2 s = 2 α ' , gives the fractional reduction in the entropy brought by imposing the size constraint R 2 /similarequal d -1 /lscript 2 s R . Under the conditions (2.23) the fluctuations (2.17) are fractionally small. More precisely, Eqs. (2.17) yield \n(∆ N ) 2 N 2 ∼ β ∼ M s M , (∆ R ) 2 R 2 ∼ β δ ∼ ( R 2 //lscript 2 s ) ( M/M s ) . (2.31) \n∼ \n∼ \nAs said above, though we worked under the (physically incorrect) assumption of d independent oscillators at each level n , we expect the result (2.29) to be correct when d /greatermuch 1. [We recall that the exact result for S in absence of size restriction is 2 π √ ( d -1) N/ 6.] Note the rough physical meanings of the auxiliary quantities β , γ and δ : β ∼ ( N ) -1 / 2 ∼ ( M/M s ) -1 , δ -1 ∼ /lscript 2 s /R 2 , γ ( N ) 1 / 2 / R 2 M/lscript 5 s /R 4 . \n∼ R Summarizing, the main result of the present section is that the number('degeneracy') of free string states of mass M and size R (within the narrow bands defined by the fluctuations (2.17)) is of the form \nD ( M,R ) ∼ exp [ c ( R ) a 0 M ] , (2.32) \nwhere a 0 = 2 π (( d -1) α ' / 6) 1 / 2 and \nc ( R ) = ( 1 -c 1 R 2 ) ( 1 -c 2 R 2 M 2 ) , (2.33) \nwith the coefficients c 1 and c 2 being of order unity in string units. [We have added, for completeness, in Eq. (2.32) the factor 1 -c 2 R 2 /M 2 which operates when one considers very 'large' string states, R 2 /greatermuch R 2 rw (as discussed below Eq. (2.3)).] The coefficient c ( R ) gives the fractional reduction in entropy brought by imposing a size constraint. Note that (as expected) this reduction is minimized when c 1 R -2 ∼ c 2 R 2 /M 2 , i.e. for R ∼ R rw ∼ /lscript s √ M/M s . [The absolute reduction in degeneracy is only a factor O (1) when R ∼ R rw .] Note also that c ( R ) → 0 both when R ∼ /lscript s and when R ∼ /lscript s ( M/M s ). [The latter value corresponding to the vicinity of the leading Regge trajectory J ∼ α ' M 2 .]", 'III. MASS SHIFT OF STRING STATES DUE TO SELF-GRAVITY': "In this section we shall estimate the mass shift of string states (of mass M and size R ) due to the exchange of the various long-range fields which are universally coupled to \n∼ R \nthe string: graviton, dilaton and axion. As we are interested in very massive string states, M /greatermuch M s , in extended configurations, R /greatermuch /lscript s , we expect that massless exchange dominates the (state-dependent contribution to the) mass shift. \nThe evaluation, in string theory, of (one loop) mass shifts for massive states is technically quite involved, and can only be tackled for the states which are near the leading Regge trajectory [18]. [Indeed, the vertex operators creating these states are the only ones to admit a manageable explicit oscillator representation.] As we consider states which are very far from the leading Regge trajectory, there is no hope of computing exactly (at one loop) their mass shifts. We shall resort to a semi-classical approximation, which seems appropriate because we consider highly excited configurations. As a starting point to derive the massshift in this semi-classical approximation we shall use the classical results of Ref. [19] which derived the effective action of fundamental strings. The one-loop exchange of g µν , ϕ and B µν leads to the effective action \nI eff = I 0 + I 1 , (3.1) \nwhere I 0 is the free (Nambu) string action ( d 2 σ 1 ≡ dσ 1 dτ 1 , γ 1 = -det γ ab ( X µ ( σ 1 , τ 1 ))) \nI 0 = -T ∫ d 2 σ 1 √ γ 1 , (3.2) \nand I 1 the effect of the one-loop interaction ( X µ 1 ≡ X µ ( σ 1 , τ 1 ) , . . . ) \nI 1 = 2 π ∫ ∫ d 2 σ 1 d 2 σ 2 G F ( X 1 -X 2 ) √ γ 1 √ γ 2 C tot ( X 1 , X 2 ) , (3.3) \nwhere G F is Feynman's scalar propagator ( ✷ G F ( x ) = -δ D ( x )), and C tot ( X 1 , X 2 ) = J ϕ ( X 1 ) · J ϕ ( X 2 ) + J g ( X 1 ) · J g ( X 2 ) + J B ( X 1 ) · J B ( X 2 ) comes from the couplings of ϕ , g µν and B µν to their corresponding world-sheet sources (indices suppressed; spin-structure hidden in the dot product). The exchange term C tot takes, in null (conformal) coordinates σ ± = τ ± σ , the simple left-right factorized form [19] \n√ γ 1 √ γ 2 C tot ( X 1 , X 2 ) = 32 G N T 2 ( ∂ + X µ 1 ∂ + X 2 µ )( ∂ -X ν 1 ∂ -X 2 ν ) . (3.4) \nHere, T = (2 π α ' ) -1 is the string tension, G N is Newton's constant 10 and ∂ ± = ∂/∂σ ± = 1 2 ( ∂ τ ± ∂ σ ). Let us define P µ ± = P µ ± ( σ ± ) by ( /lscript s = √ 2 α ' as above) \n2 ∂ ± X µ = /lscript s P µ ± , (3.5) \nso that, for an open (bosonic) string (with α µ 0 ≡ /lscript s p µ ), \nP µ ± = + ∞ ∑ -∞ α µ n e -inσ ± . (3.6) \nUsing the definition (3.5) and inserting the Fourier decomposition of G F yields \nI 1 = 4 G N π ∫ d D k (2 π ) D 1 k 2 -i ε ∫ ∫ d 2 σ 1 d 2 σ 2 ( P + ( X 1 ) · P + ( X 2 ))( P -( X 1 ) · P -( X 2 )) × e ik · ( X 1 -X 2 ) , (3.7) \nwhere one recognizes the insertion of two gravitational vertex operators V µν ( k ; X ) = P µ + ( X ) P ν -( X ) e ik · X at two different locations on the worldsheet, and with two opposite momenta for the exchanged graviton 11 . [Note that the exchanged graviton is off-shell.] It is convenient to use the Virasoro constraints ( P µ ± ( X )) 2 = 0 to replace in Eq. (3.6) P ± ( X 1 ) · P ± ( X 2 ) = -1 2 (∆ P µ ± ) 2 where ∆ P µ ± ≡ P µ ± ( X 1 ) -P µ ± ( X 2 ). It is important to note that the zero mode contribution α µ 0 drops out of ∆ P µ ± (i.e. ∆ P µ ± = ∆ P µ ± is purely oscillatory). \nWriting that the correction I 1 to the effective action I eff (which gives the vacuum persistence amplitude; see, e.g., Eq. (7) of [19]) must correspond to a phase shift -∫ δE dt = -∫ δM dX 0 cm , in the center-of-mass frame of the string, yields (with the normalization (2.5)) the link I 1 = -/lscript 2 s ∫ dτ M δM = -1 2 /lscript 2 s ∫ dτ δM 2 . Let us also define ∆ X µ ≡ X µ 1 -X µ 2 and decompose it in its zero-mode part ∆ X µ cm = /lscript 2 s p µ ( τ 1 -τ 2 ) and its oscillatory part ∆ ˜ X µ = ˜ X µ 1 -X µ 2 . Finally, the mass-shift can be read from \n˜ \n˜ ∫ dτ δM 2 = -2 G N π /lscript 2 s ∫ d D k (2 π ) D 1 k 2 -iε ∫ ∫ d 2 σ 1 d 2 σ 2 e i /lscript 2 s k · p ( τ 1 -τ 2 ) W ( k, 1 , 2) , (3.8) \n˜ \nwhere (1 and 2 being short-hands for ( τ 1 , σ 1 ) and ( τ 2 , σ 2 ), respectively) \nW ( k, 1 , 2) = (∆ P µ + (1 , 2)) 2 (∆ P ν -(1 , 2)) 2 e ik · ∆ ˜ X (1 , 2) . (3.9) \n∫ It is also important to note the good ultraviolet behaviour of Eq. (3.8). The ultraviolet limit k → ∞ corresponds to the coincidence limit ( τ 2 , σ 2 ) → ( τ 1 , σ 1 ) on the world-sheet. \nInterpreted at the quantum level, the classical result (3.8) gives (modulo some ordering problems, which are, however, fractionally negligible when considering very massive states) the mass-shift δ M 2 N of the string state | N 〉 when replacing W ( k, 1 , 2), on the right-hand side of Eq. (3.8), by the quantum average 〈 N | W ( k, 1 , 2) | N 〉 . Here, we shall mainly be interested in the real part of δ M 2 , obtained by replacing ( k 2 -iε ) -1 by the principal part of ( k 2 ) -1 (denoted simply 1 /k 2 ), i.e. the Feynman Green's function G F ( x ) by the half-retarded-halfadvanced one G sym ( x ). [The imaginary part of δ M 2 gives the decay rate, i.e. the total rate of emission of massless quanta.] As L 0 -1 is the 'Hamiltonian' that governs the τ -evolution of an open string, the vanishing of ( L 0 -1) | N 〉 for any physical state ensures that 〈 N | W ( k, 1 , 2) | N 〉 is τ -translation invariant, i.e. that it depends only on the difference τ 12 ≡ τ 1 -τ 2 , and not on the average τ ≡ 1 2 ( τ 1 + τ 2 ). This means that the double worldsheet integration d 2 σ 1 d 2 σ 2 = dτ 1 dσ 1 dτ 2 dσ 2 = dτ dτ 12 dσ 1 dσ 2 on the right-hand side of Eq. (3.8) contains a formally infinite infra-red 'volume' factor ∫ d τ which precisely cancels the integral ∫ d τ on the left-hand side to leave a finite answer for δ M 2 . \nLet us define u ≡ σ + 1 -σ + 2 , v ≡ σ -1 -σ -2 and consider the coincidence limit u → 0, v → 0. In this limit the vertex insertion factors (∆ P + ) 2 (∆ P -) 2 tend to zero like u 2 v 2 , while the Green's function blows up like [(∆ X ) 2 ] -( D -2) / 2 ∝ ( uv ) -( D -2) / 2 . The resulting integral, ∫ dudv ( uv ) -( D -6) / 2 , has its first ultraviolet pole when the space-time dimension D = d + 1 = 8. This means probably that in dimensions D ≥ 8 the exchange of massive modes (of closed strings) becomes important. Our discussion, which is limited to considering only the exchange of massless modes, is probably justified only when D < 8. \nFollowing the (approximate) approach of Section 2 we shall estimate the average mass shift δ M 2 ( R ) for string states of size R by using the grand canonical ensemble with density matrix \nρ ≡ ( Z d ( β, γ )) -1 exp ( -β N -γ : R : ) , (3.10) \nwhere the operators N and : R :, defined by Eqs. (2.7) and (2.12), belong to the Fock space built upon d sequences of string oscillators α i n (formal 'center-of-mass' oscillators). For any quantity Q (built from string oscillators) we denote the grand canonical average as 〈 Q 〉 β,γ ≡ tr ( Qρ ). Using the τ -shift invariance mentioned above, Eq. (3.8) yields \nδ M 2 β,γ = -2 G N π /lscript 2 s ∫ d d k d ω (2 π ) d +1 [ k 2 -ω 2 -iε ] ∫ dτ 12 dσ 1 dσ 2 e -i /lscript 2 s Mωτ 12 〈 W ( k, 1 , 2) 〉 β,γ , (3.11) \nwhere we have separated k µ in its center-of-mass components k 0 = ω , k i = k , and where τ 12 τ 1 -τ 2 as above. \n-We shall estimate the grand canonical average 〈 W 〉 β,γ in a semi-classical approximation in which we neglect some of the contributions linked to the ordering of the operator W , but take into account the quantum nature of the d ensity matrix ρ , Eq. (3.10). The discreteness of the Fock states built from the( a i n ) † , and the Planckian nature of ρ will play a crucial role in the calculation below. [By contrast, a purely classical calculation would be awkward and illdefined because of the problem of defining a measure on classical string configurations, and because of the Rayleigh-Jeans ultraviolet catastrophe.] To compute 〈 W 〉 β,γ it is convenient to define it as a double contraction of the coefficient of ζ 1 µ 1 ζ 2 µ 2 ζ 3 µ 3 ζ 4 µ 4 in the exponentiated version of W : \nW ζ ≡ : exp [ ζ 1 µ 1 ∆ P µ 1 + + ζ 2 µ 2 ∆ P µ 2 + + ζ 3 µ 3 ∆ P µ 3 -+ ζ 4 µ 4 ∆ P µ 4 -+ ik · ∆ X ] : (3.12) \nWe shall define our ordering of W by working with the normal ordered exponentiated operator (3.12) (and picking the term linear in ζ 1 µ 1 ζ 2 µ 2 ζ 3 µ 3 ζ 4 µ 4 ). The average 〈 W ζ 〉 β = tr ( W ζ ρ ) (where, to ease the notation, we drop the extra label γ ) can be computed by a generalization of Bloch's theorem. Namely, if A denotes any operator which is linear in the oscillators α i n , we have the results \n˜ \n〈 e A 〉 β = exp [ 1 2 〈 A 2 〉 β ] ; 〈 : e A : 〉 β = exp [ 1 2 〈 : A 2 : 〉 β ] , (3.13) \nas well as their corollaries \n〈 e A 〉 0 = exp [ 1 2 〈 A 2 〉 0 ] ; e A =: e A : exp [ 1 2 〈 A 2 〉 0 ] , (3.14) \n≡ \nwhere 〈 W 〉 0 denotes the vacuum average (obtained in the zero temperature limit β -1 → 0). The simplest way to prove these results is to use coherent-state methods [20] (see also,Ref. [15] and the Appendix 7.A of Ref. [21]). For instance, to prove the second equation (3.13) it is sufficient to consider a single oscillator and to check that (denoting q = e -/epsilon1 , with /epsilon1 = β n + γ/n (label n suppressed), so that Z = (1 -q ) -1 , and | b ) ≡ exp ( ba † ) | 0 〉 ) \n〈 e c 1 a † e c 2 a 〉 β = Z -1 tr ( e c 1 a † e c 2 a q a † a ) = Z -1 ∫ d 2 b π e -b ∗ b ( b | e c 1 a † e c 2 a q a † a | b ) = (1 -q ) ∫ d 2 b π e -b ∗ b ( b | e c 1 b ∗ e c 2 a | qb ) = (1 -q ) ∫ d 2 b π e -(1 -q ) b ∗ b e c 1 b ∗ + c 2 qb = e c 1 c 2 q/ (1 -q ) , (3.15) \nand to recognize that q/ (1 -q ) = [ e /epsilon1 -1] -1 is the Planckian mean occupation number 〈 a † a 〉 β . If we apply the second Eq. (3.13) to an expression of the type W ζ =: exp ( 4 ∑ i =1 ζ i A i + B ) :, one gets a Wick-type expansion for the coefficient (say W 1234 ) of ζ 1 ζ 2 ζ 3 ζ 4 : \nW 1234 = e 1 2 [ BB ] ([ A 1 A 2 ] [ A 3 A 4 ] + 2 terms + [ A 1 B ] [ A 2 B ] [ A 3 A 4 ] + 5 terms + [ A 1 B ] [ A 2 B ] [ A 3 B ] [ A 4 B ]) , (3.16) \nwhere [ AB ] denotes the 'thermal' contraction [ AB ] ≡ 〈 : AB : 〉 β . \n˜ 〈 W 〉 β,γ = e -1 2 〈 :( k · ∆ ˜ X ) 2 : 〉 β,γ 〈 : (∆ P + ) 2 : 〉 β,γ 〈 : (∆ P -) 2 : 〉 β,γ + . . . } , (3.17) \nThe looked-for grand canonical average of W ( k, 1 , 2), Eq. (3.9), is given by replacing B = ik · ∆ X and A 1 = A 2 = ∆ P µ + , A 3 = A 4 = ∆ P ν -in Eq. (3.16). This leads to \n{ \n} where the ellipsis stand for other contractions (which will be seen below to be subleading). The calculation of the various contractions [ AB ] ≡ 〈 : AB : 〉 β in Eqs. (3.16), (3.17) is easily performed by using the basic contractions among the oscillators a n , a † m ( n, m > 0) (which are easily derived from the definition (3.10) of the density matrix) \n〈 : a i n ( a j m ) † : 〉 β,γ = 〈 : ( a j m ) † a i n : 〉 β,γ = δ ij δ nm e /epsilon1 n -1 , (3.18) \nwhere /epsilon1 n = β n + γ/n . The other contractions [ aa ] and [ a † a † ] vanish. In terms of the α -oscillators, the basic contraction reads [ α i n α j m ] = δ ij δ 0 n + m | n | / (exp( /epsilon1 | n | ) -1), where now n and m can be negative(but not zero). Using these basic contractions, and the oscillator expansion (2.6) of ˜ X µ (and noting that, in the center-of-mass frame only the spatial components of ˜ X µ survive) one gets \nwith \n〈 : ( k · ∆ ˜ X ) 2 : 〉 β,γ = 2 k 2 /lscript 2 s ∞ ∑ n =1 x n (1 , 2) n ( e /epsilon1 n -1) , (3.19) \nx n (1 , 2) = cos 2 nσ 1 +cos 2 nσ 2 -2 cos nσ 1 cos nσ 2 cos nτ 12 . (3.20) \nSimilarly, the oscillator expansion (3.6) yields \n〈 : ∆ P + ) 2 : 〉 β,γ = 4 d ∞ ∑ n =1 np + n (1 , 2) e /epsilon1 n -1 , (3.21) \nwith \np + n (1 , 2) = 1 -cos n ( σ + 1 -σ + 2 ) = 1 -cos n ( τ 12 + σ 1 -σ 2 ) . (3.22) \nThe result for (∆ P -) 2 is obtained by changing σ + → σ -in Eq. (3.22) (i.e. σ 1 -σ 2 → σ 1 + σ 2 ). \nWe can estimate the values of the right-hand sides of Eqs. (3.19) and (3.21) by using the following 'statistical' approximation. In the parameter range discussed in Section 2 the basic sums ∑ n ± 1 ( e /epsilon1 n -1) -1 , appearing in (3.19), (3.21), see their values dominated by a large interval, ∆ n /greatermuch 1, around some n 0 /greatermuch 1, of values of n , so that one can, with a good approximation, replace the discrete sum over n by a formal continuous integral over a real parameter. In such a continuous approximation one can integrate by parts to show that any 'oscillatory' integral of the type ∫ dnf ( n ) cos nσ = [ n -1 f ( n ) sin nσ ] -∫ dnn -1 f ' ( n ) sin nσ is, because of the factors n -1 , numerically much smaller than the non-oscillatory one ∫ dnf ( n ). [Here σ denotes some combination of σ 1 and σ 2 , like 2 σ 1 , 2 σ 2 , σ 1 ± σ 2 .] Alternatively, we can say that, for generic values of σ 1 and σ 2 , one can treat in Eqs. (3.20) or (3.22) cos nσ 1 and cos nσ 2 as statistically independent random variables with zero average. Within such an approximation one can estimate (3.19) by replacing x n (1 , 2) by 1 (because cos 2 nσ 1 +cos 2 nσ 2 = 1+ 1 2 (cos 2 nσ 1 +cos 2 nσ 2 )). Similarly, one can estimate (3.21) by replacing p ± n → 1. The resulting estimates of (3.19) and (3.21) introduce exactly the grand canonical averages of the quantities : R : and N : \n1 2 〈 : ( k · ∆ ˜ X ) 2 : 〉 β,γ /similarequal k 2 〈 R 2 〉 β,γ , (3.23) \n〈 : (∆ P + ) 2 : 〉 β,γ /similarequal 〈 : (∆ P -) 2 : 〉 β,γ /similarequal 4 〈 N 〉 β,γ /similarequal 4 α ' M 2 . (3.24) \nFurthermore, one can check that the other contractions (like [∆ P + ∆ P -] or [∆ P + k · ∆ ˜ X ]) entering Eq. (3.17) are all of the 'oscillatory' type which is expected to give subleading contributions. \nInserting the results (3.23), (3.24) into Eqs. (3.17) and (3.11) leads to a trivial integral over τ 12 ( ∫ d τ 12 exp( -i /lscript 2 s Mωτ 12 ) = 2 π δ ( ω ) / ( /lscript 2 s M )) and, hence, to the following result for δ M = δ M 2 / (2 M ) \nδ M β,γ /similarequal -4 π G N M 2 ∫ d d k (2 π ) d e -k 2 R 2 k 2 -i ε . (3.25) \nThe imaginary part of δ M is easily seen to vanish in the present approximation. Finally, \n- \nδ M β,γ /similarequal -c d G N M 2 R d -2 , (3.26) \nwith the (positive 12 ) numerical constant \nequal to 1 / √ π in d = 3. \n| \n| \nc d = [ d -2 2 (4 π ) d -2 2 ] -1 , (3.27) \nThe result (3.26) was expected in order of magnitude, but we found useful to show how it approximately comes out of a detailed calculation of the mass shift which incorporates both relativistic and quantum effects. It shows clearly that perturbation theory breaks down, even at arbitrarily small coupling, for sufficiently heavy and compact strings. Let us also point out that one can give a simple statistical interpretation of the calculation (3.16) of the normal-ordered vertex operator W ( k, 1 , 2), with the basic contractions (3.18). The result of the calculation would have been the same if we had simply assumed that the oscillators a i n were classical, complex random variables with a Gaussian probability distribution ∝ exp [ -1 2 ( e /epsilon1 n -1) | a i n | 2 ] . This equivalence underlies the success of the classical random walk model of a generic excited string state. The fact that the random walk must be made of M/M s independent steps is linked to the fact that the Planckian distribution of mean occupation numbers, N n = (exp( β n + γ/n ) -1) -1 is sharply cut off when n > ∼ β -1 , i.e., from Eq. (2.28), when n > ∼ M/M s . More precisely, using the same 'statistical' approximation as above, one finds that the slope correlator 〈 : ∂ σ X i ( τ, σ 1 ) ∂ σ ˜ X j ( τ, σ 2 ) : 〉 β,γ decays quite fast when σ 2 -σ 1 > ∼ M s /M . \n˜ \n˜ -∼ Finally, let us mention that, by using the same tools as above, one can compute the imaginary part of the mass shift δ M = δ M real -i Γ / 2, i.e. the total decay rate Γ in massless quanta. The quantity Γ is, in fact, easier to define rigorously in string theory because, using ( k 2 -i ε ) -1 = PP ( k 2 ) -1 + iπ δ ( k 2 ) in (3.8), it is given by an integral where the massless quanta are all on-shell. When γ = 0 (a consistent approximation for a result dominated by n ∼ β -1 ) one can use the covariant formalism with D = d + 1 oscillators to find,after replacing a discrete sum over n by an integral over ω , \nΓ = c ' d G N M/lscript 2 s ∫ dω ω d -2 ( n e βn -1 ) 2 , (3.28) \nwhere c ' d is a numerical constant, and where n = M/lscript 2 s ω/ 2. The spectral decomposition of the total power radiated by the β -ensemble of strings is then simply deduced from (3.28) by adding a factor /planckover2pi1 ω in the integrand: \nP = c ' d G N M/lscript 2 s ∫ dω ω d -1 ( n e βn -1 ) 2 . (3.29) \nThe results (3.28), (3.29) agree with corresponding results (for closed strings) in the second reference [8] and in [22] (note, however, that the factor M 2 in the equation (3.2) of [22] should be M and that the constant contains G N and powers of /lscript s ). The integrals (3.28), (3.29) are dominated by n ∼ β -1 , i.e. ω ∼ M s . This gives for the integrated quantities: \nΓ ∼ g 2 M , P ∼ g 2 MM s . (3.30) \nThe second equation (3.30) means that the mass of a highly excited string decays exponentially, with half-evaporation time \nτ string evap ≡ M/P ∼ /lscript s g 2 . (3.31) \nLet us anticipate on the next section and note that, at the transition λ ≡ g 2 M/M s ∼ 1 between string states and black hole states, not only the mass and the entropy are (in order of magnitude at least) continuous, but also the various radiative quantities: total luminosity P , half-evaporation time τ evap , and peak of emission spectrum. Indeed, for a black hole decaying under Hawking radiation the temperature is T BH ∼ R -1 BH and \nP BH ∼ R -2 BH ∼ /lscript -2 s λ -2 / ( d -2) , τ BH evap ∼ R BH S BH ∼ /lscript s g -2 λ d/ ( d -2) . (3.32)", 'IV. ENTROPY OF SELF-GRAVITATING STRINGS': "In the present section we shall combine the main results of the previous sections, Eqs. (2.32) and (3.26), and heuristically extend them at the limit of their domain of validity. We consider a narrow band of string states that we follow when increasing adiabatically the string coupling g , starting from g = 0 13 . Let M 0 , R 0 denote the 'bare' values (i.e. for g → 0) of the mass and size of this band of states. Under the adiabatic variation of g , the mass and size, M , R , of this band of states will vary. However, the entropy S ( M,R ) remains constant under this adiabatic process: S ( M,R ) = S ( M 0 , R 0 ). We assume, as usual, that the variation of g is sufficiently slow to be reversible, but sufficiently fast to be able to neglect the decay of the states. We consider states with sizes /lscript s /lessmuch R 0 /lessmuch M 0 for which the correction factor, \nc ( R 0 ) /similarequal (1 -c 1 R -2 0 ) (1 -c 2 R 2 0 /M 2 0 ) , (4.1) \nin the entropy \nS ( M 0 , R 0 ) = c ( R 0 ) a 0 M 0 , (4.2) \nis near unity. [We use Eq. (2.32) in the limit g → 0, for which it was derived.] Because of this reduced sensitivity of c ( R 0 ) on a possible direct effect of g on R (i.e. R ( g ) = R 0 + δ g R ), the main effect of self-gravity on the entropy (considered as a function of the actual values M , R when g /negationslash = 0) will come from replacing M 0 as a function of M and R . The mass-shift result (3.26) gives δ M = M -M 0 to first order in g 2 . To the same accuracy 14 , (3.26) gives M 0 as a function of M and R : \nM 0 /similarequal M + c 3 g 2 M 2 R d -2 = M ( 1 + c 3 g 2 M R d -2 ) , (4.3) \nwhere c 3 is a positive numerical constant. \nFinally, combining Eqs. (4.1)-(4.3) (and neglecting, as just said, a small effect linked to δ g R /negationslash = 0) leads to the following relation between the entropy, the mass and the size (all considered for self-gravitating states, with g = 0) \nS ( M,R ) /similarequal a 0 M ( 1 -1 R 2 )( 1 -R 2 M 2 )( 1 + g 2 M R d -2 ) . (4.4) \n/negationslash \nFor notational simplicity, we henceforth set to unity the coefficients c 1 , c 2 and c 3 . There is no loss of generality in doing so because we can redefine /lscript s , R and g to that effect, and use the corresponding (new) string units. The main point of the present paper is to emphasize that, for a given value of the total energy M (and for some fixed value of g ), the entropy S ( M,R ) has a non trivial dependence on the radius R of the considered string state. Eq. (4.3) exhibits two effects varying in opposite directions: (i) self-gravity favors small values of R (because they correspond to larger values of M 0 , i.e. of the 'bare' entropy), and (ii) the constraint of being of some fixed size R disfavors both small ( R /lessmuch √ M ) and large ( R /greatermuch √ M ) values of R . For given values of M and g , the most numerous (and therefore most probable) string states will have a size R ∗ ( M ; g ) which maximizes the entropy S ( M,R ). Said differently, the total degeneracy of the complete ensemble of self-gravitating string states with total energy M (and no a priori size restriction) will be given by an integral (where ∆ R is the rms fluctuation of R given by Eq. (2.17)) \nD ( M ) ∼ ∫ dR ∆ R e S ( M,R ) ∼ e S ( M,R ∗ ) (4.5) \nwhich will be dominated by the saddle point R ∗ which maximizes the exponent. \nThe value of the most probable size R ∗ is a function of M , g and the space dimension d . To better see the dependence on d , let us first consider the case (which we generically assume) where the correction factors in Eq. (4.4) (parentheses on the right-hand-side) are very close to unity so that \nS ( M,R ) /similarequal a 0 M (1 -V ( R )) , (4.6) \nwhere \nV ( R ) = 1 R 2 + R 2 M 2 -g 2 M R d -2 . (4.7) \nOne can think of V ( R ) as an effective potential for R . The most probable size R ∗ must minimize V ( R ). This effective potential can be thought of as the superposition of: (i) a centrifugal barrier near R = 0 (coming from the result (2.30)), (ii) an harmonic potential (forbidding the large values of R ), and (iii) an attractive (gravitational) potential. When g 2 is small the minimum of V ( R ) will come from the competition between the centrifugal barrier and the harmonic potential and will be located around the value R -2 ∗ /similarequal R 2 ∗ /M 2 , i.e. R ∗ /similarequal √ M = R rw . This random walk value will remain (modulo small corrections) a local minimum of V ( R ) (i.e. a local maximum of S ( M,R )) as long as g 2 M/R d -2 ∗ /lessmuch R -2 ∗ , i.e. for g 2 /lessmuch g 2 0 with \ng 2 0 ≡ M d -6 2 . (4.8) \nMore precisely, working perturbatively in g 2 , the minimization of V ( R ) yields \nR ∗ /similarequal √ M ( 1 -d -2 8 g 2 g 2 0 ) . (4.9) \nNote that, when g 2 /lessmuch g 2 0 , the value of V ( R ) at this local minimum is of order V min /similarequal +2 R -2 ∗ /similarequal +2 M -1 , i.e. that it corresponds to a saddle-point entropy S ( M,R ∗ ) /similarequal a 0 M (1 -V min ) /similarequal a 0 M - O (1) which differs essentially negligibly from the 'bare' entropy a 0 M ( /greatermuch 1). To study what happens when g 2 further increases let us consider separately the various dimensions d ≥ 3. We shall see that the special value g 2 0 , Eq. (4.8), is significant (as marking a pre-transition, before the transition to the black hole state) only for d = 3. For d ≥ 4, the only special value of g 2 is the critical value \ng 2 c ∼ M -1 , (4.10) \naround which takes place a transition toward a state more compact than the usual random walk one. \nA. \nd = 3 \nLet us first consider the (physical) case d = 3, for which g 2 0 ∼ M -3 / 2 /lessmuch g 2 c ∼ M -1 . In that case, when g 2 becomes larger than g 2 0 , the (unique) local minimum of V ( R ) slowly shifts towards values of R lower than R rw and determined by the competition between the centrifugal barrier 1 /R 2 and the gravitational potential g 2 M/R . \nIn the approximation where we use the linearized form (4.6), (4.7), and where (for g 2 /greatermuch g 2 0 ) we neglect the term R 2 /M 2 , the most probable size R ∗ is \n- \nR ( d =3) ∗ /similarequal 2 g 2 M , when M -3 / 2 /lessmuch g 2 /lessmuch M -1 . (4.11) \nNote that as g 2 increases between M -3 / 2 and M -1 , the most probable size R ( d =3) ∗ smoothly interpolates between R rw and a value of order unity, i.e. of order the string length. Note also that V min /similarequal -g 2 M/ (2 R ∗ ) /similarequal -g 4 M 2 / 4 remains smaller than one when g 2 < ∼ M -1 so that the saddle-point entropy S ( M,R ∗ ) /similarequal a 0 M (1 -V min ) never differs much from the 'bare' value a 0 M . \nWhen g 2 , in its increase, becomes comparable to M -1 , the radius becomes of order one and it is important to take into account the (supposedly) more exact expression (4.4) (in which the factor (1 -R -2 ) plays the crucial role of cutting off any size R ≤ 1). If we neglect, as above, the term R 2 /M 2 (which is indeed even more negligible in the region R ∼ 1) but maximize the factored expression (4.4), we find that the most probable size R ∗ reads \nR ( d =3) ∗ /similarequal 1 + √ 1 + 3 λ 2 λ , when g 2 /greatermuch M -3 / 2 , (4.12) \nwhere we recall the definition \nλ ≡ g 2 M. (4.13) \nWhen λ /lessmuch 1, the result (4.12) reproduces the simple linearized estimate (4.11). When λ > ∼ 1, Eq. (4.12) says that the most probable size, when g 2 increases, tends to a limiting size ( R ∞ = √ 3) slightly larger than the minimal one ( R min = 1) corresponding to zero entropy. [Note that even for the formal asymptotic value R ∞ = √ 3, the reduction in entropy due to the factor 1 -R -2 is only 2 / 3.] On the other hand, the fractional self-gravity G N M/R ∗ (which measures the gravitational deformation away from flat space), or the corresponding term in Eq. (4.4), continues to increase with g 2 as \nλ R ∗ = λ 2 1 + √ 1 + 3 λ 2 . (4.14) \nThe right-hand side of Eq. (4.14) becomes unity for λ = √ 5 = 2 . 236. The picture suggested by these results is that the string smoothly contracts, as g increases, from its initial random walk size down to a limiting compact state of size slightly larger than /lscript s . For some value of λ of order unity (may be between 1 and 2; indeed, even for λ = 1 the size R ∗ = 2 and the self-gravity λ/R ∗ = 0 . 5 suggest one may still trust a compact string description) the self-gravity of this compact string state will become so strong that one expects it to collapse to a black hole state. We recall that, as emphasized in Refs. [4], [6], [7], [8], [9], [13], when λ ∼ 1, the mass of the string state matches (in order of magnitude) that of a (Schwarzschild) black hole with Bekenstein-Hawking entropy equal to the string entropy S . \nB. d = 4 \nWhen d = 4, the argument above Eq. (4.9) suggests that the random-walk size remains the most probable size up to g 2 < ∼ g 2 0 ∼ M -1 , i.e. up to λ < ∼ 1. A more accurate approximation to the most probable size R ∗ , when λ < 1, is obtained by minimizing exactly V ( R ), Eq. (4.7). This yields \nR ( d =4) ∗ /similarequal M 1 / 2 (1 -λ ) 1 / 4 , when λ < 1 . (4.15) \nThis shows that the size will decrease, but one cannot trust this estimate when λ → 1 -. To study more precisely what happens when λ ∼ 1 we must take into account the more exact factorized form (4.4). Let us now neglect the R 2 /M 2 term and consider the approximation \nS ( d =4) ( M,R ) /similarequal a 0 M ( 1 -1 R 2 )( 1 + λ R 2 ) . (4.16) \nThe right-hand side of Eq. (4.15) has a maximum only for λ > 1, in which case \nR ( d =4) ∗ /similarequal ( 2 λ λ -1 ) 1 / 2 when λ > 1 . (4.17) \nIf we had taken into account the full expression (4.4) the two results (4.15), (4.17), valid on each side of λ = 1, would have blended in a result showing that around 15 λ = 1 the most probable size continuously interpolates between R rw and a size of order /lscript s . Note that, according to Eq. (4.17), as λ becomes /greatermuch 1, R ( d =4) ∗ tends to a limiting size ( R ∞ = √ 2) slightly larger than R min = 1 (corresponding to zero entropy). When λ > 1 the fractional self-gravity of the compact string states reads \nλ R 2 ∗ = λ -1 2 . (4.18) \nAs in the case d = 3, one expects that for some value of λ strictly larger than 1, the selfgravity of the compact string state will become so strong that it will collapse to a black hole state. Again the mass, size and entropy match (in order of magnitude) those of a black hole when λ ∼ 1. The only difference between d = 4 and d = 3 is that the transition to the compact state, though still continuous, is sharply concentrated around λ = 1 instead of taking place over the extended range M -1 / 2 < ∼ λ < ∼ 1. \nC. d ≥ 5 \nWhen d ≥ 5, the argument around Eq. (4.8) shows that the random walk size R rw /similarequal √ M is a consistent local maximum of the entropy in the whole domain g 2 /lessmuch g 2 0 , i.e. for λ ≡ g 2 M /lessmuch M d -4 2 , which allows values λ /greatermuch 1. However, a second, disconnected maximum of the entropy, as function of R , could exist. To investigate this we consider again (4.4), when neglecting the R 2 /M 2 term (because we are interested in other possible solutions with small sizes): \nS ( M,R ) a 0 M /similarequal (1 -x ) (1 + λx ν ) ≡ s ( x ) , (4.19) \nwhere we have defined x ≡ R -2 and ν ≡ ( d -2) / 2. By studying analytically the maxima and inflection points of s ( x ), one finds that, in the present case where ν = ( d -2) / 2 > 1, there are two critical values λ 1 < λ 2 of the parameter λ ≡ g 2 M . The first one, \nλ 1 = ( ν +1 ν -1 ) ν -1 , R 1 = x -1 / 2 1 = ( ν +1 ν -1 ) 1 / 2 > 1 , s 1 = 1 -( ν -1 ν +1 ) 2 , (4.20) \ncorresponds to the birth (through an inflection point) of a maximum and a minimum of the function s ( R ). Because s 1 < 1 is strictly lower than the usual random walk maximum with s ( R rw ) /similarequal 1 -2 /M /similarequal 1, the local maximum near R ∼ 1 of the entropy, which starts to exist when λ > λ 1 , is, at first, only metastable with respect to R rw . However, there is a second critical value of λ , λ 2 > λ 1 , defined by \nλ 2 = ν ( ν ν -1 ) ν -1 , R 2 = x -1 / 2 2 = ( ν ν -1 ) 1 2 > 1 , s 2 = 1 . (4.21) \nWhen λ > λ 2 the local maximum near R ∼ 1 of the entropy has s ( R ) > 1, i.e. it has become the global maximum of the entropy, making the usual random walk local maximum only metastable. Therefore,when λ > λ 2 the most probable string state is a very compact state of size comparable to /lscript s . Formally, this new global maximum exists for any λ > ∼ 1 and tends, when λ →∞ , toward the limiting location R ∞ = (( ν +1) /ν ) 1 / 2 > 1, i.e. slightly (but finitely) above the minimum size R = 1. However, as in the cases d ≤ 4, the self-gravity of the stable compact string state will become strong when λ > ∼ 1, so that it is expected to collapse (for some λ c > λ 2 ) to a black hole state. As in the cases d ≤ 4, the mass, size and entropy of this compact string state match those of a black hole. The big difference with the cases d ≤ 4 is that the transition between the (stable) random walk typical configuration and the (stable) compact one is discontinuous. Our present model suggests that (when ν ≡ ( d -2) / 2 > 1) a highly excited single string system can exist, when λ > λ 2 , in two different stable typical states: (i) a dilute state of typical size R rw /similarequal √ M and typical mean density ρ ∼ M/R d rw ∼ M -ν /lessmuch 1, and (ii) a condensed state of typical size R ∼ 1 and typical mean density (using λ ∼ 1): ρ ∼ M ∼ g -2 /greatermuch 1. We shall comment further below on the value ρ ∼ g -2 of the dense state of string matter.", 'V. DISCUSSION': "Technically, the main new result of the present work is the (dimension independent) estimate 16 c ( R ) = 1 -c 1 /R 2 , with c 1 /similarequal (3 / 4) /lscript 2 s = 3 α ' / 2, of the factor giving the decrease in the entropy (2 π (( d -1) α ' / 6) 1 / 2 M ) of a narrow band of very massive (open 17 ) string states \nM /greatermuch ( α ' ) -1 / 2 , when considering only string states of size R (modulo some fractionally small grand-canonical-type fluctuations). We have also justified (by dealing explicitly with relativistic and quantum effects in a semi-classical approximation) and refined (by computing the numerical coefficient, Eq. (3.27)) the naive estimate, δM = -c d G N M 2 /R d -2 , of the mass shift of a massive string state due to the exchange of long range fields (graviton, dilaton and axion). [The exchange of these fields is expected to be the most important one both because very excited string states tend to be large, and because the corresponding interactions are attractive and cumulative with the mass.] \nConceptually, the main new result of this paper concerns the most probable state of a very massive single 18 self-gravitating string. By combining our estimates of the entropy reduction due to the size constraint, and of the mass shift we come up with the expression (4.4) for the logarithm of the number of self-gravitating string states of size R . Our analysis of the function S ( M,R ) clarifies the correspondence [6], [8], [9], [13] between string states and black holes. In particular, our results confirm many of the results of [13], but make them (in our opinion) physically clearer by dealing directly with the size distribution, in real space, of an ensemble of string states. When our results differ from those of [13], they do so in a way which simplifies the physical picture and make even more compelling the existence of a correspondence between strings and black holes. For instance, [13] suggested that in d = 5 there was a phenomenon of hysteresis, with a critical value g 2 0 ∼ M -1 / 2 for the string → black hole transition, and a different critical value g 2 c ∼ M -1 /lessmuch g 2 0 for the inverse transition: black hole → string. Also, [13] suggested that in d > 6, most excited string states would never form black holes. The simple physical picture suggested 19 by our results is the following: In any dimension, if we start with a massive string state and increase the string coupling g , a typical string state will, eventually, become more compact and will end up, when λ c = g 2 c M ∼ 1, in a 'condensed state' of size R ∼ 1, and mass density ρ ∼ g -2 c . Note that the basic reason why small strings, R ∼ 1, dominate in any dimension the entropy when λ ∼ 1 is that they descend from string states with bare mass M 0 /similarequal M (1 + λ/R d -2 ) ∼ 2 M which are exponentially more numerous than less condensed string states corresponding to smaller bare masses. \nThe nature of the transition between the initial 'dilute' state and the final 'condensed' \nN L = N R , would have complicated the definition of the grand canonical ensemble we used. We expect that our results (whichare semi-classical) apply (with some numerical changes) to open or closed superstrings. \n18 We consider states of a single string because, for large values of the mass, the single-string entropy approximates the total entropy up to subleading terms. \n19 Our conclusions are not rigourously established because they rely on assuming the validity of the result (4.4) beyond the domain ( R -2 /lessmuch 1, g 2 M/R d -2 /lessmuch 1) where it was derived. However, we find heuristically convincing to believe in the presence of a reduction factor of the type 1 -R -2 down to sizes very near the string scale. Our heuristic dealing with self-gravity is less compelling because we do not have a clear signal of when strong gravitational field effects become essential. \none depends on the value of the space dimension d . [As explained below Eq. (4.4), we henceforth set to unity, by suitable redefinitions of /lscript s , R and g , the coefficients c 1 , c 2 and c 3 .] In d = 3, the transition is gradual: when λ < M -1 / 2 the size of a typical state is R ( d =3) ∗ /similarequal M 1 / 2 (1 -M 1 / 2 λ/ 8), when λ > M 1 / 2 the typical size is R ( d =3) ∗ /similarequal (1+(1+3 λ 2 ) 1 / 2 ) /λ . In d = 4, the transition toward a condensed state is still continuous, but most of the size evolution takes place very near λ = 1: when λ < 1, R ( d =4) ∗ /similarequal M 1 / 2 (1 -λ ) 1 / 4 , and when λ > 1, R ( d =4) ∗ /similarequal (2 λ/ ( λ -1)) 1 / 2 , with some smooth blending between the two evolutions around | λ -1 | ∼ M -2 / 3 . In d ≥ 5, the transition is discontinuous (like a first order phase transition between, say, gas and liquid states). Barring the consideration of metastable (supercooled) states, on expects that when λ = λ 2 /similarequal ν ν / ( ν -1) ν -1 (with ν = ( d -2) / 2), the most probable size of a string state will jump from R rw (when λ < λ 2 ) to a size of order unity (when λ > λ 2 ). \nLet us, for definiteness, write down in more detail what happens in d = 3. After maximization over R , the entropy of a self-gravitating string is given, when M -3 / 2 /lessmuch g 2 /lessmuch M -1 , by \nBy differentiating S with respect to M , one finds the temperature of the ensemble of highly excited single string states of mass M : \nS ( M ) = S ( M,R ∗ ( M )) /similarequal a 0 M [ 1 + 1 4 ( g 2 M ) 2 ] . (5.1) \nT /similarequal T Hag ( 1 -3 4 ( g 2 M ) 2 ) , (5.2) \n∼ \nwith T Hag ≡ a -1 0 . Eq. (5.2) explicitly exhibits the modification of the Hagedorn temperature due to self-gravity (in agreement with results of [13] obtained by a completely different approach). Note that, both in Eqs. (5.1) and (5.2), the self-gravity modifications are fractionally of order unity at the transition g 2 M 1. \nOne can think of the 'condensed' state of (single) string matter, reached (in any d ) when λ ∼ 1, as an analog of a neutron star with respect to an ordinary star (or a white dwarf). It is very compact (because of self gravity) but it is stable (in some range for g ) under gravitational collapse. However, if one further increases g or M (in fact, λ = g 2 M ), the condensed string state is expected (when λ reaches some λ 3 > λ 2 , λ 3 = O (1)) to collapse down to a black hole state (analogously to a neutron star collapsing to a black hole when its mass exceeds the Landau-Oppenheimer-Volkoff critical mass). Still in analogy with neutron stars, one notes that general relativistic strong gravitational field effects are crucial for determining the onset of gravitational collapse; indeed, under the 'Newtonian' approximation (4.4), the condensed string state could continue to exist for arbitrary large values of λ . \nIt is interesting to note that the value of the mass density at the formation of the condensed string state is ρ ∼ g -2 . This is reminiscent of the prediction by Atick and Witten [23] of a first-order phase transition of a self-gravitating thermal gas of strings, near the Hagedorn temperature 20 , towards a dense state with energy density ρ ∼ g -2 (typical of a \ngenus-zero contribution to the free energy). Ref. [23] suggested that this transition is firstorder because of the coupling to the dilaton. This suggestion agrees with our finding of a discontinuous transition to the single string condensed state in dimensions ≥ 5 (Ref. [23] work in higher dimensions, d = 25 for the bosonic case). It would be interesting to deepen these links between self-gravitating single string states and multi-string states. \nAssuming the existence (confirmed by the present work) of a dense state of selfgravitating string matter with energy density ρ ∼ g -2 , it would be fascinating to be able to explore in detail (with appropriate, strong gravity tools) its gravitational dynamics, both in the present context of a single, isolated object ('collapse problem'), and in the cosmological context (problem of the origin of the expansion of the universe). \nLet us come back to the consequences of the picture brought by the present work for the problem of the end point of the evaporation of a Schwarzschild black hole and the interpretation of black hole entropy. In that case one fixes the value of g (assumed to be /lessmuch 1) and considers a black hole which slowly looses its mass via Hawking radiation. When the mass gets as low as a value 21 M ∼ g -2 , for which the radius of the black hole is of order one (in string units), one expects the black hole to transform (in all dimensions) into a typical string 22 state corresponding to λ = g 2 M ∼ 1, which is a dense state (still of radius R ∼ 1). This string state will further decay and loose mass, predominantly via the emission of massless quanta, with a quasi thermal spectrum with temperature T ∼ T Hagedorn = a -1 0 (see Eq. (3.29) and Refs. [8], [22]) which smoothly matches the previous black hole Hawking temperature. This mass loss will further decrease the product λ = g 2 M , and this decrease will, either gradually or suddenly, cause the initially compact string state to inflate to much larger sizes. For instance, if d ≥ 4, the string state will quickly inflate to a size R ∼ √ M . Later, with continued mass loss, the string size will slowly shrink again toward R ∼ 1 until a remaining string of mass M ∼ 1 finally decays into stable massless quanta. In this picture, the black hole entropy acquires a somewhat clear statistical significance (as the degeneracy of a corresponding typical string state) only when M and g are related by g 2 M ∼ 1. If we allow ourselves to vary (in a Gedanken experiment) the value of g this gives a potential statistical significance to any black hole entropy value S BH (by choosing g 2 ∼ S -1 BH ). We do not claim, however, to have a clear idea of the direct statistical meaning of S BH when g 2 S BH /greatermuch 1. Neither do we clearly understand the fate of the very large space (which could be excited in many ways) which resides inside very large classical black holes of radius R BH ∼ ( g 2 S BH ) 1 / ( d -1) /greatermuch 1. The fact that the interior of a black hole of given mass could be \n22 A check on the single-string dominance of the transition black hole → string is to note that the single string entropy ∼ M/M s is much larger than the entropy of a ball of radiation S rad ∼ ( RM ) d/ ( d +1) with size R ∼ R BH ∼ /lscript s at the transition. \narbitrarily large 23 , and therefore arbitrarily complex, suggests that black hole physics is not exhausted by the idea (confirmed in the present paper) of a reversible transition between string-length-size black holes and string states. \nOn the string side, we also do not clearly understand how one could follow in detail (in the present non BPS framework) the 'transformation' of a strongly self-gravitating string state into a black hole state. \nFinally, let us note that we expect that self-gravity will lift nearly completely the degeneracy of string states. [The degeneracy linked to the rotational symmetry, i.e. 2 J + 1 in d = 3, is probably the only one to remain, and it is negligible compared to the string entropy.] Therefore we expect that the separation δ E between subsequent (string and black hole) energy levels will be exponentially small: δ E ∼ ∆ M exp( -S ( M )), where ∆ M is the canonical-ensemble fluctuation in M . Such a δ E is negligibly small compared to the radiative width Γ ∼ g 2 M of the levels. This seems to mean that the discreteness of the quantum levels of strongly self-gravitating strings and black holes is very much blurred, and difficult to see observationally.", 'ACKNOWLEDGEMENTS': 'This work has been clarified by useful suggestions from M. Douglas, K. Gawedzki, M. Green, I. Kogan, G. Parisi, A. Polyakov, and (last but not least) M. Vergassola. We wish also to thank A. Buonanno for collaboration at an early stage and D. Gross, J. Polchinski, A. Schwarz, L. Susskind and A. Vilenkin for discussions. T.D. thanks the Theory Division of CERN, Gravity Probe B (Stanford University), and the Institute for Theoretical Physics (Santa Barbara) for hospitality. Partial support from NASA grant NAS8-39225 is acknowledged. G.V. thanks the IHES for hospitality during the early, crucial stages of this work.', 'REFERENCES': '- [1] S. Fubini and G. Veneziano, Nuovo Cim. 64 A 811 (1969); K.Huang and S. Weinberg, Phys. Rev. Lett 25 895 (1970).\n- [2] J. D. Bekenstein, Phys. Rev. D 7 2333 (1973).\n- [3] S. W. Hawking, Comm. Math. Phys. 43 199 (1975).\n- [4] M. Bowick, L. Smolin and L.C.R. Wijewardhana, Gen. Rel. Grav. 19 113 (1987).\n- [5] G. Veneziano, Europhys. Lett. 2 199 (1986).\n- [6] L. Susskind, hep-th/9309145 (unpublished).\n- [7] G. Veneziano, in Hot Hadronic Matter: Theory and Experiments , Divonne, June 1994, eds. J. Letessier, H. Gutbrod and J. Rafelsky, NATO-ASI Series B: Physics, 346 (Plenum Press, New York 1995), p. 63.\n- [8] E. Halyo, A. Rajaraman and L. Susskind, Phys. Lett. B 392 , 319 (1997); E.Halyo, B. Kol, A. Rajaraman and L. Susskind, Phys. Lett. B 401 ,15 (1997).\n- [9] G.T. Horowitz and J. Polchinski, Phys. Rev. D 55 , 6189 (1997).\n- [10] A. Strominger and C. Vafa, Phys. Lett. B 379 99 (1996).\n- [11] E.g. A. Sen, Mod. Phys. Lett. A 10 2081 (1995); C.G. Callan and J.M. Maldacena, Nucl. Phys. B 472 , 591 (1996); J.C. Breckenridge et al. Phys. Lett. B 381 423 (1996).\n- [12] P. Salomonson and B.S. Skagerstam, Nucl. Phys. B 268 ,349 (1986); Physica A 158 , 499 (1989); D. Mitchell and N. Turok, Phys. Rev. Lett. 58 , 1577 (1987); Nucl.Phys. B 294 , 1138 (1987).\n- [13] G.T. Horowitz and J. Polchinski, Phys. Rev. D 57 ,2557 (1998).\n- [14] B. Sathiapalan, Phys. Rev. D 35 , 3277 (1987); I.A. Kogan, JETP Lett. 45 , 709 (1987); J.J. Atick and E. Witten, Nucl. Phys. B 310 , 291 (1988).\n- [15] J. Scherk, Rev. Mod. Phys. 47 , 123 (1975).\n- [16] M. Karliner, I. Klebanov and L. Susskind, Int. J. Mod.Phys. A3 , 1981 (1988).\n- [17] L.D. Landau and E.M. Lifshitz, Statistical Physics , part 1, third edition (Pergamon Press, Oxford, 1980).\n- [18] H. Yamamoto, Prog. Theor. Phys. 79 , 189 (1988); K. Amano and A. Tsuchiya, Phys. Rev. D 39 , 565 (1989); B. Sundborg, Nucl. Phys. B 319 , 415 (1989); and 338 , 101 (1990).\n- [19] A. Buonanno and T. Damour, Phys. Lett. B 432 , 51(1998).\n- [20] V. Alessandrini, D. Amati, M. Le Bellac and D. Olive, Phys. Rep. C1 , 269 (1971).\n- [21] M.B. Green, J.H. Schwarz and E. Witten, Superstring Theory , Volume 1, (Cambridge University Press, Cambridge, 1987).\n- [22] D. Amati and J.G. Russo, Phys. Lett. B 454 , 207 (1999); hep-th/9901092.\n- [23] J.J. Atick and E. Witten, Nucl. Phys. B 310 , 291(1988).'}

2008PhRvD..78d1502C

Black hole thermodynamics from simulations of lattice Yang-Mills theory

2008-01-01

['-', '-', '-', '-', '-', '-', '-', '-', '-', '-', '-']

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We report on lattice simulations of 16 supercharge SU(N) Yang-Mills quantum mechanics in the ’t Hooft limit. Maldacena duality conjectures that in this limit the theory is dual to IIA string theory, and, in particular, that the behavior of the thermal theory at low temperature is equivalent to that of certain black holes in IIA supergravity. Our simulations probe the low temperature regime for N≤5 and the intermediate and high temperature regimes for N≤12. We observe ’t Hooft scaling, and at low temperatures our results are consistent with the dual black hole prediction. The intermediate temperature range is dual to the Horowitz-Polchinski correspondence region, and our results are consistent with continuous behavior there. We include the Pfaffian phase arising from the fermions in our calculations where appropriate.

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https://arxiv.org/pdf/0803.4273.pdf

{'Black hole thermodynamics from simulations of lattice Yang-Mills theory': "Simon Catterall 1 and Toby Wiseman 2 \n1 Department of Physics, Syracuse University, Syracuse, NY13244, USA and 2 Theoretical Physics, Blackett Laboratory, Imperial College London, London, SW7 2AZ, UK (Dated: March 2008) \nWe report on lattice simulations of 16 supercharge SU ( N ) Yang-Mills quantum mechanics in the 't Hooft limit. Maldacena duality conjectures that in this limit the theory is dual to IIA string theory, and in particular that the behavior of the thermal theory at low temperature is equivalent to that of certain black holes in IIA supergravity. Our simulations probe the low temperature regime for N ≤ 5 and the intermediate and high temperature regimes for N ≤ 12. We observe 't Hooft scaling and at low temperatures our results are consistent with the dual black hole prediction. The intermediate temperature range is dual to the Horowitz-Polchinski correspondence region, and our results are consistent with smooth behavior there. We include the Pfaffian phase arising from the fermions in our calculations where appropriate. \nPACS numbers: 04.60.Cf, 04.70.Dy, 11.15.Ha,11.25.Tq", 'I. INTRODUCTION': "String theory has provided remarkable insight into the quantum physics underlying black holes. Much recent progress stems from conjectured dualities, which, in an appropriate limit, relate the finite temperature low energy supergravity limit of the string theory to strongly coupled thermal field theory. The entropy of the black holes that arise in these supergravity theories can then be computed in principle by counting microstates in their dual field theories. The pioneering calculations of black hole entropy in [1, 2] are examples where the dual field theory is a 2-d conformal field theory which allows computation of the entropy despite the strong coupling. \nFor a large number N of coincident D p -branes in the 'decoupling' limit [3, 4] the dual field theory is (1 + p )-dimensional strongly coupled maximally supersymmetric SU ( N ) Yang-Mills theory, taken in the 't Hooft limit. The case of D3-branes yields the AdS-CFT correspondence. Analytic calculation of the corresponding black hole entropy of these theories has proven elusive despite interesting attempts [5]. \nHere we use lattice methods to study the thermal gauge theory and hence test these conjectured dualities. The simplest case for lattice work corresponds to D0-branes [6], where the dual is thermal 16 supercharge Yang-Mills quantum mechanics (the 'BFSS model' [7]). This theory has recently been numerically studied using a non-lattice formulation [8, 9]. Earlier analytic approaches used a variational method [10, 11]. Related zero temperature numerical works are [12, 13, 14]. \nIn this letter we simulate the super quantum mechanics in the t'Hooft limit over a range of temperature and present preliminary results. We obtain intermediate temperature results for N ≤ 12 and low temperature results for N ≤ 5. We pay particular attention to the continuum limit and the behavior of the important Pfaffian phase arising from the fermions. More details of the method and results will be given in [15].", 'II. DUALITY AND BLACK HOLES': "The type IIA string theory reduces to a supergravity theory for low energies compared to the string scale ( α ' ) -1 / 2 . In this limit the thermal theory contains black holes with N units of D0-charge. Their energy, E , is a function of their Hawking temperature, T . Defining λ = Ng s α '-3 / 2 where g s is the string coupling, we may write a dimensionless energy and temperature /epsilon1 = Eλ -1 / 3 and t = Tλ -1 / 3 . One finds provided we take N large and t /lessmuch 1 the black hole is weakly curved on string scales and the quantum string corrections are suppressed. The energy of this black hole can be precisely computed by standard methods [4] giving, \n/epsilon1 = c N 2 t 14 / 5 c = ( 2 21 3 12 5 2 7 19 π 14 ) 1 / 5 /similarequal 7 . 41 . (1) \nDuality posits that the thermodynamics of this black hole should be reproduced by the dual Yang-Mills quantum mechanics at the same temperature with g s α '-3 / 2 = g 2 Y M so λ is identified with the 't Hooft coupling. \nIn the large N limit, at high temperatures t > 1, the bound state of D0-branes is of order the size of the string scale, and hence all α ' corrections are important. One should best think of the configuration dominating the partition function as a hot gas of D0-branes bound by strings. Horowitz and Polchinski have argued that the low temperature black hole and high temperature gas are the asymptotic descriptions and intermediate temperatures smoothly interpolate between these [16].", 'III. LATTICE IMPLEMENTATION': 'The 16 supercharge SU ( N ) Yang-Mills quantum mechanics arises from dimensional reduction of N = 1 super Yang-Mills in 10-d. The 10-d gauge field reduces to the 1-d gauge field A and 9 scalars, X i , i = 1 , . . . , 9 and the 10-d Majorana-Weyl fermion to 16 single component fermions, Ψ α , α = 1 , . . . , 16. All fields transform in the adjoint of the gauge group. In order to simulate the theory we must integrate out the fermions giving rise \nto a Pfaffian. The continuum Euclidean path integral, Z = ∫ dAdX Pf ( O ) e -S bos , is then given by, \nS bos = N λ Tr ∮ R dτ { 1 2 ( D τ X i ) 2 -1 4 [ X i , X j ] 2 } O = γ τ D τ -γ i [ X i , · ] (2) \nThe γ τ , γ i are the Euclidean Majorana-Weyl gamma matrices, and we choose a representation where, γ τ = ( 0 Id 8 Id 8 0 ) . We take Euclidean time to have period R . \nWe have a choice of fermion boundary conditions. Thermal boundary conditions correspond to taking the fermions antiperiodic on the Euclidean time circle and correspond to a temperature t = λ -1 / 3 /R . We will also employ periodic fermions and the continuum partition function is then an index, with t the inverse volume. The Pfaffian is in general complex [17]. It is important in principle to include the phase of the Pfaffian in the Monte-Carlo simulation, and we discuss this later. \nWe discretize this continuum model as, \nS bos = NL 3 λR 3 L -1 ∑ a =0 Tr [ 1 2 ( D + X i ) 2 a -1 4 [ X i,a , X j,a ] 2 ] O ab = ( 0 ( D + ) ab ( D -) ab 0 ) -γ i [ X i,a , · ] δ ab (3) \nwhere we have rescaled the fields X i,a and Ψ i,α by powers of the lattice spacing a = R/L where L is the number of lattice points to render them dimensionless. We have introduced a Wilson gauge link field U a , and taken covariant difference operators ( D -W ) a = W a -U † a W a -1 U a , ( D + W ) a = U a W a +1 U † a -W a . Notice that the fermionic operator is free of doublers and is manifestly antisymmetric. This lattice action is finite in lattice perturbation theory and hence will flow without fine tuning to the correct supersymmetric continuum theory as the lattice spacing is reduced [6, 15]. \nWe use the RHMC algorithm [18] to sample configurations using the absolute value of the Pfaffian. The phase may be re-incorporated in the expectation value of an observable A by reweighting as < A > = ∑ m ( A e iφ ) ∑ m ( e iφ ) . Here e iφ ( O ) is the phase of the Pfaffian and the sum runs over all members of our phase quenched ensemble. \nWe find in practice that the RHMC simulation of the thermal theory at low temperature, t /lessorsimilar 1, exhibits an instability corresponding to the scalar fields moving out along the flat directions of the classical potential. Hence the algorithm never thermalizes and cannot be used to approximate the path integral. This has been observed before [9]. We believe this divergence is a lattice artifact that is related to the discretization of the fermion operator. In previous work [6] we have simulated the 4 supercharge quantum mechanics over a range of t , using a Weyl representation for the fermions where one obtains a real positive determinant. However, we have also tried using a Majorana representation where one obtains \nFIG. 1: Top: Plot showing /epsilon1/t verses dimensionless temperature t for the periodic SU (2) theory for various numbers of lattice points. Bottom: Plot of the Polyakov loop against temperature for the same theory. \n<!-- image --> \na Pfaffian which we have discretized in analogy with the 16 supercharge case discussed here. While no divergence of the scalars was observed in the Weyl simulations over a large range of t [6], the Majorana implementation has the same instability we observe in the 16 supercharge case for t /lessorsimilar 1. Since both representations are equivalent in the continuum limit this implies that the instability is not a property of the continuum theory as is claimed in [9] but merely an artefact of finite lattice spacing. More details will be given in [15]. \nWe find no such problem simulating the periodic theory at small t . At low temperature we expect the thermal and periodic theory to be similar, and the configurations that dominate the path integral will be similar. Hence in order to simulate the thermal theory at low temperature, t /lessorsimilar 1, we have employed a reweighting of the periodic theory. We can expect to get good results for the thermal theory at low temperature by computing expectation values using the periodic theory, and reweighting as, \n< A > T = ∑ ( P ) m ( A Pf( O T ) / | Pf( O P ) | ) ( P ) m (Pf( O T ) / | Pf( O P ) | ) (4) \nwhere ∑ ( P ) m is a sum over the phase quenched ensemble generated for the periodic theory, O P and O T are the periodic and thermal fermion operators respectively, and < . . . > T is the expectation value for the thermal theory. \n∑', 'IV. RESULTS': "We have simulated the thermal and periodic theories concentrating on the range 0 . 3 < t < 5. We have focused \nFIG. 2: Top: Plot of dimensionless energy /epsilon1/t verses dimensionless temperature t for the thermal SU (5) theory with 5 and 10 lattice points. Bottom: Plot of the cosine of the Pfaffian argument for the thermal SU (5) theory with 5 points. \n<!-- image --> \non two observables - the mean energy, /epsilon1 , and absolute value of the trace of the Polyakov loop, P . In the YangMills theory these are given by [6, 15] \n< /epsilon1/t > = 3 N 2 ( 9 2 L ( N 2 -1) -< S bos > ) P = 1 N < | Tr L -1 ∏ a =0 U a | > . (5) \nThe inclusion of 1 /N 2 , 1 /N in these definitions is to ensure these quantities are finite in the t'Hooft limit for a deconfined phase. In the periodic case, since Z is an index, it should not depend continuously on the inverse volume t , and hence in the continuum /epsilon1 = 0. \nTo check for a restoration of supersymmetry we have computed /epsilon1/t in the periodic theory for a variety of lattice sizes L = 5 , 10 and 20. The upper plot of figure 1 shows /epsilon1/t for SU (2). For large t the index /epsilon1 is already consistent with zero for L = 5, while at small t it appears to approach zero as L increases. Notice that while this quantity is a sensitive test of the restoration of supersymmetry in the continuum limit L → ∞ other observables such as P shown in the lower plot are relatively insensitive to the number of lattice sites for L ≥ 5. \nWe have also examined the continuum limit of the thermal theory. In figure 2 we show L = 5 and 10 data for the thermal energy for SU (5) (in the phase quenched approximation - which we discuss shortly). As noted above, we find a lattice instability for the thermal theory with t /lessorsimilar 1 (with some dependence on N and L ). However, \nFIG. 3: Top: A plot of the dimensionless energy /epsilon1/t verses dimensionless temperature t . Data shown is generated in two ways. For temperatures larger than t ∼ 1 we simulate the thermal theory for N = 2 , 3 , 5 , 8 , 12 with 5 points. The low temperature results are computed for N = 2 , 3 , 5 for 5 points by simulating the periodic theory, and reweighting with the appropriate combination of the thermal and periodic Pfaffians, as described in the main text. The low temperature black hole prediction is shown. Bottom: Aplot of the Polyakov loop observable P for the same cases. \n<!-- image --> \nFIG. 4: For comparison with figure 3, /epsilon1/t verses t is shown for the quenched theory for N = 5 , 12 , 30 and periodic theory with Pfaffian reweighting for N = 3 , 5, using 5 point lattices. \n<!-- image --> \nfor larger t this does not occur. As argued above we believe this is an artifact of our lattice formulation and has nothing to do with continuum physics. The points plotted in the figure are taken only from simulations where the scalar distribution remained bounded for hundreds of physical RHMC times (the observed instability sets in \nvery quickly in RHMC time, so the change in behavior is easy to identify). The plot shows that these lattice spacing effects are small and hence for the remainder of our results we show only data from L = 5 point lattices. \nIn the lower plot of figure 2 we show the mean cosine of the Pfaffian phase for the thermal SU (5) theory with L = 5 lattice sites. As expected this phase becomes more important at lower temperatures but the actual value is close to one over the range of temperatures where we can directly simulate the thermal theory. Indeed the effects of reweighting are negligible in this temperature regime. Hence for the data we present later for direct simulation of the thermal theory we use the phase quenched approximation. Since the Pfaffian is very costly to compute this allows us to work at larger N . \nWe now turn to the main results of this letter. In figure 3 we plot the energy and Polyakov loop for various N and L = 5 point lattices. For high temperature we have used direct simulation of the thermal theory (phase quenched as discussed above), and we are able to obtain results up to N = 12. At low temperatures we obtain results by a reweighting of the periodic simulations as discussed above. The results from both methods agree in the regime where they overlap t ∼ 1. \nAt very high temperatures the curves approach a constant corresponding to the result from classical equipartition assuming N 2 deconfined gluonic states (the fermions are lifted out of the dynamics by their thermal mass in this limit). In contrast for low temperatures the energy approaches zero signaling the presence of a supersymmetric vacuum at vanishing temperature. \nAs seen before [6], we see that t'Hooft scaling sets in for small N , with N = 3 already giving results close to an extrapolated large N result. The high temperature \n- [1] A. Strominger and C. Vafa, Phys. Lett. B379 , 99 (1996), hep-th/9601029.\n- [2] A. Strominger, JHEP 02 , 009 (1998), hep-th/9712251.\n- [3] J. M. Maldacena, Adv. Theor. Math. Phys. 2 , 231 (1998), hep-th/9711200.\n- [4] N. Itzhaki, J. M. Maldacena, J. Sonnenschein, and S. Yankielowicz, Phys. Rev. D58 , 046004 (1998), hepth/9802042.\n- [5] J. Kinney, J. M. Maldacena, S. Minwalla, and S. Raju (2005), hep-th/0510251.\n- [6] S. Catterall and T. Wiseman, JHEP 12 , 104 (2007), arXiv:0706.3518 [hep-lat].\n- [7] T. Banks, W. Fischler, S. H. Shenker, and L. Susskind, Phys. Rev. D55 , 5112 (1997), hep-th/9610043.\n- [8] M. Hanada, J. Nishimura, and S. Takeuchi, Phys. Rev. Lett. 99 , 161602 (2007), arXiv:0706.1647 [hep-lat].\n- [9] K. N. Anagnostopoulos, M. Hanada, J. Nishimura, and S. Takeuchi, Phys. Rev. Lett. 100 , 021601 (2008), arXiv:0707.4454 [hep-th].\n- [10] D. Kabat and G. Lifschytz, Nucl. Phys. B571 , 419 (2000), hep-th/9910001.\n- [11] D. Kabat, G. Lifschytz, and D. A. Lowe, Phys. Rev. D64 , \nasymptotics computed in [19] are also plotted and agree with our data. Our results also appear consistent with those found recently using non-lattice methods [9]. \nThere are two important physical observations. Firstly the curves appear to interpolate from high to low temperature smoothly - there is no obviously discontinuous behavior. This is to be contrasted with the quenched version of this theory which has a large N confinement/deconfinement phase transition at t /similarequal 0 . 9 [20, 21]. Since the intermediate temperature range t ∼ 1 is dual to the regime where the thermal D0-branes have a radius comparable to the string scale, we are probing the Horowitz-Polchinski correspondence regime, and seeing apparently smooth behavior there. \nSecondly, the low temperature behavior of the theory appears consistent with the prediction from supergravity, also shown in the plots. This is to be contrasted with the quenched energy curve shown for comparison in figure 4 which departs strongly from the black hole prediction at low temperature. In this figure we also show the periodic theory which shows the degree of supersymmetry breaking for these lattices is small. \nIt would be very interesting to extend these calculations to 2 and 3 dimensional Yang-Mills systems which are thought to be dual to D1 and D2 brane systems using recent lattice formulations retaining exact supersymmetry [22].", 'Acknowledgements': 'SC is supported in part by DOE grant DE-FG0285ER40237. TW is supported by a PPARC advanced fellowship and a Halliday award. Simulations were performed using USQCD resources at Fermilab. \n124015 (2001), hep-th/0105171. \n- [12] M. Campostrini and J. Wosiek, Nucl. Phys. B703 , 454 (2004), hep-th/0407021.\n- [13] J. R. Hiller, S. S. Pinsky, N. Salwen, and U. Trittmann, Phys. Lett. B624 , 105 (2005), hep-th/0506225.\n- [14] J. R. Hiller, O. Lunin, S. Pinsky, and U. Trittmann, Phys. Lett. B482 , 409 (2000), hep-th/0003249.\n- [15] S. Catterall and T. Wiseman, In preparation (2008).\n- [16] G. T. Horowitz and J. Polchinski, Phys. Rev. D55 , 6189 (1997), hep-th/9612146.\n- [17] W. Krauth, H. Nicolai, and M. Staudacher, Phys. Lett. B431 , 31 (1998), hep-th/9803117.\n- [18] M. A. Clark, A. D. Kennedy, and Z. Sroczynski, Nucl. Phys. Proc. Suppl. 140 , 835 (2005), hep-lat/0409133.\n- [19] N. Kawahara, J. Nishimura, and S. Takeuchi (2007), arXiv:0710.2188 [hep-th].\n- [20] O. Aharony, J. Marsano, S. Minwalla, and T. Wiseman, Class. Quant. Grav. 21 , 5169 (2004), hep-th/0406210.\n- [21] O. Aharony et al., JHEP 01 , 140 (2006), hep-th/0508077.\n- [22] S. Catterall, JHEP 01 , 048 (2008), arXiv:0712.2532 [hepth].'}

2007ApJ...664L..79U

Suzaku Observations of Active Galactic Nuclei Detected in the Swift BAT Survey: Discovery of a ``New Type'' of Buried Supermassive Black Holes

2007-01-01

['galaxies active', 'gamma rays', 'astronomy x rays', 'astronomy x rays', 'astrophysics']

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We present the Suzaku broadband observations of two AGNs detected by the Swift BAT hard X-ray (&gt;15 keV) survey that did not have previous X-ray data, SWIFT J0601.9-8636 and SWIFT J0138.6-4001. The Suzaku spectra reveal in both objects a heavily absorbed power-law component with a column density of N<SUB>H</SUB>~=10<SUP>23.5</SUP>-10<SUP>24</SUP> cm<SUP>-2</SUP> that dominates above 10 keV and an intense reflection component with a solid angle &gt;~2π from a cold, optically thick medium. We find that these AGNs have an extremely small fraction of scattered light from the nucleus, &lt;~0.5% with respect to the intrinsic power-law component. This indicates that they are buried in a very geometrically thick torus with a small opening angle and/or have an unusually small amount of gas responsible for scattering. In the former case, the geometry of SWIFT J0601.9-8636 should be nearly face-on as inferred from the small absorption for the reflection component. The discovery of two such objects in this small sample implies that there must be a significant number of yet unrecognized, very Compton thick AGNs viewed at larger inclination angles in the local universe, which are difficult to detect even in the currently most sensitive optical or hard X-ray surveys.

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https://arxiv.org/pdf/0706.1168.pdf

{"SUZAKU OBSERVATIONS OF ACTIVE GALACTIC NUCLEI DETECTED IN THE SWIFT /BAT SURVEY: DISCOVERY OF 'NEW TYPE' OF BURIED SUPERMASSIVE BLACK HOLES": 'YOSHIHIRO UEDA 1 , SATOSHI EGUCHI 1 , YUICHI TERASHIMA 2 , RICHARD MUSHOTZKY 3 , JACK TUELLER 3 , CRAIG MARKWARDT 3 , NEIL GEHRELS 3 , YASUHIRO HASHIMOTO 4 , STEPHEN POTTER 4 \nDraft version October 31, 2018', 'ABSTRACT': 'We present the Suzaku broad band observations of two AGNs detected by the Swift /BAT hard X-ray ( > 15 keV) survey that did not have previous X-ray data, Swift J0601.9-8636 and Swift J0138.6-4001. The Suzaku spectra reveals in both objects a heavily absorbed power law component with a column density of N H /similarequal 10 23 . 5 -24 cm -2 that dominates above 10 keV, and an intense reflection component with a solid angle > ∼ 2 π from a cold, optically thick medium. We find that these AGNs have an extremely small fraction of scattered light from the nucleus, < ∼ 0 . 5% with respect to the intrinsic power law component. This indicates that they are buried in a very geometrically-thick torus with a small opening angle and/or have unusually small amount of gas responsible for scattering. In the former case, the geometry of Swift J0601.9-8636 should be nearly face-on as inferred from the small absorption for the reflection component. The discovery of two such objects in this small sample implies that there must be a significant number of yet unrecognized, very Compton thick AGNsviewed at larger inclination angles in the local universe, which are difficult to detect even in the currently most sensitive optical or hard X-ray surveys. \nSubject headings: galaxies: active - gamma rays: observations - X-rays: galaxies - X-rays: general', '1. INTRODUCTION': 'Many observations imply the presence of a large number of heavily obscured Active Galactic Nuclei (AGNs) in the local universe (e.g., Maiolino et al. 1998; Risaliti et al. 1999). The number density of AGNs subject to absorption with a lineof-sight Hydrogen column density log N H > ∼ 23.5 cm -2 is a key parameter in understanding the accretion history of the universe (e.g., Setti & Woltjer 1989; Fabian 1999). According to synthesis models of the X-ray background (XRB) (e.g., Ueda et al. 2003; Gilli et al. 2007), much of the peak intensity of the XRB at 30 keV should be produced by these objects. In spite of the potential importance of their contribution to the growth of supermassive black holes (Marconi et al. 2004), the nature of this population of AGNs, even in the local universe, is only poorly understood due to strong biases against detecting them. In these objects, the direct emission in the UV, optical, and near IR bands as well as at E < 10 keV from the nucleus is almost completely blocked by obscuring matter, making it difficult to probe the central engine. \nHard X-ray surveys at energies above 10-15 keV provide us with an ideal opportunity to select this population of AGNs as long as the column density is less than log N H ≈ 24 . 5 cm -2 . Recent surveys performed with Swift /BAT (15-200 keV; Markwardt et al. 2005) and INTEGRAL (10-100 keV; Bassani et al. 2006; Beckmann et al. 2006; Sazonov et al. 2007), because of their relative insensitivity to absorption, are providing one of the most unbiased AGN samples in the local universe including Compton thick AGNs, i.e., those with log N H > 24 cm -2 . In fact, these surveys have started to detect hidden AGNs in the local universe located in galaxies that were previously unrecognized to contain an active nucleus at \nother wavelengths. \nTo unveil the nature of these new hard X-ray sources, follow-up observations covering a broad energy band are crucial. In this paper, we present the first results from followup observations with the Suzaku observatory (Mitsuda et al. 2007) of the AGNs Swift J0601.9-8636 and Swift J0138.64001 detected in the Swift /BAT survey. These targets are essentially randomly selected from a bright Swift AGN sample for which soft X-ray ( < 10 keV) spectroscopic observations had never been performed, and are thus reasonable representatives of unknown AGN populations selected by hard X-rays above 15 keV. In § 3, we also present an optical spectrum of Swift J0601.9-8636 taken at the South African Astronomical Observatory (SAAO) 1.9-m telescope. The cosmological parameters ( H 0, Ω m, Ω λ ) = (70 km s -1 Mpc -1 , 0.3, 0.7) (Spergel et al. 2003) are adopted throughout the paper.', '2. THE SUZAKU OBSERVATIONS AND RESULTS': "Table 1 summarizes our targets and observation log. Swift J0601.9-8636 is optically identified as the galaxy ESO 005G004 (Lauberts 1982) at z = 0 . 0062 with no previous firm evidence for AGN activity. The optical counterpart of Swift J0138.6-4001 is the galaxy ESO 297-G018 (Lauberts 1982) at z = 0 . 0252, which was identified as a narrow line AGN (Kirhakos & Steiner 1990). \nSuzaku , the 5th Japanese X-ray satellite, carries four sets of X-ray mirrors each with a focal plane X-ray CCD camera, the X-ray Imaging Spectrometer (XIS-0, XIS-1, XIS-2, and XIS-3; Koyama et al. 2007), and a non-imaging instrument called the Hard X-ray Detector (HXD; Takahashi et al. 2007), which consists of the Si PIN photo-diodes and GSO scintillation counters. The XIS and PIN simultaneously covers the energy band of 0.2-12 keV and 10-70 keV, respectively. The unique capabilities of Suzaku , high sensitivity in the 12-70 keV band and broad band coverage with good spectral resolution, are critical for studies of highly absorbed AGNs. \nWe observed Swift J0601.9-8636 and Swift J0138.6-4001 with Suzaku on 2006 April 14 and June 5 for a net exposure \nTABLE 1 TARGETS AND OBSERVATION LOG \n| Swift Optical ID redshift End Time | Start Time (UT) | Exposure a |\n|---------------------------------------------------------------|---------------------------------------------------------------|--------------|\n| J0601.9-8636 ESO 005-G004 0.0062 2006/04/13 16:24 04/14 01:52 | | 19.8 ksec |\n| | J0138.6-4001 ESO 297-G018 0.0252 2006/06/04 18:13 06/05 05:00 | 21.2 ksec | \nof 20 and 21 ksec (for the XIS), respectively. Standard analysis was made on data products, which were processed with the latest calibration (version 1.2). We detected both sources at high significance with the XIS and PIN. The XIS spectra were accumulated within a radius of 2 arcmin around the detected position. The background was taken from a source free region in the field of view. The spectra of three Front-side Illuminated CCD (hereafter FI-XIS; XIS-0, 2, and 3) are summed together, while that of Back-side Illuminated CCD (BI-XIS; XIS-1) is treated separately in the spectral fit. Examining the spectra of 55 Fe calibration source, we verify that the energy scale and resolution is accurate better than 10 eV and 60 eV levels, respectively, at 5.9 keV. For the analysis of the PIN, we utilized only data of Well units with the bias voltage set at 500 eV (W0,1,2,3 for Swift J0601.9-8636 and W1,2,3 for Swift J0138.6-4001) with the best available models of PIN background provided by the HXD team 5 . \nTo obtain the best constraint from the entire data, we perform a simultaneous fit to the spectra of XIS (FI and BI), PIN, and the archival Swift BAT, which covers the 0.2-200 keV band as a whole. The BAT spectra consist of four energy bins over the 15-200 keV range and are useful to constrain the power law index. Here we allow the relative flux normalization between Suzaku and Swift (BAT) to be a free parameter, considering possible time variability between the observations. We fixed the normalization ratio between the FI-XIS and the PIN based on the calibration result using the Crab Nebula. All the absolute fluxes quoted in this paper refer to the flux calibration of the FI-XIS. We find that the 10-50 keV PIN fluxes of Swift J0601.9-8636 and Swift J0138.64001 are 1 × 10 -11 and 4 × 10 -11 erg cm -2 s -1 , indicating time variability by a factor of 0.5 and 1.6, respectively, compared with the averaged flux measured by the Swift /BAT over the past 9 months (Tueller et al. , in preparation). \nFigure 1 shows the FI-XIS and PIN spectra unfolded for the detector response (for clarity the BI-XIS and BAT spectra are not plotted). The X-ray spectrum of Swift J0601.9-8636 below 10 keV is dominated by a hard continuum with few photons below 2 keV, consistent with the previous non-detection in soft X-rays (an upper limit of 1 . 3 × 10 -13 erg cm -2 s -1 in the 0.1-2.4 keV band by the ROSAT All Sky Survey; Voges et al. 2000). We find that the broad band spectrum can be well reproduced with a model consisting of a heavily absorbed power law with log N H /similarequal 24 cm -2 , which dominates above 10 keV, and a mildly absorbed reflection component from cold matter accompanied by a narrow fluorescence iron-K line, which dominates below 10 keV. The large column density is consistent with the observed equivalent width (EW) of the iron-K line, ≈ 1 keV (Levenson et al. 2002). Swift J0138.6-4001 shows a similar spectrum but with a smaller absorption of log N H = 23 . 7 cm -2 for both transmitted and reflected components. \nFIG. 1.- The broad-band energy spectra of (a) Swift J0601.9-8636 and (b) Swift J0138.6-4001 unfolded for the detector response in units of E 2 F ( E ), where F ( E ) is the photon spectrum. For clarity, we only plot the summed spectrum of the three FI-XIS (below 12 keV), and those of PIN (above 12 keV), while the spectral fit is performed to the whole XIS+PIN+BAT data including the BI-XIS. The crosses (black) represent the data with 1 σ statistical errors. The histograms show the best-fit model with separate components. The upper solid line (red), dot-dashed line (light blue), lower solid line (blue), and dotted line (magenta) correspond to the total, iron-K emission line, reflection component, and scattered component, respectively. \n<!-- image --> \nThe spectral model is represented as \nF ( E ) = e -σ ( E ) N Gal H [ f AE -Γ + e -σ ( E ) N H AE -Γ + e -σ ( E ) N refl H C ( E ) + G ( E )] , \nwhere N Gal H is the Galactic absorption column density fixed at 2 . 0 × 10 20 cm -2 for both targets (Dickey & Lockman 1990), N H the local absorption column density at the source redshift for the transmitted component, N refl H that for the reflected component (assumed to be the same as N H for Swift J0138.64001), and σ ( E ) the cross section of photo-electric absorption. The term C ( E ) represents the reflection component, calculated using the code in Magdziarz & Zdziarski (1995); we leave the solid angle Ω of the reflector as a free parameter by fixing the inclination angle at 60 · and cutoff energy at 300 keV, assuming Solar abundances for all elements. R ( ≡ Ω / 2 π ) > 1 means the transmission efficiency should be < ∼ 1 / R (see § 4). The term G ( E ) is the narrow iron-K emission line modelled by a Gaussian profile, where we fix the 1 σ width at 50 eV to take account of the response uncertainty (and hence, the line should be considered to be unresolved). The best-fit parameters are summarized in Table 2 with the intrinsic 2-10 keV luminosity, L 2 -10, corrected for the absorption and transmission efficiency of 1 / R . \nWe confirm that these results are robust, within the statistical error, given the systematic errors in the background es- \nTABLE 2 BEST FIT SPECTRAL PARAMETERS \n| | Swift | J0601.9-8636 | J0138.6-4001 |\n|------|-------------------------------|-------------------------|--------------------------|\n| (1) | N H (10 22 cm - 2 ) | 101 + 54 - 38 + | 46 ± 4 |\n| (2) | Γ | 1 . 95 0 . 36 0 . 33 | 1 . 66 + 0 . 16 - 0 . 04 |\n| (3) | R | - 1 . 7 + 3 . 5 - 0 . 9 | 2 . 1 + 0 . 4 - 1 . 2 |\n| (4) | N refl H (10 22 cm - 2 ) | 2 . 9 + 5 . 3 - 1 . 4 | (= N H) |\n| (5) | f scat (%) | 0 . 20 ± 0 . 11 | 0 . 23 + 0 . 23 - 0 . 16 |\n| (6) | E cen (keV) | 6 . 38 ± 0 . 02 | 6 . 38 ± 0 . 03 |\n| (7) | E.W. (keV) | 1 . 06 ± 0 . 16 | 0 . 20 ± 0 . 05 |\n| (8) | F 2 - 10 (erg cm - 2 s - 1 ) | 1 . 1 × 10 - 12 | 3 . 3 × 10 - 12 |\n| (9) | F 10 - 50 (erg cm - 2 s - 1 ) | 9 . 8 × 10 - 12 | 3 . 9 × 10 - 11 |\n| (10) | L 2 - 10 (erg s - 1 ) | 8 . 3 × 10 41 | 3 . 9 × 10 43 |\n| | χ 2 (dof) | 20.0 (27) | 101.3 (87) | \nNOTE. - (1) The line-of-sight hydrogen column density for the transmitted component; (2) The power law photon index; (3) The relative strength of the reflection component to the transmitted one, defined as R ≡ Ω / 2 π , where Ω is the solid angle of the reflector viewed from the nucleus; (4) The line-of-sight hydrogen column density for the refection component (assumed to be the same as N H for Swift J0138.6-4001); (5) The fraction of a scattered component relative to the intrinsic power law corrected for the transmission efficiency of 1 / R when R > 1; (6) The center energy of an iron-K emission line at rest frame. The 1 σ line width is fixed at 50 eV; (7) The observed equivalent width of the iron-K line with respect to the whole continuum; (8)(9) Observed fluxes in the 2-10 keV and 10-50 keV band; (10) The 2-10 keV intrinsic luminosity corrected for the absorption and transmission efficiency of 1 / R . \ntimation of the PIN detector (Kokubun et al. 2007). In the case of Swift J0138.6-4001, where photon statistics is dominated by the XIS data, we limit the allowed range of photon index to Γ = 1 . 63 -2 . 02 in the simultaneous fit, being constrained from the BAT spectrum. We have limited R < 2 . 5 for this source to avoid physical inconsistency between R and the EW of an iron-K line; otherwise (i.e., in more 'reflectiondominated' spectra), we should expect a larger EW than the observed value of ≈ 0.2 keV.", '3. OPTICAL SPECTRUM OF SWIFT J0601.9-8636': 'We performed an optical spectroscopic observation of Swift J0601.9-8636 (ESO 005-G004) during the night of 2007 March 16, using the SAAO 1.9-m telescope with the Cassegrain spectrograph. Grating six, with a spectral range of about 3500-5300 Å at a resolution around 4 Å, was used with a 2 arcsec slit placed on the center of the galaxy for a total integration time of 2400 s. To derive the sensitivity curve, we fit the observed spectral energy distribution of standard stars with low-order polynomial. The co-added, fluxcalibrated spectrum in the 4000-5500 Å range is shown in Figure 2. It reveals a rather featureless spectrum with no evidence for H β or [O III] λ 5007 emission lines, typical for this type of non-active edge-on galaxy within this spectral range. The 90% upper limit on the [O III] flux is conservatively estimated to be 3 × 10 -15 erg cm -2 s -1 , corresponding to a luminosity of 3 × 10 38 erg s -1 . This yields the ratio of the intrinsic 2-10 keV luminosity to the observed [O III] luminosity of > 2800. Although the (unknown) extinction correction for [O III] could reduce the value, the result is consistent with Swift J0601.9-8636 having an intrinsically weak [O III] emission relative to hard X-rays compared with other Seyfert galaxies (Bassani et al. 1999; Heckmann et al. 2005; Netzer et al. 2007). In particular, this object would not have been selected to be an AGN on the basis of its [O III] or H β \nFIG. 2.- The optical spectrum of the nucleus region of Swift J0601.98636 (ESO 005-G004) in the 4000-5500 Å wavelength range, tak en with the SAAO 1.9-m telescope. The arrow denotes the position of the [O III] λ 5007 line. \n<!-- image --> \nemission.', '4. DISCUSSION': "Both sources show an intense reflection component relative to the transmitted one. Using the standard reflection model (Magdziarz & Zdziarski 1995), we find that the solid angle of the reflector Ω / 2 π viewed from the nucleus exceeds unity, which is apparently unphysical if attributed only to geometry. This implies that a part of the direct emission is completely blocked by non-uniform material in the line of sight even above 10 keV. The reflection-dominated nature of the spectra of heavily obscured AGNs, if common, has an impact on the population synthesis model of the XRB, where a much weaker reflection is assumed for type 2 AGNs (Gilli et al. 2007). Another possibility is that this apparent very high reflection fraction is due to time variability, that is, the decrease of the flux in the transmitted light is echoed with a time delay corresponding to the difference in light paths between the emitter, reflector, and observers. \nIt is remarkable that both Swift J0601.9-8636 and Swift J0138.6-4001 have a very small amount of soft X-ray scattered emission, less than 0.46% of the intrinsic power law component. (If we fix N refl H = 0 in the spectral fit of Swift J0601.9-8636, then we obtain a photon index of 1 . 80 ± 0 . 29 and no significant scattered component with a 90% upper limit of 0.47%.) As far as we know these are amongst the lowest scattered fractions ever seen from an absorbed AGN (Turner et al. 1997; Cappi et al. 2006). In optically selected Seyfert 2 galaxies, the presence of prominent soft X-ray emission is common (e.g., Guainazzi et al. 2005). Such emission probably originates from the same extended gas responsible for the optical [O III] emission (Bianchi et al. 2006). This type of emission from 'classical' Seyfert 2 galaxies has always been seen in the spectra of objects well studied so far. However, this sample is dominated by optically selected Seyfert 2 galaxies, which require a scattered component to be selected. \nThe scattered fraction is proportional to both the solid angle of the scattering region as viewed from the nucleus, Ω scat, and the scattering optical depth, τ scat. Hence, the observed small scattered fraction means small Ω scat and/or small τ scat i.e., deficiency of gas in the circumnuclear environment, for some unknown reason. \nThe first possibility, which we favor as a more plausi- \nble case, indicates that these AGNs are buried in a very geometrically-thick obscuring torus. Assuming that the typical scattering fraction of 3% corresponds to the effective torus half-opening angle (see Levenson et al. 2002 for definition) θ of 45 degree, our results ( < 0 . 5%) indicate θ < ∼ 20 degree. In the case of Swift J0601.9-8636, the small absorption for the reflection component, which probably comes from the inner wall of the torus, suggests that we are seeing this source in a rather face-on geometry. Indeed, applying the formalism of Levenson et al. (2002) to the observed EW of the iron-K line, we infer that the inclination angle with respect to the axis of the disk, i , is smaller than 40 degrees if θ < 20 degree. For Swift J0138.6-4001, the presence of a high column density for the reflection component implies a more edge-on geometry than in Swift J0601.9-8636. The observed EW of 0 . 20 ± 0 . 05 keV can be explained if the torus is patchy or has a geometrical structure such that the line-of-sight column density is much smaller than that in the disk plane. \nWe infer that this type of buried AGNs is a significant fraction of the whole AGN population, although an accurate estimate of this fraction is difficult at present due to the small number statistics 6 . The observed fraction of heavily obscured AGNs with log N H > 23 . 5 is about 25% among the hard Xray ( E > 15 keV) selected AGNs (Markwardt et al. 2005). The true number density of obscured AGNs could be much larger, however. If we saw the same system of our targets at much larger inclination angles ( i /greatermuch 40 degree), the observed flux of the transmitted component would be much fainter even in hard X-rays due to the effects of repeated Compton scatterings (Wilman & Fabian 1999). Our results imply that there must be a large number of yet unrecognized, Compton thick AGNsin the local universe, which are likely to be missed even in the Swift and INTEGRAL surveys. \nThe existence of AGNs with a geometrically thick torus was predicted by Fabian et al. (1998), where the extreme obscuration was postulated to be caused by a nuclear starburst. Using the 60 µ mand 100 µ m fluxes measured by Infrared Astronomical Satellite (IRAS) , we obtain the far infrared luminosity (defined by David et al. 1992) of L FIR = 4 . 4 × 10 43 erg s -1 and 7 . 1 × 10 43 erg s -1 , and hence the ratio between the 2-10 keV to far infrared luminosities of L 2 -10 / L FIR ≈ 0 . 02 \nand ≈ 0 . 5 for Swift J0601.9-8636 and Swift J0138.6-4001, respectively. While the result of Swift J0138.6-4001 is consistent with those of the 2-10 keV selected AGNs in the local universe (Piccinotti et al. 1982) within the scatter, the small L 2 -10 / L FIR ratio of Swift J0601.9-8636 indicates a possibly significant starburst activity. However, this is not supported by the optical spectrum of this object, which apparently shows no evidence for a significant amount of star formation. The reason behind the difference between the two sources is unclear. \nBy using the unique combination of the Swift BAT survey and the Suzaku broad band spectral capabilities, we are discovering a new type of AGN with an extremely small scattering fraction. This class of object is most likely to contain a buried AGN in a very geometrically-thick torus. This population was missed in previous surveys, demonstrating the power of hard X-ray ( > 10 keV) surveys to advance our global understanding of the whole AGN population. In particular, we predict that the objects should have fainter [O III] emission luminosity relative to the hard X-ray luminosity compared with classical Seyfert 2 galaxies because much less of the nuclear flux 'leaks' out to ionize the narrow line gas. As shown above, the optical spectrum of Swift J0601.9-8636 is consistent with this prediction. This study is particularly important since the existence of numerous such objects would make surveys that rely on the [O III] emission incomplete by missing many of buried AGNs and incorrectly estimating the true AGN luminosity. \nWe thank the members of the Suzaku team for calibration efforts of the instruments, in particular Motohide Kokubun and Yasushi Fukazawa for their useful advice regarding the background of the HXD. We would also like to thank the anonymous referee for providing helpful suggestions to improve this paper. Part of this work was financially supported by Grants-in-Aid for Scientific Research 17740121 and 17740124, and by the Grant-in-Aid for the 21st Century COE 'Center for Diversity and Universality in Physics' from the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan. \n6 We note that a similar object has also been found by Comastri et al. (2007) with the Suzaku follow-up of hard X-ray ( > 10 keV) selected AGNs.", 'REFERENCES': "Bassani, L. et al. 1999, ApJS, 121, 473 \nBassani, L. et al. 2006, ApJ, 636, L65 Beckmann, V., Gehrels, N., Shrader, C. R., Soldi, S. 2006, ApJ, 638, 642 Bianchi, S., Guainazzi, M., Chiaberge, M. 2006, A&A, 448, 499 Cappi, M. et al. 2006, A&A, 113, 459 Comastri, A., Gilli, R., Vignali, C., Matt, G., Fiore, F., Iwasawa, K. 2007, in proc. of 'The Extreme Universe in the Suzaku Era', Progress of Theoretical Physics Supplement, in press (arXiv:0704.1253) David, L. 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K., Gehrels, N., Barthelmy, S. D., Mushotzky, R. F. 2005, ApJ, 633, L77 Mitsuda, K. et al. 2007, PASJ, 59, 1 Netzer, H., Mainieri, V., Rosati, P., Trakhtenbrot, B. 2007, A&A, 453, 525 Piccinotti, G., Mushotzky, R. F., Boldt, E. A., Holt, S. S., Marshall, F. E., Serlemitsos, P. J., Shafer, R. A. 1982, ApJ, 253, 485 Risaliti, G., Maiolino, R., Salvati, M. 1999, ApJ, 522, 157 Sazonov, S., Revnivtsev, M., Krivonos, R., Churazov, E., Sunyaev, R. 2007, A&A, 462, 57 Setti, G. & Woltjer, L. 1989, A&A, 224, L21 Spergel, D. N. et al. 2003, ApJS, 148, 175 Takahashi, T. et al. 2007, PASJ, 59, 35 \nTurner, T. J., George, I. M., Nandra, K., Mushotzky, R. F. 1997, ApJ, 446, 23 \nUeda, Y., Akiyama, M., Ohta, K., Miyaji, T. 2003, ApJ, 598, 886 Voges, W. et al. 2000, IAU Circular, 7432, 3 Wilman, R. J., Fabian, A. C. 1999, MNRAS, 309, 862"}

2008PhRvD..77l4047B

Exploring black hole superkicks

2008-01-01

['-', '-', '-', '-', 'methods numerical', 'waves', '-', '-', '-']

[]

Recent calculations of the recoil velocity in black-hole binary mergers have found kick velocities of ≈2500km/s for equal-mass binaries with antialigned initial spins in the orbital plane. In general the dynamics of spinning black holes can be extremely complicated and are difficult to analyze and understand. In contrast, the “superkick” configuration is an example with a high degree of symmetry that also exhibits exciting physics. We exploit the simplicity of this test case to study more closely the role of spin in black-hole recoil and find that the recoil is with good accuracy proportional to the difference between the (l=2,m=±2) modes of Ψ<SUB>4</SUB>, the major contribution to the recoil occurs within 30M before and after the merger, and that this is after the time at which a standard post-Newtonian treatment breaks down. We also discuss consequences of the (l=2,m=±2) asymmetry in the gravitational wave signal for the angular dependence of the signal-to-noise ratio and the mismatch of the gravitational wave signals corresponding to the north and south poles.

[]

https://arxiv.org/pdf/0707.0135.pdf

{'Exploring black hole superkicks': "Bernd Brugmann, Jos'e A. Gonz'alez, Mark Hannam, Sascha Husa, Ulrich Sperhake 1 1 Theoretical Physics Institute, University of Jena, 07743 Jena, Germany \nRecent calculations of the recoil velocity in black-hole binary mergers have found kick velocities of ≈ 2500 km/s for equal-mass binaries with anti-aligned initial spins in the orbital plane. In general the dynamics of spinning black holes can be extremely complicated and are difficult to analyze and understand. In contrast, the 'superkick' configuration is an example with a high degree of symmetry that also exhibits exciting physics. We exploit the simplicity of this test case to study more closely the role of spin in black-hole recoil and find that: the recoil is with good accuracy proportional to the difference between the ( l = 2 , m = ± 2) modes of Ψ 4 , the major contribution to the recoil occurs within 30 M before and after the merger, and that this is after the time at which a standard post-Newtonian treatment breaks down. We also discuss consequences of the ( l = 2 , m = ± 2) asymmetry in the gravitational wave signal for the angular dependence of the SNR and the mismatch of the gravitational wave signals corresponding to the north and south poles. \nPACS numbers: 04.25.Dm, 04.30.Db, 95.30.Sf, 98.80.Jk", 'I. INTRODUCTION': "More than forty years after Hahn and Lindquist started the numerical investigation of colliding black holes [1], a series of breakthroughs starting in 2005 [2, 3, 4] has turned the quest for stable black-hole inspiral simulations into a gold rush. \nA particular focus of the last few months has been the so-called recoil or rocket effect due to 'beamed' emission of gravitational radiation [5, 6, 7]. By momentum conservation, radiation of energy in a preferred direction corresponds to a loss of linear momentum and the black hole that results from the merger thus recoils from the center-of-mass frame with speeds of up to a few thousand km/s. The velocity of this 'kick' depends on the configuration of the system (e.g., the mass ratio and spins) and details of the merger dynamics, but not on the total mass (velocity is dimensionless in geometric units). From an astrophysical point of view, the recoil effect is particularly interesting for massive black holes with masses > 10 5 M /circledot , which exist at the center of most galaxies and may have a substantial impact on the structure and formation of their host galaxies. Observational consequences of large recoil have recently been considered in [8, 9]. \nThe largest recoil effects have so far been found [10, 11] for a particularly simple configuration suggested in [12] based on [13]: equal-mass binaries with (initially) antialigned spins in the orbital plane. Based on numerical simulations for different configurations and a postNewtonian approximation [13], an estimate of 1300 km / s had been obtained for this configuration with maximal spin [14]. The kicks found in full numerical simulations are however even larger, e.g. 2500 km/s [10] or 1800 km/s [11, 12] for non-maximally spinning black holes. This is of the order of 1% of the speed of light, and can be larger than the escape velocity of about 2000 km/s from giant elliptical galaxies. Extrapolating current numerical results for non-maximal spins to maximally spinning black holes predicts recoil velocities of up to ≈ 4000 km/s \n[11]. Smaller but still significant kick velocities have been found for several different types of black hole configurations [15, 16, 17, 18, 19, 20, 21, 22]. Estimations of the probabilities to obtain different kick velocities for different mass ratios and high spins were studied in [23]. \nThe parameter space of the inspiral of spinning black holes is very large, and although its full exploration will require numerical methods, analytical understanding and approximations will be crucial to render the task economical. The purpose of the present paper is to obtain a better understanding of the physics that leads to the large kick results recently observed, and in particular to compare with post-Newtonian approximations, and see where such approximations are accurate, and where they (currently) break down. \nWe will refer to a configuration similar to that described in [10, 11], i.e., two equal-mass black holes with spins anti-aligned and in the orbital plane, as a superkick configuration. The superkick configuration exhibits ' π symmetry', i.e. it is invariant under a rotation by an angle π about an axis perpendicular to the initial orbital plane. It follows from this symmetry that linear momentum will not be radiated in the x or y directions, but only in the z -direction. As a consequence, the center-ofmass will remain at ( x = 0 , y = 0), but can move in the z -direction. \nThe paper is organized as follows. In Sec. II we briefly summarize our numerical methods, and list the simulations we have performed. Sec. III analyses several aspects of the dynamics of the 'superkick' configurations, in particular the comparison with post-Newtonian dynamics and various aspects of the ( l = 2 , m = ± 2) asymmetry. Consequences of this asymmetry for the angular dependence of the SNR and the mismatch of the gravitational wave signals, exemplified by the extreme case of the north and south poles, are discussed in Sec. IV. The paper concludes with a discussion section and four appendices that contain post-Newtonian equations we use in this paper, and a number of small results concerning the dynamics of moving-puncture simulations.", 'II. NUMERICAL METHODS AND SUMMARY OF SIMULATIONS': "In this section we will summarize our numerical methods for evolving black-hole binary spacetimes (largely by directing the reader to the relevant references), and specify the numerical simulations we performed. The various simulations will be motivated more fully later in the paper; for now we give an overview for later reference. \nWe performed numerical simulations with the BAM [24, 25] and LEAN [26] codes, with modifications discussed in [10]. Both codes start with black-hole binary puncture initial data [27, 28] generated using a pseudospectral code [29], and evolve them with the χ -variant of the moving-puncture [3, 4] version of the BSSN [30, 31] formulation of the 3+1 Einstein evolution equations [32]. The gravitational waves emitted by the binary are calculated from the Newman-Penrose scalar Ψ 4 , and the details of this procedure for BAM and LEAN are given in [24] and [26], respectively. \nThe parameters of our simulations are summarized in Table I. Each black hole has mass M i (with mass parameter m i in the puncture data construction [28]), and the total mass is M = M 1 + M 2 . The black holes have a coordinate separation of D . In all runs the punctures are placed on the y -axis at y = ± D/ 2 and given momenta p x and spins S = 0 . 723 M 2 i = 0 . 2. The spins are aligned with the y -direction, except for the runs in the ' α -series', which are characterized by S y = ± S cos α and S x = ∓ S sin α . \nThe α -series and P -series simulations used modifications of the MI configuration described in [10]. This configuration was chosen because the results showed clean fourth-order convergence and high accuracy. We have found that the resolution requirements increase significantly for simulations of spinning black holes, and the MI configuration, with a small initial separation and therefore short evolution time, provided a convenient starting point for our study; these simulations also capture most of the important dynamics that we wish to study. \nThe α - and P -series simulations were performed with the grid setup χ η =2 [6 × 44 : 4 × 88 : 6][88 : 5 . 82] in the notation of [24], i.e., the six inner boxes had 44 3 points, the four outer boxes had 88 3 points, the resolution on the finest level is M/ 88, and the resolution at the outer boundary is 5 . 82 M . Convergence tests were performed for the α = 0 case (which is the same as the MI configuration in [10]) with inner-box sizes of 40 , 44 , 48, and corresponding resolutions. Clean fourth-order convergence of the linear momentum radiation flux dP z /dt is shown in Figure 1. Also shown is convergence in the puncture separation, which is not expected to last beyond the merger time of about t = 88 M since the separation between the two punctures inside the common apparent horizon quickly approaches zero [24]. \nFurther simulations were performed with larger initial separation and with quasi-circular orbit parameters (calculated according the prescription given in Ap- \nTABLE I: Physical parameters of the simulations performed for this paper. \n| Simulation | D | m i | p x | M | α |\n|--------------|-------|--------------|-----------------------------|---------|---------------------|\n| α -series | 6.514 | 0.363 | 0.133 1 | . 052 0 | ≤ α < 2 π δα = π/ 6 |\n| P -series | 6.514 | 0.363 | 0 . 13034 ≤ p x ≤ 0 . 13566 | 1.052 | 0 |\n| D6 | 6.0 | 0.296 | 0.1382 | 1 . 0 | 0 |\n| D8 | | 8.198 0.2875 | 0.11 | 1 . 0 | 0 | \npendix A). These are indicated D6 (for D = 6 M ) and D8 (for D = 8 . 2 M ) in the table. The D8 simulation was performed using the LEAN code, while all others were performed with BAM. The grid setup for the D6 simulations was the same as for the α - and P -series, and the convergence test referred to later used inner box sizes of 44, 48 and 52 points. The D8 simulation used a grid setup χ η =1 [2 × 133 : 1 × 155 : 2 × 133 : 3 × 67 : 9] [ 44 : 32 11 ] , where the innermost three levels with 67 points are centered around either hole and follow the motion of the puncture. \nFIG. 1: Convergence plots for the puncture separation r and the linear momentum radiation in the z -direction, dP z /dt obtained for model α = 0 of the α -series. The plots are scaled consistent with fourth-order convergence. After merger at about t = 88 M convergence in the puncture separation is lost (as expected). \n<!-- image --> \nExperimentally we have observed that the resolutions \nused in the α - and P -series simulations are not sufficient to obtain clean convergence for evolutions of spinning black holes orbiting for longer periods of time. It thus appears that the good convergence results for these particular series are largely due to the close initial separation of the black holes, which results in a rather quick merger time of about 88 M . When the black holes are placed further apart (or even making the seemingly innocuous change of choosing quasicircular orbit initial parameters for the same separation as the α -series simulations) convergence is lost before the black holes merge. We expect that fourth-order convergence would be obtained if sufficiently high resolutions were used, but the extra computational expense was not necessary for the analysis in this paper. \nIn the D6 simulations we find that the puncture separation and linear momentum radiation flux dP z /dt converge well for up to 15 M before merger, as shown in Figures 2 and 3. Note that since the waves are extracted at R ex = 50 M , we need to take into account a time lag of roughly 50 M when comparing times related to puncture motion and wave extraction. These simulations will be used only for discussions of the qualitative behavior, and for analysis at early times, when we are confident that the results are reliable. Similarly the long D8 LEAN simulation will only be used for qualitative comparison with post-Newtonian results.", 'III. ANALYSIS OF SUPERKICK DYNAMICS': 'We begin by describing the dynamics of two black holes in a superkick configuration with varying degrees of simplicity, in order to build up a clearer picture of the physics, and to motivate the simulations and analysis we have performed. In the simplest picture we draw (Section III B) the spin decouples from the orbital dynamics; a more complex picture includes spin precession effects (Section III C), and considering the dynamics in full general relativity (GR) in Section III D allows us to study the merger regime, from which most of the kick effect originates. The full GR results can then be directly compared with PN predictions, which we do in Section III E. We discuss the spin of the final black hole in Section III F.', 'A. Kick velocity and l = 2 , m = ± 2 symmetry breaking': "As noted before, the superkick configuration exhibits ' π symmetry' ( φ → φ + π ), thus linear momentum will not be radiated in the x or y directions, but radiation of linear momentum in the z direction is allowed, and the center-of-mass will only move in the z -direction. \nAs in nonspinning equal-mass binary simulations, almost all of the energy is radiated in the l = 2 , m = ± 2 modes: the maximal relative deviation of the energy in those modes from the total energies is roughly 2 %, ne- \nFIG. 2: Convergence of the puncture separation and dP z /dt as functions of time for evolutions of model D6. Results are scaled for fourth-order convergence. We see that fourth-order convergence is lost in the puncture separation at about t = 115 M , which corresponds to roughly t = 165 M in quantities from waves extracted at R ex = 50 M , which is about when we see a loss of convergence in dP z /dt . Note that we cut the plot at t ≈ 175 M when convergence is lost. \n<!-- image --> \nlecting the contribution from the junk radiation. This fact, and the symmetry discussed in the preceeding paragraph, leads us to expect that we should be able to directly relate the kick in the z -direction to the imbalance between the m = 2 and m = -2 modes, i.e., the difference in energy that is radiated toward the 'north' and 'south' hemispheres. Using the special relativistic relation | /vector p | = E between the momentum and energy of a wave packet (traveling at the speed of light), we expect a relation \np z = f × ( E 22 -E 2 -2 ) (1) \nfor the radiated momentum in the z -direction, where E 22 and E 2 -2 are the energies radiated in the l = 2 , m = ± 2 modes, and f is a geometric factor. Here 0 ≤ f < 1 expresses the fact that the radiation is smeared out in solid angle rather than sharply peaked in the direction of the poles. Neglecting all modes but l = 2 , m = ± 2, we assume a wave signal in the form Ψ 4 = κF ( t ) Y -2 22 ( θ, φ )+ λ ¯ F ( t ) Y -2 2 -2 ( θ, φ ), where κ, λ are real numbers ( κ = λ in the nonspinning equal mass case), F ( t ) is a complex time dependent function, and the Y -2 2 ± 2 are the spin-weighted \nFIG. 3: Puncture separation and dP z /dt as functions of time for the evolutions of model D6. Results from low, medium and high resolution simulations are shown. Only the highest resolution is shown for dP z /dt . \n<!-- image --> \nspherical harmonics \nY -2 2 -2 = √ 5 64 π (1 -cos θ ) 2 e -2 iφ , Y -2 22 = √ 5 64 π (1 + cos θ ) 2 e 2 iφ . (2) \nInserting this ansatz into the expressions for radiated energy and linear momentum (see e.g. Eqs. (48) and (49) in [24]) we obtain \n( \ndE dt = r 2 16 π κ 2 + λ 2 ) ∣ ∣ ∫ t ∞ F ( ˜ t ) d ˜ t ∣ ∣ ∣ 2 , (3) \n∣ ∣ \n∣ \n∣ ∣ Consequently the value of the geometric factor f can be determined as f = 2 / 3. We find this relationship satisfied to very good accuracy in our numerical evolutions, as shown in Figure (4). \n) ∣ ∣ ∣ dP z dt = 2 3 r 2 16 π ( κ 2 -λ 2 ) ∣ ∣ ∣ ∫ t ∞ F ( ˜ t ) d ˜ t ∣ ∣ ∣ 2 . (4) \n∣ \n∣ \nThe relative asymmetry in the energies emitted in the l = 2 , m = ± 2 modes, 2 E 22 / ( E 22 + E 2 -2 ) (this quantity is unity when there is no symmetry breaking) is plotted in Fig. 5, showing a maximal excess of roughly 40%. An analytic fit for extraction radius R ext = 50 M is \n2 E 22 ( E 22 + E 2 -2 ) = 1 + 0 . 416 cos(0 . 125 + α ) . (5) \nFIG. 4: Comparison of the kick velocity in km/s according to Eq. (1), for a range of angles α . Data points for the measured kick and the estimate Eq. (1) are shown, the points corresponding to the energy differences are connected. An analytical fit, v z = 2725 cos(176+ α ), to the measured kick is shown as a dashed line. Note that Eq. (1) slightly underestimates the kick, which is consistent since it neglects contributions from higher order multipoles l > 2. \n<!-- image --> \nIn the extreme case this fit corresponds to E 22 /E 2 -2 ≈ 2 . 4. For this fit the statistical error from 95 % confidence in the phase is roughly 20%, and roughly 2% for the amplitude of the oscillation. Fits corresponding to the extraction radii R ext = 30 M and 75 M give consistent results with Eq. (5) within the statistical error bars. \nFIG. 5: Excess energy in the l = 2 , m = 2 mode, 2 E 22 / ( E 22 + E 2 -2 ) plotted for extraction radii R ext = 30 M , and 50 M . The curves are the analytical fits for both extraction radii, see Eq. (5) Clearly, there is no significant dependence of this ratio on extraction radius. \n<!-- image -->", 'B. Simplest assumption: spin decouples from black-hole dynamics': 'As a first approximation of the dynamics, we may imagine that the black holes behave like (force-free) gyroscopes in flat spacetime, and as they orbit each other \ntheir individual spin vectors do not change direction. For example, if the spins were originally S 1 = (0 , S, 0) and S 2 = (0 , -S, 0), these vectors would be constant throughout the evolution. If we also assume that the spins do not noticeably influence the motion in the x -y plane, then simulations that start with the same initial separation and momenta will display the same dynamics no matter how the spins are directed in the orbital plane. This is the situation at the first-PN approximation, since spinorbit and spin-spin couplings enter at higher order. \nThis picture is surprisingly close to the observed dynamics in numerical simulations. Figure 6 shows the orbital motion in the first six simulations of the α -series, each differing only in the initial directions of the spins (i.e., S 1 , 2 = S ( ∓ sin α, ± cos α, 0) and α = 0 ...π in steps of π/ 6). The motion shows differences as α is varied (shown in the lower panels of the figure), but these differences are very small. In contrast, the resulting kick from these simulations, shown in Figure 4, displays a clear sinusoidal dependence on the angle - the kick varies from ≈ -2500 km/s to +2500 km/s. (Note that, since these values were calculated at a small radiation extraction radius R ex = 50 M , the values in the plot are systematically higher than the correct values by about 10%). Similar figures were also shown in [11]. \n<!-- image --> \nFIG. 6: The x - and y -motion of one of the punctures in six simulations from the α -series. The dynamics in the x -y plane are almost identical for all of the simulations (only one curve is actually visible in the upper panels). The small variations in the motion (measured with respect to the α = 0 simulation), are shown in the lower panels. \n<!-- image --> \nWe might conclude from these results that the final kick depends only on the initial spin magnitude and direction of each black hole. One may write an expression similar to Eq. (1) in [11], which for the superkick case reduces to \nV K = k cos( α -α 0 ) , (6) \nand determine the constants k and α 0 , which are approximately given by k ≈ 2500 km / s and α 0 ≈ 0 for our data. If this simple picture was correct, and the spins really did behave as gyroscopes in flat spacetime, then (6) would allow us to determine α 0 = 0 as the spin direction that produces the maximum kick.', 'C. The effect of spin precession during inspiral': "Needless to say, the real situation is more complicated. The picture of the black holes' spins as gyroscopes in flat spacetime is valid only at 1PN order; at higher postNewtonian orders and in full GR the spin directions evolve during the inspiral. We expect that the magnitude of the kick depends on the magnitude and direction of the spins when the black holes are close to merger. The spin configuration at merger time is a function of the initial configuration plus precession effects during the evolution. \nThe precession effects can be seen in Figure 7, which shows ( S x , S y , S z ) as a function of time for the α = 0 simulation described earlier. Here the spins were calculated from the black holes' apparent horizons, using the coordinate-based method outlined in [33] and also used in [22]. We see that the x - and y -components of the spin show noticeable precession during the last orbit, and after 80 M of evolution the spin vector has rotated by π/ 2. The z -component shows small oscillations around zero, but these are not well resolved with the current accuracy of the code. Note however by comparison with Fig. 6 that the period of the oscillation is roughly half an orbital period, which is consistent with the post-Newtonian equations in Appendix B. We will further comment on the comparison of the spin precession with PN-results in Sec. III E. After the formation of a common apparent horizon, at about t ≈ 88 M , the S x and S y quickly drop to zero, and S z jumps to its final value, corresponding to the spin of the final black hole, S z /M 2 f ≈ 0 . 7, where M f is the mass of the final black hole. We also compute the spin of the final black hole from quasi-normal ringdown in Sec. III F as S z /M 2 f ≈ 0 . 69. Note that this value of the final spin is also the value for non-spinning equal-mass inspirals, see e.g. [24]. \nAn example for the dependence of the final kick on parameters besides α , Figure 8 shows the final kick for α = 0 simulations with differing values of the initial momenta of the black holes (the P-series in Table I). Changing the initial momenta causes the merger time to change, and this means that the spins have more or less time to evolve, and are therefore in different directions when the black holes merge. The initial spin directions are, however, the same for all of these simulations. \nThis leads us back to Eq. (1) in [11], which was originally written in terms of the spin angles at merger. Above we formulated Eq. (6) in terms of the initial angle, but it would hold equally well if, instead of choosing α = α ( t = 0) we were to choose some fixed t 0 and use \nFIG. 7: Evolution of the spins S x , S y and S z of one of the black holes over the course of the α = 0 simulation. The x - and y -components of the spin show noticeable precession during the last orbit; the spin has rotated by π/ 2 after 80 M of evolution. By contrast, the z -component (lower panel) displays small oscillations around zero; these oscillations are not well resolved at the current accuracy of the code. Merger occurs around (at t ≈ 88 M ), at which time S x and S y drop to zero. The spin S z /M 2 f jumps to a final value of 0 . 723 corresponding to the merged black hole, reflecting the conversion of orbital angular momentum into the spin of the final object. \n<!-- image --> \nα = α ( t = t 0 ) in Eq. (6). Only the phase constant α 0 would change. The results shown in Figure 8 suggest that we will also see oscillatory behavior if we make a plot of the final kick versus merger time for a series of runs with the same initial spin directions. \nIn conclusion, given some superkick initial configuration we need to know both the spin angle α and the time until merger in order to predict the magnitude of the final kick from initial data.", 'D. Duration of the recoil': "Simplified models aside, we know that the final kick is due to an integration of dP z /dt over the entire evolution, and dP z /dt will be a complicated function of the instantaneous spin directions. A post-Newtonian version of this function is given in [13] and in Appendix B. Figure 9 shows dP z /dt as a function of time for three simulations in the α -series. We make several observations about these plots. The main contribution to the final kick originates \nFIG. 8: The final kick as a function of the initial momenta of the black holes for simulations with the initial spin angle α = 0. The differences between the smallest and largest merger time is only about 15 M ; these differences allow a little more or less spin precession, and therefore have a strong effect on the final recoil, which varies between 2300 and 2700 km/s. \n<!-- image --> \nfrom a time period of about 60 M . Before that time the contribution is negligible, which is even clearer in Figure 3, which shows dP z /dt for the quasi-circular model D6 which has a longer inspiral phase. We also see in Figure 9 that the function dP z /dt obtained for simulations with final kick of 2500, 0 and -2500 km / s does not only differ by a mere rescaling. Instead, the curve obtained for α = π/ 2 exhibits an additional oscillation. \nFIG. 9: Plot of dP z /dt as a function of time for the α = 0 , π/ 2 , π simulations. Most of the linear momentum is radiated over a 60 M period of time, centered roughly around the merger time. \n<!-- image --> \nWe would like to relate dP z /dt to the motion of the punctures, but this is not trivial. The radiation is extracted at some radius R ex , and plotting this as a function of retarded time r -R ex gives only a crude estimate of what is happening in the vicinity of the black holes at any given time. One could try to improve this estimate by using instead the luminosity distance (see for example [34]), but we choose to simply look at the puncture motion directly. We can calculate the coordinate acceleration of the punctures in the z -direction, a z ( t ) = d 2 z ( t ) /dt 2 . Figure 10 shows the acceleration of \none of the punctures in the α = 0 , π/ 2 , π simulations. A vertical line indicates the time at which a common apparent horizon forms, and thus gives us an indication of how much of the motion is due to effects before merger, and how much after merger. \nAlthough most of the final kick is generated before merger, Figure 10 suggests that dP z /dt is not negligible after the merger. Referring back to the plot of dP z /dt in Figure 9, the waves from the merger can be estimated to reach the radiation extraction sphere between 138 M (the time when the common apparent horizon forms, 88 M , plus the extraction radius, 50 M ) and 155 M (when the final black hole's ringdown can clearly be said to have begun, by observing that the wave amplitude has a clean exponential decay). Whatever time within this range we choose to denote as 'merger', it is clear that a significant contribution to the final kick arises after that time. \nFIG. 10: The coordinate acceleration of one of the punctures in the z -direction for the α = 0 , π/ 2 , π simulations. The vertical line represents the time of first formation of a common apparent horizon. The curves notably differ from those in Figure 3, in particular they show significant extra oscillations, while they have roughly the same amplitude The plot essentially confirms that the gauge dependence of the puncture motion in this direction is not well understood (in contrast with the orbital motion discussed in Appendix C), which prevents us from drawing quantitative conclusions about the kick velocity from the coordinate acceleration. \n<!-- image -->", 'E. Comparison with post-Newtonian predictions': "In Figure 10 we see that there is a significant contribution to the kick after the black holes merge. Before the merger, the main contribution comes from the 30 M before merger. During this time the black holes are at a separation of D < 3 . 5 M , and at this separation it is questionable whether an accurate post-Newtonian description of the radiated linear momentum is meaningful. We will now make a comparison between our numerical results and the predictions from the 2.5-PN accurate expressions in [13]. \nAppendices B, C and D summarize the techniques we use to compare post-Newtonian and numerical results. \nBriefly, Appendix B lists the expressions for the spin evolution and radiated linear momenta as found in [13]. These expressions were derived in the harmonic gauge. Our initial data are instead in the ADMTT gauge (up to 2PN accuracy [35]), and although it is not obvious how well we remain in the ADMTT gauge during evolution (but see [36] for a result that shows excellent agreement for larger separations), we would like to see how much the results differ between the two gauges. Appendix D thus gives the expressions necessary to transform the numerical quantities, which are assumed to be in the ADMTT gauge, to the harmonic gauge. To do this we need to calculate the momenta of the punctures as they evolve. Appendix C gives 2PN expressions that relate the puncture's speeds (again assumed to be in the ADMTT gauge) to momenta as given in Eq. (C2). We see in Appendix D that in fact the ADMTT → harmonic transformation makes little difference to our results over the time when the PN approximation is valid. This result may not be surprising, but quantifies any confusion that may arise when we compare results in the two gauges, and eliminates any major concern that our results may change drastically if we were to perform our simulations in the harmonic gauge. \nAs an aside, these formulas explain the speed at which the punctures move in a black-hole binary movingpuncture simulation; the puncture speeds and momenta are not related by the Newtonian formula p = mv , but instead to good accuracy by its 2PN counterpart. \nBefore considering the radiated linear momentum, we compare the post-Newtonian predictions for the spin evolution with our numerical results. Figure 11 shows the evolution of S x , S y and S z for simulation D8, compared with the predictions from Eqs. (B1). We see that there is very good qualitative agreement in S x and S y , even close to merger, which occurs at around t = 260 M . The z -component does not agree at all well with the 2.5PN prediction, but we again note that the puncture motion in the z -direction is much more gauge-dependent than the motion in the x -y plane, and it is the positions and speeds of the punctures that we use when evaluating the right-hand-sides of Eqs. (B1). Furthermore, it is not clear how well the numerical determination of the spin based on apparent horizons works in this context. Note also that the absolute error is very small. The frequency of the oscillations of the numerical simulation is rather close to the PN result, which is approximately twice the orbital frequency when precession effects are small (cmp. Appendix B). After merger, a few M after the end of the figure, S z jumps to its final value of around 0.7, and S x and S y drop to zero. \nIn the case of the radiated linear momentum flux dP z /dt and the final kick, we know that the assumptions underlying the post-Newtonian expressions break down when the black holes are very close. Eqs. (B3) diverge as 1 /r 5 as the particles' separation r → 0, so it is clear that a sensible estimate of the kick cannot be made by simply integrating this equation. What has been done \nFIG. 11: One black hole's spin as a function of time for simulation D8. Also shown is the 2.5PN prediction for the spin evolution, with the puncture dynamics { x i , v i } used in the spin evolution equations (B1). The agreement is very good for S x and S y , but poor for S z . This may once again be due to the numerical motion in the z -direction being far more gauge-dependent than the motion in the x -y plane. At late times S x and S y vanish, whereas S z is found to correspond to the angular momentum of the final black hole, compare Fig. 7 and Sec. III F. \n<!-- image --> \nin the past (see for example [37]) is to assume a cut-off separation, and integrate the post-Newtonian expression up to that point. We will now show that this approach is unlikely to give correct results in the superkick case. \nFigure 12 shows the function dP z /dt compared to the numerical values for the D8 simulation at a retarded time t -54 . 5 M , chosen to line up the early oscillations in dP z /dt , and close to a naive guess of the retarded time for the extraction radius R ex = 50 M . The post-Newtonian values of dP z /dt were calculated as follows. Eqs. (B3) \nrequires as input the positions, speeds and spins of the two black holes. Rather than integrate the full postNewtonian equations of motion, we simply enter the appropriate quantities from a numerical evolution. This allows us to compare, moment by moment, the postNewtonian and numerical predictions of dP z /dt for two particles (or black holes) with the { x i , p i , S i } configuration. \nFIG. 12: Comparison of the numerical dP z /dt with that predicted by Eqs. (B3). A time shift of 54 . 5 M was applied to the value calculated from the numerical wave extraction, to approximately take into account the wave travel time between the punctures and the wave extraction sphere by lining up the peak at t ≈ 200 M . The numerical relativity and 2.5PN results agree qualitatively at early times, but diverge quickly near merger, probably due to the 1 /r 5 term in Eq. (B3). The lower plot is a blow-up of the upper plot up to t = 225 M . \n<!-- image --> \nIn Figure 12 we once again see good qualitative agreement at early times. At late times the post-Newtonian prediction diverges, due to the 1 /r 5 term in Eqs. (B3). What is most striking about this plot is that the disagreement between numerical and post-Newtonian results becomes serious around 50 M before merger, which is just before the time when the major contribution to the recoil begins in Figures 3 and 10. This suggests that, at least in the special case of superkick configurations, if we integrate the post-Newtonian dP z /dt up to the point where its accuracy breaks down, we will grossly underestimate the value of the final kick. \nIn order to accurately estimate the value of the kick analytically, one would need to make a much more sophisticated choice of cutoff separation, and then perhaps match to a close-limit analysis. Such a procedure was \napplied in [38, 39] to nonspinning binaries, and may well be applicable in the spinning case. One may also be able to get good results from a more careful post-Newtonian analysis, as was performed (also in the nonspinning case) in [40, 41]. We would expect that the superkick case would be an extreme and particularly interesting test of such methods.", 'F. Spin of the final black hole from ringdown': 'In Sec. III C we found that the spin of the final black hole as read off from the black hole horizon is J z /M 2 ≈ 0 . 7. Here we will also determine the dimensionless Kerr spin parameter a = J z /M 2 from the quasinormal ringdown gravitational wave signal of the slowest decaying spin weighted spheroidal harmonic mode [42, 43], which we measure by projecting it onto the l = 2 , m = ± 2 spin weighted spherical harmonics . These projected l = 2 , m = ± 2 waveforms are split into amplitude and phase according to ψ 4 = A ( t ) exp( iϕ ( t )), we then perform analytical fits to the waveform for 170 ≤ t/M ≤ 230, where we see both a clean exponential decrease of the wave amplitude and a linear increase of the gravitational wave phase (corresponding to a constant frequency). Performing independent fits with a linear function for the wave phase and an exponential for the amplitude we obtain values for the complex ringdown frequency ω QNM . In order to factor out the overall mass scale we then perform a lookup of the dimensionless quantity Im( ω QNM ) / Re( ω QNM ) (i.e. essentially the inverse quality factor) in a table of QNM frequencies [44]. \nWe will consider in particular data from the α -series, see table (I). For the extraction radii R ex = 30 M , 50 M , 75 M both the l = 2 , m = ± 2 results can very well be fit with an analytic expression of the form a 0 + a 1 cos( α + ϕ 1 ) + a 2 cos( α + ϕ 2 ), see Fig. (13). At each extraction radius we get consistent results for the amplitudes a 0 , a 2 and the phase shifts ϕ 1 , ϕ 2 for the m = -2 , 2 modes, but we get the opposite sign for a 1 for the m = -2 and m = 2 modes. For a 0 we get (0 . 6963 , 0 . 6891 , 0 . 6891) ± 5 × 10 -4 (statistical error) for extraction radii R ex = (30 M, 50 M, 75 M ). For the oscillation amplitudes we get consistent results of a 1 = 0 . 004 ± 0 . 001, a 2 = -0 . 004 ± 0 . 001, with statistical errors corresponding to the 95 % confidence interval and rounded to one significant digit. We conclude that the asymptotic value of the dimensionless Kerr parameter is a/M ≈ 0 . 69, which is consistent with the value 0 . 7 we obtained from the black hole horizon. Since the oscillation a 1 cos( α + ϕ 1 ), which has the periodicity of the kick velocity, is not consistent between the m = -2 , 2 modes, and the oscillation a 2 cos( α + ϕ 2 ) is of the same size, we conclude that both may be non-physical, e.g. they could be due to gauge effects in the radiation extraction algorithm at finite radius (we suggest an alternative explanation in the next paragraph). It is plausible that such problems are more serious in the present case of large kicks, than \nwhen the black hole system does not move with respect to the center of gravity. It would be interesting to analyze the present case more carefully e.g. along the lines discussed in [45]. \nSince the final spin of the black hole is close to the value for non-spinning black hole mergers, it appears that the individual, anti-aligned spins of the black holes do not contribute to the final angular momentum, but rather cancel approximately during merger. This is worth noting since based on the PN analysis and the numerical evolutions there is a small oscillating z -component of the spin of the individual black holes. At merger time, the z -component of the black hole spins is added to the spin of the merged black holes due to orbital motion. In principle it could happen that the initially small S z is enlarged greatly (as the PN calculation becomes inaccurate), e.g. it could happen that the separate spins precess significantly towards the z -axis and add significantly to the final angular momentum of the black hole. But this does not seem to happen, at best there is a small positive or negative contribution to the final spin depending on the momentary phase of the S z oscillation during merger, e.g. as we see in Fig. 13. \nFIG. 13: The plot shows numerical results for J/M 2 = a 22 + a 2 -2 2 (points), obtained for extraction radius R ex = 50 M for the final black hole and an analytical fit (solid curve) in the text we conclude that the final Kerr spin parameter does not show significant variations (which might be further reduced by increased accuracy of the wave extraction). \n<!-- image -->', 'IV. EFFECTS OF RECOIL ON SNR AND TEMPLATE MATCH': "The gravitational recoil is essentially due to the symmetry-breaking between the dominating modes l = 2 , m = ± 2. A natural question is how this symmetry breaking is reflected in the overlap integrals of the gravitational waveforms. If the symmetry was not broken, the gravitational wave signal emitted towards the 'north pole' would provide the best template also for the south \npole. Similarly, for a sequence of waveforms that correspond to initial data that only differ in spin orientation (which we have parameterized by the angle α ), we can ask how much signal-to-noise ratio (SNR) is lost when trying to detect the gravitational wave signal corresponding to some value of α with a template corresponding to a different value of α . \nAnswering this question requires accurate waveforms, since any mismatch of waveforms can also be due to lack of numerical resolution, errors from the finite extraction radius (which can be significant, in particular because the recoil velocity creates an asymmetry of the geometry of the extraction sphere), and the contribution of the initial junk radiation. \nFor the data we are considering in this paper, the initial junk radiation cannot be separated from the main signal in a sufficiently clean fashion, neither can we obtain accurate error bars on the wave signals of the whole α -series to really settle the above questions. Nevertheless some preliminary results will illustrate the issue. \nWe define the correlation function between two time series x ( t ) and y ( t ) for a time shift τ as: \nR xy ( τ ) = ∫ ∞ -∞ x ( t ) y /star ( t -τ ) dt , (7) \nwhere a /star denotes complex conjugation. Working with Fourier transforms \nx ( t ) = ∫ ∞ -∞ ˜ x ( f ) e 2 πift df . (8) \nthe correlation function can be written as \nR xy ( τ ) = ∫ ∞ -∞ ˜ x ( f )˜ y /star ( f ) e 2 πifτ df (9) \nin terms of Fourier transforms ˜ x ( f ) , ˜ y ( f ) of the time series. The function R xy ( τ ) is thus simply the inverse Fourier transform of ˜ x ( f )˜ y /star ( f ). The value of τ for which R xy is maximal determines the time shift required to get the maximum correlation between x ( t ) and y ( t ). The self-correlation R xx is maximal for τ = 0: \nR xx (0) = ( x | x ) := ∫ ∞ -∞ | ˜ x ( f ) | 2 df . (10) \nMore generally we can define the scalar product \n( x | y ) := ∫ ∞ -∞ ˜ x ( f )˜ y /star ( f ) df . (11) \nThe 'match' which determines the efficiency of a template y to identify a signal x is defined as \nM = max τ | R xy ( τ ) | ( x | x )( y | y ) . (12) \nNote that in gravitational wave data analysis the orientation of a single detector actually reduces the signal \n√ \nto a real time series. We can now evaluate M for signals corresponding to different values of the initial spinangle α in our α -series, or for signals corresponding to different angles in the sky for a given value of α , say one with a large value of the recoil. From symmetry we expect that the mismatch 1 -M when comparing the signals corresponding to the maximal difference in the kick within the α -series ( ≈ 5000 km/s) equals the mismatch for the signals that correspond to the north and south poles for the maximal recoil case. Indeed we find a value of M ≈ 0 . 94 ± 0 . 01 both for the α = 0 case, which is close to the maximal recoil and for the maximal mismatch case within the α -series. Deviations of ± 0 . 01 here come from comparing either the full complex waveform, or just h + or h × . The uncertainties due to initial junk radiation, finite extraction radius and numerical error may however be larger than 1 %. More accurate data than presented here will be required for conclusive error estimates. Also, a detailed discussion of the dependence of the radiation signal on the angle α and the consequences for gravitational wave data analysis is beyond the scope of the present paper. \nA related question is how much brighter the source appears in the direction opposite to the recoil - in which more radiation is emitted. For the case of white noise we estimate the relative increase in SNR computing the ratios of the norm of the strain h for a given inclination angle θ to the strain measured at the south pole ( θ = π ) by computing: \nSNR( θ ) SNR( θ = π ) = √ ( h ( θ ) | h ( θ ) ( h ( θ = π ) | h ( θ = π ) . (13) \nIn Fig. 14 we plot this ratio for the close-to-extreme case member of the α -series α = 0 for h + , h × and h + -ih × . The excess of signal toward the north pole compared with the south pole is roughly 25 %.", 'V. DISCUSSION': "We have discussed 'superkick' configurations, i.e., two equal-mass black holes with spins anti-aligned and in the orbital plane, as a simple but extreme 'test case' for phenomena associated with the large recoil velocities produced by spinning black-hole binary systems. The high degree of symmetry results in the gravitational wave signal being dominated by the l = 2 , m = ± 2 spherical harmonics, i.e., the recoil is with good accuracy proportional to the difference of energies radiated into the l = 2 , m = ± 2 modes, see Figure (4). \nThe asymmetry here is rather strong and in the extreme case E 22 /E 2 -2 ≈ 2 . 4. For gravitational wave detection the ratio of the amplitude of the strain h is more interesting, in the direction opposite to the recoil we find an excess of roughly 25 % larger amplitude in the maximum recoil case. \nFIG. 14: The dependence of the expression (13) is plotted as a function of the inclination angle θ for h + , h × and h + -ih × for the near extremal member of the α -series α = 0. For comparison we also show the curve for the case without spin, when h + is symmetric around θ = π/ 2. The excess of signal toward the north pole compared with the south pole is roughly 25 %. \n<!-- image --> \nFor the large kicks one observes in the 'superkick' configuration, one should certainly worry about the accuracy of the wave extraction. For the present paper we have been interested in a qualitative discussion rather than very high accuracy, but we point out that a procedure to improve the accuracy of wave extraction via the Newman-Penrose scalar Ψ 4 at finite radius has been discussed recently in [45]. An overall improvement in the accuracy of spinning black-hole binary simulations should also be possible by employing higher-order spatial finite differencing [46] and using initial parameters based on PN inspiral calculations [47]. \nThe main emphasis of this paper has been the comparison of the dynamics with post-Newtonian predictions. We have found that the 2.5PN-accurate expressions given in [13] accurately describe the spin evolution and linear momentum radiation up to about 60 M before merger. After that time the PN estimate of dP z /dt diverges from the numerical values. It is also after that time that we find the main contribution to the final kick of the merged black hole, and this explains why it is difficult to make accurate predictions of the kick by integrating the PN equations up to a cutoff separation. In order to accurately analytically model the recoil for superkick configurations (and possibly spinning black-hole binary configurations in general) we suggest that a more sophisticated postNewtonian treatment would be necessary, or a matching of PN methods during the early inspiral with a close-limit analysis of the merger and ringdown. It has recently been found [22] that a phenomenological formula for the final \nkick, based on the angular dependence of the terms in Eq. (B3), matches numerical data reasonably well. Having found that the precise form of (B3) fails to predict the linear momentum radiation in the regime when the majority of the linear momentum is radiated, it will be interesting to see how well such a phenomenological formula works for more general configurations, or if a more detailed analytic study will suggest a more generally applicable formula.", 'Acknowledgments': "We are grateful to P. Ajith, B. Krishnan and A. Sintes for discussions regarding material in Sec. IV, and to G. Schafer for sharing insights on the post-Newtonian approach. This work was supported in part 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). Computations were performed at LRZ Munich and the Doppler and Kepler clusters at the Institute of Theoretical Physics of the University of Jena.", 'APPENDIX A: INITIAL-DATA PARAMETERS': "In the Bowen-York/puncture data that we use, we must choose parameters for the masses, separations, spins and linear momenta of the two punctures. Most of the simulations studied in this paper are based on the MI configuration in [10], for which the momenta were chosen without any attempt to have the punctures move in quasi-circular orbit (although the eccentricity in the resulting evolutions appears to be small). Often, however, one wishes to produce quasicircular orbits, and the momenta for our non-MI-based simulations are chosen to meet that requirement. In this Appendix we describe our procedure for calculating those parameters. Note that in [47] we have described a procedure to further improve the 'circularity' of the initial data by using parameters obtained from post-Newtonian inspirals. \nIn [24] we showed that a 3PN-accurate formula is sufficient to calculate initial momenta for nonspinning binaries. In the spinning case we make use of the results of Kidder [13]. They are in harmonic coordinates, but as we will see in Appendix D, the difference between harmonic coordinates and the ADMTT gauge that we expect our evolutions to be in are small. Kidder's Eq. (4.7) gives the orbital angular momentum of a binary in circular orbit as \nL = µ ( Mr ) 1 / 2 ˆ L N 1 + 2 ( M r ) -1 4 ∑ i =1 , 2 [ χ i ( ˆ L N · ˆ S i ) ( 8 M 2 i M 2 +7 η )]( M r ) 3 / 2 + [ 1 2 (5 -9 η ) -3 4 ηχ 1 χ 2 [ ( ˆ S 1 · ˆ S 2 ) -3( ˆ L N · ˆ S 1 )( ˆ L N · ˆ S 2 ) ] ]( M r ) 2 } -1 4 µ ( Mr ) 1 / 2 ∑ i =1 , 2 [ χ i ˆ S i ( 4 M 2 i M 2 + η )]( M r ) 3 / 2 , (A1) \nand the total angular momentum is J = L + S . The variables in Eq. (A1) are as follows. The black holes are separated by a distance r and have masses M i , the total mass is M = M 1 + M 2 , and the mass ratio quantities are µ = M 1 M 2 /M and η = µ/M . The black holes have spins S i , and χ i = | S i | /M 2 i . The quantity ˆ L N denotes the unit vector in the direction of the angular momentum of a Newtonian system of nonspinning particles. It need not point in the same direction as the full orbital angular momentum L of the system, and we may exploit this freedom to uniquely find a momentum P that satisfies \nL = µ ( r × P ) . (A2) \nThe specific setup of our data is as follows. The punctures are placed on the y axis and given momenta in the x direction. The orbital angular momentum therefore has only one component and that points in the z direction. In cases where the last term in Eq. (A1) has a component in the x or y directions, we tilt ˆ L N such that the last term is canceled out and L = L ˆ z . Such a case will not arise in the situations considered in this paper; the last term in Eq. (A1) will always sum to zero and we can simply write p x = ∓ L/r .", 'APPENDIX B: POST-NEWTONIAN TREATMENT OF SPINNING BINARIES': "Consider two particles with masses M 1 and M 2 , spins S 1 and S 2 , located at positions x 1 and x 2 . We define x = x 1 -x 2 and v = d x /dt . Expressions for the evolution of the spins are given up to 2.5 post-Newtonian order in harmonic coordinates by Kidder [13], \n˙ S 1 = 1 r 3 { ( L N × S 1 ) ( 2 + 3 2 M 2 M 1 ) -S 2 × S 1 +3(ˆ n · S 2 ) n × S 1 } , ˙ S 2 = 1 r 3 { ( L N × S 2 ) ( 2 + 3 2 M 1 M 2 ) -S 1 × S 2 +3(ˆ n · S 1 ) n × S 2 } , (B1) \nwhere the Newtonian orbital angular momentum is given by L N = µ ( x × v ). Note that a particular spin supplementary condition has been chosen. \nThe radiated linear momentum is in turn given by Newtonian and spin-orbit contributions, \n˙ P N = -8 105 δm M η 2 ( M r ) 4 { ˙ r ˆ n [ 55 v 2 -45˙ r 2 +12 M r ] + v [ 38˙ r 2 -50 v 2 -8 M r ]} , (B2) \n˙ P SO = -8 15 µ 2 M r 5 { 4˙ r ( v × ∆ ) -2 v 2 (ˆ n × ∆ ) -(ˆ n × v )[3 ˙ r (ˆ n · ∆ ) + 2( v · ∆ )] } , (B3) \nwhere ∆ = M ( S 2 /M 2 -S 1 /M 1 ) and δm = M 1 -M 2 . Clearly the Newtonian contribution, which was used by Fitchett [48] to provide an early estimate of the recoil from the merger of nonspinning binaries, is zero in the equal-mass case. \nTo evaluate Eqs. (B1) and (B3) one needs the particles' positions and velocities as a function of time, i.e., one needs to solve the post-Newtonian equations of motion. Alternatively, we can use the motions of the punctures calculated in our numerical simulations. This allows us to compare the precession of the spins and the radiated linear momentum with that predicted by post-Newtonian theory for the same motion. The results of this comparison are discussed in Section III E. \nIt is instructive to reduce Eqs. (B1)-(B3) to the special case of equal mass and π -symmetry. Equal masses M 1 = M 2 imply δm = 0 and hence ˙ P N = 0. Furthermore, µ = M/ 4, η = 1 / 4, and ∆ = 2( S 2 -S 1 ). \nFor π -symmetry (which implies equal masses), the positions of the punctures x i = ( x i , y i , z i ) satisfy \n( x 2 , y 2 , z 2 ) = ( -x 1 , -y 1 , + z 1 ) (B4) \nfor all times . This implies for the relative position and velocity variables \nx = (2 x 1 , 2 y 1 , 0) , v = (2 v 1 x , 2 v 1 y , 0) . (B5) \nIn words, for π -symmetry the punctures can move in unison in the z -direction, and in general z i ( t ) will describe an accelerated motion. However, the PN equations (B1)(B3) are expressed in terms of x and v , which describe the motion in coordinates in which the center of mass is at rest. Both x and v are orthogonal to the z -axis and lie \nin the z = 0 plane. In particular, v does not have a component in the z -direction, so for general z i ( t ) the center of mass frame is an accelerated frame. A related statement is that in π -symmetry the orbital plane remains orthogonal to the z -axis, there is no orbital plane precession, and L N = µ ( x × v ) is parallel to the z -axis at all times. \nFor π -symmetry the spins can be written as \nS 1 = ζ + σ, S 2 = ζ -σ, (B6) \nwhere ζ and σ are the components of the spin parallel and orthogonal to the z -axis, respectively. Hence the sum of the spins S = S 1 + S 2 points in the z -direction and the weighted spin difference ∆ = 2( S 2 -S 1 ) is orthogonal to the z -axis, \nS = 2 ζ, ∆ = -4 σ. (B7) \nFor the initial data we choose ζ = 0, but in general ζ is not constant in time. The time derivative of the spins can be written as \n˙ S i = Ω j × S i , Ω j = 1 r 3 ( 7 2 L N -S j +3(ˆ n · S j )ˆ n ) , (B8) \nwhere one part is precession about the z -axis due to the orbit-spin term, L N × S i , but the axis of precession is in general not parallel to the z -axis due to the spin-spin terms. For π -symmetry we obtain \n˙ S = -6 r 3 (ˆ n · σ )(ˆ n × σ ) = -3 r 3 | σ | 2 sin(2 α ) ˆ z , (B9) \nwhere ˆ z is the unit vector in the z -direction and α is the angle between ˆ n and σ . The z -component of the spins, ζ = S / 2, oscillates with sin(2 α ( t )). Since for negligible precession α ( t ) is equal to the orbital phase plus a phase shift, we expect two oscillations per orbit, which roughly agrees with observation, see Secs. III C and III E. For the weighted difference of the spins \n˙ ∆ = -4 ˙ σ = -7 M 2 r 2 (ˆ n × v ) × σ -4 r 3 (2 σ +3 | σ | cos( α ) ˆ n ) × ζ, (B10) \nwhich contains precession due to L N at order 1 /r 2 and the spin-spin term at order 1 /r 3 . The term ζ × σ describes a modulation of the precession about the z -axis since ζ oscillates around zero. \nFor the radiated linear momentum we note that for π -symmetry the three vectors ˆ n , v , and ∆ in (B3) are orthogonal to the z -axis, and hence ˙ P SO is parallel to the z -axis as it should be. The angle between ˆ n and v varies slowly over the entire inspiral from about π/ 2 for circular orbits to a value less than π for the plunge. The angle between the orbital vectors and ∆ oscillates with the orbital and precession time scales. Making the approximation that ˆ n · v ≈ 0 and ˆ n × v ≈ v ˆ z , we obtain \n˙ P SO ≈ -2 15 M 3 r 5 ( 7˙ rv (ˆ n · σ ) + 4 v 2 ( v v · σ ) ) ˆ z . (B11) \nEven for quasi-circular orbits where in addition we set ˙ r = 0 there will be radiation of linear momentum in the z -direction, which however averages to zero over time. \nNote that in general there are two contributions, one proportional to ˙ rv and the other to v 2 , and they are offset in phase depending on the angles between the spin and the orbital vectors. As the system approaches the plunge phase, the ˙ rv term should become as important as the v 2 effects. Note that we have not discussed the ˙ P SS spin-spin contribution at next PN order, which could also be examined for potentially large contributions near the plunge, but in numerical simulations of head-on collisions the resulting kicks have been found to be small [49]. \nThe PN expressions (B3) and the above discussion apply in the regime where the post-Newtonian approximations are valid. We see in Section III E that these expressions describe the radiation of linear momentum with reasonable accuracy up to about 50 M before merger.", 'APPENDIX C: PN CALCULATION OF PUNCTURE MOTION': "In moving-puncture simulations we can readily track the motion of the punctures and record their positions x ( t ) and velocities v ( t ) = -β ( t ). We may then be tempted to make a Newtonian analogy and guess that the puncture's momentum is P = M 1 v for a black hole with mass M 1 . However, when we compare this to the momentum specified in the initial data, the two values differ significantly. For example, evolve two equal-mass punctures with initial separation D = 8 M , P = 0 . 14, M 1 = M 2 = 0 . 5. From the numerical puncture motion we find M 1 v ≈ 0 . 075; this value disagrees with P by almost a factor of two. \nDuring a simulation the 'punctures' are at an infinite proper distance from their black holes' horizons [50, 51], and we may worry that correctly physically interpreting the punctures' motions requires a thorough investigation of the gauge and geometry of the punctures as they evolve. In fact, the punctures' motions can be understood from a simple post-Newtonian analysis. \nUp to 2PN order, the Hamiltonian for two point particles in the ADMTT gauge and center-of-mass frame has been derived in [52, 53]. \nFrom the Hamiltonian equations of motion, \n˙ x i = ∂H ∂P i , (C1) \nwhere x i is the separation vector between the two particles. At Newtonian order we recover ˙ x i = P i / (2 µ ). Up to 2PN order we have for circular orbits \n˙ x i = P i 2 µ { 1 -1 c 2 ( P 2 (1 -3 η ) 2 µ 2 + M (3 + η ) R ) + 1 c 4 ( 3 P 4 (1 -5 η +5 η 2 ) 8 µ 4 -MP 2 ( -5 + 20 η +3 η 2 ) 2 µ 2 R \n(C2) must be determined by the procedure described in Appendix C. \n+ M 2 (5 + 8 η ) R 2 )} (C2) \nThe more general expressions (removing the assumption of circular orbits), and the 3PN terms, will be omitted here for brevity. They can be readily calculated from the Hamiltonian in [53]. \nAs an example, consider the D = 8 M quasi-circular orbit parameters from the sequence presented in [54], for which P = 0 . 111 M and the orbital frequency is M Ω = 0 . 0376. From the orbital frequency we can calculate that the punctures will move at a speed ˙ x = 0 . 150, which is approximately equal to the observed value in a simulation. From the momentum, the Newtonian prediction of the speed is ˙ x N = 0 . 223, which is far too high. The 1PN prediction is ˙ x 1 PN = 0 . 126 (now the value is too small), and the 2PN and 3PN predictions are ˙ x 2 PN = 0 . 151 and ˙ x 3 PN = 0 . 149. The 2PN and 3PN predictions are both very close to the observed value. \nIn addition to providing a pleasing consistency between the dynamics observed in moving-puncture simulations and that predicted by post-Newtonian theory, this analysis is necessary when converting between ADMTT and harmonic gauges in Appendix D, where we will need to invert equations like (D1) to estimate the black holes' momenta as a function of time from the puncture motion.", 'APPENDIX D: ADMTT TO HARMONIC TRANSFORMATION': "Our initial data are, up to 2PN order, in the ADMTT gauge [35]. The post-Newtonian expressions for spin evolution and linear momentum radiation listed in Appendix B are in the harmonic gauge. Although it is not clear how closely our evolved data adhere to the ADMTT gauge, it is nonetheless useful to assume that they remain in the ADMTT gauge and transform the results to the harmonic gauge and see how different they are. \nA transformation between ADMTT and harmonic coordinates is provided up to 2PN order by Damour and Schafer [52]. If x i are the ADMTT coordinates of the i -th particle and X i are the corresponding harmonic coordinates, then the transformation for a binary system is \nX i = x i + M i { n ( 5 8 v 2 j -1 8 ( n · v j ) 2 + 7 M i 4 R + M j 4 R ) + ( 1 2 v i -7 4 v j ) ( n · v j ) } . (D1) \nThe velocities v { a,b } in Eq. (D1) are not the coordinate speeds, but are instead v i = p i /m i , and the momenta p i \nWhen the particles are far apart and moving slowly, the coordinates x i and X i will not differ much. In Figure 15, which shows results from the D6 simulation, we show the separation between the punctures in the numerical coordinates as a function of time, and in the coordinates after the transformation (D1). The coordinates differ by less than 10 % up until about 15 M before merger. After that time we do not expect the coordinate transformation (which is accurate up to only 2PN order) to be reliable. However, for most of the evolution we see that the differences between the two coordinate choices are not dramatic. A comparison of numerical and PN calculations of dP z /dt (as described in Section III E) in Figure 16 also shows that the results are similar at early times, before the PN result diverges. Note that when the puncture motion in ADMTT coordinates is used in the (harmonic) PN formula (B3), the curve is closer to the numerical result than when we use the puncture motion in harmonic coordinates. However, since this agreement occurs just before the time when the PN and numerical values seriously diverge, we do not take this agreement too seriously. \nThe main conclusion of this analysis is that the difference in results between using ADMTT and harmonic dynamical quantities in PN expressions is less than the uncertainty inherent in the PN expressions themselves. We therefore continue to use the raw numerical data, in ADMTT coordinates, for most of the analysis presented in this paper. \nFIG. 15: Coordinate separation as a function of time for the D6 simulation, comparing the numerical data (presumed to be in ADMTT coordinates), and the same data transformed to harmonic coordinates. The difference is less than 10% up until about 15 M before merger. 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2009CQGra..26j5015C

Warped AdS<SUB>3</SUB> black holes in new massive gravity

2009-01-01

['-', '-']

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We investigate stationary, rotationally symmetric solutions of a recently proposed three-dimensional theory of massive gravity. Along with BTZ black holes, we also obtain warped AdS<SUB>3</SUB> black holes, and (for a critical value of the cosmological constant) AdS<SUB>2</SUB> × S<SUP>1</SUP> as solutions. The entropy, mass and angular momentum of these black holes are computed.

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https://arxiv.org/pdf/0902.4634.pdf

{'Warped AdS 3 black holes in new massive gravity': "G'erard Cl'ement ∗ \nLaboratoire de Physique Th'eorique LAPTH (CNRS), B.P.110, F-74941 Annecy-le-Vieux cedex, France \n24 March 2009", 'Abstract': 'We investigate stationary, rotationally symmetric solutions of a recently proposed three-dimensional theory of massive gravity. Along with BTZ black holes, we also obtain warped AdS 3 black holes, and (for a critical value of the cosmological constant) AdS 2 × S 1 as solutions. The entropy, mass and angular momentum of these black holes are computed.', '1 Introduction': "In a recent paper [1], a new theory of massive gravity in three dimensions has been proposed. In this theory, as in the case of the older topologically massive gravity (TMG) [2], the linearized excitations about the Minkowski (or de Sitter or anti-de Sitter in the case of a non-vanishing cosmological constant) vacuum describe a propagating massive graviton. To the difference of TMG, which achieves this goal through the addition to the Einstein-Hilbert action of a parity-violating Chern-Simons term, new massive gravity (NMG) is a parity-preserving, higher-derivative extension of three-dimensional general relativity. The possibility of generalizing such an extension to higher dimensions has been explored in [3], with the finding that only the threedimensional model is unitary in the tree level. \nCosmological TMG admits two very different kinds of black hole solutions, BTZ black holes [4], which are discrete quotients of AdS 3 , and warped AdS 3 black holes [5, 6, 7, 8], which are discrete quotients of warped AdS 3 . Similarly to BTZ black holes, these have a four-parameter local isometry algebra, which generically is sl (2 , R ) × R , and may be generated from the corresponding vacua by local coordinate transformations [7, 8]. The ADM lapse function of these warped black holes goes to a constant value at spacelike infinity (with the ADM shift function going to zero) [5], which makes them closer in this respect to four-dimensional black holes than the BTZ black holes. But the warped AdS 3 black holes have the very special property of being intrinsically non-static, their ergosphere extending to infinity [5]. \nIt is straightforward to show that cosmological NMG also admits the BTZ black holes as solutions [1]. Because TMG and NMG have much in common, and indeed may be unified in a 'general massive gravity' model [1], the existence of warped AdS 3 black hole solutions to cosmological NMG may be conjectured. The purpose of the present paper is to contruct these black hole solutions, and to compute their mass, angular momentum, and entropy. \nIn the next section we investigate NMG with two Killing vectors, write down in compact form the dimensionally reduced field equations, and exhibit four constants of the motion. A simple ansatz reduces in the third section these fourth order derivative equations to a system of algebraic equations with two generic solutions, leading either to BTZ black holes or warped AdS 3 black holes. In the fourth section, we compute the mass, angular momentum and entropy of these black holes, which satisfy the first law of black hole thermodynamics, as well as a Smarr-like relation. Our results are \nsummarized in the Conclusion. Solutions with a horizon and without naked closed timelike curves, but which may be transformed to the vacuum by a global coordinate transformation and thus are not genuine black holes, are discussed in the Appendix.", '2 New massive gravity with two Killing vectors': "The action of the cosmological new massive gravity theory is 1 [1] \nI 3 = 1 16 πG ∫ d 3 x √ | g | [ R1 m 2 K -2Λ ] , (2.1) \nwhere R is the trace of the Ricci tensor R µν , the quadratic curvature invariant K may be defined in terms of the Schouten tensor S µν = R µν -(1 / 4) g µν R as \nK = /epsilon1 λµν /epsilon1 ρστ g λρ S µσ S ντ = R µν R µν -3 8 R 2 (2.2) \nand, for sake of generality, we will consider both signs of the real squared mass parameter m 2 . \nWe search for stationary circularly symmetric solutions of this theory using the dimensional reduction procedure of [9]. A three-dimensional metric with two commuting Killing vectors ∂ t , ∂ ϕ may be parametrized as \nds 2 = λ ab ( ρ ) dx a dx b + ζ -2 ( ρ ) R -2 ( ρ ) dρ 2 (2.3) \n( x 0 = t , x 1 = ϕ ), where R 2 = -det λ , and the scale factor ζ ( ρ ) allows for arbitrary reparametrizations of the radial coordinate ρ . The local isomorphism between the SL (2 , R ) group of transformations in the 2-Killing vector space and the Lorentz group SO (2 , 1) suggests the parametrization of the 2 × 2 matrix λ \nλ = ( T + X Y Y T -X ) , (2.4) \nsuch that R 2 ≡ X 2 is the Minkowski pseudo-norm of the 'vector' X ( ρ ) = ( T, X, Y ), \nX 2 = η ij X i X j = -T 2 + X 2 + Y 2 , (2.5) \nWe shall use the notations X · Y = η ij X i X j for the scalar product of two vectors, and \n( X ∧ Y ) i = η ij /epsilon1 jkl X k Y l (2.6) \n(with /epsilon1 012 = +1) for their wedge product. \nThe Ricci tensor components are [6] \nR a b = -ζ 2 ( ( ζRR ' ) ' 1 +( ζ/lscript ) ' ) a b , R 2 2 = -ζ ( ζRR ' ) ' + 1 2 ζ 2 ( X ' 2 ) , (2.7) \nwhere ' denotes the derivative d/dρ , and /lscript is the matrix \n/lscript = ( -L Y -L T + L X L T + L X L Y ) (2.8) \nassociated with the vector \nL ≡ X ∧ X ' . (2.9)", 'It follows that': "R = ζ 2 [ -2( RR ' ) ' + 1 2 ( X ' 2 ) ] -2 ζζ ' RR ' , K = ζ 4 [ 1 2 ( L ' 2 ) -1 4 ( RR ' ) ' ( X ' 2 ) + 5 32 ( X ' 2 ) 2 ] (2.10) + ζ 3 ζ ' [ ( L · L ' ) -1 4 RR ' ( X ' 2 ) ] + ζ 2 ζ ' 2 1 2 ( L 2 ) , \nreducing the action (2.1) to the one-dimensional form \nI 1 = ∫ dρ [ Aζ 3 + Bζ 2 ζ ' + Cζζ ' 2 + Dζ + Eζ ' +2 ζ -1 Λ ] (2.11) = ∫ dρ [( A -1 3 B ' ) ζ 3 + Cζζ ' 2 + ( D -E ' ) ζ +2 ζ -1 Λ ] (2.12) \n(up to a surface term), where \nA = 1 m 2 [ 1 2 ( X · X '' ) 2 -1 2 ( X 2 )( X '' 2 ) -1 4 ( X · X '' )( X ' 2 ) -3 32 ( X ' 2 ) 2 ] , B = 1 m 2 [ ( X · X ' )( X · X '' ) -( X 2 )( X ' · X '' ) -1 4 ( X · X ' )( X ' 2 ) ] (2.13) \n(the expression of C shall not be needed in the following, and those of D and E are obvious from (2.10)). \nRather than reducing the three-dimensional equations of massive gravity [1], it is more convenient to derive the reduced equations directly from the reduced action. Taking advantage of the reparametrization invariance of (2.3), we shall fix the gauge ζ = constant after variation of ζ . The first form \n(2.11) of the reduced action is most convenient for variation relative to X , which leads to the equations \nX ∧ ( X ∧ X '''' ) + 5 2 X ∧ ( X ' ∧ X ''' ) + 3 2 X ' ∧ ( X ∧ X ''' ) + 9 4 X ' ∧ ( X ' ∧ X '' ) -1 2 X '' ∧ ( X ∧ X '' ) (2.14) -[ 1 8 ( X ' 2 ) + m 2 ζ 2 ] X '' = 0 . \nThe wedge product X ∧ (2 . 14) can be first integrated, leading to the constancy of the super-angular momentum vector \nJ = -ζ 2 m 2 { ( X 2 )[ X ∧ X ''' -X ' ∧ X '' ] + 2( X · X ' ) X ∧ X '' + [ 1 8 ( X ' 2 ) -5 2 ( X · X '' ) ] X ∧ X ' } + X ∧ X ' (2.15) \nassociated with the SL (2 , R ) invariance of the reduced action (2.11) [13]. The second form (2.12) of the reduced action is more convenient for variation relative to ζ , leading (in the gauge ζ = constant) to the Hamiltonian constraint \nH ≡ ( X ∧ X ' ) · ( X ∧ X ''' ) -1 2 ( X ∧ X '' ) 2 + 3 2 ( X ∧ X ' ) · ( X ' ∧ X '' ) + 1 32 ( X ' 2 ) 2 + m 2 2 ζ 2 ( X ' 2 ) + 2 m 2 Λ ζ 4 = 0 . (2.16) \nThe combination 2 X · (2 . 14) -3(2 . 16) leads to the simpler scalar equation \n1 2 ( X · X '' ) 2 -1 2 ( X 2 )( X '' 2 ) -1 4 ( X ' 2 )( X · X '' ) -3 32 ( X ' 2 ) 2 -m 2 ζ 2 [ 3 2 ( X ' 2 ) + 2( X · X '' ) ] -6 m 2 Λ ζ 4 = 0 , (2.17) \nwhich is equivalent to the trace of the three-dimensional field equations [1] \nK + m 2 ( R6Λ) = 0 . (2.18)", '3 Regular black holes': 'As in the case of topologically massive gravity [10, 11, 12, 13], the equations (2.14), (2.16) presumably admit a variety of exact solutions. We shall only construct here the solutions generated from the quadratic ansatz [13] \nX = α ρ 2 + β ρ + γ , (3.1) \n(with α , β and γ linearly independent constant vectors), which in the case of TMG leads to warped AdS 3 black holes [8]. Inserting this ansatz into the vector equation (2.14), we find that the fourth and third order components vanish provided the vectors α and β are constrained by \nα 2 = 0 , ( α · β ) = 0 , . (3.2) \nIt is easy to show that the two constraints (3.2) further imply \nα ∧ β = b α , β 2 = b 2 , (3.3) \nfor some real constant b . The lower order components of (2.14) and the scalar equation (2.17) are then satisfied provided \n[ z + 17 8 b 2 -m 2 ] α = 0 , (3.4) \nz 2 + 1 4 b 2 z -3 64 b 4 -m 2 [ 3 4 b 2 -2 z ] -3 m 2 Λ = 0 , (3.5) \nwhere we have put \nand m 2 m 2 /ζ 2 , Λ ≡ Λ /ζ 2 . \n≡ \n( α · γ ) = -z , (3.6) \n≡ Equation (3.4) has two solutions. The first is α = 0 (implying z = 0), so that our ansatz (3.1) reduces to \nX = β ρ + γ , (3.7) \nwhich leads to the BTZ black hole metric \nds 2 = ( -2 l -2 ρ +M / 2) dt 2 -J dt dϕ +(2 ρ +M l 2 / 2) dϕ 2 +[4 l -2 ρ 2 -(M 2 l 2 -J 2 ) / 4] -1 dρ 2 , (3.8) \nfor ζ = 1, b 2 = 4 l -2 [13, 5]. The AdS 3 curvature parameter l -2 is obtained by solving (3.5), \nl -2 = 2 m 2 [ -1 ± √ 1 -Λ /m 2 ] (3.9) \n(compare with Eq. (29) of [1]). For the upper sign, this parameter is positive if Λ < 0 (with the restriction Λ > m 2 for m 2 < 0), and reduces as it should to l -2 = -Λ for m 2 → ∞ . The other branch of the solution (lower sign) exists only in the tachyonic case m 2 < 0 for Λ > m 2 . \nThe second solution of (3.4) is \nz = m 2 -17 8 b 2 . (3.10) \nInserting this in (3.5), we obtain the equation \nm 4 -3 b 2 m 2 + 21 16 b 4 -m 2 Λ = 0 , (3.11) \nwhich (assuming Λ /m 2 = 1) is solved by \n/negationslash \nm 2 ζ 2 b 2 = 6 ± √ 3(5 + 7Λ /m 2 ) 4(1 -Λ /m 2 ) . (3.12) \nwith b 2 /negationslash = 0. At this point, we can without loss of generality fix the scale ζ and the spatial orientation 2 so that \nb = -1 , (3.13) \nwhich enables us to make contact with the results of [8]. Squaring (3.1), we obtain \nR 2 = (1 -2 z ) ρ 2 +2( β · γ ) ρ + γ 2 = β 2 ( ρ 2 -ρ 2 0 ) , (3.14) \nwhere we have set \nz = (1 -β 2 ) / 2 , (3.15) \ntranslated the radial coordinate so that ( β · γ ) = 0, and defined γ 2 ≡ -β 2 ρ 2 0 . As shown in the Appendix, we can choose a rotating frame and a length-time scale such that \nα = (1 / 2 , -1 / 2 , 0) , β = ( ω, -ω, -1) , γ = ( z + u, z -u, -2 ωz ) ( u = β 2 ρ 2 0 / 4 z + ω 2 z ) (3.16) \n(other possible choices either can be reduced to this by a global coordinate transformation, or lead to non-black hole solutions). This choice leads to the metric \nds 2 = -β 2 ρ 2 -ρ 2 0 r 2 dt 2 + r 2 [ dϕ -ρ +(1 -β 2 ) ω r 2 dt ] 2 + 1 β 2 ζ 2 dρ 2 ρ 2 -ρ 2 0 , (3.17) \nwhere \nr 2 = ρ 2 +2 ωρ + ω 2 (1 -β 2 ) + β 2 ρ 2 0 1 -β 2 , (3.18) \nand the constants β 2 and ζ are given by \nβ 2 = 9 -21Λ /m 2 ∓ 2 √ 3(5 + 7Λ /m 2 ) 4(1 -Λ /m 2 ) , ζ -2 = 21 -4 β 2 8 m 2 . (3.19) \nThe solutions in the special cases β 2 = 1 and β 2 = 0 are given in [8] (where µ E should be replaced by ζ ). \nThe metric (3.17) represents a black hole of the warped AdS 3 type, with two horizons at ρ = ± ρ 0 , if β 2 > 0 and ρ 2 0 ≥ 0. As discussed in [8], naked closed timelike curves (CTC) do not occur provided β 2 < 1 and ω > -ρ 0 / √ 1 -β 2 . Including the limiting cases β 2 = 1 and β 2 = 0 (which are also causally regular in a certain parameter range), the necessary condition for the existence of causally regular warped AdS 3 black holes is therefore \n0 ≤ β 2 ≤ 1 (3.20) \n(implying, from the second equation (3.19), m 2 > 0). This corresponds to the upper sign in (3.19) and \n-35 m 2 289 ≤ Λ ≤ m 2 21 . (3.21) \nSpecial values are Λ = -35 m 2 / 289, corresponding to β 2 = 1, Λ = 0, corresponding to β 2 = (9 2 √ 15) / 4, and Λ = m 2 / 21, corresponding to β 2 = 0. \nThe case Λ = m 2 , for which Eq. (3.12) becomes undeterminate, deserves a special investigation. This is carried out in the Appendix, with the result that (in the framework of our quadratic ansatz) the only solution without naked CTC is AdS 2 × S 1 . \n-', '4 Entropy, mass and angular momentum': "In this section, we shall compute the various thermodynamical observables for the black holes of the previous section, and check that they obey the first law. We start with the computation of the entropy, which is a straightforward application of Wald's general formula [14, 15, 16] \nS = 2 π ∮ h dx √ γ δ L δ R µνρσ ε µν ε ρσ , (4.1) \nwhere h is the spatial section of the event horizon, γ is the determinant of the induced metric on h , L is the Lagrangian in (2.1), and ε µν is the binormal \nto h . For a stationary circularly metric of the form (2.3), this gives \nS = 4 πA h ( δ L δ R 0202 ( g 00 g 22 ) -1 ) h = A h 4 G ( 1 -1 m 2 [ ( g 00 ) -1 R 00 + g 22 R 22 -3 4 R ] h ) , (4.2) \nwhere A h = 2 πr h is the horizon 'area'. Computing the Ricci tensor components from (2.7), we obtain, on the horizon R 2 = 0, \nS = A h 4 G ( 1 + 1 2 m 2 [ ( X · X '' ) -1 4 ( X ' 2 ) ]) . (4.3) \nFor the BTZ black hole (3.7), this leads to \nS = A h 4 G ( 1 -1 2 m 2 l 2 ) , (4.4) \nwhich could also be obtained directly from the results of [17, 18, 19]. For the warped AdS 3 black holes (3.17), the result can be simplified with the aid of (3.10) to \nS = A h 2 Gm 2 . (4.5) \nIt is remarkable that, although their curvature is not constant 3 as in the BTZ case, their entropy is simply the Bekenstein-Hawking entropy renormalized by a factor 2 /m 2 = 16 / (21 -4 β 2 ), independently of the black hole parameters ρ 0 and ω . \nThe computation of the mass and angular momentum of these black holes is straightforward in the BTZ case, using e.g. the Abbott-Deser-Tekin (ADT) approach to the computation of the energy of asymptotically AdS solutions to higher curvature gravity theories [20, 21]. In the case of the warped AdS 3 black holes, an extension of the ADT approach to the case of massive gravity with non-constant curvature backgrounds, similar to that carried out in [6] for topologically massive gravity, is required. In the present work, we shall make the educated guess that, as in the case of generic threedimensional Einstein-scalar field theories [22] and TMG [6], the mass and angular momentum can be computed from \nM = -ζ 8 G ( δJ Y +∆) , J = ζ 8 G ( δJ T -δJ X ) , (4.6) \nwhere δ J is the difference between the values of the super-angular momentum (2.15) for the black hole and for the background. The term ∆, which depends on the specific theory considered, is not known in the case of new massive gravity. However we can use (4.6) to compute the angular momentum J (which does not require the knowledge of ∆), and derive the mass M by integrating the first law of black hole thermodynamics \ndM = T H dS +Ω h dJ , (4.7) \nwhere the Hawking temperature and the horizon angular velocity, computed from the metric in ADM form, are \nT H = 1 4 π ζr h ( N 2 ) ' | h , Ω h = -N ϕ | h . (4.8) \nIn the case of the BTZ black hole (3.7), we obtain \nJ = ( 1 -1 2 m 2 l 2 ) L , (4.9) \nleading (if we assume that, as in the case of TMG, ∆ = 0 for the BTZ black hole solution of new massive gravity) to \nM = ( 1 -1 2 m 2 l 2 ) M 8 G , J = ( 1 -1 2 m 2 l 2 ) J 8 G , (4.10) \nThe mass, angular momentum and entropy are renormalized by the same factor (1 -1 / 2 m 2 l 2 ) [19] so that, contrary to the case of TMG where the modification of the mass, angular momentum and entropy from the case of cosmological gravity is non trivial [5, 23]), the first law (4.7) is trivially satisfied. The integral Smarr-like relation \nM = 1 2 T H S +Ω h J (4.11) \nsatisfied by the usual BTZ black holes [24] and black hole solutions to generic three-dimensional Einstein-scalar field theories [22], as well as by BTZ black holes in TMG [25], is also satisfied by BTZ black holes in new massive gravity. \nIn the case of the warped AdS 3 black hole (3.17), we find for the superangular momentum J \nJ = -2 β 2 m 2 [ β ∧ γ + ρ 2 0 α ] , (4.12) \nleading (if the spacetime (3.17) with ρ 0 = ω = 0 is chosen as background) to \nJ = ζβ 2 4 Gm 2 [ ω 2 (1 -β 2 ) -ρ 2 0 1 -β 2 ] . (4.13) \nInserting this into the first law (4.7), together with the entropy (4.5) and the temperature and horizon angular velocity \nT H = ζβ 2 ρ 0 A h , Ω h = 2 π √ 1 -β 2 A h ( A h = 2 π √ 1 -β 2 [ ρ 0 + ω (1 -β 2 )] ) , (4.14) \nM = ζβ 2 (1 -β 2 ) 2 Gm 2 ω, (4.15) \nwe obtain \nwhich, as in the case of TMG [6], is twice the 'naive' super-angular momentum value ((4.6) with ∆ = 0). The fact that the integrability conditions for (4.7) are satisfied is a non-trivial check of our formula (4.6) for the angular momentum. These values of the mass, angular momentum and entropy satisfy the modified Smarr-like formula [8] \nM = T H S +2Ω h J (4.16) \nappropriate for warped AdS 3 black holes. Let us also note that the values (4.15), (4.13) and (4.5) for M , J and S coincide with the corresponding values for the warped black holes of gravitating Chern-Simons electrodynamics (Eq. (5.15) of [8] with λ ≡ µ E / 2 µ G = 0) renormalized by a factor 2 /m 2 .", '5 Conclusion': "In this paper, we have shown that, besides the BTZ black holes, the cosmological new massive gravity theory of [1] also admits warped AdS 3 black hole solutions, causally regular in the range (3.20), and computed their entropy, mass and angular momentum. An interesting side result of our analysis, discussed at the end of the Appendix, is that for the critical value Λ = m 2 < 0 for which the two branches of the BTZ black hole solution coincide, NMG also admits AdS 2 S 1 as a solution. \nThe fact that our values for the warped black hole mass and angular momentum satisfy non-trivially both the first law of black hole thermodynamics and the modified Smarr relation (4.11) indicates that these are very likely to be correct. This should be confirmed by a computation from first principles, via e.g. an extension of the ADT approach to the case of new \n× \nmassive gravity with non-constant curvature backgrounds, along the lines followed in [6]. \nAnother, related, problem which we feel should be addressed is that of the sign of the energy. The massive excitations of TMG [2] or of NMG [1] linearized about the Minkowski vacuum have negative energy, unless the gravitational coupling constant G is chosen to be negative. In cosmological TMGlinearized around a constant curvature background, either the massive gravitons or the BTZ black holes have negative energy, except for a critical, 'chiral' value of the Chern-Simons coupling constant [26]. Similarly, irrespective of the sign of the gravitational coupling constant, the sign of the energy of massive excitations of NMG linearized around an AdS 3 background has been found to be opposite to the sign of the mass (4.10) and entropy (4.4) of the BTZ black holes, except for the critical value m 2 l 2 = 1 / 2 [27]. However for this value the BTZ black hole mass and entropy and the energy of the massive modes all vanish, and the theory seems to be trivial. On the other hand, the entropy (4.5) of the warped AdS 3 blackholes of NMG is positive definite for G > 0, and their mass (4.15) is non-negative if ω ≥ 0. A difficult task which should be addressed is that of the linearization of the theory around an appropriate, non-constant curvature background, e.g. warped AdS 3 or the warped vacuum ρ 0 = ω = 0, and the determination of the sign of the energy of the corresponding massive excitations.", 'Appendix: Non-black hole solutions': "A generic null vector α can be parametrized by \nα = ( c, c cos α, c sin α ) , (A.1) \nwith c real and 0 ≤ α < 2 π . From (2.3) and (3.1), \ng ϕϕ ∼ c (1 -cos α ) ρ 2 ( ρ →∞ ) . (A.2) \nThis is non-negative (absence of CTC at infinity) provided either 1) α /negationslash = 0 and c > 0, or 2) α = 0. \nIn the first case ( α /negationslash = 0), transition to a rotating frame dϕ → dϕ ' = dϕ -Ω dt preserves (A.2), but transforms α Y to 0 for Ω = cot( α/ 2). The transformed null vector α ' is of the form (A.1), where α ' = π and c ' can be set to the value 1 / 2 by a combined length and time rescaling [8], leading to the parametrization (3.16). \nIn the second case ( α = 0), the sign of c remains arbitrary, while its absolute value can again be set to 1 / 2 by a similar combined rescaling. The \ngeneric vector β satisfying (3.3) with b = -1 is of the form β = ( ω, ω, 1), where ω can be transformed to 0 by transition to a co-rotating frame (which preserves α and β Y = 1) with angular velocity Ω = ω . In this frame the vectors α , β and γ are \nα = /epsilon1 (1 / 2 , 1 / 2 , 0) , β = (0 , 0 , 1) , γ = /epsilon1 ( u + z, u -z, 0) ( u = β 2 ρ 2 0 / 4 z ) , (A.3) \nwith /epsilon1 = sign( z ). This leads to the metric \nds 2 = -β 2 | 1 -β 2 | ( ρ 2 -ρ 2 0 ) dt 2 + | 1 -β 2 | [ dϕ + ρ | 1 -β 2 | dt ] 2 + 1 β 2 ζ 2 dρ 2 ρ 2 -ρ 2 0 , (A.4) \nfor all positive β 2 /negationslash = 1. This metric, similar to the rotating Bertotti-Robinson metric of [28], can be obtained from the first form of the warped AdS 3 black hole metric given in Eq. (3.16) of [8] (with c = /epsilon1 and ω = 0) by exchanging the time and angular coordinates, t ↔-ϕ . It follows that the metric (A.4) can be generated from the vacuum ((A.4) with ρ 0 = 0) by a global coordinate transformation (Eq. (6.12) of [8] with t ↔-ϕ ), and so is not a black hole. Indeed, for ρ 2 0 = -γ 2 < 0, Eq. (A.4) can be transformed to Eq. (3.3) of [7] by putting ρ = γ sinh σ and rescaling the t and ϕ coordinates, showing that this spacetime is spacelike warped AdS 3 . If, disregarding this fact, we compute the entropy (from (4.5)), and the mass and angular momentum (from (4.6), where ∆ = δJ Y is assumed), we obtain \nS = -16 π 2 √ | 1 -β 2 | κ 2 m 2 , M = 0 , J = -4 πζβ 2 | 1 -β 2 | κ 2 m 2 , (A.5) \nwhich, being independent of the (spurious) parameter ρ 0 , trivially satisfy the first law (4.7) and, together with the the temperature and horizon angular velocity \nT H = ζβ 2 ρ 0 2 π | 1 -β 2 | , Ω h == -ρ 0 | 1 -β 2 | , (A.6) \nalso satisfy the modified Smarr-like relation (4.16). \n√ \nIn the case β 2 = 0, the solution (A.4) is replaced by Eq. (3.22) of [8] with c = 1, ω = 0 and t ↔-ϕ , \nds 2 = -2 νρdt 2 + [ dϕ +( ρ + ν ) dt ] 2 + dρ 2 2 ζ 2 νρ (A.7) \n( ν > 0). This can similarly be generated from the corresponding vacuum metric by a global coordinate transformation (Eq. (6.16) of [8] with t ↔ ϕ ). \nIn the limit Λ /m 2 → 1, Eq. (3.12) still makes sense if either 1) b is fixed and the lower sign is chosen, leading in the gauge b 2 = 1 to a non-causally regular black hole with β 2 = 35 / 8, or 2) b → 0. For Λ /m 2 = 1 and b 2 = 0, Eqs. (3.4) and (3.5) are both solved by \n- \nz = m 2 . (A.8) \nNote that b 2 = 0 in (3.3) implies β ∝ α , so that the ansatz (3.1) can be reduced by a translation of ρ to \nX = α ρ 2 + γ . (A.9) \nThis leads to \nR 2 = -2 m 2 ( ρ 2 -ρ 2 0 ) , (A.10) \nwith ρ 2 0 = ( γ 2 ) / 2 m 2 , so that the metric (3.1) has the correct signature only for m 2 < 0. The constant vectors α and γ can, without loss of generality, be chosen in the form \nα = ( ± 1 / 2 , -1 / 2 , 0) , γ = ( ± ( m 2 -u ) , m 2 + u, v ) (A.11) \n(2 u = ρ 2 0 -v 2 / 2 m 2 ). In the case of the upper sign, g ϕϕ = T -X = ρ 2 -2 u < ρ 2 -ρ 2 0 , so the metric has closed timelike curves outside the horizon ρ = ρ 0 . In the case of the lower sign, g ϕϕ = T -X and g tϕ = Y are both constant, so the parameter v can be set to 0 by a frame rotation, leading to the Bertotti-Robinson-like metric (in the gauge ζ = 1) \nds 2 = -( ρ 2 -ρ 2 0 ) dt 2 -2 m 2 dϕ 2 -dρ 2 2 m 2 ( ρ 2 -ρ 2 0 ) ( m 2 < 0) , (A.12) \nwhich, for all real ρ 2 0 , is a parametrization of AdS 2 × S 1 . In this case, the contribution to the entropy from the second, quadratic term in the right-hand side of (4.3) exactly compensates the Bekenstein-Hawking entropy, leading to a vanishing net entropy. Similarly, the mass and angular momentum computed from (4.12) and (4.6) (with the same assumption as above) also vanish, \nS = 0 , M = 0 , J = 0 , (A.13) \nand both the first law and the modified Smarr formula are trivially satisfied.", 'References': "- [1] E.A. Bergshoeff, O. Hohm and P.K. Townsend, 'Massive gravity in three dimensions', arXiv:0901.1766.\n- [2] S. Deser, R. Jackiw and S. Templeton, Phys. Rev. Lett. 48 (1982) 975; Ann. Phys., NY 140 (1982) 372.\n- [3] M. Nakasone and I. Oda, 'On unitarity of massive gravity in three dimensions', arXiv:0902.3531.\n- [4] M. Ba˜nados, C.Teitelboim and J. Zanelli, Phys. Rev. Lett. 69 (1992) 1849; M. Ba˜nados, M. Henneaux, C. Teitelboim and J. Zanelli, Phys. Rev. D 48 (1993) 1506.\n- [5] K. Ait Moussa, G. Cl'ement and C. Leygnac, Class. Quantum Grav. 20 (2003) L277.\n- [6] A. Bouchareb and G. Cl'ement, Class. Quantum Grav. 24 (2007) 5581.\n- [7] D. Anninos, W. Li, M. Padi, W. Song and A. Strominger, JHEP 0903 (2009) 130.\n- [8] K. Ait Moussa, G. Cl'ement, H. Guennoune and C. Leygnac, Phys. Rev. D 78 (2008) 064065.\n- [9] G. Cl'ement, Phys. Rev. D 49 (1994) 5131.\n- [10] I. Vuorio, Phys. Lett. 163B (1985) 91; R. Percacci, P. Sodano and I. Vuorio, Ann. Phys., NY 176 (1987) 344.\n- [11] C. Aragone, Class. Quantum Grav. 4 (1987) L1; T. Dereli and R.W. Tucker, Class. Quantum Grav. 5 (1988) 951; Y. Nutku and P. Baekler, Ann. Phys., NY 195 (1989) 16; M.E. Ortiz, Ann. Phys., NY 200 (1990) 345; G. Cl'ement, Class. Quantum Grav. 9 (1992) 2615.\n- [12] Y. Nutku, Class. Quantum Grav. 10 (1993) 2657; M. Gurses, Class. Quantum Grav. 11 (1994) 2585.\n- [13] G. Cl'ement, Class. Quantum Grav. 11 (1994) L115.\n- [14] R.M. Wald, Phys. Rev. D 48 (1993) 3427.\n- [15] T. Jacobson, G. Kang and R.C. Myers, Phys. Rev. D 49 (1994) 6587.\n- [16] V. Iyer and R.M. Wald, Phys. Rev. D 50 (1994) 846.\n- [17] H. Saida and J. Soda, Phys. Lett. B 471 (2000) 358.\n- [18] B. Sahoo and A. Sen, JHEP 0607 (2006) 008.\n- [19] M.I. Park, Phys. Lett. B 663 (2008) 259.\n- [20] L.F. Abbott and S. Deser, Nucl. Phys. B 195 (1982) 76. \n- [21] S. Deser and B. Tekin, Phys. Rev. Lett. 89 (2002) 101101; Phys. Rev. D 67 (2003) 084009.\n- [22] G. Cl'ement, Phys. Rev. D 68 (2003) 024032.\n- [23] S.N. Solodukhin, Phys. Rev. D 74 (2006) 024015; Y. Tachikawa, Class. Quantum Grav. 24 (2007) 737.\n- [24] R.G. Cai, Z.J. Lu and Y.Z. Zhang, Phys. Rev. D 55 (1997) 853.\n- [25] M.I. Park, Phys. Rev. D 77 (2008) 026011.\n- [26] W. Li, W. Song and A. Strominger, JHEP 0804 (2008) 082.\n- [27] Y. Liu and Y.-W. Sun, 'Note on new massive gravity in AdS 3 ', arXiv:0903.0536; 'Consistent boundary conditions for new massive gravity in AdS 3 ', arXiv:0903.2933.\n- [28] G. Cl'ement and D. Gal'tsov, Nucl. Phys. B 619 (2001) 741."}

2008ApJ...687L..25S

Three-Dimensional Simulations of Magnetized Thin Accretion Disks around Black Holes: Stress in the Plunging Region

2008-01-01

['accretion', 'accretion disks', 'stars binaries close', 'black hole physics', 'astronomy x rays', 'astrophysics']

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We describe three-dimensional general relativistic magnetohydrodynamic simulations of a geometrically thin accretion disk around a nonspinning black hole. The disk has a thickness h/r ~ 0.05-0.1 over the radial range (2-20) GM/c<SUP>2</SUP>. In steady state, the specific angular momentum profile of the inflowing magnetized gas deviates by less than 2% from that of the standard thin disk model of Novikov and Thorne. Also, the magnetic torque at the radius of the innermost stable circular orbit (ISCO) is only ~2% of the inward flux of angular momentum at this radius. Both results indicate that magnetic coupling across the ISCO is relatively unimportant for geometrically thin disks.

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https://arxiv.org/pdf/0808.2860.pdf

{'THREE-DIMENSIONAL SIMULATIONS OF MAGNETIZED THIN ACCRETION DISKS AROUND BLACK HOLES: STRESS IN THE PLUNGING REGION': 'REBECCA SHAFEE 1 , JONATHAN C. MCKINNEY 2 , RAMESH NARAYAN 3 , ALEXANDER TCHEKHOVSKOY 3 CHARLES F. GAMMIE 4 , JEFFREY E. MCCLINTOCK 3 \nDraft version August 16, 2021', 'ABSTRACT': 'We describe three-dimensional general relativistic magnetohydrodynamic simulations of a geometrically thin accretion disk around a non-spinning black hole. The disk has a thickness h / r ∼ 0 . 05 -0 . 1 over the radial range (2 -20) GM / c 2 . In steady state, the specific angular momentum profile of the inflowing magnetized gas deviates by less than 2% from that of the standard thin disk model of Novikov & Thorne (1973). Also, the magnetic torque at the radius of the innermost stable circular orbit (ISCO) is only ∼ 2% of the inward flux of angular momentum at this radius. Both results indicate that magnetic coupling across the ISCO is relatively unimportant for geometrically thin disks. \nSubject headings: X-ray: stars - binaries: close - accretion, accretion disks - black hole physics', '1. INTRODUCTION': "The recent development of general relativistic magnetohydrodynamic (GRMHD) codes (e.g., Gammie et al. 2003; De Villiers & Hawley 2003) has finally allowed realistic numerical simulations of magnetized accretion disks around black holes (BHs). This has led to a better understanding of the inner regions of these disks (e.g. Krolik et al. 2005) and of their role in launching relativistic jets (e.g. McKinney 2006). One of the interesting new results is the recognition that magnetic fields alter the structure of the accretion flow near and inside the innermost stable circular orbit (ISCO). This may result in large deviations from the traditional picture of a vanishing torque at the ISCO (Krolik 1999; Gammie 1999; Krolik et al. 2005). \nPaczy'nski (2000) and Afshordi & Paczy'nski (2003) suggested that the zero-torque condition is likely to be a good approximation for geometrically thin disks. This was confirmed by Shafee et al. (2008) who, using a global height-integrated model, showed that modifications to the stress profile are negligibly small for disk thicknesses h less than about a tenth of the local radius r . Their work was, however, based on a hydrodynamic model with α -viscosity and did not explicitly include magnetic fields. \nNumerical MHD simulations by Krolik & Hawley (2002) using a pseudo-Newtonian potential, and by Krolik et al. (2005) using a GRMHD code, indicated that magnetic torques are indeed important inside the ISCO. In particular, these authors found no evidence for a 'stress edge,' leading them to argue that simple disk models based on the zero-torque condition (e.g., Shakura & Sunyaev 1973; Novikov & Thorne 1973, hereafter NT73) may be seriously wrong. If true this would undermine recent efforts to estimate of the spin parameters of BHs using the NT73 model (Shafee et al. 2006; McClintock et al. 2006; Davis et al. 2006; Liu et al. 2008). \nThe MHD simulations carried out so far have consid- \nered non-radiating accretion flows that are geometrically rather thick. The one exception is the recent work of Reynolds & Fabian (2008) who considered a thin disk with h / r ∼ 0 . 05 in a pseudo-Newtonian potential. In this paper, we describe global 3D GRMHD simulations of a thin disk ( h / r ∼ 0 . 05) around a non-spinning BH and compare our simulated model with the NT73 model.", '2. NUMERICAL MODEL': "Weuse units with G = c =1, e.g. the horizon is at r =2 M and the ISCO is at 6 M . We report results in Boyer-Lindquist (BL) coordinates ( t , r , θ , φ ), referred to as the coordinate frame. In the expressions below g refers to the determinant of the metric. The fluid 4-velocity and the magnetic field 4-vector (see, e.g., Anile 1989 for definitions) are given by u µ and b µ , and the rest-mass density, internal energy density, thermal pressure, and magnetic pressure as measured in the fluid comoving frame are ρ , ug , p = ( γ -1) ug with adiabatic index γ = 4 / 3, and pb = b 2 / 2. The total pressure is p tot = p + pb . \nSimulations were performed using a 3D GRMHD code HARM (Gammie et al. 2003) in Kerr-Schild coordinates using the interpolation scheme described by McKinney (2006), inversion scheme described by Noble et al. (2006) and Mignone & McKinney (2007), and other advances described by Tchekhovskoy et al. (2007). For our fiducial model we used a 3D grid with resolution 512 × 128 × 32 corresponding to: (i) 512 cells in r , logarithmically spaced from r = 1 . 8 M to 50 M with 'outflow' boundary conditions; (ii) 128 cells in θ going from 0 to π , non-uniformly spaced so that roughly half the cells are concentrated in the disk 5 ; (iii) 32 cells in φ , uniformly spaced from 0 to π/ 4. Cells at the disk equator had physical sizes roughly in the ratio of 2:1:7 in dr , rd θ , r sin( θ ) d φ , which ensured that the turbulence was roughly isotropic in Cartesian coordinates, optimal for accurately resolving the magnetorotational instability (MRI). To test convergence we also used resolutions of 256 × 64 × 16, 256 × 64 × 32, and 256 × 64 × 64. We also carried out 2D simulations with resolution of up to 2048 × 256 × 1, but we do not discuss these results because turbulence decayed on an orbital \n5 We used a grid given by θ ( x (2) ) = [ h (2 x (2) -1) + (1 -h )(2 x (2) -1) 7 + 1] π / 2 for code coordinate x (2) with h = 0 . 15, giving roughly 6 × more angular resolution compared to equation (8) with h = 0 . 3 in McKinney & Gammie (2004). \ntime-scale (i.e. Cowling's anti-dynamo theorem holds). \nWe began the simulation with an equilibrium torus (Chakrabarti 1985; De Villiers & Hawley 2003) with inner edge at r = 20 M and pressure maximum at r = 35 M and adjusted the model parameters so the torus had h / r =0 . 1 at 35 M . We found that placing the torus at a smaller radius led to results too sensitive to the initial mass distribution. We define h / r to be the density-weighted root mean square angular thickness of the disk at any given r , i.e. \n( h r ) r = ∆ θ rms = [ ∫ ∫ ( ∆ θ ) 2 ρ ( r , θ, φ ) √ -gd θ d φ ρ ( r , θ, φ ) √ -gd θ d φ ] 1 / 2 , (1) \nor written as ∆ θ rms = 〈 ∆ θ 2 〉 1 / 2 with ∆ θ ≡ θ -〈 θ 〉 . We also consider the mean density-weighted thickness: h / r = ∆ θ abs = 〈| ∆ θ . \n∫ ∫ \n〈| We embedded the torus with a weak magnetic field with β ≡ p / pb ∼ 100. The initial field consisted of two poloidal loops centered at r = 28 M and 38 M to model a disorganized field with no net flux. (During the simulation, there is no organized flux threading the disk but some organized flux threads the BH.) The field strength was randomly perturbed by 50% to seed the MRI instability. Recent GRMHD simulations (McKinney & Narayan 2007a,b; Beckwith et al. 2008a) indicate that the results for the disk (but not the jet) should be roughly independent of the initial field geometry. The MRI is initially resolved with much of the torus having 10 cells per wavelength of the fastest growing MRI mode. \n|〉 \nIn order to keep the accretion disk thin, an ad hoc cooling/heating function was added to the energy-momentum equations as a covariant source term ( -u µ dug / d τ ) with \ndug d τ = -ug -u eq τ cool , (2) \nwhere τ is the fluid proper time. The gas cooling time ( τ cool) was set to 2 π/ Ω K , where Ω K = ( r / M ) -3 / 2 M is the Keplerian frequency. Thus the gas was driven towards u eq, which we defined as that value of ug for which the specific entropy of the gas would be equal to the constant specific entropy (e.g., p /ρ γ ) of the initial solution. \nThe simulation ran for a time of 10000 M , corresponding to 108 orbits at the ISCO ( r ISCO = 6 M ) and 18 orbits at the initial inner edge of the torus. The results reported here correspond to averages computed over the period 6000 M -10000 M , when the accretion flow had reached a quasi-steady state inside a radius of about 10 M . Figure 1 shows that the simulated disk had a thickness (given by eq. 1) of h / r ∼ 0 . 06 -0 . 10 over the radius range of interest. The mean absolute thickness ∆ θ abs was smaller: ∼ 0 . 04 -0 . 07. At the ISCO, the two definitions of thickness gave h / r ∼ 0 . 08 , 0 . 06, respectively.", '3. RESULTS': "The flux of mass and specific angular momentum are given by \n˙ M ( r , t ) = -∫ ∫ ρ u r √ -gd θ d φ, (3) \n˙ L ( r , t ) ˙ M ( r , t ) = -1 ˙ M ( r , t ) ∫ ∫ ( T r φ , in -T r φ , out ) √ -gd θ d φ (4) \nrespectively, where T r φ is the r -φ component of the stressenergy tensor, and \n≡ ¯ /lscript in( r , t ) -¯ /lscript out( r , t ) , (5) \nT r φ , in = ( ρ + ug + p + b 2 ) u r u φ , T r φ , out = b r b φ . (6) \nFIG. 1.- Variation of the root mean square disk thickness h / r = ∆ θ rms (eq. 1) as a function of radius r (solid line). Simulation results were averaged over the time interval 6000 M -10000 M . Also shown is the mean absolute thickness ∆ θ abs (dashed line). The vertical dotted line corresponds to the position of the ISCO. The initial torus had its inner edge at r in = 20 M . \n<!-- image --> \nFIG. 2.- Time-averaged profile of the specific angular momentum ¯ /lscript in of the inflowing magnetized gas (solid line), compared to the idealized profile assumed in the NT73 disk model (dashed line). The horizontal dotted curve shows the net flux of angular momentum ¯ /lscript in -¯ /lscript out. This is independent of r up to ∼ 10 M , indicating the simulation has achieved steady state well-beyond the ISCO. \n<!-- image --> \nThe specific energy flux ˙ E / ˙ M is found by replacing T r φ with T r t in the above equations. In equation (5), the term ¯ /lscript in represents the specific angular momentum advected inward with the accretion flow. Similarly, ¯ /lscript out represents outflow of specific angular momentum as a result of the magnetic shear stress. In steady state, each of these terms is independent of t , and their sum is independent of both r and t . ¯ \nFigure 2 compares the time-averaged profile of ¯ /lscript in from the \nFigure 4 shows one measure of the magnitude of turbulent \n<!-- image --> \nFIG. 3.- Lower panel shows time-averaged profile of the fluid shear stress (solid line) compared to the NT73 stress (dashed line). Upper panel shows α viscosity coefficient (solid line). The non-zero stress in the plunging region causes only a minor deviation from the NT73 result for the flux of specific angular momentum. \n<!-- image --> \nsimulation, averaged over an angular range δθ = ± 0 . 2 around the disk mid-plane (2 to 3 density scale heights), with that predicted by NT73. In the latter, ¯ /lscript in is equal to the Keplerian specific angular momentum for all radii down to the ISCO; inside the ISCO, ¯ /lscript in is taken to be constant since, by assumption, no angular momentum is removed from the gas in the plunging region. The ¯ /lscript in profile from the simulation shows modest deviations from this idealized profile: (i) ¯ /lscript in is slightly sub-Keplerian outside the ISCO; (ii) ¯ /lscript in continues to decrease for a range of r inside the ISCO; (iii) ¯ /lscript in becomes essentially independent of r close to the horizon. \nThe surprising feature of Fig. 2 is that the two curves are so close to each other inside the ISCO. While it is true that the profile of ¯ /lscript in from the simulation drops inside the ISCO to a value of 3 . 39 M , this value is only 2.0% less than the NT73 value of 3 . 464 M . Thus, the magnetic coupling that technically operates inside r ISCO does not have much real effect on the accreting gas. At the ISCO ¯ /lscript out = 0 . 075 M , which is ∼ 2 . 2% of ¯ /lscript in. Further, ˙ E / ˙ M at the horizon is 0 . 940, which is only slightly different from the NT73 value of 0 . 9428, showing that the accretion efficiency is not enhanced by magnetic fields. For two-dimensional simulations of thicker disks with h / r ∼ 0 . 3, McKinney & Gammie (2004) found that ˙ L / ˙ M = 3 . 068 and ˙ E / ˙ M = 0 . 950. It appears that ˙ L / ˙ M deviates from the NT73 value roughly proportional to h / r and so thinner disks should be even closer to NT73. These results are converged since, between a resolution of 256 × 64 × 32 and 512 × 128 × 32, the value of ˙ L / ˙ M changes by less than 0 . 2% and ˙ E / ˙ M changes by less than 0 . 5%. \nFigure 3 shows the time-average of the normalized shear stress \n˜ W ≡ 10 3 ∫ ∫ T ˆ φ ˆ r √ -gd θ d φ/ (2 π r ˙ M ) , (7) \nwhere W ≡ 10 -3 ˜ W ˙ M as equation (5.6.1b) in NT73 and T ˆ φ ˆ r is the orthonormal stress-energy tensor components in the comoving frame. We restrict the integral to fluid with ut ( ρ + u + \nFIG. 4.- Time-averaged profile of ξ turb, which measures the relative magnitude of turbulent fluctuations in the accreting gas. The fluid becomes mostly laminar inside the ISCO. \np + b 2 ) /ρ > -1, i.e., only gravitationally bound fluid (no disk wind); this lowers the value of W near the horizon by 50%. Figure 3 also shows the space-time averaged viscosity parameter α ≡ T ˆ φ ˆ r / p tot that is roughly 0 . 1 -0 . 15 outside the ISCO. Between the ISCO and r = 3 M , α rises to 0 . 6 and finally drops to ∼ 0 near the horizon. There are non-vanishing stresses inside the ISCO and the 'stress edge' is near the horizon. These stresses appear due to organized magnetic flux, but evidently do not cause significant transport of angular momentum (Fig. 2) and are not expected to be a significant source of dissipation and radiation. \nAnother way of evaluating the degree of coupling is to consider, by analogy with steady state flows, the locations of 'critical points,' defined as the radii at which various volumeaveraged outgoing characteristic speeds vanish in the coordinate frame. For our GRMHD simulation, the fast magnetosonic radius is located at r fast ∼ 5 . 1 M showing that, even for this relatively thin accretion disk, magnetic fields do enhance the communication between the disk and the plunging region. Onthe other hand, the Alfvén critical point is much further out at 8 . 0 M . Since Alfvén waves play an important role in angular momentum transport, this suggests that the shear coupling via magnetic fields is weak, unlike the suggestion of Krolik (1999) and Gammie (1999). \nFor fitting disk models to observations the most useful quantity is the rate of dissipation of energy as a function of radius, since this is what determines the radiation emitted by the accretion disk. As a simple estimate, consider the standard theory of viscous disks in which the energy dissipation rate is equal to the local shear stress multiplied by rd Ω K / dr (this is not necessarily valid for an MHD fluid, and is even less valid in the plunging region). In contrast to the standard model, which assumes a vanishing torque at the ISCO, we find ¯ /lscript out = 0 . 022 ¯ /lscript in at r = r ISCO. This modification to the inner boundary condition will cause the luminosity of the disk outside the ISCO to increase by about 4%. \nfluctuations, \nξ turb = √ 〈 ( v r ) 2 〉 -〈 v r 〉 2 〈 v r 〉 , (8) \ntime and angle-averaged as for Fig. 2. We see that the gas at radii ≥ 10 M is highly turbulent due to the MRI. However, the gas flow is fairly smooth at smaller radii, suggesting that there is little dissipation of turbulent kinetic energy inside the ISCO.", '4. CONCLUSION': "For a geometrically thin accretion disk with h / r ∼ 0 . 05 -0 . 1 (Fig. 1) around a non-spinning BH, we find that the specific angular momentum profile ¯ /lscript in( r ) of the accreting magnetized gas is quite similar to that assumed in the idealized model of NT73 (Fig. 2). Specifically, ¯ /lscript in is slightly smaller than, but nearly equal to, the Keplerian value of ¯ /lscript for radii outside the ISCO, and is almost independent of r inside the ISCO. At the BH horizon, the value of ¯ /lscript in deviates from the NT73 value by only 2.0%. \nThis result suggests that the NT73 model is a good approximation for thin disks. However, fast magnetosonic waves from radii as small as 5 . 1 M can escape the plunging region and communicate magnetic stresses between the disk and the plunging regions, which means that the zero-torque condition assumed in the NT model is not perfectly valid. Nevertheless, the normalized outward flux of angular momentum ¯ /lscript out at the ISCO is only ∼ 2 . 2% of the inward flux ¯ /lscript in . Thus, the magnetic coupling across the ISCO is quantitatively weak. As a result, we estimate the luminosity of the disk to be enhanced by no more than a few percent. \nWe find that turbulent activity is pronounced at radii beyond about 10 M , and that the flow is nearly laminar inside the ISCO. This suggests that the bulk of the dissipation occurs outside the ISCO in the disk proper, not in the plunging region. In our next paper we plan to discuss the energy \ndissipation profile of 3D GRMHD simulations, which is ultimately what determines the observed radiation. Future studies should also investigate the dependence of these results on h / r , magnetic field geometry, black hole spin (e.g. Gammie et al. 2004), cooling, resistivity and viscosity, and mass distribution. \nThe results reported here are in qualitative agreement with those obtained by Reynolds & Fabian (2008). They carried out a 3D non-relativistic MHD simulation in a pseudoNewtonian potential, whereas ours is a fully relativistic GRMHD simulation. Also, their initial torus had its inner radius at the ISCO, r in = 6 M , whereas our torus initially had r in = 20 M . This might explain some differences in our results, e.g., their turbulence edge is at 5 . 5 M whereas ours is at 6 . 8 M (compare their Fig. 3 with our Fig. 4). Indeed, we find that our results are sensitive to the initial conditions if we place the inner edge of the torus much inside r in ∼ 20 M . Despite these differences the two simulations agree on the main result, viz., for a geometrically thin disk, the ISCO does behave like a physical boundary separating the disk proper from the plunging region. Beckwith et al. (2008b) suggest a larger deviation from NT73, but they considered a torus with a larger value of h / r and did not use a self-consistent dissipation model. A study of the actual dissipation in the simulation, the goal of our next paper, will hopefully resolve these differences. \nWe thank Niayesh Afshordi for stimulating discussions and useful suggestions. The simulations described in this paper were run on the BlueGene/L system at the Harvard SEAS CyberInfrastructures Lab. This work was supported in part by NASAgrantNNH07ZDA001NandNSF grantAST-0805832. JCMwassupported by NASA's Chandra Postdoctoral Fellowship PF7-80048.", 'REFERENCES': "Afshordi, N., & Paczy'nski, B. 2003, ApJ, 592, 354 \nMcKinney, J. C., & Gammie, C. F. 2004, ApJ, 611, 977 \nAnile, A. M. 1989, Cambridge and New York, Cambridge University Press, 1989, 345 Beckwith, K., Hawley, J. F., & Krolik, J. H. 2008a, ApJ, 678, 1180 Beckwith, K., Hawley, J., & Krolik, J. 2008b, ArXiv e-prints, arXiv:0801.2974 Chakrabarti, S. K. 1985, ApJ, 288, 1 \nDavis, S. W., Done, C., & Blaes, O. M. 2006, ApJ, 647, 525 \nDe Villiers, J.-P., & Hawley, J. F. 2003, ApJ, 589, 458 Gammie, C. F. 1999, ApJ, 522, 57 \nGammie, C. F., McKinney, J. C., & Tóth,G. 2003, ApJ, 589, 444 \nGammie, C. F., Shapiro, S. L., & McKinney, J. C. 2004, ApJ, 602, 312 \nKrolik, J. H. 1999, ApJL, 515, 73 \nKrolik, J. H., & Hawley, J. F. 2002, ApJ, 573, 754 \nKrolik, J. H., Hawley, J. F., Hirose, S. 2005, ApJ, 622, 1008 Liu, J., McClintock, J. E., Narayan, R., Davis, S. W., & Orosz, J. A. 2008, ApJ, 679, L37 \nMcClintock, J. E., Shafee, R., Narayan, R., Remillard, R. A., Davis, S. W., & Li, L.-X. 2006, ApJ, 652, 518 \nMcKinney, J. C., 2006, MNRAS, 367, 1797 \nMcKinney, J. C., & Narayan, R. 2007a, MNRAS, 375, 513 \nMcKinney, J. C., & Narayan, R. 2007b, MNRAS, 375, 531 \nMignone, A., & McKinney, J. C. 2007, MNRAS, 378, 1118 \nNoble S.C., Gammie C.F., McKinney, J.C., Del Zanna L., 2006,ApJ,641,626. Novikov, I. D., & Thorne, K. S. 1973, in Black Holes, ed. C. DeWitt & B. S. DeWitt (New York: Gordon and Breach), p343 Paczy'nski, B. 2000, astro-ph/0004129 \nReynolds, C. S., & Fabian, A. C. 2008, ApJ, 675, 1048 \nReynolds, C. S., & Fabian, A. C. 2008, ApJ, 675, 1048 \nShafee, R., McClintock, J. E., Narayan, R., Davis, S. W., Li, L.-X., & Remillard, R. A. 2006, ApJ, 636, L113 Shafee, R., Narayan, R., & McClintock, J. E. 2008, ApJ, 676, 549 Shakura, N. I., & Sunyaev, R. A. 1973, 337, 355 \nTchekhovskoy, A., McKinney, J. C., & Narayan, R. 2007, MNRAS, 379, 469"}

1999PhRvD..59l4007P

Thermodynamics of Reissner-Nordström-anti-de Sitter black holes in the grand canonical ensemble

1999-01-01

['-', '-', '-', '-', '-']

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The thermodynamical properties of the Reissner-Nordström-anti-de Sitter black hole in the grand canonical ensemble are investigated using York's formalism. The black hole is enclosed in a cavity with a finite radius where the temperature and electrostatic potential are fixed. The boundary conditions allow one to compute the relevant thermodynamical quantities, e.g., thermal energy, entropy, and charge. The stability conditions imply that there are thermodynamically stable black hole solutions, under certain conditions. By taking the boundary to infinity, and leaving the event horizon and charge of the black hole fixed, one rederives the Hawking-Page action and Hawking-Page specific heat. Instantons with negative heat capacity are also found.

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https://arxiv.org/pdf/gr-qc/9805004.pdf

{'Claudia S. Pe¸ca': "Departamento de F'ısica, Instituto Superior T'ecnico, Av. Rovisco Pais 1, 1096 Lisboa Codex, Portugal", "Jos'e P. S. Lemos": "Departamento de Astrof'ısica. Observat'orio Nacional-CNPq, Rua General Jos'e Cristino 77, 20921 Rio de Janeiro, Brazil and Departamento de F'ısica, Instituto Superior T'ecnico, Av. Rovisco Pais 1, 1096 Lisboa Codex, Portugal (November 26, 2024) \nThe thermodynamical properties of the Reissner-Nordstrom-anti-de Sitter black hole in the grand canonical ensemble are investigated using York's formalism. The black hole is enclosed in a cavity with finite radius where the temperature and electrostatic potential are fixed. The boundary conditions allow one to compute the relevant thermodynamical quantities, e.g. thermal energy, entropy and charge. The stability conditions imply that there are thermodynamically stable black hole solutions, under certain conditions. By taking the boundary to infinity, and leaving the event horizon and charge of the black hole fixed, one rederives the Hawking-Page action and Hawking-Page specific heat. Instantons with negative heat capacity are also found. \nPACS numbers: 04.70Bw, 04.70Dy", 'I. INTRODUCTION': "The path-integral approach to the thermodynamics of black holes was originally developed by Hawking et al. [1-3]. In this approach the thermodynamical partition function is computed from the path-integral in the saddle-point approximation, thus obtaining the thermodynamical laws for black holes. \nIn the path-integral approach we can use the three different ensembles: microcanonical, canonical and grand canonical. Due to difficulties related to stability of the black hole in the canonical ensemble, the microcanonical ensemble was originally considered [3,4]. However, further developments by York et al. [5-8] allowed to define the canonical ensemble. Effectively, by carefully defining the boundary conditions, one can obtain the partition function of a black hole in thermodynamical equilibrium. This approach was further developed to include other ensembles [10], and to study charged black holes in the grand canonical ensemble [11] and black holes in asymptotically anti-de Sitter spacetimes [12,9,13]. This approach was also applied to black holes in two [14] and three [12] dimensions. \nIn York's formalism the black hole is enclosed in a cavity with a finite radius. The boundary conditions are defined according to the thermodynamical ensemble under study. Given the boundary conditions and imposing the appropriate constraints, one can compute a reduced action suitable for doing black hole thermodynamics [11,15]. Evaluating this reduced action at its stable stationary point one obtains the corresponding classical action, which is related to a thermodynamical potential. In the canonical ensemble this thermodynamical potential corresponds to the Helmholtz free energy, while for the grand canonical ensemble the thermodynamical potential is the grand canonical potential [2,11]. From the thermodynamical potential one can compute all the relevant thermodynamical quantities and relations [16]. \nSome controversy has appeared related to the boundary conditions chosen in this formalism [12,17,18]. More precisely, Hawking and Page [17,18] fix the Hawking temperature of the black hole (i.e. the temperature defined so that the respective Euclidean metric has no conical singularity at the horizon), while York et al. [5,11,12] fix the local temperature at a finite radius, where the boundary conditions are defined. For asymptotically flat spacetimes the two formalism coincide, because at infinity the local temperature is equal to the Hawking temperature. On the contrary, for asymptotically anti-de Sitter spacetimes the two procedures disagree, since the local temperature is redshifted to zero at infinity and is equal to the Hawking temperature only in the region where spacetime has a flat metric. Louko and Winters-Hilt [13] have studied the thermodynamics of the Reissner-Nordstrom-anti-de Sitter black hole fixing a renormalized temperature at infinity that corresponds to the same procedure used in [17,18]. In this paper we have chosen to follow York's formalism [5,11,12] and study the thermodynamics of the Reissner-Nordstrom-anti-de Sitter black hole fixing the local temperature at finite radius. \nWe find that the two procedures give some identical results, e.g., in both procedures the Hawking-Bekenstein formula for the entropy [19,20] is obtained. In addition, by that taking the boundary to infinity, and leaving the event horizon and charge of the black hole fixed, we rederive the Hawking-Page action and Hawking-Page specific heat from \nYork's formalism. However, the value for the energy at infinity differs depending on which procedures one uses. In [13] it was found that the energy at infinity is equal to the mass of the black hole, a result that does not hold here. These results conform with the similarities and differences found for the Schwarzschild-anti-de Sitter black hole in [17,12]. \nIn section II we briefly introduce York's formalism. In section III we compute the reduced action for the ReissnerNordstrom-anti-de Sitter black hole and evaluate its thermodynamical quantities. In section IV we analyze the black hole solutions. In section V we study the local and global stability of these solutions. The limit where the boundary is taken to infinity mentioned above is studied in section VI. Finally some special cases are briefly referred in section VII.", 'II. THE ACTION': "The Euclidean Einstein-Maxwell [3] is given by \nI = -1 16 π ∫ M d 4 x √ g ( R -2Λ) + 1 8 π ∫ ∂ M d 3 x √ hK -1 16 π ∫ M d 4 x √ g F µν F µν -I subtr , (1) \nwhere M is a compact region with boundary ∂ M , R is the scalar curvature, Λ the cosmological constant, g the determinant of the Euclidean metrics, K the trace of the extrinsic curvature of the boundary ∂ M , h is the determinant of the Euclidean induced metrics on the boundary, F µν = ∂ µ A ν -∂ ν A µ is the Faraday tensor and I subtr is an arbitrary term that can be used to define the zero of the energy as will be seen later. \nIn order to set the nomenclature we follow [11] in this section. We consider a spherical symmetric static metric of the form [11] \nds 2 = b 2 dτ 2 + a 2 dy 2 + r 2 d Ω 2 , (2) \nwhere a , b and r are functions only of the radial coordinate y ∈ [0 , 1]. The Euclidean time τ has period 2 π . The event horizon, given by y = 0, has radius r + = r (0) and area A + = 4 πr 2 + . The boundary is given by y = 1 and at this boundary the thermodynamical variables defining the ensemble are fixed. The boundary is a two-sphere with area A B = 4 πr 2 B , where r B = r (1). We will consider the grand canonical ensemble, where heat and charge can flow in and out through the boundary to maintain a constant temperature T ≡ T ( r B ) and electrostatic potential φ ≡ φ ( r B ) at the boundary. \nWe impose a black hole topology to the metric (2), by using the conditions, b (0) = 0, \nsee [11]. \nEvaluating the action (1) for the metric (2), \nb ' a ∣ ∣ ∣ y =0 = 1 and ( r ' a ) 2 ∣ ∣ ∣ ∣ y =0 = 0, \nI = 1 2 ∫ 2 π 0 dτ ∫ 1 0 dy ( -2 r b ' r ' a -b r ' 2 a -ab +Λ abr 2 ) -1 2 ∫ 2 π 0 dτ ( b r 2 ) ' a ∣ ∣ ∣ ∣ y =0 --1 2 ∫ 2 π 0 dτ ∫ 1 0 dy r 2 ab A ' τ 2 -I subtr . (3) \nIn order to obtain the reduced action one uses the proper constraints. For the gravitational part of the action I g given in (3), the constraint used is the Hamiltonian constraint [11,15] \nG τ τ +Λ g τ τ = 8 πT τ τ . (4) \nIn addition, for the matter fields part of the action we use Maxwell equations F µν ; ν = 0. \nThe thermodynamical quantities and relations are obtained from the 'classical action' ˜ I (defined as the reduced action evaluated at its locally stable stationary points) using the well known relation between the 'classical action' and the thermodynamical potential \n˜ I = βF . (5) \nHere F is the grand canonical potential since we are considering the grand canonical ensemble. All the thermodynamical quantities can be obtained from F using the classical thermodynamical relations (see for example [16]).", 'III. THE REISSNER-NORDSTR OM-ANTI-DE SITTER BLACK HOLE': "The Reissner-Nordstrom-anti-de Sitter black hole in the grand canonical ensemble is obtained using a negative cosmological constant Λ and the boundary conditions T ≡ T ( R B ) and φ ≡ φ ( r B ), where r B is the boundary radius of the spherical cavity, T the temperature at the boundary and φ the electrostatic potential difference between the horizon and the boundary. Instead of T we can also use its inverse β . \nThe reduced action for Reissner-Nordstrom-anti-de Sitter black hole is obtained from the Euclidean EinsteinHilbert-Maxwell action I given in (1). For simplicity we split the action in two terms I = I g + I m , where I g is the gravitational term and I m the matter field term. To obtain the reduced action we use the Hamiltonian constraint (4) and the Maxwell equations. \nThe evaluation of I m is identical to the case Λ = 0 and can therefore be found in [11], \nI m = -1 2 β e φ , (6) \nwhere e is the electrical charge of the black hole and φ is the difference of potential between y = 0 and y = 1. To evaluate the gravitational term I g (3) we use as mentioned above the constraint (4) \nG τ τ +Λ g τ τ = 8 π T τ τ . (7) \nThe component of the Einstein tensor G τ τ for the metric (2) is \nG τ τ = r ' 2 a 2 r 2 -1 r 2 + 2 r ' ' a 2 r -2 a ' r ' a 3 r . (8) \nThe stress-energy tensor component T τ τ is given by \nT τ τ = 1 8 π ( A τ ' ab ) 2 = -1 8 π e 2 r 4 . (9) \nSubstituting (9) and (8) in (7) we obtain \nΛ = ( A τ ' ab ) 2 -r ' 2 a 2 r 2 + 1 r 2 -2 r ' ' a 2 r + 2 a ' r ' a 3 r . (10) \nRearranging terms in equation (10) and using (9) we obtain \n1 r 2 r ' {[ r ( r ' 2 a 2 -1 )] ' + e 2 r ' r 2 +Λ r 2 r ' } = 0 . (11) \nIntegrating and simplifying the previous equation yields \n( r ' a ) 2 = 1 -2 M r + e 2 r 2 + α 2 r 2 , (12) \nwhere 2 M is an integration constant and α 2 = -Λ / 3. The integration constant 2 M can be evaluated using the black \n2 M = r + + e 2 r + + α 2 r 3 + . (13) \nhole topology condition ( r ' a ) 2 ∣ ∣ ∣ ∣ y =0 = 0, \nThis is the known relation between the ADM mass of the Reissner-Nordstrom-anti-de Sitter black hole and its event horizon radius. \nSubstituting (10) in (3) yields \nI ∗ g = 1 2 ∫ 2 π 0 dτ ∫ 1 0 dy ( -2 r b ' r ' a -2 b r ' 2 a -2 b r r ' ' a +2 b r a ' r ' a 2 -r 2 ab A ' τ 2 ) --1 2 ∫ 2 π 0 dτ ( b r 2 ) ' a ∣ ∣ ∣ ∣ y =0 -I subtr = -∫ 2 π 0 dτ ∫ 1 0 dy ( b r r ' a ) ' -1 2 β e φ -1 2 ∫ 2 π 0 dτ b ' r 2 a ∣ ∣ ∣ ∣ y =0 -I subtr , (14) \nwhere we have used the topology conditions given in section II and to evaluate the term in A τ , we used (6), since this term is identical to I m . \nThe first term after the second equality in equation (14) can be evaluated by integrating and substituting equations (12) and (13). The respective third term is integrated and using the topology conditions gives -πr 2 + . Following this procedure, we obtain \nI ∗ g = -β r B f ( r B ; r + , e, α ) -1 2 β e φ -πr 2 + -I subtr , (15) \nwhere the inverse temperature at the boundary β is given by the proper length of the time coordinate at the boundary, β ≡ T -1 = ∫ 2 π 0 b (1) dτ = 2 π b (1) and \nf ( r B ; r + , e, α ) = √ 1 -r + r B -e 2 r + r B -α 2 r 3 + r B + e 2 r 2 B + α 2 r 2 B . (16) \nAdding (6) and (15), yields the reduced action \nI ∗ = -β r B f ( r B ; r + , e, α ) -β e φ -πr 2 + -I subtr . (17) \nThe term I subtr is of the form βE subtr , where E subtr is a constant that does not depend on β or φ , since I subtr is an arbitrary term that can be used to fix the zero of the energy but cannot affect other thermodynamical variables [5]. For convenience, we use for the zero of the energy \nE ADS = E ( r + = 0 , e = 0) = 0 , (18) \nwhere E ADS is the thermal energy of anti-de Sitter spacetime. \nTo evaluate I subtr , we compute the thermal energy of the Reissner-Nordstrom-anti-de Sitter black hole from (17) and use condition (18). The thermal energy is given by [16] \nE = F + β ( ∂F ∂β ) φ,r B -( ∂F ∂φ ) β,r B φ = ( ∂ ˜ I ∂β ) φ,r B -φ β ( ∂ ˜ I ∂φ ) β,r B = = -r B f ( r B ; r + , e, α ) -E subtr (19) \n( \nwhere F is the grand canonical potential and we have used (5). Although the reduced action I ∗ is not the classical action (therefore we cannot write I ∗ = βF ), the energy has the form given in (19). This is because the classical action ˜ I corresponds to the minimum of the reduced action and therefore the equalities ∂ ˜ I ∂β ) φ,r B = ∂I ∗ ∂β ) φ,r B ,r + ,e and \n) ) ( ∂ ˜ I ∂φ ) β,r B = ( ∂I ∗ ∂φ ) β,r B ,r + ,e hold. However, r + and e in (19) are not free parameters, they depend on the boundary conditions (i.e. on the values of β , φ and r B ) and on the cosmological constant. The functions r + = r + ( β, φ, r B , α ) and e = e ( β, φ, r B , α ) are obtained from the equilibrium conditions ∂I ∗ ∂r + = 0 and ∂I ∗ ∂e = 0 as will be seen later. \nUsing equation (18) on (19), yields \nE subtr = -r B f 0 ( r B ; α ) (20) \nwhere f 0 ( r B ; α ) = f ( r B ; 0 , 0 , α ) = √ 1 + α 2 r 2 B . Substituting (20) in (17), we finally obtain the reduced action for the Reissner-Nordstrom-anti-de Sitter black hole \nI ∗ = β r B [ f 0 ( r B ; α ) -f ( r B ; r + , e, α )] -β e φ -πr 2 + . (21) \nSimilarly substituting (20) in (19), we obtain its thermal energy \nE = r B [ f 0 ( r B ; α ) -f ( r B ; r + , e, α )] . (22) \nThe mean value of the charge is \n( \nQ = -( ∂F ∂φ ) β,r B = -1 β ( ∂ ˜ I ∂φ ) β,r B = e . (23) \nThe entropy is obtained from \nS = β 2 ( ∂F ∂β ) φ,r B = β ( ∂ ˜ I ∂β ) φ,r B -˜ I = πr 2 + , (24) \nwhere equation (5) was used. Since A + / 4 = π r 2 + , where A + is the area of the event horizon, this is the usual Hawking-Bekenstein entropy [19,20]. \nAs mentioned above, the event horizon radius r + and electric charge e of the black hole for the given boundary conditions, i.e. β , φ and r B , are obtained by evaluating the locally stable stationary points of the reduced action with respect to r + and e [11]. Effectively, once the values of β , φ and r B are held fixed by the boundary conditions, then the reduced action is a function of only r + and e , i.e. I ∗ = I ∗ ( r + , e ). The local stability conditions are then ( i ) ∇ I ∗ = 0 and ( ii ) the Hessian matrix is positive definite. The latter condition corresponds to a condition of dynamical as well as thermodynamical stability [11] and will be discussed in section V. We will start by investigating the first condition. \nThe condition of stationarity ∇ I ∗ = 0 gives \nβ = 2 π κ f ( r B ; r + , e, α ) , (25) \nwhere κ = r 2 + -e 2 +3 α 2 r 4 + 2 r 3 + is the surface gravity of the horizon, and \nφ = ( e r + -e r B ) f ( r B ; r + , e, α ) -1 . (26) \nThese are the inverse Hawking temperature and the difference in the electrostatic potential between r + and r B blueshifted from infinity to r B , respectively. \nInverting these two equations, r + and e are obtained as functions of the boundary conditions and the cosmological constant. We define the more convenient variables \nx ≡ r + r B , q ≡ e r B , β ≡ β 4 π r B , α 2 ≡ α 2 r 2 B . (27) \nIn these variables (25) and (26) are written \nβ = x √ 1 -x √ 1 -q 2 x + α 2 (1 + x + x 2 ) ( 1 -q 2 x 2 +3 α 2 x 2 ) -1 , (28) \nφ = q x √ 1 -x ( 1 -q 2 x + α 2 (1 + x + x 2 ) ) -1 / 2 . (29) \nTo invert this equations, we start by inverting equation (29) to obtain q , \nq 2 = x 2 φ 2 [1 + α 2 (1 + x + x 2 )] (1 -x + xφ 2 ) -1 . (30) \nSubstituting (30) in (28) and taking its square, we obtain a 7th degree equation in x with a double root x = 1 which is not a solution of the initial equations. Getting rid of that root, we obtain a 5th degree equation \nβ 2 ( -1 + φ 2 + α 2 φ 2 ) 2 + 4 α 2 β 2 φ 2 ( -1 + φ 2 + α 2 φ 2 ) x + ( -1 -α 2 +6 α 2 β 2 -12 α 2 β 2 φ 2 --6 α 4 β 2 φ 2 +6 α 2 β 2 φ 4 +10 α 4 β 2 φ 4 ) x 2 + ( 1 -φ 2 -α 2 φ 2 -12 α 4 β 2 φ 2 +12 α 4 β 2 φ 4 ) x 3 + + α 2 ( 9 α 2 β 2 -φ 2 -18 α 2 β 2 φ 2 +9 α 2 β 2 φ 4 ) x 4 + α 2 ( 1 -φ 2 ) x 5 = 0 . (31) \nHowever not every solution of this equation corresponds to a physical solution of a black hole. This is because the radius of the event horizon of the Reissner-Nordstrom-anti-de Sitter must obey the following condition \nr 2 + -e 2 +3 α 2 r 4 + ≥ 0 . (32) \nWhere the equality defines the extremal Reissner-Nordstrom-anti-de Sitter black hole. \nComparing (32) with (25), yields that β is real and positive. Comparing it with (26) we obtain the following condition \nφ 2 ≤ 1 + 3 α 2 r 2 + 1 + α 2 (1 + 2 r + r B +3 r 2 + ) . (33) \nIn the coordinates given in (27), the inequality (33) becomes \nφ 2 ≤ 1 + 3 α 2 x 2 1 + α 2 (1 + 2 x +3 x 2 ) . (34) \nThis is the condition that the solutions of equation (31) must obey in order to represent physical black hole solutions. Equation (31) has no known analytical solutions. However its solutions can be numerically computed and presented in graphics. This will be done in the next section.", 'IV. ANALYSIS OF THE BLACK HOLE SOLUTIONS': 'In this section we present in graphics an analysis of the solutions of equation (31) that obey condition (34). This analysis is done in two steps: ( i ) first, we analyze figures 1 to 4, that present the solutions x as functions of β and φ , for values α = 0, 0.5, 1 and 5; ( ii ) afterwards we show in figures 5 to 9 the regions with zero, one and two solutions in the space spanned by φ × α for fixed values of β .', '( i ) Analysis of figures 1 to 4:': 'α = 0: The solutions for α = 0 are presented in figure 1. These are obviously identical to the solutions of the Reissner-Nordstrom black hole, see [11]. We can see in figure 1 that for fixed φ there is a maximum of β , β max ( φ ), so that for β > β max ( φ ) there are no solutions. For β < β max ( φ ) one can have two or only one solution depending on the precise value of β . For φ = 0 (Schwarzschild) one has always two solutions for β < β max (0) ( β max (0) = 2 / √ 27). In the limiting case β → 0 (i.e. r B T → ∞ ), one finds there is a solution with x = r + /r B → 1 (as will be seen in section V this is the stable solution), see [5]. For φ = 0, still, and β > β max (0) there are no solutions (see comments at the end of this section). For 0 < φ < 1 / √ 3 one has one or two solutions up to β max ( φ ), whereas for β > β max ( φ ) there are as well no solutions. Finally for φ > 1 / √ 3 there is only one solution (again for β < β max ( φ )) corresponding to the unstable branch as will be seen in section V. Note that, for α = 0, condition (34) implies that the electrostatic potential has a maximum at φ max = 1. Notice also that in the limit β →∞ ( T → 0), the curves in figure 1 tend to the curve φ = 1, which corresponds to the extremal Reissner-Nordstrom black hole ( r + = e ). \nThe cases α = 0, which are now going to be analyzed require the following prior analysis. \nAs in the case α = 0, for α /negationslash = 0 there are solutions at T = 0 ( β = ∞ ), that correspond to the extremal black holes. This can be analytically verified by replacing β = ∞ in equation (31), from where we obtain the equation \n/negationslash \n1 -φ 2 -α 2 φ 2 -2 α 2 φ 2 x +3 α 2 (1 -φ 2 ) x 2 = 0 . (35) \nNotice this is the equation one obtains taking the equality in condition (34). In fact it corresponds to the condition of extremality of the Reissner-Nordstrom-anti-de Sitter black hole, which is in agreement with the well known fact that only the extremal black holes have zero temperature. \nEquation (35) has at least one solution that verifies 0 < x < 1 if \nα 2 < 1 6 and 1 + 3 α 2 1 + 6 α 2 ≤ φ 2 ≤ 1 1 + α 2 , 1 6 ≤ α 2 < 2 3 and 3 α 2 +6 -√ 9 α 4 +12 α 2 4 α 2 +6 ≤ φ 2 ≤ 1 1 + α 2 , α 2 ≥ 2 3 and 3 α 2 +6 -√ 9 α 4 +12 α 2 4 α 2 +6 ≤ φ 2 ≤ 1 + 3 α 2 1 + 6 α 2 . (36) \nFor these values of φ and α , the curves for fixed φ presented in the figures reach infinite β . Furthermore, equation (35) has two solutions if \n1 / 6 < α 2 < 2 / 3 and 3 α 2 +6 -√ 9 α 4 +12 α 2 4 α 2 +6 < φ 2 < 1 + 3 α 2 1 + 6 α 2 , α 2 ≥ 2 / 3 and 3 α 2 +6 -√ 9 α 4 +12 α 2 4 α 2 +6 < φ 2 < 1 1 + α 2 . (37) \nα = 0 . 5: The solutions for α = 0 . 5 are presented in figure 2. This figure presents the same properties mentioned previously for the case α = 0. Comparing 1 and 2, we verify that the maximum value of β , β max ( φ ), for which there are solutions is increasing, i.e. there are solutions for slightly lower values of the temperature at α = 0 . 5 than at α = 0. Furthermore there are solution for infinite β in the interval 0 . 83 /lessorsimilar φ /lessorsimilar 0 . 89, see (36), this can be seen using the curve β = 9 in figure 2, since it corresponds to a good approximation of infinite β . \nCondition (34) implies, for α 2 < 2 / 3, \nφ ≤ √ 1 1 + α 2 . (38) \nThis condition corresponds to the upper-limit of the interval given in (36), which in this case is φ /similarequal 0 . 89. Notice that if in equation (31) we do x = 0, we obtain precisely φ = 1 / 1 + α 2 , as can be seen in figure 2. \nα = 1: Figure 3 plots the solutions for α = 1. We can see that this figure presents similar properties to the ones obtained for lower values of α . Using (36), we can see that for φ /lessorsimilar 0 . 66 there are solutions only for β < β max ( φ ), where β max ( φ ) is finite and depends on φ . On the contrary, the curves for higher values of φ reach infinite β . In particular, for 0 . 66 /lessorsimilar φ /lessorsimilar 0 . 71 (see equation (37)) there are two solutions at low temperatures (i.e., high β ) as can be seen in figure 3, since the curve β = 9 is representative of the curves with high β . It can be seen that for β = 9 there are two solutions for φ < φ 0 , where φ 0 = 1 √ 1+ α 2 /similarequal 0 . 71 is the value of φ where x = 0 for every β . There is one \n√ \nsolution for 0 . 71 /lessorsimilar φ /lessorsimilar 0 . 76, see figure 3, where the upper-limit is imposed by condition (34). In fact, condition (34) imposes, for α 2 < 2 / 3, \nφ max = √ 1 + 3 α 2 1 + 6 α 2 . (39) \nNotice this is the upper-limit of the interval given in (36). \nα = 5: Figure 4 presents the solutions for α = 5. Using (36), we can see that for φ /lessorsimilar 0 . 19, there are solutions only for β < β max ( φ ). For 0 . 19 /lessorsimilar φ /lessorsimilar 0 . 71 there is one solution for high values of β (consider the curve β = 9). For infinite β this region is 0 . 195 /lessorsimilar φ /lessorsimilar 0 . 196, see (37). In figure 4, we can also see that for 0 . 19 /lessorsimilar φ /lessorsimilar 0 . 71 there is one solution, where the upper-limit is given by (39). \nFor higher values of α there are not new types of solutions and therefore it is not necessary to pursue our analysis.', '( ii ) Analysis of figures 5 to 9:': "In order to clarify the disposition of the number of solutions for given values of α , φ , and β , we present, in the space spanned by α and φ for fixed β , the regions with zero, one and two solutions. We do this for eleven different values of β , β = 0 , 0 . 3 , 2 / √ 27 /similarequal 0 . 38 , 1 , ∞ , see figures 5 to 9 respectively. In this figures one can see the evolution of the number of solutions as β increases. To present all possible values of α/epsilon1 [0 , ∞ [, we use in these figures the parameter a instead of α , a is defined by \na = 2 π arctan α , (40) \nsuch that 0 ≤ a 1. It is this variable that appears in the ordinate axis in figures 5 to 9. \n≤ Due to condition (34) there are no physically possible solutions on the right-hand side of figures 5 to 9. \n≤ \nAn important value of β , first studied by York [5] in connection with the Schwarzschild black hole ( φ = α = 0), is β = 2 / √ 27 /similarequal 0 . 38, i.e. β = 8 π √ 27 r B . For higher values of β , lower values of the temperature, there are no black hole solutions. This is a quantum effect and following York [5] can be understood as follows. One can associate a Compton type wavelength λ to the energy k B T of the thermal particles, by λ = /planckover2pi1 c k B T , or in Planck units λ = 1 /T = β . If this Compton wavelength is much larger than the radius r B of the cavity (or more specifically λ > 8 π √ 27 r B /similarequal 4 . 8 r B ) then the thermal particles cannot be confined within the cavity and do not collapse to form a black hole. \nBy analyzing figures 6 and 7, we can see that for nonzero cosmological constant ( α /negationslash = 0) this phenomenon starts at even lower β (higher T ). Indeed using equation (28) (with φ = 0, i.e. q = 0) we can show that to first order in α 2 \n( α 2 r 2 B << 1), York's criterion for no black hole solutions becomes \nλ = β > 8 π √ 27 r B ( 1 -5 18 α 2 r 2 B ) . (41) \nFrom equation (41) we infer that the role of the negative cosmological constant (Λ = -3 α 2 ) is to produce an effective cavity radius r eff = r B ( 1 -5 18 α 2 r 2 B ) smaller than r B . Thus for a given temperature, it is more difficult to confine the thermal particles, and harder to form black holes, in accord with the idea that a negative cosmological constant shrinks space. \nIf we extend our previous first order analysis to include the electrostatic potential φ , we obtain \nλ = β > 8 π √ 27 r B ( 1 -5 18 α 2 r 2 B +2 φ 2 ) . (42) \nWe can see that the electrostatic potential has the opposite effect of the cosmological constant (see for example figures 6, 7 and 8).", 'A. Local stability': 'As mentioned before there is a second condition of local stability that has not yet been investigated. We will follow the same procedure as given in [11]. This is the condition that the Hessian matrix of the reduced action be positive definite. For convenience in this analysis we will use the variable S = πr 2 + instead r + . The Hessian matrix of I ∗ ( S, e ) is \nI ∗ ,ij = ( I ∗ ,ee I ∗ ,eS I ∗ ,eS I ∗ ,SS ) . (43) \nThe matrix is positive definite if its pivots are all positive. The pivots of (43) are \nI ∗ ,ee (44) \nand \ndet( I ∗ ,ij ) I ∗ ,ee . (45) \nThe first condition of local stability, ∇ I ∗ = 0 yields \n( ∂I ∗ ∂e ) S = β ( ∂E ∗ ∂e ) S -β φ = 0 ⇒ φ = ( ∂E ∂e ) S , ( ∂I ∗ ∂S ) e = β ( ∂E ∗ ∂S ) e -1 = 0 ⇒ β -1 = T = ( ∂E ∂S ) e . (46) \nThese are well known thermodynamical relations [16]. \nFrom (46), we compute the second derivatives of I ∗ in the stationary points of I ∗ \n∣ \n∣ \n∂ 2 I ∗ ∂e 2 ∣ ∣ ∣ ∣ eq = β ∂ 2 E ∗ ∂e 2 ∣ ∣ ∣ ∣ eq = β ( ∂φ ∂e ) S ∂ 2 I ∗ ∂e∂S ∣ ∣ ∣ ∣ eq = β ∂ 2 E ∗ ∂e∂S ∣ ∣ ∣ ∣ eq = β ( ∂φ ∂S ) e = -1 β ( ∂β ∂e ) S ∂ 2 I ∗ ∂S 2 ∣ ∣ ∣ eq = β ∂ 2 E ∗ ∂S 2 ∣ ∣ ∣ eq = -1 β ( ∂β ∂S ) e . (47) \n∣ ∣ where eq means quantities evaluated at equilibrium, i.e. at the stationary points of the reduced action I ∗ . The first pivot (44) is simply the first of these equations. The second pivot (45) is easily obtained from (47) \ndet I ∗ ,ij I ∗ ,ee = -1 β ( ∂β ∂S ) e + 1 β ( ∂β ∂e ) S ( ∂φ ∂S ) e /( ∂φ ∂e ) S = -1 β [ ( ∂β ∂S ) e + ( ∂β ∂e ) S ( ∂e ∂S ) φ ] = -1 β ( ∂β ∂S ) φ = 1 C φ,r B , (48) \nwhere C φ,r B is the heat capacity at constant φ and r B . Notice that in all this calculation we have implicitly held r B constant, since in the grand canonical ensemble the dimension of the system is held constant. \nImposing positive pivots for local stability yields \n( ∂φ ∂e ) S,r B ≥ 0 , C φ,r B ≥ 0 . (49) \nThese conditions are identical to the classical thermodynamical stability conditions [16]. Therefore one can conclude that the dynamical stability conditions given in (46) and (49) are identical to the thermodynamical stability conditions [11]. \nFor the Reissner-Nordstrom-anti-de Sitter black hole, the pivots obtained from (47) and (48) are \n( ∂φ ∂e ) S,r B = ( 1 r + -1 r B )( 1 -r + r B -e 2 r + r B -α 2 r + 3 r B + e 2 r B 2 + α 2 r B 2 ) 1 / 2 + + e 2 r B ( 1 r + -1 r B ) 2 ( 1 -r + r B -e 2 r + r B -α 2 r + 3 r B + e 2 r B 2 + α 2 r B 2 ) 3 / 2 , (50) \nwhich is positive, and \nC φ,r B = 4 πr + 3 ( r -r + )( r + 2 -e 2 +3 α 2 r + 4 )[1 + α 2 ( r B 2 + r + r B + r + 2 )] × × { e 4 + r + 3 [(6 α 2 r + 2 -2)( r B + α 2 r B 3 ) + 3 r + +2 α 2 r + 3 +3 α 4 r + 5 ]+ + 2 e 2 r + [ -2 r + + r B (1 + α 2 ( r B 2 -2 r + r B -2 r + 2 ))] -1 . (51) \nThe numerator of C φ,r B is positive, therefore the condition C φ,r B > 0 is verified if the denominator in (51) is positive. Using (27) and (30), we obtain the following condition of stability for the solutions of equation (31) \n} \n[ -2 (1 + α 2 ) + 3 x +6 α 2 (1 + α 2 ) x 2 +2 α 2 x 3 +3 α 4 x 5 ] (1 -x + xφ 2 ) 2 --[ 2 φ 2 (1 + α 2 (1 + x + x 2 ))( -1 + 2 x + α 2 ( -1 + 2 x +2 x 2 )) ] (1 -x + xφ 2 ) + + φ 4 x (1 + α 2 (1 + x + x 2 )) 2 > 0 . (52) \nBy numerical computation, we can verify the solutions that obey this condition. We have found that the lower branches of the curves presented in figures 1 to 4 correspond to unstable solutions, while the upper branches correspond to stable solutions. Therefore we can say that in general only the solutions with the higher value of x , that is with higher event horizon radius, are stable. \nIn more detail we can distinguish three cases: ( i ) for low α (see figures 1 and 2), and for values of β and φ for which there are one solution, it corresponds to a lower branch and therefore this solutions is unstable; ( ii ) for high values of α (see figure 4), whenever there is only one solution, it corresponds to an upper branch and therefore this solution is stable; ( iii ) whenever there are two solutions, for given values of β, φ, α , the smaller one is an instanton (i.e. it is an unstable solution and dominates the semi-classical evaluation of the rate of nucleation of black holes [22]) and the one with larger event horizon radius is a stable black hole. \nThese three cases can easily be distinguished in figures 5 to 9. In these figures there are in general two separated regions with one solution. The region with lower values of a , i.e. lower values of α , corresponds to case ( i ) and these are unstable solutions. The region with higher values of a corresponds to case ( ii ) and these are stable solutions. Obviously, the region with two solutions in each figure 5 to 9, corresponds to case ( iii ), i.e. one of those solution is stable, the one with higher value of x , and the other unstable.', 'B. Global stability': 'The stable solutions computed in the previous subsection are not necessarily global minimum of the action. In this case they will not dominate the partition function and the zero-loop approximation cannot be considered accurate [6,11]. \nAs the reduced action given in equation (21) grows without bound in the directions where r + or e tend to infinity, the global minimum must be found either at the local minimum or at r + = e = 0. The action tends to zero as r + and e tend to zero, therefore the condition of global stability of the stable solutions is that the classical action I must be negative. \nIf the classical action is positive, the partition function is dominated by points near the origin. But these points do not correspond to a black hole in thermal equilibrium. In this case, the black hole, that corresponds to the stable solution of the reduced action, is metastable. \nWe verified numerically which boundary conditions (given by the values of β , φ and α ) correspond to globally stable black holes, i.e. to solutions that dominate the partition function. We can also see, using a simple argument, that for β < 8 27 all locally stable solutions are also globally stable. Indeed, York [6] has shown that for the Schwarzschild black hole the condition of global stability is β < 8 27 and since the classical action I is a decreasing function of φ and α , this condition is still a bound for globally stability for all φ and α . However for β > 8 27 , there is always a certain region of φ × α , for which the locally stable solutions do not correspond to global minima of the action. We show these regions in figures 5 to 9, where the regions for which the solutions do not correspond to a global minimum of the action, i.e. the metastable solutions, are shaded. In particular figures 5 and 9 do not present a shaded region because all the stable solutions are dominant in the limit β → 0 as said above and for β → ∞ the region with metastable solutions is too thin to be presented in graphic.', 'VI. THE R B →∞ LIMIT AND THE HAWKING-PAGE SOLUTIONS': "One can study the case where the boundary goes to infinity. There are two different ways for taking this limit: ( i ) fixing the horizon radius r + and the charge e of the black hole; ( ii ) fixing the boundary conditions, i.e. fixing β and φ . In this section we will study only the pure Reissner-Nordstrom-anti-de Sitter cases, i.e. the cases with α, φ /negationslash = 0. Other cases, with either α or φ equal to zero are considered in the next section. \nWe will start by studying the first case: (i) Fixing the black hole solution, i.e. fixing r + and e and taking the limit r B →∞ , we obtain from equations (25) and (26) that the temperature T = β -1 and electrostatic potential φ go to zero as T ∼ r 2 + -e 2 +3 α 2 r 4 + 4 π α r 3 + r -1 B and φ ∼ e αr + r -1 B , respectively. In this case the thermal energy goes to zero as / \nE ∼ M ( 1 + α 2 r 2 B ) -1 / 2 , where M is the mass of the black hole given in (13). As also previously found in [12], the thermal energy at infinity is not equal to the ADM mass of the black hole, due to the the fact that the spacetime is not asymptotically flat. Note that in [13,17] it is found that E = M due to a different definition of the temperature of the ensemble. \nTo determine the stability of the black hole solutions in this limit, we compute the heat capacity, from equation (51) and obtain \nC φ = 2 π r 2 + ( r 2 + -e 2 +3 α 2 r 4 + ) e 2 -r 2 + +3 α 2 r 4 + . (53) \nThe numerator in (53) is necessarily positive due to condition (32), that r + must obey to represent the event horizon radius. Therefore, the stability condition for these solutions is simply e 2 -r 2 + +3 α 2 r 4 + > 0. Notice that equation (53) is a generalization to e /negationslash = 0 of the heat capacity found by Hawking and Page in [17]. Computing the action in this limit yields \nI = π r 2 + ( r 2 + -e 2 -α 2 r 4 + ) r 2 + -e 2 +3 α 2 r 4 + . (54) \nThis is precisely Hawking-Page action [17,13], which here we have recovered from York's formalism taking the appropriate limit. \nFrom equation (54), we can verify the global stability of the black hole by imposing that I < 0. Therefore we conclude that the solution given by ( r + , e ) corresponds to a globally stable black hole if conditions r 2 + -e 2 -α 2 r 4 + < 0 and (32) are both verified. These conditions are similar to those found in [13]. \nWe conclude that taking different boundary conditions, i.e. choosing to fix the boundary conditions in the boundary at infinity (as done here) or in the region where spacetime is flat like Hawking and Page [17], yields a different value for the energy only. Furthermore this difference is so that all the other physical quantities (like the action, entropy, mean-value of the charge and heat capacity) remain the same. \nOn the other hand, we can take a different limit: (ii) we can fix the boundary conditions, i.e. β and φ , when taking the limit. This is done by recovering the variables β and α in equation (31), using (27). Taking the limit r B →∞ , we obtain the equation α 2 β 2 ( φ 2 (1 + 2 x +3 x 2 ) -3 x 2 ) 2 -x 2 (1 + x + x 2 ) (1 -x + φ 2 x ) = 0. This is a 5th degree equation in x . For fixed β , φ and α and taking the limit r B →∞ , we obtain a solution r + that tends to infinity like r + ∼ c + r B , where c + is a constant that depends only on β , φ and α . Considering now equation (30), we can see that the charge e goes to infinity as e ∼ c e r 2 B , where again c e is a constant that depends only on β , φ and α . Therefore the entropy (24) and the mean-value of the charge (23) both go to infinity as r 2 B . In this limit the action, the thermal energy and the heat capacity, given in (21), (22) and (51) respectively, also diverge as r 2 B . The heat capacity is always positive, which means these solutions are stable. The action is always positive, therefore the solutions are not globally stable and represent metastable black holes.", 'VII. COMMENTS ON SPECIAL CASES': 'Several black holes may be considered as special cases of the Reissner-Nordstrom-anti-de Sitter black hole. \n- (I) putting φ = 0 and Λ = 0, we obtain the Schwarzschild black hole studied in [5]. There are two solutions for β < β max = 2 / √ 27. This solutions can be computed analytically, since equation (31) becomes a 3rd degree equation for Λ = 0. Only the solution with higher event horizon radius, i.e. higher mass, is stable.\n- In the limit r B → ∞ the unstable solution, i.e. the solution with lower value of the horizon radius, goes to r + = 1 / (4 πr + ), while the stable solution goes to infinity as r + ∼ r B [5].\n- (II) putting Λ = 0 we obtain the Reissner-Nordstrom black hole. This has been studied in [11]. There are one or two solution for β < β max . These as stated above can be computed analytically. Once again only the solution with higher event horizon radius, when it exists, is stable.\n- In the limit r B → ∞ , the black hole horizon radius is r + = β 4 π ( 1 -φ 2 ) and the charge is given by e = φr + . The thermal energy of these solutions is equal to their ADM mass E = M . The heat capacity is negative C φ = -2 πr 2 + . therefore the solutions are unstable.\n- (III) putting φ = 0 we obtain the Schwarzschild-anti-de Sitter. This black hole has been studied before in [17,12]. This black hole has two solutions for β < β max , and again only the one with higher event horizon radius is stable.\n- The limit r B →∞ can be taken in two ways: ( i ) fixing the temperature and the cosmological constant, there are one unstable solution that tends to zero as r + ∼ β 4 πα r -1 B and one stable solution that tends to infinity as 3 2 2\n- r + ∼ cr B , where c is the solution of equation c 3 + ( 3 αβ 4 π ) c 2 -1 = 0; ( ii ) fixing the horizon radius r + , the temperature goes to zero as T ∼ 1+3 α 2 r 2 + 4 παr + r -1 B , these solutions are stable if 3 α 2 r 2 + > 1, see [12].\n- (IV) the extremal cases require special care [23-25] and were not studied in any detail in this paper.', 'VIII. CONCLUSIONS': "We have studied the thermodynamics of the Reissner-Nordstrom-anti-de Sitter black hole in York's formalism. In the grand canonical ensemble where the temperature and the electrostatic potential are fixed at a boundary with finite radius, we have found one or two black hole solutions depending on the boundary conditions and the value of the cosmological constant. In general when there are two solutions, one is stable, the one with larger event horizon radius, and the other is an instanton. On the other hand, the cases with a single solution can correspond either to a stable or unstable solution. We have found that both high values of the cosmological constant and low temperatures favor stable solutions.", 'ACKNOWLEDGMENTS': "C.S.P. acknowledges a research grant from JNICT FMRH/BIC/1535/95. \n- [1] J. B. Hartle and S. W. Hawking, Phys. Rev. D 13 , 2188 (1976).\n- [2] G. W. Gibbons and S. W. Hawking, Phys. Rev. D 15 , 2752 (1977).\n- [3] S. W. Hawking, in General Relativity: An Einstein Centenary Survey , edited by S. W. Hawking and W. Israel (Cambridge University Press, Cambridge, 1979).\n- [4] S. W. Hawking, Phys. Rev. D 13 , 191 (1976).\n- [5] J. W. York, Phys. Rev. D 33 , 2091 (1986).\n- [6] B. F. Whiting and J. W. York, Phys. Rev. Lett. 61 , 1336 (1988).\n- [7] B. F. Whiting, Class. Quantum Grav. 7 , 15 (1990).\n- [8] J. D. Brown and J. W. York, in The Black Hole 25 Years After , edited by C. Teitelboim and J. Zanelli (Plenum, New York, in press), gr-qc/9405024.\n- [9] J. D. E. Creighton, 'Gravitational Calorimetry', PhD. thesis, (Waterloo, 1996), gr-qc/9610038.\n- [10] J. D. Brown et al. , Class. Quantum Grav. 7 , 1433 (1990).\n- [11] H. W. Braden, J. D. Brown, B. F. Whiting, and J. W. York, Phys. Rev. D 42 , 3376 (1990).\n- [12] J. D. Brown, J. Creighton, and R. B. Mann, Phys. Rev. D 50 , 6394 (1994).\n- [13] J. Louko and S. N. Winters-Hilt, Phys. Rev. D 54 , 2647 (1996).\n- [14] J. P. S. Lemos, Phys. Rev. D 54 , 6206 (1996).\n- [15] J. W. York, Physica A 158 , 425 (1989).\n- [16] H. B. Callen, Thermodynamics and an Introduction to Thermostatistics (John Wiley & Sons, New York, 1985).\n- [17] S. W. Hawking and D. N. Page, Commun. Math. Phys. 87 , 577 (1983).\n- [18] D. N. Page, in Black Hole Physics , edited by V. D. Sabbata and Z. Zhang (Kluwer Academic, Dordrecht, 1992).\n- [19] S. W. Hawking, Commun. Math. Phys. 43 , 199 (1975).\n- [20] J. D. Bekenstein, Phys. Rev. D 7 , 2333 (1973).\n- [21] G. W. Gibbons and S. W. Hawking, Commun. Math. Phys. 66 , 291 (1979).\n- [22] D. J. Gross, M. J. Perry and L. G. Yaffe, Phys. Rev. D 25 , 330 (1982).\n- [23] S. W. Hawking, G. T. Horowitz, and S. F. Ross, Phys. Rev. D 51 , 4302 (1995).\n- [24] O. B. Zaslavskii, Phys. Rev. Lett. 76 , 2211 (1996).\n- [25] A. Ghosh and P. Mitra, Phys. Rev. Lett. 78 , 1858 (1997). \nFIG. 1. Solutions of equation (31) for α = 0 (Reissner-Nordstrom) as a function of the electrostatic potential at the boundary φ for fixed values of β = 0 . 1 , 0 . 3 , 0 . 6 , 0 . 9 , 3 , 9. The stable solutions correspond to the upper branch of the curves. This means that when there are 2 solutions for given values of β and φ , only the solution with higher value of x is stable. \n<!-- image --> \nFIG. 2. Solutions of equation (31) for α = 0 . 5 as a function of the electrostatic potential at the boundary φ for fixed values of β = 0 . 1 , 0 . 3 , 0 . 6 , 0 . 9 , 3 , 9. Notice that φ /lessorsimilar . 89, as imposed by condition (34). The stable solutions correspond to the upper branch of the curves. This means that when there are 2 solutions for given values of β and φ , only the solution with higher value of x is stable. \n<!-- image --> \nFIG. 3. Solutions of equation (31) for α = 1 as a function of the electrostatic potential at the boundary φ for fixed values of β = 0 . 1 , 0 . 3 , 0 . 6 , 0 . 9 , 3 , 9. The maximum value of φ for which there are solutions, φ max /similarequal 0 . 76 is imposed by condition (34). The stable solutions correspond to the upper branch of the curves. This means that when there are 2 solutions for given values of β and φ , only the solution with higher value of x is stable. \n<!-- image --> \nFIG. 4. Solutions of equation (31) for α = 5 as a function of the electrostatic potential at the boundary φ for fixed values of β = 0 . 1 , 0 . 3 , 0 . 6 , 0 . 9 , 3 , 9. The maximum value of φ for which there are solutions, φ max /similarequal 0 . 71, is imposed by condition (34). The stable solutions correspond to the upper branch of the curves. This means that when there are 2 solutions for given values of β and φ , only the solution with higher value of x is stable. \n<!-- image --> \nFIG. 5. Number of solutions of equation (31) with β → 0, in the space φ × a , where a is given by equation (40). There is one black hole solution in the confined region and also for a = 1 (i.e. infinite cosmological constant), for φ < √ 0 . 5. There are two solutions for φ = 0, i.e. for the Schwarzschild-anti-de Sitter black hole. \n<!-- image --> \nFIG. 6. Number of solutions of equation (31) with β = 0 . 3, in the space φ × a , where a is given by equation (40) 0 - means that there are zero solutions in this region, i.e. there are no solutions of black holes in thermodynamical equilibrium for this set of values of β , φ and a . 1 - means that there is one solution. 2 - means that there are two solutions. In the shaded region the stable solutions are not globally stable and therefore represent metastable black holes (see subsection V B). \n<!-- image --> \nFIG. 7. Number of solutions of equation (31) with β = 2 √ 27 /similarequal 0 . 38, in the space φ × a , where a is given by equation (40) (see caption of figure 6). \n<!-- image --> \nFIG. 8. Number of solutions of equation (31) with β = 1, in the space φ × a , where a is given by equation (40) (see caption of figure 6). \n<!-- image --> \nFIG. 9. Number of solutions of equation (31) with β = ∞ , in the space φ × a , where a is given by equation (40). Notice that for β → ∞ , equation (31) becomes (35) which corresponds to the extremal Reissner-Nordstrom-anti-de Sitter black hole, see discussion following equation (35) (see caption of figure 6). \n<!-- image -->"}

2004PhRvD..70j4016C

Excision boundary conditions for black-hole initial data

2004-01-01

['-', '-', '-', '-', 'methods numerical', '-', '-', '-', '-']

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We define and extensively test a set of boundary conditions that can be applied at black-hole excision surfaces when the Hamiltonian and momentum constraints of general relativity are solved within the conformal thin-sandwich formalism. These boundary conditions have been designed to result in black holes that are in quasiequilibrium and are completely general in the sense that they can be applied with any conformal three-geometry and slicing condition. Furthermore, we show that they retain precisely the freedom to specify an arbitrary spin on each black hole. Interestingly, we have been unable to find a boundary condition on the lapse that can be derived from a quasiequilibrium condition. Rather, we find evidence that the lapse boundary condition is part of the initial temporal gauge choice. To test these boundary conditions, we have extensively explored the case of a single black hole and the case of a binary system of equal-mass black holes, including the computation of quasicircular orbits and the determination of the innermost stable circular orbit. Our tests show that the boundary conditions work well.

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https://arxiv.org/pdf/gr-qc/0407078.pdf

{'Excision boundary conditions for black hole initial data': 'Gregory B. Cook ∗ \nDepartment of Physics, Wake Forest University, Winston-Salem, North Carolina 27109', 'Harald P. Pfeiffer †': 'Theoretical Astrophysics, California Institute of Technology, Pasadena, California 91125 (Dated: September 7, 2018) \nWe define and extensively test a set of boundary conditions that can be applied at black hole excision surfaces when the Hamiltonian and momentum constraints of general relativity are solved within the conformal thin-sandwich formalism. These boundary conditions have been designed to result in black holes that are in quasiequilibrium and are completely general in the sense that they can be applied with any conformal three-geometry and slicing condition. Furthermore, we show that they retain precisely the freedom to specify an arbitrary spin on each black hole. Interestingly, we have been unable to find a boundary condition on the lapse that can be derived from a quasiequilibrium condition. Rather, we find evidence that the lapse boundary condition is part of the initial temporal gauge choice. To test these boundary conditions, we have extensively explored the case of a single black hole and the case of a binary system of equal-mass black holes, including the computation of quasi-circular orbits and the determination of the inner-most stable circular orbit. Our tests show that the boundary conditions work well. \nPACS numbers: 04.20.-q, 04.25.Dm, 04.70.Bw, 97.80.-d', 'I. INTRODUCTION': "The simulation of black-hole systems necessarily starts with the specification of initial data. In order for such simulations to yield astrophysically relevant results, the initial data must be constructed to be astrophysically realistic. Achieving this is the goal of efforts being made to improve black-hole, and in particular black-hole binary, initial data. It has become clear that all of the freely specifiable pieces of the initial data, including the boundary conditions, must be chosen carefully to respect the physical content of any system we wish to simulate. In this paper we will focus on the boundary conditions that are required when a black hole's interior is excised from the initial-data domain. \nIn Ref. [1], one of the authors proposed a set of boundary conditions that were intended to yield a black hole that was in quasiequilibrium. These conditions were chosen to be consistent with the desire to create a binary system fully in quasiequilibrium. It is reasonable to expect such a system will be astrophysically realistic if the black holes in the binary are sufficiently far apart and in a nearly circular orbit. In this paper, we refine and extensively test these boundary conditions in the cases of a single black hole and a pair of equal-mass black holes in a binary system. \nThe most significant refinement of the quasiequilibrium boundary conditions over the original version in Ref. [1] is to the procedure for specifying the spin of each black hole. The analysis below shows that the spin must \nbe chosen in a very specific way in order to be compatible with the assumptions of quasiequilibrium. Fortunately, the procedure still allows for a completely arbitrary specification of the spin and this procedure is compatible with any choice of the conformal three-geometry. \nA significant result from our tests on the original set of quasiequilibrium boundary conditions is that the proposed lapse boundary condition is not viable. We will show below that this boundary condition is degenerate when combined with the other quasiequilibrium boundary conditions. Furthermore, the nature of this degeneracy can be easily understood. We conjecture that the boundary condition on the lapse is not fixed by quasiequilibrium considerations but is, rather, a part of the initial temporal gauge choice. Below, we provide analytical and numerical evidence to support this conjecture. \nFor a single black hole, the quasiequilibrium boundary conditions allow for the construction of initial data that yield true stationary spacetimes. Doing so, however, requires that all of the freely specifiable data be chosen in a way that is compatible with a stationary black hole. In particular, it requires that the conformal threegeometry be chosen correctly. Unfortunately, there is still no general prescription for choosing an appropriate conformal three-geometry. Because of this, we have chosen to perform all of our tests on a flat conformal threegeometry. By testing the boundary conditions for the three cases of a single static, spinning, and boosted black hole, we will be able to test both the quasiequilibrium boundary conditions and the effect that the assumption of conformal flatness has on the resulting initial data. For equal-mass black hole binaries, we will extensively test the special cases of corotating and irrotational black holes. The numerical results we obtain will be compared with post-Newtonian results and previous numerical re- \nsults for both cases. \nWe begin in Sec. II with a review of the conformal thinsandwich decomposition of the constraints, and then derive the quasiequilibrium boundary conditions in Sec. III. In Secs. IV and V we apply the boundary conditions to the cases of a single black hole and to equal-mass black hole binary systems. Finally, in Sec. VI we further explore the effectiveness of these boundary conditions.", 'II. THE CONFORMAL THIN-SANDWICH DECOMPOSITION': "In this work, we will use the standard 3+1 decomposition with the interval written as \nd s 2 = -α 2 d t 2 + γ ij (d x i + β i d t )(d x j + β j d t ) , (1) \nwhere γ ij is the 3-metric induced on a t = const. spatial hypersurface, α is the lapse function, and β i is the shift vector. The extrinsic curvature of the spatial slice, K ij , is defined by \nK µν ≡ -1 2 γ δ µ γ ρ ν L n γ δρ , (2) \nwhere L n denotes the Lie derivative along the unit normal to the spatial slice, n µ . Einstein's equations, in vacuum, then reduce to four sets of equations. Two are evolution equations for the spatial metric and extrinsic curvature: \n∂ t γ ij = -2 αK ij +2 ¯ ∇ ( i β j ) , (3) \nand \n∂ t K ij = -¯ ∇ i ¯ ∇ j α + α [ ¯ R ij -2 K i/lscript K /lscript j + KK ij ] + β /lscript ¯ ∇ /lscript K ij +2 K /lscript ( i ¯ ∇ j ) β /lscript . (4) \nThe remaining two are the constraint equations \n¯ R + K 2 -K ij K ij = 0 (5) \nand \n¯ ∇ j ( K ij -γ ij K ) = 0 . (6) \nHere, ¯ ∇ i , ¯ R ij , and ¯ R are, respectively, the covariant derivative, Ricci tensor, and Ricci scalar associated with the spatial metric γ ij . Finally, the trace of the extrinsic curvature is denoted K K i i . \nThe task of constructing initial data for a Cauchy evolution via Einstein's equations requires that we decompose the constraints in such a way that we can specify how the constrained, gauge, and dynamical degrees of freedom are associated with the initial data. In this paper, we are primarily interested in initial data associated with systems in quasiequilibrium. Because of this, it is natural to use the the conformal thin-sandwich decomposition of the constraints[2, 3]. This decomposition is \n≡ \nparticularly useful in this situation because quasiequilibrium is a dynamical concept, and this decomposition retains a close connection to dynamics that is lost in most other decompositions of the constraints (cf Refs. [1, 4]). \nThe conformal thin-sandwich decomposition employs a York-Lichnerowicz conformal decomposition of the metric and various other quantities[5, 6, 7]. The conformal factor, ψ , is defined via \nγ ij ≡ ψ 4 ˜ γ ij , (7) \nwhere ˜ γ ij is a 'conformal metric'. The time derivative of the conformal metric is introduced by the definitions \n˜ u ij ≡ ∂ t ˜ γ ij , (8) \n˜ γ ij ˜ u ij ≡ 0 . (9) \nFrom this, it follows that the tracefree extrinsic curvature A ij ≡ K ij -1 3 γ ij K takes the form \nA ij = ψ -10 2˜ α [ ( ˜ L β ) ij -˜ u ij ] , (10) \nwhere ˜ α ≡ ψ -6 α is the conformal lapse function, and ˜ u ij = ˜ u kl ˜ γ ik ˜ γ jl . Furthermore, ( ˜ L V ) is the conformalKilling (or longitudinal) operator acting on a vector, defined by \n( ˜ L V ) ij ≡ 2 ˜ ∇ ( i V j ) -2 3 ˜ γ ij ˜ ∇ k V k , (11) \nwhere ˜ ∇ k is the covariant derivative compatible with ˜ γ ij . Notice that this decomposition of K ij incorporates the kinematical variables of the 3+1 decomposition, that is, the shift vector β i and the lapse function α through the conformal lapse ˜ α . It also includes the trace-free time derivative of the conformal metric, ˜ u ij . Below, the conformal tracefree extrinsic curvature will be useful, \n˜ A ij ≡ ψ 10 A ij = 1 2˜ α [ ( ˜ L β ) ij -˜ u ij ] . (12) \nWithin the conformal thin-sandwich formalism, one must specify ˜ γ ij , ˜ u ij , K , and ˜ α . With these quantities defined, the Hamiltonian (5) and momentum (6) constraints take the form of a coupled set of elliptic equations that determine ψ and β i . In terms of our conformally decomposed variables, the Hamiltonian constraint (5) can be written \n˜ ∇ 2 ψ -1 8 ψ ˜ R -1 12 ψ 5 K 2 + 1 8 ψ -7 ˜ A ij ˜ A ij = 0 , (13) \nwhere ˜ R is the Ricci scalar associated with ˜ γ ij , and the momentum constraint (6) as \n˜ ∇ j ( 1 2˜ α ( ˜ L β ) ij ) -2 3 ψ 6 ˜ ∇ i K -˜ ∇ j ( 1 2˜ α ˜ u ij ) = 0 . (14) \nThe freely-specified data includes the conformal metric, ˜ γ ij , and its time derivative, ˜ u ij = ∂ t ˜ γ ij , as well as the trace of the extrinsic curvature, K , and the conformal lapse, ˜ α . It is possible, and desirable, to make the \nset of freely-specified data more symmetric by choosing to specify the time derivative of the extrinsic curvature instead of the conformal lapse. This is possible because these two quantities are related by the trace of Eq. (4). The resulting equation is an elliptic equation for the conformal lapse that is coupled to both the Hamiltonian and momentum constraints, (13) and (14). This equation can be written as \n˜ ∇ 2 ( ψ 7 ˜ α ) -( ψ 7 ˜ α ) ( 1 8 ˜ R + 5 12 ψ 4 K 2 + 7 8 ψ -8 ˜ A ij ˜ A ij ) = -ψ 5 ( ∂ t K -β k ˜ ∇ k K ) . (15) \n≡ \nThe statement made earlier, that the conformal thinsandwich decomposition has a close connection to dynamics is now clear. Not only does this decomposition incorporate the kinematical variables of the 3+1 splitting, but fully half of the freely-specifiable data consist of time derivatives of fundamental fields. In particular, we are free to choose the conformal metric and the trace of the extrinsic curvature (˜ γ ij and K ) and the time derivatives of these fields ( ∂ t ˜ γ ij ˜ u ij and ∂ t K ). \nRemaining to be determined are the conformal factor, ψ , the conformal lapse, ˜ α , and the shift vector, β i . These are determined by solving Eqs. (13), (14), and (15) as a coupled set of elliptic equations. Formulating a wellposed system requires that we impose boundary condition. Typically, these systems are solved under the assumption that the spacetime is asymptotically flat. If we let r denote a coordinate radius measured from the location of the center of energy of the system, then as r →∞ we have that \nψ ∣ r →∞ = 1 , (16a) \n∣ ∣ Ω 0 is the orbital angular velocity of a binary system, or the rotational angular velocity of a single object, as measured at infinity. The boundary condition on the shift is chosen so that the time coordinate, t µ = αn µ + β µ , is helical and tracks the rotation of the system[1, 8, 9]. If we wish to consider systems with one or more black holes, and if we excise the interior of the black hole to avoid difficulties with singularities, then we will also need to impose boundary conditions on the excision surfaces. This is the topic of the next section. \n∣ →∞ β i ∣ r = ( Ω 0 × r ) i , (16b) \n∣ ∣ ∣ \n∣ α ∣ r →∞ = ˜ α ∣ r →∞ = 1 . (16c) \n∣", 'III. BLACK-HOLE EXCISION BOUNDARY CONDITIONS': "The physical content of initial data depends on the choices made for the initial-data decomposition scheme, freely specifiable data, and the boundary conditions. Therefore, it is important to choose boundary conditions that are motivated by, or at least compatible with, the sort of initial data that we wish to construct. \n→∞ \nThe first attempts to impose boundary conditions on black hole excision boundaries were based on topological arguments[10, 11, 12, 13]. By demanding that the initial-data hypersurface consist of two identical (isometric) asymptotically flat hypersurfaces connected together at a number of spherical excision surfaces (one for each black hole), it is possible to show that the surfaces where the isometric sheets connect are fixed point sets of the isometry. This condition is enough to determine either Dirichlet or Neumann boundary conditions at the excision surface for any fields that are present. \nBoundary conditions based on this idea have been used successfully for generating general black-hole initial data using various initial-data decompositions[14, 15, 16, 17]. Their first use in conjunction with the conformal thinsandwich decomposition[9, 18] was only partially successful due to an unavoidable constraint violation. The difficulties with this approach were outlined in Ref. [1], where an alternative approach of using quasiequilibrium boundary conditions was first outlined. In this section, we will refine, and in subsequent sections test, this approach. \nIn constructing initial data on a spacelike hypersurface, we cannot have knowledge of the event horizon that is typically used to define the surface of a black hole. However, we can identify the apparent horizon of a black hole, defined as the outermost marginally outer-trapped surface. A marginally outer-trapped surface (MOTS), in turn, is a surface on which the expansion, θ , of the family of outgoing null rays, k µ , vanishes everywhere. \nIn this paper, we are interested in the situation in which each black hole is in quasiequilibrium. The assumptions required to enforce this are essentially the same as those required of an 'isolated horizon'(cf [19, 20, 21]). To ensure that the black hole is in quasiequilibrium, we enforce the following conditions. First, we demand that the expansion θ , vanish on the excision surface, S , thus forcing the boundary to be an apparent horizon: \nθ ∣ S = 0 . (17) \n∣ Next, we require that the shear σ µν of the outgoing null rays also vanish on the excision boundary, \n∣ \nσ µν ∣ S = 0 . (18) \n∣ Consider the family of null geodesics threading the apparent horizon to the future of our initial-data slice that are tangent to k µ on S . Raychaudhuri's equation for null congruences, \n∣ \nL k θ = -1 2 θ 2 -σ µν σ µν + ω µν ω µν -R µν k µ k ν , (19) \ntogether with Eqs. (17) and (18) are sufficient to imply that \nL k θ ∣ S = 0 . (20) \n∣ \n∣ That is, initially, the apparent horizon will evolve along k µ . Note that ω µν is the twist of the congruence, which \nvanishes because the congruence is surface forming. Also, we assume that there in no matter on , so R µν = 0. \nWhile conditions (17), (18) and (20) are coordinate independent, our next and final demand breaks precisely this coordinate freedom: We demand that the coordinate location of the apparent horizon does not move initially in an evolution of the initial data. \nS \nAs we show in the subsequent sections, the requirements listed so far yield four conditions that can be imposed on the initial data at the excision boundary. However, there are five coupled initial-data equations that must be solved in the conformal thin-sandwich approach. When quasiequilibrium black-hole boundary conditions were first derived, a fifth condition was considered[1]. In particular, the condition that L k ' θ = 0 was considered, where ' θ is the expansion of a family of ingoing null rays, ' k µ . As we will show below, this fifth condition cannot be used as a boundary condition even though it is satisfied for a stationary black hole ! \nIn the remainder of this section, we will derive boundary conditions for black hole excision surfaces based on the demands outlined above. A good portion of the following derivation appeared previously[1]. However, because of a change in notation, and more importantly in a few sign conventions, we include the full derivation below.", 'A. Geometry of the excision boundary': "We demand that the excision boundary surface, S , be a spacelike 2-surface with topology S 2 and define s i to be the outward pointing unit vector normal to the surface. In this case, we define outward with respect to the black hole (not the domain), so that s i points toward infinity. The 4-dimensional generalization of s i has components s µ = [0 , s i ] obtained from the condition that s µ n µ = 0. \nThe metric, h ij , induced on S by γ ij is given by \nh ij ≡ γ ij -s i s j . (21) \nWe also define the extrinsic curvature, H ij , of S embedded in the 3-dimensional spatial hypersurface as \nH ij ≡ h k i h /lscript j ¯ ∇ ( k s /lscript ) = 1 2 h k i h /lscript j L s h k/lscript . (22) \nNaturally associated with S are two sets of null vectors: a set of outgoing null rays, k µ , and a set of ingoing null rays, ' k µ , defined by \nk µ ≡ 1 √ 2 ( n µ + s µ ) and ' k µ ≡ 1 √ 2 ( n µ -s µ ) . (23) \nAssociated with each set of null rays is an extrinsic curvature of S as embedded in the full 4-dimensional manifold. These are defined as \nΣ µν ≡ 1 2 h α µ h β ν L k g αβ , (24) \n' Σ µν ≡ 1 2 h α µ h β ν L ' k g αβ , (25) \nwhere g µν is the full spacetime metric. Because these tensors Σ µν and ' Σ µν are spatial, we will use spatial indices below. To simplify the definitions that follow, we will introduce various projections of K ij along and normal to the excision boundary S : \nJ ij ≡ h k i h /lscript j K k/lscript , (26) \n≡ J ≡ h ij J ij = h ij K ij . (28) \nJ i ≡ h k i s /lscript K k/lscript , (27) \nWe can then simplify Eqs. (24) and (25) to \nΣ ij = -1 √ 2 ( J ij -H ij ) and ' Σ ij = -1 √ 2 ( J ij + H ij ) . (29) \nNow, we define the expansion of outgoing null rays, θ , and ingoing null rays, ' θ , via \nθ ≡ h ij Σ ij = -1 √ 2 ( J -H ) , (30) \n' θ ≡ h ij ' Σ ij = -1 √ 2 ( J + H ) . (31) \nFinally, we define the shear of the outgoing null rays, σ ij , and ingoing null rays, ' σ ij , via \nσ ij ≡ Σ ij -1 2 h ij θ, (32) \n' σ ij ≡ ' Σ ij -1 2 h ij ' θ. (33)", 'B. Quasiequilibrium boundary conditions': 'With the definitions of Sec. III A, we can now evaluate the demands we made earlier in Sec. III and translate them into boundary conditions for the conformal thinsandwich equations. In order to express these as useful boundary conditions, we must write them in terms of the variables of the conformal thin-sandwich approach. We must also make connection with the global notion of quasiequilibrium that is closely associated with an approximate helical Killing vector. \nA spacetime that is in true equilibrium is said to be stationary and has two Killing vectors of interest: a timelike Killing vector, ∂/∂t 0 , and a spatial Killing vector associated with rotational symmetry, ∂/∂φ 0 . If Ω 0 denotes the angular velocity of a spinning object or system as measured at infinity, then the linear combination ∂/∂t 0 +Ω 0 ∂/∂φ 0 is referred to as the helical Killing vector . If a system, such as a binary, is in a state of quasiequilibrium, there are in general no vector fields similar to ∂/∂t 0 or ∂/∂φ 0 that are even close to being Killing vectors. But there will be a helical vector field that is an approximate Killing vector of the spacetime. If we let this approximate Killing vector field define our time vector, t µ , and thus our time coordinate t , then we will have ∂/∂t ≈ 0 for fields in this spacetime. Within the 3+1 decomposition, we write the time vector as \nt µ = αn µ + β µ . (34) \nOur desire for t µ to represent an approximate helical Killing vector is, therefore, the reason for our condition on the shift at infinity, Eq. (16b). \nNow we consider the demand that the apparent horizon should initially not move in an evolution of the quasi equilibrium initial data. Because of Eq. (20), the apparent horizon initially coincides with the null surface generated by k µ . In order for the coordinates to track this null surface, the time-vector of the evolution, t µ , must lie in this null surface. This requires that \n∣ \nt µ k µ ∣ ∣ S = 0 . (35) \n∣ Substituting Eqs. (34) and (23) into Eq. (35), and recalling that the shift vector is spatial, β µ n µ = 0, yields \n∣ \nα ∣ S = β i s i ∣ S . (36) \n∣ ∣ This equation is often referred to as the Killing-horizon condition. We split the shift vector into its component normal to the surface, β ⊥ , and a vector tangent to the surface, β ‖ i , defined by \n∣ \nβ ⊥ ≡ β i s i , (37) \nβ ‖ i ≡ h i j β j . (38) \nWith these definitions, we see that Eq. (36) is a condition on the normal component of the shift, \n∣ \nβ ⊥ ∣ S = α ∣ S . (39) \n∣ ∣ The component of the shift tangential to the excision surface S , β ‖ i , remains unconstrained so far. This makes sense, because fixing the location of a surface does not restrict motion within this surface. We can gain insight into the relevance of β ‖ i by considering a stationary Kerr black hole. \n∣ \nThe Kerr spacetime has two Killing vectors of interest: A timelike Killing vector, ∂/∂t 0 , and a spatial Killing vector associated with rotational symmetry, ∂/∂φ 0 . The null generators of the horizon are given by \nk = ∂ ∂t 0 +Ω H ∂ ∂φ 0 , (40) \nwhere Ω H is the angular frequency of the horizon. If we introduce a fiducial helical Killing vector, \n/lscript = ∂ ∂t 0 +Ω ∂ ∂φ 0 , (41) \nfor some Ω, then, of course, on the horizon, \n/lscript = k +(Ω -Ω H ) ∂ ∂φ 0 . (42) \nNow consider a hypersurface through Kerr to which ∂/∂φ 0 is always tangent, e.g., the usual Kerr-Schild slice. To make the connection with the usual 3+1 decomposition straightforward, we normalize k and /lscript by choosing \ntheir time components to be one ( k t = /lscript t = 1). If we choose the vector /lscript as the time vector of an evolution, then the last term in Eq. (42) corresponds precisely to β ‖ i - this term is tangent both to the horizon and to the hypersurface. For the choice β ‖ i = 0, the last term in Eq. (42) would be absent, i.e. Ω = Ω H . In this case the black hole is corotating with the coordinate system, and the generators of the horizon, k µ do not twist relative to the helical Killing vector, /lscript µ . Conversely, a non-rotating black hole (Ω H = 0) would have a tangential shift of \nβ ‖ i = Ω ( ∂ ∂φ 0 ) i . (43) \nFor a binary black hole in quasiequilibrium, neither ∂/∂t 0 nor ∂/∂φ 0 exist as separate Killing vectors. However, based on the discussion above, we expect that β ‖ i is still connected to the rotation of the black hole. In the case β ‖ i = 0, the horizon generators have no twist relative to the helical Killing vector (which coincides with the time-vector), corresponding to corotating black holes. Any rotation, β ‖ i ∝ ( ∂/∂φ S ) i , where ( ∂/∂φ S ) i lies in the surface S would impart additional rotation on the black hole. Below we will make these notions precise. \nHaving obtained a boundary condition on β ⊥ , we now turn our attention to Eqs. (17) and (18). We need to consider the horizon boundary S in the conformal space. The conformal transformation on γ ij (7) induces a natural conformal weighting for h ij and for the unit normal to S , \nh ij ψ 4 ˜ h ij , (44) \ns i ≡ ψ -2 ˜ s i . (45) \n≡ \nIf we also define ˜ D i as the covariant derivative compatible with ˜ h ij , then without loss of generality, we can express the expansion of the outgoing null rays, θ , as \nθ = ψ -2 √ 2 ( ˜ h ij ˜ ∇ i ˜ s j +4˜ s k ˜ ∇ k ln ψ -ψ 2 J ) , (46) \nand the shear of the outgoing null rays, σ ij , as \nσ ij = 1 √ 2 ( H ij -1 2 h ij H ) ( 1 -β ⊥ α ) -1 √ 2 ψ -4 α [ ˜ D ( i β ‖ j ) -1 2 ˜ h ij ˜ D k β ‖ k (47) -1 2 ( ˜ h i k ˜ h j /lscript ˜ u k/lscript -1 2 ˜ h ij ˜ h k/lscript ˜ u k/lscript ) ] . \nIt is now clear how to obtain the remaining boundary conditions. By applying condition (17) to Eq. (46), we obtain a boundary condition that forces an excision boundary to be an apparent horizon (or MOTS). The condition is \n˜ s k ˜ ∇ k ln ψ ∣ ∣ ∣ S = -1 4 ( ˜ h ij ˜ ∇ i ˜ s j -ψ 2 J )∣ ∣ ∣ S , (48) \nand it takes the form of a nonlinear Robin-type boundary condition on the conformal factor, ψ . Finally, if we recall the condition from Eq. (39) and that we have chosen ˜ u ij = 0 everywhere, and apply condition (18) to Eq. (47), we obtain a condition that restricts the form of β ‖ i . The condition is that \n˜ D ( i β ‖ j ) ∣ ∣ S -1 2 ˜ h ij ˜ D k β ‖ k ∣ ∣ ∣ S = 0 . (49) \n∣ \n∣ This shows that the components of the shift that are associated with the spin of the black hole must be proportional to a conformal Killing vector of the conformal metric, ˜ h ij , defined on the 2-dimensional excision surface. \nThis condition is quite remarkable. Recall that any 2-surface that is topologically S 2 is conformally equivalent to the unit 2-sphere. If you consider a unit 2-sphere embedded in a flat 3-dimensional Euclidean space, then there is a family of rotational Killing vectors, ξ i , associated with any rotation axis passing through the center of the 2-sphere. Because ξ i ˆ n i = 0 on the 2-sphere, where ˆ n i is the unit normal vector on the 2-sphere, we see that ξ i trivially form a family of 2-dimensional vectors tangent to the 2-sphere and that these are Killing vectors of the metric on the unit 2-sphere. But, the Killing vectors associated with any metric are also conformal Killing vectors of any metric conformally related to it. So, if ϕ represents a conformal transformation such that ϕ 4 ˜ h ij is the metric of the unit 2-sphere, then ξ i will satisfy the conformal Killing equation on ˜ h ij . Thus, \nβ ‖ i = Ω r ξ i (50) \nwill satisfy Eq. (49), with Ω r being an arbitrary parameter. The freedom left in Eq. (50) is precisely what is required to parameterize an arbitrary spin on the black hole. The parameter Ω r is associated with the magnitude of the rotation or spin of the black hole, whereas the axis of rotation of ξ i is related to the orientation of the spin. Of course, Ω r does not correspond directly to the rotational angular velocity of the black hole. From the discussion leading to Eq. (43), it is clear that Ω r = 0 corresponds to a black hole that is corotating with an approximate helical Killing vector t µ and thus represents a black hole that is rotating as seen by an inertial observer at infinity. In order to construct a black hole that is not rotating as seen from infinity, we need to choose a shift that is similar in form to Eq. (43). It seems reasonable to choose Ω r = Ω 0 and to pick the conformal Killing vector, ξ i from the family that corresponds to rotation about an axis that is perpendicular to the plane of the orbit. \nTo summarize, the quasiequilibrium conditions defined in Eqs. (17) and (18) define boundary conditions on the conformal factor, ψ , via Eq. (48) and on the shift vector, β i , via Eqs. (37), (38), (39), and (50). These total to four of the five necessary boundary conditions for solving the coupled elliptic equations associated with the conformal thin-sandwich equations. Missing is a condition on the conformal lapse, ˜ α .', 'C. Boundary conditions on the lapse function': "A possible boundary condition on the lapse was derived in Ref. [1]. The condition was essentially based on the reasonable quasiequilibrium condition that L k ' θ = 0. Notice that this is the change in the expansion of ingoing null rays, ' k µ , with respect to the outgoing null congruence. The resulting boundary condition takes the form \n∣ \nα BC ≡ ( J ˜ s i ˜ ∇ i α -E α )∣ ∣ S = 0 , (51) \n∣ where E is a nonlinear operator that is elliptic within the surface S (see Eq. (84) of Ref. [1] for a precise description). This condition is satisfied on the horizon of a stationary black hole and it seemed to supply a reasonable boundary condition for the lapse to be used in conjunction with the previously defined boundary conditions for the conformal factor and shift vector. \nWe implemented the full set of boundary conditions within the code described in Ref. [22] and attempted to solve the full set of conformal thin-sandwich equations for the case of a single nonrotating black hole. It became clear immediately that the iterative solutions would not converge in general. Interestingly, if the analytic solution for an isolated black hole was supplied for the starting point of the iterations, then the equations and boundary conditions were satisfied to truncation error, but the iterations were at best only marginally stable. Furthermore, if any one of the boundary conditions on ψ , β ⊥ , or α were replaced by an arbitrary Dirichlet or Neumann boundary condition, then the iterative solution was convergent to a solution representing a static black hole and the omitted boundary condition was satisfied . \nThis clearly indicates that the set of boundary conditions including Eq. (51) is degenerate and leads to an ill-posed elliptic system. We can understand the nature of the degeneracy by considering the family of timeindependent maximal slicings of Schwarzschild[23, 24, 25]. The line element for the spatial metric, lapse, and shift vector are \nd s 2 = ( 1 -2 M R + C 2 R 4 ) -1 d R 2 + R 2 dΩ 2 , (52a) \nβ R = C R 2 √ 1 -2 M R + C 2 R 4 , (52c) \nα = √ 1 -2 M R + C 2 R 4 , (52b) \nwhere R is the usual Schwarzschild 'areal' radial coordinate, M is the mass of the black hole, and C is a constant parametrizing the family of maximal slicings. In spherical coordinates, the extrinsic curvature takes the form \nK i j = C R 3 -2 0 0 0 1 0 0 0 1 . (52d) \nFor 0 ≤ C/M 2 < √ 27 16 ≈ 1 . 299 the maximal spatial slice extends from spatial infinity, through the black-hole interior, and to the second spatial infinity of the maximally extended Schwarzschild geometry. When C/M 2 = 0, we recover the standard Schwarzschild maximal slice that passes through the bifurcation point. And, for C/M 2 > √ 27 16 , the maximal spatial slice extends from spatial infinity, through the black-hole horizon, and ends on the singularity. \nIf desired, the family of maximal slicings of Schwarzschild can be be rewritten in terms of an isotropic radial coordinate, r . It is then easy to verify that the boundary conditions (48), (39), and (51) are satisfied on the horizon for any value of C/M 2 . So, we see that while the set of boundary conditions proposed in Ref. [1] hold for a time-independent configuration, they do not uniquely fix the spatial slicing. It is in this way that they are degenerate. \nIf we consider the value of the lapse on the horizon, we find that \nα ( r H ) = 1 4 ( C M 2 ) , (53) \nWhere r H denotes the location of the horizon in isotropic coordinates. Note that r H /negationslash = M/ 2 unless C/M 2 = 0. If, instead of using the lapse boundary condition (51), we simply fix a Dirichlet value for the lapse, then we find that we have effectively chosen a value of C/M 2 and thus a particular maximal slicing of Schwarzschild. Similarly, we find that the family of mixed boundary conditions \n∣ \n∂α ∂r ∣ ∣ ∣ r H = A α r ∣ ∣ r H (54) \n∣ corresponds to a slicing choice of C/M 2 = √ A 2 +4 -A , where A is any real number. It is clear that any reasonable choice of a Dirichlet, Neumann, or mixed boundary condition on the lapse will uniquely fix a particular maximal slicing of Schwarzschild and effectively break the degeneracy. \n∣ \nAs mentioned previously, our numerical investigations have shown that we could also have chosen to use the lapse boundary condition (51) and instead fix either ψ or β ⊥ on the horizon via a Dirichlet, Neumann, or mixed boundary condition. One reason to choose to replace the lapse condition (51), as opposed to the boundary conditions on the conformal factor (48) or the shift (39), is that the lapse boundary condition is much more complex. However, there is a more fundamental reason to choose to replace the lapse condition. The degeneracy that we must eliminate is in the choice of the spatial slice which is a choice of the initial temporal gauge. The lapse function fixes the evolution of the temporal gauge. Therefore it seems reasonable that we should consider the choice of the lapse boundary condition as part of the initial temporal gauge choice. It is customary to view the choice of the trace of the extrinsic curvature as fixing the initial \ntemporal gauge. However, this is apparently not sufficient within the conformal thin-sandwich approach when interior boundaries are present. \nThis last assertion, that the lapse boundary condition must be chosen as part of the initial temporal gauge choice, is supported by the behavior of a very special solution of Einstein's equations - the static Schwarzschild solution. However the same generic behavior is seen in a broad class of examples as we will outline below.", 'IV. QUASIEQUILIBRIUM SOLUTIONS FOR A SINGLE BLACK HOLE': "As we have seen in Sec. II, the conformal thin sandwich equations require specification of free data, which are the conformal metric ˜ γ ij and its time derivative ˜ u ij , as well as the mean curvature K and its time derivative ∂ t K . Moreover, boundary conditions are required on the variables being solved for, the conformal factor ψ , the shift β i and the lapse α . The quasiequilibrium approximation fixes a good portion of these choices, namely ˜ u ij = ∂ t K = 0, as well as the the following boundary conditions at the excised regions: The apparent horizon condition Eq. (48) on ψ , the null horizon condition Eq. (39) on β ⊥ , and Eq. (50) on β ‖ i . At spatial infinity, the boundary conditions are straightforward and are given by Eqs. (16). \nBefore the conformal thin-sandwich equations can be solved, we have to choose the remaining quantities which are not fixed by the quasiequilibrium framework. These are ˜ γ ij , K , and an inner boundary condition on the lapse α . Furthermore, we have to choose the shape of the excised regions, S . As argued above, and as we confirm below, the lapse boundary condition and K are part of the temporal gauge choice. It thus remains to choose ˜ γ ij and the shape of the excised regions, S . \nS In this work, we will assume that the conformal threegeometry is flat, and we will always excise exact spheres. These choices are not motivated by physical considerations, and they will affect the quality of the quasiequilibrium solutions we obtain in this paper. For example, the Kerr spacetime does not admit conformally flat slices [26, 27]. Therefore, when we solve for a rotating black hole, our initial-data sets will not exactly represent a Kerr black hole, but will rather correspond to a perturbed Kerr black hole, which will settle down to Kerr. We stress that this failure of the initial-data sets constructed here to represent Kerr is not inherent in the quasiequilibrium method, but is caused by our choices for ˜ γ ij and S . With the appropriate choices for ˜ γ ij and S , the quasiequilibrium method can reproduce exactly any time-independent solution of Einsteins equations. Indeed, for single black holes, better choices for ˜ γ ij and S are easily obtained from stationary analytic solutions of Einstein's equations, for example based on Kerr-Schild coordinates. While such a choice certainly leads to single black hole initial-data sets closer to the true Kerr metric, it is not clear how to generalize to binary black hole \nconfigurations. A widely used approach superposes single black hole quantities to construct binary black hole initial data (e.g., [16, 28, 29, 30, 31]). However, because of the nonlinear nature of Einsteins equations, and since often, the black holes are separated by only a few Schwarzschild radii, the superposition introduces uncertainties that may be large [22] and that have not yet been adequately quantified. In this paper, rather than using superposition, we start with choices for ˜ γ ij and S that are not optimal for single black hole spacetimes, but that are equally well suited for binary black hole configurations. We then use the single black hole solutions to quantify the effects of our approximation. \nWe solve the conformal thin-sandwich equations with the pseudo-spectral collocation method described in [32]. For the single black hole spacetimes, typically, two spherical shells are employed. The inner one ranges from the excised sphere to a radius of ∼ 20 and distributes gridpoints exponentially in radius. The outer shell has an outer radius of typically ∼ 10 10 , and employs an inverse mapping in radius, which is well adapted to the 1 /r -falloff of many quantities. The fifth elliptic equation (for the lapse function) is coded in the form of Eq. (15), i.e. as an equation for ψ 7 ˜ α = αψ . Therefore, we formulate the lapse boundary condition as a condition on αψ . Finally, we note that we always solve the three-dimensional initial value equations, even in cases which have spherical or cylindrical symmetries like the single black hole solutions.", 'A. Spherical symmetry': "We begin by solving for spherically symmetric initialdata sets that contain one black hole. In spherical symmetry, the assumption of conformal flatness is no restriction, because any spherically symmetric metric can be made conformally flat through an appropriate radial coordinate transformation. For example, a hypersurface through the Schwarzschild spacetime of constant KerrSchild coordinate time has the induced metric \nds 2 = ( 1 + 2 M R ) dR 2 + R 2 d Ω 2 , (55) \nwhere R denotes the areal radius. Here, the coordinate transformation [1] \nr = R 4 ( 1 + √ 1 + 2 M R ) 2 e 2 -2 √ 1+2 M/R , (56) \nbrings the induced metric into conformally flat form, \nds 2 = ψ 4 KS ( r 2 dr 2 + r 2 d Ω 2 ) , (57) \nwith ψ KS = R/r . \nIn order to support our claim that K and the lapse boundary condition merely represent a coordinate choice, \n√ \nwe solve the quasiequilibrium equations for several, essentially arbitrary, choices for these quantities. For the mean curvature, we choose \nK = 0 , (58a) \nK = 2 M r 2 , (58b) \nK = K KS ≡ 2 M R 2 ( 1 + 2 M R ) -3 / 2 ( 1 + 3 M R ) , (58c) \nwhere in the last case, R is given implicitly by Eq. (56). Equation (58c) represents the mean curvature for a KerrSchild slice of the Schwarzschild spacetime with mass M . \nFor the lapse boundary condition at the excised spheres, we use \ndαψ dr ∣ ∣ ∣ = 0 , (59a) \n∣ αψ ∣ ∣ S = 1 √ 2 ψ KS ∣ S . (59d) \n∣ S ∣ S ∣ \n∣ ∣ dαψ dr ∣ ∣ ∣ = αψ 2 r ∣ ∣ ∣ , (59b) \n∣ ∣ ∣ ∣ αψ ∣ S = 1 2 , (59c) \n∣ \n∣ S \n∣ The last condition, Eq. (59d) is correct for the KerrSchild slice. \nWe now compute twelve initial-data sets, combining any of the choices for K with any of the lapse boundary conditions. In order to fully recover the Kerr-Schild slice for the choices (58c) and (59d), we set the radius of the excised sphere to \nr exc = r ∣ R =2 M = 1 2 ( 1 + √ 2 ) 2 e 2 -2 √ 2 M. (60) \n∣ For each of the initial-data sets, we compute the residual in the Hamiltonian and momentum constraints, Eqs. (5) and (6). We compute ADM quantities of the initial-data set by the standard integrals at infinity in Cartesian coordinates, \n∣ \nE ADM = 1 16 π ∫ ∞ ( γ ij,j -γ jj,i ) d 2 S i , (61) \nFor the x -component of the linear ADM-momentum, ξ = ˆ e x in Eq. (62). The choice ξ = x ˆ e y -y ˆ e x yields the z -component of the ADM-like angular momentum as defined by York [33]. We also compute the irreducible mass \n∞ J ( ξ ) = 1 8 π ∫ ∞ ( K ij -γ ij K ) ξ j d 2 S i . (62) \nM irr = √ A AH 16 π , (63) \nwhere we have approximated the area of the (unknown) event horizon by the area A AH of the apparent horizon, and the Komar mass, \nM K = 1 4 π ∫ r = ∞ γ ij ( ¯ ∇ i α -β k K ik ) d 2 S j . (64) \nFIG. 1: Solution of the quasiequilibrium equations recovering the Kerr-Schild slicing of Schwarzschild. Plotted are the maximum values of Hamiltonian and momentum constraints and of time-derivatives, as well as the deviation of M irr , E ADM and M K from the analytical answer M . N r is the radial number of collocation points in each of the two spherical shells. \n<!-- image --> \nFinally, we evaluate the time-derivative of K ij by Eq. (4), evaluate the residual α BC of the quasiequilibrium lapse condition, Eq. (51), and compute \n∂ t ln ψ = -1 6 ( αK -¯ ∇ i β i ) , (65) \nwhich follows from the trace of Eq. (3). \nFigure 1 presents a convergence plot for one of the twelve cases, Eqs. (58c) and (59d). This case recovers the usual Kerr-Schild slice with mass M . The residual of the Hamiltonian and momentum constraints decrease exponentially with resolution, and the three different masses M irr , E ADM and M K all converge to the expected result, M . Furthermore, the time-derivatives exponentially converge to zero, and ADM linear and angular momenta converge to zero, too. The vanishing timederivatives indicate that the quasiequilibrium method constructs lapse and shift along the timelike Killing vector of the Schwarzschild spacetime. \nFor different choices of K or for the lapse BC, we find that the masses are no longer exactly unity. However, for all choices of K and lapse BC, we find to within truncation error, that the three masses agree, \nM irr = E ADM = M K , (66) \nand that all time-derivatives (and the lapse condition Eq. [51]) vanish: \n∂ t ψ = ∂ t K ij = α BC = 0 . (67) \nTABLE I: Spherically symmetric quasiequilibrium initial-data sets. Given are the irreducible mass, ADM-energy and Komar mass (these three quantities are found to be identical to within truncation error). The last column gives an upper bound on the deviation from zero of Hamiltonian and momentum constraints, time-derivatives of ψ and K ij , as well as the lapse condition α BC \n| K | Lapse BC | M irr , E ADM , M K | H , M i , ' ∂ t ' < 10 - 10 - 10 |\n|-------------------|-----------------------|-----------------------|------------------------------------|\n| 0 | ( αψ ) ' = 0 ' | 1 . 48079275 | |\n| | ( αψ ) = αψ/ (2 r ) | 1 . 61967937 | < 10 - 10 |\n| | αψ = 1 / 2 | 1 . 65726413 | < 10 |\n| | αψ = ( αψ ) KS | 1 . 28974831 | < 10 - 10 |\n| 2 M/r 2 | ( αψ ) ' = 0 | 0 . 68281 | < 10 - 9 |\n| | ( αψ ) ' = αψ/ (2 r ) | 0 . 73571 | < 10 - 9 |\n| | αψ = 1 / 2 | 0 . 99176 | < 10 - 8 |\n| | αψ = ( αψ ) KS | 0 . 77233 | < 10 - 9 |\n| K KS ( αψ ) ' = 0 | 0 | . 9942475 < | 10 - 9 |\n| K KS ( αψ ) ' = 0 | ( αψ ) ' = αψ/ (2 r ) | 1 . 091637 | < 10 - 9 |\n| K KS ( αψ ) ' = 0 | αψ = 1 / 2 | 1 . 295099 | < 10 - 9 |\n| | αψ = ( αψ ) KS | 1 . 0000000 | < 10 - 10 | \nThese findings are summarized in Table I. These runs indicate that any (reasonable) choice for the mean curvature K and the lapse BC recovers a slice through Schwarzschild with time-vector along the timelike Killing vector. \nFor the maximal slices, K = 0, the different lapse BCs choose different parameters C/M 2 in the family of maximal slicings, Eqs. (52a)-(52d). The boundary conditions in Eqs. (59a)-(59d) correspond, respectively, to C/M 2 = 2 / 3( √ 13 -1), C/M 2 = 4 / 3, C/M 2 ≈ 1 . 2393, and C/M 2 ≈ 2 . 4905. Based on the results of Table I, we conjecture that for any (reasonable) function K ( r ), there exists a one-parameter family of spherically symmetric slicings, which extend from the horizon to spatial infinity. \nIn situations with less symmetry like binary black holes, we prefer Neumann or Robin boundary conditions on the lapse (Eqs. [59a] and [59b]), because they allow the lapse on the horizon to respond to tidal deformations. Furthermore, as Table I confirms that the choice of mean curvature plays a marginal role, we will concentrate on the most obvious choice, maximal slicing K = 0, below. \nIn summary, the quasiequilibrium method is singularly successful for spherically symmetric spacetimes: There exists a natural choice for ˜ γ ij (the flat metric), that, together with any (reasonable) choices for K and the lapseboundary condition yields a slice though Schwarzschild and the timelike Killing vector that results in a completely time-independent evolution.", 'B. Single rotating black holes': 'In Sec. III we explained that the tangential component of the shift-vector on the excised surface induces a rota- \nFIG. 2: Spinning single black hole initial-data sets. \n<!-- image --> \ntion on the hole, cf. Eq. (50). We now test this assertion by constructing initial-data sets for single rotating black holes. \nWe set the mean curvature and the lapse BC by Eqs. (58a) and (59a), and we continue to use the conformal flatness approximation, ˜ γ ij = f ij , and choose the excised region to be a coordinate sphere centered on the origin with radius r exc = 0 . 8594997. This value ensures a unit-mass black hole in the limit of no rotation for the choice of K and lapse BC. The shift boundary conditions encode the rotation. At infinity, we set β i = 0; on the horizon, Eq. (50) implies \nβ ‖ i = Ω j r x k /epsilon1 ijk , (68) \nwhere x k is the Cartesian coordinate separation of points on S to the center of the excised sphere. We choose Ω i r parallel to the z-axis, and solve the initial value equations for different magnitudes of Ω r . For each solution, we compute the diagnostics mentioned in Sec. IV A. Figure 2 presents the ADM-energy, irreducible mass and angular momentum of the obtained data sets. We see that, for small Ω r , the angular momentum increases linearly with Ω r , as expected. \nAs discussed early in Sec. IV, our assumption of conformal flatness will necessarily introduce some errors when solving for a rotating black hole, because the Kerr metric does not admit conformally flat slices[26, 27]. Very interesting are therefore measures of the deviation of the quasiequilibrium initial-data sets to a slice through the exact Kerr spacetime. \nOne such quantity is the maximum amount of energy that can potentially be radiated to infinity, \nE rad = √ E 2 ADM -P 2 ADM -√ M 2 irr + J 2 ADM 4 M 2 irr . (69) \nFIG. 3: Single rotating black hole initial-data sets: Deviations from the exact Kerr metric. \n<!-- image --> \nFor a stationary spacetime, E rad = 0. Another interesting question is how closely Ω r of Eq. (68) corresponds to the angular frequency of the horizon. For a Kerr black hole with angular momentum J ADM and total mass E ADM , the angular frequency of the horizon is given by [34] \nso that \nΩ H = J ADM /E 3 ADM 2 + 2 √ 1 -( J ADM /E 2 ADM ) 2 , (70) \n∆Ω ≡ E ADM (Ω r -Ω H ) (71) \nmeasures the deviation of Ω r from the angular frequency of the horizon. Figure 3 presents these quantities. The maximum radiation content, E rad , is proportional to ( M irr Ω r ) 4 . For the binary black hole data sets we construct below in Sec. V, the relevant angular frequency is the orbital angular frequency. We find below, that at the innermost stable circular orbit, M irr Ω 0 ∼ 0 . 11. From Fig. 3, we find for this angular frequency, E rad /E ADM ≈ 2 · 10 -4 . This indicates that, when conformal flatness is assumed, we should not expect the fractional error induced in E ADM for systems with non-vanishing angular momentum to be larger than ∼ 10 -3 . Furthermore, at M irr Ω r = 0 . 11, ∆Ω ≈ 0 . 0025. From this we expect that the rotational state of each black hole, within the binary black hole configurations below, should deviate by at most one or two percent from the intended values for corotating or irrotational holes. In Fig. 3 we also plot some time-derivatives assuming the initial data are evolved with the constructed gauge α, β i . These timederivatives are proportional to ( M irr Ω r ) 2 . However, their interpretation is more difficult, due to their gauge dependence and the difficulty of finding a meaningful normalization. \nFIG. 4: Single boosted black hole initial-data sets \n<!-- image -->', 'C. Single boosted black holes': 'A boosted single black hole in a comoving coordinate system appears time-independent. A well-known example is the boosted Kerr-Schild form of a Kerr black hole. In such comoving coordinates, the shift does not vanish at infinity, but approaches the boost velocity of the black hole, \n∣ \nβ i ∣ r →∞ = v i . (72) \n∣ We apply now the quasiequilibrium formalism to construct boosted black holes in comoving coordinates by using Eq. (72) as the boundary condition on the shift at the outer boundary. At the excised sphere, we set β ‖ i ∣ ∣ S = 0. Furthermore, we assume again conformal flatness, use Eqs. (58a) and (59a) to fix the mean curvature and the lapse BC, and excise a coordinate sphere with radius r exc . The remaining free parameter is the magnitude of the boost velocity, v . Figure 4 presents the ADM-energy, irreducible mass and P ADM /E ADM as a function of the boost-velocity. P ADM /E ADM is linear in v for small v , as it should be. As v approaches unity, E ADM strongly increases. Figure 5 presents measures of how faithfully these initial-data sets represent a boosted stationary black hole. The maximum radiation content E rad grows as v 4 . At the ISCO we can estimate that v ∼ 0 . 4 where we find E rad ≈ 10 -3 E ADM . In order to measure how well the special relativistic relation v = P/E is satisfied, we define \n∆ v ≡ v -P ADM E ADM ; (73) \nwe find that ∆ v ∝ v 3 for small v , with ∆ v ≈ 0 . 01 for v = 0 . 4. \nFIG. 5: Single boosted black hole initial-data sets: Deviations from exact time-independence. \n<!-- image -->', 'V. QUASI-CIRCULAR ORBITS FOR BLACK-HOLE BINARIES': 'In the case of a single black hole, if appropriate choices for the freely specifiable data and boundary conditions are made, then an exact equilibrium solution of the initial-data equations can be found. However for binary black-hole configurations, no such true equilibrium or stationary state exists. This is a much more stringent test of the quasiequilibrium boundary conditions. In this section, we will examine the solutions for the case of equal-mass black-hole binaries that are either corotating or irrotational. \nWe will consider binary configurations over a range of separations. A black hole is represented in the coordinate system by an excised 2-surface. The relative coordinate sizes of various excised surfaces parameterize the relative sizes of the resulting physical black holes. The coordinate separations of the holes parameterize their physical separation. These coordinate sizes and separations are measured in the coordinates associated with the chosen conformal metric. In this work, all excised surfaces are the surfaces of coordinate spheres. Furthermore, for the simple cases of corotating and irrotational binaries, equal-mass black holes are obtained by choosing excision surfaces for the two holes that have equal radii. \nBefore solving the initial-data equations, we must make choices for the freely specifiable data. In all cases, we will make use of the quasiequilibrium assumptions on the free data that ˜ u ij = 0 and ∂ t K = 0. We also continue to use the approximation that the conformal three-geometry is flat (ie. that ˜ γ ij is a flat metric). The remaining free data is the trace of the extrinsic curvature, K . For this, we will consider two choices: maximal \nslicing with K = 0, and a non-maximal slicing based on Eddington-Finkelstein slicing. \nIn addition to the freely specifiable data, we must also fix the boundary conditions on the excision surfaces that correspond to the surface of each black hole and at the outer boundary of the computational domain. The outer boundary conditions were discussed at the end of Sec. II. The boundary condition on the shift, Eq. (16b), contains a free parameter, Ω 0 , that determines the orbital angular velocity of the system. The value of this parameter is chosen by demanding that the ADM and Komar masses of the system must be equal[1, 9, 18]. This is a quasiequilibrium condition that is satisfied by a single value of Ω 0 and places the binary in a nearly circular orbit. \nFor the excision boundaries, we will use the apparent horizon condition given by Eq. (48) as a boundary condition of the conformal factor ψ , and we will use Eq. (39) to fix the component of the shift that is normal to the excision surface. Boundary conditions on the components of the shift that are tangent to the excision surface depend on our choice for the spins of the black holes. For the case of corotation, we will demand that β ‖ i = 0. However, the irrotational case requires a somewhat more complicated choice. \nThe condition of quasiequilibrium requires that we choose the tangential components of the shift so that they have the form given in Eq. (50). For irrotational black holes in a binary, it is reasonable to choose the conformal Killing vector ξ i so that it represents rotation about an axis that is orthogonal to the plane of the orbit. If we let Ω i r represent an angular velocity vector that is orthogonal to the plane of the orbit, and if we use Cartesian coordinates for our flat conformal metric, then Eq. (50) can be written as \nβ ‖ i = Ω j r x k ± /epsilon1 ijk , (74) \nwhere x i ± ≡ x i -C i ± and C i ± is the Cartesian coordinate location of the center of either of two excision spheres. Finally, we take the magnitude of Ω i r to be equal to the orbital angular velocity of the binary system as measured at infinity, Ω 0 . We note that there is no rigorous proof that these choices lead to an irrotational binary system. However, as argued in Sec. III B, especially the paragraphs leading to Eq. (43), these choices seem reasonable. \nFinally, we must choose a boundary condition on the lapse at the excision boundaries. For all choices of the black-hole spins and choices for K , we repeat the computations for three different lapse boundary conditions, namely Eqs. (59a)-(59c). \nThe conformal thin-sandwich equations are solved with the pseudo-spectral collocation method described in [32]. The computational domain consists of one inner spherical shell around each excised sphere, which overlap 43 rectangular blocks, which in turn overlap an outer spherical shell extending to r out = 10 9 . Fig. 6 shows convergence of this solver with spatial resolution for one typical configuration (separation d = 9, K = 0, corotating black \nFIG. 6: Convergence of the elliptic solver for the binary black hole configurations. Plotted are the constraint violations in Hamiltonian and momentum constraint, and differences to the highest resolution solve. N is the cube-root of the total number of grid-points. \n<!-- image --> \nholes). The calculations below are performed at a resolution comparable to N = 60, so that the discretization errors in E ADM and M K should be about 10 -6 .', '1. Corotating binary systems': "We now compute initial-data sets corresponding to a binary black hole system in a quasi-circular orbit for many different separations. Figure 7 shows the binding energy E b of the binary system as a function of the total angular momentum of the system for all the lapse boundary conditions. The binding energy is defined as E b ≡ E ADM -m where E ADM is the total (ADM) energy of the system and m = m 1 + m 2 is the total mass of the system. For the quasiequilibrium numerical results described in this paper, we take m 1 | 2 ≡ √ A 1 | 2 16 π as the irreducible mass of each individual black hole and A 1 | 2 are the areas of the apparent horizon of each hole. The reduced mass is defined as µ m 1 m 2 m . \n≡ \nWe note that the choice of the lapse boundary condition has very little effect on the solutions. This is consistent with out assertion that the choice of the lapse boundary condition is part of the initial temporal gauge choice. The inset in Fig. 7 shows a magnified view of the region where the black holes are closest to each other. Even in this region, the result of the different lapse boundary conditions are nearly indistinguishable. Because of this, subsequent plots displaying corotating maximal slicing \nFIG. 7: Constant M irr sequence of corotating equal-mass black holes. Maximal slicing is used in these cases, and three different excision boundary conditions for the lapse were used. \n<!-- image --> \nresults will only display one of these sequences. \nFigure 8 shows a comparison of the same data to analogous results obtained by Grandcl'ement et al.[18] (labeled CO:HKV-GGB in figure legends), and effective one-body post-Newtonian results as reported in Ref. [35]. First, second, and third post-Newtonian (PN) results are displayed (labeled CO:EOB-1PN, CO:EOB-2PN, and CO:EOB-3PN respectively in figure legends). The 3PN results correspond to the approach labeled '3PN corot. ¯ A ( u, ˆ a 2 )' in Table I of Ref. [35]. \nThere is good agreement between all of the results at large separation. Also, it appears that the PN results are converging toward the quasiequilibrium numerical results, even when the black holes are quite close to each other as seen in the figure's inset. We also see that the numerical results obtained by Grandcl'ement et al.[18] (hereafter GGB) differ only slightly from the quasiequilibrium results. As discussed in Ref. [1], the numerical solutions obtained by GGB must violate the constraints. The agreement seen in Fig. 8 lends support to the belief that the violation of the constraints is, in some sense, small and has a small impact on the physical content of the data. \nAnother method for comparing data for the circular orbits of compact binaries is to examine the location of the inner-most stable circular orbit (ISCO). The ISCO is not a well defined concept in general. However, in situations where the dissipative effects of radiation reaction have been eliminated, an ISCO becomes more meaningful. The ISCO is defined in terms of a minimum of some appropriate energy. For corotating binary systems, true stationary configurations can exist (although they contain an infinite amount of energy in the form of gravi- \nFIG. 8: Constant M irr sequence of corotating equal-mass black holes. Comparison of post-Newtonian EOB sequences with numerical maximal slicing results from HKV and this paper. \n<!-- image --> \ntational radiation, see Ref. [36]). In this case, the minimum of the total energy can be rigorously associated with the onset of a secular instability[37, 38]. In the absence of true stationary configurations, this 'turning point' method is still used to define the ISCO. \nIn order to locate a turning point, one must have a sequence of binary configurations spanning a range of separations. How this sequence is constructed is not uniquely defined. The ambiguity arises because of the lack of a fixed fundamental length scale in the problem. In the first work to construct sequences of black-hole binary initialdata sets representing circular orbits[39], the total mass of the black hole (defined in terms of the Christodoulou mass formula[40]) was used to normalize the sequences. \nGGB suggest another approach based on the thermodynamic identity[36] \nd E ADM = Ω 0 d J ADM . (75) \nThis identity should be satisfied by a true stationary sequence of corotating black holes. Let s denote some parameter along a sequence of initial-data sets, and let e ( s ), j ( s ), and ω ( s ) denote the numerical values for dimensionless versions of the total energy, total angular momentum, and orbital angular velocity at location s along the sequence. We are free to define a fundamental length scale χ ( s ) along the sequence in any way we like, so long as we define the dimensionful total energy E ADM ( s ), total angular momentum J ADM ( s ), and orbital angular velocity Ω 0 ( s ) consistently via, \nE ADM ( s ) ≡ χ ( s ) e ( s ) , (76) \nJ ADM ( s ) χ 2 ( s ) j ( s ) , (77) \nΩ 0 ( s ) ≡ χ -1 ( s ) ω ( s ) . (78) \n≡ \nEnforcing the identity (75) is sufficient to determine the change in χ ( s ) between two points on the sequence. If we integrate along the sequence from a point s 1 to another point s 2 , then we find that \nχ ( s 2 ) = χ ( s 1 ) exp { -∫ s 2 s 1 e ' ( s ) -ω ( s ) j ' ( s ) e ( s ) -ω ( s ) j ( s ) ds } , (79) \nwhere a prime denotes differentiation along the sequence. \nFIG. 9: Plot of the irreducible mass of one black hole when the sequence of solutions is normalized to maintain d E = Ωd J . Three different choices for the lapse boundary condition are shown for both corotating (upper plot) and irrotational (lower plot) black holes in an equal-mass binary. The vertical dashed line shows the approximate location of the ISCO defined by a minimum in E b . The vertical dotted line shows the approximate location of the ISCO defined by a minimum in E ADM . In the irrotational case, this latter ISCO line is off the plot to the right. \n<!-- image --> \nIf the sequence is normalized via Eq. (75), then the irreducible mass of one black hole, M irr = 1 2 m , is not necessarily constant along the sequence. The top half of Fig. 9 shows M irr for a corotating quasiequilibrium equal-mass binary as a function of the orbital angular velocity. The length scale has been normalized so that M irr = 1 / 2 at infinite separation. We confirm the finding of GGB that M irr is nearly constant along the sequence. While there is a clear increase in the mass as the separation decreases, this increase is small and appears to be of roughly the same order of magnitude as the differences due to using different lapse boundary conditions. As we will see later, this behavior is not mirrored in the irrotational data. \nYet another approach for normalizing the sequence is to demand that the individual irreducible masses associated with the apparent horizons remain constant. This is a particularly convenient normalization for numerical \nwork since it relies on a well defined and easily measured geometric quantity. Each of these choices for normalizing an initial data sequence can effect the location of the ISCO, so it is important to use a consistent definition when comparing data. \nIt is also important to clearly define which energy is being extremised when using a turning-point method to locate the ISCO. We can consider using the minimum in either the ADM energy E ADM or the binding energy E b along any sequence to define the ISCO. From the definition of the binding energy as E b ≡ E ADM -m , we see that the minima will not necessarily agree if m varies along the sequence. \nFor the PN sequences, the ISCO is defined as the minimum in the binding energy E b along sequences where the irreducible masses of the black holes remain fixed. For the PN sequences, Eq. (75) is identically satisfied as well, so this is equivalent to finding the minimum in E ADM . This is not true for the quasiequilibrium numerical data. \nTABLE II: Parameters of the ISCO configuration for corotating equal-mass black holes computed with the maximal slicing condition. Results are given for three different choices of the lapse boundary condition and two choices for the definition of the location of the ISCO. For comparison, the lower part of the table lists results of Refs. [18, 35, 41]; 'PN standard' [41] represents a post-Newtonian expansion in the standard form without use of the EOB-technique. \n| Lapse BC | ISCO min. | m Ω 0 | E b /m | J/m 2 |\n|----------------------|--------------|---------|----------|---------|\n| d( αψ ) d r = 0 | ADM | 0.105 | -0.0165 | 0.844 |\n| | E b | 0.107 | -0.0165 | 0.844 |\n| αψ = 1 2 | ADM | 0.106 | -0.0165 | 0.843 |\n| | E b | 0.107 | -0.0165 | 0.843 |\n| d( αψ ) d r = αψ 2 r | ADM | 0.106 | -0.0165 | 0.843 |\n| | E b | 0.107 | -0.0165 | 0.843 |\n| HKV-GGB | HKV-GGB | 0.103 | -0.017 | 0.839 |\n| 1PN EOB | 1PN EOB | 0.0667 | -0.0133 | 0.907 |\n| 2PN EOB | 2PN EOB | 0.0715 | -0.0138 | 0.893 |\n| 3PN EOB | 3PN EOB | 0.0979 | -0.0157 | 0.86 |\n| 1PN standard | 1PN standard | 0.5224 | -0.0405 | 0.621 |\n| 2PN standard | 2PN standard | 0.0809 | -0.0145 | 0.882 |\n| 3PN standard | 3PN standard | 0.0915 | -0.0153 | 0.867 | \nTable II displays the dimensionless orbital angular velocity, binding energy, and total angular momentum of the ISCO for corotating equal-mass black holes on a maximal slice. For the quasiequilibrium data defined in this paper, two definitions of the ISCO are listed. One uses the minimum in E ADM along sequences where condition (75) is satisfied to define the ISCO. The alternative method uses a minimum in E b along sequences where M irr is held fixed. However, we note that the minima in E b along sequences that satisfy (75) are numerically indistinguishable from the latter. The ISCO for the GGB data (listed as HKV-GGB in the table) is defined as the minima in E b along a sequence where M irr remains fixed. Recall that the definition of the ISCO for the PN data is consistent with either definition used for \nthe quasiequilibrium data. In addition to the effective one-body (EOB) PN data displayed previously in Fig. 8, we also include 'standard' PN results for the ISCO as reported in Ref. [41]. \nFor corotating quasiequilibrium data, there is very little difference in the results for the two definitions of the ISCO. As we will see later, this is not true for the irrotational configurations (see Table III). In that case, it is clear that only the definition in terms of E b is consistent with the PN data. \nFIG. 10: ISCO configuration for three different choices of the lapse boundary condition for equal-mass corotating and irrotational black holes computed with the maximal slicing condition. For comparison, results of Refs. [18, 35, 39, 41] are included. For post-Newtonian calculations the size of the symbol indicates the order, the largest symbol being 3PN. 'PN standard' [41] represents a PN-expansion in the standard form without use of the EOB-technique (only 2PN and 3PN are plotted). \n<!-- image --> \nFigure 10 plots binding energy versus orbital angular velocity for the ISCO obtained for all three lapse boundary conditions for the corotating quasiequilibrium data, as well as the corotating results from GGB and PN results. All the numerical results are computed on a maximal slice. We see that the results for the different lapse boundary conditions are essentially indistinguishable. We also see that the PN results converge roughly toward the numerical quasiequilibrium results. While we would not expect the quasiequilibrium numerical results to agree with any of the individual PN results, we might expect the GGB result to agree within numerical error. All of our numerical results using different lapse boundary conditions are essentially indistinguishable. Furthermore, if we use a lapse boundary condition that approaches a Dirichlet value of zero on the excision surface, the resulting set of boundary conditions is equivalent to those used by GGB. The difference seen between \nthe GGB ISCO and our results may well be due to the regularization procedure introduced by GGB. \nFIG. 11: Constant M irr sequence of corotating equal-mass black holes. Comparison of post-Newtonian EOB sequences with numerical maximal slicing results from HKV and this paper. \n<!-- image --> \nFIG. 12: ISCO configuration for three different choices of the lapse boundary condition for equal-mass corotating and irrotational black holes computed with the maximal slicing condition. Symbols as in Fig. 10. \n<!-- image --> \nThere are three rigorously defined gauge-invariant global quantities associated with a black-hole binary system: the ADM energy E ADM , the total angular momentum J , and the orbital angular velocity as seen at infinity \nΩ 0 . Figures 8 and 10 plotted the binding energy E b (directly related to the ADM energy) as a function of J . Figures 11 and 12 plot E b as a function of Ω 0 for the same set of sequences and for the ISCO. And, Figs. 13 and 14 plot J as a function of Ω 0 for the same set of sequences and for the ISCO. \nFIG. 13: Constant M irr sequence of corotating equal-mass black holes. Comparison of post-Newtonian EOB sequences with numerical maximal slicing results from HKV and this paper. \n<!-- image --> \nFIG. 14: ISCO configuration for three different choices of the lapse boundary condition for equal-mass corotating and irrotational black holes computed with the maximal slicing condition. Symbols as in Fig. 10. \n<!-- image -->", '2. Irrotational binary systems': "For the case of irrotational black holes, Fig. 15 shows the binding energy E b of the binary system as a function of J for all the lapse boundary conditions. As with the corotating black holes, we again see that the choice of the lapse boundary condition has very little effect on the sequence. In the inset to the figure we see that the effect is largest at small separations and that the differences due to varying the lapse boundary condition are somewhat larger than in the corotating case. \nFIG. 15: Constant M irr sequence of irrotational equal-mass black holes. Maximal slicing is used in these cases, and three different excision boundary conditions for the lapse were used. \n<!-- image --> \nPerhaps the most striking difference between the corotating and irrotational sequences is that the extrema in E b versus J are much less 'sharp' in the irrotational sequences. If we consider sequences, either corotating or irrotational, that are normalized so that d E ADM = Ω 0 d J is satisfied, then the extremum in E ADM will necessarily coincide with the extremum in J leading to a very sharp cusp in a plot of these quantities. However, because M irr is not necessarily constant along a sequence with this normalization, the extremum in E b will not necessarily coincide with that of J . Thus, we should certainly not expect to see a sharp cusp in either Fig. 7 or Fig. 15. \nAnother difference between the corotating and irrotational sequences can be seen if Fig. 9. Here we see that, if we demand that the thermodynamic identity (75) be satisfied along the sequence, then the variation in M irr is 20 times larger in the irrotational sequences than in the corotating sequences. For the corotating sequences, the variation in M irr due to differences in the lapse boundary condition was comparable to the average variation. For the irrotational sequences, the effect of different lapse boundary conditions is clearly negligible. Furthermore, \nwe note that M irr is decreasing as the binary separation decreases. This behavior is unphysical, as the irreducible mass never decreases; therefore, M irr should also not decrease during the insipiral of a binary black hole. \nFIG. 16: Constant M irr sequence of irrotational equal-mass black holes. Comparison of post-Newtonian EOB sequences with numerical maximal slicing results from Conformal Image and this paper. \n<!-- image --> \nFigure 16 shows a comparison of our irrotational data to the effective one-body post-Newtonian results for irrotational holes as reported in Ref. [35]. First, second, and third post-Newtonian results are displayed (labeled IR:EOB-1PN, IR:EOB-2PN, and IR:EOB-3PN respectively in figure legends). The 3PN results correspond to the approach labeled '3PN corot. ¯ A ( u, 0)' in Table I of Ref. [35]. Also plotted in this figure is the first sequence of numerical initial-data solutions for an equal-mass black-hole binary in quasicircular orbit, obtained from inversion-symmetric initial data using an effective potential approach[39] (labeled IR:Conf. Imaging/Eff. Pot. or IVP-conf in figure legends). \nAgain, there is good agreement between all of the results at large separation. Also, it appears that the PN results are converging toward the irrotational quasiequilibrium numerical results, even when the black holes are quite close to each other as seen in the figure's inset. We also see that the early numerical results obtained from the conformal-imaging data[39] differ only slightly from the quasiequilibrium results up the location of the ISCO in the quasiequilibrium sequence. However, the conformal-imaging sequence extends to much smaller separations before encountering its ISCO. \nTable III displays the ISCO data for irrotational equalmass black holes on a maximal slice. Again, the ISCO is defined in two ways for the QE-sequence data. However, unlike the corotating case, the two definitions disagree \nTABLE III: Parameters of the ISCO configuration for irrotational equal-mass black holes computed with the maximal slicing condition. Results are given for three different choices of the lapse boundary condition and two choices for the definition of the location of the ISCO. Layout as in Table II. \n| Lapse BC | ISCO min. | m Ω 0 | E b /m | J/m 2 |\n|----------------------|--------------|---------|----------|---------|\n| d( αψ ) d r = 0 | ADM | 0.144 | -0.0146 | 0.761 |\n| | E b | 0.101 | -0.0181 | 0.767 |\n| αψ = 1 2 | ADM | 0.148 | -0.0145 | 0.76 |\n| | E b | 0.103 | -0.0181 | 0.765 |\n| d( αψ ) d r = αψ 2 r | ADM | 0.145 | -0.0146 | 0.761 |\n| | E b | 0.101 | -0.0181 | 0.766 |\n| Conf. Imag. | Conf. Imag. | 0.166 | -0.0225 | 0.744 |\n| 1PN EOB | 1PN EOB | 0.0692 | -0.0144 | 0.866 |\n| 2PN EOB | 2PN EOB | 0.0732 | -0.015 | 0.852 |\n| 3PN EOB | 3PN EOB | 0.0882 | -0.0167 | 0.82 |\n| 1PN standard | 1PN standard | 0.5224 | -0.0405 | 0.621 |\n| 2PN standard | 2PN standard | 0.1371 | -0.0199 | 0.779 |\n| 3PN standard | 3PN standard | 0.1287 | -0.0193 | 0.786 | \ndramatically. It is clear that the ISCO, when defined as a minimum of E b along a sequence where M irr is held fixed, is consistent with the results from the PN data. This is fortunate given the fact that M irr decreases as the binary separation decreases along sequences where Eq. (75) is satisfied. However, it is unclear why these E ADM defined ISCOs disagree so significantly from the PN data. The E b -defined ISCO data for the various irrotational sequences is plotted with the corotating data in Figs. 10, 12, and 14. \nFIG. 17: Constant M irr sequence of irrotational equal-mass black holes. Comparison of post-Newtonian EOB sequences with numerical maximal slicing results from Conformal Image and this paper. \n<!-- image --> \nFigures 16 and 10 plotted the binding energy E b as a \nfunction of J for the irrotational sequences. Figures 17 and 12 plot E b as a function of Ω 0 for the same set of sequences and for the ISCO. And, Figs. 18 and 14 plot J as a function of Ω 0 for the same set of sequences and for the ISCO. \nFIG. 18: Constant M irr sequence of irrotational equal-mass black holes. Comparison of post-Newtonian EOB sequences with numerical maximal slicing results from Conformal Image and this paper. \n<!-- image -->", 'B. Non-maximal slicing': 'For the case of sequences of corotating or irrotational QE initial data obtained on maximal slices (i.e. K =0), the choice of the lapse boundary condition seemed to have very little effect on gauge invariant quantities. This lends support to our assertion that the choice of the lapse boundary condition is part of the initial temporal gauge freedom. To further test this assertion, we should consider varying other aspects of the freely specifiable data. Quasiequilibrium considerations demand that we choose ˜ u ij = 0 and ∂ t K = 0. This leaves us the options of choosing a non-flat conformal metric or a non-maximal slice. \nWhile changing either can affect the content of the dynamical degrees of freedom in the initial data, the conformal metric is more closely tied to these dynamical degrees of freedom and to the spatial gauge freedom. The choice of the trace of the extrinsic curvature is usually thought of as fixing the initial temporal gauge freedom, although as we have seen from the example of the family of maximal slices of Schwarzschild in Sec. III C, this is not always sufficient to fix this aspect of the gauge freedom. In any case, it seems reasonable that the best choice is to vary K . \nA convenient choice for a non-maximal slicing is one based on a stationary black hole in Kerr-Schild coordinates. For the case of a non-charged, non-spinning black hole these are also referred to as ingoing EddingtonFinkelstein coordinates (cf. Ref. [42]). The spatial metric is given by Eq. (55), and after the coordinate transformation (56) it becomes conformally flat. The trace of the extrinsic curvature for the Kerr-Schild slicing of Schwarzschild is given by Eq. (58c). For a binary system, we use a linear combination of two copies of Eq. (58c), each centered at the location of one of the black holes, to define K . \nFIG. 19: Constant M irr sequence of corotating equal-mass black holes. Eddington-Finkelstein slicing is used in these cases, and three different excision boundary conditions for the lapse were used. \n<!-- image --> \nFigures 19 and 20 display the results for both corotating and irrotational sequences of equal-mass black-hole binaries based on the Kerr-Schild-like slicing described above. In both cases, the same three lapse boundary conditions used for the maximal slicing solutions were again used. When the holes are at large separation, the different lapse boundary conditions cause little variation in the results. However, when the holes are close together, the different lapse boundary conditions cause significant variation in the sequences. From this example it seems that the choice of the lapse boundary condition may have a significant effect on QE solutions of the conformal thin-sandwich equations. However, this example may be somewhat misleading. \nMaximal slicing ( K = 0) is based on a global geometric concept that does not depend on the separation of the black holes in a binary. For an isolated black hole, the Kerr-Schild slicing also has a geometric interpretation. However, a linear combination of the traces of the extrinsic curvatures for individual black holes does not \nFIG. 20: Constant M irr sequence of irrotational equal-mass black holes. Eddington-Finkelstein slicing is used in these cases, and three different excision boundary conditions for the lapse were used. \n<!-- image --> \nretain this geometrical meaning. Thus, as we vary the separation of the black holes in the non-maximal slicing sequences, we are also effectively varying the slicing condition. This effect is weak when the holes are at large separation, but becomes significant when the black holes are close together. \nIn constructing meaningful sequences, everything in the construction of the individual models should be held fixed except for the separation. For maximal slicing, the various choices of the lapse boundary condition choose a particular slice from among a family of maximal slices. However in the of the Kerr-Schild based slicing, is seems likely that the functional form of K as a function of separation, and the form of the lapse boundary condition conspire to define a different slicing condition for each initial-data model considered. Therefore, it seems reasonable to question the validity of these non-maximal slicing sequences. More importantly, we should be cautious in attributing undue significance to the choice of the lapse boundary condition based on this example.', 'VI. DISCUSSION': "In this paper, we have refined the QE boundary conditions defined originally in Ref. [1] and explored both single and binary black-hole configurations. The original motivation in deriving these boundary conditions was to provide conditions that would be consistent with quasiequilibrium configurations. The binary black hole initial-data sets constructed in Sec. V are intended to be in quasiequilibrium. While the individual black holes in \na binary cannot be in true equilibrium, it would be useful to determine if they are roughly in equilibrium. \nOne measure of this is to see how well the QE lapse condition (51) is satisfied. For stationary black holes, this equation holds. However, in the general case, it does not. Equation (51) defines α BC as the error in this boundary condition when applied on the excision boundary of one of the holes. \nFIG. 21: Plot of the residual of the quasiequilibrium boundary condition on the lapse (51) of one black hole. Three different choices for the lapse boundary condition are shown for corotating black holes in an equal-mass binary. The upper plot shows the average of the residual over the boundary surface. The lower plot shows the L 2 -norm of the residual. The vertical dashed line shows the approximate location of the ISCO defined by a minimum in E b . The vertical dotted line shows the approximate location of the ISCO defined by a minimum in E ADM . \n<!-- image --> \nFigure 21 shows both the average value of α BC and the L 2 norm of α BC as a function of Ω 0 for a corotating equal-mass binary. Figure 22 shows the same information for an irrotational equal-mass binary. At large separations (small Ω 0 ), we should expect that each black hole is nearly in equilibrium. For the corotating sequences, m Ω 0 ≈ 0 . 01 corresponds to a proper separation between the horizons of approximately 20 m . At this separation, | α BC | L 2 ≈ 0 . 0003. At the ISCO separation, m Ω 0 ≈ 0 . 11 corresponding to a proper separation between the horizons of approximately 4 . 5 m , and | α BC | L 2 has increased by a factor of approximately 20. As we might expect, the level of violation of the QE lapse boundary condition increases steadily as the separation between the holes decreases. However, there is no dramatic increase in the violation near the ISCO. For the irrotational sequences, | α BC | L 2 begins at m Ω 0 ≈ 0 . 01 at a level approximately twice as large as that of the corotating sequence. Near the ISCO, it has increased by a factor of approximately \nFIG. 22: Plot of the residual of the quasiequilibrium boundary condition on the lapse (51) of one black hole. Three different choices for the lapse boundary condition are shown for irrotational black holes in an equal-mass binary. The upper plot shows the average of the residual over the boundary surface. The lower plot shows the L 2 -norm of the residual. The vertical dashed line shows the approximate location of the ISCO defined by a minimum in E b . \n<!-- image --> \nIt seems that the rate of increase in the violation of the QE lapse boundary condition for the irrotational sequences is faster than that seen in the corotating sequences. This is not too surprising when we recall that a true stationary binary configuration can only be achieved for corotating binaries[36]. As with the corotating sequences, the level of violation of the QE lapse boundary condition increases steadily as the separation between the holes decreases, and there is no dramatic increase in the violation near the ISCO. \nAnother indicator of whether or not each black hole in the binary is in equilibrium is given by the value of ∂ t ψ as evaluated on the apparent horizon. We can express the time derivative of the conformal factor on any closed surface as \n∂ t ln ψ = 1 4 [ ˜ D k β ‖ k +4 β ‖ k ˜ D k ln ψ (80) -1 2 ˜ h k/lscript ˜ u k/lscript + √ 2 θ -( α -β ⊥ ) H ] . \nClearly, when the QE boundary conditions in Eqs. (17) and (39) are imposed on the excision surface, the last two terms in Eq. (80) vanish. Furthermore, in constructing QE configurations, we have also demanded that ˜ u ij = 0 globally. Therefore, the only terms that are possibly nonzero on the excision surface are those that involve β ‖ i . \nFor corotating binaries, β ‖ i = 0 and we find that the time derivative of the conformal factor vanishes identi- \nly on the excision surface. This is confirmed in our numerical results as shown in the upper half of Fig. 23. There, we see that ∂ t ln ψ = 0 to roundoff error. For irrotational binaries, the QE conditions require that we take β ‖ i proportional to a conformal Killing vector of ˜ h ij . This implies that β ‖ i will also be a conformal Killing vector of h ij . Unfortunately, the operator acting on β ‖ in Eq. (80) is not the conformal Killing operator, but rather \n˜ D k β ‖ k +4 β ‖ k ˜ D k ln ψ = D k β ‖ k . (81) \nWhile this would vanish if β ‖ k were a Killing vector of h ij , it will not vanish if β ‖ k is only a conformal Killing vector of h ij , as it will be unless the configuration is truly stationary. Again, this is confirmed in our numerical results where we find that ∂ t ln ψ ∼ 10 -5 when m Ω ∼ 0 . 01 and grows monotonically as the binary approaches the ISCO. These results are shown in the lower half of Fig. 23. \nFIG. 23: Plot of the L 2 -norm of ∂ t ln ψ as evaluated on the excision boundary of one black hole. The upper half of the figure shows the results for corotating black holes in an equal-mass binary, while the lower half shows the results for irrotational black holes. For both cases, three different choices for the lapse boundary condition are shown. In the upper half of the figure, the vertical dashed line shows the approximate location of the ISCO defined by a minimum in E b . The vertical dotted line shows the approximate location of the ISCO defined by a minimum in E ADM . In the lower half of the figure, the vertical dashed line shows the approximate location of the ISCO defined by a minimum in E b . \n<!-- image --> \nThe quasiequilibrium boundary conditions we have derived and tested in this paper are extremely general. Within the conformal thin-sandwich approach, they will work for any number of black holes that are to be considered in quasiequilibrium, or 'isolated'. In this paper, we have used several different choices for K , but maintained \nthe assumption of conformal flatness. We emphasize that this is not a limitation of the boundary conditions which can, in fact, be used with any viable choice for the conformal three-geometry specified by ˜ γ ij . Furthermore, for binary systems, we have only considered the special cases of corotating and irrotational black holes. Again, this is not a limitation of the boundary conditions which can, in principle, produce any desired spin on the individual black holes. \nIt has been pointed out that the boundary conditions we have derived are precisely those required to construct a black hole satisfying the isolated-horizon conditions[19, 20, 21, 43]. This is not surprising since the physical notions underlying an isolated horizon and a black hole in quasiequilibrium are essentially the same thing. It seems likely that the unified approach offered by the isolated-horizon framework will prove useful in further understanding the physical content of the binary black hole initial data constructed with the quasiequilibrium boundary conditions and to further understand the role of the lapse boundary condition. In fact, during the final stages of the preparation of this manuscript, we became aware of a paper by Jaramillo et al.[44] that makes the connection between our quasiequilibrium boundary conditions and isolated horizons more precise. This paper argues that the lapse boundary condition, Eq. (51) derived previously in Ref. [1], could be problematic, as we have found and discussed, and shows that weakly isolated horizon considerations do not restrict the lapse boundary condition when constructing initial data, consistent with our findings here. Furthermore, they suggest an alternate boundary condition on the lapse, based on a Lie-derivative along the null-generators of the horizon. It is not immediately clear that this proposed boundary condition can work. For the case of a static black hole, we have shown that essentially any boundary condition on the lapse, when combined with the quasiequilibrium boundary conditions, will yield a valid static slice of Schwarzschild. It may well be that this proposed boundary condition is degenerate similar to Eq. (51) (cf. our discussion in Sec. III C). \nClearly, additional work is required to fully understand the boundary conditions we have derived, and in particular the proper role of the lapse boundary condition. However, it is also clear that obtaining appropriate boundary conditions is not the final issue in the quest to construct astrophysically realistic binary black hole initial data. The most pressing issue is the question of how to make a realistic choice for the conformal three-geometry. While the errors introduced by the assumption of conformal flatness are not 'grave', it is clear that we must find a way to allow the physics to dictate the conformal three-geometry instead of choosing it a priori . The approach along these directions outlined in Ref. [45] (see also Ref. [46]) is clearly promising.", 'Acknowledgments': "The authors are grateful to J. Isenberg, L. Lindblom, V. Moncrief, N. ' O Murchadha, M. Scheel, M. Shibata, and S. Teukolsky for illuminating discussions. The authors are grateful to K. Thorne and L. Lindblom for their hospitality during the Caltech Visitors Program in the Numerical Simulation of Gravitational Wave Sources in the spring of 2003, during which part of this work was performed. This work was supported in part by NSF grants PHY-0140100 to Wake Forest University, PHY9900672 to Cornell University, and PHY-0244906 and PHY-0099568 to the California Institute of Technology. Computations were performed on the Wake Forest University DEAC Cluster.", 'APPENDIX A: COROTATING SEQUENCE': 'In this Appendix, we list the numerical results for corotating equal-mass black holes assuming conformal flatness, maximal slicing, and using Eq. (59b) for the lapse boundary condition on both excision surfaces. The data has been scaled so that the sequence satisfies Eq. (75) by following the procedure outlined in Eqs. (76)-(79). In order to maintain accuracy in the scaling, the maximum coordinate separation between successive models was ∆ d = 0 . 05. Data in the given tables can be easily rescaled to construct sequences with M irr held constant. \nIn Table IV, d is the coordinate separation of the centers of the excised regions. M irr is the irreducible mass associated with one of the black holes. E ADM is the ADM energy of the system. Ω 0 is the orbital angular velocity of the binary system as measured at infinity. E b is the binding energy of the system defined as E b ≡ E ADM -2 M irr . J ADM is the total ADM angular momentum of the binary system as measured at infinity. Finally, /lscript is the minimum proper separation between the two excision surfaces as measured on the initial-data slice.', 'APPENDIX B: IRROTATIONAL SEQUENCE': "In this Appendix, we list the numerical results for irrotational equal-mass black holes assuming conformal flatness, maximal slicing, and using Eq. (59b) for the lapse boundary condition on both excision surfaces. The data has been scaled so that the sequence satisfies Eq. (75) by following the procedure outlined in Eqs. (76)-(79). In order to maintain accuracy in the scaling, the maximum coordinate separation between successive models was ∆ d = 0 . 05. Data in the given tables can be easily rescaled to construct sequences with M irr held constant. \nIn Table V, d is the coordinate separation of the centers of the excised regions. M irr is the irreducible mass associated with one of the black holes. E ADM is the ADM energy of the system. Ω 0 is the orbital angular velocity of \nTABLE IV: Sequence of corotating equal-mass black holes on a maximal slice. The length scale is set so that the ADM mass of the binary at infinite separation is 1. The ISCO is at separation d = 8 . 28. \n| d | M irr E ADM - | 1 Ω 0 E b J ADM /lscript |\n|-----|----------------------------------|-----------------------------------------------------------|\n| 40 | | 0.5000000 -0.0058296 0.01090 -0.0058296 1.2280 21.81 |\n| 35 | | 0.5000001 -0.0065815 0.01327 -0.0065816 1.1655 19.17 |\n| 30 | | 0.5000003 -0.0075478 0.01665 -0.0075483 1.1005 16.52 |\n| 25 | | 0.5000006 -0.0088277 0.02175 -0.0088289 1.0332 13.83 |\n| 20 | | 0.5000013 -0.0105789 0.03012 -0.0105816 0.9647 11.11 |\n| 19 | | 0.5000016 -0.0110046 0.03246 -0.0110079 0.9511 10.56 |\n| 18 | | 0.5000020 -0.0114600 0.03511 -0.0114639 0.9376 10.00 |\n| 17 | | 0.5000024 -0.0119466 0.03814 -0.0119514 0.9243 9.444 |\n| 16 | | 0.5000030 -0.0124654 0.04164 -0.0124715 0.9113 8.882 |\n| 15 | | 0.5000039 -0.0130163 0.04571 -0.0130240 0.8986 8.316 |\n| | | 14.5 0.5000044 -0.0133032 0.04800 -0.0133119 0.8925 8.031 |\n| 14 | | 0.5000050 -0.0135969 0.05049 -0.0136068 0.8865 7.745 |\n| | | 13.5 0.5000059 -0.0138966 0.05320 -0.0139080 0.8807 7.458 |\n| 13 | | 0.5000065 -0.0142009 0.05617 -0.0142140 0.8752 7.169 |\n| | | 12.5 0.5000076 -0.0145079 0.05942 -0.0145231 0.8699 6.879 |\n| 12 | | 0.5000088 -0.0148151 0.06300 -0.0148328 0.8649 6.587 |\n| | | 11.5 0.5000104 -0.0151187 0.06696 -0.0151395 0.8602 6.293 |\n| 11 | | 0.5000123 -0.0154139 0.07135 -0.0154385 0.8559 5.997 |\n| | | 10.5 0.5000147 -0.0156939 0.07625 -0.0157232 0.8521 5.699 |\n| 10 | | 0.5000177 -0.0159493 0.08174 -0.0159847 0.8489 5.398 |\n| 9.5 | | 0.5000217 -0.0161675 0.08792 -0.0162108 0.8463 5.095 |\n| 9 | | 0.5000268 -0.0163311 0.09492 -0.0163846 0.8445 4.788 |\n| 8.9 | | 0.5000280 -0.0163553 0.09644 -0.0164112 0.8442 4.726 |\n| 8.8 | | 0.5000293 -0.0163761 0.09799 -0.0164346 0.8440 4.665 |\n| 8.7 | | 0.5000306 -0.0163934 0.09959 -0.0164546 0.8438 4.603 |\n| 8.6 | 0.5000321 -0.0164068 0.1012 | -0.0164709 0.8437 4.540 |\n| 8.5 | 0.5000336 -0.0164160 0.1029 | -0.0164832 0.8436 4.478 |\n| 8.4 | 0.5000352 -0.0164209 0.1047 | -0.0164913 0.8436 4.416 |\n| | 8.35 0.5000361 -0.0164215 0.1055 | -0.0164936 0.8436 4.384 |\n| 8.3 | 0.5000369 -0.0164209 0.1064 | -0.0164948 0.8436 4.353 |\n| | 8.28 0.5000373 -0.0164203 0.1068 | -0.0164949 0.8436 4.341 |\n| 8.2 | 0.5000388 -0.0164159 0.1083 | -0.0164934 0.8436 4.290 |\n| 8.1 | 0.5000407 -0.0164054 0.1102 | -0.0164869 0.8437 4.227 |\n| 8 | 0.5000428 -0.0163890 0.1121 | -0.0164747 0.8439 4.164 | \n- [1] G. 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2005MNRAS.359.1469M

MCG-6-30-15: long time-scale X-ray variability, black hole mass and active galactic nuclei high states

2005-01-01

['black hole physics', 'galaxies active', 'galaxies', 'astronomy x rays', 'astronomy x rays', 'astrophysics']

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We present a detailed study of the long time-scale X-ray variability of the Seyfert 1 Galaxy MCG-6-30-15, based on eight years of frequent monitoring observations with the Rossi X-ray Timing Explorer. When combined with the published short-time-scale XMM-Newton observations, we derive the power-spectral density (PSD) covering six decades of frequency from ~10<SUP>-8</SUP> to ~10<SUP>-2</SUP> Hz. As with NGC 4051, another narrow-line Seyfert 1 galaxy (NLS1), we find that the PSD of MCG-6-30-15 is a close analogue of the PSD of a galactic black hole X-ray binary system (GBH) in a `high' rather than a `low' state. As with NGC 4051 and the GBH Cygnus X-1 in its high state, a smoothly bending model is a better fit to the PSD of MCG-6-30-15, giving a derived break frequency of 7.6<SUP>+10</SUP><SUB>-3</SUB>× 10<SUP>-5</SUP> Hz. Assuming linear scaling of the break frequency with black hole mass, we estimate the black hole mass in MCG-6-30-15 to be ~2.9<SUP>+1.8</SUP><SUB>-1.6</SUB>× 10<SUP>6</SUP>M<SUB>solar</SUB>. <P />Although, in the X-ray band, it is one of the best observed Seyfert galaxies, there has as yet been no accurate determination of the mass of the black hole in MCG-6-30-15. Here we present a mass determination using the velocity dispersion (M<SUB>BH</SUB>-σ<SUB>*</SUB>) technique and compare it with estimates based on the width of the Hα line. Depending on the calibration relationship assumed for the M<SUB>BH</SUB>-σ<SUB>*</SUB> relationship, we derive a mass of between 3.6 and 6 × 10<SUP>6</SUP>M<SUB>solar</SUB>, consistent with the mass derived from the PSD. <P />Using the newly derived mass and break time-scale, and revised reverberation masses for other active galactic nuclei (AGN) from Peterson et al., we update the black hole mass-break-time-scale diagram. The observations are still generally consistent with narrow-line Seyfert 1 galaxies having shorter break time-scales, for a given mass, than broad-line AGN, probably reflecting a higher accretion rate. However, the revised, generally higher, masses (but unchanged break time-scales) are also consistent with perhaps all of the X-ray bright AGN studied so far being high-state objects. This result may simply be a selection effect, based on their selection from high-flux X-ray all-sky catalogues, and their consequent typically high X-ray/radio ratios, which indicate high-state systems.

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https://arxiv.org/pdf/astro-ph/0503100.pdf

{'I M M c Hardy, 1 K F Gunn, 1 P Uttley, 1 M R Goad, 1': '1 Department of Physics and Astronomy, The University, Southampton SO17 1BJ \nAccepted for publication in Mon. Not. R. Astron. Soc.', 'ABSTRACT': "We present a detailed study of the long-timescale X-ray variability of the Narrow Line Seyfert 1 Galaxy (NLS1) MCG-6-30-15, based on eight years of frequent monitoring observations with the Rossi X-ray Timing Explorer . When combined with the published short timescale XMM-Newton observations, we derive the powerspectral density (PSD) covering 6 decades of frequency from ∼ 10 -8 to ∼ 10 -2 Hz. As with NGC4051, another NLS1, we find that the PSD of MCG-6-30-15 is a close analogue of the PSD of a Galactic Black Hole X-ray binary system (GBH) in a 'high' rather than a 'low' state. As with NGC4051 and the GBH Cygnus X-1 in its high state, a smoothly bending model is a better fit to the PSD of MCG-6-30-15, giving a derived break frequency of 7 . 6 +10 -3 × 10 -5 Hz. Assuming linear scaling of break frequency with black hole mass, we estimate the black hole mass in MCG-6-30-15 to be ∼ 2 . 9 +1 . 8 -1 . 6 × 10 6 M /circledot . \n. Although, in the X-ray band, it is one of the best observed Seyfert galaxies, there has as yet been no accurate determination of the mass of the black hole in MCG-630-15. Here we present a mass determination using the velocity dispersion ( M BH -σ ∗ ) technique and compare it with estimates based on the width of the H α line. Depending on the calibration relationship assumed for the M BH -σ ∗ relationship, we derive a mass between 3.6 and 6 × 10 6 M /circledot , consistent with the mass derived from the PSD. \nUsing the newly derived mass and break timescale, and revised reverberation masses for other AGN from Peterson et al. (2004), we update the black hole mass/break timescale diagram. The observations are still generally consistent with narrow line Seyfert 1 galaxies having shorter break timescales, for a given mass, than broad line AGN, probably reflecting a higher accretion rate. However the revised, generally higher, masses (but unchanged break timescales) are also consistent with perhaps all of the X-ray bright AGN studied so far being high state objects. This result may simply be a selection effect, based on their selection from high-flux X-ray all sky catalogues, and their consequent typically high X-ray/radio ratios, which indicate high state systems. \nKey words: black hole physics - galaxies:active - galaxies:individual:MCG-6-30-15 - X-rays:binaries - X-rays:galaxies", '1 INTRODUCTION': "It is now reasonably well established that the X-ray powerspectral densities (PSDs) of AGN are broadly similar to those of galactic black hole X-ray binary systems (GBHs) (M c Hardy 1988; Edelson & Nandra 1999; Uttley et al. 2002; Markowitz et al. 2003; M c Hardy et al. 2004). The PSDs are described by powerlaws of the form P ( ν ) ∝ ν α , where P ( ν ) is the power at frequency ν , but α varies with frequency (see Section 3.2.1 for details). At high frequencies the PSDs of both AGN and GBHs are steep (slope, α, ∼ -2) but below a break frequency, ν B , typically at a few Hz for GBHs, they flatten to a slope of α ∼ -1. To first order, the break \ntimescale scales linearly with the black hole mass but there is still considerable uncertainty in the mass/break timescale relationship and hence in our understanding of the physical similarities between AGN and GBHs. \nThe uncertainty arises largely because GBHs occur in several distinct states (e.g. see McClintock & Remillard 2003). Most commonly they are found in the so-called 'low' state where their X-ray fluxes are low and their medium energy (2-10 keV) X-ray spectra are hard. The second most common state is the 'high' state where their fluxes are high and their 2-10 keV spectra are soft. The PSDs of GBHs in these two states are quite different. In the low state, there is a second break, about a decade below the high frequency \nbreak (Nowak et al. 1999). Below the lower frequency break the PSD flattens further to a slope of zero. In the high state there is no second break and the PSD continues with slope ∼ -1 to very low frequencies (Cui et al. 1997). In addition the break from slope ∼ -2 to slope ∼ -1 occurs at a higher frequency in the high state than in the low state ( ∼ 15Hz cf ∼ 3Hz in the best studied GBH Cyg X-1). Before comparing AGN with GBHs it is therefore important to know whether the AGN is a high or low state system. \nBased on their 2-10 keV X-ray spectra, it has generally been assumed that AGN are the equivalent of low state GBHs. However, although the high frequency ( ν > 10 -6 Hz) parts of many AGN PSDs are now reasonably well determined, the lower frequencies are, in general, not well determined and so it is not usually possible to be sure whether AGN are the analogues of low or high state GBHs. In general it is not possible to determine whether there is a second, lower frequency, break or not. The difficulty in determining the low frequency shape of AGN PSDs has been in obtaining well sampled lightcurves stretching over sufficiently long timescales. For an assumed linear scaling of break timescale with mass, we require well sampled lightcurves of a few years duration to detect the second, lower, break in a black hole of mass ∼ 10 6 M /circledot . \nPrior to the launch of RXTE in November 1995 it was not possible to obtain long timescale lightcurves of sufficient quality but, since early 1996 we (M c Hardy et al. 1998; Uttley et al. 2002; M c Hardy et al. 2004), and others (Edelson & Nandra 1999; Markowitz et al. 2003), have been monitoring a small sample of AGN, typically observing each AGN once every 2 days, and so are able to determine the shape of the low frequency PSD to high precision. Using these observations we (M c Hardy et al. 2004) recently showed that the PSD of the narrow line Seyfert 1 (NLS1) galaxy NGC 4051 was, in fact, identical to that of a high, rather than low state GBH and so provided the first definite confirmation of a high state AGN. Markowitz et al. (2003) plotted the black hole mass against the timescales associated with well measured PSD breaks, some from slope ∼ -2 to ∼ -1 and some from slope ∼ -1 to ∼ 0. They claimed that there was a linear relationship between the two parameters. With a somewhat larger sample we (M c Hardy et al. 2004) plotted the black hole mass against the timescale associated with the specific PSD break from slope ∼ -2 to ∼ -1 for a sample of AGN. We found that the best fit to all of the AGN together had a slope flatter than unity and did not extrapolate to either the high or low state break timescales of Cyg X-1. However we noted that the break timescales associated with broad line Seyfert 1 galaxies were, for a given black hole mass, generally longer than those for those objects usually classed as NLS1s. NLS1s are probably not a distinct class of object but, more likely, just lie at one end of a spectrum of AGNproperties, characterised perhaps by a higher accretion rate. However taking the crude broad/narrow line distinction which is commonly used in the literature, we note that a linear BH-mass/break-timescale relationship fits broad line AGN and Cyg X-1 in the low state, and a displaced linear relationship fits the NLS1s and Cyg X-1 in the high state. We therefore suggested that there is no fixed break timescale/mass relationship which fits all AGN but that the break timescale/mass relationship may actually vary in re- \ne or more other factors, which might be accretion rate or black hole spin. \nIn order to test and clarify the above hypothesis, it is necessary to place AGN on the break timescale/mass plane with high precision and to determine whether they are high or low state analogues. There are very few AGN for which the X-ray observations are extensive enough to make the latter determination possible and a particularly important object is therefore the Seyfert galaxy MCG-6-30-15. Although MCG-6-30-15 is not always described as an NLS1, it has many of the properties commonly associated with NLS1s, ie rapid X-ray variability and relatively narrow permitted emission lines (FWHM 1700 km s -1 (Pineda et al. 1980) c.f. the 2000 km s -1 limit which is usually quoted for NLS1s). In this paper we therefore class MCG-6-30-15 as a NLS1, like NGC4051. \nMCG-6-30-15 is one of the best studied X-ray bright Seyfert galaxies. It was the galaxy in which the first detection of a relativistically broadened X-ray iron line was made, providing direct evidence for the existence of a massive black hole (Tanaka et al. 1995). We have monitored it extensively with the Rossi X-ray Timing Explorer ( RXTE ) and have presented its long timescale PSD (Uttley et al. 2002), based on 3 years of observations (1996-1999). From that early, limited, dataset Uttley et al. (2002) were able to show that a simple unbroken powerlaw was not a good fit to the PSD (at greater than 99 percent rejection confidence) but that a broken powerlaw did provide a good fit (rejection confidence 33 percent). However it was not possible to determine, with any accuracy, the low frequency PSD slope. If they assumed a powerlaw of slope -1, Uttley et al. were able to derive a break frequency in the PSD at ∼ few × 10 -5 Hz and to estimate the high frequency powerlaw slope ( ∼ -2). Thus Uttley et al. were unable to distinguish a high from a low state system, or measure the break timescale with high precision, although they preferred the high state interpretation as a low state interpretation implied a luminosity close to the Eddington limit. Vaughan et al. (2003) have subsequently published a detailed study of the short timescale X-ray variability of MCG-6-30-15 using observations from XMM-Newton . These observations are insufficient, on their own, to determine the low frequency PSD slope but, by also assuming a low frequency PSD slope of -1, they have determined the break frequency to be ∼ 1 × 10 -4 Hz. \nIn this paper (Section 2) we present our full RXTE observations, of 8 years duration, and consisting of over 800 separate observations, compared to 100 in Uttley et al. (2002). In Section 3 we combine these RXTE observations with the published XMM-Newton observations and derive a PSD of excellent quality covering over 6 decades of frequency from < 10 -8 to > 10 -2 Hz. We determine accurately the shape of the PSD below the ∼ 10 -4 Hz break and we thereby show that MCG-6-30-15 is, like NGC4051 (M c Hardy et al. 2004), the analogue of a high state GBH. Using our improved determination of the low frequency PSD shape, we slightly refine the break timescale. \nThe most widely accepted method of determining black hole masses in Seyfert galaxies is that of reverberation mapping (e.g. Peterson 2001) but this technique has not yet been applied to MCG-6-30-15. The best current estimate of the BH mass in MCG-6-30-15 is ∼ 1 × 10 6 M /circledot (Uttley et al. 2002). \nFigure 1. Long Term RXTE 2-10 keV lightcurve of MCG-6-30-15. Each data point represents an observation of ∼ 1 ks. \n<!-- image --> \nThat mass is based on an estimated bulge mass of 3 × 10 9 M /circledot (Reynolds 2000) and on the correlation between black hole mass and galactic bulge mass presented by Wandel (1999), where he claims that Seyfert galaxies have a lower black hole mass, for a given bulge mass, than was claimed in the original relationship of Magorrian et al. (1998). In a later paper Wandel (2002) revises the black hole mass/bulge ratio to 0.0015 and shows that Seyfert galaxies fit the same relationship as normal galaxies. Using the revised ratio, and the estimated bulge mass from Reynolds (2000), the revised black hole mass would be 4 . 5 × 10 6 M /circledot . \nAn upper limit to the black hole mass of 10 7 M /circledot has been derived by Morales & Fabian (2002), using a method based on balancing the radiative and gravitational forces acting on outflowing warm absorber clouds. However it has recently become clear that the successful technique of determining black hole masses from the central velocity dispersion of the galaxy bulge can be applied to active, as well as quiescent, galaxies by observing the width of the Ca ii triplet in the far red, where the AGN continuum is not dominant (Ferrarese et al. 2001). In Section 4 we report the determination of the black hole mass in MCG-6-30-15 from central velocity dispersion measurements. For comparison (Section 5) we also estimate the BH mass from the width of the H β emission line and the continuum flux at 5100 ˚ A, using empirical relations from Kaspi et al. (2000). With greater error, we also estimate (Section 6) the BH mass using the 'photoionisation' method (Wandel et al. 1999). \nIn Section 7 we discuss the implications of our results for the comparison of AGN and GBHs.", '2.1 RXTE': "The RXTE observations discussed here consist of our own monitoring observations, which have continued since 1996, and two 'long looks' in August 1997 and July 1999 which we have obtained from the RXTE archive. The monitoring observations cover timescales from less than a day to ∼ few years and the long looks cover timescales from ∼ days to ∼ minutes.", '2.1.1 Monitoring Observations': "The RXTE monitoring observations, of typically ∼ 1 ks duration, were made and analysed in exactly the same way as for NGC4051 (M c Hardy et al. 2004). Prior to 2000, we used a quasi-logarithmic sampling pattern, covering all timescales, but in order to improve the S/N on our resulting PSDs, we then increased our coverage. From 2000 onwards, our typical observation frequency has been once every two days. In addition, to sample properly the higher frequencies, we observed once every 6 hours for 2 months in 2000. Each observation, of ∼ 1ksec duration, contributes one flux point to the monitoring lightcurve. We do not attempt to split the monitoring data into higher time resolution bins. \nThe observations were made with the Proportional Counter Array (PCA) which consists of 5 Xenon-filled proportional counter units (PCUs). We extracted the Xenon 1 (top) layer data from all PCUs that were switched on during the observation as layer 1 provides the highest S/N for photons in the energy range 2-20 keV where the flux from AGN \nFigure 2. The 2-10 keV RXTE lightcurve of MCG-6-30-15 covering the two-month period of four observations per day. Each data point again represents an observation of ∼ 1 ks. \n<!-- image --> \nis strongest. We used FTOOLS v4.2 for the reduction of the PCA data. We used standard 'good time' data selection criteria, i.e. target elevation > 10 · , pointing offset < 0 . 01 · , time since passage through the South Atlantic Anomaly of > 30min and standard threshold for electron contamination. We calculated the model background in the PCA with the tool PCABACKEST v2.1 using the L7 model for faint sources. PCA response matrices were calculated individually for each observation using PCARSP V2.37, taking into account temporal variations of the detector gain and the changing number of detectors used. Fluxes in the 2-10 keV band were then determined using XSPEC, fitting a simple powerlaw with variable slope but with absorption fixed at the Galactic level of 4 . 06 × 10 20 cm -2 (Elvis et al. 1989). The errors in the flux are scaled directly from the observed errors in the measured count rate. \nAs with NGC4051, we produce a lightcurve in flux units, rather than raw observed counts/sec, so that we may use together data from periods when the number, and gain, of the PCUs changes. The resulting lightcurve, from 1996 to 2004, is presented in Fig. 1. \nFrom the monitoring observations we make three lightcurves. The 6-hr sampled lightcurve consists of the 2 months of observations 4 times per day. The 2-d sampled lightcurve consists of the period from 2000 onwards where sampling is once every 2 days. The total lightcurve consists of all of the observations since 1996, but when used in the PSD analysis it is always heavily binned to 28 or 56-d resolution.", '2.1.2 Long Looks': 'There have been two long looks with RXTE , one in August 1997 and the other in July 1999. The total duration of each observation was about 9 days, consisting of continuous segments of ∼ 3ksec, separated by gaps of approximately equal length caused by earth occultation. The August 1997 obser- \nFigure 3. RXTE background-subtracted lightcurve of MCG-630-15 in the 2-10 keV energy band, with 128s time bins. The lightcurves are shown as histogram segments. Gaps are left where there are no data. Bottom panel is the August 1997 observation and top panel is the July 1999 observation. In both cases the time is measured since the start of the observation. \n<!-- image --> \ntions have been published by Lee et al. (2000) who have carried out a preliminary analysis of their variability properties. Lee et al. (2000) claim a break in the powerspectrum at ∼ 4 -5 × 10 -6 Hz. Nowak & Chiang (2000) have analysed the same dataset, together with ASCA observations, and claim that the resultant PSD resembles that of a GBH in a low state. The claim that the PSD is flat below 10 -5 Hz, has a slope of approximately -1 from 10 -5 Hz to ∼ 10 -4 Hz and, above ∼ 10 -4 Hz it steepens to a slope of -2. Neither Lee et al. or Nowak and Chiang carry out simulations to determine the validity of their PSDs. Uttley et al. (2002) do carry out simulations and conclude that that although there is a break in the PSD of MCG-6-30-15, the model of Nowak and Chiang can be rejected at 99 percent confidence and that there is significant power in the PSD below 10 -5 Hz. A time-series analysis of the July 1999 observations has not yet been published. \nThe two long looks were analysed in exactly the same way as the monitoring observations, except that we retained 16s time resolution. As the July 1999 lightcurve has not previously been published, we include it here (Fig. 3) and, for the convenience of the reader, we also include the August 1997 lightcurve which has previously been published by Lee et al. \nFigure 4. XMM-Newton background-subtracted lightcurve of MCG-6-30-15 in the 0.1-10 keV energy band, with 20s time bins, using data from the PN CCDs. \n<!-- image -->', '2.2 XMM-Newton': 'The XMM-Newton timing observations of MCG-6-30-15 have been discussed extensively by Vaughan et al. (2003) to which we refer readers for a detailed discussion. For the convenience of readers, we reproduce their lightcurve here (Fig 4).', '3 PSD DETERMINATION': 'The method used to determine the PSD is the same Monte Carlo simulation-based modelling technique (Uttley et al. 2002), psresp , which we employed in the analysis of the combined XMM-Newton and RXTE observations of NGC4051 (M c Hardy et al. 2004). This method is able to take account of non-uniform sampling and gaps in the data.', '3.1 Long Timescale PSD': 'A variety of datasets are available for determination of the overall PSD but the RXTE monitoring observations provide the only information on timescales longer than ∼ days. We therefore begin by fitting a simple power law to the PSD from the combined 6-hr, 2-d and total RXTE lightcurves. We retain the intrinsic 6-hr and 2-d resolution for the first two lightcurves but bin the total lightcurve up to 28-d resolution. \nThe resulting PSD is well fitted (fit probability = 68 percent) by a slope of 0 . 9 ± 0 . 15 (90 percent confidence intervals are used throughout this paper). In Uttley et al. (2002) we describe how we determine the goodness of fit from the simulations. A fit probability of 68 percent means that the model is rejected at 32 percent confidence. The fit is shown in Fig. 5. For a GBH in the low-hard state, we do not expect the region of slope ∼ -1 to extend for more than one or 1.5 decades, before flattening to a slope near zero. Although this fit would not be very sensitive to breaks near to the end of the frequency spectral range covered, nonetheless the fit to a simple power law, of slope close to -1 over approximately three decades, from ∼ 10 -8 to ∼ 10 -5 Hz, suggests that MCG-6-30-15 is the analogue of an XRB in a high-soft state rather than in a low-hard state. \nFigure 5. Long timescale RXTE PSD of MCG-6-30-15. The PSD is well fitted (P=68 percent) by a simple powerlaw of slope 0 . 9 ± 0 . 25 over three decades. The underlying undistorted model PSD is shown by the smooth continuous line. The distorted model for each of the three datasets, together with its errors, is shown by the individual points. The observed dirty PSD is given by the jagged lines. \n<!-- image -->', '3.2 Combined Long and Short Timescale PSD': '3.2.1 RXTE monitoring observations and XMM-Newton 4-10 keV observations \nIn order to determine properly the overall long and short timescale PSD, and hence measure any break frequency and high frequency PSD slope, and refine measurements of the low frequency PSD slope, we must combine the RXTE monitoring observations with observations which sample shorter timescales. \nBoth the RXTE long looks and the XMM-Newton long observations provide information on shorter timescales. The RXTE long looks provide a good determination of the PSD on timescales shorter than the length of the individual segments ( ∼ 3000s). However on timescales between ∼ 3000s and ∼ few days, the PSD is distorted by the many gaps in the datasets. Our psresp analysis software is able to cope with those gaps, but they do give the software considerable work to do as the dirty PSD differs a good deal from any likely underlying true PSD. Thus the errors are larger than if there are fewer gaps. The XMM-Newton data are continuous and so the XMM-Newton PSD suffers negligibly from distortion. However the sensitivity of XMM-Newton is less than that of RXTE in the 2-10 keV range, which is the only energy band in which the long timescale PSD is determined. Nonetheless, the long XMM-Newton observations of MCG6-30-15, which total approximately three times as long as those of NGC4051, do allow a reasonable determination of the high frequency PSD. \nAs discussed in M c Hardy et al. (2004), the mean photon energy of the 4-10 keV XMM-Newton band is approximately the same as that of the RXTE 2-10 keV band. In order to derive a PSD of uniform energy at all frequencies we have therefore included the 4-10 keV XMM-Newton PSD with the RXTE monitoring datasets described above in our \nPSD simulation process. The high-soft state PSD of Cyg X1 is better described by a bending powerlaw model than a sharply broken powerlaw model, and the PSD of NGC4051 is also slightly better fit by the bending powerlaw model \nP ( ν ) = Aν α L ( 1 + ( ν ν b ) ( -α H + α L ) ) -1 \nwhere α L and α H are the low and high frequency powerlaw slopes respectively and ν b is the bend, or break, frequency. As stated in the Introduction, here we use the convention that a slope decreasing towards high frequencies will be defined by P ( ν ) ∝ ν α where α is a negative number, eg α = -2. 1 \nWe fit this model to the combined RXTE and XMMNewton observations and the result is shown in Figs. 6 and 7. In Fig. 6 we plot the log of power vs. the log of frequency, as in Fig. 5. In both of these figures the observed dirty PSDs from the various constituent lightcurves are given by the jagged lines. The model, distorted by the effects of sampling, red noise leak and aliasing, is given by the points with errorbars, and the underlying, undistorted, model is given by the smooth dashed line. In Fig. 7 we unfold the observed dirty PSD from the distorting effect of the sampling in order to produce the closest approximation that we can to the true underlying PSD. Thus displacements of the observed PSD from the model distorted PSD are translated into displacements from the best-fit underlying model PSD. The technique is identical to that used to deconvolve energy spectra from count rate (ie instrumental) spectra in standard Xray spectral fitting. In this case the error is translated onto the observed datapoints and the underlying best-fit model is shown as a continuous line. As in standard X-ray spectral fitting, the position of the datapoints does depend on the shape of the assumed, best-fitting, model. Note that in Fig. 7 we plot frequency × power so that a horizontal line would represent Power ( ν ) ∝ ν -1 and equal power in each decade. Similar plots for NGC4051, NGC3516 and Cyg X-1 are shown in Fig.18 of M c Hardy et al. (2004). \nThe combined PSD is well fitted (P=45 percent) by this model with low frequency slope α L = -0 . 8 +0 . 4 -0 . 16 , high frequency slope α H = -1 . 98 +0 . 32 -0 . 40 and break frequency ν B = 6 . 0 +10 -5 × 10 -5 Hz. A sharply broken powerlaw, although tolerable, is not such a good fit (P=14 percent) but the best fit parameters are similar to those of the smoothly bending model ( α L = -0 . 8, α H = -1 . 74 and ν B = 3 . 2 × 10 -5 Hz). As is often the case, the break frequency is slightly lower for the sharply bending model than for the smoothly bending model but, in this case, the errors are not well determined. (Note that our software performs a simple grid search so the values given here are those of the minimum point on the grid and so may differ very slightly from the optimum values which may lie slightly off any particular grid point.) The confidence contours for these parameters are shown in Figs. 8, 9 and 10. \n1 This convention is used throughout the Tables and text of M c Hardy et al. (2004). Note that when we defined the formula for a bending powerlaw in M c Hardy et al. (2004)(top of second column, page 788), we forgot to make the formula consistent with the rest of the text and Tables and so, only in that one formula, a slope decreasing towards high frequencies is defined by P ( ν ) ∝ ν -α where α is a positive number, eg α = +2. \nFigure 6. Combined 2-10 keV RXTE and 4-10 keV XMMNewton PSD of MCG-6-30-15. The lines and datapoints are as described in Fig. 5. \n<!-- image --> \nFigure 7. Unfolded combined 2-10 keV RXTE and 4-10 keV XMM-Newton PSD of MCG-6-30-15. Here the errors have been translated onto the datapoints and the continous line represents the best-fit underlying PSD. Note that here we plot frequency × Power. See text for details. \n<!-- image --> \nThe break frequency for ,¸ assuming a high state, bending powerlaw model, is 22 . 9 ± 1 . 5Hz (M c Hardy et al. 2004) (or 13 . 9 ± 0 . 8Hz for a sharply breaking high state model). Assuming linear scaling of black hole mass with frequency, and a black hole mass of 10 M /circledot for ¸ (Herrero et al. 1995), we derive a black hole mass for MCG-6-30-15 of 3 . 6 +18 -2 × 10 6 M /circledot . \nWenote that, although consistent within the errors with the break frequency derived by Vaughan et al. (2003) using purely XMM-Newton observations, our break frequency is slightly lower. The reason for the difference is that Vaughan et al. were unable to measure the PSD slope below the break and so assumed a value of -1. However our data show that the slope is actually slightly flatter, which leads to a lower break frequency. \nFigure 10. 68 percent, 90 percent and 99 percent confidence contours for low and high frequency slopes, α L and α H respectively for bending powerlaw fit to the combined RXTE and XMMNewton 4-10 keV PSD shown in Fig. 6. Note we again plot -α L and -α H . \n<!-- image --> \nFigure 8. 68 percent, 90 percent and 99 percent confidence contours for high frequency slope, α H , and break frequency, ν B , for bending powerlaw fit to the combined RXTE and XMM-Newton 4-10 keV PSD shown in Fig. 6. Note we plot -α H . \n<!-- image --> \nFigure 9. 68 percent, 90 percent and 99 percent confidence contours for low frequency slope, α L , and break frequency, ν B , for bending powerlaw fit to the combined RXTE and XMM-Newton 4-10 keV PSD shown in Fig. 6. Note we plot -α L \n<!-- image -->', '3.2.2 RXTE monitoring observations and low energy XMM-Newton observations': 'In M c Hardy et al. (2004) we refined our determination of the break frequency in NGC4051 by combining RXTE monitoring observations with continuous XMM-Newton observations in the 0.1-2 keV band, where the break is better defined. We assumed that the break frequency was independent of energy. We have carried out the same procedure here (again taking proper account of the different PSD normalisations between the 0.1-2 and 2-10 keV bands) and find that α L = -0 . 8 +0 . 2 -0 . 1 , α H = -2 . 5 +0 . 3 -0 . 4 and ν B = 7 . 6 +10 -3 × 10 -5 Hz. The fit probability is 67 percent. Apart from the steeper high energy slope which is well known at low energies \n(Vaughan et al. 2003; M c Hardy et al. 2004), the other fit parameters are very similar to those obtained when using the RXTE and XMM-Newton 4-10 keV observations. Thus within the errors, we find no evidence for a change of break frequency with photon energy. The implied black hole mass is more tightly constrained by the smaller errors on the break frequency and is 2 . 9 +1 . 8 -1 . 6 × 10 6 M /circledot . \nWe fitted a sharply breaking powerlaw model to the combined RXTE and XMM-Newton 0.1-2 keV data. As with the combined RXTE and 4-10 keV XMM-Newton data, the fit parameters are almost exactly the same as for the bending powerlaw model, but the fit probability is worse (19 percent). The fact that a bending powerlaw is a better fit than a sharply bending powerlaw to the PSD of MCG-6-30-15, as it is to the PSD of i¸n its high state, strengthens our conclusion that MCG-6-30-15 is the analogue of a GBH in the high, rather than low, state.', '3.2.3 RXTE monitoring observations and RXTE long looks': 'We have also determined the overall PSD shape by including the RXTE long looks binned up to 128s time resolution with the RXTE monitoring observations, and also including a very high frequency PSD ( > 10 -3 Hz) made from the individual segments of the RXTE long looks. The fit parameters are approximately the same as in the combined RXTE and XMM-Newton fit ( α L = -0 . 8 +0 . 5 -0 . 15 , α H = -2 . 1 +0 . 4 -0 . 6 and ν B = 5 . 0 +8 -4 × 10 -5 Hz). The fit is formally good (P=74 percent) but the errors are rather large because of the difficulty of coping with the many gaps in the RXTE long look lightcurves.', '3.3 Search for a second, lower frequency, break': 'We searched for a second, lower frequency break, using the RXTE and XMM-Newton 4-10 keV data. We assumed a low \nFigure 11. Confidence contours for the high and possible low break frequencies. The 68 percent contour is solid, the 90 percent contour is dashed and the 99 percent contour is dot-dashed. The straight dashed lines, with labels of 10 and 100 respectively, indicate those ratios between the high and possible low break frequencies. The fits assume a break below the low frequency break of 0 and an intermediate slope between the low and high frequency breaks of 0.8. Both break frequencies were allowed to vary, as was the slope above the upper break, which was measured at 1 . 95 ± 0 . 3. \n<!-- image --> \nstate PSD model, fixing the slope below the lower break at 0, and the slope between the lower and upper breaks at 0.8, i.e. the value we measure in the same frequency range assuming a high state model. We allowed the two break frequencies and the slope above the upper break to vary. The best fit frequency for the higher break and for the slope above the higher break was the same as in our high state fit and the best fit for the lower break frequency (6 × 10 -9 Hz) was almost at the end of the fitting range. The fit probability was 43 percent. In Fig. 11 we show the 68 percent, 90 percent and 99 percent confidence ranges for the two break frequencies. As the typical ratio of the two break frequencies for a low state GBH is between 10 and 30, we plot, as dashed straight lines, the limits corresponding to a ratios of 10 and of 100. The best fit ratio is over 1000, but a ratio of 100 is just within the edge of the 90 percent confidence region. \nWe repeated the above analysis after fixing the intermediate slope at -1 . 0 and obtained almost identical frequency ratios although the high break frequency was then about half a decade higher in frequency. \nOn the basis of its PSD, the above analysis strongly suggest that MCG-6-30-15 is the analogue of a high state GBH, although a low state cannot be entirely ruled out.', '4 MASS DETERMINATION: ABSORPTION LINE VELOCITY DISPERSION': 'As discussed in the Introduction, the mass of the black hole in MCG-6-30-15 is not well determined. In particular there have been no reverberation mapping observations, and no measurement of the stellar velocity dispersion, the two techniques widely regarded as giving the most reliable measure- \nment of black hole masses. However a reliable measurement of the mass in MCG-6-30-15 is important for any discussion of mass/timescale scalings in AGN and for any discussions of whether AGN are in low or high states. In this section we therefore present stellar velocity dispersion measurements from which we estimate the black hole mass. In subsequent sections (Sec 5 and Sec 6) we present secondary determinations of the black hole mass, based on the width of the H β emission line and on photoionisation calculations. We find that all these optically based mass determination methods give consistent answers.', '4.1 Observations': "The observations of MCG-6-30-15 described here were taken by the service programme of the 3.6m Anglo-Australian Telescope on 2002 June 5. The RGO Spectrograph was used, with the 25cm camera plus the 1200R grating (blazed at 7500 ˚ A). The detector is an EEV2 CCD, windowed to 600 × 4096 pixels for the science observations and 150 × 4096 pixels for the standard stars. A slit width of 1 '' (0.15mm) was chosen, resulting in a spectrum of resolution R ∼ 9500 covering a wavelength range of 1000 ˚ A. The slit was placed along the major axis of the galaxy, at a position angle of 116 · . \nThe measured widths of the Ca ii triplet lines are used to estimate the black hole mass. At the redshift of MCG-630-15, z = 0 . 00775, the Ca ii lines are shifted to λλ 8566 , 8610 and 8731 ˚ A, and therefore the central wavelength was chosen to be 8620 ˚ A. The total wavelength coverage of 1000 ˚ A, spanning the wavelength range 8120 -9120 ˚ A, allows accurate determination of the continuum on either side of the lines, and therefore correct measurement of the line widths. \nObservations of late-type giant stars, which have very narrow lines, and which dominate the stellar light of nearby galaxy bulges, were performed in order to measure the instrumental broadening and provide templates for the crosscorrelation analysis. The K0 iii standard stars HD117927 and HD118131 were observed, with the same set-up and wavelength coverage as MCG-6-30-15, since the rest wavelength of the Ca ii triplet ( λλ 8498 , 8542 , 8662 ˚ A) lies within the wavelength range covered. \nDedicated flat field exposures were taken before and after the science and standard star observations, for accurate de-fringing, and argon and neon arc spectra were taken for the wavelength calibration.", '4.2 Data Reduction': "The raw data were reduced with IRAF using standard bias subtraction, flat fielding and cosmic ray removal techniques. At the red wavelengths observed here, the EEV2 CCD suffers from fringing at a level of ∼ 5 percent. A comparison of the extracted science spectra with equivalent spectra taken from the flat field frames showed that the fringes were efficiently removed from the science frames, and did not contribute any spurious features. Curvature in the spatial direction was rectified using bright night sky lines in the object frames. The narrow window used for the standard star observations meant that in this case, rectification was unnecessary. The continua of MCG-6-30-15 and the standard stars \nFigure 12. Normalised spectra of MCG-6-30-15 (top panel) and a template star (bottom), with the position of the calcium triplet feature and broad O i λ 8446 emission line marked. \n<!-- image --> \nwere sufficiently bright that curvature of the spectra in the dispersion direction could be traced and rectified. The total exposure time for MCG-6-30-15 was 1 hr, broken into two 30min integrations. Wavelength calibration was performed using the arc exposures bracketing each science frame. This calibration was checked by measuring the resulting wavelength of night sky emission lines, and no further correction was required. \nThe nuclear spectrum of MCG-6-30-15 was extracted from the central 5.6 pixels of the trace, which corresponds to a 2 . 4 '' × 1 '' aperture. To be compatible with the M BH -σ ∗ relation defined by Ferrarese & Merritt (2000), the length of this aperture was chosen to correspond to r e / 8, where r e is the effective radius of the galaxy, measured from the profile of the spectral data, compressed along the dispersion direction. We note however that the choice of extraction aperture has only a very small effect on the results obtained (Ho, private communication). Sky subtraction was performed using regions either side of the galaxy, 50 pixels (21 . 5 '' ) wide, at a distance of 50 pixels (21 . 5 '' ) from the centre of the galaxy. The resulting spectrum is shown in Fig. 12, normalised by a spline fit to the continuum to emphasize the emission and absorption features. \nIn a similar analysis, Filippenko & Ho (2003) detect Paschen emission lines in the spectrum of NGC 4395, adjacent to each of the calcium absorption features. For MCG6-30-15, there is perhaps a suggestion of a broad Paschen 15 line at ∼ 8615 ˚ A, but it is not conclusive. Even if present, the strength of such a line is very small compared to those detected in NGC4395, and as such, would not significantly affect our results.", '4.3 Data Analysis': 'The calcium absorption features in a galaxy spectrum are broadened compared to those of an individual star due to the bulk motion of the stars in the galaxy. The extent of this Doppler broadening is related to the mass of the central black hole, giving rise to the observed tight relationship between the two quantities. The region of the galaxy spectrum of MCG-6-30-15 containing the calcium triplet is therefore \ncross-correlated with the equivalent region of the K0 iii stellar templates using the IRAF task fxcor . The task fits a spline function to the continuum of each input spectrum, and cross-correlates the resulting features, giving as output a velocity measurement, due to the redshift of MCG-6-3015, and a velocity width, due to the Doppler effect of the motion of stars in the galaxy caused by the gravitational influence of the black hole. \nThe width of the cross-correlation peak function calculated by fxcor measures the combined Doppler and instrumental broadening, therefore in order to ascertain the Doppler broadening alone, the stellar templates themselves were broadened using a Gaussian of various widths, and cross-correlated with the unbroadened templates. The width of the Gaussian which gave rise to a cross-correlation peak width equal to that of MCG-6-30-15 provides our measurement of the true velocity dispersion of the galaxy. We found that broadening the stellar templates by 11 pixels gave the best match to the cross-correlation peak width of MCG-6-30-15. The velocity dispersion of the spectra is 8 . 5 kms -1 pix -1 , giving a velocity dispersion for MCG-630-15 of σ = 93 . 5 km s -1 . \nThe statistical errors on this measurement are very small, and therefore do not provide a realistic estimate of the true errors involved. Ferrarese et al. (2001) estimate that the systematic uncertainties involved in making such measurements are of order 15 percent. We therefore undertook a number of tests in order to ascertain the likely error in our result. First, since we have three standard star observations (two of HD117927 and one of HD118131), each spectrum was broadened in turn and cross-correlated with the two unbroadened stellar templates to estimate the variation due to the use of differing standards. The results usually varied by less than one percent, and never more than two percent. Secondly, to assess the possible impact of the proximity of the O i line to the first feature in the calcium triplet, we restrict the region of the spectra used in the cross-correlation analysis to that containing only two of the three calcium absorption lines. By changing the wavelength range over which the cross-correlation analysis is performed, a maximum variation of ± 4 percent in the peak width was measured. Thirdly, we vary the amount of smoothing of the stellar templates required to match the result from MCG-6-30-15, and find that all the results of the above tests are easily bracketed by taking a Gaussian width of 11 ± 1 pixels, which is equivalent to σ = 93 . 5 ± 8 . 5 kms -1 ( ± 9 . 1 percent). We therefore take this as our conservative estimate of the error in our measurement, and note that this is small compared to the uncertainty in the relationship between this quantity and the inferred black hole mass, as described in the following section.', '4.4 Black Hole Mass Estimate': 'The largest source of error in estimating the mass of the black hole from the width of the absorption lines is not uncertainty in the measurement of the width but uncertainty in the M BH -σ relation itself. The two main groups concerned with deriving the relationship use slightly different observational methods and fit slightly different relationships to the resultant data (see Merritt & Ferrarese 2001; Tremaine et al. 2002, for full discussions). Recently \nGreene et al. (2004) have shown that AGN with very low black hole masses (down to ∼ 10 4 M /circledot ) fit reasonably well onto the relationship given by Tremaine et al. (2002), i.e., \nM BH = 1 . 35 +0 . 20 -0 . 18 × 10 8 M /circledot ( σ 200 km s -1 ) 4 . 02( ± 0 . 32) . \nOur observations then imply M BH = 6 . 3 +3 . 0 -2 . 0 × 10 6 M /circledot . As can be seen from Fig. 4 of Greene et al. (2004), the spread in the datapoints implies an uncertainty in the derived mass of at least 50 per cent. \nAn alternative version of this relationship has been given by Merritt & Ferrarese (2001). Using the most recent version of this alternative relationship (Ferrarese 2002) i.e. \nM BH = 1 . 66( ± 0 . 32) × 10 8 M /circledot ( σ 200 kms -1 ) 4 . 58( ± 0 . 52) , \nwe derive M BH = 5 . 1 +3 . 8 -2 . 4 × 10 6 M /circledot , which is entirely consistent with the mass derived using the relationship of Tremaine et al. (2002).', '5 MASS DETERMINATION: EMISSION LINE WIDTH': 'An alternative estimate for M BH can be obtained using the empirical relationships found by Kaspi et al. (2000) in reverberation studies of nearby Seyfert 1 galaxies and quasars between M BH , the velocity dispersion of the broad emissionline gas, V FWHM , 2 and the effective size, R BLR , of the broad line emitting region, i.e.: \nM BH = 1 . 464 × 10 5 ( R BLR lt -days ) ( V FWHM 10 3 km s -1 ) 2 M /circledot . \nR BLR is determined from the measured delay between the continuum and emission-line variations and is related to the galaxy subtracted continuum luminosity at λ 5100 ˚ A by \nR BLR = 32 . 9 +2 -1 . 9 ( λL λ (5100 ˚ A) 10 44 erg s -1 ) 0 . 7 . \nReynolds et al. (1997) discuss in detail the optical spectrum of MCG-6-30-15 and, in their Fig. 4, derive an estimate of the intrinsic non-stellar optical flux (ie νF ν , or λF λ ) at 5100 ˚ Aof ∼ 6 × 10 -11 ergs cm -2 s -1 . Assuming the standard flat cosmology of H 0 = 75 km s -1 Mpc -1 and Λ matter = 0 . 3, with z = 0 . 007749, this flux equates to a rest-frame luminosity of λL λ = 7 . 2 × 10 42 erg s -1 and yields R BLR = 5 . 2 lt-day. \nWe note that there is considerable scatter in the relationship between R BLR and λF λ at 5100 ˚ A. Vestergaard (2002) repeated the empirical study of Kaspi et al. (2000) using the linear regression techniques of Akritas & Bershady (1996) and obtained a slightly different relationship. However the derived value of R BLR , ie 5.36 lt-day is, within the errors, identical to the value derived using the relationship of Kaspi et al. (2000). Vestergaard (2002) derives the relationship using only broad line Seyfert 1 galaxies. The narrow \n2 While one might expect V FWHM of the root mean square profile to better represent the velocity dispersion of the variable part of the BLR, and thus show a better correspondence with R BLR , in practice there is little difference in the derived masses (Kaspi et al. 2000). \nFigure 13. HST/STIS spectrum of MCG-6-30-15, showing broad and narrow H β components, and [O iii ] λλ 4959 , 5007 ˚ A emission lines. \n<!-- image --> \nline Seyfert galaxy, NGC4051, is an outlier and is not used in deriving the relationship, although it is used by Kaspi et al. (2000). \nUsing archival HST/STIS data for MCG-6-30-15 (Fig. 13) we have measured (rest frame) V FWHM = 32 . 9 ± 1 . 9 ˚ A ( ≡ 2020 ± 120 km s -1 ) which, together with R BLR , gives a virial mass for the black hole in MCG-6-30-15 of 3 × 10 6 M /circledot .', '6 MASS DETERMINATION: PHOTOIONISATION': 'The size of the BLR may also be found from photoionization calculations, but with relatively large uncertainty. The calculated BLR size, combined with the velocity at FWHM of the H β line profile can then be used to provide an estimate for the virial mass of the central black hole. From a sample of 17 Seyfert 1 galaxies and 2 quasars, Wandel et al. (1999) found an approximately linear relationship between photoionization mass estimates and reverberation mass estimates, suggesting that photoionization calculations might providing a route for mass determinations in systems with only single epoch spectral observations. The empirical relationship derived by Wandel et al. (1999) is: \nM BH = 2 . 8 × 10 6 f ( L 44 Un 10 ) 1 / 2 ( V FWHM 10 3 km s -1 ) 2 M /circledot . \nHere U is the ionization parameter (the dimensionless ratio of photon to gas density), n 10 is the electron density in units of 10 10 cm -3 , and f is the product f k f 1 / 2 L E -1 / 2 , where f k is a factor relating the effective velocity dispersion to the projected velocity dispersion, f L relates the observed luminosity L to the ionizing luminosity, and E = E/ 1 ryd. If we assume a BLR ionization parameter and gas density typical for nearby Seyfert 1 galaxies ( U = 0 . 1, n 10 = 10), and a weighted f -value ∼ 1 . 45 Wandel et al. (1999) we derive a black hole mass of 4 . 5 × 10 6 M /circledot .', '7.1 Summary of Mass Determinations For MCG-6-30-15': 'Although considerable systematic uncertainties and large assumptions (eg in n 10 and U ) are involved in their derivations, the various optical measurements of the black hole mass in MCG-6-30-15, including the revised estimate based on the bulge mass, are in reasonable agreement. All values lie between ∼ 3 and ∼ 6 × 10 6 M /circledot . We cannot tell which measurement is correct and so, as a working value, we take the middle of the range, ie ∼ 4 . 5 × 10 6 M /circledot and adopt an error equal to the spread in the measurements (ie 3 × 10 6 M /circledot ). We note that although there are again large uncertainties in the mass derived from the PSD, the most tightly constrained mass, ie that derived from a combination of RXTE and low energy XMM-Newton observations and assuming linear scaling of break timescale with mass from Cyg X-1 in the high state, ie ∼ 2 . 9 +1 . 8 -1 . 6 × 10 6 M /circledot , is in good agreement with the mass as determined by optical methods. \nThe average 2-10 keV X-ray flux of MCG-6-30-15 from our RXTE monitoring is 5 . 9 × 10 -11 ergs cm -2 s -1 , which corresponds to a luminosity of ∼ 7 × 10 42 ergs s -1 . For an assumed X-ray/bolometric correction of 27 (Padovani & Rafanelli 1988; Elvis et al. 1994), the bolometric luminosity is 1 . 9 × 10 44 ergs s -1 , implying that, for a mass of ∼ 4 . 5 × 10 6 M /circledot , MCG-6-30-15 is radiating at ∼ 40 percent of its Eddington luminosity. For narrow line Seyfert galaxies, the bolometric correction should probably be less than 27 but probably still greater than 10, so MCG-6-30-15 is almost certainly radiating at > 10 percent of its Eddington luminosity. A low state interpretation of the PSD which, although very unlikely, cannot be ruled out entirely, implies a black hole mass of ∼ 5 × 10 5 M /circledot , thus requiring a superEddington luminosity.', '7.2 X-ray Selected AGN and High State PSDs': "Using our newly derived black hole mass, and slightly refined break timescale, we plot (Fig. 14) MCG-6-30-15 on a revised version of the break timescale/black hole mass diagram which we presented in M c Hardy et al. (2004). Although bending, rather than sharply breaking, powerlaws fit the PSDs best, we still use timescales derived from sharp breaks in this diagram as we do not yet have timescales derived from bending powerlaw fits to all of the AGN. Like NGC4051, MCG-6-30-15 also sits above the high state line. The break timescales are taken from the compilation in M c Hardy et al. (2004) with the addition of NGC4395 from Vaughan et al. (2005), with mass estimate from Filippenko & Ho (2003). Kraemer et al. (1999) present UV and optical spectral of NGC4395 and show broad, although rather weak, wings to Hβ . They estimate FWHM ∼ 1500 km s -1 , but do not give an error. We are therefore unsure whether to class it as a broad or narrow line Seyfert galaxy and so mark it by an open triangle in Fig. 14. We also include NGC3227 from Uttley and M c Hardy(in preparation). Uttley and M c Hardyalso discuss NGC5506 in considerable detail but we do not include it here as its mass is highly uncertain. \nWhere available (ie Fairall9, NGC3227, NGC3783, NGC4051, NGC4151, NGC5548) we use reverberation \nmasses from the compilation of Peterson et al. (2004). No reverberation masses are available for the other AGN, ie the narrow line Seyfert 1s, and so masses are derived from stellar velocity dispersion measurements. For some objects alternative masses are available from the emission line width (eg 6 . 3 × 10 5 M /circledot for Mkn766 from Botte et al. 2005) but to reduce the number of variables we restrict ourselves to the velocity dispersion mass (3 . 5 × 10 6 M /circledot for Mkn766, using the calibration of Tremaine et al. 2002). Current work (e.g. Ferrarese et al. 2001; Peterson et al. 2004) shows that masses derived from reverberation mapping and velocity dispersion are consistent. An exception is Akn564. As in some other NLS1s, the Ca ii triplet lines in Akn564 are in emission (van Groningen 1993), rather than absorption, and so cannot be used to determine the black hole mass. In this case the width of the [O III] 5007 emission line is used as a substitute for the width of the stellar absorption lines in order to estimate the black hole mass (Botte et al. 2004). The width of the [O III] 5007 emission line does correlate, although with considerable scatter (Boroson 2003), with the width of the Ca ii triple stellar absorption lines. However it has been noted (Botte et al. 2005) that the [O III] lines tend to be wider than the Ca absorption lines. Therefore, assuming that the Ca absorption lines represent the black hole mass more accurately, the [O III] line will typically give an overestimate of the black hole mass, hence the upper limit on Akn564 in Fig. 14. \nDue to recalibration to better fit the M BH -σ ∗ relationship, masses in the sample of Peterson et al. (2004) have generally increased from the earlier estimates used in M c Hardy et al. (2004). Also, the black hole mass estimate for NGC4051 from Peterson et al. (2004) is higher than the estimate from Shemmer et al. (2003) which we used in M c Hardy et al. (2004). Here, for consistency, we use the black hole mass values from Peterson et al. (2004) wherever available. Thus all AGN from the Peterson et al. (2004) sample have moved further away from the low-state line and towards the high state line. With the exception of NGC4151, all AGN now lie above the low state line. Although the 'broad line' Seyfert galaxies are, in general, closer to the low state line than the 'narrow line' Seyfert 1s, it is possible that all AGN considered here might be high state systems. Thus although ˙ m may well have an important effect on determining the break timescale, the accretion rate in all AGN studied here may be high enough to make them 'high state' systems. Indeed Peterson et al. (2004) show that the average accretion rate for the current sample of AGN is roughly 10 percent of the Eddington rate, which exceeds the ∼ 2 percent rate which typifies the transition to the high state in galactic X-ray binary systems (Maccarone 2003). For completeness we also include the break timescale of NGC3227 from Uttley and M c Hardy(2005, in preparation). NGC3227 is an interesting galaxy, having broad permitted lines, a hard X-ray spectrum, and a high state PSD. It is discussed in detail by Uttley and M c Hardy(2005, in preparation). \nAn additional indicator of the 'state' of an accreting black hole is given by the ratio of its X-ray to radio luminosity (e.g. Gallo et al. 2003; Fender 2001, 2003). For Galactic black hole systems, a high X-ray/radio ratio signifies a high accretion rate and a 'high' state system. A low X-ray/radio ratio signifies a low accretion rate and a jet-dominated 'low' state system. The core radio flux of MCG-6-30-15 is very \nlow, ∼ 1mJy at 5GHz (Ulvestad & Wilson 1984), and very similar to that of NGC4051 (M c Hardyet al, in preparation). The X-ray fluxes, and hence the X-ray/radio ratios, of both galaxies are high, again indicating 'high' state systems.", '7.2.1 A Possible Selection Effect': 'It is interesting to note that the few AGN with sufficiently good PSDs to distinguish between high and low states (NGC4051, MCG-6-30-15, NGC3227), all have high state PSDs. The AGN which we, and others (e.g. Markowitz et al. 2003), have been monitoring are mainly taken from the Xray bright AGN which are visible in the bright-flux limit, all-sky, X-ray catalogues (e.g. McHardy et al. 1981). As Xray flux rather than, eg, radio flux, is the only selection criterion in such surveys, there is a strong selection effect towards selecting high state AGN. \nAGN with low state X-ray PSDs are likely to be found amongst those AGN which have relatively more luminous radio emission (cf Merloni et al. 2003). As X-ray emission will not be the only parameter on which we select such AGN, we may expect them to be, in general, fainter in X-rays than those studied presently. Such AGN should be present in, eg, the sample of radio and X-ray bright objects selected from the ROSAT and VLA FIRST all-sky catalogues (Brinkmann et al. 2000). \nThe observations from RXTE presented here, and elsewhere (e.g. Markowitz et al. 2003; M c Hardy et al. 2004), demonstrate the tremendous importance of long timescale (years/decades) X-ray monitoring observations for our understanding of AGN. 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2020MNRAS.493.1888T

The relationship between black hole mass and galaxy properties: examining the black hole feedback model in IllustrisTNG

2020-01-01

['galaxies evolution', 'galaxies general', 'galaxies star clusters', '-']

[]

Supermassive black hole feedback is thought to be responsible for the lack of star formation, or quiescence, in a significant fraction of galaxies. We explore how observable correlations between the specific star formation rate (sSFR), stellar mass (M<SUB>star</SUB>), and black hole mass (M<SUB>BH</SUB>) are sensitive to the physics of black hole feedback in a galaxy formation model. We use the IllustrisTNG simulation suite, specifically the TNG100 simulation and 10 model variations that alter the parameters of the black hole model. Focusing on central galaxies at z = 0 with M<SUB>star</SUB> &gt; 10<SUP>10</SUP> M<SUB>⊙</SUB>, we find that the sSFR of galaxies in IllustrisTNG decreases once the energy from black hole kinetic winds at low accretion rates becomes larger than the gravitational binding energy of gas within the galaxy stellar radius. This occurs at a particular M<SUB>BH</SUB> threshold above which galaxies are found to sharply transition from being mostly star forming to mostly quiescent. As a result of this behaviour, the fraction of quiescent galaxies as a function of M<SUB>star</SUB> is sensitive to both the normalization of the M<SUB>BH</SUB>-M<SUB>star</SUB> relation and the M<SUB>BH</SUB> threshold for quiescence in IllustrisTNG. Finally, we compare these model results to observations of 91 central galaxies with dynamical M<SUB>BH</SUB> measurements with the caveat that this sample is not representative of the whole galaxy population. While IllustrisTNG reproduces the observed trend that quiescent galaxies host more massive black holes, the observations exhibit a broader scatter in M<SUB>BH</SUB> at a given M<SUB>star</SUB> and show a smoother decline in sSFR with M<SUB>BH</SUB>.

[]

https://arxiv.org/pdf/1906.02747.pdf

{'The relationship between black hole mass and galaxy properties: Examining the black hole feedback model in IllustrisTNG': "Bryan A. Terrazas 1 glyph[star] , Eric F. Bell 1 , Annalisa Pillepich 2 , Dylan Nelson 3 , Rachel S. Somerville 4 , Shy Genel 4 , 5 , Rainer Weinberger 6 , M'elanie Habouzit 4 , Yuan Li 4 , 7 , Lars Hernquist 6 , and Mark Vogelsberger 8 \n- 1 Department of Astronomy, University of Michigan, Ann Arbor, MI 48109, USA\n- 2 Max-Planck-Institut fur Astronomie, Konigstuhl 17, 69117 Heidelberg, Germany\n- 3 Max-Planck-Institut fur Astrophysik, Karl-Schwarzschild-Str. 1, D-85748, Garching, Germany\n- 4 Center for Computational Astrophysics, Flatiron Institute, 162 Fifth Ave., New York, NY 10027, USA\n- 5 Columbia Astrophysics Laboratory, Columbia University, 550 West 120th Street, New York, NY 10027, USA\n- 6 Institute for Theory and Computation, Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA\n- 7 Astronomy Department, University of California, Berkeley, CA 94720, USA\n- 8 Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA \nJune 10, 2019", 'ABSTRACT': 'Supermassive black hole feedback is thought to be responsible for the lack of star formation, or quiescence, in a significant fraction of galaxies. We explore how observable correlations between the specific star formation rate (sSFR), stellar mass (M star ), and black hole mass (M BH ) are sensitive to the physics of black hole feedback in a galaxy formation model. We use the IllustrisTNG simulation suite, specifically the TNG100 simulation and ten model variations that alter the parameters of the black hole model. Focusing on central galaxies at z = 0 with M star > 10 10 M glyph[circledot] , we find that the sSFR of galaxies in IllustrisTNG decreases once the energy from black hole kinetic winds at low accretion rates becomes larger than the gravitational binding energy of gas within the galaxy stellar radius. This occurs at a particular M BH threshold above which galaxies are found to sharply transition from being mostly star-forming to mostly quiescent. As a result of this behavior, the fraction of quiescent galaxies as a function of M star is sensitive to both the normalization of the M BH -M star relation and the M BH threshold for quiescence in IllustrisTNG. Finally, we compare these model results to observations of 91 central galaxies with dynamical M BH measurements with the caveat that this sample is not representative of the whole galaxy population. While IllustrisTNG reproduces the observed trend that quiescent galaxies host more massive black holes, the observations exhibit a broader scatter in M BH at a given M star and show a smoother decline in sSFR with M BH . \nKey words: galaxies: general - galaxies: evolution - galaxies: star formation', '1 INTRODUCTION': "In the last 10 billion years, the amount of new star formation in the Universe has decreased substantially. Observational surveys of galaxy populations at different epochs have also shown a gradual increase in the number of galaxies whose light is dominated by older stellar populations signaling a lack of new stars in these systems (e.g., Bell et al. 2004; \nc © 2019 The Authors \nBaldry et al. 2004; Williams et al. 2009; Muzzin et al. 2013; Ilbert et al. 2013; Tomczak et al. 2014). Studies of quiescence, i.e. the state of reduced star formation activity, in galaxies have aimed at pinpointing a physical mechanism responsible for this behavior. Many of these studies, including this work, focus on central galaxies situated at the centres of their dark matter haloes, since satellites undergo additional environmental processes that may affect their star formation activity. Yet, clear identification of the mechanisms behind quiescence in these central galaxies has proved elusive, as \ngalaxy evolution is complex and many physical processes act in concert to shape the properties of galaxies. \nOne observational approach to understanding quiescence in central galaxies focuses on measuring correlations between galaxy properties and quiescence. For example, galaxies with high stellar masses (e.g., Kauffmann et al. 2003), bulge-dominated morphologies (e.g., Bell et al. 2012; Bluck et al. 2014), high halo masses (e.g., Wang et al. 2018), high S'ersic indices (e.g., Cheung et al. 2012), high central stellar surface densities (e.g., Woo et al. 2015), and more massive black holes (e.g., Terrazas et al. 2016, 2017; Mart'ınNavarro et al. 2018) have a higher likelihood of being quiescent. Several mechanisms attempting to explain these correlations have been proposed, such as supernovae/stellar winds (e.g., Tang & Wang 2005; Conroy et al. 2015), gravitational heating (e.g., Johansson et al. 2009), morphological quenching (e.g., Martig et al. 2009), halo mass quenching (e.g., Dekel & Birnboim 2006; Cattaneo et al. 2006; Birnboim et al. 2007), and central black hole feedback (e.g., Kauffmann & Haehnelt 2000; Di Matteo et al. 2005; Croton et al. 2006; Bower et al. 2006; Somerville et al. 2008; Fabian 2012). \nIn a cosmological context, gas that falls into a dark matter halo is able to lose energy and cool via dissipative processes to form a disc of cold gas at the bottom of the halo's potential well (e.g. Silk 1977; White & Rees 1978; Fall & Efstathiou 1980; Katz & Gunn 1991). The gas that accumulates in the disc fragments, cools, and condenses into molecular clouds which eventually collapse into new stars. \nOne natural feedback channel that limits star formation is stellar feedback from star formation itself. Radiation from stars, stellar winds, and supernovae regulate the production of new stars by limiting the amount of cold dense gas (e.g., Silk 2003; Springel & Hernquist 2003; Hopkins et al. 2011). This form of feedback, however, is insufficient for producing quiescence in massive galaxies. Recent models show that the gas expelled from galaxies by stellar feedback is reincorporated to provide fuel for continued star formation (e.g., Dubois et al. 2016; Pontzen et al. 2017; Su et al. 2018; Choi et al. 2018), a behavior that is more efficient for more massive systems (Oppenheimer et al. 2010; Bower et al. 2012; Christensen et al. 2016; Muratov et al. 2015). Thus, a picture emerges of galaxies undergoing a cycle of ejection and reincorporation that regulates cooling and star formation via stellar feedback similar to various 'bathtub' models that have been proposed (Bouch'e et al. 2010; Dutton et al. 2010; Dav'e et al. 2012; Peng & Maiolino 2014; Dekel & Mandelker 2014; Birrer et al. 2014). \nIn this physically-motivated framework, quiescence can be defined as the consequence resulting from the disruption of this cycle. This disruption can take the form of 'ejective' and/or 'preventative' feedback (see Section 3.3 of Somerville & Dav'e 2015 for a full review). Ejective feedback pushes gas out of galaxies which may otherwise continue forming stars. Preventative feedback prevents galaxies from accumulating star-forming gas from the cooling of the circumgalactic medium. While both processes likely play a role in suppressing star formation, any proposed mechanism for long-term quiescence must be at least partly preventative in nature since the accretion of new gas would re-establish star formation in the galaxy. \nA popular mechanism for producing quiescence in recent physics-based galaxy formation models is black hole \nfeedback (Bower et al. 2006; Croton et al. 2006; Sijacki et al. 2007; Somerville et al. 2008; Guo et al. 2010; Booth & Schaye 2011; Vogelsberger et al. 2014a; Henriques et al. 2015; Somerville & Dav'e 2015; Schaye et al. 2015; Dubois et al. 2016; McCarthy et al. 2017; Bower et al. 2017; Weinberger et al. 2018). Theoretically, accretion onto a black hole has the potential to release an enormous amount of energy into the surrounding medium. Observationally, activity from central supermassive black holes has been seen in many forms. \nWhen black holes undergo episodes of high accretion rate feedback, the radiation pressure created by the luminous accretion disc is thought to result in large-scale outflows. Many X-ray luminous active galactic nuclei (AGN) are observed to host ionized gas outflows in support of this idea (Heckman et al. 1981; Crenshaw et al. 2010; Villar-Mart'ın et al. 2011; Cicone et al. 2014; Woo et al. 2016; Rupke et al. 2017). At lower accretion rates, black holes are thought to produce jets from their accretion discs which propel lowdensity buoyant bubbles into the atmospheres of galaxies. Indeed, large extended radio-emitting lobes that create cavities within the surrounding hot gas visible in X-ray maps have been observed around massive galaxies in various environments from small groups to large clusters (e.g., Bˆırzan et al. 2004; McNamara & Nulsen 2007; Fabian 2012; Shin et al. 2016; Werner et al. 2019). Some galaxies have been observed to exhibit signs of both a bright X-ray point source in conjunction with extended radio emission, further complicating the issue of the accretion physics that could produce these varied effects on the gas within and around galaxies (Komossa et al. 2006; Yuan et al. 2008; Berton et al. 2015; Coziol et al. 2017). More recently, some galaxies with low luminosity AGN have been observed to produce bisymmetric winds of ionized gas which are thought to affect star formation in the galaxy (Cheung et al. 2016; Penny et al. 2018). Finally, our own Milky Way galaxy also hosts evidence of a possible relic from black hole feedback in the form of Fermi bubbles (Su et al. 2010; Guo & Mathews 2012). \nThe black hole accretion physics occurring at parsecscales and producing these observed forms of black hole activity is not well understood. As a result, there is substantial freedom in the subgrid physics used between different cosmological simulations to model the effects black holes may have on the surrounding gas. This can result in important differences in the star formation histories, gas content, and stellar mass distributions of simulated galaxy populations which can then be compared to observational data (e.g., Terrazas et al. 2016, 2017; Bluck et al. 2016, 2019). As such, understanding the details of how small-scale subgrid physics affects large-scale galaxy population statistics in models can improve our understanding of the feedback mechanisms that may produce quiescence in the real Universe. \nWith these considerations in mind, the central challenge this work seeks to address is understanding how observed galaxy correlations between specific star formation rate (sSFR), stellar mass (M star ), and black hole mass (M BH ) at z = 0 can be interpreted in light of the results from a physical model of black hole feedback. In this work, we use the IllustrisTNG simulation suite (TNG; Springel et al. 2018; Marinacci et al. 2018; Naiman et al. 2018; Pillepich et al. 2018b; Nelson et al. 2018a) in order to explore this question, focusing on central galaxies with M star > 10 10 M glyph[circledot] . TNG uses black hole feedback in order to suppress the SFRs \nand M star content of galaxies with M star glyph[greaterorsimilar] 10 10 M glyph[circledot] (Weinberger et al. 2017, 2018). Thus, as we will show, the sSFR, M star , and M BH of galaxies are causally related to one another through the model's mechanism for quiescence. \nOne important feature of TNG for our purposes is the availability of dozens of TNG model variations, first introduced in Pillepich et al. (2018a). We use these model variations to study the impact of different parameter choices, as is similarly done in Weinberger et al. (2017). By using the model variations explicitly related to black hole feedback, we can explore how changes to the model can affect TNG's galaxy population statistics. We aim to link these differences in observable correlations to the effects of black hole feedback on the physical properties of the gas within the galaxy and in the surrounding circumgalactic medium. Thus, our work will show how these observed correlations may be physically interpreted in the real Universe. \nWe organize our results as follows: Section 2 describes the TNG simulation suite, the model variations, the black hole physics model, and our definitions of galaxy properties in the model. Section 3 describes the necessary conditions for quiescence in TNG using three of the model variations. Section 4 describes the effects of black hole feedback on gas and outlines a phenomenological framework for the physics of quiescence in TNG. Section 5 compares the model results to observational correlations, illuminating how the sSFR, M star , and M BH can encode information on the physics behind quiescence within the context of black hole feedback. Section 6 uses model variations to illuminate how observables may be affected by changes in the way black hole feedback operates. In Section 7 we reflect on our results and present future outlooks. Section 8 summarizes our findings and contains our concluding remarks.", '2 THE ILLUSTRIS TNG SIMULATION SUITE': 'The IllustrisTNG project is a large-scale cosmological and gravo-magneto-hydrodynamical simulation suite of galaxy formation in a ΛCDM Universe (Springel et al. 2018; Marinacci et al. 2018; Naiman et al. 2018; Pillepich et al. 2018b; Nelson et al. 2018a). TNG is the descendant of the original Illustris project (Vogelsberger et al. 2014a,b; Genel et al. 2014; Sijacki et al. 2015), modifying and adding numerous features with the goal to improve the agreement between the simulation and observational results by providing a comprehensive physical model of galaxy formation from the early Universe to the present day. The simulation uses the Arepo code to solve the equations of ideal magnetohydrodynamics and self-gravity on a moving, unstructured mesh (Springel 2010; Pakmor et al. 2011, 2016). For more information on the numerical aspects of the TNG model, we refer the reader to Pillepich et al. (2018a) and Weinberger et al. (2017). \nThe simulation suite contains three simulation volumes TNG50, TNG100, and TNG300 sized at 51.7 3 , 110.7 3 , and 302.6 3 comoving Mpc 3 volumes, respectively. In this work, we will be focusing on results from TNG100 as it has roughly the same resolution (within a factor of 2) as the model variations which we will use extensively in this paper and which are detailed in Section 2.2. This will allow a comparison without the complications of resolution effects present in TNG300 (see Appendix B). TNG100 has 2 × 1820 3 ini- \nial resolution elements with a baryonic mass resolution of 1 . 4 × 10 6 M glyph[circledot] and gravitational softening length of 0.74 kpc at z = 0. The cosmological parameters of the model are based on Planck Collaboration et al. (2016) with a matter density Ω M, 0 = 0 . 3089, baryon density Ω b, 0 = 0 . 0486, dark energy density Ω Λ , 0 = 0 . 6911, Hubble constant H 0 = 67.74 km/s/Mpc, power spectrum normalization factor σ 8 = 0 . 8159, and spectral index n s = 0 . 9667. \nTNG models the physics of primordial and metal-line gas cooling, magnetic fields, star formation, stellar evolution and feedback, chemical enrichment, and black hole growth and feedback. There are several key differences between the original Illustris model and TNG which are described in Pillepich et al. (2018a). For our purposes, the most relevant modification is how black hole feedback operates at low accretion rates. Instead of the thermal bubble model implemented by the original Illustris model and described in Sijacki et al. (2007, 2015), TNG adopts a kinetic wind model that inputs kinetic energy originating at the black hole into nearby gas particles (Weinberger et al. 2017). The primary motivation for changing the physical prescription for black hole feedback was to prevent the ejection of large amounts of gas from the haloes of intermediate to high mass galaxies. The amount of halo gas around many of these galaxies in Illustris is much lower than observations suggest (see Figure 10 in Genel et al. 2014), yet the mass and state of the interstellar medium gas is such that a clear colour bimodality as compared to SDSS data is not present (see Figure 14 in Vogelsberger et al. 2014a). This led to the implementation of a modified form of feedback.', '2.1 The formation, growth, and feedback of black holes in TNG': "Here we provide a brief description of the black hole model in TNG. For full details, see Weinberger et al. (2017). \nBlack holes are placed at the centre of a halo's potential well with a seed mass of M seed = 1 . 18 × 10 6 M glyph[circledot] once a halo grows past a threshold mass of M h = 7 . 38 × 10 10 M glyph[circledot] . Once seeded, black holes grow either through accretion at the Eddington-limited Bondi accretion rate or by merging with other black holes during a galaxy merger. Additionally, black holes are made to stay at the potential minimum of their host subhaloes at each global integration time step in order to avoid numerical effects that may displace them. \nTNG employs a M BH -dependent Eddington ratio threshold for determining whether a black hole provides either pure thermal or pure kinetic mode feedback energy to the galaxy. Kinetic mode feedback in TNG is turned on when a galaxy's black hole accretion rate drops below an Eddington ratio of \nχ = min [ χ 0 ( M BH M piv ) β , χ max ] , (1) \nwhere M BH is the black hole mass and the parameters for the fiducial model are χ 0 = 0 . 002, M piv = 10 8 M glyph[circledot] , β = 2, and χ max = 0 . 1 (refer to Equation 5 and Figure 6 of Weinberger et al. (2017)). \nApart from χ max , which is chosen to be the canonical, observationally-suggested value of 0.1 for the Eddington ratio of black holes in quasars (Yu & Tremaine 2002), none of the other parameter values were adopted based on empirical \nTable 1. The list of model variations we use for this work. Columns: (1) model number, (2) model name, (3) parameter changed, (4) fiducial model value, (5) changed value, (6) short description of the alteration. See Section 2.1 for more information on each of the model parameters in column (3). \n| Model No. (1) | Name (2) | Parameter changed (3) | Fiducial value (4) | Changed value (5) | Description (6) |\n|-----------------|---------------|-------------------------|------------------------|--------------------------|--------------------------------------------------------|\n| 0000 | FiducialModel | n/a | n/a | n/a | Same physics as the larger TNG volumes |\n| 2201 | NoBHs | n/a | n/a | n/a | Black holes turned off |\n| 3000 | NoBHwinds | χ 0 | 0.002 | 0 | Thermal mode at all ˙ M BH |\n| 3101 | LowKinEff | ε f , kin , max | 0.2 | 0.05 | Lower gas coupling efficiency for BH kinetic feedback |\n| 3102 | HighKinEff | ε f , kin , max | 0.2 | 0.8 | Higher gas coupling efficiency for BH kinetic feedback |\n| 3103 | HighThermEff | ε f , high | 0.1 | 0.4 | Higher gas coupling efficiency for BH thermal feedback |\n| 3104 | LowThermEff | ε f , high | 0.1 | 0.05 | Lower gas coupling efficiency for BH thermal feedback |\n| 3301 | HighChi0 | χ 0 | 0.002 | 0.008 | Higher Eddington ratio threshold normalization |\n| 3302 | LowChi0 | χ 0 | 0.002 | 0.0005 | Lower Eddington ratio threshold normalization |\n| 3403 | HighMpiv | M piv | 1e8 M glyph[circledot] | 4e8 M glyph[circledot] | Higher pivot mass in Eddington ratio threshold |\n| 3404 | LowMpiv | M piv | 1e8 M glyph[circledot] | 2.5e7 M glyph[circledot] | Lower pivot mass in Eddington ratio threshold | \nevidence or other theoretical models of black hole physics and accretion. The parameters have been chosen to ensure that the TNG model returns the observed stellar mass content of massive haloes and the observed location of the knee in the stellar mass function at the current epoch. \nThermal mode feedback energy is parameterized as ˙ E thermal = ε f , high ε r ˙ M BH c 2 , where ε f , high is the fraction of thermal energy that couples to the surrounding gas (set to 0.1 for the fiducial model) and ε r is the black hole radiative efficiency (set to 0.2 for the fiducial model). Kinetic mode feedback energy is parameterized as ˙ E kinetic = ε f , kinetic ˙ M BH c 2 , where ε f , kinetic is the fraction of kinetic energy that couples to the surrounding gas (set to a maximum value of ε f , kin , max = 0.2 for the fiducial model, see Equation 9 of Weinberger et al. 2017 for a full description). Once the black hole accumulates enough energy in this mode to reach a minimum energy threshold, E inj , min , the gas immediately surrounding the black hole receives a momentum kick in a random direction away from the black hole.", '2.2 Model Variations': 'Along with TNG100, there are dozens of smaller simulations sized at 36.9 3 comoving Mpc 3 volumes that vary individual parameters of the model (Pillepich et al. 2018a). These simulations have 2 × 512 3 initial resolution elements with roughly the same baryonic mass resolution (2 . 4 × 10 6 M glyph[circledot] ) and gravitational softening length (0.74 kpc at z = 0) as TNG100. Each of these simulations have identical cosmological initial conditions resulting in similar dark matter structures and galaxy placement throughout the volume. This allows a galaxy-by-galaxy comparison of the model variations, providing us with a powerful tool to explore the effects of each component of the model relevant for quiescence. \nIn Table 1 we describe the model variations that we use in this work. The first row describes a model with the same physics as TNG100 but with the same initial conditions, resolution, and volume as all other model variations. We will refer to this run as the FiducialModel simulation. Each of the other model variations alter the black hole feedback model described in Section 2.1. These variations will affect how galaxies populate the M BH -M star -sSFR parameter space, as we will show in later sections. The details of each altered \nparameter in the third column are given in Section 2.1, with a brief description of the change in the sixth column. We will refer to each simulation according to their names in the second column of Table 1.', '2.3 Definitions of physical properties in TNG': "We calculate the M star and SFR within the galaxy radius, defined as twice the stellar half mass radius. SFRs are calculated by averaging the star formation activity in the last 200 Myr in order to reasonably compare our results to observational SFR indicators. Our results remain qualitatively the same whether we use instantaneous SFRs or those averaged over 50, 100, or 200 Myr. \nWe define star-forming and quiescent galaxies to be those with sSFRs above or below 10 -11 yr -1 , respectively. This provides a consistent separation between galaxies on and off the star forming main sequence for both observations and simulations at z = 0. \nTNG100 and the model variations are sensitive to 200 Myr-averaged SFRs above ∼ 10 -2 . 5 M glyph[circledot] yr -1 , below which star formation is unresolved (see Figure 2 in Donnari et al. 2019). This value is related to the mass resolution limit of a single stellar particle in TNG100 and the model variations. Since we focus on galaxies with M star > 10 10 M glyph[circledot] , any values for sSFR < 10 -12 . 5 yr -1 will be taken as an upper limit. This ensures that all sSFRs above this limit are resolved. Upper limits will be shown as a value chosen from a gaussian distribution centred at sSFR = 10 -12 . 5 yr -1 in order to visualize what would be observed as non-detections. \nBlack hole parameters are taken for the most massive black hole at the centre of the galaxy. Halo properties such as gas cooling rates ( C halo ), gas masses (M gas , halo ), and dark matter halo masses (M DM , halo ) are the sum values for each cell within the radius at which the density within is 200 times the critical density of the Universe, R 200c . Halo gas cooling times (t cool ) are the average of gas cells within R 200c . \nAdditionally, our analysis will use a measure of the gravitational binding energy of gas within the galaxy, which we define as \nE bind , gal ( < r gal ) = 1 2 ∑ g( < r gal ) m g φ g , (2) \nwhere r gal is twice the stellar half mass radius, g indexes gas cells in the simulation, m g is the mass of the gas cell, and φ g is the gravitational potential felt at the gas cell's position within the galaxy. This value represents the energy needed to unbind those gas cells from the galaxy's position to infinity, taking into account not only the mass within the galaxy but also the mass within the halo. We choose to calculate the binding energy of gas within this radius in order to match the radius within which the SFR and M star are calculated.", '3 THE NECESSARY CONDITIONS FOR QUIESCENCE': "We begin by exploring the necessary conditions under which quiescence can occur for central galaxies with M star > 10 10 M glyph[circledot] in the context of the TNG model. In Figure 1, we show histograms of the number of galaxies as a function of sSFR for three model variations at z = 0. Due to their smaller box size, there are very few galaxies with M DM , halo > 10 13 M glyph[circledot] . We also plot galaxies from TNG100 (renormalized to match the box size of the FiducialModel run) in gray to show the results from a larger sample size with more massive haloes. \nIn light green, we show the TNG model variation with no black holes, where the only feedback channel is stellar feedback. This simulation results in a distribution of galaxies centred at sSFR ∼ 10 -10 yr -1 , denoting the existence of the star forming main sequence. \nIn dark green, we show the model variation with black holes included where thermal mode black hole feedback operates for all accretion rates (the NoBHwinds model in Table 1). In TNG, a black hole accreting in the thermal mode injects pure thermal energy into the surrounding gas particles. Weinberger et al. (2018) show that the total amount of thermal energy released in this mode can be large yet has little effect on cooling and the galaxy's SFR. Thermally injected energy, as implemented for black hole thermal mode feedback in TNG, is more likely to be immediately radiated away rather than have any lasting impact on the thermodynamic properties of the gas, especially for dense gas where the cooling times are short (Navarro & White 1993; Katz et al. 1996). This is likely due, at least in part, to a numerical effect resulting from the limited resolution of the simulation and leading to the 'over-cooling' problem (e.g., Springel & Hernquist 2002). The pileup of galaxies at sSFR ∼ 10 -10 yr -1 illustrates the inability of thermal energy injection, at least in the continuous fashion with which it is implemented within the TNG model, to prevent gas cooling and star formation in TNG galaxies. \nThe black histogram shows galaxies in the FiducialModel simulation, where kinetic mode black hole feedback is turned on at low accretion rates. This black histogram along with the TNG100 results in gray show a significant population of quiescent galaxies at low sSFRs, most of which have upper limit values placed at sSFR ∼ 10 -12 . 5 yr -1 . These results show that the existence of a quiescent population in TNG depends on the model's implementation of low accretion rate kinetic mode feedback from the central black hole. Similar conclusions have been made in previous studies of TNG both in terms of colour (Weinberger et al. 2017; Nelson et al. 2018a) and sSFR (Weinberger et al. 2018). \nFigure 1. Histograms of sSFR for central galaxies with M star > 10 10 M glyph[circledot] at z = 0 for the model variations with no black holes (light green), thermal mode at all accretion rates (dark green), and both thermal and kinetic modes included as in the fiducial model (black). The number of galaxies shown in each of these histograms is roughly the same. Due to the smaller box size, there are very few galaxies with M DM , halo > 10 13 M glyph[circledot] . TNG100 scaled to the volume of the smaller box simulations is shown in gray. The star forming main sequence is present in all simulations but a quiescent population only appears in the models with kinetic mode black hole feedback. \n<!-- image -->", '4 THE PHYSICS OF QUIESCENCE FROM BLACK HOLE-DRIVEN KINETIC WINDS': 'In this section we describe how black hole-driven kinetic winds employed by TNG affect the gas within and around galaxies in order to produce quiescence.', '4.1 Comparing TNG with semi-analytic approaches to quiescence': "Semi-analytic models of galaxy formation often use prescriptions of gas cooling and heating rates in order to determine the net rate of cold gas accreted onto a galaxy (e.g., De Lucia & Blaizot 2007; Somerville et al. 2008; Guo et al. 2010; Henriques et al. 2015). In these models, once the heating rate from black hole feedback equals or exceeds the halo gas cooling rate, cold gas accretion onto the galaxy halts and star formation subsequently shuts off. This describes a method to model preventative black hole feedback, where fuel for star formation is prevented from reaching the galaxy. \nIn order to test whether a similar physical scenario occurs for TNG galaxies, we show the sSFR as a function of the ratio between the instantaneous energy released from black hole-driven kinetic feedback, ˙ E kinetic (see Section 2.1), and the instantaneous cooling rate of the gas halo, C halo 1 , at \nFigure 2. sSFR as a function of the ratio between the instantaneous black hole wind energy injection rate, ˙ E kinetic , and the instantaneous halo gas cooling rate, C halo . The vertical dotted line shows where these two energy rates equal. The grayscale heatmap shows the distribution of galaxies in TNG100 and the black points show galaxies in the FiducialModel simulation at z = 0. The histogram shows the distribution of sSFR for galaxies with no black hole wind energy injection, ˙ E kinetic = 0, for the TNG100 (gray, scaled to the volume of the FiducialModel run) and FiducialModel (black) simulations. While kinetic winds are required for quiescence, comparing the kinetic energy injection rate to the halo gas cooling rate is a poor indicator of a galaxy's star formation properties. \n<!-- image --> \nz = 0. The distribution of TNG100 galaxies is shown as a grayscale heatmap and FiducialModel galaxies are shown as black points. The vertical dotted line indicates where the two rates are equal. Since many galaxies are not in the kinetic mode (with ˙ E kinetic = 0) and therefore cannot be represented on this plot, we show the distribution of their sSFRs as a histogram to the left of the main plot for both the TNG100 (gray, scaled to the volume of the FiducialModel run) and FiducialModel (black) simulations. \nWe find that all galaxies that are not currently experiencing kinetic mode black hole feedback (with ˙ E kinetic = 0) are actively forming stars (left panel histogram in Figure 2). The main panel of this figure shows that galaxies must be experiencing kinetic mode black hole feedback in order to be quiescent, in agreement with our results from Section 3. \nHowever, a significant number of galaxies releasing black hole-driven kinetic winds are star-forming. In fact, whether these galaxies are star-forming or quiescent does not correlate tightly with whether the black hole wind energy greatly exceeds or falls below the cooling rate of the \nthreshold is reached (Springel & Hernquist 2003). As such, cooling rates for these cells are excluded in the calculation of the halo cooling rate. Star forming cells account for a negligible amount of mass in the gas halo. \ngas halo. As such, the TNG simulation cannot be easily described using the logic employed by semi-analytic models. We note that the comparison between these 'heating' and 'cooling' rates in TNG differ in detail to those analytically determined in semi-analytic approaches. This is due to the fact that these rates in TNG are sensitive to internal gas hydrodynamics whereas in semi-analytic models these rates are calculated using global galaxy and halo properties. Even so, this analysis indicates differences between how black hole feedback energy is transferred to the surrounding gas to produce quiescence in TNG and how it is transferred in purely preventative semi-analytic black hole feedback models.", '4.2 Gas distributions in TNG galaxies': "A useful approach for understanding the effects of black hole feedback on the gas within and around galaxies in TNG is to visualize and quantify the distribution of this gas. We do this using the model variations described in Section 2.2. Figure 3 shows the gas column densities of three quiescent galaxies of increasing mass at z = 0 in the FiducialModel simulation (left images) and their direct counterparts in the NoBHwinds simulation (right images). Their radial density profiles are shown in the rightmost panels. We match these galaxies between the two model variations by the position of its dark matter halo and by ensuring that at least half of the dark matter particles have the same IDs. Each panel is at the same spatial scale of 300 × 300 kpc. The circle at the centre of each figure depicts twice the stellar half mass radius of the galaxy from its centre in order to show where a galaxy's visible matter would lie in the image. \nWe find that there is much less dense gas in the central regions of the galaxies undergoing kinetic feedback. The discs of these galaxies are extended and disturbed, showing that black hole kinetic winds produce outflows that push gas out of galaxies, in agreement with recent TNG50 results described in Nelson et al. (2019). Additionally, the density of the gas extending past the disc and into the gas halo is also depleted. We find that the same galaxies in the NoBHwinds simulation are star-forming, have retained dense gas within the galaxy's radius, and have centrally peaked radial density profiles (dotted lines in the right hand panels).", '4.3 Overcoming gravitational binding energies': 'The results from Figure 3 strongly suggest that black holedriven kinetic winds drive gas out of the galaxy, producing a form of ejective feedback. Therefore, we choose to test whether the gravitational binding energy of the gas within the galaxy (defined in Section 2.3) can be a useful parameter for characterizing quiescence in the TNG model. \nWe compare this value to the time-integrated amount of black hole-driven wind energy that has been released into the gas particles near the black hole at each time-step, ∫ ˙ E kinetic dt 2 . Nelson et al. (2018a) and Weinberger et al. (2018) have demonstrated that quiescent galaxies have released more of this cumulative black hole wind energy relative to star-forming galaxies. \nFigure 3. Left panels : The leftmost galaxy images show the projected (across 300 kpc) gas column density distributions of three quiescent galaxies in the FiducialModel simulation at z = 0 ordered by increasing mass from top to bottom. The images to the right show galaxies identified to be the centrals of the same haloes as those on the left but in the NoBHwinds simulation. The values on the lower right indicate the sSFR of each galaxy. Each image is 300 × 300 kpc in size in order to directly compare the distribution of gas. The red circles indicate the galaxy radius, defined as twice the stellar half mass radius. Right panels: The radial gas density distributions of the galaxies in the FiducialModel (solid lines) and NoBHwinds (dotted lines) simulations. The vertical red lines indicate the galaxy radius. The density of gas in the central regions of galaxies experiencing kinetic winds is depleted by orders of magnitude. \n<!-- image --> \nThe top panel of Figure 4 shows the sSFR as a function of the ratio between the cumulative black hole wind energy, ∫ ˙ E kinetic dt , and the gravitational binding energy of gas within the galaxy, E bind , gal , in the TNG100 (gray heatmap) and FiducialModel (black points) simulations at z = 0. We find that star-forming galaxies lie to the left of the vertical dotted line, showing that they exhibit cumulative black hole wind energies that fall below the binding energy of the gas. Galaxies with intermediate sSFRs between sSFR ∼ 10 -11 -10 -12 yr -1 exhibit cumulative black hole wind energies that exceed the binding energy of the gas (to the right of the vertical dotted line) by up to a factor of 100. Above an energy ratio of 100, galaxies host very low sSFRs shown as upper limits at ∼ 10 -12 . 5 yr -1 . \nIt is important to note that the binding energy of the gas decreases as gas leaves the system since the total mass of the system decreases and there is less gas to push out. The high ratios of cumulative black hole wind energy to gravi- \ntational binding energy exceeding a factor of 1000 tend to have very low gas masses and therefore low binding energies. While this is true, we verify that sSFR correlates only slightly with the binding energy of gas within the galaxy, and that the cumulative black hole wind energy drives most of the correlation seen in the top panel of Figure 4. The fact that sSFR begins to drop when these two energies equal indicates that gas is being gravitationally unbound from the central galaxy and pushed into the circumgalactic medium by black hole-driven kinetic winds. \nThe bottom panel of Figure 4 shows that galaxies with cumulative black hole wind energies lower than the binding energy of the gas have M BH glyph[lessorsimilar] 10 8 . 2 M glyph[circledot] , whereas those with higher ratios have M BH glyph[greaterorsimilar] 10 8 . 2 M glyph[circledot] . This indicates that black hole winds are effective at removing gas and producing quiescence for a majority of galaxies once the black hole exceeds the threshold mass at ∼ 10 8 . 2 M glyph[circledot] (shown as a horizontal dotted line). \nFigure 4. sSFR (top) and M BH (bottom) as a function of the ratio between the cumulative kinetic energy released from black hole feedback, ∫ ˙ E kinetic dt , and the binding energy of the gas within twice the stellar half mass radius, E bind , gal . The vertical dotted line shows where these two energies equal. The grayscale heatmap shows the distribution of galaxies in TNG100 and the black points show galaxies in the FiducialModel simulation at z = 0. The sSFR decreases once the cumulative energy from black hole-driven winds exceeds the binding energy of gas in the galaxy. This drop in sSFR occurs when M BH exceeds 10 8 . 2 M glyph[circledot] shown as a horizontal dotted line in the bottom panel. \n<!-- image --> \nWe also note that the most massive black holes (M BH glyph[greaterorsimilar] 10 9 M glyph[circledot] ) tend to have lower ratios of black hole wind energy to binding energy, whereas less massive black holes (with M BH still glyph[greaterorsimilar] 10 8 . 2 M glyph[circledot] ) can have much higher ratios. This is likely due to the fact that more massive black holes live in more massive galaxies and haloes where the gravitational potential is much deeper. As such, many of the galaxies with intermediate sSFRs (between ∼ 10 -11 -10 -12 yr -1 ) are hosted by some of the most massive haloes, whereas those with very low sSFRs (shown as upper limits at ∼ 10 -12 . 5 ) are those in lower mass haloes (with M BH still glyph[greaterorsimilar] 10 8 . 2 M glyph[circledot] ). We expand on this result in the following section.', '4.4 The effects of black hole-driven kinetic feedback on the interstellar and circumgalactic media': "The bottom panel of Figure 4 indicates that there exists a M BH threshold for quiescence (also see Figure 5 in Weinberger et al. 2018). In this section, we explicitly explore the gas properties of galaxies in TNG as a function of M BH . Figure 5 shows the average gas density within the galaxy (top panel), the average cooling time of the entire gas halo 3 (middle panel), and the ratio of total halo gas mass to dark matter halo mass (bottom panel) as a function of M BH for galaxies in the NoBHwinds (green crosses) and FiducialModel (black points) simulations at z = 0. The distribution of galaxies in TNG100 is shown as a grayscale heatmap. \nWithout kinetic winds (green crosses), the average gas density within twice the stellar half mass radius is similar across galaxies with different M BH (top panel). We verify that this is because galaxy gas masses correlate closely with their radii when they are star-forming. The average cooling time of these galaxies' gas haloes is a fairly flat function of M BH at low masses and gradually rises for more massive black holes that live in more massive haloes where cooling becomes less efficient (middle panel). Additionally, the total halo gas masses of these galaxies correlate with dark matter halo mass, producing a scattered, flat relation as a function of black hole mass (bottom panel). \nIntroducing kinetic winds significantly alters these gas properties for galaxies above the M BH threshold for quiescence at ∼ 10 8 . 2 M glyph[circledot] , as is seen in all three panels of Figure 5 (black points and grayscale heatmap). Once feedback from kinetic winds becomes effective, there is a sharp decrease in the average gas density within galaxies, a sharp increase in the average cooling time of halo gas, and a sharp decrease in the fractions of halo gas mass to dark matter halo mass compared to the NoBHwinds model. \nIn the bottom panel we also show the median values from the original Illustris model with a dashed gray line. This demonstrates that while TNG retains more gas in massive haloes on average, it still removes large quantities of gas from within the halo's radius once black hole winds are produced at the M BH threshold for quiescence. The lowest halo gas fractions (with M gas , halo /M DM , halo ∼ 0) are seen in galaxies with M BH near the threshold mass at ∼ 10 8 . 2 M glyph[circledot] . The gas fraction increases with M BH thereafter. This is likely because M BH correlates closely with M star (as we will show in Figure 7), and therefore M DM , halo , for TNG galaxies. The feedback from black holes in more massive galaxies must overcome a larger potential well in order to expel gas from the galaxy's halo. \nTo illustrate this, Figure 6 shows the ratio between the cumulative energy from black hole winds and the gravitational binding energy of the halo gas as a function of M BH . The binding energy of the halo gas is calculated by replacing r gal with r halo in Eqn. 2 (see Section 2.3). The horizontal dotted line shows where the black hole kinetic wind energy equals the halo gas binding energy. Galaxies hosting black holes just over the M BH threshold for quiescence (vertical \nFigure 5. The average gas density within the galaxy radius (top panel), the average cooling time of the halo gas (middle panel), and the ratio of halo gas mass to dark matter halo mass (bottom panel) as a function of M BH for galaxies in the TNG100 (grayscale heatmap), FiducialModel (black), and NoBHwinds (green) simulations at z = 0. Including black hole winds in TNG abruptly lowers the density of gas within the galaxy, increases the cooling time of the halo gas, and reduces the amount of halo gas for galaxies with M BH glyph[greaterorsimilar] 10 8 . 2 M glyph[circledot] . The most massive black holes live in the most massive haloes, where affecting gas kinetically is more difficult due to the halo's deep potential well. As a result, the galaxies whose SFRs are most affected by kinetic mode feedback are the least massive galaxies that have M BH glyph[greaterorsimilar] 10 8 . 2 M glyph[circledot] . \n<!-- image --> \nFigure 6. The ratio between the cumulative energy from black hole winds, ∫ ˙ E kinetic dt , and the binding energy of gas within the halo radius, E bind , halo as a function of M BH for the TNG100 (grayscale heatmap) and FiducialModel (black points) simulations at z = 0. The horizontal dotted line indicates where the black hole wind energy equals the halo gas binding energy and the vertical dotted line indicates the M BH threshold for quiescence (see Section 4.3). The black hole wind energy exceeds the halo gas binding energy by the largest amount just above the M BH threshold for quiescence. \n<!-- image --> \ndotted line) have the highest ratios of black hole wind energy to halo gas binding energy. This coincides with the lowest gas mass fractions in the lower panel of Figure 5. This indicates that a significant fraction of gas is evacuated from many of the haloes once black hole wind energies exceed not only the binding energy of the gas in the galaxy (Figure 4) but also of the gas in the halo. \nProgressively higher mass black holes live in more massive haloes with deeper gravitational potentials. In Figure 6 as well as in the lower panel of Figure 4, the ratio between the energy from black hole winds and the binding energy of gas in both the galaxy and the halo decreases for more massive black holes. The decrease in this ratio with black hole mass coincides with an increase in the halo gas fraction in the lower panel of Figure 5. This behavior indicates that black hole winds are less effective at removing gas at higher mass regimes.", '5 COMPARISONS TO OBSERVATIONS': 'In this section we evaluate the extent to which the kinetic wind model in TNG can produce results which agree with currently available observational correlations related to quiescence. Here we use the observational diagnostics and data from Terrazas et al. (2016, 2017) in order to examine and compare the relationship between M star , M BH , and sSFR in TNG. We aim to link the phenomenological description of quiescence we put forth for this model in Section 3 and 4 to these observable properties of simulated galaxies since these properties are causally tied to one another in this model as a result of the black hole feedback prescriptions. \nFigure 7. Top panels: The M BH -M star (left), sSFR-M BH (centre), and sSFR-M star (right) parameter spaces for TNG100 galaxies in blue and red heatmaps and the observational sample in gray and black circles, for star-forming and quiescent galaxies, respectively, at z = 0. An uncertainty of 0.15 dex is added to sSFR and M star values for TNG100 galaxies. Observational uncertainties for M BH in TNG100 are taken into account by convolving them with the errors from the T17 data. Bottom panels: sSFR as a function of M BH in bins of M star for TNG100 galaxies and the T17 observational sample. \n<!-- image -->', '5.1 Observational sample': "The 91 galaxies in Terrazas et al. (2016, 2017, hereafter T17) represent a diverse collection of local galaxies (distances within ∼ 150 Mpc) with various morphologies, SFRs, colours, and environments (isolated, group, or cluster) with M star > 10 10 M glyph[circledot] . The sample represents central galaxies defined as the most massive galaxy within ∼ 1 Mpc - that have a dynamical M BH measurement using stellar dynamics (45 galaxies), gas dynamics (15), masers (12), or reverberation mapping (18). Black hole mass measurements were taken from Saglia et al. (2016) and van den Bosch (2016). \nT17 measures the stellar masses using 2MASS K s -band luminosities (Huchra et al. 2012) with a single M star / L K s ratio of 0.75 since mass-to-light ratios for L K s do not vary substantially (Bell & de Jong 2001). Their scatter in mass-tolight ratios are ∼ 0.15 dex, thus they assume the same value for their uncertainty. SFRs are measured following Kennicutt & Evans (2012) where T17 use IRAS far-infrared flux measurements to estimate a total infrared luminosity. This SFR estimate is sensitive to star formation during the past ∼ 100 Myr. SFRs are assumed to have an uncertainty of 0.15 dex following Bell (2003). Any SFRs with no far-infrared detection, fluxes below the detection limits of IRAS, or detections below sSFR < 10 -12 . 5 yr -1 are taken as upper limits. We define star-forming and quiescent galaxies to be those with sSFRs above or below 10 -11 yr -1 , respectively, just as we do with TNG galaxies. \nWe note that these data are not representative of the \nentire galaxy population at z = 0 due to the requirement that their M BH be large enough and their host galaxies close enough to detect their gravitational sphere of influence. The various detection methods used for this sample make it difficult to fully understand the biases associated with the sample. For example, black hole masses detected with reverberation mapping or masers are more likely to be in star-forming galaxies where there is enough gas to produce the emission required to use these measurement techniques. Black holes measured with stellar dynamics, however, are likely in systems that are close enough to resolve the gravitational sphere of influence and that have low enough gas masses in their nuclear regions to allow starlight to dominate. These factors make it difficult to compare these observations directly with the simulation results, since the selection function and bias for the observational sample with regards to its sSFR, M star , and M BH distributions are not well understood. \nOne may decide to use a sample with proxies instead, with the advantage that a complete, representative sample may be obtained. However, proxies for M BH such as velocity dispersion, bulge mass, or S'ersic index exhibit intrinsic scatter that likely indicates differences in the physics that set these properties (e.g., Gultekin et al. 2009; Beifiori et al. 2012; Shankar et al. 2016; Terrazas et al. 2016; Mart'ınNavarro et al. 2016, 2018). Using proxies for M BH would also introduce unknown uncertainties that likely depend on M star . While the number density of galaxies at a given sSFR, M star , and M BH are uncertain for this sample, our primary \nconcern for this study is accuracy. Using dynamical black hole masses also ensures that the measurements for M BH and M star are independent from one another. Finally, while the sample may not be representative quantitatively, it contains a large diversity of galaxy properties ranging 4 orders of magnitude in sSFR, 2 orders of magnitude in M star , and 4 orders of magnitude in M BH . This results from the variety of M BH measurement techniques that sample distinct parts of the whole galaxy population. As such, we opt to use highquality M BH estimates instead of proxies in order to reliably understand how galaxy properties correlate with M BH .", '5.2 Results': 'Figure 7 shows the M BH -M star (top left), sSFR-M BH (top centre), and sSFR-M star (top right) parameter spaces for the TNG100 simulation at z = 0 (star-forming: blue heatmap, quiescent: red heatmap), and the observational data from T17 (star-forming: gray circles, quiescent: black circles). On the lower right corner of each panel we indicate the 0.15 dex uncertainties on the sSFR and M star of the observational sample. We convolve the simulation quantities for sSFR and M star with these uncertainties in order to more fairly compare the observations with the simulation results. Similarly, we convolve the simulation quantities for M BH with uncertainties from the observed M BH sample. The bottom panels show sSFR as a function of M BH in bins of M star , whose value is denoted above each plot. \nThe top leftmost panel of Figure 7 shows that TNG produces a tight M BH -M star relation, even with added observational uncertainties that increase the scatter. While TNG is in good agreement with the qualitative result that quiescent galaxies host more massive black holes, quantitatively the distributions of star-forming and quiescent galaxies differ between TNG and the observations. Namely, the observational data show significant scatter in M BH as a function of total M star . We note that the relation shown in the top leftmost panel of Figure 7 differs from canonical M BH -galaxy scaling relation studies in that a total M star is preferred in this work over a bulge-only M star . \nThe scatter in the observed M BH -M star relation correlates well with the sSFR of these galaxies, where quiescent galaxies host more massive black holes than star-forming galaxies. TNG does not easily produce galaxies with the M BH and M star demographics of star-forming galaxies in the T17 sample. For example, the Milky Way (included in the T17 sample) with M star ≈ 10 10 . 7 M glyph[circledot] and M BH ≈ 10 6 . 7 M glyph[circledot] is not represented in TNG100. Instead, TNG predicts that the distribution of star-forming galaxies lies far to the left of where they lie in the observational sample. \nAdditionally, the boundary between star-forming and quiescent galaxies on the top leftmost plot showing the M BH -M star relation for TNG is flat, representing a M BH threshold for quiescence at ∼ 10 8 . 2 M glyph[circledot] , as was discussed in Section 4. The slope of the boundary in the observational data is positive with a value of ∼ 1. T17 note that this slope represents lines of constant M BH /M star and that the sSFR decreases perpendicular to these lines moving towards more massive black holes. As such, while quiescence does correlate with M BH in the observational sample, a threshold for quiescence does not exist at a particular M BH value as it does in TNG. We show this more explicitly in the top middle panel of \nFigure 7 where sSFR is shown as a function of M BH . There is a sharp transition between star-forming and quiescent galaxies at the M BH threshold for quiescence at ∼ 10 8 . 2 M glyph[circledot] for TNG galaxies. Galaxies having very low sSFRs shown as upper limits at sSFR glyph[similarequal] 10 -12 . 5 yr -1 , make up 73% of the total quiescent galaxy population in TNG. This produces a strong apparent bimodality in sSFR for the simulated galaxies (in agreement with TNG results shown and discussed in Donnari et al. 2019) as a result of the black hole feedback model at low accretion rates. \nThe observational data show a scattered, more gradual decrease in sSFR as a function of M BH compared to TNG. Namely, there is no indication in the observational data that there is a sharp M BH threshold where galaxies mostly exhibit very low sSFRs ( glyph[similarequal] 10 -12 . 5 yr -1 ). T17 measure significant, intermediate sSFR values ( ∼ 10 -11 -10 -12 yr -1 ) for many galaxies with massive black holes glyph[greaterorsimilar] 10 8 . 2 M glyph[circledot] . Empirically, the sSFR of the observational sample is a tighter and more smoothly declining function of M BH /M star rather than M BH alone, resulting in the scattered relationship between sSFR and M BH . \nWe note that low SFRs in early-type galaxies are notoriously difficult to measure (e.g. Crocker et al. 2011). The sSFRs from T17 are measured using far-infrared IRAS luminosities in order to avoid contamination from possible AGN. They also used ultraviolet luminosities from GALEX to check the robustness of their values, finding consistent results using different techniques. Additionally, recent work indicates the existence of a population of galaxies with low but significant levels of star formation (Oemler et al. 2016), especially when measured in the infrared (Eales et al. 2017). The morphologies and colours of galaxies at intermediate sSFRs in the T17 sample are also visually more disc-like and bluer compared to the more spheroidal and redder galaxies at lower sSFRs, providing further evidence for the robustness of the measured difference between galaxies at intermediate ( glyph[similarequal] 10 -11 -10 -12 yr -1 ) compared to low ( glyph[similarequal] 10 -12 . 5 yr -1 ) sSFRs in the T17 sample. \nIn order to more clearly show the relationship between sSFR and M BH /M star in the observational sample, we plot sSFR as a function of M BH in four bins of M star in the bottom panels of Figure 7. These plots show a gradual decrease in sSFR as a function of M BH in each M star bin for the observational sample (see Terrazas et al. 2017 for a full quantitative discussion on this empirical relationship). TNG adequately reproduces the M BH distribution of the quiescent population in all but the lowest M star bin. However, TNG and the observational data differ significantly for the starforming population, where TNG predicts much higher M BH values compared to the observational data at each M star bin. Additionally, TNG shows a much narrower distribution of M BH values at each M star bin than in the observations due to the tightness of their M BH -M star relation. \nThe top right plot shows the sSFR as a function of M star . The strong bimodality in TNG galaxies as a result of black hole winds is evident, particularly between M star = 10 10 -11 M glyph[circledot] . Additionally, the most massive galaxies in TNG exhibit low yet significant amounts of sSFRs moreso than galaxies at lower masses. We discussed this in Section 4.4, where black hole wind energy was less effective against a more massive potential well where the gas can be \nFigure 8. The M BH -M star relation at z = 0 for the LowThermEff (light blue and red points) and HighThermEff (dark red and blue points) model variations. The top and right panels show the quiescent fraction as a function of M star and M BH , respectively, for the LowThermEff (gray) and HighThermEff (black) simulations. The main difference between the two models is the normalization of the M BH -M star relation since thermal mode feedback primarily regulates M BH growth. Both simulations have the same M BH threshold for quiescence (black arrow) yet the change in normalization alters the quiescent fraction as a function of M star . \n<!-- image --> \nmore readily retained due to the higher binding energies in these galaxies. \nThe sSFR-M star parameter space also highlights the biases in the T17 sample. There is no star forming main sequence, as is seen in more representative samples of galaxies in this parameter space (e.g., Chang et al. 2015). However, this panel also shows how the sample spans a large range of M star and sSFR.', '6 IMPLICATIONS OF THE BLACK HOLE FEEDBACK MODEL ON OBSERVATIONAL DIAGNOSTICS': 'In this section, we explore how the distribution of galaxies on the sSFR-M star -M BH parameter space is sensitive to changes in the physics of quiescence using the TNG model variations.', '6.1 The coevolution of M BH and M star': "The strong correlation between M BH and M star paired with a M BH threshold for quiescence produces a fairly narrow range of M star where both quiescent and star-forming galaxies can coexist with comparable numbers in TNG. The quiescent fraction rises from 0.3 to 0.7 between M star = 10 10 . 25 -10 . 55 M glyph[circledot] (in general agreement with TNG results discussed in Donnari et al. 2019), giving a 0.3 dex range in M star . As a consequence of these features, the distribution of star formation as a function of a galaxy's M star is particularly sensitive to the normalization of the M BH -M star relation. \nFigure 9. The Eddington ratio as a function of M BH for the TNG100 population of galaxies (grayscale heatmap) along with the Eddington ratio threshold that determines whether a black hole is producing kinetic or thermal feedback energy (red solid line) at z = 0. The median distributions of the LowMpiv, HighChi0, LowChi0, and HighMpiv model variations at z = 0 are also plotted (blue lines). These models alter the threshold described in Eqn. 1 (shown as blue dotted lines). \n<!-- image --> \nWe show this in Figure 8 where we plot the M BH -M star relations at z = 0 for the LowThermEff (light blue and red) and HighThermEff (dark blue and red) model variations, which show two simulations where thermal mode feedback is less and more efficient, respectively. The blue and red colours indicate whether galaxies are star-forming or quiescent, i.e. above or below sSFR = 10 -11 yr -1 . The top and right panels show the quiescent fraction as a function of M star and M BH for the LowThermEff (gray) and HighThermEff (black) simulations. \nIn TNG, the thermal mode primarily regulates the growth of the black hole (see Fig. 7 in Weinberger et al. 2018 and the discussion therein for a detailed description of M BH growth in TNG during different accretion phases). A high efficiency of thermal mode feedback more readily suppresses a black hole's growth since it transfers thermal energy to the gas immediately surrounding it, reducing the amount of cold gas that can accrete. As a result, while the M BH threshold for quiescence does not change as a result of different thermal mode efficiencies (see right hand panel of Figure 8), the distribution of star formation amongst galaxies at different M star (i.e., the quiescent fraction as a function of M star ) does change. The top panel of Figure 8 shows that the quiescent fraction significantly increases at different M star values as a result of the change in the normalization of the M BH -M star relation. This indicates changes to the extent of the star forming main sequence on a sSFR-M star plot. \nWe find, however, that the observational sample from T17 exhibits a larger intrinsic scatter in the M BH -M star relation that is not produced by TNG, as discussed in Sec- \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFigure 10. The M BH -M star relation for HighMpiv, LowChi0, HighChi0, and LowMpiv model variations at z = 0. The quiescent fraction as a function of M star and M BH is shown on top and to the right of each plot, respectively. As the Eddington ratio threshold between thermal and kinetic mode black hole feedback ( χ ) increases in units of the fiducial model's value from left to right, the black hole mass at which galaxies are able to become quiescent decreases. \n<!-- image --> \ntion 5.2. We confirm that lowering the seed M BH only increases the scatter for low M star glyph[lessorsimilar] 10 10 . 2 M glyph[circledot] , above which the scatter remains small. This discrepancy likely comes from the particular circumstances in which black holes coevolve and grow with their host galaxies in TNG (Li et al. in prep). The broad scatter in the observational sample's M BH -M star relation suggests that black holes can exhibit different mass assembly histories in different galaxies. If M BH is truly an indicator of quiescence, then the intrinsic scatter in how galaxies can host black holes of different masses will largely affect the observed relationship between sSFR and M star .", '6.2 The M BH threshold for quiescence': 'The top middle panel of Figure 7 shows that for TNG galaxies with M star > 10 10 M glyph[circledot] there exists a M BH threshold for quiescence at ∼ 10 8 . 2 M glyph[circledot] where below this mass most ( > 90%) of these galaxies are star-forming and above this mass most ( > 90%) are quiescent. The majority of these quiescent central galaxies (73%) have very low sSFRs shown as upper limits at ∼ 10 -12 . 5 yr -1 , demonstrating that black hole kinetic winds are extremely effective at preventing star formation in most galaxies undergoing this form of black hole feedback in TNG. In this section, we show that the M BH threshold at ∼ 10 8 . 2 M glyph[circledot] is sensitive to the parameter choices that constitute the TNG fiducial model. \nThe four model variations that illustrate this point are the HighMpiv, LowMpiv, HighChi0, LowChi0 models (see Table 1). In Section 2.1, we describe how TNG uses a M BH -dependent Eddington ratio threshold (Eqn. 1) for determining whether a black hole is releasing thermal or kinetic mode feedback energy. Each of the four model variations we analyze in this section changes the Eddington-ratio threshold, χ , by a certain factor, η : \nχ = min [ ηχ fid , χ max ] , (3) \nwhere χ fid is the fiducial M BH -dependent Eddington ratio threshold (see Eqn. 1), χ max = 0 . 1 as described in Section 2.1, and η = 1 16 , 1 4 , 4 , 16 for the HighMpiv, LowChi0, HighChi0, and LowMpiv model variations, respectively. These variations change the normalization of the threshold. \nIn Figure 9 we show the Eddington ratio as a function of M BH for these four model variations (median distributions \nshown as solid blue lines), explicitly showing the changes to the Eddington ratio threshold between thermal and kinetic mode feedback (shown as dotted blue lines). The variation in the Eddington ratio threshold changes how galaxies populate this parameter space. \nThe changes to the normalization of the Eddington ratio threshold change the M BH at which black hole kinetic wind energy can begin to accumulate. Since kinetic winds from black holes are necessary to produce quiescence in TNG (see Section 3), changes in the normalization of χ result in changes to the M BH threshold for quiescence since χ is M BH -dependent. \nThis behavior is shown in Figure 10, where the M BH -M star relation is shown for each model variation at z = 0. From left to right, the M BH threshold at which galaxies become mostly quiescent moves down as the normalization of χ increases. The quiescent fraction as a function of M star and M BH are shown on the top and right panels of each plot, respectively. As the M BH threshold for quiescence decreases, the M star at which galaxies begin exhibiting quiescence also decreases, until χ = 16 χ fid (rightmost panel) where most of the population above M star = 10 10 M glyph[circledot] is quiescent and above this threshold. \nWe also note that the number of galaxies at high M BH and M star decreases as χ increases from left to right in Figure 10. This is because increasing the χ value makes it easier for black holes to satisfy the conditions necessary for kinetic mode feedback to be turned on. Namely, black holes do not need to be as massive or accrete as inefficiently as would be necessary with a lower χ value. This increases the amount of time that galaxies spend in the kinetic mode since lower mass systems can become quiescent earlier. We have shown in Section 3 that kinetic winds are necessary to produce quiescence and therefore largely stop the growth of galaxies via in situ star formation. Concurrently, kinetic winds also largely halt the growth of the black hole as can be seen in Figure 9, where the Eddington ratio decreases dramatically once galaxies enter the kinetic mode (Weinberger et al. 2018; Habouzit et al. 2019). Therefore, massive black holes and massive galaxies become rarer in the model variations with larger χ values due to the inefficient growth of M BH and M star for these galaxies. \nWe note that the M BH threshold for quiescence at ∼ 10 8 . 2 M glyph[circledot] emerges from the ensemble of parameter choices \nthat produce a realistic z = 0 galaxy population. TNG model parameters were calibrated in order approximately return the shape and amplitudes of certain observables mostly related to the stellar mass content of z = 0 galaxies (see Pillepich et al. 2018a). While the model variations shown in Figure 10 produce unrealistic galaxy populations that would violate the observational constrains described in Pillepich et al. (2018a), they demonstrate the sensitivity of galaxy properties to changes in the M BH threshold. \nWhile different TNG model variations exhibit different behaviors in the sSFR-M BH -M star parameter space, we note that the binding energy correlation described in Section 4.3 continues to hold. Namely, for all model variations, the accumulated energy from black hole-driven kinetic winds must exceed the binding energy of gas within the galaxy in order for quiescence to be produced in each model (See Appendix A). Differences in the model variations occur when the galaxy properties required to meet this condition are changed. This occurs, for example, by allowing lower mass black holes to accumulate kinetic feedback energy (as in this section), or by changing the way black holes occupy galaxies of different M star (as in Section 6.1).', '7 DISCUSSION': 'This work aims to illuminate how the effects of black hole feedback on gas within and around galaxies can alter the observable properties of sSFR, M BH , and M star in the context of a physical model of galaxy formation. We have used the IllustrisTNG simulation suite and its model variations to characterize and assess the black hole model and the resulting physics of quiescence for central galaxies with M star > 10 10 M glyph[circledot] . In this section, we discuss how our results may aid in understanding the link between black hole feedback and quiescence in the real Universe, and how other approaches to modeling this feedback in current and future simulations may benefit from the type of analysis we do in this work.', '7.1 Assessing models using the sSFR-M BH -M star parameter space': "We turn our attention first to the sharp transition from a mostly star-forming to a mostly quiescent galaxy population above a particular M BH threshold at ∼ 10 8 . 2 M glyph[circledot] in TNG. We compare this to a sample of 91 galaxies with dynamical M BH measurements from T17. We note that this sample is not representative of the entire galaxy population due to the biases associated with various dynamical M BH measurement techniques. \nThe current data available, however, do not indicate the existence of a particular M BH threshold for quiescence as is seen in TNG (see the middle panel of Figure 7). Instead, these data show that the sSFR is a smoothly decreasing function of the M BH /M star ratio of these galaxies (for a full discussion see Terrazas et al. 2017). This empirical relationship provides a constraint for models that use black hole feedback in order to produce quiescence. Future work using more representative samples of dynamical M BH measurements with an understanding of their selection biases will be important for more precisely quantifying the observed correlations between sSFR, M star , and M BH found in T17. \nAnother point of discussion centres on the fact that TNG produces a M BH -M star relation with significantly less intrinsic scatter compared to the observational data from T17 (See the left panel of Figure 7). Earlier studies of black hole demographics were heavily biased towards the most massive black holes and mostly took into account those within quiescent, early-type galaxies and a few bulge components of late-type galaxies (Magorrian et al. 1998; Ferrarese & Merritt 2000; Gebhardt et al. 2000). The tight correlation between black holes and the galaxy properties of their early-type components led to the use of samples like these (e.g., Figure 2 of McConnell & Ma 2013, showing the M BH -M bulge relation for early-type galaxies) to assess a model's black hole demographics for the entirety of the galaxy population, early-types or otherwise. The small scatter seen in TNG's M BH -M star relation is a common feature in other galaxy formation models for similar reasons (e.g., Section 3.3 of Volonteri et al. 2016). \nWhile recent studies of black hole demographics using M BH measurements are still biased due to the specifics of each detection method, they incorporate a more diverse set of galaxies and show a significantly more pronounced intrinsic scatter in the relationship between M BH and M star (Reines & Volonteri 2015; Savorgnan et al. 2015; Terrazas et al. 2016, 2017; Davis et al. 2018; Sahu et al. 2019). Namely, they show that late-type, star-forming galaxies lie below the canonical M BH -M star relations for early-type, quiescent galaxies, increasing the scatter in the relation for the whole population (see also Li et al. in prep). Models need to take both the large scatter and its correlation with star formation properties into account when assessing whether their black hole demographics adequately represent the whole galaxy population. \nThe coevolution of M star and M BH is difficult to constrain, especially since the processes governing the growth of these two properties function at spatial and temporal scales that differ by several orders of magnitude. Observations show that M BH accretion rates differ for galaxies of different masses and star formation properties (Aird et al. 2012, 2017, 2018). Additionally, studies have shown that different black hole seeding mechanisms can affect the evolution of both the black hole and its host galaxy (Wang et al. 2019). These factors are likely important for determining the intrinsic scatter between M star and M BH . \nUncertainties on how to model accretion onto a black hole and how this relates to large-scale gas accretion onto its host galaxy also exacerbates the issue of linking M star and M BH in simulations. Many studies argue that BondiHoyle accretion (Hoyle & Lyttleton 1939; Bondi & Hoyle 1944) is an oversimplification, assuming the gas immediately surrounding the black hole has zero angular momentum. This assumption fails to take into account the fact that non-spherical accretion is an important factor for growing black holes (e.g., Hopkins & Quataert 2011). While BondiHoyle accretion is an attractive model due to its simplicity, it may prove to be an important limitation when attempting to reproduce large-scale galaxy properties. Other models for black hole growth, such as the gravitational torque model (Hopkins & Quataert 2011; Angl'es-Alc'azar et al. 2015, 2017a,b), will provide different avenues for exploring possible black hole-galaxy coevolutionary scenarios once \nthey are fully incorporated into a large-scale cosmological model of galaxy formation that includes black hole feedback. \nBlack hole feedback itself can also affect the growth of black holes in galaxies. In Section 6.1, we show that the thermal mode feedback implemented at high accretion rates in TNG regulates M BH growth and plays a role in setting the normalization of the M BH -M star relation. Figure 9 also shows that the Eddington ratio drops to low values once the distribution of galaxies on this plot crosses the threshold where kinetic mode feedback takes effect (Weinberger et al. 2018). Habouzit et al. (2019) explore the black hole population and the the properties of AGN in TNG, showing that their kinetic wind feedback implementation may be too effective at suppressing M BH growth compared to observational constraints. Additionally, TNG assumes constant coupling ( ε f , high , ε f , kinetic ) and radiative ( ε r ) efficiencies for black hole feedback (see Section 2.1 and Weinberger et al. 2018). The values for these efficiencies likely vary across the galaxy population and would increase the scatter in the M BH -M star relation (Bustamante & Springel 2019). \nStellar feedback can also slow M BH growth by reducing the amount of cold gas present near the black hole, particularly in shallow potential wells during the early growth of the black hole (Habouzit et al. 2017; McAlpine et al. 2017; Pillepich et al. 2018a). At the galaxy mass scales relevant to this work with M star > 10 10 M glyph[circledot] , stellar feedback is also expected to play an important role in setting the mass scale above which quiescent galaxies begin to dominate the galaxy population. Bower et al. (2017) and Henriques et al. (2019) both demonstrate this by showing that black hole activity can be triggered once stellar feedback becomes ineffective at preventing cooling from the gaseous halo to the galaxy in progressively more massive systems. \nThe prescriptions for black hole accretion, the implementation of stellar and black hole feedback, and the complex interactions between feedback's effect on the availability of gas for M BH and M star growth all contribute to the characteristics of simulated M BH -M star relations. In this work, we show that if M BH is causally related to quiescence, then the shape, normalization, and scatter of the M BH -M star relation will determine the M star distributions of star-forming and quiescent galaxies (e.g., in TNG compared to Illustris, Donnari et al. 2019, Li et al. in prep). This is a fundamental observational diagnostic for models. As such, it is essential that the the observed scatter in the M BH -M star relation and its correlation with its host galaxy's sSFR are reproduced for any model where M BH correlates with quiescence. This will largely determine the stellar mass distribution of the star forming main sequence and the galaxies that lie off of it, as we describe in Section 6.", '7.2 Characterizing the impact of black hole kinetic winds in TNG': "In Section 4.3, we show that quiescent galaxies have black hole wind energies that exceed the binding energy of the gas in the galaxy. This supports a framework where black hole kinetic winds gravitationally unbind gas from the galaxy above a M BH threshold of ∼ 10 8 . 2 M glyph[circledot] . \nFollowing our analysis in Section 4.3 and visually indicated by Figure 3, black hole winds push dense gas out of galaxies in TNG, effectively producing quiescence. How- \never, high density, cooler gas is more difficult to gravitationally unbind or heat with feedback energy than gas that is more diffuse. Other studies confirm that energy from these winds in TNG galaxies affects not only the cold gas but also the more diffuse gas that preferentially resides at high scale heights above the galaxy and within the circumgalactic medium (Nelson et al. 2018b; Kauffmann et al. 2019, Zinger et al. in prep). \nThis likely explains the behavior seen in the bottom panel of Figure 5, where galaxies just above the M BH threshold for quiescence exhibit an abrupt decrease in halo gas mass. More massive black holes that live in more massive haloes are able to retain a larger fraction of their gas since it is increasingly difficult to push gas out of a deeper potential well. Davies et al. (2019) and Oppenheimer et al. (2019) independently found similar behavior in the EAGLE simulation, where the gas fraction decreases when the ratio of black hole feedback energy and halo binding energy is high. \nFigures 4 and 8 in Pillepich et al. (2018a) and Figure 11 in Weinberger et al. (2017) show rough agreement with observational constraints of halo gas masses around galaxies with dark matter halo masses > 10 13 M glyph[circledot] , showing an improvement compared to the original Illustris model where halo gas masses were severely underestimated. However, while there is evidence to suggest that halo gas masses must be suppressed by feedback in order to agree with cluster mass estimates (e.g., Choi et al. 2015, 2017; Barnes et al. 2017), the evacuation of glyph[greaterorsimilar] 80% of halo gas for a large population of intermediate mass galaxies undergoing black hole feedback should be corroborated before the agreement between TNG and observations is taken at face value. Upcoming work will compare the the observational X-ray signatures of the gaseous atmospheres in TNG haloes with observational data (Pop et al. in prep, Truong et al. in prep). \nWithin the framework of ejective and preventative feedback, TNG's kinetic wind model produces quiescence through both mechanisms. The idea that TNG produces quiescence through sustained ejective feedback is supported by our analysis in Section 4.1 where we find that quiescence in TNG cannot be easily described using the logic of balancing heating and cooling rates employed by purely preventative semi-analytic models. However, Figure 5 shows that the cooling time in galaxies exhibiting black hole winds is significantly greater than in the model variation where no black hole winds are present. Weinberger et al. (2017) demonstrate that these winds are able to thermalize at large distances from the black hole, indicating that the kinetic mode black hole feedback mechanism employed in TNG is partly preventative, resulting in the increase of the gas halo's average cooling time. This is further supported by results from Nelson et al. (2019) who show the thermal properties of gas around a massive galaxy in TNG50 undergoing kinetic black hole feedback (see their Figure 2 and Zinger et al. in prep). Additionally, the ejective nature of TNG's black hole kinetic winds likely aids the process that leads to quiescence since it reduces the mass of the reservoir from which star-forming gas is accreted onto the galaxy.", '7.3 Black hole feedback prescriptions in models': "Another key issue for large-volume simulations is the need to implement subgrid physics that models the small-scale \nphenomena important for galaxy evolution. One crucial approximation is how energy from feedback is transferred to the surrounding gas. Feedback from both stars and black holes has been modeled using a variety of different subgrid physics. The details of these subgrid physics will inevitably affect observable galaxy properties and the transfer of feedback energy to the gas within and around galaxies. \nIn the case of TNG, we have described a two-mode black hole feedback model separating the forms of energy injection at high- and low-accretion rates, corresponding to radiatively efficient and inefficient accretion. High- and lowaccretion phases have been modeled in TNG as an injection of pure thermal energy or pure kinetic energy, respectively. The low-accretion phase injection of kinetic winds in TNG was motivated by recent observational evidence of strong outflows in galaxies hosting inefficiently accreting black holes (Cheung et al. 2016; Wylezalek et al. 2017; Penny et al. 2018). Theoretically, this implementation was also motivated by the wind launching mechanism from inefficient accretion onto black holes put forth by studies such as Blandford & Begelman (1999). \nIn this work, we show that thermal mode black hole feedback in TNG regulates M BH growth (Section 6.1) whereas kinetic mode feedback suppresses M BH and, most notably for this work, M star growth (Sections 3 and 4). The abrupt change in galaxy and halo gas properties at the particular M BH threshold we describe in this work (e.g., Figure 5) indicates an abrupt change in physical processes affecting galaxies that are undergoing kinetic mode as opposed to thermal mode feedback. Similar two-mode black hole feedback models have been implemented in a number of other large-volume simulations (e.g. Croton et al. 2006; Somerville et al. 2008; Vogelsberger et al. 2014a; Henriques et al. 2015; Dubois et al. 2016). \nHowever, this dichotomy between feedback modes is likely an oversimplification of the physics that occurs in the real Universe. As discussed in Section 1, there is an abundance of observational evidence for supermassive black hole activity in galaxies. However, the interpretation of this evidence for constructing a generalizable black hole feedback model for all forms of accretion is not straightforward. For example, radio jets and lobes, generally attributed to low accretion rate feedback, have also been seen in galaxies that are producing large-scale outflows from supposedly high accretion rates (e.g., Komossa et al. 2006; Berton et al. 2018). \nWhile TNG produces winds through low rates of accretion, some higher resolution zoom-in simulations that explicitly include kinetic feedback from high rates of accretion have shown that this process can also effectively eject cold gas from galaxies, depress the central density of the circumgalactic medium, and reduce star formation in galaxies over long timescales (e.g., Choi et al. 2012, 2015, 2017, 2018). This is supported by observational evidence of outflows in AGN and quasars (e.g., Heckman et al. 1981; Villar-Mart'ın et al. 2011; Cicone et al. 2014). These types of zoom-in studies take advantage of their smaller volumes in order to incorporate multiple avenues of feedback energy transfer from black holes (e.g., kinetic, thermal, and/or radiative) simultaneously that may more comprehensively model the complex interactions between black holes and the ambient medium (e.g., Bourne & Sijacki 2017; Mukherjee et al. 2018; Brennan et al. 2018). \nAs such, a consensus on how black hole feedback should be implemented in cosmological simulations within the limitations of finite resolution has not yet been reached. These complications need to be considered when assessing models that simplify black hole feedback physics into low- and high-accretion rate modes. This dichotomy is likely missing important physical recipes that may shape the way galaxies grow in simulations.", '8 CONCLUSIONS': 'We explore the effects of black hole feedback on the properties of central galaxies with M star > 10 10 M glyph[circledot] in the context of the IllustrisTNG simulation suite. In particular, we use TNG100 and ten model variations to assess how observable correlations between the sSFR, M star , and M BH of these galaxies are sensitive to changes in the physics model. We also connect these correlations to the effects of black hole feedback on the distribution of gas within and around galaxies. Finally, we compare results from TNG with observational data of galaxies with dynamical M BH measurements. We highlight our main results below: \n- · TNG requires low accretion rate black hole feedback in the form of kinetic winds in order to produce a quiescent galaxy population (Section 3, Figure 1, and also shown in Weinberger et al. 2017).\n- · A decline in the sSFR of simulated galaxies is seen when the accumulated black hole wind energies exceed the gravitational binding energies of the gas within galaxies, ∫ ˙ E kinetic dt > E bind , gal (top panel of Figure 2). This behavior is seen for all model variations we examine in this work (See Appendix A). This provides strong evidence that black hole-driven kinetic winds push cold gas out of the galaxy to produce quiescence in TNG (Section 4.3, Figure 3).\n- · Simulated galaxies with black hole wind energies that fall below the binding energy of gas within the galaxy have black hole masses below a M BH threshold of ∼ 10 8 . 2 M glyph[circledot] . Those that have wind energies exceeding the binding energies host black holes above this M BH threshold (bottom panel of Figure 2). This produces a sharp decrease in the amounts of interstellar and circumgalactic gas at this M BH threshold (Section 4.4, Figure 5).\n- · Below the M BH threshold at ∼ 10 8 . 2 M glyph[circledot] , most ( > 90%) simulated central galaxies with M star > 10 10 M glyph[circledot] are starforming and above this mass most ( > 90%) are quiescent. 73% of the quiescent population have very low sSFRs (with upper limits at 10 -12 . 5 yr -1 ), indicating that black hole winds are extremely effective at suppressing star formation once low-accretion rate feedback takes effect in TNG (also see Weinberger et al. 2018).\n- · We compare TNG to observational data of 91 central galaxies with dynamical M BH measurements from Terrazas et al. (2016, 2017), with the caveat that these data are not representative of the entire galaxy population. We find that TNG qualitatively reproduces the result that quiescent galaxies host more massive black holes than star-forming galaxies. However, the M BH -M star relation for TNG produces a much smaller scatter compared to what is seen in the observational data (Section 5.2, top left panel of Figure 7). Additionally, these observational data, while incomplete, show a smoother decline in sSFR as a function of M BH , \nshowing no indication of an abrupt suppression of sSFR at a particular M BH threshold (Section 5.2, top middle and bottom panels of Figure 7). \n- · The distribution of star-forming and quiescent galaxies across M star parameter space in TNG is sensitive to both the normalization of the M BH -M star relation and the M BH at which black holes produce kinetic winds. We show that the normalization of the M BH -M star relation depends on the efficiency of thermal mode feedback on regulating M BH growth. \nThese results demonstrate that the relationship between M BH , M star , and sSFR is a powerful tool for exploring the physics of quiescence within the context of black hole feedback. We show that if the M BH and sSFR of galaxies are causally linked, as is the case in TNG, then the way M BH pairs with galaxies of different M star will determine the star formation properties of the entire central galaxy population. We find important differences between the results from TNG and the observational data for galaxies with dynamical M BH measurements from Terrazas et al. (2016, 2017). These differences illuminate the importance of taking into account the scattered relationship between M BH and M star and its dependence on sSFR found in current samples of galaxies with dynamical M BH measurements.', 'ACKNOWLEDGEMENTS': "B.A.T. is supported by the National Science Foundation Graduate Research Fellowship under Grant No. DGE1256260. The Flatiron Institute is supported by the Simons Foundation. The flagship simulations of the IllustrisTNG project used in this work have been run on the HazelHen Cray XC40-system at the High Performance Computing Center Stuttgart as part of project GCS-ILLU of the Gauss centres for Supercomputing (GCS). Ancillary and test runs of the project were also run on the Stampede supercomputer at TACC/XSEDE (allocation AST140063), at the Hydra and Draco supercomputers at the Max Planck Computing and Data Facility, and on the MIT/Harvard computing facilities supported by FAS and MIT MKI. 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F., Franx M., van Dokkum P., Labb'e | |\n| Woo J.-H., Bae H.-J., Son D., Karouzos M., 2016, ApJ, 817, 108 | Woo J.-H., Bae H.-J., Son D., Karouzos M., 2016, ApJ, 817, 108 | \nWylezalek D., et al., 2017, MNRAS, 467, 2612 Yu Q., Tremaine S., 2002, MNRAS, 335, 965 Yuan W., Zhou H. Y., Komossa S., Dong X. B., Wang T. G., Lu H. L., Bai J. M., 2008, ApJ, 685, 801 \nvan den Bosch R., 2016, preprint, ( arXiv:1606.01246 )", 'APPENDIX A: THE PHYSICS OF QUIESCENCE IN THE MODEL VARIATIONS': 'Here we append further evidence in support of our phenomenological framework presented in Section 4.3. This framework describes a physically-motivated picture of quiescence in TNG, where a majority of galaxies shut off their star formation once black hole-driven winds accumulate enough energy to push gas out of their host galaxy. This framework comes from the fact that the sSFR of a galaxy declines once the cumulative energy from black hole-driven winds becomes larger than the binding energy of gas in the galaxy. \nThe model variations listed in Table 1 alter the parameters of the black hole physics model in TNG, producing significantly different galaxy populations. This is described in Section 6 and shown in Figures 9, 8, and 10. Even so, the evidence for our phenomenological framework holds true for every model variation we assess in this work. In Figure A1, we show the sSFR as a function of the ratio between accumulated black hole wind energy and the binding energy of gas in the galaxy (the same parameter space as the top panel of Figure 4). Each panel shows the distribution of TNG100 galaxies as a grayscale heatmap, while the individual galaxies in each model variation are shown as red crosses. The fiducial TNG model is shown in the large panel on the left, whereas the eight model variations that include black holedriven kinetic winds are to the right. \nDespite the differences in galaxy population statistics in each model variation, we find that galaxies in all of the simulations reproduce the same relation on these plots as the fiducial model runs. The only differences are where galaxies lie along the distribution compared to one another. For example, in Section 6.2 and Figure 10 we show that most of the galaxies in the HighMpiv model variation are star-forming since the criteria for producing kinetic winds is only satisfied by the most massive black holes and therefore the most massive galaxies. For the LowMpiv model variation, the criteria for producing kinetic winds is satisfied by all galaxies with M star > 10 10 M glyph[circledot] . In Figure A1, the distribution of galaxies in these two model variations reflect this behavior since in the LowMpiv model all black holes produce enough accumulated energy to overcome the binding energy of the gas within the galaxy. Namely, all galaxies in this simulation lie to the right of the vertical dotted line where these two energies equal. The HighMpiv model shows galaxies that have not yet produced enough black hole wind energy to do this and as such also populate the star-forming region of this plot to the left of the vertical dotted line. \nThe NoBHs and NoBHwinds simulations do not have a value for ∫ ˙ E kinetic dt . However, we have shown in Figure 1 that they do not produce a population of quiescent galaxies, further supporting our phenomenological framework.', 'APPENDIX B: RESOLUTION EFFECTS IN TNG300 COMPARED WITH TNG100': 'In this work, we exclusively use results from TNG100 in order to directly compare with the model variations that have roughly the same resolution. Here we describe resolution effects present in TNG300 that further motivate the exclusive use of TNG100 for the purposes of our study. \nTNG300 is one of the flagship simulations of the IllustrisTNG project with a 302.6 3 comoving Mpc 3 volume. TNG300 has 2 × 2500 3 initial resolution elements with a baryonic mass resolution of 1 . 1 × 10 7 M glyph[circledot] and gravitational softening length of 1.48 kpc at z = 0. This represents approximately an order of magnitude decrease in mass resolution and an increase in the gravitational softening length by a factor of 2 compared to TNG100. \nIn order to differentiate between the effects of a smaller/larger box size in comparison to higher/lower resolution, we use four TNG runs: two with the same resolution as the fiducial TNG100 but different box volumes (TNG1001 sized at 110.7 3 Mpc 3 and FiducialModel sized at 36.9 3 Mpc 3 ) and two with the same resolution as TNG300 but different box volumes (TNG300-1 sized at 302.6 3 Mpc 3 and TNG100-2 sized at 110.7 3 Mpc 3 ). In Figures B1 and B2 the black and red lines indicate the simulations with higher and lower resolution, respectively. Larger and smaller box sizes are indicated with solid and dashed lines, respectively. \nIn Section 4, we describe how quiescence occurs above a particular M BH threshold for TNG galaxies. Figure B1 shows the fraction of quiescent galaxies as a function of M BH for these four simulations. Regardless of box size, those simulations with lower resolution produce up to 20% more quiescent galaxies at M BH glyph[lessorsimilar] 10 8 . 2 and glyph[greaterorsimilar] 10 9 M glyph[circledot] . This indicates that the feedback present in the lower resolution runs is more effective at suppressing star formation and producing quiescence despite the fact that they have the same subgrid prescriptions. Additionally, star formation itself may also be less efficient at lower resolution. \nOur analysis in Section 5.2 indicates that the scatter in the M BH -M star relation is much smaller than the scatter seen in the observations. To determine whether the scatter would increase for a larger box simulation, we show the median M BH -M star relations and their 10 and 90 percentile distributions around their medians for the four simulations we discuss in this section. We convolve these M star and M BH values with observational errors just as we do in Section 5. \nRegardless of box size, those simulations with lower resolution produce a slightly larger scatter than the higher resolution simulations at M star glyph[lessorsimilar] 10 10 . 4 M glyph[circledot] , indicating that this increase in scatter is a resolution effect. Above this mass, the scatter is similarly small for all simulations. We also show the observational data for 91 galaxies with dynamical M BH measurements to show that in all these models, the scatter does not easily reproduce the broad distribution of galaxies in the Terrazas et al. (2017) sample. \nThis paper has been typeset from a T E X/L A T E X file prepared by the author. \nFigure A1. The sSFR as a function of the ratio between the cumulative energy from black hole-driven kinetic winds and the binding energy of gas in the galaxy for all model variations listed in Table 1 (red crosses). The gray heatmaps are the same in each panel and show the distribution of galaxies in TNG100. All models reproduce the same relation as the fiducial model on this plot. \n<!-- image --> \nFigure B1. The fraction of quiescent galaxies as a function of M BH in TNG100-1 (black, solid line), FiducialModel (black, dashed line), TNG300-1 (red, solid line), and TNG100-2 (red, dashed line). The black lines represent simulations run at higher resolution than those represented by the red lines. The dashed lines represent smaller volume simulations than those represented by the solid lines. Lower resolution TNG runs produce up to 20% more quiescent galaxies at M BH glyph[lessorsimilar] 10 8 . 2 and glyph[greaterorsimilar] 10 9 M glyph[circledot] . \n<!-- image --> \nFigure B2. The median M BH -M star relation for TNG100-1 (black, solid line), FiducialModel (black, dashed line), TNG300-1 (red, solid line), and TNG100-2 (red, dashed line). The 10 and 90 percentiles below and above the median are shown as translucent regions matching the colour of their median lines. Similar to Figure B1, black and gray colours represent simulations run at higher resolution than those represented by red and the dashed lines represent smaller volume simulations than those represented by the solid lines. Lower resolution TNG runs produce a slightly more scattered relation at M star glyph[lessorsimilar] 10 10 . 4 M glyph[circledot] . Above this mass, the scatter is consistent between simulations. The observational sample of galaxies with dynamical M BH measurements are shown as gray circles and its M star error bar is shown in the lower right corner. \n<!-- image -->'}

2019JCAP...07..048D

The ineludible non-Gaussianity of the primordial black hole abundance

2019-01-01

['-', '-', '-', '-']

[]

We study the formation of primordial black holes when they are generated by the collapse of large overdensities in the early universe. Since the density contrast is related to the comoving curvature perturbation by a nonlinear relation, the overdensity statistics is unavoidably non-Gaussian. We show that the abundance of primordial black holes at formation may not be captured by a perturbative approach which retains the first few cumulants of the non-Gaussian probability distribution. We provide two techniques to calculate the non-Gaussian abundance of primordial black holes at formation, one based on peak theory and the other on threshold statistics. Our results show that the unavoidable non-Gaussian nature of the inhomogeneities in the energy density makes it harder to generate PBHs. We provide simple (semi-)analytical expressions to calculate the non-Gaussian abundances of the primordial black holes and show that for both narrow and broad power spectra the gaussian case from threshold statistics is reproduced by increasing the amplitude of the power spectrum by a factor Script O(2÷ 3).

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https://arxiv.org/pdf/1904.00970.pdf

{'V. De Luca, a G. Franciolini, a A. Kehagias, b M. Peloso, c,d A. Riotto, a,e C. Unal f': "- a D'epartement de Physique Th'eorique and Centre for Astroparticle Physics (CAP), Universit'e de Gen'eve, 24 quai E. Ansermet, CH-1211 Geneva, Switzerland\n- b Physics Division, National Technical University of Athens, 15780 Zografou Campus, Athens, Greece\n- c Dipartimento di Fisica e Astronomia 'G. Galilei', Universit'a degli Studi di Padova, via Marzolo 8, I-35131, Padova, Italy \nd INFN, Sezione di Padova, via Marzolo 8, I-35131, Padova, Italy \n- e CERN, Theoretical Physics Department, Geneva, Switzerland \nf CEICO, Institute of Physics of the Czech Academy of Sciences (CAS), Na Slovance 1999/2, 182 21 Prague, Czechia \nE-mail: [email protected], [email protected], [email protected], [email protected], [email protected], [email protected] \nAbstract. We study the formation of primordial black holes when they are generated by the collapse of large overdensities in the early universe. Since the density contrast is related to the comoving curvature perturbation by a nonlinear relation, the overdensity statistics is unavoidably non-Gaussian. We show that the abundance of primordial black holes at formation may not be captured by a perturbative approach which retains the first few cumulants of the non-Gaussian probability distribution. We provide two techniques to calculate the non-Gaussian abundance of primordial black holes at formation, one based on peak theory and the other on threshold statistics. Our results show that the unavoidable non-Gaussian nature of the inhomogeneities in the energy density makes it harder to generate PBHs. We provide simple (semi-)analytical expressions to calculate the non-Gaussian abundances of the primordial black holes and show that for both narrow and broad power spectra the gaussian case from threshold statistics is reproduced by increasing the amplitude of the power spectrum by a factor O (2 ÷ 3).", 'Contents': '| 1 | Introduction | 1 |\n|----------------------------------------------------------|--------------------------------------------------------------------------------------------------------------|-----|\n| 2 | A simple criterion to show that intrinsic non-Gaussianity matters | 4 |\n| 3 | The non-Gaussian probability from peak theory | 6 |\n| 3.1 | Spiky peaks of the curvature perturbation may be confused with peaks of the overdensity for large thresholds | 6 |\n| 3.2 | The calculation of the probability from peak theory | 8 |\n| 3.3 | The log-normal power spectrum | 10 |\n| 3.4 | Broad power spectrum | 10 |\n| 4 The non-Gaussian probability from threshold statistics | | 11 |\n| 4.1 | Spiky power spectrum | 13 |\n| 4.2 | Log-normal power spectrum | 15 |\n| 4.3 | Broad power spectrum | 15 |\n| 5 | Conclusions | 16 |\n| | A The cumulants for a narrow power spectrum | 17 |\n| | B Spiky peaks in the curvature perturbation versus peaks in overdensity: | a |\n| | numerical treatment | 19 |\n| | C Analytic integration of the PBH abundance for spiky power spectra using threshold statistics | 20 |\n| | D Peaks versus thresholds | 22 |', '1 Introduction': 'Since the first detection of gravitational waves originated by the merging of two ∼ 30 M glyph[circledot] black holes [1], the idea that Primordial Black Holes (PBHs) might form a considerable fraction of the dark matter [2-4] has attracted again much interest [5] (see Ref. [6] for a recent review). A popular mechanism for the formation of PBHs is the scenario in which PBHs are originated from the enhancement of the curvature power spectrum at a given short length scale due to some features [6]. If the power spectrum of the curvature perturbation is enhanced during inflation to values ∼ 10 -2 on small scales and subsequently transferred to radiation during the reheating process, PBHs may form from sizeable fluctuations if the latter overcome the counter effect of the radiation pressure. \nSince the perturbation of fixed comoving size does not collapse till it re-enters the cosmological horizon, the size of a PBH at formation is related to the horizon length and its mass M is approximately the mass contained in such a horizon volume. Fluctuations collapse immediately after horizon re-entry to form PBHs if they are sizeable enough. We indicate by δ the overdensity and by σ 2 δ its variance \nσ 2 δ = ∫ d 3 k (2 π ) 3 W 2 ( k, R H ) P δ ( k ) , (1.1) \nwhere P δ is the overdensity power spectrum, R H being the comoving horizon length R H = 1 /aH , H is the Hubble rate and a the scale factor. The quantity W ( k, R H ) is a window function, for which we choose a top-hat in real space. Under the assumption that the density contrast is a linear quantity obeying gaussian statistics, threshold statistics (or Press-Schechter) predicts that the primordial mass fraction β ( M ) of the universe stored into PBHs at the formation time is given by 1 \nP G ( δ > δ c ) = β ( M ) = ∫ δ c d δ √ 2 π σ δ e -δ 2 / 2 σ 2 δ . (1.2) \nHere δ c is the threshold for formation of the PBHs which quantifies how large the overdensity perturbations must be and depends on the shape of the power spectrum [7, 8, 10]. By defining \nν c = δ c σ δ , (1.3) \nthe Gaussian mass fraction can be well approximated by ( ν c ∼ > 5) \nβ th G glyph[similarequal] √ 1 2 πν 2 c e -ν 2 c / 2 . (1.4) \nThis expression for the PBH mass fraction comes about when identifying the PBHs with regions whose overdensity is above a given threshold, hence the name of threshold statistics. \nAlternatively, one can identify the PBHs with the local maxima of the overdensity, and one may use peak theory [11] to compute their mass fraction. In such a case one has [12] 2 \nβ pk G glyph[similarequal] 1 3 π ( 〈 k 2 〉 3 ) 3 / 2 R 3 H ( ν 2 pk -1) e -ν 2 pk / 2 with 〈 k 2 〉 = 1 σ 2 δ ∫ d 3 k (2 π ) 3 k 2 P δ ( k ) , (1.5) \nwhere now [8] \nν pk = δ c pk σ δ , (1.6) \nand δ c pk is to be identified with the critical value of the overdensity at the center of the peak above which an initial perturbation eventually collapses into a PBH [8, 10]. Notice that here we follow Refs. [7, 8] and do not introduce a window function for the peak theory. Indeed, for the examples we will discuss the window function is not strictly necessary because they are characterised by a well-defined scale in momentum space and the corresponding distribution is already smooth on length scales smaller than that characteristic scale. Also, in the case of peak theory a typical length pops out automatically, that is the scale R ∗ . \nThe gaussian expressions (1.4) and (1.5) make already manifest the essence of the problem we are going to discuss in this paper. PBHs are generated through very large, but rare fluctuations. Therefore, their mass fraction at formation is extremely sensitive to changes in the tail of the fluctuation distribution and therefore to any possible non-Gaussianity in the density contrast [13-26]. This implies that non-Gaussianities need to be accounted for as they can alter the initial \n2 We differ slightly from the corresponding expression in Ref. [12]. First by a factor of 3 to account for the fact that one counts the number density of peaks at superhorizon scales, but the PBHs formed once the overdensity crosses the horizon at a slightly later time [8] (see also section 3). Secondly, by the fact that we define the mass going into PBH to be M = (4 π/ 3) ρR 3 H , where ρ is the background radiation density. More importantly, we use here the definition (1.6) for the critical value ν pk . We will give more details in section 3. At the gaussian level, peak theory gives a PBH abundance which is systematically larger than the one provided by the threshold statistics [12]. \nmass fraction of PBHs in a dramatic way. For instance, the presence of a primordial local nonGaussianity in the comoving curvature perturbation can significantly alter the number density of PBHs through mode coupling [27-32]. \nIn this paper we will be dealing with a source of non-Gaussianity which is unavoidably generated by the non-linear relation among the overdensity δ ( glyph[vector]x, t ) ( t is the cosmic time) and the comoving curvature perturbation ζ ( glyph[vector]x ). It is important to stress that this non-linear relation makes the overdensity non-Gaussian even if the curvature perturbation is gaussian. In this sense, the non-Gaussianity we will discuss here is ineludible. \nLet us briefly discuss where this non-linearity relation comes from. As we mentioned above, in the early radiation-dominated universe, the PBHs are generated when highly overdense regions gravitationally collapse directly into a black hole. Before collapse, the comoving sizes of such regions are larger than the horizon length and the separate universe approach can be applied [33]. One therefore expands at leading order in spatial gradients of the various observables, e.g. the overdensity. At this stage, the slicing and the threading of the spacetime manifold are to be fixed. For instance, the so-called comoving gauge seems appropriate as it has been adopted to perform numerical relativity simulations to describe the formation of PBHs and to calculate the threshold for PBH formation [7]. \nIn the comoving slicing, the overdensity turns out to be [33] \nδ ( glyph[vector]x, t ) = -8 9 a 2 H 2 e -5 ζ ( glyph[vector]x ) / 2 ∇ 2 e ζ ( glyph[vector]x ) / 2 = -4 9 1 a 2 H 2 e -2 ζ ( glyph[vector]x ) ( ∇ 2 ζ ( glyph[vector]x ) + 1 2 ∂ i ζ ( glyph[vector]x ) ∂ i ζ ( glyph[vector]x ) ) . (1.7) \nAs the universe expands, the overdensity grows. Regions where it becomes of order unity eventually stop expanding and collapse. This happens when the comoving scale of such a region becomes of the order of the horizon scale. Even though the gradient expansion approximation breaks down, it has been used to obtain an acceptable criterion for the PBH formation (that is to compute the overdensity threshold) and this approximation has been confirmed to hold by nonlinear numerical studies [6, 34]. \nThe standard procedure in the literature is to expand the relation (1.7) to first-order in ζ \nδ ( glyph[vector]x, t ) = -4 9 a 2 H 2 ∇ 2 ζ ( glyph[vector]x ) (1.8) \nand to relate the power spectrum of the overdensity to the one of the curvature perturbation by the relation \nP δ ( k, t ) = 16 81 k 4 a 4 H 4 P ζ ( k ) . (1.9) \nThe question is to what extent this is a good approximation given the fact that even tiny changes (percent level) in the square root of the overdensity variance are exponentially amplified in the PBH mass fraction. \nTo get the feelings of the numbers, let us roughly estimate the impact of the exponential e -2 ζ ( glyph[vector]x ) . Calling k glyph[star] the typical momentum of the perturbation, from Eq. (1.8) we get \nζ glyph[similarequal] 9 a 2 H 2 4 k 2 glyph[star] δ glyph[similarequal] 9 a 2 H 2 4 k 2 glyph[star] δ c glyph[similarequal] 0 . 15 , (1.10) \nwhere we have taken the threshold δ c glyph[similarequal] 0 . 5 and k glyph[star] glyph[similarequal] 2 . 7 aH [8]. This gives e -2 ζ ( glyph[vector]x ) glyph[similarequal] 0 . 7. This looks as a small change, but in fact it has an exponentially large effect in the mass fraction when the corresponding overdensity variance is calculated. \nThe goal of this paper is to deal with the intrinsically non-Gaussian nature of the overdensity onto the mass fraction of PBHs. First of all, we will provide a simple argument to convince \nthe reader that the non-Gaussianity introduced by the non-linear relation (1.7) between the overdensity and the gaussian curvature perturbation has an impact on the PBH mass fraction which may not be accounted for by a perturbative approach. Based on this finding, we will proceed by computing the mass fraction taking into account such intrinsic non-Gaussianity. We will do so by using two methods. \nSince PBHs may be thought to originate from peaks, that is, from maxima of the local overdensity, we will resort to peak theory [11] to calculate the probability of formation of the PBHs. This method is based on the fact that for high values of the overdensity at the peaks, their location can be confused with the location of the peaks in the comoving curvature perturbation as long as such peaks are sufficiently spiky, that is if their curvature (proportional to the second spatial derivatives) is large enough at the center of the peak [7]. \nAlternatively, we will use the non-Gaussian threshold statistics and provide an exact expression for the probability to form PBHs. Both methods indicate that the inevitable non-Gaussian nature of the overdensity makes more difficult to generate PBHs, independently from the shape of the power spectrum. \nLet us also add a cautionary note. The intrinsic non-Gaussianity of the overdensity changes also the shape of the profile of the peaks which eventually give rise to PBHs upon collapse. Since the threshold depends on the shape of the overdensity, such non-Gaussianity influences as well the threshold value. This will be discussed in a separate publication [35]. \nThe paper is organised as follows. In section 2 we offer a simple criterion to show that the intrinsic non-Gaussianity cannot be described by perturbative methods. Sections 3 and 4 will describe the two methods mentioned above. Section 5 contains our conclusions. The paper contains as well several appendices for the technical details.', '2 A simple criterion to show that intrinsic non-Gaussianity matters': 'In order to establish if the intrinsic non-Gaussianity introduced by the non-linear relation (1.7) is relevant, we start from the non-Gaussian threshold statistics developed in Ref. [36] and refined in Ref. [24] by means of a path-integral approach. We do not report all the details here and the interested reader is refereed to those references for more details. We do not use here the window function which would introduce painful, but useless technicalities without changing the conclusions. Suffice to say that the probability of having the overdensity larger than a given threshold can be viewed as the one-point function of the threshold quantity \nP ( δ > δ c ) = 〈 Θ( δ -ν c σ δ ) 〉 = ∫ [ Dδ ( glyph[vector]x )] P [ δ ( glyph[vector]x )]Θ ( δ ( glyph[vector]x ) -ν c σ δ ) , (2.1) \nwhere Θ( x ) is the Heaviside function. By defining the connected correlators of the overdensity as \n〈 δ ( glyph[vector]x 1 ) · · · δ ( glyph[vector]x n ) 〉 c = ξ n ( glyph[vector]x 1 , · · · , glyph[vector]x n ) , (2.2) \none finds that, in the limit of large ν c , the threshold statistics is given by [24, 36] \nP ( δ > δ c ) = β ( M ) = 1 √ 2 πν 2 c exp { -ν 2 c / 2 + ∞ ∑ n =3 ( -1) n n ! ξ n (0) ( δ c /σ 2 δ ) n } , (2.3) \nwhere the label 0 means that the correlators are computed at equal points. To see under which circumstances the non-Gaussianity of the overdensity alters the predictions of the gaussian primordial abundance of PBHs in a significant way, we define dimensionless quantities, the cumulants, \nby the relations \nS n = ξ n (0) ( ξ 2 (0)) n -1 = n -times 〈 ︷ ︸︸ ︷ δ ( glyph[vector]x ) · · · δ ( glyph[vector]x ) 〉 c σ 2( n -1) δ . (2.4) \nFollowing Ref. [24] we may define the fine-tuning ∆ n to be the response of the PBH abundance to the introduction of the n -th cumulant as \n∆ n = dln β ( M ) dln S n . (2.5) \nEach cumulant allows to express the non-Gaussian PBH abundance in terms of the gaussian abundance as \nβ th NG ( M ) β th G ( M ) = e ∆ n . (2.6) \nThis implies that the PBH abundance is exponentially sensitive to the non-Gaussianity unless ∆ n is in absolute value smaller than unity \n| ∆ n | ∼ < 1 . (2.7) \nInspecting Eqs. (1.4) and (2.3), we see that \n| ∆ n | = 1 n ! ( δ c σ δ ) 2 | S n | δ n -2 c . (2.8) \nThis tells us that intrinsic non-Gaussianity in the overdensity alters exponentially the gaussian prediction for the PBH abundance unless \n| S n | ∼ < ( σ δ δ c ) 2 n ! δ n -2 c . (2.9) \nTo investigate how restrictive this condition is, we take the simplest case possible, i.e. a very narrow power spectrum for the comoving curvature perturbation which we approximate by a Dirac delta \nP ζ ( k ) = 2 π 2 k 3 P ζ ( k ) and P ζ ( k ) = A s k glyph[star] δ D ( k -k glyph[star] ) . (2.10) \nHere A s is the amplitude of the power spectrum and k glyph[star] is the characteristic scale of the power spectrum. Its relation with the cosmological horizon at formation R H has to be fixed running numerical simulations [8, 10]. For the case at hand, it is given by k glyph[star] glyph[similarequal] 2 . 7 /R H (more comments on this later on). We do not report all the technical details here, which can be found in Appendix A, where we have consistently calculated the variance, the skewness S 3 and the kurtosis S 4 up to third-order in perturbation theory (in the power spectrum P ζ , that is up to A 3 s ). We get \n〈 δ 2 〉 c = σ 2 δ = c 2 glyph[star] k 4 glyph[star] A s ( 1 + 133 6 A s + 511 3 A 2 s ) , 〈 δ 3 〉 c = -c 3 glyph[star] k 6 glyph[star] 12 A 2 s ( 1 + 3889 108 A s ) , 〈 δ 4 〉 c = 240 c 4 glyph[star] k 8 glyph[star] A 3 s , c glyph[star] k 2 glyph[star] = 4 9 ( k glyph[star] aH ) 2 glyph[similarequal] 3 . 2 . (2.11) \nOne can check that the criterion (2.9) for the skewness (kurtosis) gives the lower bound \nA s ∼ > 6 . 0 (4 . 0) · 10 -3 , (2.12) \nwhere we have taken δ c = 0 . 5. We now impose the condition that the PBHs form at most the totality of dark matter, which provides an upper bound on their mass fraction given by \nβ ∼ < 1 . 3 × 10 -9 ( M M glyph[circledot] ) 1 / 2 . (2.13) \nFor instance, for PBH masses around the interesting value of 10 -12 M glyph[circledot] [38, 39], one would get from the gaussian mass fraction (1.4) β ∼ 10 -15 , ν c glyph[similarequal] 8 and therefore A s glyph[similarequal] 3 . 7 · 10 -4 . This figure violates the bound required (2.12) to neglect the non-Gaussianity by one order of magnitude. More importantly, the kurtosis does not provide a bound which is much weaker than the skewness. This signals the breaking of the perturbative approach and calls for a more refined treatment. \nThe same conclusion can be obtained in the case where the power spectrum of the comoving curvature perturbation is parametrised by a log-normal shape of the form \nP ζ ( k ) = A g √ 2 πσ exp [ -ln 2 ( k/k glyph[star] ) 2 σ 2 ] . (2.14) \nUsing the results in Appendix A, one finds the following (for σ = 0 . 2) \n〈 δ 2 〉 c = σ 2 δ = 1 . 4 · c 2 glyph[star] k 4 glyph[star] A g ( 1 + 20 A g +150 A 2 g ) , 〈 δ 3 〉 c = -18 · c 3 glyph[star] k 6 glyph[star] A 2 g (1 + 34 A g ) , 〈 δ 4 〉 c = 400 · c 4 glyph[star] k 8 glyph[star] A 3 g . (2.15) \nThe criterion (2.9) in this case results in a lower bound \nA g ∼ > 3 . 8 (2 . 2) · 10 -3 , (2.16) \nfor the skewness and kurtosis respectively, while requiring again β ∼ 10 -15 for M ∼ 10 -12 M glyph[circledot] gives A g = 2 . 5 · 10 -4 . Again we do not see signs of convergence in the perturbative approach.', '3 The non-Gaussian probability from peak theory': 'Having shown that perturbation theory fails to provide the probability for PBH formation, we first resort to peak theory [11]. As we already mentioned in the introduction, PBHs trace the peaks of the radiation density field on superhorizon scales where the number of peaks per comoving volume is constant. Notice that we are dealing with peaks of the overdensity rather than the peaks of the curvature perturbation. This is because one cannot impose any constraint on the value of the gravitational potential (or curvature perturbation) on superhorizon scales because constant gravitational potentials cannot lead to any observable effect. Nevertheless, one can start from the following important point: large threshold peaks of the overdensity may be identified within a Hubble volume with the peaks of the curvature perturbation if the Laplacian of the curvature perturbation (that is the curvature of the peak) at the peak is large enough [7]. More in details, one can show that if the value of δ is comparable to the threshold value at a peak, one can find the associated peak of ζ well inside the horizon patch and centered at the peak of δ as long as the peaks in ζ is spiky enough. Let us elaborate about this point in the next subsection.', '3.1 Spiky peaks of the curvature perturbation may be confused with peaks of the overdensity for large thresholds': 'The argument given in Ref. [7] is as follows. Let us consider the nonlinear expression (1.7) relating δ and ζ on superhorizon scales and in radiation domination \nδ ( glyph[vector]x, t ) = -4 9 a 2 H 2 e -2 ζ ( glyph[vector]x ) [ ∇ 2 ζ ( glyph[vector]x ) + 1 2 ∂ i ζ ( glyph[vector]x ) ∂ i ζ ( glyph[vector]x ) ] . (3.1) \nWe can expand the comoving curvature perturbation ζ ( glyph[vector]x ) for points glyph[vector]x around the peak position glyph[vector]x pk of the overdensity 3 δ ( glyph[vector]x, t ) \nζ ( glyph[vector]x ) = ζ ( glyph[vector]x pk ) + ∂ i ζ ( glyph[vector]x pk )( x i -x i pk ) + 1 2 ∂ i ∂ j ζ ( glyph[vector]x pk )( x i -x i pk )( x j -x j pk ) . (3.2) \nAround such a peak we can also write \nδ ( glyph[vector]x pk , t ) glyph[similarequal] -4 9 a 2 H 2 e -2 ζ ( glyph[vector]x pk ) ∇ 2 ζ ( glyph[vector]x pk ) , (3.3) \nwhere we neglected the second term in the square bracket since its contribution is of higher order in ζ with respect to (3.3). \nSince the peak amplitude of the overdensity must be larger than some critical value δ c pk , we deduce that the curvature of the peak in ζ is bounded from above \n-∇ 2 ζ ( glyph[vector]x pk ) > 9 a 2 H 2 4 e 2 ζ ( glyph[vector]x pk ) δ c pk . (3.4) \nThis is what we meant by saying that the peaks in ζ must be spiky enough. Now, the peak in ζ is located in glyph[vector]y pk such that ∂ i ζ ( glyph[vector]y pk ) = 0, or \n∂ i ζ ( glyph[vector]x pk ) + ∂ i ∂ j ζ ( glyph[vector]x pk )( y i pk -x i pk ) = 0 or ( y i pk -x i pk ) = -( ζ -1 ) i j ( glyph[vector]x pk ) ∂ j ζ ( glyph[vector]x pk ) , (3.5) \nwhere we have used in the last passage the notation ∂ i ∂ j ζ ( glyph[vector]x pk ) = ζ ij ( glyph[vector]x pk ). Performing a rotation of the coordinate axes to be aligned with the principal axes of the constantζ ellipsoids gives the eigenvalues of the shear tensor ζ ij to be equal to -σ 2 λ i , where σ 2 is the characteristic root-meansquare variance of the components of ζ ij (that of ∂ i ζ is σ 1 ) and \nλ i glyph[similarequal] γν 3 , ν = ζ ( glyph[vector]x pk ) σ 0 , γ = σ 2 1 σ 0 σ 2 and σ 2 j = ∫ k 2 d k 2 π 2 P ζ ( k ) k 2 j = ∫ d k k P ζ ( k ) k 2 j . (3.6) \nThe crucial point is now that the moments σ 2 j are typically much smaller than ( aH ) j (because of the presence of the amplitude of the power spectrum). From Eq. (3.4), we deduce that \n-∇ 2 ζ ( glyph[vector]x pk ) ∼ λ i σ 2 > 9 a 2 H 2 4 e 2 ζ ( glyph[vector]x pk ) δ c pk glyph[greatermuch] σ 2 (3.7) \nand therefore λ i ∼ γν glyph[greatermuch] 1 (the probability to have negative eigenvalues is small for large curvatures around the peak [11]) . This implies \n| y i pk -x i pk | glyph[similarequal] | σ 1 /σ 2 λ i | glyph[lessmuch] | σ 1 /σ 2 | ∼ < 1 /aH, (3.8) \nwhere in the last equality we have used the fact that σ 1 /σ 2 glyph[similarequal] k -1 glyph[star] ∼ < R H . Therefore the high overdensity peaks in δ lie close to the peaks of the curvature perturbation (i.e. within the Hubble volume) if the latter are characterised by a large second derivatives at the origin of the peak. This statement if of course valid in the probabilistic sense. \nSince some approximations have been made along the way, in Appendix B the reader can find a numerical simulation we have performed to support this result.', '3.2 The calculation of the probability from peak theory': "If the argument above is correct, one can associate the number of rare peaks in the overdensity with the number of peaks in the curvature perturbation which are spiky enough, see Eq. (3.7). Therefore, expanding around the peak location glyph[vector]x pk of ζ (where ∂ i ζ ( glyph[vector]x pk ) = 0) we can write \nδ ( glyph[vector]x pk , t ) = -4 9 a 2 H 2 e -2 ζ ( glyph[vector]x pk ) [ ∇ 2 ζ ( glyph[vector]x pk ) + 1 2 ∂ i ζ ( glyph[vector]x pk ) ∂ i ζ ( glyph[vector]x pk ) ] = -4 9 a 2 H 2 e -2 ζ ( glyph[vector]x pk ) ∇ 2 ζ ( glyph[vector]x pk ) = 4 9 a 2 H 2 e -2 σ 0 ν xσ 2 , (3.9) \nwhere \nν = ζ ( glyph[vector]x pk ) σ 0 and x = -∇ 2 ζ ( glyph[vector]x pk ) σ 2 . (3.10) \nSince the number of peaks (if spiky enough) in ζ is approximately the number of peaks in δ , we can use the expression (A.14) of Ref. [11] to find the number of peaks of the overdensity \nN pk ( ν, x )d ν d x = e -ν 2 / 2 (2 π ) 2 R 3 ∗ f ( x ) exp[ -( x -x ∗ ) 2 / 2(1 -γ 2 )] [2 π (1 -γ 2 )] 1 / 2 d ν d x, (3.11) \nwhere \nR ∗ = √ 3 σ 1 σ 2 , γ = σ 2 1 σ 0 σ 2 , and x ∗ = γν, (3.12) \nand f ( x ) is provided by the expression \nf ( x ) = ( x 3 -3 x ) 2 [ erf ( x √ 5 2 ) +erf ( x 2 √ 5 2 ) ] + √ 2 5 π [( 31 x 2 4 + 8 5 ) e -5 x 2 8 + ( x 2 2 -8 5 ) e -5 x 2 2 ] . (3.13) \nThus the number density of non-Gaussian peaks of the overdensity above a given threshold δ c pk is simply given by \nN pk = ∫ ∞ -∞ d ν ∫ ∞ x c δ ( ν ) d x e -ν 2 / 2 (2 π ) 2 R 3 ∗ f ( x ) exp[ -( x -x ∗ ) 2 / 2(1 -γ 2 )] [2 π (1 -γ 2 )] 1 / 2 , (3.14) \nwhere \nx c δ ( ν ) glyph[similarequal] 9 a 2 H 2 4 σ 2 e 2 σ 0 ν δ c pk (3.15) \naccounts for the fact that only large enough Laplacian values at the peak of the curvature perturbation have to be accounted for, see Eq. (3.4). Notice that if we take the lower limit (3.15) at ν = 0, x c δ (0) glyph[similarequal] (9 a 2 H 2 / 4 σ 2 ) δ c pk , we automatically reproduce the gaussian case. We have checked numerically that in such a case, the peak theory abundance of PBHs obtained from the number density (3.14) with x c δ (0) reproduces the abundance (1.5) within a factor of order unity. This gives us extra confidence that identifying large threshold peaks in δ with the spiky enough peaks in ζ is a correct procedure. From the expression above one can see that the narrower is the power spectrum (that is the closer to unity is the parameter γ ) the more the integrand is peaked at the value x glyph[similarequal] x ∗ glyph[similarequal] ν . \nWe conclude that the non-liner relation between the curvature perturbation and the overdensity makes it harder to generate PBHs, independently from the shape of the curvature perturbation power spectrum. \nFrom the knowledge of the number density of peaks N pk we can compute the mass fraction of PBHs β at the time of the formation t f . Since PBHs trace the peaks of the radiation density field on superhorizon scales and since the number of peaks per comoving volume is constant, the number of enough sizeable peaks on superhorizon scales provides the number of PBHs formed once the overdensity has crossed the horizon and one has properly rescaled it to the formation time [8, 10]. \nThe next question is therefore what defines the horizon crossing. In cosmology we are used to the concept of the horizon crossing associated to a given comoving wavelength k -1 and we say that horizon crossing takes place when k = aH . In the case of PBHs, the large inhomogeneities have characteristic profiles in coordinate space and therefore it is not immediate to associate to them a given wavelength or momentum. The procedure we will follow is the one adopted to define the threshold for collapse [10]. Suppose the overdensity has an average profile in real space given by [11] 4 \nδ ( r, t ) = δ pk ξ 2 ( r, t ) σ 2 δ ( t ) , (3.16) \nwhere ξ 2 ( r, t ) is the two-point correlator. One can define a scale r m through the relation \nr 3 m = ∫ r m 0 d r δ ( r, t ) r 2 δ ( r m , t ) . (3.17) \nThis scale is relevant since one can show that the threshold for PBH formation is given by [8, 10] \nδ c pk = δ c 3 σ 2 δ ( t m ) ξ 2 ( r m , t m ) , (3.18) \nwhere δ c = 3 δ ( r m , t m ), since r m is precisely the scale at which the compaction function C glyph[similarequal] 2 δM/ar (being δM the overmass generated by the averaged curvature perturbation) is maximised [10]. Such a maximum is located at distances larger than the cosmological horizon. It is then natural to define the 'horizon crossing' as the time at which 5 a ( t m ) H ( t m ) r m = 1. Numerical simulations must provide a relation between the scale r m and the characteristic momentum appearing in the power spectrum of the curvature perturbation. \nThe mass fraction at formation time (that is when the horizon forms) from peak theory will then be \nβ pk NG = M ( R H ) ρ f a 3 m a 3 f N pk , (3.19) \nwhere M ( R H ) is the mass of the PBH associated with the horizon size 6 R H , \nM ( R H ) = 4 π 3 ρ m R 3 H ( t m ) , (3.20) \nand ρ f and ρ m are the background radiation energy densities at the time of formation and horizon crossing, respectively. Numerical simulations show that the ratio a f /a m is rather independent from the shape of the power spectrum and ∼ 3 [8]. We therefore have \nβ pk NG glyph[similarequal] 3 · 4 π 3 R 3 H N pk . (3.21) \nFigure 1 . Mass fraction β pk as a function of A g for log-normal power spectrum (PS) computed using peak theory for both the gaussian and the non-Gaussian case. \n<!-- image -->", '3.3 The log-normal power spectrum': 'We assume a power spectrum of the form \nP ζ ( k ) = A g √ 2 πσ exp [ -ln 2 ( k/k glyph[star] ) 2 σ 2 ] , (3.22) \nwhere changing the value of σ changes the broadness of the power spectrum. For the case at hand it turns out that [8] \na m H m = 1 R H = 1 2 . 7 k glyph[star] (3.23) \nand one has to choose the critical value δ c pk = 1 . 16 corresponding to δ c = 0 . 51 [8, 10]. \nIn Fig. 1 we plot the mass fraction for various values of σ as a function of A g . We see that the inclusion of the intrinsic non-Gaussian effects systematically lowers the PBH abundance (having kept fixed the amplitude of the power spectrum of the curvature perturbation). Said in other words, keeping the amplitude of the fluctuations fixed, it is more difficult to generate PBHs. This will remain true also using the threshold statistics, as we show in the next Section. Quantitatively, in the case considered, for the usual value of β ∼ 10 -15 necessary for PBHs to be all the dark matter in the universe for masses of the order of 10 -12 M glyph[circledot] , we find that in the gaussian case the value of the amplitude is consistent with the one reported in Ref. [8] once the difference in the normalisation of the power spectrum is taken into account, while the non-Gaussian abundance is suppressed.', '3.4 Broad power spectrum': 'We also consider a broad power spectrum, that is a top-hat function with amplitude A t as \nP ζ ( k ) = A t Θ( k max -k ) Θ( k -k min ) (3.24) \nwhere Θ stands again for the Heaviside step function and k max glyph[greatermuch] k min , such that the scale k min in practice does not participate in the PBH formation [8]. In this case one finds k max glyph[similarequal] 3 . 5 /r m , δ c is again 0.51, and δ c pk glyph[similarequal] 1 . 22 [8] and the variances are obtained by putting a m H m as the infrared cut-off since the unphysical long wavelength modes should be disregarded. Fig. 2 shows the mass \nFigure 2 . Mass fraction β pk as a function of A t for the broad (top-hat) power spectrum computed using peak theory for both the gaussian and the non-Gaussian case. In this plot (and in the following) we show in the horizontal axes the value of the amplitude of the power spectrum and its corresponding root of the variance σ 0 . \n<!-- image --> \nfraction as a function of A t . 7 As predicted, both for narrow spectra and broad ones, the intrinsic non-Gaussianity in the overdensity makes it harder to produce PBHs.', '4 The non-Gaussian probability from threshold statistics': 'In this section we present an alternative way to calculate the non-Gaussian probability to form PBHs which does not rely on the fact that spiky peaks of the curvature perturbation coincide with peaks of the overdensity for large thresholds. The price to pay is that we will be dealing with the threshold statistics (the threshold being identified with δ c [8]). This might be not a great sacrifice as regions characterised by large thresholds are likely to be regions of maxima of the overdensity [40]. The gain is that the expressions we are going to obtain are exact. \nLet us consider again the curvature perturbation ζ ( glyph[vector]x ) as a random field. Following the notation of the Appendix A of Ref. [11], we define \nζ i = ∂ i ζ, ζ ij = ∂ i ∂ j ζ. (4.1) \nThe correlations of these fields are provided by the expressions \n〈 ζζ 〉 = σ 2 0 , (4.2) \n〈 ζζ ij 〉 = -σ 2 1 3 δ ij , (4.3) \n〈 ζζ i 〉 = 0 , (4.4) \n〈 ζ i ζ i 〉 = σ 2 1 3 δ ij , (4.5) \n〈 ζ ij ζ kl 〉 = σ 2 2 15 ( δ ij δ kl + δ ik δ jl + δ il δ jk ) , (4.6) \n〈 ζ i ζ jk 〉 = 0 . (4.7) \nThese variances will be computed numerically using the Fourier transform of the top-hat window function in real space, that is \nσ 2 j = ∫ k 2 d k 2 π 2 W 2 ( k, r m ) P ζ ( k ) k 2 j , W ( k, r m ) = 3 sin( kr m ) -kr m cos( kr m ) ( kr m ) 3 . (4.8) \nThe matrix -ζ ij can be diagonalized with eigenvalues σ 2 λ i , ordered such that λ 1 ≥ λ 2 ≥ λ 3 . Thus we define \nx = -∇ 2 ζ σ 2 = λ 1 + λ 2 + λ 3 , y = λ 1 -λ 3 2 , z = λ 1 -2 λ 2 + λ 3 2 . (4.9) \nIntroducing again ν = ζ ( glyph[vector]x ) /σ 0 , the correlations become \n〈 ν 2 〉 = 1 , 〈 x 2 〉 = 1 , 〈 xν 〉 = γ, 〈 y 2 〉 = 1 / 15 , 〈 z 2 〉 = 1 / 5 (4.10) \nand all the others are zero. The joint gaussian probability distribution for these variables is provided by the expression (from now on we will label η i ≡ ζ i ) \nP ( ν, glyph[vector]η , x, y, z )d ν d 3 η d x d y d z = N | 2 y ( y 2 -z 2 ) | e -Q d ν d x d y d z d 3 η σ 3 0 (4.11) \nas a function of \nand \n2 Q = ν 2 + ( x -x ∗ ) 2 (1 -γ 2 ) +15 y 2 +5 z 2 + 3 glyph[vector]η · glyph[vector]η σ 2 1 (4.12) \nx ∗ = γν, γ = σ 2 1 σ 0 σ 2 , N = (15) 5 / 2 32 π 3 6 σ 3 0 σ 3 1 (1 -γ 2 ) 1 / 2 . (4.13) \nThe variables y and z are unconstrained and we integrate them out. With the ordering of the eigenvalues previously defined, we see that the variable z lies in the range [ -y, y ], while y ≥ 0. The result is therefore given by 8 \nP ( ν, glyph[vector]η , x )d ν d 3 η d x = Ce -Q 2 d ν d 3 η d x, (4.14) \nwhere we have defined \nand \nC = 6 √ 3 8 π 5 / 2 √ 2(1 -γ 2 ) σ 3 1 (4.15) \n2 Q 2 = ν 2 + ( x -x ∗ ) 2 (1 -γ 2 ) + 3 glyph[vector]η · glyph[vector]η σ 2 1 . (4.16) \nWe can then write the δ as a function of these variables as \nδ ( glyph[vector]x, t ) = -4 9 a 2 H 2 e -2 ζ ( glyph[vector]x ) [ ∇ 2 ζ ( glyph[vector]x ) + 1 2 ζ i ( glyph[vector]x ) ζ i ( glyph[vector]x ) ] = 4 9 a 2 H 2 e -2 νσ 0 [ xσ 2 -1 2 glyph[vector]η · glyph[vector]η ] . (4.17) \nNow we perform the change of variables: \nx δ = x, glyph[vector]η δ = glyph[vector]η , ν = 1 2 σ 0 ln [ 4 ( x δ σ 2 -1 2 glyph[vector]η δ · glyph[vector]η δ ) 9 a 2 H 2 δ ] . (4.18) \nThe argument of the logarithm is positive for x δ > glyph[vector]η δ · glyph[vector]η δ / 2 σ 2 . The Jacobian of the transformation is given by \nJ = ∣ ∣ ∣ ∣ 1 2 δσ 0 ∣ ∣ ∣ ∣ . (4.19) \nTherefore the distribution in terms of the new variables is given by \nP ( δ, glyph[vector]η δ , x δ )d δ d 3 η δ d x δ = De -Q 3 Θ( x δ σ 2 -η 2 δ / 2)d δ d 3 η δ d x δ (4.20) \nwhere we have defined \nD ( δ ) = CJ = 6 √ 3 8 π 2 √ 2 π (1 -γ 2 ) σ 3 1 ∣ ∣ ∣ ∣ 1 2 δσ 0 ∣ ∣ ∣ ∣ (4.21) \nand \n2 Q 3 = 1 4 σ 2 0 ln 2 [ 4 ( x δ σ 2 -1 2 glyph[vector]η δ · glyph[vector]η δ ) 9 a 2 H 2 δ ] + 1 (1 -γ 2 ) { x δ -γ 2 σ 0 ln [ 4 ( x δ σ 2 -1 2 glyph[vector]η δ · glyph[vector]η δ ) 9 a 2 H 2 δ ]} 2 + 3 glyph[vector]η δ · glyph[vector]η δ σ 2 1 . (4.22) \nFinally, since the probability distribution is only a function of the modulus glyph[vector]η δ · glyph[vector]η δ = η 2 δ , one can change variable as d 3 η δ = η 2 δ sin θ δ d η δ d θ δ d φ δ and perform the integration on the angles which trivially results in \nP ( δ, η δ , x δ )d δ d η δ d x δ = 4 πη 2 δ De -Q 3 Θ( x δ σ 2 -η 2 δ / 2)d δ d η δ d x δ . (4.23) \nFinally we get \nβ th NG = 4 π ∫ δ c d δ ∫ ∞ 0 d η δ η 2 δ ∫ ∞ η 2 δ / 2 σ 2 d x δ D ( δ, x δ , η δ ) e -Q 3 . (4.24) \nThis is an exact result, no approximations have been made at this stage 9 .', '4.1 Spiky power spectrum': 'In the limit of γ glyph[similarequal] 1, i.e. for power spectra whose width is very narrow (typical of the PBHs), we can simplify our expressions dramatically. First of all, from Eq. (4.16) one sees that the distribution in x δ becomes a Dirac delta centered in x ∗ glyph[similarequal] ν . We then obtain \nP ( δ, η δ , x δ )d δ d η δ d x δ = 4 πη 2 δ Ee -Q 4 δ D ( x δ -1 2 σ 0 ln [ 4 ( x δ σ 2 -1 2 η 2 δ ) 9 a 2 H 2 δ ]) Θ( x δ σ 2 -η 2 δ / 2)d δ d η δ d x δ (4.25) \nwhere \nE = 6 √ 3 8 π 2 σ 3 1 1 2 δσ 0 and 2 Q 4 = 1 4 σ 2 0 ln 2 [ 4 ( x δ σ 2 -1 2 η 2 δ ) 9 a 2 H 2 δ ] + 3 η 2 δ σ 2 1 . (4.26) \nThen, to perform the integral in d η δ , we rewrite the Dirac delta as \nδ D ( x δ -1 2 σ 0 ln [ 4 ( x δ σ 2 -1 2 η 2 δ ) 9 a 2 H 2 δ ]) = δ D ( η δ -η c δ ) · ∣ ∣ ∣ ∣ ∣ 9 a 2 H 2 δσ 0 e -2 σ 0 x δ √ 8 σ 2 x δ -18 a 2 H 2 δe 2 σ 0 x δ ∣ ∣ ∣ ∣ ∣ , (4.27) \nwhere \nη c δ = √ 2 σ 2 x δ -9 2 a 2 H 2 δe 2 σ 0 x δ (4.28) \nand where we have chosen the positive root since η δ is always positive. The root imposes the condition \n2 σ 2 x δ -9 2 a 2 H 2 δe 2 σ 0 x δ > 0 , (4.29) \nFigure 3 . Mass fraction β th in the case of a spiky power spectrum as a function of A s , for the nonGaussian and the gaussian cases computed using the threshold statistics. \n<!-- image --> \nwhich is solved by ( W 0 and W -1 are the so-called principal and negative branches of the Lambert function) \nx -( δ ) = -1 2 σ 0 W 0 ( -9 a 2 H 2 σ 0 δ 2 σ 2 ) < x δ < -1 2 σ 0 W -1 ( -9 a 2 H 2 σ 0 δ 2 σ 2 ) = x + ( δ ) , (4.30) \nwith the requirement that 10 \n0 < δ < 1 e · 2 σ 2 9 a 2 H 2 σ 0 = δ + . (4.31) \nAfter integrating in d η δ , we find that the joint probability is \nP ( δ, x δ )d δ d x δ = 54 √ 3 8 π √ 2 a 3 H 3 σ 3 1 √ 4 x δ σ 2 a 2 H 2 -9 δe 2 σ 0 x δ × exp [ -1 2 x 2 δ +2 σ 0 x δ -3 a 2 H 2 4 σ 2 1 ( 4 x δ σ 2 a 2 H 2 -9 δe 2 σ 0 x δ ) ] d δ d x δ . (4.32) \nThis means that the threshold probability is \nβ th NG = ∫ δ + δ c d δ ∫ x + ( δ ) x -( δ ) d x δ 54 √ 3 8 π √ 2 a 3 H 3 σ 3 1 √ 4 x δ σ 2 a 2 H 2 -9 δe 2 σ 0 x δ × exp [ -1 2 x 2 δ +2 σ 0 x δ -3 a 2 H 2 4 σ 2 1 ( 4 x δ σ 2 a 2 H 2 -9 δe 2 σ 0 x δ ) ] , (4.33) \nwhere the higher extremum of integration in δ is due to (4.31). In Fig. 3 one can find the comparison of the gaussian and non-Gaussian mass functions computed using the threshold statistics for a spiky power spectrum. To proceed further and provide more analytical insights, we notice that the integration over x δ in Eq. (4.33) is highly dominated by the lower extremum of integration \nx -( δ ). As we show in Appendix C, the integrand in this region is very well approximated by \nβ th NG glyph[similarequal] 54 √ 6 8 π a 2 H 2 σ 1 / 2 2 σ 3 1 ∫ δ + δ c d δ √ 1 -2 σ 0 x -( δ ) exp [ x -( δ ) { 4 σ 0 σ 2 1 +6 σ 2 + x -( δ ) [ σ 2 1 -12 σ 0 σ 2 ]} 2 σ 2 1 ] · ∫ x + ( δ ) x -( δ ) d x δ √ x δ -x -( δ ) exp [ -3 σ 2 + x -( δ ) [ σ 2 1 -6 σ 0 σ 2 ] σ 2 1 x δ ] . (4.34) \nSince the integral in the second line is highly dominated by the lower extremum of integration, we can set x + ( δ ) →∞ and perform the integration analytically, obtaining (for γ glyph[similarequal] 1) \nβ th NG glyph[similarequal] 54 8 √ 3 2 π a 2 H 2 σ 2 ∫ 2 σ 2 9 a 2 H 2 σ 0 e δ c d δ √ 1 -2 σ 0 x -( δ ) { σ 0 x -( δ ) + 3 [1 -2 σ 0 x -( δ )] } 3 / 2 e -x -( δ ) [ x -( δ ) -4 σ 0 ] 2 . (4.35) \nIn Appendix C we show that this expression is extremely accurate for the case of a Dirac delta power spectrum of the curvature perturbation. \nWe can perform the final integral (4.35) by changing the variable of integration from δ to x -( δ ). The lower and higher extrema of integration then become, respectively, x -( δ c ) and 1 / 2 σ 0 . The integrand is highly dominated by the region around the lower extremum, so that we can send the higher extremum to infinity. We can also evaluate all the integrand, apart from the exponential factor exp( -x 2 -( δ ) / 2) at the lower extremum. The integral of this exponent can be then done analytically, and its result (the complementary error function) can be expanded in the limit of large argument. This leads to \nβ th NG glyph[similarequal] 6 √ 3 2 π (1 -2 σ 0 x c ) 3 / 2 2 x c (3 -5 σ 0 x c ) 3 / 2 e -x 2 c 2 , x c ≡ x -( δ c ) = -1 2 σ 0 W 0 ( -9 a 2 H 2 σ 0 δ c 2 σ 2 ) . (4.36) \nThe accuracy of this result is shown in Figure 10 of Appendix C, performed for the case of a Dirac delta power spectrum of the curvature perturbation, where it is compared with a two-dimensional numerical integration of the starting expression (4.33).', '4.2 Log-normal power spectrum': 'We assume again a power spectrum of the form \nP ζ ( k ) = A g √ 2 πσ exp [ -ln 2 ( k/k glyph[star] ) 2 σ 2 ] . (4.37) \nThen, one can integrate Eq. (4.24) numerically to get the mass fraction. In Fig. 4 we plot the beta for various values of σ as a function of A g .', '4.3 Broad power spectrum': 'We also consider a broad power spectrum, that is a top-hat with amplitude A t as \nP ζ ( k ) = A t Θ( k max -k ) Θ( k -k min ) (4.38) \nwhere Θ stands for the Heaviside step function and k max glyph[greatermuch] k min . Again, the parameters used are k max glyph[similarequal] 3 . 5 /r m , δ c = 0 . 51 [8] and, to disregard unphysical long wavelength modes, variances are obtained by choosing a m H m as the infrared cut-off. The results are presented in Fig. 5. \nWe conclude that threshold statistics confirms what we found in peak theory: independently from the power spectrum, non-Gaussian abundances are smaller than the gaussian ones. \nWe also see that the difference between the gaussian and the non-Gaussian cases in terms of the amplitude of the power spectrum is about a factor (2 ÷ 3), the same for the Dirac delta case. This is the shift one should adopt if insisting in using the gaussian expressions. \nFigure 4 . Mass fraction β th as a function of A g for the log-normal power spectrum (PS) computed using threshold statistics for both the gaussian and the non-Gaussian case. \n<!-- image --> \nFigure 5 . Mass fraction β th as a function of A t for the broad (top-hat) power spectrum computed using threshold statistics for both the gaussian and the non-Gaussian case. \n<!-- image -->', '5 Conclusions': 'In this paper we have discussed the impact of the non-Gaussianity arising from the non-linear relation between the density contrast and the curvature perturbation when dealing with PBH abundances. We have proposed two different methods to deal with such unavoidable and intrinsic non-Gaussianity, providing simple analytical expressions for the abundance to take it into account. \nThe first method is based on peak theory and on the realisation that the number of peaks in the overdensity is approximately equal to the number of peaks in the curvature perturbation as long as one restricts her/himself to those peaks having large spatial second derivatives at the peak location. \nThe second method relies on the threshold statistics and contains no approximations. Both methods show that the intrinsic non-Gaussianity makes it harder to generate PBHs. In particular, if one insists in adopting the gaussian expression for the abundance coming from threshold \nstatistics, one has simply to increase the amplitude of the power spectrum by a factor 11 O (2 ÷ 3). \nOur findings do not alleviate the differences between peak theory and threshold statistics in the computation of the abundance, already present at the gaussian level [12]. \nOur results can be surely improved along some directions. It would be important to have a full non-Gaussian extension of peak theory. More importantly, the intrinsic non-Gaussianity of the overdensity is expected to change the shape of the profile of the peaks which eventually give rise to PBHs upon collapse. Since the threshold δ c pk depends on the shape of the overdensity, such non-Gaussianity might change as well the value of δ c pk . We leave this study for a future publication [35].', 'Acknowledgments': 'We thank I. Musco for many fruitful discussions. We thank C. Byrnes, I. Musco and S. Young for sharing their draft [42] with us and for useful interactions. M.P. and C.U. thank Geneva Theoretical Physics group for their hospitality during this project. V. DL. thanks the Galileo Galilei Institute for Theoretical Physics (Florence, Italy) for the nice hospitality during the realisation of this project. A.R., V. DL., and G.F. are supported by the Swiss National Science Foundation (SNSF), project The non-Gaussian Universe and Cosmological Symmetries , project number: 200020-178787. C.U. is supported by European Structural and Investment Funds and the Czech Ministry of Education, Youth and Sports (Project CoGraDS- CZ.02.1.01/0.0/0.0/15 003/0000437) and would like to acknowledge networking support by the COST Action GWverse CA16104.', 'A The cumulants for a narrow power spectrum': 'In this Appendix we derive the relations (2.11) of the main text. We start from Eq. (1.7), that we need to expand as a power series of ζ . We denote by δ n the term that is of O ( ζ n ) \nδ 1 = -c glyph[star] ∂ i ∂ i ζ, δ n = c glyph[star] ( -1) n 2 n -1 ( n -1)! ζ n -2 ( ζ ∂ i ∂ i ζ -n -1 4 ∂ i ζ ∂ i ζ ) , n = 2 , 3 , 4 , . . . . (A.1) \nUsing the convention \nζ ( glyph[vector]x ) = ∫ d 3 p (2 π ) 3 e iglyph[vector]p · glyph[vector]x ζ ( glyph[vector] p ) , (A.2) \nfor the Fourier transform of the curvature perturbation, and symmetrizing over the momenta p i of the Fourier modes, the above relations can be cast in the form \nδ 1 (0) = c glyph[star] ∫ d 3 p (2 π ) 3 p 2 ζ ( glyph[vector] p ) , δ n (0) = c glyph[star] ( -2) n -1 n ! n ∏ k =1 [∫ d 3 p k (2 π ) 3 ζ ( glyph[vector] p k ) ] n ∑ i =1 p 2 i -1 2 n -1 ∑ i =1 n ∑ j = i +1 glyph[vector] p i · glyph[vector] p j , n = 2 , 3 , 4 , . . . . (A.3) \nWe are interested in computing the connected 2-, 3- and 4-point correlation functions of δ (0) = ∑ ∞ n =1 δ n , where by connected we mean terms that cannot be factorized as products of smallerorder correlation functions. Under the assumption of Gaussianity of the curvature ζ , all the correlators can be broken down to the products of the two-point function of ζ , \n〈 ζ ( glyph[vector] p ) ζ ( glyph[vector]q ) 〉 = P ζ ( p ) (2 π ) 3 δ (3) ( glyph[vector] p + glyph[vector]q ) = 2 π 2 p 3 P ζ ( p ) (2 π ) 3 δ (3) ( glyph[vector] p + glyph[vector]q ) . (A.4) \nThe practical effect of computing a connected, rather than a full, correlator is that some of the contractions are not included. To give just one example, we have \n〈 δ 2 2 (0) 〉 c = 〈 δ 2 2 (0) 〉 -〈 δ 2 (0) 〉 2 = c 2 glyph[star] ∫ d 3 p 1 d 3 p 2 d 3 q 1 d 3 q 2 (2 π ) 12 [ p 2 1 + p 2 2 -glyph[vector] p 1 · glyph[vector] p 2 2 ][ q 2 1 + q 2 2 -glyph[vector]q 1 · glyph[vector]q 2 2 ] 〈 ζ ( glyph[vector] p 1 ) ζ ( glyph[vector] p 2 ) ζ ( glyph[vector]q 1 ) ζ ( glyph[vector]q 2 ) 〉 c , (A.5) \nwith \n〈 ζ ( glyph[vector] p 1 ) ζ ( glyph[vector] p 2 ) ζ ( glyph[vector]q 1 ) ζ ( glyph[vector]q 2 ) 〉 c = 〈 ζ ( glyph[vector] p 1 ) ζ ( glyph[vector]q 1 ) 〉 〈 ζ ( glyph[vector] p 2 ) ζ ( glyph[vector]q 2 ) 〉 + 〈 ζ ( glyph[vector] p 1 ) ζ ( glyph[vector]q 2 ) 〉 〈 ζ ( glyph[vector] p 2 ) ζ ( glyph[vector]q 1 ) 〉 , (A.6) \nwith the omission of the 〈 ζ ( glyph[vector] p 1 ) ζ ( glyph[vector] p 2 ) 〉 〈 ζ ( glyph[vector]q 1 ) ζ ( glyph[vector]q 2 ) 〉 term. \nMore in general, we note that the first cumulants are related to the full correlators by \n〈 δ (0) 〉 c = 〈 δ (0) 〉 , 〈 δ 2 (0) 〉 c = 〈 δ 2 (0) 〉 -〈 δ (0) 〉 2 , 〈 δ 3 (0) 〉 c = 〈 δ 3 (0) 〉 -3 〈 δ (0) 〉 〈 δ 2 (0) 〉 +2 〈 δ (0) 〉 3 , 〈 δ 4 (0) 〉 c = 〈 δ 4 (0) 〉 -4 〈 δ (0) 〉 〈 δ 3 (0) 〉 -3 〈 δ 2 (0) 〉 2 +12 〈 δ (0) 〉 2 〈 δ 2 (0) 〉 -6 〈 δ (0) 〉 4 . (A.7) \nIt is worth noting that only the first cumulant is affected by the average of δ . In fact, the expressions (A.7) show that a shift δ → δ + C , where C is a constant, only affects the first cumulant, 〈 δ 〉 c →〈 δ 〉 c + C , while the higher cumulants are unchanged. \nWorking up to cubic order in the power of ζ , we compute \n〈 δ 2 (0) 〉 c = 〈 δ 2 1 (0) 〉 c + 〈 δ 2 2 (0) + 2 δ 1 (0) δ 3 (0) 〉 c + 〈 δ 2 3 (0) + 2 δ 2 (0) δ 4 (0) + 2 δ 1 (0) δ 5 (0) 〉 c , 〈 δ 3 (0) 〉 c = 3 〈 δ 2 1 (0) δ 2 (0) 〉 c + 〈 3 δ 2 1 (0) δ 4 (0) + 6 δ 1 (0) δ 2 (0) δ 3 (0) + δ 3 2 (0) 〉 c , 〈 δ 4 (0) 〉 c = 〈 6 δ 2 1 (0) δ 2 2 (0) + 4 δ 3 1 (0) δ 3 (0) 〉 c , (A.8) \nwhere we have kept together terms that are of the same order in P ζ . We note that the last expression does not contain the contraction of δ 4 1 (0) as it has no connected component. \nThe evaluations of the correlators in (A.8) is tedious, but straightforward. We expand the various terms according to (A.3) and we then split the correlators in sums of connected products \nof 〈 ζ ( glyph[vector] p ) ζ ( glyph[vector]q ) 〉 . Half of the integrals over momenta are then removed with the Dirac delta functions arising from Eq. (A.4). We divide the remaining half into integrals over the magnitude of the momenta and the angles. We encounter the following nontrivial angular integrals \n∫ dΩ ˆ p 1 dΩ ˆ p 2 dΩ ˆ p 3 (ˆ p 1 · ˆ p 2 ) 2 = 64 π 3 3 , ∫ dΩ ˆ p 1 dΩ ˆ p 2 dΩ ˆ p 3 (ˆ p 1 · ˆ p 2 ) (ˆ p 1 · ˆ p 3 ) = 0 , ∫ dΩ ˆ p 1 dΩ ˆ p 2 dΩ ˆ p 3 (ˆ p 1 · ˆ p 2 ) (ˆ p 1 · ˆ p 3 ) (ˆ p 2 · ˆ p 3 ) = 64 π 3 9 . (A.9) \nThe explicit evaluations then give \n〈 δ 2 (0) 〉 c = c 2 glyph[star] ∫ d p p 3 P ζ ( p ) + c 2 glyph[star] ∫ d p 1 p 1 d p 2 p 2 P ζ ( p 1 ) P ζ ( p 2 ) [ 4 p 4 1 +4 p 4 2 + 85 6 p 2 1 p 2 2 ] + c 2 glyph[star] ∫ d p 1 d p 2 d p 3 p 1 p 2 p 3 P ζ ( p 1 ) P ζ ( p 2 ) P ζ ( p 3 ) [ 32 3 ( p 4 1 + p 4 2 + p 4 3 ) + 415 9 ( p 2 1 p 2 2 + p 2 1 p 2 3 + p 2 2 p 2 2 ) ] , 〈 δ 3 (0) 〉 c = -6 c 3 glyph[star] ∫ d p 1 p 1 d p 2 p 2 P ζ ( p 1 ) P ζ ( p 2 ) p 2 1 p 2 2 [ p 2 1 + p 2 2 ] -c 3 glyph[star] ∫ d p 1 d p 2 d p 3 p 1 p 2 p 3 [ 46 ( p 2 1 + p 2 2 ) ( p 2 2 + p 2 3 ) ( p 2 1 + p 2 3 ) + 577 9 p 2 1 p 2 2 p 2 3 ] P ζ ( p 1 ) P ζ ( p 2 ) P ζ ( p 3 ) , 〈 δ 4 (0) 〉 c = c 4 glyph[star] ∫ d p 1 d p 2 d p 3 p 1 p 2 p 3 [ 16 ( p 4 1 p 4 2 + p 4 1 p 4 3 + p 4 2 p 4 3 ) +64 p 2 1 p 2 2 p 2 3 ( p 2 1 + p 2 2 + p 2 3 )] P ζ ( p 1 ) P ζ ( p 2 ) P ζ ( p 3 ) . (A.10) \nIn the case of a very narrow power spectrum of the curvature perturbation, that can be approximated by a Dirac delta function as in Eq. (2.10), these expressions give the results (2.11) reported in the main text.', 'B Spiky peaks in the curvature perturbation versus peaks in overdensity: a numerical treatment': 'We start from the relation between δ and ζ \nδ ( glyph[vector]x, t ) = -4 9 1 a 2 H 2 e -2 ζ ( glyph[vector]x ) ( ∇ 2 ζ ( glyph[vector]x ) + 1 2 ∂ i ζ ( glyph[vector]x ) ∂ i ζ ( glyph[vector]x ) ) ≡ 4 9 1 a 2 H 2 δ r ( glyph[vector]x, t ) . (B.1) \nOne can simulate numerically a realisation of the gaussian random field ζ ( glyph[vector]x ) in a n -dimensional box of dimensions N which is discretised using a grid of N n points with a spacing ∆ x = 1 between them in all directions. We choose to present the analysis in a 2-dimensional space ( n = 2) since the results can be more easily depicted. We set the parameters of the perturbation assuming a narrow power spectrum described by a log-normal function as \nP ζ ( k ) = 0 . 01 exp [ -ln 2 ( k/k glyph[star] ) 2 · 0 . 1 2 ] . (B.2) \nThe variance of the field turns out to be σ 2 0 = 2 . 5 · 10 -3 . The characteristic momentum has been chosen to be k glyph[star] = 0 . 2 / ∆ x . The realisation of the field ζ ( glyph[vector]x ) and the corresponding field δ r ( glyph[vector]x ) can be seen in Fig. 6. There the stars indicate the location of the spiky peaks in ζ and the peaks in δ r , showing the location correspondence. The color code is the same as in Fig. 7. \n<!-- image --> \nFigure 6 . A depiction of the two-dimensional simulation. Left: gaussian field ζ ( x, y ) . Right: density contrast δ r ( x, y ) found using the relation in Eq. (B.1) . The stars indicate the location of the spiky peaks in ζ and the peaks in δ r , showing the location correspondence. The color code is the same as in Fig. 7. \n<!-- image --> \nIn Fig. 7 one can find an analysis of the field values obtained in the simulation. More in detail, each point of the plot represents a peak in ζ with the corresponding values of the rescaled amplitude ν and the curvature x . The red, cyan and yellow lines correspond to lower bounds on x > x c δ ( ν ) in terms of the absolute maximum of the density contrast δ max in the simulation as \nx c δ ( ν ) = 9 a 2 H 2 4 σ 2 e 2 σ 0 ν δ max , (B.3) \nwith δ max = 0 . 4. This bound corresponds to the condition (3.15). With red, cyan and yellow dots we highlight the points which, at the same positions, have a peak in δ , with x satisfying the corresponding lower limits. Green dots are peaks in ζ as well, but they do not satisfy these conditions. This shows the correspondence between peaks of ζ and peaks of δ , provided the condition (3.15) is met. We expect that this correspondence will be even more satisfied when rarer events are simulated. We also checked that, by extending the simulation to three dimensions, and these findings are confirmed. \nThese results strongly indicates that, assuming condition (3.15), peaks in δ are located at the positions of peaks in ζ .', 'C Analytic integration of the PBH abundance for spiky power spectra using threshold statistics': 'In this appendix we derive the expressions (4.34) and (4.35) of the main text. We start from Eq. (4.33). One can verify that the integration over x δ of this equation is highly dominated by the lower extremum x -( δ ) (from now on, in this appendix, we do not write the dependence of x -on δ to shorten the notation). We therefore perform an expansion of the integrand for x δ glyph[similarequal] x -that allows us to perform the integration analytically. We expand the expression in the square root and in the exponent by linearising the exponential in x δ -x - \n4 x δ σ 2 a 2 H 2 -9 δe 2 σ 0 x δ glyph[similarequal] 4 x δ σ 2 a 2 H 2 -9 δe 2 σ 0 x -[1 + 2 σ 0 ( x δ -x -)] = 4 (1 -2 σ 0 x -) σ 2 a 2 H 2 ( x δ -x -) , (C.1) \nFigure 7 . A plot with field values of ν and x (corresponding to ζ and -∇ 2 ζ ) in a position of the grid. See the text for a more detailed explanation of the color code. All points are peaks in ζ , but only those spiky enough are also peaks of δ , as predicted. \n<!-- image --> \nwhere the second line has been obtained exploiting the fact that x -satisfies (exactly) δ e 2 σ 0 x -= 4 σ 2 x -/ 9 a 2 H 2 . We also approximate the first two terms in the exponent of Eq. (4.33) as \n-1 2 x 2 δ +2 σ 0 x δ glyph[similarequal] x -( x -+4 σ 0 ) 2 -x -x δ , (C.2) \nwhere we have linearised the first term on the left-hand side to first order in x δ -x -, while in the second term we simply put x δ = x -(since this term is highly subdominant). With these approximations, the expression (4.33) reduces to the form (4.34) written in the main text. \nThe integration over x δ in Eq. (4.34) is highly dominated by the lower extremum of integration, and we can set x + → ∞ . In this way the integration can be done analytically, leading to \nβ th NG glyph[similarequal] 18 8 1 √ 2 π a 2 H 2 σ 2 ( 3 σ 0 σ 2 σ 2 1 ) 3 / 2 ∫ δ + δ c d δ √ 1 -2 σ 0 x -( σ 0 x -+ 3 σ 0 σ 2 σ 2 1 (1 -2 σ 0 x -) ) 3 / 2 e -x -( x --4 σ 0 ) 2 (C.3) \nRecalling that these results are valid for γ ≡ σ 2 1 σ 0 σ 2 glyph[similarequal] 1 then leads to the expression (4.35) written in the main text. \nIn the case of a Dirac delta power spectrum of the curvature perturbation ζ , see Eq. (2.10), we have σ i = w √ A s k i glyph[star] , where w = W ( k glyph[star] , r m ). Recalling that k glyph[star] glyph[similarequal] (27 / 10) a m H m , the probability distribution reduces to \nP ( δ, x δ ) glyph[similarequal] 25 9 π √ 3 w 3 A 3 / 2 s √ 1 -ˆ x -e ( 12+8 w 2 As -11ˆ x -) ˆ x -8 w 2 As √ ˆ x -ˆ x -e -6 -5ˆ x -4 w 2 As ˆ x , (C.4) \nwhere on the right-hand side we have defined ˆ x ≡ 2 w √ A s x δ and ˆ x -≡ 2 w √ A s x -= -W 0 ( -50 δ/ 81) (which is the expression of the first root in eq. (4.30) in the present case). Figure 8 confirms the validity of this result. The probability in the figure is shown for ˆ x glyph[similarequal] ˆ x -glyph[similarequal] 0 . 54 (for the value of δ chosen in the figure), while ˆ x + glyph[similarequal] 1 . 67. We note that indeed this expression is highly dominated by the lower bound ˆ x glyph[similarequal] ˆ x -(this extends also for the values of ˆ x not shown in the figure). \nThe integration over x δ of this expression leads to \nβ th NG glyph[similarequal] ∫ 81 / 50 e δ c d δ ∫ x + x -d x δ P ( δ, x δ ) glyph[similarequal] 50 9 √ 3 π 1 w √ A s ∫ 81 / 50 e δ c d δ √ 1 -ˆ x -(6 -5 ˆ x -) 3 / 2 e -ˆ x 2 -8 w 2 As +ˆ x -, (C.5) \nFigure 8 . Validity of the analytic result (C.4), in the case of a Dirac delta power spectrum of ζ . We show the normalised probability ¯ P ≡ P/ ( 25 / 9 π √ 3 w 3 A 3 / 2 s ) for δ = 0 . 51 and A s = 6 · 10 -3 . \n<!-- image --> \n<!-- image --> \nFigure 9 . Validity of the analytic result (C.5) in the case of a Dirac delta power spectrum of ζ . This result is compared with the exact numerical integration of the expression (4.33). Left: we fix A s = 6 · 10 -3 and we vary δ . Right: we fix δ = 0 . 51 and we vary A s . \n<!-- image --> \nwhere we stress that ˆ x -depends on δ . The higher extremum of integration is the upper bound in Eq. (4.31) written in the present context. This result is extremely accurate, as we show in Figure 9. \nThe expression (C.5) can be integrated, proceeding as we did in the main text to obtain the result (4.36) from (4.35). We obtain \nβ th NG glyph[similarequal] 12 √ 3 π ( 1 -ˆ x c 6 -5ˆ x c ) 3 / 2 w √ A s ˆ x c e -ˆ x 2 c 8 w 2 As , ˆ x c ≡ ˆ x -( δ c ) = -W 0 ( -50 δ c 81 ) . (C.6) \nThis expression also follows immediately from (4.36), in the limit of Dirac delta power spectrum of the curvature perturbation, and noting that x c = ˆ x c / 2 σ 0 = ˆ x c / 2 w √ A s . The high accuracy of this result is shown in Figure 10, where we compare it with a fully numerical two-dimensional integration of the starting expression (4.33).', 'D Peaks versus thresholds': 'In the past literature PBHs have been identified either with peaks or with thresholds of the superhorizon overdensity, where by thresholds one means those regions in real space where the \n<!-- image --> \nFigure 10 . Validity of the analytic result (C.6) in the case of a Dirac delta power spectrum of ζ . This result is compared with the exact two-dimensional numerical integration of the expression (4.33). Left panel: we fix A s = 6 · 10 -3 and we vary δ c . Right panel: we fix δ c = 0 . 51 and we vary A s . \n<!-- image --> \nvalue of the density contrast is larger than a given threshold, in our case the critical value δ c . Regions characterised by large thresholds of the overdensity are indeed probable to be also local extrema. We first find the average threshold statistics profile δ ( r ) of the density contrast δ ( r ) at a given distance r from the point r = 0 (therefore without threshold) in the following way \nδ ( r ) = 〈 δ ( r ) | δ 0 > νσ δ 〉 = ∫ ∞ -∞ d δ ( r ) δ ( r ) P ( δ ( r ) | δ 0 > νσ δ ) , (D.1) \nwhere \nP ( δ ( r ) | δ 0 > νσ δ ) = P ( δ ( r ) , δ 0 > νσ δ ) P ( δ 0 > νσ δ ) (D.2) \nand δ 0 = δ (0). If both δ ( r ) and δ 0 are Gaussian variables, one can calculate the above quantity by recalling that P ( δ ( r ) , δ 0 ) is constructed in the standard way through the covariance matrix \nP ( δ ( r ) , δ 0 ) = 1 2 π √ det C exp ( -glyph[vector] δ T C -1 glyph[vector] δ/ 2 ) glyph[vector] δ T = ( δ 0 , δ ( r )) , C = ( σ 2 δ ξ 2 ( r ) ξ 2 ( r ) σ 2 δ ) , (D.3) \nwhere \nξ 2 ( r ) = 〈 δ ( glyph[vector]r ) δ ( glyph[vector] 0) 〉 (D.4) \nis the two-point correlator in coordinate space. From these expressions we derive \nP ( δ ( r ) , δ 0 > νσ δ ) = e -δ 2 ( r ) / 2 σ 2 δ 2 √ 2 πσ δ ( 1 + Erf [ ( ξ 2 ( r ) δ ( r ) -νσ 3 δ ) σ δ √ 2 det C ]) , P ( δ 0 > νσ δ ) = 1 2 Erfc ( ν/ √ 2 ) , (D.5) \nwhere Erfc( x ) is the complementary error function. Combining the different terms we finally get \nδ ( r ) = ξ 2 ( r ) σ δ √ 2 π e -ν 2 / 2 Erfc ( ν/ √ 2 ) . (D.6) \nUsing the expansion for large values of the argument \nErfc ( x glyph[greatermuch] 1) ≈ e -x 2 x √ π , (D.7) \nwe can finally evaluate the average δ ( r ) at distance r from the threshold for ν glyph[greatermuch] 1 \nδ ( r ) glyph[similarequal] ν ξ 2 ( r ) σ δ . (D.8) \nTaking ν = δ pk /σ δ one finds \nδ ( r ) = δ pk ξ 2 ( r ) σ 2 δ , (D.9) \nwhich is exactly the average profile derived in peak theory [11] 12 . This already suggests that large thresholds overdensity should correspond to extrema. To have further evidence, we follow Ref. [40] and consider the curvature of the large threshold regions. The mean value of the second derivative of δ ( r ) in any random direction at r = 0 is (by expanding the density contrast around the origin in powers of r and taking the mean value of it) with δ (0) = δ pk \n〈 d 2 δ ( r ) d r 2 ∣ ∣ ∣ ∣ r =0 〉 = δ pk σ 2 δ d 2 ξ 2 ( r ) d r 2 ∣ ∣ ∣ ∣ r =0 . (D.10) \nThe scatter of the second derivative from its mean value is found by averaging over all d 2 δ ( r ) / d r 2 | r =0 and δ pk , yet keeping δ (0) = δ pk , \nΣ 2 2 = 〈 [ d 2 δ ( r ) d r 2 ∣ ∣ ∣ ∣ r =0 -δ pk σ 2 δ d 2 ξ 2 ( r ) d r 2 ∣ ∣ ∣ ∣ r =0 ] 2 〉 = d 4 ξ 2 ( r ) d r 4 ∣ ∣ ∣ ∣ r =0 -1 σ 2 δ ( d 2 ξ 2 ( r ) d r 2 ∣ ∣ ∣ ∣ r =0 ) 2 . (D.11) \nWe then get \n∣ ∣ ∣ ∣ 1 Σ 2 〈 d 2 δ ( r ) d r 2 ∣ ∣ ∣ ∣ r =0 〉 ∣ ∣ ∣ ∣ ∼ ( δ pk σ 2 δ σ 2 δ r 2 m )( σ 2 δ r 4 m ) -1 / 2 = δ pk σ δ , (D.12) \nwhere we have taken ξ 2 (0) ∼ σ 2 δ and assumed that the profile varies over a characteristic scale r m . 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2017PhRvD..96b3012T

Determining the population properties of spinning black holes

2017-01-01

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There are at least two formation scenarios consistent with the first gravitational-wave observations of binary black hole mergers. In field models, black hole binaries are formed from stellar binaries that may undergo common envelope evolution. In dynamic models, black hole binaries are formed through capture events in globular clusters. Both classes of models are subject to significant theoretical uncertainties. Nonetheless, the conventional wisdom holds that the distribution of spin orientations of dynamically merging black holes is nearly isotropic while field-model black holes prefer to spin in alignment with the orbital angular momentum. We present a framework in which observations of black hole mergers can be used to measure ensemble properties of black hole spin such as the typical black hole spin misalignment. We show how to obtain constraints on population hyperparameters using minimal assumptions so that the results are not strongly dependent on the uncertain physics of formation models. These data-driven constraints will facilitate tests of theoretical models and help determine the formation history of binary black holes using information encoded in their observed spins. We demonstrate that the ensemble properties of binary detections can be used to search for and characterize the properties of two distinct populations of black hole mergers.

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https://arxiv.org/pdf/1704.08370.pdf

{'Determining the population properties of spinning black holes': 'Colm Talbot a \nSchool of Physics and Astronomy, Monash University, Clayton, Victoria 3800, Australia', 'Eric Thrane': 'School of Physics and Astronomy, Monash University, Clayton, Victoria 3800, Australia and OzGrav: The ARC Centre of Excellence for Gravitational-wave Discovery, Hawthorn, Victoria 3122, Australia (Dated: August 29, 2017) \nThere are at least two formation scenarios consistent with the first gravitational-wave observations of binary black hole mergers. In field models, black hole binaries are formed from stellar binaries that may undergo common envelope evolution. In dynamic models, black hole binaries are formed through capture events in globular clusters. Both classes of models are subject to significant theoretical uncertainties. Nonetheless, the conventional wisdom holds that the distribution of spin orientations of dynamically merging black holes is nearly isotropic while field-model black holes prefer to spin in alignment with the orbital angular momentum. We present a framework in which observations of black hole mergers can be used to measure ensemble properties of black hole spin such as the typical black hole spin misalignment. We show how to obtain constraints on population hyperparameters using minimal assumptions so that the results are not strongly dependent on the uncertain physics of formation models. These data-driven constraints will facilitate tests of theoretical models and help determine the formation history of binary black holes using information encoded in their observed spins. We demonstrate that the ensemble properties of binary detections can be used to search for and characterize the properties of two distinct populations of black hole mergers.', 'I. INTRODUCTION': "At present, merging black holes are the only directly detected source of gravitational waves [1-10]. A variety of mechanisms by which black hole binaries can form have been proposed. These mechanisms might yield significantly different distributions of the intrinsic parameters of binaries [11]. In this work we focus on the distribution of spin orientations to probe black hole binary formation mechanisms. We consider two mechanisms which are expected to dominate, the field and dynamical models (see, e.g., [12] for a detailed review). \nIn dynamical models, the binary forms when two black holes become gravitationally bound in dense stellar environments such as globular clusters [13]. Due to mass segregation such clusters arrange themselves with more massive objects being found in the center and less massive objects on the outside. This means that binaries are expected to have mass ratios close to unity [14]. It is expected that the spins of the two companions will be isotropically oriented [11]. \nThe distribution of spin orientations in field models is subject to more theoretical uncertainty (e.g., [15]). In field models, a stellar binary forms and the components of the binary then coevolve. Although such stars are expected to form with their angular momenta aligned with the total angular momentum of the binary, there are exceptions (e.g., [16, 17]). If binaries are formed with misaligned spins, tidal interactions and mass transfer processes between the stars can align the angular momenta \nof the stars with the total angular momentum of the binary (e.g., [18, 19]). When the first star explodes in a supernova and collapses to form a black hole, a natal kick may be imparted on the two companions due to asymmetry of the explosion (e.g., [20]), increasing misalignment between spin and angular momentum vectors. The subsequent evolution of the secondary, possibly involving a common envelope phase, can reverse this misalignment [21]. This is followed by the supernova of the secondary, which may give each black hole another kick and some additional degree of misalignment. The net effect is to leave the population of black hole spin orientations distributed about the angular momentum vector of the binary with some unknown typical misalignment angle [11, 22-24]. \nFollowing the formation of the black hole binary (either through dynamical capture or common evolution) the spin orientation of nonaligned spinning black holes changes due to precession. Isotropic spin orientation distributions are expected to remain isotropic throughout such evolution [25]. However, anisotropic distributions, such as those predicted by field models, may change significantly [26-28]. Here, we are interested in the distribution of spin orientations at the moment the binary enters LIGO's observing band. We therefore measure our spin orientations at f ref = 20Hz. Advanced LIGO's observing band will eventually extend down to 10 Hz, but we use 20 Hz here for the sake of convenience. One may use the spin orientation at f ref to reverse engineer the spin alignment distribution at the moment of formation, but this is not our present goal. \nIn this paper, we use Bayesian hierarchical modeling (e.g., [29]) and model selection to infer the parameters describing the distribution of spins of black hole bina- \nries. We construct a mixture model, which treats the fraction of dynamical mergers, the fraction of isolated binary mergers, and the typical spin misalignment of the primary and secondary black holes as free parameters. We apply the model to simulated data (including noise) to show that we can both detect the presence of distinct populations, and also measure hyperparameters describing typical spin misalignment. \nOur method builds on a body of research using gravitational waves to study the ensemble properties of compact binaries. In [30], it was shown that Bayesian model selection can be used to distinguish between formation channels using nonparametrized mass distributions. Clustering was used in [31] to show that model-independent statements about the existence of distinct mass subpopulations can be made with an ensemble of detections. In [32, 33], it was shown that the spin magnitude distribution can be used to determine whether observed merging black holes formed through hierarchical mergers of smaller black holes. Hierarchical merger models predict an isotropic distribution of black hole spin orientations since all binaries form through dynamical capture. \nVitale et al. [34] showed that model selection can be used to distinguish between models which predict mutually exclusive spin orientations of merging compact binaries, both binary black holes and neutron star black hole binaries. In order to generate two distinct populations with different spin distributions, binaries were generated with random spin angles. Those with tilt angles (between the black hole spin and the Newtonian orbital angular momentum) < 10 · were considered to be a fieldlike binary while those with tilt angles > 10 · were considered to be dynamiclike. The authors showed that, after ∼ 100 detections, one can recover the proportion of binaries in each population to within 10% at 1 σ . \nStevenson et al. [35] used Bayesian hierarchical modeling to recover the proportion of binaries taken from a set of four populations distributed according to astrophysically motivated, spin orientation distributions with fixed spin magnitudes ( a i = 0 . 7). Unlike [34], the populations overlap so that even precise knowledge of a binary's spin parameters does not provide certain knowledge about its parent population. Of the four populations, three are different distributions predicted by population synthesis models of isolated binary evolution and the fourth is the isotropic distribution predicted for dynamic formation. They achieve a similar result to Vitale et al. , measuring the relative proportion of different populations at the ∼ 10% level after 100 events. They also demonstrate that their two 'extreme hypotheses' (perfect alignment and isotropy) can be ruled out at > 5 σ after as few as five events if they are not good descriptions of nature. \n∼ \nWe build on these studies by employing a (hyper)parametrized model of the spin orientation distribution for the field model in order to measure not just the fraction of binaries from different populations, but also properties of the field model. In particular, we aim to measure the typical black hole misalignment for black \nhole binaries formed in the field. The advantage of this approach is that our modeling employs a broadly accepted idea from theoretical modeling (black holes in field binaries should be somewhat aligned) without assuming less certain details about the size of the misalignment. Since our model is agnostic with respect to the detailed physics of binary formation and subsequent evolution, the resulting methodology is robust against theoretical bias and provides a measurement of black hole spin misalignment for binaries formed in the field. \nThe remainder of the paper is organized as follows. In the next section we review how the properties of merging binary black holes are recovered from observed data and briefly discuss the current observational results. We then introduce a useful parametrization to describe an admixture of field and dynamical black hole mergers. We follow this with a description hierarchical inference. We then present the results of a proof-of-principle study using simulated data. We introduce a new tool for visualizing spin orientations, spin maps. Finally, closing thoughts are provided.", 'II. GRAVITATIONAL-WAVE PARAMETER ESTIMATION': "In order to determine the parameters describing the sources of gravitational waves Θ from gravitational-wave strain data h , we employ Bayesian inference. Merging binary black hole waveforms are described by 15 parameters: two masses { m 1 , m 2 } , two three-dimensional spin vectors { S 1 , S 2 } , and seven additional parameters to specify the position and orientation of the source relative to Earth. It is possible that in both the field and dynamical formation models the presence of a third companion will induce eccentricity when the binary enters LIGOs observing band through Lidov-Kozai cycles [3640]. However, we consider only circular binaries. Most gravitational-wave parameter estimation results obtained to date have been obtained using the Bayesian parameter estimation code LALInference [41]. For our study we use the LALInference implementation of nested sampling [42]. We employ reduced order modeling and reduced order quadrature [43] to limit the computational time of the analysis. \nPerforming parameter estimation over this 15dimensional space is computationally intensive. In order to maximize the efficiency sampling this highdimensional space, the effect of the two spin vectors on the waveform is approximately represented using two spin parameters [44], \nχ eff = a 1 cos( θ 1 ) + qa 2 cos( θ 2 ) 1 + q χ p =max ( a 1 sin( θ 1 ) , ( 4 q +3 4 + 3 q ) qa 2 sin( θ 2 ) ) . (1) \nHere ( a 1 , a 2 ) are the dimensionless spin magnitudes, q = m 1 /m 2 < 1 is the mass ratio and ( θ 1 , θ 2 ) are the angles \nbetween the spin angular momenta and the Newtonian orbital angular momentum of the binary. The variable χ eff is 'the effective spin parameter.' When χ eff > 0, the binary merges at a higher frequency than for χ eff = 0 and hence spends more time in the observing band [45]. Similarly, binaries with χ eff < 0 spend less time in the observing band. The variable χ p describes the precession of the binary, which is manifest as a long-period modulation of the signal [46]. \nUsing numerical relativity to compute all of the waveforms necessary for parameter estimation is computationally prohibitive. Parameter estimation therefore relies on 'approximants,' which can be used for rapid waveform estimation. We use the IMRPhenomP approximant [47], which has been used in many recent parameter estimation studies, including parameter estimation for recently observed binaries (e.g., [2, 7, 10]). IMRPhenomP approximates a generically precessing binary waveform using χ eff and χ p . Parameter estimation of the confirmed binary black hole detections, GW150914 [2, 5], GW151226 [3] and GW170104 [10], yield (slightly) informative posterior distributions for χ eff . However, the posterior distributions for χ p show no significant deviation from the prior. \nThe observed distribution of these two effective spin parameters will depend on the mass and spin magnitude distributions of black holes. The distributions are expected to differ for binaries formed through different mechanisms [22]. We do not consider these effects. Instead we work directly with the spin orientations of each black hole. For our purposes, it will be useful to define two additional variables: \nz 1 =cos( θ 1 ) z 2 =cos( θ 2 ) . (2) \nInstead of working with χ eff and χ p , we work with distributions of z 1 , z 2 . We note that z i ≈ 1 corresponds to aligned spin while z i ≈ -1 corresponds to antialigned spin and z i = 0 corresponds to black holes spinning in the orbital plane.", 'III. MODELS': 'For the purpose of this work we ignore the detailed formation history used in population synthesis studies. Instead, we introduce a simple parametrization designed to capture the salient features of the field and dynamic models. More sophisticated parametrizations are possible and will (eventually) be necessary to accurately describe realistic populations. However, we believe this is a suitable starting point given current theoretical uncertainty. \nWe hypothesize that the distribution of { z 1 , z 2 } can be approximated as an admixture of two populations. The first population is described by a truncated Gaussian peaked at ( z 1 , z 2 ) = (1 , 1) with width ( σ 1 , σ 2 ). This is our \nproxy for the population formed in the field. The Gaussian shape mimics the form of distributions predicted by population synthesis models, which are clustered about z = 1 with some unknown spread. The second population is uniform in ( z 1 , z 2 ), this represents the dynamically formed population. The relative abundances of each population are given by ξ (field) and 1 -ξ (dynamic). Thus, according to our parametrization, the true distribution of black hole mergers can be approximately described as follows: \np 0 ( z 1 , z 2 ) = 1 4 p 1 ( z 1 , z 2 ) = 2 π 1 σ 1 e -( z 1 -1) 2 / 2 σ 2 1 erf ( √ 2 /σ 1 ) 1 σ 2 e -( z 2 -1) 2 / 2 σ 2 2 erf ( √ 2 /σ 2 ) (3) \np ( z 1 , z 2 ) = (1 -ξ ) p 0 + ξp 1 (4) \nHere, p 0 ( z 1 , z 2 ) is the true dynamic-only distribution, p 1 ( z 1 , z 2 ) is the true field-only distribution, and p ( z 1 , z 2 ) is the true distribution for all black hole binaries. These distributions depend on three hyperparameters: two widths ( σ 1 , σ 2 ) and one fraction ξ . \nFor each of our population hyperparameters { σ 1 , σ 2 , ξ } , we choose uniform prior distributions between 0 and 1. For ξ this covers the full allowed range of values. For σ , this prior is chosen to be consistent with the most conservative estimates on spin misalignments predicted by field models (isotropically distributed kicks with the same velocity distribution as neutron stars, isotropic full kicks in [11]). In Fig. 1, we plot p 1 for various values of σ . \nThere are two interesting limiting cases. We note that p 1 ( z | σ ) → δ ( z -1) as σ → 0. This corresponds to perfect alignment of black hole spins. We also note that p 1 ( z | σ ) → p 0 as σ →∞ . Thus, depending on the choice of prior, the dynamical model is degenerate with the field model evaluated at one point in hyperparameter space. A consequence of this limiting behavior is that it is far more difficult to distinguish samples drawn from a broad aligned distribution ( σ = 1), than an almost perfectly aligned distribution ( σ = 0 . 01). It is simple to extend this model to include more terms describing additional subpopulations or alter the form of the existing terms to better fit physically motivated distributions.', 'IV. BAYESIAN HIERARCHICAL MODELING': "Bayesian hierarchical modeling involves splitting a Bayesian inference problem into multiple stages. In the case of merging compact binaries these steps are as follows: \n- i Perform gravitational-wave parameter estimation as described above. We adopt priors that are uniform in spin magnitude and isotropic in spin orientations.\n- ii Assume the population from which events are drawn is described by hyperparameters Λ. Calculate a like- \nFIG. 1. The distribution of z for our field model proxy with varying σ ; see Eq. 3. By sending σ → 0, we obtain perfect alignment and by sending σ → ∞ , we obtain an isotropic distribution. \n<!-- image --> \nihood function for the data given Λ by marginalizing over the parameters for individual events Θ. \n- iii Combine multiple events to derive a joint likelihood for Λ.\n- iv Use the joint likelihood to derive posterior distributions for Λ, which, in turn, may be used to construct Bayes factors or odds ratios comparing different population models and confidence intervals on hyperparameters. \nStep (i) produces a set of n k posterior samples { Θ i } , sampled according to the likelihood of the binary having each set of parameters, p (Θ | h ). This step is computationally expensive and requires the application of a specialized tool such as LALInference . In Step (ii), we estimate Λ using the posterior samples { z i } . Our likelihood requires marginalization over z , for each event. Since LALInference approximates the posterior for Θ with a list of posterior sample points, the marginalization integral over ( z 1 , z 2 ) can be approximated by summing the probability of each sample in the LALInference posterior chain for our population model (see, e.g., [48, Chapter 29] for details). \nStep (iii): To combine data from N events, we multiply the likelihoods: \nL k ( h k | Λ) = ∫ dz 1 dz 2 p ( z 1 , z 2 | h k ) p ( z 1 , z 2 | Λ) = 1 n k n k ∑ α =1 p ( z α 1 , z α 2 | Λ) (5) \nL ( { h k }| Λ) = N ∏ k =1 L k ( h k | Λ) . (6) \nHere, L k ( h k | Λ) is the likelihood function for the k th event with strain data h k . The joint likelihood function \nL ( { h k }| Λ) combines data from all N measurements to arrive at the best possible constraints on Λ. \nStep (iv): At last, we arrive at the posterior distribution for Λ, p (Λ |{ h k } ). Combining the joint likelihood L ( { h k }| Λ) with a prior distribution for the hyperparameters Λ, π (Λ | H ), for a particular population model, H , we obtain \np (Λ |{ h k } ) = L ( { h k }| Λ) π (Λ | H ) Z ( { h k }| H ) = π (Λ | H ) Z ( { h k }| H ) N ∏ k =1 1 n k n k ∑ α =1 p ( z α 1 , z α 2 | Λ) ∝ N ∏ k =1 n k ∑ α =1 p ( z α 1 , z α 1 | Λ) . (7) \nHere, Z ( { h k }| H ) is the Bayesian evidence for the data from N observations { h k } , for a model H , which is given by marginalizing over the hyperprior space \nZ ( { h k }| H ) = ∫ d Λ L ( { h k }| Λ , H ) π (Λ | H ) . (8) \nFrom our (hyper)posterior distribution p (Λ |{ h k } ), we construct confidence intervals for our hyperparameters. \nThe odds ratio of two models is: \nO i j = Z ( { h k }| H i ) p ( H i ) Z ( { h k }| H j ) p ( H j ) . (9) \nWe use the odds ratio to select between different models. Here, the p ( H i ) are the prior probabilities assigned to each model. In our study, we assign equal probabilities to each model. Thus, the odds ratio is equivalent to the Bayes factor: \nB i j = Z ( { h k }| H i ) Z ( { h k }| H j ) . (10) \nWe impose a somewhat arbitrary, but commonly used threshold of | ln( B ) | > 8 ( ∼ 3 . 6 σ ) to define the point at which one model is significantly preferred over another. \nNow that we have derived a number of statistical tools, it is worthwhile to pause and consider what astrophysical questions we can answer with them. \n- i. If p ( σ 1 , σ 2 |{ h k } ) excludes σ 1 = σ 2 = ∞ , then it necessarily follows that p ( ξ |{ h k } ) excludes ξ = 0, and we may infer that at least some binaries merge through fieldlike models.\n- ii. If p ( ξ |{ h k } ) excludes ξ = 1, we may infer that not all binaries can be formed via fieldlike models.\n- iii. If both ξ = 0 and ξ = 1 are excluded, then we may infer the existence of at least two distinct populations.\n- iv. If the ( σ 1 , σ 2 ) posterior distribution p ( σ 1 , σ 2 |{ h k } ) excludes σ 1 = σ 2 = 0, we may infer that not all binaries are perfectly aligned. \nIn this way we can distinguish between different formation channels or specific models, i.e., perfect alignment in case (iv). \nWe employ Bayes factors to compare our population models. We calculate evidences for three hypotheses: \n- i. Z dyn - Dynamic formation only, ξ = 0.\n- ii. Z field - Field formation only, ξ = 1.\n- iii. Z mix - Mixture of field and dynamic, ξ ∈ [0 , 1]. \nWe then define three Bayes' factors to compare these three hypotheses: \n- i. B mix field = Z mix /Z field .\n- ii. B mix dyn = Z mix /Z dyn .\n- iii. B field dyn = Z field /Z dyn . \nIn the next section, we apply these tools to a variety of simulated data sets in order to show under what circumstances we can measure various hyperparameters and carry out model selection.", 'V. SIMULATED POPULATION STUDY': "We use a simulated population to test our models. For the sake of simplicity, we construct a somewhat contrived population in which every binary shares some parameters corresponding to the best-fit parameters of GW150914: \n- · ( m 1 , m 2 ) = (35 M glyph[circledot] , 30 M glyph[circledot] ).\n- · d L = 410Mpc.\n- · ( a 1 , a 2 ) = (0 . 6 , 0 . 6). \nHere, d L is luminosity distance and ( a 1 , a 2 ) are the black hole spin magnitudes. The remaining extrinsic parameters (sky position and source orientation) are sampled from isotropic distributions. We emphasize that the distance and mass and spin magnitude distributions are not representative of the full population of black hole binaries, which is poorly constrained. These distributions represent a subset of GW150914-like events, chosen for illustrative purposes. In reality, for every GW150914-like event, there are likely to be a large number of more distant (and possibly lower mass) events, which contribute relatively less information about spin. \nWe inject 160 binary merger signals into simulated Gaussian noise corresponding to Advanced LIGO at design sensitivity [49, 50]. Of these, we generate 80 distributed according to p 0 and 80 distributed according to p 1 ; see Eq. (3). The injected values of ( z 1 , z 2 ) are shown in Fig. 2. The red diamonds correspond to the p 0 dynamical model and the blue circles to the p 1 fieldlike model. From these we construct 'universes' summarized in Table I. Each universe contains a different mixture of field and dynamical binaries. In every universe, ( σ 1 , σ 2 ) = (0 . 3 , 0 . 5). \nTABLE I. Hyperparameters describing different simulated universes. Here, ξ is the proportion drawn from our aligned model and ( σ 1 , σ 2 ) describe the typical misalignment angle; see Eq. (3). \n| Universe | ξ | σ 1 | σ 2 |\n|------------|-----|-------|-------|\n| A | 0 | N/A | N/A |\n| B | 0.1 | 0.3 | 0.5 |\n| C | 0.5 | 0.3 | 0.5 |\n| D | 0.9 | 0.3 | 0.5 |\n| E | 1 | 0.3 | 0.5 | \nFIG. 2. Simulated spin misalignment parameters ( z 1 , z 2 ) for the different populations of binary black holes used in our study. Red diamonds are drawn from the isotropic distribution p 0 while the blue circles are drawn from the aligned distribution, p 1 ( z 1 , z 2 | σ 1 , σ 2 = 0 . 3 , 0 . 5); see Eq. (3). \n<!-- image --> \nFor each universe, we present the results of the methods described above. In Fig. 3, we plot the 1 σ (dark), 2 σ (lighter), and 3 σ (lightest) confidence regions as a function of the number of GW150914-like events. In Fig. 4, we plot the three Bayes factors defined in Eq. (10) as a function of the number of GW150914-like events. Each row in Fig. 3 and panel in Fig. 4 represents a different universe. \nFirst we consider universe A, consisting of only dynamically formed binaries, ξ = 0; see the top row of Fig. 3. Since all binaries form dynamically in this universe, σ is undefined. We see that after O (1) event we rule out ξ = 1 at 3 σ (the hypothesis that all binaries form in the field). \nNext we consider universe E in which all events are drawn from the aligned model, ξ = 1; see the bottom row of Fig. 3 and the bottom panel of Fig. 4. For this universe, σ 1 = 0 . 3, σ 2 = 0 . 5. We rule out ξ = 0 (dynamical only) at 3 σ after O (1) event. The Bayes factors also rule out all binaries forming dynamically after glyph[lessorsimilar] 10 events. \nThe threshold | ln( B ) | = 8 is shown by the dashed line. After 80 events, the 1 σ confidence intervals for σ 1 and σ 2 have shrunk to ∼ 30% and the 1 σ confidence interval for ξ has shrunk to 3%. The Bayes factor comparing the two-population hypothesis to the purely field hypothesis B mix field (the blue line in the bottom panel of Fig. 4) does not strongly favor field-only formation. \nUniverses B, C and D are mixtures of the field and dynamical populations. Of these, B and D have only 10% drawn from the subdominant population. We recover marginally weaker constraints than the corresponding single population universes. The hypothesis that all binaries form through the dominant mechanism is disfavored at 1 σ after a few tens of events for universes B and D, establishing a weak preference for the presence of two distinct populations. For some realizations we can rule out both one component models after 80 events, however generally we see a subthreshold preference for the mixture model. This is unsurprising since each one-population model is a subset of our two-population model. For universe C, an equal mixture of events drawn from the field and dynamical populations. Both ξ = 0 and ξ = 1 are excluded at 3 σ after tens of events establishing the presence of two distinct subpopulations. \nFor all five universes, the presence of a perfectly aligned component ( σ = 0) is excluded after fewer than 20 events. For many realizations this number is < 5. For universes B, C and D (consisting of a mixture of field and dynamical mergers), we can rule out the entire population forming from one of the two channels after 10-40 GW150914like events. When there is a large contribution from the aligned model, we observe that the allowed region for σ 1 becomes small faster than the allowed region for σ 2 . There are two effects, which explain this. First, the secondary black hole's spin has a less significant effect on the waveform [51-53]. The spin orientation of the secondary is therefore less well constrained for each event. This translates to a larger uncertainty for σ 2 compared to σ 1 . Second, the width of the distribution of spin tilts is broader for the secondary black holes. This broader distribution is intrinsically more difficult to resolve.", 'VI. SPIN MAPS': "In addition to our hierarchical analysis, we present a visualization tool for the distribution of spin orientations. We introduce 'spin maps': histograms of posterior spin orientation probability density, averaged over many events, and plotted using a Mollweide projection of the sphere defining the spin orientation, see Fig. 5. The maps use HEALPix [54]. For each posterior sample the latitude is the spin tilt of the primary black hole, θ 1 , and the longitude the difference in azimuthal angles of the two black holes, ∆Φ. The difference in azimuthal angles may give information about the history of the binary, specifically by identifying spin-orbit resonances at ∆Φ = 0 , π [2628, 55-57]. These resonances, if detected, would appear \nas bands of constant longitude. We do not utilize azimuthal angle in this work and our injected distributions are isotropic in ∆Φ. In the future, it would also be interesting to produce ensemble spin disk plots (e.g., Fig. 5 of [2]), showing the spin magnitude and orientation for a population of binaries. \nThe spin maps in Fig. 5 include contributions from 80 events for universes A and C (see Table I). This simple representation is useful because it provides qualitative insight into the distribution of spins and helps us to see trends and patterns that might not be obvious from our likelihood formalism. The north pole on these maps corresponds to spin aligned with the total angular momentum of the binary. We see the preference for the spin to be aligned with the angular momentum vector of the binary by the clustering in the northern hemisphere.", 'VII. DISCUSSION': "The physics underlying the formation of black hole binaries is poorly constrained both theoretically and observationally. We do not know which of the proposed mechanisms is the main source of binary mergers: preferentially aligned mergers formed in the field versus randomly aligned mergers formed dynamically. We are also not confident in the predicted characteristics of binaries formed through either channel. We therefore create a simple (hyper)parametrization, describing the ensemble properties of black hole binaries. We demonstrate that we can measure hyperparameters describing the spin properties of an ensemble of black hole mergers with multiple populations. Previous work by Vitale et al. [34] and Stevenson et al. [35] demonstrated that the fraction of binaries drawn from different populations can be inferred after O(10) events. We show that after a similar number of events, the shape of the spin-orientation distribution can be inferred using a simple hyperparametrization. We reproduce the finding from Stevenson et al. , that O (1) event is required to distinguish an isotropically oriented distribution, ξ = 0, from a perfectly aligned distribution, ξ = 1, σ 1 = σ 2 = 0. After fewer than 40 GW150914-like events we can determine the properties of the dominant formation mechanism for all of our considered scenarios. We also introduce the concept of spin maps, which provide a tool for visualizing the distribution of spin orientations from an ensemble of detections. \nOne limitation of our study is that, for the sake of simplicity, we employ a population of binaries with masses, distance, and spins fixed to values consistent with GW150914. The advantage of this simple model is that we are able to isolate the effect of spin orientation by holding other parameters fixed. The disadvantage is that the GW150914-like population is not a realistic description of nature. By changing from a population of binaries at a fixed distance to a population distributed uniformly in comoving volume, more events will be required for measurement of population hyperparameters. \nFIG. 3. In each panel we plot 1 σ (dark shading), 2 σ (medium shading), and 3 σ (light shading) confidence for different hyperparameters as a function of the number of events N . Each column represents a different hyperparameter: σ 1 (left), σ 2 (middle), and ξ (right). Each row represents a different universe; see Table I. From top to bottom, the universes are A, B, C, D, and E. The dashed line indicates the true hyperparameter values. The highest likelihood values of the three parameters after 80 events are shown on each panel along with the width of the 1 σ confidence interval. \n<!-- image --> \nFIG. 4. Log Bayes factors as a function of the number of GW150914-like events. The dot-dashed red line shows B field dyn comparing the pure-field hypothesis to the pure-dynamical hypothesis. The dashed green line shows B mix dyn comparing the twopopulation hypothesis to the dynamical hypothesis. The solid blue line shows B mix field comparing the two-population hypothesis to the pure-field hypothesis. The dashed lines denotes | ln( B ) | = 8, our threshold for distinguishing between models. Each panel is a different universe. The top panel is universe C (equal mixture of field and dynamic). With glyph[lessorsimilar] 40 events, there is a strong preference for the two-component hypothesis over the pure-dynamic hypothesis. After ∼ 50 events there is a preference for the two-component hypothesis over the pure-field hypothesis. The center panel is universe D (majority field with some dynamic). With glyph[lessorsimilar] 10 events, there is a strong preference for the two-component and pure-field hypotheses over the pure-dynamic hypothesis. There is a preference for the correct two-population hypothesis over the pure-field hypothesis. The bottom panel is universe E (pure field). With glyph[lessorsimilar] 10 events, there is a strong preference for the two-component and field hypotheses over the dynamic hypothesis. There is a marginal preference for the correct field hypothesis over the two-population hypothesis. \n<!-- image --> \nFIG. 5. Spin maps: maps of posterior spin orientation probability density averaged over many realizations. The latitude is the spin tilt of the primary (more massive) black hole. The longitude is the angle between the projection of the black hole spins onto the orbital plane. The color bar is the number of posterior samples per 5 deg 2 HEALPix bin. The left panel shows a spin map for 80 events drawn from universe A (see, Table I) in which every binary merges dynamically. The right panel shows 80 events drawn from universe C drawn in which, on average, 50% of the events are drawn from the dynamical population while 50% are drawn from the field population with ( σ 1 , σ 2 ) = (0 . 3 , 0 . 5); see, Eq. 3. The presence of a preferentially oriented population is seen as clustering around the north pole. \n<!-- image --> \nThis is because most events, coming from the edge of the visible volume, will contribute only marginally to our knowledge of these hyperparameters. We assume fixed spin magnitudes of a 1 = a 2 = 0 . 6. For a binary with aligned spins, this would imply χ eff = 0 . 6. Based on recent LIGO detections, this might be optimistic. For GW151226, χ eff = 0 . 21 +0 . 20 -0 . 10 . For all other observed events, χ eff is consistent with 0. This implies either that the observed black holes are not spinning rapidly or that the merging black holes observed so far possess significantly misaligned spins [6, 10, 58]. 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2014EPJC...74.2836M

<inline-formula id="IEq1"><mml:math><mml:mi>P</mml:mi></mml:math></inline-formula>–<inline-formula id="IEq2"><mml:math><mml:mi>V</mml:mi></mml:math></inline-formula> criticality of topological black holes in Lovelock–Born–Infeld gravity

2014-01-01

['-', 'black hole physics', '-', '-', '-', '-']

[]

To understand the effect of third order Lovelock gravity, <inline-formula id="IEq5"><mml:math><mml:mi>P</mml:mi></mml:math></inline-formula>–<inline-formula id="IEq6"><mml:math><mml:mi>V</mml:mi></mml:math></inline-formula> criticality of topological AdS black holes in Lovelock–Born–Infeld gravity is investigated. The thermodynamics is further explored with some more extensions and in some more detail than the previous literature. A detailed analysis of the limit case <inline-formula id="IEq7"><mml:math><mml:mrow><mml:mi mathvariant="italic">β</mml:mi><mml:mo stretchy="false">→</mml:mo><mml:mi>∞</mml:mi></mml:mrow></mml:math></inline-formula> is performed for the seven-dimensional black holes. It is shown that, for the spherical topology, <inline-formula id="IEq8"><mml:math><mml:mi>P</mml:mi></mml:math></inline-formula>–<inline-formula id="IEq9"><mml:math><mml:mi>V</mml:mi></mml:math></inline-formula> criticality exists for both the uncharged and the charged cases. Our results demonstrate again that the charge is not the indispensable condition of <inline-formula id="IEq10"><mml:math><mml:mi>P</mml:mi></mml:math></inline-formula>–<inline-formula id="IEq11"><mml:math><mml:mi>V</mml:mi></mml:math></inline-formula> criticality. It may be attributed to the effect of higher derivative terms of the curvature because similar phenomenon was also found for Gauss–Bonnet black holes. For <inline-formula id="IEq12"><mml:math><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math></inline-formula>, there would be no <inline-formula id="IEq13"><mml:math><mml:mi>P</mml:mi></mml:math></inline-formula>–<inline-formula id="IEq14"><mml:math><mml:mi>V</mml:mi></mml:math></inline-formula> criticality. Interesting findings occur in the case <inline-formula id="IEq15"><mml:math><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math></inline-formula>, in which positive solutions of critical points are found for both the uncharged and the charged cases. However, the <inline-formula id="IEq16"><mml:math><mml:mi>P</mml:mi></mml:math></inline-formula>–<inline-formula id="IEq17"><mml:math><mml:mi>v</mml:mi></mml:math></inline-formula> diagram is quite strange. To check whether these findings are physical, we give the analysis on the non-negative definiteness condition of the entropy. It is shown that, for any nontrivial value of <inline-formula id="IEq18"><mml:math><mml:mi mathvariant="italic">α</mml:mi></mml:math></inline-formula>, the entropy is always positive for any specific volume <inline-formula id="IEq19"><mml:math><mml:mi>v</mml:mi></mml:math></inline-formula>. Since no <inline-formula id="IEq20"><mml:math><mml:mi>P</mml:mi></mml:math></inline-formula>–<inline-formula id="IEq21"><mml:math><mml:mi>V</mml:mi></mml:math></inline-formula> criticality exists for <inline-formula id="IEq22"><mml:math><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math></inline-formula> in Einstein gravity and Gauss–Bonnet gravity, we can relate our findings with the peculiar property of third order Lovelock gravity. The entropy in third order Lovelock gravity consists of extra terms which are absent in the Gauss–Bonnet black holes, which makes the critical points satisfy the constraint of non-negative definiteness condition of the entropy. We also check the Gibbs free energy graph and "swallow tail" behavior can be observed. Moreover, the effect of nonlinear electrodynamics is also included in our research.

[]

https://arxiv.org/pdf/1401.0785.pdf

{'P -V Criticality of Topological Black Holes in Lovelock-Born-Infeld Gravity': "Jie-Xiong Mo a,b ∗ , Wen-Biao Liu a † a Department of Physics, Institute of Theoretical Physics, Beijing Normal University, Beijing, 100875, China b Institute of Theoretical Physics, Zhanjiang Normal University, Zhanjiang, 524048, China \nTo understand the effect of third order Lovelock gravity, P -V criticality of topological AdS black holes in Lovelock-Born-Infeld gravity is investigated. The thermodynamics is further explored with some more extensions and details than the former literature. A detailed analysis of the limit case β →∞ is performed for the seven-dimensional black holes. It is shown that for the spherical topology, P -V criticality exists for both the uncharged and charged cases. Our results demonstrate again that the charge is not the indispensable condition of P -V criticality. It may be attributed to the effect of higher derivative terms of curvature because similar phenomenon was also found for Gauss-Bonnet black holes. For k = 0, there would be no P -V criticality. Interesting findings occur in the case k = -1, in which positive solutions of critical points are found for both the uncharged and charged cases. However, the P -v diagram is quite strange. To check whether these findings are physical, we give the analysis on the non-negative definiteness condition of entropy. It is shown that for any nontrivial value of α , the entropy is always positive for any specific volume v . Since no P -V criticality exists for k = -1 in Einstein gravity and Gauss-Bonnet gravity, we can relate our findings with the peculiar property of third order Lovelock gravity. The entropy in third order Lovelock gravity consists of extra terms which is absent in the Gauss-Bonnet black holes, which makes the critical points satisfy the constraint of non-negative definiteness condition of entropy. We also check the Gibbs free energy graph and the 'swallow tail' behavior can be observed. Moreover, the effect of nonlinear electrodynamics is also included in our research. \nPACS numbers: 04.70.Dy, 04.70.-s", 'I. INTRODUCTION': "Gravity in higher dimensions has attained considerable attention with the development of string theory. Concerning the effect of string theory on gravitational physics, one may construct a low energy effective action which includes both the Einstein-Hilbert Lagrangian (as the first order term) and higher curvature terms. However, this approach may lead to field equations of fourth order and ghosts as well. This problem has been solved by a particular higher curvature gravity theory called Lovelock gravity [1]. The field equation in this gravity theory is only second order and the quantization of Lovelock gravity theory is free of ghosts [2]. In this context, it is of interest to investigate both the black hole solutions and their thermodynamics in Lovelock gravity [3]-[29]. Moreover, it is natural to consider the nonlinear terms in the matter side of the action while accepting the nonlinear terms on the gravity side [3]. Motivated by this, Ref. [3] presented topological black hole solutions in Lovelock-Born-Infeld gravity. Both the thermodynamics of asymptotically flat black holes for k = 1 and the thermodynamics of asymptotically AdS rotating black branes with flat horizon were detailedly investigated there. However, concerning the charged topological AdS black holes in Lovelock-Born-Infeld gravity, only the temperature was given in Ref. [3]. Ref. [18] further studied their entropy and specific heat at constant charge. However, the expression of entropy seems incomplete for k is missing. And the thermodynamics in the extended space needs to be further explored. Probing this issue is important because it is believed that the physics of black holes in higher dimensions is essential for one to understand a full theory of quantum gravity. \nAs is well known, phase transition is a fascinating phenomenon in the classical thermodynamics. Over the past decades, phase transitions of black holes have aroused more and more attention. The pioneer phase transition research of AdS black holes can be traced back to the discovery of the famous Hawking-Page phase transition between the Schwarzschild AdS black hole and thermal AdS space [30]. Recently, a revolution in this field is led by P -V criticality research [31]-[44] in the extended phase space. Kubizˇn'ak et al. [31] perfectly completed the analogy between charged AdS black holes and the liquid-gas system first observed by Chamblin et al. [45, 46]. The approach of treating the \nL 2 \ncosmological constant as thermodynamic pressure and its conjugate quantity as thermodynamic volume is essential with the increasing attention of considering the variation of the cosmological constant in the first law of black hole thermodynamics recently [47]-[53] . \nHere, we would like to investigate the thermodynamics and phase transition of charged topological AdS black holes in Lovelock-Born-Infeld gravity in the extended phase space. Some related efforts have been made recently. P -V criticality of both four-dimensional [32] and higher dimensional [44] Born-Infeld AdS black holes have been investigated. A new parameter called Born-Infeld vacuum polarization was defined to be conjugated to the BornInfeld parameter [32]. And it was argued that this quantity is required for the consistency of both the first law of thermodynamics and the Smarr relation. Moreover, Cai et al. [40] studied the P -V criticality of Gauss-Bonnet AdS black holes. It was found that no P -V criticality can be observed for Ricci flat and hyperbolic Gauss-Bonnet black holes. However, for the spherical case, P -V criticality can be observed even when the charge is absent, implying that the charge may not be the indispensable factor for the existence of P -V criticality. Such an interesting result motivates us to probe further the third order Lovelock gravity to explore whether it is a peculiar property due to the higher derivative terms of curvature. So we would mainly investigate their effects on the P -V criticality. Moreover, we will probe the combined effects of higher derivative terms of curvature and the nonlinear electrodynamics. \nIn Sec. II, the solutions of charged topological AdS black holes in Lovelock-Born-Infeld gravity will be briefly reviewed and their thermodynamics will be further investigated. In Sec. III, a detailed study will be carried out in the extended phase space for the limit case β →∞ so that we can concentrate on the effects of third order Lovelock gravity. In Sec. IV, the effects of nonlinear electrodynamics will also be included. In the end, a brief conclusion will be drawn in Sec. V.", 'II. THERMODYNAMICS OF CHARGED TOPOLOGICAL BLACK HOLES IN LOVELOCK-BORN-INFELD GRAVITY': 'The action of third order Lovelock gravity with nonlinear Born-Infeld electromagnetic field is [3] \nwhere \nI G = 1 16 π ∫ d n +1 x √ -g ( -2Λ + L 1 + α 2 L 2 + α 3 L 3 + L ( F ) ) , (1) \n1 = R, (2) \nL = R µνγδ R µνγδ -4 R µν R µν + R 2 , (3) L 3 = 2 R µνσκ R σκρτ R ρτ µν +8 R µν σρ R σκ ντ R ρτ µκ +24 R µνσκ R σκνρ R ρ µ \nL ( F ) = 4 β 2 ( 1 -√ 1 + F 2 2 β 2 ) . (5) \n+3 RR µνσκ R σκµν +24 R µνσκ R σµ R κν +16 R µν R νσ R σ µ -12 RR µν R µν + R 3 , (4) \nIn the above action, β , α 2 and α 3 are Born-Infeld parameter, the second and third order Lovelock coefficients respectively while L 1 , L 2 , L 3 and L ( F ) are Einstein-Hilbert, Gauss-Bonnet, the third order Lovelock and Born-Infeld Lagrangians respectively. Considering the case \nα 2 = α ( n -2)( n 3) , (6) \nRef. [3] derived the ( n +1)-dimensional static solution as \n-α 3 = α 2 72 ( n -2 4 ) , (7) \nds 2 = -f ( r ) dt 2 + dr 2 f ( r ) + r 2 d Ω 2 , (8) \n- \nwhere \nwhere \nη = ( n -1)( n -2) q 2 2 β 2 r 2 n -2 . (13) \nThe Hawking temperature has been derived in Ref. [3] as \nT = ( n -1) k [3( n -2) r 4 + +3( n -4) kαr 2 + +( n -6) k 2 α 2 ] + 12 r 6 + β 2 (1 -√ 1 + η + ) -6Λ r 6 + 12 π ( n -1) r + ( r 2 + + kα ) 2 . (14) \nHowever, only the Hawking temperature is not enough to discuss the P -V criticality in the extended phase space. So we would like to calculate other relevant quantities. \nSolving the equation f ( r ) = 0, one can obtain the parameter m in terms of the horizon radius r + as \nm = r n + 3 α -1 + ( r 2 + + kα ) 3 r 6 + + 12 αβ 2 [ 1 -Λ 2 β 2 -√ 1 + η + ( n -1) Ϝ ( η ) η n -2 ] n ( n -1) . (15) \n Then the mass of ( n +1)-dimensional topological AdS black holes can be derived as \nM = ( n -1)Σ k 16 π m = ( n -1)Σ k r n + 48 πα -1 + ( r 2 + + kα ) 3 r 6 + + 12 αβ 2 [ 1 -Λ 2 β 2 -√ 1 + η + ( n -1) Ϝ ( η ) η n -2 ] n ( n -1) , (16) \nThe entropy can be calculated as \nwhere Σ k denotes the volume of the ( n -1)-dimensional hypersurface mentioned above. \nS = ∫ r + 0 1 T ( ∂M ∂r + ) dr = Σ k ( n -1) r n -5 + 4 ( r 4 + n -1 + 2 kr 2 + α n -3 + k 2 α 2 n -5 ) . (17) \nNote that the above integration is accomplished under the condition of n > 5. For n /lessorequalslant 5 the integration is divergent. So in this paper, we would mainly investigate the case of n = 6, which corresponds to the seven-dimensional black holes. The third term of the entropy in Eq. (17) does not appear in the expression of the entropy of Gauss-Bonnet black holes [40]. Our result also extends the expression in Ref. [18] where k was missing. \nIn the extended phase space, one may identify the pressure of the black hole as [31] \nP = -Λ 8 π . (18) \nf ( r ) = k + r 2 α (1 -g ( r ) 1 / 3 ) , (9) \n- \ng ( r ) = 1 + 3 αm r n -12 αβ 2 n ( n 1) [ 1 -√ 1 + η -Λ 2 β 2 + ( n -1) η ( n 2) Ϝ ( η ) ] , (10) \n- \n d Ω 2 denotes the line element of ( n -1)-dimensional hypersurface with constant curvature ( n -1)( n -2) k and Ϝ ( η ) denotes the hypergeometric function as follow \n√ d Ω 2 = dθ 2 1 + n -1 ∑ i =2 i -1 ∏ j =1 sin 2 θ j dθ 2 i k =1 dθ 2 1 +sinh 2 θ 1 dθ 2 2 +sinh 2 θ 1 n -1 ∑ i =3 i -1 ∏ j =2 sin 2 θ j dθ 2 i k =-1 n -1 ∑ i =1 dφ 2 i k =0 . (11) \nϜ ( η ) = 2 F 1 ([ 1 2 , n -2 2 n -2 ] , [ 3 n -4 2 n -2 ] , -η ) , (12) \nAnd the mass of black holes should be interpreted as enthalpy rather than the internal energy. In this context, the Gibbs free energy can be derived through \nG = H -TS = M -TS. (19) \nAfter tedious calculation, we can obtain \nG = Σ k r n -6 + 48 πα ( r 2 + + kα ) 2 { ( n -1) r 6 + ( r 2 + + kα ) 2 [ -1 + ( r 2 + + kα ) 3 r 6 + + 12 αβ 2 ( 1 -Λ 2 β 2 -√ 1 + η + ( n -1) Ϝ ( η ) η n -2 ) n ( n -1) ] -α ( r 4 + n -1 + 2 kr 2 + α n -3 + k 2 α 2 n -5 ) [ ( n -1) k ( 3( n -2) r 4 + +3( n -4) kαr 2 + +( n -6) k 2 α 2 ) -6Λ r 6 + +12 r 6 + β 2 (1 -√ 1 + η + ) ]} . (20) \nImitating the approach of Refs. [32, 40], the first law of thermodynamics in the extended phase space can be rewritten as \ndM = TdS +Φ dQ + V dP + A dα + B dβ, (21) \nwhere A and B denote the quantities conjugated to the Lovelock coefficient and Born-Infeld parameter respectively. And they can be obtained as \n-- \n] \n[ \nA = ( ∂M ∂α ) S,Q,P,β = k 2 ( n -1) r n -6 + (3 r 2 + +2 kα )Σ k 48 π -1 2 k ( n -1) r n -5 + T ( r 2 + n -3 + kα n -5 ) Σ k , (22) B = ( ∂M ∂β ) S,Q,P,α = Σ k r -n + 8 nπβ { 2 r 2 n + β 2 ( 2 -√ 4 + 2( n -1)( n -2) q 2 r 2 -2 n + β 2 ) +( n -2)( n -1) q 2 r 2 + 2 F 1 ([ 1 2 , n 2 2 n 2 , 3 n -4 2 n -2 ] , -( n -1)( n -2) q 2 2 β 2 r 2 n -2 )} . (23) \nComparing Eq. (22) with Gauss-Bonnet black holes in Ref. [40], one may find extra terms due to the third order Lovelock gravity. Note that Eq. (21) is limited to the case of charged topological black holes in Lovelock Born-Infeld gravity in which the second and the third order Lovelock coefficients are related via the Lovelock coefficient α . For a general case and a nice physical interpretation of the quantity conjugated to the Lovelock coefficient, see Ref. [15], where the Smarr relation and the first law of thermodynamics in Lovelock gravity was thoroughly investigated and it was shown that the conjugate quantity Ψ ( k ) to the Lovelock coefficient b k consists of three terms related to mass, entropy and the anti-symmetric Killing-Lovelock potential respectively.', 'III. P -V CRITICALITY OF A LIMIT CASE': "To concentrate on the effects of the third order Lovelock gravity, we would like to investigate an interesting limit case in this section and leave the issue of nonlinear electrodynamics to be further investigated in Sec. IV. \nWhen β →∞ , the Born-Infeld Lagrangian reduces to the Maxwell form and Ϝ ( η ) → 1. So one can have \ng ( r ) → 1 + 3 αm r n + 6 α Λ n ( n -1) -3 αq 2 r 2 n -2 . (24) \nAnd the temperature for this limit case can be simplified as \nT = ( n -1) k [3( n -2) r 4 + +3( n -4) kαr 2 + +( n -6) k 2 α 2 ] -6Λ r 6 + -3( n -2)( n -1) q 2 r 8 -2 n + 12 π ( n -1) r + ( r 2 + + kα ) 2 . (25) \nSubstituting Eq. (18) into Eq. (25), one can find the expression for P as \nP = n -1 48 π [ 12 πT r + + 24 kπαT r 3 + + 12 k 2 πα 2 T r 5 + + 3 k (2 -n ) r 2 + + 3 k 2 α (4 -n ) r 4 + -k 3 ( n -6) α 2 r 6 + +3( n -2) q 2 r 2 -2 n + ] . (26) \nWe can identify the specific volume v as \nv = 4 r + n -1 . (27) \nThen Eq. (26) can be transformed into \nP = T v + 32 kTα ( n -1) 2 v 3 + 256 k 2 Tα 2 ( n -1) 4 v 5 -k ( n -2) ( n -1) πv 2 -16 k 2 ( n -4) α ( n -1) 3 πv 4 -256 k 3 ( n -6) α 2 3( n -1) 5 πv 6 + 16 n -2 ( n -2) q 2 π ( n -1) 2 n -3 v 2 n -2 . (28) \nThe possible critical point should satisfy the following conditions \n∂P ∂v = 0 , (29) \n∂ 2 P ∂v 2 = 0 . (30) \nFirstly, we would focus on the spherical case corresponding to k = 1. The equation of state reads \nP = T v + 32 Tα ( n -1) 2 v 3 + 256 Tα 2 ( n -1) 4 v 5 -( n -2) ( n -1) πv 2 -16( n -4) α ( n -1) 3 πv 4 -256( n -6) α 2 3( n -1) 5 πv 6 + 16 n -2 ( n -2) q 2 π ( n -1) 2 n -3 v 2 n -2 . (31) \nWhen q = 0 , n = 6, Eqs. (29) and (30) can be analytically solved and the corresponding physical quantities can be obtained as \nT c = 1 π √ 5 α , v c = 4 √ α √ 5 , P c = 17 200 πα , P c v c T c = 17 50 . (32) \nWe can see clearly that the critical temperature is inversely proportional to √ α while the critical specific volume is proportional to it. The critical pressure is inversely proportional to α . However, the ratio P c v c T c is independent of the parameter α . Our results demonstrate again that P -V criticality may exist even in the uncharged case. That may be attributed to the effect of higher derivative terms of curvature. \nWhen q /negationslash = 0 , n = 6, one can obtain the corresponding physical quantities at the critical point as listed in Table I by solving Eqs. (29) and (30) numerically. From Table I, one can find that there exists only one critical point for all the cases studied. And the physical quantities at the critical point T c , v c , P c depend on both the charge and the parameter α which is related to the second and the third order Lovelock coefficients. With the increasing of α or q , both T c and P c decrease while v c increases. However the ratio P c v c T c decreases with α but increases with q . \nTABLE I: Critical values for k = 1 , n = 6 , β →∞ \n| q | α | T c | v c | P c | P c v c T c |\n|-----|-----|---------|---------|---------|---------------|\n| 0.5 | 1 | 0.14213 | 1.80992 | 0.02691 | 0.343 |\n| 2 | 1 | 0.13989 | 1.97347 | 0.02553 | 0.36 |\n| 1 | 1 | 0.14154 | 1.85884 | 0.02653 | 0.348 |\n| 1 | 0.5 | 0.19287 | 1.53461 | 0.04727 | 0.376 |\n| 1 | 2 | 0.10062 | 2.53773 | 0.01351 | 0.341 | \nTo witness the P -V criticality behavior more intuitively, we plot the P -v diagram in Fig. 1. When the temperature is less than the critical temperature T c , the isotherm can be divided into three branches. Both the large radius branch and the small radius branch are stable corresponding to a positive compression coefficient while the medium radius branch is unstable corresponding to a negative compression coefficient. The phase transition between the small black hole and the large black hole is analogous to the van der Waals liquid-gas phase transition. Figs. 1(a), 1(b), 1(c) and 1(d) show the impact of the charge on the P -V criticality while Figs. 1(c), 1(e), and 1(f) show the effect of α . The comparisons are in accord with the analytical results in Table I. We also plot both the two-dimensional and three dimensional Gibbs free energy graph for q = 0 , n = 6 in Fig. 2 and for the case q = 1 , n = 6 in Fig. 3. Below the critical temperature, the Gibbs free energy graphs display the classical swallow tail behavior implying the occurrence of the first order phase transition. Above the critical temperature, there is no swallow tail behavior. \n<!-- image --> \n<!-- image --> \nFIG. 1: P vs. v for (a) n = 6 , α = 1 , q = 0, (b) n = 6 , α = 1 , q = 0 . 5, (c) n = 6 , α = 1 , q = 1, (d) n = 6 , α = 1 , q = 2, (e) n = 6 , α = 0 . 5 , q = 1 and (f) n = 6 , α = 2 , q = 1 \n<!-- image --> \nFIG. 2: (a) G vs. T for k = 1 , n = 6 , α = 1 , q = 0, ' P = 0 . 015 < P c , Blue curve', ' P = 0 . 02 < P c , Black curve', ' P = P c = 0 . 02706, Red curve', ' P = 0 . 04 > P c , Purple curve' (b) G vs. P and T for k = 1 , n = 6 , α = 1 , q = 0 \n<!-- image --> \nFIG. 3: (a) G vs. T for k = 1 , n = 6 , α = 1 , q = 1, ' P = 0 . 015 < P c , Blue curve', ' P = 0 . 02 < P c , Black curve', ' P = P c = 0 . 02653, Red curve', ' P = 0 . 04 > P c , Purple curve' (b) G vs. P and T for k = 1 , n = 6 , α = 1 , q = 1 \n<!-- image --> \nSecondly, we would discuss the k = 0 case corresponding to Ricci flat topology. The equation of state reads \nP = T v + 16 n -2 ( n -2) q 2 π ( n -1) 2 n -3 v 2 n -2 . (33) \nFor n = 6, utilizing Eq. (33), one can obtain \n∂P ∂v = -T v 2 -524288 q 2 390625 πv 11 , (34) \nwhich is always negative for nontrivial temperature. So there would be no P -V criticality for k = 0. \nP = T v -32 Tα ( n -1) 2 v 3 + 256 Tα 2 ( n -1) 4 v 5 + ( n -2) ( n -1) πv 2 -16( n -4) α ( n -1) 3 πv 4 + 256( n -6) α 2 3( n -1) 5 πv 6 + 16 n -2 ( n -2) q 2 π ( n -1) 2 n -3 v 2 n -2 . (35) \nThirdly, we would investigate the k = -1 case corresponding to hyperbolic topology. The equation of state reads \nSimilarly, when q = 0 , n = 6, Eqs. (29) and (30) can be analytically solved and the corresponding physical quantities can be obtained as \nT c = 1 2 π √ α , v c = 4 √ α 5 , P c = 5 8 πα , P c v c T c = 1 . (36) \nFIG. 4: P vs. v for (a) k = -1 , n = 6 , α = 1 , q = 0, (b) k = -1 , n = 6 , α = 1 , q = 1 \n<!-- image --> \nWhen q /negationslash = 0 , n = 6, one can obtain the numerical solutions of Eqs. (29) and (30) as listed in Table II. These results are quite different from those in former literature which demonstrated that P -V criticality only exists in the k = 1 case for topological black holes in both Einstein gravity and Gauss-Bonnet gravity [31, 40]. \nTABLE II: Critical values for k = -1 , n = 6 , β →∞ \n| q | α | T c | v c | P c | P c v c T c |\n|-----|-----|---------|---------|---------|---------------|\n| 0.5 | 1 | 0.33836 | 1.07752 | 0.22718 | 0.723 |\n| 2 | 1 | 0.72658 | 1.31811 | 0.35029 | 0.635 |\n| 1 | 1 | 0.469 | 1.19335 | 0.26507 | 0.674 |\n| 1 | 0.5 | 1.88727 | 1.01514 | 1.12928 | 0.607 |\n| 1 | 2 | 0.18669 | 1.38663 | 0.10503 | 0.78 | \nTo gain an intuitive picture, we plot the P -v diagram in Fig. 4, which shows strange behaviors different from van der Waals liquid-gas phase transition. The isotherm at the critical temperature is quite similar to the van der Waals liquid-gas system. However, for the uncharged case in Fig. 4(a), the isotherm below or above the critical temperature both behave as the coexistence phase which is similar to the behaviors of van der Waals liquid-gas system below the critical temperature. For the charged case in Fig. 4(b), the 'phase transition' picture is quite the reverse of van der Waals liquid-gas phase transition. Above the critical temperature the behavior is 'van der Waals like' while the behavior is 'ideal gas like' below the critical temperature. This process is achieved by lowering the temperature rather than increasing the temperature. We also plot the Gibbs free energy in Fig. 5 and 'swallow tail' behavior can be observed. \nThe results above are so strange that motivates us to check whether they are physical. The non-negative definiteness of entropy demands that \nr 4 + n -1 + 2 kr 2 + α n -3 + k 2 α 2 n -5 ≥ 0 . (37) \nIn fact, when n = 6, the L.H.S. of the above inequality can be obtained by utilizing Eq. (27) as \n125 v 4 256 -25 v 2 α 24 + α 2 . (38) \nDenoting v 2 as x , one can consider the equation \n125 x 2 256 -25 αx 24 + α 2 = 0 , (39) \nFIG. 5: G vs. T for (a) k = -1 , n = 6 , α = 1 , q = 0, ' P = 0 . 15 < P c , Blue curve', ' P = P c = 0 . 19894, Red curve', ' P = 0 . 24 > P c , Black curve' (b) k = -1 , n = 6 , α = 1 , q = 1,' P = 0 . 25 < P c , Blue curve', ' P = P c = 0 . 26507, Red curve', ' P = 0 . 29 > P c , Black curve',' P = 0 . 32 > P c , Purple curve' \n<!-- image --> \nwith the discriminant as \n∆ = ( 25 α 24 ) 2 -4 × α 2 × 125 256 = -125 α 2 144 . (40) \nNote that for any nontrivial value of α , the discriminant of Eq. (39) is always negative, implying that the values of entropy are always positive for any specific volume v .", 'IV. INCLUSION OF THE NONLINEAR ELECTRODYNAMICS': "In this section, we would like to take into account the effect of non-linear electrodynamics to complete the analysis of topological AdS black holes in Lovelock-Born-Infeld gravity. \nUtilizing Eqs. (13) and (18), Eq. (14) can be rewritten as \nP = T v + 32 kTα ( n -1) 2 v 3 + 256 k 2 Tα 2 ( n -1) 4 v 5 -k ( n -2) ( n -1) πv 2 -16 k 2 ( n -4) α ( n -1) 3 πv 4 -256 k 3 ( n -6) α 2 3( n -1) 5 πv 6 -β 2 4 π { 1 -√ 1 + 2 4 n -5 ( n -2)( n -1) q 2 [( n -1) v ] 2 -2 n β 2 } . (41) \nSimilarly, we would discuss the k = 1 case corresponding to spherical topology first. The equation of state reads \nP = T v + 32 Tα ( n -1) 2 v 3 + 256 Tα 2 ( n -1) 4 v 5 -( n -2) ( n -1) πv 2 -16( n -4) α ( n -1) 3 πv 4 -256( n -6) α 2 3( n -1) 5 πv 6 -β 2 4 π { 1 -√ 1 + 2 4 n -5 ( n -2)( n -1) q 2 [( n -1) v ] 2 -2 n β 2 } . (42) \nOne can obtain the corresponding physical quantities at the critical point as listed in Table III by solving Eqs. (29) and (30) for the case n = 6 numerically. As is shown, the physical quantities at the critical point T c , v c , P c depend on the charge, the Lovelock coefficient α and the Born-Infeld parameter β . With the increasing of α or q , both T c and P c decrease while v c increases. However the ratio P c v c T c decreases with α but increases with q . These observations are similar to the limit case β →∞ . With the increasing of β , T c , P c decrease while v c and the ratio P c v c T c increase. However, only slight differences can be observed concerning the impact of nonlinear electrodynamics. That may be attributed to the parameter region we choose. Readers who are interested in the 'Schwarzschild like' behavior of \nTABLE III: Critical values for different dimensions for k = 1 , n = 6 \n| β | q | α | T c | v c | P c | P c v c T c |\n|------|-----|-----|----------|---------|----------|---------------|\n| 10 | 1 | 1 | 0.141541 | 1.85884 | 0.026528 | 0.34839 |\n| 0.5 | 1 | 1 | 0.141545 | 1.85829 | 0.026531 | 0.34832 |\n| 1 | 1 | 1 | 0.141542 | 1.85871 | 0.026529 | 0.34838 |\n| 1 | 0.5 | 1 | 0.142126 | 1.80991 | 0.02691 | 0.343 |\n| 1 | 2 | 1 | 0.139898 | 1.97286 | 0.02554 | 0.36 |\n| 1 | 1 | 0.5 | 0.192905 | 1.53258 | 0.0473 | 0.376 |\n| 1 | 1 | 2 | 0.100617 | 2.53773 | 0.01351 | 0.341 | \nFIG. 6: P vs. v for (a) k = 1 , n = 6 , α = 1 , β = 1 , q = 1 and (b) k = -1 , n = 6 , α = 1 , β = 1 , q = 1 \n<!-- image --> \nBorn-Infeld black holes can read the interesting paper Ref. [32]. For an intuitive understanding, we plot the P -v diagram in Fig. 6(a) and show the effect of the parameter q and α in Fig. 7. \nSecondly, we would discuss the k = 0 case corresponding to Ricci flat topology. The equation of state reads \nP = T v -β 2 4 π { 1 -√ 1 + 2 4 n -5 ( n -2)( n -1) q 2 [( n -1) v ] 2 -2 n β 2 } . (43) \nThere would be no P -V criticality because P monotonically decreases with v . \nP = T v -32 Tα ( n -1) 2 v 3 + 256 Tα 2 ( n -1) 4 v 5 + ( n -2) ( n -1) πv 2 -16( n -4) α ( n -1) 3 πv 4 + 256( n -6) α 2 3( n -1) 5 πv 6 -β 2 4 π { 1 -√ 1 + 2 4 n -5 ( n -2)( n -1) q 2 [( n -1) v ] 2 -2 n β 2 } . (44) \nThirdly, we would discuss the k = -1 case corresponding to hyperbolic topology. The equation of state reads \nNumerical solutions of Eqs. (29) and (30) are listed in Table IV and we also plot the P -v diagram in Fig. 6(b), in which similar strange behavior is also observed. Note that the entropy analysis also holds because the entropy in Eq. (17) is independent of β . So we would not repeat the analysis here.", 'V. CONCLUSIONS': "Till now, the topological AdS black holes in Lovelock-Born-Infeld gravity are investigated in the extended phase space. The black hole solutions are reviewed while their thermodynamics is further explored in the extended phase \nFIG. 7: Isotherm at the critical temperature for (a) k = 1 , n = 6 , β = 1 , q = 1 and (b) k = 1 , n = 6 , β = 1 , α = 1 \n<!-- image --> \nTABLE IV: Critical values for different dimensions for k = -1 , n = 6 \n| β | q | α | T c | v c | P c | P c v c T c |\n|------|-----|-----|----------|---------|---------|---------------|\n| 10 | 1 | 1 | 0.468887 | 1.19315 | 0.26504 | 0.674 |\n| 0.5 | 1 | 1 | 0.424488 | 1.1147 | 0.25296 | 0.664 |\n| 1 | 1 | 1 | 0.457384 | 1.17317 | 0.26182 | 0.672 |\n| 1 | 0.5 | 1 | 0.333977 | 1.06663 | 0.22621 | 0.722 |\n| 1 | 2 | 1 | 0.690585 | 1.2827 | 0.33877 | 0.629 |\n| 1 | 1 | 0.5 | 1.4985 | 0.92425 | 0.9427 | 0.581 |\n| 1 | 1 | 2 | 0.186104 | 1.38311 | 0.10496 | 0.78 | \nspace. We calculate the entropy by integration and find that the result in former literature [18] was incomplete. Treating the cosmological constant as pressure, we rewrite the first law of thermodynamics for the specific case in which the second order and the third order Lovelock coefficients are related by the Lovelock coefficient α . The quantity conjugated to Lovelock coefficient and the Born-Infeld parameter respectively are calculated. Comparing our results of the above quantities with those in former literature of Gauss-Bonnet black holes [40], we find that there exist extra terms due to the third order Lovelock gravity. In order to make the phase transition clearer, the Gibbs free energy is also calculated. \nTo figure out the effect of the third order Lovelock gravity on the P -V criticality, a detailed analysis of the limit case β →∞ has been performed. Since the entropy is convergent only when n > 5, our investigation is carried out in the case of n = 6, corresponding to the seven-dimensional black holes. It is shown that for the spherical topology, P -V criticality exists even when q = 0. The critical physical quantities can be analytically solved and they vary with the parameter α . However, the ratio of P c v c T c is independent of the parameter α . Our results demonstrate again that the charge is not the indispensable condition of P -V criticality. It may be attributed to the effect of higher derivative terms of curvature because similar phenomenon was also found for Gauss-Bonnet black holes [40]. For q /negationslash = 0, it is shown that the physical quantities at the critical point T c , v c , P c depends on both the charge and the parameter α . With the increasing of α or q , both T c and P c decrease while v c increases. However the ratio P c v c T c decreases with α but increases with q . Similar behaviors as van der Waals liquid-gas phase transition can be observed in the P -v diagram and the classical swallow tail behaviors can be observed in both the two-dimensional and three-dimensional graph of Gibbs free energy. These observations indicate that phase transition between small black holes and large black holes take place when k = 1. For k = 0, no critical point can be found and there would be no P -V criticality. Interesting findings occur in the case k = -1, in which positive solutions of critical points are found for both the uncharged and charged case. However, the P -v diagram is very strange. For the uncharged case, the isotherms below or above the critical temperature both behave as the coexistence phase which is similar to the behaviors of van der Waals liquid-gas system below the critical temperature. For the charged case, the 'phase transition' picture is quite the reverse of \nvan der Waals liquid-gas phase transition. Above the critical temperature the behavior is 'van der Waals like' while the behavior is 'ideal gas like' below the critical temperature. This process is achieved by lowering the temperature rather than increasing the temperature. To check whether these findings are physical, we perform analysis on the non-negative definiteness condition of entropy. It is shown that for any nontrivial value of α , the entropy is always positive for any specific volume v . We relate the findings in the case k = -1 with the peculiar property of the third order Lovelock gravity. Because the entropy in the third order Lovelock gravity consists of extra terms which is absent in the Gauss-Bonnet black holes, which makes the critical points satisfy the constraint of non-negative definiteness condition of entropy. We also check the Gibbs free energy graph and 'swallow tail' behavior can be observed. \nMoreover, the effect of nonlinear electrodynamics is included in our work. Similar observations are made as the limit case β →∞ and only slight differences can be observed when we choose different values of β . That may be attributed to the parameter region we choose. More interesting findings concerning the 'Schwarzschild like' behaviors can be found in the former literature [32] and we would not repeat them here because our main motivation is to investigate the impact of the third order Lovelock gravity on the P -V criticality in the extend phase space.", 'Acknowledgements': "This research is supported by the National Natural Science Foundation of China (Grant Nos.11235003, 11175019, 11178007). It is also supported by 'Thousand Hundred Ten' Project of Guangdong Province and Natural Science Foundation of Zhanjiang Normal University under Grant No. QL1104. \n- [1] D. Lovelock, The Einstein tensor and its generalizations , J. Math. Phys. (N.Y.) 12 (1971)498.\n- [2] D. G. Boulware and S. Deser, String-Generated Gravity Models , Phys. Rev. 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2015PhRvD..92h3014M

Rotating black holes in Einstein-dilaton-Gauss-Bonnet gravity with finite coupling

2015-01-01

['-', '-', '-', 'black hole physics', '-', '-', '-']

[]

Among various strong-curvature extensions of general relativity, Einstein-dilaton-Gauss-Bonnet gravity stands out as the only nontrivial theory containing quadratic curvature corrections while being free from the Ostrogradsky instability to any order in the coupling parameter. We derive an approximate stationary and axisymmetric black hole solution of this gravitational theory in closed form, which is of fifth order in the black hole spin and of seventh order in the coupling parameter of the theory. This extends previous work that obtained the corrections to the metric only to second order in the spin and at the leading order in the coupling parameter, and allows us to consider values of the coupling parameter close to the maximum permitted by theoretical constraints. We compute some quantities which characterize this solution, such as the dilaton charge, the moment of inertia, and the quadrupole moment, and its geodesic structure, including the innermost stable circular orbit and the epicyclic frequencies for massive particles. The latter provides a valuable tool to test general relativity against strong-curvature corrections through observations of the electromagnetic spectrum of accreting black holes.

[]

https://arxiv.org/pdf/1507.00680.pdf

{'Rotating black holes in Einstein-dilaton-Gauss-Bonnet gravity with finite coupling': "Andrea Maselli, 1 Paolo Pani, 2, 3 Leonardo Gualtieri, 2 and Valeria Ferrari 2 \n1 Center for Relativistic Astrophysics, School of Physics, Georgia Institute of Technology, Atlanta, GA 30332, USA. 2 Dipartimento di Fisica, Universit'a di Roma 'La Sapienza' & Sezione INFN Roma1, P.A. Moro 5, 00185, Roma, Italy. 3 CENTRA, Departamento de F'ısica, Instituto Superior T'ecnico, Universidade de Lisboa, Avenida Rovisco Pais 1, 1049 Lisboa, Portugal. \nAmong various strong-curvature extensions of general relativity, Einstein-dilaton-Gauss-Bonnet gravity stands out as the only nontrivial theory containing quadratic curvature corrections while being free from the Ostrogradsky instability to any order in the coupling parameter. We derive an approximate stationary and axisymmetric black hole solution of this gravitational theory in closed form, which is of fifth order in the black hole spin and of seventh order in the coupling parameter of the theory. This extends previous work that obtained the corrections to the metric only to second order in the spin and at the leading order in the coupling parameter, and allows us to consider values of the coupling parameter close to the maximum permitted by theoretical constraints. We compute some quantities which characterize this solution, such as the dilaton charge, the moment of inertia and the quadrupole moment, and its geodesic structure, including the innermost stable circular orbit and the epicyclic frequencies for massive particles. The latter provides a valuable tool to test general relativity against strong-curvature corrections through observations of the electromagnetic spectrum of accreting black holes.", 'I. INTRODUCTION': "Observational and experimental tests of general relativity (GR) [1] have mostly probed the weakfield/slow-motion regimes of the theory, while a number of strong-field, relativistic GR predictions still remain elusive and difficult to verify [2, 3]. Furthermore, a series of long-lasting problems in Einstein's theory - such as the accelerated expansion of the Universe, dark matter, the nature of curvature singularities, and the quest for an ultraviolet completion of GR - have motivated strong efforts to develop extended theories of gravity which would modify GR in its most extreme regimes while conforming with current weak-field observations [4]. \nBlack holes (BHs) are genuine strong-field predictions of GR and have no analog in Newtonian theory. Thus, they are natural candidates to test gravity in the strong-field regime. Future networks of electromagnetic detectors [5-8] and ground-based gravitational-wave detectors [9, 10] will allow us to measure some crucial properties of BHs, such as their shadows, the location of the event horizon, and of the innermost stable circular orbit (ISCO). This information will be instrumental to test the Kerr hypothesis , according to which all stationary astrophysical BHs are uniquely described by the Kerr family and are, thus, characterized by only two parameters: their mass and angular momentum (see, e.g., Ref. [11] and references therein). \nIn recent years, several modified theories of gravity have been proposed. They can be divided in \nvarious categories, each one lifting some of the fundamental principles (Lorentz invariance, weak and strong equivalence principles, massless spin-2 mediators, etc.) upon which Einstein's theory is uniquely built [4]. From this and other classifications, it emerges that one of the simplest and best motivated ways to modify GR consists of including a fundamental scalar field which is nonminimally coupled to the metric tensor. In order to modify the strong-curvature regime, it is natural to couple this scalar field in the gravitational action to terms quadratic in the curvature tensor. Such couplings can also be interpreted as the first terms in the expansion in all possible curvature invariants, as suggested by low-energy effective string theories [12]. Generally, a quadratic curvature term in the action leads to field equations of third (or higher) order, which are subject to Ostrogradsky's instability [13]. Therefore, these theories should be considered as effective, i.e., truncations of a theory with further terms in the action, which are neglected in the perturbative regime. \nIt should also be mentioned that quadratic curvature terms are crucial, not only to modify the strong-curvature regime of GR, but also to affect the behavior of stationary BHs. Indeed, standard scalar-tensor theories (in which one or more scalar fields are included in the gravitational sector of the action) satisfy the so-called no-hair theorems; i.e., stationary, vacuum BHs are the same as in GR [14-16] (but see [17-19] for possible violations of these theorems). When quadratic curvature terms are included in the action instead, stationary BH solutions are different. \nWe shall consider a member of this family of modified-gravity theories, Einstein-dilaton-GaussBonnet (EDGB) theory, in which a scalar field (the dilaton) is coupled to the Gauss-Bonnet invariant [12, 20] \nR 2 GB = R αβγδ R αβγδ -4 R αβ R αβ + R 2 (1) \nin the action. EDGB gravity is one of the best motivated alternatives to GR. Indeed, it is the only theory of gravity with quadratic curvature terms in the action, whose field equations are of second differential order for any coupling, and not just in the weak-coupling limit which is assumed in the effective-field-theory approach [4]. As a consequence, EDGB gravity is ghost-free; i.e., it avoids the Ostrogradsky instability [13]. Furthermore, as mentioned above, the higher-curvature coupling - which modifies the strong-curvature regime of gravity - violates the hypothesis of the BH no-hair theorems so that BH solutions in EDGB gravity are different from those predicted by GR and provide the ideal arena for genuine strong-field tests of the Kerr hypothesis. Finally, the EDGB term naturally arises in low-energy effective string theories [21]. \nIn this work, we construct an analytical, perturbative solution of EDGB theory, which describes a slowly rotating BH endowed with a scalar field. To this aim, we extend the formalism developed in [22, 23] up to fifth order in the BH (dimensionless) spin parameter χ = J/M 2 , where J and M are the angular momentum and the Arnowitt-Deser-Misner mass of the solution, respectively. \nAnalytical BH solutions of EDGB theory in the small-coupling limit have been investigated in [24, 25], where stationary, spherically symmetric configurations where found. Approximate, stationary, and axisymmetric solutions to linear and quadratic order in the BH spin were obtained in [26] and [27], respectively. Both of these works considered a weak-field expansion of the coupling between the scalar field and the Gauss-Bonnet invariant R 2 GB in terms of a dimensionful coupling constant α . Exact numerical solutions were constructed to zeroth [20] and first order [28] in the spin and also for arbitrary values of the angular momentum [29, 30]. Although exact in α , such solutions are of limited practical use (for instance, for Monte Carlo data analysis) because they require a numerical integration for each set of parameters. On the other hand, numerical solutions are necessary in regimes where the slow-spin expansion does not converge and are, therefore, complementary to our analysis. \nOur results extend the study carried out so far. In particular, we go beyond the analysis of Ref. [27], where a BH solution was obtained to \nsecond order both in the spin and in the coupling parameter. Indeed, we compute the metric tensor and the scalar field up to O ( ζ 7 , χ 5 ) , where ζ ≡ α/M 2 , and α is the EDGB coupling constant. We use this expansion to derive the main features of the solution, such as the geometry of the event horizon and of the ergoregion. Furthermore, we study the geodesic structure of this solution by computing the ISCO and the epicyclic frequencies (see, e.g., Refs. [31-33]) consistent with our approximation scheme. We compare these quantities with those obtained in [34], where a numerical solution was derived, which is exact in the coupling parameter (i.e., with no perturbative expansion in ζ ) and approximate to linear order in the BH spin. We find relative errors at most of the order of 1% for the maximum value of ζ allowed by theoretical constraints for the existence of BH solutions ζ glyph[lessorsimilar] 0 . 691 [20] and much smaller for less extreme couplings. \nThe results of this paper can be useful to devise tests of GR in the strong-field regime through astrophysical observations of BHs. For instance, we have shown [34] that observations of quasi-periodic oscillations of accreting BHs, with the sensitivity of recently proposed large-area X-ray space telescopes (e.g., [6, 7]), allow us to set constraints on the parameter space of EDGB theory, thus, probing the strong-field regime of gravity (see, also, Ref. [35] for a recent study). However, since BH solutions in EDGB theory (for finite α ) were only known at first order 1 in the spin parameter χ , in [34] we only considered BHs with very slowrotation rate, for which the deviations from GR are expected to be small. \nThis paper is organized as follows. In Sec. II we derive our solution of the EDGB field equations, describing rotating BHs up to O ( ζ 7 , χ 5 ) . In Sec. III we study this solution, computing its geometrical properties, the location of the ISCO, and the azimuthal and epicyclic frequencies. We also estimate the accuracy of our approximation in the determination of these quantities and how our results improve on the existing literature. In particular, we discuss how the spin correction to the azimuthal and epicyclic frequencies can affect possible tests of GR based on observations of accreting BHs, such as those discussed in [34]. Finally, in Sec. IV we draw our conclusions.", 'II. SPINNING BLACK HOLES IN EINSTEIN-DILATON-GAUSS-BONNET THEORY': 'In this section we derive the spacetime metric and scalar field describing rotating BHs in EDGB theory, up to O ( ζ 7 , χ 5 ) .', 'A. EDGB gravity': 'Einstein-dilaton-Gauss-Bonnet theory is defined by the following action [12, 20]: \nS = 1 2 ∫ d 4 x √ -g [ R -1 2 ∂ µ Φ ∂ µ Φ+ αe Φ 4 R 2 GB ] , (2) \nwhere g < 0 is the metric determinant, Φ is a scalar field coupled to the Gauss-Bonnet invariant (1) and α > 0 is the coupling constant [20]. Since we are interested in BH solutions, in the action above we have neglected matter fields. We use geometric units G = c = 1: with this choice, the scalar field Φ is dimensionless and α has the dimensions of a length squared. \nThe field equations of EDGB gravity are found by varying the action (2) with respect to g µν and Φ: \nG µ ν = 1 2 ∂ µ Φ ∂ ν Φ -1 4 g µ ν ∂ α Φ ∂ α Φ -α K µ ν , (3) S ≡ 1 √ -g ∂ µ ( √ -g∂ µ Φ) + α 4 e Φ R 2 GB = 0 , (4) \nwhere G µν = R µν -1 2 g µν R is the Einstein tensor, \nK µν = 1 8 ( g µρ g νλ + g µλ g νρ ) glyph[epsilon1] kλαβ ×∇ γ ( glyph[epsilon1] ργµν R µναβ ∂ k e Φ ) , (5) \nand glyph[epsilon1] µναβ is the Levi-Civita tensor, with glyph[epsilon1] 0123 = -( -g ) -1 / 2 . Note that - by virtue of the GB combination entering the action (2) - the equations are of second differential order, and, therefore, this theory is free from the Ostrogradsky instability [13]. Indeed, EDGB gravity is a particular case [36] of Horndeski gravity - the most general scalar-tensor theory with second-order field equations [37]. This special subcase is the only one known to date in which regular, stationary, asymptotically flat, hairy BH solutions other than GR ones are found [38]. Furthermore, EDGB gravity can be obtained from the low-energy expansion of the bosonic sector of heterotic string theory [12, 21], in such case the coupling α is related to the string tension. \nIn order to simplify our notation, in the next sections we shall introduce the modified Einstein \ntensor ˜ G µ ν = G µ ν -T µ ν , where \nT µ ν = 1 2 ∂ µ Φ ∂ ν Φ -1 4 g µ ν ∂ α Φ ∂ α Φ -α K µ ν , (6) \nis the effective stress-energy tensor for the dilaton.', 'B. Static BH solutions': "Since the EDGB coupling constant has the dimensions of the inverse of the curvature tensor, it is natural to expect that in this theory the strongest deviations from GR will come from physical systems involving high curvature, such as BHs, neutron stars and the early Universe. We focus here on BH solutions and, in particular, on rotating BH geometries that are obtained through a slow-rotation expansion around a static background solution. \nThe exact BH background solution (first derived in [20]) is described by the static, spherically symmetric line element \nds 2 = -e Γ( r ) dt 2 + e -Λ( r ) dr 2 + r 2 d Ω 2 , (7) \nand by a spherically symmetric scalar field, Φ = φ ( r ). The field equations (3) and (4) supplied by the metric ansatz (7) reduce to a set of differential equations for the scalar field and for the functions Γ and Λ. Indeed, Eq. (4) yields \nφ '' + φ ' ( Γ ' -Λ ' 2 + 2 r ) = αe φ 2 r 2 ( Γ ' Λ ' e -Λ + +(1 -e -Λ ) [ Γ '' + Γ ' 2 (Γ ' -Λ ' ) ]) , (8) \nwhile the t -t , r -r and θ -θ components of ˜ G µ ν = 0 reduce to \n[ 1 + αe φ 2 r φ ' (1 -3 e -Λ ) ] Λ ' = φ ' 2 r 4 + 1 -e Λ r + + αe φ r (1 -e -Λ )( φ '' + φ ' 2 ) , (9) [ 1 + αe φ 2 r φ ' (1 -3 e -Λ ) ] Γ ' = φ ' 2 r 4 + e Λ -1 r , (10) Γ '' + ( Γ ' 2 + 1 r ) (Γ ' -Λ ' ) = -φ ' 2 2 + αe φ -Λ r × × [ φ ' Γ '' +Γ ' ( φ '' + φ ' 2 ) + Γ ' φ ' 2 ( φ ' -3Λ ' ) ] . (11) \nNote that Eqs. (9)-(11) are not all independent and that the r -r component can be solved analytically, yielding \ne Λ = -β + √ β 2 -4 γ 2 , (12) \nwhere \nβ = φ ' 2 r 2 4 -1 -Γ ' ( r + e φ φ ' 2 ) , γ = 3 2 Γ ' φ ' e φ . (13) \nThe remaining two independent equations can be written as \nφ '' = -d 1 d , Γ '' = -d 2 d , (14) \nwhere the radial functions d, d 1 and d 2 are given in Appendix A of [20]. The Arnowitt-Deser-Misner mass M and the dilatonic charge D can be read off the asymptotic behavior of the metric and of the dilaton field, \ng tt →-1 + 2 M/r + . . . (15) \nφ →D /r + . . . (16) \nIt turns out that for each value of M , there is only one solution describing a static BH. In other words, the scalar field is a 'secondary hair': the dilatonic charge D is not an independent parameter but is determined in terms of the BH mass M . \nThe field equations are invariant under the rescaling φ → φ + ˆ φ and r → re ˆ φ/ 2 (or, equivalently, M → Me ˆ φ/ 2 and D → D e ˆ φ/ 2 ) where ˆ φ is a constant. We fix this freedom by requiring that φ → 0 at spatial infinity; this means that at infinity, the Gauss-Bonnet invariant appears in the action (2) multiplied by the constant α/ 4. \nAs noted in [20] static BH solutions in EDGB gravity exist only if \ne φ h ≤ r h α √ 6 , (17) \nwhere φ h is the value of the scalar field computed at the horizon r h . As shown in [28], by requiring that φ → 0 at spatial infinity, Eq. (17) can be recast in the form \n0 ≤ α M 2 glyph[lessorsimilar] 0 . 691 . (18) \nThus, smaller BHs would correspond to a more stringent bound on α . \nPresently, the tightest observational bound on the EDGB coupling parameter (obtained by the orbital decay of X-ray binaries) is α glyph[lessorsimilar] 47 M 2 glyph[circledot] [39]. As discussed in [34], this upper bound is weaker than the theoretical constraint (17) for BHs with M glyph[lessorsimilar] 8 . 2 M glyph[circledot] . For such BHs, the entire range (18) is phenomenologically allowed. \nSolutions of Eqs. (14) have been solved numerically in Ref. [20], while an analytical static BH solution has been derived to second order in α/M 2 in Refs. [24, 25].", 'C. Spinning BH solutions': "To describe slowly rotating BH solutions we extend the approach developed by Hartle [22, 23], in which spin corrections to the static solutions are introduced within a perturbative framework. The procedure described in this section is generic and can be applied also to other theories and different spinning solutions. \nLet us start with the most general solution for a stationary, axially symmetric spacetime 2 which is given by \nds 2 = -H 2 dt 2 + Q 2 dr 2 + r 2 K 2 [ dθ 2 + +sin 2 θ ( dϕ -Ldt ) 2 ] , (19) \nwhere H,Q,K, and L are functions of ( r, θ ). The ansatz (19) can be expanded perturbatively in the spin around the static solution \nds 2 = -e Γ [1 + 2 h ( r, θ )] dt 2 + e -Λ [1 + 2 m ( r, θ )] dr 2 + r 2 [1 + 2 k ( r, θ )][ dθ 2 +sin 2 θ ( dϕ -ˆ ω ( r, θ ) dt ) 2 ] , (20) \nwhere the functions ˆ ω , h , m , and k can be expanded in a complete basis of orthogonal functions according to their symmetry properties as \nˆ ω = N χ -q ∑ n =1 , 3 , 5 ,... n ∑ l =1 , 3 , 5 ,... χ n ω ( n ) l ( r ) S l ( θ ) , (21) \nh = N χ -p ∑ n =2 , 4 ,... n ∑ l =0 , 2 , 4 ,... χ n h ( n ) l ( r ) P l (cos θ ) , (22) \nm = N χ -p ∑ n =2 , 4 ,... n ∑ l =0 , 2 , 4 ,... χ n m ( n ) l ( r ) P l (cos θ ) , (23) \nk = N χ -p ∑ n =2 , 4 ,... n ∑ l =0 , 2 , 4 ,... χ n k ( n ) l ( r ) P l (cos θ ) , (24) \nwhere P l are the Legendre polynomials, S l = -1 sin θ dP l (cos θ ) dθ (note that P 0 = S 1 = 1), and p (respectively q ) is zero when the order N χ of the spin expansion is even (respectively odd), whereas p = 1 (resp. q = 1) otherwise. The radial functions ( ω ( n ) l , h ( n ) l , m ( n ) l , k ( n ) l ) are of the order O ( χ n ). Note that, since the metric (20) is invariant under the rescaling r → f ( r ), the functions k ( n ) 0 ( r ) can be set to zero without loss of generality [22, 23]. \nBecause the dilaton field transforms as a scalar under rotation, we expand it as \nΦ( r ) = φ ( r ) + N χ -p ∑ n =2 , 4 ,... n ∑ l =0 , 2 , 4 ,... χ n φ ( n ) l ( r ) P l (cos θ ) , (25) \nwhere φ is the background static solution and the radial functions φ ( n ) l are of the order O ( χ n ). \n1. O ( χ ) corrections \nRotating BH solutions in EDGB gravity have been investigated to linear order in the spin angular momentum in Refs. [26, 28]. At this order, the metric (20) reduces to the static case with a nonvanishing gravitomagnetic term described by ˆ ω ( r, θ ) = ω (1) 1 ≡ ω ( r ) [see Eq. (21)]: \nds 2 = -e Γ( r ) dt 2 + e -Λ( r ) dr 2 + r 2 d Ω 2 -2 r 2 ω ( r ) sin 2 θdtdϕ. (26) \nFrom ˜ G t ϕ = 0, it is easy to show that ω satisfies the second-order equation [28]: \nω '' + [ 2 r 2 e Λ -2 αrφ ' e φ ] -1 ( -αe φ [2 φ '' r + φ ' (6 + 2 φ ' r -Γ ' r -3Λ ' r )] -e Λ r [ -8 + r (Γ ' +Λ ' )] ) ω ' = 0 , (27) \nwhere the coefficient of ω ' depends on the nonspinning solution. The BH angular momentum can be read off the asymptotic behavior of the gyromagnetic term, \nω ( r ) → 2 J r 3 , (28) \nat large distance. \n2. O ( χ n ) corrections: n ≥ 2 and even \nReplacing the metric ansatz (20) into the field equations and using the decomposition in Legendre polynomials, a set of ordinary differential equations can be obtained, at each order in the spin expansion. The equations are inhomogeneous with source terms given by the lower-order functions. \nAt each given order n ≥ 2 with even n , the equations are found from E 1 ≡ ˜ G tt = 0, E 2 ≡ ˜ G rr = 0, E 3 ≡ ˜ G θθ + (sin θ ) -2 ˜ G ϕϕ = 0, and E 4 ≡ S for the scalar equation (4), each contracted with a Legendre polynomial, \n∫ π 0 dθ sin θP l (cos θ ) E i ( r, θ ) = 0 , (29) \nwhere i = 1 , 2 , 3 , 4 and l = 0 , 2 , 4 , ..., n . Because of the symmetry properties of the field equations \nand of the background, this procedure gives a set of purely radial, inhomogeneous, ordinary differential equations for h ( n ) l , m ( n ) l , k ( n ) l , and φ ( n ) l with l = 0 , 2 , 4 , . . . , n (we recall that k ( n ) 0 = 0). \n3. O ( χ n ) corrections: n ≥ 3 and odd \nSimilarly, at a given order n ≥ 3 (with odd n ) in the spin expansion, a set of radial equations for the gravitomagnetic terms can be obtained by contracting ˜ G tϕ = 0 with the (axisymmetric) vector spherical harmonics, namely \n∫ π 0 dθ sin θ dP l (cos θ ) d cos θ ˜ G tϕ = 0 (30) \nwith l = 1 , 3 , 5 , ..., n . Again, this procedure yields a set of purely radial, inhomogeneous, ordinary differential equations for ω ( n ) l with l = 1 , 3 , 5 , . . . , n .", 'D. Small-coupling approximation': 'The set of equations presented above provides a full description of the BH solution at any perturbative order in the spin, but generic (i.e., nonperturbative) in the EDGB coupling. However, such equations are cumbersome and it is impractical to solve them numerically. More importantly, the theoretical constraint (18) shows that the dimensionless coupling parameter has to be smaller than unity. This motivates a small-coupling approximation [26-28], in which the field equations are solved perturbatively in α/M 2 glyph[lessmuch] 1 to some desired order. Actually, because we are interested in the regime α/M 2 glyph[lessorsimilar] 1 (the maximum value 3 of this parameter is 0 . 691), we shall compute terms of relatively high order in this expansion. \nTo simplify the notation, we introduce the dimensionless parameter \nζ = α M 2 . (31) \nAs a result of our approximation scheme, we expand all quantities, such as the metric functions \nand the scalar field, in terms of the two parameters ζ , χ . For example, \ng µν = g (0 , 0) µν + N χ ∑ i =1 N ζ ∑ j =1 χ i ζ j g ( i,j ) µν , (32) \nwhere the double superscript ( i,j ) denotes the order of the expansion in the BH spin parameter and in the EDGB coupling parameter, respectively; g (0 , 0) µν is the Schwarzschild metric. In practice, using the spin decomposition previously discussed, we simply expand the set of radial variables glyph[vector] f = { Γ , Λ , φ, ω ( n ) l , m ( n ) l , h ( n ) l , k ( n ) l , φ ( n ) l } as \nf = N ζ ∑ j =0 ζ j f ( j ) , (33) \nwhere f ( j ) are radial functions which do not depend on the coupling parameter ζ . By replacing these expressions into the field equations derived in Sec. II C, and solving them order by order in ζ , we obtain the desired expansion for the metric tensor and the scalar field. Remarkably, this procedure yields an analytical solution. We compute the explicit solution up to O ( ζ 7 , χ 5 ), but the procedure can be straightforwardly extended to higher order both in ζ and in χ . \nSolving the differential equations at each order in χ and ζ yields some integration constants, which are uniquely fixed by requiring that \n- 1. the metric is asymptotically flat, and the scalar field vanishes at spatial infinity;\n- 2. there exists an event horizon, where perturbations are regular;\n- 3. the physical mass and angular momentum of the BH are given by M and M 2 χ , as measured by an observer at spatial infinity. In particular, the bare mass of the O ( ζ 0 ) solution acquires some corrections to each order in ζ , which are reabsorbed in the physical mass M . \nWe note that only one of the two integration constants appearing in the solution of the scalar field at each order in ζ is fixed by requiring regularity outside the horizon, while the metric is regular for each value of the remaining constants. Although this is not evident in the Schwarzschild coordinates adopted here, it can be nonetheless checked by computing some curvature invariants. However, the remaining integration constants can all be reabsorbed in the definitions of the physical mass and angular momentum, so that the final solution truncated at a given order depends only on two parameters, as in the Kerr case. \nThe explicit expressions of the metric tensor and the scalar field up to O ( ζ 7 , χ 5 ) are quite long and are available in a Mathematica notebook provided in the Supplemental Material. For completeness, the explicit Kerr metric to O ( χ 5 ) in the Hartle-Thorne coordinates is given in Appendix A.', 'III. GEOMETRICAL AND GEODESIC PROPERTIES OF THE SOLUTION': 'Here we study the properties of the analytical solution we have derived. To this aim, we compute some geometrical and geodesic quantities which characterize the spinning EDGB BH solution to O ( ζ 7 , χ 5 ).', 'A. Event horizon, ergosphere, intrinsic curvature, and dilaton charge': "The event horizon is given by the largest root r = r h of the equation (cf. e.g., [40]) g φφ g tt -g 2 tφ = 0, which yields the following power expansion in terms of ζ and χ : \nr h M = 7 ∑ i =0 ζ i ( a i + b i χ + c i χ 2 + d i χ 3 + e i χ 4 + f i χ 5 ) , (34) \nwhere the coefficients ( a i , b i , c i , d i , e i , f i ) are listed 4 in Table II of Appendix B. As in the Kerr case, the horizon radius r h does not depend on the angular coordinates. Nonetheless, its intrinsic geometry as computed by considering a spatial section dt = 0 at r = r h - is nonspherical. Indeed, \nds 2 t =const ,r = r h = g θθ ( r = r h , θ ) d Ω 2 , (35) \nand since g θθ explicitly depends on θ , the intrinsic geometry is nonspherical. For the line element (35), the curvature radius is \nR intr = 2 g θθ -cot θg ' θθ g 2 θθ + g ' 2 θθ g 3 θθ -g '' θθ g 2 θθ , (36) \nwhere (only in the above formula) a prime denotes differentiation with respect to θ , and for our solution is \nM 2 R intr = 7 ∑ i =0 ζ i [ l i + χ 2 ( m i + n i cos 2 θ ) + + χ 4 ( p i + q i cos 2 θ + u i cos 4 θ )] ; (37) \nthis is constant only when ζ = 0 = χ . Hereafter, we adopt the same expansion of Eqs. (34) and (37) for other physical quantities. The numerical values of the coefficients of these expansions are given in Appendix B, whereas their exact form is provided in the Supplemental Material Mathematica notebook. \nThe location of the ergosphere is given by the largest root of g tt = 0, \nr ergo M = 7 ∑ i =0 ζ i [ l i + χ 2 ( m i + n i cos 2 θ ) + + χ 4 ( p i + q i cos 2 θ + u i cos 4 θ )] , (38) \nwhere the only nonvanishing spin corrections correspond to even powers of χ . \nFinally, the dilaton charge D can be extracted from the leading-order large-distance behavior of \nthe dilaton field Φ →D /r and reads \nD M = 7 ∑ i =1 ζ i ( a i + b i χ + c i χ 2 + e i χ 4 ) , (39) \nwhere the coefficients d i and f i identically vanish.", 'B. Moment of inertia': 'The moment of inertia is defined as I = J/ Ω h , where J is the BH angular momentum and Ω h is the angular velocity at the horizon of locally nonrotating observers, \nΩ h = -lim r → r h g tϕ g ϕϕ . (40) \nIn our case we obtain \nI M 3 = 4 -0 . 2625000 ζ 2 -0 . 1721966 ζ 3 -0 . 1458764 ζ 4 -0 . 1409996 ζ 5 -0 . 1474998 ζ 6 -0 . 1627298 ζ 7 -χ 2 [1 -0 . 2359276 ζ 2 -0 . 2175544 ζ 3 -0 . 2431079 ζ 4 -0 . 2776072 ζ 5 -0 . 3283860 ζ 6 + -0 . 3984877 ζ 7 ] + χ 4 [0 . 25 -0 . 1170266 ζ 2 -0 . 04956483 ζ 3 +0 . 01732049 ζ 4 +0 . 09842336 ζ 5 + +0 . 2055222 ζ 6 +0 . 3503737 ζ 7 ] , (41) \nwhere again the only nonvanishing spin corrections correspond to even powers of χ .', 'C. Quadrupole moment': 'According to the BH no-hair theorems, the quadrupole moment (as well as the higher-order multipole moments) of any regular, stationary, asymptotically flat BH in GR is uniquely determined by its mass M and angular momentum J [4143]. A deformed Kerr geometry as the one just discussed, does not necessarily possess this unique no-hair property. Since the dilaton charge of this solution is not an independent parameter, the multipole moments of an EDGB BH can all be written in terms of M and J , but the relations among them will change with respect to Kerr. The ζ corrections to the BH quadrupole moment are, thus, relevant to test the Kerr hypothesis [2-4]. \nTo compute the quadrupole moment, we follow the general approach described in [44], in which the multipole moments of an asymptotically flat geometry are read off the asymptotic behavior of the metric. This approach requires the metric to be expressed in asymptotically Cartesian and mass- \ncentered (ACMC) coordinates. In particular, in order to extract the quadrupole moment, the metric has to be ACMC-2, i.e., g tt and g ij ( i, j glyph[negationslash] = t ) should not contain any angular dependence up to O (1 /r 2 ) terms. In our case, the coordinate transformation that enforces such property is \nr → r + χ 2 M 2 2 r [ 1 + M r -2 M 2 r 2 + M (6 M -r ) r 2 cos 2 θ ] , θ → θ + χ 2 M 3 r 3 sin θ cos θ , \nand does not involve the EDGB coupling ζ or spin corrections higher than second order. In the new ACMC-2 coordinates, the g tt component reads \ng tt = -1 + 2 M r + √ 3 2 r 3 [ Q 20 Y 20 + +( l = 0 pole)] + O ( M 4 r 4 ) , (42) \nwhere Y 20 is the ( l = 2 , m = 0) spherical harmonic, and Q 20 is the m = 0 mass quadrupole moment. From our explicit solution, we obtain to order O ( ζ 7 , χ 5 ), \nQ 20 = -√ 64 π 15 χ 2 M 3 [( 1 + 0 . 1061619 ζ 2 +0 . 07524246 ζ 3 +0 . 07459416 ζ 4 +0 . 07756926 ζ 5 + +0 . 08553316 ζ 6 +0 . 09805643 ζ 7 ) -χ 2 ζ 2 ( 0 . 0308519 + 0 . 0408857 ζ +0 . 0638894 ζ 2 + +0 . 0866408 ζ 3 +0 . 116314 ζ 4 +0 . 154763 ζ 5 )] . (43) \nInterestingly, the O ( χ 4 ) corrections to the quadrupole moment are proportional to ζ 2 ; i.e., they vanish in the GR limit. For ζ ∼ 0 . 4, the O ( ζ 2 , χ 2 ) correction to the quadrupole moment relative to the Kerr case is about 1 . 7%, whereas the O ( ζ 3 , χ 2 ) correction is approximately 0 . 5%. Finally, for ζ ∼ 0 . 4 and χ ∼ 0 . 6, the O ( ζ 2 , χ 4 ) correction is approximately 0 . 1%. \nWe remark that the quadrupole moment of spinning EDGB BHs has been computed numerically in [30]. Our solution has the advantage of giving this quantity in analytical form.', 'D. Geodesics and epicyclic frequencies': "We shall now consider timelike geodesics in the slowly rotating EDGB BH spacetime. We assume a minimally coupled test particle and restrict to equatorial orbits, for which θ = π/ 2 and dθ = 0. We first compute stable circular orbits; then, by considering small perturbations of these orbits, we derive the epicyclic frequencies ω r and ω θ (see, e.g., Refs. [31-33]). For a stationary-axisymmetric spacetime, the ISCO corresponds to the radius at which the second derivative of the effective potential \nV ( r ) = 1 g rr ( E 2 g ϕϕ +2 E Lg tϕ + L 2 g tt g 2 tϕ -g tt g ϕϕ -1 ) (44) \nvanishes. Here, we have introduced the particlespecific energy and angular momentum E and L [45], given by \nE = -g tt + g tϕ ω ϕ √ -g tt -2 g tϕ ω ϕ -g ϕϕ ω 2 ϕ , (45) \nL = g tϕ + g ϕϕ ω ϕ √ -g tt -2 g tϕ ω ϕ -g ϕϕ ω 2 ϕ , (46) \nwhere ω ϕ is the azimuthal angular velocity \nω ϕ = -g tϕ,r + √ g 2 tϕ,r -g tt,r g ϕϕ,r g ϕϕ,r . (47) \nSolving V '' ( r ) = 0 order by order, we obtain the ISCO radius up to O ( ζ 7 , χ 5 ): \nr ISCO M = 7 ∑ i =0 ζ i ( a i + b i χ + c i χ 2 + d i χ 3 + + e i χ 4 + f i χ 5 ) . (48) \nOrbits with radius r > r ISCO are stable. Under a small perturbation, a massive particle orbiting in one of these stable, circular orbits oscillates with radial and vertical frequencies given by [31-33] \nω 2 r = ( g tt + ω ϕ g tφ ) 2 2 g rr ∂ 2 U ∂r 2 ∣ ∣ ∣ ∣ l , (49) \nω 2 θ = ( g tt + ω ϕ g tφ ) 2 2 g θθ ∂ 2 U ∂θ 2 ∣ ∣ ∣ ∣ l . (50) \nThese are the epicyclic frequencies. Here U = g tt -2 lg tϕ + l 2 g ϕϕ , with l = L/ E being the ratio between the particle angular momentum and its energy [33]. The full expressions for ω r , ω θ , as well as for ω ϕ as functions of ( r, M, χ, ζ ) and up to order O ( ζ 7 , χ 5 ) are available in the Mathematica notebook provided as Supplemental Material. We explicitly show here their values at the ISCO: \nMω ϕ ∣ ∣ ISCO = 7 ∑ i =0 ζ i ( a i + b i χ + c i χ 2 + d i χ 3 + + e i χ 4 + f i χ 5 ) , (51) \nMω θ ∣ ∣ ISCO = 7 ∑ i =0 ζ i ( a i + b i χ + c i χ 2 + d i χ 3 + + e i χ 4 + f i χ 5 ) , (52) \nwhereas ω r ∣ ∣ ISCO = 0 as in the Kerr case.", 'E. Comparison with previous results': 'As a check, we can compare our results with those derived in [27], where the metric of the EDGB spinning BH was found to O ( ζ 2 , χ 2 ) in Boyer-Lindquist coordinates. A direct comparison of the metric coefficients is not possible, since the BH solutions have been derived on different charts. However, we can overcome this problem by computing the Kretschmann invariant K = \nK ( r, θ ) = 48 M 2 r 6 + 144 M 2 r 8 [ (1 -8 cos 2 θ ) + M r sin 2 θ +2 M 2 r 2 (3 cos 2 θ -1) ] -ζ 2 r 4 [ 2 M 3 r 3 + M 4 r 4 +144 M 5 r 5 + +14 M 6 r 6 + 128 5 M 7 r 7 -1680 M 8 r 8 ] + ζ 2 χ 2 r 4 [ M 3 r 3 + 54431 1750 M 4 r 4 + 12846 175 M 5 r 5 + 77047 1225 M 6 r 6 + -348909 350 M 7 r 7 -304938 175 M 8 r 8 -28023 35 M 9 r 9 + 359468 35 M 10 r 10 + 53848 5 M 11 r 11 -21984 M 12 r 12 + + ( -80334 875 M 4 r 4 -19638 175 M 5 r 5 -234816 1225 M 6 r 6 + 1448877 350 M 7 r 7 + 711114 175 M 8 r 8 + 92679 35 M 9 r 9 + -2168052 35 M 10 r 10 -59544 5 M 11 r 11 +65952 M 12 r 12 ) cos 2 θ ] . (53) \nReplacing the explicit expression for r h in Eq. (34), we find that on the horizon \nK ( r h , π/ 2) = 3 4 M 4 + 9 χ 2 8 M 4 + + ζ 2 M 4 [ 327 1280 + 404023 χ 2 784000 ] . (54) \nThis result coincides with the Kretschmann scalar derived in [27] and evaluated at the event horizon on the equatorial plane in Boyer-Lindquist coordinates. Finally, we have verified the agreement between the expression for Mω ϕ at the ISCO which is also a gauge invariant quantity - obtained from the metric derived in Ref. [27], and the same expression obtained truncating the expression in Eq. (51) to O ( ζ 2 , χ 2 ).', 'F. Accuracy of the expansion': "In this section, we estimate the accuracy of our perturbative scheme. In particular, we estimate the truncation error arising from neglecting O ( ζ 8 ) terms in the expansion. To this aim, we compare our results with those obtained in Refs. [20, 28, 34], where a solution for slowly rotating BHs in the EDGB theory has been derived at first order in χ and is 'exact' in ζ (i.e., with no perturbative expansion in ζ ). To be consistent, we neglect terms of the order O ( χ 2 ) in Eqs. (34), (48), (51) and (52). \nIn Fig. 1, we compare the dilatonic charge computed in [20, 28] nonperturbatively in ζ , with the expression in Eq. (39), for χ = 0, truncated at various orders of ζ . As expected, for ζ glyph[lessmuch] 0 . 2, higher-order corrections are negligible, but they contribute significantly as ζ → 0 . 691. To O ( ζ 7 ), the deviation from the exact result is about 1% for ζ ∼ 0 . 6 and is as large as 5% for ζ ∼ 0 . 691. \nIn contrast, the O ( ζ 2 ) truncation differs by about 30% as ζ increases to its maximum value. \nFigure 1. Top panel: Dilatonic charge as a function of ζ = α/M 2 computed in [20, 28] (gray markers) compared with the expression in Eq. (39) truncated at O ( ζ 2 ), O ( ζ 5 ), and O ( ζ 7 ) and for vanishing spin. Bottom panel: Relative discrepancy between the perturbative and nonperturbative estimates of the dilatonic charge, as a function of α for various truncations. \n<!-- image --> \nLikewise, for the set of quantities f = { r h , r ISCO , Mω ϕ | ISCO , Mω θ | ISCO } , we compute the relative error \nglyph[epsilon1] n = f ( n, 1) ¯ f -1 , (55) \nwhere ¯ f represents the exact quantity (nonperturbative in ζ ) [28, 34]. We estimate glyph[epsilon1] n at various \norders of approximation in ζ , for different values of the BH spin parameter. In Table I, we show the largest relative errors obtained for all considered quantities, at different levels of accuracy, in the limiting case ζ = 0 . 691 (left) and ζ = 0 . 576 (right). We remark that ζ = 0 . 691 is an extreme situation, since for slightly smaller values of ζ (i.e., ζ = 0 . 576) the deviations are much smaller. \nFigure 1 and Table I show that our analytical solution approaches the exact solution of [20, 28] as the value of n increases, i.e., when we consider more and more terms in the small-coupling expansion. In particular, for r ISCO , Mω ϕ | ISCO , Mω θ | ISCO , the relative errors (for n = 7) are always smaller than 1% for any value of ζ , even for the maximum allowed value, ζ ∼ 0 . 691. For the horizon, the threshold above which glyph[epsilon1] n =7 > 0 . 01 is lower, namely ζ ∼ 0 . 55.", 'G. Are spin corrections important?': 'The analysis presented in the Sec. III F shows that the metric expanded in powers of ζ , which we derived in a closed, analytic form, is a very good approximation of the exact numerical result: it reproduces the most relevant geodesic quantities within 1% for the maximum value ζ ∼ 0 . 691 and within 0 . 3% for ζ ∼ 0 . 576. It is, therefore, justified to adopt such higher-order perturbative expansion as a starting point to devise strong-field tests of gravity. \nIn Ref. [34], we studied the deviations of the azimuthal and epicyclic frequencies in a slowly rotating EDGB BH to first order in the spin. However, deviations from the Kerr case should increase with higher values of the spin. Indeed, as the spin increases, the ISCO gets closer to the horizon, and, therefore, observables from orbits near the ISCO probe a region of higher curvature, where the deviations should be larger. \nIn Fig. 2, we confirm this claim by showing the deviations of the horizon and ISCO locations and, most important, of the azimuthal frequency ω ϕ and angular epicyclic frequency ω θ at the ISCO, relative to their values computed using the Kerr metric approximated at O ( χ 5 ), and as functions of ζ and χ . For a fixed value of ζ , the percentual errors are systematically larger as the spin increases, reaching up to 7% for χ = 0 . 6. This large value of the spin parameter should be considered as an extrapolation. Indeed, our results neglect terms of the order O ( χ 6 ), which introduce corrections of roughly 5% for χ ∼ 0 . 6. \nOur perturbative solution is also useful to estimate the convergence properties of the expansion. From the coefficients listed in Table II, we can compute the ratio of the O ( χ n ) and O ( χ n -1 ) \ncorrections for a given quantity. For the angular epicyclic frequency Mω θ | ISCO , this ratio is roughly (0 . 41 , 0 . 39 , 0 . 37) for n = (3 , 4 , 5), in the extreme case χ ∼ 0 . 5 and ζ ∼ 0 . 5. Therefore, the fifth-order spin correction is about 20% of the quadratic one. Other quantities show a similar behavior. Clearly, the convergence improves for smaller values of χ , whereas it is almost insensitive to the values of the EDGB coupling ζ . \nFinally, we note that the percentual error of the horizon location is almost insensitive to the spin, whereas the epicyclic frequencies are much more sensitive to this parameter.', 'IV. CONCLUDING REMARKS': 'With the advent of precision measurements of the spectrum of accreting compact objects, it is of utmost importance to devise tests of gravity that use these measurements to probe the geometry near compact objects. To this aim, we have considered a specific modified theory - namely, EDGB gravity - as a case study. This theory has some appealing theoretical features; for example, it is free from instabilities and circumvents the BH no-hair theorems. Furthermore, it modifies GR precisely in the strong-curvature regions, while passing all current solar-system and binary-pulsar tests [4]. \nSpinning BHs in this theory have been studied in the past, both numerically [20, 28-30] and analytically [24-27] to leading order in the coupling parameter. Numerical solutions have the advantage of being general, but they are impractical for some applications, for example for Monte Carlo simulations spanning a high-dimensional parameter space. Approximate analytical solutions can be very useful for this purpose, although they are usually perturbative. \nHere, as a first step to develop precision tests of gravity based on geodesic motion near stationary BHs, we have constructed an analytical, approximate solution of EDGB theory describing a deformed Kerr BH. The solution is valid to fifth order in the spin and to seventh order in the coupling parameter, thus, extending previous solutions that are valid only to quadratic order in the coupling and the spin. With the analytical solution at hand, it is straightforward to compute various quantities of interest. We have presented the corrections to the horizon and ergoregion location, moment of inertia and quadrupole moment relative to the Kerr metric, as well as the charge of the dilaton field that characterizes this solution. For a given value of the coupling ζ , the solution depends only on the mass M and on the dimensionless angular momentum χ , while the dilaton charge is fixed in terms of M . In addition, we have computed some geodesic quant- \nTable I. Left: the relative error glyph[epsilon1] n [cf. Eq. (55)] between different quantities listed in the first column, computed through the solution derived in [28], nonperturbative in ζ , and compared with our perturbative results truncated at O ( ζ n ). We consider the maximum value of ζ allowed for BH solutions in EDGB gravity, ζ = 0 . 691, and different values of the BH spin parameter. Right: Same for ζ = 0 . 576. \n| | χ | glyph[epsilon1] n =2 (%) | glyph[epsilon1] n =4 (%) | glyph[epsilon1] n =6 (%) | glyph[epsilon1] n =7 (%) | | χ | glyph[epsilon1] n =2 (%) | glyph[epsilon1] n =4 (%) | glyph[epsilon1] n =6 (%) | glyph[epsilon1] n =7 (%) |\n|-------------|------|----------------------------|----------------------------|----------------------------|----------------------------|-------------|------|----------------------------|----------------------------|----------------------------|----------------------------|\n| r h /M | 0 | 5.9 | 4.45 | 3.72 | 3.48 | r h /M | 0 | 1.33 | 0.52 | 0.24 | 0.17 |\n| r ISCO /M | 0 | 1 | 0.58 | 0.43 | 0.39 | r ISCO /M | 0 | 0.32 | 0.093 | 0.038 | 0.026 |\n| r ISCO /M | 0.05 | 1.11 | 0.65 | 0.49 | 0.44 | | 0.05 | 0.37 | 0.12 | 0.055 | 0.042 |\n| r ISCO /M | 0.1 | 1.23 | 0.72 | 0.54 | 0.49 | | 0.1 | 0.42 | 0.14 | 0.074 | 0.059 |\n| Mω ϕ | ISCO | 0 | 1.36 | 0.79 | 0.59 | 0.53 | Mω ϕ | ISCO | 0 | 0.44 | 0.13 | 0.053 | 0.036 |\n| Mω ϕ | ISCO | 0.05 | 1.56 | 0.95 | 0.72 | 0.66 | | 0.05 | 0.56 | 0.23 | 0.14 | 0.13 |\n| Mω ϕ | ISCO | 0.1 | 1.88 | 1.22 | 0.98 | 0.91 | | 0.1 | 0.8 | 0.44 | 0.35 | 0.33 |\n| Mω θ | ISCO | 0.05 | 1.53 | 0.92 | 0.69 | 0.63 | Mω θ | ISCO | 0.05 | 0.54 | 0.2 | 0.12 | 0.1 |\n| Mω θ | ISCO | 0.1 | 1.78 | 1.13 | 0.88 | 0.81 | | 0.1 | 0.71 | 0.36 | 0.27 | 0.25 | \nFigure 2. Left panel: The percentual error in the horizon radius r h and in the ISCO r ISCO for our perturbative result [Eqs. (34) and (48)], relative to the Kerr solution expanded at O ( χ 5 ), as a function of the EDGB coupling parameter α . We consider three values of the BH spin parameter χ = (0 . 2 , 0 . 4 , 0 . 6). Right panel: Same as the left panel but for the epicyclic frequencies Eqs. (51) and (52) evaluated at the ISCO. \n<!-- image --> \nies, namely, the ISCO location and the azimuthal and epicyclic frequencies as functions of M , χ , ζ , and the orbital radius r . \nWhen truncated at first order in the BH spin, our solution reproduces the most relevant geodesic quantities obtained in [20, 28] with a numerical approach, within 1% for the maximum value ζ ∼ 0 . 691, and within 0 . 3% for ζ ∼ 0 . 576. The accuracy of the solution grows dramatically for smaller values of the coupling. These results indicate that our perturbative solution is a good approximation of the exact numerical results. \nIn a future publication, we will extend the analysis of Ref. [34], which studied how observations of quasiperiodic oscillations in the spectrum of ac- \ncreting BHs can be used to constrain EDGB theory in the strong-field regime, to larger values of the BH spin. A similar analysis can also be performed for other tests based on stationary BHs, for example, those based on the broadened iron line (e.g., [46, 47]) or the continuum fitting method (e.g., [48]; see also, [35]). On the technical side, our perturbative approach is generic: it can be applied to any order in ζ and in χ , as well as to other modified-gravity theories and different spinning solutions. \nA.M. is supported by the NSF Grants No. 1205864, No. 1212433, and No. 1333360. P.P. was supported by the European Community through the Intra-European Marie Curie Contract No. AstroGRAphy-2013-623439 and by FCTPortugal through the Project No. IF/00293/2013. \nThis work was partially supported by the NRHEP Grant No. 295189 FP7-PEOPLE-2011-IRSES.', 'Appendix A: Kerr metric in the Hartle-Thorne approximation': 'In this appendix, we show the form of the BH solution in the Hartle-Thorne approximation for α = 0, i.e., the slowly rotating Kerr BH in GR, up to the fifth order in the BH spin angular momentum, \ng tt = 1 -2 M r + J 2 [( 2 Mr 3 -2 r 4 -4 M r 5 ) P 2 (cos θ ) + 2 cos 2 θ r 4 ] + J 4 { 2 5 M 2 r 6 -12 M 5 r 9 + 11 5 Mr 7 + 6 5 r 8 + [ 146 7 r 8 -16 7 M 2 r 6 + 44 M 7 r 9 + 46 7 Mr 7 -cos(2 θ ) ( 4 Mr 7 + 8 r 8 )] P 2 (cos θ ) -( 8 5 Mr 7 + 24 5 r 8 ) cos(2 θ ) + sin 2 ( θ ) ( 8 3 M 2 r 6 -8 15 Mr 7 -48 5 r 8 ) S 3 ( θ ) + ( 66 35 M 2 r 6 -2 M 3 r 5 -192 M 35 r 9 + 316 35 Mr 7 -48 7 r 8 ) P 4 (cos θ ) } , (A1) \ng rr = -r r -2 M + 2 J 2 r 2 ( r -2 M ) [ 1 rM ( r -5 M ) ( r -2 M ) P 2 (cos θ ) + 1 ] + J 4 (2 M -r ) 3 [ 152 M 2 5 r 7 + 9 5 M 2 r 3 -264 M 5 r 6 -59 5 Mr 4 + 196 5 r 5 + ( -1464 M 2 7 r 7 + 52 7 M 2 r 3 + 1496 M 7 r 6 -242 7 Mr 4 -106 7 r 5 ) P 2 (cos θ ) + ( 2 M 3 r 2 + 2112 M 2 35 r 7 -358 35 M 2 r 3 -4512 M 35 r 6 -8 35 Mr 4 + 2616 35 r 5 ) P 4 (cos θ ) ] , (A2) \ng θθ = -r 2 + J 2 ( 2 Mr + 4 r 2 ) P 2 (cos θ ) -J 4 [( 4 7 M 2 r 4 + 26 7 Mr 5 + 68 7 r 6 ) P 2 (cos θ ) + ( 2 M 3 r 3 + 162 35 M 2 r 4 + 24 35 Mr 5 + 24 35 r 6 ) P 4 (cos θ ) ] , (A3) \ng ϕϕ = g θθ sin θ 2 , (A4) \ng tϕ = 2 J r -J 3 [ 4 5 Mr 4 + 12 5 r 5 + ( 4 Mr 4 + 8 r 5 ) P 2 (cos θ ) + ( 2 3 M 2 r 3 -2 15 Mr 4 -12 5 r 5 ) S 3 ( θ ) ] + J 5 { 24 5 Mr 8 + 72 5 r 9 -6 7 M 3 r 6 -73 35 M 2 r 7 -( 2 15 M 2 r 7 + 2 Mr 8 + 28 5 r 9 ) S 3 ( θ ) + [ 96 35 M 2 r 7 + ( 4 3 M 3 r 6 + 12 5 M 2 r 7 -16 3 Mr 8 -48 5 r 9 ) S 3 ( θ ) + 108 7 Mr 8 + 1016 35 r 9 ] P 2 (cos θ ) + ( 4 M 3 r 6 + 324 35 M 2 r 7 + 48 35 Mr 8 + 48 35 r 9 ) P 4 (cos θ ) + 2 5 ( 1 M 4 r 5 + 1 7 M 3 r 6 -44 7 M 2 r 7 + 8 r 9 ) S 5 ( θ ) } . (A5)', 'Appendix B: Coefficients of the small coupling': "In the following table, we show the numerical coefficients of various analytic expansions presen- \nted in the previous sections, as functions of the BH spin angular momentum and the EDGB coupling parameter. For the sake of clarity all the coefficients are rounded to the seventh digit. The exact \nexpressions are available in a Supplemental Mater- \nial Mathematica notebook. \n- [1] C. M. Will, Living Rev.Rel. 17 , 4 (2014), arXiv:1403.7377 [gr-qc].\n- [2] D. Psaltis, Living Reviews in Relativity 11 (2008), 10.12942/lrr-2008-9.\n- [3] N. Yunes and X. Siemens, Living Reviews in Relativity 16 , 9 (2013), arXiv:1304.3473 [gr-qc].\n- [4] E. Berti, E. Barausse, V. Cardoso, L. Gualtieri, P. Pani, U. Sperhake, L. C. Stein, N. Wex, K. Yagi, et al. , ArXiv e-prints (2015), arXiv:1501.07274 [gr-qc].\n- [5] F. Eisenhauer et al. , in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series , Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 7013 (2008) p. 2, arXiv:0808.0063.\n- [6] K. C. Gendreau, Z. Arzoumanian, and T. Okajima, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series , Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 8443 (2012) p. 13.\n- [7] M. Feroci, L. Stella, M. Van der Klis, T.-L. Courvoisier, M. Hernanz, R. Hudec, A. Santangelo, D. Walton, A. Zdziarski, D. Barret, et al. , Experimental Astronomy 34 , 415 (2012).\n- [8] A. E. Broderick, T. Johannsen, A. Loeb, and D. Psaltis, Astrophysical Journal 784 , 7 (2014), arXiv:1311.5564 [astro-ph.HE].\n- [9] F. Acernese et al. , Class. Quantum Grav. 25 , 114045 (2008).\n- [10] G. M. Harry and the LIGO Scientific Collaboration, Class. Quantum Grav. 27 , 084006 (2010).\n- [11] D. L. Wiltshire, M. Visser, and S. M. 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Schiappa, Class.Quant.Grav. 24 , 361 (2007), arXiv:hep-th/0605001 [hep-th].\n- [22] J. B. Hartle, Astrophys.J. 150 , 1005 (1967).\n- [23] J. B. Hartle and K. S. Thorne, Astrophys.J. 153 , 807 (1968).\n- [24] S. Mignemi and N. Stewart, Phys.Rev. D47 , 5259 (1993), arXiv:hep-th/9212146 [hep-th].\n- [25] N. Yunes and L. C. Stein, Phys.Rev. D83 , 104002 (2011), arXiv:1101.2921 [gr-qc].\n- [26] P. Pani, C. F. Macedo, L. C. Crispino, and V. Cardoso, Phys.Rev. D84 , 087501 (2011), arXiv:1109.3996 [gr-qc].\n- [27] D. Ayzenberg and N. Yunes, Phys.Rev. D90 , 044066 (2014), arXiv:1405.2133 [gr-qc].\n- [28] P. Pani and V. Cardoso, Phys.Rev. D79 , 084031 (2009), arXiv:0902.1569 [gr-qc].\n- [29] B. Kleihaus, J. Kunz, and E. Radu, Phys.Rev.Lett. 106 , 151104 (2011), arXiv:1101.2868 [gr-qc]. \n[30] B. Kleihaus, J. Kunz, and S. Mojica, Phys.Rev. D90 , 061501 (2014), arXiv:1407.6884 [gr-qc]. \n- [31] R. M. Wald, General relativity (University of Chicago press, 2010).\n- [32] A. Merloni, M. Vietri, L. Stella, and D. Bini, Mon.Not.Roy.Astron.Soc. 304 , 155 (1999), astroph/9811198.\n- [33] M. A. Abramowicz and W. Kluzniak, Gen.Rel.Grav. 35 , 69 (2003), arXiv:gr-qc/0206063 [gr-qc].\n- [34] A. Maselli, L. Gualtieri, P. Pani, L. Stella, and V. Ferrari, Astrophys.J. 801 , 115 (2015), arXiv:1412.3473 [astro-ph.HE].\n- [35] C. J. Moore and J. R. Gair, Phys. Rev. D92 , 024039 (2015), arXiv:1507.02998 [gr-qc].\n- [36] T. Kobayashi, M. Yamaguchi, and J. Yokoyama, Prog.Theor.Phys. 126 , 511 (2011), arXiv:1105.5723 [hep-th].\n- [37] G. W. Horndeski, Int.J.Theor.Phys. 10 , 363 (1974).\n- [38] T. P. Sotiriou and S.-Y. Zhou, Phys.Rev.Lett. 112 , 251102 (2014), arXiv:1312.3622 [gr-qc].\n- [39] K. Yagi, Phys.Rev. D86 , 081504 (2012), arXiv:1204.4524 [gr-qc].\n- [40] E. Poisson, A Relativist's Toolkit: The Mathematics of Black-Hole Mechanics (Cambridge University Press, 2004).\n- [41] B. Carter, Phys. Rev. Lett. 26 , 331 (1971).\n- [42] S. W. Hawking and G. F. R. 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Numerical values of the coefficients of the expressions (34),(39),(48), (51)-(52) (left panel), and of the ergosphere and intrinsic curvature radius (38)], (37) (right panel). \n| | r h /M | r ISCO /M | D /M | Mω ϕ | ISCO | Mω θ | ISCO | | r ergo /M | M 2 R intr |\n|---------|---------------|---------------|---------------|---------------------------|------------------------------|---------|-------------------------------------|-------------------------|\n| a 0 | 2.000000 | 6.000000 | 0 | 0.06804138 | 0.06804138 | l 0 | 2.000000 | 0.5000000 |\n| b 0 | 0 | -3.265986 | 0 | 0.05092593 | 0.04166667 | m 0 | 0 | 0 |\n| c 0 | -0.2500000 | -0.2962963 | 0 | 0.03717075 | 0.02488551 | n 0 | | -0.06640625 -0.05859375 |\n| d 0 | 0 | -0.1436429 | 0 | 0.02797068 | 0.01521776 | p 0 | -0.2500000 | -0.3750000 |\n| e 0 | -0.07812500 | -0.08957762 | 0 | 0.02176680 | 0.009504151 | q 0 | -0.04687500 | -0.2343750 |\n| f 0 | 0 | -0.06362468 | 0 | 0.01744794 | 0.005997155 | u 0 | 0.03515625 | 0.1054688 |\n| a 1 | 0 | 0 | 0.5000000 | 0 | 0 | l 1 | 0 | 0 |\n| b 1 | 0 | 0 | 0 | 0 | 0 | m 1 | 0 | 0 |\n| c 1 | 0 | 0 | -0.1250000 | 0 | 0 | n 1 | 0 | 0 |\n| d 1 | | | 0 | 0 | 0 | | | |\n| e | 0 | 0 | -0.06250000 | 0 | 0 | p 1 | 0 | 0 |\n| 1 f | 0 0 | 0 0 | 0 | 0 | 0 | q 1 u | 0 | 0 0 |\n| 1 a 2 | -0.07656250 | -0.1047904 | 0.1520833 | 0.001563316 | 0.001563316 | 1 l 2 | 0 -0.07656250 | 0.03828125 |\n| b 2 | 0 | -0.1201586 | 0 | 0.003733697 | 0.003267414 | m 2 | 0.02239583 | -0.02182674 |\n| | -0.005273438 | 0.01442503 | -0.06562500 | 0.003913488 | 0.002948313 | | | |\n| c 2 d 2 | 0 | 0.03108340 | 0 | 0.003160772 | 0.002041599 | n 2 p 2 | -0.02390784 -0.01729976 -0.02766927 | -0.1164568 |\n| e 2 | 0.0007317631 | 0.02534504 | -0.01282676 | 0.002252579 | 0.001211371 | q 2 | 0.003249614 -0.02236320 | |\n| f 2 | 0 | 0.03275054 | 0 | 0.001261845 | 0.0005010288 | u 2 | 0.02138998 | 0.1302572 |\n| a 3 | -0.05482722 | -0.05057329 | 0.09658358 | 0.0007597788 | 0.0007597788 | l 3 | -0.05482722 | 0.02741361 |\n| b 3 | 0 | -0.06564501 | 0 | 0.001933365 | 0.001708284 | m 3 | 0.02660141 | -0.02038983 |\n| c 3 | 0.003374719 | 0.02695641 | -0.06503722 | 0.001759287 | 0.001360011 | n 3 | -0.02726616 0.001655395 | |\n| d 3 | 0 | 0.03707053 | 0 | 0.0009886223 | 0.0006954496 | p 3 | -0.02322669 -0.08694771 | |\n| e 3 | -0.003268537 | 0.008889195 | 0.006444151 | 0.0005021134 | 0.0004182817 | q 3 | 0.006734437 0.01982431 | |\n| f 3 | 0 | 0.01125766 | 0 | 0.0001173107 | 0.0002713766 | u 3 | 0.01726319 | 0.1306188 |\n| a 4 | -0.05139096 | -0.03780985 | 0.08178788 | 0.0005969829 | 0.0005969829 | l 4 | -0.05139096 | 0.02789366 |\n| b 4 | | -0.05431288 | | 0.001640950 | 0.001457307 | m | 0.03391131 | -0.02622298 |\n| c 4 | 0 0.007351713 | 0.03876245 | 0 -0.06717236 | 0.001340086 | 0.001049576 | 4 n 4 | -0.04523654 | 0.01648181 |\n| d 4 | 0 | 0.05336676 | 0 | 0.0003245650 | 0.0002579998 | p 4 | -0.02655960 | -0.1007545 |\n| e 4 | -0.01308955 | -0.0002670172 | | 0.02912691 -0.00008270490 | 0.0001598391 | q 4 | 0.01024171 | 0.05758840 |\n| f 4 | 0 | 0.00005147144 | | -0.0002712206 | 0.0002346154 | u 4 | 0.02190527 | 0.1906275 |\n| a 5 | | | 0 | 0.0005272127 | 0.0005272127 | l 5 | | 0.02948113 |\n| | -0.05266569 | -0.03321722 | 0.07886477 | | | | -0.05266569 | -0.03204816 |\n| b 5 | 0 | -0.04890924 | 0 | 0.001475785 | 0.001313209 | m 5 | 0.04318465 | |\n| c 5 | 0.01223578 | 0.05011869 | -0.07508136 | 0.0009815575 | 0.0007719423 | n 5 | -0.06780117 | 0.03771938 |\n| d 5 | 0 | 0.06775197 | 0 | -0.0003132155 | -0.0001796837 | p 5 | -0.03094887 | -0.1122069 |\n| e 5 | -0.02668707 | -0.01513785 | 0.05662227 | | -0.0005664351 -0.00003872048 | q 5 | 0.01536541 | 0.1033644 |\n| f 5 | 0 | -0.02079280 | 0 | -0.0003557189 | 0.0003967294 | u 5 | 0.02574869 | 0.2495574 |\n| a 6 | -0.05753945 | -0.03250101 | 0.08245910 | 0.0005165825 | 0.0005165825 | l 6 | -0.05753945 | 0.03296015 |\n| b 6 | 0 | -0.04824270 | 0 | 0.001458090 | 0.001298530 | m 6 | 0.05574249 | -0.04017469 |\n| c 6 | 0.01812429 | 0.06363000 | -0.08788829 | 0.0007567590 | 0.0005928172 | n 6 | -0.1009268 | 0.06678447 |\n| d 6 | 0 | 0.08594476 | 0 | -0.0008980528 | -0.0005936325 | p 6 | -0.03761820 | -0.1310279 |\n| e 6 | -0.04690661 | -0.03465520 | 0.09290842 | -0.0009679387 | -0.0001794779 | q 6 | 0.02256511 | 0.1634876 |\n| f 6 | 0 | -0.04866939 | 0 | -0.0002465314 | 0.0007018829 | u 6 | 0.03145506 | 0.3323836 |\n| a 7 | -0.06565095 | -0.03416370 | 0.09098999 | 0.0005425110 | 0.0005425110 | l 7 | -0.06565095 | 0.03820390 |\n| b 7 | 0 | -0.05069519 | 0 | 0.001533246 | 0.001365759 | m 7 | 0.07278610 | -0.05103945 |\n| c 7 | 0.02592898 | 0.08041305 | -0.1064184 | 0.0005835871 | 0.0004504112 | n 7 | -0.1481975 | 0.1073399 |\n| d 7 | 0 | 0.1086434 | 0 | -0.001525555 | -0.001044296 | p 7 | -0.04685712 | -0.1565866 |\n| e 7 | -0.07619661 | -0.06177540 | 0.1427116 | -0.001328310 | -0.0002654697 | q 7 | 0.03308656 | 0.2435265 |\n| f 7 | 0 | -0.08762124 | 0 | 0.0001137773 | 0.001223565 | u 7 | 0.03891437 | 0.4418871 |"}

2013JPhA...46u4001A

Black holes in three dimensional higher spin gravity: a review

2013-01-01

['-', '-']

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We review recent progress in the construction of black holes in three dimensional higher spin gravity theories. Starting from spin-3 gravity and working our way toward the theory of an infinite tower of higher spins coupled to matter, we show how to harness higher spin gauge invariance to consistently generalize familiar notions of black holes. We review the construction of black holes with conserved higher spin charges and the computation of their partition functions to leading asymptotic order. In view of the anti-de Sitter/conformal field theory (CFT) correspondence as applied to certain vector-like conformal field theories with extended conformal symmetry, we successfully compare to CFT calculations in a generalized Cardy regime. A brief recollection of pertinent aspects of ordinary gravity is also given. <P />This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Higher spin theories and holography’.

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https://arxiv.org/pdf/1208.5182.pdf

{'Black holes in three dimensional higher spin gravity: A review': 'Martin Ammon, Michael Gutperle, Per Kraus and Eric Perlmutter 1 \nDepartment of Physics and Astronomy University of California, Los Angeles, CA 90095, USA', 'Abstract': 'We review recent progress in the construction of black holes in three dimensional higher spin gravity theories. Starting from spin-3 gravity and working our way toward the theory of an infinite tower of higher spins coupled to matter, we show how to harness higher spin gauge invariance to consistently generalize familiar notions of black holes. We review the construction of black holes with conserved higher spin charges and the computation of their partition functions to leading asymptotic order. In view of the AdS/CFT correspondence as applied to certain vector-like conformal field theories with extended conformal symmetry, we successfully compare to CFT calculations in a generalized Cardy regime. A brief recollection of pertinent aspects of ordinary gravity is also given. \nJuly 2012', 'Contents': '| 1. Introduction . . . . . | . . . . . . . . . | . . . . . . . . . . . . | | | | | | | | | | . . . | . | | | | . | . | . | . . | 1 | | 6 |\n|----------------------------------------------------------------|-------------------------------------------------------------------------|---------------------------|-------|----------|-----|---------|---------|-------|-----|-------|-----|---------|-------|-----|---------|-----|-------|-----|----------|-------|------|-------|-------|\n| 2. Black holes in three dimensional gravity | . . | . . . . . | . . | . | . | | | | | . | | . . . . | . . | . . | . | | . | . . | . | | . | . | 6 |\n| 2.3. Charged BTZ black holes | 2.1. Action, boundary conditions, and Virasoro symmetry . . . . . . . . | . . . . . . . . . . . . | | | | | | | . | . . | . . | . | . | . . | . . | 10 | | | | | . | | |\n| | | . . | | | | | | | | | . | . . | . | . . | . | . | | 11 | 11 | 11 | 11 | 11 | 11 |\n| 2.4. Chern-Simons formulation of three dimensional AdS gravity | | . . . | | | | | | | | | | . . . . | . . | . . | . | . . | 13 | 13 | 13 | 13 | 13 | 13 | 13 |\n| 2.5. Black hole entropy in the Chern-Simons formulation ⊕ | | | | . . . . | | | . | | . | . . | | . | . | . | . | | 15 | 15 | 15 | 15 | 15 | 15 | 15 |\n| 3. Black holes in sl ( N,R ) sl ( N,R | ⊕ sl (3 , R ) higher spin gravity and | ) higher spin gravity | W 3 | symmetry | | | | | . | | | | | | . | | | | . | | | | |\n| 3.1. Review of sl (3 , R ) | . . . . . . | | | . | . . | | . | | | . | . | . . | . . | . | . . . . | . | . | . | . 15 . . | 19 | 19 | 19 | 19 |\n| 3.2. Black holes with higher spin charge | | | | . | . | | . | . | . | . | . | | . | . | | . | | . | . | 22 | 22 | 22 | 22 |\n| 3.3. Holonomy and integrability | . . . . . | . | | . | . | . . | . | . . | . | | | . | . . | . . | . . | . | 23 | 23 | 23 | 23 | 23 | 23 | 23 |\n| 3.4. Finding the black hole gauge | . . . . . | | . . . | . . . | | . . | . | . | | | . | . | . | . | | . | | . | . 26 | | | | |\n| 3.5. Black hole thermodynamics . | . . . | . | . | . | . . | . . . | . | | . | . | . | . | | . | | . | | . | 28 | 28 | 28 | 28 | 28 |\n| 3.6. Other black holes in spin-3 gravity | | . . | . . . | . . | . | . . | . . | | . | . . | . | . | . | | . | . | . | | . 29 | . 29 | . 29 | . 29 | . 29 |\n| 3.7. Black hole entropy from the Euclidean action | . | | . | . | . . | . . . | . . . . | | | | . | . . | . | . | . | | | | . | . | . | . | . |\n| 4. hs[ λ ] black holes . . . . . . | . . . | . . | . | . . | . | . . | . . | . | | | . | . . . | . . | | 30 | 30 | 30 | 30 | 30 | 30 | 30 | 30 | 30 |\n| 4.1. hs[ λ ] higher spin gravity | . . . . | . | . . | . | . | . | | . | . | . | . . | . . | . . | . | | | . . . | | . | . | . | . | . |\n| . 4.2. Building the hs[ λ ] black hole | . . . . | . | . | . | . | . . . | | . | . | | . | . . | | . | . | | | . . | . | . | . . | 30 33 | 30 33 |\n| | . . . | . . | . | . . | . | . . | | | . . | . . | | . . | | . . | . | | . . | . | . . | . 36 | . 36 | . 36 | . 36 |\n| 4.3. Black hole partition function . . . . | . | . . | | . | . | . . | | . | | . | . | | | | | | | | | | | | |\n| 4.4. Matching to CFT . . | . . . . | | . . | . . | . . | . . | . | | . | . . . | . | . | . | . . | . . | . | . | 38 | 38 | 38 | 38 | 38 | 38 |\n| 5. Discussion . . . . . Appendix A. Some details on hs[ λ ] | . . . . . . . . . . . | . . . . . | . . . | . . . | . | . . . . | . | . . . | . | . . | . . | . . | . . . | 41 | | | . | | . | | . | | 43 |', '1. Introduction': "Black holes play a central role in both classical and quantum gravitational physics. In classical general relativity black holes constitute an important class of exact solutions of Einstein's equations, where questions such as the causal structure of spacetime and the nature of singularities can be addressed. There is overwhelming evidence that black holes exist as astrophysical objects in binary systems and at the center of galaxies. \nThe discovery of Hawking [1] that black holes radiate as black bodies leads to the black hole information paradox, whereby in the process of black hole formation and evaporation pure states seem to evolve into mixed states and hence information (and consequently unitary time evolution in quantum mechanics) is not preserved. Any theory of quantum gravity has to resolve this clash between classical gravity and quantum mechanics. String theory has made important steps in this direction. First, string theory provides a microscopic accounting of the Bekenstein-Hawking entropy of certain near extremal black holes found in superstring theories [2]. Second, the AdS/CFT correspondence [3,4,5] provides a conceptual resolution of the information paradox for black holes in asymptotically anti-de Sitter spacetimes. As time evolution on the CFT side is unitary, the time evolution on the dual bulk gravity side must be unitary too. These arguments give credence to the point of \nview that the information of an evaporating black hole is somehow encoded in the outgoing Hawking radiation. \nThe entropy counting in string theory generically involves a continuation from weak to strong coupling and works most reliably for black holes in theories which enjoy nonrenormalization due to a large amount of supersymmetry. One needs to go beyond supersymmetric examples in order to obtain a microscopic counting of the entropy for Schwarzschild black holes. Unfortunately this goal has not yet been achieved. It is therefore useful to consider black holes in simplified theories and toy models, where supersymmetry and non renormalization theorems are not necessary to perform calculations reliably. \nOne such example is three dimensional gravity. The main simplifying feature of this theory is that a graviton in three dimensions does not have any propagating degrees of freedom. However, in asymptotically AdS space the theory has nontrivial dynamics given by the dual two-dimensional CFT which is localized at the boundary of AdS 3 . A reflection of the topological nature of three dimensional gravity is the fact that with a negative cosmological constant Einstein gravity can be reformulated as a Chern-Simons gauge theory with gauge algebra sl (2 , R ) ⊕ sl (2 , R ) [6,7]. The diffeomorphism and local Lorentz symmetries of the gravitational theory are reinterpreted as gauge transformations of the Chern-Simons theory. In [8] it was discovered that the generators of the diffeormorphism transformations that preserve the asymptotic form of the AdS metric form the Virasoro algebra, to be viewed as the algebra of conformal symmetries in the dual CFT. In the Chern-Simons formulation of the theory the Virasoro generators manifest themselves in terms of Hamiltonian reduction of the gauge connection. \nIn three dimensional gravity with a negative cosmological constant BTZ black hole solutions [9,10] can be constructed as orbifolds of the AdS 3 vacuum. These black hole solutions are simpler than their higher dimensional cousins as they are locally still given by the AdS 3 vacuum. The BTZ black hole nevertheless has a horizon, finite entropy and nonzero Hawking temperature. In the limit of high temperature the entropy of the black hole is related to the central charge of the CFT via the Cardy formula. The simplicity of the BTZ black hole allows one to address difficult conceptual questions related to black holes. Two examples are the nature of entanglement of degrees of freedom inside and outside the horizon [11] and whether it is possible to access the region behind the black hole horizon by correlation functions of operators localized on the boundary [12,13]. \nIn the AdS/CFT correspondence bulk calculations can be most easily performed in the weak coupling, small curvature regime where semi-classical gravity methods are sufficient. However, in this limit we lose several characteristic features of string theory such as the non-locality due to the finite extent of strings, the infinite tower of fields of ever increasing spin and the generalization of diffeomorphism invariance of gravity to a very large gauge symmetry which is manifest in string field theory. \nAn interesting theory has been developed over the last 20 years by Vasiliev and collaborators which is in some sense located 'in-between' Einstein gravity and full fledged string theory. It is an extension of ordinary gravity with a negative cosmological constant \ncoupled to an infinite tower of massless fields of higher spin in three or more dimensions. The classical equations of motion are explicitly known, and are nonlocal as well as highly nonlinear. Note however that at present no action is known which reproduces the equations of motion. The no-go theorems in flat space are evaded due to the absence of an S-matrix in AdS space; see, e.g., [14] for a modern discussion. The higher spin theory has a large higher spin gauge symmetry which generalizes the spin 2 gauge symmetry of ordinary gravity. A partial list of the original papers is [15,16,17,18,19], see also the reviews [20,21], which contain more comprehensive lists of references. There exists some speculation that the higher spin theories might be related to a tensionless limit of (topological) string theory [22]. These questions as well as the question of whether classical Vasiliev theory theory can be gauge fixed and quantized while preserving the higher spin symmetries remain open at present. \nThere has been a renewed interest in Vasiliev theory due to the duality conjecture that relates the large N limit of vector models in three dimensions to higher spin gravity theories in AdS 4 . The most developed form of this conjecture is due to Klebanov and Polyakov [23], stating that Vasiliev theory in AdS 4 is dual to the large N limit of the three dimensional O ( N ) vector model. The massless higher spin fields in the bulk holographically realize the infinite set of higher spin conserved currents in the (free) UV boundary theory. An impressive amount of evidence for the duality has been obtained in recent years [24,25,26,27,28,29,30], including the calculation of three point functions in the bulk [31,32,33]. A proof of the duality relation has recently been obtained with some assumptions which are very reasonable from the point of view of AdS/CFT [34,35]. Other examples of dualities between higher spin theories in AdS 4 and vector like theories in three dimensions were presented in [36,37,38]. \nIn the present review we will focus on black holes in higher spin gravity in three dimensions. A significant simplification over higher spin theories in four or more dimensions is the fact that the theory can be implemented as a Chern-Simons gauge theory [39,40,41]. In its simplest realization the sl (2 , R ) ⊕ sl (2 , R ) Chern Simons gauge group is replaced by sl ( N,R ) ⊕ sl ( N,R ). Ordinary gravity is contained as a subsector of the theory and realized by the principal embedding of sl (2 , R ) inside sl ( N,R ). The theory describes massless interacting fields of spin s = 2 , 3 , · · · , N . The simplest case N = 3 describes ordinary gravity coupled to a massless spin three field [41,42] and exemplifies a general feature of the higher spin gravity theories: the gauge transformations include coordinate transformations as well as well as spin 3 gauge transformations under which the metric g µν transforms in a nontrivial way. Consequently, diffeomorphism invariant quantities such as the Ricci scalar are not gauge invariant under the higher spin gauge transformations. Similarly, global aspects of the spacetime such as its causal structure and the existence of a horizon are also not gauge invariant in higher spin gravity. \nThe three dimensional Vasiliev theory is a one parameter family of theories based on an infinite-dimensional gauge algebra hs[ λ ] [43]. For these theories it is possible to consistently couple propagating matter fields to the infinite set of higher spin fields. Note \nthat there appears to be little understanding of how to consistently couple matter to higher spin theories for finite dimensional gauge algebras such as sl ( N,R ). \nThe asymptotic symmetries of the higher spin gravity can most easily be obtained by Hamiltonian reduction [44,45,46,41,47,48]. This construction generalizes the realization of the Virasoro algebra for the sl (2 , R ) case of pure gravity. Instead of the Virasoro algebra one obtains W N algebras as the asymptotic symmetry of the dual CFTs. Note that just like pure gravity the higher spin gravity theories based on sl ( N,R ) have no propagating bulk degrees of freedom. Similarly for the hs[ λ ] theory, the asymptotic symmetry is a particular W -algebra known as W [ λ ]. \nAconcrete connection between bulk and boundary is given by a conjectured AdS/CFT duality due to Gaberdiel and Gopakumar [49,50]. This conjecture relates the three dimensional Vasiliev theory to a large N limit of W N minimal models. The W N minimal model CFT has a representation in terms of a coset theory and the 't Hooft like limit is defined as follows: \n∞ \nsu ( N ) k ⊕ su ( N ) 1 su ( N ) k +1 , k, N →∞ , λ = N k + N fixed (1 . 1) \nwhere the 't Hooft coupling λ is identified with the deformation parameter of the higher spin algebra hs[ λ ]. The central charge of the CFT is given in the large N limit by \nc ∼ N (1 -λ 2 ) . (1 . 2) \nThe fact that the central charge grows like N instead of N 2 makes this duality analogous to the Klebanov-Polyakov dual of vector models in AdS 4 . A very appealing feature of this example of a AdS/CFT duality is that the coset dual CFT is very well under control. In addition both the bulk and boundary have a huge symmetry group which strongly constrains the dynamics. \nA considerable amount of evidence for this conjecture has been obtained recently. For example the symmetries and the spectrum of the bulk and boundary CFT have been matched in [51,50,48]. The calculation of the 1-loop contribution to the bulk partition function utilizing heat-kernel methods [52,53] agrees with the large N limit of the CFT partition function of the coset theories (1.1). In [54,55,56,57] certain classes of correlation functions were calculated both in the bulk and boundary theory and were found to match. Some open questions remain, such as the exact structure of the symmetry algebra of the hs[ λ ] theory at the quantum level, i.e. for finite central charge as well as the identification of certain light states which are present in the CFT with bulk configurations. For recent progress towards resolving these questions, and a slight modification of the original conjecture of [49], see [50,58]. \nIn [59,60] the Gaberdiel-Gopakumar conjecture has been generalized to other coset CFTs such as so (2 N ) k ⊕ so (2 N ) 1 /so (2 n ) k +1 coset, which is conjectured to be dual to a Vasiliev theory with massless fields with even spin only. Furthermore versions of the \nconjecture which relate superconformal cosets and supersymmetric higher spin theories were put forward in [61,62,63,64,65,66,67,68]. \nBlack holes are important laboratories to explore both perturbative and non perturbative aspects of this duality further. An immediate conceptual challenge arises due to the existence of the higher spin gauge symmetries. In ordinary gravity the global causal structure of a spacetime, in particular the existence of a black hole horizon and singularities are diffeomorphism invariant properties of the spacetime. In higher spin gravity however these properties are not invariant under higher spin gauge transformations and consequently it is not at all obvious how we define a black hole in higher spin gravity. \nSince it is possible to canonically embed sl (2 , R ) as a sub algebra into both sl ( N,R ) and hs[ λ ], a solution of ordinary gravity can always be embedded into a solution of higher spin gravity. Consequently, the BTZ black hole solution of ordinary three dimensional gravity is always a solution of higher spin gravity. However the presence of higher spin fields raises the question of how to generalize the BTZ black hole to a black hole which carries conserved higher spin charges. \nWe would like to find a gauge invariant characterization of what constitutes a black hole in the higher spin gravity. In [69] a criterion for what constitutes a black hole was presented, which states that the holonomies of the gauge connections around the contractible Euclidean time cycle have to be identical to the holonomies of the gauge connection of the embedded BTZ black hole. This condition leads to sensible black hole thermodynamics, i.e. the black hole has a free energy which satisfies integrability conditions that are equivalent to the first law of thermodynamics. In [70,71] it was shown that if the holonomy conditions are satisfied, there exists a higher spin gauge transformation which makes the solution manifestly a black hole, i.e. there is a horizon and the higher spin fields are smooth on the horizon. On the dual CFT side the solution describes the the finite temperature partition function with a nonzero chemical potential for the higher spin W charge. Consequently, the holographic calculation of the partition functions produces nontrivial predictions on the CFT side [72,73], which can be checked by direct CFT calculations [74] when a CFT with the right symmetries and large central charge is available. \nThe structure of this review is as follows. In section 2 we give a brief review of ordinary three dimensional gravity, its Chern-Simons formulation, the BTZ black hole solution and the AdS 3 /CFT 2 correspondence. While this material is well known, it makes the review self contained and emphasizes similarities in the construction of the Chern-Simons formulation of ordinary gravity and higher spin gravity as well as the construction of BTZ black hole and the higher spin black holes. In section 3 we describe the construction of black holes carrying higher spin charge in sl ( N,R ) ⊕ sl ( N,R ) higher spin gravity. The simplest case N = 3 is discussed in detail. In section 4 the construction of black holes is generalized to the hs[ λ ] ⊕ hs[ λ ] higher spin gravity. An important check of the proposal is performed by comparing the asymptotic form of the gravity partition function with CFT. In section 5 we conclude with a list of open problems and directions for future research.", '2. Black holes in three dimensional gravity': 'Three dimensional gravity has no propagating degrees of freedom; the Einstein equations fix the metric to be locally flat in the absence of a cosmological constant, or locally (A)dS in the presence of a (negative) positive cosmological constant. Nevertheless, the theory is far from trivial. In particular, the case with negative cosmological constant admits black holes and figures importantly in the AdS/CFT correspondence. In this section we review the main aspects of this story, and set the stage for the generalization to the higher spin theory. We begin by working in the metric formulation, and then show how to recover the same results in the Chern-Simons formulation. Some relevant pedagogical references include [75,76,41].', '2.1. Action, boundary conditions, and Virasoro symmetry': "The action for pure gravity with negative cosmological constant is \nI = 1 16 πG ∫ d 3 x √ g ( R -2 /lscript 2 ) + I bndy . (2 . 1) \nWe work in Euclidean signature. The boundary terms contained in I bndy are needed to ensure a proper variational principle, and will be displayed below. \nOne solution of the equations of motion is AdS 3 in global coordinates, \nds 2 = (1 + r 2 //lscript 2 ) dt 2 + dr 2 1 + r 2 //lscript 2 + r 2 dφ 2 . (2 . 2) \nAs noted above, this is in fact the general solution of the field equations up to coordinate transformations and global identifications. The physical content of the theory, at least at the classical level, thus corresponds to understanding the effect of these coordinate transformations and global identifications. \nTo define the theory we need to impose boundary conditions at infinity. These take the form of falloff conditions on the metric components. We'd like to allow for the most general falloff conditions compatible with a well-defined variational principle. It is convenient to write the line element in the form \nds 2 = dρ 2 + g ij ( x k , ρ ) dx i dx j , i, j = 1 , 2 . (2 . 3) \nWe then demand that g ij takes the Fefferman-Graham [77] form as ρ →∞ , \ng ij ( x k , ρ ) dx i dx j = e 2 ρ//lscript g (0) ij ( x k ) + g (2) ij ( x k ) + . . . . (2 . 4) \nThis defines g (0) ij as the conformal boundary metric; in the AdS/CFT correspondence the CFT lives on a space with this metric. By allowing for a general g (0) ij we arrive at the notion of an asymptotically, locally, AdS 3 spacetime. See, e.g., [78] for more details. \nOur variational principle is defined by the demand that the action be stationary under any variation that respects the Einstein equations and the falloff conditions (2.4), with g (0) ij fixed. With g (0) ij allowed to vary, we will have the on-shell variation \nδI = 1 2 ∫ d 2 x √ g (0) T ij δg (0) ij , (2 . 5) \nwhich defines the boundary stress tensor T ij [79]. These requirements fix the boundary terms in the action to be \nI bndy = 1 8 πG ∫ ∂ M d 2 x √ g ( Tr K -1 /lscript ) +anomaly term , (2 . 6) \nwhere K ij = 1 2 ∂ η g ij is the extrinsic curvature. The anomaly term is needed to cancel a logarithmic divergence, but it will not be needed here. The integral in (2.6) is performed at a fixed η , whose value is taken to infinity at the end of the computation, with a finite result. The boundary stress tensor works out to be \nT ij = 1 8 πG/lscript ( g (2) ij -Tr( g (2) ) g (0) ij ) , (2 . 7) \nwhere the trace is defined by raising an index with g (0) ij . \nWe now consider the case that the conformal boundary is a cylinder, as in (2.2). We work in complex coordinates, w = φ + it//lscript , so that g (0) ww = g (0) ww = 0 and g (0) ww = 1 / 2. The asymptotic symmetry group is obtained by considering coordinate transformations that preserve the form of g (0) ij . This is the case for the following infinitesimal transformations [8]: \nw → w + /epsilon1 ( w ) -/lscript 2 2 e -2 ρ//lscript ∂ 2 w /epsilon1 ( w ) w → w + /epsilon1 ( w ) -/lscript 2 2 e -2 ρ//lscript ∂ 2 w /epsilon1 ( w ) ρ → ρ -/lscript 2 ( ∂ w /epsilon1 ( w ) + ∂ w /epsilon1 ( w ) ) , (2 . 8) \nwhere /epsilon1 ( w ) and /epsilon1 ( w ) are arbitrary functions. These coordinate transformations leave g (0) ij fixed but act nontrivially on g (2) ij , and so transform the stress tensor as \nT ww → T ww +2 ∂ w /epsilon1 ( w ) T ww + /epsilon1 ( w ) ∂ w T ww -c 24 π ∂ 3 w /epsilon1 ( w ) (2 . 9) \nc = 3 /lscript 2 G . (2 . 10) \nwith \nThe analogous result holds for T ww . This is the transformation law for a stress tensor in a two-dimensional conformal field theory, with c being the central charge. \nBy following the rules of the AdS/CFT correspondence we can compute boundary correlation functions of the stress tensor. Following precisely the same logic as in CFT (e.g., see [80]), the conformal symmetry implied by (2.9) leads to OPE relations 2 \nT ww ( w ) T ww (0) ∼ c 8 π 2 w 4 + 1 πw 2 T ww (0) + 1 2 πw ∂ w T ww (0) . (2 . 11) \nAlternatively, if we define the mode generators L n via \nL n = ∮ dw e -inw T ww (2 . 12) \nthen the generators will obey the Virasoro algebra \n[ L m , L n ] = ( m -n ) L m + n + c 12 ( m 3 -m ) δ m + n . (2 . 13) \nAt the level of classical gravity (2.13) is a statement about Poisson brackets, while in the dual CFT it is an operator statement. The zero mode generators are related to mass and angular momentum as \nL 0 = M/lscript -J 2 , L 0 = M/lscript + J 2 . (2 . 14)", '2.2. The BTZ black hole [9]': "The metric of the rotating Euclidean BTZ black is \nds 2 = ( r 2 -r 2 + )( r 2 -r 2 -) r 2 /lscript 2 dt 2 + /lscript 2 r 2 ( r 2 -r 2 + )( r 2 -r 2 -) dr 2 + r 2 ( dφ + i r + r -/lscriptr 2 dt ) 2 . (2 . 15) \nTo obtain a real Euclidean signature metric, r -is taken to be purely imaginary. The global identifications that lead from pure AdS 3 to BTZ may be found in [9]. In Lorentzian signature, the BTZ solution is a black hole with inner/outer horizons at r = r ± . Besides the 2 π periodicity of the angular coordinate φ , smoothness of the Euclidean signature metric at r = r + requires that we impose the further identification \nw ∼ = w +2 πτ , τ = i/lscript r + + r -, (2 . 16) \nwhere w is defined as before, w = φ + it//lscript . With these identifications, the conformal boundary of the BTZ black hole is seen to be a torus of modular parameter τ . \nThe BTZ solution can be written in the Fefferman-Graham form (2.4) as \nds 2 = dρ 2 +8 πG/lscript ( L dw 2 + L dw 2 ) + ( /lscript 2 e 2 ρ//lscript +(8 πG ) 2 LL e -2 ρ//lscript ) dwdw . (2 . 17) \nThe constants L and L are rescaled versions of the Virasoro zero modes, \nL = 1 2 π L 0 , L = 1 2 π L 0 , (2 . 18) \nwhich are in turn related to r ± as \nL 0 = ( r + -r -) 2 16 G/lscript , L 0 = ( r + + r -) 2 16 G/lscript . (2 . 19) \nOther useful relations are \nτ = ik 2 1 √ 2 πk L , τ = -ik 2 1 √ 2 πk L . (2 . 20) \nThe area law gives the black hole entropy as \nS = A 4 G = 2 π √ c 6 L 0 +2 π √ c 6 L 0 , (2 . 21) \nwhich exhibits the famous fact that the entropy is given by Cardy's formula [81]. \nIt is illuminating to obtain the entropy from the Euclidean action, which is related to the thermodynamic quantities as \n-I = S +2 πiτL 0 -2 πiτL 0 , (2 . 22) \nComputing the Euclidean action directly for the BTZ black hole will recover (2.21). We can instead use modular invariance to simplify the computation. In three dimensional gravity, modular transformations are coordinate transformations that act on the boundary so as to change the modular parameter as \nτ → τ ' = aτ + b cτ + d , ad -bc = 1 . (2 . 23) \nIn fact, we can find a coordinate transformation such that the BTZ metric goes over into the pure AdS 3 metric (2.2) with modular parameter τ ' = -1 /τ [82]. The Euclidean action of this solution is easily worked out as \nI = iπc 12 ( τ ' -τ ' ) = -iπc 12 ( 1 τ -1 τ ) . (2 . 24) \nSince the Euclidean action is invariant under coordinate transformations, this is therefore the action for the BTZ black hole. Writing the result in the form (2.22) yields the entropy formula (2.21). \nA major virtue of the modular transformation approach is that it can be applied to cases where the area law is no longer valid, such as when higher derivative terms are present in the action. All such corrections can be absorbed into the central charge c , and taking this into account Cardy's formula continues to be valid [83].", '2.3. Charged BTZ black holes': 'Our main focus in this review is the construction and study of black holes endowed with higher spin charge. A simpler version of this problem, free from the complications of higher spin gravity, involves black holes carrying a spin-1 charge. These have a rather simple description when we add to our theory a pure Chern-Simons U(1) gauge field. \nTo our previous action we now add \nI CS = i ˆ k 8 π ∫ AdA -ˆ k 16 π ∫ d 2 x √ gA i A i , (2 . 25) \nwhere A is a 1-form. The second term is a boundary term (the metric that appears in this term is that induced on the boundary), introduced for the same reasons as in the pure gravity case [84]. We impose boundary conditions such that in Fefferman-Graham coordinates A ρ vanishes and \nA i ( ρ, x j ) = A (0) i ( x j ) + . . . , ρ →∞ . (2 . 26) \nSee [85] for a discussion of more general boundary conditions, including the case where a Maxwell term is present. The on-shell variation of the action then takes the form \nδI CS = i 2 π ∫ d 2 w √ g (0) J w δA (0) w (2 . 27) \nwhere the boundary current is \nJ w = 1 2 J w = i ˆ k 2 A (0) w . (2 . 28) \nA (0) w plays the role of a source conjugate to the current J w . In particular, to incorporate A (0) w on the CFT side of the AdS/CFT correspondence we add to the CFT action the term \nI CFT → I CFT + i 2 π ∫ d 2 wJ w A (0) w . (2 . 29) \nNote that in the absence of a source the current is holomorphic, ∂ w J w = i ˆ k 2 ∂ w A (0) w = 0, where in the last step we used the Chern-Simons equations of motion, namely that the field strength vanishes. An anti-holomorphic current is obtained by starting with (2.25), but flipping the sign of the first term. The modes of the current, defined as, \nJ n = ∮ dw 2 πi e -inw J w (2 . 30) \nfurnish a U (1) Kac-Moody algebra [84]. \nSince the field strength vanishes by the Chern-Simons equations of motion, the gauge field is locally pure gauge. However, in the presence of a black hole the gauge field can have a nontrivial holonomy around the horizon, and this translates into an electric charge carried by the black hole. The BTZ boundary torus has two non-contractible cycles, defined by the two identifications w ∼ = w + 2 π and w ∼ = w + 2 πτ . The former is noncontractible when extended into the bulk (it goes around the horizon), while the latter is contractible. A smooth gauge field must have trivial holonomy around the contractible cycle, which imposes the condition τA w + τA w = 0. Subject to this constraint, we can choose an arbitrary flat connection, and then the black hole charge is given by J 0 . \nThe black hole entropy is still given by the usual area law, and the BTZ metric is clearly unaffected by the addition of a flat U(1) connection. However, the boundary term appearing in (2.25) depends on the metric, and this induces a shift in the stress tensor. So the formula for the entropy is modified when expressed in terms of the Virasoro and U(1) charges, and is given by [84] \nS = 2 π √ c 6 ( L 0 -1 ˆ k J 2 0 ) + 2 π √ c 6 L 0 , (2 . 31) \nwhere we include just a holomorphic current.', '2.4. Chern-Simons formulation of three dimensional AdS gravity': "An alternative formulation of three dimensional gravity is as a Chern-Simons theory [7,6]. We now review this story and show how to recover the results found in the metric formulation. \nWe consider the action \nS = S CS [ A ] -S CS [ A ] (2 . 32) \nwhere \nS CS [ A ] = k 4 π ∫ Tr ( A ∧ dA + 2 3 A ∧ A ∧ A ) . (2 . 33) \nThe 1-forms A and A take values in the Lie algebra of SL (2 , R ). The Chern-Simons level k will eventually be related to ratio of the AdS 3 radius /lscript and the Newton constant G as \nk = /lscript 4 G . (2 . 34) \nFor convenience, we henceforth set /lscript = 1. \nThe Chern-Simons equations of motion correspond to vanishing field strengths, \nF = dA + A ∧ A = 0 , F = dA + A ∧ A = 0 . (2 . 35) \nTo relate these to the Einstein equations we introduce a vielbein e and spin connection ω as \nA = ω + e , A = ω -e . (2 . 36) \nThe flatness conditions (2.35) then become Einstein's equations in first order form. The metric is obtained from the vielbein in the usual fashion \ng µν = 2Tr( e µ e ν ) . (2 . 37) \nTo define asymptotically AdS boundary conditions it is convenient to choose an explicit basis for the sl (2 , R ) generators. We take \nL 1 = ( 0 0 -1 0 ) , L -1 = ( 0 1 0 0 ) , L 0 = ( 1 / 2 0 0 -1 / 2 ) (2 . 38) \nwhich obey \n[ L i , L j ] = ( i -j ) L i + j . (2 . 39) \nThe BTZ metric (2.17) is reproduced starting from \nA = ( e ρ L 1 -2 π L k e -ρ L -1 ) dz + L 0 dρ A = ( e ρ L -1 -2 π L k e -ρ L 1 ) dz -L 0 dρ (2 . 40) \nwhere we're now using z = φ + it rather than w as our complex coordinate. Noting that the dependence on the BTZ charges ( L , L ) arises purely through subleading terms at large ρ , we define an asymptotically AdS 3 connection to be one which differs from (2.40) by terms that go to zero for large ρ . \nUpon using the freedom to make gauge transformations (see [41] for a proof), asymptotically AdS 3 connections can be taken to have the form \nA = b -1 a ( z ) b + b -1 db , A = ba ( z ) b -1 + bdb -1 (2 . 41) \nwith \nand \nThe coefficient functions ( L ( z ) , L ( z )) are nothing but the components of the boundary stress tensor, \nb = e ρL 0 , (2 . 42) \na ( z ) = ( L 1 -2 π k L ( z ) L -1 ) dz a ( z ) = ( L -1 -2 π k L ( z ) L 1 ) dz . (2 . 43) \nT zz = L , T zz = L . (2 . 44) \nThis can be established in a variety of ways. One approach is to work out the metric corresponding to (2.43) and then read off the stress tensor components from (2.7). Alternatively, \nwe can study the asymptotic symmetries directly in the Chern-Simons formulation, and use this to identify ( ( z ) , L ( z )) as the currents associated with conformal symmetries [41,69]. \nL \nL Focussing on the unbarred connection, infinitesimal gauge transformations act as \nδA = d Λ+[ A, Λ] . (2 . 45) \nTo preserve the structure in (2.41) we take \nΛ = b -1 λ ( z ) b . (2 . 46) \nFurther, the form of a ( z ) is (2.43) is preserved by taking \nλ ( z ) = /epsilon1 ( z ) L 1 -∂ z /epsilon1 ( z ) L 0 + ( 1 2 ∂ 2 z /epsilon1 ( z ) -2 π k L /epsilon1 ( z ) ) L -1 , (2 . 47) \nwhich acts as \nδ L = /epsilon1 ( z ) ∂ z L +2 ∂ z /epsilon1 ( z ) L k 4 π ∂ 3 z /epsilon1 ( z ) . (2 . 48) \nComparing with (2.9) shows that L is indeed transforming as a stress tensor under the asymptotic symmetries. \nNow let us return to the BTZ solution as expressed by the connections (2.40). In the Chern-Simons formulation the metric is a derived concept, and the physical content of a solution is far from obvious just by inspecting the connection. For instance, the Euclidean BTZ spacetime (2.17) smoothly caps off at the horizon, e 2 ρ + = 2 π k √ LL , whereas from the Chern-Simons perspective there is no obvious reason we should restrict the range of ρ . \nIn the metric formulation the relation between the charges ( L , L ) and the modular parameter τ is fixed by demanding the absence of a conical singularity at the horizon. We need a corresponding condition expressed in Chern-Simons language; i.e. in terms of the connection and not the metric. The idea is to focus on the holonomy of the connection around the Euclidean time circle. This holonomy should be trivial if we want to say that this circle smoothly closes off at the horizon. The holonomy of A is, up to conjugation by b , \nH = e 2 πτa z . (2 . 49) \n2 πτa z has eigenvalues ± √ 8 π 3 L k τ . If we use the relations (2.20) we find that the eigenvalues are ± iπ , so that, H = diag( -1 , -1). Since this H is in the center of SL (2 , R ), it indeed represents a trivial holonomy.", '2.5. Black hole entropy in the Chern-Simons formulation': 'Next we turn to computing the black hole entropy. As reviewed in the next section, with suitable attention paid to boundary terms, it is possible to compute the black hole free energy via the Euclidean Chern-Simons action [73], but a more direct approach is to \nuse the first law of thermodynamics, which is how we proceed here. We can define the black hole partition function as \nZ ( τ, τ ) = Tr [ e 4 π 2 iτ ˆ L4 π 2 iτ ˆ L ] = e S +4 π 2 iτ L4 π 2 iτ L , (2 . 50) \nwhere the operator expression written in the first line is essentially a mnemonic for present purposes, but can be thought of as the fundamental definition of the partition function in some putative microscopic theory, while the second line represents the saddle point approximation, which is what we use here. Factors of 2 π arise from the rescaling in (2.18) as well as from the 2 πτ identification. \nFrom the definition of Z it follows that \nL = -i 4 π 2 ∂ ln Z ∂τ , L = i 4 π 2 ∂ ln Z ∂τ (2 . 51) \nS = ln Z -4 π 2 iτ ˆ L +4 π 2 iτ ˆ L . (2 . 52) \nand \nSince expressions for ( L , L ) in terms of ( τ, τ ) have already been fixed by the holonomy condition, namely as (2.20), we can integrate (2.51) to find ln Z , and then use (2 . 52) to compute S . Doing so, we of course reproduce the previous result (2.21). \nWe now consider a more general case in which additional charges are present. Consider a charge Q with corresponding conjugate potential α . In such a case we can define the partition function \nZ ( τ, α ) = Tr [ e 4 π 2 i ( τ ˆ L + α ˆ Q ) ] = e S +4 π 2 i ( τ L + αQ ) , (2 . 53) \nwhere we suppress the dependence on τ for clarity. Now, suppose we are given a black hole solution and we wish to determine Z , and hence S . As in the case above, a first step is to determine expressions for L and Q in terms of τ and α . These relations will be fixed by demanding appropriate smoothness conditions at the horizon. There is an important consistency condition that any such smoothness condition must respect if an underlying partition function is to exist. Namely, since \nL = -i 4 π 2 ∂ ln Z ∂τ , Q = -i 4 π 2 ∂ ln Z ∂α , (2 . 54) \nthe existence of Z implies \n∂ L ∂α = ∂Q ∂τ . (2 . 55) \nConversely, given expressions L ( τ, α ) and Q ( τ, α ), in order for Z to exist the integrability condition (2.55) must be obeyed. In the case of higher spin black holes there are some \nsubtleties in defining what precisely is meant by a solution being smooth; for black holes an important check on any proposal is that it leads to a charge assignments compatible with the integrability condition. \nWe will determine the thermodynamic properties of higher spin black holes using the approach just outlined. The alternative approach [73] (see also [86]) based on evaluating the Euclidean Chern-Simons action with appropriate boundary terms leads to the same final result.', '3. Black holes in sl ( N,R ) ⊕ sl ( N,R ) higher spin gravity': 'In the last section we reviewed that Einstein gravity with a negative cosmological constant can be reformulated as a sl (2 , R ) ⊕ sl (2 , R ) Chern-Simons theory. In a series of recent papers generalizations to Einstein gravity with a negative cosmological constant coupled to higher spin fields were discussed. While there is no straightforward generalization within the metric formulation we can easily generalize the Chern-Simons formulation. In particular in [39,40] it was shown that a sl ( N,R ) ⊕ sl ( N,R ) Chern-Simons theory corresponds to Einstein gravity coupled to N -2 symmetric tensor fields of spin s = 3 , 4 , . . . , N . For simplicity, we restrict ourselves to N = 3 corresponding to Einstein gravity with a spin-3 field and study this theory in more detail. In particular, we construct black hole solutions with spin-3 charge.', '3.1. Review of sl (3 , R ) ⊕ sl (3 , R ) higher spin gravity and W 3 symmetry': "In this section we first review the formulation of three dimensional higher spin gravity in terms of sl (3 , R ) ⊕ sl (3 , R ) Chern-Simons theory. As in the case of Einstein gravity with a negative cosmological constant we use the action S = S CS [ A ] -S CS [ ¯ A ] where the Chern-Simons action is given by equation (2.33). The 1-forms A and A take values in the Lie algebra sl (3 , R ) and satisfy the flatness conditions \nF = dA + A ∧ A = 0 , F = dA + A ∧ A = 0 . (3 . 1) \nWhenever necessary, we use the following basis of sl (3 , R ) generators 3 \nL 1 = 0 0 0 1 0 0 0 1 0 , L 0 = 1 0 0 0 0 0 0 0 -1 , L -1 = 0 -2 0 0 0 -2 0 0 0 , \nW 2 = 0 0 0 0 0 0 2 0 0 , W 1 = 0 0 0 1 0 0 0 -1 0 , W 0 = 2 3 1 0 0 0 -2 0 0 0 1 , (3 . 2) \nW -1 = 0 -2 0 0 0 2 0 0 0 , W -2 = 0 0 8 0 0 0 0 0 0 , \nwhich satisfy the following commutation relations \n[ L i , L j ] = ( i -j ) L i + j , [ L i , W m ] = (2 i -m ) W i + m , [ W m , W n ] = -1 3 ( m -n )(2 m 2 +2 n 2 -mn -8) L m + n . (3 . 3) \nNote that the generators L i generate a sl (2 , R ) subalgebra of sl (3 , R ) . Under this sl (2 , R ) the generators W m form a spin two multiplet. This is called the principal embedding of sl (2 , R ) into sl (3 , R ) which we use in this section. Note that this is not the only possible inequivalent embedding of sl (2 , R ) into sl (3 , R ): instead of ( L 1 , L 0 , L -1 ) we can consider the sl (2 , R ) subalgebra generated by ( W 2 / 4 , L 0 / 2 , W -2 / 4) . \n- \n-Instead of using A and A we can also determine the vielbein e and the spin connection ω according to equation (2.36). Expanding e and ω in a basis of 1-forms dx µ , the spacetime metric g µν and spin-3 field ϕ µνγ are identified as 4 \ng µν = 1 2 Tr( e µ e ν ) , ϕ µνγ = 1 3! Tr( e ( µ e ν e γ ) ) (3 . 4) \nwhere ϕ µνγ is totally symmetric as indicated. Restricting to the sl (2 , R ) subalgebra generated by L i , the flatness conditions (3.2) can be seen to be equivalent to Einstein's equations \nfor the metric g µν with a torsion free spin-connection [41]. More generally, we find equations describing a consistent coupling of the metric to the spin-3 field. \nActing on the metric and spin-3 field, the sl (3 , R ) ⊕ sl (3 , R ) gauge symmetries of the Chern-Simons theory turn into diffeomorphisms along with spin-3 gauge transformations (the Chern-Simons gauge transformation also include frame rotations, which leave the metric and spin-3 field invariant). Under diffeomorphisms, the metric and spin-3 field transform according to the usual tensor transformation rules. The spin-3 gauge transformations are less familiar, as they in general act nontrivially on both the metric and spin-3 field. It is worth noting, though, that if we ignore the spin-3 gauge invariance, then we can view the theory as a particular diffeomorphism invariant theory of a metric and a rank-3 symmetric tensor field. In [87] an action for a metric-like formulation of the sl (3 , R ) ⊕ sl (3 , R ) Chern-Simons theory was formulated which is valid up to quadratic order in the spin-3 field. \nLet us consider the following connection: \nA ( z ) = ( e ρ L 1 -2 π k e -ρ L ( z ) L -1 -π 2 k e -2 ρ W ( z ) W -2 ) dz + L 0 dρ A ( z ) = ( e ρ L -1 -2 π k e -ρ L ( z ) L 1 -π 2 k e -2 ρ W ( z ) W 2 ) dz -L 0 dρ . (3 . 5) \nSetting W ( z ) = 0 and W ( z ) = 0 we recover the connection (2.40). Moreover, since A z = 0 , A ρ = L 0 and \nA -A AdS ∼ O (1) as ρ →∞ (3 . 6) \nwith analogous conditions for A, the connection (3.5) is asymptotically AdS [41]. \nAs explained in section 2 it is convenient to use a gauge transformation of the form (2.41) and (2.42) to obtain \na ( z ) = ( L 1 -2 π k L ( z ) L -1 -π 2 k W ( z ) W -2 ) dz ¯ a ( z ) = ( L -1 -2 π k L ( z ) L 1 -π 2 k W ( z ) W 2 ) dz . (3 . 7) \nThe coefficients L ( z ) and L ( z ) are the components of the energy momentum tensor as identified in equation (2.44). The new coefficient functions W and W correspond to the spin-3 currents as we will see now by analyzing the asymptotic symmetry algebra. \nGiven the connection (3.5), the asymptotic symmetry algebra is obtained by finding the most general gauge transformation (2.44) preserving the asymptotic conditions (3.5). The functions appearing in the connection (3.5), L , L , W and W , transform under these gauge transformations. Expanding in modes, one thereby arrives at two copies of the classical W 3 algebra [88]. \nAlternatively (see section 4 of [69] for details), one can translate these variations into \nan operator product expansion for the symmetry currents, 5 \nT ( z ) T (0) ∼ 3 k z 4 + 2 z 2 T (0) + 1 z ∂T (0) T ( z ) W (0) ∼ 3 z 2 W (0) + 1 z ∂ W (0) W ( z ) W (0) ∼ -5 k π 2 1 z 6 -5 π 2 1 z 4 T (0) -5 2 π 2 1 z 3 ∂T (0) -3 4 π 2 1 z 2 ∂ 2 T (0) -1 6 π 2 1 z ∂ 3 T (0) -8 3 kπ 2 1 z T (0) ∂T (0) -8 3 kπ 2 1 z 2 T (0) 2 (3 . 8) \nand the result is the classical version of the W 3 algebra. In order to write the W 3 algebra as in [88] we identify the central charge c as c = 6 k and we define the operator V used in [88] by V = 2 πi √ 10 W . The T W OPE identifies W as a spin-3 current, i.e., as a dimension (3 , 0) primary operator. The same analysis for the barred connection gives rise to an anti-holomorphic W 3 algebra with a dimension (0 , 3) current. \nThere is another way to show that sl (3 , R ) ⊕ sl (3 , R ) Chern-Simons theory give rise to W 3 symmetry algebra. Instead of considering the asymptotic symmetry algebra we consider the Ward identities which are the bulk field equations for spin-3 gravity in the presence of sources for the spin-3 operators. \nFor simplicity let us consider the holomorphic part only corresponding to the connection a. We consider a chemical potential µ ( z, z ) for the spin-3 operator which is given by a term -µ ( z, z ) W 2 dz in the connection a. We will justify this assignment a posteriori by deriving the Ward-Identities. For a consistent ansatz we do not only need a term of the form -µW 2 dz in the connection but we have to consider \nThe flatness condition determines w i as a function of µ, L , W and z -derivatives thereof. Moreover L and W are subject to \na = ( L 1 -2 π k L L -1 -π 2 k W W -2 ) dz -( µW 2 + w 1 W 1 + w 0 W 0 + w -1 W -1 + w -2 W -2 + w -3 L -1 ) dz . (3 . 9) \n∂ z L = 3 W ∂ z µ +2 ∂ z W µ ∂ z W = k 12 π ∂ 5 z µ -2 3 ∂ 3 z L µ -3 ∂ 2 z L ∂ z µ -5 ∂ z L ∂ 2 z µ -10 3 L ∂ 3 z µ + 64 π 3 k ( L µ∂ z L + L 2 ∂ z µ ) . (3 . 10) \nThese equations are the Ward identities as we will see. Note that L and W are now z dependent which can be traced back to the singular terms in the OPE (3.8). Let us understand this in more detail. Therefore we compute \n∂ z 〈 T ( z, z ) 〉 µ , ∂ z 〈W ( z, z ) 〉 µ (3 . 11) \nwhere 〈· · ·〉 µ denotes an insertion of e ∫ µ W inside the expectation value. Expanding in powers of µ , and using \n∂ z ( 1 z ) = 2 πδ (2) ( z, z ) (3 . 12) \nas well as the OPE's (3.8) we obtain (3.10) if we identify T = -2 π L . This justifies a posteriori the AdS/CFT dictionary for introducing the chemical potential. Moreover, we have seen that the Ward identities can be derived from the equations of motion of spin-3 gravity. \nSo far we considered only the principal embedding of sl (2 , R ) into sl (3 , R ) giving rise to W 3 as asymptotic symmetry. As already explained there is another inequivalent embedding of sl (2 , R ) into sl (3 , R ). This embedding gives rise to another W -algebra known as W (2) 3 , sometimes referred to as the Polyakov-Bershadsky algebra [44,45]. More details can be found in section 2.3 of [70] and [89].", '3.2. Black holes with higher spin charge': "Let us now consider black holes with higher spin charge within sl (3 , R ) ⊕ sl (3 , R ) Chern-Simons theory. In [69] the following solution was proposed to represent black holes carrying spin-3 charge: \na = ( L 1 -2 π k L L -1 -π 2 k W W -2 ) dz -µ ( W 2 -4 π L k W 0 + 4 π 2 L 2 k 2 W -2 + 4 π W k L -1 ) dz ¯ a = -( L -1 -2 π k L L 1 -π 2 k W W 2 ) dz -¯ µ ( W -2 -4 π L k W 0 + 4 π 2 L 2 k 2 W 2 + 4 π W k L 1 ) dz . (3 . 13) \nThe structure of this solution is easy to understand. Let us focus on the a -connection. As in (3.7), to add energy and charge density to the W 3 vacuum we should add to a z terms involving L -1 and W -2 , as seen in the top line of (3.13). For black holes, which represent states of thermodynamic equilibrium, the energy and charge should be accompanied by their conjugate thermodynamic potentials, which are temperature and spin-3 chemical potential. We incorporate the former via the periodicity of imaginary time, while the latter correspond to a µW 2 term in a z . The remaining a z terms appearing in (3.13) are then fixed by the equations of motion. In fact we can rewrite the component a z of the connection (3.13) as \na z = -2 µ [ ( a z ) 2 -1 3 Tr( a 2 z ) ] . (3 . 14) \nIn order to solve the flatness condition (3.1) we have to show [ a z , a z ] = 0 which is indeed satisfied by (3.14). The ansatz for the connection a z given in (3.14) can be generalized to more complicated cases, including black holes in hs[ λ ] higher spin theories. \nLet us from now on restrict ourselves to the nonrotating case \nL = L , W = -W , ¯ µ = -µ , (3 . 15) \nand study the connection (3.13) in more detail. If we set µ = W = 0, then the connections become those corresponding to a BTZ black hole asymptotic to the W 3 vacuum. From the standpoint of this CFT, the µW 2 dz term represents a chemical potential for spin-3 charge, and the W W -2 dz term gives the value of the spin-3 charge. This solution is therefore interpreted as a generalization of the BTZ black hole to include nonzero spin-3 charge and chemical potential. \nThe general non-rotating solution, i.e. the connection (3.13) imposing the conditions (3.15), can be thought of as depending on four free parameters: three of these are ( L , W , µ ), and the fourth is the inverse temperature β , corresponding to the periodicity of imaginary time, t ∼ = t + iβ . However, we expect that there should only be a two-parameter family of physically admissible solutions: once one has specified the temperature and chemical potential the values of the energy and charge should be determined thermodynamically as a function of α and τ where τ is the modular parameter of the torus given by \nτ = iβ 2 π , (3 . 16) \nand α is the potential related to the chemical potential µ by \nα = ¯ τµ, ¯ α = τ ¯ µ . (3 . 17) \nWhich conditions do the functions L = L ( τ, α ) and W = W ( τ, α ) satisfy? As discussed in section 2.4 for the uncharged BTZ solution the relation between the energy and the temperature is obtained by demanding the absence of a conical singularity at the horizon in Euclidean signature. The analogous procedure in the presence of spin-3 charge is more subtle. For the charged spin-3 black hole solution, we would like to impose the following three conditions: \n- (i) The Euclidean geometry is smooth and the spin-3 field is nonsingular at the horizon.\n- (ii) In the limit µ → 0 the solution goes smoothly over to the BTZ black hole. In particular, we want that W → 0 .\n- (iii) The charge assignment L = L ( τ, α ) and W = W ( τ, α ) should arise from an underlying partition function and therefore, as discussed in section 2.5, the charge assignment should obey the integrability condition \n∂ L ∂α = ∂ W ∂τ . (3 . 18) \nLet us first discuss the condition (i) . The metric (3.4) associated with the gauge connection (3.14) is given by \nds 2 = dρ 2 -F ( ρ ) dt 2 + G ( ρ ) dφ 2 (3 . 19) \nwith \nF ( ρ ) = ( 2 µe 2 ρ + π k W e -2 ρ -8 π 2 k 2 µ L 2 e -2 ρ ) 2 + ( e ρ -2 π k L e -ρ + 4 π k µ W e -ρ ) 2 G ( ρ ) = ( e ρ + 2 π k L e -ρ + 4 π k µ W e -ρ ) 2 +4 ( µe 2 ρ + π 2 k W e -2 ρ + 4 π 2 k 2 µ L 2 e -2 ρ ) 2 + 4 3 ( 4 π k ) 2 µ 2 L 2 . (3 . 20) \nThe metric reduces to BTZ in the uncharged limit, but the surprise is that for generic charge, F ( ρ ) and G ( ρ ) are both positive definite quantities. The horizon occurs where g tt vanishes, and since the radial dependence of the metric is simply dρ 2 , we need g tt to have a double zero at the horizon in order for the Euclidean time circle to smoothly pinch off. One solution is given by W = 0 and µ = 0, which is the nonrotating BTZ black hole. There is a second solution provided \nk +32 µ 2 π ( µ W-L ) = 0 (3 . 21) \nis satisfied. The temperature is then determined by demanding the absence of a conical singularity at the horizon, which is located at e ρ + = √ (2 π L4 πµ W ) /k , \n∣ \nβ = 2 π √ 2 -g '' tt ∣ ∣ ∣ ∣ ρ = ρ + = 2 √ 2 πkµ √ (3 k -32 π L µ 2 )(16 π L µ 2 -k ) . (3 . 22) \nThe two equations (3.21) and (3.22) determine the charge assignments L = L ( τ, α ) and W = W ( τ, α ) . However, these charge assignments do not satisfy the conditions (ii) and (iii) . Although these charge assignments are the only one for which the geometry of the connection (3.13) has a horizon, the charge assignments are not consistent. \nFor any other charge assignment, F never vanishes and there is no event horizon since this geometry possesses a globally defined timelike Killing vector. At large positive and negative ρ , both F and G have leading behavior e 4 | ρ | , corresponding to an AdS 3 metric of radius 1 / 2, and so the metric (3.20) describes a traversible wormhole connecting two asymptotic AdS 3 geometries. \nOne may therefore ask what, if anything, this solution has to do with black holes. It is at this point that we should remember that the metric of the spin-3 theory is not invariant under higher spin gauge transformations and the connection (3.13) after a suitable gauge transformation represents a smooth black hole whose charge assignment satisfies the conditions (ii) and (iii) . In order to find this charge assignment, we have to consider the holonomy ω around the time circle which is given by \nω = 2 π ( τa z + ¯ τa z ) . (3 . 23) \nIn [69] it was proposed that the eigenvalues of the holonomy ω take the fixed values (0 , 2 πi, -2 πi ). It is convenient to recast the conditions on its eigenvalues in the form \nTr( ω 2 ) = -8 π 2 , Tr( ω 3 ) = 0 . (3 . 24) \nWe will see in section 3.3 that the holonomy condition (3.24) gives rise to a charge assignment which satisfies the conditions (ii) and (iii) . Moreover, in section 3.4 we find an explicit gauge transformation of the connection (3.13) such that the geometry associated with the gauge transformed connection is indeed a smooth black hole, whose smoothness enforces the holonomy condition (3.24).", '3.3. Holonomy and integrability': 'In this section we evaluate the holonomy condition for the connection (3.13) and show that the consistency condtions (ii) an (iii) are indeed satisfied. \nThe holonomy condition (3.24) for the connection (3.13), become explicitly \n0 = -2048 π 2 µ 3 L 3 +576 πkµ L 2 -864 πkµ 2 WL +864 πkµ 3 W 2 -27 k 2 W 0 = 256 π 2 µ 2 L 2 +24 πk L72 πkµ W + 3 k 2 τ 2 (3 . 25) \ntogether with the same formulas with unbarred quantities replaced by their barred versions. We have replaced µ by α using (3.20). \nTo verify the integrability conditions we proceed as follows. First solve the second equation for W and differentiate with respect to τ to get an expression for ∂ W ∂τ in terms of ∂ L ∂τ . Next, insert the solution for W into the first equation, and then differentiate with respect to α , and solve to get an expression for ∂ L ∂α . Similarly, differentiate with respect to τ to get an expression for ∂ L ∂τ . Substituting the latter into our previous expression for ∂ W ∂τ , we find that it precisely equals ∂ L ∂α , which is the desired integrability condition. \nIt will be convenient to define dimensionless versions of the charge and the chemical potential by \nζ = √ k 32 π L 3 W , γ = √ 2 π L k µ . (3 . 26) \nRewriting (3.25) in terms of the quantities (3.26) we obtain \n1728 γ 3 ζ 2 -(432 γ 2 +27) ζ -128 γ 3 +72 γ = 0 ( 1 + 16 3 γ 2 -12 γζ ) Lπk 2 β 2 = 0 . (3 . 27) \nLet us solve (3.27) for the charge ζ and the inverse temperature β. \nSolution of equations (3.25) for the charge ζ and inverse temperature β yields \nζ = 1 + 16 γ 2 -( 1 -16 3 γ 2 ) √ 1 + 128 3 γ 2 128 γ 3 β = √ πk 2 L √ 1 + 16 3 γ 2 -12 γζ . (3 . 28) \nIn equation (3.28) we singled out a particular branch of ζ to ensure that for γ → 0 also ζ → 0 , i.e. that condition (ii) defined in section 3.2 is satisfied. While the uncharged BTZ limit corresponds to γ, ζ → 0 the solution (3.28) limits the values ζ and γ to \nζ ≤ ζ max = √ 4 27 , γ ≤ γ max = √ 3 16 . (3 . 29) \nFor given L we thus have a maximal spin-3 charge W given by \nW 2 max = 128 π 27 k L 3 . (3 . 30) \nNote that these maximal values can be seen only by the holonomy conditions (3.25) and not directly from the gauge connection (3.13) or its resulting geometry.', '3.4. Finding the black hole gauge': "In this section we want to find a suitable gauge transformation to turn the connection (3.13) into a smooth black hole. In particular we will see that the smoothness conditions are indeed equivalent to the holonomy conditions (3.25). The ρ -dependent connections corresponding to (3.13) are denoted by A and ¯ A. We refer to this connection as in the wormhole gauge . We then consider new connections A and ¯ A related to A and ¯ A by sl (3 , R ) gauge transformations: \nA = g -1 ( ρ ) A ( ρ ) g ( ρ ) + g -1 ( ρ ) dg ( ρ ) ¯ A = g ( ρ ) ¯ A ( ρ ) g -1 ( ρ ) -dg ( ρ ) g -1 ( ρ ) (3 . 31) \nwith g ( ρ ) ∈ sl (3 , R ). The relative gauge transformation for ¯ A versus A is taken to maintain a non-rotating ansatz. The metric and spin-3 field corresponding to ( A, A ) will take the form \nds 2 = g ρρ ( ρ ) dρ 2 + g tt ( ρ ) dt 2 + g φφ ( ρ ) dφ 2 ϕ αβγ dx α dx β dx γ = ϕ φρρ ( ρ ) dφdρ 2 + ϕ φtt ( ρ ) dφdt 2 + ϕ φφφ ( ρ ) dφ 3 . (3 . 32) \nWe demand that this solution describe a smooth black hole with event horizon at ρ = ρ + , or at r = 0 with \nr = ρ -ρ + . (3 . 33) \nAssuming that g rr (0) > 0, as will be the case, this first of all requires g tt (0) = g ' tt (0) = 0 and g φφ (0) > 0, so that after rotating to imaginary time the metric expanded around r = 0 will look locally like R 2 × S 1 : \nds 2 ≈ g rr (0) dr 2 -1 2 g '' tt (0) r 2 dt 2 E + g φφ (0) dφ 2 . (3 . 34) \nIn order for the metric to avoid a conical singularity at r = 0 we need to identify t E ∼ = t E + β with \nβ = 2 π √ 2 g rr (0) -g '' tt (0) . (3 . 35) \nHaving done so, we can switch to Cartesian coordinates near r = 0 and the metric will be smooth. \nThe same smoothness considerations apply to the spin-3 field. Noticing the parallel structure, we see that we should demand ϕ φtt (0) = ϕ ' φtt (0) = 0, and \nβ = 2 π √ 2 ϕ φrr (0) -ϕ '' φtt (0) (3 . 36) \nwith the same β as in (3.35). \nThere is still one more condition to impose to ensure that the solution is completely smooth at the horizon. If we work in Cartesian coordinates ( x, y ) around r = 0, we should demand that all functions are infinitely differentiable with respect to both x and y . If this is not the case then some curvature invariant (or spin-3 quantity) involving covariant derivatives will diverge. Given the rotational symmetry, this condition implies that the series expansion of all functions should only involve non-negative even powers of r . We impose this by demanding that all functions be smooth at the horizon, and even under reflection about the horizon: \ng rr ( -r ) = g rr ( r ) , g tt ( -r ) = g tt ( r ) , g φφ ( -r ) = g φφ ( r ) ϕ φrr ( -r ) = ϕ φrr ( r ) , ϕ φtt ( -r ) = ϕ φtt ( r ) , ϕ φφφ ( -r ) = ϕ φφφ ( r ) . (3 . 37) \nIn [70] it was shown that the symmetry conditions (3.37) can be enforced by demanding \ne t ( -r ) = -h ( r ) -1 e t ( r ) h ( r ) e φ ( -r ) = h ( r ) -1 e φ ( r ) h ( r ) e r ( -r ) = h ( r ) -1 e r ( r ) h ( r ) (3 . 38) \nwith h ( r ) ∈ sl (3 , R ), and similar conditions on the spin-connection. The BTZ solution has h ( r ) = 1 , so we can think of these conditions as a 'twist' of the BTZ vielbein reflection \nsymmetries. In addition, h (0) = 1 , implying that e t (0) = 0, a feature of the BTZ solution that persists in the spin-3 case. 6 \nTo gain some insight into the form of g ( r ) and h ( r ) we can start with the BTZ solution and then carry out the gauge transformation perturbatively in the charge. These considerations lead us to the ansatz \ng ( r ) = e F ( r )( W 1 -W -1 )+ G ( r ) L 0 h ( r ) = e H ( r )( W 1 + W -1 ) (3 . 39) \nfor some functions F, G and H . Perturbation theory suggests that this ansatz gives the unique solution to our problem, although we have not proven this. On the other hand, having assumed the ansatz (3.39) the remaining analysis definitely has a unique solution. \nEven after assuming this ansatz, finding a solution that satisfies all the smoothness conditions involves a surprisingly large amount of complicated algebra requiring extensive use of Mathematica and Maple. More details can be found in the appendix of [70]. As we have already mentioned, a crucial point is that the solution to our problem requires that the holonomy conditions (3.24) or equivalently (3.27) for the connection (3.13) are obeyed; therefore we can also derive the holonomy conditions by requiring the existence of a smooth black hole solution. \nHere we just present the final form for the transformed metric. It will be convenient to use the variables (3.26) instead of the charges and the chemical potential and to introduce the parameter C by \nζ = C -1 C 3 / 2 . (3 . 40) \nThe metric takes the form (3.32) with \ng rr = ( C -2)( C -3) ( C -2 -cosh 2 ( r ) ) 2 g tt = -( 8 π L k )( C -3 C 2 ) ( a t + b t cosh 2 ( r ) ) sinh 2 ( r ) ( C -2 -cosh 2 ( r ) ) 2 g φφ = ( 8 π L k )( C -3 C 2 ) ( a φ + b φ cosh 2 ( r ) ) sinh 2 ( r ) ( C -2 -cosh 2 ( r ) ) 2 + ( 8 π L k ) (1 + 16 3 γ 2 +12 γζ ) . (3 . 41) \nThe coefficients a t,φ and b t,φ are functions of γ and C , given by \na t = ( C -1) 2 ( 4 γ -√ C ) 2 , a φ = ( C -1) 2 ( 4 γ + √ C ) 2 , b t = 16 γ 2 ( C -2)( C 2 -2 C +2) -8 γ √ C (2 C 2 -6 C +5) + C (3 C -4) , b φ = 16 γ 2 ( C -2)( C 2 -2 C +2) + 8 γ √ C (2 C 2 -6 C +5) + C (3 C -4) . (3 . 42) \nThe t and φ coefficients are related by flipping the sign of γ , though this is not a bonafide sign flip of the charge, under which C would also transform. \nWith these results in hand, we can verify that the smoothness condition of the black hole implies the holonomy condition. Demanding a smooth horizon via (3.35) and (3.36) and using the definition (3.40), the resulting equations are indeed equivalent to (3.27) which are the holonomy condition (3.24) evaluated for the gauge connection (3.13). \nTherefore we showed that the black hole with spin-3 charge given by the connection (3.13) satisfies the three conditions (i), (ii) and (iii) defined in section 3.2. Let us now explore the black hole thermodynamics.", '3.5. Black hole thermodynamics': "Having satisfied the integrability condition, we know that if we now compute the entropy from the partition function it is guaranteed to be consistent with the first law of thermodynamics. For black holes in Einstein-Hilbert gravity, we can of course directly compute the entropy in terms of the area of the event horizon. But in the present context we do not know a priori whether the entropy is related to the area in this way, in particular due to the nontrivial spin-3 field. We instead base our entropy computation on demanding adherence to the first law. \nExpressed as a function of ( L , W ) the entropy S obeys the following thermodynamic relations: \nThe analogous barred relations hold as well. Since the entropy breaks up into a sum of an unbarred piece plus a barred piece with identical structure, in the following we just focus on the unbarred part and add the two parts at the end. \nτ = i 4 π 2 ∂S ∂ L , α = i 4 π 2 ∂S ∂ W . (3 . 43) \nWe can use dimensional analysis to write the entropy in terms of an unknown function of the dimensionless ratio ζ 2 ∼ W 2 / L 3 . With some foresight, it proves convenient to write \nwith \nS = 2 π √ 2 πk L f ( y ) (3 . 44) \ny = 27 2 ζ 2 = 27 k W 2 64 π L 3 . (3 . 45) \nDemanding agreement with the BTZ entropy imposes f (0) = 1. Using (3.43) and plugging into the second line of (3.25) we arrive at the following differential equation \n36 y 2 -y ) ( f ' ) 2 + f 2 -1 = 0 . (3 . 46) \n( \n) This equation also implies the first equation in (3.25). The solution with the correct boundary condition is \nf ( y ) = cos θ , θ = 1 6 arctan ( √ y (2 -y ) 1 -y ) . (3 . 47) \nThe physical range of y is given by 0 ≤ y ≤ 2, and we choose a branch of the arctangent such that 0 θ ≤ π 6 . \n≤ Our final result for the entropy, including both sectors, is thus \nS = 2 π √ 2 πk L f ( 27 k W 2 64 π L 3 ) +2 π √ 2 πk L f ( 27 k W 2 64 π L 3 ) . (3 . 48) \nThe function f ( y ) takes a simple form upon plugging in for ζ as a function of C , using (3.40). This step yields \nwith \nSurprisingly, taking the cosine of this angle yields the simple expression \nθ = 1 6 arctan ( Λ( C ) √ 1 -3 4 C ) (3 . 49) \nΛ( C ) ≡ 6 √ 3 C ( C -1)( C -3) (2 C -3)( C 2 -12 C +9) . (3 . 50) \nf ( y ) = √ 1 -3 4 C . (3 . 51) \nAs with other quantities in our analysis, we see that the entropy is most simply expressed in terms of C . The extremal and zero charge limits are recovered upon inspection. \nIt is of course natural to wonder if the black hole entropy can be expressed in terms of a geometrical property of the horizon. There is of course no reason to expect that the Bekenstein-Hawking area law holds, since the spin-3 field is nonzero at the horizon, and indeed one easily checks that S /negationslash = A/ 4 G . In [87] a metric like formulation of the sl (3 , R ) ⊕ sl (3 , R ) higher spin theory was developed and the Wald entropy formula [90] was applied to calculate the entropy of the black hole. At present there is a numerical discrepancy between the entropies obtained in (3.48) and [87], which remains to be understood. \nLet us also comment on the case of maximal value for the spin-3 charge (3.30)(for constant energy). Approaching this maximal value, f ( y ) behaves near y = 2 as \nf ( y ) ∼ √ 3 4 + √ 2 12 √ 2 -y + · · · (3 . 52) \n≤ \nand τ diverges according to (3.43), corresponding to vanishing chiral temperature. On the other hand, the entropy is finite, attaining a relative value of √ 3 4 compared to the entropy at W = 0. This behavior is to be contrasted with that of the BTZ black hole, or its charged generalizations with respect to bulk U (1) gauge fields [76]. In those cases, whenever the temperature of one chiral sector goes to zero, so too does the entropy associated with that sector. For extremal BTZ black holes the entropy is carried entirely by the sector at nonzero temperature, whereas here a zero temperature sector can contribute to the entropy. \nGiven the entropy, τ and α can be computed using (3.43), and from there we compute the partition function according to \nln Z = S +4 π 2 i τ L + α W-¯ τ L-¯ α W ) . (3 . 53) \n( \n) This partition function should match the asymptotic behavior of the partition function of any candidate CFT dual to the higher spin theory in the bulk. \nWe briefly note that this entire story has been generalized without complication to the sl (4 , R ) ⊕ sl (4 , R ) case [91].", '3.6. Other black holes in spin-3 gravity': 'In the last sections we built a higher spin black hole in spin-3 gravity. In particular we proposed a holonomy condition which allowed us to define the charge assignments in agreement with the integrability condition. We also saw that the holonomy condition is equivalent to the smoothness condition of the Euclidean black hole in the appropriate gauge. \nIn [71] another embedding of the sl (2 , R ) subalgebra was considered, the diagonal embedding. The asymptotic symmetry algebra of the diagonal embedding is the classical W (2) 3 ×W (2) 3 algebra. In the W (2) 3 algebra apart from the energy momentum tensor T, we have two weight 3 / 2 primaries G ± as well as a weight one current U. It turns out that in this theory it is particularly simple to construct black holes. A consistent truncation of that theory is general relativity coupled to a pair of Chern-Simons gauge fields. The connection of the BTZ black hole carrying charge under these gauge fields are given by [71] \nThe corresponding metric is given by \na = [ W 2 + wW -2 -qW 0 ] dz -η 2 W 0 dz , ¯ a = -[ W -2 + wW 2 -qW 0 ] dz + η 2 W 0 dz . (3 . 54) \nds 2 = dρ 2 -4 ( e 2 ρ -we -2 ρ ) 2 dt 2 +4 ( e 2 ρ + we -2 ρ ) 2 dφ 2 (3 . 55) \n( and the pair of gauge fields, χ and ¯ χ read \nχ = -qdz -η 2 dz , ¯ χ = qdz + η 2 dz . (3 . 56) \nCalculating the holonomies around the Euclidean thermal circle and imposing the holonomy condition we can relate q to η as well as w to the inverse temperature β, \nη = 2 q , β = 1 8 √ w . (3 . 57) \nNote that with this identification the Euclidean geometry is smooth and the time component of the gauge fields vanish at the horizon. Using the integrability condition we can derive a formula for the entropy. The dependence of the entropy on the charges q and w is exactly what it is expected from a R-charged BTZ black hole (2.31).', '3.7. Black hole entropy from the Euclidean action': 'In this section we briefly discuss an approach to computing the black entropy via the Euclidean Chern-Simons action [73]. An action based approach is very useful if one wants to include quantum corrections. In ordinary Einstein-Hilbert gravity, it is well known that after including suitable boundary terms the Euclidean action yields the correct black hole free energy [92]. The situation is more subtle in the Chern-Simons formulation, mainly due to the fact that the action is not invariant under gauge transformations that extend to the boundary. \nThe approach of [73] is to consider sl ( N,R ) Chern-Simons theory on the Euclidean solid torus (recall that this is the topology of the Euclidean BTZ solution). They focus on non-rotating solutions, with φ and t being the coordinates that parametrize the noncontractible and contractible cycles of the solid torus. Consider a connection a = a t dt + a φ dφ obeying the holonomy conditions. The authors define charges in terms of the sl ( N,R ) Casimir operators as \nQ 2 = 1 2 Tr( a 2 φ ) , Q 3 = 1 3 Tr( a 3 φ ) , . . . , Q N = 1 N Tr( a N φ ) . (3 . 58) \nConjugate potentials σ n are identified in an analogous manner as in (3.14), \na t = σ 2 a φ + σ 3 ( a 2 φ -I N Tr( a 3 φ ) ) + . . . + σ N ( a N -1 φ -I N Tr( a N -1 φ ) ) . (3 . 59) \nThe idea is to work out the on-shell variation of the action, and then add terms such that we have δI ∼ ∑ n Q n δσ n . The resulting action evaluated on-shell can then be identified with the black hole free energy in an ensemble in which the potentials { σ n } are held fixed. In [73] they find that a suitable action is \nI = I 0 -N ∑ n =2 k ( n -2) σ n Q n , (3 . 60) \nwhere I 0 is the original Chern-Simons action. The on-shell variation of this action takes the desired form \nδI = 2 k N ∑ n =2 Q n δσ n , (3 . 61) \nand its on-shell value is \nI = -2 k N ∑ n =2 ( n -1) σ n Q n . (3 . 62) \nThis gives a rather simple result for the free energy in the ensemble with fixed { σ n } . In this ensemble the charges { Q n } are functions of { σ n } , obtained by solving the holonomy constraints.', '4. hs[ λ ] black holes': 'Having constructed smooth black holes in sl (3 , R ) gravity, we turn to the more challenging case of doing so in the three dimensional Vasiliev theory containing an infinite tower of higher spins s ≥ 2. In addition to being interesting in its own right, it is this theory as opposed to the sl ( N,R ) theories that is most likely to play some role in the possible relation of higher spin gravity to string theory. The full Vasiliev theory has various ingredients which altogether describe the nonlinear coupling of the higher spin fields to matter, consistent with higher spin gauge invariance. We will not make use of most of them here: to make higher spin black holes, all we need from the theory is that its higher spin sector can be cast as two copies of Chern-Simons theory with connections valued in the infinite-dimensional Lie algebra hs[ λ ]. In particular, the matter fields of the Vasiliev theory are set to zero in the black hole background. For more details on the full theory, see [93], as well as recent summaries in [54] and [56]. \nAfter describing hs[ λ ] in some detail, we will apply the logic of the previous chapter to a satisfying end, emerging with the asymptotic black hole partition function and hence a generalized Cardy formula for spin-3 charged black holes. Our results and their CFT counterparts [74] lend support to the minimal model duality proposal. \nIn most of the following, we focus on the unbarred Chern-Simons connection with the barred sector implied. We work with the connection a = a z dz + a z dz stripped of ρ -dependence as in (2.41), and the ρ -dependence is restored by conjugation with a matrix we introduce momentarily.', '4.1. hs[ λ ] higher spin gravity': "The hs[ λ ] Lie algebra is spanned by generators labeled by two integers: a spin and a mode index. We use the notation of [48], in which a generator is represented as \nV s m , s ≥ 2 , | m | < s . (4 . 1) \nThe commutation relations are \n[ V s m , V t n ] = s + t -| s -t |-1 ∑ u =2 , 4 , 6 ,... g st u ( m,n ; λ ) V s + t -u m + n (4 . 2) \nwith structure constants given in Appendix A. The parameter λ labels inequivalent algebras; while its appearance enters via generalized hypergeometric functions, these simplify to polynomials in λ 2 when evaluated for integer ( s, m ). \nThe polynomial appearance of λ 2 is obvious upon constructing hs[ λ ] as a subspace of a quotient of the universal enveloping algebra of sl (2 , R ) by its quadratic Casimir. For an sl (2 , R ) with canonical commutation relation (2.39), the Casimir C 2 is fixed as \nC 2 = ( L 0 ) 2 -1 2 ( L 1 L -1 + L -1 L 1 ) = 1 4 ( λ 2 -1) . (4 . 3) \nThe hs[ λ ] generators are then constructed from sl (2 , R ) generators as 7 \nV s m = ( -1) m + s -1 ( m + s -1)! (2 s -2)! [ L -1 , . . . [ L -1 , [ L -1 , ︷︷ s -1 -m L s -1 1 ]] ] (4 . 4) \n︸ \n︷︷ which generate hs[ λ ] upon modding out by the ideal formed by (4.3) and dropping the identity element, which is formally V 1 0 . \n︸ \nThe generators with s = 2 form an sl (2 , R ) subalgebra, and the remaining generators transform simply under the adjoint sl (2 , R ) action as \n[ V 2 m , V t n ] = ( m ( t -1) -n ) V t m + n . (4 . 5) \nThese sl (2 , R ) generators will be relevant in the construction of the BTZ solution. In addition we can move from a to A connections a la (2.41) by conjugation with \nb = e ρV 2 0 (4 . 6) \ngiving generators with mode index m a factor of e mρ . \nIn contrast to the sl ( N,R ) algebras, hs[ λ ] has only a single sl (2 , R ) subalgebra, nor does it have any other nontrivial subalgebra: any commutator of two generators each with spin s > 2 will produce a generator with spin t > s . However, at λ = N an ideal forms, consisting of all generators with spins s > N , and one recovers sl ( N,R ) upon modding out by this ideal. This feature is manifest, for instance, in the simple low-spin commutator \n[ V 3 2 , V 3 -2 ] = 8 V 4 0 -4 5 ( λ 2 -4) V 2 0 (4 . 7) \nupon setting λ = 2. In this manner we can view the sl ( N,R ) gravity theories as limiting cases of hs[ λ ] gravity 8 . \nWhen λ = 1 / 2, this algebra is isomorphic to hs(1,1) whose commutator can be written as the antisymmetric part of the Moyal product. Similarly, the general λ commutation relations (4.2) can be realized as a star commutator \n[ V s m , V t n ] = V s m /star V t n -V t n /star V s m (4 . 8) \nif we define the associative product \nV s m /star V t n ≡ 1 2 s + t -| s -t |-1 ∑ u =1 , 2 , 3 ,... g st u ( m,n ; λ ) V s + t -u m + n . (4 . 9) \nThis is known as the 'lone star product' [43], and (4.8) follows upon using the fact that \ng st u ( m,n ) = ( -1) u +1 g ts u ( n, m ) . (4 . 10) \nIn the remainder of this chapter, all multiplication of hs[ λ ] generators is done using this lone star product. \nOne can define a bilinear trace by picking out the V 1 0 element of any lone star product, up to some normalization: \nTr( V s m V t n ) ∝ g st s + t -1 ( m,n ; λ ) δ st δ m, -n . (4 . 11) \nExplicitly, this structure constant can be written rather simply as \ng ss 2 s -1 ( m, -m ; λ ) = ( -1) m 2 3 -2 s Γ( s + m )Γ( s -m ) (2 s -1)!!(2 s -3)!! s -1 ∏ σ =1 ( λ 2 -σ 2 ) . (4 . 12) \nNote that the trace automatically factors out the ideal when λ = N : when we compute the holonomies of our black hole, this will allow easy comparison to the sl ( N,R ) cases. To interface with the previous chapter's sl (3 , R ) results and the conventions of [69,70], we choose to normalize our trace as \nTr( V s m V s -m ) = 12 ( λ 2 -1) g ss 2 s -1 ( m, -m ; λ ) (4 . 13) \nwhich implies the s = 2 traces \nTr( V 2 1 V 2 -1 ) = -4 , Tr( V 2 0 V 2 0 ) = 2 . (4 . 14) \nIn [48], it was shown that the asymptotic symmetry group of hs[ λ ] gravity with generalized AdS boundary conditions (3.6) is the infinite-dimensional W -algebra known as W ∞ [ λ ]. To show this in the unbarred sector, say, one follows (2.43) by writing an hs[ λ ]-valued connection in highest-weight gauge, as \na z = V 2 1 -2 π k L ( z ) V 2 -1 + ∞ ∑ s =3 J ( s ) ( z ) V s 1 -s (4 . 15) \nwhere the J ( s ) ( z ) are spins currents, and demanding that an infinitesimal gauge transformation leave the form of the connection intact. W ∞ [ λ ] is a nonlinear algebra with OPEs now known in closed form [42]. It contains hs[ λ ] not as a proper subalgebra, but only in the infinite central charge limit in which the nonlinear terms drop out; from this perspective, hs[ λ ] is known as the 'wedge subalgebra' of W ∞ [ λ ]. Expanding the currents in modes, \nJ ( s ) ( z ) = ∑ m ∈ Z J ( s ) m z m + s , (4 . 16) \nthe hs[ λ ] generators are identified as the 'wedge modes' \nJ ( s ) m = V s m , | m | < s . (4 . 17) \nTo conclude this subsection, the black holes we will construct are dual to asymptotically high temperature states of generic CFTs with W ∞ [ λ ] symmetry, deformed by spin-3 chemical potential. Not only will this be a useful perspective as we compute in the bulk, but we will later discuss the dual CFT computation [74] that yields perfect agreement with the bulk results.", '4.2. Building the hs[ λ ] black hole': 'There are two main steps to constructing higher spin black holes in a generic theory of higher spin gravity with Lie algebra G containing at least one sl (2 , R ) subalgebra. These are preceded by a zeroth step, which is to write down the BTZ solution using generators of an sl (2 , R ) subalgebra of G , and to compute its Euclidean time circle holonomy eigenvalues. The two steps are: \n- (i) Write down the higher spin black hole connection with some nonzero higher spin chemical potentials.', '(ii) Make the black hole smooth.': 'To find the right connection, we recall the sl (3 , R ) black hole connection (3.13) in wormhole gauge: Ward identities showed that the leading term in a z represents a nonzero spin-3 chemical potential, and flatness fixed the rest of a z to be a simple traceless function of a z . We are led to generalize this to the G case: an unbarred connection for a higher spin black hole with nonzero spins chemical potential µ s can be written, in wormhole gauge, as \n[ \n] where multiplication is determined by the chosen representation of the bulk Lie algebra, and the barred connection is as usual implied. To restore the ρ -dependence, one conjugates \na z = a BTZ z +(higher spin charges) a z ∼ µ s ( a z ) s -1 -trace ] (4 . 18) \nas in (2.41) by b = e ρL 0 , where L 0 is the diagonal element of the sl (2 , R ) subalgebra. Note that a z is simply the highest-weight gauge connection, as in (4.15), with constant charges. \nAs a quick check, this will clearly give the correct asymptotic falloff at large ρ for the chemical potential term: the leading piece of ( a z ) s -1 will carry e ( s -1) ρ dependence. In the G =hs[ λ ] case, for instance, this is manifestly true because \n( V 2 1 ) s -1 = V s s -1 (4 . 19) \nas is obvious from (4.4). \nAs we have emphasized repeatedly in earlier sections, in order to make the black hole smooth , we must fix all charges in terms of the potentials ( µ s , τ ) consistent with an integrability condition (2.55), where now Q ∼ J ( s ) and α ∼ α s ∝ µ s . To do so, we return to the BTZ holonomy condition: computing the Euclidean time circle holonomy matrix ω once again, \nω = 2 π ( τa z + ¯ τa z ) (4 . 20) \nwe demand that its eigenvalues equal those of the BTZ solution. One conveniently expresses this condition as \nTr( ω n ) = Tr( ω n BTZ ) , n = 2 , 3 , . . . , rk( ω ) . (4 . 21) \nAlthough the connections (4.18) are in wormhole gauge, the sl (3 , R ) case encourages the perspective that somewhere on the gauge orbit of this solution lies a solution with a manifest black hole metric. Its smooth Euclidean horizon as seen by the tower of higher spin fields exactly reproduces the holonomy equations (4.21). We assume this to be true in what follows. \nLet us also comment that (4.21) works directly on the level of the matrix ω , that is, we do not exponentiate to determine the holonomy H . With regard to constructing smooth black hole solutions, this approach suffices to capture the essential feature of the BTZ black hole; in the case where the exponentiation of G to a Lie group is not understood, neither is the notion of a trivial holonomy. This is the case with G = hs[ λ ]. Nevertheless we continue to abuse language by referring to (4.21) as a holonomy condition. 9 \nWithout further ado, let us apply this algorithm to construct the hs[ λ ] black hole with spin-3 chemical potential. First, the smooth BTZ solution is \na z = V 2 1 + 1 4 τ 2 V 2 -1 a z = 0 (4 . 22) \nwhere we have used (2.20). The BTZ holonomy matrix ω BTZ is \nω BTZ = 2 πτ ( V 2 1 + 1 4 τ 2 V 2 -1 ) . (4 . 23) \nAll oddn traces vanish. The lowest evenn traces are \nTr( ω 2 BTZ ) = -8 π 2 Tr( ω 4 BTZ ) = 8 π 4 (3 λ 2 -7) \n5 Tr( ω 6 BTZ ) = -8 π 6 7 (3 λ 4 -18 λ 2 +31) \n. \nThe simplest higher spin black hole has spin-3 chemical potential, so following (4.18) our ansatz is \na z = V 2 1 -2 π L k V 2 -1 -N ( λ ) π W 2 k V 3 -2 + J a z = -µN ( λ ) ( a z /star a z -2 π L 3 k ( λ 2 -1) ) (4 . 25) \nwhere \nJ = J (4) V 4 -3 + J (5) V 5 -4 + . . . (4 . 26) \nallows for an infinite series of higher-spin charges. The solution is accompanied by the analogous barred connection. N ( λ ) is a normalization factor, \nN ( λ ) = √ 20 ( λ 2 -4) (4 . 27) \nchosen to simplify comparison to the sl (3 , R ) results of the previous section. In particular, truncating all spins s > 3 gives a solution with the same generator normalizations and bilinear traces as the spin-3 black hole of the sl (3 , R ) theory. \nSuppressing the dependence on barred quantities, we think of this black hole as a saddle point contribution to the partition function \nZ ( τ, α ) = Tr [ e 4 π 2 i ( τ L + α W ) ] (4 . 28) \nwhere we continue to define the potential as \nα = τµ (4 . 29) \nwhere τ is the modular parameter of the boundary torus. This assignment will once again be justified upon solving the holonomy equations, as the charges will satisfy the integrability condition \n∂ L ∂α = ∂ W ∂τ . (4 . 30) \n(4 . 24) \nIt is instructive to compare the hs[ λ ] black hole to the sl (3 , R ) black hole. There are three novel infinities here: \n- · The number of holonomy equations (4.21), by virtue of the infinite dimensionality of hs[ λ ]. Morally, we are fixing smoothness at the black hole horizon for the metric and an infinite tower of higher spin fields.\n- · The UV behavior of the tower of metric-like higher spin fields. While there is no unambiguous prescription for how to write these fields in terms of traces over vielbeins beyond low spins [42], the spins field will generically involve traces over s vielbeins. Recalling (2.36) and our connection (4.25) (and its barred partner), metric-like higher spin fields grow with increasingly large powers of e ρ at successively higher spins. As with the metric, such behavior is gauge-dependent, and in contrast to the terms in the ChernSimons connection it is not clear how to physically intepret these fields; but their rapid UV growth bears noting.\n- · The number of nonzero higher spin charges J ( s ) , as a generic solution of the infinite set of holonomy equations would dictate. This is particularly clear from a W ∞ [ λ ] perspective: thinking of taking operator expectation values \n〈 J ( s ) 〉 α = Tr [ J ( s ) e 4 π 2 i ( τ L + α W ) ] Tr [ e 4 π 2 i ( τ L + α W ) ] (4 . 31) \nand expanding perturbatively in α , the coupled nature of the W ∞ [ λ ] OPEs guarantees that all charges are nonzero. In particular, one uncovers the following simple perturbative expansion, \nL = L 0 + α 2 L 2 + . . . W = α W 1 + α 3 W 3 + . . . J ( s ) = α s -2 J ( s ) s -2 + α s J ( s ) s + . . . . (4 . 32) \nThe polynomial degree and interdependence of the holonomy equations will force us to work perturbatively in α .', '4.3. Black hole partition function': 'The holonomy matrix ω for the black hole connection (4.25) is \nω = 2 π [ τa z + αN ( λ ) ( a z /star a z -2 π L 3 k ( λ 2 -1) ) ] . (4 . 33) \nDetails on the mechanics of solving equations (4.21), and the equations themselves, are given in [72]. Suffice it to say here that this is a highly overconstrained problem which \nnevertheless has a solution. Using the perturbative expansions (4.32), the charges through O ( α 8 ) are given as \nL = -k 8 πτ 2 + 5 k 6 πτ 6 α 2 -50 k 3 πτ 10 λ 2 -7 λ 2 -4 α 4 + 2600 k 27 πτ 14 5 λ 4 -85 λ 2 +377 ( λ 2 -4) 2 α 6 -68000 k 81 πτ 18 20 λ 6 -600 λ 4 +6387 λ 2 -23357 ( λ 2 -4) 3 α 8 + . . . W = -k 3 πτ 5 α + 200 k 27 πτ 9 λ 2 -7 λ 2 -4 α 3 -400 k 9 πτ 13 5 λ 4 -85 λ 2 +377 ( λ 2 -4) 2 α 5 + 32000 k 81 πτ 17 20 λ 6 -600 λ 4 +6387 λ 2 -23357 ( λ 2 -4) 3 α 7 + . . . J (4) = 35 9 τ 8 1 λ 2 -4 α 2 -700 9 τ 12 2 λ 2 -21 ( λ 2 -4) 2 α 4 + 2800 9 τ 16 20 λ 4 -480 λ 2 +3189 ( λ 2 -4) 3 α 6 + . . . J (5) = 100 √ 5 9 τ 11 1 ( λ 2 -4) 3 / 2 α 3 -400 √ 5 27 τ 15 44 λ 2 -635 ( λ 2 -4) 5 / 2 α 5 + . . . J (6) = 14300 81 τ 14 1 ( λ 2 -4) 2 α 4 + . . . . (4 . 34) \nThe charges ( L , W ) manifestly satisfy the integrability condition; integrating either one of them, we recover the left-moving black hole partition function through O ( α 8 ): \nln Z ( τ, α ) = iπk 2 τ [ 1 -4 3 α 2 τ 4 + 400 27 λ 2 -7 λ 2 -4 α 4 τ 8 -1600 27 5 λ 4 -85 λ 2 +377 ( λ 2 -4) 2 α 6 τ 12 + 32000 81 20 λ 6 -600 λ 4 +6387 λ 2 -23357 ( λ 2 -4) 3 α 8 τ 16 ] + . . . . (4 . 35) \nThe leading term is the BTZ term (2.24), using ln Z ( τ, α ) = -I ( τ, α ) and c = 6 k . This result reduces to the sl (3 , R ) results of the previous section at λ = 3, and has also been checked against independent calculations for λ = 4 , 5 , 1 2 . \nThis is to be viewed as a Cardy formula for higher spin black holes with nonzero spin-3 chemical potential in CFTs with W ∞ [ λ ] symmetry. It applies in the asymptotic regime \nτ → 0 , α → 0 , α τ 2 fixed . (4 . 36) \nGiven that we construct this solution in perturbation theory around the BTZ black hole, it is only sensible to think of this solution as a black hole too, now augmented by higher spin degrees of freedom contributing to its entropy. It is a delicate question whether this formula holds away from infinite temperatures, and for what CFTs. In the absence of a one-loop computation which has yet to be done, it is unclear when corrections to this result from perturbative excitations outside the black hole become important. If it holds \nfor finite but large temperatures, any potential match to a CFT will involve more than its vacuum structure. \nWhat is clear, however, is that at large enough temperatures, this result gives the higher spin charged black hole saddle point contribution to the Euclidean path integral. We now give more evidence for this from holography.', '4.4. Matching to CFT': "One might ask whether the partition function of the black hole in the sl (3 , R ) theory -which we computed exactly in the classical limit - can be reproduced from some dual CFT calculation, but there are no known examples of unitary CFTs with W 3 symmetry and large central charge. The same goes for the series of W N minimal models at finite N . But there are such examples of CFTs with W [ λ ] symmetry. \nA simple realization of W ∞ [ λ ] symmetry at λ = 1 is given by a theory of free, singlet complex bosons. The holomorphic and anti-holomorphic spins currents are bilinear in the bosons, with s derivatives peppered about. Accordingly, the symmetry algebra (also known [43] as W PRS ∞ ) can be written in a linear basis, unlike the W ∞ [ λ ] algebra for all other values of λ . One can derive the exact partition function (4.28) in this theory at asymptotically high temperatures τ 2 → 0, which is \n∞ \nln Z ( τ, α ) = -3 ik 2 πτ ∫ ∞ 0 dx [ ln ( 1 -e -x + iaα τ 2 x 2 ) +ln ( 1 -e -x -iaα τ 2 x 2 )] (4 . 37) \na = √ 5 3 π 2 . (4 . 38) \nwhere \nExpanding in α gives the result (4.35) at λ = 1. We notice that (4.37) is nonanalytic at α = 0, so its series expansion has zero radius of convergence, as expected. A similar computation can be done at λ = 0 with free fermions. \nThe reason these computations match the bulk result is due to symmetry. These free CFTs are not dual to Vasiliev gravity, if only on account of their U ( N ) symmetry; to reproduce (4.35), apparently all that matters is that they furnish the asymptotic symmetry of the hs[ λ ] gravity theory. The partition function can be viewed in CFT as a perturbative sum of torus correlation functions. At asymptotically high temperatures, we can relate each term by a modular S transformation to the low temperature behavior, which is in turn determined by the vacuum symmetry algebra alone. \nThis is exactly the philosophy discussed in the introduction for computing the BTZ partition function, indeed it follows purely from the presence of 2d conformal symmetry, extended or not. For some more insight, we return to the other example discussed in section 2, the charged BTZ black hole. On the CFT side, the symmetry is Virasoro plus a U(1) current algebra; the left-moving partition function \nZ ( τ, z ) = Tr [ e 2 πiτL 0 e 2 πizQ ] (4 . 39) \nwith U(1) charge Q and conjugate potential z transforms under a modular S -transformation as \nZ ( τ ' = -1 /τ, z ' = -z/τ ) = e πic 3 z 2 τ Z ( τ, z ) (4 . 40) \nwhere c is the central charge [84]. The entropy of the charged black hole quoted in (2.31) follows entirely from (4.40) and the on-shell action in global AdS with a flat U(1) connection, which can be computed from the action (2.25). \nIn the case of the higher spin black hole with spin-3 chemical potential, the left-moving W ∞ [ λ ] CFT partition function is \nZ ( τ, α ) = Tr [ e 2 πiτL 0 e 2 πiαW 0 ] (4 . 41) \nwith spin-3 charge W 0 and conjugate potential α . Unlike the case of a spin-1 current insertion, we do not know the modular transformation property of this object. One can instead proceed perturbatively in α , carrying out modular transformations term-by-term. This calculation was carried out in [74] for arbitrary λ through O ( α 6 ), to perfect agreement with (4.35). \nShifting notation 10 to that of [74], we consider the partition function \nZ (ˆ τ, α ) = Tr ( ˆ q L 0 -c 24 y W 0 ) , ˆ q = e 2 πi ˆ τ , y = e 2 πiα (4 . 42) \nso that ˆ τ → 0 is the high temperature limit. Expanding in α , \nZ ( ˆ τ, α ) = Tr ( ˆ q L 0 -c 24 ) + (2 πiα ) 2 2! Tr ( ( W 0 ) 2 ˆ q L 0 -c 24 ) + (2 πiα ) 4 4! Tr ( ( W 0 ) 4 ˆ q L 0 -c 24 ) + · · · . (4 . 43) \nOur goal is to compute the modular transformation properties of these traces and evaluate in the vacuum. Thus we compute connected 2 n -point correlators of W currents on the plane and map them to the infinite cylinder. \nLet us present some of the relevant technology; the reader is referred to [74] for the full scope of the computation, which includes new results on modular properties of traces with higher spin zero mode insertions for arbitrary spin. We specialize to the spin-3 Virasoro primary with conformal dimension 3. Generally, one can express traces of zero modes over some representation r in terms of contour integrals of torus amplitudes F r , which are function of the spin-3 current W and τ , as follows: \nTr r ( ( W 0 ) n q L 0 -c 24 ) = 1 (2 πi ) n n ∏ j =1 ∮ dz j z j F r (( W 1 , z 1 ) , . . . , ( W n , z n ); τ ) (4 . 44) \nwhere W 0 is defined by the mode expansion \nW 0 = 1 2 πi ∮ dzW ( z ) z 2 . (4 . 45) \nThe torus amplitude F r is doubly periodic under identifications z j ∼ e 2 πi z j ∼ qz j , and its modular transformation properties are known: in particular, under an S -transformation, we have [94] \nF r (( W 1 , z 1 ) , . . . , ( W n , z n ); ˆ τ = -1 /τ ) = τ 3 n ∑ s S rs F s (( W 1 , z τ 1 ) , . . . , ( W n , z τ n ); τ ) (4 . 46) \nwhere S rs is the modular S -matrix. Therefore we can obtain the high temperature trace (4.44) via contour integrals of vacuum torus amplitudes. We are interested only in the partition function at asymptotically high temperatures, which means that we take all traces in the vacuum representation of W ∞ [ λ ] (i.e. r = 0 in the above), ignoring subleading corrections. Consequently, we would like to simplify (4.46) as much as we can in terms of traces over zero modes which annihilate the vacuum. \nTo actually evaluate these integrals, we want to exchange currents W in the F r for zero modes W 0 and boil everything down to taking vacuum expectation values. We can do just this with the torus amplitude recursion relation \nF (( W 1 , z 1 ) , . . . , ( W n , z n ); τ ) = F ( W 0 ; ( W 2 , z 2 ) , . . . , ( W n , z n ); τ ) + n ∑ j =2 ∑ m ∈ N 0 P m +1 ( z j z 1 , q ) F ( ( W 2 , z 2 ) , . . . , ( W 1 [ m ] W j , z j ) , . . . , ( W n , z n ); τ ) (4 . 47) \nwhere the P m +1 are Weierstrass functions defined in [74], and the quantity W [ m ] represents a linear combination of modes of W , \nW [ m ] = (2 πi ) -m -1 ∞ ∑ j = m c (3 , j, m ) W j -2 (4 . 48) \nwhere the constants c (3 , j, m ) are defined by the generating function \n(log(1 + z )) m (1 + z ) 2 = ∞ ∑ j = m c (3 , j, m ) z j . (4 . 49) \nIn words, for every current in a torus amplitude, we can replace it by a zero mode as long as we add new terms encoding the interactions between currents. The first term on the right hand side of (4.47) can be written as a trace over spin-3 zero modes and vertex operators V ( W j , z j ), \nF ( W 0 ; ( W 2 , z 2 ) , . . . , ( W n , z n ); τ ) = n ∏ j =2 z 3 j Tr ( W 0 V ( W 1 , z 1 ) . . . V ( W n , z n ) q L 0 -c/ 24 ) . (4 . 50) \nFurther reducing these to traces over zero modes, the second term on the right hand side of (4.47) can be reduced by repeated application of (4.47). Similarly, the first term can be reduced with help of a recursion relation derived in [74] that is the analog of (4.47), now for torus amplitudes with zero mode insertions. Using these recursion relations, one can eventually boil down the trace (4.44) to contour integrals of Weierstrass functions and their derivatives, multiplied by vacuum n -point correlators of W currents. Precisely which correlators appear is determined by the series (4.48) and the powers of z required for nonzero Weierstrass integrals. \nThe preceding recursion relations and modular S -transformation properties generalize to insertions of Virasoro primary fields of arbitrary positive integral conformal dimension, and to arbitrary modular transformations [74]. \nAn interesting aspect of this calculation is that the nonlinear parts of W ∞ [ λ ] OPEs are crucial for the success of the matching to the bulk partition function, despite the initial expectation that they are irrelevant in the large c limit. These enter at O ( α 6 ), where the CFT computation becomes rather hefty; this is in contrast to the bulk calculation using black hole holonomy which appears to encode the same information in a more efficient way. A direct understanding of how and 'why' the bulk holonomy condition captures these features of the CFT would be desirable.", '5. Discussion': 'We close this review by presenting a - by no means complete - list of open problems and avenues for future research. \nWhile one motivation for the study of higher spin theories is to obtain solvable models without relying on supersymmetry, it would nevertheless be worthwhile to consider the supersymmetric generalizations of the higher spin gravities and the Gaberdiel-Gopakumar AdS 3 /CFT 2 duality. In particular, it would be interesting to investigate the existence and properties of BPS black holes in these theories; some work in this direction was recently initiated in [67,68]. \nOne of the main results of the work described in this review is that holonomy provides us with the correct criterion for what constitutes an admissible smooth geometry. The holonomy conditions work for the black hole as well as the conical deficit/surplus geometries [58]. It would be very interesting to find an independent derivation of the holonomy conditions from a more geometric point of view in higher spin gravity. \nFor hs[ λ ] higher spin gravity the construction of black holes carrying higher spin charge utilizes a perturbation expansion in the higher spin charge α . As discussed in section 4 there have been impressive checks of this proposal by relating the perturbative α expansion of the bulk partition function to the corresponding result in the CFT. However, the perturbation expansion produces a small charge perturbation around the uncharged BTZ black hole. Whether the charged black hole solution can be generalized to finite values of α is an open problem, as the perturbation series for the special values λ = 0 , 1 (and quite likely for \ngeneral values of λ ) is not convergent. Mathematically, the calculation of the holonomy of a black hole with finite charge would entail the construction of an exponential map for the infinite dimensional algebra hs[ λ ]. Whether such a map is possible to construct, i.e. whether its possible to construct an analog of the notion of a Lie group of hs[ λ ], is an open question. \nThe fact that matter can be consistently coupled to the hs[ λ ] higher spin gravity provides us in principle with a probe of the black hole geometry which is not gauge dependent. One can calculate scalar two-point correlators perturbatively in α in the black hole background of section 4, which should help clarify the extent to which this solution can rightfully be called a black hole. This is work in progress [95]; for now we note only that the pure gauge nature of the higher spin fields plays a crucial role in efficiently computing the correlator. \nThe simplicity of the black hole solution in three dimensional higher spin gravity can be attributed to the Chern-Simons like formulation of the higher spin dynamics. In three dimensions all massless higher spin fields do not carry any propagating degrees of freedom and hence even the hs[ λ ] theory is almost topological (apart from matter which can be consistently coupled). Furthermore the large amount of symmetry makes it feasible to perform calculations of AdS/CFT correlators in the black hole background, which can help to address many basic questions of black hole physics, such as the formation and evaporation and the information paradox. \nOn the other hand, in four dimensions it is quite complicated to construct nontrivial solutions of Vasiliev theory for several reasons. First, the analog of the simple BTZ black hole solution is missing in four dimensions. Second, the dynamics are considerably more complicated due to the fact that the higher spin fields are not topological anymore. Third, there is no known analog of the holonomy condition described above, which diagnoses the existence of a black hole in a gauge invariant manner. \nExamples of solutions in four dimensional Vasiliev theory include [96,97,98,99], and those potentially related to black holes include [100,101,102]. However the black hole character of the solutions can only be seen in a weak field approximation in the gravitational sector. Recently the finite temperature behavior of vector models on the sphere in the large N limit was studied on the field theory side in [103]. It was found that the phase transition occurs at very high temperatures of order √ N and it was suggested that this behavior of the free energy cannot be described by a black hole in the bulk higher spin gravity as expected by the standard AdS/CFT philosophy. \nAn interesting question is whether it is possible to generalize the lessons learned from three dimensional higher spin gravity to higher dimensional higher spin theories. It may be the case that the advances in lower dimensions help us to construct the elusive black hole solutions in four dimensional higher spin gravity.', 'Acknowledgments': 'This work was supported in part by NSF grant PHY-07-57702.', 'Appendix A. Some details on hs[ λ ]': 'The hs[ λ ] structure constants are given as \ng st u ( m,n ; λ ) = q u -2 2( u -1)! φ st u ( λ ) N st u ( m,n ) (A.1) \nwhere \nN st u ( m,n ) = u -1 ∑ k =0 ( -1) k ( u -1 k ) [ s -1 + m ] u -1 -k [ s -1 -m ] k [ t -1 + n ] k [ t -1 -n ] u -1 -k φ st u ( λ ) = 4 F 3 [ 1 2 + λ , 1 2 -λ , 2 -u 2 , 1 -u 2 3 2 -s , 3 2 -t , 1 2 + s + t -u ∣ ∣ ∣ ∣ 1 ] (A.2) \n∣ \nWe make use of the descending Pochhammer symbol, \n[ a ] n = a ( a -1) ... ( a -n +1) . (A.3) \nq is a normalization constant that can be scaled away by taking V s m → q s -2 V s m . As in much of the existing literature, we choose to set q = 1 / 4. \nWe note a handful of useful properties of the structure constants: \nφ st u ( 1 2 ) = φ st 2 ( λ ) = 1 N st u ( m,n ) = ( -1) u +1 N ts u ( n, m ) N st u (0 , 0) = 0 , u even N st u ( n, -n ) = N ts u ( n, -n ) . 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2016MNRAS.463.2323W

Production of the entire range of r-process nuclides by black hole accretion disc outflows from neutron star mergers

2016-01-01

['accretion', 'accretion disks', 'dense matter', 'gravitational waves', 'neutrinos', 'nuclear reactions;nucleosynthesis;abundances', 'nuclear reactions;nucleosynthesis;abundances', 'nuclear reactions;nucleosynthesis;abundances', '-', '-', '-', '-', 'nuclear reactions;nucleosynthesis;abundances']

[]

We consider r-process nucleosynthesis in outflows from black hole accretion discs formed in double neutron star and neutron star-black hole mergers. These outflows, powered by angular momentum transport processes and nuclear recombination, represent an important - and in some cases dominant - contribution to the total mass ejected by the merger. Here we calculate the nucleosynthesis yields from disc outflows using thermodynamic trajectories from hydrodynamic simulations, coupled to a nuclear reaction network. We find that outflows produce a robust abundance pattern around the second r-process peak (mass number A ∼ 130), independent of model parameters, with significant production of A &lt; 130 nuclei. This implies that dynamical ejecta with high electron fraction may not be required to explain the observed abundances of r-process elements in metal poor stars. Disc outflows reach the third peak (A ∼ 195) in most of our simulations, although the amounts produced depend sensitively on the disc viscosity, initial mass or entropy of the torus, and nuclear physics inputs. Some of our models produce an abundance spike at A = 132 that is absent in the Solar system r-process distribution. The spike arises from convection in the disc and depends on the treatment of nuclear heating in the simulations. We conclude that disc outflows provide an important - and perhaps dominant - contribution to the r-process yields of compact binary mergers, and hence must be included when assessing the contribution of these systems to the inventory of r-process elements in the Galaxy.

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https://arxiv.org/pdf/1607.05290.pdf

{'Production of the entire range of r -process nuclides by black hole accretion disc outflows from neutron star mergers': "Meng-Ru Wu 1 ? , Rodrigo Fern'andez 2 ; 3 , Gabriel Mart'ınez-Pinedo 1 ; 4 , Brian D. Metzger 5 \n- 1 Institut fur Kernphysik, Technische Universitat Darmstadt, 64289 Darmstadt, Germany\n- 2 Department of Physics, University of California, Berkeley, CA 94720, USA\n- 3 Department of Astronomy and Theoretical Astrophysics Center, University of California, Berkeley, CA 94720, USA\n- 4 GSI Helmholtzzentrum fur Schwerionenforschung, Planckstr. 1, 64291 Darmstadt, Germany\n- 5 Department of Physics and Columbia Astrophysics Laboratory, Columbia University, New York, NY, 10027, USA \nReceived / Accepted", 'ABSTRACT': 'We consider r -process nucleosynthesis in outflows from black hole accretion discs formed in double neutron star and neutron star - black hole mergers. These outflows, powered by angular momentum transport processes and nuclear recombination, represent an important - and in some cases dominant - contribution to the total mass ejected by the merger. Here we calculate the nucleosynthesis yields from disc outflows using thermodynamic trajectories from hydrodynamic simulations, coupled to a nuclear reaction network. We find that outflows produce a robust abundance pattern around the second r -process peak (mass number A GLYPH<24> 130), independent of model parameters, with significant production of A < 130 nuclei. This implies that dynamical ejecta with high electron fraction may not be required to explain the observed abundances of r -process elements in metal poor stars. Disc outflows reach the third peak ( A GLYPH<24> 195) in most of our simulations, although the amounts produced depend sensitively on the disc viscosity, initial mass or entropy of the torus, and nuclear physics inputs. Some of our models produce an abundance spike at A = 132 that is absent in the Solar system r -process distribution. The spike arises from convection in the disc and depends on the treatment of nuclear heating in the simulations. We conclude that disc outflows provide an important and perhaps dominant - contribution to the r -process yields of compact binary mergers, and hence must be included when assessing the contribution of these systems to the inventory of r -process elements in the Galaxy. \nKey words: accretion, accretion discs - dense matter - gravitational waves - nuclear reactions, nucleosynthesis, abundances - neutrinos - stars: neutron', '1 INTRODUCTION': "Approximately half of the elements with mass number A > 70, and all of the transuranic nuclei, are formed by the rapid neutron capture process (the r -process; Burbidge et al. 1957, Cameron 1957). The astrophysical site of this process has been under debate for more than 50 yrs (see, e.g., Qian & Wasserburg 2007, Arnould et al. 2007, Sneden et al. 2008, Thielemann et al. 2011 for reviews). Neutrino-driven outflows from proto neutron stars (NSs) following core collapse supernovae have for long been considered the prime candidate site (Meyer et al. 1992; Woosley et al. 1994; Qian & Woosley 1996). However, state-of-the-art calculations find thermodynamic conditions that are at best marginal for the r -process, especially when extending up to the heaviest third-peak elements with \nmass number A GLYPH<24> 195 (e.g. Mart'ınez-Pinedo et al. 2012; Roberts et al. 2012; Mart'ınez-Pinedo et al. 2014). \nProspects for a successful r -process in neutrino-driven outflows may be improved if the proto-NS is born with a strong magnetic field and very rapid rotation (e.g. Thompson et al. 2004; Metzger et al. 2007; Vlasov et al. 2014). If the supernova itself is MHD-powered, additional magnetocentrifugal acceleration could substantially reduce the electron fraction of the outflow compared to its value in the purely neutrino-driven case favouring the occurrence of an r -process (e.g. Burrows et al. 2007, Winteler et al. 2012, Nishimura et al. 2015). However, current simulations of MHDpowered supernova explosions need further improvements, especially considering the role of instabilities on the jet structure which manifest in three dimensions (Mosta et al. 2014). \nThe coalescence of double NS (NS-NS) and NS-black hole (NS-BH) binaries (Lattimer & Schramm 1974) provides an al- \nternative r -process source. Numerical simulations of these events show that a robust outcome of the merger is the ejection of GLYPH<24> 10 GLYPH<0> 4 -10 GLYPH<0> 1 M GLYPH<12> of highly neutron-rich matter on the dynamical time (e.g. Hotokezaka et al. 2013, Bauswein et al. 2013; see Lehner & Pretorius 2014 and Baiotti & Rezzolla 2016 for recent reviews). Estimates show that NS-NS / NS-BH mergers could contribute a sizable fraction of the total production of r -process elements in the Galaxy, depending on the uncertain merger rates. At the same time, previous arguments against mergers being dominant r -process sites based on Galactic chemical evolution and the observed prompt enrichment of r -process nuclei in metal poor stars (e.g. Argast et al. 2004) have been challenged (e.g., van de Voort et al. 2015; Shen et al. 2015; Hirai et al. 2015). Additional evidence supporting the presence of a 'high yield' r -process site - like an NS-NS / NS-BH merger - includes the discovery of highly r -process enriched stars in the ultra-faint dwarf galaxy Reticulum II (Ji et al. 2016), and the abundance of the short-lived isotope 244 Pu on the sea floor (Wallner et al. 2015; Hotokezaka et al. 2015). \nNucleosynthesis in NS-NS / NS-BH mergers has also received a recent surge of interest due to the realization that the radioactive decay of the r -process ejecta can power a thermal transient (a 'kilonova'; e.g., Li & Paczy'nski 1998, Metzger et al. 2010, Roberts et al. 2011, Barnes & Kasen 2013, Tanaka & Hotokezaka 2013), which could serve as a promising electromagnetic counterpart to the gravitational waves (Metzger & Berger 2012). The detection of a possible kilonova following the Swift GRB 130603B (Berger et al. 2013; Tanvir et al. 2013) highlights the potential of kilonovae as both a unique diagnostic of physical processes at work during the merger and a direct probe of the formation of r -process nuclei (see, e.g., Rosswog 2015, Fern'andez & Metzger 2016, Tanaka 2016 for recent reviews). \nPrevious work on the r -process in NS-NS / NS-BH mergers has been focused primarily on the dynamical ejecta that is unbound promptly during the immediate aftermath of the merger (e.g. Meyer 1989, Freiburghaus et al. 1999, Goriely et al. 2005). Earlier simulations that did not include weak interactions have shown this unbound matter to be highly neutron-rich, with an electron fraction Ye GLYPH<24> < 0 : 1, su GLYPH<14> ciently low to produce a robust abundance pattern for heavy nuclei with A GLYPH<24> > 130 as the result of fission cycling (e.g., Goriely et al. 2011, Korobkin et al. 2012, Bauswein et al. 2013, Mendoza-Temis et al. 2015). More recently, a number of merger calculations that include the e GLYPH<11> ects of e GLYPH<6> captures and neutrino irradiation in full general-relativity have shown that the dynamical ejecta can have a wider electron fraction distribution( Ye GLYPH<24> 0 : 1 GLYPH<0> 0 : 4) than models without weak interaction e GLYPH<11> ects (Sekiguchi et al. 2015; Foucart et al. 2015; Radice et al. 2016). As a result, lighter r -process elements with 90 GLYPH<24> < A GLYPH<24> < 130 are generated in addition to third-peak elements (Wanajo et al. 2014). It is important to keep in mind, however, that the light element yields in these calculations are dependent on the assumed dense-matter equation of state and on the details of the neutrino transport employed, in addition to the NS radii and the binary mass ratio. \nIn addition to ejecting material dynamically, NS-NS / NS-BH mergers result in the formation of an accretion disc around the central remnant (e.g., Oechslin & Janka 2006); with the latter being a promptly formed BH or a longer-lived hypermassive NS (HMNS) (e.g., Shibata & Ury¯u 2000). In both cases, the accretion disc can generate outflows on time-scales much longer than the orbital time (e.g., Metzger et al. 2008; Lee et al. 2009; Metzger et al. 2009), and with a contribution to the total mass ejection that can be comparable to, or even larger than that from the dynamical ejecta (Fig / 1, see also Fern'andez & Metzger 2016). A relatively massive disc \nFigure 1. Mass ejected dynamically during a compact binary merger versus that ejected in disc outflows. Each point corresponds to the result of a single time-dependent NS-NS (triangles) and BH-NS (squares) simulation. Shown are models by Hotokezaka et al. (2013) (blue), Oechslin & Janka (2006) (green, upper limits shown by arrows), Just et al. (2015) (brown), East et al. (2012) (red), and Foucart et al. (2014) (orange). The mass unbound in disc outflows is estimated to be 10 per cent of the mass of the remnant disc, based on calculations of the subsequent accretion disc evolution (e.g., FM13). Dashed lines show total ejecta mass contours (dynamical + disc winds) of 0 : 01 M GLYPH<12> and 0 : 1 M GLYPH<12> , bracketing the range necessary to explain the Galactic production rate of heavy r -process nuclei GLYPH<24> 5 GLYPH<2> 10 GLYPH<0> 7 M GLYPH<12> yr GLYPH<0> 1 (Qian 2000), given the allowed range of the rates of NS-NS mergers 2 [4 ; 61] Myr GLYPH<0> 1 (99% confidence) calculated based on the population of Galactic binaries (Kim et al. 2015). In reality, the ejecta mass range required to reproduce the Galactic abundances is uncertain by greater than an order of magnitude, due to systematic uncertainties on the merger rate and depending on the precise atomic mass range under consideration (e.g., Bauswein et al. 2014). See also Fern'andez & Metzger (2016). \n<!-- image --> \n( GLYPH<24> 0 : 1 M GLYPH<12> ) can be formed following a NS-NS merger, as part of the process by which the HMNS sheds angular momentum outwards prior to collapsing into a BH (e.g. Shibata & Taniguchi 2006). Long-term hydrodynamic simulations of the disc evolution show that a significant fraction of the initial disc mass ( GLYPH<24> 5 GLYPH<0> 20%, corresponding to GLYPH<24> 0 : 01 M GLYPH<12> ) is unbound in outflows powered by heating from angular momentum transport and nuclear recombination, on a timescale of GLYPH<24> > 1 s (Fern'andez & Metzger 2013, hereafter FM13; Just et al. 2015, Fern'andez et al. 2015). As the result of weak interactions, the electron fraction of the disc outflows lies in the range Ye GLYPH<24> 0 : 2 GLYPH<0> 0 : 4, generally higher than that of the dynamical ejecta, but still su GLYPH<14> ciently low to achieve the r -process (Just et al. 2015). \nAlthough most of the previous work on merger disc wind nucleosynthesis has focused on parametrized outflows powered by neutrino heating (e.g. McLaughlin & Surman 2005; Surman et al. 2008; Caballero et al. 2012; Surman et al. 2014), in analogy with proto-NS winds, time-dependent models of the long-term disc evolution show that neutrino heating is sub-dominant relative to viscous heating in driving most of the disc outflow when a BH sits at the centre (FM13, Just et al. 2015). Neutrino heating is much more important if the merger produces a long-lived HMNS (Dessart et al. 2009; Metzger & Fern'andez 2014; Perego et al. 2014; Martin et al. 2015), resulting in a larger ejecta mass with higher electron fraction, depending on the uncertain lifetime of such a remnant. To date, the only fully time-dependent nucleosynthesis study of longterm outflows from BH accretion discs was carried out by Just et al. (2015), who found that disc outflows can generate elements from \nA GLYPH<24> 80 to the actinides, with the contribution above A = 130 being sensitive to system parameters. \nIn this paper we further investigate nucleosynthesis in the outflows from NS-NS / NS-BH merger remnant accretion discs around BHs, by applying a nuclear reaction network on thermodynamic trajectories extracted from fully time-dependent, long-term hydrodynamic simulations of disc outflows. Our aim is to carry out a systematic study of the dependence of the r -process production on system parameters such as disc mass or viscosity, and on additional ingredients such as nuclear physics inputs to the reaction network or the feedback from nuclear heating on the disc dynamics. Our main conclusion is that disc outflows from NS binary mergers can in principle produce both the light and heavy r -process elements, without necessarily requiring additional contributions from the dynamical ejecta. \nThe paper is organized as follows. Section 2 describes the hydrodynamic models, extraction of thermodynamic trajectories, and the properties of the nuclear reaction network. Results and analysis are presented in section 3. Finally, section 4 summarizes our findings and discusses broader astrophysical implications.", '2.1 Disc evolution and thermodynamic trajectories': 'We evolve NS merger remnant accretion discs around BHs using the approach described in FM13, Metzger & Fern\'andez (2014), and Fern\'andez et al. (2015). The equations of hydrodynamics are solved numerically using FLASH3 (Fryxell et al. 2000; Dubey et al. 2009). The public version of the code has been modified to include the equation of state of Timmes & Swesty (2000) with abundances of nucleons and GLYPH<11> particles ( 4 He) in nuclear statistical equilibrium (NSE), angular momentum transport due to an GLYPH<11> viscosity (Shakura & Sunyaev 1973), and the pseudo-Newtonian potential of Artemova et al. (1996). In addition, neutrino emission and absorption due to charged-current weak interactions on nucleons are included through energy- and lepton number source terms using a leakage scheme for cooling and a disc-adapted lightbulb approach for selfirradiation, with simple optical-depth corrections (see FM13 and Metzger & Fern\'andez 2014 for details). \nDiscs start from an equilibrium initial condition and are evolved in axisymmetric spherical polar coordinates over thousands of orbits at the initial density maximum (physical time of several seconds). The majority of the mass loss is produced once the disc reaches the advective state, in which viscous heating and nuclear recombination forming GLYPH<11> particles are unbalanced by neutrino cooling (Metzger et al. 2009). Heating due to neutrino absorption makes only a minor contribution to the total mass ejection if a BH sits at the centre (FM13, Just et al. 2015). Total ejected masses can range from a few percent to more than GLYPH<24> 20% of the initial disc mass for a highly-spinning BH. \nThermodynamic trajectories of the outflow are obtained by inserting passive tracer particles in the hydrodynamic simulations. For each model, 10 4 equal mass particles are initially placed in the disc with random positions that follow the mass distribution (Fig. 2a). Each particle records thermodynamic variables including density, temperature, electron fraction and nucleon abundances, as well as neutrino and viscous source terms. Some of these quantities are used as input for the r -process nuclear reaction network ( x 2.3), while others are retained to check for consistency. \nIn order to assess the e GLYPH<11> ects of additional nuclear heating after \nTable 1. Hydrodynamic disc models evolved. Columns from left to right are: initial torus mass, black hole mass, radius of initial torus density peak, initial electron fraction, initial entropy, viscosity parameter, BH spin parameter, and amplitude of parametrized nuclear heating (equation 6). \n| Model | M t0 ( M | M BH GLYPH<12> ) | R 0 (km) | Ye 0 | s 0 ( kB / b) | GLYPH<11> | GLYPH<31> | " |\n|----------------|------------|--------------------|------------|--------|-----------------|-------------|-------------|------|\n| S-def | 0.03 | 3 | 50 | 0.10 | 8 | 0.03 | 0 | 0 |\n| GLYPH<31> 0.8 | | | | | | | 0.8 | |\n| m0.01 | 0.01 | 3 | 50 | 0.10 | 8 | 0.03 | 0 | 0 |\n| m0.10 | 0.10 | | | | | | | |\n| r75 | 0.03 | 3 | 75 | | | | | |\n| M10 | | 10 | 150 | | | | | |\n| y0.05 | | 3 | 50 | 0.05 | | | | |\n| y0.15 | | | | 0.15 | | | | |\n| s6 | | | | 0.10 | 6 | | | |\n| s10 | | | | | 10 | | | |\n| GLYPH<11> 0.01 | | | | | 8 | 0.01 | | |\n| GLYPH<11> 0.10 | | | | | | 0.10 | | |\n| " 1.0 | 0.03 | 3 | 50 | 0.10 | 8 | 0.03 | 0 | 1.0 |\n| " 0.1 | | | | | | | | 0.1 |\n| " 10.0 | | | | | | | | 10.0 | \nthe GLYPH<11> -recombination on the disc dynamics, we carry out a few simulations in which we include this extra source term. This heating is parametrized analytically as a function of temperature by analysing an ensemble of trajectories in our baseline model ( x 3.4). \nRelative to the BH accretion disc models of FM13, we have corrected an error in the treatment of the nuclear binding energy of GLYPH<11> particles (as described in Metzger & Fern\'andez 2014), which results in slightly lower mass ejection than reported in FM13. Since Fern\'andez et al. (2015) we have also corrected an error in the computation of the charged-current weak interaction rates, whereby the neutron-proton mass di GLYPH<11> erence was missing in the argument of the Fermi-Dirac integrals of the neutrino rates but not the antineutrino rates (e.g., Bruenn 1985). After correcting this error, we obtain a slightly less neutron rich outflow distribution, with a tail towards higher Ye . Nonetheless the average properties of the outflow remain relatively unchanged.', '2.2 Models Evolved': 'Table 1 presents all the models studied in this paper. Following FM13, the baseline configuration (model S-def) consists of a BH with mass M BH = 3 M GLYPH<12> , and an equilibrium torus with disc mass M t0 = 0 : 03 M GLYPH<12> , radius at density peak R 0 = 50 km, constant specific angular momentum, constant initial entropy s 0 = 8 kB per baryon, constant electron fraction Ye 0 = 0 : 1, viscosity parameter GLYPH<11> = 0 : 03 and BH spin GLYPH<31> = 0. We also include a model ( GLYPH<31> 0 : 8) with BH spin parameter GLYPH<31> = 0 : 8 to bracket the range of outcomes expected for an NS-NS merger. \nThe remaining models vary one parameter at a time, mirroring the simulations in FM13. In addition, we evolve three models that include nuclear heating after GLYPH<11> recombination in the hydrodynamics ( x 2.1). All models are evolved for 3000 orbits measured at R 0, which amounts to several seconds of physical time, in order to approach convergence in mass ejection (the disc material left over interior to r = 2 GLYPH<2> 10 9 cm amounts to a few percent of the initial disc mass). \n<!-- image --> \nFigure 2. Left: initial density field (grey) and particle distribution in model S-def. Particles that reach a distance of 2 GLYPH<2> 10 9 cm from the BH are considered to be ejected (red), with most of the remainder being accreted (blue). Right: paths followed by ejected particles, showing the convective character of the disc outflow (only 20% of the outflow trajectories are shown, for clarity). \n<!-- image -->', '2.3 Nuclear Reaction Network': 'The initial disc temperature is of the order of a few MeV, so that nearly all the ejecta consist initially of neutrons and protons. Once the temperature drops to GLYPH<24> 10 GK, most of the free protons recombine with neutrons to form 4 He, as favoured by NSE under neutronrich conditions ( Ye GLYPH<24> 0 : 2). The nuclear binding energy released by forming 4 He plays an important role in unbinding the material and is included in the hydrodynamic simulations as described in Sec. 2.1. \nIn order to follow the change in abundances of nuclear species in the disc outflow, we use a large nuclear reaction network for all tracer particles that reach a distance of 2 GLYPH<2> 10 9 cm from the centre of the BH. The nuclear reaction network includes 7360 nuclei from nucleons to 313 Ds, and reaction rates including GLYPH<11> -decays, GLYPH<12> -decays, charged-particle reactions, neutron captures and their inverse photo-dissociation interactions, as well as spontaneous, GLYPH<12> -delayed, and neutron-induced fission and the corresponding fission yields (see Mendoza-Temis et al. 2015, for a detailed description of the nuclear physics input used). In addition, we include charged-current neutrino interactions on free nucleons as described in Sec. 2.1. \nFor each tracer particle, we start the network calculation from the last moment when the temperature equals 10 GK. For a small fraction ( GLYPH<24> < 10%) of the particles the temperature is always below 10 GK, because they are initially located at the outer edge of the disc (e.g. Fig. 2a). For these particles, we instead follow the nucleosynthesis evolution from the beginning of the disc simulation. The initial composition of each trajectory is determined by NSE given the temperature, density, and Ye . The density evolution is obtained at early times by interpolating the values provided by the simulation. Once the network evolution reaches the end of the disc simulation time t f , we assume homologous expansion such that GLYPH<26> ( t ) = GLYPH<26> ( t f ) GLYPH<2> ( t f = t ) 3 subsequently. The temperature evolution is interpolated from the simulation as long as the temperature satisfies T > Ts GLYPH<17> 6 GK. For T < Ts , the temperature is evolved with the network following the same method used in Freiburghaus et al. \n(1999) by including the energy sources from the viscous and the neutrino heating for t < t f provided by the hydrodynamic simulations, and the nuclear energy release due to reactions calculated directly by the network. We have checked that the final overall abundances are not sensitive to the choice of Ts . This suggests that, differently to what happens in dynamical ejecta, r -process heating in disc ejecta does not strongly influence the abundances because the total energy released by this channel is much smaller than the internal energy of the ejecta. This is consistent with the findings of Just et al. (2015). However, we discuss in Sec. 3.4 the potential impact of nuclear heating on the disc dynamics and on specific abundance features. \nSince the r -process typically involves the nuclear physics properties of very neutron-rich nuclei that are not experimentally known, the outcome of nucleosynthesis calculations is subject to theoretical uncertainties in the modelling of these properties. In the case of dynamical ejecta, it has been shown that both the neutron-capture and GLYPH<12> -decay rates can substantially influence the positions and heights of the r -process peaks (e.g., Mendoza-Temis et al. 2015, Eichler et al. 2015). Our standard set of calculations uses neutron-capture and photo-dissociation rates based on nuclear masses from the Finite Range Droplet Model (FRDM) (Moller et al. 1995) supplemented by the GLYPH<12> -decay rates of Moller et al. (2003) for all disc models listed in Table 1. To address the so far unexplored impact of nuclear physics on disc ejecta, we have performed calculations using neutron capture and photodissociation rates based on the Duflo-Zuker mass model with 31 parameters (DZ31; Duflo & Zuker 1995) and GLYPH<12> -decay rates from Marketin et al. (2016), see Sec. 3.5.', '3 RESULTS': "Neutrino cooling in the disc is strong during the first GLYPH<24> 10 GLYPH<0> 100 ms, largely balancing viscous heating. As the disc spreads, neutrino cooling shuts o GLYPH<11> and energy deposition by viscous heating and GLYPH<11> recombination make the disc highly convective. The resulting out- \nflow peaks on a time-scale of GLYPH<24> 1 s, measured at a large enough radius that a significant fraction of the material will become gravitationally unbound from the system upon expansion ( r GLYPH<24> 10 9 cm). Weak interactions drive Ye towards its equilibrium value, which in the inner disc can be as high as GLYPH<24> 0 : 4. However, most of this material is accreted on to the BH. The bulk of the wind material arises from regions of the disc midplane and on its back side relative to the BH (Fig. 2a), and are thus only moderately influenced by neutrino interactions. A component close to the BH at high altitude is mixed into the wind, increasing the average electron fraction, particularly when the BH spin in high (Fern'andez et al. 2015). \nDuring the initial expansion at high temperature (10 GK GLYPH<24> > T GLYPH<24> > 5 GK), charged-particle reactions dominate the nucleosynthetic flow to form seed nuclei. At T GLYPH<25> 5 GK, seed nuclei abundances peak at A GLYPH<25> 50 ( N GLYPH<25> 30) and A GLYPH<25> 80 ( N GLYPH<25> 50). Heavier nuclei are only synthesized when T GLYPH<24> < 5 GK by subsequent neutron capture processes and GLYPH<12> -decays. Given the importance of this temperature threshold, it is useful to examine di GLYPH<11> erent quantities when this point is reached. For each trajectory, we denote the value of a given quantity X at T = 5 GK by \nX 5 GLYPH<17> X GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> T = 5 GK : (1) \nIf T = 5 GK is reached multiple times due to convective motions (Fig. 2b), X 5 is chosen as the value at the last time when T = 5 GK. It is also informative to measure quantities in the disc outflow at large radius, where the e GLYPH<11> ects of the BH and convection are less likely to a GLYPH<11> ect the properties of the outflow. We thus denote the value of a given quantity X at r = 10 9 cm by \nX 9 GLYPH<17> X GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> r = 10 9 cm : (2) \nFinally, we denote the average of a quantity over all trajectories by \n¯ Xs GLYPH<17> P i Xs ; i N ; (3) \nwhere N is the total number of trajectories, i labels each trajectory, and the subscript s can be 5 or 9. Note that since all trajectories have the same mass, this is a mass average.", '3.1 Ejecta properties and nucleosynthesis in the baseline model': "The r -process abundance distribution obtained with the baseline model S-def is shown in Fig. 3. A breakdown of the thermodynamic- and kinematic properties of the ejecta as a function of various quantities is illustrated in Fig. 4. The particleaveraged properties are also summarized in Table 2. The disc outflow has a broad distribution of electron fraction spanning the range 0 : 1 6 Ye ; 5 6 0 : 4, with a mean at ¯ Ye ; 5 ' 0 : 24. The entropy peaks at s 5 = 10 kB per baryon, with a long tail extending out to GLYPH<24> 30 kB per baryon, and the neutron-to-seed ratio Rn = s ; 5 lies in the range 20-100. In terms of kinematics, most of the outflow reaches 5 GK within 2 s and its angular distribution peaks at GLYPH<24> GLYPH<6> 45 GLYPH<14> -60 GLYPH<14> from the disc equatorial plane. Ejecta velocities lie in the range (0 : 2-1 : 6) GLYPH<2> 10 9 cm s GLYPH<0> 1 , much slower than the typical values for the early phase of the dynamical ejecta (see, e.g., Just et al. 2015). With such a broad distribution of Ye ; 5, the resulting average nucleosynthetic abundances are in overall agreement with the Solar system r -process distribution from A GLYPH<24> 80 to U and Th, as shown in Fig. 3, despite a large spread among individual trajectories. \nIn order to clarify the origin of the spread in abundances for individual particles, we bin trajectories into intervals of electron \nFigure 3. Abundances at 1 Gyr as a function of mass number A for the baseline model S-def. Shown are the average abundances over all trajectories (thick red curve) and results for individual particles (thin grey curves). Black dots show the Solar system r -abundances (Cowan et al. 1999) scaled to match the averaged abundances around the second peak. \n<!-- image --> \nFigure 4. Mass distribution of the disc outflow from the baseline model S-def, as a function of electron fraction Ye ; 5, neutron-to-seed ratio Rn = s ; 5, entropy s 5, time t 5, polar angle GLYPH<18> 9, and radial velocity v 9. Subscripts have the meanings defined in equations (1) and (2). \n<!-- image --> \nfraction Ye ; 5, entropy s 5, time t 5, polar angle GLYPH<18> 9, and radial velocity v 9, with the results shown in Fig. 5. The nucleosynthesis results are most sensitive to the electron fraction (see Fig. 5a). Most of the abundances above the rare-Earth peak ( A GLYPH<24> > 150) come from ejecta with Ye ; 5 GLYPH<24> < 0 : 2. Trajectories with Ye ; 5 in the range 0.1-0.3 contribute to the production of the second peak ( A ' 130). As for the abundances around A GLYPH<24> < 110, they are dominantly from the ejecta with Ye ; 5 GLYPH<24> > 0 : 25. Fission plays a role only for ejecta with Ye GLYPH<24> < 0 : 15 and consequently represents a minor contribution to the final abundances. \nFigure 5. Dependence of the abundances on di GLYPH<11> erent ejecta properties, for the baseline model S-def. Each panel shows average abundances as a function of mass number, binned into intervals of the variable shown in the upper-left corner: electron fraction (a), entropy (b), time at T = 5 GK (c), polar angle (d), radial velocity (e), and ejection time (f), with subscripts defined in equations (1) and (2). The interval values are shown in the corresponding legends. \n<!-- image --> \nInterestingly, the nucleosynthesis outcome is not sensitive to the entropy intervals in the ejecta as shown in Fig. 5b. This is mainly because the di GLYPH<11> erent entropy bins have nearly the same electron fraction distribution, with only a minor trend of lower entropy having lower peaked Ye ; 5 distributions. As a result, we do not recover the entropy dependence obtained in parametrized nucleosynthesis studies such as Lippuner & Roberts (2015). The abundance patterns are rather insensitive to other quantities such as the polar angle GLYPH<18> 9 of each particle (see Fig. 5c-e). \nThe disc outflow is strongly convective out to late times (Fig. 2b), unlike the ejecta from the neutrino-driven wind in corecollapse supernovae or the dynamical ejecta of NS-NS / NS-BH mergers, for which the density and temperature of the ejecta monotonically decrease as the medium expands. Particles in the disc outflow may in fact go through several convective cycles before being ejected at temperatures of GLYPH<24> 1 GK. To quantify the e GLYPH<11> ect of this process on the nucleosynthesis, we define the ejection time t ej of each particle as the time in the simulation after which the radial position only increases monotonically with time. This quantity is shown in Fig. 6 as a function of t 5 for all particles in the S-def model, with each point colored by the electron fraction Ye ; 5. \n; Fig. 6 clarifies several aspects of the ejecta behaviour. First, there is a substantial amount of material having much larger t ej than t 5 for t 5 GLYPH<24> < 1 s. Before ejection, these particles are subject to strong convective motions in low-temperature regions ( T < 5 GK) at a late phase in the disc evolution. Secondly, the electron fraction Ye ; 5 of trajectories ejected at the earlier times is higher. This occurs because neutrino irradiation plays a larger role in unbinding this early ejecta and thus naturally raises its Ye ; 5 to higher values. Consequently, the rare-earth and third peak abundances are much lower for matter being ejected at earlier time than for later ejecta, as shown in Fig. 5f. In particular, the abundance of Lanthanides can di GLYPH<11> er by almost a factor of 10 compared to the latest particles. This is consistent with the results of Fern'andez et al. (2015), who \nFigure 6. Ejection time t ej as a function of time t 5 (equation 1), for all trajectories from the baseline model S-def. Each point is colored by the electron fraction at time t 5. \n<!-- image --> \non the basis of the Ye distribution of the ejecta suggested that the wind contains a Lanthanide-free 'skin' - even in the case of a nonspinning BH - which can generate a small amount of blue optical emission if not obstructed by neutron-rich dynamical ejecta (Kasen et al. 2015). \nNote also that a very strong A = 132 abundance spike around the second peak is formed for ejecta with 2 : 0 < t ej < 10 s. This feature and its relation to convective motions is the subject of x 3.3.", '3.2 Dependence on model parameters': 'The e GLYPH<11> ect of changing various disc parameters on the average nucleosynthetic abundances of the outflow is shown in Fig. 7 and summarized in Table 2. We note that the Solar system r -abundances \nTable 2. Average ejecta properties for models listed in Table 1. Columns from left to right show model name, electron fraction, entropy, time, expansion times-cale GLYPH<28> exp = ¯ r = ¯ vr , radius, velocity, final average mass number, percentage of ejected mass, percentage of ejected tracer particles that contain both YA = 132 > 10 GLYPH<0> 3 and YA = 128(136) < 0 : 1 YA = 132, and the total mass fraction of lanthanides and actinides (57 6 Z 6 71 and 89 6 Z 6 103), X Lan + Act, at t = 1 d. Subscripts and averages are defined in equations (1)-(3). Only trajectories that reach r = 2 GLYPH<2> 10 9 cm at the end of the simulation are considered. \n| Model | ¯ Ye ; 5 | ¯ s 5 ( kB / b) | ¯ t 5 (s) | ¯ GLYPH<28> exp ; 5 (ms) | ¯ r 5 (10 7 cm) | ¯ vr ; 9 10 9 cm s GLYPH<0> 1 | h ¯ A i f | M ej = Mt 0 % | 132-peak % | X Lan + Act % |\n|----------------|------------|-------------------|-------------|----------------------------|-------------------|---------------------------------|-------------|-----------------|--------------|-----------------|\n| S-def | 0.237 | 19.9 | 0.65 | 81 | 3.84 | 0.81 | 119 | 5.67 | 11.8 | 4.3 |\n| GLYPH<31> 0.8 | 0.282 | 19.9 | 0.81 | 82 | 4.56 | 1.04 | 103 | 18.05 | 13.2 | 2.01 |\n| m0.01 | 0.202 | 20.4 | 0.43 | 89 | 3.18 | 0.91 | 137 | 5.32 | 16.5 | 8.32 |\n| m0.10 | 0.261 | 19.1 | 1.04 | 104 | 4.9 | 0.93 | 109 | 5 | 10.6 | 3.02 |\n| M10 | 0.228 | 20.2 | 0.53 | 227 | 6.48 | 0.62 | 119 | 6.29 | 24.2 | 5.49 |\n| r75 | 0.231 | 17.6 | 0.77 | 117 | 4.94 | 0.73 | 122 | 14.8 | 22.3 | 5.4 |\n| s10 | 0.26 | 18.8 | 0.6 | 91 | 4.29 | 0.75 | 107 | 7.19 | 16.4 | 2.34 |\n| s6 | 0.203 | 20.8 | 0.73 | 96 | 3.75 | 0.86 | 136 | 4.22 | 13 | 0.91 |\n| GLYPH<11> 0.01 | 0.321 | 26.1 | 2.66 | 100 | 2.88 | 0.7 | 91 | 1.36 | 0 | 0.25 |\n| GLYPH<11> 0.10 | 0.206 | 18.1 | 0.16 | 52 | 4.56 | 1.23 | 130 | 13.53 | 2.2 | 8.85 |\n| y0.05 | 0.23 | 20.5 | 0.72 | 110 | 3.93 | 0.91 | 122 | 5 | 13.6 | 5.12 |\n| y0.15 | 0.252 | 19.4 | 0.66 | 102 | 4.44 | 0.87 | 112 | 5.72 | 19.2 | 3.26 |\n| " 1.0 | 0.23 | 14.9 | 0.41 | 111 | 4.28 | 0.85 | 118 | 11.39 | 5.1 | 4.4 |\n| " 0.1 | 0.228 | 17.5 | 0.58 | 95 | 3.9 | 0.77 | 122 | 6.08 | 15.3 | 4.77 |\n| " 10.0 | 0.216 | 14.5 | 0.28 | 131 | 4.24 | 2.38 | 123 | 16.11 | 0 | 5.09 | \nFigure 7. Dependence of average disc outflow abundances on disc mass (a), BH mass and disc radius (b), initial entropy (c), viscosity parameter (d), initial electron fraction (e), and BH spin (f). Model parameters are shown in Table 1, with nucleosynthesis results summarized in Table 2. Note that the Solar system r -abundances are scaled to match the second peak abundances of S-def model only. When comparing to abundances of other models, they should be further re-scaled. \n<!-- image --> \nshown in Fig. 7 are scaled to match the second peak abundances of S-def model only. When comparing to abundances of other models, they should be further re-scaled. \nThe largest impact of changing disc parameters is obtained when varying the GLYPH<11> viscosity parameter (see Fig. 7d). Since the disc evolution times-cale is governed by the viscous time t visc / GLYPH<11> GLYPH<0> 1 , models with lower (higher) viscosity have a longer (shorter) ¯ t 5 relative to the S-def model. The corresponding ¯ Ye ; 5 is raised to higher \n(lower) values by weak interactions. As a result, model GLYPH<11> 0.10 ( ¯ Ye ; 5 = 0 : 206) has about a factor of 2 higher abundances of third peak and heavier elements compared to S-def ( ¯ Ye ; 5 = 0 : 237). At the other end, model GLYPH<11> 0.01 ( ¯ Ye ; 5 = 0 : 321) displays a relatively large first peak ( A GLYPH<25> 80) while the abundances at and above the second peak are reduced by roughly an order of magnitude. \nThe initial disc mass and entropy have a moderate impact (see Fig. 7a and c). For discs with initially larger (smaller) entropy, the \ntemperature is higher (lower). As a result, higher (lower) positron capture rates give rise to a higher (lower) value of ¯ Ye ; 5. Similarly, for a disc with initially larger (smaller) mass, ¯ Ye ; 5 is also higher (lower) because the disc temperature is higher (lower) for fixed entropy, disc radius, and BH mass due to the higher (lower) density. Smaller changes are obtained when varying the initial electron fraction (see Fig. 7e). Higher (lower) initial Ye leads to a slightly larger (smaller) third peak and trans-lead abundances. This insensitivity originates in the fact that, while the outflow itself does not achieve GLYPH<12> -equilibrium (c.f. Fig. 6 of Fern\'andez & Metzger 2013), weak interactions nonetheless modify Ye significantly from its initial value. \nRegarding BH spin, a higher value of this parameter results in a smaller innermost stable circular orbit, leading to more gravitational energy release in the form of neutrinos and viscous heating (e.g., Fern\'andez et al. 2015). As a result, ¯ Ye ; 5 is higher in model GLYPH<31> 0 : 8 than in the baseline S-def model, slightly hindering the nucleosynthesis of heavy nuclei for both the second and third peaks while enhancing the first peak abundances as shown in Fig. 7f. \nChanging the mass of the BH and radius of the disc do not lead to significant di GLYPH<11> erences given our parameter choices. Fig. 7b shows that there are minor di GLYPH<11> erences between models M10 and S-def. The similarities between them can be attributed to the very similar compactness, M BH = R 0 = 0 : 07 and 0 : 06 M GLYPH<12> km GLYPH<0> 1 for M10 and S-def, respectively, which should correlate with the disc temperatures (and thus strength of weak interactions) if the medium is in near hydrostatic equilibrium. In contrast, model r75 has a somewhat lower compactness 0 : 04 M GLYPH<12> km GLYPH<0> 1 and shows a slightly larger abundance of elements with A > 130. \nThe dependence of the ejecta mass distribution on flow quantities - such as polar angle and velocity - for models with di GLYPH<11> erent parameters is generally similar to that for the baseline model S-def shown in Fig. 4. \nWe conclude that some nucleosynthesis features - particularly abundances of elements with A GLYPH<24> < 130 - are quite robust against variations in astrophysical disc parameters. On the other hand, the abundances of rare-earth peak, third peak and heavier elements are more sensitive to those parameters that alter either the disc evolution time-scale or the temperature.', '3.3 Anomalously high abundance of A = 132': 'Fig. 7 shows that all models we have explored, except GLYPH<11> 0.01 and GLYPH<11> 0.10, display a peculiarity in the abundances at the second peak: the abundance of A = 132 elements ( YA = 132) exceeds that of the second Solar system r -process peak at A = 128 GLYPH<0> 130 ( YA = 128 GLYPH<0> 130) We find that this spike in YA = 132 arises primarily from late-time ejecta (see, e.g., Fig. 5f). Table 2 shows that material satisfying YA = 132 > 10 GLYPH<0> 3 and YA = 128(136) < 0 : 1 YA = 132 can contribute more than 10% to the total ejected mass. This would cause a discrepancy with the observed solar abundance ratio YA = 132 = YA = 130 GLYPH<25> 0 : 5 if we were to assume that the disc ejecta dominates the galactic r -process production of second peak elements. \nThe origin of this anomaly can be traced back to late-time, low-temperature convection in the disc outflow. For ejecta satisfying t ej GLYPH<29> t 5, neutron-capture processes operate once the temperature decreases to GLYPH<24> < 2-3 GK. Once neutrons are exhausted, the produced nuclei at N = 82, mainly isotones with A = 128-130, start to decay to the stability valley. If convection brings matter back to regions of T GLYPH<24> > 2-3 GK on time-scales shorter than the half-lives of long lived Sn isotopes: 128 Sn ( t 1 = 2 = 59 : 07 min), 129 Sn ( t 1 = 2 = 2 : 23 min), and 130 Sn ( t 1 = 2 = 3 : 42 min), additional neutron \ncaptures can occur due to the release of neutrons by photodissociation. This results in a pile up of material in the double-magic nucleus 132 Sn that itself has a GLYPH<12> -decay half-life of t 1 = 2 = 39 : 7 s. \n= The model with the smallest viscosity parameter ( GLYPH<11> 0.01) does not display this anomaly because on one hand, the strength of convection is strongly reduced relative to the baseline value. Correspondingly, most of the ejecta from this model satisfy t ej GLYPH<24> t 5. On the other hand, most of the ejecta in this model contains Ye ; 5 GLYPH<24> > 0 : 25 so that little nuclei with A > 130 can be produced. At the opposite end, the model with the highest viscosity ( GLYPH<11> 0.10) also displays a much smaller abundance anomaly than S-def. In this case, if the viscous time is shorter than the r -process time scale GLYPH<24> < 1 s, each convection episode results in an incomplete r -process that does not use all available neutrons. Only the very last ejection episode results in a complete r -process. Therefore, only little trajectories contain the abundance anomaly. \nThe above discussion illustrates the close interplay between convection and r -process nucleosynthesis. Convection operating on time-scales longer than the r -process time-scale of GLYPH<24> < 1 s will reheat the r -process products during their decay back to stability and produce anomalies in the abundance distribution.', '3.4 E GLYPH<11> ect of nuclear heating': 'The hydrodynamic disc models discussed so far do not selfconsistently include the nuclear energy released by chargedparticle reactions involving nuclei heavier than 4 He, in addition to the energy released by the r -process. These contributions are only included in post-processing with the nuclear network and hence can only a GLYPH<11> ect the temperature. Charged-particle reactions can release a nuclear energy GLYPH<24> 3 MeV / nucleon through processes like 20n + 15 GLYPH<11> ! 80 Zn. This can give rise to a heating rate GLYPH<24> 10 19 erg s GLYPH<0> 1 , assuming that such reactions occur on a time-scale of GLYPH<24> 0 : 1 s. It can become comparable to the viscous heating rate in the same temperature regime, and has the potential to change not only the nucleosynthesis but also the disc dynamics. The energy released during the r -process is smaller in comparison, amounting to GLYPH<24> 10 18 erg s GLYPH<0> 1 , but can potentially also play a role at late times. Given the abundance anomaly discussed in the previous section, and its sensitivity to convection in the disc, we explore here the extent to which additional sources of heating can introduce changes in the dynamics and nucleosynthesis results. \nIn order to include these additional heating channels into hydrodynamic models, we have parametrized it out of trajectories from the S-def model using a method similar to that described in Just et al. (2015). From our post-processed nucleosynthesis calculation, we estimate the particle-averaged nuclear heating rate h ˙ q i as a function of the average temperature h T i by calculating \nh ˙ q (˜ t ) i GLYPH<17> P i ˙ qi (˜ t ) N (4) \nand \nh T (˜ t ) i GLYPH<17> P i Ti (˜ t ) N ; (5) \nwhere ˙ qi (˜ t ) = ˙ qi ( t GLYPH<0> t 0 ; i ) and Ti (˜ t ) = Ti ( t GLYPH<0> t 0 ; i ) are the nuclear energy generation rate and temperature of each trajectory i , respectively, and t 0 ; i is the time at which the network integration begins for each trajectory ( x 2.3). Following Korobkin et al. (2012), we find that the function h ˙ q i ( h T i ) can be approximated by \nh ˙ q i ( h T i ) = 2 : 5 GLYPH<2> 10 19 " " 1 GLYPH<25> arctan h T i 1 : 1 GK !# 5 = 2 erg g GLYPH<0> 1 s GLYPH<0> 1 ; (6) \nFigure 8. Nuclear heating rate as a function of temperature for all trajectories from the baseline model S-def (grey curves). Also plotted is an analytic fit to the average heating rate (equation 6) for three choices of the heating amplitude " , as shown in the legend. \n<!-- image --> \nwith " \' 1 : 0. Fig. 8 compares the result from equation (6) with the heating rates from individual particles in the S-def model. \nWe evolve three additional hydrodynamic models that include this additional heating rate using a range of values for the heating amplitude " = f 0 : 1 ; 1 ; 10 g to compensate for the crudeness of the approximation (models " 0.1, " 1.0 and " 10.0 in Table 1). The heating term is added to the energy equation once T 6 6 GK if vr > 0. The latter condition is used to prevent heating in flows that move back towards the BH. \nTable 2 shows that the additional heating can enhance the ejected mass up to a factor of 2 when using " GLYPH<24> > 1 : 0. The angular distribution can also be a GLYPH<11> ected. In the model " 10.0, the mass ejection becomes more isotropic rather than peaks at high latitudes. \nNucleosynthesis results are shown in Fig. 9. The abundances of third peak and heavier elements are slightly increased relative to the baseline model S-def. Interestingly, the A = 132 abundance anomaly is reduced in model " 1.0 and completely vanishes in model " 10.0, as shown in the inset of Fig. 9. For the " 1.0 model, this suppression occurs because the disc evolution becomes faster as a result of the extra heating, so that the amount of late-time ejecta a GLYPH<11> ected by convection is reduced. For model " 10.0, the nuclear heating is so strong that convection is strongly suppressed. \nOur results indicate that including all nuclear heating sources may be necessary to fully understand the ejecta dynamics and the details of the nucleosynthesis outcome. Alternatively, the dynamics of the disc when transporting angular momentum via magnetohydrodynamic stresses might be di GLYPH<11> erent enough to erase the A = 132 abundance anomaly without the need for enhanced nuclear heating. The properties of convective motion in a purely hydrodynamical GLYPH<11> -disc may di GLYPH<11> er substantially from those of MHD turbulence (e.g., Balbus & Hawley 2002).', '3.5 Impact of nuclear physics input': 'In order to explore the e GLYPH<11> ect of varying the nuclear physics inputs on the outflow nucleosynthesis, we have performed two additional network calculations on the baseline model S-def using di GLYPH<11> erent neutron-capture and photo-dissociation rates, as well as GLYPH<12> -decay rates. Alternative neutron capture and photodissociation rates are computed with the nuclear mass model of Duflo-Zuker with 31 pa- \nFigure 9. Dependence of the average disc outflow abundances on the amplitude " of the parametrized radioactive heating rate (equation 6), relative to the baseline model S-def (c.f. Table 1). The inset shows the amplitude of the anomalous abundance peak at A = 132 in comparison with the solar r -abundances including observational uncertainties. \n<!-- image --> \nFigure 10. Average abundances in the outflow from the baseline model Sdef for di GLYPH<11> erent nuclear mass models (top) and GLYPH<12> -decay rates (bottom). See x 3.5 for details. \n<!-- image --> \nrameters (DZ31; Duflo & Zuker 1995), as in Mendoza-Temis et al. (2015). To explore the impact of GLYPH<12> -decay rates, we alternatively employ those of Marketin et al. (2016) and Moller et al. (2003). \nThe results of these additional calculations are shown in Fig. 10. The first and second r -process peaks are robust against changes in nuclear physics inputs, as they are produced primarily by trajectories with high Ye ; 5, whose nucleosynthesis paths are close to the stability valley. For these trajectories, the relative change of neutron separation energy is small between the di GLYPH<11> erent mass models, and the predictions from two sets of GLYPH<12> -decay rates are similar. In contrast, nuclear physics inputs can strongly a GLYPH<11> ect the abundances \nFigure 11. Comparison of average elemental abundances in the outflows from the baseline model S-def and models with di GLYPH<11> erent initial entropy (s6 and s10) to observed abundances in three metal-poor stars: CS22892-052 (Sneden et al. 2003), HD160617 (Roederer & Lawler 2012), and HD122563 (Roederer et al. 2012). Abundances are re-scaled to the Eu abundance of CS22892-052. Here log " ( Z ) = log( NZ = N 1) + 12, where NZ is the abundance of an element with charge number Z . \n<!-- image --> \nof rare-earth and third peak elements - produced by low Ye ; 5 ejecta - to a similar or even larger degree than the change due to astrophysical parameters discussed in x 3.2. In particular, the troughs after the A GLYPH<24> 130 peak and before the A GLYPH<24> 195 peak may be due to deficiencies in the FRDM mass model (Winteler et al. 2012). A more detailed analysis of the e GLYPH<11> ect of nuclear physics inputs will be reported separately.', '3.6 Comparison with abundances of metal-poor stars': 'We have thus far restricted the comparison of disc nucleosynthesis abundances to the Solar system r -process distribution. Observations of r -process abundances in metal poor stars ([Fe / H] = log( N Fe = N H) GLYPH<3> GLYPH<0> log( N Fe = N H) GLYPH<12> GLYPH<24> < GLYPH<0> 2 : 5) suggest that the r -process operated at early times in the Galaxy (Sneden et al. 2008). While a number of metal-poor stars show robust relative abundances in the region around the rare-Earth peak (56 GLYPH<24> < Z GLYPH<24> < 72) that are similar to the solar r -abundance distribution, larger relative variations exist among lighter elements (30 GLYPH<24> < Z GLYPH<24> < 50). Moreover, the ratio between light-to-heavy elemental abundances can vary from star to star by more than a factor of 10. This has lead to suggestions of the existence of more than one astrophysical site producing the r -process elements in the early Galaxy (Qian & Wasserburg 2007; Sneden et al. 2008). \nFig. 11 shows the comparison of elemental abundances from three models (S-def, s6, s10) to the observed abundances in three metal-poor stars: CS22892-052 (Sneden et al. 2003), HD160617 (Roederer & Lawler 2012), and HD122563 (Honda et al. 2006, Roederer et al. 2012), normalized to the Eu abundance of CS22892052. These three stars show large di GLYPH<11> erences in the ratio of lightto-heavy elemental abundances. If we consider disc outflows alone, changing the initial entropy of the disc can lead to qualitative agreement with the observations in these three metal-poor stars across the whole range of neutron capture elements. As shown in Sec. 3.2, changing other disc parameters such as the initial disc mass or the adopted GLYPH<11> -viscosity can also result in similar or larger changes in abundances. This suggests that if the initial conditions in NSNS / NS-BH mergers lead to variations in the remnant configura- \ntions, the resulting disc outflows may account for di GLYPH<11> erent observed ratios of light and heavy elemental r -process abundances in metal poor stars. Therefore, the observed variations may not only be due to di GLYPH<11> erent combinations of dynamical and disc ejecta (as suggested by Just et al. 2015), but may arise due to variations of the merger remnant disc properties.', '4 SUMMARY AND DISCUSSION': "We have studied the production of r -process elements in the disc outflows from remnant accretion tori around BHs formed in NS-NS / NS-BH mergers. We used tracer particles in long-term, time dependent hydrodynamic simulations of these discs to record thermodynamic and kinematic quantities. The resulting trajectories were then post-processed with a dynamic r -process nuclear reaction network. Our results can be summarized as follows: \n- 1. - Outflows from merger remnant discs around BHs can robustly generate light r -process elements with A GLYPH<24> < 130, regardless of the astrophysical parameters of the disc or the nuclear physics inputs employed (Fig. 5, Table 2).\n- 2. - The yield of elements with A > 130 is most sensitive to the type and magnitude of the angular momentum transport process (Fig. 7d) and on the nuclear physics inputs employed in the nuclear reaction network (Fig. 10). Given the physics employed in our hydrodynamic models, the GLYPH<11> viscosity parameter is a key factor determining the abundance of third peak and heavier elements. Other parameters such as the disc mass or initial entropy have a relatively smaller impact on the abundances.\n- 3. - We have identified a spike in the abundance of A = 132 elements that arises whenever the disc outflow is highly convective, as is the case when using reasonable choices for the disc parameters (Fig. 5f). This feature can be erased if the disc evolution is fast or if the heating rate in the disc is very low, so that convection is suppressed.\n- 4. - Inclusion of energy deposition from charged-particle reactions beyond 4 He recombination can a GLYPH<11> ect the ejecta dynamics and nucleosynthesis to the point where the A = 132 abundance anomaly disappears (Fig. 9). Alternatively, the processes responsible for controlling angular momentum transport and the thermodynamics of the disc (e.g. MHD and neutrino transport) can have a sensitive nucleosynthetic impact. The properties of convective motions in an hydrodynamical GLYPH<11> GLYPH<0> disc may di GLYPH<11> er substantially from those of MHDturbulence (Balbus & Hawley 2002).\n- 5. - The comparison with abundances observed in metal-poor stars shows that if the disc outflows contribute dominantly to the NS-NS / BS-BH ejecta, di GLYPH<11> erent initial configurations of the disc may account for the variation of light-to-heavy abundance ratio seen in these stars. \nOur results together with those of Just et al. (2015) show that disc outflows are fundamental to understand r -process nucleosynthesis in mergers. For cases in which little dynamical ejecta is generated, the disc outflow alone can contribute substantially to the heavy r-process element enrichment , even while producing proportionally more elements with A < 130. This result reinforces the general view that NS-NS / NS-BH mergers are the primary astro- \nphysical site for heavy r -process elements. It is also important because the presence of highYe dynamical ejecta in NS-NS mergers is uncertain theoretically, due in part to the dependence of shockheated polar ejecta on the NS radius and equation of state. \nThe early ejection of material with high Ye in the disc outflow has the potential to generate a blue peak in the kilonova, since it will generally reside on the outer layers of the ejecta (Fern'andez et al. 2015; Kasen et al. 2015). This nonetheless depends on the viewing directions close to the rotation axis to be relatively free of Lanthanide-rich dynamical ejecta material, which is usually the case for BH-NS mergers (Fern'andez et al. 2015; Kasen et al. 2015). \nThe net kilonova contribution of systems studied is Lanthanide-rich, as inferred from Table 2. The work of Kasen et al. (2013) shows that even a mass fraction of GLYPH<24> 10 GLYPH<0> 2 in Lanthanides can increase the optical opacity by at least an order of magnitude relative to that of iron-like elements. Nearly all of our models have a Lanthanide mass fraction bigger than 1% and thus while an outer Lanthanide-free 'skin' is usually obtained, the bulk of the wind will lead to an infrared transient. More promising in this respect is the possible onset of a neutron-powered precursor if a small fraction of material escapes quickly enough to freeze-out before neutrons are captured by heavy seeds (Metzger et al. 2015). Simulations at much higher resolution than currently available are needed to resolve this question. A long-lived NS remnant can also increase the quantity of highYe ejecta, producing a more prominent blue component of the kilonova (Metzger & Fern'andez 2014). \nA crucial improvement to our calculation would involve obtaining trajectories from an accretion disc outflow in which angular momentum transport - and the associated energy dissipation - is carried out by MHD stresses. Such a calculation can di GLYPH<11> er from ours in a number of ways. First, the change in the entropy due to viscous heating in an GLYPH<11> -viscosity model is likely di GLYPH<11> erent in an MHD disc, with the associated change in the equilibrium Ye that the weak interactions try to achieve. Secondly, the amount of mass ejected and the associated Ye distribution can change, altering the relative amounts of heavy- and light r -process elements in the outflow composition. Finally, the kinematic properties of the wind can change, in particular the velocity, which controls the expansion time, and the angular distribution, which is associated with the level of neutrino irradiation of the ejecta (material on the equatorial plane is more e GLYPH<11> ectively shadowed from neutrinos than material leaving at high latitudes). \nOur calculations would also benefit from better neutrino transport, although the magnitude of the di GLYPH<11> erence introduced can be comparable to that due to the spin of the BH (e.g., compare the results of Fern'andez et al. 2015 and Just et al. 2015). A more thorough, self-consistent treatment of nuclear heating would also make our calculations more realistic. In this respect, it is worth noting that the results of Just et al. (2015) do not appear to show the abundance spike at A = 132 that we obtain in many of our models. We surmise that this di GLYPH<11> erence arises due to their inclusion of a single species of heavy nucleus ( 54 Mn) in the equation of state, which partially accounts for the energy production beyond GLYPH<11> formation. \nFinally, it is important to consider the combined evolution of disc and dynamical ejecta in computing the net r -process yield from NS-NS / NS-BH mergers. Part of the dynamical ejecta is gravitationally bound, and mixes with the accretion disc, increasing the neutron-richness of the disc and therefore lowering the peak of the Ye distribution of the disc outflow (Fern'andez et al. 2015).", 'ACKNOWLEDGEMENTS': 'Wegratefully thank Lutz Huther for his help during the initial stage of this work. We also thank Almudena Arcones, Kenta Hotokezaka for helpful discussions, Friedel Thielemann, Andreas Bauswein, Thomas Janka, Oliver Just, Luciano Rezzolla, Luke Roberts, and Ian Roederer for their valuable comments. MRW and GMP acknowledge support from the Deutsche Forschungsgemeinschaft through contract SFB 1245, the Helmholtz Association through the Nuclear Astrophysics Virtual Institute (VH-VI-417), and the BMBF-Verbundforschungsprojekt number 05P15RDFN1. RF acknowledges support from the University of California O GLYPH<14> ce of the President, and from NSF grant AST-1206097. BDM gratefully acknowledges support from NASA grants NNX15AU77G (Fermi), NNX15AR47G (Swift), and NNX16AB30G (ATP), NSF grant AST-1410950, and the Alfred P. Sloan Foundation. The software used in this work was in part developed by the DOE NNSA-ASC OASCR Flash Center at the University of Chicago. This research used resources of the National Energy Research Scientific Computing Center (repository m2058), which is supported by the O GLYPH<14> ce of Science of the U.S. Department of Energy under Contract No. DEAC02-05CH11231. Some computations were performed at Carver and Edison . 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2018JCAP...07..032B

Primordial black holes from inflation and quantum diffusion

2018-01-01

['-', '-', '-']

[]

Primordial black holes as dark matter may be generated in single-field models of inflation thanks to the enhancement at small scales of the comoving curvature perturbation. This mechanism requires leaving the slow-roll phase to enter a non-attractor phase during which the inflaton travels across a plateau and its velocity drops down exponentially. We argue that quantum diffusion has a significant impact on the primordial black hole mass fraction making the classical standard prediction not trustable.

[]

https://arxiv.org/pdf/1804.07124.pdf

{'Primordial Black Holes from Inflation and Quantum Diffusion': 'M. Biagetti a ∗ , G. Franciolini b † , A. Kehagias c ‡ and A. Riotto b § \na Institute of Physics, Universiteit van Amsterdam, Science Park, 1098XH Amsterdam, The Netherlands \nb Department of Theoretical Physics and Center for Astroparticle Physics (CAP) \n24 quai E. Ansermet, CH-1211 Geneva 4, Switzerland \nc Physics Division, National Technical University of Athens, 15780 Zografou Campus, Athens, Greece', 'Abstract': 'Primordial black holes as dark matter may be generated in single-field models of inflation thanks to the enhancement at small scales of the comoving curvature perturbation. This mechanism requires leaving the slow-roll phase to enter a non-attractor phase during which the inflaton travels across a plateau and its velocity drops down exponentially. We argue that quantum diffusion has a significant impact on the primordial black hole mass fraction making the classical standard prediction not trustable.', 'I. Introduction': "The interest in the physics of Primordial Black Holes (PBHs) and the possibility that they form all (or a fraction) of the dark matter in the universe has risen up [1-10] again after the discovery of two ∼ 30 M glyph[circledot] black holes through the gravitational waves generated during their merging [11]. A standard mechanism to account for the generation of the PBHs is through the boost of the curvature perturbation R at small scales [12-14]. Such an enhancement can occur either within single-field models of inflation, see Refs [15-19] for some recent literature, or through some spectator field [20-23] which could be identified the Higgs of the Standard Model [24]. \nIn order for this primordial mechanism to occur, one needs an enhancement of the power spectrum of the curvature perturbation from its ∼ 10 -9 value at large scales to ∼ 10 -2 on small scales. Subsequently, these large perturbations are communicated to radiation during the reheating process after inflation and they may give rise to PBHs upon horizon re-entry if they are sizeable enough. If we indicate by P R the comoving curvature power spectrum, a region of size the Hubble radius may collapse and form a PBH if the corresponding square root of the variance σ R smoothed with a high-pass \nfilter on the same length scale (comoving momenta larger than the inverse of the comoving Hubble) is larger than some critical value R c . Its exact value is sensitive to the equation of state upon horizon re-entry and it is about 0.086 for radiation [25]. However, larger values have been adopted in the literature [26-28]. We will later on use the common representative value c glyph[similarequal] 1 . 3. \nR \nβ prim ( M ) = ∫ ∞ R c d R √ 2 π σ R e -R 2 / 2 σ 2 R . (1.1) \nglyph[similarequal] Under the (strong) hypothesis that the curvature perturbation obeys a Gaussian statistics, the primordial mass fraction β prim ( M ) of the universe occupied by PBHs formed at the time of formation reads \nIt corresponds to a present dark matter abundance made of PBH of masses M given by (neglecting accretion) [28] \n( Ω DM ( M ) h 2 0 . 12 ) glyph[similarequal] ( β prim ( M ) 7 · 10 -9 ) ( γ 0 . 2 ) 1 / 2 ( 106 . 75 g ∗ ) 1 / 4 ( M glyph[circledot] M ) 1 / 2 , (1.2) \nwhere γ < 1 is a parameter accounting for the efficiency of the collapse and g ∗ is the effective number of degrees of freedom. Imposing PBHs to be the dark matter, values γ < 1 require larger values of β prim and therefore to be conservative we impose γ = 1 [28] \nβ prim ( M ) ∼ > 3 · 10 -9 ( M M glyph[circledot] ) 1 / 2 . (1.3) \nFor a mass of the order of 10 -15 M glyph[circledot] we find [28] \nσ R ∼ > 0 . 16 . (1.4) \nNow, in single-field models of inflation the power spectrum of the comoving curvature perturbation is given by (we set the Planckian mass equal to one from now on) [29] \nP 1 / 2 R ( k ) = ( H 2 πφ ' ) , φ ' = d φ d N , (1.5) \nwhere N is the number of e-folds, the prime denotes differentiation with respect to N , and H is the Hubble rate. The generation of PBHs requires the jumping within a few e-folds ∆ N of the value of the power spectrum of about seven orders of magnitude from its value on CMB scales. Without even specifying the single-field model of inflation, one may conclude that there must be a violation of the slow-roll condition as φ ' must change rapidly with time. This may happen when the inflaton field goes through a so-called non-attractor phase (dubbed also ultra-slow-roll) [30-39] in the scalar potential, thus producing a sizeable resonance in the power spectrum of the curvature perturbation. \nWhen the inflaton experiences a plateau in its potential, since φ ' must be extremely small, a short non-attractor period is achieved during which the equation of motion of the inflaton background φ reduces to \nφ '' +3 φ ' + V ,φ H 2 glyph[similarequal] φ '' +3 φ ' = 0 , (1.6) \nwhere ,φ denotes differentiation with respect to the inflaton field φ with potential V ( φ ). The comoving curvature perturbation increases, due to its decaying mode which in fact is growing, as \nφ ' ∼ e -3 N and P 1 / 2 R ∼ e 3 N . (1.7) \nIt is this exponential growth which helps obtaining large fluctuations in the curvature perturbation and the formation of PBHs upon horizon re-entry during the radiation phase. This is also the reason why the power spectrum should be quoted at the end of inflation and not, as usually done in slow-roll, at Hubble crossing: its value at the instant of Hubble crossing differs by a significant factor from the asymptotic value at late times. Not respecting these rules might lead to an incorrect estimate of the power spectrum and consequently of the PBH abundance at formation, see e.g. Ref. [40] and the subsequent discussions in Refs. [28,36]. \nPutting aside the strong sensitivity of the PBH mass fraction at formation to possible nonGaussianities [14, 41-50] which is common to all mechanisms giving rise to PBHs through sizeable perturbations (we will however devote Appendix C for some considerations about non-Gaussianity where we will show that the δN formalism [51] can help in assessing the role of non-Gaussianity for those perturbations generated during the non-attractor phase) in this paper we are interested in another issue, the role of quantum diffusion. One might be reasonably suspicious that during the non-attractor phase the quantum diffusion becomes relevant [49]. The reason is the following. The stochastic equation of motion for the classical inflaton field takes into account that each Hubble time the inflaton field receives kicks of the order of ± ( H/ 2 π ) [52] \nφ '' +3 φ ' + V ,φ H 2 = ξ, (1.8) \nwhere ξ is a Gaussian random noise with \n〈 ξ ( N ) ξ ( N ' ) 〉 = 9 H 2 4 π 2 δ ( N -N ' ) . (1.9) \nDuring the non-attractor phase, V ,φ needs to be tiny enough to allow φ ' to promptly decrease thus violating slow-roll. For the same reason, one needs to make sure that quantum jumps are not significant in this case. One could try to impose the condition \n2 πV ,φ 3 H 3 ∼ > 1 . (1.10) \nto be satisfied during the non-attractor phase. In slow-roll, the condition (1.10) could be exactly expressed in terms of the power spectrum and the latter would be required to be smaller than unity, thereby giving a direct constraint on a physical observable. However, during the non-attractor phase, the bound (1.10) is not directly expressed in terms of the power spectrum (1.5), making the comparison with physical observables more difficult. One might naively think that during the non-attractor phase the noise is not relevant if φ ' is larger than ( H/ 2 π ) [17]. However, this is not correct for two reasons. First because, as we will see, the relevant noise to be evaluated is the one for the inflaton velocity. Secondly, and above all, because this is not the right criterion to evaluate the strong impact of quantum diffusion onto the PBH mass fraction. \nWe will first elaborate on the computation of the power spectrum during the non-attractor phase in order to understand some basic features of the power spectrum itself, e.g. its time evolution and the location of its peak. This will be useful for the considerations about the quantum diffusion. We will then use the so-called Kramers-Moyal equation [53] to assess the impact of quantum diffusion. \nThe Kramers-Moyal equation is the suitable starting point as it highlights the importance of the inflaton velocity and it is a generalisation of the Fokker-Planck equation. Indeed, in general the Kramers-Moyal equation contains an infinite number of derivatives with respect to the inflaton \nfield after having integrated out the velocity, while the Fokker-Planck equation is a truncation of the Kramers-Moyal equation by retaining only two spatial derivatives of the inflaton field. This is not an irrelevant point: Pawula's theorem [54] tells us that if we set to zero some coefficient c n of the higher derivative terms with respect to the inflaton field with n ≥ 3, then all the coefficients of the higher derivatives are zero. It is therefore not consistent to keep some higher derivative term unless all of them are kept. This means that, whenever the Kramers-Moyal equation may not be solved exactly, a numerical approach is useful to solve it without applying an unjustified truncation. \nWe will present both analytical and numerical results to quantify the impact of quantum diffusion on the PBH abundance. In the simplest case of non-attractor phase with an approximately linear or quadratic potential, the system can be solved analytically. For a linear potential, the stochastic motion of the system is characterised by the fact that the variance of the velocity φ ' rapidly converges towards a stationary value. For a quadratic potential, the spread in the velocity varies with time, although slowly, away from the stationary point. \nHowever, for more complicated situations, e.g. if during the non-attractor phase the inflaton goes through an inflection point, a numerical analysis is called for and we will show that the spread in the velocity grows with time. If this growth is too large, classicality as well as information on the PBH abundance is lost. We will propose two criteria to be respected in order to neglect the quantum diffusion and we will see that the curvature perturbation is severely constrained from above in order to avoid an undesirable spreading of the velocity wave packet. We will also argue that the capability of the standard (that is classical) calculation to predict the correct dark matter abundance in terms of PBH is severely challenged by the presence of quantum diffusion. \nThis paper is organised as follows. In section II we discuss the computation of the curvature perturbation during the non-attractor phase. We then start our study of the quantum diffusion, both analytically and numerically, in section III and IV respectively. In sections V and VI we offer two criteria to assess the importance of the diffusion. Finally, in section VII we offer our conclusions. \nThe paper contains several Appendices. Appendix A deals with the curvature perturbation, the Schwarzian derivative and the dual transformation; appendix B with the study of the evolution of the comoving curvature perturbation from the non-attractor phase back to the slow-roll phase; appendix C offers some consideration about non-Gaussianity.", 'II. The comoving curvature perturbation and the nonattractor phase': 'In this section we offer some considerations about the curvature perturbation generated thanks to the non-attractor phase. We start by some analytical considerations and then we will proceed with a more realistic example.', 'Non-attractor: some analytical considerations': "We are interested in the curvature perturbation for those modes leaving the Hubble radius deep in the non-attractor phase. We suppose that the non-attractor phase starts when the inflaton field acquires the value φ 0 and ends when it becomes equal to φ glyph[star] . We also assume that the non-attractor phase is preceded and followed by slow-roll phases, see Fig. 1. During these phases η = -φ '' /φ ' \npasses from a tiny value (slow-roll) to 3 (non-attractor) back to small values (slow-roll). To treat \nFigure 1: A representative behaviour of the inflaton potential during the various phases, highlighting the possibility of quantum diffusion during the non-attractor phase. \n<!-- image --> \nthe problem analytically, we suppose that during the non-attractor phase we may Taylor expand the inflaton potential as \nV ( φ ) glyph[similarequal] V 0 ( 1 + √ 2 glyph[epsilon1] V ( φ -φ 0 ) ) + · · · for φ glyph[star] < φ < φ 0 , (2.1) \nwhere √ 2 glyph[epsilon1] V = V ,φ /V 0 is the slow-roll parameter. Of course, the potential during the non-attractor phase may be more complex, but its linearisation captures the main features. \nComputing everything in terms of the number of e-folds N and defining N = 0 the beginning of the non-attractor phase with initial conditions φ 0 and d φ/ d N | 0 = Π 0 and setting Π( N ) = φ ' ( N ), the solution of the equations of motion leads to \nφ ( N ) = φ 0 + 1 3 (Π 0 -Π) -√ 2 glyph[epsilon1] V N, Π( N ) = √ 2 glyph[epsilon1] V ( e -3 N -1 ) +Π 0 e -3 N . (2.2) \nWe see that if the inflaton field starts with a large velocity from the preceeding slow-roll phase, there is a period over which the velocity of the inflaton field decays exponentially. Depending on the duration of the non-attractor phase, the velocity may or may not attain its slow-roll asymptote given by -√ 2 glyph[epsilon1] V . Indicating by Π glyph[star] the value of the velocity at the end of the non-attractor phase, the final value of the curvature perturbation at the end of the non-attractor phase is given by \nR glyph[star] = -( δφ Π ) glyph[star] , δφ ( k ) = H √ 2 k 3 . (2.3) \nThe corresponding power spectrum is flat. This might come as a surprise as slow-roll is badly violated, but in fact its a direct consequence of the dual symmetry described in Ref. [55] (see also Refs. [56-58]). We elaborate extensively on this point in Appendix A. In a nutshell and alternatively, one can show \nit in the following way. Using the conformal time τ and setting R = u/z , z = ( a d φ/ d τ ) / H , where H is the Hubble rate in conformal time, one can write the equation for the function u as (cfr. Eq. (A.2)) \n1 u d 2 u d τ 2 = 2 a 2 H 2 ( 1 + 5 2 glyph[epsilon1] + glyph[epsilon1] 2 -2 glyph[epsilon1]η -1 2 V ,φφ H 2 ) -k 2 , (2.4) \nwhere glyph[epsilon1] and η are the slow-roll parameters defined in Eq. (A.3) and a is the scale factor. Since during the non-attractor phase glyph[epsilon1] glyph[lessmuch] 1, η glyph[similarequal] 3 and the potential is very flat, the right-hand side of the previous equation is approximated on super-Hubble scales to 2 /τ 2 glyph[similarequal] (d 2 a/ d τ 2 ) /a and therefore u ∝ a . It provides the standard solution for the mode function of the curvature perturbation \nR k = H (d φ/ d τ ) √ 2 k 3 (1 + ikτ ) e -ikτ . (2.5) \nThis is the standard slow-roll solution with the crucial exception that the inflaton velocity changes rapidly with time. Notice also that the expression (2.3) can be extended at the end of inflation. This is possible if the transition from the non-attractor phase to the subsequent slow-roll phase (if any) is sudden, i.e. the velocity during the subsequent slow-roll phase is much bigger than Π glyph[star] . Under these circumstances, the power spectrum does not have time to change and remains indeed (2.3) till the end of inflation [39]. We give more details in Appendix B. \nSo far, we have discussed the perturbation associated to the modes which leave the Hubble radius deep in the non-attractor phase. However, the peak of the curvature perturbation is in fact reached for those modes which leave the Hubble radius during the sudden transition from the slow-roll phase into the non-attractor phase. During this transition the (would-be) slow-roll parameter η = -Π ' / Π jumps from a tiny value to 3. \nTo see what happens, we model the parameter η as η glyph[similarequal] 3 θ ( τ -τ 0 ), where we have now turned again to conformal time τ . If so, and if we indicate by glyph[epsilon1] + the slow-roll parameter during the slow-roll phase preceding the non-attractor phase and assume it to be constant in time, one has \nand [39] \nR k = H √ 2 glyph[epsilon1] + k 3 (1 + ikτ ) e -ikτ for τ < τ 0 , (2.6) \n( τ τ 0 ) 3 R k = α k H √ 2 k 3 (1 + ikτ ) e -ikτ + β k H √ 2 k 3 (1 -ikτ ) e ikτ for τ > τ 0 , (2.7) \nwhere we have taken into account that immediately after the beginning of the non-attractor phase the curvature perturbation increases as the inverse cubic power of the conformal time (cfr. Eq. (2.2)). Imposing continuity of the two functions together of their derivatives, one obtains a power spectrum at the end of inflation \nP R = g ( -kτ 0 ) P R glyph[star] , g ( x ) = 1 2 x 6 ( 9 + 18 x 2 +9 x 4 +2 x 6 +3( -3 + 7 x 4 ) cos 2 x -6 x (3 + 4 x 2 -x 4 ) sin 2 x ) . (2.8) \nThe function g ( x ) is O ( x 4 ) for x glyph[similarequal] 0, has a maximum of about 2.5 around x glyph[similarequal] 3 and oscillates rapidly around 1 for x glyph[greatermuch] 1. We can conclude that the power spectrum has the following shape: it increases, reaches a peak, and then decreases a bit till a plateau is encountered. This is in good agreement with what obtained, for instance, in Refs. [57,58]. The amplitude of the peak is about 2.5 times larger than \nthe plateau in correspondence of the modes which leave the Hubble radius during the non-attractor phase \nP R pk glyph[similarequal] 2 . 5 P R glyph[star] = 2 . 5 ( H 2 π Π ) glyph[star] . (2.9) \nOf course, given the assumption of sudden transition of η from tiny values to 3 around τ 0 and having assumed glyph[epsilon1] + constant, we expect this number to change by a factor O (1) depending upon the exact details. The linearisation of the potential is an approximation, but it captures the main features of the final result. For more complicated situations, for instance if during the non-attractor phase the inflation crosses an inflection point, one expects again a peak in the curvature perturbation for that mode leaving the Hubble radius at the sudden transition between the slow-roll and the non-attractor phase. However, one does not expect a significant plateau following the maximum as the non-attractor phase is typically very short. This also implies that the exact amplitude of the peak depends on the fine details of the transition.", "An example: Starobinsky's model.": "In order to assess the quality of our findings we can consider Starobinsky's model [59] which is characterised by a potential with two linear regions \n( \nV ( φ ) glyph[similarequal] V 0 1 + √ 2 glyph[epsilon1] + ( φ -φ 0 ) + · · · for φ > φ 0 , (2.10) \n) \nwhere glyph[epsilon1] ± are the slow-roll parameter during the slow-roll phase and the non-attractor phase, respectively. In fact, to deal with the problem numerically we have parametrised the discontinuity in the potential as \n√ V ( φ ) glyph[similarequal] V 0 ( 1 + √ 2 glyph[epsilon1] -( φ -φ 0 ) ) + · · · for φ < φ 0 , (2.11) \nV ( φ ) = V 0 ( 1 + 1 2 ( √ 2 glyph[epsilon1] + -√ 2 glyph[epsilon1] -) ( φ -φ 0 ) tanh ( φ -φ 0 δ ) + 1 2 ( √ 2 glyph[epsilon1] + + √ 2 glyph[epsilon1] -) ( φ -φ 0 ) ) , (2.12) \nwhere δ determines the size of the region in which the potential smoothly changes slope. If δ glyph[lessmuch] 1 the potential during the non-attractor phase becomes exactly linear. We will comment on the effect of varying δ on the quantum diffusion in Sec. IV. \nIf glyph[epsilon1] + glyph[greatermuch] glyph[epsilon1] -, a prolonged non-attractor phase is obtained during which \nφ ' H = -√ 2 glyph[epsilon1] --( √ 2 glyph[epsilon1] + -√ 2 glyph[epsilon1] -)( τ/τ 0 ) 3 . (2.13) \nThe inflaton velocity at the beginning of the non-attractor phase is -√ 2 glyph[epsilon1] + , then it quickly decays reaching a maximum (recall velocities are negative) at ( τ max /τ 0 ) 3 glyph[similarequal] 2 √ glyph[epsilon1] + /glyph[epsilon1] -and then it reaches the value -√ 2 glyph[epsilon1] + . The corresponding power spectrum in Fig. 2 illustrates three relevant points: the fact that the power spectrum reaches a plateau and becomes scale-independent, the amplitude of the plateau is reproduced by the standard slow-roll formula (see Appendix A) and finally that the formula (2.8) provides a good fit to the numerical result. \n<!-- image --> \nFigure 2: On the left, the Starobinsky's potential. On the right, the power spectrum in Starobinsky's model (not normalised at the CMB on large scales) obtained for glyph[epsilon1] + /glyph[epsilon1] -= 10 8 and as a function of N corresponding to k = aH . We have arbitrarily set N = 0 at the time at which η reaches 3. \n<!-- image -->", 'Non-attractor: more physical cases': 'For a more realistic potential, like the one in Ref. [18], we have seen that the spread in the velocities at the end of the non-attractor phase is as large as 〈 (∆Π) 2 〉 1 / 2 /H glyph[similarequal] 0 . 4. To assess the impact of the quantum diffusion on the PBH abundance, we have proceeded as follows. We have set the parameters of the model as in section II, see Tab. 1, in such a way to reproduce the right abundance for the PBHs to be dark matter and for the potential to be consistent with the CMB constraints on the power spectrum at the reference scale of k CMB = 0 . 05 Mpc -1 (the spectral index and the tensor to scalar ratio computed in the slow-roll region are as well in agreement with current data). \nThe PBH abundance has been calculated using the density contrast ∆( glyph[vector]x ) = (4 / 9 a 2 H 2 ) ∇ 2 ζ ( glyph[vector]x ) with threshold ∆ c glyph[similarequal] 0 . 45 [26] where the variance is defined as \nσ 2 ∆ ( R H ) = 16 81 ∫ ∞ 0 dln q ( qR H ) 4 W 2 ( qR H ) P R ( q ) , (6.8) \nwhere W ( qR H ) is a Gaussian window function smoothing out the density contrast on the comoving horizon length R H = 1 /aH . The Gaussian approximation of the primordial mass fraction \nβ prim ( M ) glyph[similarequal] σ ∆ √ 2 π ∆ c e -∆ 2 c / 2 σ 2 ∆ , (6.9) \ngives β prim (10 -15 M glyph[circledot] ) glyph[similarequal] 3 · 10 -16 and therefore the right dark matter abundance. We have then included the quantum diffusion, run 10 4 realisations of the stochastic background evolution and for each of them we have calculated the primordial PBH abundance β qd prim . \nOur results show that ln β qd prim is approximately Gaussian distributed around the value of β cl prim computed using the classical inflaton evolution, and with a standard deviation σ β qd prim , see Fig. 11. This is only an approximation because there is a small skewness shifting the average slightly away from its classical value. This means that β qd prim is nearly distributed as a log-normal distribution. Extending what we have done previously, we can introduce a fine-tuning parameter defined to be \n∆ qd = ln β qd prim β cl prim . (6.10) \nThis quantity is distributed like a Gaussian and is a measure of how close the distribution of the PBH mass fraction is peaked around the classical value. Therefore ∆ qd is (nearly) centered around zero and within pσ β qd prim it acquires values \n<!-- image --> \nFigure 11: The probability density of ∆ qd , which is nearly Gaussian distributed around the classical value determined ignoring quantum diffusion, and of β qd prim for model in Ref. [18]. The results are derived from 10 4 realisations of the stochastic evolution. \n<!-- image --> \n<!-- image --> \nFigure 12: The probability density of ∆ qd , which is nearly Gaussian distributed around the classical value determined ignoring quantum diffusion, and of β qd prim for model in Ref. [19]. The results are derived from 10 4 realisations of the stochastic evolution. \n<!-- image --> \n-pσ β qd prim ∼ < ∆ qd ( p ) ∼ < pσ β qd prim . (6.11) \nThe values of ∆ qd ( p ) are summarised in Tab. 2. Notice that the range is not totally symmetric because of the small skewness. We observe that the criterion | ∆ qd | ∼ < 1 is grossly violated and the value of β qd prim ( M ) violently deviates from the classical value due to the effect of quantum diffusion on the evolution of the background. In other words the values of the PBH mass fraction violently fluctuate around an average which is very different from the classical value thought to be needed to get the right abundance of the dark matter in the form of PBH. \nTable 2: Detailed values of the ∆ qd ( p ) as defined in Eq. (6.11) and their corresponding values of β prim for models [18,19]. \n| Model [18] | p = 1 | p = - 1 | p = 2 | p = - 2 | p = 3 | p = - 3 |\n|-----------------|------------------|------------------|------------------|------------------|------------------|------------------|\n| ∆ qd ( p ) | - 6 . 49 | 6 . 88 | - 13 . 17 | 13 . 56 | - 19 . 85 | 20 . 25 |\n| β qd prim ( p ) | 2 . 13 · 10 - 13 | 3 . 33 · 10 - 19 | 1 . 70 · 10 - 10 | 4 . 16 · 10 - 22 | 1 . 36 · 10 - 7 | 5 . 20 · 10 - 25 |\n| Model [19] | p = 1 | p = - 1 | p = 2 | p = - 2 | p = 3 | p = - 3 |\n| ∆ qd ( p ) | - 4 . 20 | 4 . 36 | - 8 . 48 | 8 . 64 | - 12 . 76 | 12 . 92 |\n| β qd prim ( p ) | 7 . 33 · 10 - 15 | 1 . 41 · 10 - 18 | 5 . 29 · 10 - 13 | 1 . 95 · 10 - 20 | 3 . 81 · 10 - 11 | 2 . 70 · 10 - 22 | \nSimilar results are obtained for the model in Ref. [19], they are shown in Fig. 12. For the sake of comparison we have used the same reference values as in Ref. [19]. We notice that the dispersion of β prim ( M ) is less prominent. However, this is only due to the fact that in Ref. [19] a smaller threshold, ∆ c glyph[similarequal] 0 . 3, has been adopted, leading to smaller values of the variances to reproduce the right amount of dark matter in the form of PBHs. As a consequence, the impact of quantum diffusion is relatively smaller. Still, the criterion is violated as we can see from Tab. 2. We also remark that higher values of ∆ c = (0 . 4 -0 . 7) [26] are used in the literature and therefore even larger values of ∆ qd will be obtained. \nOur results make us confident that, while in principle conclusions might depend on the exact values of the square root of the variance σ ∆ and threshold ∆ c , the corresponding | ∆ qd | will in general be too large. This is because changing the parameters of the model to get new variances with some new thresholds does not reduce significantly the spread of ln β prim . Therefore, while our results are specific of the models we have considered, we believe the conclusions apply to any model where the inflaton field crosses a plateau with an inflection point in order to generate a spike in the power spectrum and give rise to PBHs. \nWe expect therefore that the standard (classical) picture to evaluate the dark matter abundance in terms of PBHs is significantly altered.', 'III. The non-attractor phase and quantum diffusion': "Let us now come back to the role of quantum diffusion. If too large, quantum diffusion causes a loss of information as the curvature perturbation may not be reconstructed any longer at late times in terms of classical trajectories [17]. Different scales mix and the corresponding amplitude will be left undetermined for an observer at late times. Since quantum diffusion becomes more and more relevant \nas the field slows down and consequently the power spectrum grows, this clearly creates an issue and one expects a upper bound on the curvature perturbation in order for the quantum diffusion to be irrelevant. \nSince the power spectrum is fixed by the inverse of the velocity of the inflaton field at the end of the non-attractor phase, we expect that, if the spread of the distribution of velocities caused by the stochastic motion is too large, then along most of the trajectories the perturbation will be either too large or to too small to generate PBHs in the allowed range of masses. One has therefore to find the amount of dispersion undergone by the velocity of the inflaton field. \nLet us also notice that the power spectrum is growing during the non-attractor phase after the corresponding wavelength leaves the Hubble radius and therefore the issue of the quantum diffusion becomes more relevant at the end of the non-attractor phase. We will therefore discuss the criterion at the end of such a phase, where one expects the strongest constraints. \nThe stochastic equation (1.8) can be written as an Ornstein-Uhlenbeck process \nd φ d N = Π , dΠ d N +3Π+ V ,φ H 2 = ξ, 〈 ξ ( N ) ξ ( N ' ) 〉 = Dδ ( N -N ' ) , D = 9 H 2 4 π 2 , (3.1) \nwhere D is the diffusion coefficient. We may write the Kramers-Moyal (KM) equation for the corresponding probability P ( φ, Π , N ) as [53] \n∂P ∂N = -∂ ∂φ (Π P ) + ∂ ∂ Π [ V , Π P + V ,φ H 2 P ] + D 2 ∂ 2 ∂ Π 2 P, (3.2) \nwhere \nV (Π) = 3 2 Π 2 . (3.3) \nThe initial condition for the probability can be taken to be \nP ( φ, Π , 0) = δ D ( φ -φ 0 ) δ D (Π -Π 0 ) , (3.4) \nas we assume that during the preceding slow-roll phase the motion is purely along classical trajectories.", 'Generic potential': "We re-write the KM equation as \n∂P ∂N = L KM P = ( L rev + L ir ) P, L rev = -Π ∂ ∂φ + V ,φ H 2 ∂ ∂ Π , L ir = ∂ ∂ Π ( 3Π + D 2 ∂ ∂ Π ) . (3.5) \nThe stationary solution for the operator L ir is proportional to exp( -3Π 2 /D ) and we can generate a Hermitian operator \nwhere \nL ir = exp ( 3Π 2 2 D ) L ir exp ( -3Π 2 2 D ) = L † ir = -3 a † a, (3.6) \na = √ D 6 ∂ ∂ Π + Π 2 √ 6 D , a † = -√ D 6 ∂ ∂ Π + Π 2 √ 6 D (3.7) \nare the annihilation and creator operators with [ a, a † ] = 1. To take advantage of this procedure, we also redefine the operator \nL rev = exp ( 3Π 2 2 D + ρ V H 2 6 D ) L rev exp ( -3Π 2 2 D -ρ V H 2 6 D ) = -aA -a † ˆ A, (3.8) \nwhere ρ is an arbitrary constant \nA = √ D 6 ∂ ∂φ -ρ V H 2 √ 6 D , ˆ A = √ D 6 ∂ ∂φ +(1 -ρ ) V H 2 √ 6 D , (3.9) \nwith [ A, ˆ A ] = V φφ /H 2 . Now, the orthonormalised eigenfunctions of the operator L ir , that is \nL ir φ n (Π) = -3 nφ n (Π) , a † aφ n (Π) = nφ n (Π) , (3.10) \nare \nSince the operator L KM is of the form \nor \nφ n (Π) = ( a † ) n φ 0 (Π) / √ n ! , φ 0 (Π) = exp ( -3Π 2 / 2 D ) √ D/ 6 √ 2 π . (3.11) \nL KM = exp ( -3Π 2 2 D -ρ V H 2 6 D ) ( L ir + L rev ) exp ( 3Π 2 2 D + ρ V H 2 6 D ) φ -1 0 (Π) , (3.12) \nwe can expand the probability as [53] \nL KM = -φ 0 (Π)exp ( -ρ V H 2 6 D ) ( 3 a † a + aA + a † ˆ A ) exp ( ρ V H 2 6 D ) φ -1 0 (Π) , (3.13) \nP = φ 0 (Π)exp ( -ρ V H 2 6 D ) ∑ n ≥ 0 c n ( φ, N ) φ n (Π) , (3.14) \nso that the distribution in the inflaton field is only given by the first term of the expansion \n∫ dΠ P = exp ( -ρ V H 2 6 D ) c 0 ( φ, t ) , (3.15) \nwhere nevertheless the coefficients c n satisfy the so-called Brinkman's hierarchy \n∂c n ∂N = -√ n ˆ Ac n -1 -3 nc n -√ n +1 Ac n +1 (3.16) \nand it is equivalent to the KM equation. This equation contains an infinite number of terms. For pedagogical purposes, let us truncate though the system by setting c n = 0 for n ≥ 3, so that the Brinkman's hierarchy reduces to \n∂c 0 ∂N + Ac 1 = 0 , ∂c 1 ∂N + ˆ Ac 0 +3 c 1 = 0 . (3.17) \nFor a large friction term one can neglect the term ∂c 1 /∂N , and we could eliminate c 1 in favour of c 0 . Setting ρ = 0, we find \n∂c 0 ∂N = -Ac 1 = 1 3 A ˆ Ac 0 = 1 3 H 2 ∂ ∂φ ( V ,φ c 0 ) + 1 2 D 9 ∂ 2 c 0 ∂φ 2 , (3.18) \nwhich is the standard Fokker-Planck equation. Had not we dropped the term ∂c 1 /∂N , we could have eliminated c 1 and get the equation for c 0 \n∂ 2 c 0 ∂N 2 +3 ∂c 0 ∂N = 1 H 2 ∂ ∂φ ( V ,φ c 0 ) + D 6 ∂ 2 c 0 ∂φ 2 , (3.19) \nwhich is Brinkman's equation. Retaining the coefficients c n with n ≥ 3 will introduce spatial derivatives higher than two. We find here what we mentioned in the introduction, that the KM contains an infinite tower of spatial derivatives of the effective probability of the inflaton field and, due to Pawula's theorem, it is not consistent to drop derivatives higher than two. In this sense, the Fokker-Planck equation is not the correct starting point. \nIn the case of a linear potential the operators A and ˆ A commute, while for a quadratic potential their commutator is a constant and the analysis is made it easier. We will consider these cases next.", 'Linear potential': 'Assuming that σ R glyph[similarequal] P 1 / 2 R pk ∼ ( H/ 2 π Π glyph[star] ), the PBH mass fraction has an average induced by quantum diffusion equal to \n〈 β prim ( M ) 〉 = ∫ dΠ P Π (Π) β prim ( M ) | Gaussian . (6.2) \nUsing a Gaussian distribution for Π glyph[star] with spread 〈 (∆Π) 2 〉 1 / 2 , we get \n〈 β prim ( M ) 〉 = H 3 σ R √ 2 π ( H 2 +4 π 2 R 2 c 〈 (∆Π) 2 〉 ) 3 / 2 R c e -H 2 R 2 c 2 ( H 2 +4 π 2 R 2 c 〈 (∆Π) 2 〉 ) σ 2 R . (6.3) \nNotice that the average value of the PBH primordial abundance gets shifted with respect to the expression (6.1) precisely because the distribution of the inflaton velocity has a nonvanishing width. \nWe may define a fine-tuning parameter ∆ qd defined through the ratio of the averaged mass fraction in the presence of diffusion and the mass fraction in the absence of diffusion as \n〈 β prim ( M ) 〉 β prim ( M ) = e ∆ qd . (6.4) \nEssentially, this fine-tuning parameter says how far is the average of the distribution of β prim ( M ) from the classical value computed in the absence of quantum diffusion. We find that \n∆ qd glyph[similarequal] -R 2 c 2 σ 2 R ( ε 1 + ε ) , ε = -4 π 2 R 2 c 〈 (∆Π) 2 〉 H 2 . (6.5) \nImposing that the calculation is done in the absence of diffusion is trustable requires | ∆ qd | ∼ < 1, or \n〈 (∆Π) 2 〉 1 / 2 H ∼ < σ R √ 2 π R 2 c glyph[similarequal] 10 -2 ( σ R 0 . 1 ) ( 1 . 3 R c ) 2 . (6.6) \nFor a linear potential this bound is violated by the fact that the spread in the velocity is √ D/ 6 /H glyph[similarequal] 0 . 2. One might think to reduce the fine-tuning by, for instance, decrease the value of R c , however one should also recall that in order to get the right amount of dark matter in the form of PBH, β prim glyph[similarequal] 10 -16 , one needs σ R / R c ∼ 1 / 8 and therefore decreasing R c leads to a strong decrease in σ R . Alternatively, one can fix the spread in the velocity to be √ D/ 6 /H glyph[similarequal] 0 . 2 and, imposing | ∆ qd | ∼ < 1, find a lower bound on the square root of the variance \nσ R ∼ > 2 √ 3 R 2 c glyph[similarequal] 2 ( R c 1 . 3 ) 2 , (6.7) \nwhich signals the difficulty of avoiding the impact of the quantum noise.', 'Linear plus quadratic potential': 'Our considerations can be extended beyond the linear order in the potential. Let us expand the potential including the quadratic order \nV ( φ ) = V 0 [ 1 + √ 2 glyph[epsilon1] V ( φ -φ 0 ) + 1 2 η V ( φ -φ 0 ) 2 ] + · · · , (3.30) \nwhere η V = V ,φφ / 3 H 2 parametrises the second derivative of the potential. The equation of motion leads to a classical value \n〈 Π( N ) 〉 = Π( N ) glyph[similarequal] Π 0 e -3 N ( 1 + 1 3 η V + √ 2 glyph[epsilon1] V Π 0 ) -Π 0 e -η V N ( 1 3 η V + √ 2 glyph[epsilon1] V Π 0 ) . (3.31) \nIn particular, if one has a potential where φ 0 corresponds to a minimum and only the quadratic piece is there in the Taylor expansion one finds \n〈 Π( N ) 〉 = Π( N ) glyph[similarequal] Π 0 e -3 N ( 1 + 1 3 η V ) -Π 0 3 η V e -η V N . (3.32) \nIn order to simplify the problem, we notice that in the stochastic equation of motion of the inflaton field \nd 2 φ d N 2 +3 d φ d N +3 √ 2 glyph[epsilon1] V +3 η V ( φ -φ 0 ) = ξ, (3.33) \none can shift the field Π by an amount -3 √ 2 glyph[epsilon1] V and φ by an amount -3 √ 2 glyph[epsilon1] V N in order to get rid of the constant force. The problem reduces for this shifted field to the following set of equations (we do not redefine the fields to avoid cluttering notation) \nd φ d N = Π , dΠ d N +3Π+3 η V φ = ξ. (3.34) \nThe solution of these equations is again given in Eq. (3.20). This time however \nφ ( N ) = [exp( -A N )] φφ φ 0 +[exp( -A N )] φ Π Π 0 , Π( N ) = [exp( -A N )] Π φ φ 0 +[exp( -A N )] ΠΠ Π 0 , A = ( 0 -1 3 η V 3 ) . (3.35) \nAlso, by defining \none obtains [53] \nλ 1 , 2 = 1 2 ( 3 ± √ 9 -12 η ) , λ 1 glyph[similarequal] 3 , λ 2 glyph[similarequal] η V , (3.36) \nM φφ = D 2( λ 1 -λ 2 ) 2 [ λ 1 + λ 2 λ 1 λ 2 + 4 λ 1 + λ 2 ( e -( λ 1 + λ 2 ) N -1 ) -1 λ 1 e -2 λ 1 N -1 λ 2 e -2 λ 2 N ] , M φ Π = D 2( λ 1 -λ 2 ) 2 ( e -λ 1 N -e -λ 2 N ) 2 , M ΠΠ = D 2( λ 1 -λ 2 ) 2 [ λ 1 + λ 2 + 4 λ 1 λ 2 λ 1 + λ 2 ( e -( λ 1 + λ 2 ) N -1 ) -λ 1 e -2 λ 1 N -λ 2 e -2 λ 2 N ] , (3.37) \nM φφ = D 18 [ 1 3 + 1 η V + 4 3 ( e -3 N -1 ) -1 3 e -6 N -1 η V e -2 η V N ] , M φ Π = D 18 ( e -3 N -e -η V N ) 2 , M ΠΠ = D 18 [ 3 + 4 η V ( e -3 N -1 ) -3 e -6 N -η V e -2 η V N ] . (3.38) \nor \nAt large times and for small η V they reduce to \nM φφ = D 18 [ 1 η V ( 1 -e -2 η V N ) -1 ] , M φ Π = D 18 e -2 η V N , M ΠΠ = D 18 ( 3 -η V e -2 η V N ) . (3.39) \nFor η V > 0, i.e. for a harmonically bound state, in the large time limit one obtains a stationary solution. However, for η V < 0, i.e. for an inverted parabolic potential, the force felt by the inflaton is repulsive. In both cases, the width of the distribution of the inflaton velocities obtained integrating out over all possible values of the inflaton field reads \n〈 (∆Π) 2 〉 glyph[similarequal] D 18 ( 3 -η V e -2 η V N ) . (3.40) \nIn the model of Ref. [18,19] the plateau is in fact a region around an inflection point between a minimum and a maximum so that η V changes sign from positive to negative (if the minimum is encountered first). Being the dynamics more complex than what described above, we should expect deviations of order unity from our estimate.', 'IV. Numerical analysis of quantum diffusion': 'In this section we present the numerical studies we performed in order to check the validity of our analytical findings. We have numerically solved the system (3.1) with the available Mathematica routines for the solution of stochastic differential equations. We focus only on the inflaton velocity since the perturbations are sensitive to it. The spread in the inflaton field, which acquires typically Planckian values (at least in the vast majority of the literature) is irrelevant. At any rate, we have numerically checked that our numerical results coincide with this statement. We were able to test the robustness of our numerical implementation for the case of the linear potential, for which we have the analytical solution, Eq. (3.20).', "Linear potential and Starobinsky's model": "We start by checking the solution of the KM equation in the case of a linear potential. Since we are interested in the dispersion of Π glyph[star] around its classical value, we recover numerically its variance among many realizations of the stochastic evolution. \nIn Fig. 6, one can see the comparison between the prediction (3.27) and the numerical results. The numerical results, obtained integrating over the inflaton field positions (whose spread is however tiny with respect to the average classical position), fully reproduce the analytical results (to the extent that the red line of the fit and the green one representing the theory overlap perfectly). We have also repeated our analysis for Starobinsky's model [59] we have introduced in section II. The results are in Fig. 7 which show that the spread of the velocity approaches ( D/ 6) 1 / 2 . For smaller values of δ , the agreement with the linear potential result would be extended to the whole non-attractor phase, but the choice of δ is limited by numerical precision. \n<!-- image --> \nFigure 6: On the left, the numerical, the fit to numerical and the theoretical prediction for the probability P Π (Π glyph[star] , N glyph[star] ) for the case of a linear potential with glyph[epsilon1] V = 10 -7 . On the right, the classical evolution of Π( N ) together with the spread 〈 (∆Π) 2 〉 1 / 2 which stabilises at ( D/ 6) 1 / 2 for N glyph[greaterorsimilar] 1. It was checked numerically that changing the initial condition for Π(0) by order of magnitudes does not give rise to significant modification of 〈 (∆Π) 2 〉 1 / 2 for N < 1. Furthermore, the plateau's value is not sensitive to the initial conditions, confirming the analytical result in Eq. (3.27). \n<!-- image --> \n<!-- image --> \nFigure 7: The numerical, the fit to numerical and the theoretical prediction for the probability P Π (Π glyph[star] , N glyph[star] ) for the case of Starobinsky's model with δ = 0 . 01 and glyph[epsilon1] + /glyph[epsilon1] -= 10 8 . On the right, the classical evolution of Π( N ) together with the spread 〈 (∆Π) 2 〉 1 / 2 which tends towards ( D/ 6) 1 / 2 . We have used the same parameters as in section II and performed 5 · 10 4 realisations of the stochastic evolution. \n<!-- image -->", 'More physical cases': 'For the more realistic models discussed in Refs. [18,19] our results are provided in Figs. 8 and 9. As we have noticed, the distribution is well-fitted by a Gaussian with spread 〈 (∆Π) 2 〉 ( N ). As already mentioned, one needs to take the value of the curvature perturbation at the peak at the end of the non-attractor phase since the corresponding mode does not change in time afterwards. Therefore, taking into account that for both cases P 1 / 2 R pk glyph[similarequal] 7( H/ 2 π Π glyph[star] ), we obtain \nδ Π glyph[star] H glyph[similarequal] 1 . 4 , (5.7) \nwhile \n〈 (∆Π) 2 〉 1 / 2 H glyph[similarequal] 0 . 4 , (5.8) \nwhich is comfortably smaller than (5.7). The criterion is well satisfied thanks to the boost the power spectrum gets at the peak with respect to the power spectrum calculated for the wavelength leaving the Hubble radius deep in the non-attractor phase. However, as we will see next, this does not seem enough for the quantum diffusion not to have an impact on the PBH abundance.', 'V. A criterion for the quantum diffusion': 'As previously discussed, the crucial quantity is the spread of the velocity ∆Π of the inflaton field for the various trajectories. If the spread of the probability distribution 〈 (∆Π) 2 〉 1 / 2 is smaller than the size δ Π glyph[star] of the region over which the perturbation is of the order of P 1 / 2 R pk , then an insignificant part of the wave packet goes out the region where the curvature perturbation is P 1 / 2 R pk and most of the trajectories will have the same curvature perturbation ∼ P 1 / 2 R pk . We impose therefore the criterion that the spread of the probability distribution is still within the region where P 1 / 2 R ∼ P 1 / 2 R pk , see Fig. 10, \nFigure 10: A representative behaviour of the quantum diffusion issue. The spread of the inflaton velocity probability 〈 (∆Π) 2 〉 1 / 2 has to be smaller than the distance between the origin and the average value of the velocity. \n<!-- image --> \n〈 (∆Π) 2 〉 1 / 2 H glyph[lessmuch] δ Π glyph[star] H . (5.1)', 'VI. A stronger criterion for the quantum diffusion': 'The presence of sizeable quantum diffusion enters in another relevant consideration and provides a stronger criterion. Assume a Gaussian form for the PBH mass function \nβ prim ( M ) glyph[similarequal] σ R √ 2 π R c e -R 2 c / 2 σ 2 R . (6.1) \nSuppose one fine-tunes the parameters of the inflaton potential to produce the right amount of PBH as dark matter, but without accounting for the quantum diffusion and therefore the spread of the inflaton velocities. \nPractitioners of the production of PBHs as dark matter in single-field models of inflation know that a considerable fine-tuning is needed in any model to produce the right amount of dark matter in the form of PBHs. Any deviation from the fine-tuned set of parameters due to the uncertainty caused by the quantum diffusion will lead to huge variations of the PBH primordial mass fraction (as well as the ignorance on the non-Gaussian corrections do). Let us take therefore into account the spread now on the PBH mass fraction itself.', 'VII. Conclusions': 'There is a lot of interest in the cosmology community for the possibility that the dark matter is formed by PBHs. Their origin might be ascribed to the same mechanism giving rise to the CMB anisotropies and large-scale scale structure, i.e. a period of inflationary accelerated expansion during the early stages of evolution of the universe. In single-field models the power spectrum of the curvature perturbation might increase at small scales if the inflaton crosses a region which is flat enough and various models in the literature have been proposed recently. \nIn this paper we have discussed the role of quantum diffusion in the determination of the final abundance of PBHs. Quantum diffusion necessarily acquires importance when the force induced by the inflaton potential becomes tiny during the dynamics of the inflaton field. We have analysed both analytically and numerically the impact of diffusion and concluded that in realistic models it can significantly affect the capability of making a firm prediction of the PBH abundance. This is because the velocity of the inflaton field turns out to be distributed around its classical value with a spread which has an exponential impact on the PBH mass fraction. \nWhile by itself the mass fraction does not say anything about the spatial distribution of the PBHs, \nwe expect that different regions of the universe upon PBH formation would be populated with different relative abundances, thus changing the prediction for how much dark matter there is or its subsequent evolution.', 'Acknowledgments': "We thank M. Cicoli, F. G. Pedro, and G. Tasinato for many interactions about their work (Refs. [18] and [19] respectively) and feedback on our draft. We also thank A. Linde for interactions about the PBH formation probability. We also thank C. Germani for disscussions. A.R. is supported by the Swiss National Science Foundation (SNSF), project Investigating the Nature of Dark Matter , project number: 200020-159223. A.K. thanks the Cosmology group at the D'epartement de Physique Th'eorique at the Universit'e de Gen'eve for the kind hospitality and financial support.", 'Appendix A: the curvature perturbation, the Schwarzian derivative and the dual transformation': "In this Appendix we elaborate further the issue of why the power spectrum during the non-attractor phase is indeed flat. This Appendix does not contain some new material with respect to the literature. \nOur starting point is the equation for the curvature perturbation on comoving hypersurfaces R \nR '' +2 z ' z R ' + k 2 R = 0 , (A.1) \nwhere for convenience the prime denotes in this Appendix and in the following one the conformal time derivative d / d τ and z = a ˙ φ/H (the dot denotes the cosmic time derivative). The function z satisfies the following equation \nz '' z = 2 a 2 H 2 ( 1 + glyph[epsilon1] -3 2 η + glyph[epsilon1] 2 -2 glyph[epsilon1]η + 1 2 η 2 + 1 2 ξ 2 ) = 2 a 2 H 2 ( 1 + 5 2 glyph[epsilon1] + glyph[epsilon1] 2 -2 glyph[epsilon1]η -1 2 V ,φφ H 2 ) , (A.2) \nwhere \nglyph[epsilon1] = -˙ H H 2 , η = -¨ φ H ˙ φ , ξ 2 = 3( glyph[epsilon1] + η ) -η 2 -V ,φφ H 2 . (A.3) \nAs long as slow-roll is attained, one can make use of the corresponding slow-roll parameters deduced from the form of the potential \nglyph[epsilon1] V = 1 2 ( V ,φ V ) 2 , η V = 1 3 V ,φφ H 2 , ξ 2 V = 3 glyph[epsilon1] V -η 2 V , (A.4) \nwhere the dynamics around Hubble crossing is dominated by the exponentially growing friction term proportional to R ' , and the solution to Eq. (A.1) is well approximated by \nR ( τ ) = constant and R ' ( τ ) aH ∼ ( τ τ k ) 2 , (A.5) \nwhere τ k indicates the value of the conformal time at which the comoving wavelength ∼ 1 /k leaves the comoving Hubble radius. \nIn the case in which one is interested in the generation of PBH from sizeable curvature fluctuations at small scales, a violent departure from the slow-roll must occur. In particular, if after Hubble crossing the friction term proportional to z ' /z changes its sign from positive to negative, it may become a driving term. This can have significant effects on modes which leave or have left the Hubble radius during this transient and non-attractor epoch and thus induce a growth of the curvature perturbations [57, 58]. A necessary, but not sufficient condition, to have PBH generation from single-field models is therefore the presence of a transient period for which \nz ' z = aH (1 + glyph[epsilon1] -η ) < 0 . (A.6) \nDuring such stage the function z reaches a local extremum (a maximum or a minimum depending upon the sign of ˙ φ ) at some time whenever \n1 + glyph[epsilon1] -η = 0 , (A.7) \nSince glyph[epsilon1] is always positive, the presence of a transient stage implies that η must be at least unity, signalling a breakdown of the slow-roll conditions. \nIn order to simplify the problem of dealing with a non-attractor phase necessary to generate a large amount of PBHs, we start by noticing that, upon the redefinition \nR = ˜ z z ˜ R , (A.8) \n˜ R '' +2 ˜ z ' z ˜ R ' + k 2 ˜ R = 0 , (A.9) \nthe quantity ˜ R satisfies the same equation of R \nas long as \n˜ The transformation from z to ˜ z which satisfies the relation (A.10) has been nicely worked out in Ref. [55] and called a dual transformation. It reads \nz '' z = ˜ z '' z . (A.10) \n˜ \n˜ z ( τ ) = C 1 z ( τ ) + C 2 z ( τ ) ∫ τ d τ ' z 2 ( τ ' ) . (A.11) \n˜ In fact this transformation is a property inferred from the so-called Schwarzian derivative [60], which we briefly summarise in the next subsection \n˜", 'The Schwarzian derivative': "Given a function f ( τ ), the Schwarzian derivative is defined as \nf ↦→ S [ f ] = f ''' f ' -3 2 ( f '' f ' ) 2 . (A.12) \nA property of the Schwarzian is that it is invariant under the transformation \nthat is \n˜ f = af + b c f + d , ad -bc glyph[negationslash] = 0 , (A.13) \nS [ f ] = S [ f ] . (A.14) \nNote that the symmetry (A.14) is just SL(2 , R ) up to a rescaling of f . Consider now a differential equation \n˜ \nu '' + q ( τ ) u = 0 . (A.15) \nq ( τ ) = 1 2 S [ f ] , (A.16) \nf ( τ ) = ∫ τ d τ ' u 2 ( τ ' ) . (A.17) \nu = 1 √ f ' , (A.18) \nIt can easily be seen that \nwhere \nIndeed, from Eq. (A.17) we find that \nand therefore, \nu '' = -1 2 S [ f ] u, (A.19) \nwhich is nothing else than Eq.(A.15) and Eq. (A.16). Now, since the Schwarzian is invariant under the transformation (A.13), we have that \nwhere \nu '' u = -1 2 S [ f ] = -1 2 S [ ˜ f ] = ˜ u '' ˜ u , (A.20) \n˜ Then, using Eqs. (A.13), (A.18) and (A.21) we find that \n˜ u = 1 √ ˜ f ' . (A.21) \n˜ u = u ( C 1 + C 2 f ) , C 1 = d √ ad -bc , C 2 = c √ ad -bc , (A.22) \nwhich, by using Eq.(A.17), is written as \nu ( τ ) = C 1 u ( τ ) + C 2 u ( τ ) ∫ τ d τ ' u 2 ( τ ) , (A.23) \nwhich is nothing else than the dual transformation found in Ref. [55]. \n˜ \nGoing back to the transformation (A.11), the power spectrum of the comoving curvature perturbation at the end of inflation reads \nP R ∣ ∣ ∣ end of inflation = ˜ z z P ˜ R ∣ ∣ end of inflation , (A.24) \n∣ from which we deduce that the power spectrum of the comoving curvature perturbation R is flat, a property that is inherited by the power spectrum of ˜ R for which the dynamics is of the slow-roll nature. The question now is what is the most suitable dual transformation to perform in order to simplify the computation of the power spectrum for those modes which exit the Hubble radius during the non-attractor phase and which are ultimately responsible for the production of PBHs when these curvature perturbations re-enter the Hubble radius during the radiation phase. \n∣ \nAs we have mentioned already several times, the production of PBHs may originated from the enhancement of the curvature power spectrum below a certain length scale. This can be achieved by a temporary abandonment of the slow-roll condition. When the inflaton field follows slow-roll and ˙ φ is approximately constant, the function z = a ˙ φ/H grows \nz ∼ 1 τ during slow-roll . (A.25) \nDuring the non-attractor phase, when the inflaton field experiences an approximately flat potential and V ,φ can be neglected, it satisfies the equation of motion \nφ '' +2 H φ ' glyph[similarequal] 0 ( H = aH ) , (A.26) \nand consequently φ ' ∼ τ 2 , or \nz = a ˙ φ H = φ ' H ∼ τ 2 during the non-attractor phase . (A.27) \nIt is this rapid fall of z which allows the possibility of enhancing the power spectrum. When the non-attractor phase is over, the slow-roll conditions are attained again and one recovers the behaviour in Eq. (A.25). In terms of the friction term z ' /z one has \nz ' z glyph[similarequal] aH { 1 during slow-roll , -2 during the non-attractor phase . (A.28) \nLet us now use the dual transformation (A.11) where we choose the lower limit of the integral to be τ 0 , the initial conformal time for the non-attractor phase. In such a case, we find \n∫ \n˜ z ' ˜ z = z ' z + C 2 z 1 C 1 z + C 2 z τ τ 0 d τ ' /z 2 ( τ ' ) . (A.29) \n˜ We can now choose C 1 = 1 and compute this expression during the non-attractor phase for those modes which enter the Hubble radius during the non-attractor phase \n1 aH ˜ z ' ˜ z = -2 + C 2 aHz · 1 z + C 2 z ∫ τ τ 0 d τ ' /z 2 ( τ ' ) . (A.30) \nSetting a = a 0 ( τ 0 /τ ) and z ( τ ) = z 0 ( τ/τ 0 ) 2 , we obtain \n1 aH ˜ z ' z = -2 + C 2 a 0 z 0 H ( τ 0 τ ) 1 z 0 ( τ/τ 0 ) 2 -( C 2 / 3 z 0 )( τ 2 0 /τ ) . (A.31) \n˜ \nwhere in the last passage we have neglected the subleading term ∼ 1 /τ 3 0 . Taking -τ < -τ 0 (recall that τ < 0) and recalling that a 0 = -1 / ( Hτ 0 ), one finally obtains \n1 aH ˜ z ' ˜ z glyph[similarequal] -2 + 3 = 1 (during the non-attractor phase) . (A.32) \n˜ \nz ( τ ) = z ( τ ) + C 2 z ( τ ) ∫ τ τ 0 d τ ' z 2 ( τ ' ) , (A.33) \n˜ This demonstrates that the choice \nmaps the non-attractor phase into a slow-roll phase for the curvature perturbation ˜ R and one can conclude that the power spectrum of the curvature perturbation for those modes entering the Hubble radius during the non-attractor phase is dictated by a slow-roll dynamics and therefore is flat. Its amplitude is however magnified by a factor ˜ z ( τ e ) /z ( τ e ). \n˜ To elaborate further and find a useful prescription, let us consider, as we also did in the main text, Starobinsky's model [59] where the inflaton field reaches a non-attractor phase after a slow-roll era (and eventually enters afterwards another slow-roll phase).", 'Slow-roll phase before the non-attractor phase': "If we indicate by φ 0 the moment at which the first slow-roll phase ends and the non-attractor phase starts, we can Taylor expand the inflaton potential as \nV ( φ ) glyph[similarequal] V 0 ( 1 + √ 2 glyph[epsilon1] + ( φ -φ 0 ) ) + · · · for φ > φ 0 , (A.34) \nwhere glyph[epsilon1] + is the slow-roll parameter during the first slow-roll phase. The corresponding parameter z reads \n3 H φ ' = -V ,φ a 2 and z + ( τ ) glyph[similarequal] -a 0 2 glyph[epsilon1] + ( τ 0 /τ ) , (A.35) \nhaving indicated τ 0 and a 0 is the conformal time and the scale factor when φ = φ 0 , respectively. \n√", 'Non-attractor phase': "For φ < φ 0 the potential is Taylor expanded as \nThe dynamics leads to \nand \nV ( φ ) glyph[similarequal] V 0 ( 1 + √ 2 glyph[epsilon1] -( φ -φ 0 ) ) + · · · for φ < φ 0 . (A.36) \n3 H φ ' V 0 a 2 = -√ 2 glyph[epsilon1] --( √ 2 glyph[epsilon1] + -√ 2 glyph[epsilon1] -)( τ/τ 0 ) 3 (A.37) \nz -( τ ) glyph[similarequal] -a 0 ( √ 2 glyph[epsilon1] -( τ 0 /τ ) + ( √ 2 glyph[epsilon1] + -√ 2 glyph[epsilon1] -)( τ/τ 0 ) 2 ) . (A.38) \nIf glyph[epsilon1] + glyph[greatermuch] glyph[epsilon1] -, there is a prolonged non-attractor phase where the second term in the above equation dominates over the first one. It is easy to show that z -reaches a maximum at the point \n( τ 0 /τ m ) 3 = 2 √ 2 glyph[epsilon1] + -√ 2 glyph[epsilon1] -√ 2 glyph[epsilon1] -glyph[similarequal] 2 √ glyph[epsilon1] + glyph[epsilon1] -, (A.39) \ncorresponding to z -( τ m ) ≈ ( glyph[epsilon1] -/glyph[epsilon1] + ) 1 / 3 z ( τ 0 ) and causing a sizeable change in R on super-Hubble scales if z -( τ m ) is very tiny. Notice that the smallness of glyph[epsilon1] -parametrises the duration of the non-attractor phase from τ 0 to τ glyph[star] . Let us now consider the duality transformation (A.11) with again lower limit τ 0 in the integral. We deduce \nz -( τ ) = -C 1 a 0 √ 2 glyph[epsilon1] -( τ 0 /τ ) + a 3 0 √ 2 glyph[epsilon1] -( √ 2 glyph[epsilon1] --√ 2 glyph[epsilon1] + ) C 1 +3 C 2 H 3 a 2 0 √ 2 glyph[epsilon1] -( τ/τ 0 ) 2 . (A.40) \nWe are free to choose C 2 such that the second term on the right-hand side of Eq. (A.40) vanishes, which happens for \n˜ \nC 2 = a 3 0 √ 2 glyph[epsilon1] -( √ 2 glyph[epsilon1] + -√ 2 glyph[epsilon1] -) 3 H 3 C 1 , (A.41) \nand hence \n˜ z -( τ ) = -C 1 a 0 √ 2 glyph[epsilon1] -( τ 0 /τ ) . (A.42) \nC 1 = √ glyph[epsilon1] + glyph[epsilon1] -. (A.43) \n˜ We are also free to make the dual transformation only for φ < φ 0 and therefore we match z + with the new ˜ z -at τ 0 and find \nTherefore we have a single slow-roll parameter \nz = z + = z -= -a 0 √ 2 glyph[epsilon1] + ( τ 0 /τ ) , (A.44) \n˜ \nP 1 / 2 ˜ R = 3 H 3 2 πV 0 √ 2 glyph[epsilon1] + ∣ ∣ ∣ k = aH , (A.45) \nthroughout all the evolution. This implies that the power spectrum for the ˜ R not only remains constant after Hubble crossing, but also can be computed using the slow-roll approximations and it reads \n∣ even during the non-attractor phase. The power spectrum therefore evolves as \nP 1 / 2 R ( τ ) = ˜ z ( τ ) z ( τ ) P 1 / 2 ˜ R = ˜ z -( τ ) z -( τ ) P 1 / 2 ˜ R = 1 2 glyph[epsilon1] -/ 2 glyph[epsilon1] + +(1 -√ 2 glyph[epsilon1] -/ 2 glyph[epsilon1] + )( τ/τ 0 ) 3 P 1 / 2 ˜ R , (A.46) \n√ \n√ Defining by τ glyph[star] the end of the non-attractor phase and computing the power spectrum just after τ glyph[star] (recall that the conformal time is negative and therefore -τ glyph[star] glyph[lessmuch] -τ 0 )) we find that immediately after the non-attractor phase \nP 1 / 2 R ( τ ∼ > τ glyph[star] ) = √ 2 glyph[epsilon1] + √ 2 glyph[epsilon1] -P 1 / 2 ˜ R = 3 H 3 2 πV 0 √ 2 glyph[epsilon1] -. (A.47) \nAt the end of the non-attractor phase therefore one finds \nP 1 / 2 R ( τ ∼ > τ glyph[star] ) = ( H 2 π Π glyph[star] ) = 3 H 3 2 πV ,φ ∣ ∣ ∣ k = aH , (A.48) \n∣ \nwhich provides the prescription to compute the power spectrum of the curvature perturbations for those modes crossing the Hubble radius deep in the non-attractor phase. By tuning the slope of the potential one can in principle obtain a large enhancement of the power spectrum. \nA few comments are in order. The last passage in Eq. (A.48) is valid only if the subsequent slow-roll phase starts when the velocity of the inflaton field has already settled to its slow-roll value proportional to √ 2 glyph[epsilon1] -. We remind the reader that the power spectrum does not further evolve during the subsequent transition between the non-attractor phase and the second slow-roll phase [39]. The prescription (A.48) was already proposed in Ref. [56] (see also Refs. [55, 57, 58]) to deal with the singular case in which ˙ φ = 0. In this sense the results of this long Appendix are not new, but we have given an alternative and maybe more intuitive derivation. Furthermore, the prescription (A.48) can be used for those modes which exit the Hubble radius deep in the non-attractor phase and predicts a flat power spectrum as the dual ˜ R experiences a slow-roll dynamics. Said in other words, the power spectrum must be flat since ˜ z '' / ˜ z = z '' /z glyph[similarequal] 2 a 2 H 2 up to small correction O ( glyph[epsilon1] V ). If one wishes to compute the abundance of PBHs using single-field models where a non-attractor phase is necessary, the corresponding power spectrum of the comoving curvature perturbation can be computed by simply evaluating it at Hubble crossing, even during the non-attractor phase, as long as one makes use of the slow-roll relation ˙ φ = -V ,φ / 3 H ; one can then account for the modes leaving the Hubble radius when η grows fast from tiny values to 3 using Eq. (2.8). Finally, the prescription is based on the fact that the non-attractor phase is long enough for the dynamics to established. If the plateau is short in field space, the inflaton field may arrive at it with an excessive kinetic energy and roll away of it in a Hubble time or so.", 'Appendix B: from the non-attractor back to the slow-roll phase': "The modes which have crossed the Hubble radius during the non-attractor phase are on super-Hubble scales during the eventual subsequent transition to a slow-roll phase with larger slope in the potential. To see what happens to these modes we follow Ref. [39] and model again the potential during the transition as \nV ( φ ) glyph[similarequal] V 0 ( 1 + √ 2 glyph[epsilon1] glyph[star] ( φ -φ glyph[star] ) + · · · , (B.1) \n( where we have defined φ glyph[star] the field value at the end of the attractor phase. The equation of motion for the inflaton field during the transition epoch reads \n) \nφ '' +3 H φ ' +3 a 2 √ 2 glyph[epsilon1] glyph[star] = 0 , (B.2) \nwhose solution for initial velocity Π glyph[star] leads to \nz ( τ ) = -Π glyph[star] 18 ( 3(6 + h )( τ/τ glyph[star] ) 2 -3 h τ glyph[star] τ ) , h = 6 √ 2 glyph[epsilon1] glyph[star] / Π glyph[star] . (B.3) \nThe solution for the super-Hubble scale comoving perturbation during the transient epoch reads \nR ( τ ) = C 1 + C 2 ∫ τ d τ z 2 ( τ ' ) = C 1 + C 2 12 τ glyph[star] 2(6 + h ) glyph[epsilon1] -( -h +(6 + h ) τ 3 ) . (B.4) \nThis solution needs to be matched now with the solution for τ < τ glyph[star] which (apart from the standard ( H/ 2 π )(1 / √ 2 k 3 ) scales like ( τ glyph[star] /τ ) 3 / Π glyph[star] . Matching the perturbations and their derivatives at τ glyph[star] , one gets \nR| end of inflation = ( H 2 π √ 2 k 3 ) 6 + h h Π glyph[star] . (B.5) \nWe see that if the non-attractor phase is followed by another slow-roll phase for which | h | glyph[greatermuch] 1, the curvature perturbations associated to the modes which are on super-Hubble scales during the transition will keep be enhanced as 1 / √ 2 glyph[epsilon1] -and the prescription (A.48) remains valid for those modes exiting the Hubble radius deep in the non-attractor phase [39].", 'Appendix C: the role of non-Gaussianities': "As we mentioned in the introduction, PBHs are born as large, but rare fluctuations of the curvature perturbation. As such, their abundance is extremely sensitive to the non-linearities of the curvature perturbation. A formalism particularly useful when dealing with non-linearities is the so-called δN formalism [51], where the scalar field fluctuations are quantized on the flat slices and R = -δN , being N the number of e-folds. The formalism is based on the assumption that on super-Hubble scales, each spatial point of the universe has an independent evolution and the latter is well approximated by the evolution of an unperturbed universe. \nLet us suppose that during the entire non-attractor phase the inflaton velocity decays exponentially. If so \nN ( φ, Π) = -1 3 ln [ Π Π+3 H ( φ -φ glyph[star] ) ] = -1 3 ln Π Π glyph[star] , (C.1) \nwhere φ glyph[star] is again the value of the field at the end of the non-attractor phase. Notice that we have retained the dependence on Π since slow-roll is badly violated. In the relation (D.3) we have followed the notation of Ref. [39] and defined N = 0 to be the end of the attractor phase, so that N < 0 and \nWe therefore find that \nφ ( N ) = φ glyph[star] + Π glyph[star] 3 ( 1 -e -3 N ) and Π( N ) = Π glyph[star] e -3 N . (C.2) \nR = -δ N = -N + N = -1 3 ln ( 1 + δ Π glyph[star] Π glyph[star] ) , (C.3) \nwhere the overlines indicate the corresponding background values. One can safely neglect the perturbation δ Π as it decays exponentially fast. On the other hand, by using the relation \nΠ glyph[star] = 3[ φ ( N ) -φ glyph[star] ] + Π( N ) , (C.4) \nwe see that up to irrelevant constants, \nR = -1 3 ln ( 1 + 3 δφ Π glyph[star] ) , δφ < -Π glyph[star] 3 . (C.5) \nThe crucial point now is that the dynamics of δφ is the one of a massless perturbation in de Sitter and to a very good approximation its behaviour is Gaussian. The non-Gaussianity in the curvature perturbation arises because of the non-linear mapping between δφ and 1 . \nThe fact that P ( δφ ) is Gaussian considerably simplifies the computation: the primordial mass fraction β prim ( M ) of the universe occupied by PBHs at formation time is dictated by probability conservation, \nor \nR \nP ( R ) = ∣ ∣ ∣ ∣ d δφ d R ∣ ∣ ∣ ∣ P [ δφ ( R )] , (C.6) \nP ( R ) = | Π glyph[star] | √ 2 π σ δφ exp [ -3 RΠ 2 glyph[star] 18 σ 2 δφ ( 1 -e -3 R ) 2 ] , (C.7) \nwhere we have written the Gaussian distribution of δφ as 2 \nP ( δφ ) = 1 √ 2 π σ δφ e -( δφ ) 2 / 2 σ 2 δφ , (C.8) \nwith \nFor 3 R c ∼ < 1, we obtain \ni.e., a Gaussian with variance \nσ 2 δφ = ∫ k dln p P δφ ( p ) . (C.9) \nP ( R ) ≈ Π glyph[star] √ 2 π σ δφ e -Π 2 glyph[star] R 2 / 2 σ 2 δφ , (C.10) \nσ 2 R = σ 2 δφ Π 2 glyph[star] . (C.11) \nOn the other hand, assuming now 3 R c ∼ > 1, we obtain \nβ prim ( M ) = ∫ R c d R P ( R ) glyph[similarequal] 1 2 erf ( Π glyph[star] 3 √ 2 σ δφ ) -1 2 erf ( Π glyph[star] (1 -e -3 R c ) 3 √ 2 σ δφ ) glyph[similarequal] -Π glyph[star] e -3 R c 3 √ 2 π σ δφ e -Π 2 glyph[star] / 18 σ 2 δφ , (C.12) \n1 A few comments. The non-Gaussianity during the non-attractor phase is not washed out by the subsequent transition to a slow-roll phase. This is because such a transition is sudden [39] as the velocity during the nonattractor phase must be much smaller than the one during the subsequent slow-roll phase to generate PBHs. The non-Gaussianity we are dealing with here is not the non-Gaussianity in the squeezed configuration which peaks when one of the wavelengths is much larger than the other two. This non-Gaussianity is not observable by a local observer testing a region much smaller than the long wavelength [63]. We are instead referring to that non-Gaussianity which arises at the same small wavelengths where the density perturbations are sizeable. In the limit of a spiked power spectrum centered around a given momentum k pk , the non-Gaussianity will be peaked at equilateral configurations. \n2 Sometimes the Gaussian probability is multiplied by a factor of 2 to account for the fact that one deals with a first time-passage problem [62]. We do not put it here as there is no general consensus of this factor. Quantitatively, it does not make a big difference though. \nto be confronted to the Gaussian result (6.9). The probability is clearly non- Gaussian. We can estimate σ δφ as well to be of the order of ( H/ 2 π )∆ N . We obtain \nβ prim ( M ) glyph[similarequal] e -3 R c 3 R glyph[star] ∆ N e -1 / (3 √ 2 R glyph[star] ∆ N ) 2 . (C.13) \nTo obtain the same primordial mass fraction, non-Gaussianity seems to require a smaller R glyph[star] . We write 'seems' as the curvature perturbation is not the best variable to study the PBH mass function. As written in the main text, the density contrast ∆( glyph[vector]x ) = (4 / 9 a 2 H 2 ) ∇ 2 ζ ( glyph[vector]x ) (during radiation) is the good variable [26]. This however will make things more difficult to analyse because of the complication arising from taking the laplacian of the expression (C.3). One possible, but not entirely satisfactory, way out might to evaluate the density contrast at Hubble re-entry, i.e. setting k = aH . In such a case, one could relate the density contrast to the curvature perturbation through the relation ∆( glyph[vector]x ) = (4 / 9) R ( glyph[vector]x ).", 'Appendix D: Comment on 1807.09057': 'After the publication of this paper, Ref. [64] appeared with some comments about our findings. Here we respond to them. This Appendix can be considered as an independent part of this work and therefore some concepts of the main text might be repeated. \nFirst of all, to the best of our understanding, Ref. [64] just contains the demonstration that one can compute the curvature perturbation in the non-attractor phase using the stochastic approach instead of adopting the standard computation in curved spacetime quantum field theory. This result differs from that in Ref. [65] and this is reassuring, as the standard linear computation has been our starting point 3 . In our paper, however, we have not used the stochastic approach to compute the perturbations, but to investigate the role of quantum diffusion on the observables. We did not assume the validity of the stochastic approach to compute the perturbations, the latter being derived by the standard field theory techniques (as in the large majority of the literature on the non-attractor phase). \nThe stochastic approach to study the cosmological perturbations focus on the behaviour of the perturbations on large scales under the action of the short modes which are integrated out from the action. These long mode perturbations are then treated classically under the action of a stochastic noise and give rise to a given power spectrum. In our approach, we do not focus on the perturbations, but on the effect of the noise onto the background observables. \nOur results have therefore little to do with those in Ref. [64]. In fact, the importance of diffusion in the determination of the primordial PBH abundance has been already discussed in Ref. [49] where it was also shown to be crucial (their analysis is restricted to slow-roll. However in the limit of extreme flatness of the potential during the non-attractor phase one can apply the duality discussed in Ref. [55] and in our Appendix A to map the problem into a slow-roll one). \nNevertheless, let us provide some comments about the criticisms raised in Ref. [64]. This will also allow us to discuss some clarifications/considerations. \nThe point raised in Ref. [64] is that the power spectrum is not a stochastic quantity and therefore may not be used to calculate the impact of quantum diffusion onto the PBH abundance. However, the smoothed power spectrum (the one which enters in the calculation of the abundance of PBHs) is a stochastic quantity once one specifies the scale at which the average is operated and in the presence of long-mode perturbations with wavelengths larger than the size of the region where the averaged is performed. This is nicely explained, for instance, in Ref. [66]. Let us consider two counter-examples to the statement of Ref. [64]. First, the computation of the celebrated Maldacena consistency relation relating the power spectra of the curvature perturbation to the bispectrum in the squeezed limit. It is well-known that such a result may be obtained simply taking the power spectrum, computed on a small box, and average it over a bigger volume containing the long-mode perturbation. The very simple result that the power spectrum correlates with the long mode shows that it is not a function, but a stochastic quantity. Similarly, in order to compute the local halo bias in the presence of primordial non-Gaussianity one exploits the fact that the variance of the density contrast is stochastic quantity in the presence of long-mode perturbations. \nAs we explain in the main text, one way of producing PBHs in the early universe is to generate an enhancement of the power spectrum of the curvature perturbation during inflation, more specifically during the non-attractor. These large perturbations re-enter the horizon during the radiation era and may collapse to form PBHs on comoving scales that left the Hubble radius about (20 -30) e-folds before the end of inflation. \nAt the end of the non-attractor phase the curvature perturbation (in the flat gauge) is \nR = -H δφ ˙ φ glyph[star] (D.1) \nThe calculation of the curvature perturbation till the end of the non-attractor can be performed by using the δN formalism [39]. Let us suppose that during the entire non-attractor phase the inflaton velocity decays exponentially so that \nφ ( N ) = φ glyph[star] + ˙ φ glyph[star] 3 H 1 -e -3 N ) and ˙ φ ( N ) = ˙ φ glyph[star] e -3 N . (D.2) \n( \n) Here φ glyph[star] and ˙ φ glyph[star] are the values of the inflaton field and its velocity at the end of the non-attractor phase, respectively and we have set N = 0 to be the end of the attractor phase, so that N < 0. Then \nN ( φ, ˙ φ ) = -1 3 ln ˙ φ ˙ φ glyph[star] . (D.3) \nOn the other hand, by using the relation (D.2) and expanding at first-order one finds the expression (D.1). Now, in this paper we have followed the same logic which has been neatly explained in Ref. [13]. The δ N method consists of three steps [13]: \n- · First of all, to find an inflationary trajectory for any point in the ( φ, ˙ φ ) space and to calculate the number of e-folds N ( φ, ˙ φ ) for this trajectory.\n- · The position of the point ( φ, ˙ φ ) has to be perturbed by adding to it inflationary jumps. This provides the perturbation of the number of e-folds δ N , which is directly related to the density perturbations. \n- · The third step and most relevant for us (usually not performed in slow-roll single-field models) comes from the fact that the resulting density perturbation for a given N (i.e. for a given wavelength) will depend on the place ( φ, ˙ φ ) the trajectory come from. Thus the remaining step is to evaluate the probability that for a given number of e-folds N till the end of the non-attractor phase (usually till the end of inflation in slow-roll models) the field was at any particular point ( φ, ˙ φ ). This is because for an observer restricted to her/his own Hubble radius during inflation, the classical value of the field is not given only by the zero mode, but also by the sum of the modes with wavelength larger than the Hubble length. This was the essence of the results of Ref. [49] which indeed found that the PBH abundance is different when quantum diffusion is present. The necessity of such third sanity-check step was also stressed in Ref. [66] (even though with no reference to PBHs). \nAs pointed out also in Ref. [13], this third step can be performed by using the stochastic approach which tells what is the probability to find a given value of the inflaton field and its velocity at a given point as a function of time. \nIn the main text we have pointed out that during the non-attractor phase the role of quantum diffusion on the coarse-grained field ˙ φ may become relevant and changes the value measured by observers restricted to their own Hubble patches. In other words, we have pointed out that the last and third sanity-check step described above is necessary. Usually in slow-roll single-field models probabilities are basically Gaussian functions peaked around the classical values and tiny widths and one neglects the third step (even though for the problem of the PBH abundance is indeed necessary [49]). \nIn order to compute the variance of the sizeable curvature perturbation upon horizon re-entry, which will eventually give rise to PBHs by collapse, one usually considers the classical evolution of the homogeneous fields φ and ˙ φ and the effect of perturbations about the classical trajectories on a given scale. However, in the extreme case in which diffusion overcomes the classicality, one may not estimate the curvature perturbation in terms of the classical trajectories [13]. Luckily, in the case at hand, quantum diffusion never becomes more relevant than its classical evolution. Nevertheless, even tiny differences may have an impact on the final abundance of the PBHs as it is exponentially sensitive to the variance deduced from power spectrum of . \nIn order to calculate the probability distribution for the field ˙ φ we have used the stochastic approach which amounts to assuming an average quantum diffusion per Hubble volume per Hubble time of the order of H/ 2 π . The velocity of the inflaton field becomes also a stochastic variable and the corresponding variance characterises the dispersion of the classical trajectories due to quantum fluctuations. \nR \nTaking for simplicity the case of constant potential during the non-attractor phase, the stochastic equations are (we report them here) \nH ∂ ∂ N 〈 (∆ φ ) 2 〉 = -2 〈 ∆ φ ∆ ˙ φ 〉 , H ∂ ∂ N 〈 ∆ φ ∆ ˙ φ 〉 = -〈 (∆ ˙ φ ) 2 〉 +3 〈 ∆ φ ∆ ˙ φ 〉 , ∂ ∂ N 〈 (∆ ˙ φ ) 2 〉 = 6 〈 (∆ ˙ φ ) 2 〉 -H 2 D, (D.4) \nwhere D = (3 H/ 2 π ) 2 and we have indicated with ∆ φ = φ -φ ( N ) and ∆ ˙ φ = ˙ φ -˙ φ ( N ). This set of \nequations shows that, even if the inflaton field and its velocity are taken to be homogeneous till one e-fold before the end of the non-attractor phase, it is unavoidable that at the end of it the inflaton field receives kicks of the order of \n∆ φ glyph[similarequal] H 2 π (D.5) \nand the velocity of the order of \n∆ ˙ φ glyph[similarequal] √ 3 H 2 2 √ 2 π . (D.6) \nIn slow-roll one does not worry about these kicks, as the classical value of the velocity is given by ˙ φ 2 = 2 glyph[epsilon1]H 2 M 2 p glyph[greatermuch] (∆ ˙ φ ) 2 glyph[similarequal] H 4 , where glyph[epsilon1] ∼ 10 -2 is a slow-roll parameter and M p the reduced Planck mass. This effect is totally negligible. However, at the end of the non-attractor phase glyph[epsilon1] is much smaller, glyph[epsilon1] ∼ 10 -8 . The variance of the inflaton velocity is not negligible when recalling that tiny changes of the curvature perturbations are exponentially inflated when computing the PBH abundance. \nNotice that the variance of the inflaton velocity reaches the value (D.6) after one e-fold or so, and remains the same on all coarse-grained lengths. In particular, assuming that the peak of the perturbation is reached, say, 20 e-folds before the end of inflation, our current universe will contain, at the time of formation of the PBHs, about exp(40 · 3) = exp(120) Hubble volumes. In any of them the velocity has tiny differences due to the variance (D.6). Since the probability to form a PBH in any of each patches is Eq. (1.1), having a not fully fixed R leads to different values of σ 2 R and therefore β in all the patches. One point to stress is that the kicks (D.6) are kicks of the short modes leaving the Hubble radius each Hubble time and what they do is to change the infrared long modes of the inflaton velocity whose cumulative effect is measured by the local observer as the background inflaton velocity. \nThus, while the final word is certainly given by the calculation of the exact probability of the comoving curvature perturbation and the corresponding KM-like equation (see for example Ref. [67] for the slow-roll case), in order to avoid any analytical approximation, we have numerically constructed different realisations of the comoving curvature perturbation by solving the corresponding equation of motion (A.1) for any given wavenumber, one for each random trajectory identified by the corresponding coarse-grained background values. We have then deduced the mean and the variance of the abundance of the PBHs. 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2006PhRvD..73b4017A

Geometry of higher-dimensional black hole thermodynamics

2006-01-01

['-', '-', '-', '-', '-', '-', '-']

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We investigate thermodynamic curvatures of the Kerr and Reissner-Nordström (RN) black holes in spacetime dimensions higher than four. These black holes possess thermodynamic geometries similar to those in four-dimensional spacetime. The thermodynamic geometries are the Ruppeiner geometry and the conformally related Weinhold geometry. The Ruppeiner geometry for a d=5 Kerr black hole is curved and divergent in the extremal limit. For a d≥6 Kerr black hole there is no extremality but the Ruppeiner curvature diverges where one suspects that the black hole becomes unstable. The Weinhold geometry of the Kerr black hole in arbitrary dimension is a flat geometry. For the RN black hole the Ruppeiner geometry is flat in all spacetime dimensions, whereas its Weinhold geometry is curved. In d≥5 the Kerr black hole can possess more than one angular momentum. Finally we discuss the Ruppeiner geometry for the Kerr black hole in d=5 with double angular momenta.

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https://arxiv.org/pdf/hep-th/0510139.pdf

{'Geometry of Higher-Dimensional Black Hole Thermodynamics': 'Jan E. ˚ Aman ∗ and Narit Pidokrajt † \nQuantum and Field Theory group, Department of Physics, AlbaNova Stockholm University, SE-106 91 Stockholm, Sweden. ‡ \n(Dated: November 26, 2024) \nWe investigate thermodynamic curvatures of the Kerr and Reissner-Nordstrom (RN) black holes in spacetime dimensions higher than four. These black holes possess thermodynamic geometries similar to those in four dimensional spacetime. The thermodynamic geometries are the Ruppeiner geometry and the conformally related Weinhold geometry. The Ruppeiner geometry for d = 5 Kerr black hole is curved and divergent in the extremal limit. For d ≥ 6 Kerr black hole there is no extremality but the Ruppeiner curvature diverges where one suspects that the black hole becomes unstable. The Weinhold geometry of the Kerr black hole in arbitrary dimension is a flat geometry. For RN black hole the Ruppeiner geometry is flat in all spacetime dimensions, whereas its Weinhold geometry is curved. In d ≥ 5 the Kerr black hole can possess more than one angular momentum. Finally we discuss the Ruppeiner geometry for the Kerr black hole in d = 5 with double angular momenta. \nPACS numbers: 04.50.+h, 04.70.Dy', 'I. INTRODUCTION': 'The Hessian matrix of the thermodynamic entropy is known as the Ruppeiner metric [1]. It is a metric defined on the state space by \ng R ij = -∂ i ∂ j S ( M,N a ) , (1) \nwhere S is the entropy, M denotes the energy and N a are other extensive variables of the system. There have been a number of results indicating that this geometry measures the underlying statistical mechanics of the system. In particular for systems with no statistical mechanical interactions ( e.g. the ideal gas), the Ruppeiner geometry is flat and vice versa . Furthermore it appears that a divergent Ruppeiner curvature indicates a phase transition [1, 2, 3]. There is another metric which is defined as the Hessian of the energy (mass), it is known as the Weinhold metric [4, 5] \ng W ij = ∂ i ∂ j M ( S, N a ) . (2) \nThe Ruppeiner and Weinhold metrics are related to each other [6, 7] via \nds 2 R = 1 T ds 2 W . (3) \nSince black holes are regarded as thermodynamic systems [8], it is then natural to investigate their thermodynamic geometries. Previous studies [9, 10] suggest that the thermodynamic geometry of the black holes has a pattern from which one may possibly deduce physical insights. In [9] it is found that the divergence of the Ruppeiner curvature of the Kerr black hold indicates a phase transition. \nAs a generalization of our previous work [9], we will apply the Ruppeiner thermodynamic theory to the higher dimensional black hole solutions which were first derived by Myers and Perry in 1986 [11], in hopes to obtain further structure of black holes and the Ruppeiner theory itself. As it turns out, we obtain interesting results which may be a justification for a possible application of the Ruppeiner theory to black hole solutions that exist in various gravity theories e.g. string theory.', 'II. REISSNER-NORDSTR OM BLACK HOLE': "The Reissner-Nordstrom black hole is a solution of the Einstein equation coupled to the Maxwell field. In arbitrary spacetime dimension it is given by \nds 2 = -V dt 2 + V -1 dr 2 + r 2 d Ω 2 ( d -2) (4) \nwhere d Ω 2 ( d -2) is the line element on the ( d -2) unit sphere with d being a spacetime dimension. The volume of the ( d -2) unit sphere is given by \nΩ ( d -2) = 2 π d -1 2 Γ( d -1 2 ) . (5) \nV is a function of mass and charge given in terms of parameters µ and q \nV = 1 -µ r d -3 + q 2 r 2( d -3) (6) \nwhere \nand \nq = √ 8 πG ( d -2)( d -3) Q. (8) \nAn event horizon of the RN black hole is where V = 0, which can be solved analytically in arbitrary dimension. For the sake of tidiness and simplicity, we set Newton's gravitational constant to be G = Ω 2 ( d -2) / 16 π in order to eliminate all the π 's under the square root in (9). There are two roots, one of which is an outer horizon, r + while the other is called a Cauchy horizon \nr ± = ( µ 2 ± µ 2 √ 1 -4 q 2 µ 2 ) 1 / ( d -3) . (9) \nIt is obviously seen that \nr d -3 + + r d -3 -= µ and r d -3 + r d -3 -= q 2 . (10) \nThis solution develops a singularity when q 2 > µ 2 / 4 with the singularity at r = 0. When q 2 < µ 2 / 4, we have the outer event horizon as in (9). We note that µ and q are the ADM mass and electric charge of the black hole respectively [11]. When expressed in \nµ = 16 πGM ( d -2)Ω ( d -2) (7) \nterms of the mass, charge and dimensionality of the RN black hole, we obtain \nr d -3 + = M Ω ( d -2) 2( d -2) ( 1 + √ 1 -d -2 2( d -3) Q 2 M 2 ) . (11) \nIn arbitrary dimension the black hole becomes extremal when \nQ 2 M 2 = 2( d -3) d -2 . (12) \nThe area of the event horizon of the RN black hole is thus given by \nA = Ω ( d -2) r ( d -2) + . (13) \nThe entropy of the hole [12] takes the form \nS = k B A 4 G /planckover2pi1 = k B 4 G Ω ( d -2) r ( d -2) + , (14) \nwith /planckover2pi1 = 1 for simplicity. We can further introduce Boltzmann's constant to absorb the π 's and other numbers in the following way, \nk B = [2( d -2)] d -2 d -3 4 π Ω 1 d -3 ( d -2) . (15) \nHence we obtain the entropy function as \nS = r ( d -2) + = r d -3 + ) d -2 d -3 . (16) \n( \n) Explicitly in terms of mass and charge, it reads \nS = ( M + M √ 1 -d -2 2( d -3) Q 2 M 2 ) d -2 d -3 . (17) \nWe have learned from our previous work [9] that for the RN black hole, it is simpler to work in Weinhold coordinates. An inversion of (17) gives M as \nM = S d -3 d -2 2 + d -2 4( d -3) Q 2 S d -3 d -2 . (18) \nIn d = 4 it takes a simple form \nM = √ S 2 ( 1 + Q 2 S ) . (19) \nThe Weinhold metric of the RN black hole is diagonalizable in any dimension by choosing the new coordinate \nu = √ d -2 2( d -3) Q S d -3 d -2 , (20) \nFIG. 1: The state space of the four-dimensional Reissner-Nordstrom black holes shown as a wedge in a flat Minkowski space. Note that the curves of constant entropy reach the edge the wedge. \n<!-- image --> \nwhere -1 ≤ u ≤ 1. Thus we obtain the diagonal Weinhold metric as \nds 2 W = S -d -1 d -2 [ -1 2 d -3 ( d -2) 2 (1 -u 2 ) dS 2 + S 2 du 2 ] . (21) \nThis metric is curved and Lorentzian. By means of conformal transformation (3), we obtain the diagonalized Ruppeiner metric for the d -dimensional RN black hole in the new coordinates as \nds 2 R = -dS 2 ( d -2) S + 2( d -2) S ( d -3) du 2 1 -u 2 , (22) \nwhich is a flat metric. The black hole's temperature is given by \nT = ∂M ∂S = d -3 2( d -2) 1 -u 2 S 1 / ( d -2) . (23) \nFurthermore, we can introduce new coordinates so that the metric in (22) can be written in Rindler coordinates as \nds 2 = -dτ 2 + τ 2 dσ 2 , (24) \nusing \nτ = 2 √ S d -2 and sin σ √ 2( d -3) d -2 = u. (25) \nIt is readily seen that σ lies within the following interval \n-d -2 2 √ 2( d -3) π ≤ σ ≤ d -2 2 √ 2( d -3) π. (26) \nThis can be turned into Minkowski coordinates t and x via the following coordinate transformations \nt = τ cosh σ, x = τ sinh σ, (27) \nsuch that \nds 2 = -dt 2 + dx 2 = -dτ 2 + τ 2 dσ 2 . (28) \nHence we obtain a Rindler wedge whose opening angle depends on the dimensionality of the RN black hole, i.e. \ntanh -( d -2) π 2 2( d -3) ≤ x t ≤ tanh ( d -2) π 2 √ 2( d -3) . (29) \n√ \n√ For d = 4 the resulting wedge is shown in FIG. 1. It is noticeable that the opening angle of the wedge of the RN black hole grows and reaches the lightcone as d →∞ . We represent the entropy function of the d = 4 RN black hole in the Minkowskian coordinates as \nS = 1 2 ( t 2 -x 2 ) . (30) \nCurves of constant S are segments of hyperbolas.", 'III. KERR BLACK HOLE': "The electrically charged rotating black hole is known as the Kerr-Newman black hole [13, 14]. The limiting case where the electric charge is zero is known as the Kerr solution. In higher dimensional spacetime there can be more than one angular momentum in the Kerr solution. For the Kerr black hole with a single nonzero spin [11, 15, 16, 17], we obtain the outermost event horizon by solving the equation \nr 2 + + a 2 -µ r d -5 + = 0 . (31) \nWe take our liberty to set the Newton's constant G = Ω ( d -2) / 4 π for the sake of simplification. The area of the event horizon is given by \nA = Ω ( d -2) r d -4 + ( r 2 + + a 2 ) . (32) \nThe ADM mass of the hole is defined by \nµ = 4 M d -2 . (33) \nThe angular momentum per unit mass is dimensiondependent, namely \na = d -2 2 J M . (34) \nBy setting k B = 1 /π we obtain the entropy function of the Kerr black hole in d -dimension as \nS = r d -4 + ( r 2 + + a 2 ) = r + µ. (35) \nWith further algebraic manipulation, we find that even though an explicit entropy function in arbitrary dimension cannot be obtained, we can still work in \nd dimensions via the Weinhold metric. The mass function in arbitrary d can be written in terms of S and J as \nM = d -2 4 S d -3 d -2 ( 1 + 4 J 2 S 2 ) 1 / ( d -2) . (36) \nThe Weinhold metric g W ij = ∂ i ∂ j M ( S, J ) of the d -dimensional Kerr black hole takes a complicated form, but it is found to be flat as anticipated a priori based on our previous work [9]. It takes the form \nds 2 W = λ ( [ -48( d -5) J 4 +24 S 2 J 2 -( d -3) S 4 ] dS 2 + [ 64( d -5) J 3 S -16( d -1) JS 3 ] dSdJ + [ -32( d -4) J 2 S 2 +8( d -2) S 4 ] dJ 2 ) . (37) \nwhere \nλ = 1 4( d -2)( S 2 +4 J 2 ) 2 d -5 d -2 S d +1 d -2 . (38) \nThis metric can be brought into a diagonal form via coordinate transformations \nu = J S (39) \nand \nτ = √ d -2 d -3 S d -3 2( d -2) (1 + 4 u 2 ) 1 2( d -2) . (40) \nThe Weinhold metric in a diagonal form now reads: \nds 2 W = -dτ 2 + 2( d -3) ( d -2) (1 -4 d -5 d -3 u 2 ) (1 + 4 u 2 ) 2 τ 2 du 2 . (41) \nThis metric is a flat metric. In d = 4 we can write it in Rindler coordinates as \nds 2 W = -dτ 2 + τ 2 dσ 2 (42) \nby using \nu = 1 2 sinh 2 σ. (43) \nIn four dimensional spacetime we have the extremal limit along J/M 2 = 1 hence u is bounded by \n| u | ≤ 1 2 ⇔| σ | ≤ 1 2 sinh -1 1 ≈ 0 . 4406 . (44) \nFIG. 2: The state space of d = 4 Kerr black holes shown as a wedge in a flat Minkowski space. The slope of the wedge measures approximately 80 · from the x -axis. Curves of constant entropy give causal structure to the state space of the black hole. \n<!-- image --> \nBy using (27) we obtain the wedge of the state space of the Kerr black hole (see FIG. 2) in a flat Minkowski space whose edge is bounded by \n-√ √ 2 -1 √ 2 + 1 ≤ x t ≤ √ √ 2 -1 √ 2 + 1 . (45) \nIn five dimensions, the extremal limit is given by \nJ 2 /M 3 = 16 / 27 and -∞≤ u ≤ ∞ where \nu = 1 2 tan √ 3 σ. (46) \nHence we obtain a wedge with a different opening angle since σ falls in the range \n| σ | ≤ 1 √ 3 arctan ∞ = π 2 √ 3 ≈ 0 . 9069 . (47) \nThe opening angle of the wedge for the d = 5 Kerr black hole is wider than that of d = 4. Remarkably, the wedge of the d ≥ 6 Kerr black hole fills the entire light cone. This is because for black holes in d ≥ 6 there are no extremal limits. It is noteworthy that there is a causal structure of state space, but it is determined by curves of constant entropy rather than by the lightcone itself. The curves of constant entropy for d = 4 Kerr black hole in Minkowskian coordinates are given by \nS = ( t 2 -x 2 ) 4 4( t 2 + x 2 ) 2 . (48) \nDifferentiation of the mass function (36) with respect to the entropy gives the temperature of the Kerr black hole in arbitrary d as \nT = ( d -3) ( 1 + 4 d -5 d -3 J 2 S 2 ) 4 S 1 d -2 1 + 4 J 2 S 2 ) d -3 d -2 . (49) \n( \n) This temperature shrinks to zero at extremality for d = 4 , 5. According to [11, 16] for d ≥ 6 there is no extremal limit, the black hole's temperature does not vanish but reaches minimum and starts to behave differently as T ∼ r -1 + . In any dimension, we obtain the Ruppeiner metric by using the conformal relation (3). It is found to be a curved metric with the curvature scalar of the form \nR = -1 S 1 -12 d -5 d -3 J 2 S 2 ( 1 -4 d -5 d -3 J 2 S 2 )( 1 + 4 d -5 d -3 J 2 S 2 ) . (50) \nIn d = 4 the curvature scalar diverges along the curve 4 J 2 = S 2 which is consistent with the previous result [9]. The Ruppeiner curvature scalar in (50) is valid in any dimension higher than three. In d = 5 the curvature is reduced to \nR = -1 S (51) \nwhich diverges in the extremal limit of the d = 5 Kerr black hole. For d ≥ 6 we have a curvature blow-up but not in the limit of extremal black hole, rather at \n4 J 2 = d -3 d -5 S 2 . (52) \nThis is where Emparan and Myers [16] suggest that the Kerr black hole becomes unstable and changes its behavior to be like a black membrane. Note that in d = 5 for some values of the parameters, there exist 'black ring' solutions [18] whose entropy is larger than that of the black hole studied in this paper. A careful observation of (51) indicates that nothing special happens to the Gibbs surface of the Kerr black hole ' a la Myers-Perry.", 'IV. MULTIPLE-SPIN KERR BLACK HOLE': "This is a case where the entropy of the black hole is a function of three parameters, namely the function of mass and two spins. Another example of the threeparameter thermodynamic geometry can be found in [9] where the Ruppeiner and Weinhold geometries of the Kerr-Newman black hole were investigated. The general Kerr metric in arbitrary dimension d is available in, e.g. [11, 16, 19]. The black holes have ( d -1) / 2 angular momenta if d is odd and ( d -2) / 2 if d is even. The multiple-spin Kerr black hole's metric in Boyer-Lindquist coordinates for odd d is given by \nds 2 = -d ¯ t 2 +( r 2 + a 2 i )( dµ 2 i + µ 2 i d ¯ φ 2 i ) + µr 2 Π F ( d ¯ t + a i µ 2 i d ¯ φ i ) 2 + Π F Π -µr 2 dr 2 , (53) \nwhere \nd ¯ t = dt -µr 2 Π -µr 2 dr, (54) \nd ¯ φ i = dφ i + Π Π -µr 2 a i r 2 + a 2 i dr, (55) \nwith the constraint \nµ 2 i = 1 . (56) \nThe functions Π and F are defined as follows: \nΠ = ( d -1) / 2 ∏ i =1 ( r 2 + a 2 i ) , F = 1 -a 2 i µ 2 i r 2 + a 2 i . (57) \nThe metric is slightly modified for even d [11]. The event horizons in the Boyer-Linquist coordinates will occur where g rr = 1 /g rr vanishes. They are the largest roots of \nΠ -µr = 0 even d (58) \nΠ -µr 2 = 0 odd d. (59) \nThe areas of the event horizon are given by \nA = Ω ( d -2) r + ∏ i ( r 2 + + a 2 i ) odd d, (60) \nA = Ω ( d -2) ∏ i ( r 2 + + a 2 i ) even d. (61) \nIn d = 5 there can be only two angular momenta associated with the Kerr black hole, thus the area of the event horizon reads \nA = 2 π 2 r + ( r 2 + + a 2 1 )( r 2 + + a 2 2 ) . (62) \nThe temperature of the d = 5 Kerr black hole with two spins is the Hawking temperature T = κ/ 2 π where the surface gravity κ is given by \nκ = r + ( 1 r 2 + + a 2 1 + 1 r 2 + + a 2 2 ) -1 r + . (63) \nSince there are two angular momenta, hence two angular velocities are associated with this black hole, \nΩ a 1 = a 1 r 2 + + a 2 1 , Ω a 2 = a 2 r 2 + + a 2 2 . (64) \nThe first law of thermodynamics for this black hole takes the form [19] \ndM = TdS +Ω a 1 dJ a 1 +Ω a 2 dJ a 2 . (65) \nThe entropy of the d = 5 Kerr black hole with double spins is given by \nS = k B A 4 G = k B 4 G 2 π 2 r + ( r 2 + + a 2 1 )( r 2 + + a 2 2 ) . (66) \nWe can choose k B and G such that S simplifies as \nS = 1 r + ( r 2 + + a 2 1 )( r 2 + + a 2 2 ) , (67) \nwhere r + is the largest root of \n( r 2 + a 2 1 )( r 2 + a 2 2 ) -µr 2 = 0 , (68) \nwhere µ is the ADM mass defined in (33) with d = 5 and a i = 3 J i / 2 M . The temperature of the d = 5 double-spin Kerr black hole reaches zero in the extremal limit which is given by \na 1 + a 2 = √ µ (69) \nor explicitly in terms of mass and the two spins as \nJ 1 + J 2 = 4 M 3 / 2 3 √ 3 . (70) \nSince solving for the entropy function directly is rather complicated, we thus use the same procedure as in the case of the single-spin Kerr black hole and obtain the mass as a function of entropy and two angular momenta as \nM = 3 S 2 / 3 4 ( 1 + 4 J 2 1 S 2 ) 1 3 ( 1 + 4 J 2 2 S 2 ) 1 3 . (71) \nThe Hessian of M with respect to the entropy and two angular momenta yields the Weinhold metric, which is found to be curved. The curvature scalar of the Weinhold metric takes the form \nTABLE I: Geometry of higher-dimensional black hole thermodynamics. \n| Spacetime dimension Black hole family Ruppeiner Weinhold | | | |\n|------------------------------------------------------------|------------------|--------|--------|\n| d = 4 | Kerr | Curved | Flat |\n| d = 4 | RN | Flat | Curved |\n| d = 5 | Kerr | Curved | Flat |\n| d = 5 | double-spin Kerr | Curved | Curved |\n| d = 5 | RN | Flat | Curved |\n| d = 6 | Kerr | Curved | Flat |\n| d = 6 | RN | Flat | Curved |\n| any d | Kerr | Curved | Flat |\n| any d | RN | Flat | Curved | \nR Weinhold = 16 3 S 2 3 ( S 8 +3 S 6 J 2 1 +3 S 6 J 2 2 +4 S 4 J 2 1 J 2 2 +64 J 4 1 J 4 2 ) ( S 2 +4 J 2 1 ) 1 3 ( S 2 +4 J 2 2 ) 1 3 ( S 2 -4 J 1 J 2 ) 2 ( S 2 +4 J 1 J 2 ) 2 . (72) \nWe next transform it into the Ruppeiner metric via the conformal relation with an inverse temperature as a conformal factor. The temperature of the doublespin Kerr black hole in five dimensions is given by \nThe Ruppeiner curvature scalar of the double-spin Kerr black hole in five dimensions reads \nT = 1 2 S 5 / 3 ( S 2 +4 J 1 J 2 )( S 2 -4 J 1 J 2 ) ( S 2 +4 J 2 1 ) 2 / 3 ( S 2 +4 J 2 2 ) 2 / 3 . (73) \nR Ruppeiner = -S 8 +20 S 6 J 2 1 +20 S 6 J 2 2 +256 S 4 J 2 1 J 2 2 +192 J 4 1 J 2 2 S 2 +192 J 2 1 J 4 2 S 2 -256 J 4 1 J 4 2 2 S ( S 2 +4 J 2 1 )( S 2 +4 J 2 2 )( S 2 -4 J 1 J 2 )( S 2 +4 J 1 J 2 ) . (74) \nNote that both the Weinhold and Ruppeiner curvature scalars are divergent at \nJ 1 J 2 = S 2 4 , (75) \nwhich is the extremal limit of the d = 5 double-spin Kerr black hole. Note also that this curvature scalar does not vanish either in the limit of J 1 = 0 or J 2 = 0.", 'V. DISCUSSION': 'The geometry of black hole thermodynamics in higher dimensional spacetime has a pattern similar \nto the previous study in d = 4 spacetime. Since microstructures of black holes are unknown, we cannot yet conclude our findings along the same line as those done for the ideal gas [2]. Examination of our examples shows that when a flat thermodynamic curvature arises, it is the Hessian of a function of the form \nψ ( x, y ) = x a f ( x y ) (76) \nwith a a constant. We have checked that such a function always gives a flat thermodynamic metric, regardless of a , and regardless of the function f . \nIt is worthwhile to observe that inclusion of a cosmological constant leads to curved thermody- \nnamic geometries [9]; see [20] for a discussion of the higher dimensional cases. We note that the calculations of the three-parameter thermodynamic curvature scalars are best achievable by utilization of computer programs for algebraic computations such as CLASSI [21] and GRTensor [22] for Maple.', 'VI. SUMMARY AND OUTLOOK': 'In this paper, we study thermodynamic geometries of the black hole families and obtain some interesting results, coinciding with our previous findings. We speculate some sort of duality between entropy and mass of the black hole, which is somewhat corresponding to the very distinction between a rotation parameter and an electric charge. \n- [1] G. Ruppeiner, Rev. Mod. Phys. 67 , 605. (1995)\n- [2] J. D. Nulton and P. Salomon, Phys. Rev. A31 , 2520. (1985)\n- [3] D.A. Johnston, W. Janke and R. Kenna, Acta Phys. Pol. B34 , 4923. (2003)\n- [4] F. Weinhold, J. Chem. Phys. 63 , 2479. (1975)\n- [5] S. Ferrara, G. Gibbons and R. Kallosh, Nucl. Phys. B500 , 75. (1997)\n- [6] R. Mruga/suppressla, Physica 125A , 631. (1984)\n- [7] P. Salamon, J. D. Nulton and E. Ihrig, J. Chem. Phys. 80 , 436. (1984)\n- [8] S. W. Hawking, Phys. Rev. Lett. 26 , 1344. (1971)\n- [9] J. E. ˚ Aman, I. Bengtsson and N. Pidokrajt, Gen. Rel. Grav. 35 , 1733. (2003)\n- [10] R-G Cai and J-H Cho, Phys. Rev. D60 , 067502. (1999)\n- [11] R. C. Myers and M. J. Perry, Ann. Phys. (N.Y.) 172 , 304. (1986)\n- [12] H. Falcke and F. W. Hehl (eds.), The Galactic Black Hole: Lectures on General Relativity and Astrophysics (IoP, Bristol 2003)\n- [13] C. W. Misner, K.S. Thorne and J.A. Wheeler, Gravitation (Freeman, San Francisco, 1973)\n- [14] V. P. Frolov and I. D. Novikov, Black Hole Physics: Basic Concepts and New Developments \nWe summarize our results in the TABLE I. Furthermore, we have seen that the Ruppeiner curvature, in all the systems we have studied so far, behaves in a physically very suggestive way. The question why we have this pattern may be answered by the quantum theory of gravity in the future.', 'Acknowledgments': "We would like to thank Ingemar Bengtsson for encouraging and insightful discussions as well as valuable comments on this manuscript. N.P. wishes to thank his parents for financial support. \nAdded Note: Ref. [23] contains very relevant information about the five-dimensional case. \n(Kluwer, Dordrecht, 1998) \n- [15] G. T. Horowitz, gr-qc/0507080, to appear in Kerr spacetime: Rotating Black Holes , eds. S. Scott, M. Visser, and D. Wiltshire (Cambridge University Press).\n- [16] R. Emparan and R. C. Myers, JHEP 09 (2003) 025.\n- [17] V. Frolov and D. Stojkovi'c, Phys. Rev. D68 , 064011. (2003)\n- [18] R. Emparan and H. S. Reall, Phys. Rev. Lett. 88 , 101101. (2002)\n- [19] G. W. Gibbons, M.J. Perry and C.N. Pope, Class. Quant. Grav. 22 , 1503. (2005)\n- [20] B. M. N. Carter and I. P. Neupane, Phys. Rev. D72 , 043534. (2005).\n- [21] J. E. ˚ Aman, Manual for CLASSI: Classification Programs for Geometries in General Relativity , Department of Physics. Stockholm University, Technical Report, Provisional edition. Distributed with the sources for SHEEP and CLASSI. (2002)\n- [22] http://grtensor.phy.queensu.ca/, cited on October 9, 2005.\n- [23] G. Arcioni and E. Lozano-Tellechea, Phys. Rev. D72 , 104021. (2005)"}

2007ApJ...667..704V

Black Hole Spin and Galactic Morphology

2007-01-01

['black hole physics', 'cosmology theory', 'galaxies evolution', 'galaxies quasars', 'astrophysics']

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We investigate the conjecture by Sikora, Stawarz, and Lasota that the observed active galactic nuclei (AGNs) radio loudness bimodality can be explained by the morphology-related bimodality of black hole spin distribution in the centers of galaxies: central black holes (BHs) in giant elliptical galaxies may have (on average) much larger spins than black holes in spiral/disk galaxies. We study how accretion from a warped disk influences the evolution of black hole spins and conclude that within the cosmological framework, where the most massive BHs have grown in mass via merger-driven accretion, one indeed expects most supermassive black holes in elliptical galaxies to have on average higher spin than black holes in spiral galaxies, where random, small accretion episodes (e.g., tidally disrupted stars, accretion of molecular clouds) might have played a more important role.

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https://arxiv.org/pdf/0706.3900.pdf

{'BLACK-HOLE SPIN AND GALACTIC MORPHOLOGY': 'Marta Volonteri 1 , Marek Sikora 2,3 & Jean-Pierre Lasota 4,5 Draft version October 30, 2018', 'ABSTRACT': 'We investigate the conjecture by Sikora, Stawarz & Lasota (2007) that the observed AGN-radioloudness bimodality can be explained by the morphology - related bimodality of black-hole spin distribution in the centers of galaxies: central black holes in giant elliptical galaxies may have (on average) much larger spins than black holes in spiral/disc galaxies. We study how accretion from a warped disc influences the evolution of black hole spins and conclude that within the cosmological framework, where the most massive BHs have grown in mass via merger driven accretion, one indeed expects most supermassive black holes in elliptical galaxies to have on average higher spin than black holes in spiral galaxies, where random, small accretion episodes (e.g. tidally disrupted stars, accretion of molecular clouds) might have played a more important role. \nSubject headings: cosmology: theory - black holes - galaxies: evolution - quasars: general', '1. INTRODUCTION': "It has been known for many years that the radio loudness of AGN hosted by disc galaxies is on average three orders of magnitude lower than the radio loudness of AGN hosted by giant ellipticals (see Xu et al. 1999 and references therein). However, as as shown by HST observations, such a galaxy morphology - radio-loudness correspondence is not 'one-to-one': both radio-quiet and radio-loud very luminous quasars are hosted by giant ellipticals (Floyd et al. 2004). On the other hand the popular version of the so-called 'spin paradigm' asserts that powerful relativistic jets are produced in AGN with fast rotating black holes (Blandford 1990), implying that BHs rotate slowly in radio-quiet quasars, which represent the majority of quasars. However, such conjecture, at least in its basic interpretation, is in conflict with the high average BH spin in quasars deduced from the high average radiation efficiency of quasars using the 'So/suppressltan argument' (So/suppressltan 1982; Wang et al. 2006 and references therein). \nParallel studies of radio-emission from X-ray binaries showed that at high accretion rates production of jets is intermittent (Gallo et al. 2003) and that this intermittency can be related to transitions between two different accretion modes (Livio et al. 2003). This inspired Ulvestad & Ho (2001), Merloni et al. (2003), Nipoti et al. (2005), and Kording et al. (2006) to postulate the existence of a similar intermittency of jet production in quasars and formulate an 'accretion paradigm' according to which the radio-loudness is entirely related to the states of accretion discs. However, Sikora et al. (2007) found that on the radio-loudness - Eddington-ratio plane \nAGN form two parallel sequences that occurrence of cannot be explained by the 'accretion-paradigm' (see also Terashima & Wilson 2003; Chiaberge et al. 2005; and Panessa et al. 2007). Sikora et al. (2007) therefore proposed a revised version of the 'spin paradigm', suggesting that giant elliptical galaxies host, on average, black holes with spins larger than those hosted by spiral/disc galaxies. \nThis morphology-related radio dichotomy breaks down at high accretion rates where the dominant fraction of luminous quasars hosted by elliptical galaxies is radio quiet. This radio-quietness occurs in quasars with high spin values. In such systems with high accretion rates the intermittency is related to the conditions of production of collimated jets, in agreement with what is found in Xray binaries, and with the 'So/suppressltan argument'. It should be emphasized that even if the production of powerful relativistic jets is conditioned by the presence of fast rotating BHs, it also depends on the accretion rate and on the presence of disc MHD winds required to provide the initial collimation of the central Poynting flux dominated outflow. \nIn this article we will examine under which condition the cosmological evolution of BHs in galaxies may lead to low spins in disc galaxies and high spins in more massive ellipticals. \nElectronic address: [email protected], [email protected], [email protected] \n1 Department of Astronomy, University of Michigan, 500 Church Street, Ann Arbor, MI, USA \n- 2 Nicolaus Copernicus Astronomical Center, Bartycka 18, 00-716 Warszawa, Poland\n- 3 Department of Physics and Astronomy, University of Kentucky, Lexington, KY, USA\n- 4 Institut d'Astrophysique de Paris, UMR 7095 CNRS, Universit'e Pierre et Marie Curie, 98bis Bd Arago, 75014 Paris, France\n- 5 Astronomical Observatory, Jagiellonian University, ul. Orla 171, 30-244 Krak'ow, Poland \nTo put our investigation in the relevant context we will first recall why the value of black-hole's spin might be of fundamental importance for relativistic jet launching. Assuming that relativistic jets are powered by rotating black holes through the Blandford-Znajek mechanism, Blandford (1990) suggested that the efficiency of jet production is determined by the dimentionless black hole spin, ˆ a ≡ J h /J max = c J h /GM 2 BH , where J h is the angular momentum of the black hole. If true, this could explain the very wide range of radio-loudness of AGN that look very similar in many other aspects by attributing it to a corresponding black-hole spin distribution. This so called 'spin paradigm' was explored by Wilson & Colbert (1995), who assumed that the black-hole spin evolution is determined mainly by mergers. They claimed that mergers of black holes, following mergers of galaxies, \nlead to a broad, 'bottom-heavy' distribution of the spin, consistent with a distribution of quasar radio-loudness. However, this claim was challenged by Hughes & Blandford (2003), who showed that mergers cannot produce the required fraction of black holes with high spins and concluded that accretion of matter is essential in determining black-hole spins. In this case, however, as noticed earlier by Moderski & Sikora (1996) and Moderski, Sikora & Lasota (1998; hereafter MSL), one encounters the difficulty of maintaining a sufficient number of black holes at the required low spin, the spin-up by accretion discs being so efficient. MSL could match the distribution of radio-loudness with the spin distribution only by feeding holes with very small randomly oriented accretion events, i.e. by accretion events forming co-rotating and counter-rotating discs with the same probability. \nMSL also addressed the problem of the spin overflipping due to the Bardeen-Petterson effect. When an accretion disc does not lie in the equatorial plane of the BH, that is, when the angular momentum of the accretion disc is misaligned with respect to the direction of J h , the dragging of inertial frame causes a precession that twists the disc plane due to the coupling of J h with the angular momentum of matter in the disc. The torque tends to align the angular momentum of the matter in the disc with that of the black hole, causing thus the inclination angle between the angular momentum vectors to decrease with decreasing distance from the BH, forcing the inner parts of the accretion disc to rotate in the equatorial plane of the BH (Bardeen & Petterson 1975). Sustained accretion from a twisted disc would align the BH spin (and the innermost equatorial disc) with the angular momentum vector of the disc at large radii (Scheuer & Feiler 1996). If the disc was initially counter-rotating with respect to the BH, a complete overflip would eventually occur, and then accretion of co-rotating material would act to spin up the BH (Bardeen 1970). \nMSL concluded that the Bardeen-Petterson effect can be neglected because the alignment time (10 7 years; Rees 1978) is longer than the duration of a single accretion event. Later, however, a series of papers put into doubt the validity of Rees's estimate (Scheuer & Feiler 1996, Natarayan & Pringle 1998). This framework was recently investigated by Volonteri et al. (2005) who argue that the lifetime of quasars is long enough that angular momentum coupling between black holes and accretion discs through the Bardeen-Petterson effect effectively forces the innermost region of accretion discs to align with black-hole spins (possibly through spin flips), and hence all AGN black-holes should have large spins. \nRecently King et al. (2005) pointed out that under some conditions the alignment torque can lead to disc-hole counter-alignment reactivating the debate. The counter-alignment process was numerically simulated by Lodato & Pringle (2006). \nWe here re-analyze the alignment problem in view of all these latest results. We explore what are the likely outcomes of accretion episodes that grow black holes along the cosmic history, and determine under which conditions black holes in disc galaxies end up having low spins.", '2. ASSUMPTIONS': "The dynamics involving a spinning black hole accreting matter from a thin disc whose angular momentum is \nnot aligned with the spin axis has been studied in a number of papers (e.g., Papaloizou & Pringle 1983; Pringle 1992; Scheuer & Feiler 1996; Natarajan & Pringle 1998). A misaligned disc is subject to the Lense-Thirring precession, which tends to align the inner parts of the disc with the the angular momentum of the black hole. The outer regions of the disc are initially inclined with respect to the hole's axis, with a transition between alignment and misalignment occurring in between at the so-called 'warp radius' (see below). The direction of the angular momentum of the infalling material changes direction as it passes through the warp. In our calculations we will assume that the black-hole spin evolution is determined by accretion only. Volonteri et al. (2005) have shown that BH mergers play a sub-dominant role in the global spin evolution.", '2.1. Viscosities': "Despite many efforts the problem of warped discs, especially in the non-linear regime, has yet to be solved. Therefore the characteristic scales of the problem are subject to several uncertainties. The main quantity of interest is the 'warp radius' R w defined as the radius at which the timescale for radial diffusion of the warp is comparable to the local dragging-of-inertial frame ('Lense-Thirring' in the weak-field approximation) time (ˆ acR 2 s /R 3 ) -1 (Wilkins 1972). The timescale for the warp radial diffusion can be written as \nt w ≈ R 2 w ν 2 (1) \nwhere ν 2 is a viscosity characterizing the warp propagation which can be different from the accretion driving viscosity, ν 1 , which is responsible for the transfer of the component of the angular momentum parallel to the spin of the disc. The relation between ν 1 and ν 2 is the main uncertainty of the problem, assuming of course that such two-viscosity description is adequate at all. Describing ν 1 by the Shakura-Sunyaev parameter α one can show (Papaloizou & Pringle 1983) that the regime in which H/R < α /lessmuch 1 ( H being the disc thickness) one has ν 1 /ν 2 ≈ α 2 . In such a case the accretion time t acc ≈ R w /ν 1 would be much longer than the warp diffusion time t w . \nHowever, such a description can be questioned on several grounds. First, is the α /lessmuch 1 appropriate for highrate accretion onto AGN black-holes? There are no reliable estimates of this parameter for AGN but outbursts of LMXBs suggest that in hot accretion discs α ∼ > 0 . 1 (see e.g. Dubus et al. 2001). In such a case ν 1 is comparable to ν 2 (Kumar & Pringle 1985). Second, even if the two viscosities are different is t w the relevant time for black-hole re-alignment? This is not clear since this latter process is very dissipative and could be controlled by accretion and not warp propagation.", '2.2. Relevant radii': 'During the accretion process, the angular momentum of the disc at the warp location sums up with that of the black hole, so the angle between the angular momentum of the outer disc and the BH spin changes. King et al. (2005) suggest that the condition of alignment or counter-alignment can be expressed as a function of the \nangular momenta of the hole and of the disc: J h and J d . The counter-alignment condition depends on the ratio 0 . 5 J d /J h , to be compared with the cosine of the inclination angle, φ . If cos φ < -0 . 5 J d /J h , the counteralignment condition is satisfied. King et al. (2005), however, leave the definition of J d vague, indeed they suggest that J d is the angular momentum of the disc inside a certain radius R J such that J d ( R J ) = J h . First, this is not a useful operational definition, because in this case cos φ < -0 . 5 J d /J h = -0 . 5, is a static condition, which does not depend on the properties of the black hole or of the accretion disc. Second, matter contained within radius R J cannot transfer all its angular momentum to R w but only a fraction √ R w /R J J d ( R J ). Therefore a more natural radial scale in the problem is the warp radius R w and in the following we will assume that J d = J d ( R w ) (note that we share this choice with Lodato & Pringle 2006).', '3. METHOD': "We explore the dependence of the alignment timescale in a Shakura-Sunyaev disc on: viscosity ν 2 /ν 1 , black hole mass M BH , misalignment angle, Eddington ratio, accreted mass m . Some articles on this subject use the solution of Collin-Souffrin & Dumont (1990), however in the view of the basic uncertainties of the problem we decided to use the less refined solution of Shakura & Sunyaev 1973. \nAssuming a Shakura-Sunyaev disc ('middle region'), the warp radius (in units of the Schwarzschild radius R S ) can be expressed as: \nR w R s = 3 . 6 × 10 3 ˆ a 5 / 8 ( M BH 10 8 M /circledot ) 1 / 8 × f -1 / 4 Edd ( ν 2 ν 1 ) -5 / 8 α -1 / 2 . (2) \nwhere f Edd ˙ Mc 2 /L Edd . \nWe can then define the accretion timescale: \n≡ \nt acc = R 2 w ν 1 = 3 × 10 6 yr α -3 / 2 ( ν 2 ν 1 ) -7 / 8 ˆ a 7 / 8 × f -3 / 4 Edd ( M BH 10 8 M /circledot ) 11 / 8 (3) \n(where ν 1 = αH 2 Ω K was used), and the timescale for warp propagation: \nt w = ν 1 ν 2 t acc . (4) \nThe ratio of angular momenta of the disc at R w , defining M d ( R w ) = ˙ Mt acc ( R w ), and of the black hole is: \nJ d J h = M d ˆ aM BH ( R w R s ) 1 / 2 = 2 × 10 -9 f Edd × ( t acc 1 y )( R w R s ) 1 / 2 ˆ a -1 . (5) \nFig. 1.Evolution of the misalignment angle between the angular momentum vector of the outer accretion disc and the BH spin (top panel), and of the magnitude of the BH spin (bottom panel), due to accretion of aligned material (spin-up) or counter-aligned material (spin-down). The initial BH mass is M BH0 = 5 × 10 6 M /circledot , the initial spin ˆ a = 0 . 5, ν 2 = ν 1 , α = 0 . 1, and the accretion rate is f Edd = 1. The four curves show different initial misalignment angles (top to bottom: φ = 3 , 2 , 1 , 0 . 5 radians. ) \n<!-- image -->", '4.1. Single accretion episodes': "We first discuss the behavior of the disc+BH system during the alignment process. The scheme we adopt is as follows: \n- 1. for a BH with initial mass M BH0 determine the initial conditions: warp radius, R w , timescale for warp propagation t w = R 2 w /ν 2 ( ν 2 is chosen either coincident with ν 1 , or ν 2 = ν 1 /α 2 ), accretion timescale for material at the warp radius, t acc = R 2 w /ν 1 , angular momentum of the hole and of the disc at R w , J h and J d .\n- 2. using the King et al. (2005) condition for misalignment determine if the BH and the inner disc are aligned or counter-aligned (counter-aligned if cos φ < -0 . 5 J d /J h ).\n- 3. over timesteps ∆ t = t acc ( R w ) compute the necessary quantities at the end of every step: increase in black hole mass due to accretion, new BH spin (following Bardeen 1970; where the counter-alignment or alignment conditions are taken into consideration, i.e., BHs can be spun down or up), new J h , new R w , new J d , new angle between J h and J d (vectorial sum). In every timestep the disc within R w is consumed. \nFigures 1 and 2 give examples of single accretion episodes for different initial angles between the angular momentum vector of the outer (not warped) portions of the accretion disc and the black hole spin. They show the evolution of the spin magnitude and inclination as computed for ν 2 = ν 1 (Fig.1) and ν 2 /ν 1 = 1 /α 2 (Fig.2). \nAs we can see, the alignment timescale is basically independent of the misalignment angle. To modify significantly the BH spin, one has to bring to R w an amount of angular momentum comparable to J h . Therefore, if J h > J d ( R w ), then: \nt align /similarequal J h J d ( R w ) t acc ( R w ) (6) \nSince J h ∝ ˆ aM BH √ R S , J d ( R w ) ∝ M d ( R w ) √ R w , and M d ( R w ) = ˙ Mt acc ( R w ), Equation 6 gives (Rees 1978): \nt align /similarequal ˆ a M BH ˙ M ( R s R w ) 1 / 2 . (7) \nDefining the mass accreted during t align as m align = t align ˙ M , one gets: \nm align /similarequal M BH ˆ a ( R s R w ) 1 / 2 . (8) \nTherefore a series of many randomly oriented accretion events with accreted mass m /lessmuch m align should result in black-hole's spin oscillating around zero. For the opposite case of m /greatermuch m align the black hole will be spun-up to large positive spins; for m ∼ M BH the hole will be spun-up to ˆ a ∼ 1. Let us notice finally that since it is reasonable to assume that m align /lessmuch M BH the existence of AGN hosting black holes with ˆ a ∼ -1 is rather unlikely. \nFig. 2.As in Figure 1, but ν 2 /ν 1 = 1 /α 2 , α = 0 . 1. \n<!-- image --> \nOur calculations show that it is difficult to avoid high spin for the most massive black holes. For large BH masses, the accretion timescale is very long, and consequently the warp radius and the angular momentum within the warp radius, J d , are large. If J d > 2 J h , then the value | 0 . 5 J d /J h | > 1 and the counter-alignment condition cannot be satisfied for any angle. This condition corresponds to: \nM BH , max > 6 . 2 × 10 8 M /circledot α 28 / 23 × \n( ν 2 ν 1 ) 19 / 23 f -2 / 23 Edd ˆ a -3 / 23 . (9) \nIf ν 2 = ν 1 , M BH , max is of order 10 7 -10 8 M /circledot for most sensible choices of α and f Edd . In this case the most massive black holes force accretion to occur from aligned discs, therefore causing a systematic spin-up. If the warp propagation is instead better described by ν 2 = ν 1 /α 2 , M BH , max becomes exceedingly high and large accretion events can still act to spin down the black hole, provided m<m align . \nThe condition expressed in Equation 9 is true only if there is enough mass to fill the warp radius, that is if the total mass of the disc is larger than: \nM d , min > 6 . 5 × 10 5 M /circledot ( M BH 10 8 M /circledot ) 19 / 8 α -3 / 2 × f 1 / 4 Edd ˆ a 7 / 8 ( ν 2 ν 1 ) -7 / 8 . (10) \nIf the mass to be accreted by the BH in an episode is smaller than M d , min , then J d /lessmuch J h , and both alignment or counter-alignment can happen.", '4.2. Multi-accretion events': 'We then run a series of simulations in which we explore different parameters. We start with a small BH, M BH0 = 10 5 M /circledot , and have it grown by a series of accretion episodes. The accreted mass m is randomly extracted from only one of two different distributions: (1) a distribution flat in m , with m < 0 . 1 M BH , (2) a distribution flat in m , with m < 0 . 01 M BH . The angle φ is extracted from a flat distribution 0 < φ < π at the beginning of every accretion episode. Every simulation is composed by a large number of accretion episodes, until one of the following conditions are met: M BH > 10 9 M /circledot or t tot > 10 10 years, that is the total simulation time (total time a BH accretes to grow from the initial M BH0 = 10 5 M /circledot mass to its final mass) is shorter than the age of the universe. During an episode where the BH accretes counter-aligned material, the BH is spun down. If the black hole is spun-down until its spin is zero, any subsequently accreted matter acts to spin the BH up again, although the direction of the spin axis is now reversed and aligned with the angular momentum of the disc. \nWe run 100 simulations for every parameter sets choice, and we trace the spins at the end of every accretion episode, for all the accretion episodes in the simulations. \nWe have explored a wide range of parameters, and we summarize here our findings. We have varied the accretion rate, from f Edd = 0 . 05 to f Edd = 1. If the accretion rate is low, the main caveat is that black holes do not reach high masses within the Hubble time, however, the efficiency of alignment is not strongly dependent on f Edd (see Equation 8). \nWe have considered different black hole spins at birth, from ˆ a = 10 -3 to ˆ a = 0 . 9. After the BHs have changed their initial mass by about one order of magnitude, the distributions are indistinguishable from each other. During the first few e-foldings, however, the spin distribution is peaked around the black hole spin at birth. \nFig. 3.Distribution of BH spins in different mass ranges. The accreted mass m is randomly extracted from a distribution flat in m , with m < 0 . 1 M BH . Initial mass M BH0 = 10 5 M /circledot , initial spin ˆ a = 10 -3 , α = 0 . 03, f Edd = 0 . 1. Solid line: ν 2 = ν 1 , dashed line: ν 2 = ν 1 /α 2 . \n<!-- image --> \nWe have also varied the viscosity parameter α (see section 2.1), and the relation between the viscosity characterizing the warp propagation ( ν 2 ) with respect to the viscosity responsible for the transfer of the component of the angular momentum parallel to the spin of the disc ( ν 1 ). When α is varied, but ν 2 /ν 1 is kept fixed, the differences between the spin distributions are not large. A smaller α skews the distribution towards higher spins (cf Equations 2, 8). \nFig. 4.As in Figure 3, but with m randomly extracted from a distribution flat in m , with m< 0 . 01 M BH . \n<!-- image --> \nOne of the main parameters influencing the spin distribution is the relation between ν 2 and ν 1 . If ν 2 /ν 1 = 1, after the BHs have reached m BH ∼ 10 6 M /circledot , the spin dis- \nion is dominated by rapidly spinning black holes. Equation 9 also shows that the most massive black holes force accretion to occur from aligned discs, therefore causing a systematic spin-up, unless very small parcels of material are accreted at every single accretion episode. If the warp propagation is instead better described by a high ν 2 = ν 1 /α 2 , M BH , max becomes exceedingly high, and all sorts of accretion events can still act to spin down the black hole, provided m<m align . \nIn fact, we confirm the results by MSL, that is that the main parameter governing the distribution of BH spins is the amount of material accreted in a single accretion episode. This result is clear from Figures 3, 4 which refer to different choices for the distributions of m . Only if the mass accreted in one episode is smaller than m align , the distribution of black hole spins can remain flat. \nIn the next section we discuss the likelihood of different m distributions in the light of evolutionary models for the BH population in a hierarchical cosmology.', '4.3. Merger driven accretion': "We first present an evolutionary track for BH spins, where a BH grows by a sequence of randomly oriented accretion episodes in a merger driven scenario. The BH mass evolutionary tracks are extracted from semianalytical simulations of BH growth that have been shown to reproduce the evolution of the BH population as traced by the luminosity function of quasars (Marulli et al. 2006, 2007; Volonteri, Salvaterra & Haardt 2006). We focus here on two specific tracks, for a putative BH in an 'elliptical' (E) galaxy, and one in a putative 'disc' (D) galaxy (Fig 5). Here the morphological classification is purely based on the frequency of major mergers, i.e., mergers between comparable mass galaxy systems which are believed to contribute mainly to the spherical component of galaxies. A BH hosted in an 'elliptical' galaxy should have experienced a major accretion event in connection with the last high-redshift major merger, which formed the elliptical galaxy as we see it now. Afterwards, the galaxy (BH) has not grown in mass due to merger driven star formation (accretion). In the case of the BH hosted in a 'disc' galaxy, a small number of minor mergers might have happened after the last major mergers. These minor mergers are believed to be responsible for re-building the galaxy disc. In conjunction with these minor mergers, a small infall of gas can produce a relatively minor accretion episode onto the BH as well. \nAlong the evolutionary tracks for our BHs, we trace the joint evolution of accretion onto the BH, the dynamics of the accretion disc, and the consequences on the spin. The scheme we adopt is similar to the one described in Section 4.1. During an episode where the BH accretes counter-aligned material, the BH is spun down, until the spin is zero, and subsequently any accreted matter acts to spin the BH up again, although the direction of the spin axis is now reversed and aligned with the angular momentum of the disc. \nIn case 'E', the last accretion episode caused a large increase in the BH mass, following the major merger which created the elliptical itself (Hopkins & Hernquist 2006). During this episode the spin increased significantly as well, up to very high values. Let us remind here that in the extreme event of a maximally-rotating hole spun down by retrograde accretion, the BH is braked af- \nFig. 6.Spin of BHs at z = 0 in 'elliptical' (upper panel) or 'disc' (lower panel) galaxies. \n<!-- image --> \nFig. 5.Growth of BHs in putative 'elliptical' (upper panel) or 'disc' (lower panel) galaxy ( only merger driven accretion events are considered). The initial spin ˆ a = 10 -3 , ν 2 = ν 1 /α 2 , α = 0 . 03, and the accretion rate is f Edd = 0 . 1. \n<!-- image --> \nter its mass has increased by the factor √ 3 / 2. Any mass accreted afterwards spins up the black hole, and if the final mass increase is by a factor 3, the BH will end up maximally-rotating again. \nLet us now consider a 'D'-type evolution. A BH would experience a series of small accretion episodes (triggered possibly by minor mergers), extending for a longer period of times. If these episodes are uncorrelated, that is if the inflow during a given episode is not aligned with the orientation of the spin of the BH, the randomization of the angle φ over the (few) accretion episodes tends to spin down the BH. \nWe run a statistical sample of 'E' and 'D' track, for BHs hosted in large (i.e., Andromeda-size systems) galaxies. We find that, if only the merger driven evolution is taken into account, BHs in 'elliptical' galaxies are left with large spins. BHs in 'disc' galaxies have, on average, slightly lower spins, however the distribution is still peaked at large values (Figure 6).", '4.4. Short-lived accretion events': "Even minor mergers tend to trigger inflows of matter which are too large to lead to the series of short lived accretion events necessary to leave BHs with small spins (cfr. the discussion in MSL). Moreover, several observations suggest that single accretion events last /similarequal 10 5 years in Seyfert galaxies, while the total activity lifetime (based on the fraction of disc galaxies that are Seyfert) is 10 8 -10 9 years (e.g., Kharb et al. 2006; Ho et al. 1997). This suggests that accretion events are very small and very 'compact'. \nA type of random event which leads to short-lived accretion episodes is the tidal disruption of stars. One expects discs formed by stellar debris to form with a random orientation. Stellar disruptions would therefore contribute to the spin-down of BHs. Let us consider the maximal influence that feeding via tidal disruption of stars can have on spinning down a BH. The number of \ntidal disruptions of solar type stars in an isothermal cusp per billion years can be written as: \n. \nN ∗ = 4 × 10 5 ( σ 60 km / s )( M BH 10 6 M /circledot ) -1 (11) \nAssuming that BH masses scale with the velocity dispersion, σ , of the galaxy (we adopt here the Tremaine et al. 2002 scaling), we can derive the relative mass increase for a BH in 1 billion years: \nM ∗ M BH = 0 . 37 ( M BH 10 6 M /circledot ) -9 / 8 . (12) \nThe maximal level of spin down would occur assuming that all the tidal disruption events form counterrotating discs, leading to retrograde accretion (note that the mass of the debris disc is much smaller than M d , min , cfr. Eq. 10, so that counter-alignment is allowed for any BH mass). Eq. 12 shows that a small (say 10 6 M /circledot ) BH starting at ˆ a = 0 . 998 would be spun down completely, on the other hand the spin of a larger (say 10 7 M /circledot ) BH would not be changed drastically. \nEarly type discs typically host faint bulges characterized by steep density cusps, both inside (Bahcall & Wolf 1976; Merritt & Szell 2006) and outside (Faber et al. 1997) the sphere of influence of the BH. In this environment, the rate of stars which are tidally disrupted by BHs (Hills 1975; Rees 1988) less massive than 10 8 M /circledot 6 is non negligible (Milosavljevic et al. 2006). Since in elliptical galaxies the central relaxation timescale is typically longer than the Hubble time, and the central density profile often displays a shallow core, tidal disruption of stars is unlikely to play a dominant role. \nAn additional feeding mechanism might be at work in gas-rich galaxies with active star formation. Compact self-gravitating cores of molecular clouds (MC) can occasionally reach subparsec regions, and may do it with \nFig. 7.Evolution of BH spins due to accretion of molecular clouds cores. We assume a lognormal distribution for the mass function of molecular clouds (peaked at log( M MC / M /circledot ) = 4, with a dispersion of 0.75. The initial spin of the BHs is 0.998. Upper curve: the initial BH mass is 10 7 M /circledot , lower curve: the initial BH mass is 10 8 M /circledot \n<!-- image --> \nrandom directions provided that the galactic disc is much thicker than the spatial scale of the BH gravity domination region (Shlosman, private communication). Although the rate of such events is uncertain, we can adopt the estimates of Kharb et al. (2006), and assume that about 10 4 of such events happen. We can further assume a lognormal distribution for the mass function of MC close to galaxy centers (based on the Milky Way case, e.g., Perets, Hopman & Alexander 2006). We do not distinguish here giant MC and clumps, and, for illustrative purpose we assume a single lognormal distribution peaked at log( M MC / M /circledot ) = 4, with a dispersion of 0.75. \nFig 7 shows the possible effect that accretion of molecular clouds can have on spinning BHs. The result is, on the whole, similar to that produced by minor mergers of black holes (Hughes & Blandford 2003), that is a spin down in a random walk fashion. The larger the BH mass, the more effective the spin down. \nIn a gas-poor elliptical galaxy, however, substantial populations of molecular clouds are lacking (e.g. Sage et al. 2007), thus hampering the latter mechanism for short lived accretion events proposed.", '5. DISCUSSION AND CONCLUSIONS': "We have investigated the evolution of BH spins driven by accretion from discs with angular momentum vectors that can be misaligned with respect to the spin axis. We have assumed that accretion discs can be described by Shakura-Sunyaev α -discs, and that when the angular momentum of the accretion disc is not aligned with the spin of the BHs, the disc itself is warped. The inner portions of the discs experience Lense-Thirring torque, which tends to align the inner parts of the disc. The timescale of the Lense-Thirring precession increases faster with distance from a BH than the timescale of warp propagation, and they equate at the so-called 'warp radius', where a transition occurs from alignment to \nmisalignment. King et al. (2005) pointed out that for highly misaligned discs, counter-alignment, rather than alignment can occur. The co- or counter-alignment of the accretion discs has important consequences on the spin of BHs. A black hole accreting from prograde orbits (i.e., alignment case) is spun up by the coupling between the angular momentum of the infalling material and its spin (Bardeen 1970). If, instead, an initially spinning hole accretes from retrograde orbits (i.e., counter-alignment case), it is spun down. An initially non-rotating BH gets spun up to a maximally-rotating state (ˆ a = 1) after reaching the mass M BH = √ 6 M BH0 . A maximally-rotating hole (ˆ a = 1) gets spun down by retrograde accretion to ˆ a = 0 after reaching the mass M BH = √ 3 / 2 M BH0 . A 180 · flip of the spin of an extreme-Kerr hole will occur after M BH = 3 M BH0 . \nIt is therefore necessary that accretion episodes increase the mass of a BH by less than M BH = √ 6 M BH0 , in order to keep the spin at low values, if accretion preferentially occurs from prograde orbits. Natarajan & Pringle (1998) suggested that accretion indeed occurs from aligned discs (i.e. prograde orbits), as the timescale for disc alignment is much shorter than the timescale of the BH mass growth by M BH = √ 6 M BH0 . King et al. (2005) suggested, however, that when the initial misalignment angle is large and m is sufficiently small, counter-alignment, rather than alignment, occurs and BHs can be spun down in a large fraction of the accretion episodes. \nWe have quantified here the likelihood of counteralignment and spin down as claimed by King et al. (2005). We identify two main parameters influencing the distribution of BH spins: the distribution of the accreted mass, m , with respect to the mass of the BH, and the relation between ν 2 and ν 1 , where ν 2 is the viscosity characterizing the warp propagation, and ν 1 , which is responsible for the transfer of the component of the angular momentum parallel to the spin of the disc. ν 2 can in principle differ from ν 1 . \nIf the accreted mass, m , is much smaller than the mass of the BH (e.g., m< 0 . 01 M BH ), the distribution of black hole spins is flat, as the timescale for spin overflipping due to the Bardeen-Petterson effect is longer than the timescale to accrete the whole m . If instead m /similarequal M BH , BHs can align with the angular momentum of the accretion disc, and accrete enough mass to be spun up. In this case the distribution of BH spins is dominated by rapidly rotating systems. \nUnderstanding if the description of the warp propagation is correctly described by a different viscosity with respect to the one responsible for the radial propagation of the angular momentum is beyond the scope of this work. We have therefore explored a wide range of possible viscosities, and we simply report here our results. If ν 2 /ν 1 = 1 the timescale for alignment is short, and the spin of a BH increases rapidly. If the warp propagation is instead better described by a high ν 2 = ν 1 /α 2 , a substantial fraction of black holes of all masses can have small spins, provided m M BH0 . \nHowever, both semi-analytical models of the cosmic BH evolution (Volonteri et al. 2005) and simulations of merger driven accretion (di Matteo et al. 2005) show that most BHs increase their mass by an amount /greatermuch m align , \n/lessmuch \nif the evolution of the LF of quasars is kept as a constraint. These high m values are likely characteristic of the most luminous quasars and most massive black holes - especially at high redshift. We expect therefore that bright quasars at z > 3 have large spins, in contrast with the suggestion of King & Pringle (2006). High spins in bright quasars are also indicated by the high radiative efficiency of quasars, as deduced from observations applying the So/suppressltan argument (So/suppressltan 1982; Wang et al. 2006 and references therein). \nIf the mass of a BH need to reach 10 9 M /circledot by z = 3, or even more strikingly, by z = 6, so that they can represent the engines of quasars with luminosity L > 10 46 erg s -1 , BHs need to grow from typical seed masses (e.g. Madau & Rees 2001, Koushiappas et al. 2003, Begelman, Volonteri & Rees 2006, Lodato & Natarajan 2006) by at least 3-4 orders of magnitude in 10 8 -10 9 years. The necessity of long and continuous accretion episodes implies therefore that, for these BHs, m m align . \nSmaller BHs, powering low luminosity Active Galactic Nuclei, can instead grow by accreting smaller packets of material, such as tidally disrupted stars (for BHs with mass < 2 × 10 6 M /circledot , Milosavljevic et al. 2006), or possibly molecular clouds (Hopkins & Hernquist 2006). For these black holes the spin distribution is more probably flat, or skewed towards low values. This latter result is in agreement with Sikora, Stawarz & Lasota (2006), who find that disc galaxies tend to be weaker radio sources with respect to elliptical hosts. In the hierarchical framework we might expect that the BH hosted by an elliptical galaxy had, as last major accretion episode, a large increase in its mass following the major merger which created the elliptical itself (Hopkins & Hernquist 2006). During this episode the spin increased significantly as well, possibly up to very high values. Subsequently the black hole might have grown by swallowing the occasional molecular cloud, or by tidally disrupting stars. If the total contribution of these random episodes represents a small fraction of the BH mass, the spin is, however, kept at high values. \nBlack holes in spiral galaxies, on the other hand, probably had their last major merger (i.e., last major accretion episode), if any, at high redshift, so that enough time elapsed for the galaxy disc to reform. Most of the latest growth of the BH should have happened through minor events, which have likely contributed to the BH spin down. \nOur results are supported also by the recent finding by Capetti & Balmaverde (2006; 2007) that radio bimodality correlates with bimodality of stellar brightness profiles in galactic nuclei. The inner regions of radio loud galaxies display shallow cores (star deficient). Cores, in turn, are preferentially reside in giant ellipticals (see Lauer et al. 2007 and references therein). Radio quiet galaxies, including nearby low luminosity Seyferts, have instead power-law (cuspy) brightness profiles and preferentially reside in SO and spiral galaxies. \nHence, noting that core nuclei result from merging BHs following galaxy mergers (Ebisuzaki et al. 1991, Milosavljevic & Merritt 2001, Milosavljevic et al. 2002, Ravindranath et al. 2002, Volonteri et al. 2003), Balmaverde & Capetti's discovery is consistent with our conjecture that spin bimodality is determined by diverse evolutionary tracks of BH spins in disc galaxies (random small mass accretion events) and giant elliptical galaxies (massive accretion events which follow galaxy mergers). Both tidal disruption of stars, and accretion of gaseous clouds is unlikely in shallow, stellar dominated galaxy cores. Therefore it is conceivable that the observed morphology-related bimodality of AGN radioloudness results from bimodality of central black-holes spin distribution. \nM.S. thanks the Dept. of Physics and Astronomy, University of Kentucky at Lexington, for hospitality during his stay. This research was supported in part by the National Science Foundation under Grant No. PHY9907949.", 'REFERENCES': "Kharb, P., O'Dea, C. P., Baum, S. A., Colbert, E. J. M., & Xu, C. 2006, ApJ, 652, 177 King, A. R. & Pringle J. E. 2006, MNRAS, in press King, A. R., Lubow, S. H., Ogilvie, G. I., & Pringle, J. E. 2005, MNRAS, 363, 49 \n- Kording, E.G., Jester, S., & Fender, R. 2006, MNRAS, 372, 1366 Koushiappas S. M., Bullock J. S., Dekel A., 2004, MNRAS, 354, 292\n- Kumar S. & Pringle, J. E. 1985, MNRAS, 213, 435\n- Lauer, T. R. et al. 2006, astro-ph/0606739\n- Lodato, G. & Pringle, J. E. 2006, MNRAS, 368, 1196\n- Madau, P., & Rees, M. 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2018PhRvD..97j4062P

Black hole shadow in an expanding universe with a cosmological constant

2018-01-01

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We analytically investigate the influence of a cosmic expansion on the shadow of the Schwarzschild black hole. We suppose that the expansion is driven by a cosmological constant only and use the Kottler (or Schwarzschild-de Sitter) spacetime as a model for a Schwarzschild black hole embedded in a de Sitter universe. We calculate the angular radius of the shadow for an observer who is comoving with the cosmic expansion. It is found that the angular radius of the shadow shrinks to a nonzero finite value if the comoving observer approaches infinity.

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https://arxiv.org/pdf/1804.04898.pdf

{'Black hole shadow in an expanding universe with a cosmological constant': 'Volker Perlick, 1, ∗ Oleg Yu. Tsupko, 2, † and Gennady S. Bisnovatyi-Kogan 2, 3, ‡ \n1 ZARM, University of Bremen, 28359 Bremen, Germany \n2 Space Research Institute of Russian Academy of Sciences, Profsoyuznaya 84/32, Moscow 117997, Russia 3 National Research Nuclear University MEPhI (Moscow Engineering Physics Institute), Kashirskoe Shosse 31, Moscow 115409, Russia \n(Dated: April 16, 2018) \nWe analytically investigate the influence of a cosmic expansion on the shadow of the Schwarzschild black hole. We suppose that the expansion is driven by a cosmological constant only and use the Kottler (or Schwarzschild-deSitter) spacetime as a model for a Schwarzschild black hole embedded in a deSitter universe. We calculate the angular radius of the shadow for an observer who is comoving with the cosmic expansion. It is found that the angular radius of the shadow shrinks to a non-zero finite value if the comoving observer approaches infinity. \nPACS numbers: 04.20.-q - 98.62.Sb - 98.62.Mw - 98.35.Jk', 'I. INTRODUCTION': "In recent years strong evidence for the existence of supermassive black holes at the centers of most galaxies has been accumulated. According to theory, an observer should see such a black hole as a dark disk, known as the 'shadow' of the black hole, in the sky against a backdrop of light sources. Attempts to actually observing the shadow of the black-hole candidates at the center of our own galaxy and at the center of M87 are under way, see the homepages of the Event Horizon Telescope (http://eventhorizontelescope.org) and of the BlackHoleCam (http://blackholecam.org). \nFor the simplest case of a non-rotating black hole, the shadow is a circular disk in the sky. If the black hole is uncharged, it is to be modelled by the Schwarzschild metric. For a static observer in the spacetime of a Schwarzschild black hole, the angular radius of the shadow was calculated in a seminal paper by Synge [1]. (Synge calculated what he called the 'escape cone' of light which is just the complement in the sky of what we now call the shadow.) For a rotating black hole, the shadow is no longer circular but rather flattened on one side, as a consequence of the 'dragging' of lightlike geodesics by the black hole. The shape of the shadow of a Kerr black hole for a stationary observer at a large distance was first calculated by Bardeen [2]. More generally, an analytical formula for the shape and the size of the shadow of a black hole of the Pleba'nski-Demia'nski class, for an observer anywhere in the domain of outer communication, was derived by Grenzebach et al.[3, 4]. In this paper the observer's fourvelocity was assumed to be a linear combination of ∂ t and ∂ ϕ and in the plane spanned by the two principal null directions; with this result at hand, the shadow can then be calculated for observers with any other four-velocities with the help of the standard aberration formula, see \nGrenzebach [5] for details. For the case of the Kerr metric, which is contained as a special case in the work by Grenzebach et al., Tsupko [6] worked out an approximate formula that allows to extract the spin of the black hole from the shape of the shadow. \nIn all these works, the black hole is assumed to be eternal, i.e, the spacetime is assumed to be time independent. Then, of course, a static or stationary observer will see a time-independent shadow. Actually, we believe that we live in an expanding universe. This gives rise to the question of how the shadow depends on time. Also, in an expanding universe the dependence of the shadow on the momentary position of the observer will no longer be given by the formulas for a static or stationary black hole. Of course, for the black-hole candidates at the center of our own galaxy and at the centers of nearby galaxies the effect of the cosmological expansion is tiny. However, for galaxies at a larger distance the influence on the angular diameter of the shadow may be considerable. In any case, calculating this influence is an interesting question from a conceptual point of view. This is the purpose of the present paper. We restrict to the simplest model of a black hole in an expanding universe, viz. to the Kottler spacetime (also known as the Schwarzschild-deSitter spacetime). This spacetime, which was found by Kottler [7] in 1918, describes a Schwarzschild-like (i.e., nonrotating and uncharged) black hole embedded in a deSitter universe. More precisely, the Kottler metric depends on two parameters, m and Λ, both of which are assumed to be positive with 9Λ m 2 < 1. It is a spherically symmetric solution of Einstein's field equation for vacuum with a cosmological constant. Near the center the spacetime geometry is similar to a Schwarzschild black hole with mass parameter m , and far away from the center it is similar to a deSitter universe with cosmological constant Λ. We admit that, according to the concordance model of cosmology, the deSitter universe is a good model only for the late stage of our universe, whereas for the present and earlier stages of our universe the influence of matter cannot be neglected. Nonetheless, we believe that it is instructive to consider this model because it allows to \ndetermine the influence of the cosmological expansion on the shadow for the case that this expansion is driven by the cosmological constant only. \nThe Kottler metric admits a timelike Killing vector field. Observers whose worldlines are integral curves of this Killing vector field see a static (i.e., timeindependent) spacetime geometry. We refer to them as to the static observers in the Kottler spacetime. When we consider the Kottler spacetime as a model for a black hole embedded in an expanding universe, we are not interested in these static observers, but rather in observers that are comoving with the cosmic expansion. However, the existence of the static observers gives us a useful tool for calculations: We may first consider the shadow as it is seen by a static observer. This was calculated for the Schwarzschild black hole without a cosmological constant by Synge [1], as was already mentioned above, and generalized to the case of a Kottler black hole by Stuchl'ık and Hled'ık [8]. From these results we can then calculate the angular radius of the shadow for an observer that is comoving with the cosmic expansion by applying the standard aberration formula. \nIn this paper we want to concentrate on the influence of the cosmic expansion, as driven by the cosmological constant, on the shadow. Therefore, we simplify all other aspects as far as possible. In particular, we consider a black hole that is characterized by its mass only, i.e., it is nonspinning and carries no (electric, magnetic, gravitomagnetic, ... ) charges. It is certainly possible to consider, more generally, a Pleba'nski-Demia'nski black hole, which may be spinning and carrying various kinds of charges, and to transform the above-mentioned results of Grenzebach et al. [3, 4] with the help of the aberration formula to an observer that is comoving with the cosmic expansion. Then, however, it would be difficult to disentangle the influence of the various parameters on the result and to extract the effect of the Λ-driven expansion. Also, it would be possible to take the influence of a plasma onto the light rays into account. The shadow in a plasma for a static or stationary observer was calculated for nonrotating and rotating black holes by Perlick, Tsupko and Bisnovatyi-Kogan [9, 10], cf. [11]. Again, we will not do this because here we want to concentrate on the effect of the cosmic expansion driven by a cosmological constant. \nAs a starting point for our calculations we need the equation for lightlike geodesics in the Kottler spacetime, written in coordinates adapted to the static observers. It is well known that the set of solution curves of this differential equation is independent of Λ, see Islam [12]. It was widely believed that, as a consequence, Λ has no influence on the lensing features. However, it was realized by Rindler and Ishak [13] that this is not true: Although the coordinate representation of the lightlike geodesics is unaffected by Λ, the cosmological constant does influence the lensing features because it changes the angle measurements. Therefore it should not come as a surprise that also the angular radius of the shadow does depend on Λ. When changing to the observers that are \ncomoving with the cosmic expansion we have to apply the aberration formula. A detailed study of this formula in the Kottler spacetime was brought forward recently by Lebedev and Lake [14, 15] and we will comment on the relation of our work to theirs in an appendix. \nThe paper is organized as follows. In Section II we calculate the shadow in the Kottler spacetime for a static observer. The results are not new, but we have to repeat them here because we want to use them later. Section III contains the main results of this paper: Here we calculate the shadow in the Kottler spacetime as it is seen by an observer that is comoving with the cosmic expansion. An approximation for these results is given in Section IV for the case that the observer is far away from the black hole. We conclude with a discussion of our results in Section V. In an appendix we point out how our work is related to the above-mentioned work by Lebedev and Lake. - Throughout the paper, we use Einstein's summation convention for greek indices taking values 0,1,2,3. Our choice of signature is ( -, + , + , +).", 'II. SHADOW IN THE KOTTLER SPACETIME AS SEEN BY A STATIC OBSERVER': "The Kottler metric is the unique spherically symmetric solution to Einstein's vacuum field equation with a cosmological constant. In its standard form it reads \ng µν dx µ dx ν = -f ( r ) c 2 dt 2 + dr 2 f ( r ) + r 2 d Ω 2 (1) \nwhere \nf ( r ) = 1 -2 m r -Λ 3 r 2 , d Ω 2 = sin 2 ϑdϕ 2 + dϑ 2 . (2) \nm is the mass parameter, \nm = GM c 2 (3) \nwhere M is the mass of the central object and Λ is the cosmological constant. (As usual, G is Newton's gravitational constant and c is the vacuum speed of light). We assume throughout that \n0 < Λ < 1 9 m 2 . (4) \nThen the Kottler metric has two event horizons, given by the zeros of the function f ( r ), an inner one at a radius r H1 and an outer one at a radius r H2 where 2 m<r H1 < 3 m<r H2 < ∞ . The region between the two horizons is called the domain of outer communication because any two observers in this region may communicate with each other without being hindered by a horizon. In this region the function f ( r ) is positive, i.e., the vector field ∂ t is timelike. As a consequence, the integral curves of the vector field ∂ t may be interpreted as the worldlines \nof observers. Since ∂ t is a Killing vector field, these observers see a time-independent universe. As mentioned already in the introduction, we will refer to them as to the static observers in the Kottler spacetime. For the following it is crucial that the static observers exist only in the domain of outer communication. \nFIG. 1. (COLOR ONLINE) Carter-Penrose diagram of the Kottler spacetime. The picture shows only the part of spacetime that is of relevance to us: The domain of outer communication I , the black-hole region II and the region beyond the (future) cosmological horizon III . A signal (i.e., a future-oriented causal worldline) that starts somewhere in the domain of outer communication may do one of three things: (i) It may stay inside I forever, approaching future timelike infinity i + ; examples are the circular lightlike geodesics at r = 3 m . (ii) It may cross the black-hole horizon and end up in the singularity at r = 0; examples are the ingoing radial lightlike geodesics. (iii) It may cross the cosmological horizon and go to future null infinity I + ; examples are the outgoing radial lightlike geodesics. - The Carter-Penrose diagram of the (maximal) Kottler spacetime was first determined by Gibbons and Hawking [16]. \n<!-- image --> \nThe horizon at r = r H 1 consists of a future inner horizon that separates the domain of outer communication from a black-hole region and of a past inner horizon that separates it from a white-hole region. (For literature on white holes see e.g. [17-19].) Similarly, the horizon at r = r H 2 consists of a future outer horizon and a past outer horizon. In this paper we are interested in the shadow of the black hole. It is constructed under the assumption that there are light sources only in the domain of outer communication. As the light emitted from such a light source can never reach one of the two past horizons, the regions beyond the past horizons will be of no relevance for us. We will be concerned only with the domain of outer communication, tagged I in Fig. 1, and to the regions beyond the future horizons, tagged II and III in Fig. 1. We will refer to the future inner horizon as to the black-hole horizon and to the future outer horizon as to the (future) cosmological horizon . \nBefore introducing moving observers in the next section, we will now calculate the shadow as it is momentarily seen by a static observer at a spacetime point ( t O , r O , ϑ O = π/ 2 , ϕ O = 0) in the domain of outer com- \nication. Because of the symmetry, it is no restriction to place the observer in the equatorial plane and it suffices to consider lightlike geodesics in the equatorial plane. Geodesics in the equatorial plane derive from the Lagrangian \nL ( x, ˙ x ) = 1 2 ( -f ( r ) c 2 ˙ t 2 + ˙ r 2 f ( r ) + r 2 ˙ ϕ 2 ) . (5) \nThe t and ϕ components of the Euler-Lagrange equation give us two constants of motion, \nE = f ( r ) c 2 ˙ t , L = r 2 ˙ ϕ. (6) \nFor lightlike geodesics we have \n-f ( r ) c 2 ˙ t 2 + ˙ r 2 f ( r ) + r 2 ˙ ϕ 2 = 0 . (7) \nSolving for ˙ r 2 / ˙ ϕ 2 = ( dr/dϕ ) 2 and inserting (6) yields the orbit equation for lightlike geodesics, \n( dr dϕ ) 2 = r 4 ( E 2 c 2 L 2 + Λ 3 -1 r 2 + 2 m r 3 ) . (8) \nWe see that Λ can be absorbed into a new constant of motion C = E 2 / ( c 2 L 2 ) + Λ / 3, i.e., that the set of all lightlike geodesics is independent of Λ in the chosen coordinate representation. This, however, does not mean that Λ has no influence on the lensing features because angle measurements do depend on Λ, see Rindler and Ishak [13]. \nBy evaluating the equations dr/dϕ = 0 and d 2 r/dϕ 2 = 0 we find that there is a circular lightlike geodesic at radius r = 3 m and that the constants of motion for this circular light ray satisfy \nE 2 c 2 L 2 = 1 27 m 2 -Λ 3 . (9) \nThis circular light ray is unstable in the sense that a slight perturbation of the initial direction in the equatorial plane gives a light ray that moves away from the circle at r = 3 m and crosses one of the two horizons. If we take all three spatial dimensions into account, we find that there is such an unstable circular light ray in any plane through the origin. These circular light rays fill the photon sphere at r = 3 m . \nFor constructing the shadow we consider all light rays that go from the position of the static observer at ( t O , r O , ϑ O = π/ 2 , ϕ O = 0) into the past. They leave the observer at an angle θ with respect to the radial line that satisfies \ntan θ = lim ∆ x → 0 ∆ y ∆ x , (10) \nsee Fig. 2. From the Kottler metric (1) we read that ∆ x and ∆ y satisfy, in the desired limit, \nFIG. 2. Definition of the angle θ . \n<!-- image --> \ntan θ = r dϕ ( 1 -2 m r -Λ 3 r 2 ) -1 / 2 dr ∣ ∣ ∣ ∣ ∣ ∣ ∣ r = r O . (11) \nExpressing dr/dϕ with the help of the orbit equation (8) results in \ntan 2 θ = r O -2 m -Λ 3 r 3 O ( E 2 c 2 L 2 + Λ 3 ) r 3 O -r O +2 m . (12) \nBy elementary trigonometry, \nsin 2 θ = 1 -2 m r O -Λ 3 r 2 O E 2 c 2 L 2 r 2 O . (13) \nThe shadow is constructed in the following way, see Fig. 3. We assume that there are light sources everywhere in the domain of outer communication but not between the observer and the black hole. Each point in the observer's sky corresponds to a light ray issuing from the observer position into the past. We assign darkness (respectively brightness) to those directions which correspond to light rays that go to the horizon at r H 1 (respectively to the horizon at r H 2 ). The boundary of the shadow corresponds to light rays that spiral asymptotically towards circular lightlike geodesics at r = 3 m . Therefore, the angular radius of the shadow is found be equating E 2 /L 2 to the constant of motion that corresponds to the circular light ray at r = 3 m . Substituting from (9) into (13) yields the angular radius θ stat of the shadow as it is seen by a static observer, \nsin 2 θ stat = 1 -2 m r O -Λ 3 r 2 O ( 1 27 m 2 -Λ 3 ) r 2 O . (14) \nθ stat varies from 0 (bright sky) to π (dark sky) when the observer position r O varies from r H 2 to r H 1 . For r O = 3 m we have θ stat = π/ 2, i.e., half of the sky is dark, see Fig. 4. \nEq. (14) is equivalent to a result found by Stuchl'ık and Hled'ık [8]. For Λ → 0, (14) reduces of course to the formula for the shadow of a Schwarzschild black hole which \nwas first calculated by Synge [1]. The word 'shadow' is used neither by Synge nor by Stuchl'ık and Hled'ık. They calculated what they called the 'escape cone' of light which is the complement of the shadow.", 'III. SHADOW IN THE KOTTLER SPACETIME AS SEEN BY AN OBSERVER COMOVING WITH THE EXPANDING UNIVERSE': "We will now turn to the shadow as it is seen by an observer who is comoving with the cosmic expansion. To that end we introduce on the Kottler spacetime a new coordinate system ( ˜ t, ˜ r, ˜ ϑ = ϑ, ˜ ϕ = ϕ ) which is related to the old coordinate system by \nr = ˜ r e H 0 ˜ t ( 1 + m 2˜ r e -H 0 ˜ t ) 2 , (15) \nt = ˜ t + ∫ ˜ r e H 0 ˜ t w 0 H 0 ( 1 + m 2 w ) 6 wdw c 2 ( 1 -m 2 w ) 2 -H 2 0 w 2 ( 1 + m 2 w ) 6 (16) \nwhere \nH 0 = √ Λ 3 c (17) \nand w 0 is an integration constant that has to be chosen appropriately. If we differentiate (15) and (16), we find the relation between the coordinate differentials, \ndr = e H 0 ˜ t ( 1 -m 2 4˜ r 2 e -2 H 0 ˜ t )( d ˜ r + ˜ r H 0 d ˜ t ) , (18) \nc dt = (19) \n( 1 -m 2˜ r e -H 0 ˜ t ) 2 c d ˜ t + H 0 c ˜ re 2 H 0 ˜ t ( 1 + m 2˜ r e -H 0 ˜ t ) 6 d ˜ r ( 1 -m 2˜ r e -H 0 ˜ t ) 2 -H 2 0 c 2 ˜ r 2 e 2 H 0 ˜ t ( 1 + m 2˜ r e -H 0 ˜ t ) 6 . \nInserting these expressions into (1) gives us the Kottler metric in the new coordinates, \n˜ g µν d ˜ x µ d ˜ x ν = -( 1 -m 2˜ r e -H 0 ˜ t ) 2 ( 1 + m 2˜ r e -H 0 ˜ t ) -2 c 2 d ˜ t 2 + e 2 H 0 ˜ t ( 1 + m 2˜ r e -H 0 ˜ t ) 4 ( d ˜ r 2 + ˜ r 2 d Ω 2 ) . (20) \nIn this coordinate system, observers on ˜ t lines see an exponentially expanding universe with a (timeindependent) Hubble constant H 0 . We call them the comoving observers , where 'comoving' refers to the cosmic expansion. The twiddled coordinates are known as the McVittie coordinates , referring to 1933 work by McVittie [20] on a more general class of spacetimes, although for the Kottler metric Robertson [21] had used these coordinates already in 1928. For H 0 → 0 the Kottler spacetime \nFIG. 3. (COLOR ONLINE) Formation of the shadow as seen by a static observer in the Kottler spacetime. The Kottler metric has a black hole event horizon at r H 1 and a cosmological event horizon at r H 2 . The observer is at radial coordinate r O . Without loss of generality, we consider light rays in the equatorial plane and we assume that the observer is located on the x -axis. If the observer 'emits light rays into the past', some of them go towards the horizon at r H 1 while others, after approaching the black hole, go towards the horizon at r H 2 . The borderline cases between these two classes are light rays which asymptotically spiral towards the photon sphere at r = 3 m which is filled with unstable circular light orbits. In the case of light sources distributed everywhere in the domain of outer communication but not between the black hole and the observer, the cone bounded by light rays that spiral towards the photon sphere will be empty, so the observer will see the shadow as a black disk of angular radius θ stat . We have extended the tangents to the initial directions of these light rays in the coordinate picture by straight dashed lines up to the plane x = 0. This dashed cone has no coordinate-independent meaning, but it shows that application of the naive Euclidean formula tan θ stat = 3 m/r O gives an angular radius of the shadow that is smaller than the correct one. Also note that the Euclidean formula is independent of Λ whereas the correct one, given by (14), is not. \n<!-- image --> \nFIG. 4. (COLOR ONLINE) Angular radius θ stat of the shadow plotted against the observer position r O . The picture is for √ Λ / 3 = H 0 /c = 0 . 15 m -1 . The dashed (red) lines mark the horizons at r = r H1 and r = r H2 . \n<!-- image --> \nin the Robertson-McVittie representation (20) reduces to the Schwarzschild spacetime in isotropic coordinates while for m → 0 it reduces to the steady-state universe , i.e., to one half of the deSitter spacetime in RobertsonWalker coordinates adapted to a spatially flat slicing. \nIf solved for the differentials of the twiddled coordinates, (18) and (19) can be expressed as \nd ˜ t = dt -H 0 rdr c 2 √ 1 -2 m r ( 1 -2 m r -H 2 0 r 2 c 2 ) , (21) \nd ˜ r ˜ r = √ 1 -2 m r dr r ( 1 -2 m r -H 2 0 r 2 c 2 ) -H 0 dt . (22) \nThis transformation can be equivalently rewritten in terms of the Gaussian basis vector fields as \n∂ ∂ ˜ t = ( 1 -2 m r ) ( 1 -2 m r -H 2 0 r 2 c 2 ) ∂ ∂t + H 0 r √ 1 -2 m r ∂ ∂r , (23) \n˜ r ∂ ∂ ˜ r = H 0 r 2 c 2 ( 1 -2 m r -H 2 0 r 2 c 2 ) ∂ ∂t + r √ 1 -2 m r ∂ ∂r . (24) \nWe want to find the angular radius θ comov of the shadow as it is seen by a comoving observer. We have calculated in (14) the angular radius θ stat of the shadow for a static observer. The angle θ comov we are looking for is related to θ stat by the standard aberration formula \nsin 2 θ comov = ( 1 -v 2 c 2 ) sin 2 θ stat ( 1 -v c cos θ stat ) 2 (25) \nwhere v is the 3-velocity of the comoving observer with respect to the static observer at the same observation event. Here we have to be careful when expressing cos θ stat with \nFIG. 5. (COLOR ONLINE) Worldlines of the comoving observers in the r -t coordinate system. As in Fig. 4, we have chosen √ Λ / 3 = H 0 /c = 0 . 15 m -1 . The worldlines of the comoving observers are shown here in the region between the two horizons which are, again, marked by dashed (red) lines. This corresponds to the region I in Fig. 1. If extended beyond the cosmological horizon, the worldlines of the comoving observers fill the regions I and III in Fig. 1 and terminate at I + . \n<!-- image --> \nthe help of our formula (14) for sin 2 θ stat : We know from the preceding section that θ stat lies between π/ 2 and π for r H 1 < r O < 3 m and that it lies between 0 and π/ 2 for 3 m<r O < r H 2 . Therefore, we rewrite (25) as \nsin 2 θ comov = ( 1 -v 2 c 2 ) sin 2 θ stat ( 1 ± v c √ 1 -sin 2 θ stat ) 2 (26) \nwhere we have to choose the upper sign in the domain r H 1 < r O < 3 m and the lower sign in the domain 3 m < r O < r H 2 . \nThe 3-velocity v has to be calculated from the specialrelativistic equation \ng µν U µ stat U ν comov = -c 2 √ 1 -v 2 c 2 (27) \nwhere U µ stat ∂/∂x µ is the four-velocity vector of the static observer and U µ comov ∂/∂x µ is the four-velocity vector of the comoving observer. The former is proportional to ∂/∂t while the latter is proportional to ∂/∂ ˜ t , \nU µ stat ∂ ∂x µ = N stat ∂ ∂t , (28) \nU µ comov ∂ ∂x µ = N comov ∂ ∂ ˜ t = (29) \nN comov ( 1 -2 m r ) ( 1 -2 m r -H 2 0 r 2 c 2 ) ∂ ∂t + H 0 r √ 1 -2 m r ∂ ∂r , \nwhere in the last equality we have used (23). The factors N stat and N comov follow from the normalization condition, \n-c 2 = g µν U µ stat U ν stat = -c 2 N 2 stat ( 1 -2 m r -H 2 0 r 2 c 2 ) , (30) \n-c 2 = g µν U µ comov U ν comov = -c 2 N 2 comov ( 1 -2 m r ) , (31) \nhence (28) and (29) yield \nU µ stat ∂ ∂x µ = 1 √ 1 -2 m r -H 2 0 r 2 c 2 ∂ ∂t , (32) \nU µ comov ∂ ∂x µ = √ 1 -2 m r ( 1 -2 m r -H 2 0 r 2 c 2 ) ∂ ∂t + H 0 r ∂ ∂r . (33) \nInserting these expressions for U µ stat and U ν comov into (27) results in \n1 -v 2 c 2 = 1 -2 m r -H 2 0 r 2 c 2 1 -2 m r (34) \nwhich is equivalent to \nv = H 0 r √ 1 -2 m r . (35) \nFrom (34) we read that v tends to c if one of the two horizons is approached; this is clear because on the horizons the worldlines of the static observers become lightlike. Between the two horizons, v is decreasing from c to a local minimum at the photon sphere and then increasing again to c , see Fig. 6. \nWe can now calculate θ comov by inserting (14) and (35) with r = r O into (26). After some elementary algebra we find \nsin θ comov = √ 27 m r O √ 1 -2 m r O √ 1 -27 H 2 0 m 2 c 2 ∓ √ 27 mH 0 c √ 1 -27 m 2 r 2 O ( 1 -2 m r O ) . (36) \nThis equation makes sense for all momentary observer positions r O with r H 1 < r O < ∞ , although for the \nFIG. 6. (COLOR ONLINE) Three-velocity v of a comoving observer relative to a static observer at the same event, plotted as a function of the radius coordinate r O . As in the preceding pictures, we have chosen √ Λ / 3 = H 0 /c = 0 . 15 m -1 and the dashed (red) lines mark the horizons. \n<!-- image --> \nderivation it was assumed that r H 1 < r O < r H 2 . This reflects the fact that the worldlines of the comoving observers may be analytically extended beyond the cosmological horizon. In (36) we have to choose the upper sign in the domain r H 1 < r O < 3 m and the lower sign in the domain 3 m < r O < ∞ ; for r O = 3 m the term with the ∓ sign is equal to zero. (36) gives us the angular radius of the shadow as it is seen by a comoving observer on his way from the inner horizon through the outer horizon to infinity. Recall that a comoving observer has a constant twiddled radius coordinate, ˜ r O = constant; hence, when we express r O in terms of ˜ r O and ˜ t O with the help of (15) we get from (36) the angle θ comov as a function of the time coordinate ˜ t O . \nIf one of the horizons is approached, \n1 r O √ 1 -2 m r O → H 0 c . (37) \nFor the inner horizon, we have to use the upper sign in (36). Then (37) yields \nsin θ comov → 0 for r O → r H 1 . (38) \nThe angle θ comov itself goes to π . For the outer horizon, however, we have to use the lower sign in (36). Then (37) yields \nsin θ comov → 2 √ 27 H 0 m c √ 1 -27 H 2 0 m 2 c 2 for r O → r H 2 . (39) \nMoreover, from (36) with the lower sign we read that \nsin θ comov → √ 27 H 0 m c for r O →∞ . (40) \nWhen the comoving observer starts at the inner horizon, the shadow covers the entire sky, θ comov = π . On his way out to infinity, the shadow monotonically shrinks to a finite value given by (40), see Fig. 7. Nothing particular happens when the observer crosses the outer horizon. Note that the (future) cosmological horizon is an event horizon for all observers who stay forever in the domain of outer communication, in particular for the static observers, but not for the comoving observers. This can be clearly seen from Fig. 1: Even after crossing this horizon a comoving observer can receive light signals from region I . \nAccording to eq. (40) the angular radius θ comov of the shadow of very distant black holes is determined by the cosmological constant and of course, by the mass of the black hole. With a value of Λ ≈ 1 . 1 × 10 -46 km -2 , which is in agreement with present day observations, (17) gives us a Hubble time of H -1 0 ≈ 5 × 10 17 s. Upon inserting this value into (40) we find for a supermassive black hole of 10 10 Solar masses in the limit r O →∞ an angular radius of θ comov ≈ 0 . 1 microarcseconds. Present-day VLBI technology allows to resolve angles of a few dozen microarcseconds, so a resolution of 0.1 microarcseconds cannot be achieved at the moment but it could come into reach within one or two decades. Also, the existence of black holes with masses of more than 10 10 Solar masses, for which the shadow would be bigger, cannot be ruled out. Note, however, that this line of argument does not necessarily imply that the shadows of very distant black holes will become observable with VLBI instruments in a few years' time. Firstly, we have to keep in mind that our calculation was done in a universe where the cosmic expansion is driven by the cosmological constant only. In a realistic model of the universe, taking the matter content into account, the Hubble 'constant' is a function of time; the chosen value of the Hubble time, H -1 0 ≈ 5 × 10 17 s is a reasonably good approximation for the present time (and an even better approximation for later times, when the cosmological constant dominates even more over matter), but at earlier times the Hubble time had different values. So one would have to repeat our calculation in a universe with a time-dependent Hubble 'constant' to see how the matter content influences our result. Secondly, for the observability of the shadow it is necessary not only that the angular radius of the shadow is big enough but also that there are sufficiently bright light sources that can serve as a backdrop against which the shadow can be observed. This requires calculating, for a realistic model of our universe, the influence of the spacetime geometry on the surface brightness of distant light sources.", 'IV. SHADOW FOR OBSERVERS AT LARGE DISTANCES': 'In the preceding sections we have calculated the shadow for any possible observer position, i.e. r H 1 < r O < r H 2 for static observers and r H 1 < r O < ∞ \nFIG. 7. (COLOR ONLINE) Angular radius θ comov of the shadow plotted against the observer position r O . As before, we have chosen √ Λ / 3 = H 0 /c = 0 . 15 m -1 and the dashed (red) lines mark the horizons. \n<!-- image --> \nfor comoving observers. In this section we want to derive approximate formulas for the case that the observer is far away from the black hole, r O glyph[greatermuch] m . Physically this means that over a large part of a light ray to the observer the effect of the cosmic expansion dominates over the gravitational attraction by the black hole. Clearly, for a static observer the condition r O glyph[greatermuch] m can be satisfied only if r H 2 glyph[greatermuch] m . No such restriction is necessary for comoving observers. Therefore, we will consider the cases of static and comoving observers separately.', 'Static observer': 'As a preliminary note, we want to discuss an important difference between the black-hole shadow in Schwarzschild and Kottler spacetimes that arises from the fact that the former is asymptotically flat whereas the latter is not. In the case of the Schwarzschild metric, the angular radius of the shadow (as seen by a static observer) can be written as \n(Schwarzschild) sin 2 θ stat = ( 1 -2 m r O ) b 2 cr r 2 O , (41) \nwhere b cr is the critical value of the impact parameter b = cL/E corresponding to photons on unstable circular orbits filling the photon sphere. In the Schwarzschild metric the radius of the photon sphere equals 3 m and \n(Schwarzschild) b cr = 3 √ 3 m, (42) \nsee (9) with Λ = 0. \nWith increasing distance r O , both the sine of the angular radius of the shadow and the angular radius itself tend to zero. This is because the denominator of the fraction in (41) increases while the factor in brackets in the numerator tends to unity. Therefore, for large distances the angular size of the shadow can be written as \n(Schwarzschild) sin 2 θ stat ≈ b 2 cr r 2 O , r O glyph[greatermuch] m. (43) \nThis approach reduces the determination of the angular size of the shadow at large distances to the calculation of the critical value of the impact parameter: knowing the critical impact parameter, one gets an approximate value for sin θ stat after dividing by r O . Bardeen [2] has used this approach for the more general case of the Kerr metric. In this case the shadow is not circular; its shape for distant observers is determined by two impact parameters. Accordingly, the angular radii of the shadow can be approximately found by dividing these impact parameters by the (Boyer-Lindquist) radius coordinate r O of the observer. \nThis method works for metrics that are asymptotically flat at infinity. The Kottler spacetime, however, is not asymptotically flat; the metric coefficient f ( r ) does not tend to unity for large r . In this metric the angular radius of the shadow (as seen by a static observer) can be written as \n(Kottler) sin 2 θ stat = ( 1 -2 m r O -Λ 3 r 2 O ) b 2 cr r 2 O , (44) \nwhere the critical value of the impact parameter b = cL/E is given by (9), \n(Kottler) b cr = 3 √ 3 m (1 -9Λ m 2 ) 1 / 2 . (45) \nThis value of the critical impact parameter for the Kottler metric is well known, see e.g. [22, 23]. \nFor Λ glyph[negationslash] = 0 the dependence of the shadow size on r O is very different from the Schwarzschild case. With increasing r O , the denominator of the fraction in (44) increases, while the factor in brackets in the numerator tends to zero if r O approaches its maximal value r H 2 . Therefore for the Kottler spacetime the determination of the angular size of the shadow at large distances does not reduce to the calculation of the critical value of the impact parameter: \n(Kottler) sin 2 θ stat glyph[negationslash]≈ b 2 cr r 2 O , r O glyph[greatermuch] m. (46) \nNote that in the above argument we implicitly assume that Λ is sufficiently small such that r H 2 glyph[greatermuch] m because otherwise the condition r O glyph[greatermuch] m could not hold for a static observer. \nLet us now approximate formula (14) for static observers at large distances, r O glyph[greatermuch] m . As this requires r H 2 glyph[greatermuch] m , the equation for the outer horizon \n1 -2 m r H 2 -Λ 3 r 2 H 2 = 0 (47) \ncan be approximated by \n1 -Λ 3 r 2 H 2 ≈ 0 , r 2 H 2 ≈ 3 Λ . (48) \nCombining (48) with the condition that r H 2 glyph[greatermuch] m , we obtain a restriction on the value of Λ: \nΛ m 2 glyph[lessmuch] 1 . (49) \nWith r O glyph[greatermuch] m and (49), eq. (14) for the angular size of the shadow for static observers can be simplified to \nsin 2 θ stat ≈ 27 m 2 r 2 O ( 1 -Λ 3 r 2 O ) for r O glyph[greatermuch] m. (50) \nComoving observer \nIn the case of comoving observers, the condition r O glyph[greatermuch] m does not require any restriction on r H 2 because such observers can exist both inside and outside the cosmological horizon. \nFor r O glyph[greatermuch] m , eq. (36) for the angular size of the shadow for comoving observers is simplified to \nsin θ comov ≈ √ 27 m r O (√ 1 -27 H 2 0 m 2 c 2 + H 0 r O c ) . (51) \nHere we have to choose the + sign in (36) because the condition r O glyph[greatermuch] m implies that r O > 3 m . For r O → ∞ we recover, of course, (40). \nIf we want to apply the approximation formula (51) for comoving observers near r H 2 we need to assume that r H 2 glyph[greatermuch] m . As we already know, this requires (48) and (49) which read, in terms of H 0 , \nr H 2 ≈ c H 0 , H 2 0 m 2 c 2 glyph[lessmuch] 1 . (52) \nThen we obtain from (51) the approximate formula \nsin θ comov ≈ 2 √ 27 H 0 m c for r O ≈ r H 2 glyph[greatermuch] m. (53)', 'V. CONCLUSIONS': 'In this paper we have calculated the angular radius of the shadow for an observer that is comoving with the cosmic expansion in Kottler (Schwarzschild-deSitter) spacetime. As far as we know, the shadow for a comoving observer in an expanding universe was not calculated before. The resulting expression is presented in formula (36). \nQuite generally, the cosmic expansion has a magnifying effect on the shadow. This is in agreement with the wellknown fact that the image of an object is magnified by aberration if the observer moves away from the object. Moreover, it is found that the shadow shrinks to a finite value if the comoving observer approaches infinity, see formula (40). As a consequence, even the most distant black holes have a shadow whose angular radius is bigger than the bound given by (40). \nThe magnification effect caused by a cosmological constant of Λ ≈ 10 -46 km -2 is rather strong: for a black hole of 10 10 Solar masses we found that even in the limit r O →∞ the angular radius of the shadow is not smaller than θ comov ≈ 0 . 1 microarcseconds. This is only two orders of magnitude beyond the resolvability of presentday VLBI technology. However, there are two caveats. Firstly, our calculations where done in the Kottler spacetime in which the cosmic expansion is driven by the cosmological constant only. It has to be checked how our results are to be modified in a more realistic spacetime model, taking the matter content of the universe into account. Secondly, the shadow can be observed only if there is a backdrop of sufficiently bright light sources against which the shadow can be seen as a dark disk. Therefore, when doing the calculations in a realistic model of our universe one would also have to estimate the influence of the spacetime geometry on the surface brightness of light sources. \nNote that a comoving observer in the Kottler spacetime can exist behind the cosmological event horizon, in contrast to a static observer, and that he can see the shadow until he ends up at future null infinity. Simplified approximative formulas for distant observers, both static and comoving, are presented in Section IV. \nIn an Appendix we demonstrate that our results for the angular size of the shadow can be also obtained by using formulas for the deflection angle in Kottler spacetime derived by Lebedev and Lake [14, 15].', 'ACKNOWLEDGEMENTS': 'O. Yu. T. is grateful to Dmitri Lebedev for useful conversations. O. Yu. T. and G. S. B.-K. express their gratitude to C. Lammerzahl and his group for warm hospitality during their visit of ZARM, University of Bremen. The work of O. Yu. T. and G. S. B.-K. was partially supported by the Russian Foundation for Basic Research Grant No. 17-02-00760. V. P. gratefully acknowledges support from the DFG within the Research Training Group 1620 Models of Gravity .', 'APPENDIX: DERIVATION OF THE ANGULAR SIZE OF THE SHADOW USING RESULTS OF LEBEDEV AND LAKE': "Here we show how to obtain formulas (26), (36) and (51) using results from Lebedev and Lake [14] (cf. [15]) on the deflection of light in the Kottler (SchwarzschilddeSitter) spacetime. \n- (i) Formula (128) from [14] is: \ncos( α radial ) = √ f ( r 0 ) r 2 0 -f ( r ) r 2 + (√ f ( r 0 ) r 2 0 + √ f ( r 0 ) r 2 0 -f ( r ) r 2 ) ( U r 2 f ( r ) -U r √ f ( r ) √ 1 + U r 2 f ( r ) ) ( √ f ( r 0 ) r 2 0 √ 1 + U r 2 f ( r ) -√ f ( r 0 ) r 2 0 -f ( r ) r 2 U r √ f ( r ) )( √ 1 + U r 2 f ( r ) -U r √ f ( r ) ) . (54) \nHere α radial is the angle, as measured by a radially moving observer in the Kottler spacetime, between a radial light ray and a light ray with r 0 as radial coordinate of the point of closest approach. The observer's radial coordinate is r and the observer's four-velocity is U = ( U t , U r , 0 , 0). In this appendix we follow Lebedev and Lake and choose units such that c = 1. Then the function f ( r ) is \nf ( r ) = 1 -2 m r -H 2 0 r 2 . (55) \nNote that in our notation the observer's radial coordinate is denoted r O which should not be confused with the r 0 of Lebedev and Lake. \nTo rederive the formula for the sine of the angular radius of the shadow, sin θ comov , we have to choose the minimal coordinate distance as r 0 = 3 m , the observer's position as r = r O , and the observer's four-velocity as U r = H 0 r O , see (33). With these substitutions α radial in (54) gives us θ comov . \nTo rewrite (54) in a more compact way, we use the equation \nU t = 1 √ f ( r O ) √ 1 + U r 2 f ( r O ) , (56) \nand we introduce the notation \nw 1 ≡ √ 1 -h 2 (3 m ) h 2 ( r O ) , w t ≡ √ f ( r O ) U t , w r ≡ U r √ f ( r O ) . (57) \nHere the function h ( r ) is defined by \nh 2 ( r ) = r 2 f ( r ) = r 2 1 -2 m r -H 2 0 r 2 , (58) \nsimilar to our previous work [9]. \nThe quantities w 1 , w t , w r are introduced for convenience only and have no specific physical meaning. In particular, they are not the covariant components of any four-vector. Note that the expression h 2 (3 m ) /h 2 ( r O ) coincides with sin 2 θ stat from formula (14). With this notation the expression (54) takes the following form (compare with eq. (129) of [14]): \ncos θ comov = w 1 +(1 + w 1 ) w r ( w r -w t ) ( w t -w 1 w r )( w t -w r ) . (59) \nFrom U µ U µ = -1 we find that w 2 t -w 2 r = 1, hence \n( w t -w 1 w r )( w t -w r ) = 1 + (1 + w 1 ) w r ( w r -w t ) . (60) \nThis allows us to rewrite (59) as \ncos θ comov = w 1 + z w 1 + z w , z w ≡ (1 + w 1 ) w r ( w r -w t ) . (61) \nAs a consequence, \nsin 2 θ comov = 1 -( w 1 + z w ) 2 (1 + z w ) 2 = 1 + 2 z w -w 2 1 -2 w 1 z w (1 + z w ) 2 \n= (1 -w 2 1 )( w t -w r ) 2 (1 + z w ) 2 = 1 -w 2 1 ( w t -w 1 w r ) 2 . (62) \nNote that the numerator 1 -w 2 1 coincides with sin 2 θ stat from formula (14). \nFrom these results we can re-obtain a formula for the shadow in the form of (26) in the following way. We substitute U r = H 0 r O into (56) and (57) and obtain: \nw t = 1 √ 1 -v 2 , w r = v √ 1 -v 2 . (63) \nHere we have introduced for compactness the variable v in the same way as in (34) and (35). With these expressions, we can transform formula (62) to (26) with v given by (35). \nLebedev and Lake assume that the radial coordinate of the observer is bigger than the radial coordinate of the point of the closest approach of the light ray. In our problem this means that r O > 3 m . Therefore we get from their approach eq. (26) only with the minus sign in the denominator. If r H 1 < r O < 3 m we have to use eq. (26) with the plus sign because cos θ stat < 0 in this case. \n(ii) If we want to obtain a formula for the shadow in the form of (36), we can perform the following transformation: \nsin θ comov = sin θ stat w t ± w 1 w r = sin θ stat ( w t ∓ w 1 w r ) w 2 t -w 2 1 w 2 r = = sin θ stat ( w t ∓ w 1 w r ) 1 + w 2 r sin 2 θ stat . (64) \nBy substituting U r = H 0 r O into (56) and (57) we recover (36). \n(iii) Our approximative formula (51) for the size of the shadow as seen by a distant observer can also be derived using formula (132) from [14]: \ncos( α comoving ) = √ f ( r 0 ) r 2 0 -f m =0 ( r ) r 2 -√ f ( r 0 ) r 2 0 √ Λ 3 r √ f ( r 0 ) r 2 0 -√ f ( r 0 ) r 2 0 -f m =0 ( r ) r 2 √ Λ 3 r , (65) \nwhere \nf m =0 ( r ) = 1 -Λ 3 r 2 . (66) \nThen the four-velocity of a comoving observer in static coordinates is \nU µ comoving = ( 1 f m =0 ( r ) , √ Λ 3 r, 0 , 0 ) . (67) \nSubstituting r 0 = 3 m , r = r O and √ Λ 3 = H 0 we rewrite \n- [1] J. L. Synge, The escape of photons from gravitationally intense stars, Mon. Not. R. Astron. Soc. 131 , 463 (1966).\n- [2] J. M. Bardeen, Timelike and null geodesics in the Kerr metric, in Black Holes , ed. by C. DeWitt and B. DeWitt (Gordon and Breach, New York, 1973), p. 215.\n- [3] A. Grenzebach, V. Perlick, and C. Lammerzahl, Photon regions and shadows of Kerr-Newman-NUT black holes with a cosmological constant, Phys. Rev. D 89 , 124004 (2014).\n- [4] A. Grenzebach, V. Perlick, and C. Lammerzahl, Photon regions and shadows of accelerated black holes. Int. J. Mod. Phys. D 24 , 1542024 (2015).\n- [5] A. Grenzebach, Aberrational effects for shadows of black holes, in E quations of Motion in Relativistic Gravity, ed. by D. Puetzfeld, C. Lammerzahl, and B. Schutz (Springer, Heidelberg, 2015), pp. 823.\n- [6] O. Yu. Tsupko, Analytical calculation of black hole spin using deformation of the shadow, Phys. Rev. D 95 , 104058 (2017).\n- [7] F. Kottler, Uber die physikalischen Grundlagen der Einsteinschen Gravitationstheorie, Ann. Phys. (Berlin) 361 , 401 (1918).\n- [8] Z. Stuchl'ık and S. Hled'ık, Some properties of the Schwarzschild-de Sitter and Schwarzschild-anti-de Sitter spacetimes, Phys. Rev. D 60 , 044006 (1999)\n- [9] V. Perlick, O. Yu. Tsupko, and G. S. Bisnovatyi-Kogan, Influence of a plasma on the shadow of a spherically symmetric black hole, Phys. Rev. D 92 , 104031 (2015).\n- [10] V. Perlick and O. Yu. Tsupko, Light propagation in a plasma on Kerr spacetime: Separation of the HamiltonJacobi equation and calculation of the shadow, Phys. Rev. D 95 , 104003 (2017).\n- [11] G. Bisnovatyi-Kogan and O. Tsupko, Gravitational lensing in presence of plasma: Strong lens systems, black hole \ncos θ comov = w 1 -H 0 r O 1 -w 1 H 0 r O , (68) \nwhere \nw 1 = √ 1 -9 m 2 f m =0 ( r O ) r 2 O f (3 m ) . (69) \nBy applying the transformation \nsin θ comov = √ 1 -w 2 1 √ 1 -H 2 0 r 2 O 1 -w 1 H 0 r O = (70) \n= √ 1 -w 2 1 √ 1 -H 2 0 r 2 O (1 + w 1 H 0 r O ) 1 -w 2 1 H 2 0 r 2 O \nand simplifying sin θ comov with r O glyph[greatermuch] m , we recover (51). \nlensing and shadow, Universe 3 , 57 (2017) \n- [12] J. N. Islam, The cosmological constant and classical tests of general relativity, Phys. Lett. A 97 , 239 (1983).\n- [13] W. Rindler and M. Ishak, Contribution of the cosmological constant to the relativistic bending of light revisited, Phys. Rev. D 76 , 043006 (2007).\n- [14] D. Lebedev and K. Lake, On the influence of the cosmological constant on trajectories of light and associated measurements in Schwarzschild de Sitter space, eprint arXiv:1308.4931 (2013).\n- [15] D. Lebedev and K. Lake, Relativistic aberration and the cosmological constant in gravitational lensing I: Introduction, eprint arXiv:1609.05183 (2016).\n- [16] G. W. Gibbons and S. W. Hawking, Cosmological event horizons, thermodynamics, and particle creation, Phys. Rev. D 15 , 2738 (1977)\n- [17] I. D. Novikov, Delayed explosion of a part of the Friedman universe and quasars, Soviet Astronomy 8 , 857 (1965).\n- [18] Yu. Ne'eman, Expansion as an energy source in quasistellar radio sources, Astrophys. J. 141 , 1303 (1965).\n- [19] K. Lake, White holes, Nature 272 , 599 (1978).\n- [20] G. C. McVittie, The mass-particle in an expanding universe, Mon. Not. Roy. Astron. Soc. 93, 325 (1933)\n- [21] H. P. Robertson, On relativistic cosmology, Philos. Mag. J. Sci., Series 7, 5, 835 (1928)\n- [22] K. Lake and R. C. Roeder, Effects of a nonvanishing cosmological constant on the spherically symmetric vacuum manifold, Phys. Rev. D 15 , 3513 (1977).\n- [23] Z. Stuchl'ık, The motion of test particles in black-hole backgrounds with non-zero cosmological constant, Bull. Astr. Inst. Czech. 34 , 129 (1983)."}

1998ApJ...492L..53C

Evidence for Frame Dragging around Spinning Black Holes in X-Ray Binaries

1998-01-01

['black hole physics', 'astronomy x rays', 'black hole physics', 'astronomy x rays', 'astrophysics', '-']

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In the context of black hole spin in X-ray binaries, we propose that certain types of quasi-periodic oscillations (QPOs) observed in the light curves of black hole binaries (BHBs) are produced by X-ray modulation at the precession frequency of accretion disks, because of relativistic dragging of inertial frames around spinning black holes. These QPOs tend to be relatively stable in their centroid frequencies. They have been observed in the frequency range of a few hertz to a few hundred hertz for several black holes with dynamically determined masses. By comparing the computed disk precession frequency with that of the observed QPO, we can derive the black hole spin, given its mass. When applying this model to GRO J1655-40, GRS 1915+105, Cyg X-1, and GS 1124-68, we found that the black holes in GRO J1655-40 and GRS 1915+105, the only known BHBs that occasionally produce superluminal radio jets, spin at a rate close to the maximum limit, while Cyg X-1 and GS 1124-68, typical (persistent and transient) BHBs, contain only moderately rotating ones. Extending the model to the general population of black hole candidates, the fact that only low-frequency QPOs have been detected is consistent with the presence of only slowly spinning black holes in these systems. Our results are in good agreement with those derived from spectral data, thus strongly supporting the classification scheme that we proposed previously for BHBs.

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https://arxiv.org/pdf/astro-ph/9710352.pdf

{'No Header': 'Published in ApJ Letters (1998, 492, L53)', 'EVIDENCE FOR FRAME-DRAGGING AROUND SPINNING BLACK HOLES IN X-RAY BINARIES': 'Wei Cui 1 , S. N. Zhang 2 , 3 , and Wan Chen 4 , 5', 'ABSTRACT': 'In the context of black hole spin in X-ray binaries, we propose that certain type of quasi-period oscillations (QPOs) observed in the light curves of black hole binaries (BHBs) are produced by X-ray modulation at the precession frequency of accretion disks, due to relativistic dragging of inertial frames around spinning black holes. These QPOs tend to be relatively stable in their centroid frequencies. They have been observed in the frequency range of a few to a few hundred Hz for several black holes with dynamically determined masses. By comparing the computed disk precession frequency with that of the observed QPO, we can derive the black hole angular momentum, given its mass. When applying this model to GRO J1655-40, GRS 1915+105, Cyg X-1, and GS 1124-68, we found that the black holes in GRO J1655-40 and GRS 1915+105, the only known BHBs that occasionally produce superluminal radio jets, spin at a rate close to the maximum limit, while Cyg X-1 and GS 1124-68, typical (persistent and transient) BHBs, contain only moderately rotating ones. Extending the model to the general population of black hole candidates, the fact that only low-frequency QPOs have been detected is consistent with the presence of only slowly spinning black holes in these systems. Our results are in good agreement with those derived from spectral data, thus strongly support the classification scheme that we proposed previously for BHBs. \nSubject headings: black hole physics - X-rays: Stars', '1. Introduction': "It has long been argued that a spinning black hole may provide an ideal laboratory for testing the theory of general relativity (GR). Recently, Zhang, Cui, & Chen (1997; Paper I hereafter) have successfully derived the angular momentum of accreting black holes (BHs) in several Galactic X-ray binaries, by carefully modeling the soft disk component in the observed X-ray spectra for high luminosity states. They found that superluminal jet sources contain BHs spinning close to the extreme theoretical limit while most others contain only slowly rotating ones. For BHBs, the relativistic dragging of inertial frames around a spinning BH should cause the accretion disk to precess, if it does not coalign with the BH equatorial plane, thus may produce direct observable effects. \nRecently, Stella & Vietri (1997) proposed that for low-mass X-ray binaries that contain an accreting neutron star (LMXBNs) the observed QPOs in the range of tens of Hz are the manifestation of disk precession due to the 'frame dragging' (FD) effect. They were able to account for such QPOs for several atoll and Z sources, with the help of recent discoveries of kHz QPOs in LMXBNs and progress in theoretical interpretation of such phenomena. Inspired by their work, we quickly realized that the same may also apply to certain type of QPOs that seem to be unique to BHBs. \nThough not as commonly seen as in LMXBNs, QPOs have been observed in several BHBs over a frequency range from mHz to roughly ten Hz (review by van der Klis 1995, and references therein). With much improved timing resolution of the RXTE instrumentation (Bradt, Rothschild, & Swank 1993), the upper range has recently been pushed up to hundreds of Hz (Remillard et al. 1997). The QPO models that invoke magnetic effects for LMXBNs probably do not apply to BHBs, because of the less importance of the magnetic field, which is maintained by the accretion disk, in such systems. Consequently, accretion disks in BHBs can probably extend all the way (or very close) to the last stable orbit, under certain conditions. Therefore, we expect GR effects to be stronger in BHBs than in LMXBNs. \nOne type of QPO in BHBs is of particular interest. The QPO was first reliably established in GRS 1915+105, a superluminal jet source, with the detection of now famous 67 Hz QPO (Morgan, Remillard, & Greiner 1997). One unique characteristic of this QPO is the constancy of its centroid frequency (which moves around by only a few Hz). The QPO is clearly a transient event, and is only present in certain spectral states. The origin of such a QPO is still unknown. Shortly after this discovery, a similar QPO was detected at ∼ 300 Hz for GRO J1655-40, another superluminal jet source (Remillard et al. 1997). This led to the speculation that such phenomenon may be common for BHBs and may be related to some fundamental properties of these sources. In retrospect, we may have seen similar QPOs in other BHBs before as well, such as Cyg X-1 (Cui et al. 1997) and GS 1124-68 (Belloni et al. 1997), although the QPOs are not as stable (the centroid frequency can vary by up to a factor of two). A more stable QPO was also seen at 6 Hz for GX 339-4, a BH candidate (BHC), in the very high state (Miyamoto et al. 1991). It seems natural \nto ask if all of these QPOs are indeed similar in origin and what are the physical processes that produce them. In this letter, we propose that they are the result of disk precession due to the FD effect. When applying this model to several known BHs with dynamically determined masses, we can derive their angular momenta. We show that the results are consistent with what we derived from spectral data in Paper I, thus providing further support to the classification scheme that we proposed for BHBs.", '2. Accretion Disk and Frame Dragging Effect': "As in Paper I, we assume a geometrically thin, optically thick accretion disk (Novikov & Thorne 1973; Shakura & Sunyaev 1973), which now may be tilted with respect to the equatorial plane of a spinning BH. Further more, we assume that in the high luminosity state (i.e., not the advection dominated state; e.g., Narayan & Yi 1995) the inner disk edge extends to the last stable orbit. Of course, with a central X-ray source, radiation pressure may have effects on the inner disk structure. We will delay the discussion of such effects until the next section. \nThe radius of the last (marginally) stable orbit of a test particle is a function of the BH angular momentum (Bardeen, Press, & Teukolsky 1972), \nr last = r g { 3 + A 2 ± [(3 -A 1 )(3 + A 1 +2 A 2 )] 1 / 2 } , (1) \nwhere A 1 = 1+(1 -a 2 ∗ ) 1 / 3 [(1 + a ∗ ) 1 / 3 +(1 -a ∗ ) 1 / 3 ], A 2 = (3 a 2 ∗ + A 2 1 ) 1 / 2 ; a ∗ = a/r g , with a = J/Mc ( J is the BH angular momentum and M the mass) and r g = GM/c 2 ; the lower and upper signs are for prograde orbits (i.e., in the same sense as the BH spin, i.e., a ∗ > 0) and retrograde orbits ( a ∗ < 0), respectively. In the presence of an accretion disk, a ∗ takes values in the range from -0.998-+0.998 (Thorne 1974), with ± 1 being the absolute theoretical limits. The event horizon of a Kerr BH is at r h = r g +( r 2 g -a 2 ) 1 / 2 = r g [1 + (1 -a 2 ∗ ) 1 / 2 ], so the disk may extend all the way to the horizon, r last ( a ∗ =+1)= r g , or be expelled to r last ( a ∗ =-1)=9 r g , as compared to the canonical Schwarzschild case, r last ( a ∗ =0)=6 r g . \nAs discussed in Paper I, X-ray emission from the disk is from the hot innermost region. The X-ray spectrum from such a disk can be described as a 'diluted' blackbody, but with a varying temperature as a function of radius. The effective temperature of the disk peaks at an annulus slightly beyond r last at r peak = r last /η , where η varies only slowly from 0.62 to 0.76 as a ∗ goes from -1 to +1. That is where most of the disk X-ray emission comes from. \nDue to the FD effect, the tilted orbital plane of a test particle must precess around the same axis and in the same direction as the BH spin (Lense & Thirring 1918; Wilkins 1972). For simplicity, we specialize to only circular orbits, which are most relevant to accretion processes in X-ray binaries. First, we define a node as a point where a non-equatorial orbit intersects the equatorial plane. Then, the nodal precession frequency can be expressed as \nν FD = ν orb ∆Ω 2 π , (2) \nwhere ν orb is the orbital frequency, and ∆Ω is an angle by which the nodes of a circular orbit are dragged per revolution. In the weak field limit, Lense and Thirring (1972) have derived ∆Ω / 2 π = 2( | a | /c )( GM/r 3 ) 1 / 2 , where r is the radius of test particle orbit. In this case, the orbital frequency is simply that of Keplerian motion, i.e., ν orb = (1 / 2 π )( GM/r 3 ) 1 / 2 . Substituting these into Eq. (2), we derive the Lense-Thirring (LT) precession frequency \nν LT = 6 . 45 × 10 4 | a ∗ | ( M M /circledot ) -1 ( r r g ) -3 Hz. (3) \nFor BHBs that contain extremely spinning BHs ( a ∗ /similarequal +1), the accretion disk can extend very close to the event horizon (at r /similarequal r g ), where weak field approximation clearly breaks down. The exact problem has been solved analytically, by Wilkins (1972), to derive the allowed ranges for constants of motion, E , Φ, and Q , which are respectively the energy, the component of angular momentum along the BH spin axis, and a non-negative quantity related to the θ velocity (with Q = 0 specifying equatorial orbits). The requirement for a stable circular orbit provides two equations that allow the expression of E and Φ in terms of Q (and of course r and a ∗ ) (cf. Wilkins 1972). Therefore, unlike the weak-field limit, for a given a ∗ it is no longer true in general that ∆Ω depends only on the orbital radius. \nThe orbital frequency of a test particle around a spinning BH also deviates from that of Keplerian motion at small radii. Bardeen et al. (1972) have shown that the frequency is given by \nν orb = 3 . 22 × 10 4 ( M M /circledot ) -1 ( r r g ) 3 / 2 + a ∗ -1 Hz. (4) \nWith corrections to both terms in Eq. (2) in the case of strong field, we can compute ν FD as a function of r , given a ∗ , Q , and BH mass. As an example, we plot the results in Fig. 1, for cases where a ∗ = +0 . 5 , +0 . 95 , +1, and Q = 0 . 01 (i.e., slightly off the equatorial plane). For comparison, we have also plotted the weak field approximation. As expected, strong-field effects are only important for extremely spinning BHs. Since the bulk of disk emission is produced at r peak , we have derived the precession frequency at this radius, as shown in Fig. 2. It is interesting to note that the frequency is always below ∼ 8 Hz (derived for a 3 M /circledot BH) for systems with retrograde disks, and at low frequencies there are two solutions to the BH angular momentum for a given mass.", '3. Disk Precession and QPOs': "We now propose that the 'stable' QPOs observed in BHBs are simply X-ray modulation at the disk precession frequency. First, we apply this hypothesis to micro-quasars, GRO J1655-40 and GRS 1915+105, since we know from Paper I that both may contain nearly maximal rotating BHs. The BH mass of GRO J1655-40 was determined to a high accuracy, 7 . 02 ± 0 . 22 M /circledot (Orosz \n& Bailyn 1997). If the observed QPO frequency (300 Hz) is indeed the disk precession frequency at r peak , we derive a ∗ = +0 . 95, in excellent agreement with the most probably value from Paper I. The same can be applied to GRS 1915+105, although the BH mass is not reliably determined in this case. The 67 Hz QPO implies a ∗ /similarequal +0 . 65 for a 3 M /circledot BH or +0.95 for a 30 M /circledot one. Compared with the spectral results from Paper I, a self-consistent solution for the BH mass should be around 30 M /circledot . While the same value of a ∗ for both sources may purely be a numerical coincidence, it strongly hints at the presence of an rapidly spinning BH in GRS 1915+105 and extraordinary similarities between the two. \nThen, we examine more typical BHBs. A strong QPO was observed for GS 1124-68 during the very high state (VHS) and a transition from the high to low state (Belloni et al. 1997). Within each observation, the QPO frequency remained quite stable, but it varied in the range 5-8 Hz between observations. The QPO observed during the transition (at 6.7 Hz) appears to be of the same origin. In VHS, the mass accretion rate is thought to be near the Eddington limit (van der Klis 1995), thus radiation pressure may play an important role in shaping up the inner disk structure. Depending on the exact details of the physical processes involved, VHS might not be a long-term stable state as far as the accretion disk is concerned. The inner edge of the optically thick disk may sometimes be pushed outward (Misra & Melia 1997), consequently the disk precession frequency becomes smaller. We would then expect an anti-correlation between the QPO frequency and X-ray luminosity; there seems to be a hint of such relationship for GS 1124-68 (Belloni et al. 1997). From Eq. (3) it is clear that disk precession frequency is a sensitive function of position ( r ) - it only takes a variation of ∼ 17% in radius to account for the frequency variation observed. The BH mass of the source was measured to be ∼ 6 . 3 M /circledot (Orosz et al. 1996), so a QPO frequency of 8 Hz would imply a ∗ = +0 . 35, which is moderate compared to that of superluminal jet sources. A similar QPO was also observed for GX 339-4 also in VHS (Miyamoto et al. 1991). The measured QPO frequency clusters around 6 Hz. It should be noted, however, that evidence for the source being a BHB is at best circumstantial - no dynamical BH mass measurement is available. Nevertheless, the source has long been considered a BHC, because of the similarities to Cyg X-1 in its X-ray properties. \nRecently, a QPO has been detected in Cyg X-1 only during the transitions between its hard and soft states (Cui et al. 1997), perhaps similar to that in GS 1124-68 observed during the transition. The QPO frequency varied in the range 4 - 9 Hz, showing strong correlation with X-ray spectral shape. This is consistent with the fact that the evolution of the accretion disk was not completed yet during these episodes; consequently the innermost disk may still be in the process of settling down to its final stable configuration. Given the BH mass 10 M /circledot (Herrero et al. 1995), a QPO frequency of 9 Hz indicates a ∗ = +0 . 48, which is lower than what we inferred (+0.75) from the spectral data for the soft state. This is, nonetheless, an encouraging result considering the uncertainties (on the BH mass, the inclination angle, and so on) involved in the calculations. If the spectral state transitions are indeed caused by flip-flops of the disk, as we suggested in Paper I, we might see a QPO at ∼ 2 Hz in the hard state (for a ∗ = -0 . 48). Of course, the detection is not \nguaranteed since mechanisms for producing X-ray modulation might only be present under certain conditions. For instance, no apparent QPOs have been detected in the soft state (Cui et al. 1997). \nOnly low-frequency QPOs (at a few to tens of mHz) have been detected for a few other BHCs (van der Klis 1995). This seems to be consistent with most BHBs containing only slowly rotating BHs (see Paper I), although it is not clear if any of these QPOs are due to disk precession.", '4. Discussion': "We summarize the results for known BHBs in Table 1. Although the results are computed for a specific Q value ( Q = 0 . 01), they are actually quite insensitive to Q . For example, we have calculated the angular momentum for GRO J1655-40 as a function of allowed Q values (0 - 2) for stable circular orbits. Over the entire range, the result varies only by roughly 1%. \nOur results suggest that the FD effect is quite significant in BHBs and is manifested in the presence of relatively stable QPOs in these systems. We emphasize that the strength of the disk precession model lies in the fact that it can account for certain type of QPOs in all BHBs. Another model invoking trapped g -mode oscillations near the inner edge of the disk can also quantitatively account for the 300 Hz QPO for GRO J1655-40 and 67 Hz QPO for GRS 1915+105 (Nowak et al. 1997; Paper I), but not those in other BHBs, such as Cyg X-1 and GS 1124-68 (cf. Perez et al. 1997). Therefore, we consider the disk precession model being more natural for BHBs. The model provides a direct way of measuring BH angular momentum from the observed QPO, once the mass is determined, independent of measurements based on spectral information. It is the consistency between the two types of measurements (see Table 1) that gives us confidence in the model. \nInferring BH angular momentum directly from QPOs avoids the need for information on binary orbital inclination, thus does not suffer from the large uncertainty associated with inclination angles, which is built into spectroscopic measurements (Paper I). Compared with Paper I, a slightly lower value for a ∗ , from the 67 Hz QPO for GRS 1915+105, might suggest an actual inclination angle slightly larger than 70 · for the system, while a higher a ∗ for GS 1124-68 might be due partly to a smaller inclination (than that adopted in Paper I). \nThe results show that the BHs in superluminal jet sources spin near the maximum limit, while those in 'normal' BHBs only moderately or slowly rotate. This supports the idea that jet formation may be closely related to the BH spin. Some models suggest that jets are formed by the ejection of matter from the inner disk region and are collimated by the magnetic field maintained by the disk (e.g. Blandford & Payne 1982; Shu et al. 1995; Ustyugova et al. 1995; Kudoh & Shibata 1997). If so, the disk precession model would naturally predict the precession of jets at the same frequency in these systems. Future observations will shed light on this issue. \nWhy do most BHBs only contain slowly or moderately spinning BHs? There are two important spin-down mechanisms: accretion from retrograde disks onto BHs (Moderski & Sikora \n1996) and the Blandford-Znajek extraction of rotational energy of spinning BHs (Blandford & Znajek 1977; Moderski, Sikora, & Lasota 1997). It has been shown that it is much easier to spin down a BH by accreting from a retrograde disk than to spin it up from a prograde disk (Thorne 1974; Moderski & Sikora 1996). For instance, a BH must accrete about 20% of its initial mass from a retrograde disk to decelerate from maximum to zero spin, while it takes roughly 180% of its current mass from a prograde disk to accelerate it back up to maximum spin. This process alone can cause a uniform initial distribution of BH angular momentum to evolve into an extremely non-uniform one with most systems being relatively slowly rotating; the Blandford-Znajek process would only speed up such evolution, despite being relatively inefficient (Moderski, Sikora, & Lasota 1997). In reality, mass accretion is probably inadequate in spinning up those born slow rotators to the extreme limit, due to such factors as low average accretion rate, disk flip-flop, or the lack of available mass from companion stars. This implies that for systems like GRO J1655-40 and GRS 1915+105 BHs are likely formed with high angular momentum. \nOne key element to the disk precession model is the requirement of misalignment of the disk with the equatorial plane of a spinning BH. Such misalignment could be the result of the Pringle instability (Pringle 1996; also see Stella & Vietri 1997 for detailed discussion). However, the Bardeen-Petterson effect (Bardeen & Petterson 1975) should lead to the equatorial configuration for the innermost disk, thus the precession might cease for the X-ray emitting region (although the time scale for it to happen seems to be quite long; see Scheuer & Feiler 1996). This might be the reason why the QPOs of interest only occur during transitional or unstable periods, when the inner disk region experiences significant changes, and why they only last for a limited time. Moreover, the disk precession only provides a natural frequency for QPOs, and physical mechanisms are still needed to produce X-ray modulations. At present we have no definitive knowledge about the mechanisms, but the modulation could be caused by any kind of asymmetry or inhomogeneity in a precessing disk, as well as by the varying projected area of the disk ring and/or perhaps an occulted (by the disk) central hard X-ray emitting region (cf. Stella & Vietri 1997). \nFinally, it is interesting to compare BHBs with atoll sources that first drew Stella and Vietri's attention. For atoll sources, the magnetic field is thought to be very weak (e.g., van der Klis 1995), it may hardly affect the dynamics of accretion flows. Consequently, these system should resemble BHBs in their X-ray properties including QPOs. Indeed, many similarities between the two have been observed, such as hard power-law tails in the X-ray spectra (e.g., Zhang et al. 1996), which used to be considered one of the defining signatures for BHCs. It is therefore of no surprise that the disk precession model works well in both types of systems. However, no QPOs at the Keplerian frequency of the inner disk edge (appearing as kHz QPOs in LMXBNs) are present in BHBs. This might be related to the lack of magnetic field in BHBs, if the interaction between magnetosphere and accretion disk is responsible for introducing X-ray modulation. The situation becomes more complicated for LMXBNs with relatively strong field (such as Z sources), since in those cases precise knowledge about the field strength is required to determine the location of the inner disk boundary. Recent RXTE discoveries of kHz QPOs in Z sources seem to provide such \ninformation (Stellar & Vietri 1997). \nWe thank Luigi Stella and Mario Vietri for their inspiring work that triggered this investigation, and the referee for prompt and helpful comments. Cui acknowledges support by NASA through Contract NAS5-30612. SNZ is partially supported by NASA Grants NAG5-3681, NAG5-4411, and NAG5-4423.", 'REFERENCES': "Bardeen, J. M., Press, W. H., & Teukolsky, S. A. 1972, ApJ, 178, 347. \nBardeen, J. M., & Petterson, J. A. 1975, ApJ, 195, L65. \nBelloni, T., van der Klis, M., Lewin, W. H. G., van Paradijs, J., Dotani, T., Mitsuda, K., & Miyamoto, S. 1997, A&A, 322, 857. \nBlandford, R. D., & Znajek, R. L. 1977, MNRAS, 179, 433. \nBlandford, R. D., & Payne, D. G. 1982, MNRAS, 199, 883. \nBradt, H. V., Rothschild, R. E., & Swank, J. H. 1993, A&AS, 97, 355 \nCui, W., Zhang, S. N., Focke, W., & Swank, J. 1997, ApJ, 484, 383. \nLense, J., & Thirring, H. 1918, Physik, Z. 19, 156 \nKudoh, T., & Shibata, K. 1997, ApJ, 476, L632. \nMisra, R., & Melia, F. 1997, ApJ, 484, 848. \nMiyamoto, S., Kimura, K., Kitamoto, S., Dotani, T., & Ebisawa, K. 1991, ApJ, 383, 784. \nModerski, R., & Sikora, M. 1996, A&AS, 120, 591 \nModerski, R., Sikora, M., & Lasota, J.-P. 1997, Proceedings of the Conference 'Relativistic Jets in Astrophysics', Cracow, in press (astro-ph/9706263). \nMorgan, E., Remillard, R., & Greiner, J. 1997, ApJ, 482, 993. \nNarayan, R. & Yi, I. 1995, ApJ, 444, 231. \nHerrero, A., Kudritzki, R. P., Gabler, R., Vilchez, J. M., & Gabler, A. 1995, A&A, 297, 556 \nNovikov, I.D. & Thorne, K.S. 1973, in Black Holes , ed. C. DeWitt and B. DeWitt (New York: Cordon & Breach), p. 343-450. \nNowak, M. A., Wagoner, R. V., Begelman, M. C., & Lehr, D. E. 1997, ApJ, 477, L91. \nOrosz, J.A., Bailyn, C.D. 1997, 477, 876. \nOrosz, J.A. et al. 1996, ApJ, 468, 380. \nPerez, C. A., Silbergleit, A. S., Wagoner, R. V., & Lehr, D. E. 1997, ApJ, 476, 589. \nPringle, J. E. 1996, MNRAS, 281, 357. \nR. A. Remillard, E. H. Morgan, J. E. McClintock, C. D. Bailyn, J. A. Orosz, & J. Greiner 1997, Proceedings of 18th Texas Symposium on Relativistic Astrophysics, Eds. A. Olinto, J. Frieman, and D. Schramm (Chicago: World Scientific Press), Dec. 1996, in press (astro-ph/9705064) \nScheuer, P. A. G., & Feiler, R. 1996, MNRAS, 282, 291. \nShakura, N.I. & Sunyaev, R.A. 1973, A&A, 24, 337. \nShu, F., Najita, J., Ostriker, E., & Shang, H. 1995, ApJ, 455, L155. \nStella, L., & Vietri, M. 1997, ApJ, in press. \nThorne, K.S., 1974, ApJ, 191, 507. \nUstyugova, G. V., Kildoba, A. V., Romanova, A. V., Chechetkin, V. M., & Lovelace, R. V. E. 1995, ApJ, 439, L39. \nvan der Klis, M. 1995, in 'X-ray Binaries', eds. W. H. G. Lewin, J. van Paradijs, & E. P. J. van den Heuvel (Cambridge U. Press, Cambridge) p. 252. \nWilkins, D. C. 1972, Phys. Rev. D., 5, 814. \nZhang, S. N., et al. 1996, A&AS, 120C, 279. \nZhang, S. N., Cui, W., & Chen, W. 1997, ApJ, 482, L155 (Paper I). \nTable 1. Inferred Black Hole Angular Momentum ∗ \n| Source | Mass ( M /circledot ) | QPO Frequency (Hz) | a ∗ | References |\n|--------------|-------------------------|----------------------|---------------------|--------------|\n| GRO J1655-40 | 7 | 300 | +0 . 95 (+0 . 93) | 1, 2 |\n| GRS 1915+105 | 30 | 67 | +0 . 95 ( ∼ +1) | 3, 4 |\n| GS 1124-68 | 6 . 3 | 8 | +0 . 35 ( - 0 . 04) | 5, 6 |\n| Cyg X-1 | 10 | 9 | +0 . 48 (+0 . 75) | 7, 8 | \nFig. 1.- Disk precession frequency (multiplied by black hole mass), as a function of distance from the last stable orbit, for different BH angular momentum (assuming Q = 0 . 01; see text). The weak-field limits are also shown in dash-dotted line for comparison. \n<!-- image --> \nFig. 2.- Disk precession frequency at where the disk emission peaks ( r peak ), as a function of the dimensionless specific angular momentum ( a ∗ ) of Kerr black holes (assuming Q = 0 . 01). \n<!-- image -->"}

1997PThPS.128....1M

Chapter 1. Black Hole Perturbation

1997-01-01

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In this chapter, we present analytic calculations of gravitational waves from a particle orbiting a black hole. We first review the Teukolsky formalism for dealing with the gravitational perturbation of a black hole. Then we develop a systematic method to calculate higher order post-Newtonian corrections to the gravitational waves emitted by an orbiting particle. As applications of this method, we consider orbits that are nearly circular, including exactly circular ones, slightly eccentric ones and slightly inclined orbits off the equatorial plane of a Kerr black hole and give the energy flux and angular momentum flux formulas at infinity with higher order post-Newtonian corrections. Using a different method that makes use of an analytic series representation of the solution of the Teukolsky equation, we also give a post-Newtonian expanded formula for the energy flux absorbed by a Kerr black hole for a circular orbit.

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https://arxiv.org/pdf/gr-qc/9712057.pdf

{'Black Hole Perturbation': 'Yasushi Mino 1 , 2 , Misao Sasaki 1 , Masaru Shibata 1 , Hideyuki Tagoshi 3 and Takahiro Tanaka 1 \n1 Department of Earth and Space Science, Osaka University, Toyonaka 560, Japan 2 Department of Physics, Kyoto University, Kyoto 606, Japan 3 National Astronomical Observatory, Mitaka, Tokyo 181, Japan', 'Abstract': 'In this chapter, we present analytic calculations of gravitational waves from a particle orbiting a black hole. We first review the Teukolsky formalism for dealing with the gravitational perturbation of a black hole. Then we develop a systematic method to calculate higher order post-Newtonian corrections to the gravitational waves emitted by an orbiting particle. As applications of this method, we consider orbits that are nearly circular, including exactly circular ones, slightly eccentric ones and slightly inclined orbits off the equatorial plane of a Kerr black hole and give the energy flux and angular momentum flux formulas at infinity with higher order post-Newtonian corrections. Using a different method that makes use of an analytic series representation of the solution of the Teukolsky equation, we also give a postNewtonian expanded formula for the energy flux absorbed by a Kerr black hole for a circular orbit.', '§ 1. Introduction': "In this chapter, we review recent progress in the analytic calculations of gravitational waves from a particle orbiting a black hole using a systematic post-Newtonian expansion method. There has been substantial activities in this field recently and there is a diversity of literature. Here we are mostly concerned with the actual calculations of the gravitational waves from an orbiting particle and we intend to make this chapter as self-contained as possible. We do not, however, discuss much about implications of the results to actual astrophysical situations. \nIn the black hole perturbation approach, one considers gravitational waves from a particle of mass µ orbiting a black hole of mass M assuming µ /lessmuch M . Although this method is restricted to the case when µ /lessmuch M , one can calculate very high order post-Newtonian corrections to gravitational waves using a relatively simple algorithm in contrast with the standard post-Newtonian analysis. This is because the fully relativistic effect of the spacetime curvature is naturally taken into account in the basic perturbation equation. We can also calculate numerically the gravitational waves without assuming the slow motion of its source. Then, we can easily investigate the convergence of the post-Newtonian expansion by comparing the result of the postNewtonian approximation with the fully relativistic one. In this sense, the black hole perturbation method gives a very important test of the post-Newtonian expansion. \nFurther, since the effect of the spacetime curvature is naturally taken into account, we can easily investigate interesting relativistic effects such as tails of gravitational waves. \nWe consider the post-Newtonian wave forms and luminosity which are expanded by v/c , where v is the order of the orbital velocity. The lowest order of the gravitational waves are given by the Newtonian quadrupole formula. We call the postNewtonian formulas for the wave forms and luminosity which contain terms up to O (( v/c ) n ) beyond the Newtonian quadrapole formula as the n/ 2PN formulas. \nLet us first briefly give a historical review. The gravitational perturbation equation of a black hole using the Newman-Penrose formalism 1) was derived by Bardeen and Press 2) for the Schwarzschild black hole, and by Teukolsky 3) for the Kerr black hole. By using these equations, many numerical calculations of gravitational waves induced by the presence of a test particle have been done. We do not list up all such works. Here, we only refer to three articles; Breuer 4) , Chandrasekhar 5) , and Nakamura, Oohara, and Kojima 6) . \nOn the other hand, analytic calculations of gravitational waves produced by the motion of a test particle have not been developed very much until recently. This direction of research was first done by Gal'tsov, Matiukhin and Petukhov 7) in which they considered a case when a particle moves a slightly eccentric orbit around a Schwarzschild black hole, and calculated the gravitational waves up to 1PN order. Then, Poisson 8) considered a case of circular orbit around a Schwarzschild black hole and calculated the wave forms and luminosity to 1.5PN order at which the tail effect appears. Cutler, Finn, Poisson and Sussman 9) also worked on the same problem numerically by using the least square fitting method, and obtained a formula for the luminosity to 2.5PN order. Subsequently, a highly accurate numerical calculation was carried out by Tagoshi and Nakamura 10) . They obtained the formulas for the luminosity to 4PN order numerically by using the least square fitting method. They found the log v terms in the luminosity formula at 3PN and 4PN orders. They showed that, although the convergence of the post-Newtonian expansion is slow, the luminosity formula which is accurate to 3.5PN order will be good enough to represent the orbital phase evolution of coalescing compact binaries accurately. After that, Sasaki 11) found an analytic method and obtained the formulas which are needed to calculate the gravitational waves to 4PN order. Then, Tagoshi and Sasaki 12) obtained the gravitational wave forms and luminosity to 4PN order analytically, and confirmed the results of Tagoshi and Nakamura. These calculations were extended to 5.5PN order by Tanaka, Tagoshi, and Sasaki 13) . \nIn the case of orbits around a Kerr black hole, Poisson calculated the 1.5PN order corrections to the wave forms and luminosity due to the rotation of the black hole and showed that the result agrees with the standard post-Newtonian effect due to spin-orbit coupling 14) . Then, Shibata, Sasaki, Tagoshi and Tanaka 15) calculated the luminosity to 2.5PN order. They calculated the luminosity from a particle in circular orbit with small inclination from the equatorial plane. They used the SasakiNakamura equation as well as the Teukolsky equation. This analysis was extended to 4PN order by Tagoshi, Shibata, Tanaka and Sasaki 16) in which the orbit of the test particle was restricted to circular ones on the equatorial plane. The analysis in the \ncase of slightly eccentric orbit on the equatorial plane was also done by Tagoshi 17) to 2.5PN order. \nTanaka, Mino, Sasaki, and Shibata 18) considered the case when a spinning particle moves a circular orbits near the equatorial plane around a Kerr black hole, and derived the luminosity formula to 2.5PN order including the linear order effect of the particle's spin. They used the equations of motion of Papapetrou's 19) and the energy momentum tensor of the spinning particle given by Dixon 20) . \nThe absorption of gravitational waves into the black hole horizon, appearing at 4PN order in the Schwarzschild case, was calculated by Poisson and Sasaki in the case when a test particle is in a circular orbit 21) . The black hole absorption in the case of rotating black hole appears at 2.5PN order 22) . Recently a new analytic method to solve the homogeneous Teukolsky equation was found by Mano, Suzuki, and Takasugi 23) . Using this method, the black hole absorption in the case of rotating black hole was calculated by Tagoshi, Mano, and Takasugi 24) to 6.5PN order beyond the quadrupole formula. \nIf gravity is not described by the Einstein theory but by the Brans-Dicke theory, there will appear scalar type gravitational waves as well as transverse-traceless gravitational waves. Such scalar type gravitational waves were calculated by Ohashi, Tagoshi and Sasaki 25) in a case when a compact star is in a circular orbit on the equatorial plane around a Kerr black hole. \nThe organization of this chapter is as follows. We review the Teukolsky formalism for the black hole perturbation in section 2 and formulate a post-Newtonian expansion method of the Teukolsky equation in section 3. Then we turn to the evaluation of gravitational waves by an orbiting particle in the rest of sections. \nFirst we consider circular orbits. In section 4, we calculate the gravitational wave luminosity from a test particle in circular orbit around a Schwarzschild black hole to 5.5PN order, based on Tanaka, Tagoshi, and Sasaki 13) . This is the highest postNewtonian order achieved so far. Based on this result, we investigate the convergence property of the post-Newtonian expansion in section 5. In section 6, we consider circular orbits on the equatorial plane around a Kerr black hole and calculate the luminosity to 4PN order, based on Tagoshi, Shibata, Tanaka, and Sasaki 16) . We find the luminosity contains the terms which describe the effect of not only spinorbit coupling but also the effect of higher multipole moments of the Kerr black hole. \nNext we consider slightly noncircular orbits. In section 7, we calculate the O ( e 2 ) corrections to the 4PN energy and angular momentum flux formulas in the case of a slightly eccentric orbit around a Schwarzschild black hole, where e is the eccentricity. In section 8, we consider a slightly eccentric orbit on the equatorial plane around a Kerr black hole and evaluate the O ( e 2 ) corrections to 2.5PN order, based on Tagoshi 17) . Then in section 9, we calculate the gravitational waves induced by a test particle in circular orbit with small inclination from the equatorial plane around a Kerr black hole and evaluate the 2.5PN energy and angular momentum fluxes, based on Shibata, Sasaki, Tagoshi, Tanaka 15) . In section 10, we discuss the adiabatic orbital evolution around a Kerr black hole under radiation reaction and show that circular orbits will remain circular under adiabatic radiation reaction but \nthe stability of circular orbits can only be examined by an explicit evaluation of the backreaction force. \nIn section 11, we consider the effect of the spin of a particle. We first give a general formalism to treat the gravitational radiation from a spinning particle orbiting a Kerr black hole. Then we calculate the 2.5PN luminosity formula with the first order corrections of the spin for circular orbits which are slightly inclined due to the spin of the particle. \nFinally, in section 12, we review a calculation of the black hole absorption based on Tagoshi, Mano, and Takasugi 24) . The black hole absorption effect appears at O ( v 5 ) relative to the Newtonian quadrupole luminosity for a Kerr black hole, while at O ( v 8 ) for a Schwarzschild black hole. We show the energy absorption rate to O ( v 8 ) beyond the lowest order for the Kerr case, i.e., O ( v 13 ) or 6.5PN order beyond the Newtonian quadrupole luminosity. \nSince many of the calculations encountered in this black hole perturbation approach are lengthy, various subsidiary equations and formulas are deferred to appendices A to J. In the rest of this chapter, we use the units of c = G = 1.", '§ 2. Teukolsky formalism': "In terms of the conventional Boyer-Lindquist coordinates, the metric of a Kerr black hole is expressed as \nds 2 = -∆ Σ ( dt -a sin 2 θ dϕ ) 2 + sin 2 θ Σ [ ( r 2 + a 2 ) dϕ -adt ] 2 + Σ ∆ dr 2 + Σdθ 2 , (2 . 1) \nwhere Σ = r 2 + a 2 cos 2 θ and ∆ = r 2 -2 Mr + a 2 . In the Teukolsky formalism 3) , the gravitational perturbations of a Kerr black hole are described by a NewmanPenrose quantity ψ 4 = -C αβγδ n α ¯ m β n γ ¯ m δ , where C αβγδ is the Weyl tensor, n α = (( r 2 + a 2 ) , ∆, 0 , a ) / (2 Σ ) and m α = ( ia sin θ, 0 , 1 , i/ sin θ ) / ( √ 2( r + ia cos θ )). \nWe decompose ψ 4 into Fourier-harmonic components according to \n- \n( r -ia cos θ ) 4 ψ 4 = ∑ /lscriptm ∫ dωe -iωt + imϕ -2 S /lscriptm ( θ ) R /lscriptmω ( r ) , (2 . 2) \nThe radial function R /lscriptmω and the angular function s S /lscriptm ( θ ) satisfy the Teukolsky equations with s = -2 as \n∆ 2 d dr ( 1 ∆ dR /lscriptmω dr ) -V ( r ) R /lscriptmω = T /lscriptmω , (2 . 3) \n[ 1 sin θ d dθ { sin θ d dθ } -a 2 ω 2 sin 2 θ -( m -2cos θ ) 2 sin 2 θ +4 aω cos θ -2 + 2 maω + λ ] -2 S /lscriptm = 0 . (2 . 4) \nThe potential V ( r ) is given by \nV ( r ) = -K 2 +4 i ( r -M ) K ∆ +8 iωr + λ, (2 . 5) \nwhere K = ( r 2 + a 2 ) ω -ma and λ is the eigenvalue of -2 S aω /lscriptm . The angular function s S /lscriptm ( θ ) is the spin-weighted spheroidal harmonic which may be normalized as \n∫ π 0 | -2 S /lscriptm | 2 sin θdθ = 1 . (2 . 6) \nThe source term T /lscriptmω is specified later. Here we only mention that for orbits of our interest, which are bounded, T /lscriptmω has support in a compact range of r . \nWe define two kinds of homogeneous solutions of the radial Teukolsky equation: \nR in /lscriptmω → { B trans /lscriptmω ∆ 2 e -ikr ∗ for r → r + r 3 B ref /lscriptmω e iωr ∗ + r -1 B inc /lscriptmω e -iωr ∗ for r → + ∞ , (2 . 7) \nwhere k = ω -ma/ 2 Mr + and r ∗ is the tortoise coordinate defined by \nR up /lscriptmω → { C up /lscriptmω e ikr ∗ + ∆ 2 C ref /lscriptmω e -ikr ∗ for r → r + , C trans /lscriptmω r 3 e iωr ∗ for r → + ∞ (2 . 8) \nr ∗ = ∫ dr ∗ dr dr = r + 2 Mr + r + -r -ln r -r + 2 M -2 Mr -r + -r -ln r -r -2 M , (2 . 9) \nwhere r ± = M ± √ M 2 -a 2 , and for definiteness, we have fixed the integration constant. \nWe solve the radial Teukolsky equation by using the Green function method. A solution of the Teukolsky equation which has purely outgoing property at infinity and has purely ingoing property at the horizon is given by \nR /lscriptmω = 1 W /lscriptmω { R up /lscriptmω ∫ r r + dr ' R in /lscriptmω T /lscriptmω ∆ -2 + R in /lscriptmω ∫ ∞ r dr ' R up /lscriptmω T /lscriptmω ∆ -2 } , (2 . 10) \nwhere the Wronskian W /lscriptmω is given by \nW /lscriptmω = 2 iωC trans /lscriptmω B inc /lscriptmω . (2 . 11) \nThen, the solution has an asymptotic property at the horizon as \nR /lscriptmω ( r → r + ) → B trans /lscriptmω ∆ 2 e -ikr ∗ 2 iωC trans /lscriptmω B inc /lscriptmω ∫ ∞ r + dr ' R up /lscriptmω T /lscriptmω ∆ -2 ≡ ˜ Z H /lscriptmω ∆ 2 e -ikr ∗ , (2 . 12) \nThe solution at infinity is also expressed as \nR /lscriptmω ( r →∞ ) → r 3 e iωr ∗ 2 iωB inc /lscriptmω ∫ ∞ r + dr ' T /lscriptmω ( r ' ) R in /lscriptmω ( r ' ) ∆ 2 ( r ' ) ≡ ˜ Z ∞ /lscriptmω r 3 e iωr ∗ , (2 . 13) \nHere and in the following sections except for section 12, we focus on the gravitational waves emitted to infinity. Hence ˜ Z ∞ /lscriptmω will be simply denoted as ˜ Z /lscriptmω . The gravitational waves absorbed into the black hole horizon will be treated separately in section 12. \nNow let us discuss the general form of the source term T /lscriptmω . It is given by \nT /lscriptmω = 4 ∫ dΩdtρ -5 ρ -1 ( B ' 2 + B '∗ 2 ) e -imϕ + iωt -2 S aω /lscriptm √ 2 π , (2 . 14) \nB ' 2 = -1 2 ρ 8 ρL -1 [ ρ -4 L 0 ( ρ -2 ρ -1 T nn )] -1 2 √ 2 ρ 8 ρ∆ 2 L -1 [ ρ -4 ρ 2 J + ( ρ -2 ρ -2 ∆ -1 T mn )] , B '∗ 2 = -1 4 ρ 8 ρ∆ 2 J + [ ρ -4 J + ( ρ -2 ρT mm )] -1 2 √ 2 ρ 8 ρ∆ 2 J + [ ρ -4 ρ 2 ∆ -1 L -1 ( ρ -2 ρ -2 T mn )] , (2 . 15) \nwhere \nwith \nρ = ( r -ia cos θ ) -1 , L s = ∂ θ + m sin θ -aω sin θ + s cot θ, (2 . 16) J + = ∂ r + iK/∆. \nIn the above, T nn , T mn and T mm are the tetrad components of the energy momentum tensor ( T nn = T µν n µ n ν etc.), and the bar denotes the complex conjugation. \nWe consider T µν of a monopole particle of mass µ . The case of a spinning particle will be discussed in section 11 separately. The energy momentum tensor takes the form, \nT µν = µ Σ sin θdt/dτ dz µ dτ dz ν dτ δ ( r -r ( t )) δ ( θ -θ ( t )) δ ( ϕ -ϕ ( t )) . (2 . 17) \nwhere z µ = ( t, r ( t ) , θ ( t ) , ϕ ( t ) ) is a geodesic trajectory and τ = τ ( t ) is the proper time along the geodesic. The geodesic equations in Kerr geometry are given by \nΣ dθ dτ = ± [ C -cos 2 θ { a 2 (1 -E 2 ) + l 2 z sin 2 θ }] 1 / 2 ≡ Θ ( θ ) , Σ dϕ dτ = -( aE -l z sin 2 θ ) + a ∆ ( E ( r 2 + a 2 ) -al z ) ≡ Φ, Σ dt dτ = -( aE -l z sin 2 θ ) a sin 2 θ + r 2 + a 2 ∆ ( E ( r 2 + a 2 ) -al z ) ≡ T, Σ dr dτ = ± √ R, (2 . 18) \nwhere E , l z and C are the energy, the z -component of the angular momentum and the Carter constant of a test particle, respectively. ∗ ) Σ = r 2 + a 2 cos 2 θ and \nR = [ E ( r 2 + a 2 ) -al z ] 2 -∆ [( Ea -l z ) 2 + r 2 + C ] . (2 . 19) \nUsing Eq. (2 . 18), the tetrad components of the energy momentum tensor are expressed as \nT nn = µ C nn sin θ δ ( r -r ( t )) δ ( θ -θ ( t )) δ ( ϕ -ϕ ( t )) , T mn = µ C mn sin θ δ ( r -r ( t )) δ ( θ -θ ( t )) δ ( ϕ -ϕ ( t )) , (2 . 20) T mm = µ C mm sin θ δ ( r -r ( t )) δ ( θ -θ ( t )) δ ( ϕ -ϕ ( t )) , \nwhere \nC nn = 1 4 Σ 3 ˙ t [ E ( r 2 + a 2 ) -al z + Σ dr dτ ] 2 , C mn = -ρ 2 √ 2 Σ 2 ˙ t [ E ( r 2 + a 2 ) -al z + Σ dr dτ ] [ i sin θ ( aE -l z sin 2 θ )] , (2 . 21) C mm = ρ 2 2 Σ ˙ t [ i sin θ ( aE -l z sin 2 θ )] 2 , \nand ˙ t = dt/dτ . Substituting Eq. (2 . 15) into Eq. (2 . 14) and performing integration by part, we obtain \nT /lscriptmω = 4 µ √ 2 π ∫ ∞ -∞ dt ∫ dθe iωt -imϕ ( t ) × [ -1 2 L † 1 { ρ -4 L † 2 ( ρ 3 S ) } C nn ρ -2 ρ -1 δ ( r -r ( t )) δ ( θ -θ ( t )) + ∆ 2 ρ 2 √ 2 ρ ( L † 2 S + ia ( ρ -ρ ) sin θS ) J + { C mn ρ -2 ρ -2 ∆ -1 δ ( r -r ( t )) δ ( θ -θ ( t )) } + 1 2 √ 2 L † 2 { ρ 3 S ( ρ 2 ρ -4 ) ,r } C mn ∆ρ -2 ρ -2 δ ( r -r ( t )) δ ( θ -θ ( t )) -1 4 ρ 3 ∆ 2 SJ + { ρ -4 J + ( ρρ -2 C mm δ ( r -r ( t )) δ ( θ -θ ( t )) )} ] , (2 . 22) \nwhere \nL † s = ∂ θ -m sin θ + aω sin θ + s cot θ, (2 . 23) \nand S denotes -2 S aω /lscriptm ( θ ) for simplicity. \nWe further rewrite Eq. (2 . 22) as \nT /lscriptmω = µ ∫ ∞ -∞ dte iωt -imϕ ( t ) ∆ 2 [ ( A nn 0 + A mn 0 + A mm 0 ) δ ( r -r ( t )) + { ( A mn 1 + A mm 1 ) δ ( r -r ( t )) } ,r + { A mm 2 δ ( r -r ( t )) } ,rr ] , (2 . 24) \nwhere \nA nn 0 = -2 √ 2 π∆ 2 C nn ρ -2 ρ -1 L + 1 { ρ -4 L + 2 ( ρ 3 S ) } , A mn 0 = 2 √ π∆ C mn ρ -3 [ ( L + 2 S ) ( iK ∆ + ρ + ρ ) -a sin θS K ∆ ( ρ -ρ ) ] , A mm 0 = -1 √ 2 π ρ -3 ρC mm S [ -i ( K ∆ ) ,r -K 2 ∆ 2 +2 iρ K ∆ ] , A mn 1 = 2 √ π∆ ρ -3 C mn [ L + 2 S + ia sin θ ( ρ -ρ ) S ] , A mm 1 = -2 √ 2 π ρ -3 ρC mm S ( i K ∆ + ρ ) , A mm 2 = -1 √ 2 π ρ -3 ρC mm S. (2 . 25) \nInserting Eq. (2 . 24) into Eq. (2 . 13), we obtain ˜ Z /lscriptmω as \n˜ Z /lscriptmω = µ 2 iωB inc /lscriptmω ∫ ∞ -∞ dte iωt -imϕ ( t ) W /lscriptmω , (2 . 26) \nwhere \nW /lscriptmω = [ R in /lscriptmω { A nn 0 + A mn 0 + A mm 0 } -dR in /lscriptmω dr { A mn 1 + A mm 1 } + d 2 R in /lscriptmω dr 2 A mm 2 ] r = r ( t ) . (2 . 27) \nIn this paper, we focus on orbits which are either circular (with or without inclination) or eccentric but confined on the equatorial plane. In either case, the frequency spectrum of T /lscriptmω becomes discrete. Accordingly ˜ Z /lscriptmω in Eq. (2 . 12) or (2 . 13) takes the form, \nThen, in particular, ψ 4 at r →∞ is obtained from Eq. (2 . 2) as \n˜ Z /lscriptmω = ∑ n δ ( ω -ω n ) Z /lscriptmω . (2 . 28) \nψ 4 = 1 r ∑ /lscriptmn Z /lscriptmω n -2 S aω n /lscriptm √ 2 π e iω n ( r ∗ -t )+ imϕ . (2 . 29) \nAt infinity, ψ 4 is related to the two independent modes of gravitational waves h + and h × as \nψ 4 = 1 2 ( h + -i h × ) . (2 . 30) \nFrom Eqs.(2 . 29) and (2 . 30), the luminosity averaged over t /greatermuch ∆t , where ∆t is the characteristic time scale of the orbital motion (e.g., a period between the two consecutive apastrons), is given by \n〈 dE dt 〉 = ∑ /lscript,m,n ∣ ∣ ∣ ∣ Z /lscriptmω n ∣ ∣ ∣ ∣ 2 4 πω 2 n ≡ ∑ /lscript,m,n ( dE dt ) /lscriptmn . (2 . 31) \nIn the same way, the time-averaged angular momentum flux is given by \n〈 dJ z dt 〉 = ∑ /lscript,m,n m ∣ ∣ ∣ ∣ Z /lscriptmω n ∣ ∣ ∣ ∣ 2 4 πω 3 n ≡ ∑ /lscript,m,n ( dJ z dt ) /lscriptmn = ∑ /lscript,m,n m ω n ( dE dt ) /lscriptmn . (2 . 32)", '§ 3. Post-Newtonian expansion of the ingoing wave solutions': 'We consider the case when a test particle of mass µ is in an orbit which is nearly circular around a Kerr black hole of mass M /greatermuch µ and describe a method to calculate the ingoing wave Teukolsky functions R in /lscriptmω which are necessary to evaluate the 4PN formulas for the gravitational waves energy and angular momentum fluxes emitted to infinity. In the Schwarzschild case, we shall derive the 5.5PN luminosity formula in section 4. A method to calculate the ingoing wave solutions in this case is separately discussed in Appendix D because it is considerably more complicated than the method explained in this section. \nUsing non-dimensional variables in the Teukolsky equation, we can see that the Teukolsky equation is expressed in terms of three basic variables, z ≡ ωr , /epsilon1 ≡ 2 Mω and aω ≡ q/epsilon1/ 2 where q ≡ a/M . In order to calculate the gravitational waves induced by a particle, we need to know the explicit form of the source terms T /lscriptmω ( r ). They will be given in the proceeding sections for specified orbits. Here it is sufficient to note that they have support only around r = r 0 where r 0 is the orbital radius for a circular orbit or the mean radius in the case of an eccentric orbit (with small eccentricity). Hence from Eq. (2 . 13), what we need to know are the ingoing wave functions R in ( r ) around r = r 0 , and their incident amplitudes B inc . Note that we do not need the transmission amplitudes B trans to evaluate the gravitational waves at infinity. This fact considerably simplifies the calculations. Since we treat a test particle in a bound orbit which is nearly circular, the contribution of ω to the Teukolsky functions comes from ω ∼ mΩ ϕ , where Ω ϕ ∼ ( M/r 3 0 ) 1 / 2 is the orbital angular frequency. We will evaluate R in by setting three basic variables to be z = ωr 0 ∼ m ( M/r 0 ) 1 / 2 ∼ v , /epsilon1 ∼ 2 m ( M/r 0 ) 3 / 2 ∼ v 3 , and aω ∼ qm ( M/r 0 ) 3 / 2 ∼ v 3 . Here, we have introduced a parameter v ≡ ( M/r 0 ) 1 / 2 which represents the magnitude of the orbital velocity. \nWe assume that v is much smaller than the velocity of light; v /lessmuch 1. Consequently, we also assume that /epsilon1 /lessmuch v /lessmuch 1. This relation is the basic assumption in obtaining the homogeneous solutions below. \nNow we calculate the ingoing wave solutions which are necessary to calculate the luminosity to O ( v 8 ) beyond the lowest order for the Kerr case. The method is mainly based on Shibata et al. 15) and Tagoshi et al. 16) . An extension to O ( v 11 ) calculations done by Tanaka, Tagoshi and Sasaki 13) for the Schwarzschild case is given in Appendix D. \nFirst, we discuss the angular solutions. The angular solutions are the spinweighted spheroidal harmonics. The angular equation (2 . 4) contains only one small parameter aω . It is straightforward to calculate the spin-weighted spheroidal harmonic -2 S /lscriptm and its eigenvalue λ by expanding the solution in power of aω . It can be done by the usual perturbation method 26) , 16) , 15) . It is also possible to obtain them by using an expansion by means of Jacobi functions 27) . The method and the results are given explicitly in Appendix A. Here we only show the eigenvalue λ which is used to calculate the radial functions. The eigenvalue λ is given by \nλ = λ 0 + aωλ 1 + a 2 ω 2 λ 2 + O (( aω ) 3 ) (3 . 1) \nwhere \nλ 0 = /lscript ( /lscript +1) \n-λ 1 = -2 m /lscript ( /lscript +1) + 4 /lscript ( /lscript +1) , \n2 = ( /lscript -1)( /lscript +2) , \nλ 2 = -2( /lscript +1)( c /lscript +1 /lscriptm ) 2 +2 /lscript ( c /lscript -1 /lscriptm ) 2 + 2 3 -2 3 ( /lscript +4)( /lscript -3)( /lscript 2 + /lscript -3 m 2 ) /lscript ( /lscript +1)(2 /lscript +3)(2 /lscript -1) , (3 . 2) \nwith \nc /lscript +1 /lscriptm = 2 ( /lscript +1) 2 [ ( /lscript +3)( /lscript -1)( /lscript + m +1)( /lscript -m +1) (2 /lscript +1)(2 /lscript +3) ] 1 / 2 , c /lscript -1 /lscriptm = -2 /lscript 2 [ ( /lscript +2)( /lscript -2)( /lscript + m )( /lscript -m ) (2 /lscript +1)(2 /lscript -1) ] 1 / 2 . (3 . 3) \nNext we consider the homogeneous solution R in . We assume ω > 0 below. The solution for ω < 0 can be obtained from the one for ω > 0 by using the symmetry of the homogeneous Teukolsky equation which implies R /lscript,m,ω = R /lscript, -m, -ω . Here, we do not treat the Teukolsky equation directly. Instead, we transform the homogeneous Teukolsky equation to the Sasaki-Nakamura equation 28) , which is given by \n[ d 2 dr ∗ 2 -F ( r ) d dr ∗ -U ( r ) ] X /lscriptmω = 0 . (3 . 4) \nThe function F ( r ) is given by \nF ( r ) = η ,r η ∆ r 2 + a 2 , (3 . 5) \nwhere \nwith \nc 0 = -12 iωM + λ ( λ +2) -12 aω ( aω -m ) , c 1 = 8 ia [3 aω -λ ( aω -m )] , c 2 = -24 iaM ( aω -m ) + 12 a 2 [1 -2( aω -m ) 2 ] , c 3 = 24 ia 3 ( aω -m ) -24 Ma 2 , c 4 = 12 a 4 . (3 . 7) \nThe function U ( r ) is given by \nU ( r ) = ∆U 1 ( r 2 + a 2 ) 2 + G 2 + ∆G ,r r 2 + a 2 -FG, (3 . 8) \nwhere \nG = -2( r -M ) r 2 + a 2 + r∆ ( r 2 + a 2 ) 2 , U 1 = V + ∆ 2 β [( 2 α + β ,r ∆ ) ,r -η ,r η ( α + β ,r ∆ )] , α = -i Kβ ∆ 2 +3 iK ,r + λ + 6 ∆ r 2 , β = 2 ∆ ( -iK + r -M -2 ∆ r ) . (3 . 9) \nWhen we set a = 0, this transformation becomes the Chandrasekhar transformation 29) for the Schwarzschild black hole. The Sasaki-Nakamura equation was originally introduced, for the inhomogeneous case, to make the potential term short-ranged and to make the source term well-behaved at infinity. It is not necessary to perform this transformation in this case, since we are interested only in bound orbits. Nevertheless we choose to do this because the lowest order solution becomes the spherical Bessel function and we can apply the post-Newtonian expansion techniques developed for the Schwarzschild case by Poisson 8) and Sasaki 11) . \nThe relation between R /lscriptmω and X /lscriptmω is \nR /lscriptmω = 1 η {( α + β ,r ∆ ) χ /lscriptmω -β ∆ χ /lscriptmω,r } , (3 . 10) \nwhere χ /lscriptmω = X /lscriptmω ∆/ ( r 2 + a 2 ) 1 / 2 . Conversely, we can express X /lscriptmω in terms of R /lscriptmω as \nwhere J -= ( d/dr ) -i ( K/∆ ). Then the asymptotic behavior of the ingoing-wave solution X in which corresponds to Eq. (2 . 7) is \nX /lscriptmω = ( r 2 + a 2 ) 1 / 2 r 2 J -J -[ 1 r 2 R /lscriptmω ] , (3 . 11) \nX in /lscriptmω → { A ref /lscriptmω e iωr ∗ + A inc /lscriptmω e -iωr ∗ for r ∗ →∞ A trans /lscriptmω e -ikr ∗ for r ∗ →-∞ . (3 . 12) \nChapter 1 \nBlack Hole Perturbation \nη = c 0 + c 1 /r + c 2 /r 2 + c 3 /r 3 + c 4 /r 4 , (3 . 6) \nThe coefficient A inc , A ref and A trans are respectively related to B inc , B ref and B trans , defined in Eq. (2 . 7), by \nB inc /lscriptmω = -1 4 ω 2 A inc /lscriptmω , (3 . 13) \nB ref /lscriptmω = -4 ω 2 c 0 A ref /lscriptmω , (3 . 14) \nB trans /lscriptmω = 1 d /lscriptmω A trans /lscriptmω , (3 . 15) \n(3 . 16) \nwhere c 0 is given in Eq. (3 . 7) and \nd /lscriptmω = √ 2 Mr + [(8 -24 iMω -16 M 2 ω 2 ) r 2 + +(12 iam -16 M +16 amMω +24 iM 2 ω ) r + -4 a 2 m 2 -12 iamM +8 M 2 ] . \nNow we introduce the variable z ∗ as \nz ∗ = z + /epsilon1 [ z + z + -z -ln( z -z + ) -z -z + -z -ln( z -z -) ] = ωr ∗ + /epsilon1 ln /epsilon1 , (3 . 17) \nwhere z = ωr and z ± = ωr ± . To solve for X in , we set \nwhere \nX in /lscriptmω = √ z 2 + a 2 ω 2 ξ /lscriptm ( z ) exp ( -iφ ( z )) , (3 . 18) \nφ ( z ) = ∫ dr ( K ∆ -ω ) = z ∗ -z -/epsilon1 2 mq 1 z + -z -ln z -z + z -z -. (3 . 19) \nWith this choice of the phase function, the ingoing wave boundary condition at horizon reduces to that ξ /lscriptm is regular and finite at z = z + . \nInserting Eq. (3 . 18) into Eq. (3 . 4) and expanding it in powers of /epsilon1 = 2 Mω , we obtain \nL (0) [ ξ /lscriptm ] = /epsilon1L (1) [ ξ /lscriptm ] + /epsilon1Q (1) [ ξ /lscriptm ] + /epsilon1 2 Q (2) [ ξ /lscriptm ] + /epsilon1 3 Q (3) [ ξ /lscriptm ] + /epsilon1 4 Q (4) [ ξ /lscriptm ] + O ( /epsilon1 5 ) , (3 . 20) \nwhere L (0) , L (1) , Q (1) and Q (2) are differential operators given by \nL (0) = d 2 dz 2 + 2 z d dz + ( 1 -/lscript ( /lscript +1) z 2 ) , (3 . 21) \nQ (1) = iqλ 1 2 z 2 d dz -4 imq l ( l +1) z 3 , (3 . 23) \nL (1) = 1 z d 2 dz 2 + ( 1 z 2 + 2 i z ) d dz -( 4 z 3 -i z 2 + 1 z ) , (3 . 22) \nwhere \n( ( ( \nThe formulas for Q (2) , Q (3) and Q (4) are very complicated, and they are given explicitly in Appendix B. Note that, when we set a = 0, all Q ( n ) vanish. \nBy expanding ξ /lscriptm in terms of /epsilon1 as \nξ /lscriptm = ∞ ∑ n =0 /epsilon1 n ξ ( n ) /lscriptm ( z ) , (3 . 24) \nwe obtain from Eq. (3 . 20) the iterative equations, \nL (0) [ ξ ( n ) /lscriptm ] = z -2 W ( n ) /lscriptm , (3 . 25) \nW (0) /lscriptm = 0 , (3 . 26) \n) W (3) /lscriptm = z 2 L (1) [ ξ (2) /lscriptm ] + Q (1) [ ξ (2) /lscriptm ] + Q (2) [ ξ (1) /lscriptm ] + Q (3) [ ξ (0) /lscriptm ] ) , (3 . 29) \nW (1) /lscriptm = z 2 L (1) [ ξ (0) /lscriptm ] + Q (1) [ ξ (0) /lscriptm ] ) , (3 . 27) \n) W (4) /lscriptm = z 2 ( L (1) [ ξ (3) /lscriptm ] + Q (1) [ ξ (3) /lscriptm ] + Q (2) [ ξ (2) /lscriptm ] + Q (3) [ ξ (1) /lscriptm ] + Q (4) [ ξ (0) /lscriptm ] ) . (3 . 30) \n) W (2) /lscriptm = z 2 L (1) [ ξ (1) /lscriptm ] + Q (1) [ ξ (1) /lscriptm ] + Q (2) [ ξ (0) /lscriptm ] ) , (3 . 28) \nAs /epsilon1 = 2 GMω if we recover G , the above expansion corresponds to the postMinkowski expansion of the vacuum Einstein equations. \nThe iterative equations (3 . 25) have been obtained by expanding Eq. (3 . 4) in powers of /epsilon1 by regarding z = ωr as the independent variable. Since the horizon is at z = z + = O ( /epsilon1 ), this procedure implicitly assumes that /epsilon1 /lessmuch z . Consequently, we cannot apply the above expansion near the horizon where the ingoing wave boundary condition is to be imposed. To implement the boundary condition correctly, we have to consider a series solution of ξ /lscriptm which is valid near the horizon as well as in the range /epsilon1 /lessmuch z and match it to the series solution of the form (3 . 24). Recently this matching problem has been rigorously solved by Mano, Suzuki, and Takasugi 23) for the original Teukolsky equation. However, for our present purpose, it is sufficient to resort to a simple power-counting argument, by which it is possible to implement the boundary condition of ξ /lscriptm at the horizon to the behaviors of ξ ( n ) /lscriptm at z /greatermuch /epsilon1 for n 2 /lscript (for n ≤ 2 /lscript +1 in the Schwarzschild case; see Appendix D). \n≤ \n≤ Since the ingoing wave boundary condition is that ξ /lscriptm is regular at horizon, if we introduce an independent variable x := ( z -z + ) //epsilon1 we can expand ξ /lscriptm near the horizon as \nThis means we have \nξ /lscriptm = ∞ ∑ n =0 /epsilon1 n ξ { n } /lscriptm ( x ) , (3 . 31) \nξ /lscriptm (0) = ξ { 0 } /lscriptm (0) ∞ ∑ n =0 /epsilon1 n C n , (3 . 32) \nwhere C n = ξ { n } /lscriptm (0) /ξ { 0 } /lscriptm (0). In other words, ξ /lscriptm (0) should have a well-defined limit for /epsilon1 → 0 except for the overall normalization factor ξ { 0 } /lscriptm (0). Keeping this property in mind, let us consider the boundary conditions for Eqs. (3 . 26) - (3 . 30). \nThe general solution to Eq. (3 . 26) is immediately obtained as \nξ (0) /lscriptm = α (0) /lscriptm j /lscript + β (0) /lscriptm n /lscript , (3 . 33) \nwhere j /lscript and n /lscript are the spherical Bessel functions. The coefficients α (0) /lscriptm and β (0) /lscriptm are to be determined by the boundary condition. For convenience, we normalize the solution X in such that the incident amplitude A inc is of order unity. Then both α (0) /lscriptm and β (0) /lscriptm must be of order unity. Since j /lscript ( z ) ∼ z /lscript and n /lscript ( z ) ∼ z -/lscript -1 for z /lessmuch 1, the latter is O ( /epsilon1 2 /lscript +1 ) larger than the former near the horizon where z = O ( /epsilon1 ). Hence Eq. (3 . 32) implies we should set β (0) /lscriptm = 0. As for the value of α (0) /lscriptm , since it only contributes to the overall normalization of X in , we set α (0) /lscriptm = 1 for convenience. \nInspection of Eqs. (3 . 27) - (3 . 30) reveals that the solution ξ ( n ) /lscriptm behaves as z /lscript -n plus the homogeneous solution α ( n ) /lscriptm j /lscript + β ( n ) /lscriptm n /lscript for z → 0. As for α ( n ) /lscriptm ( n ≥ 1), they simply contribute to renormalizations of α (0) /lscriptm . Hence we put them to zero. As for β ( n ) /lscriptm , from the same argument as given above, we find they may become non-zero only for n ≥ 2 /lscript + 1. Since /lscript ≥ 2 and /epsilon1 = O ( v 3 ), this implies that the near zone contribution of n /lscript ∼ z -/lscript -1 , which is O ( v -(2 /lscript +1) ) greater than the lowest order term j /lscript , to the gravitational waves emitted to infinity may arise only at O ( v 10 ) beyond the quadrupole order. Since the post-Newtonian corrections we shall consider for the Kerr case are those up to O ( v 8 ), we set β ( n ) /lscriptm = 0 and solve the iterative equations (3 . 25) to O ( /epsilon1 4 ) with the boundary conditions that ξ ( n ) /lscriptm ∼ z /lscript -n at z → 0. We note that, in the Schwarzschild case which is discussed separately in Appendix D, these boundary conditions turn out to be appropriate for n ≤ 2 /lscript +1; i.e., up to one power of /epsilon1 higher than the Kerr case. \nTo calculate ξ ( n ) /lscriptm for n ≥ 1, we rewrite Eqs. (3 . 27) - (3 . 30) in the indefinite integral form by using the spherical Bessel functions as \nξ ( n ) /lscriptm = n /lscript ∫ z dzj /lscript W ( n ) /lscriptm -j /lscript ∫ z dzn /lscript W ( n ) /lscriptm . (3 . 34) \nThe calculation is straightforward but tedious. All the formulas which are needed to calculate the above integration to obtain ξ ( n ) /lscriptm for n ≤ 2 are shown in Appendix of Sasaki 11) . They are recapitulated in an alternative way in Appendix D. Using those formulas, we have for n = 1, \nξ (1) /lscriptm = ( /lscript -1)( /lscript +3) 2( /lscript +1)(2 /lscript +1) j /lscript +1 -( /lscript 2 -4 2 /lscript (2 /lscript +1) + 2 /lscript -1 /lscript ( /lscript -1) ) j /lscript -1 + R /lscript, 0 j 0 + /lscript -2 ∑ m =1 ( 1 m + 1 m +1 ) R /lscript,m j m -2 D nj /lscript + ij /lscript ln z + imq 2 ( /lscript 2 +4 /lscript 2 (2 /lscript +1) ) j /lscript -1 + imq 2 ( ( /lscript +1) 2 +4 ( /lscript +1) 2 (2 /lscript +1) ) j /lscript +1 , (3 . 35) \nwhere D nj /lscript is an extension of the spherical Bessel function defined in subsection D.2.1 \nof Appendix D; \nD nj /lscript = 1 2 [ j /lscript Si(2 z ) -n /lscript (Ci(2 z ) -γ -ln 2 z )] , (3 . 36) \nwhere γ = 0 . 5772 · · · is the Euler constant, Ci( x ) = -∫ ∞ x dt cos t/t and Si( x ) = ∫ x 0 dt sin t/t , and R m,k is a polynomial of the inverse power of z defined by \nfor m>k and it is defined by \nR m,k = z 2 ( n m j k -j m n k ) = -[( m -k -1) / 2] ∑ r =0 ( -1) r ( m -k -1 -r )! Γ ( m + 1 2 -r ) r ! ( m -k -1 -2 r )! Γ ( k + 3 2 + r ) ( 2 z ) m -k -1 -2 r , (3 . 37) \nR m,k = -R k,m . (3 . 38) \nfor m<k . \nNext we consider ξ (2) /lscriptm . From Eqs.(3 . 34) and (3 . 35), we obtain ξ (2) /lscriptm as \nξ (2) /lscriptm = f (2) /lscript + ig (2) /lscript + k (2) /lscriptm ( q ) , (3 . 39) \nwhere f (2) /lscript and g (2) /lscript are the real and imaginary parts of ξ (2) /lscriptm in the Schwarzschild limit, respectively, and k (2) /lscriptm ( q ) is the correction term due to non-vanishing q = a/M . For /lscript = 2 and /lscript = 3, f (2) /lscript are given by \nf (2) 2 = -389 70 z 2 j 0 -113 420 z j 1 + 1 7 z j 3 +4 D nnj 2 -5 3 z D nj 2 + 10 3 D nj 1 + 6 z D nj 0 + 107 105 D nj -3 -107 210 j 2 ln z -1 2 j 2 (ln z ) 2 , f (2) 3 = 1 4 z j 4 + 323 49 z j 2 -5065 294 z 2 j 1 -( 1031 588 z + 445 14 z 3 ) j 0 + 65 6 z 2 n 0 -65 6 z n 1 -3 z D nj 3 + 13 3 D nj 2 + 9 z D nj 1 + 30 z 2 D nj 0 -13 21 D nj -4 +4 D nnj 3 -13 42 j 3 ln z -1 2 j 3 (ln z ) 2 , (3 . 40) \nwhere D nnj /lscript is defined in Appendix D, Eq. (D . 20). As explained in subsection D.1 of Appendix D, the term g (2) /lscript is given for any /lscript as \ng (2) /lscript = -1 z j /lscript + f (1) /lscript ln z. (3 . 41) \nThe term k (2) /lscriptm ( q ) is given for /lscript = 2 and /lscript = 3 by \nk (2) 2 m = 191 i 180 mqj 0 -m 2 q 2 j 0 30 -mqj 1 10 \n-68 i 63 mqj 2 -q 2 392 j 2 -73 m 2 q 2 1764 j 2 + 7 mqj 3 180 -i 72 q 2 j 3 + i 324 m 2 q 2 j 3 + 11 i 420 mqj 4 -q 2 j 4 392 -71 m 2 q 2 j 4 8820 + 13 i 6 mqn 1 + mq ( -j 1 5 -13 j 3 90 ) ln z + imq ( -2 5 D nj 1 -13 45 D nj 3 ) , (3 . 42) \nk (2) 3 m = 3527 i 840 mqj 1 -2 m 2 q 2 j 1 315 -mqj 2 36 -5 i 504 q 2 j 2 + 5 i 2268 m 2 q 2 j 2 -379 i 360 mqj 3 -q 2 j 3 360 -7 m 2 q 2 j 3 720 + 3 mqj 4 160 -i 140 q 2 j 4 + i 1120 m 2 q 2 j 4 + 97 i 5040 mqj 5 -q 2 j 5 360 -17 m 2 q 2 j 5 5040 -103 i 48 mqn 0 + 25 i 8 mqn 2 -( 13 mqj 2 126 + 5 mqj 4 56 ) ln z + imq ( -13 63 D nj 2 -5 28 D nj 4 ) . (3 . 43) \nAs noted previously, the source term T /lscriptmω has support only around r = r 0 , hence around z = ωr 0 = O ( v ). Therefore, to evaluate the source integral, we only need X in at z = O ( v ) /lessmuch 1, apart from the value of the incident amplitude A inc . Hence the post-Newtonian expansion of X in corresponds to the expansion not only in terms of /epsilon1 = O ( v 3 ), but also of z by assuming /epsilon1 /lessmuch z /lessmuch 1. In order to evaluate the gravitational wave luminosity to O ( v 8 ) beyond the leading order, we must calculate the series expansion of ξ ( n ) /lscriptm in powers of z for n ≤ 6 -/lscript for each 2 ≤ /lscript ≤ 6. This follows from a simple power counting. The leading order contribution of the /lscript -th pole is O ( v 2( /lscript -2) ) smaller than that of the quadrupole, while the n -th post-Minkowski terms are O ( /epsilon1 n z -n ) = O ( v 2 n ) relative to the lowest order terms in the near-zone. Hence the leading term of ξ ( n ) /lscriptm contributes at O ( v 2( /lscript -2)+2 n ) and O ( v 8 ) is attained for /lscript + n = 6 (see Appendix C of Ref. 15)). \nTo evaluate A inc , we need to know the asymptotic behavior of ξ ( n ) /lscriptm at infinity. Since the accuracy of A inc we need is O ( /epsilon1 2 ), we do not have to calculate ξ (3) /lscriptm and ξ (4) /lscriptm in closed analytic form. We need only the series expansion formulas for ξ (3) /lscriptm and ξ (4) /lscriptm around z = 0, which are easily obtained from Eq. (3 . 34). This is also true for ξ (2) 4 m for /lscript = 4. Inserting ξ ( n ) /lscriptm into Eq. (3 . 18) and expanding it by z and /epsilon1 assuming /epsilon1 /lessmuch z /lessmuch 1, we obtain \nξ (3) 2 m = -q 2 30 z -i 30 z mq 3 + -i 30 + 7 mq 180 -i 60 m 2 q 2 + mq 3 36 -m 3 q 3 90 -mq ln z 30 -i 30 m 2 q 2 ln z + z ( 319 6300 + 100637 i 441000 mq -q 2 180 + 17 m 2 q 2 1134 + 83 i 5880 mq 3 -61 i 13230 m 3 q 3 + ln z 15 -106 i 1575 mq ln z -i 30 mq (ln z ) 2 ) \nwhere \n+ O ( z 2 ) , (3 . 44) \nξ (4) 2 m = q 4 80 z 2 + O ( z -1 ) , (3 . 45) \nξ (3) 3 m = -i 1260 mq + m 2 q 2 1890 -i 1260 mq 3 -i 3780 m 3 q 3 + O ( z ) , (3 . 46) \nξ (2) 4 m = ( 1 1764 -11 i 15120 mq + q 2 10584 -19 m 2 q 2 105840 ) z 2 + O ( z 3 ) . (3 . 47) \nInserting these ξ ( n ) /lscriptm into Eq. (3 . 18) and expanding the result in terms of /epsilon1 = 2 Mω , we obtain the post-Newtonian expansion of X in . The transformation from X in to R in is done by using Eq. (3 . 10). \nNext, we consider A inc to O ( /epsilon1 2 ). Using the relation j /lscript +1 ∼-j /lscript -1 ∼ ( -1) /lscript + n n 2 n -/lscript at z ∼ ∞ , etc., we obtain the asymptotic behaviors of ξ (1) /lscriptm and ξ (2) /lscriptm at z ∼ ∞ as \nξ (1) /lscriptm ∼ p (1) /lscriptm j /lscript +( q (1) /lscriptm -ln z ) n /lscript + ij /lscript ln z, (3 . 48) ξ (2) /lscriptm ∼ p (2) /lscriptm + q (1) /lscriptm ln z -(ln z ) 2 ) j /lscript +( q (2) /lscriptm -p (1) /lscriptm ln z ) n /lscript \n( \n) + ip (1) /lscriptm j /lscript ln z + i ( q (1) /lscriptm -ln z ) n /lscript ln z (3 . 49) \np (1) /lscriptm = -π 2 , (3 . 50) \nψ ( /lscript ) = /lscript -1 ∑ k =1 1 k -γ, (3 . 52) \nq (1) /lscriptm = 1 2 [ ψ ( /lscript ) + ψ ( /lscript +1) + ( /lscript -1)( /lscript +3) /lscript ( /lscript +1) ] -ln 2 -2 imq /lscript 2 ( /lscript +1) 2 , (3 . 51) \nfor any /lscript and \np (2) 2 m = 457 γ 210 -γ 2 2 + 5 π 2 24 -i 18 γ mq + 457 ln 2 210 -γ ln 2 -i 18 mq ln 2 -(ln 2) 2 2 , (3 . 53) \nq (2) 2 m = -457 π 420 + γ π 2 + 5 mq 36 + i 36 mπq -i 72 q 2 + i 324 m 2 q 2 + π ln 2 2 , (3 . 54) \np (2) 3 m = 52 γ 21 -γ 2 2 + 5 π 2 24 -i 72 γ mq + 52 ln 2 21 \n-γ ln 2 -i 72 mq ln 2 -(ln 2) 2 2 , (3 . 55) \nq (2) 3 m = -26 π 21 + γ π 2 + 67 mq 1440 + i 144 mπq + i 360 q 2 -17 i 12960 m 2 q 2 + π ln 2 2 . (3 . 56) \nThen noting that exp( -iφ ) ∼ exp( -i ( z ∗ -z )) at z ∼ ∞ , the asymptotic form of X in is expressed as \nX in = √ z 2 + a 2 ω 2 exp( -iφ ) { f (0) /lscriptm + /epsilon1ξ (1) /lscriptm + /epsilon1 2 ξ (2) /lscriptm + ... } \n∼ e -iz ∗ ( zh (2) /lscript e iz ) [ 1 + /epsilon1 ( p (1) /lscriptm + iq (1) /lscriptm ) + /epsilon1 2 ( p (2) /lscriptm + iq (2) /lscriptm ) ] + e iz ∗ ( zh (1) /lscript e -iz ) [ 1 + /epsilon1 ( p (1) /lscriptm -iq (1) /lscriptm ) + /epsilon1 2 ( p (2) /lscriptm -iq (2) /lscriptm ) ] , (3 . 57) \nwhere h (1) /lscript and h (2) /lscript are the spherical Hankel functions of the first and second kinds, respectively, which are given by \nh (1) /lscript = j /lscript + in /lscript → ( -1) /lscript +1 e iz z , h (2) /lscript = j /lscript -in /lscript → ( -1) /lscript +1 e -iz z . (3 . 58) \nFrom these equations, noting ωr ∗ = z ∗ -/epsilon1 ln /epsilon1 , we obtain \nThe corresponding incident amplitude B inc for the Teukolsky function is obtained from Eq. (3 . 13). \nA inc = 1 2 i /lscript +1 e -i/epsilon1 ln /epsilon1 [ 1 + /epsilon1 ( p (1) /lscriptm + iq (1) /lscriptm ) + /epsilon1 2 ( p (2) /lscriptm + iq (2) /lscriptm ) + ... ] . (3 . 59)', '§ 4. Gravitational waves to O ( v 11 ) in Schwarzschild case': "In this section we consider a circular orbit around a Schwarzschild black hole and derive the 5.5PN formula for the energy flux emitted to infinity. In this case, we can take the orbit to lie on the equatorial plane ( θ = π/ 2) without loss of generality. Then E and l z are given by setting R ( r 0 ) = ∂R/∂r ( r 0 ) = 0 where R is given by Eq. (2 . 19). This gives \nE = ( r 0 -2 M ) / √ r 0 ( r 0 -3 M ) , l z = √ Mr 0 / √ 1 -3 M/r 0 , (4 . 1) \nwhere r 0 is the orbital radius. The angular frequency is given by Ω ϕ = ( M/r 3 0 ) 1 / 2 . Defining s b /lscriptm by \n0 b /lscriptm = 1 2 [( /lscript -1) /lscript ( /lscript +1)( /lscript +2)] 1 / 2 0 Y /lscriptm ( π 2 , 0 ) Er 0 r 0 -2 M , -1 b /lscriptm = [( /lscript -1)( /lscript +2)] 1 / 2 -1 Y /lscriptm ( π 2 , 0 ) l z r 0 , -2 b /lscriptm = -2 Y /lscriptm ( π 2 , 0 ) l z Ω ϕ , (4 . 2) \nwhere s Y /lscriptm ( θ, ϕ ) are the spin-weighted spherical harmonics 30) , ˜ Z /lscriptmω is found to take the form \n˜ Z /lscriptmω = Z /lscriptm δ ( ω -mΩ ϕ ) , (4 . 3) \nZ /lscriptm = π iωr 2 0 B in /lscriptω -0 b /lscriptm -2 i -1 b /lscriptm ( 1 + i 2 ωr 2 0 / ( r 0 -2 M ) ) + i -2 b /lscriptm ωr 0 (1 -2 M/r 0 ) -2 ( 1 -M/r 0 + 1 2 iωr 0 ) R in /lscriptm \nwhere \nChapter 1 \nBlack Hole Perturbation \n+ [ i -1 b /lscriptm --2 b /lscriptm ( 1 + iωr 2 0 / ( r 0 -2 M ) )] r 0 R in /lscriptω ' ( r 0 ) + 1 2 -2 b /lscriptm r 2 0 R in /lscriptω '' ( r 0 ) , (4 . 4) \nwhere prime denotes the derivative with respect to the radial coordinate r . In terms of the amplitudes Z /lscriptm , the gravitational wave luminosity at infinity is given by \ndE dt = ∞ ∑ /lscript =2 /lscript ∑ m = -/lscript | Z /lscriptm | 2 2 πω 2 , (4 . 5) \nwhere ω = mΩ ϕ . Since the dominant frequency of the gravitational waves at infinity is 2 Ω ϕ , an observationally relevant post-Newtonian parameter is x ≡ ( MΩ ϕ ) 1 / 3 . We mention that our post-Newtonian expansion parameter is defined by v := ( M/r 0 ) 1 / 2 . In the case of a circular orbit around a Schwarzschild black hole, however, we have v = x . Hence the parameter v is directly related to the observable frequency in the present case. \nFollowing the method given in section 3, instead of directly calculating R in /lscriptω from the homogeneous Teukolsky equation, we calculate the corresponding Regge-Wheeler function X in /lscriptω first and then transform it to R in /lscriptω . The homogeneous Regge-Wheeler equation, which is given by setting a = 0 in Eq. (3 . 4), takes the form 31) , \n[ d 2 dr ∗ 2 + ω 2 -V ( r ) ] X /lscriptω ( r ) = 0 , (4 . 6) \nwhere \nThe transformation (3 . 10) reduces to 32) \nV ( r ) = ( 1 -2 M r )( /lscript ( /lscript +1) r 2 -6 M r 3 ) . (4 . 7) \nR /lscriptmω = ∆ c 0 ( d dr ∗ + iω ) r 2 ∆ ( d dr ∗ + iω ) rX /lscriptω , (4 . 8) \nwhere c 0 , defined in Eq. (3 . 7), reduces to c 0 = ( /lscript -1) /lscript ( /lscript +1)( /lscript +2) -12 iMω . The inverse transformation (3 . 11) reduces to \nX /lscriptω = r 5 ∆ ( d dr ∗ -iω ) r 2 ∆ ( d dr ∗ -iω ) R /lscriptω r 2 . (4 . 9) \nThe asymptotic forms of X in is the same as given in Eq. (3 . 12) except that now we have k = ω . The coefficients A inc , A ref and A trans are also respectively related to B inc , B ref and B trans as before. (See Eqs. (3 . 13), (3 . 14) and (3 . 15).) Note that the coefficient that appears in Eq. (3 . 15) now reduces to \nd /lscriptmω = 16(1 -2 iMω )(1 -4 iMω ) . (4 . 10) \nCorresponding to Eq. (3 . 17), we introduce the variable z ∗ = z + /epsilon1 ln( z -/epsilon1 ). Then Eq. (3 . 18) reduces to \nX in /lscript = e -i ( z ∗ -z ) zξ /lscript ( z ) = e -i/epsilon1 ln( z -/epsilon1 ) zξ /lscript ( z ) , (4 . 11) \nand Eq. (3 . 20) becomes L (0) [ ξ /lscript ] = /epsilon1L (1) [ ξ /lscript ]. Thus Eq. (3 . 25) simplifies considerably to become \nL (0) [ ξ ( n ) /lscript ] = L (1) [ ξ ( n -1) /lscript ] . (4 . 12) \nIt may be worthwhile to note that the left hand side can be expressed concisely as \nL (1) [ ξ ( n -1) /lscript ] = e -iz d dz [ 1 z 3 d dz ( e iz z 2 ξ ( n -1) /lscript ) ] . (4 . 13) \nThe calculations to O ( /epsilon1 2 ) are already done in section 3. When we consider the gravitational wave luminosity to O ( v 11 ), we need to calculate A inc to O ( /epsilon1 3 ) for /lscript = 2 and 3 and to O ( /epsilon1 2 ) and for /lscript = 4. Thus we need the closed analytic forms of ξ (3) /lscript for /lscript = 2 and 3 and ξ (2) 4 . The latter can be obtained in the same way as in the previous section. The procedure to obtain ξ (3) /lscript is explained in detail in Appendix D. \nThe real parts of ξ (3) /lscript , f (3) /lscript , for /lscript = 2 and 3 are given as \nf (3) 2 = 214 F 2 , 0 [ z (ln z ) j 0 ] 105 -107 D nj -4 630 -457 D nj -2 70 -2629 D nj 0 630 + 16949 D nj 2 4410 -107(ln z ) D nj 2 105 +(ln z ) 2 D nj 2 -2 D nj 4 49 -12 D nnj -1 -18 D nnj 1 + 2 D nnj 3 3 -8 D nnnj 2 -197 j -3 126 + 2539 j -1 3780 + 107(ln z ) j -1 70 + 3(ln z ) 2 j -1 2 + 21 j 1 100 + 349(ln z ) j 1 140 + 9(ln z ) 2 j 1 4 -457 j 3 1050 + 29(ln z ) j 3 252 -(ln z ) 2 j 3 12 + j 5 504 , (4 . 14) \nf (3) 3 = 26 F 3 , 0 [ z (log z ) j 0 ] 21 + 13 D nj -5 98 + 14075 D nj -3 882 -1424 D nj -1 63 -2511 D nj 1 70 + 269 D nj 3 70 -13(log z ) D nj 3 21 +(log z ) 2 D nj 3 -D nj 5 18 +60 D nnj -2 +34 D nnj 0 -710 D nnj 2 21 + 6 D nnj 4 7 -8 D nnnj 3 + 75 j -4 28 -19 j -2 56 -65(log z ) j -2 14 -15(log z ) 2 j -2 2 -2789 j 0 3780 -221(log z ) j 0 84 -17(log z ) 2 j 0 4 -7495 j 2 5292 + 4867(log z ) j 2 1764 + 355(log z ) 2 j 2 84 -10963 j 4 32340 + 15(log z ) j 4 196 -3(log z ) 2 j 4 28 + 4 j 6 1485 , (4 . 15) \nwhere the definitions of the functions D nnj /lscript etc. are given in Eqs. (D . 20) and (D . 24) of Appendix D. The imaginary parts g (3) /lscript are expressed in terms of f (1) /lscript and f (2) /lscript as given in Eq. (D . 9). As for the real part of ξ (2) 4 , f (2) 4 , it is calculated to be \nf (2) 4 = 56 165 z j 5 + ( -5036 33 z 4 + 30334 1155 z 2 ) j 4 + ( 35252 33 z 5 -30334 165 z 3 + 14401 3465 z ) j 3 -( 5036 11 z 5 + 45461 693 z 3 + 36287 9240 z ) j 1 + ( 140 z 3 -5 18 z ) n 0 -49 6 z n 2 \n-21 5 z D nj 4 + 149 30 D nj 3 -10 3 z D nj 2 + 105 z 2 D nj 1 + 210 z 3 D nj 0 -20 z D nj 0 + 1571 3465 D nj -5 +4 D nnj 4 -1571 6930 j 4 ln z -1 2 j 4 (ln z ) 2 . (4 . 16) \nUsing the analysis given in subsection D.4.2 of Appendix D, the above results readily give us the asymptotic forms of ξ (3) /lscript ( /lscript = 2 , 3) and ξ (2) 4 at z →∞ , from which the amplitudes A inc to the required order are calculated. The results are \nA inc 2 = -1 2 ie -i/epsilon1 (ln 2 /epsilon1 + γ ) exp [ i { /epsilon1 5 3 -/epsilon1 2 107 420 π + /epsilon1 3 ( 29 648 -107 1260 π 2 + ζ (3) 3 ) + · · · }] × 1 -/epsilon1 π 2 + /epsilon1 2 ( 25 18 + 5 24 π 2 + 107 210 ( γ +ln2) ) + /epsilon1 3 ( -25 36 π -107 420 ( γ +ln2) π -π 3 16 ) + · · · , A inc 3 = 1 2 e -i/epsilon1 (ln 2 /epsilon1 + γ ) exp [ i { /epsilon1 13 6 -/epsilon1 2 13 84 π + /epsilon1 3 ( -29 810 -13 252 π 2 + ζ (3) 3 ) + · · · }] × 1 -/epsilon1 π 2 + /epsilon1 2 ( 169 72 + 5 24 π 2 + 13 42 ( γ +ln2) ) + /epsilon1 3 ( -169 144 π -13 84 ( γ +ln2) π -π 3 16 ) + · · · , A inc 4 = 1 2 ie -i/epsilon1 (ln 2 /epsilon1 + γ ) exp [ i { /epsilon1 149 60 -i/epsilon1 2 1571 13860 π + · · · }] × [ 1 -/epsilon1 π 2 + /epsilon1 2 ( 22201 7200 + 5 24 π 2 + 1571 6930 ( γ +ln2) ) + · · · ] . (4 . 17) \nThe corresponding amplitudes B inc are readily obtained from Eq. (3 . 13). \nAs in the previous section, from Eqs. (4 . 11) and (4 . 8), it is also straightforward to obtain the near-zone post-Newtonian expansion of X in and hence of R in , assuming z /lessmuch 1. As discussed there, we need the series expansion formulas for R in for 2( n + /lscript -2) ≤ 11, hence for n ≤ 7 -/lscript for each 2 ≤ /lscript ≤ 7. The resulting R in for 2 ≤ /lscript ≤ 7 which are necessary to calculate the luminosity to O ( v 11 ) are given in Appendix E. \nFinally, from Eq. (4 . 5), we obtain the luminosity to O ( v 11 ) as \n〈 dE dt 〉 = ( dE dt ) N 1 -1247 336 v 2 +4 π v 3 -44711 9072 v 4 -8191 π 672 v 5 + ( 6643739519 69854400 -1712 γ 105 + 16 π 2 3 -3424 ln 2 105 -1712 ln v 105 ) v 6 -16285 π 504 v 7 + ( -323105549467 3178375200 + 232597 γ 4410 -1369 π 2 126 \n+ 39931 ln 2 294 -47385 ln 3 1568 + 232597 ln v 4410 ) v 8 + ( 265978667519 π 745113600 -6848 γ π 105 -13696 π ln 2 105 -6848 π ln v 105 ) v 9 + ( -2500861660823683 2831932303200 + 916628467 γ 7858620 -424223 π 2 6804 -83217611 ln 2 1122660 + 47385 ln 3 196 + 916628467ln v 7858620 ) v 10 + ( 8399309750401 π 101708006400 + 177293 γ π 1176 + 8521283 π ln 2 17640 -142155 π ln 3 784 + 177293 π ln v 1176 ) v 11 , (4 . 18) \nwhere ( dE/dt ) N is the Newtonian quadrupole luminosity given by \n( dE dt ) N = 32 µ 2 M 3 5 r 5 0 = 32 5 ( µ M ) 2 v 10 . (4 . 19) \nTo compare the above result with those obtained previously by the standard postNewtonian method, we note that v = x ≡ ( MΩ ϕ ) 1 / 3 in the present case. Then we find our result agrees with the standard post-Newtonian results up to O ( x 5 ) 33) , 34) , 35) , 36) , 37) , 38) in the limit µ/M /lessmuch 1. The contributions to the luminosity from individual /lscript modes are given in Appendix E.", '§ 5. Convergence of the post-Newtonian expansion': 'Using the results obtained in the previous section, we compare the formula for the gravitational wave flux with the corresponding numerical results and investigate the accuracy of the post-Newtonian expansion. \nA high precision numerical calculation of gravitational waves from a particle in a circular orbit around a Schwarzschild black hole has been performed by Tagoshi and Nakamura. 10) Since no assumption was made about the velocity of the test particle, their results are correct relativistically in the limit µ /lessmuch M . In that work, dE/dt was calculated only for /lscript = 2 ∼ 6. Here, for the orbital radius r 0 ≤ 100 M , we calculate dE/dt again for all modes of /lscript = 2 ∼ 6 and for /lscript = 7 with odd m . The estimated accuracy of the calculation is about 10 -11 , which turns out to be accurate enough for the present purpose. As for the radius r 0 > 100 M , we use the data calculated by Tagoshi and Nakamura 10) which contain modes from /lscript = 2 to 6. \nIn Figs. 1 and 2, we show the error in the post-Newtonian formulas as a function of the orbital radius r . The error of the post-Newtonian formula is defined as \n∣ \n∣ \n∣ ∣ where ( dE/dt ) n and ( dE/dt ) denote the ( n/ 2)PN formula and the numerical result, respectively. In the plot of Fig.2, only the contributions from /lscript = 2 to 6 are included \nerror = ∣ ∣ ∣ ∣ 1 -( dE dt ) n / ( dE dt ) ∣ ∣ ∣ ∣ , (5 . 1) \nFig. 1. The error of the post-Newtonian formulas as functions of the Schwarzschild radial coordinate r for 6 ≤ r/M ≤ 100. Contributions from /lscript = 2 to 7 modes are included. \n<!-- image --> \nin both the post-Newtonian formulas and the numerical data. We can see that, at small radius less than r ∼ 10 M , the error of the 1PN and 2.5PN formulas are larger than the other formulas. On the other hand, the Newtonian and the 2PN formulas are very accurate within this radius. This is because those formulas coincide with the exact one accidentally at a radius between 6 M and 10 M . The error of each post-Newtonian formula at the inner most stable circular orbit, r = 6 M , becomes as follows; 12% (Newtonian), 66% (1PN), 8 . 6% (1.5PN), 3 . 4% (2PN), 42% (2.5PN), 11% (3PN), 5 . 4% (3.5PN), 17% (4PN), 8 . 4% (4.5PN), 6 . 5% (5PN), 4 . 1% (5.5PN). As is expected, the errors of the post-Newtonian formulas decrease almost linearly up to r ∼ 5000 M in a log-log plot. This fact also suggests that the numerical data have accuracy of at least 10 -18 at r ∼ 5000 M . \n∼ \n∼ In order to examine exactly to what order the post-Newtonian formulas are needed to do accurate estimation of the parameters of a binary, using data from laser interferometers, we must evaluate the systematic error produced by incorrect templates. However, here we simply calculate the total cycle of gravitational waves \nFig. 2. The same figure for 100 ≤ r/M < 5000. Contributions only from /lscript = 2 to 6 are included in both the post-Newtonian formulas and in the numerical data. \n<!-- image --> \nfrom a coalescing binary in a laser interferometer band and evaluate the error produced by the post-Newtonian formulas. It has been suggested that whether the error in the total cycle is less than unity or not gives a useful guideline to examine the accuracy of the post-Newtonian formulas as templates 39) (see also Ref. 40)). \nWe ignore the finite mass effect in the post-Newtonian formulas and interpret M as the total mass and µ as the reduced mass of the system. The total cycle N of gravitational waves from an inspiraling binary is calculated by using the postNewtonian energy loss formula, ( dE/dt ) n , and the orbital energy formula ( dE/dv ) n which is truncated at n/ 2PN order as \nN ( n ) = ∫ v i v f dv Ω ϕ π ( dE/dv ) n | ( dE/dt ) n | , (5 . 2) \nwhere v i = ( M/r i ) 1 / 2 , v f = ( M/r f ) 1 / 2 , and r i and r f are the initial and final orbital separations of the binary. We define the relative difference of cycle ∆N ( n ) as ∆N ( n ) ≡ | N ( n ) -N ( n -1) | . We adopt r f = 6 M and r i is the one at which the \nfrequency of wave is 10Hz and which is given by r i /M 347( M /circledot /M ) 2 / 3 . \nThe results for typical binary systems are given in Table I. We only show the \n∼ \nTable I. The relative difference of cycle ∆N ( n ) for typical coalescing compact binaries. The last line shows the cycle calculated by Newtonian quadrupole formula. \n| n | (1.4 M /circledot ,1.4 M /circledot ) | (10 M /circledot ,10 M /circledot ) | (1.4 M /circledot ,10 M /circledot ) | (1.4 M /circledot ,70 M /circledot ) |\n|-------|-----------------------------------------|---------------------------------------|----------------------------------------|----------------------------------------|\n| 2 | 356 | 54 | 216 | 212 |\n| 3 | 228 | 60 | 208 | 296 |\n| 4 | 11 | 5 | 15 | 31 |\n| 5 | 12 | 7 | 20 | 53 |\n| 6 | 11 | 8 | 22 | 75 |\n| 7 | 1.2 | 1 | 2.6 | 10 |\n| 8 | 0.12 | 0.14 | 0.3 | 2.2 |\n| 9 | 0.82 | 0.8 | 1.9 | 8.9 |\n| 10 | 0.09 | 0.08 | 0.2 | 0.87 |\n| 11 | 0.03 | 0.03 | 0.07 | 0.4 |\n| N (0) | 16000 | 600 | 3578 | 898 | \nresults for q = 0. The cases for q /negationslash = 0 are investigated in Shibata et al. 15) and Tagoshi et al. 16) . This table suggests that we need the 3PN ∼ 4PN formula to obtain accurate wave forms for typical binaries whose total mass are less than 20 M /circledot . Although the required post-Newtonian order is very high and it has not been achieved yet in the standard post-Newtonian analysis, this results show that the post-Newtonian approximation is applicable to the inspiral phase of coalescing compact binaries. In this sense, we can be optimistic. \nOn the other hand, the convergence for the case of neutron star -black hole binaries, whose mass is above several ten M /circledot , is very slow. This is because r i /M become smaller for a larger mass black hole, and the higher relativistic correction becomes more important. From Table I, we might think that N ( n ) converges at n = 11 even for ( m 1 , m 2 ) = (1.4 M /circledot ,70 M /circledot ). However this is not true. Note that Table I shows only the relative difference between the post-Newtonian approximated cycles. If we calculate the difference between the post-Newtonian formula and the fully relativistic one, we find that the 5.5PN formula is not accurate enough for the case ( m 1 , m 2 ) = (1.4 M /circledot ,70 M /circledot ), as pointed out by Tanaka, Tagoshi and Sasaki 13) . \nFinally we comment on the initial frequency. The above results are obtained by setting the initial frequency to 10Hz. However, it may be difficult to observe gravitational wave at this frequency because of the seismic noise. If we set the initial frequency higher than 10Hz, the error ∆N becomes slightly smaller. But since this dependence of ∆N on the initial frequency is very weak, the above results are insensitive to the variation of the initial frequency.', '§ 6. Circular orbit on the equatorial plane around a rotating black hole': 'In this section, we consider a circular orbit on the equatorial plane of a Kerr black hole and calculate the 4PN luminosity formula. \nWe define the orbital radius as r = r 0 . As in section 4, we have C = 0, and E \nand l z are determined by R ( r 0 ) = 0 and ∂R/∂r | r = r 0 = 0 as \nE = 1 -2 v 2 + qv 3 (1 -3 v 2 +2 qv 3 ) 1 / 2 , l z = r 0 v (1 -2 qv 3 + q 2 v 4 ) (1 -3 v 2 +2 qv 3 ) 1 / 2 , (6 . 1) \nwhere v = ( M/r 0 ) 1 / 2 . From these, we can easily obtain ϕ ( t ) as \nϕ ( t ) = Ω ϕ t ; Ω ϕ = M 1 / 2 r 3 / 2 0 [ 1 -qv 3 + q 2 v 6 + O ( v 9 ) ] . (6 . 2) \nWhen a = 0, this becomes Ω ϕ = ( M/r 3 0 ) 1 / 2 . \nThe rest of the calculation is almost the same as in section 4. The amplitude of the Teukolsky function ˜ Z ∞ /lscriptmω at infinity is expressed as \n˜ Z ∞ /lscriptmω = µ 2 πδ ( ω -mΩ ) 2 iωB inc [ R in { A nn 0 + A ¯ mn 0 + A ¯ m ¯ m 0 } -dR in dr { A ¯ mn 1 + A ¯ m ¯ m 1 } + d 2 R in dr 2 A ¯ m ¯ m 2 ] r = r 0 ,θ = π/ 2 , ≡ δ ( ω -mΩ ) Z ∞ /lscriptmω , (6 . 3) \nwhere A nn 0 , etc. are given by Eq. (2 . 25). \nThe total luminosity up to O ( v 8 ) is expressed as \n〈 dE dt 〉 = ( dE dt ) N { 1 + ( q -independent terms) -73 q 12 v 3 ++ 33 q 2 16 v 4 + 3749 q 336 v 5 -( 169 π q 6 + 3419 q 2 168 ) v 6 + ( 83819 q 1296 + 65 π q 2 8 -151 q 3 12 ) v 7 + ( 3389 π q 96 -124091 q 2 9072 17 q 4 16 ) v 8 } , (6 . 4) \nwhere ( dE/dt ) N is the Newtonian quadrupole luminosity, Eq. (4 . 19), and the q -independent terms are identical to those in Eq. (4 . 18). \nFrom an observational point of view, it is more convenient to express the luminosity in terms of the variable x ≡ ( MΩ ϕ ) 1 / 3 . Using the relation between v ≡ ( M/r 0 ) 1 / 2 and x given by Eq. (6 . 2), the luminosity is expresses as \n〈 dE dt 〉 = ˜ ( dE dt ) N ( 1 + ( q -independent terms) -11 q 4 x 3 + 33 q 2 16 x 4 -59 q 16 x 5 + ( -65 π q 6 + 611 q 2 504 ) x 6 + ( 162035 q 3888 + 65 π q 2 8 -71 q 3 24 ) x 7 \nwhere \nand the q -independent terms are again identical to those in Eq. (4 . 18) with the replacement v → x . The partial luminosities for individual modes are given in Appendix F. The spin dependent term at O ( v 3 ) agrees with the standard postNewtonian result 41) . \nChapter 1 \nBlack Hole Perturbation \n+ ( 359 π q 14 + 22667 q 2 4536 + 17 q 4 16 ) x 8 ) , (6 . 5) \n˜ ( dE dt ) N = 32 5 ( µ M ) 2 x 10 , (6 . 6) \nHere it is interesting to investigate the origin of some of the spin-dependent terms. As an example, we consider the mode /lscript = | m | = 2. The luminosity from the /lscript = | m | = 2 modes is given by \nWe can derive some of the spin-dependent terms in the above formula from the quadrupole formula 42) ; dE/dt = (32 / 5) µ 2 ˆ r 4 Ω 6 ϕ , where ˆ r is the orbital radius of a test particle in harmonic coordinates. If multipole moments of the black hole exist, the orbital radius changes due to the influence of those moments. The mass and mass current multipole moments of a Kerr black hole is given by M l + iS l = M ( ia ) l . We can express the orbital frequency of the test particle in harmonic coordinates. We find that the dominant effect of the multipole moments of a Kerr black hole to dE/dt can be expressed as \n( dE dt ) 2 , 2 + ( dE dt ) 2 , -2 = ˜ ( dE dt ) N 1 + ( q -independent terms) -8 q 3 x 3 +2 q 2 x 4 + 52 q 27 x 5 + ( -32 π q 3 -1817 q 2 567 ) x 6 + ( 364856 q 11907 +8 π q 2 -8 q 3 3 ) x 7 + ( 208 π q 27 + 105022 q 2 9261 + q 4 ) x 8 . (6 . 7) \n〈 dE dt 〉 = ˜ ( dE dt ) N { 1 -8 3 S 1 M 2 x 3 -2 M 2 M 3 x 4 + ( 4 S 3 M 4 -4 3 M 2 S 1 M 5 ) x 7 + ( -3 2 M 2 2 M 6 + 5 2 M 4 M 5 ) x 8 } . (6 . 8) \nThe terms in this expression agree with the corresponding terms of our result such as ( -8 / 3) qx 3 , 2 q 2 x 4 , ( -8 / 3) q 3 x 7 and q 4 x 8 . Thus, we may interpret the term 2 q 2 x 4 as the effect of the quadrupole moment. The terms ( -8 / 3) q 3 x 7 and q 4 x 8 are not due to a single multipole moment, but to combined effects of the multipole moments.', '§ 7. Slightly eccentric orbit around a Schwarzschild black hole': 'In this section, we present post-Newtonian formulas of gravitational waves from a particle in slightly eccentric orbits around a Schwarzschild black hole. We derive the 4PN formulas of the energy and angular momentum fluxes to O ( e 2 ) where e is the eccentricity of the orbit. \nThe solution of the geodesic equations for slightly eccentric orbits has been given by Apostolatos et al. 43) . Here we briefly sketch the derivation of it. Since we may consider the orbit to lie on the equatorial plane without loss of generality, we put θ = π/ 2 and C = 0 in the geodesic equations (2 . 18). Then we define a slightly eccentric orbit as follows: First of all, we assume that E and l z are set to be such that R ( r ) /r 4 , which plays the role of an effective potential for the radial motion, has the minimum at r = r 0 and that the maximum value of the orbital radius is at r = r 0 (1 + e ), where e /lessmuch 1. Thus, the following conditions hold; \n∂ ( R/r 4 ) ∂r ( r = r 0 ) = 0 and R ( r = r 0 (1 + e )) = 0 . (7 . 1) \nFrom these equations, E and l z are expressed in terms of r 0 and e as \nE 2 = (1 -2 v 2 ) 2 1 3 v 2 + v 2 (1 -6 v 2 ) 1 -3 v 2 e 2 -2 v 2 (1 -7 v 2 ) 1 3 v 2 e 3 + O ( e 4 ) , (7 . 2) \n- \n- \nwhere v ≡ √ M/r 0 . For convenience, we also define Ω ≡ v 3 /M . Then expanding the geodesic equations in powers of e , the solution is found to be 43) \n-l z = M v √ 1 -3 v 2 , (7 . 3) \nr ( t ) = r 0 [1 + e cos Ω r t + e 2 { q 1 ( v )(1 cos Ω r t ) + q 2 ( v )(1 -cos 2 Ω r t ) ] + O ( e 3 ) , (7 . 4) \n{ -ϕ ( t ) = Ω ϕ t -ep 1 ( v ) sin Ω r t + e 2 { p 2 ( v ) sin Ω r t + p 3 ( v ) sin 2 Ω r t } + O ( e 3 ) , (7 . 5) \n- \n} \nΩ r = Ω (1 6 v 2 ) 1 / 2 , (7 . 6) \nΩ ϕ = Ω [1 -f ( v ) e 2 ] ; f ( v ) = 3(1 -3 v 2 )(1 -8 v 2 ) 2(1 -2 v 2 )(1 6 v 2 ) , (7 . 7) \n- \n- \n-q 1 ( v ) = 1 -7 v 2 1 -6 v 2 , q 2 ( v ) = 1 -11 v 2 +26 v 4 2(1 6 v 2 )(1 -2 v 2 ) , (7 . 8) \n- \n- \n--p 1 ( v ) = 2(1 -3 v 2 ) (1 2 v 2 ) √ 1 -6 v 2 , p 2 ( v ) = 2(1 -3 v 2 )(1 -7 v 2 ) (1 2 v 2 )(1 -6 v 2 ) 3 / 2 , \n--p 3 ( v ) = 5 -64 v 2 +250 v 4 -300 v 6 4(1 -2 v 2 ) 2 (1 -6 v 2 ) 3 / 2 . (7 . 9) \n- \nAs is well known, since Ω r = Ω ϕ , the orbit does not close. \n/negationslash \nwhere \nwhere \nNow we evaluate the source term of the Teukolsky equation. In the present case, A nn 0 etc. given in Eqs. (2 . 25) reduce to \nA nn 0 = -E 2 √ 2 ∆ S (0) /lscriptm ( 1 + r 2 ∆ dr dt ) 2 , (7 . 10) \nA ¯ m ¯ m 0 = l 2 z 2 √ 2 πE S (2) /lscriptm ( 2 iω ( r -M ) ∆r 2 -ω 2 ∆ ) , (7 . 12) \nA ¯ mn 0 = il z √ 2 π S (1) /lscriptm ( iω ∆ + 2 r 3 )( 1 + r 2 ∆ dr dt ) , (7 . 11) \nA ¯ mn 1 = il z √ 2 πr 2 S (1) /lscriptm ( 1 + r 2 ∆ dr dt ) , (7 . 13) \nA ¯ m ¯ m 1 = l 2 z √ 2 πE S (2) /lscriptm ( iω r 2 + ∆ r 5 ) , (7 . 14) \nA ¯ m ¯ m 2 = l 2 z ∆ 2 √ 2 πEr 4 S (2) /lscriptm , (7 . 15) \nS (0) /lscriptm = L † 1 L † 2 S /lscriptm ( θ = π/ 2) , (7 . 16) \nS (1) /lscriptm = L † 2 S /lscriptm ( θ = π/ 2) , (7 . 17) \nS (2) /lscriptm = S /lscriptm ( θ = π/ 2) . (7 . 18) \nThen noting that the orbits of our interest have the properties, \nr ( t + ∆t r ) = r ( t ) , dϕ dt ∣ ∣ ∣ t = t + ∆t r = dϕ dt ∣ ∣ ∣ ∣ t = t , (7 . 19) \n∣ \nwhere ∆t r = 2 π/Ω r , Eq. (2 . 26) can be rewritten as \n˜ Z /lscriptmω = µ 2 iωB inc /lscriptmω ∫ ∞ -∞ dte iωt -imϕ ( t ) W /lscriptmω = µ 2 iωB inc /lscriptmω 2 π ∆t r ∫ ∆t r 0 dte iωt -imϕ ( t ) W /lscriptmω ∑ n δ ( ω -ω n ) , (7 . 20) \nwhere \nω n = nΩ r + mΩ ϕ ( n = 0 , ± 1 , ± 2 , . . . ) . (7 . 21) \nWe see that ˜ Z /lscriptmω takes the form as given in Eq. (2 . 28) with Z /lscriptmω given by \nZ /lscriptmω = µΩ r 2 iωB inc /lscriptmω ∫ ∆t r 0 dte iωt -imϕ ( t ) W /lscriptmω . (7 . 22) \nWhen Z /lscriptmω n are obtained, the energy and angular momentum fluxes averaged over t /greatermuch ∆t r are calculated by using Eqs. (2 . 31) and (2 . 32), respectively. Here, we \nshow these fluxes accurate to O ( e 2 ) and to O ( v 8 ) beyond Newtonian: \n〈 dE dt 〉 = ˙ E (0) + e 2 ˙ E (2) + O ( e 3 ) , (7 . 23) \n〈 dJ z dt 〉 = ˙ J (0) z + e 2 ˙ J (2) z + O ( e 3 ) . (7 . 24) \nWe note that ˙ J (0) z = ˙ E (0) /Ω where Ω = v 3 /M . We have already given the 5.5PN formula for ˙ E (0) in section 4, Eq. (4 . 18). Hence our task here is to evaluate the O ( e 2 ) corrections. From the forms of r ( t ) and ϕ ( t ) given in Eqs. (7 . 5), we readily see that Z /lscriptmω n = O ( e | n | ) for n = ± 1 , ± 2. Thus, we only need the modes n = 0 , ± 1: As for the n = 0 modes, we must retain terms up to O ( e 2 ), while for the n = ± 1 modes, we only need terms up to O ( e ). Then the 4PN formulas for ˙ E (2) and ˙ J (2) z are found as \ne 2 ˙ E (2) PN = e 2 ( dE dt ) N × 37 24 -65 v 2 21 + 1087 π v 3 48 -465337 v 4 9072 -118607 π v 5 1344 -17328779 π v 7 48384 + ( 98546617999 69854400 -65056 γ 315 + 608 π 2 9 + 1712 ln 2 315 -234009 ln 3 560 -65056 ln v 315 ) v 6 + ( -6653525574791 2118916800 + 118015 γ 98 -34093 π 2 126 -1035547 ln 2 4410 + 3986901 ln 3 1120 + 118015 ln v 98 ) v 8 , (7 . 25) \nand \ne 2 ˙ J (2) z PN = e 2 Ω ( dE dt ) N × -5 8 + 749 v 2 96 + 49 π v 3 8 -232181 v 4 6048 + 773 π v 5 336 -300637 π v 7 1008 + ( 8017536229 12700800 -19367 γ 210 + 181 π 2 6 + 20009 ln 2 210 -78003 ln 3 280 -19367 ln v 210 ) v 6 + ( -12713730793 61122600 + 3463711 γ 8820 -14675 π 2 252 -2312441 ln 2 980 + 35449083 ln 3 15680 + 3463711 ln v 8820 ) v 8 . (7 . 26) \nIn Appendix G, we show each ( /lscript, m,n ) component of the energy and angular momentum fluxes. Note that there was an error in the coefficients of the e 2 v 4 terms in Ref. 17). This error is corrected in Eqs. (7 . 25) and (7 . 26) above. \nTo express the energy and angular momentum fluxes in terms of the variable x = ( MΩ ϕ ) 1 / 3 , we use Eq. (7 . 7). To O ( e 2 ), it can be easily solved for v as \nv = x [ 1 + 1 3 f ( x ) e 2 + O ( e 3 ) ] . (7 . 27) \nThen to O ( x 8 ) the energy and angular momentum fluxes are expressed as \n〈 dE dt 〉 = ˜ ( dE dt ) N 1 + ( e -independent terms) + e 2 ( 157 24 -6781 x 2 168 + 2335 π x 3 48 -14929 x 4 189 -773 π x 5 3 + 156066596771 x 6 69854400 -106144 γ x 6 315 + 992 π 2 x 6 9 -80464 x 6 ln 2 315 -234009 x 6 ln 3 560 -106144 x 6 ln x 315 -32443727 π x 7 48384 -3045355111074427 x 8 671272842240 + 507208 γ x 8 245 -31271 π 2 x 8 63 -151336 x 8 ln 2 441 + 12887991 x 8 ln 3 3920 + 507208 x 8 ln x 245 ) , (7 . 28) \nand \n〈 dJ dt 〉 = ˜ ( dJ dt ) N 1 + ( e -independent terms) + e 2 ( 23 8 -3259 x 2 168 + 209 π x 3 8 -1041349 x 4 18144 -785 π x 5 6 + 91721955203 x 6 69854400 -41623 γ x 6 210 + 389 π 2 x 6 6 -24503 x 6 ln 2 210 -78003 x 6 ln 3 280 -41623 x 6 ln x 210 -91565 π x 7 168 -105114325363 x 8 72648576 + 696923 γ x 8 630 -4387 π 2 x 8 18 -7051 x 8 ln 2 10 + 3986901 x 8 ln 3 1960 + 696923 x 8 ln x 630 ) , (7 . 29) \nwhere ˜ ( dJ/dt ) N is the Newtonian angular momentum flux expressed in terms of x , \nand the e -independent terms in both 〈 dE/dt 〉 and 〈 dJ/dt 〉 are the same and are given by the terms in the case of circular orbit, Eq. (4 . 18), with the replacement v x . \n˜ ( dJ z dt ) N = 32 5 ( µ M ) 2 Mx 7 , (7 . 30) \n→ \nFinally, we consider the stability of circular orbits. We note that the following relation holds: \n〈 dE dt 〉 -Ω ϕ 〈 dJ z dt 〉 = ∑ /lscript,m mΩ r 4 π ( | Z (+) /lscriptm | 2 ω 3 + -| Z ( -) /lscriptm | 2 ω 3 -) e 2 + O ( e 4 ) ≡ H ( v ) e 2 + O ( e 4 ) , (7 . 31) \nwhere ω ± = mΩ ϕ ± Ω r and Z ( ± ) /lscriptm = Z /lscriptmω ± . Here H ( v ) is an important quantity which determines the stability of circular orbits under the radiation reaction. Assuming the adiabatic evolution of the orbit, the evolution equations for r 0 and e due to the gravitational radiation reaction are written as 43) \nµ dr 0 dt = -2 M (1 -3 v 2 ) 3 / 2 v 4 (1 -6 v 2 ) ˙ E (0) + O ( e 2 ) , (7 . 32) \n-µ d ln e dt = -(1 -2 v 2 )(1 -3 v 2 ) 1 / 2 v 2 (1 -6 v 2 ) [ g ( v ) ˙ E (0) + G ( v ) ] + O ( e ) , (7 . 33) \ng ( v ) = 2 -27 v 2 +72 v 4 -36 v 6 2(1 2 v 2 ) 2 (1 -6 v 2 ) , (7 . 34) \n- \n-G ( v ) = ˙ E (2) -Ω ˙ J (2) z = H ( v ) -f ( v ) ˙ E (0) . (7 . 35) \nUsing Eqs.(7 . 25) and (7 . 26), the 4PN formula of G ( v ) is calculated as \nG PN ( v ) = ( dE dt ) N [ 13 6 -2441 v 2 224 + 793 π v 3 48 -234131 v 4 18144 -121699 π v 5 1344 -414029 π v 7 6912 + ( 36300112493 46569600 -72011 γ 630 + 673 π 2 18 -56603 ln 2 630 -78003 ln 3 560 -72011 ln v 630 ) v 6 + ( -18638348721901 6356750400 + 7157639 γ 8820 -17837 π 2 84 + 1085179 ln 2 1764 + 20367531 ln 3 15680 -133120 ln 4 441 + 7157639 ln v 8820 ) v 8 ] . (7 . 36) \nNote that [ g ( v ) ˙ E (0) + G ( v )] / ˙ E (0) → 19 / 6 for v → 0; i.e., in the Newtonian limit, the radiation reaction always reduces the eccentricity 44) . By a numerical calculation, \nwhere \nApostolatos et al. 43) found that there exists a critical radius r c below which the circular orbit becomes unstable; r c /similarequal 6 . 6792 M . On the other hand, we find the use of the 4PN formulas for ˙ E (0) and G ( v ) gives r c ∼ 7 . 38 M . This indicates that a much higher order PN formula will be necessary to determine r c with good accuracy.', '§ 8. Slightly eccentric orbit around a rotating black hole': "In this section, we consider a slightly eccentric orbit on the equatorial plane of a Kerr black hole and calculate the leading order corrections of the eccentricity to the energy and angular momentum fluxes up to O ( v 5 ) beyond Newtonian. The calculation is parallel to the one given in the previous section. \nWe consider the motion of a particle in the equatorial plane θ = π/ 2, hence we have C = 0. We define the radius r = r 0 as the one at which the potential R/r 4 is minimum; ∂ ( R/r 4 ) /∂r | r = r 0 = 0. We define the eccentricity e such that r = r 0 (1 + e ) is a turning point of the radial motion at which R ( r = r 0 (1 + e )) = 0. We assume e /lessmuch 1. Using these definitions of r 0 and e , E and l z are expressed as \nE = E (0) + eE (1) + e 2 E (2) + O ( e 3 ) , l z = l (0) z + el (1) z + e 2 l (2) z + O ( e 3 ) , \nwhere E ( n ) and l ( n ) z ( n = 0 , 1 , 2) are given by \nE (0) = 1 -2 v 2 + qv 3 (1 -3 v 2 +2 qv 3 ) (1 / 2) , \nE (2) = v 2 (1 -3 v 2 + qv 3 + q 2 v 4 )( -1 + 6 v 2 -8 qv 3 +3 q 2 v 4 ) 2(1 3 v 2 +2 qv 3 ) 3 / 2 ( -1 + 2 v 2 -q 2 v 4 ) , \nE (1) = 0 , \nl (0) z = r 0 v (1 -2 qv 3 + q 2 v 4 ) (1 -3 v 2 +2 qv 3 ) (1 / 2) , \nl (1) z = 0 , \nl (2) z = q r 0 v 5 ( q -3 v + qv 2 + q 2 v 3 )( -1 + 6 v 2 -8 qv 3 +3 q 2 v 4 ) 2(1 -3 v 2 +2 qv 3 ) 3 / 2 ( -1 + 2 v 2 -q 2 v 4 ) , \nwhere v = ( M/r 0 ) 1 / 2 . The post-Newtonian expansions of E ( n ) and l ( n ) z up to the required order are \nE = 1 -M 2 r 0 + 3 M 2 8 r 2 0 -qM 5 / 2 r 5 / 2 0 + e 2 ( M 2 r 0 -5 M 2 4 r 2 0 + 3 qM 5 / 2 r 5 / 2 0 ) + O ( v 6 ) , (8 . 1) l z = ( Mr 0 ) 1 / 2 [ 1 + 3 M 2 r 0 -3 qM 3 / 2 r 3 / 2 0 + ( 27 8 + q 2 ) M 2 r 2 0 -15 qM 5 / 2 2 r 5 / 2 0 + e 2 ( q 2 M 2 2 r 2 0 -3 qM 5 / 2 2 r 5 / 2 0 ) + O ( v 6 ) ] . (8 . 2) \n- \nNow we solve the geodesic equations for a slightly eccentric orbit. The radial equation is \n( dr dt ) 2 = R T 2 . (8 . 3) \nWe expand r ( t ) as \nr ( t ) = r 0 [ 1 + er (1) ( t ) + e 2 r (2) ( t ) + O ( e 3 ) ] , (8 . 4) \nand R/T 2 in terms of e and v using Eq. (8 . 1) and (8 . 2). Collecting terms of the equal order in e , we obtain \n( dr (1) dt ) 2 = Ω 2 r (1 -( r (1) ) 2 ) , (8 . 5) \n1 Ω 2 r dr (1) dt dr (2) dt = -r (1) r (2) + q 0 + q 1 r (1) + q 2 ( r (1) ) 2 , (8 . 6) \nand \nwhere Ω r , q 0 , q 1 and q 2 are given in the post-Newtonian series forms as \nΩ r = M 1 / 2 r 3 / 2 0 [ 1 -3 M r 0 + 3 qM 3 / 2 r 3 / 2 0 -(9 + 3 q 2 ) M 2 2 r 2 0 + 15 qM 5 / 2 r 5 / 2 0 + O ( v 6 ) ] , (8 . 7) \nq 1 = 2 M r 0 [ 1 + 2 M r 0 -3 qM 3 / 2 r 3 / 2 0 + 4 M 2 r 2 0 -6 qM 5 / 2 r 5 / 2 0 + O ( v 6 ) ] , (8 . 9) \nq 0 = -1 + M r 0 -2 qM 3 / 2 r 3 / 2 0 + ( 6 + q 2 ) M 2 r 2 0 -20 qM 5 / 2 r 5 / 2 0 + O ( v 6 ) , (8 . 8) \nq 2 = 1 -3 M r 0 + 2 qM 3 / 2 r 3 / 2 0 -(10 + q 2 ) M 2 r 2 0 + 26 qM 5 / 2 r 5 / 2 0 + O ( v 6 ) . (8 . 10) \nWe obtain r (1) ( t ) from Eq. (8 . 5) as \nr (1) ( t ) = cos Ω r t, (8 . 11) \nwhere we set r ( t = 0) = r 0 (1 + e ). Substitution of Eq. (8 . 11) into Eq. (8 . 6) and yields after integration \nr (2) ( t ) = q 3 (1 -cos Ω r t ) + q 4 (1 -cos 2 Ω r t ) , (8 . 12) \nwhere q 3 = q 0 and q 4 = q 2 / 2. \nIn the same way, we can solve the angular motion ϕ ( t ). From Eq. (2 . 18), we have dϕ/dt = Φ/T , which can be expanded in terms of e using Eqs. (8 . 1), (8 . 2), (8 . 4), (8 . 11) and (8 . 12). Integrating the resulting equation, we obtain \n- \nϕ ( t ) = Ω ϕ t + ep 1 sin Ω r t + e 2 p 2 sin Ω r t + e 2 p 3 sin 2 Ω r t + O ( e 3 ) , (8 . 13) \nwhere \nand \nwhere \nand \np 1 = -2 -4 M r 0 + 6 qM 3 / 2 r 3 / 2 0 -(17 + q 2 ) M 2 r 2 0 + 48 qM 5 / 2 r 5 / 2 0 + O ( v 6 ) , p 2 = 2 + 2 M r 0 -2 qM 3 / 2 r 3 / 2 0 + (1 -q 2 ) M 2 r 2 0 + 6 qM 5 / 2 r 5 / 2 0 + O ( v 6 ) , p 3 = 5 4 + M 4 r 0 -2 qM 3 / 2 r 3 / 2 0 -(9 + 7 q 2 ) M 2 8 r 2 0 + 59( -1 + q 2 ) M 3 8 r 3 0 + O ( v 6 ) , (8 . 14) \nΩ ϕ = ( M r 0 ) 1 / 2 [ 1 -q M 3 / 2 r 0 3 / 2 + e 2 ( -3 2 + 9 M 2 r 0 -9 qM 3 / 2 2 r 3 / 2 0 + 3 ( 6 + q 2 ) M 2 r 2 0 -60 qM 5 / 2 r 5 / 2 0 ) + O ( v 6 ) ] . (8 . 15) \nAs in the case of the previous section, the fact that Ω r /negationslash = Ω ϕ implies that these eccentric orbits are not closed. \nUsing the above solution of the geodesic equations, we evaluate the source term of the Teukolsky equation. We set θ = π/ 2 in the expressions of A nn 0 etc. in Eqs. (2 . 25). Again, parallel to the discussion in section 7, the orbits of our interest have the properties, \nwhere ∆t r = 2 π/Ω r . Hence Eq. (2 . 26) reduces to the form, \nr ( t + ∆t r ) = r ( t ) , dϕ dt ∣ ∣ ∣ t = t + ∆t r = dϕ dt ∣ ∣ ∣ ∣ t = t , (8 . 16) \n∣ \n˜ Z /lscriptmω = Z /lscriptmω δ ( ω -ω n ) , (8 . 17) \nω n = nΩ r + mΩ ϕ , ( n = 0 , ± 1 , ± 2 , . . . ) , (8 . 18) \nZ /lscriptmω = µΩ r 2 iωB inc /lscriptmω ∫ ∆t r 0 dte iωt -imϕ ( t ) W /lscriptmω , (8 . 19) \nwith W /lscriptmω given by Eq. (2 . 27). \nUsing the solution of the geodesic equation for r ( t ), we expand W /lscriptmω in terms of e . The result takes the form, \nW /lscriptmω = f 0 + e ( f 1 r (1) + f 2 dr (1) dt ) + e 2 ( f 3 r (2) + f 4 dr (2) dt + f 5 ( r (1) ) 2 + f 6 ( dr (1) dt ) 2 + f 7 r (1) dr (1) dt + f 8 + O ( e 3 ) , (8 . 20) \nwhere f 0 ∼ f 8 are time-independent coefficients. Inserting this form to Eq. (8 . 19) we obtain \nZ /lscriptmω n = µπ iω n B inc /lscriptmω n [{ f 0 + e 2 ( f 5 2 + f 8 -m 2 p 2 1 4 f 0 +( q 3 + q 4 ) f 3 + imΩ r p 1 2 f 2 + Ω 2 r 2 f 6 )] δ n, 0 + e ( f 1 2 + mp 1 2 f 0 -iΩ r 2 f 2 ) δ n, 1 + e ( f 1 2 -mp 1 2 f 0 + iΩ r 2 f 2 ) δ n, -1 + e 2 ( 3 f 5 4 + f 8 + f 1 mp 1 4 -f 0 m 2 p 1 2 8 + f 0 mp 3 2 + f 3 q 3 + f 3 q 4 2 -i 4 f 7 w + i 4 f 2 mp 1 w + i f 4 q 4 w + f 6 w 2 4 ) δ n, 2 + e 2 ( 3 f 5 4 + f 8 -f 1 mp 1 4 -f 0 m 2 p 1 2 8 -f 0 mp 3 2 + f 3 q 3 + f 3 q 4 2 + i 4 f 7 w + i 4 f 2 mp 1 w -i f 4 q 4 w + f 6 w 2 4 ) δ n, -2 ] , (8 . 21) \nwhere δ n,n ' is the Kronecker delta. We see from this equation that Z /lscriptmω n = O ( e | n | ) just as in the Schwarzschild case. Therefore we only need to retain the n = 0, ± 1 modes to evaluate the luminosity up to O ( e 2 ). \nWe calculate the energy and angular momentum fluxes to O ( v 5 ) beyond the quadrupole formula and to O ( e 2 ) in the eccentricity. The time-averaged energy and angular momentum fluxes are given by Eqs. (2 . 31) and (2 . 32), respectively. In order to express the post-Newtonian corrections to the luminosity, we define η /lscriptmn as \n( dE dt ) /lscriptmn ≡ 1 2 ( dE dt ) N η /lscript,m,n , (8 . 22) \nwhere ( dE/dt ) N is the Newtonian quadrupole luminosity given by Eq. (4 . 19). In the following, we show η /lscriptmn for m ≥ 0 modes. η /lscript,m,n for m < 0 are obtained from the symmetry η /lscript,m,n = η /lscript, -m, -n , which follows from the property of ω n in the present case, given by Eq. (7 . 21). \nFor /lscript = 2, the 2.5PN formulas for η /lscript,m,n are found to be ∗ ) \nη 2 , 2 , 0 = 1 -107 v 2 21 +4 π v 3 -6 q v 3 + 4784 v 4 1323 +2 q 2 v 4 -428 π v 5 21 + 4216 q v 5 189 \n+ e 2 ( -10 + 932 v 2 21 -46 π v 3 +84 q v 3 -14270 v 4 147 -23 q 2 v 4 + 4748 π v 5 21 + 2675 q v 5 189 ) , η 2 , 2 , 1 = e 2 ( 729 64 -3645 v 2 64 + 2187 π v 3 32 -3645 q v 3 32 + 24057 v 4 256 + 2187 q 2 v 4 64 -6561 π v 5 16 + 9477 q v 5 112 ) , η 2 , 2 , -1 = e 2 ( 9 64 + 1041 v 2 448 + 9 π v 3 32 -153 q v 3 32 + 2224681 v 4 112896 + 99 q 2 v 4 64 + 615 π v 5 112 -27857 q v 5 336 ) , η 2 , 1 , 0 = v 2 36 -q v 3 12 -17 v 4 504 + q 2 v 4 16 + π v 5 18 -793 q v 5 9072 + e 2 ( -2 v 2 9 + 2 q v 3 3 + 93 v 4 112 -q 2 v 4 2 -19 π v 5 36 -27113 q v 5 18144 ) , η 2 , 1 , 1 = e 2 ( 4 v 2 9 -4 q v 3 3 -172 v 4 63 + q 2 v 4 + 16 π v 5 9 + 2794 q v 5 567 ) , η 2 , 0 , ± 1 = e 2 ( 1 96 -145 v 2 672 + π v 3 48 + 3 q v 3 16 + 282521 v 4 169344 -3 q 2 v 4 32 -83 π v 5 168 -1255 q v 5 504 ) , \nand η 2 , 1 , -1 becomes O ( v 6 ). Putting together the above results, we obtain ( dE/dt ) /lscript ≡ ∑ mn ( dE/dt ) /lscriptmn for /lscript = 2 as \n( dE dt ) 2 = ( dE dt ) N { 1 -1277 v 2 252 +4 π v 3 -73 q v 3 12 + 37915 v 4 10584 + 33 q 2 v 4 16 -2561 π v 5 126 + 201575 q v 5 9072 + e 2 ( 37 24 -2581 v 2 252 + 1087 π v 3 48 -211 q v 3 6 + 325393 v 4 21168 + 105 q 2 v 4 8 -29857 π v 5 168 + 11293 q v 5 672 )} . (8 . 23) \nFor /lscript = 3, we obtain \nη 3 , 3 , 0 = 1215 v 2 896 -1215 v 4 112 + 3645 π v 5 448 -1215 q v 5 112 \n+ e 2 ( -10935 v 2 448 + 37665 v 4 256 -142155 π v 5 896 + 134865 q v 5 448 ) , η 3 , 3 , 1 = e 2 ( 640 v 2 21 -46720 v 4 189 + 5120 π v 5 21 -1280 q v 5 3 ) , η 3 , 3 , -1 = e 2 ( 15 v 2 14 + 1055 v 4 126 + 30 π v 5 7 -435 q v 5 14 ) , η 3 , 2 , 0 = 5 v 4 63 -40 q v 5 189 + e 2 ( -65 v 4 63 + 520 q v 5 189 ) , η 3 , 2 , 1 = e 2 ( 3645 v 4 1792 -1215 q v 5 224 ) , η 3 , 2 , -1 = e 2 ( 5 v 4 1792 -5 q v 5 672 ) , η 3 , 1 , 0 = v 2 8064 -v 4 1512 + π v 5 4032 -17 q v 5 9072 + e 2 ( -v 2 4032 + 65 v 4 16128 -π v 5 1152 + 199 q v 5 36288 ) , η 3 , 1 , 1 = e 2 ( v 2 126 -23 v 4 126 + 2 π v 5 63 + 122 q v 5 567 ) , η 3 , 0 , ± 1 = e 2 ( v 4 2688 -q v 5 1008 ) , \nand η 3 , 1 , -1 becomes O ( v 6 ). Thus we obtain \n( dE dt ) 3 = ( dE dt ) N { 1367 v 2 1008 -32567 v 4 3024 + ( 16403 π 2016 -896 q 81 ) v 5 + e 2 ( 1801 v 2 252 -78509 v 4 864 + ( 40083 π 448 -8913 q 56 ) v 5 )} . (8 . 24) \nFor /lscript = 4, we have \nη 4 , 4 , 0 = 1280 v 4 567 -37120 e 2 v 4 567 , η 4 , 4 , 1 = 48828125 e 2 v 4 580608 , η 4 , 4 , -1 = 32805 e 2 v 4 7168 , η 4 , 2 , 0 = 5 v 4 3969 -25 e 2 v 4 3969 , η 4 , 2 , -1 = 5 e 2 v 4 254016 , \nη 4 , 0 , ± 1 = e 2 v 4 225792 , \nand η 4 , 2 , 1 becomes O ( v 6 ). Hence we have \n( dE dt ) 4 = ( dE dt ) N { 8965 v 4 3969 + 2946739 e 2 v 4 127008 } . (8 . 25) \nFinally, gathering all the above results, we have the luminosity up to O ( v 5 ) as \n〈 dE dt 〉 = ( dE dt ) N { 1 -1247 v 2 336 +4 π v 3 -73 q v 3 12 -44711 v 4 9072 + 33 q 2 v 4 16 -8191 π v 5 672 + 3749 q v 5 336 + e 2 ( 37 24 -65 v 2 21 + 1087 π v 3 48 -211 q v 3 6 -465337 v 4 9072 + 105 q 2 v 4 8 -118607 π v 5 1344 -95663 q v 5 672 )} . (8 . 26) \nIf we set q = 0, the e 2 correction terms in the above formula completely agree with the corresponding terms in Eq. (7 . 25) in the previous section. \nTo compare our results with those derived in the standard post-Newtonian method, it is convenient to change the parameter from v to x ≡ ( MΩ ϕ ) 1 / 3 . The relation between v and x is given by \nv = x [ 1 + q 3 x 3 + e 2 { 1 2 -3 2 x 2 + 8 3 qx 3 -6 x 4 -q 2 x 4 + 31 2 qx 5 }] . (8 . 27) \nThen we obtain \n〈 dE dt 〉 = ˜ ( dE dt ) N { 1 -1247 336 x 2 -11 4 qx 3 +4 x 3 π -44711 9072 x 4 + 33 16 x 4 q 2 -59 16 qx 5 -8191 672 x 5 π + ( 157 24 -6781 168 x 2 -2009 72 qx 3 + 2335 48 x 3 π -14929 189 x 4 + 281 16 x 4 q 2 -2399 56 qx 5 -773 3 x 5 π ) e 2 } , (8 . 28) \nwhere ˜ ( dE/dt ) N is the quadrupole flux expressed in terms of x , Eq. (6 . 6). We find that the terms which are proportional to e 2 agree with the formulas derived by Peters and Mathews 45) at leading order, Galt'sov et al. 7) and Blanchet and Schafer at v 2 order 46) , Blanchet and Schafer at v 3 order for q = 0 47) and Shibata at v 3 order for q = 0 48) , if we expand their formulas by e assuming e 1 and µ/M /lessmuch 1. \n/negationslash \n/lessmuch \n/lessmuch From Eq. (2 . 32), the partial mode contributions to the angular momentum fluxes for /lscript = 2, 3 and 4 are calculated to be \n( dJ z dt ) 2 = ( dJ z dt ) N { 1 -1277 v 2 252 +4 π v 3 -61 q v 3 12 + 37915 v 4 10584 + 33 q 2 v 4 16 \n-2561 π v 5 126 + 22229 q v 5 1296 + e 2 ( -5 8 + 137 v 2 24 + 49 π v 3 8 -57 q v 3 4 -235675 v 4 14112 + 203 q 2 v 4 32 -20437 π v 5 504 -164449 q v 5 4536 )} . ( dJ z dt ) 3 = ( dJ z dt ) N { 1367 v 2 1008 -32567 v 4 3024 + ( 16403 π 2016 -88049 q 9072 ) v 5 + e 2 ( 67 v 2 32 -66497 v 4 2016 + ( 43193 π 1008 -1675571 q 18144 ) v 5 )} , ( dJ z dt ) 4 = ( dJ z dt ) N { 8965 v 4 3969 + 478195 e 2 v 4 42336 } , \nwhere ( dJ z /dt ) N is defined by \n( dJ z dt ) N = 32 µ 2 M 5 / 2 5 r 7 / 2 0 = 32 5 ( µ M ) 2 Mv 7 . (8 . 29) \nTotal angular momentum luminosity is then given by \n〈 dJ z dt 〉 = ( dJ z dt ) N { 1 -1247 v 2 336 + ( 4 π -61 q 12 ) v 3 + ( -44711 9072 + 33 q 2 16 ) v 4 + ( -8191 π 672 + 417 q 56 ) v 5 + e 2 ( -5 8 + 749 v 2 96 + ( 49 π 8 -57 q 4 ) v 3 + ( -232181 6048 + 203 q 2 32 ) v 4 + ( 773 π 336 -28807 q 224 ) v 5 )} . (8 . 30) \nThe e 2 terms in the above also agree with the corresponding terms in Eq. (7 . 26). The angular momentum flux expressed in terms of x = ( MΩ ϕ ) 1 / 3 is given by \n〈 dJ z dt 〉 = ˜ ( dJ z dt ) N { 1 -1247 336 x 2 -11 4 qx 3 +4 x 3 π -44711 9072 x 4 + 33 16 x 4 q 2 -59 16 qx 5 -8191 672 x 5 π + ( 23 8 -3259 168 x 2 -371 24 qx 3 + 209 8 x 3 π -1041349 18144 x 4 + 171 16 x 4 q 2 -243 8 qx 5 -785 6 x 5 π ) e 2 } , (8 . 31) \nwhere ( dE/dt ) N is the Newtonian flux expressed in terms of x , Eq. (7 . 30).", '§ 9. Circular orbit with small inclination from the equatorial plane': "˜ \nIn this section, we consider the case of a circular orbit at r = r 0 with small inclination from the equatorial plane. We evaluate 〈 dE/dt 〉 and 〈 dJ z /dt 〉 to O ( v 5 ) beyond Newtonian. \nwhere \nIn this case, the orbital plane precesses around the symmetric axis. The degree of precession is determined by the value of the Carter constant C . If r 0 and C are given, the energy E and the z -component of the angular momentum l z are obtained by the two equations, R = 0 and ∂R/∂r = 0, where R is a function defined by Eq. (2 . 19). We introduce a dimensionless parameter y defined by \ny = C Q 2 ; Q 2 = l 2 z + a 2 (1 -E 2 ) . (9 . 1) \nWe assume y is a small number. Since Q 2 ∼ l 2 z and C ∼ l 2 x + l 2 y , this is physically equivalent to assuming l 2 x + l 2 y /lessmuch l 2 z . Since we do not need the exact expressions for E and l z in terms of r 0 and y , we show them to the first order in y as well as to O ( v 5 ). They are given by \nE = 1 -M 2 r 0 + 3 M 2 8 r 2 0 -M 3 / 2 a r 5 / 2 0 (1 -y 2 ) + O ( v 6 ) , (9 . 2) l z = ( Mr 0 ) 1 / 2 ( 1 -y 2 ) + 3 M 2 r 0 (1 -y 2 ) -3 M 1 / 2 a r 3 / 2 0 (1 -y ) + 27 M 2 8 r 2 0 (1 -y 2 ) + a 2 r 2 0 (1 -2 y ) -15 M 3 / 2 a 2 r 5 / 2 0 (1 -y ) + O ( v 6 ) , (9 . 3) \nwhere note that a = Mq ( q | < 1). \n| \n| To solve the geodesic equations under the assumption y /lessmuch 1, we first set θ = π/ 2 + y 1 / 2 θ ' and consider the geodesic equation for θ . It then becomes \n( dθ ' dτ ) 2 = 1 Σ 2 [ Q 2 -sin 2 ( y 1 / 2 θ ' ) y { a 2 (1 -E 2 ) + l 2 z cos 2 ( y 1 / 2 θ ' ) } ] . (9 . 4) \nSince the right hand side of Eq. (9 . 4) contains only even-functions of y 1 / 2 θ ' , we can solve it iteratively by expanding θ ' as \nθ ' = θ (0) + yθ (1) + y 2 θ (2) + · · · . (9 . 5) \nThis method is similar to the one we have used in section 7 or 8. However, here we only consider the lowest order solution θ (0) . This means we take into account the effect of inclination up to O ( y ), as seen from the structure of the geodesic equations. The equation for θ (0) is \nor dividing it by ( dt/dτ ) 2 , \n( dθ (0) dτ ) 2 = Q 2 Σ 2 (1 -θ 2 (0) ) , (9 . 6) \n( dθ (0) dt ) 2 = Q 2 σ 2 (1 -θ 2 (0) ) , (9 . 7) \nσ ≡ -a ( aE -l z ) + a 2 + r 2 0 ∆ ( r 0 ) { E ( r 2 0 + a 2 ) -al z } . (9 . 8) \nThen the solution is easily obtained as \nθ (0) = sin( Ω θ t ); Ω θ = Q σ , (9 . 9) \nwhere we have chosen θ (0) = 0 at t = 0. Thus we have \nθ = π 2 + y 1 / 2 sin( Ω θ t ) . (9 . 10) \nNote that the solution (9 . 10) implies that the inclination angle θ i is indeed given by θ i = y 1 / 2 in the present approximation. \nNext, we consider the geodesic equation for ϕ . Taking account of the terms up to O ( y ), it becomes \ndϕ dt = κ σ [ 1 + ( l z κ -a 2 E σ ) yθ 2 (0) ] = Ω ϕ -y Ω 2 2 cos(2 Ω θ t ) , (9 . 11) \nwhere \nand \nκ ≡ -( aE -l z ) + a ∆ ( r 0 ) { E ( r 2 0 + a 2 ) -al z } , (9 . 12) \nΩ ϕ = κ σ + 1 2 yΩ 2 , Ω 2 = κ σ ( l z κ -a 2 E σ ) . (9 . 13) \nThe solution to Eq. (9 . 11) with ϕ = 0 at t = 0 is \nϕ = Ω ϕ t -y Ω 2 4 Ω θ sin(2 Ω θ t ) . (9 . 14) \nNote that Ω ϕ /negationslash = Ω θ . This means the precession of a test particle orbit around the spin axis of the black hole. Specifically, to the order required for the present purpose, we have \nΩ ϕ = M 1 / 2 r 3 / 2 0 [ 1 -M 1 / 2 a r 3 / 2 0 + 3 2 y ( M 1 / 2 a r 3 / 2 0 -a 2 r 2 0 ) + O ( v 6 ) ] , Ω θ = M 1 / 2 r 3 / 2 0 [ 1 -3 M 1 / 2 a r 3 / 2 0 + 3 a 2 2 r 2 0 + O ( v 6 ) + O ( y ) ] . (9 . 15) \nWe see that Ω ϕ -Ω θ → 2 Ma/r 3 0 for r 0 → ∞ and y → 0, which is just the LenseThirring precessional frequency 49) . \nNow we are ready to calculate the source integral for the amplitude ˜ Z /lscriptmω . Analogous to the case of an eccentric orbit considered in section 7 or 8, Eq. (2 . 26) can be simplified further by noting that the orbits of our interest have the properties, \nθ ( t + ∆t θ ) = θ ( t ) , ϕ ( t + ∆t θ ) = ϕ ( t ) + ∆ϕ, (9 . 16) \nwhere ∆t θ is the orbital period of the motion in the θ -direction and ∆ϕ is the phase advancement during ∆t θ . In other words, we have \nΩ θ = 2 π ∆t θ , Ω ϕ = ∆ϕ ∆t θ . (9 . 17) \nThen we obtain \nwhere \nand \n˜ Z /lscriptmω = ∑ n δ ( ω -ω n ) Z /lscriptmω n , (9 . 18) \nω n = nΩ θ + mΩ ϕ ( n = 0 , ± 1 , ± 2 , · · · ) , (9 . 19) \nZ /lscriptmω n = µΩ θ 2 iω n B inc /lscriptmω n ∫ ∆t 0 dte iωt -imϕ ( t ) W /lscriptmω n , (9 . 20) \nwith W /lscriptmω n being given by Eq. (2 . 27). \nLet us discuss the final form of Z /lscriptmω n . In the present case, up to O ( y ) the integrand W /lscriptmω n has the form, \nW /lscriptmω n = g 0 + y 1 / 2 g 1 θ (0) + yg 2 θ 2 (0) + y 1 / 2 g 3 dθ (0) dt + yg 4 θ (0) dθ (0) dt + yg 5 ( dθ (0) dt ) 2 + O ( y 3 / 2 ) , (9 . 21) \nwhere g 0 ∼ g 5 are complicated functions of r 0 . Using an approximation, \ne iω n t -imϕ ( t ) = e inΩ θ t ( 1 + y mΩ 2 8 Ω θ ( e 2 iΩ θ t -e -2 iΩ θ t ) + O ( y 3 / 2 ) ) , (9 . 22) \nwe have \n∫ ∆t 0 dte iωt -imϕ ( t ) W /lscriptmω n = 2 π Ω θ [ { δ n 0 + y mΩ 2 8 Ω θ ( δ n, -2 -δ n, 2 ) } g 0 + y 1 / 2 1 2 i ( δ n, -1 -δ n, 1 ) g 1 + y 1 4 (2 δ n, 0 -δ n, -2 -δ n, 2 ) g 2 + y 1 / 2 Ω θ 2 ( δ n, -1 + δ n, 1 ) g 3 + y Ω θ 4 i ( δ n, -2 -δ n, 2 ) g 4 + y Ω 2 θ 4 (2 δ n, 0 + δ n, -2 + δ n, 2 ) g 5 ] + O ( y 3 / 2 ) . (9 . 23) \nThus the amplitude Z /lscriptmω n is found to have the form, \nZ /lscriptmω n = [ ( Z 0 , 0 + yZ 0 , 2 ) δ n, 0 + y 1 / 2 ( Z 1 , 1 δ n, 1 + Z 1 , -1 δ n, -1 ) + y ( Z 2 , 2 δ n, 2 + Z 2 , -2 δ n, -2 ) + O ( y 3 / 2 ) ] , (9 . 24) \nwhere Z i,j are functions of r 0 . Here, it is worth noting the symmetry of Z /lscriptmω n . The spin weighted spheroidal harmonics have a property -2 S aω n /lscriptm ( θ ) = ( -1) /lscript -2 S aω -n /lscript -m ( π -θ ). Then, from Eqs. (9 . 20) and (9 . 22), we have Z /lscript -mω -n = ( -1) n + /lscript Z /lscriptmω n . \nNow we evaluate the energy and angular momentum fluxes at infinity. The energy and angular momentum fluxes averaged over t /greatermuch ∆t θ are give by Eqs. (2 . 31) and (2 . 32), respectively. Then we see from Eq.(9 . 24) that the n = ± 2 modes contribute to the luminosity at O ( y 2 ). Thus, when we calculate the luminosity to O ( y ), we need to include only the n = 0 , ± 1 modes. In order to express the post-Newtonian corrections to the luminosity, we define η /lscriptmn as \n( dE dt ) /lscriptmn ≡ 1 2 ( dE dt ) N η /lscriptmn , (9 . 25) \nwhere ( dE/dt ) N is the Newtonian quadrupole luminosity given by Eq. (4 . 19). \nFor /lscript = 2, the results are as follows. If | m + n | > 2 or m + n = 0, η /lscriptmn becomes of O ( v 6 ) or higher. The remaining η /lscriptmn which contribute to the 2.5PN luminosity formula are given by \nη 2 ± 2 0 = 1 -107 21 v 2 +4 πv 3 -6 qv 3 + 4784 1323 v 4 +2 q 2 v 4 -428 π 21 v 5 + 4216 189 qv 5 + y ( -1 + 170 21 v 2 -4 πv 3 +15 qv 3 -4784 1323 v 4 -11 q 2 v 4 + 428 π 21 v 5 -13186 189 qv 5 ) , η 2 ± 2 ∓ 1 = y ( 1 36 v 2 -17 504 v 4 + π 18 v 5 + 17 1134 qv 5 ) , η 2 ± 1 0 = 1 36 v 2 -1 12 qv 3 -17 504 v 4 + 1 16 q 2 v 4 + π 18 v 5 -793 9072 qv 5 + y ( -5 72 v 2 + 1 8 qv 3 + 85 1008 v 4 -1 32 q 2 v 4 -5 π 36 v 5 + 13931 18144 qv 5 ) , η 2 ± 1 ± 1 = y ( 1 -170 21 v 2 +4 πv 3 -12 qv 3 + 4784 1323 v 4 + 11 2 q 2 v 4 -428 π 21 v 5 + 11078 189 qv 5 ) , η 2 0 ± 1 = y ( 1 24 v 2 -1 12 qv 3 -17 336 v 4 + 1 24 q 2 v 4 + π 12 v 5 -745 1008 qv 5 ) . (9 . 26) \n( dE dt ) 2 = ( dE dt ) N { 1 -1277 252 v 2 +4 πv 3 -73 12 qv 3 ( 1 -y 2 ) + 37915 10584 v 4 + 33 16 q 2 v 4 -527 96 q 2 v 4 y -2561 π 126 v 5 + 201575 9072 qv 5 ( 1 -y 2 )} . (9 . 27) \nPutting together the above results, we obtain ( dE/dt ) /lscript ≡ ∑ mn ( dE/dt ) /lscriptmn for /lscript = 2 as \nFor /lscript = 3, the non-trivial η /lscriptmn are given by \nη 3 ± 3 0 = 1215 896 v 2 -1215 112 v 4 + 3645 π 448 v 5 -1215 112 qv 5 + y ( -3645 1792 v 2 + 3645 224 v 4 -10935 π 896 v 5 + 3645 112 qv 5 ) , η 3 ± 3 ∓ 1 = 5 42 v 4 y, \nη 3 ± 2 0 = 5 63 v 4 -40 189 qv 5 + y ( -20 63 v 4 + 100 189 qv 5 ) , η 3 ± 2 ± 1 = y ( 3645 1792 v 2 -3645 224 v 4 + 10935 π 896 v 5 -6075 224 qv 5 ) , η 3 ± 2 ∓ 1 = y ( 5 16128 v 2 -5 3024 v 4 + 5 π 8064 v 5 + 25 18144 qv 5 ) , η 3 ± 1 0 = 1 8064 v 2 -1 1512 v 4 + π 4032 v 5 -17 9072 qv 5 + y ( -11 16128 v 2 + 11 3024 v 4 -11 π 8064 v 5 + 95 9072 qv 5 ) , η 3 ± 1 ± 1 = y ( 25 126 v 4 -80 189 qv 5 ) , η 3 0 ± 1 = y ( 1 2688 v 2 -1 504 v 4 + π 1344 v 5 -11 1008 qv 5 ) . (9 . 28) \nThe other η /lscriptmn are of O ( v 6 ) or higher. Then we obtain \n( dE dt ) 3 = ( dE dt ) N { 1367 1008 v 2 -32567 3024 v 4 + 16403 π 2016 v 5 -896 81 qv 5 ( 1 -y 2 )} . (9 . 29) \nFor /lscript = 4, we have \nη 4 ± 4 0 = 1280 567 v 4 (1 -2 y ) , η 4 ± 3 ± 1 = 2560 567 v 4 y, η 4 ± 3 ∓ 1 = 5 1134 v 4 y, η 4 ± 2 0 = 5 3969 v 4 (1 -8 y ) , η 4 ± 1 ± 1 = 5 882 v 4 y, (9 . 30) \nand the others are of O ( v 6 ) or higher. Hence we obtain \n( dE dt ) 4 = ( dE dt ) N × 8965 3969 v 4 . (9 . 31) \nFinally, gathering all the terms, the total energy flux up to O ( v 5 ) is found to be \n〈 dE dt 〉 = ( dE dt ) N ( 1 -1247 336 v 2 +4 πv 3 -73 12 qv 3 ( 1 -y 2 ) -44711 9072 v 4 + 33 16 q 2 v 4 -527 96 q 2 v 4 y -8191 π 672 v 5 + 3749 336 qv 5 ( 1 -y 2 ) ) . (9 . 32) \nUsing the above results for η /lscriptmn , the time-averaged angular momentum flux is calculated from Eq. (2 . 32). The partial mode contributions of the /lscript = 2, 3 and 4 modes are calculated to give \n( dJ z dt ) 2 = ( dJ z dt ) N [ 1 -y 2 -1277 252 v 2 ( 1 -y 2 ) +4 πv 3 ( 1 -y 2 ) \n-qv 3 ( 61 12 -61 8 y ) + 37915 10584 v 4 ( 1 -y 2 ) + q 2 v 4 ( 33 16 -229 32 y ) -πv 5 2561 126 ( 1 -y 2 ) + qv 5 ( 22229 1296 -27809 864 y ) ] , ( dJ z dt ) 3 = ( dJ z dt ) N [ 1367 1008 v 2 ( 1 -y 2 ) -32567 3024 v 4 ( 1 -y 2 ) + πv 5 16403 2016 ( 1 -y 2 ) -qv 5 ( 88049 9072 -9817 756 y ) ] , ( dJ z dt ) 4 = ( dJ z dt ) N [ 8965 3969 v 4 ( 1 -y 2 ) ] , (9 . 33) \nwhere ( dJ z /dt ) N is defined in Eq. (8 . 29). The total angular momentum flux is then given by \n〈 dJ z dt 〉 = ( dJ z dt ) N [ ( 1 -y 2 ) -1247 336 v 2 ( 1 -y 2 ) +4 πv 3 ( 1 -y 2 ) -61 12 qv 3 ( 1 -y 2 ) -44711 9072 v 4 ( 1 -y 2 ) + q 2 v 4 ( 33 16 -229 32 y ) -8191 672 πv 5 ( 1 -y 2 ) + qv 5 ( 417 56 -4301 224 y ) ] . (9 . 34) \nWe note that the result is proportional to (1 -y/ 2) in the limit q → 0. This is simply because the orbital plane is slightly tilted from the equatorial plane by an angle θ i ∼ y 1 / 2 , hence dJ z /dt ∼ ( dJ tot /dt ) cos θ i .", '§ 10. Adiabatic backreaction': "In the preceding sections, we have evaluated the energy flux 〈 dE/dt 〉 and the z -component of the angular momentum flux 〈 dJ z /dt 〉 emitted to infinity by a particle for various cases. By emitting gravitational waves, a particle orbit will suffer from radiation reaction. In the limit of small µ/M , the reaction time scale will be much longer than the characteristic orbital time scale; t react ∼ M 2 /µ /greatermuch ∆t . Hence the evolution of the orbit will be well described by the adiabatic backreaction. \nIn the case of orbits around a Schwarzschild black hole or orbits confined on the equatorial plane around a Kerr black hole, it is straightforward to calculate the evolutionary path under radiation reaction because the orbits are completely specified by the energy E and the z -component of the angular momentum l z , hence their time derivatives can be simply evaluated by equating them with -〈 dE/dt 〉 and -〈 dJ z /dt 〉 , respectively. However, once we consider motions off the equatorial plane of a Kerr black hole, the orbits cannot be specified by E and l z alone but the specification of the Carter constant C becomes necessary. Unlike E or l z , since C is not associated with the Killing vector of the spacetime, one cannot calculate the radiation reaction to C by simply calculating the gravitational waves at infinity. This implies that we have to derive a local radiation reaction force term to the geodesic equation by evaluating the metric perturbations around the particle, as is done in the \nderivation of radiation reaction force in the standard post-Newtonian method. For almost Newtonian orbits, applying a post-Newtonian radiation reaction force, Ryan derived the evolution equation for the Carter constant 50) . However, no relativistic treatment has been done so far. This is a challenging issue. An approach to this issue is discussed in Chapter 7. \nIn this section, instead of attacking this very difficult problem, we discuss some general properties of the adiabatic radiation reaction in a restricted class of orbits. Namely we consider orbits which are circular or those having small eccentricity. We clarify the conditions for circular orbits to remain circular under radiation reaction. A detailed discussion on this matter has been given by Kennefick and Ori 51) . We give a less detailed but more general discussion below. \nWe recall that the radial velocity u r = dr/dτ is written in terms of the first integrals of motion in the test particle limit as \n( Σu r ) 2 = R ( I i , r ) , (10 . 1) \nwhere I i = ( E,l z , C ) and R ( I i , r ) is independent of θ and φ . First let us consider orbits which are circular in the test particle limit. These orbits are determined by the conditions, \nR ( I i , r ) = 0 , ∂R ∂r ( I i , r ) = 0 . (10 . 2) \nEliminating r from these equations gives an implicit relation among I i 's. For example, \nf ( I i ) = R ( I i , r ( I i )) = 0 , (10 . 3) \nwhere r ( I i ) is obtained by solving the second of Eq. (10 . 2) for r . This equation determines a two-dimensional hypersurface S in the 3-dimensional space M of I i 's. The adiabatic evolution of an orbit is characterized by slow evolution of I i , i.e., ˙ I i = O ( µ ), where µ is the mass of the particle. Then a necessary condition for circular orbits to remain circular under radiation reaction is that we have ˙ f ( I i ) = f ,i ˙ I i = O ( µ 2 ). In other words, the vector ˙ I i on S is tangent to S to O ( µ ). This condition can be shown to hold by the following theorem. \nTheorem : If the radiation reaction to the r -component of the acceleration is of order µ ; a r := du r /dτ = O ( µ ), i.e., the r -component of the radiation reaction force is well-defined and finite, then for orbits which are circular in the test particle limit; i.e., u r = O ( µ ), the radiation reaction to I i is constrained by the equation, \n∂R ∂I i ( I i , r ) ˙ I i = O ( µ 2 ) , (10 . 4) \nwhere the argument r is to be replaced by r ( I i ) after differentiation. \nProof : It is almost trivial. Just taking the τ -derivatives of both hand sides of Eq. (10 . 1) gives Eq. (10 . 4). Q.E.D. \nThus, since \n˙ f ( I i ) = ( ∂R ∂I i ( I i , r ( I i )) + ∂R ∂r ( I i , r ( I i )) ∂r ∂I i ) ˙ I i , (10 . 5) \nand the second term in the parentheses vanishes by definition, we have ˙ f = O ( µ 2 ). \nThis theorem alone, however, does not mean that circular orbits remain circular, since we have constrained the first integrals to be those for circular orbits from the beginning. Let us explain the reason. Since we may regard ˙ I i a vector field in M , what we need for circular orbits to remain circular is the regularity of ˙ I i in the vicinity of the hypersurface S . In other words, if the vector field ˙ I i is not differentiable on S , an orbit on S may spontaneously deviates away from S . A simple illustrative example is the case ˙ I ⊥ = √ I ⊥ at I ⊥ = 0 where I ⊥ is the component of I i perpendicular to S . \n˙ C = ( 2 ∆ ( E ( r 2 + a 2 ) -al z )( r 2 + a 2 ) -2 a ( Ea -l z ) ) ˙ E + ( -2 a ∆ ( E ( r 2 + a 2 ) -al z ) +2( Ea -l z ) ) ˙ l z . (10 . 6) \nThus, provided ˙ I i is regular in an open neighborhood of S , the above theorem implies that a circular orbit in the test particle limit remains circular under adiabatic radiation reaction. In this case, the radiation reaction to the Carter constant, ˙ C , is determined by the radiation reaction to the energy, ˙ E , and the z -component of the angular momentum, ˙ l z . Specifically we have \nYet this is not the end of the story. What we have shown is that ˙ I i lies on S . But if ˙ I i slightly off the hypersurface S is diverging away from S , circular orbits will be unstable. Thus the condition for the stability of circular orbits is that S is an attractor plane of the vector field ˙ I i . However, the notion of divergence or convergence of a vector depends on the metric of the space M , but we have no guiding principle to determine the metric. This implies that the notion of the attractor or the stability is ambiguous. \nNevertheless, extrapolating from the case of Newtonian orbits, there seems to exist a natural choice of the metric. Namely, as the distance of the orbit from the hypersurface S of circular orbits, we define the eccentricity of an orbit as given in sections 7 and 8. With this choice of the metric, let us consider the adiabatic radiation reaction problem in more specific terms. \nLet us parametrize an orbit in terms of the mean radius r 0 , the eccentricity e and the square root of the Carter constant y := C 1 / 2 , instead of the energy E , the angular momentum l z and the Carter constant C . The mean radius r 0 is defined by the equation, \n0 = R ' ( I i , r 0 ) , (10 . 7) \nwhere the prime denotes the partial derivative with respect to r . This definition says that ˙ r is maximum at r = r 0 . The eccentricity e is defined by setting the maximum radius to r = r 0 (1 + e ), i.e., \n0 = R ( I i , r 0 (1 + e )) , (10 . 8) \nThis definition guarantees that e = 0 corresponds to a circular orbit. Assuming e /lessmuch 1, the above equation can be expanded in powers of e as \n0 = R ( I i , r 0 ) + 1 2 R '' ( I i , r 0 )( r 0 e ) 2 + 1 3! R (3) ( I i , r 0 )( r 0 e ) 3 + 1 4! R (4) ( I i , r 0 )( r 0 e ) 4 + · · · , (10 . 9) \nwhere R ( n ) is the n -th derivative of R with respect to r . The parameters ( r 0 , e, y ) are chosen because the geodesic trajectory x µ = z µ ( τ ) allows perturbative expansion in powers of e and y at least for e /lessmuch 1 and y /lessmuch 1. Therefore the first integrals of motion I i = ( E,l z , C ) will be regular functions of the parameters ( r 0 , e, y ). On the other hand, if we consider ( r 0 , e, y ) as functions of I i , it should be noted that e is not a regular function of I i in a neighborhood of circular orbits because of the absence of a term linear in e in the right hand side of Eq. (10 . 9). \nNow we consider the adiabatic evolution of e under the radiation reaction. Taking the τ -derivative of Eqs. (10 . 7) and (10 . 9), we get \n0 = R ' 0 ,i ˙ I i + R '' 0 ˙ r 0 , (10 . 10) \n0 = R 0 ,i I i + R '' 0 ( r 0 ˙ r 0 e 2 + r 2 0 e ˙ e ) + 1 2 ( R '' 0 ,i ˙ I i + R (3) 0 ˙ r 0 ) r 2 0 e 2 + 1 2 R (3) 0 r 3 0 e 2 ˙ e + 1 3! R (4) 0 r 4 0 e 3 ˙ e + O ( e 3 , ˙ ee 4 ) , (10 . 11) \nwhere R ' 0 ,i = ∂ 2 R/∂I i ∂r etc. Equation (10 . 10) determines ˙ r 0 as \n˙ r 0 = -R ' 0 ,i R '' 0 ˙ I i . (10 . 12) \nSubstituting this into Eq. (10 . 11), we obtain the expression for ˙ e as \n˙ e = 1 er 2 0 R '' 0 -R 0 ,i + e 2 r 0 R (3) 0 R '' 0 R 0 ,i + e 2 -1 2 r 0 2 R '' 0 ,i + ( r 0 + 1 2 r 2 0 R (3) 0 R '' 0 ) R ' 0 ,i + r 2 0 1 6 R (4) 0 R '' 0 -1 4 ( R (3) 0 R '' 0 ) 2 R 0 ,i + O ( e 3 ) ˙ I i . (10 . 13) \nSince the trajectory z µ ( τ ) is assumed to be analytic in ( r 0 , e, y ), it is reasonable to further assume that ˙ I i are regular functions of ( r 0 , e, y ). Then we can expand ˙ I i with respect to e as \n˙ I i ( r 0 , e, y ) = ˙ I i (0) ( r 0 , y ) + e ˙ I i (1) ( r 0 , y ) + e 2 ˙ I i (2) ( r 0 , y ) + · · · . (10 . 14) \nThen we obtain \n˙ e = 1 r 2 0 R '' 0 -1 e R 0 ,i ˙ I i (0) + ( -R 0 ,i ˙ I i (1) + 1 2 r 0 R ''' 0 R '' 0 R 0 ,i ˙ I i (0) ) + e -R 0 ,i ˙ I i (2) + 1 2 r 0 R ''' 0 R '' 0 R 0 ,i ˙ I i (1) + { -1 2 r 0 2 R '' 0 ,i + ( r 0 + 1 2 r 0 2 R ''' 0 R '' 0 ) R ' 0 ,i + r 2 0 ( 1 6 R (4) 0 R '' 0 -1 4 ( R ''' 0 R '' 0 ) 2 ) R 0 ,i } ˙ I i (0) \n+ O ( e 2 ) . (10 . 15) \nAs shown by the theorem above, Eq. (10 . 4), the leading term of order e -1 vanishes provided the radiation reaction force is finite: \nR 0 ,i ˙ I i (0) = 0 . (10 . 16) \nThe next order term determines whether the circular orbit remains circular or not. If it does not vanish, the eccentricity will spontaneously develop as \n˙ e = -R 0 ,i ˙ I i (1) . (10 . 17) \nHere the regularity of ˙ I i comes into play. As noted above, e is singular on the hypersurface S . Hence if ˙ I i is regular on S , ˙ I (1) ( r 0 , y ) should vanish. By a detailed analysis, it is shown in Ref. 51) that this is indeed the case. The physical reason is rather simple: If one considers a slightly eccentric orbit, there appears a frequency of wobbling motion due to the eccentricity, say Ω e . In general the ratio of Ω e to the frequency of the motion in the θ or ϕ direction is an irrational number. Hence the part of the metric perturbation which is proportional to e will have frequencies that are integer multiples of Ω e , and the same property is shared by the corresponding term of the backreaction force linear in e . Since any sinusoidal oscillation has zero mean when averaged over time longer than its period, this implies there will be no term linear in e in the adiabatic expression of ˙ I i . \nThus we have \n˙ e = [ -R 0 ,i ˙ I i (2) + { -1 2 r 0 2 R '' 0 ,i + ( r 0 + 1 2 r 2 0 R ''' 0 R '' 0 ) R ' 0 ,i } ˙ I i (0) ] e + O ( e 2 ) , (10 . 18) \nand circular orbits will remain circular under radiation reaction. As for the stability of circular orbits, whether the eccentricity decreases or increases is determined by the sign of the coefficient of e in the right hand side. Thus it is necessary to calculate the radiation reaction to the Carter constant to determine the stability. As mentioned in the beginning of this section, this is a challenging issue. Finally, we should again note that the meaning of stability does depend on the definition of the eccentricity, i.e., how we define the distance from the hypersurface of circular orbits S .", '§ 11. Spinning particle': 'So far we have considered only a monopole particle orbiting a black hole. However, in a realistic binary system of compact bodies such as a neutron star-neutron star, black hole-neutron star or black hole-black hole binary, both bodies may have non-negligible spin angular momenta. Hence it is desirable to take into account not only the spin of a black hole but also the spin of a particle in the calculations of gravitational waves from a particle orbiting a black hole. \nTo incorporate the spin of a particle, one must know (1) the equations of motion and (2) the energy momentum tensor of a spinning particle. Fortunately, we know \nthat (1) have been derived by Papapetrou 19) , Dixon 20) and Wald 52) and (2) has also been derived by Dixon 20) . Hence, by using the expression for the energy momentum tensor of a spinning particle as the source term in the Teukolsky formalism 3) , we can calculate the gravitational waves emitted by a spinning small mass particle orbiting a rotating black hole. One may regard this particle as a model of a small Kerr black hole, but it may be appropriate here to give a word of caution. A Kerr black hole of mass µ and the spin parameter S , where S is defined so that µS gives the spin angular momentum, has quadrupole ( /lscript = 2) and higher multipole moments ( /lscript > 2) proportional to µS /lscript as well. Since we neglect the contributions of these higher multipole moments here, our treatment will be valid only up to O ( S ) if we regard the particle as a Kerr black hole. To incorporate the contributions of all higher multipole moments to represent the Kerr black hole is a future problem to be investigated. \nHere we review the results obtained by Tanaka et al. 18) . We concentrate on the leading effect due to the spin of the small mass particle. We consider a class of circular orbits which stay near the equatorial plane with the inclination solely due to the spin of the particle, i.e., those orbits which would be confined in the equatorial plane if the spin were zero. Then we calculate the gravitational wave luminosity to O ( v 5 ) with linear corrections due to the spin.', '11.1. Equation of Motion and Source Term of a spinning particle': "To give the source term of the Teukolsky equation, we need to solve the equations of motion of a spinning particle and also to give an expression for the energy momentum tensor. In this section we give the necessary expressions, following Refs. 20), 52), 53). \nNeglecting the effect of the higher multipole moments, the equations of motion of a spinning particle are given by \nD dτ p µ ( τ ) = -1 2 R µ νρσ ( z ( τ )) v ν ( τ ) S ρσ ( τ ) , D dτ S µν ( τ ) = 2 p [ µ ( τ ) v ν ] ( τ ) , (11 . 1) \nwhere v µ ( τ ) = dz µ ( τ ) /dτ , τ is a parameter which is not necessarily the proper time of the particle, and, as we will see later, the vector p µ ( τ ) and the antisymmetric tensor S µν ( τ ) represent the linear and spin angular momenta of the particle, respectively. Here D/dτ denotes the covariant derivative along the particle trajectory. \nWe do not have the evolution equation for v µ ( τ ) yet. In order to determine v µ ( τ ), we need to impose a supplementary condition which determines the center of mass of the particle 20) , \nS µν ( τ ) p ν ( τ ) = 0 . (11 . 2) \nThen one can show that p µ p µ = const . and S µν S µν = const . along the particle trajectory 52) . Therefore we may set \np µ = µu µ , u µ u µ = -1 , S µν = /epsilon1 µν ρσ p ρ S σ , p µ S µ = 0 , \nS 2 = S µ S µ = 1 2 µ 2 S µν S µν , (11 . 3) \nwhere µ is the mass of the particle, u µ is the specific linear momentum, and S µ is the specific spin vector with S its magnitude. Note that if we use S µ instead of S µν in the equations of motion, the center of mass condition (11 . 2) will be replaced by the condition \np µ S µ = 0 . (11 . 4) \nSince the above equations of motion are invariant under reparametrization of the orbital parameter τ , we can fix τ to satisfy \nu µ ( τ ) v µ ( τ ) = -1 . (11 . 5) \nThen, from Eqs. (11 . 1), (11 . 2) and (11 . 5), v µ ( τ ) is determined as 20) \nv µ ( τ ) -u µ ( τ ) = 1 2 ( µ 2 + 1 4 R χξζη ( z ( τ )) S χξ ( τ ) S ζη ( τ ) ) -1 S µν ( τ ) R νρσκ ( τ ) u ρ ( τ ) S σκ ( τ ) . (11 . 6) \nWith this equation, the equations of motion (11 . 1) completely determine the evolution of the orbit and the spin. Note that v µ = u µ + O ( S 2 ), hence v µ and u µ are identical to each other to O ( S ). \nAs for the energy momentum tensor, Dixon 20) gives it in terms of the Dirac delta-function on the tangent space at x µ = z µ ( τ ). For later convenience, in this paper we use an equivalent but alternative form of the energy momentum tensor, given in terms of the Dirac delta-function on the coordinate space: \nT αβ ( x ) = ∫ dτ p ( α ( x, τ ) v β ) ( x, τ ) δ (4) ( x -z ( τ )) √ -g -∇ γ ( S γ ( α ( x, τ ) v β ) ( x, τ ) δ (4) ( x -z ( τ )) √ -g ) , (11 . 7) \nwhere v α ( x, τ ), p α ( x, τ ) and S αβ ( x, τ ) are bi-tensors which are spacetime extensions of v µ ( τ ), p µ ( τ ) and S µν ( τ ) which are defined only along the world line, x µ = z µ ( τ ). ∗ ) To define v α ( x, z ( τ )), p α ( x, z ( τ )) and S αβ ( x, z ( τ )) we introduce a bi-tensor ¯ g α µ ( x, z ) which satisfies \nlim x → z ¯ g α µ ( x, z ( τ )) = δ α µ , lim x → z ∇ β ¯ g α µ ( x, z ( τ )) = 0 . (11 . 8) \nFor the present purpose, further specification of ¯ g α µ ( x, z ) is not necessary. Using this bi-tensor ¯ g α µ ( x, z ), we define p α ( x, τ ), v α ( x, τ ) and S αβ ( x, τ ) as \np α ( x, τ ) = ¯ g α µ ( x, z ( τ ) ) p µ ( τ ) , \nv α ( x, τ ) = ¯ g α µ ( x, z ( τ ) ) v µ ( τ ) , S αβ ( x, τ ) = ¯ g α µ ( x, z ( τ ) ) ¯ g β ν ( x, z ( τ ) ) S µν ( τ ) . (11 . 9) \nIt is easy to see that the divergence free condition of this energy momentum tensor gives the equations of motion (11 . 1). Noting the relations, \n∇ β ¯ g α µ ( x, z ( τ )) δ (4) ( x, z ( τ )) = 0 , v α ( x ) ∇ α δ (4) ( x, z ( τ ) ) √ -g = -d dτ δ (4) ( x, z ( τ ) ) √ -g , (11 . 10) \nthe divergence of Eq. (11 . 7) becomes \n∇ β T αβ ( x ) = ∫ dτ ¯ g α µ ( x, z ( τ ) ) δ (4) ( x -z ( τ ) ) √ -g ( d dτ p µ ( τ ) + 1 2 R µ νσκ ( z ( τ ) ) v ν ( τ ) S σκ ( τ ) ) + 1 2 ∫ dτ ∇ β ¯ g α µ ( x, z ( τ ) ) ¯ g β ν ( x, z ( τ ) ) δ (4) ( x -z ( τ ) ) √ -g × ( d dτ S µν ( τ ) -2 p [ µ ( τ ) v ν ] ( τ ) ) . (11 . 11) \nSince the first and second terms in the right-hand side must vanish separately, we obtain the equations of motion (11 . 1). \nIn order to clarify the meaning of p µ and S µν , we consider the volume integral of this energy momentum tensor such as ∫ Σ ( τ 0 ) ¯ g µ α T αβ dΣ β , where we take the surface \n∫ Σ ( τ 0 ) ¯ g µ α T αβ dΣ β = ∫ d 4 x √ -g ∂τ ∂x β δ ( τ ( x ) -τ 0 )¯ g µ α T αβ ( x ) = ∫ dτ ' δ ( τ ' -τ 0 ) [ p µ + p [ µ v ν ] u ν -1 2 Du ν dτ S νµ ] = p µ ( τ 0 ) . (11 . 12) \nΣ ( τ 0 ) to be perpendicular to u α ( τ 0 ). It is convenient to introduce a scalar function τ ( x ) which determines the surface Σ ( τ 0 ) by the equation τ ( x ) = τ 0 , and ∂τ/∂x β = -u β at x = z ( τ 0 ). Then we have \nwhere we used the center of mass condition and the equation of motion for S µν . We clearly see p µ indeed represents the linear momentum of the particle. \nIn order to clarify the meaning of S µν , following Dixon 20) , we introduce the relative position vector \nX µ := -g µν ∂ ν σ ( x, z ) , (11 . 13) \nwhere σ ( x, z ) is the squared geodetic interval between z and x defined by using the parametric form of a geodesic y ( u ) joining z = y (0) and x = y (1) as \nσ ( x, z ) := 1 2 ∫ 1 0 g αβ dy α du dy β du du. (11 . 14) \nThen noting the relations \nit is easy to see that \nlim x → z X µ = 0 , lim x → z X µ ,β = δ µ β , (11 . 15) \nS µν = 2 ∫ Σ τ 0 X [ µ ¯ g ν ] α T αβ dΣ β . (11 . 16) \nNow that the meaning of S µν is manifest. From the above equation, it is also easy to see that the center of mass condition (11 . 2) is the generalization of the Newtonian counter part, \nwhere ρ is the matter density. \n∫ d 3 xρ ( x ) x i = 0 , (11 . 17) \nBefore closing this subsection, we mention several conserved quantities of the present system. We have already noted that p µ p µ = -µ 2 and S µ S µ = S 2 are constant along the particle trajectory on an arbitrary spacetime. There will be an additional conserved quantity if the spacetime admits a Killing vector field ξ µ , \nξ ( µ ; ν ) = 0 . (11 . 18) \nNamely, the quantity \nQ ξ := p µ ξ µ -1 2 S µν ξ µ ; ν , (11 . 19) \nis conserved along the particle trajectory 20) . It is easy to verify that Q ξ is conserved by directly using the equations of motion.", '11.2. Circular orbits near the equatorial plane': 'Let us consider circular orbits around a Kerr black hole with a fixed BoyerLindquist radial coordinate, r = r 0 . We consider a class of orbits that would stay on the equatorial plane if the particle were spinless. Hence we assume that ˜ θ := θ -π/ 2 ∼ O ( S/M ) /lessmuch 1. Under this assumption, we write down the equations of motion and solve them up to the linear order in S . \nIn order to find a solution representing a circular orbit, it is convenient to introduce the tetrad frame defined by \ne 0 µ = ( √ ∆ Σ , 0 , 0 , -a sin 2 θ √ ∆ Σ ) , e 1 µ = ( 0 , √ Σ ∆ , 0 , 0 ) , e 2 µ = ( 0 , 0 , √ Σ, 0 ) , e 3 µ = ( -a √ Σ sin θ, 0 , 0 , r 2 + a 2 √ Σ sin θ ) , (11 . 20) \nwhere Σ = r 2 + a 2 cos 2 θ , and e a µ = ( e a t , e a r , e a θ , e a ϕ ) for a = 0 ∼ 3. Hereafter, we use the Latin letters to denote the tetrad indices. \nFor convenience, we introduce ω 1 ∼ ω 6 to represent the tetrad components of the spin coefficients, ω c ab = e µ a e ν b e c ν ; µ , near the equatorial plane: \nω 0 01 = ω 1 00 = ω 1 + O ( ˜ θ 2 ) , ω 1 = a 2 -Mr r 2 ∆ 1 / 2 , ω 0 31 = ω 1 30 = ω 0 13 = ω 3 10 = ω 1 03 = -ω 3 01 = ω 2 + O ( ˜ θ 2 ) , ω 2 := a r 2 ω 1 22 = -ω 2 21 = ω 1 33 = -ω 3 31 = ω 3 + O ( ˜ θ 2 ) , ω 3 := ∆ 1 / 2 r 2 , ω 0 02 = ω 2 00 = ω 1 12 = -ω 2 11 = ˜ θ ω 4 + O ( ˜ θ 2 ) , ω 4 := -a 2 r 3 , ω 0 32 = ω 2 30 = -ω 0 23 = -ω 3 20 = ω 2 03 = -ω 3 02 = ˜ θ ω 5 + O ( ˜ θ 2 ) , ω 5 := -a∆ 1 / 2 r 3 , ω 2 33 = -ω 3 32 = ˜ θ ω 6 + O ( ˜ θ 2 ) , ω 6 := -( r 2 + a 2 ) r 3 . (11 . 21) \nSince the following relation holds for an arbitrary vector f µ , \ne a µ D dτ f µ = d dτ f a -ω a bc v b f c , \nthe tetrad components of Df µ /dτ along a circular orbit are given explicitly as \ne 0 µ D dτ f µ = ˙ f 0 -( Af 1 + ˜ θCf 2 ) + O ( ˜ θ 2 ) , e 1 µ D dτ f µ = ˙ f 1 -( Af 0 + Bf 3 + Ef 2 ) + O ( ˜ θ 2 ) , e 2 µ D dτ f µ = ˙ f 2 -( ˜ θCf 0 + ˜ θDf 3 -Ef 1 ) + O ( ˜ θ 2 ) , e 3 µ D dτ f µ = ˙ f 3 -( -Bf 1 -˜ θDf 2 ) + O ( ˜ θ 2 ) , (11 . 22) \nwhere A , B , C , D and E are defined by ∗ ) \nA := ω 1 v 0 + ω 2 v 3 , B := ω 2 v 0 + ω 3 v 3 , C := ω 4 v 0 + ω 5 v 3 , D := ω 5 v 0 + ω 6 v 3 , E := ω 3 v 2 , (11 . 23) \nand we have assumed that v 1 = 0 and v 2 = O ( ˜ θ ). \nFor convenience, we rewrite the equations of motion by changing the spin variable. Instead of the spin tensor, we introduce a unit vector parallel to the spin, ζ a , defined by \nζ a := S a S = -1 2 µS /epsilon1 a bcd u b S cd , (11 . 24) \nor equivalently by \nS ab = µS/epsilon1 ab cd u c ζ d , (11 . 25) \nwhere /epsilon1 abcd is the completely antisymmetric symbol with the sign convention /epsilon1 0123 = 1. As noted in the previous subsection, if we use the spin vector as an independent variable, the center of mass condition (11 . 2) is replaced by Eq. (11 . 4), that is \nζ a u a = 0 . (11 . 26) \nThen the equations of motion reduce to \ndu a dτ = ω a bc v b u c -SR a , dζ a dτ = ω a bc v b ζ c -Su a ζ b R b , (11 . 27) \nwhere \nR a := R ∗ a bcd v b u c ζ d = 1 2 µS R a bcd v b S cd , (11 . 28) \nand R ∗ abcd = 1 2 R abef /epsilon1 ef cd is the right dual of the Riemann tensor. It will be convenient to write explicitly the tetrad components of R ∗ abcd . Since ˜ θ = O ( S ), we only need R ∗ abcd at O ( ˜ θ 0 ). Then the non-vanishing components of R ∗ abcd are given by \n-1 2 R ∗ 0123 = -R ∗ 0213 = R ∗ 0312 = R ∗ 1203 = -R ∗ 1302 = -1 2 R ∗ 2301 = -M r 3 + O ( ˜ θ 2 ) . (11 . 29) \nAlthough we do not need them, we note that the following components are not identically zero but are of O ( ˜ θ ). \nR ∗ 1212 , R ∗ 1313 , R ∗ 1010 , R ∗ 2323 , R ∗ 2020 , and R ∗ 3030 . \nFurther, we may set u µ = v µ in the equations of motion (11 . 27).', '11.2.1. Lowest Order in S': 'We first solve the equations of motion for a circular orbit at r = r 0 at the lowest order in S . For notational simplicity, we omit the suffix 0 of r 0 in the following. For the class of orbits we have assumed, we have v 1 = 0 and v 2 = O ( ˜ θ ).Then the non-trivial equations are \nd dτ v 1 = Av 0 + Bv 3 = 0 , (11 . 30) \nd dτ ζ 2 = 0 , d dτ ζ 0 ζ 1 ζ 3 = 0 A 0 A 0 B 0 -B 0 ζ 0 ζ 1 ζ 3 . (11 . 31) \nThe equation (11 . 30) determines the rotation velocity of the orbital motion. By setting ξ := v 3 /v 0 , we obtain the equation \nω 1 +2 ω 2 ξ + ω 3 ξ 2 = 0 , (11 . 32) \nwhich is solved to give \nξ = ± √ Mr -a √ ∆ . (11 . 33) \nThe upper (lower) sign corresponds to the case that v 3 is positive (negative). Then, with the aid of the normalization condition of the four momentum, v µ v µ = -1 + O ( S 2 ), we find \nv 0 = 1 1 -ξ 2 , v 3 = ξ √ 1 -ξ 2 . (11 . 34) \n√ \n√ Note that, in this case, the orbital angular frequency Ω ϕ is given by a well known formula, \n√ \nΩ ϕ = ± M r 3 / 2 ± √ Ma . (11 . 35) \nOn the other hand, the equations of spin (11 . 31) are solved to give \nζ 2 = -ζ ⊥ , ζ 0 ζ 1 ζ 3 = ζ ‖ α sin( φ + c 1 ) + βc 2 cos( φ + c 1 ) -β sin( φ + c 1 ) -αc 2 . (11 . 36) \nwhere ζ ⊥ , ζ ‖ , c 1 and c 2 are constants, and \nα = A √ B 2 -A 2 = ∓ v 3 , β = B √ B 2 -A 2 = ± v 0 , φ = Ω p τ, Ω p = √ B 2 -A 2 = √ M r 3 . (11 . 37) \nThe supplementary condition v a ζ a = 0 requires that c 2 = 0. The condition ζ a ζ a = 1 implies ζ 2 ⊥ + ζ 2 ‖ = 1. Further since the origin of the time τ can be chosen arbitrarily, we set c 1 = 0. Thus, we obtain \nζ 2 = -ζ ⊥ , ζ 0 ζ 1 ζ 3 = ζ ‖ α sin φ cos φ -β sin φ . (11 . 38) \nHere, we should note that Ω p = Ω ϕ in general if a = 0 or S = 0 (see below).', '11.2.2. First order in S': '/negationslash \n/negationslash \n/negationslash \nHaving obtained the leading order solution with respect to S , we now turn to the equations of motion up to the linear order in S . We assume that the spin vector components are expressed in the same form as were in the leading order but consider corrections of O ( S ) to the coefficients α, β and Ω p . As we have noted, Eq. (11 . 6) tells us that v a can be identified with u a to O ( S ). In order to write down the equations of motion up to the linear order in S , we need the explicit form of R a , which can be evaluated by using the knowledge of the lowest order solution. They are given as \nR 0 = R 3 = O ( ˜ θ ) , \nR 1 = 3 M r 3 v 0 v 3 ζ 2 + O ( ˜ θ ) , R 2 = 3 M r 3 v 0 v 3 ζ 1 + O ( ˜ θ ) . (11 . 39) \nFirst we consider the orbital equations of motion. With the assumption that v 1 = 0 and v 2 = O ( ˜ θ ), the non-trivial equations of the orbital motion are \n˙ v 1 = Av 0 + Bv 3 SR 1 = 0 , (11 . 40) \n˙ v 2 = ( Cv 0 + Dv 3 ) ˜ θ -SR 2 . (11 . 41) \n- \nThe first equation gives the rotation velocity as before, while the second equation determines the motion in the θ -direction. \nAgain using the variable ξ = v 3 /v 0 , Eq. (11 . 40) is rewritten as \nω 1 +2 ω 2 ξ + ω 3 ξ 2 +3 S ⊥ M r 3 ξ = 0 , (11 . 42) \nwhere S ⊥ := Sζ ⊥ . The solution of this equation is \nξ = ( ± √ Mr -a √ ∆ )( 1 ∓ 3 S ⊥ √ M 2 r 3 / 2 ) + O ( S 2 ) . (11 . 43) \nUsing the relations (11 . 34), it immediately gives v 0 and v 3 . From the definition of the tetrad, we have the following relations, \nv 0 = √ ∆ Σ [ dt dτ -a sin 2 θ dϕ dτ ] , v 3 = sin θ √ Σ [ -a dt dτ +( r 2 + a 2 ) dϕ dτ ] . (11 . 44) \nThus, the orbital angular velocity observed at infinity is calculated to be \nΩ ϕ := dϕ dt = a + ξ √ ∆ r 2 + a 2 + aξ √ ∆ + O ( ˜ θ 2 ) = ± √ M r 3 / 2 ± a √ M [ 1 -3 S ⊥ 2 ± √ Mr -a r 2 ± a √ Mr ] + O ( ˜ θ 2 ) . (11 . 45) \nIn order to solve the second equation (11 . 41), we note that v 2 = √ Σ ˙ θ /similarequal r ˙ ˜ θ and \nCv 0 + Dv 3 = -M r 2 1 + 2 ξ 2 1 -ξ 2 + O ( S ) . (11 . 46) \nThen we find that Eq. (11 . 41) reduces to \nr ¨ ˜ θ = -M r 2 1 + 2 ξ 2 1 -ξ 2 ˜ θ -3 S ‖ M r 3 ξ 1 -ξ 2 cos φ, (11 . 47) \nwhere S ‖ = Sζ ‖ . This equation can be solved easily by setting ˜ θ = θ 0 cos φ . Recalling that Ω 2 p = M/r 3 + O ( S ), we obtain \nθ 0 = -S ‖ rξ . (11 . 48) \nThus we see that the orbit will remain in the equatorial plane if S ‖ = 0, but deviates from it if S ‖ = 0. We note that there exists a degree of freedom to add a homogeneous \n- \n/negationslash \nsolution of Eq. (11 . 47), whose frequency, Ω θ = √ M r 3 1 + 2 ξ 2 1 ξ 2 , is different from Ω p \nand which corresponds to giving a small inclination angle to the orbit, indifferent to the spin. Here, we only consider the case when this homogeneous solution to ˜ θ is zero, i.e., those orbits which would be on the equatorial plane if the spin were zero. Schematically speaking, the orbits under consideration are those with the total angular momentum J being parallel to the z -direction, which is sum of the orbital and spin angular momentum J = L + S (see Fig. 3)). \nFig. 3. A schematic picture of the precession of orbit and spin vector, to the leading order in S . The vector J represents the total angular momentum of the particle. The vector L is orthogonal to the orbital plane and reduces to the orbital angular momentum in the Newtonian limit. In the relativistic case, however, these vectors should not be regarded as well-defined. \n<!-- image --> \nNext we consider the evolution of the spin vector. To the linear order in S , the equations to be solved are \n˙ ζ 0 = Aζ 1 + Cζ 2 ˜ θ -Sv 0 ζ a R a , ˙ ζ 1 = Aζ 0 + Bζ 3 + Eζ 2 , ˙ ζ 2 = ( Cζ 0 + Dζ 3 ) ˜ θ -Eζ 1 , \n˙ ζ 3 = -Bζ 1 -Dζ 2 ˜ θ -Sv 3 ζ a R a . (11 . 49) \nThe third equation is written down explicitly as \n˙ ζ 2 = -¯ θζ ‖ κ sin φ cos φ, (11 . 50) \nwhere \nThus we find \nκ := αD -βC -Ω p ω 3 r. (11 . 51) \nζ 2 = -ζ ⊥ + θ 0 ζ ‖ κ 4 Ω p cos 2 φ. (11 . 52) \nSince the spin vector S a is itself of O ( S ) already, the effect of the second term is always unimportant as long as we neglect corrections of O ( S 2 ) to the orbit. \nThe remaining three equations determine α , β and Ω p . Corrections of O ( S ) to α and β are less interesting because they remain to be small however long the time passes. On the other hand, the correction to Ω p will cause a big effect after a sufficiently long lapse of time because it appears in the combination of Ω p τ . The small phase correction will be accumulated to become large. Hence, we solve Ω p to the next leading order. Eliminating ζ 0 and ζ 3 from these three equations, we obtain \n[ ( B 2 -A 2 ) -Ω 2 p ] = S ⊥ ξ ( AC -BD r -Ω 2 p ω 3 ) . (11 . 53) \nThen after a straightforward calculation, we find \nΩ 2 p = M r 3 { 1 -3 S ⊥ r 3 / 2 ± √ M ( 2 r 2 -3 Mr + a 2 ) + ar 1 / 2 ( M -r ) r 2 -3 Mr ± 2 a √ Mr } . (11 . 54) \nAs noted above, Ω p /negationslash = Ω ϕ for S ⊥ /negationslash = 0. The difference Ω p -Ω ϕ gives the angular velocity of the precession of the spin vector, as depicted in Fig. 3.', '11.3. Gravitational waves and energy loss rate': 'We now proceed to the calculation of the source terms in the Teukolsky equation and evaluate the gravitational wave flux. For this purpose, we must write down the expression of the energy momentum tensor of the spinning particle explicitly. We rewrite the tetrad components of the energy momentum tensor in the following way: \nT ab = µ ∫ dτ { u ( a v b ) δ (4) ( x -z ( τ )) √ -g -e ( a ν e b ) ρ ∇ µ S µν v ρ δ (4) ( x -z ( τ )) √ -g } = µ ∫ dτ [ u ( a v b ) + ω ( a dc v b ) S dc -ω ( a dc S b ) d v c ] δ (4) ( x -z ( τ )) √ -g -1 √ -g ∂ µ ( S µ ( a v b ) δ (4) ( x -z ( τ )) ) =: µ ∫ dτ { A ab δ (4) ( x -z ( τ )) √ -g + 1 √ -g ∂ µ ( B µab δ 4 ( x -z ( τ )) ) } . (11 . 55) \nThe last line gives the definition of A ab and B µab . Then the source term of the Teukolsky equation is given by Eq. (2 . 14) with Eqs. (2 . 15). \nAs we will see shortly, the terms proportional to S ‖ in the energy momentum tensor do not contribute to the energy or angular momentum fluxes at linear order in S . In other words, the energy and angular momentum fluxes are the same for all orbits having the same S ⊥ . Thus, we ignore these terms in the following discussion. Further we recall that the particle can stay in the equatorial plane if S ‖ = 0. Hence we fix θ = π/ 2 in the following calculations. \nUsing the formula (2 . 13), we obtain the amplitude of gravitational waves at infinity as \nwhere \n˜ Z ∞ /lscriptmω = ˜ Z nn /lscriptmω + ˜ Z ¯ mn /lscriptmω + ˜ Z ¯ m ¯ m /lscriptmω , (11 . 56) \n˜ Z nn /lscriptmω = i √ 2 π ωB in /lscriptmω δ ( ω -mΩ ) ( dt dτ ) -1 [ A nn -iωB t nn + imB ϕ nn -B r nn ∂ ∂r ] × [ L † 1 ρ -4 ( L † 2 ρ 3 -2 S aω /lscriptm )] θ = π/ 2 1 r∆ R in /lscriptmω ∣ ∣ ∣ ∣ ∣ ∣ r = r 0 , ˜ Z ¯ mn /lscriptmω = i √ π ωB in /lscriptmω δ ( ω -mΩ ) ( dt dτ ) -1 [ A ¯ mn -iωB t ¯ mn + imB ϕ ¯ mn -B r ¯ mn ∂ ∂r ] × ( L † 2 -2 S aω /lscriptm ) θ = π/ 2 1 √ ∆ [ 2 ∂ ∂r -2 iK ∆ -4 r ] R in /lscriptmω ∣ ∣ ∣ ∣ ∣ ∣ r = r 0 , ˜ Z ¯ m ¯ m /lscriptmω = i √ π ωB in /lscriptmω δ ( ω -mΩ ) ( dt dτ ) -1 [ A ¯ m ¯ m -iωB t ¯ m ¯ m + imB ϕ ¯ m ¯ m -B r ¯ m ¯ m ∂ ∂r ] × ( -2 S aω /lscriptm ) θ = π/ 2 [ ∂ 2 ∂r 2 -2 ( 1 r + iK ∆ ) ∂ ∂r -( iK ∆ ) ,r + 2 iK ∆r -K 2 ∆ 2 ] R in /lscriptmω ∣ ∣ ∣ ∣ ∣ ∣ r = r 0 , (11 . 57) \nand \nA nn = 1 4 1 1 -ξ 2 { 1 -S ⊥ ((2 ω 1 + ω 3 ) ξ + ω 2 ) } , B µ nn = 1 4 r S ⊥ 1 1 -ξ 2 ( r 2 + a 2 √ ∆ ξ + a, -√ ∆ξ, 0 , a √ ∆ ξ +1 ) , A ¯ mn = i 4 √ 2 1 1 -ξ 2 { 2 ξ -S ⊥ ( ω 1 ξ 2 -4 ω 2 ξ -ω 3 )} , B µ ¯ mn = i 4 √ 2 r S ⊥ 1 1 -ξ 2 ( r 2 + a 2 √ ∆ ξ 2 + aξ, -√ ∆ (1 + ξ 2 ) , 0 , a √ ∆ ξ 2 + ξ ) , A ¯ m ¯ m = -1 2 1 1 -ξ 2 { ξ 2 + S ⊥ ( ω 2 (1 + 2 ξ 2 ) + ω 3 ξ )} , B µ ¯ m ¯ m = 1 2 r S ⊥ 1 1 -ξ 2 ( 0 , √ ∆ξ, 0 , 0 ) . (11 . 58) \nThe Lorentz factor dt/dτ which appears in Eqs. (11 . 57) is calculated from Eqs. (11 . 44) as \n√ In general, as we have seen in the preceding sections, when the orbit is quasiperiodic the Fourier components of gravitational waves will have a discrete spectrum; \ndt dτ = 1 r √ 1 -ξ 2 ( aξ + r 2 + a 2 √ ∆ ) . (11 . 59) \n˜ Z /lscriptmω = ∑ n δ ( ω -ω n ) Z /lscriptmω n . (11 . 60) \nThen the time-averaged energy flux and the z -component of the angular momentum flux are given by the formulas (2 . 31) and (2 . 32), respectively. In the present case, since we may regard the orbits to be on the equatorial plane, the index n degenerates to the angular index m and ω n is simply given by mΩ ϕ ( n = m ). Hence we eliminate the index n in the following discussion. Here we mention the effect of nonzero S ‖ . If we recall that all the terms which are proportional to S ‖ have the time dependence of e ± iΩ p τ , we find that they give the contribution to the side bands. That is to say, their contributions in ˜ Z /lscriptmω are all proportional to δ ( ω -mΩ ± Ω p ). Then, since the energy and angular momentum fluxes are quadratic in Z /lscriptmω n , they are not affected by the presence of S ‖ as long as we are working only up to linear order in S . \nAs before, in order to express the post-Newtonian corrections to the energy flux, we define η /lscriptmω as \nwhere ( dE/dt ) N is the Newtonian quadrupole formula defined by Eq. (4 . 19). \n( dE dt ) /lscriptm = 1 2 ( dE dt ) N η /lscriptm , (11 . 61) \nWe calculate η /lscriptm up to 2.5PN order. Keeping the S -dependent terms, the results are \nη ( s ) 2 ± 2 = ( -19 3 v 3 9 qv 4 + 2134 63 v 5 ) ˆ s, η ( s ) 2 ± 1 = ( -1 12 v 3 -1 8 qv 4 -535 1008 v 5 ) ˆ s, η ( s ) 3 ± 3 = -10935 896 v 5 ˆ s, η ( s ) 3 ± 2 = 20 63 v 5 ˆ s, η ( s ) 3 ± 1 = -1 8064 v 5 ˆ s, (11 . 62) \nwhere q = a/M and ˆ s := S ⊥ /M . The rest of η ( s ) /lscriptm are all of higher order. We should mention that if we regard the spinning particle as a model of a black hole or neutron star, S is of order µ . Therefore the correction due to S is small compared with the S -independent terms in the test particle limit µ/M /lessmuch 1. \nPutting all together, we obtain \n〈 dE dt 〉 = ( dE dt ) N 1 -1247 336 v 2 + ( 4 π -73 12 q -25 4 ˆ s ) v 3 \n+ ( -44711 9072 + 33 16 q 2 + 71 8 q ˆ s ) v 4 + ( -8191 672 π + 3749 336 q + 2403 112 ˆ s ) v 5 . (11 . 63) \nSince v is defined in terms of the coordinate radius of the orbit, the expansion with respect to v does not have a clear gauge-invariant meaning. In particular, for the purpose of the comparison with the standard post-Newtonian calculations it is better to write the result by means of the angular velocity observed at infinity. Using the post-Newtonian expansion of Eq. (11 . 45) \nMΩ ϕ = v 3 ( 1 -( 3 2 ˆ s + q ) v 3 + 3 2 q ˆ sv 4 + O ( v 6 ) ) , (11 . 64) \nEq. (11 . 63) can be rewritten as \n〈 dE dt 〉 = ˜ ( dE dt ) N 1 -1247 336 x 2 + ( 4 π -11 4 q -5 4 ˆ s ) x 3 + ( -44711 9072 + 33 16 q 2 + 31 8 q ˆ s ) x 4 + ( -8191 672 π + 59 16 q -13 16 ˆ s ) x 5 , (11 . 65) \nwhere x = ( MΩ ϕ ) 1 / 3 and ˜ ( dE/dt ) N is the Newtonian quadrupole formula expressed in terms of x , Eq. (6 . 6). Since there is no sideband contribution in the present case, the angular momentum flux is simply given by 〈 dJ z /dt 〉 = Ω -1 ϕ 〈 dE/dt 〉 GW . The result (11 . 65) is consistent with the one obtained by the standard post-Newtonian approach 41) , 54) to the 2PN order in the limit µ/M → 0. The ˆ s -dependent term of order x 5 is the one which is newly obtained by the black hole perturbation approach 18) .', '§ 12. Black hole absorption': "When a particle moves around a Kerr black hole, it radiates gravitational waves. Some of those waves are absorbed by the black hole. We calculate such absorption of gravitational waves induced by a particle of mass µ in circular orbit on the equatorial plane around a Kerr black hole of Mass M . \nThe post-Newtonian approximation of the absorption of gravitational waves into the black hole horizon was first calculated by Poisson and Sasaki in the case when a test particle is in a circular orbit around a Schwarzschild black hole 21) . In this case, the effect of the black hole absorption is found to appear at O ( v 8 ) compared to the flux emitted to infinity and it turns out to be negligible for the orbital evolution of coalescing compact binaries in the near future laser interferometer's band. On the other hand, the black hole absorption appears at O ( v 5 ) if a black hole is rotating. That calculation was done by Tagoshi, Mano, and Takasugi 24) . \nIn order to calculate the post-Newtonian expansion of ingoing gravitational waves into a Schwarzschild black hole, Poisson and Sasaki 21) used two type of representations of a solution of the homogeneous Teukolsky equation. One is expressed in terms of the spherical Bessel functions which can be used at large radius, and the other is expressed in terms of a hypergeometric function which can be used near the horizon. Then two type of expressions are matched at some region where both formulas can be applied. They got formulas for a solution of the Teukolsky equation which can be used to calculate ingoing gravitational waves to O ( v 13 ), although they gave formulas for ingoing waves only to O ( v 8 ). \nHere, we first review the method found by Mano, Suzuki and Takasugi 23) , since it is the only existing method by which higher order post-Newtonian terms of the gravitational waves absorbed into a rotating black hole can be calculated. We note that this method is also the only existing method that can be used to calculate the gravitational waves emitted to infinity to an arbitrarily high post-Newtonian order. Then we calculate the energy flux absorbed into the horizon to O ( v 13 ), i.e., O ( v 8 ) beyond the lowest order flux absorbed into the horizon, for circular orbits on the equatorial plane of a Kerr black hole.", '12.1. Analytic solutions of the homogeneous Teukolsky equation': 'Analytic series solutions of the homogeneous Teukolsky equation were found by Mano, Suzuki and Takasugi 23) and various properties of the solution were discussed by Mano and Takasugi 55) . Here, we follow the notation of Ref. 55) except that we focus on the case of spin weight s = -2. In this method, the solution of the radial Teukolsky equation (2 . 3) is represented by two kinds of expansion. One is given by a series of hypergeometric functions and the other by a series of Coulomb wave functions. The former is convergent at horizon and the latter at infinity. Then the matching of these two solutions are done exactly in the overlapping region of convergence. \nFirst we consider the solution expressed in terms of hypergeometric functions. The solution which satisfies the ingoing wave boundary condition at horizon is expressed as \nR in ν = e i/epsilon1κx ( -x ) 2 -i/epsilon1 + (1 -x ) i/epsilon1 -× ∞ ∑ n = -∞ a ν n F ( n + ν +1 -iτ, -n -ν -iτ, 3 -2 i/epsilon1 + ; x ) , (12 . 1) \nwhere F ( a, b, c, x ) is the hypergeometric function and \nx = -ω ( r -r + ) /epsilon1κ , κ = √ 1 -q 2 , τ = /epsilon1 -mq κ , /epsilon1 ± = /epsilon1 ± τ 2 . (12 . 2) \nThe coefficients a ν n obey a three terms recurrence relation, \nα ν n a ν n +1 + β ν n a ν n + γ ν n a ν n -1 = 0 , (12 . 3) \nα ν n = i/epsilon1κ ( n + ν -1 + i/epsilon1 )( n + ν -1 -i/epsilon1 )( n + ν +1+ iτ ) ( n + ν +1)(2 n +2 ν +3) , \nwhere \nwhere \nwhere \nβ ν n = -λ -2 + ( n + ν )( n + ν +1) + /epsilon1 2 + /epsilon1 ( /epsilon1 -mq ) + /epsilon1 ( /epsilon1 -mq )(4 + /epsilon1 2 ) ( n + ν )( n + ν +1) , γ ν n = -i/epsilon1κ ( n + ν +2+ i/epsilon1 )( n + ν +2 -i/epsilon1 )( n + ν -iτ ) ( n + ν )(2 n +2 ν -1) . (12 . 4) \nThe series converges if ν satisfies the equation, \nR n ( ν ) L n -1 ( ν ) = 1 , (12 . 5) \nwhere R n ( ν ) and L n ( ν ) are the continued fractions defined by \nR n ( ν ) ≡ a ν n a ν n -1 = -γ ν n β ν n + α ν n R n +1 ( ν ) , L n ( ν ) ≡ a ν n a ν n +1 = -α ν n β ν n + γ ν n L n -1 ( ν ) . (12 . 6) \nThe range of convergence is 0 ≤ ( -x ) < ∞ for physical x . We can prove that if ν is a solution to Eq. (12 . 5), then so is -ν -1. Since we can set n in Eq. (12 . 5) to an arbitrary integer, a convenient choice is to put n = 1. Further, for convenience, we may set a ν 0 = a -ν -1 0 = 1. It then follows that we have \na -ν -1 -n = a ν n . (12 . 7) \nThis implies R in ν = R in -ν -1 ≡ R in . Consequently, for ( -x ) > 0, Eq. (12 . 1) can be rewritten as \nR in = e i/epsilon1κ ( R ν 0 + R -ν -1 0 ) , (12 . 8) \nR ν 0 = e -i/epsilon1κ ˜ x (˜ x ) ν + i/epsilon1 + (˜ x -1) -s -i/epsilon1 + × ∞ ∑ n = -∞ Γ (3 -2 i/epsilon1 + ) Γ (2 n +2 ν +1) Γ ( n + ν +1 -iτ ) Γ ( n + ν +3 -i/epsilon1 ) a ν n × ˜ x n F ( -n -ν -iτ, -n -ν +2 -i/epsilon1, -2 n -2 ν ; 1 ˜ x ) , (12 . 9) \n˜ x = 1 -x = ω ( r -r -) /epsilon1κ . (12 . 10) \nSince ν -( -ν -1) = 2 ν +1 is not an integer in general, the solutions R ν 0 and R -ν -1 0 form a pair of independent solutions. \nThe other series solution which is convergent at infinity is expressed in terms of the Coulomb wave functions; 56) \nR ν C = ˜ z ( 1 -/epsilon1κ ˜ z ) 2 -i/epsilon1 + ∞ ∑ n = -∞ ( -i ) n ( ν -1 -i/epsilon1 ) n ( ν +3+ i/epsilon1 ) n a ν n F n + ν (˜ z ) , (12 . 11) \nwhere ˜ z = ω ( r -r -) = /epsilon1κ ˜ x , ( a ) n = Γ ( n + a ) /Γ ( a ), and F n + ν ( z ) is the Coulomb wave function given by \nF n + ν ( z ) = e -iz (2 z ) n + ν z Γ ( n + ν +3+ i/epsilon1 ) Γ (2 n +2 ν +2) \n× Φ( n + ν +3+ i/epsilon1, 2 n +2 ν +2;2 iz ) , \nwhere Φ( a, b, z ) is the regular confluent hypergeometric function 57) . A crucial observation made by Mano, Suzuki and Takasugi 23) is that the coefficients a ν n obey the same recurrence relation as that for the hypergeometric type solution, Eq. (12 . 3). The series (12 . 11) converges in the range ˜ z > /epsilon1κ if ν is a solution of Eq. (12 . 5). The solution R ν C can be decomposed into a pair of solutions, a purely incoming wave at infinity R ν + and a purely outgoing wave at infinity R ν -. Explicitly, we have \nR ν C = R ν + + R ν -, (12 . 12) \nwhere \nR ν + = 2 ν e -π/epsilon1 e iπ ( ν +3) Γ ( ν +3+ i/epsilon1 ) Γ ( ν -1 -i/epsilon1 ) e -i ˜ z ˜ z ν + i/epsilon1 + (˜ z -/epsilon1κ ) 2 -i/epsilon1 + × ∞ ∑ n = -∞ i n a ν n (2˜ z ) n Ψ( n + ν +3+ i/epsilon1, 2 n +2 ν +2;2 i ˜ z ) , (12 . 13) \n× ( ν -1 -i/epsilon1 ) n ( ν +3+ i/epsilon1 ) n a ν n (2˜ z ) n Ψ( n + ν -1 -i/epsilon1, 2 n +2 ν +2; -2 i ˜ z ) , (12 . 14) \nR ν -= 2 ν e -π/epsilon1 e -iπ ( ν -1) e i ˜ z ˜ z ν + i/epsilon1 + (˜ z -/epsilon1κ ) 2 -i/epsilon1 + ∞ ∑ n = -∞ i n \nwhere Ψ( a, b, z ) is the irregular confluent hypergeometric function 57) . By definition, the upgoing solution R up is given by \nR up = R ν -. (12 . 15) \nWe see that the above two kinds of solutions are convergent in the common region 1 < ˜ x < ∞ . Then comparing the asymptotic behaviors of R ν 0 and R ν C for ˜ x → ∞ , we find that they have the same characteristic exponent, ∼ ˜ x ν , hence describe the same solution up to the normalization factor. Therefore by comparing each power of ˜ x we have \nR ν 0 = K ν R ν C , (12 . 16) \nwhere \nK ν = (2 /epsilon1κ ) -ν -2 -˜ r 2 2 i ˜ r Γ (3 -2 i/epsilon1 + ) Γ (˜ r +2 ν +1) Γ (˜ r +2 ν +2) Γ ( ν +1 -iτ ) Γ ( ν +3 -i/epsilon1 ) Γ (˜ r + ν +3+ i/epsilon1 ) × Γ ( ν +1+ iτ ) Γ ( ν -1 + i/epsilon1 ) Γ (˜ r + ν +1+ iτ ) Γ (˜ r + ν -1 + i/epsilon1 ) × ( ∞ ∑ n =˜ r (˜ r +2 ν +1) n ( n -˜ r )! ( ν -1 -i/epsilon1 ) n ( ν +3+ i/epsilon1 ) n a ν n ) × ( ˜ r ∑ n = -∞ ( -1) n (˜ r -n )!(˜ r +2 ν +2) n ( ν -1 -i/epsilon1 ) n ( ν +3+ i/epsilon1 ) n a ν n ) -1 , (12 . 17) \nwhere ˜ r can be any integer and K ν is independent of the choice of ˜ r . \nThe gravitational wave absorbed into the black hole is expressed by Eq. (2 . 12). Hence we need to know the amplitudes B inc and B trans of R in and C trans of R up , defined in Eq. (2 . 7). The asymptotic ingoing amplitude at horizon, B trans , of R in is readily obtained from Eq. (12 . 1) as \nB trans = ( /epsilon1κ ω ) 2 s e i/epsilon1 + ln κ ∞ ∑ n = -∞ a ν n . (12 . 18) \nSimilarly, the asymptotic outgoing amplitude at infinity, C trans , of R up is obtained from Eq. (12 . 14) as \nC trans =2 1+ i/epsilon1 e -π/epsilon1/ 2 e -π 2 i ( ν -1) ω 3 e i/epsilon1 ln /epsilon1 × [ ∞ ∑ n = -∞ ( ν -1 -i/epsilon1 ) n ( ν +3+ i/epsilon1 ) n ( -1) n a ν n ] . (12 . 19) \nOn the other hand, a bit of work is necessary to obtain the asymptotic incoming amplitude at infinity, B inc , of R in . Setting the asymptotic behavior of R ν C at z →∞ as \nR ν C → A ν + z -1 e -i ( z + /epsilon1 ln z ) + A ν -z 3 e i ( z + /epsilon1 ln z ) , (12 . 20) \nwe find \nA ν + = 2 -3 -i/epsilon1 e i ( π/ 2)( ν +3) e -π/epsilon1/ 2 Γ ( ν +3+ i/epsilon1 ) Γ ( ν -1 -i/epsilon1 ) ∞ ∑ n = -∞ a ν n , A ν -= 2 1+ i/epsilon1 e -i ( π/ 2)( ν -1) e -π/epsilon1/ 2 ∞ ∑ n = -∞ ( -1) n ( ν -1 -i/epsilon1 ) n ( ν +3+ i/epsilon1 ) n a ν n . (12 . 21) \nBecause of Eq. (12 . 7), A ν ± and A -ν -1 ± are related to each other as \nA -ν -1 + = -ie -iπν sin π ( ν + i/epsilon1 ) sin π ( ν -i/epsilon1 ) A ν + , A -ν -1 -= ie iπν A ν -. (12 . 22) \nWith the help of the above relations, we find from Eqs. (12 . 8) and (12 . 16) the asymptotic amplitudes of R in at infinity as \nB inc = e i/epsilon1κ ω [ K ν -ie -iπν sin π ( ν + i/epsilon1 ) sin π ( ν -i/epsilon1 ) K -ν -1 ] A ν + e -i/epsilon1 ln /epsilon1 , B ref = e i/epsilon1κ ω 3 [ K ν + ie iπν K -ν -1 ] A ν -e i/epsilon1 ln /epsilon1 . (12 . 23) \nSo far our discussion has been on exact analytic series expressions for the homogeneous Teukolsky functions. Now we consider their post-Minkowski expansion by assuming /epsilon1 /lessmuch 1. Provided we set a ν 0 = 1, we see from Eqs. (12 . 4) that α ν n , γ ν n = O ( /epsilon1 ) and β ν n = O (1) unless the value of ν is such that the denominator in the expression of α ν n or γ ν n happens to vanish or β ν n happens to vanish in the limit /epsilon1 → 0. Except for \nsuch an exceptional case, it is easy to see from Eq. (12 . 3) that the order of a ν n in /epsilon1 increases as | n | increases. Thus the series solution naturally gives the post-Minkowski expansion. \nFor the moment, let us assume that the above mentioned exceptional case does not happen for n = 0. Then we have R 1 ( ν ) = O ( /epsilon1 ) and L 0 ( ν ) = O (1 //epsilon1 ). This implies β ν 0 + γ ν 0 L -1 ( ν ) = O ( /epsilon1 2 ). Then assuming L -1 ( ν ) = O ( /epsilon1 ), we must have β ν 0 = O ( /epsilon1 2 ). Using the expansion of λ given by Eq. (3 . 1), we then find ν = /lscript + O ( /epsilon1 2 ) or ν = -/lscript -1 + O ( /epsilon1 2 ). Since we know that -ν -1 is a solution if ν is so, we may take the solution ν = /lscript + O ( /epsilon1 2 ) without loss of generality. Then the assumptions that R 1 ( ν ) = 1 /L 0 ( ν ) = O ( /epsilon1 ) and L -1 ( ν ) = O ( /epsilon1 ) are justified. Further it is easily seen that R n ( ν ) = O ( /epsilon1 ) for all n > 0. On the other hand, for n < 0, α ν n = O (1) at n = -/lscript -1 and β ν n = O ( /epsilon1 2 ) at n = -2 /lscript -1. Thus we have L -/lscript -1 ( ν ) = O (1) and L -2 /lscript -1 ( ν ) = O (1 //epsilon1 ). To summarize, we have \nR n ( ν ) = a ν n a ν n -1 = O ( /epsilon1 ) for all n > 0 , L -/lscript -1 ( ν ) = a ν -/lscript -1 a ν -/lscript = O (1) , L -2 /lscript -1 ( ν ) = a ν -2 /lscript -1 a ν -2 /lscript = O (1 //epsilon1 ) , L n ( ν ) = a ν n a ν n +1 = O ( /epsilon1 ) for all the other n < 0 . (12 . 24) \nWith the above results, the post-Minkowski expansion of the homogeneous Teukolsky functions can be obtained with arbitrary accuracy by solving Eq. (12 . 5) to a desired order and by summing up the terms to a sufficiently large | n | . For our present purpose, we need ν which is accurate to O ( /epsilon1 2 ). Solving Eq. (12 . 5) to this order, we find \nν = /lscript + /epsilon1 2 2 /lscript +1 -2 -4 /lscript ( /lscript +1) + ( /lscript -1) 2 ( /lscript +3) 2 (2 /lscript +1)(2 /lscript +2)(2 /lscript +3) -( /lscript -2) 2 ( /lscript +2) 2 (2 /lscript -1)2 /lscript (2 /lscript +1) . (12 . 25) \nInterestingly, ν is found to be independent of the azimuthal eigenvalue m to O ( /epsilon1 2 ). \nThe post-Newtonian expansion in the near zone is given by further assuming /epsilon1 /lessmuch z /lessmuch 1 in the series solution (12 . 11) and expand it in powers z . For evaluation of the black hole absorption, we need the post-Newtonian expansion of R up which is obtained from Eq. (12 . 14). The explicit post-Newtonian formula for R up and the asymptotic amplitudes B inc , B trans and C trans to O ( /epsilon1 2 ) are given in Appendix H.', '12.2. Absorption rate to O ( v 8 )': 'In this subsection, we evaluate the energy absorption rate by a black hole. The energy flux formula is given by Teukolsky and Press 58) as \n( dE hole dtdΩ ) = ∑ /lscriptm ∫ dω 2 S 2 /lscriptm 2 π 128 ωk ( k 2 +4˜ /epsilon1 2 )( k 2 +16˜ /epsilon1 2 )(2 Mr + ) 5 | C | 2 | ˜ Z H /lscriptmω | 2 , (12 . 26) \nwhere ˜ /epsilon1 = κ/ (4 r + ) and \n| C | 2 = ( ( λ +2) 2 +4 aωm -4 a 2 ω 2 )[ λ 2 +36 aωm -36 a 2 ω 2 ] +(2 λ +3)(96 a 2 ω 2 -48 aωm ) + 144 ω 2 ( M 2 -a 2 ) . (12 . 27) \nThe calculation of ˜ Z H /lscriptmω is parallel to the calculation of ˜ Z ∞ /lscriptmω except that R in is replaced by R up . The solution of the geodesic equations are given in section 6. Using that solution, we have the amplitude of the Teukolsky function at the horizon in Eq. (2 . 12) as \n˜ Z H /lscriptmω = 2 πB trans δ ( ω -mΩ ) 2 iωB inc C trans [ R up /lscriptmω { A nn 0 + A ¯ mn 0 + A ¯ m ¯ m 0 } -dR up /lscriptmω dr { A ¯ mn 1 + A ¯ m ¯ m 1 } + d 2 R up /lscriptmω dr 2 A ¯ m ¯ m 2 ] r = r 0 ,θ = π/ 2 , ≡ δ ( ω -mΩ ) Z H /lscriptm . (12 . 28) \nFrom Eq. (12 . 26) and (12 . 28), the time averaged energy absorption rate becomes \n〈 dE dt 〉 H = ∑ /lscriptm [ 128 ωk ( k 2 +4˜ /epsilon1 2 )( k 2 +16˜ /epsilon1 2 )(2 Mr + ) 5 | C | 2 | Z H /lscriptm | 2 ] ω = mΩ ≡ ∑ /lscriptm ( dE dt ) /lscript,m . (12 . 29) \nAs in the case of the Teukolsky function at infinity, we can show that ¯ Z H /lscript, -m, -ω = ( 1) /lscript Z H /lscript,m,ω . Then, from Eq. (12 . 29), we have ( dE/dt ) /lscript, -m = ( dE/dt ) /lscript,m . \nIn order to express the post-Newtonian corrections to the black hole absorption, we define η H /lscript,m as \n( dE dt ) /lscript,m ≡ 1 2 ( dE dt ) N v 5 η H /lscript,m , (12 . 30) \nwhere ( dE/dt ) N is the Newtonian quadrupole luminosity at infinity, Eq. (4 . 19). In Appendix I, we show η /lscriptm . \nThe total absorption rate to O ( v 8 ) beyond the lowest order is given by \n〈 dE dt 〉 H = ( dE dt ) N v 5 -3 4 q 3 -1 4 q + ( -q -33 16 q 3 ) v 2 + ( 7 2 q 4 +2 qB 2 + 1 2 +6 q 3 B 2 + 85 12 q 2 +3 q 4 κ + 1 2 κ + 13 2 κq 2 ) v 3 + ( -4651 336 q 3 -43 7 q -17 56 q 5 ) v 4 + ( 569 24 q 2 + 371 48 q 4 +18 q 3 B 2 -3 4 q 3 B 1 +2 κ +2+ 33 4 q 4 κ +6 qB 2 + 163 8 κq 2 + qB 1 ) v 5 \n- \nwhere \n+ ( -2718629 44100 q -4 B 2 + 428 105 γ q + 2 3 π 2 q + 428 105 q ln 2 -4 qC 2 -12 q 3 C 2 -36 q 4 B 2 -56 q 2 B 2 + 428 35 q 3 γ + 428 35 q 3 ln 2 + 2 q 3 π 2 + 428 105 q ln κ + 428 105 qA 2 + 428 35 q 3 ln κ +6 q 7 κ + 428 35 q 3 A 2 -8 qB 2 2 -24 q 3 B 2 2 + 856 105 q ln v + 856 35 q 3 ln v -4 B 2 κ -32 q 3 κ -31 q κ +57 q 5 κ + q 5 κ + q 4 B 1 -2 3 κq -7 6 κq 3 -4 3 q 2 B 1 -48 q 2 B 2 κ +28 q 4 B 2 κ -24 q 3 C 2 κ -8 qC 2 κ +24 q 6 B 2 κ -2400247 19600 q 3 + 299 16 q 5 ) v 6 + ( 225 28 q 5 B 3 -41 28 κq 6 + 86 7 + 8741 56 q 2 + 3485 42 q 4 + 167 112 q 6 + 86 7 κ + 45 56 qB 3 -9 28 q 5 B 1 + 899 168 qB 1 -803 224 q 3 B 1 + 1665 224 q 3 B 3 + 2372 21 q 3 B 2 -16 7 q 5 B 2 + 719 12 q 4 κ + 22201 168 κq 2 + 796 21 qB 2 ) v 7 + ( -20542807 88200 q -12 B 2 -2 B 1 + 1061 35 γ q + 13 6 π 2 q + 995 21 q ln 2 -12 qC 2 -36 q 3 C 2 -308 3 q 4 B 2 -1496 9 q 2 B 2 + 12197 140 q 3 γ + 3873 28 q 3 ln 2 + 47 8 q 3 π 2 + 1391 105 q ln κ + 428 35 qA 2 + 5029 140 q 3 ln κ + 37 6 q 7 κ + 1284 35 q 3 A 2 -24 qB 2 2 -72 q 3 B 2 2 + 4574 105 q ln v + 8613 70 q 3 ln v -12 B 2 κ -341 4 q 3 κ -637 6 q κ + 741 4 q 5 κ + 73 6 q 4 B 1 -qC 1 -283 18 q 2 B 1 + 3 2 q 3 B 1 2 + 107 105 qA 1 -107 140 q 3 A 1 -3 q 6 B 1 κ + 13 2 q 4 B 1 κ -3 2 q 2 B 1 κ -2 qC 1 κ + 3 2 q 3 C 1 κ -2 qB 1 2 + 3 4 q 3 C 1 -2 B 1 κ -144 q 2 B 2 κ +84 q 4 B 2 κ -72 q 3 C 2 κ -24 qC 2 κ +72 q 6 B 2 κ -2945984497 6350400 q 3 + 1385 24 q 5 + 25 252 q 7 ) v 8 . (12 . 31) \nA n = 1 2 [ ψ (0) ( 3 + niq √ 1 -q 2 ) + ψ (0) ( 3 -niq √ 1 -q 2 )] , \nB n = 1 2 i [ ψ (0) ( 3 + niq √ 1 -q 2 ) -ψ (0) ( 3 -niq √ 1 -q 2 )] , C n = 1 2 [ ψ (1) ( 3 + niq √ 1 -q 2 ) + ψ (1) ( 3 -niq √ 1 -q 2 )] , (12 . 32) \n( dE dt ) H = ( dE dt ) N ( v 8 + O ( v 10 ) ) , (12 . 33) \nand ψ ( n ) ( z ) is the polygamma function. We see that the absorption effect starts at O ( v 5 ) beyond the quadrapole formula in the case q /negationslash = 0, while for q = 0, the above formula reduced to \nas was found by Poisson and Sasaki 21) . We note that the leading terms in 〈 dE/dt 〉 H are negative for q > 0, i.e., the black hole loses the energy if the particle is corotating. This is because of the superradiance for modes with k < 0. In Appendix J, we also show 〈 dE/dt H written in terms of x ≡ ( MΩ ϕ ) 1 / 3 . \n〈 ≡ It is not manifest from Eq. (12 . 31) that if it has a finite limit for | q | → 1. But by using the formulas, \n〉 \nlim q →± 1 ψ (0) ( 3 + niq √ 1 -q 2 ) = ln n -ln κ + i q | q | π 2 , (12 . 34) \nwe obtain the limit of 〈 dE/dt 〉 H as \nlim q →± 1 ψ ( k ) ( 3 + niq √ 1 -q 2 ) = 0 , ( k /negationslash = 0) , (12 . 35) \nlim q →± 1 ( dE dt ) H = ( dE dt ) N v 5 -q | q | -49 16 q | q | v 2 + ( 4 π + 133 12 ) v 3 -6817 336 q | q | v 4 + ( 535 16 + 97 8 π ) v 5 + ( 3424 105 q | q | ln(2) + 1712 105 γ q | q | + 3424 105 q | q | ln( v ) -3647533 22050 q | q | -289 6 q | q | π -16 / 3 π 2 q | q | ) v 6 + ( 84955 336 + 55873 672 π ) v 7 ( 14077 60 q | q | ln(2) + 16441 140 γ q | q | + 34987 210 q | q | ln( v ) -193 12 π 2 q | q | -4057965601 6350400 q | q | -1289 9 q | q | π ) v 8 . (12 . 36)', 'Appendix A': "Spheroidal harmonics \nIn this Appendix, we describe the expansion of the spheroidal harmonics -2 S aω /lscriptm to O (( aω ) 2 ). \nThe spheroidal harmonics of spin weight s = -2 obey the equation, \n[ 1 sin θ d dθ { sin θ d dθ } -a 2 ω 2 sin 2 θ -( m -2cos θ ) 2 sin 2 θ +4 aω cos θ -2 + 2 maω + λ ] -2 S aω /lscriptm = 0 . (A . 1) \nWe expand -2 S aω /lscriptm and λ as \n-2 S aω /lscriptm = -2 P /lscriptm + aωS (1) /lscriptm +( aω ) 2 S (2) /lscriptm + O (( aω ) 3 ) , λ = λ 0 + aωλ 1 + a 2 ω 2 λ 2 + O (( aω ) 3 ) , (A . 2) \nwhere -2 P /lscriptm are the spherical harmonics of spin weight s = -2. We set the normalizations of -2 P /lscriptm and -2 S aω /lscriptm as \n∫ π 0 | -2 P /lscriptm | 2 sin θdθ = ∫ π 0 | -2 S aω /lscriptm | 2 sin θdθ = 1 . (A . 3) \nInserting Eq. (A . 2) into Eq. (A . 1) and collecting the terms of the same order to ( aω ) 2 , we obtain \n[ L 0 + λ 0 ] -2 P /lscriptm = 0 , (A . 4) \nL \n-[ L 0 + λ 0 ] S (2) /lscriptm = -(4 cos θ +2 m + λ 1 ) S (1) /lscriptm -( λ 2 -sin 2 θ ) -2 P /lscriptm , (A . 6) \n[ 0 + λ 0 ] S (1) /lscriptm = -(4 cos θ +2 m + λ 1 ) -2 P /lscriptm (A . 5) \nL 0 ≡ 1 sin θ d dθ ( sin θ d dθ ) -( m -2cos θ ) 2 sin 2 θ -2 . (A . 7) \nwhere \nThe lowest order equation (A . 4) says we have λ 0 = ( /lscript 1)( /lscript +2). \nThe first order correction to the eigenvalue, λ 1 , is obtained by multiplying Eq. (A . 5) by -2 P /lscriptm from the left hand side and integrating it over θ . The result is \n- \nλ 1 = -2 m /lscript ( /lscript +1) + 4 /lscript ( /lscript +1) . (A . 8) \nS (1) /lscriptm = ∑ /lscript ' c /lscript ' /lscriptm -2 P /lscript ' m . (A . 9) \nTo obtain S (1) /lscriptm , we set \nWe insert this into Eq. (A . 5), multiply it by -2 P /lscript ' m and integrate it over θ . Then noting the normalization of the spheroidal harmonics, we have \nc /lscript ' /lscriptm = 4 ( /lscript ' -1)( /lscript ' +2) -( /lscript -1)( /lscript +2) ∫ d (cos θ ) -2 P /lscript ' m cos θ -2 P /lscriptm , /lscript ' /negationslash = /lscript , 0 , /lscript ' = /lscript . \nHence c /lscript ' /lscriptm is non-zero only for /lscript ' = /lscript ± 1, and we obtain \nc /lscript +1 /lscriptm = 2 ( /lscript +1) 2 [ ( /lscript +3)( /lscript -1)( /lscript + m +1)( /lscript -m +1) (2 /lscript +1)(2 /lscript +3) ] 1 / 2 , c /lscript -1 /lscriptm = -2 /lscript 2 [ ( /lscript +2)( /lscript -2)( /lscript + m )( /lscript -m ) (2 /lscript +1)(2 /lscript -1) ] 1 / 2 . \nThe next order equation can be solved similarly. The second order correction to the eigenvalue, λ 2 , is obtained by multiplying Eq. (A . 6) by -2 P /lscriptm from the left hand side and integrating it over θ . We find \nλ 2 = -4 ∫ d (cos θ ) -2 P /lscriptm cos θ S (1) /lscriptm + ∫ d (cos θ ) -2 P /lscriptm sin 2 θ -2 P /lscriptm = -2( /lscript +1)( c /lscript +1 /lscriptm ) 2 +2 /lscript ( c /lscript -1 /lscriptm ) 2 +1 -∫ d (cos θ ) -2 P /lscriptm cos 2 θ -2 P /lscriptm , (A . 10) \nwhere the last integral becomes \n∫ d (cos θ ) -2 P /lscriptm cos 2 θ -2 P /lscriptm = 1 3 + 2 3 ( /lscript +4)( /lscript -3)( /lscript 2 + /lscript -3 m 2 ) /lscript ( /lscript +1)(2 /lscript +3)(2 /lscript -1) . \nAs before, to obtain S (2) /lscriptm , we set \nS (2) /lscriptm = ∑ /lscript ' d /lscript ' /lscriptm -2 P /lscript ' m . (A . 11) \nInserting Eqs. (A . 9) and (A . 11) into Eq. (A . 6), multiplying it by -2 P /lscript ' m and integrate it over θ , we obtain \nd /lscript ' /lscriptm = 1 λ 0 ( /lscript ) -λ 0 ( /lscript ' ) [ -(2 m + λ 1 ( /lscript )) ( c /lscript +1 /lscriptm δ /lscript ' ,/lscript +1 + c /lscript -1 /lscriptm δ /lscript ' ,/lscript -1 ) -4 c /lscript +1 /lscriptm ∫ d (cos θ ) -2 P /lscript ' m cos θ -2 P /lscript +1 m -4 c /lscript -1 /lscriptm ∫ d (cos θ ) -2 P /lscript ' m cos θ -2 P /lscript -1 m + ∫ d (cos θ ) -2 P /lscript ' m sin 2 θ -2 P /lscriptm ] , (A . 12) \nfor /lscript ' = /lscript . The integrals in this equation are given by 59) , 60) \n∫ d (cos θ ) -2 P /lscript ' m cos θ -2 P /lscriptm = √ 2 /lscript +1 2 /lscript ' +1 < /lscript, 1 , m, 0 | /lscript ' , m >< /lscript, 1 , 2 , 0 | /lscript ' , 2 >, ∫ d (cos θ ) -2 P /lscript ' m sin 2 θ -2 P /lscriptm = 2 3 δ /lscript ' ,/lscript -2 3 √ 2 /lscript +1 2 /lscript ' +1 < /lscript, 2 , m, 0 | /lscript ' , m >< /lscript, 2 , 2 , 0 | /lscript ' , 2 >, \n/negationslash \nwhere < j 1 , j 2 , m 1 , m 2 | J, M > is a Clebsch-Gordan coefficient. For /lscript = 2 and 3, the non-vanishing d /lscript ' /lscriptm ( /lscript ' = /lscript ) are given explicitly as \nd /lscript +1 /lscriptm = m 324 √ 7 (3 -m ) 1 / 2 (3 + m ) 1 / 2 , \n/negationslash \nd /lscript +2 /lscriptm = 11 1764 √ 3 (3 -m ) 1 / 2 (3 + m ) 1 / 2 (4 -m ) 1 / 2 (4 + m ) 1 / 2 , \nfor /lscript = 2, and \nd /lscript +1 /lscriptm = m 120 √ 21 (4 -m ) 1 / 2 (4 + m ) 1 / 2 , d /lscript +2 /lscriptm = 1 180 √ 11 (4 -m ) 1 / 2 (4 + m ) 1 / 2 (5 -m ) 1 / 2 (5 + m ) 1 / 2 , d /lscript -1 /lscriptm = -m 324 √ 7 (3 -m ) 1 / 2 (3 + m ) 1 / 2 , \nfor /lscript = 3. As for d /lscript /lscriptm , it is determined by the normalization of -2 S aω /lscriptm , i.e., \n1 = ∫ d (cos θ ) | -2 S /lscriptm | 2 = ∫ d (cos θ ) { ( -2 P /lscriptm ) 2 +2 aω ∑ /lscript ' c /lscript ' /lscriptm -2 P /lscript ' m -2 P /lscriptm + ( aω ) 2 ∑ /lscript ' /lscript '' c /lscript ' /lscriptm c /lscript '' /lscriptm -2 P /lscript ' m -2 P /lscript '' m + 2( aω ) 2 ∑ /lscript ' d /lscript ' /lscriptm -2 P /lscript ' m -2 P /lscriptm + O ( ( aω ) 3 ) } = 1 + ( aω ) 2 ∑ /lscript ' ( c /lscript ' /lscriptm ) 2 +2( aω ) 2 d /lscript /lscriptm + O (( aω ) 3 ) . \nThen we have \nd /lscript /lscriptm = -1 2 { ( c /lscript +1 /lscriptm ) 2 + ( c /lscript -1 /lscriptm ) 2 } . (A . 13) \nAppendix B The operators Q (2) , Q (3) and Q (4) \nIn this Appendix, we show the operators Q ( n ) for n = 2, 3 and 4 which appear in Eq. (3 . 20). \nQ (2) = [( -28 i m q -32 i m q /lscript +8 i /lscript mq +4 i /lscript 2 mq -13 q 2 -6 q 2 /lscript -12 /lscript q 2 -/lscript 2 q 2 +6 /lscript 3 q 2 +2 /lscript 4 q 2 +8 m 2 q 2 + 32 m 2 q 2 /lscript 2 + 8 m 2 q 2 /lscript ) 1 z 4 + ( 16 mq + 24 mq /lscript 2 + 20 mq /lscript -8 /lscript mq -4 /lscript 2 mq -14 i q 2 -16 i q 2 /lscript + 4 i /lscript q 2 +2 i /lscript 2 q 2 +2 i λ 1 mq 2 -4 i λ 1 mq 2 /lscript 2 + \n2 i λ 1 mq 2 /lscript -4 i m 2 q 2 + 56 i m 2 q 2 /lscript 2 -4 i m 2 q 2 /lscript ) 1 z 3 + ( 24 i m q /lscript 2 + 17 q 2 2 + 10 q 2 /lscript -13 /lscript q 2 4 -9 /lscript 2 q 2 4 -3 /lscript 3 q 2 4 -/lscript 4 q 2 4 -λ 2 q 2 2 -3 /lscript λ 2 q 2 4 + /lscript 2 λ 2 q 2 4 + 3 /lscript 3 λ 2 q 2 4 + /lscript 4 λ 2 q 2 4 -2 λ 1 mq 2 + 4 λ 1 mq 2 /lscript 2 -2 λ 1 mq 2 /lscript -24 m 2 q 2 /lscript 2 ) 1 z 2 ] 1 ( /lscript +1) 2 ( /lscript 2 + /lscript -2) + [( -24 i λ 0 mq -4 i λ 0 2 mq +4 i λ 0 3 mq -12 λ 0 q 2 -6 λ 0 2 q 2 + 24 λ 0 m 2 q 2 -4 λ 0 2 m 2 q 2 ) 1 z 3 + ( 24 λ 0 mq -12 i λ 0 q 2 -2 i λ 0 2 q 2 + 2 i λ 0 3 q 2 +2 i λ 0 2 λ 1 mq 2 +24 i λ 0 m 2 q 2 ) 1 z 2 ] 1 ( λ 0 +2) 2 λ 2 0 d dz -q 2 4 z 2 d 2 dz 2 (B . 1) \nQ (3) /lscript =2 = [( i 2 mq + 5 q 2 8 -5 m 2 q 2 9 + 11 i 24 mq 3 -11 i 54 m 3 q 3 ) 1 z 4 + ( -( mq ) 24 + 5 i 48 q 2 -65 i 216 m 2 q 2 -mq 3 2 + 16 m 3 q 3 81 ) 1 z 3 + ( i 24 mq -q 2 48 + 17 m 2 q 2 216 -65 i 378 mq 3 + 17 i 252 m 3 q 3 ) 1 z 2 ] d dz + [( i m q + 7 q 2 4 -25 m 2 q 2 18 + 4 i 3 mq 3 -29 i 54 m 3 q 3 ) 1 z 5 + ( -5 mq 12 + 19 i 24 q 2 -103 i 108 m 2 q 2 -19 mq 3 12 + 101 m 3 q 3 162 ) 1 z 4 + ( -q 2 8 + 41 m 2 q 2 108 -127 i 189 mq 3 + 601 i 2268 m 3 q 3 ) 1 z 3 + ( -( mq ) 24 -i 48 q 2 + 17 i 216 m 2 q 2 + λ 3 q 3 8 + 65 mq 3 378 -17 m 3 q 3 252 ) 1 z 2 ] , (B . 2) \nQ (3) /lscript =3 = [( i 10 mq + q 2 8 -7 m 2 q 2 180 + i 60 mq 3 -7 i 540 m 3 q 3 ) 1 z 4 + ( -7 mq 600 -83 i 1200 q 2 -7 i 1350 m 2 q 2 -23 mq 3 720 + 19 m 3 q 3 3240 ) 1 z 3 + ( i 1200 mq -7 q 2 1200 + 23 m 2 q 2 5400 -7 i 2160 mq 3 + 11 i 12960 m 3 q 3 ) 1 z 2 ] d dz \n+ [( i 5 mq + q 2 2 -31 m 2 q 2 180 + 23 i 120 mq 3 -5 i 108 m 3 q 3 ) 1 z 5 + ( -8 mq 75 + 109 i 1200 q 2 -79 i 1350 m 2 q 2 -25 mq 3 144 + 31 m 3 q 3 810 ) 1 z 4 + ( -13 i 1200 mq + 19 q 2 300 + 17 m 2 q 2 1800 -19 i 540 mq 3 + 29 i 4320 m 3 q 3 ) 1 z 3 + ( -( mq ) 1200 -7 i 1200 q 2 + 23 i 5400 m 2 q 2 + λ 3 q 3 8 + 7 mq 3 2160 -11 m 3 q 3 12960 ) 1 z 2 ] . (B . 3) \nQ (4) /lscript =2 = [( -3 q 2 8 + 3 m 2 q 2 8 -23 i 24 mq 3 + 43 i 108 m 3 q 3 -5 q 4 32 + 4 m 2 q 4 9 -79 m 4 q 4 648 ) 1 z 5 + ( -( mq ) 8 -3 i 32 q 2 + 5 i 72 m 2 q 2 + 77 mq 3 96 -263 m 3 q 3 648 -19 i 96 q 4 + 79 i 108 m 2 q 4 -137 i 648 m 4 q 4 ) 1 z 4 + ( -i 96 mq -5 q 2 192 + 73 m 2 q 2 864 + 13 i 108 mq 3 -i 9 m 3 q 3 + 25 q 4 252 -109 m 2 q 4 252 + 680 m 4 q 4 5103 ) 1 z 3 + ( -( mq ) 96 -i 192 q 2 + 11 i 288 m 2 q 2 + 5 mq 3 108 -m 3 q 3 3888 + 7 i 432 q 4 + i 72 λ 3 mq 4 -1349 i 13608 m 2 q 4 + 509 i 15309 m 4 q 4 ) 1 z 2 ] d dz + [( -3 q 2 4 + 7 m 2 q 2 12 -47 i 24 mq 3 + 19 i 27 m 3 q 3 -7 q 4 16 + 65 m 2 q 4 72 -71 m 4 q 4 324 ) 1 z 6 + ( -( mq ) 4 -7 i 16 q 2 + 3 i 8 m 2 q 2 + 29 mq 3 12 -659 m 3 q 3 648 -29 i 48 q 4 + 49 i 27 m 2 q 4 -53 i 108 m 4 q 4 ) 1 z 5 + ( -5 i 48 mq + 5 q 2 96 + 41 m 2 q 2 432 + 2287 i 3024 mq 3 -2039 i 4536 m 3 q 3 + 101 q 4 288 -3991 m 2 q 4 3024 + 15853 m 4 q 4 40824 ) 1 z 4 + ( -i 32 q 2 + 53 i 432 m 2 q 2 -2 mq 3 27 + 431 m 3 q 3 3888 + 349 i 3024 q 4 + i 72 λ 3 mq 4 -7235 i 13608 m 2 q 4 + 2549 i 15309 m 4 q 4 ) 1 z 3 \n+ ( -i 96 mq + q 2 192 -11 m 2 q 2 288 + 5 i 108 mq 3 -i 3888 m 3 q 3 -7 q 4 432 + λ 4 q 4 16 -λ 3 mq 4 72 + 1349 m 2 q 4 13608 -509 m 4 q 4 15309 ) 1 z 2 ] . (B . 4) \nAppendix C 4PN formulas for R in /lscriptm \nIn this Appendix, we show the post-Newtonian expansion of R in in the near zone, where z = ωr /lessmuch 1, for a Kerr black hole which are needed to evaluate gravitational waves at infinity to O ( v 8 ). For convenience, we recover the indices /lscriptm on R in and give the formulas for ωR in /lscriptm . \nωR in 2 m = z 4 30 + i 45 z 5 -11 z 6 1260 -i 420 z 7 + 23 z 8 45360 + i 11340 z 9 -13 z 10 997920 -i 598752 z 11 + 59 z 12 311351040 + /epsilon1 ( -z 3 15 -i 60 mqz 3 -i 60 z 4 + mqz 4 45 -41 z 5 3780 + 277 i 22680 mqz 5 -31 i 3780 z 6 -7 mqz 6 1620 + 17 z 7 5670 -61 i 54432 mqz 7 + 41 i 54432 z 8 + 47 mqz 8 204120 -1579 z 9 10692000 + 703 i 17962560 mqz 9 ) + /epsilon1 2 ( z 2 30 + i 40 mqz 2 + q 2 z 2 60 -m 2 q 2 z 2 240 -i 60 z 3 -mqz 3 30 + i 90 q 2 z 3 -i 120 m 2 q 2 z 3 + 7937 z 4 55125 -53 i 9072 mqz 4 -101 q 2 z 4 35280 + 4213 m 2 q 2 z 4 635040 + 4673 i 55125 z 5 -13 mqz 5 2835 -5 i 63504 q 2 z 5 + 3503 i 1143072 m 2 q 2 z 5 -1665983 z 6 55566000 -1777 i 544320 mqz 6 -q 2 z 6 5040 -643 m 2 q 2 z 6 653184 -107 z 4 ln z 6300 -107 i 9450 z 5 ln z + 1177 z 6 ln z 264600 ) + /epsilon1 3 (( -i 180 mq -q 2 60 + m 2 q 2 240 -i 144 mq 3 + i 1440 m 3 q 3 ) z + ( i 120 + 2 mq 135 -i 360 q 2 + 19 i 1440 m 2 q 2 + 11 mq 3 1080 -m 3 q 3 540 ) z 2 + z 3 ( -10933 49000 -578569 i 7938000 mq -677 q 2 52920 -529 m 2 q 2 63504 \n+ 317 i 63504 mq 3 -167 i 84672 m 3 q 3 + 107 ln z 3150 + 107 i 12600 mq ln z )) + /epsilon1 4 ( -i 720 mq + m 2 q 2 2880 + i 288 mq 3 -i 2880 m 3 q 3 + q 4 480 -m 2 q 4 720 + m 4 q 4 11520 ) . (C . 1) ωR in 3 m = z 5 630 + i 1260 z 6 -z 7 3780 -i 16200 z 8 + 29 z 9 2494800 + i 554400 z 10 -47 z 11 194594400 + /epsilon1 ( -z 4 252 -i 1890 mqz 4 -i 756 z 5 + 11 mqz 5 22680 + 19 i 90720 mqz 6 -i 9450 z 7 -mqz 7 16200 + 647 z 8 14968800 -247 i 17962560 mqz 8 ) + /epsilon1 2 ( z 3 315 + i 945 mqz 3 + q 2 z 3 1260 -m 2 q 2 z 3 15120 + i 2520 z 4 -17 mqz 4 15120 + i 2160 q 2 z 4 -31 i 272160 m 2 q 2 z 4 + 81409 z 5 11113200 -313 i 907200 mqz 5 -41 q 2 z 5 226800 + 617 m 2 q 2 z 5 8164800 -13 z 5 ln z 26460 ) + /epsilon1 3 ( -z 2 1260 -i 1680 mqz 2 -q 2 z 2 840 + m 2 q 2 z 2 10080 -i 5040 mq 3 z 2 ) . (C . 2) ωR in 4 m = z 6 11340 + i 28350 z 7 -13 z 8 1247400 -i 467775 z 9 + 71 z 10 194594400 + /epsilon1 ( -z 5 3780 -i 45360 mqz 5 -11 i 136080 z 6 + mqz 6 64800 + 131 z 7 18711000 + 697 i 124740000 mqz 7 ) + /epsilon1 2 ( z 4 3528 + i 18144 mqz 4 + q 2 z 4 21168 -m 2 q 2 z 4 635040 ) . (C . 3) ωR in 5 m = z 7 207900 + i 623700 z 8 -z 9 2316600 + /epsilon1 ( -z 6 59400 -i 1039500 mqz 6 ) . (C . 4) ωR in 6 m = z 8 4054050 . (C . 5) \nThe ingoing Regge-Wheeler functions to O ( /epsilon1 3 ) \nIn this Appendix, we present a method to calculate the ingoing Regge-Wheeler functions to O ( /epsilon1 3 ) which are needed to calculate the luminosity to O ( v 11 ) beyond Newtonian in the Schwarzschild case. In the Schwarzschild limit, q → 0, and solving Eq. (3 . 25) recursively is in principle straightforward. Since the general homogeneous solution to the left-hand side of it is given by a linear combination of the spherical Bessel functions j /lscript and n /lscript , one can immediately write the integral expression for ξ ( n ) /lscript . Noting that j /lscript n ' /lscript -n /lscript j ' /lscript = 1 /z 2 , we have \nξ ( n ) /lscript = n /lscript ∫ dz j /lscript W ( n ) -j /lscript ∫ dz n /lscript W ( n ) , (D . 1) \nwhere the source term W ( n ) in the Schwarzschild case is given by \nW ( n ) = z 2 L (1) [ ξ ( n -1) ] = e -iz d dz [ 1 z 3 d dz ( e iz z 2 ξ ( n -1) /lscript ( z ) ) ] . (D . 2) \nWe perform the above indefinite integral and set the appropriate boundary condition by examining the asymptotic behavior at z → 0 order by order.", 'D.1. General remarks': 'Let us first consider the boundary conditions of ξ ( n ) /lscript . In the Schwarzschild case, the original Regge-Wheeler ingoing wave function X in /lscript is related to ξ in /lscript as \nX in /lscript = ze -i/epsilon1 ln( z -/epsilon1 ) ξ in /lscript , (D . 3) \nand it has the asymptotic behavior given by Eq. (3 . 12), which in the present case reduces to ∗ \nThus, noting that z ∗ = z + /epsilon1 ln( z -/epsilon1 ), the boundary condition of ξ in /lscript is that it is regular at z ∗ →-∞ ( z → /epsilon1 ). To implement this boundary condition to ξ in /lscript , we need a different series expansion of it; a series in terms of the variable x := r/ 2 M = z//epsilon1 around x = 1. We write this expansion as \nX in /lscript → { A ref /lscript e iωr ∗ + A inc /lscript e -iωr ∗ for z ∗ →∞ A trans /lscript e -iωr ∗ for z ∗ →-∞ . (D . 4) \nξ in /lscript = ξ { 0 } /lscript ( x ) + /epsilon1ξ { 1 } /lscript ( x ) + /epsilon1 2 ξ { 2 } /lscript ( x ) + · · · . (D . 5) \nOn the other hand, there are two independent solutions expanded by functions written in terms of z . We denote the solution whose zeroth order is given by j /lscript ( n /lscript ) as ξ j/lscript ( ξ n/lscript ). Then the general solution is given by c j ξ j/lscript + c n ξ n/lscript . As can be shown by using the result of Poisson and Sasaki 21) , ξ { 0 } /lscript does not have terms matched with ξ n/lscript . A term matched with ξ n/lscript first appears from ξ { 1 } /lscript . For x →∞ ( ↔ /epsilon1 /lessmuch z ), ξ { 0 } /lscript ( x ) behaves as x /lscript = z /lscript /epsilon1 -/lscript ∼ /epsilon1 -/lscript ξ j/lscript On the other hand, /epsilon1ξ { 1 } /lscript contains the term that behaves as /epsilon1x -/lscript -1 = /epsilon1 /lscript +2 z -/lscript -1 ∼ /epsilon1 /lscript +2 ξ n/lscript . Therefore, in the sense of the post-Minkowskian expansion, the inner boundary condition affects the ingoing wave solution at and beyond O ( /epsilon1 2 /lscript +2 ). However, in the post-Newtonian sense, since we evaluate ξ /lscript in the \nnear zone, i.e., for z = O ( v ), the contribution from ξ n/lscript becomes O ( v 4 /lscript +5 ) relative to ξ j/lscript . Since /lscript ≥ 2, we find that the inner boundary condition affects the homogeneous solution at and beyond O ( v 13 ) in the near zone. ∗ ) \nSince j /lscript = O ( z /lscript ) as z → 0, we have X in /lscript → O ( /epsilon1 /lscript +1 ) e -iz ∗ , or A trans /lscript = O ( /epsilon1 /lscript +1 ). On the other hand, from the asymptotic behavior of j /lscript at z = ∞ , we find A inc /lscript and A ref /lscript are of order unity. Then using the Wronskian argument, we obtain \n| A inc /lscript | - | A ref /lscript | = | A trans /lscript | 2 | A inc /lscript | + | A ref /lscript | = O ( /epsilon1 2 /lscript +2 ) . (D . 6) \n-To see this explicitly, let us decompose ξ ( n ) /lscript into the real and imaginary parts: \nThus | A inc /lscript | = | A ref /lscript | until we go to O ( /epsilon1 2 /lscript +2 ) or more. This fact implies that we can make A inc /lscript and A ref /lscript to be complex conjugate to each other to O ( /epsilon1 2 /lscript +1 ). Hence the imaginary part of X in /lscript , which reflects the boundary condition at horizon, appears at O ( /epsilon1 2 /lscript +2 ) because the Regge-Wheeler equation is real. This is consistent with the argument given in the above paragraph. Provided we choose the phase of X in /lscript in this way, Im ( ξ ( n ) /lscript ) for a given n ≤ 2 /lscript +1 is completely determined in terms of Re ( ξ ( r ) /lscript ) for r n -1. \nξ ( n ) /lscript = f ( n ) /lscript + ig ( n ) /lscript . (D . 7) \nInserting this expression into Eq. (D . 3) and expanding the result with respect to /epsilon1 by assuming z /greatermuch /epsilon1 , we find \nHence we must have \nX in /lscript = e -i/epsilon1 ln( z -/epsilon1 ) z ( j /lscript + /epsilon1 ( f (1) /lscript + ig (1) /lscript ) + /epsilon1 2 ( f (2) /lscript + ig (2) /lscript ) + /epsilon1 3 ( f (3) /lscript + ig (3) /lscript ) + · · · ) = z j /lscript + /epsilon1f (1) /lscript + /epsilon1 2 ( f (2) /lscript + g (1) /lscript ln z -1 2 j /lscript (ln z ) 2 ) + /epsilon1 3 ( f (3) -1 2 (ln z ) 2 f (1) /lscript + ln z z j /lscript +ln zg (2) /lscript -1 z g (1) /lscript ) + · · · + iz /epsilon1 ( g (1) /lscript -j /lscript ln z ) + /epsilon1 2 ( g (2) /lscript + 1 z j /lscript -f (1) /lscript ln z ) + /epsilon1 3 ( g (3) /lscript -1 2 (ln z ) 2 g (1) /lscript -ln z f (2) /lscript + 1 z f (1) + ( 1 2 z 2 + 1 6 (ln z ) 3 ) j /lscript ) + · · · . (D . 8) \ng (1) /lscript = j /lscript ln z , g (2) /lscript = -1 z j /lscript + f (1) /lscript ln z , g (3) /lscript = ( 1 3 (ln z ) 3 -1 2 z 2 ) j /lscript -1 z f (1) /lscript +ln z f (2) /lscript , · · · . (D . 9) \n≤ \nFor completeness, we also give the relation between the functions f ( n ) /lscript and the conventional post-Newtonian expansion of X in /lscript : \nX in /lscript = ∞ ∑ n =0 /epsilon1 n X ( n ) /lscript ; X (0) /lscript = zf (0) /lscript = zj /lscript , X (1) /lscript = zf (1) /lscript , X (2) /lscript = z ( f (2) /lscript + 1 2 j /lscript (ln z ) 2 ) , X (3) /lscript = z ( f (3) /lscript + 1 2 f (1) /lscript (ln z ) 2 -ln z z j /lscript ) , · · · . (D . 10) \nNow we turn to the asymptotic behavior at z = ∞ . Let the asymptotic form of f ( n ) /lscript be \nThen noting Eq. (D . 9) and the equality e -i/epsilon1 ln( z -/epsilon1 ) = e -iz ∗ e iz , the asymptotic form of X in /lscript is expressed as \nf ( n ) /lscript → P ( n ) /lscript j /lscript + Q ( n ) /lscript n /lscript as z →∞ ( n = 1 , 2 , 3) . (D . 11) \nX in /lscript → 1 2 e -iz ∗ ( zh (2) /lscript e iz ) 1 + /epsilon1 { P (1) /lscript + i ( Q (1) /lscript +ln z )} + /epsilon1 2 {( P (2) /lscript -Q (1) /lscript ln z ) + i ( Q (2) /lscript + P (1) /lscript ln z )} + /epsilon1 3 { ( f (3) /lscript -Q (2) /lscript ln z ) + i ( Q (3) /lscript + P (2) /lscript ln z + 1 3 (ln z ) 3 )} + · · · + 1 2 e iz ∗ ( zh (1) /lscript e -iz ) e -2 i/epsilon1 ln( z -/epsilon1 ) 1 + /epsilon1 { P (1) /lscript -i ( Q (1) /lscript -ln z )} + /epsilon1 2 {( P (2) /lscript + Q (1) /lscript ln z ) -i ( Q (2) /lscript -P (1) /lscript ln z )} + /epsilon1 3 { ( P (3) /lscript + Q (2) /lscript ln z ) -i ( Q (3) /lscript -P (2) /lscript ln z -1 3 (ln z ) 3 )} + · · · . (D . 12) \nUsing the asymptotic behavior of h (1) /lscript and h (2) /lscript given in Eq. (3 . 58), the incident amplitude A inc /lscript can be readily extracted out as \nA inc /lscript = 1 2 i /lscript +1 e -i/epsilon1 ln /epsilon1 [ 1 + /epsilon1 { P (1) /lscript + i ( Q (1) /lscript +ln z )} + /epsilon1 2 {( P (2) /lscript -Q (1) /lscript ln z ) + i ( Q (2) /lscript + P (1) /lscript ln z )} + /epsilon1 3 { ( P (3) /lscript -Q (2) /lscript ln z ) + i ( Q (3) /lscript + P (2) /lscript ln z + 1 3 (ln z ) 3 )} + · · · ] , (D . 13) \nwhere note that the definition of r ∗ , Eq. (2 . 9), in the limit q → 0 is \nωr ∗ = ω ( r +2 M ln r -2 M 2 M ) = z ∗ -/epsilon1 ln /epsilon1, (D . 14) \nwhich gives rise to the phase -i/epsilon1 ln /epsilon1 of A inc /lscript . \nAn important point to be noted in the above expression for A inc /lscript is that it contains ln z -dependent terms. Since A inc /lscript should be constant, P ( n ) /lscript and Q ( n ) /lscript should contain appropriate ln z -dependent terms which exactly cancel the ln z -dependent terms in the formula (D . 13).', 'D.2. Basic formalism for iteration': 'Here we derive the formulas necessary to perform the iteration scheme.', 'D.2.1. Definitions': 'We introduce the following functions, \nB jj := ∫ z 0 zj 0 j 0 dz = -1 2 C, B nj := ∫ z 0 zn 0 j 0 dz = -1 2 S, B jn := ∫ z 0 zj 0 n 0 dz = -1 2 S, B nn := ∫ z z ∗ zn 0 n 0 dz = -B jj +ln z, (D . 15) \nwhere \nS = ∫ 2 z 0 dy sin y y = -∫ ∞ x dy sin y y + π 2 , C = ∫ 2 z 0 dy cos y -1 y = -∫ ∞ 2 z dy cos y y -γ -ln 2 z , (D . 16) \nand the lower bound z ∗ of the integral for the definition of B nn is adjusted so as to make B nn equal to the last expression of the line. \nAs an extension of these integral sinusoidal functions, we further introduce the following functions: \nB jJ := ∫ z z ∗ dzzj 0 D J 0 , (D . 17) \nB nJ := ∫ z z ∗ dzzn 0 D J 0 , (D . 18) \nwhere J stands for a sequence of j and n , say, J = jnnj , and we have also introduced an extension of the spherical Bessel functions by \nD j /lscript := j /lscript , D n /lscript := n /lscript , (D . 19) \nand \nD nJ /lscript := n /lscript B jJ -j /lscript B nJ , D jJ /lscript := j /lscript B jJ + n /lscript B nJ . (D . 20) \nWe adopt the following rule to determine the lower bound of the integrals in Eq. (D . 18). Whenever we can put z ∗ = 0, we do so, which is always possible when the sequence \nJ ends with j . On the other hand, in the case when J ends with n , there may appear in the integrand the square of n 0 which causes logarithmic divergence if we set z ∗ = 0. In such cases, we use the relation, \nn 2 0 = 1 z 2 -j 2 0 , (D . 21) \nto replace n 2 0 with the right hand side and extract out the logarithmically divergent term due to 1 /z 2 . Then we set z ∗ = 0 for the j 2 0 term, while we set z ∗ = 1 for the 1 /z 2 term so as to make the resulting logarithmic term zero at z = 1. For J of two indices, this is how we have defined B nn in Eq. (D . 15). For J of three indices, this applies to B njn . Specifically it is given by \nB njn = B jjj -B jj ln z + 1 2 (ln z ) 2 . (D . 22) \nFor convenience, in what follows we call B J the generalized integral sinusoidal functions and D J /lscript the generalized spherical Bessel functions. \nNote that all the B J whose J end with n can be expressed in terms of those whose J end with j . For example, for J of three indices, we have \nB jjn = -B nnj +ln zB nj , B jnn = 2 B jjj + B nnj -ln zB jj , B nnn = 2 B njj -B jnj -ln zB nj , (D . 23) \ntogether with Eq. (D . 22). Using these relations, we can express all the D J /lscript whose J end with n in terms of those whose J end with j . \nFurther we introduce the following indefinite integral operator, \nF k,/lscript [ X ] := n k ∫ dzj /lscript X -j k ∫ dzn /lscript X, (D . 24) \nfor a function X . Note that Eq. (D . 1) is expressed in terms of this operator as \nξ ( n ) /lscript = F /lscript,/lscript [ W ( n ) ] . (D . 25) \nWe also introduce the following operator, \nH J k [ Y ] := n k ∫ dzD J 0 Y -j k ∫ dzD J 0 ˜ Y , (D . 26) \nwhere Y stands for a linear combination of the generalized Bessel functions with the coefficients given by linear combinations of z m (ln z ) n ( m ≤ 1, n ≥ 0) and ˜ Y denotes the quantity which is obtained by replacing j , n , D jJ and D nJ with n , j , D nJ and D jJ , respectively, in the expression of Y . By definition we see that \nF /lscript, 0 [ z m (ln z ) n j 0 ] = H j /lscript [ z m (ln z ) n j 0 ] . (D . 27)', 'D.2.2. Basic formulas': 'The spherical Bessel functions satisfy the recursion relation, \nζ m -1 + ζ m +1 = 2 m +1 z ζ m , (D . 28) \nwhere ζ m = j m or n m . Note that \nn m = ( -1) m +1 j -m -1 , j m = ( -1) m n -m -1 . (D . 29) \nThe same recursion relation holds for the generalizes spherical Bessel functions, \nD ζJ m -1 + D ζJ m +1 = 2 m +1 z D ζJ m , (D . 30) \nwhere ζ = j or n . Further the relations the same as Eqs. (D . 29) hold for D ζJ m , \nD nJ m = ( -) m +1 D jJ -m -1 , D jJ m = ( -) m D nJ -m -1 . (D . 31) \nThe derivative recursion relation for the spherical Bessel functions is \nd dz ζ /lscript = 1 2 /lscript +1 { /lscriptζ /lscript -1 -( /lscript +1) ζ /lscript +1 } , (D . 32) \nand this extends to the generalized spherical Bessel functions as \nd dz D nζJ /lscript = 1 2 /lscript +1 { /lscriptD nζJ /lscript -1 -( /lscript +1) D nζJ /lscript +1 } + R /lscript, 0 z D ζJ 0 , d dz D jζJ /lscript = 1 2 /lscript +1 { /lscriptD jζJ /lscript -1 -( /lscript +1) D jζJ /lscript +1 } -R /lscript, -1 z D ζJ 0 . (D . 33) \nUseful integral formulas for the spherical Bessel functions are \n∫ dz ζ m ζ ∗ n = z 2 ( m -n )( m + n +1) ( ζ m ζ ∗ n +1 + ζ m -1 ζ ∗ n ) -z m -n ζ m ζ ∗ n , ( m /negationslash = n, -n -1) , ∫ dz ζ /lscript ζ ∗ /lscript = 1 2 /lscript +1 { ∫ dz ζ 0 ζ ∗ 0 -z ( ζ 0 ζ ∗ 0 +2 /lscript -1 ∑ m =1 ζ m ζ ∗ m + ζ /lscript ζ ∗ /lscript )} , ∫ dz zζ m ζ ∗ n = ∫ dz zζ m -1 ζ ∗ n -1 -z 2 m + n ( ζ m -1 ζ ∗ n -1 + ζ m ζ ∗ n ) , ∫ dz zζ /lscript ζ ∗ /lscript = ∫ dz zζ 0 ζ ∗ 0 -z 2 2 { ζ 0 ζ ∗ 0 + /lscript -1 ∑ m =1 ( 1 m + 1 m +1 ) ζ m ζ ∗ m + 1 /lscript ζ /lscript ζ ∗ /lscript } , ∫ dz zζ /lscript ζ ∗ /lscript -1 = ∫ dz zζ 0 ζ ∗ -1 -z 2 { ζ 0 ζ ∗ -1 + /lscript -1 ∑ m =1 4 m 4 m 2 -1 ζ m ζ ∗ m -1 + 1 2 /lscript -1 ζ /lscript ζ ∗ /lscript -1 } , ∫ dz zζ /lscript ζ ∗ /lscript +1 = ∫ dz zζ -1 ζ ∗ 0 -z 2 { ζ -1 ζ ∗ 0 + /lscript ∑ m =1 4 m 4 m 2 -1 ζ m -1 ζ ∗ m + 1 2 /lscript +1 ζ /lscript ζ ∗ /lscript +1 } , (D . 34) \nwhere ζ m or ζ ∗ m stands for j m or n m . \nThe following polynomial of 1 /z plays an important role in the calculations: \nR m,k = z 2 ( n m j k -j m n k ) = -[( m -k -1) / 2] ∑ r =0 ( -1) r ( m -k -1 -r )! Γ ( m + 1 2 -r ) r ! ( m -k -1 -2 r )! Γ ( k + 3 2 + r ) ( 2 z ) m -k -1 -2 r , (D . 35) \nfor m>k and \nR m,k = -R k,m . (D . 36) \nfor m < k . By construction, a recursion formula similar to that satisfied by the spherical Bessel functions holds: \nR m,k -1 + R m,k +1 = 2 k +1 z R m,k . (D . 37) \nNote that the indices m and k can be negative as well. As examples, we write down the explicit forms for some special cases: \nR k,k = 0 , R k,k +1 = 1 , R k,k -1 = -1 , R k,k -2 = -2 k -1 z . (D . 38)', 'D.2.3. Source terms': 'The source term can be rewritten as \nW ( n ) /lscript = z 2 [ d 2 dz 2 + 2 z d dz + ( 1 -/lscript ( /lscript +1) z 2 ) ] ξ ( n -1) /lscript z + [ d dz -2 z + /lscript ( /lscript +1) -4 z ] ξ ( n -1) /lscript + i [ 2 z d dz +1 ] ξ ( n -1) /lscript . (D . 39) \nThe contribution to ξ ( n ) /lscript from the first term is given by z -1 ξ ( n -1) /lscript . So we focus on the second and third terms. Note that the operators of the second and third terms have opposite parities; the second term is odd while the third term is even under the transformation z →-z . To perform the integration of these terms, we introduce the concept of the standard form of source terms as follows. \nFirst consider the first iteration, n = 1. Since ξ (0) /lscript = j /lscript and since we only need to calculate the real part of ξ (1) /lscript , we only need to consider the second term. Using the recursion relations (D . 28) and (D . 32), it may be rewritten in the form, \nα 0 zj /lscript + β 0 j /lscript -1 + β 1 j /lscript +1 . (D . 40) \nThen using the integral formulas (D . 34), the integrals F /lscript,/lscript [ zj /lscript ] and F e /lessmuch ,/lscript [ j /lscript ± 1 ] are readily evaluated to give \nF /lscript,/lscript [ zj /lscript ] = D nj /lscript -1 2 { R /lscript, 0 j 0 + /lscript -1 ∑ m =1 ( 1 m + 1 m +1 ) R /lscript,m j m } , F /lscript,/lscript [ j k ] = -1 ( /lscript -k )( /lscript + k +1) j k ( k = /lscript ± 1) . (D . 41) \nThe real part of ξ (1) /lscript is expressed in terms of these functions, while the imaginary part is (ln z ) j /lscript as given by Eq. (D . 9). The result is Eq. (3 . 35) with q = 0. \nAt the second iteration, n = 2, we insert the real part of ξ (1) /lscript to the second term in Eq. (D . 39) and the imaginary part i (ln z ) j /lscript to the third term, to evaluate the real part of ξ (2) /lscript . Let us focus on the contribution of the terms of the form R /lscript,m j m in ξ (1) /lscript for the moment. Since R /lscript,m are polynomials in 1 /z , we cannot apply the integral formulas (D . 34) directly. So, by using the recursion relation (D . 28) we get rid of the inverse powers of z . Then we find the corresponding source term may be expressed in the form, \nz (˜ α -j /lscript -1 + ˜ α + j /lscript +1 ) + ∑ m /negationslash =0 ˜ β m j /lscript +2 m . (D . 42) \nSimilarly, at the third iteration, n = 3, the terms in ξ (2) /lscript having the form z k j n ( k ≤ 0) will give rise to the source term which can be written in the form, \nz ( α 0 j /lscript + α -j -/lscript -2 + α + j -/lscript ) + ∑ m /negationslash = -/lscript β m j /lscript +2 m -1 , (D . 43) \nwhich is a generalization of Eq. (D . 40). Because the operator of the second term in Eq. (D . 39) has the odd parity, the source terms for n = 2 and 3 take different forms. We call Eqs. (D . 42) and (D . 43) the standard forms. For convenience we call the former the even standard form and the latter the odd standard form. Now turning to the terms with D J m or (ln z ) j m , since they satisfy the same recursion relation as j m do, the same idea can be extended to them in a natural sense. The standard form for them is then defined by Eqs. (D . 42) and (D . 43) with j m replaced by D J m or (ln z ) j m . Note that D nj /lscript at n = 2 plays an analogous role of j /lscript at n = 1. Hence the odd standard form of D nj m appears at n = 2. On the other hand, since the second and third terms in Eq. (D . 39) have opposite parities, the parity of the standard form of (ln z ) j m is equal to that of j m . \nTo summarize, the source term at the second iteration consists of the standard forms of \nj m : even , (ln z ) j m : even , D nj m : odd . (D . 44) \nThe integration of these terms can be done by using the formulas given in subsection D.3 below. The resulting ξ (2) /lscript are given by Eqs. (3 . 40) for /lscript = 2, 3 and Eq. (4 . 16) for /lscript = 4. Then we find there appear new types of the source term at the third iteration, which are \nzD nnj /lscript , D nnj /lscript ± 1 , z (ln z ) 2 j /lscript , (ln z ) 2 j /lscript ± 1 , z (ln z ) D nj /lscript ± 1 , (D . 45) \nin addition to the opposite parity terms of Eq. (D . 44), \nj m : odd , (ln z ) j m : odd , D nj m : even . (D . 46)', 'D.3. Reduction of integrals': 'In this subsection, we reduce the expressions F /lscript,/lscript [source terms] to those written in terms of D J m . For this purpose we need to evaluate integrals such as \nF k,/lscript [ zD nJ /lscript ] := n k ∫ dz zj /lscript D nJ /lscript -j k ∫ dz zn /lscript D nJ /lscript . (D . 47) \nAs an example let us show how this is evaluated. Using the basic integral formulas (D . 34), we integrate it by part as \nF k,/lscript [ zD nJ /lscript ]= n k ∫ dz zj 0 D nJ 0 -j k ∫ dz zn 0 D nJ 0 + j k ( -z 2 2 ){ n 0 j 0 + /lscript -1 ∑ m =1 ( 1 m + 1 m +1 ) n m j m + 1 /lscript n /lscript j /lscript } -n k ( -z 2 2 ){ j 0 j 0 + /lscript -1 ∑ m =1 ( 1 m + 1 m +1 ) j m j m + 1 /lscript j /lscript j /lscript } B nJ - j k ( -z 2 2 ){ n 0 n 0 + /lscript -1 ∑ m =1 ( 1 m + 1 m +1 ) n m n m + 1 /lscript n /lscript n /lscript } -n k ( -z 2 2 ){ j 0 n 0 + /lscript -1 ∑ m =1 ( 1 m + 1 m +1 ) j m n m + 1 /lscript j /lscript n /lscript } B jJ -n k ∫ dz ( -z 2 2 ){ j 0 n 0 + /lscript -1 ∑ m =1 ( 1 m + 1 m +1 ) j m n m + 1 /lscript j /lscript n /lscript } zj 0 D J 0 -∫ dz ( -z 2 2 ){ j 0 j 0 + /lscript -1 ∑ m =1 ( 1 m + 1 m +1 ) j m j m + 1 /lscript j /lscript j /lscript } zn 0 D J 0 + j k ∫ dz ( -z 2 2 ){ n 0 n 0 + /lscript -1 ∑ m =1 ( 1 m + 1 m +1 ) n m n m + 1 /lscript n /lscript n /lscript } zj 0 D J 0 -∫ dz ( -z 2 2 ){ n 0 j 0 + /lscript -1 ∑ m =1 ( 1 m + 1 m +1 ) n m j m + 1 /lscript n /lscript j /lscript } zn 0 D J 0 = D nnJ k + 1 2 { R 0 k D nJ 0 + /lscript -1 ∑ m =1 ( 1 m + 1 m +1 ) R mk D nJ m + 1 /lscript R /lscriptk D nJ /lscript } + 1 2 n k ∫ zdz ( /lscript -1 ∑ m =1 ( 1 m + 1 m +1 ) R m 0 j m + 1 /lscript R /lscript 0 j /lscript ) D J 0 -1 2 j k ∫ zdz ( /lscript -1 ∑ m =1 ( 1 m + 1 m +1 ) R m 0 n m + 1 /lscript R /lscript 0 n /lscript ) D J 0 = D nnJ k + 1 2 { R 0 k D nJ 0 + /lscript -1 ∑ m =1 ( 1 m + 1 m +1 ) R mk D nJ m + 1 /lscript R /lscriptk D nJ /lscript } + 1 2 H J k [ /lscript -1 ∑ m =1 ( 1 m + 1 m +1 ) R m 0 j m + 1 /lscript R /lscript 0 j /lscript ] . (D . 48) \nThe reduction of the H J k term in the last expression is done similarly. In the following, we give formulas for each type of the source terms.', 'D.3.1. j m -terms': 'The source terms have the standard form (D . 42) or (D . 43). Using Eqs. (D . 34), their integrals are evaluated as \nF /lscript,/lscript [ ζ m ] = -1 ( /lscript -m )( /lscript + m +1) ζ m ( m /negationslash = /lscript, -/lscript -1) , (D . 49) \nF /lscript,/lscript [ zj /lscript -1 ] = -D nn /lscript -{ -R /lscript, 0 n 0 + /lscript -1 ∑ m =1 4 m 4 m 2 -1 R /lscript,m j m -1 } , (D . 51) \nF /lscript,/lscript [ zζ /lscript ] = D nζ /lscript -1 2 { R /lscript, 0 ζ 0 + /lscript -1 ∑ m =1 ( 1 m + 1 m +1 ) R /lscript,m ζ m } , (D . 50) \nF /lscript,/lscript [ zn /lscript -1 ] = D nj /lscript -{ R /lscript, 0 j 0 + /lscript -1 ∑ m =1 4 m 4 m 2 -1 R /lscript,m n m -1 } , (D . 52) \nwhere ζ represents j or n . Note also that rather general formulas, \nF /lscript,/lscript [ zζ /lscript +1 ] = -D jζ /lscript -{ R /lscript, -1 ζ 0 + /lscript ∑ m =1 4 m 4 m 2 -1 R /lscript,m -1 ζ m } , (D . 53) \nF k,/lscript [ ζ m ] = 1 ( /lscript -m )( /lscript + m +1) { R k,/lscript ζ m +1 + R k,/lscript -1 ζ m } -1 /lscript m R k,/lscript z ζ m ( m /negationslash = /lscript, -/lscript -1) , (D . 54) \nF k,/lscript [ zζ /lscript ] = D nζ k -1 2 { R k, 0 ζ 0 + /lscript -1 ∑ m =1 ( 1 m + 1 m +1 ) R k,m ζ m + 1 /lscript R k,/lscript ζ /lscript } , (D . 55) \nhold.', 'D.3.2. (ln z ) j m -terms': 'The source terms are in the form (D . 40) or (D . 42) with j m replaced by (ln z ) j m . First we give the general formula for F /lscript,/lscript [(ln z ) j m ]. Using the first formula in Eqs. (D . 34), we obtain \nF /lscript,/lscript [(ln z ) j m ] = 1 ( /lscript -m )( /lscript + m +1) ( -ln z + 1 ( /lscript + m +1) ) j m + 1 /lscript -m F /lscript,/lscript [ -2 z /lscript + m +1 j m +1 + j m ] , (D . 56) \nfor m = /lscript, -/lscript 1. \n/negationslash \n-Next we consider the remaining term of the odd parity. With the aid of the second formula in Eqs. (D . 34), after the integration by part, we have \n- \nF /lscript,/lscript [ z (ln z ) j /lscript ] = F /lscript, 0 [ z (ln z ) j 0 ] -ln z 2 { R /lscript, 0 j 0 + /lscript -1 ∑ m =1 ( 1 m + 1 m +1 ) R /lscript,m j m } + ( /lscript ∑ m =1 1 m ) D nj /lscript -1 4 { 2 ( /lscript ∑ m =1 1 m ) -1 } R /lscript, 0 j 0 \n- \n-1 4 /lscript -1 ∑ m =1 ( 1 m + 1 m +1 ) 2 /lscript ∑ k = m +1 1 k + 1 m ( m +1) R /lscript,m j m . (D . 57) \n To evaluate the first term, we use the following trick. Note that \nB jnJ = ∫ z 0 dz zj 0 ( n 0 B jJ -j 0 B nJ ) = ∫ z 0 dz zn 0 ( j 0 B jJ + n 0 B nJ ) -∫ z 0 dz z B nJ = B njJ -(ln z ) B nJ + ∫ z 0 dz (ln z ) dB nJ dz , (D . 58) \nwhere we assumed J do not end with n , i.e., J = n, jn, · · · . In the same way, \n/negationslash \nB jjJ = -B nnJ +(ln z ) B jJ -∫ z 0 dz (ln z ) dB jJ dz . (D . 59) \nThus we obtain \n∫ z 0 dz (ln z ) dB nJ dz = B jnJ -B njJ +(ln z ) B nJ , ∫ z 0 dz (ln z ) dB jJ dz = -B jjJ -B nnJ +(ln z ) B jJ . (D . 60) \nUsing Eqs. (D . 60), we find \nF /lscript, 0 [ z (ln z ) j 0 ] = -D njj /lscript -D jnj /lscript +(ln z ) D nj /lscript . (D . 61) \nAs for the remaining terms of the even parity, we have \nF /lscript,/lscript [ z (ln z ) j /lscript +1 ] = -D nnj /lscript + D jjj /lscript -ln zD jj /lscript -R /lscript, -1 (ln z ) j 0 -ln z /lscript ∑ m =1 4 m 4 m 2 -1 R /lscript,m -1 j m \n-R /lscript, -1 (ln z ) j 0 +ln z /lscript -1 ∑ m =1 4 m 4 m 2 -1 R /lscript,m j m -1 \n+ F /lscript, -1 [ zj 0 ] + /lscript ∑ m =1 4 m 4 m 2 -1 F /lscript,m -1 [ zj m ] + 1 2 /lscript +1 F /lscript,/lscript [ zj /lscript +1 ] , F /lscript,/lscript [ z (ln z ) j /lscript -1 ] = D jjj /lscript -D nnj /lscript -(ln z ) D jj /lscript + 1 2 (ln z ) 2 j /lscript \n-F /lscript, 0 [ zn 0 ] + /lscript -1 ∑ m =1 4 m 4 m 2 -1 F /lscript,m [ zj m -1 ] + 1 2 /lscript +1 F /lscript,/lscript [ zj /lscript -1 ] . (D . 62)', 'D.3.3. (ln z ) 2 j m -terms': 'The source terms we have to evaluate are z (ln z ) 2 j /lscript and (ln z ) 2 j /lscript ± 1 . Using the first formula in Eqs. (D . 34), we obtain \nF /lscript,/lscript [(ln z ) 2 j m ] = 1 ( /lscript -m )( /lscript + m +1) { [ -(ln z ) 2 + 2ln z /lscript + m +1 -2 ( /lscript + m +1) 2 ] j m \n+2 F /lscript,/lscript [ (ln z ) ( -2 zj m +1 +( /lscript + m +1) j m ) + 2 zj m +1 /lscript + m +1 ] } , (D . 63) \nfor m = /lscript, /lscript -1. \n/negationslash \n- \n-Also, with the aid of the second formula in Eqs. (D . 34), we find \nF /lscript,/lscript [ z (ln z ) 2 j /lscript ] = F /lscript, 0 [ z (ln z ) 2 j 0 ] -(ln z ) 2 2 { R /lscript, 0 j 0 + /lscript -1 ∑ m =1 ( 1 m + 1 m +1 ) R /lscript,m j m } + F /lscript, 0 [ z (ln z ) j 0 ] + /lscript -1 ∑ m =1 ( 1 m + 1 m +1 ) F /lscript,m [ z (ln z ) j m ] + 1 /lscript F /lscript,/lscript [ z (ln z ) j /lscript ] . (D . 64) \nThe evaluation of the first term in the above equation is done as follows. Using a technique similar to the one used to derive Eqs. (D . 60), we obtain \n∫ z 0 dz (ln z ) 2 dB nJ dz = 2[ -B nnnJ -B jjnJ + B njjJ -B jnjJ ] +2(ln z ) [ B jnJ -B njJ ] + (ln z ) 2 B nJ , ∫ z 0 dz (ln z ) 2 dB jJ dz = 2[ -B jnnJ + B njnJ + B jjjJ + B nnjJ ] -2(ln z ) [ B nnJ + B jjJ ] + (ln z ) 2 B jJ . (D . 65) \nUsing these, we can rewrite the first term as \nF /lscript, 0 [ z (ln z ) 2 j 0 ] = 2 ( -D nnnj /lscript + D jjnj /lscript + D njjj /lscript + D jnjj /lscript ) -2(ln z ) ( D jnj /lscript + D njj /lscript ) +(ln z ) 2 D nj /lscript . (D . 66) \nWe need one more formula to evaluate (D . 64): \nF /lscript,m [ z (ln z ) j m ] = F /lscript, 0 [ z (ln z ) j 0 ] -1 4 m 2 R /lscript,m j m -ln z 2 { R /lscript, 0 j 0 + m -1 ∑ k =1 ( 1 k + 1 k +1 ) R /lscript,k j k + 1 m R /lscript,m j m } + ( m ∑ k =1 1 k ) D nj /lscript -1 4 { 2 ( m ∑ k =1 1 k ) -1 } R /lscript, 0 j 0 -1 4 m -1 ∑ k =1 ( 1 k + 1 k +1 ) 2 m ∑ p = k +1 1 p + 1 k ( k +1) R /lscript,k j k . (D . 67)', 'D.3.4. D nJ k -terms': 'The source terms are the even and odd standard forms of D nj m , and zD nnj /lscript and D nnj /lscript ± 1 . \nNecessary formulas are \nF /lscript,/lscript [ zD nJ /lscript +1 ] = -D jnJ /lscript -R /lscript, -1 D nJ 0 -/lscript ∑ m =1 4 m 4 m 2 -1 R /lscript,m -1 D nJ m \n+ H J /lscript [ z { /lscript ∑ m =1 4 m 4 m 2 -1 R m, 0 j m -1 + R /lscript +1 , 0 2 /lscript +1 j /lscript }] , F /lscript,/lscript [ zD nJ /lscript -1 ] = D njJ /lscript -R /lscript, 0 D jJ 0 -/lscript -1 ∑ m =1 4 m 4 m 2 -1 R /lscript,m D nJ m -1 + D nJ /lscript + H J /lscript [ z { /lscript -1 ∑ m =2 4 m 4 m 2 -1 R m -1 , 0 j m + R /lscript -1 , 0 2 /lscript -1 j /lscript }] , F /lscript,/lscript [ D nJ m ] = -1 ( /lscript -m )( /lscript + m +1) D nJ m -H J /lscript [ z ( /lscript -m )( /lscript + m +1) ( R m +1 , 0 j /lscript + R m, 0 j /lscript -1 ) -R m, 0 /lscript -m j /lscript ] . F /lscript,/lscript [ zD nJ /lscript ] = D nnJ /lscript + 1 2 { R 0 ,/lscript D nJ 0 + /lscript -1 ∑ m =1 ( 1 m + 1 m +1 ) R m,k D nJ m } + H J /lscript [ z 2 ( /lscript -1 ∑ m =1 ( 1 m + 1 m +1 ) R m, 0 j m + 1 /lscript R /lscript, 0 j /lscript )] , F /lscript,/lscript [ D nJ /lscript +1 ] = 1 2( /lscript +1) D nJ /lscript +1 + H J /lscript [ -z 2( /lscript +1) ( R 0 ,/lscript +2 j /lscript + R 0 ,/lscript +1 j /lscript -1 ) + R 0 ,/lscript +1 j /lscript ] , F /lscript,/lscript [ D nJ /lscript -1 ] = -1 2 /lscript D nJ /lscript -1 + H J /lscript [ z 2 /lscript ( R 0 ,/lscript j /lscript + R 0 ,/lscript -1 j /lscript -1 ) -R 0 ,/lscript -1 j /lscript ] , (D . 68) \nIn order to evaluate the terms involving the integral operator H J /lscript , we recast its argument into the form, \nz ( α -2 j -2 + α -1 j -1 + α 0 j 0 + α 1 j 1 ) + ∑ n /negationslash =0 , -1 β n j n . (D . 69) \nThen all the necessary terms can be easily evaluated as \nH j /lscript [ j m ] = 1 m ( m +1) ( -R /lscript,m +1 j 0 + R /lscript,m n 0 ) + R /lscript,m mz j 0 ( m /negationslash = 0 , -1) , H j /lscript [ zj 1 ] = -D nn /lscript + R /lscript, 0 n 0 -R /lscript, 1 j 0 , H j /lscript [ zj 0 ] = D nj /lscript , H j /lscript [ zj -1 ] = -D jj /lscript , H j /lscript [ zj -2 ] = -D jn /lscript -j 0 R /lscript, -2 + j -1 R /lscript, -1 , (D . 70) \nand \nH nj /lscript [ zj 1 ] = D njj /lscript -R /lscript, 0 z D nj 0 , H nj /lscript [ zj 0 ] = D nnj /lscript , H nj /lscript [ zj -1 ] = -D jnj /lscript , H nj /lscript [ zj -2 ] = (unnecessary for our present calculation) , H nj k [ j m ] = 1 m ( m +1) ( R k,m D nj 1 + R k,m -1 D nj 0 ) -1 mz R k,m D nj 0 + 1 m ( m +1) H j k [ zj m ] . (D . 71) \nThe last term H j k [ zj m ] can be reduced recursively to those given in Eq. (D . 70). \nD.3.5. \nz (ln z ) D nj m -terms \nWhat we need to evaluate are F /lscript,/lscript [ z (ln z ) D nj /lscript ± 1 ]. \nIt is evaluated as \nF /lscript,/lscript [ z (ln z ) D nj /lscript +1 ] = F /lscript, -1 [ z (ln z ) D nj 0 ] -(ln z ) { R /lscript, -1 D nj 0 + /lscript ∑ m =1 4 m 4 m 2 -1 R /lscript,m -1 D nj m } + F /lscript, -1 [ zD nj 0 ] + /lscript ∑ m =1 4 m 4 m 2 -1 F /lscript,m -1 [ zD nj m ] + 1 2 /lscript +1 F /lscript,/lscript [ zD nj /lscript +1 ] + H j /lscript [ z (ln z ) { /lscript ∑ m =1 4 m 4 m 2 -1 R m, 0 j m -1 + 1 2 /lscript +1 R /lscript +1 , 0 j /lscript }] , (D . 72) \nFirst we consider F /lscript,/lscript [ z (ln z ) D nj /lscript +1 ] . \nwhere the first term is further expressed in terms of D J /lscript as \nThe other unknown terms in Eq. (D . 72) are also evaluated as \nF /lscript, -1 [ z (ln z ) D nj 0 ] = -D nnnj /lscript + D jjnj /lscript -(ln z ) D jnj /lscript . (D . 73) \nF /lscript,m -1 [ zD nj m ] = -D jnj /lscript -{ R /lscript, -1 D nj 0 + m -1 ∑ k =1 4 k 4 k 2 -1 R /lscript,k -1 D nj k + R /lscript,m -1 2 m -1 D nj m } + H j /lscript [ z { m -1 ∑ k =1 4 k 4 k 2 -1 R k, 0 j k -1 + R m, 0 2 m -1 j m -1 }] ( m ≥ 1) , F /lscript, -1 [ zD nj 0 ] = -D jnj /lscript , H j /lscript [ z (ln z ) j 0 ] = F /lscript, 0 [ z (ln z ) j 0 ] = -D njj /lscript -D jnj /lscript +(ln z ) D nj /lscript , H j /lscript [(ln z ) j m ] = (ln z ) { 1 m ( m +1) ( -R /lscript,m +1 j 0 + R /lscript,m n 0 ) + R /lscript,m mz j 0 } + H j /lscript [ 2 z m ( m +1) j m +1 ] + 2 m +1 m 2 ( m +1) 2 ( R /lscript,m +1 j 0 -R /lscript,m n 0 ) -R /lscript,m m 2 z j 0 . (D . 74) \nF /lscript,/lscript [ z (ln z ) D nj /lscript -1 ] = F /lscript, 0 [ z (ln z ) D jj 0 ] -(ln z ) { R /lscript, 0 D nj -1 + /lscript -1 ∑ m =1 4 m 4 m 2 -1 R /lscript,m D nj m -1 } + F /lscript, 0 [ zD nj -1 ] + /lscript -1 ∑ m =1 4 m 4 m 2 -1 F /lscript,m [ zD nj m -1 ] + 1 2 /lscript -1 F /lscript,/lscript [ zD nj /lscript -1 ] + H j /lscript [ z (ln z ) ( j 0 + /lscript -1 ∑ m =2 4 m 4 m 2 -1 R m -1 , 0 j m + 1 2 /lscript -1 R /lscript -1 , 0 j /lscript )] , (D . 75) \nIn the same way, we can evaluate F /lscript,/lscript [ z (ln z ) D nj /lscript -1 ] as \nwhere the unknown terms in the right hand side are evaluated as \nF /lscript, 0 [ z (ln z ) D jj 0 ] = -D njjj /lscript -D jnjj /lscript +(ln z ) D njj /lscript , \nF /lscript,m [ zD nj m -1 ] = D njj /lscript -{ R /lscript, 0 D nj -1 + m -1 ∑ k =1 4 k 4 k 2 -1 R /lscript,k D nj k -1 + R /lscript,m 2 m -1 D nj m -1 } + H j /lscript [ z ( j 0 + m -1 ∑ k =1 4 k 4 k 2 -1 R k -1 , 0 j k + R m -1 , 0 2 m -1 j m )] ( m ≥ 1) , F /lscript, 0 [ zD nj -1 ] = D njj /lscript . (D . 76)', 'D.4. The asymptotic behavior': "Using the results of the preceding subsection, we obtain the analytic expression for ξ (3) /lscript . The real part of it is given in Eq. (4 . 14) for /lscript = 2 and in Eq. (4 . 15) for /lscript = 3, while the imaginary part is determined by Eq. (D . 9). To obtain A inc /lscript to O ( /epsilon1 3 ), we then see that all what we need to evaluate are the asymptotic behaviors of D nj /lscript , D nnj /lscript , D nnnj /lscript and F /lscript, 0 [ z (ln z ) j 0 ]. Although the last of these can be expressed in terms of D J /lscript as given by Eq. (D . 61), we find it is easier to evaluate the integral directly as it is. \nHere in order to evaluate the asymptotic behaviors of these functions, we first give necessary basic formulas. Then we evaluate the asymptotic behavior of all the necessary B J and F /lscript, 0 [ z (ln z ) j 0 ]. In this subsection x = 2 z . \nD.4.1. Basic asymptotic formulas \nFirst, we give the most basic formulas: \nS = ∫ x 0 dy sin y y → π 2 , C = ∫ x 0 dy cos y -1 y →-ln x -γ, ∫ x 0 dy sin y y ln y →-π 2 γ, ∫ x 0 dy cos y -1 y ln y →-1 2 (ln x ) 2 -π 2 24 + γ 2 2 , ∫ x 0 dy sin y y (ln y ) 2 → π 2 γ 2 + π 3 24 , ∫ x 0 dy cos y -1 y (ln y ) 2 →-1 3 (ln x ) 3 -γ 3 3 + γπ 2 12 -2 3 ζ (3) , (D . 77) \nwhere ζ ( z ) is the Riemann zeta function and ζ (3) = 1 . 202 . \nThere appear several expressions to be estimated that diverge if we evaluate them term by term. But they give finite results when combined together. They are \n· · · \n∫ x 0 dy y +1 { ln( y +1) -ln y } → π 2 6 , ∫ x 0 dy y +1 { (ln( y +1)) 2 -(ln y ) 2 } → 0 , ∫ x 0 dy y +1 { ln( y +2) -ln( y +1) } → π 2 12 , ∫ x 0 dy y +1 { (ln( y +2)) 2 -(ln( y +1)) 2 } → 2 φ (3 , 1 / 2) + π 2 6 ln 2 -1 3 (ln 2) 3 . \n∫ x 0 dy y +2 ln( y +1) { ln( y +2) -ln( y +1) } →-φ (3 , 1 / 2) -π 2 12 ln 2 + 1 6 (ln 2) 3 + 3 2 ζ (3) , (D . 78) \nwhere φ ( a, b ) represents the modified zeta function. We mention that the φ (3 , 1 / 2) terms are found to cancel out in the final expression for A inc /lscript . \nThere are several formulas which require multiple integrations. They are evaluated as \n∫ x 0 dy y [∫ ∞ y du u (cos( u -y ) -cos u ) ] → π 2 6 , (D . 79) \n∫ x 0 dy y ln y [∫ ∞ y du u (cos( u -y ) -cos u ) ] →-π 2 γ 6 , (D . 80) \n∫ x 0 dy y ln y [∫ ∞ y du u (cos( u + y ) -cos u ) ] → φ (3 , 1 / 2) + π 2 12 ln 2 -1 6 (ln 2) 3 + π 2 γ 12 , (D . 81) \n∫ x 0 dy y ln y [∫ ∞ y du u (sin( u -y ) -sin u ) ] → π 3 12 , (D . 82) \n∫ x 0 dy y ln y [∫ ∞ y du u (sin( u + y ) -sin u ) ] →-π 3 24 , (D . 83) \n∫ x 0 dy y [∫ ∞ y du u (sin( u -y ) -sin u ) ] → 0 . (D . 84) \nThese are obtained by changing the variable from u to u ' = u y -1, performing the dy -integration first and using the formulas (D . 78). \nNext we give several formulas that contain S and C . Let us recall the definition of S and C : \nC = ∫ x 0 dy y (cos y -1) = -∫ ∞ x dy y cos y -ln x -γ. (D . 86) \nS = ∫ x 0 dy y sin y = -∫ ∞ x dy y sin y + π 2 , (D . 85) \nBy using the integration by part, we have \n∫ x 0 dy ln y ( sin y y S + cos y -1 y C ) = π 2 ∫ x 0 dy sin y y ln y -∫ x 0 dy cos y -1 y ( γ ln y +(ln y ) 2 ) -∫ x 0 dy ln y y [∫ ∞ y du u (cos( u -y ) -cos u ) ] , (D . 87) \nwhose asymptotic behavior is determined by Eqs. (D . 77) and (D . 80). Similarly we have \n∫ x 0 dy ln y ( sin y y S -cos y -1 y C ) = π 2 ∫ x 0 dy sin y y ln y + ∫ x 0 dy cos y -1 y ( γ ln y +(ln y ) 2 ) + ∫ x 0 dy ln y y [∫ ∞ y du u (cos( u + y ) -cos u ) ] , (D . 88) \nand \n∫ x 0 dy ln y ( sin y y C ± cos y -1 y S ) = -∫ x 0 dy sin y y ln y (ln y + γ ) ± π 2 ∫ x 0 dy cos y -1 y ln y ∓ ∫ x 0 dy ln y y [∫ ∞ y du u (sin( u ± y ) -sin u ) ] , (D . 89) \nwhose asymptotic behaviors are determined by Eqs. (D . 77), (D . 81), (D . 82) and (D . 83). Then we obtain \n∫ x 0 dy y ( S 2 + C 2 ) → π 2 4 (ln x + γ ) + 1 3 (ln x + γ ) 3 -4 3 ζ (3) , (D . 90) \n∫ x 0 dy y ( S 2 -C 2 ) → π 2 4 ln x -1 3 (ln x + γ ) 3 + π 2 γ 4 + 4 3 ζ (3) -2 ( φ (3 , 1 / 2) + π 2 12 ln 2 -1 6 (ln 2) 3 ) . (D . 91) \nThese two formulas are obtained by reducing them to the forms to which Eqs. (D . 87) and (D . 88) can be applied, respectively. \nFinally, we present two complicated formulas. The first is \nI 1 := ∫ x 0 dy y ( -2sin ySC +(cos y -1)( S 2 -C 2 ) ) → 2 ( φ (3 , 1 / 2) + π 2 12 ln 2 -1 6 (ln 2) 3 ) -7 3 ζ (3) -1 4 π 2 (ln x + γ ) + 1 3 (ln x + γ ) 3 (D . 92) \nwhere we have used the equalities, \nSC = ∫ ∞ 0 sin( t +2) x t +2 ln( t +1) dt + π 2 C -(ln x + γ ) S + π 2 (ln x + γ ) , (D . 93) \nS 2 -C 2 = -2 ∫ ∞ 0 cos( t +2) x t +2 ln( t +1) dt + πS -π 2 4 +2(ln x + γ ) C +(ln x + γ ) 2 , (D . 94) \nand applied the formulas (D . 77), (D . 87) and the last one of Eqs. (D . 78). The second is \nI 2 := ∫ x 0 dy y ( 2(cos y -1) SC +sin y ( S 2 -C 2 ) ) → π 2 ( (ln x + γ ) 2 + π 2 4 ) , (D . 95) \nwhich is obtained in the same way by applying the formulas (D . 77), (D . 84) and (D . 89). \nD.4.2. The asymptotic behavior of B J \nAs we have mentioned, what we have to evaluate are the asymptotic behaviors of D nj /lscript , D nnj /lscript , D nnnj /lscript and F /lscript, 0 [ z (ln z ) j 0 ]. Hence, recalling the definition of D J /lscript , we need to evaluate B nj , B jj , B nnj , B jnj , B nnnj and B jnnj in addition to F /lscript, 0 [ z (ln z ) j 0 ]. \nThe formulas for B J with two indices are given by the first two equations of (D . 77): \nB nj = -1 2 S →-π 4 , B jj = -1 2 C → 1 2 (ln x + γ ) . (D . 96) \nAs for F /lscript, 0 [ z (ln z ) j 0 ], its asymptotic behavior is directly evaluated as \nF /lscript, 0 [ z (ln z ) j 0 ] = n /lscript ∫ z 0 dzz (ln z ) j 2 0 -j /lscript ∫ z 0 dzz (ln z ) j 0 n 0 → 1 4 ( (ln z ) 2 + π 2 12 -( γ +ln2) 2 ) n /lscript -π 4 ( γ +ln2) j /lscript . (D . 97) \nThe formulas for B J with three indices are given by \nB jnj = 1 4 ∫ x 0 dy ( sin y y C -cos y -1 y S ) → π 8 (ln x + γ ) , (D . 98) \nwhere we have used Eqs. (D . 77) and (D . 84), and \nB nnj = -1 4 ∫ x 0 dy ( cos y +1 y C + sin y y S ) →-1 8 ( 5 12 π 2 -(ln x + γ ) 2 ) , (D . 99) \nwhere we have used Eqs. (D . 77). \nFor B J with four indices, we have \nB jnnj = -1 2 ∫ x 0 dy sin y y B jnj + 1 2 ∫ x 0 dy cos y -1 y B nnj = -1 2 [ SB jnj -CB nnj ] + 1 8 ∫ x 0 dy ( sin y y C -cos y -1 y S ) S + 1 8 ∫ x 0 dy ( sin y y S + cos y +1 y C ) C \n= -1 2 [ SB jnj -CB nnj ] -1 8 I 1 + 1 4 ∫ x 0 dy C 2 y → 1 24 [ 5 8 π 2 (ln x + γ ) -1 2 (ln x + γ ) 3 -ζ (3) ] , (D . 100) \nwhere we have used Eqs. (D . 90), (D . 91) and (D . 92), and \nB nnnj = 1 2 ∫ x 0 dy sin y y B nnj + 1 2 ∫ x 0 dy cos y +1 y B jnj = ∫ x 0 dyB jnj + 1 2 [ CB jnj + SB nnj ] -1 8 ∫ x 0 dy ( sin y y C -cos y -1 y S ) C + 1 8 ∫ x 0 dy ( sin y y S + cos y +1 y C ) S = ln xB jnj -1 4 ∫ x 0 dy ln y ( sin y y C -cos y -1 y S ) + 1 2 [ CB jnj + SB nnj ] + 1 8 I 2 + 1 4 ∫ x 0 dy SC y → π 32 [ -1 4 π 2 +(ln x + γ ) 2 ] , (D . 101) \nwhere we have used Eqs. (D . 89) and (D . 95). We should note that Eq. (D . 89) has been also used in evaluating the term 1 4 ∫ x 0 dy SC y . \nWith these results, the asymptotic incoming amplitudes A inc /lscript to the required order are obtained, which are given in Eqs. (4 . 17) in the text. \nAppendix E 5.5PN formulas for R in /lscript and ( dE/dt ) /lscriptm in the Schwarzschild case \nIn this Appendix, we show the post-Newtonian expansion of R in which are necessary to evaluate the 5.5PN gravitational wave luminosity in the case of a Schwarzschild black hole. Then we show each ( /lscript, m )-mode contribution to the total luminosity from a particle in circular orbit. \nFor convenience, we give the formulas for c 0 ωR in /lscript , where we recover the index /lscript on R in and c 0 = ( /lscript -1) /lscript ( /lscript +1)( /lscript +2) -6 i/epsilon1 . \nc 0 ωR in 2 = 4 z 4 5 + 8 i 15 z 5 -22 z 6 105 -2 i 35 z 7 + 23 z 8 1890 + 2 i 945 z 9 -13 z 10 41580 -i 24948 z 11 + 59 z 12 12972960 + i 2162160 z 13 -83 z 14 1945944000 -i 277992000 z 15 + -8 z 3 5 -3 i 5 z 4 -8 z 5 63 -13 i 90 z 6 + 109 z 7 1890 + 341 i 22680 z 8 -9403 z 9 3118500 -293 i 594000 z 10 + 38963 z 11 567567000 + 75529 i 9081072000 z 12 /epsilon1 \n+ 4 z 2 5 + 123317 z 4 36750 + 231479 i 110250 z 5 -889954 z 6 1157625 -454499 i 2315250 z 7 + 215321483 z 8 5501034000 + 35106811 i 5501034000 z 9 -214 z 4 ln z 525 -428 i 1575 z 5 ln z + 1177 z 6 ln z 11025 + 107 i 3675 z 7 ln z -2461 z 8 ln z 396900 -107 i 99225 z 9 ln z /epsilon1 2 + -66823 z 3 12250 -99851 i 55125 z 4 -504569 z 5 694575 -2488639 i 3969000 z 6 + 428 z 3 ln z 525 + 107 i 350 z 4 ln z + 428 z 5 ln z 6615 + 1391 i 18900 z 6 ln z /epsilon1 3 + ( 471487 z 2 220500 -263 i 1260 z 3 -214 z 2 ln z 525 ) /epsilon1 4 , (E . 1) c 0 ωR in 3 = 4 z 5 21 + 2 i 21 z 6 -2 z 7 63 -i 135 z 8 + 29 z 9 20790 + i 4620 z 10 -47 z 11 1621620 -i 294840 z 12 + 23 z 13 64864800 + i 29937600 z 14 + -10 z 4 21 -53 i 315 z 5 + z 6 210 -i 90 z 7 + 751 z 8 155925 + 1483 i 1247400 z 9 -23 z 10 102960 -367 i 10810800 z 11 /epsilon1 + 8 z 3 21 + i 14 z 4 + 40337 z 5 46305 + 79099 i 185220 z 6 -12562147 z 7 91683900 -4840537 i 157172400 z 8 -26 z 5 ln z 441 -13 i 441 z 6 ln z + 13 z 7 ln z 1323 + 13 i 5670 z 8 ln z /epsilon1 2 + ( -2 z 2 21 -182981 z 4 92610 -3753697 i 5556600 z 5 + 65 z 4 ln z 441 + 689 i 13230 z 5 ln z ) /epsilon1 3 , (E . 2) c 0 ωR in 4 = 2 z 6 63 + 4 i 315 z 7 -13 z 8 3465 -8 i 10395 z 9 + 71 z 10 540540 + i 54054 z 11 -37 z 12 16216200 -i 4054050 z 13 \n+ -2 z 5 21 -4 i 135 z 6 + 142 z 7 51975 -31 i 51975 z 8 + 929 z 9 2702700 + 8 i 96525 z 10 /epsilon1 + 5 z 4 49 + 97 i 4410 z 5 + 958223891 z 6 6051137400 + 239560304 i 3781960875 z 7 \n + -20 z 3 441 -i 196 z 4 /epsilon1 3 , (E . 3) \n-1571 z 6 ln z 218295 \n-3142 i 1091475 z 7 \nln z /epsilon1 2 \nc 0 ωR in 5 = 2 z 7 495 + 2 i 1485 z 8 -7 z 9 19305 -i 15015 z 10 + 17 z 11 1621620 + i 737100 z 12 + -7 z 6 495 -67 i 17325 z 7 + 1831 z 8 4054050 -43 i 4054050 z 9 /epsilon1 + 28 z 5 1485 + 59 i 14850 z 6 /epsilon1 2 , (E . 4) \n + -32 z 7 19305 -163 i 405405 z 8 /epsilon1, (E . 5) \nc 0 ωR in 6 = 8 z 8 19305 + 16 i 135135 z 9 -4 z 10 135135 -2 i 405405 z 11 \n c 0 ωR in 7 = 8 z 9 225225 + 2 i 225225 z 10 . (E . 6) \nNext we show the contribution of each ( /lscript, m )-mode to the gravitational wave luminosity to O ( v 11 ) in the case of a circular orbit around a Schwarzschild black hole. We set \nwhere ( dE/dt ) N is the Newtonian quadrupole luminosity, Eq. (4 . 19). In the present case we have η /lscript,m = η /lscript, -m . Hence we show only the modes m> 0. \n〈 dE dt 〉 = 1 2 ( dE dt ) N ∑ /lscript,m η /lscript,m , (E . 7) \nη 2 , 2 = 1 -107 v 2 21 +4 π v 3 + 4784 v 4 1323 -428 π v 5 21 + v 6 ( 99210071 1091475 -1712 γ 105 + 16 π 2 3 -3424 ln 2 105 -1712 ln v 105 ) + 19136 π v 7 1323 + v 8 ( -27956920577 81265275 + 183184 γ 2205 -1712 π 2 63 + 366368 ln 2 2205 + 183184 ln v 2205 ) \n+ v 9 ( 396840284 π 1091475 -6848 γ π 105 -13696 π ln 2 105 -6848 π ln v 105 ) + v 10 ( 187037845924 6257426175 -8190208 γ 138915 + 76544 π 2 3969 -16380416 ln 2 138915 -8190208 ln v 138915 ) + v 11 ( -111827682308 π 81265275 + 732736 γ π 2205 + 1465472 π ln 2 2205 + 732736 π ln v 2205 ) , (E . 8) η 2 , 1 = v 2 36 -17 v 4 504 + π v 5 18 -2215 v 6 254016 -17 π v 7 252 + v 8 ( 15707221 26195400 -107 γ 945 + π 2 27 -107 ln 2 945 -107 ln v 945 ) -2215 π v 9 127008 + v 10 ( -6435768121 57210753600 + 1819 γ 13230 -17 π 2 378 + 1819 ln 2 13230 + 1819 ln v 13230 ) + v 11 ( 15707221 π 13097700 -214 γ π 945 -214 π ln 2 945 -214 π ln v 945 ) , (E . 9) η 3 , 3 = 1215 v 2 896 -1215 v 4 112 + 3645 π v 5 448 + 243729 v 6 9856 -3645 π v 7 56 + v 8 ( 25037019729 125565440 -47385 γ 1568 + 3645 π 2 224 -47385 ln 2 1568 -47385 ln 3 1568 -47385 ln v 1568 ) + 731187 π v 9 4928 + v 10 ( -2074855555161 1381219840 + 47385 γ 196 -3645 π 2 28 + 47385 ln 2 196 + 47385 ln 3 196 + 47385 ln v 196 ) + v 11 ( 75111059187 π 62782720 -142155 γ π 784 -142155 π ln 2 784 -142155 π ln 3 784 -142155 π ln v 784 ) , (E . 10) η 3 , 2 = 5 v 4 63 -193 v 6 567 + 20 π v 7 63 + 86111 v 8 280665 -772 π v 9 567 + v 10 ( 960188809 178783605 -1040 γ 1323 + 80 π 2 189 -2080 ln 2 1323 -1040 ln v 1323 ) \n+ 344444 π v 11 280665 , (E . 11) η 3 , 1 = v 2 8064 -v 4 1512 + π v 5 4032 + 437 v 6 266112 -π v 7 756 + v 8 ( -1137077 50854003200 -13 γ 42336 + π 2 6048 -13 ln 2 42336 -13 ln v 42336 ) + 437 π v 9 133056 + v 10 ( -38943317051 5034546316800 + 13 γ 7938 -π 2 1134 + 13 ln 2 7938 + 13 ln v 7938 ) + v 11 ( -1137077 π 25427001600 -13 γ π 21168 -13 π ln 2 21168 -13 π ln v 21168 ) , (E . 12) η 4 , 4 = 1280 v 4 567 -151808 v 6 6237 + 10240 π v 7 567 + 560069632 v 8 6243237 -1214464 π v 9 6237 + v 10 ( 36825600631808 88497884475 -25739264 γ 392931 + 81920 π 2 1701 -25739264 ln 2 130977 -25739264 ln v 392931 ) + 4480557056 π v 11 6243237 , (E . 13) η 4 , 3 = 729 v 6 4480 -28431 v 8 24640 + 2187 π v 9 2240 + 620077923 v 10 246646400 -85293 π v 11 12320 , (E . 14) η 4 , 2 = 5 v 4 3969 -437 v 6 43659 + 20 π v 7 3969 + 7199152 v 8 218513295 -1748 π v 9 43659 + v 10 ( 9729776708 619485191325 -25136 γ 2750517 + 80 π 2 11907 -50272 ln 2 2750517 -25136 ln v 2750517 ) + 28796608 π v 11 218513295 , (E . 15) η 4 , 1 = v 6 282240 -101 v 8 4656960 + π v 9 141120 + 7478267 v 10 139848508800 -101 π v 11 2328480 , (E . 16) η 5 , 5 = 9765625 v 6 2433024 -2568359375 v 8 47443968 + 48828125 π v 9 1216512 + 7060478515625 v 10 25904406528 -12841796875 π v 11 23721984 , (E . 17) η 5 , 4 = 4096 v 8 13365 -18231296 v 10 6081075 + 32768 π v 11 13365 , (E . 18) η 5 , 3 = 2187 v 6 450560 -150903 v 8 2928640 + 6561 π v 9 225280 + 600654447 v 10 2665062400 -452709 π v 11 1464320 , (E . 19) η 5 , 2 = 4 v 8 40095 -15644 v 10 18243225 + 16 π v 11 40095 , (E . 20) η 5 , 1 = v 6 127733760 -179 v 8 2490808320 + π v 9 63866880 \n| 290803 v | 10 - 179 π v 11 | (E . 21) |\n|--------------------------------|--------------------------------------------------|------------|\n| + 971415244800 | 1245404160 , 10 | |\n| 26244 v 8 2965572 | v + 314928 π v 11 , | (E . 22) |\n| 3575 - | 25025 3575 | |\n| 244140625 v 10 , | (E . | 23) |\n| 435891456 8 | - 4063232 v 10 22297275 + 1048576 π v 11 9555975 | |\n| 131072 v | , (E | . 24) |\n| 9555975 59049 v 10 , | | (E . 25) |\n| 98658560 4 v 8 - 4 v 10 | 16 π v 11 5733585 , | |\n| 5733585 v 10 | | (E . 26) |\n| 7192209024 | 495495 + | |\n| , | , | (E . 27) |\n| 96889010407 v 10 | 96889010407 v 10 | (E . 28) |\n| 7116595200 , 6103515625 v 10 , | 7116595200 , 6103515625 v 10 , | |\n| 181330845696 1594323 v 10 | | (E . 29) |\n| 205209804800 , v 10 | 205209804800 , v 10 | (E . 30) |\n| 5983917907968 , | 5983917907968 , | (E . 31) |", 'Appendix F': '4PN luminosity in terms of the orbital frequency \nHere we present the ( /lscript, m )-mode contributions to the gravitational wave luminosity for a circular orbit on the equatorial plane around a Kerr black hole. In stead of v ≡ ( M/r 0 ) 1 / 2 , the formulas are expressed in terms of the parameter x ≡ ( MΩ ϕ ) 1 / 3 , where Ω ϕ is the orbital angular frequency, which is more relevant in the actual analysis of observed gravitational wave signals. We express the partial mode contributions as \n〈 dE dt 〉 ≡ 16 5 ( µ M ) 2 x 10 ∑ /lscript,m η /lscript,m . (F . 1) \nSince η /lscript,m = η /lscript, -m , we show only the modes m> 0 below. \nη 2 , 2 = 1 -107 x 2 21 + ( 4 π -8 q 3 ) x 3 + ( 4784 1323 +2 q 2 ) x 4 + ( -428 π 21 + 52 q 27 ) x 5 + ( 99210071 1091475 -1712 γ 105 + 16 π 2 3 -32 π q 3 -1817 q 2 567 \n-3424 ln 2 105 -1712 ln x 105 ) x 6 + ( 19136 π 1323 + 364856 q 11907 +8 π q 2 -8 q 3 3 ) x 7 + ( -27956920577 81265275 + 183184 γ 2205 -1712 π 2 63 + 208 π q 27 + 105022 q 2 9261 + q 4 + 366368 ln 2 2205 + 183184 ln x 2205 ) x 8 . (F . 2) η 2 , 1 = x 2 36 -q x 3 12 + ( -17 504 + q 2 16 ) x 4 + ( π 18 + 215 q 9072 ) x 5 + ( -2215 254016 -π q 6 + 313 q 2 1512 ) x 6 + ( -17 π 252 -18127 q 190512 + π q 2 8 -7 q 3 24 ) x 7 + ( 15707221 26195400 -107 γ 945 + π 2 27 + 215 π q 4536 + 44299 q 2 95256 + q 4 16 -107 ln 2 945 -107 ln x 945 ) x 8 . (F . 3) η 3 , 3 = 1215 x 2 896 -1215 x 4 112 + ( 3645 π 448 -1215 q 224 ) x 5 + ( 243729 9856 + 3645 q 2 896 ) x 6 + ( -3645 π 56 + 41229 q 1792 ) x 7 + ( 25037019729 125565440 -47385 γ 1568 + 3645 π 2 224 -3645 π q 112 -236925 q 2 14336 -47385 ln 2 1568 -47385 ln 3 1568 -47385 ln x 1568 ) x 8 . (F . 4) η 3 , 2 = 5 x 4 63 -40 q x 5 189 + ( -193 567 + 80 q 2 567 ) x 6 + ( 20 π 63 + 982 q 1701 ) x 7 + ( 86111 280665 -160 π q 189 + 80 q 2 189 ) x 8 . (F . 5) η 3 , 1 = x 2 8064 -x 4 1512 + ( π 4032 -25 q 18144 ) x 5 + ( 437 266112 + 17 q 2 24192 ) x 6 + ( -π 756 + 2257 q 435456 ) x 7 + ( -1137077 50854003200 -13 γ 42336 + π 2 6048 \n-25 π q 9072 + 12863 q 2 3483648 -13 ln 2 42336 -13 ln x 42336 ) x 8 . (F . 6) \nη 4 , 4 = \n+ ( 560069632 6243237 + 5120 q 2 567 ) x 8 . (F . 7) \n1280 x 4 567 \n-151808 x 6 6237 \n+ ( 10240 π 567 \n-20480 q 1701 ) x 7 \nη 4 , 3 = 729 x 6 4480 -729 q x 7 1792 + ( -28431 24640 + 3645 q 2 14336 ) x 8 . (F . 8) \nη 4 , 2 = \n+ ( 7199152 218513295 + 200 q 2 27783 ) x 8 . (F . 9) \n5 x 4 3969 -437 x 6 43659 \n+ ( 20 π 3969 -170 q 11907 ) x 7 \nη 4 , 1 = x 6 282240 -q x 7 112896 + ( -101 4656960 + 5 q 2 903168 ) x 8 . (F . 10) \nη 5 , 5 = 9765625 x 6 2433024 -2568359375 x 8 47443968 . (F . 11) \nη 5 , 4 = 4096 x 8 13365 . (F . 12) \nη 5 , 3 = 2187 x 6 450560 -150903 x 8 2928640 . (F . 13) \nη 5 , 2 = 4 x 8 40095 . (F . 14) \nη 5 , 1 = x 6 127733760 -179 x 8 2490808320 . (F . 15) \nη 6 , 6 = 26244 x 8 3575 . (F . 16) \nη 6 , 4 = 131072 x 8 9555975 . (F . 17) \nη 6 , 2 = 4 x 8 5733585 . (F . 18)', 'Appendix G': '4PN luminosity for a slightly eccentric orbit in the Schwarzschild case \nWe define the partial mode contribution η /lscript,m,n as \n〈 dE dt 〉 = 1 2 ( dE dt ) N ∑ /lscript,m,n η /lscript,m,n , (G . 1) \nwhere ( dE/dt ) N is the Newtonian quadrupole luminosity, Eq. (4 . 19), and η /lscript, -m, -n = η /lscript,m,n because of the symmetry in the Teukolsky equation. Up to O ( v 8 ), η /lscript, 1 , -1 do not appear irrespective of /lscript , and other modes are as follows. \nFor /lscript = 2, \nη 2 , 2 , 0 = 1 -107 v 2 21 +4 π v 3 + 4784 v 4 1323 -428 π v 5 21 + 99210071 v 6 1091475 -1712 γ v 6 105 + 16 π 2 v 6 3 + 19136 π v 7 1323 -27956920577 v 8 81265275 + 183184 γ v 8 2205 -1712 π 2 v 8 63 -3424 v 6 ln(2) 105 + 366368 v 8 ln(2) 2205 -1712 v 6 ln( v ) 105 + 183184 v 8 ln( v ) 2205 + e 2 ( -10 + 932 v 2 21 -46 π v 3 -14270 v 4 147 + 4748 π v 5 21 -516582901 v 6 363825 + 22256 γ v 6 105 -208 π 2 v 6 3 -189502 π v 7 441 + 405725734982 v 8 127702575 -352672 γ v 8 315 + 3296 π 2 v 8 9 + 44512 v 6 ln(2) 105 -705344 v 8 ln(2) 315 + 22256 v 6 ln( v ) 105 -352672 v 8 ln( v ) 315 ) , (G . 2) \nη 2 , 1 , 0 = \nv 2 36 -17 v 4 504 + π v 5 18 -2215 v 6 254016 -17 π v 7 252 + 15707221 v 8 26195400 -107 γ v 8 945 + π 2 v 8 27 -107 v 8 ln(2) 945 -107 v 8 ln( v ) 945 + e 2 ( -2 v 2 9 + 93 v 4 112 -19 π v 5 36 + 60667 v 6 84672 + 169 π v 7 84 \n+ 1177 γ v 8 945 \n-11 π 2 v 8 27 \n+ 169 π v 7 84 -1877507981 v 8 419126400 + 1177 v 8 ln(2) 945 + 1177 v 8 ln( v ) 945 ) , \n(G . 3) \nη 2 , 2 , 1 = e 2 ( 729 64 -3645 v 2 64 + 2187 π v 3 32 + 24057 v 4 256 -6561 π v 5 16 + 9067629321 v 6 3449600 -234009 γ v 6 560 + 2187 π 2 v 6 16 + 102789 π v 7 128 -545827954239 v 8 44844800 + 234009 γ v 8 80 -15309 π 2 v 8 16 -234009 v 6 ln(2) 560 + 234009 v 8 ln(2) 80 -234009 v 6 ln(3) 560 + 234009 v 8 ln(3) 80 -234009 v 6 ln( v ) 560 + 234009 v 8 ln( v ) 80 ) , (G . 4) \nη 2 , 2 , -1 = e 2 ( 9 64 + 1041 v 2 448 + 9 π v 3 32 + 2224681 v 4 112896 + 615 π v 5 112 + 11918100443 v 6 93139200 -321 γ v 6 560 + 3 π 2 v 6 16 + 3083119 π v 7 56448 + 181180743580847 v 8 228843014400 -50611 γ v 8 3920 + 473 π 2 v 8 112 -321 v 6 ln(2) 560 -50611 v 8 ln(2) 3920 \n) ) \n-321 v 6 ln( v ) 560 -50611 v 8 ln( v 3920 , (G . 5) η 2 , 1 , 1 = e 2 ( 4 v 2 9 -172 v 4 63 + 16 π v 5 9 + 10300 v 6 1323 -856 π v 7 63 + 145520063 v 8 3274425 -6848 γ v 8 945 + 64 π 2 v 8 27 -13696 v 8 ln(2) 945 -6848 v 8 ln( v ) 945 ) , (G . 6) η 2 , 0 , 1 = e 2 ( 1 96 -145 v 2 672 + π v 3 48 + 282521 v 4 169344 -83 π v 5 168 -776764633 v 6 139708800 -107 γ v 6 2520 + π 2 v 6 72 + 384203 π v 7 84672 + 6362456713 v 8 3467318400 + 20009 γ v 8 17640 -187 π 2 v 8 504 -107 v 6 ln(2) 2520 + 20009 v 8 ln(2) 17640 -107 v 6 ln( v ) 2520 + 20009 v 8 ln( v ) 17640 ) . (G . 7) \nFor /lscript = 3, \nη 3 , 3 , 0 = 1215 v 2 896 -1215 v 4 112 + 3645 π v 5 448 + 243729 v 6 9856 -3645 π v 7 56 + 25037019729 v 8 125565440 -47385 γ v 8 1568 + 3645 π 2 v 8 224 -47385 v 8 ln(2) 1568 -47385 v 8 ln(3) 1568 -47385 v 8 ln( v ) 1568 + e 2 ( -10935 v 2 448 + 37665 v 4 256 -142155 π v 5 896 -4428189 v 6 9856 + 455625 π v 7 448 -81628707987 v 8 17937920 + 142155 γ v 8 224 -10935 π 2 v 8 32 + 142155 v 8 ln(2) 224 + 142155 v 8 ln(3) 224 + 142155 v 8 ln( v ) 224 ) , (G . 8) η 3 , 2 , 0 = 5 v 4 63 -193 v 6 567 + 20 π v 7 63 + 86111 v 8 280665 + e 2 ( -65 v 4 63 + 947 v 6 189 -290 π v 7 63 -442003 v 8 40095 ) , (G . 9) η 3 , 1 , 0 = v 2 8064 -v 4 1512 + π v 5 4032 + 437 v 6 266112 -π v 7 756 -1137077 v 8 50854003200 -13 γ v 8 42336 + π 2 v 8 6048 -13 v 8 ln(2) 42336 -13 v 8 ln( v ) 42336 + e 2 ( -v 2 4032 + 65 v 4 16128 -π v 5 1152 -11063 v 6 798336 + 5 π v 7 448 + 614545391 v 8 30512401920 + 65 γ v 8 42336 -5 π 2 v 8 6048 + 65 v 8 ln(2) 42336 + 65 v 8 ln( v ) 42336 ) , (G . 10) η 3 , 3 , 1 = e 2 ( 640 v 2 21 -46720 v 4 189 + 5120 π v 5 21 + 2135648 v 6 2673 -408320 π v 7 189 \nη 3 , 3 , -1 = e 2 ( 15 v 2 14 \n+ 485145507664 v 8 59594535 -532480 γ v 8 441 + 40960 π 2 v 8 63 -532480 v 8 ln(2) 441 -532480 v 8 ln(4) 441 -532480 v 8 ln( v ) 441 ) , (G . 11) \n+ 1055 v 4 126 \n+ 30 π v 5 7 \n+ 1530967 v 6 37422 \n+ 2515 π v 7 63 \nη 3 , 2 , 1 = e 2 ( 3645 v 4 1792 -13851 v 6 896 + 10935 π v 7 896 + 2808837 v 8 49280 ) , (G . 13) \n+ 5409381833 v 8 19864845 -520 γ v 8 49 + 40 π 2 v 8 7 -1040 v 8 ln(2) 49 -520 v 8 ln( v ) 49 ) , (G . 12) \nη 3 , 2 , -1 = e 2 ( 5 v 4 1792 + 1609 v 6 24192 + 5 π v 7 896 + 4153669 v 8 5132160 ) , (G . 14) \nη 3 , 0 , 1 = e 2 ( v 4 2688 -55 v 6 6048 + π v 7 1344 + 203417 v 8 2395008 ) . (G . 16) \nη 3 , 1 , 1 = e 2 ( v 2 126 -23 v 4 126 + 2 π v 5 63 + 6437 v 6 4158 -7 π v 7 9 -1688449328 v 8 297972675 -104 γ v 8 1323 + 8 π 2 v 8 189 -208 v 8 ln(2) 1323 -104 v 8 ln( v ) 1323 ) , (G . 15) \nFor /lscript = 4, \nη 4 , 4 , 0 = 1280 v 4 567 -151808 v 6 6237 + 10240 π v 7 567 + 560069632 v 8 6243237 + e 2 ( -37120 v 4 567 + 3284224 v 6 6237 -312320 π v 7 567 -60497401856 v 8 31216185 ) , (G . 17) \nη 4 , 3 , 0 = 729 v 6 4480 -28431 v 8 24640 + e 2 ( -2187 v 6 640 + 2062341 v 8 98560 ) , (G . 18) \nη 4 , 2 , 0 = \n+ e 2 ( -25 v 4 3969 + 2743 v 6 43659 -130 π v 7 3969 -27410 v 8 77077 ) , (G . 19) \n5 v 4 3969 -437 v 6 43659 \n+ 20 π v 7 3969 \n+ 7199152 v 8 218513295 \nη 4 , 1 , 0 = v 6 282240 -101 v 8 4656960 + e 2 ( -v 6 56448 + 4951 v 8 18627840 ) , (G . 20) \nη 4 , 4 , 1 = e 2 ( 48828125 v 4 580608 -11767578125 v 6 12773376 + 244140625 π v 7 290304 + 3640732421875 v 8 874266624 ) , (G . 21) \nη 4 , 4 , -1 = e 2 ( 32805 v 4 7168 + 321489 v 6 22528 + 98415 π v 7 3584 + 251783118603 v 8 6314147840 ) , (G . 22) \nη 4 , 3 , -1 = e 2 ( 2 v 6 35 + 24 v 8 35 ) , (G . 24) \nη 4 , 3 , 1 = e 2 ( 2048 v 6 315 -216064 v 8 3465 ) , (G . 23) \nη 4 , 2 , 1 = e 2 ( 295245 v 8 495616 ) , (G . 25) \nη 4 , 1 , 1 = e 2 ( 2 v 6 2205 -28 v 8 1485 ) , (G . 27) \nη 4 , 2 , -1 = e 2 ( 5 v 4 254016 + 5501 v 6 11176704 + 5 π v 7 127008 + 5568858419 v 8 895030456320 ) , (G . 26) \nη 4 , 0 , 1 = e 2 ( v 4 225792 -113 v 6 709632 + π v 7 112896 + 1469592037 v 8 596686970880 ) . (G . 28) \nFor /lscript = 5, \nη 5 , 5 , 0 = 9765625 v 6 2433024 -2568359375 v 8 47443968 + e 2 ( -419921875 v 6 2433024 + 336376953125 v 8 189775872 ) , (G . 29) \nη 5 , 4 , 0 = 4096 v 8 13365 -131072 e 2 v 8 13365 , (G . 30) \nη 5 , 3 , 0 = 2187 v 6 450560 -150903 v 8 2928640 + e 2 ( -2187 v 6 40960 + 5736501 v 8 11714560 ) , (G . 31) \nη 5 , 2 , 0 = 4 v 8 40095 -32 e 2 v 8 40095 , (G . 32) \nη 5 , 5 , 1 = e 2 ( 19683 v 6 88 -17498187 v 8 5720 ) , (G . 34) \nη 5 , 1 , 0 = v 6 127733760 -179 v 8 2490808320 + e 2 ( v 6 25546752 -43 v 8 9963233280 ) , (G . 33) \nη 5 , 5 , -1 = e 2 ( 512 v 6 33 -54272 v 8 6435 ) , (G . 35) \nη 5 , 4 , -1 = e 2 ( 19683 v 8 56320 ) , (G . 37) \nη 5 , 4 , 1 = e 2 ( 48828125 v 8 2737152 ) , (G . 36) \nη 5 , 3 , 1 = e 2 ( 512 v 6 13365 -16384 v 8 173745 ) , (G . 38) \nη 5 , 2 , -1 = e 2 ( v 8 2566080 ) , (G . 40) \nη 5 , 3 , -1 = e 2 ( 11 v 6 9720 + 13 v 8 1080 ) , (G . 39) \nη 5 , 1 , 1 = e 2 ( v 6 13860 -1091 v 8 540540 ) , (G . 41) \nη 5 , 0 , 1 = e 2 ( v 8 9580032 ) . (G . 42) \nNote that up to O ( v 8 ), η 5 , 2 , 1 does not appear. \nFor /lscript = 6, \nη 6 , 6 , 0 = 26244 v 8 3575 -314928 e 2 v 8 715 , (G . 43) \nη 6 , 4 , 0 = 131072 v 8 9555975 -524288 e 2 v 8 1911195 , (G . 44) \nη 6 , 2 , 0 = 4 v 8 5733585 + 16 e 2 v 8 5733585 , (G . 45) \nη 6 , 6 , 1 = e 2 678223072849 v 8 1186099200 , \n(G . 46) \nη 6 , 6 , -1 = e 2 244140625 v 8 5271552 , (G . 47) \nη 6 , 4 , 1 = e 2 244140625 v 8 782825472 , (G . 48) \nη 6 , 4 , -1 = e 2 964467 v 8 80537600 , \n(G . 49) \nη 6 , 2 , 1 = e 2 6561 v 8 32215040 , (G . 50) \nη 6 , 2 , -1 = e 2 5 v 8 4696952832 , \n(G . 51) \nη 6 , 0 , 1 = e 2 v 8 2739889152 . (G . 52) \nNote that up to O ( v 8 ), η 6 , 5 ,k , η 6 , 3 ,k and η 6 , 1 ,k do not appear. \nNow, we define η (0) /lscript , η (2) /lscript , and η (2) z/lscript as \n〈 dE dt 〉 = ( dE dt ) N ∑ /lscript ( η (0) /lscript + η (2) /lscript e 2 ) , (G . 53) \n〈 dJ z dt 〉 = 1 Ω ( dE dt ) N ∑ /lscript ( η (0) /lscript + η (2) z/lscript e 2 ) . (G . 54) \nThen, \nη (0) 2 = 1 -1277 v 2 252 +4 π v 3 + 37915 v 4 10584 -2561 π v 5 126 + 76187 π v 7 5292 + v 6 ( 2116278473 23284800 -1712 γ 105 + 16 π 2 3 -3424 ln(2) 105 -1712 ln( v ) 105 ) + v 8 ( -2455920939443 7151344200 + 548803 γ 6615 -5129 π 2 189 + 219671 ln(2) 1323 + 548803 ln( v ) 6615 ) , (G . 55) η (2) 2 = 37 24 -2581 v 2 252 + 1087 π v 3 48 + 346561 v 4 21168 -29857 π v 5 168 + 35639309 π v 7 84672 + v 6 ( 93579660049 69854400 -65056 γ 315 + 608 π 2 9 + 1712 ln(2) 315 -234009 ln(3) 560 -65056 ln( v ) 315 ) + v 8 ( -3395552510663 416078208 + 4730363 γ 2646 -221045 π 2 378 + 2914573 ln(2) 4410 + 234009 ln(3) 80 + 4730363 ln( v ) 2646 ) , (G . 56) η (0) 3 = 1367 v 2 1008 -32567 v 4 3024 + 16403 π v 5 2016 + 152122 v 6 6237 -13991 π v 7 216 + v 8 ( 5712521850527 28605376800 -79963 γ 2646 + 6151 π 2 378 -79963 ln(2) 2646 -47385 ln(3) 1568 -79963 ln( v ) 2646 ) (G . 57) η (2) 3 = 1801 v 2 252 -78509 v 4 864 + 40083 π v 5 448 + 8163047 v 6 21384 -1894997 π v 7 1728 + v 8 ( 446664927141403 114421507200 -771979 γ 1323 + 59383 π 2 189 -87347 ln(2) 147 + 142155 ln(3) 224 -532480 ln(4) 441 -771979 ln( v ) 1323 ) , (G . 58) η (0) 4 = 8965 v 4 3969 -84479081 v 6 3492720 + 23900 π v 7 1323 + 51619996697 v 8 582702120 , (G . 59) η (2) 4 = 2946739 v 4 127008 -58555205 v 6 155232 + 107560723 π v 7 338688 + 5187619686371 v 8 2330808480 , (G . 60) η (0) 5 = 1002569 v 6 249480 -3145396841 v 8 58378320 , (G . 61) \nη (2) 5 = 2491525 v 6 37422 -150181214159 v 8 116756640 , (G . 62) \nη (0) 6 = 210843872 v 8 28667925 , (G . 63) \nη (2) 6 = 142651028551 v 8 802701900 , (G . 64) \nand \nη (2) z 2 = -5 8 + 137 v 2 24 + 49 π v 3 8 -235675 v 4 14112 -20437 π v 5 504 + 883609 π v 7 14112 + v 6 ( 303218627 470400 -19367 γ 210 + 181 π 2 6 + 20009 ln(2) 210 -78003 ln(3) 280 -19367 ln( v ) 210 ) + v 8 ( -1553872210987 488980800 + 3197053 γ 4410 -29879 π 2 126 -2649641 ln(2) 13230 + 234009 ln(3) 140 + 3197053 ln( v ) 4410 ) , (G . 65) η (2) z 3 = 67 v 2 32 -66497 v 4 2016 + 43193 π v 5 1008 + 2711543 v 6 18144 -2203487 π v 7 4032 + v 8 ( 321229428757 133358400 -586079 γ 1764 + 45083 π 2 252 -1842685 ln(2) 5292 + 1848015 ln(3) 3136 -133120 ln(4) 147 -586079 ln( v ) 1764 ) , (G . 66) η (2) z 4 = 478195 v 4 42336 -64132457 v 6 317520 + 5239355 π v 7 28224 + 670696042069 v 8 537878880 , (G . 67) η (2) z 5 = 1778041 v 6 45360 -300193429 v 8 374220 , (G . 68) η (2) z 6 = 113949013 e 2 v 8 980100 . (G . 69)', 'Appendix H': 'Asymptotic amplitudes and \nR up \nIn this Appendix, we show the asymptotic amplitudes B inc , B trans and C trans and the post-Newtonian expansion of R up which are used in section 12 to evaluate the black hole absorption rate to O ( v 13 ) relative to the quadrupole energy flux at infinity in section. \n(a) /lscript = 2 \nB inc = 1 ω 1 κ 4 /epsilon1 4 e 1 2 πi ( ν +3) e i/epsilon1κ e -i/epsilon1 ln /epsilon1 \n -2 \n 4 + 56 -144 15 γ ψ \n 15 + 15 iγ -25 12 mq -15 8 π -15 4 iψ (0) (3 + imq κ ) -15 4 i ln(2) + 125 16 i /epsilon1 1089 56 γ + 725 2352 κ 2 + 1089 112 ln(2) -15 2 γ 2 + 12625 21168 m 2 q 2 + 35 32 π 2 535 imq + 15 4 iγ π -125 32 iπ + 25 6 iγ mq + 25 24 mπq -15 2 γ ln(2) -2 (0) (3 + imq κ ) -15 4 ln(2) ψ (0) (3 + imq κ ) + 107 56 ln( /epsilon1 ) + 107 56 ln( κ ) -20573 960 -15 8 (ln(2)) 2 + 1089 112 ψ (0) (3 + imq κ ) -15 8 ( ψ (0) (3 + imq κ ) ) 2 -15 8 ψ (1) (3 + imq κ ) + 25 12 imq ln(2) + 25 12 imqψ (0) (3 + imq κ ) + 15 8 iπ ln(2) + 15 8 iπ ψ (0) (3 + imq κ ) -15 4 ψ (1) (3 + imq κ ) κ -1 /epsilon1 2 . (H . 1) B trans = ( ω /epsilon1κ ) 4 e i/epsilon1 + ln κ 1 + ( 5 6 iκ -5 18 mq ) /epsilon1 + ( 325 7938 m 2 q 2 + 5 18 -15 49 κ 2 -85 378 iκ mq ) /epsilon1 2 . (H . 2) C trans = ω 3 2 i/epsilon1 e -π 2 i ( ν -1) e i/epsilon1 ln /epsilon1 2 + ( -1 3 iκ -π + 1 9 mq ) /epsilon1 + 1 4 π 2 -1 189 iκ mq + 2 49 q 2 -1 3 imq -11 3969 m 2 q 2 -1 18 mπq + 1 6 iκ π -67 441 /epsilon1 2 . (H . 3) R up = -3 i z -3 + 3 2 iz + 1 2 z 2 -1 8 iz 3 + 31 40 z 4 + 43 80 iz 5 -117 560 z 6 -769 13440 iz 7 + /epsilon1 ( -3 2 i + mq ) 1 z 2 + ( -3 γ + κ +3 iπ -1 6 imq ) 1 z -9 4 i +3 iγ -iκ +3 π + 1 3 mq + ( -5 2 + 3 2 γ -1 2 κ -3 2 iπ -1 4 imq ) z + ( 85 48 i -1 2 iγ + 1 6 iκ -1 2 π -7 72 mq ) z 2 + ( -13 60 -1 8 γ + 1 24 κ + 1 8 iπ -269 720 imq ) z 3 \n+ ( -1559 2400 i + 1 40 iγ + 31 120 iκ -3 8 π + 49 180 mq + 4 5 i ln(2) + 4 5 i ln( z ) ) z 4 + /epsilon1 2 ( -3 4 i -3 28 iκ 2 + mq + 11 56 im 2 q 2 ) z -3 + ( -1 -3 2 γ + 1 2 κ + 1 14 κ 2 + 3 2 iπ -7 12 imq -iγ mq + 1 3 iκ mq -mπq -31 504 m 2 q 2 ) z -2 + ( 183 28 i -107 70 iγ + 3 2 iγ 2 -iγ κ -4 49 iκ 2 +3 γ π -κπ -7 4 iπ 2 -1 4 mq -1 6 γ mq + 19 252 κmq + 1 6 imπ q + 781 21168 im 2 q 2 -107 70 i ln(2) -107 70 i ln( z ) ) z -1 + 1791 280 -529 140 γ + 3 2 γ 2 + 3 4 κ -γ κ -11 392 κ 2 + 9 4 iπ -3 iγ π + iκ π -7 4 π 2 + 13 24 imq -1 3 iγ mq + 23 252 iκ mq -1 3 mπq -563 21168 m 2 q 2 -107 70 ln(2) -107 70 ln( z ) + ( -13751 3360 i + 457 140 iγ -3 4 iγ 2 -5 6 iκ + 1 2 iγ κ + 17 4704 iκ 2 + 5 2 π -3 2 γ π + 1 2 κπ + 7 8 iπ 2 + 2 3 mq -1 4 γ mq + 37 504 κmq + 1 4 imπ q + 751 28224 im 2 q 2 + 107 140 i ln(2) + 107 140 i ln( z ) ) z + /epsilon1 3 ( -3 8 i -9 56 iκ 2 + 3 4 mq + 15 112 κ 2 mq + 33 112 im 2 q 2 -3 112 m 3 q 3 ) z -4 + ( -1 -3 4 γ + 1 4 κ -1 7 κ 2 -3 28 γ κ 2 + 1 28 κ 3 + 3 4 iπ + 3 28 iκ 2 π -157 168 imq -iγ mq + 1 3 iκ mq -5 84 iκ 2 mq -mπq + 41 504 m 2 q 2 + 11 56 γ m 2 q 2 -11 168 κm 2 q 2 -11 56 im 2 π q 2 -23 1008 im 3 q 3 ) z -3 + ( -1 3 iκ mπ q + 31 504 iγ m 2 q 2 -1 72 iκ m 2 q 2 + iγ mπ q -1 2 γ 2 mq + 7 12 mπ 2 q + 31 504 m 2 π q 2 + 107 210 mq ln(2) + 107 210 mq ln( z ) + 2335 42336 im 2 q 2 -1 2 iγ κ + 1 3 γ κ mq + 7 12 imπ q -25 168 iκ q 2 -1 14 iγ κ 2 -239 14112 κ 2 mq + π + 1269 280 i + 3 2 γ π -1 2 κπ -1 14 κ 2 π + 25 504 mq 3 + 69 196 iκ 2 + 33 140 iγ -31 168 iκ -107 140 i ln(2) -107 140 i ln( z ) + 3 4 iγ 2 \n. \n(b) /lscript = 3 \nB inc = 1 ω 1 κ 5 /epsilon1 5 e 1 2 πi ( ν +3) e i/epsilon1κ e -i/epsilon1 ln /epsilon1 945 2 κ 3 κ + imq \n--+180 iψ (0) ( 3 κ + imq κ ) κ -60 ψ (0) ( 3 κ + imq κ ) mq -60 i /epsilon1 . (H . 5) \n-63 8 κ (3 κ + imq ) 2 90 κπ +30 imqπ +180 iκ ln(2) -60 mq ln(2) +45 κmq +15 im 2 q 2 +360 iκ γ 120 γ mq 591 iκ +197 mq \nC trans = ω 3 2 i/epsilon1 e -π 2 i ( ν -1) e i/epsilon1 ln /epsilon1 ( 2 + ( -π + 1 36 mq -2 3 iκ ) /epsilon1 ) . (H . 7) \n B trans = ( ω /epsilon1κ ) 4 e i/epsilon1 + ln κ ( 1 + ( -11 72 mq + 2 3 iκ ) /epsilon1 ) . (H . 6) \nR up = -45 i z 2 \n-30 z -1 + 15 2 i + 5 8 iz 2 \n+ /epsilon1 2 135 8 κmq -75 2 iκ 2 +135 iκ ( -1 2 + 1 2 κ ) -135 4 ( -1 2 + 1 2 κ ) mq + 15 8 im 2 q 2 -45 i ( -( -1 2 + 1 2 κ ) 2 +2 ( -1 + κ )( -1 2 + 1 2 κ )) z -4 . \n+ /epsilon1 ( 45 4 mq -45 iκ +90 i ( -1 2 + 1 2 κ )) z -3 + ( 45 iπ +30 + 15 2 κ -45 γ -15 8 imq ) z -2 + ( 30 π -45 i + 35 24 mq +30 iγ -5 iκ ) z -1 \n-7 8 iπ 2 -1 8 iκ 3 -31 420 γ mq + 103 504 κmq -421 210 mq -239 127008 m 3 q 3 ) z -2 + /epsilon1 4 ( ( -3 16 i -9 56 iκ 2 -1 112 iκ 4 + 1 2 mq + 15 56 κ 2 mq + 33 112 im 2 q 2 + 11 252 iκ 2 m 2 q 2 -3 56 m 3 q 3 -23 8064 im 4 q 4 ) z -5 ) (H . 4) \n(c) /lscript = 4 \n(H . 8) \nB inc = 1 ω 1 κ 6 /epsilon1 6 e 1 2 πi ( ν +3) e i/epsilon1κ e -i/epsilon1 ln /epsilon1 { 79380 κ 2 (3 κ + imq ) (4 κ + imq ) } . (H . 9) \nC trans = ω 3 2 i/epsilon1 e -π 2 i ( ν -1) e i/epsilon1 ln /epsilon1 (2 + O ( /epsilon1 )) . (H . 11) \nB trans = ( ω /epsilon1κ ) 4 e i/epsilon1 + ln κ (1 + O ( /epsilon1 )) . (H . 10) \nR up = -630 i z 3 . (H . 12)', 'Appendix I': 'Energy absorption by a Kerr black hole \nHere we give the ( /lscript, m )-components of the energy absorption rate to O ( v 8 ) beyond the lowest order for the Kerr black hole, that is O ( v 13 ) relative to the quadrupole luminosity at infinity. \nη H 2 , 2 = -1 4 q -3 4 q 3 + v 2 ( -3 4 q -9 4 q 3 ) + ( 27 4 q 2 +2 qB 2 + 15 4 q 4 +6 q 3 B 2 + 13 2 κq 2 +3 q 4 κ + 1 2 κ + 1 2 ) v 3 + ( -199 42 q -593 42 q 3 + 2 7 q 5 ) v 4 + ( 721 36 q 2 +6 qB 2 + 127 12 q 4 +18 q 3 B 2 + 39 2 κq 2 +9 q 4 κ + 3 2 κ + 3 2 ) v 5 + ( -607076 11025 q -4 B 2 + 428 105 γ q + 2 3 π 2 q + 428 105 q ln 2 -4 qC 2 -12 q 3 C 2 -36 q 4 B 2 -56 q 2 B 2 + 428 35 q 3 γ + 428 35 q 3 ln 2 +2 q 3 π 2 + 428 105 q ln κ + 428 105 qA 2 + 428 35 q 3 ln κ +6 q 7 κ + 428 35 q 3 A 2 -8 qB 2 2 -24 q 3 B 2 2 + 856 105 q ln v + 856 35 q 3 ln v -4 B 2 κ -32 q 3 κ -31 q κ +57 q 5 κ -48 q 2 B 2 κ +28 q 4 B 2 κ -24 q 3 C 2 κ -8 qC 2 κ +24 q 6 B 2 κ -11883052 99225 q 3 + 548 27 q 5 ) v 6 + ( 16747 126 q 2 + 15893 189 q 4 -155 126 q 6 +123 κq 2 + 1142 21 q 4 κ -8 7 κq 6 + 199 21 κ + 199 21 + 796 21 qB 2 + 2372 21 q 3 B 2 -16 7 q 5 B 2 ) v 7 \n+ ( -761349 3920 q -12 B 2 + 3076 105 γ q +2 π 2 q + 4868 105 q ln 2 -12 qC 2 -36 q 3 C 2 -308 3 q 4 B 2 -1496 9 q 2 B 2 + 3076 35 q 3 γ + 4868 35 q 3 ln 2 + 6 q 3 π 2 + 428 35 q ln κ + 428 35 qA 2 + 1284 35 q 3 ln κ + 46 3 q 7 κ + 1284 35 q 3 A 2 -24 qB 2 2 -72 q 3 B 2 2 + 872 21 q ln v + 872 7 q 3 ln( v ) -12 B 2 κ -272 3 q 3 κ -833 9 q κ + 1511 9 q 5 κ -144 q 2 B 2 κ +84 q 4 B 2 κ -72 q 3 C 2 κ -24 qC 2 κ +72 q 6 B 2 κ -140529967 317520 q 3 + 46465 756 q 5 -191 588 q 7 ) v 8 . (I . 1) η H 2 , 1 = v 2 ( 3 16 q 3 -1 4 q ) + ( -1 4 q 4 + 1 3 q 2 ) v 3 + ( 1 12 q 5 + 8 9 q 3 -4 3 q ) v 4 + ( -137 48 q 4 -3 4 q 3 B 1 + 265 72 q 2 + qB 1 -3 4 q 4 κ + 1 2 κ + 7 8 κq 2 + 1 2 ) v 5 + ( -382 63 q + 865 756 q 3 + 2321 1008 q 5 -2 3 κq -7 6 κq 3 + q 5 κ + q 4 B 1 -4 3 q 2 B 1 ) v 6 + ( 8 3 + 4643 252 q 2 -32 9 q 3 B 1 -1405 3024 q 6 -65 18 q 4 κ -123269 9072 q 4 + 44 9 κq 2 -1 3 q 5 B 1 -1 3 κq 6 + 16 3 qB 1 + 8 3 κ ) v 7 + ( -6292409 176400 q -2 B 1 + 107 105 γ q + 1 6 π 2 q + 107 105 q ln 2 -107 140 q 3 γ -107 140 q 3 ln 2 -1 8 q 3 π 2 + 107 105 q ln κ -107 140 q 3 ln κ -55 6 q 7 κ + 214 105 q ln v -107 70 q 3 ln v + 65 12 q 3 κ -245 18 q κ + 625 36 q 5 κ + 73 6 q 4 B 1 -qC 1 -283 18 q 2 B 1 + 3 2 q 3 B 1 2 + 107 105 qA 1 -107 140 q 3 A 1 -3 q 6 B 1 κ + 13 2 q 4 B 1 κ -3 2 q 2 B 1 κ -2 qC 1 κ + 3 2 q 3 C 1 κ -2 qB 1 2 + 3 4 q 3 C 1 -2 B 1 κ + 381643 6350400 q 3 + 6439 378 q 5 -11 168 q 7 ) v 8 . (I . 2) η H 3 , 3 = ( -75 112 q 5 -555 896 q 3 -15 224 q ) v 4 + ( -45 112 q -1665 448 q 3 -225 56 q 5 ) v 6 + ( 225 28 q 5 B 3 + 375 112 q 6 + 15 112 + 1905 448 κq 2 + 2055 224 q 4 κ + 15 112 κ + 1665 224 q 3 B 3 + 45 56 qB 3 + 10995 896 q 4 + 2055 448 q 2 ) v 7 \n+ ( -17697 896 q 3 -18315 896 q 5 + 125 112 q 7 -481 224 q ) v 8 . (I . 3) \nη H 3 , 1 = ( -1 224 q + 59 8064 q 3 -1 336 q 5 ) v 4 + ( 767 12096 q 3 -13 504 q 5 -13 336 q ) v 6 + ( 1 84 q 5 B 1 + 1 56 qB 1 + 31 4032 κq 2 -55 2016 q 4 κ + 1 84 κq 6 + 109 3024 q 6 -6395 72576 q 4 -59 2016 q 3 B 1 + 1 112 + 185 4032 q 2 + 1 112 κ ) v 7 + ( -1 336 q 7 -431 2016 q + 25105 72576 q 3 -3271 24192 q 5 ) v 8 (I . 5) \nη H 3 , 2 = ( 25 189 q 5 -5 63 q -110 567 q 3 ) v 6 + ( 5 42 q 2 + 55 189 q 4 -25 126 q 6 ) v 7 + ( 25 336 q 7 -40 63 q -14485 9072 q 3 + 205 216 q 5 ) v 8 (I . 4) \nη H 4 , 4 = -5 2268 v 8 q ( 9 + 7 q 2 )( 3 q 2 +1 )( 15 q 2 +1 ) . (I . 6) \nη H 4 , 2 = -5 63504 v 8 q ( 5 q 2 -9 )( 3 q 2 -4 )( 3 q 2 +1 ) . (I . 7) \nAppendix J ( dE/dt ) H in terms of the orbital frequency \nIn this Appendix, we describe the absorption rate 〈 dE/dt 〉 H by a Kerr black hole in terms of x ≡ ( MΩ ϕ ) 1 / 3 . Using the relation, \nwe have \nv = x ( 1 + 1 3 qx 3 + 2 9 q 2 x 6 + O ( x 9 ) ) , (J . 1) \n( dE dt ) H = 32 5 ( µ M ) 2 x 10 x 5 -1 4 q -3 4 q 3 + ( -q -33 16 q 3 ) x 2 + ( 2 qB 2 + 1 2 + 13 2 κq 2 + 35 6 q 2 -1 4 q 4 + 1 2 κ +3 q 4 κ +6 q 3 B 2 ) x 3 + ( -43 7 q -17 56 q 5 -4651 336 q 3 ) x 4 + ( 433 24 q 2 -95 24 q 4 +2 -3 4 q 3 B 1 +2 κ + 33 4 q 4 κ +6 qB 2 \n```\n+18 q 3 B 2 + 163 8 κq 2 + qB 1 ) x 5 + ( -2586329 44100 q -4 B 2 -1640747 19600 q 3 +19 q 5 κ + 428 105 γ q + 2 3 π 2 q + 428 105 q ln(2) -4 qC 2 -12 q 3 C 2 -44 q 2 B 2 + 428 35 q 3 γ + 428 35 q 3 ln(2) +2 q 3 π 2 + 428 105 q ln( κ ) + 428 105 qA 2 + 428 35 q 3 ln( κ ) + 6 q 7 κ + 428 35 q 3 A 2 -8 qB 2 2 -24 q 3 B 2 2 -4 B 2 κ -32 q 3 κ -31 q κ +57 q 5 κ + q 4 B 1 -4 3 q 2 B 1 + 7 3 κq + 227 6 κq 3 + 455 16 q 5 -48 q 2 B 2 κ +28 q 4 B 2 κ -24 q 3 C 2 κ -8 qC 2 κ +24 q 6 B 2 κ + 856 105 q ln( x ) + 856 35 q 3 ln( x ) ) x 6 + ( 19687 168 q 2 -145 336 q 6 -4729 1008 q 4 + 899 168 qB 1 -41 28 κq 6 + 45 56 qB 3 -803 224 q 3 B 1 + 1665 224 q 3 B 3 + 86 7 + 719 12 q 4 κ + 796 21 qB 2 + 86 7 κ -16 7 q 5 B 2 + 225 28 q 5 B 3 -9 28 q 5 B 1 + 22201 168 κq 2 + 2372 21 q 3 B 2 ) x 7 ( -19366807 88200 q -12 B 2 -2 B 1 -2062220497 6350400 q 3 -qC 1 +55 q 5 κ + 1061 35 γ q + 13 6 π 2 q + 995 21 q ln(2) -12 qC 2 -36 q 3 C 2 + 52 3 q 4 B 2 -1136 9 q 2 B 2 + 12197 140 q 3 γ + 3873 28 q 3 ln(2) + 47 8 q 3 π 2 + 1391 105 q ln( κ ) + 428 35 qA 2 + 5029 140 q 3 ln( κ ) + 37 6 q 7 κ + 1284 35 q 3 A 2 -24 qB 2 2 -72 q 3 B 2 2 -12 B 2 κ -341 4 q 3 κ -637 6 q κ + 741 4 q 5 κ + 43 6 q 4 B 1 -163 18 q 2 B 1 + 3 2 q 3 B 1 2 + 107 105 qA 1 -107 140 q 3 A 1 -2 qB 1 2 + 3 4 q 3 C 1 -2 B 1 κ + 40 3 κq + 815 6 κq 3 + 1265 18 q 5 + 25 252 q 7 -144 q 2 B 2 κ +84 q 4 B 2 κ -72 q 3 C 2 κ -24 qC 2 κ +72 q 6 B 2 κ -3 q 6 B 1 κ + 13 2 q 4 B 1 κ -3 2 q 2 B 1 κ -2 qC 1 κ + 3 2 q 3 C 1 κ\n``` \n+ 4574 105 q ln( x ) + 8613 70 q 3 ln( x ) ) x 8 . 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2011ApJ...742..107B

The Relation between Black Hole Mass and Host Spheroid Stellar Mass Out to z ~ 2

2011-01-01

['accretion', 'accretion disks', 'black hole physics', 'galaxies active', 'galaxies evolution', 'galaxies quasars', '-']

[]

We combine Hubble Space Telescope images from the Great Observatories Origins Deep Survey with archival Very Large Telescope and Keck spectra of a sample of 11 X-ray-selected broad-line active galactic nuclei in the redshift range 1 &lt; z &lt; 2 to study the black-hole-mass-stellar-mass relation out to a look-back time of 10 Gyr. Stellar masses of the spheroidal component (M <SUB>sph, sstarf</SUB>) are derived from multi-filter surface photometry. Black hole masses (M <SUB>BH</SUB>) are estimated from the width of the broad Mg II emission line and the 3000 Å nuclear luminosity. Comparing with a uniformly measured local sample and taking into account selection effects, we find evolution in the form M <SUB>BH</SUB>/M <SUB>sph, sstarf</SUB>vprop(1 + z)<SUP>1.96 ± 0.55</SUP>, in agreement with our earlier studies based on spheroid luminosity. However, this result is more accurate because it does not require a correction for luminosity evolution and therefore avoids the related and dominant systematic uncertainty. We also measure total stellar masses (M <SUB>host, sstarf</SUB>). Combining our sample with data from the literature, we find M <SUB>BH</SUB>/M <SUB>host, sstarf</SUB>vprop(1 + z)<SUP>1.15 ± 0.15</SUP>, consistent with the hypothesis that black holes (in the range M <SUB>BH</SUB> ~ 10<SUP>8-9</SUP> M <SUB>⊙</SUB>) pre-date the formation of their host galaxies. Roughly, one-third of our objects reside in spiral galaxies; none of the host galaxies reveal signs of interaction or major merger activity. Combined with the slower evolution in host stellar masses compared to spheroid stellar masses, our results indicate that secular evolution or minor mergers play a non-negligible role in growing both BHs and spheroids.

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https://arxiv.org/pdf/1102.1975.pdf

{'THE RELATION BETWEEN BLACK HOLE MASS AND HOST SPHEROID STELLAR MASS OUT TO Z ∼ 2': 'Vardha N. Bennert 1,2 , Matthew W. Auger 1 , Tommaso Treu 1,3 , Jong-Hak Woo 4 , Matthew A. Malkan 5 Draft version November 8, 2018', 'ABSTRACT': 'We combine Hubble Space Telescope images from the Great Observatories Origins Deep Survey with archival Very Large Telescope and Keck spectra of a sample of 11 X-ray selected broad-line active galactic nuclei in the redshift range 1 < z < 2 to study the black hole mass - stellar mass relation out to a lookback time of 10 Gyrs. Stellar masses of the spheroidal component ( M sph ,/star ) are derived from multi-filter surface photometry. Black hole masses ( M BH ) are estimated from the width of the broad MgII emission line and the 3000 ˚ A nuclear luminosity. Comparing with a uniformly measured local sample and taking into account selection effects, we find evolution in the form M BH / M sph ,/star ∝ (1+ z ) 1 . 96 ± 0 . 55 , in agreement with our earlier studies based on spheroid luminosity. However, this result is more accurate because it does not require a correction for luminosity evolution and therefore avoids the related and dominant systematic uncertainty. We also measure total stellar masses ( M host ,/star ). Combining our sample with data from the literature, we find M BH / M host ,/star ∝ (1+ z ) 1 . 15 ± 0 . 15 , consistent with the hypothesis that black holes (in the range M BH ∼ 10 8 -9 M /circledot ) predate the formation of their host galaxies. Roughly one third of our objects reside in spiral galaxies; none of the host galaxies reveal signs of interaction or major merger activity. Combined with the slower evolution in host stellar masses compared to spheroid stellar masses, our results indicate that secular evolution or minor mergers play a non-negligible role in growing both BHs and spheroids. \nSubject headings: accretion, accretion disks - black hole physics - galaxies: active - galaxies: evolution - quasars: general', '1. INTRODUCTION': "Active Galactic Nuclei (AGNs) are thought to represent an integral phase in the formation and evolution of galaxies during which the central supermassive black hole (BH) is growing through accretion. The empirical relations between BH mass ( M BH ) and the properties of the host galaxy (e.g., Ferrarese & Merritt 2000; Gebhardt et al. 2000; Marconi & Hunt 2003; Haring & Rix 2004) have been explained by a combination of AGN feedback (e.g., Volonteri et al. 2003; Ciotti & Ostriker 2007; Di Matteo et al. 2008; Hopkins et al. 2009) and hierarchical assembly of M BH and stellar mass through galaxy merging (e.g., Peng 2007; Jahnke et al. 2011). \nThe great interest in the origin of the scaling relations is reflected in the flood of observational studies, focusing on their cosmic evolution (e.g., Treu et al. 2004; Walter et al. 2004; Shields et al. 2006; McLure et al. 2006; Peng et al. 2006; Woo et al. 2006; Salviander et al. 2007; Treu et al. 2007; Woo et al. 2008; Jahnke et al. 2009; Bennert et al. 2010; Decarli et al. 2010; Merloni et al. 2010), with the majority pointing to a scenario in which BH growth precedes bulge assembly. \nHowever, many high-redshift studies to date are based on monochromatic Hubble Space Telescope (HST) imag- \n- 1 Department of Physics, University of California, Santa Barbara, CA 93106; [email protected]; [email protected]\n- 3 Packard Fellow\n- 2 Current address: Physics Department, California Polytechnic State University, San Luis Obispo, CA 93407; [email protected] 3\n- 4 Astronomy Program, Department of Physics and Astronomy, Seoul National University, Korea; [email protected]\n- 5 Department of Physics and Astronomy, University of California, Los Angeles, CA 90095; [email protected] \ning, determining only the luminosity of the host spheroid and not its stellar mass. This is acceptable at z ∼ 0 . 5 (e.g., Bennert et al. 2010), where the stellar populations of bulges are fairly well known and their luminosities can be passively evolved to zero redshift with uncertainties smaller than other sources of error. In contrast, at z > 1, the stellar populations of bulges are an uncharted territory, particularly for AGN hosts which are believed to be connected with major mergers and may have undergone recent episodes of star formation (e.g., Kauffmann et al. 2003; S'anchez et al. 2004). The uncertainty on the conversion from observed luminosity to equivalent z = 0 luminosity can be comparable to the evolutionary signal (e.g., Peng et al. 2006). \nAn exception is the study by Merloni et al. (2010) who estimate total stellar masses ( M host ,/star ) and AGN luminosities by fitting spectral-energy distribution (SED) models to multi-band data from the rest-frame ultraviolet to the rest-frame mid-infrared for a sample of 89 broad-line AGN (BLAGN) hosts at 1 < z < 2 . 2. Estimating M BH from broad MgII emission, they find that black holes of a given mass reside in less massive hosts at higher redshift with a modest evolutionary slope. However, Merloni et al. (2010) are unable to distinguish between M host ,/star and the stellar mass of the central bulge component of the host ( M sph ,/star ). Such a difference may be important when studying the evolution of the scaling relations: there are indications that the relations between M BH and total host-galaxy luminosity (Bennert et al. 2010) and stellar mass (Jahnke et al. 2009) may not be evolving, or at least not as rapidly as the relations between M BH and spheroid properties. \nIn this paper, we study the cosmic evolution of the M BH -M sph ,/star and M BH -M host ,/star relations for a sample of \n11 BLAGNs (1 < z < 2; lookback time: 8-10 Gyrs) selected from the Great Observatories Origins Deep Survey (GOODS) fields, taking into account selection effects. M sph ,/star and M host ,/star are derived from the deep multi-filter HST images. M BH is estimated using the width of the broad MgII emission line, measured from existing spectra and the 3000 ˚ A nuclear luminosity. We use a local comparison sample of Seyfert-1 galaxies (Bennert et al. 2011) for which all relevant quantities were derived following the same procedures adopted for the distant sample to minimize potential systematic bias. Our strategy allows us to address two major limitations of previous studies: eliminate uncertainties due to luminosity evolution and determine the evolution of the spheroidal component of the host. \nThroughout the paper, we assume a Hubble constant of H 0 = 70kms -1 Mpc -1 , Ω Λ = 0.7 and Ω M = 0.3. Note that all magnitudes are AB.", '2.1. Sample Selection': "The high-redshift sample consists of AGNs in GOODSN (Treister et al. 2009) and GOODS-S (Trouille et al. 2008; Silverman et al. 2010), selected based on their X-ray emission using the Chandra Deep Field North and South (CDF-N/CDF-S) survey and spectroscopically confirmed to be BLAGNs. We select all 11 objects within 1 < z < 2 for which archival Very Large Telescope (VLT) and Keck spectra covering the broad MgII line exist (Table 1). \nBy design, all objects have deep HST/Advanced Camera for Surveys (ACS) images in four different broad-band filters (B=F435W, V=F606W, i =F775W, z =F850LP) (Giavalisco et al. 2004). Color images are shown in Figure 1. The total exposure times range between 5,000 and 25,000 sec, depending on the filter and the image region. The reduced data are taken from the v2.0 data release. 6 The spatial resolution is approximately 0 . '' 1 full-width-at-half-maximum (FWHM), which at z = 1 . 3 (our average redshift) corresponds to 0.84 kpc; thus, our data have higher spatial resolution than Sloan Digital Sky Survey (SDSS) images at z = 0 . 05 (1 . '' 4 = 1.37 kpc). Overall, the AGN host galaxies look like typical ellipticals or spirals, without any signs of merger activity. \nOur local comparison sample consists of 25 Seyfert1 galaxies selected from SDSS (0 . 02 < z < 0 . 1; M BH > 10 7 M /circledot ) for which all relevant quantities were derived following the same procedures adopted for the distant sample to minimize potential systematic bias (Bennert et al. 2011).", '2.2. Surface Photometry': "We perform two-dimensional surface photometry using the code 'Surface Photometry and Structural Modeling of Imaging Data' (SPASMOID) developed by one of us (M.W.A.). The code allows a joint multi-band analysis of surface brightness models, thus superceding the functionality of GALFIT (Peng et al. 2002), and is described in detail in Bennert et al. (2011). The point-spread function (PSF) of the HST/ACS optics is modeled using the \nclosest bright star to a given object. We impose a Gaussian prior on the AGN colors with the mean given by the quasar composite spectrum from Vanden Berk et al. (2001) redshifted to the AGN redshift and with a σ of 0.2 mag. We model the host galaxy by either a single de Vaucouleurs (1948) profile or by a de Vaucouleurs (1948) profile plus an exponential profile to account for a disk. Depending on the images and residuals, we decide whether a given object is best fitted by three components (PSF, spheroid, disk) or two components (PSF + spheroid), as described by Treu et al. (2007); Bennert et al. (2010, 2011). A disk component is evident in 4/11 objects. In all four cases, we can only clearly detect the bulge in the z band. \nTo probe the reliability of our AGN-host-galaxy decompositions when using the blue restframe wavelengths covered by the GOODS images, we tested the effect of bandpass shifting. Given the host-galaxy morphology and level of activity of the sample studied here, a local sample of Seyfert galaxies is a suitable comparison sample for this test. (Schawinski et al. 2010 also concluded that 'moderate luminosity AGN host galaxies at z /similarequal 2 and z /similarequal 0 are remarkably similar'.) We thus repeated the analysis of our local sample of AGN host galaxies (Bennert et al. 2011), but now using only ug SDSS photometry (instead of griz). We are able to recover the photometry of the bulge and point source to within 0.1 mags, i.e. smaller than our adopted systematic uncertainty, demonstrating that bandpass shifting is not a concern within our level of precision. Moreover, this is a conservative estimation, since the GOODS images at z /similarequal 1 . 3 not only cover wavelengths comparable to ug rest frame (F775W and F850LP), but additionally also shorter wavelengths (F606W and F435W), thus effectively providing more information to disentangle point source and bulge. Furthermore, as already pointed out above, the GOODS images of z /similarequal 1 . 3 objects have even higher resolution than SDSS images at /similarequal 0 . 5.", '2.3. Stellar Mass': 'From the resulting magnitudes (Table 2), stellar masses are estimated using a Bayesian stellar-mass estimation code (Auger et al. 2009) assuming a Chabrier initial mass function (IMF) (Table 1). We impose conservative uncertainties of 0.3 dex on the masses of the bulges (disks) for the bulge-dominated (disk-dominated) hosts. The masses for the bulge components of the diskdominated hosts are estimated by using the z -band massto-light ratios of the bulge-dominated hosts in our sample that are at similar redshifts; we therefore add in quadrature a 0.3 dex uncertainty, yielding a total stellar mass uncertainty of 0.4 dex for these objects. For two of our objects, Schawinski et al. (2010) report stellar masses based upon template fits to the integrated light. Our results agree within the uncertainties (assumed to be 0.2 dex for Schawinski et al. 2010).', '2.4. BH Mass': "Black hole masses are estimated via the empirically calibrated photo-ionization method ('virial method') (e.g., Wandel et al. 1999; Vestergaard & Peterson 2006; McGill et al. 2008), by combining the FWHM of the broad MgII λ 2798 ˚ A emission line and the 3000 ˚ A AGN \nCO-EVOLUTION OF SPHEROIDS AND BLACK HOLES. \n3 \nFig. 1.Deep HST/ACS color images (B, V, i, z), 3 '' × 3 '' . \n<!-- image --> \nTABLE 1 Sample Properties, BH Masses, and Stellar Masses \n| ID | R.A. (J2000) | Decl. (J2000) | z | Ref. | FWHM MgII (kms - 1 ) | λL 3000 (10 44 erg s - 1 ) | M BH | M sph ,/star | M disk ,/star |\n|------------------|----------------|-----------------|-------|---------|------------------------|------------------------------|--------|----------------|-----------------|\n| (1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10) |\n| J033252 - 275119 | ID.88 | ID.8 | 1.227 | S10/S04 | 16208 | 0.43 | 9.01 | 9.83 | 10.50 |\n| J033243 - 274914 | ID.24 | ID.2 | 1.900 | T09/S04 | 16381 | 1.77 | 9.31 | 10.64 | · · · |\n| J033239 - 274601 | ID.09 | ID.8 | 1.220 | T09/S04 | 4344 | 5.38 | 8.39 | 10.54 | · · · |\n| J033226 - 274035 | ID.50 | ID.5 | 1.031 | S10/S04 | 2430 | 9.51 | 8.00 | 9.53 | 10.75 |\n| J033225 - 274218 | ID.17 | ID.8 | 1.617 | S10/S04 | 4744 | 1.64 | 8.22 | 10.61 | · · · |\n| J033210 - 274414 | ID.91 | ID.9 | 1.615 | S10/S04 | 5852 | 2.02 | 8.45 | 10.45 | · · · |\n| J033200 - 274319 | ID.36 | ID.7 | 1.037 | S10/L05 | 3602 | 1.08 | 7.90 | 9.62 | · · · |\n| J033229 - 274529 | ID.98 | ID.9 | 1.218 | S10/M05 | 5308 | 4.33 | 8.52 | 10.71 | · · · |\n| J123553+621037 | ID.13 | ID.3 | 1.371 | T08/W04 | 5441 | 2.32 | 8.41 | 9.99 | 10.84 |\n| J123618+621115 | ID.58 | ID.0 | 1.021 | T08/W04 | 6988 | 1.19 | 8.49 | 9.29 | 10.95 |\n| J123707+622147 | ID.46 | ID.9 | 1.450 | T08/W04 | 10654 | 1.57 | 8.91 | 10.74 | · · · | \nNote . - Column 1: target ID (based on R.A. and Decl.). Column 2: Right Ascension. Column 3: Declination. Column 4: redshift (taken from Team Keck Redshift Survey (TKRS) (Wirth et al. 2004) for GOODS-N). Column 5: reference for catalog from which objects were selected/ reference for origin of spectra. S10=Silverman et al. (2010), T09=Treister et al. (2009), T08=Trouille et al. (2008); S04=Szokoly et al. (2004), L05=Le F'evre et al. (2005), M05=Mignoli et al. (2005), W04=Wirth et al. (2004). Column 6: FWHM of broad MgII. Column 7: rest-frame luminosity at 3000 ˚ A (fiducial error 0.1 dex). Column 8: log M BH /M /circledot (uncertainty: 0.5 dex). Column 9: stellar spheroid mass log M sph ,/star /M /circledot (Chabrier IMF; uncertainty 0.3 dex for ellipticals, 0.4 dex for spirals). Column 10: stellar disk mass log M disk ,/star /M /circledot (Chabrier IMF; uncertainty 0.3 dex). \nTABLE 2 Results from Surface Photometry \n| Object | PSF | PSF | PSF | PSF | Spheroid | Spheroid | Spheroid | Spheroid | Disk | Disk | Disk | Disk |\n|------------------|-------------|-------------|-------------|-------------|-------------|-------------|-------------|-------------|--------------|--------------|--------------|--------------|\n| (1) | B (mag) (2) | V (mag) (3) | i (mag) (4) | z (mag) (5) | B (mag) (6) | V (mag) (7) | i (mag) (8) | z (mag) (9) | B (mag) (10) | V (mag) (11) | i (mag) (12) | z (mag) (13) |\n| J033252 - 275119 | 25.03 | 23.84 | 23.40 | 23.47 | · · · | · · · | · · · | 23.86 | 24.12 | 23.49 | 22.84 | 22.18 |\n| J033243 - 274914 | 22.53 | 23.09 | 22.98 | 23.10 | 25.30 | 24.68 | 23.67 | 23.37 | · · · | · · · | · · · | · · · |\n| J033239 - 274601 | 21.33 | 21.08 | 21.14 | 21.38 | 24.17 | 24.32 | 23.06 | 22.10 | · · · | · · · | · · · | · · · |\n| J033226 - 274035 | 20.26 | 20.04 | 20.04 | 20.01 | · · · | · · · | · · · | 23.27 | 22.22 | 21.87 | 21.22 | 20.63 |\n| J033225 - 274218 | 25.38 | 23.87 | 22.84 | 22.50 | 25.08 | 24.61 | 23.84 | 23.08 | · · · | · · · | · · · | · · · |\n| J033210 - 274414 | 24.24 | 23.19 | 22.61 | 22.74 | 24.13 | 24.10 | 23.55 | 22.96 | · · · | · · · | · · · | · · · |\n| J033200 - 274319 | 22.81 | 22.42 | 22.45 | 22.47 | 43.90 | 25.33 | 24.00 | 23.04 | · · · | · · · | · · · | · · · |\n| J033229 - 274529 | 21.24 | 21.31 | 21.42 | 21.76 | 23.80 | 23.32 | 22.49 | 21.66 | · · · | · · · | · · · | · · · |\n| J123553+621037 | 22.55 | 22.12 | 22.15 | 22.33 | · · · | · · · | · · · | 24.04 | 23.62 | 23.20 | 22.45 | 21.83 |\n| J123618+621115 | 22.77 | 22.27 | 22.62 | 22.81 | · · · | · · · | · · · | 23.87 | 23.15 | 22.37 | 21.40 | 20.79 |\n| J123707+622147 | 23.16 | 22.97 | 22.69 | 22.55 | 24.95 | 24.66 | 23.46 | 22.57 | · · · | · · · | · · · | · · · | \nNote . - Column 1: target ID. Columns 2-5: extinction-corrected B, V, i, and z PSF magnitudes (uncertainty: 0.2 mag). Columns 6-9: extinction-corrected B, V, i, and z spheroid magnitudes (uncertainty: 0.2 mag). Columns 10-13: extinction-corrected B, V, i, and z disk magnitudes (uncertainty: 0.2 mag). \ncontinuum luminosity (McGill et al. 2008): \nlog M BH = 6 . 767 + 2 log ( FWHM MgII 1000 kms -1 ) +0 . 47 log ( λL 3000 10 44 erg s -1 ) \nThe AGN luminosity is derived from the PSF magnitudes in the filter closest to rest frame 3000 ˚ A, and extrapolated based on the assumed AGN SED of Vanden Berk et al. (2001) ( § 2.2; Table 1). \nThe nominal uncertainty of M BH using this method is 0.4 dex. However, for some spectra, the low signalto-noise (S/N) makes the FWHM measurements uncertain by up to ∼ 50%, conservatively estimated. Moreover, the spectra are not of sufficient quality to remove the Fe emission which can result in overestimating the width of MgII by up to 0.03 dex (in FWHM, McGill et al. 2008; see, however, Merloni et al. 2010). We therefore adopt an uncertainty of 0.5 dex. Note that while we used uniform priors for both the 3000 ˚ A luminosity and the black hole mass in our analysis, employing more informed pri-", '3. RESULTS AND DISCUSSION': '4/11 AGNs are clearly hosted by late-type spiral galaxies, while the rest seem to be spheroid dominated. Keeping in mind the small-number statistics, the fraction of disk-dominated host galaxies (36 ± 17%) is lower than what has been found by Schawinski et al. (2010) (80 ± 10%) for 20 X-ray selected AGNs at a comparable redshift (1 . 5 < z < 3) imaged by HST/Wide Field Camera 3 (WFC3; F160W) with 1-2 orbits integration time. One difference is that our objects have higher X-ray luminosities (0.5-8keV; 43 . 5 < log L X < 44 . 5, mean=44.2, compared to 42 < log L X < 44, mean=43.1) which might explain why we find a larger fraction of elliptical host galaxies. \nInterestingly, none of the objects shows clear signs of interactions or merger activity, while at redshifts of z = 0 . 4 -0 . 6, 32 ± 9% of Seyfert-1s are hosted by interacting/merging galaxies (Bennert et al. 2010). However, we cannot exclude that some of these low surface-brightness features might have been missed (see, e.g. Bennert et al. 2008). Schawinski et al. (2010) also do not report interactions/mergers but their images are significantly shallower than ours. Star-forming galaxies at a redshift of z ∼ 2, on the other hand, show a 33 ± 6% fraction of interacting or merging systems (Forster-Schreiber et al. 2009). Schawinski et al. (2010) interpret their high fraction of spiral galaxies as a sign that secular evolution may play a non-negligible role in growing spheroids and black holes. Our findings, including the lack of merger activity, are consistent with such a scenario. \nFigure 2 (left) shows the M BH -M sph ,/star relation, including a sample of 18 inactive galaxies and the local AGNs from Bennert et al. (2011). In Figure 2 (right), we show the M BH -M host ,/star relation, again including the local AGNs from Bennert et al. (2011) and additionally, the 89 AGNs from Merloni et al. (2010) (10/89 with upper limits only; subtracting 0.255 dex to convert their total stellar masses from Salpeter to Chabrier IMF, Bruzual & Charlot 2003). Note that for all comparison samples, M BH were estimated using the same recipe adopted here. \nIn Figure 3, we show the offset in log M BH as a function of constant M sph ,/star (left panel) and M host ,/star (right panel) with respect to the z = 0 relations (see § 2.5). For comparison, the offset in log M BH as a function of constant stellar spheroid luminosity (left panel) and total luminosity (right panel) from Bennert et al. (2010) is overplotted. Taking into account selection effects ( § 2.5), we find significant evolution in M BH / M sph ,/star ( ∝ (1 + z ) 1 . 96 ± 0 . 55 ), consistent with (but with larger uncertainties) what we reported previously for the evolution of the M BH -L sph relation ( M BH / L sph ∝ (1+ z ) 1 . 4 ± 0 . 2 ; Bennert et al. 2010). The agreement between the stellar mass and luminosity evolution suggests that the passive luminosity correction is appropriate, although modeling luminosity evolution rather than stellar masses may increase the scatter. \nFor total stellar masses, including the Merloni et al. (2010) data, the evolutionary trend can be described as M BH / M host ,/star ∝ (1 + z ) 1 . 15 ± 0 . 15 , in agreement with what has been found by Merloni et al. (2010) within the uncertainties. This evolution is slower than the one for spheroid masses ( β = 1 . 96 ± 0 . 55) in line with recent studies (Jahnke et al. 2009; Bennert et al. 2010). It in- \nTABLE 3 Evolving M BH -M ∗ Scaling Relations \n| Model | α | β | γ | σ |\n|--------------------------|------|-----------------|-------|-----------------|\n| M a sph , ∗ M a host , b | 1.09 | 1 . 96 ± 0 . 55 | -0.48 | 0 . 36 ± 0 . 1 |\n| ∗ | 1.12 | 1 . 68 ± 0 . 53 | -0.68 | 0 . 35 ± 0 . 1 |\n| M host , ∗ | 1.12 | 1 . 15 ± 0 . 15 | -0.68 | 0 . 16 ± 0 . 06 |\n| M c host , ∗ | 1.12 | 1 . 11 ± 0 . 16 | -0.68 | 0 . 17 ± 0 . 07 | \nNote . -a Fitted only using the 11 objects presented here. b Fitted using the objects presented here and the objects from Merloni et al. (2010). c Fitted only using data from Merloni et al. (2010). \nors from the quasar luminosity function of Richards et al. (2006) or the black hole mass function of Kelly et al. (2010) yield negligible changes to our inference.', '2.5. M BH -M /star Evolution': "Following and expanding on work by Treu et al. (2007) and Bennert et al. (2010), we model the evolution of the offset of the M BH -M sph ,/star and M BH -M host ,/star scaling relations by assuming a model of the form \nlog M BH -8 = α [log M ∗ -10] + β log [1 + z ] + γ + σ \nwhere α is the slope of the relations at z = 0 and is assumed not to evolve, γ is the intercept of the relations at z = 0, and σ is the intrinsic scatter which is also assumed to be non-evolving. Here β describes the evolution of the scaling relation (with β = 0 implying no evolution). We impose δ -function priors of α = 1 . 09 (Bennert et al. 2011) and γ = -0 . 48 for the M sph , ∗ relation and α = 1 . 12 (Haring & Rix 2004) and γ = -0 . 68 for the M Host , ∗ relation; the priors on γ were determined by fitting to the local AGNs from Bennert et al. (2011) while keeping the slope fixed to the noted values. A normal distribution prior is used for the intrinsic scatter with mean 0.4 and variance 0.01 and we employ a broad uniform prior for β . (Note that, strictly speaking, the variable σ accounts for both the intrinsic scatter in the relationship and the (much smaller) uncertainty on γ .) We use the z = 1 . 0 -1 . 2 'elliptical' stellar mass function from Ilbert et al. (2010) to place priors on the stellar masses. Furthermore, we include a prior on the black hole masses that models our selection effects by using a hard cutoff at the low mass end. This cutoff is determined from the data and models lower limit of black hole masses observable in each considered set of data. \nThe relation above is first fitted using the 11 galaxies in this sample. The lower limit for the black hole masses assumed for the high redshift objects is 10 7 . 4 M /circledot . Merloni et al. (2010) have independently tried to infer the evolution of the M BH -M host ,/star relation, but their analysis is somewhat different than ours (e.g. IMFs, local comparison samples, definition of offset, treatment of upper limits and selection effects). We therefore also fit the relation using the Merloni et al. (2010) data (adjusted to a Chabrier IMF), and we impose a limiting black hole mass of 10 7 . 3 for these data. The results of our inference are shown in Table 3. Given that the different fits to the M BH -M host ,/star relation (Merloni et al. data only, our data only, both combined) result in the same β within the uncertainties, we adopt the one for the combined sample in the following. \ndicates that the amount by which at least some of the distant AGN host galaxies have to grow their bulge component in order to fall on the local BH mass scaling relations is contained within the galaxy itself. It can thus be considered as another evidence that secular evolution and/or minor mergers play a non-negligible role in growing spheroids through a redistribution of stars from disk to bulge. The deduced evolution is either in line with or slightly faster than what has been predicted by theoretical studies (for a detailed comparison, see, e.g. Bennert et al. 2010; Lamastra et al. 2010).", '4. CONCLUSIONS': 'We determine spheroid and total stellar masses for the host galaxies of 11 X-ray selected BLAGNs (1 < z < 2) in GOODS. In combination with M BH estimated via the virial method from the broad MgII emission line as measured from archival VLT and Keck spectra and the 3000 ˚ A nuclear luminosity, we study the evolution of the M BH -M /star scaling relation out to a lookback time of 10 Gyrs. Using a uniformly measured local comparison sample and taking into account selection effects, we find evolution of the correlations consistent with BH growth preceding galaxy assembly, confirming and extending the results of previous studies (e.g., Merloni et al. 2010; Decarli et al. 2010; Bennert et al. 2010). \nOur results show that a significant fraction (4/11) of AGNs at z=1-2 are hosted by spiral galaxies. None of the galaxies show evidence for recent major merger interaction, contrary to the general assumption that BHs and spheroids grow predominantly through major mergers, a scenario which might hold true only for the most luminous AGNs. The evolution we find for the M BH -total stellar mass relation is slower than the one for spheroid stellar masses in line with recent studies (Jahnke et al. 2009; Bennert et al. 2010). Combined, our results indicate that secular evolution and/or minor mergers play a \nnon-negligible role in growing both BHs and spheroids. \nOur study demonstrates the feasibility of obtaining stellar masses of AGN host galaxies out to lookback times of 10 Gyrs based on deep multicolor HST photometry. This approach has the great advantage of being independent of the luminosity evolution correction - the dominant source of systematic uncertainty in previous studies at comparable redshifts (e.g. Peng et al. 2006; Bennert et al. 2010). Furthermore, we can distinguish between spheroid and total host galaxy mass, which is not possible based on SED fitting (e.g., Merloni et al. 2010). \nSample size is a major limitation of this work, allowing us to constrain only average evolution and preventing us from investigating, e.g, mass-dependent trends or correlations between evolution and morphology. Follow-up of BLAGN hosts imaged by existing and upcoming multicolor HST surveys (e.g. CANDLES) is needed to gather larger samples and address theses remaining issues. \nWe thank Knud Jahnke, Andrea Merloni, and Kevin Schawinski for discussions, and Brandon Kelly for providing the quasar BHMF. We thank the anonymous referee for a careful reading of the manuscript and valuable suggestions. VNB, MWA, and TT acknowledge support by the NSF through CAREER award NSF-0642621, and by the Packard Foundation through a Packard Fellowship. JHW acknowledges support by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (2010-0021558). This research has made use of the public archive of the SDSS and the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration.', 'REFERENCES': "Auger, M. W., Treu, T., Bolton, A. S., Gavazzi, R., Koopmans, L. V. E., Marshall, P. J., Bundy, K., & Moustakas, L. A 2009, ApJ, 705, 1099 Bennert, N., Canalizo, G., Jungwiert, B., Stockton, A., Schweizer, F., Peng, C. Y., Lacy, M. 2008, ApJ, 677, 846 Bennert, V. N., Treu, T., Woo, J.-H., Malkan, M. A., Le Bris, A., Auger, M. W., Gallagher, S., & Blandford, R. D. 2010, ApJ, 708, 1507 Bennert, V. N., Auger, M. W., Treu, T., Woo, J.-H., & Malkan, M. A. 2011, ApJ, accepted Bentz, M. C., Peterson, B. M., Pogge, R. W., & Vestergaard, M. 2009, ApJ, 694, 166 Bruzual, G. & Charlot, S. 2003, MNRAS, 344, 1000 Ciotti, L., & Ostriker, J. 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(2010) (blue filled squares; 10 with upper limits indicated by arrows). \n<!-- image --> \n<!-- image --> \nFig. 3.Left panel: Offset in log M BH as a function of constant M sph ,/star (our objects: red filled pentagons) with respect to the fiducial local relation of AGNs (black filled circles). The offset in log M BH as a function of constant stellar spheroid luminosity from Bennert et al. (2010) is overplotted (green open symbols), corresponding to AGNs at different redshifts, (left to right: z /similarequal 0.08, reverberation-mapped AGN from Bennert et al. 2010; Bentz et al. 2009; z /similarequal 0.4 from Bennert et al. 2010; Treu et al. 2007 z /similarequal 0.6 from Bennert et al. 2010, z /similarequal 1.8 from Bennert et al. 2010; Peng et al. 2006). The best linear fit derived here is overplotted as dotted line ( M BH / M sph ,/star ∝ (1 + z ) 1 . 96 ± 0 . 55 ; dashed lines: 1 σ range). Right panel: The same as in the left panel as an offset in log M BH as a function of constant total host galaxy mass (luminosity for Bennert et al. 2010). The Merloni et al. (2010) sample is overplotted (blue filled squares). The lines correspond to M BH / M host ,/star ∝ (1 + z ) 1 . 15 ± 0 . 15 . \n<!-- image --> \nSzokoly, G. P., et al. 2004, ApJS, 155, 271 Treister, E., et al. 2009, ApJ, 693, 1713 Treu, T., Malkan, M. A., & Blanford, R. D. 2004, ApJ, 615, L97 Treu, T., Woo, J.-H., Malkan, M. A., & Blanford, R. D. 2007, ApJ, 667, 117 \nTrouille, L., Barger, A. J., Cowie, L. L., Yang, Y., & Mushotzky, R. F. 2008, ApJS, 179, 1 Vanden Berk, D. E. et al. 2001, AJ, 122, 549 Vestergaard, M. & Peterson, B. M. 2006, ApJ, 641, 689 Volonteri, M., Haardt, F., & Madau, P. 2003, ApJ, 582, 559 \nWalter, F., Carilli, C., Bertoldi, F., Menten, K., Cox, P., Lo, K. Y., Fan, X., & Strauss, M. A. 2004, ApJ, 615, L17 Wandel, A., Peterson, B. M., & Malkan, M. A. 1999, ApJ, 526, 579 Wirth, G. 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2007PhRvD..75j6001H

Local bulk operators in AdS/CFT correspondence: A holographic description of the black hole interior

2007-01-01

['-', '-', '-', '-', '-', '-', '-']

[]

To gain insight into how bulk locality emerges from the holographic conformal field theory (CFT), we reformulate the bulk-to-boundary map in as local a way as possible. In previous work, we carried out this program for Lorentzian anti-de Sitter (AdS), and showed the support on the boundary could always be reduced to a compact region spacelike separated from the bulk point. In the present work the idea is extended to a complexified boundary, where spatial coordinates are continued to imaginary values. This continuation enables us to represent a local bulk operator as a CFT operator with support on a finite disc on the complexified boundary. We treat general AdS in Poincaré coordinates and AdS<SUB>3</SUB> in Rindler coordinates. We represent bulk operators inside the horizon of a Banados-Teitelboim-Zanelli (BTZ) black hole and we verify that the correct bulk two-point functions are reproduced, including the divergence when one point hits the BTZ singularity. We comment on the holographic description of black holes formed by collapse and discuss locality and holographic entropy counting at finite N.

[]

https://arxiv.org/pdf/hep-th/0612053.pdf

{'Local bulk operators in AdS/CFT: A holographic description of the black hole interior': "Alex Hamilton, 1 Daniel Kabat, 1 Gilad Lifschytz, 2 and David A. Lowe 3 \n1 Department of Physics Columbia University, New York, NY 10027 USA hamilton, [email protected] \n2 Department of Mathematics and Physics and CCMSC University of Haifa at Oranim, Tivon 36006 ISRAEL [email protected] \n3 Department of Physics Brown University, Providence, RI 02912 USA [email protected] \nTo gain insight into how bulk locality emerges from the holographic conformal field theory, we reformulate the bulk to boundary map in as local a way as possible. In previous work, we carried out this program for Lorentzian AdS, and showed the support on the boundary could always be reduced to a compact region spacelike separated from the bulk point. In the present work the idea is extended to a complexified boundary, where spatial coordinates are continued to imaginary values. This continuation enables us to represent a local bulk operator as a CFT operator with support on a finite disc on the complexified boundary. We treat general AdS in Poincar'e coordinates and AdS 3 in Rindler coordinates. We represent bulk operators inside the horizon of a BTZ black hole and we verify that the correct bulk two point functions are reproduced, including the divergence when one point hits the BTZ singularity. We comment on the holographic description of black holes formed by collapse and discuss locality and holographic entropy counting at finite N .", '1 Introduction': "The AdS/CFT correspondence relates string theory in an asymptotically anti-de Sitter (AdS) background to a conformal field theory (CFT) living on the boundary of AdS [1, 2, 3, 4]. The main observable of interest in the original work on AdS/CFT was the boundary S-matrix. In the present work we will focus instead on how one might recover approximate local bulk quantities from the CFT. Even with interactions included, one can hope to recover quasi-local observables in the gravitational theory [5]. In this paper we study this in detail, generalizing our earlier work [6, 7]. We develop the map at leading order in 1 /N where we can treat the gravity theory semiclassically and work with free scalar fields. \nThe original bulk to boundary map of Euclidean AdS [2, 3] and its Lorentzian generalization [8, 9] has been reformulated and studied further in [10, 11, 12, 13, 14]. In these works, one can view the construction of a local bulk operator as an integral over the entire boundary of AdS. Vanishing of commutators of local bulk operators at spacelike separation relies on delicate cancellations in this approach [12]. The strategy we will adopt, following our earlier work [6, 7], is to reformulate the bulk to boundary map so that the support on the boundary is as small as possible. \nIt is worth emphasizing the physical relevance of our approach. By representing bulk operators as operators on the boundary with compact support in fact with support that is as small as possible - we can have bulk operators whose dual boundary operators are spacelike separated. Such bulk operators will manifestly commute with each other just by locality of the boundary theory. This statement will continue to hold at finite N . Moreover we will find interesting applications of this basis of operators to the study of black hole interiors and singularities, as well as holographic entropy counting. \nWe will use the following framework developed in [6, 7]. The first of these works [6] mainly considered two-dimensional AdS space, and showed the boundary support of a local bulk operator could be reduced to points spacelike separated from the bulk point. This was generalized to the higher dimensional case in [7]. In Lorentzian AdS, a free bulk scalar field φ is dual \nto a non-local operator in the CFT, via a correspondence \nφ ( x, Z ) ↔ ∫ dx ' K ( x ' | x, Z ) O ( x ' ) . (1) \nHere Z is a radial coordinate in AdS which vanishes at the boundary, x represents coordinates along the boundary, and O ( x ' ) is a local operator in the CFT. A similar approach has been considered previously in [10, 11, 12, 14]. We refer to the kernel K as a smearing function. This correspondence can be used inside correlation functions, for example \n〈 φ ( x 1 , Z 1 ) φ ( x 2 , Z 2 ) 〉 SUGRA = ∫ dx ' 1 dx ' 2 K ( x ' 1 | x 1 , Z 1 ) K ( x ' 2 | x 2 , Z 2 ) 〈O ( x ' 1 ) O ( x ' 2 ) 〉 CFT . \nTo construct smearing functions one begins with a field in Lorentzian AdS that satisfies the free wave equation and has normalizable fall-off near the boundary of AdS, 1 \nφ ( x, Z ) ∼ Z ∆ φ 0 ( x ) as Z → 0 . \nThe parameter ∆ is related to the mass of the field. The boundary field φ 0 ( x ) is dual to a local operator in the CFT 2 \nφ 0 ( x ) ↔O ( x ) . (2) \nWe will construct smearing functions that let us solve for the bulk field in terms of the boundary field \nφ ( x, Z ) = ∫ dx ' K ( x ' | x, Z ) φ 0 ( x ' ) . (3) \nK should not be confused with the standard bulk-to-boundary propagator [3], since our smearing functions generate normalizable solutions to the Lorentzian equations of motion. Using the duality (2), we obtain the relation between bulk and boundary operators given in (1). \nSolving for the bulk field in terms of the boundary field is not a standard Cauchy problem: since the 'initial conditions' are specified on a timelike hypersurface we have neither existence nor uniqueness theorems. In global AdS it was shown that, although the smearing function is not unique, one can always construct a smearing function which has support on the boundary at points which are spacelike separated from the bulk point [6, 7]. It is then interesting to see if a stronger statement can be made. Can we further reduce the support on the boundary? This was studied in [7], where smearing functions for AdS 3 were constructed in accelerating Rindler coordinates. It was shown that smearing functions can only be constructed by analytically continuing the boundary coordinates to complex values, since the naive expression derived from mode sums was divergent. This continuation leads to a well-defined smearing function with compact support on the complexified boundary of the Rindler patch; it can be thought of as arising from a retarded Green's function in de Sitter space. Moreover the support shrinks to a point as the bulk point approaches the boundary. In this way we recover the expected relation (2). \nIt thus seems the most economical description of local bulk physics in AdS/CFT requires the use of complexified boundary coordinates. Complexified coordinates also appeared in [15], and have been used to study the region inside horizons in [16, 17, 18, 19, 20]. For other approaches to recovering bulk physics see [21, 22, 23]. \nAn outline of this paper is as follows. In section 2 we extend the work of [7] and use complexified boundary coordinates to construct compact smearing functions in AdS spacetimes of general dimension in two ways. First we work in Poincar'e coordinates and perform a Poincar'e mode sum, then we Wick rotate to de Sitter space and use a retarded Green's function. In section 3 we translate our AdS 3 results into Rindler coordinates and show that we recover bulk correlators inside the Rindler horizon. After these preliminaries we develop applications of this new formulation of the bulk/boundary map to black holes and to holographic entropy counting. In section 4 we argue that the Rindler smearing functions can also be used in a BTZ spacetime and we show how the BTZ singularity manifests itself in the conformal field theory. In section 5 we discuss local operators inside the horizon of an AdS black hole formed by collapse, where there is only a single asymptotic AdS region. This provides evidence that our results will generalize to time-dependent situations. Finally in section 6 we explain how the number of degrees of \nfreedom is reduced at finite N and how this leads to a new perspective on holographic entropy counting.", "2 Poincar'e coordinates": "In this section we construct a compact smearing function for a generaldimensional AdS spacetime. We obtain the result in two ways: by performing the Poincar'e mode sum in section 2.2, and by Wick rotating to de Sitter space in section 2.3.", '2.1 Preliminaries': "We will work in AdS D in Poincar'e coordinates, with metric \nds 2 = R 2 Z 2 ( -dT 2 + | dX | 2 + dZ 2 ) . (4) \nHere R is the AdS radius. The coordinates range over 0 < Z < ∞ , -∞ < T < ∞ , and X ∈ R d -1 where d = D -1. An AdS-invariant distance function is provided by \nσ ( T, X, Z | T ' , X ' , Z ' ) = 1 2 ZZ ' ( Z 2 + Z ' 2 + | X -X ' | 2 -( T -T ' ) 2 ) . (5) \nWe consider a free scalar field of mass m in this background. Normalizable solutions to the free wave equation ( -/square + m 2 ) φ = 0 can be expanded in a complete set of modes \nφ ( T, X, Z ) = ∫ | ω | > | k | dωd d -1 k a ωk e -iωT e ik · X Z d/ 2 J ν ( √ ω 2 -k 2 Z ) . (6) \nThe Bessel function has order ν = ∆ -d/ 2 where ∆ = d 2 + √ d 2 4 + m 2 R 2 is the conformal dimension of the corresponding operator. \nIn Poincar'e coordinates we define the boundary field by \nφ Poincare 0 ( T, X ) = lim Z → 0 1 Z ∆ φ ( T, X, Z ) (7) = 1 2 ν Γ( ν +1) ∫ | ω | > | k | dωd d -1 k a ωk e -iωT e ik · X ( ω 2 -k 2 ) ν/ 2 . \nNote that \na ωk = 2 ν Γ( ν +1) (2 π ) d ( ω 2 -k 2 ) ν/ 2 ∫ dTd d -1 Xe iωT e -ik · X φ Poincare 0 ( T, X ) . (8) \nSubstituting this back into the bulk mode expansion (6), we obtain an expression for the bulk field in terms of the boundary field, namely \nφ ( T, X, Z ) = ∫ dT ' d d -1 X ' K ( T ' , X ' | T, X, Z ) φ Poincare 0 ( T ' , X ' ) (9) \nwhere \nK ( T ' , X ' | T, X, Z ) = 2 ν Γ( ν +1) (2 π ) d ∫ | ω | > | k | dωd d -1 k e -iω ( T -T ' ) e ik · ( X -X ' ) Z d/ 2 J ν ( √ ω 2 -k 2 Z ) / ( ω 2 -k 2 ) ν/ 2 . (10) \nOne could proceed to evaluate this integral representation for K along the lines of [12, 6, 7]. However one generically obtains a smearing function with support on the entire boundary of the Poincar'e patch. 3 In the following we will improve on this by constructing smearing functions that make manifest the property that local bulk operators go over to local boundary operators as the bulk point approaches the boundary. \nSuch smearing functions require complexifying the boundary spatial coordinates X . We will establish this in two ways: in section 2.2, for fields in AdS 3 , by starting with the mode sum (9) and performing a suitable analytic continuation, and again in section 2.3, for fields in general dimensional AdS, by Wick rotating to de Sitter space and using a retarded de Sitter Green's function.", "2.2 Poincar'e mode sum": "Consider a field in AdS 3 . The Poincar'e mode sum (9) reads \nφ ( T, X, Z ) = 2 ν Γ( ν +1) 4 π 2 ∫ | ω | > | k | dωdk ZJ ν ( √ ω 2 -k 2 Z ) ( ω 2 -k 2 ) ν/ 2 × (∫ dT ' dX ' e -iω ( T -T ' ) e ik ( X -X ' ) φ Poincare 0 ( T ' , X ' ) ) \nThe Poincar'e boundary field has no Fourier components with | ω | < | k | , so provided we perform the T ' and X ' integrals first we can subsequently integrate over ω and k without restriction. Thus \nφ ( T, X, Z ) = 2 ν Γ( ν +1) ∫ dωdk e -iωT e ikX ZJ ν ( √ ω 2 -k 2 Z ) ( ω 2 -k 2 ) ν/ 2 φ Poincare 0 ( ω, k ) (11) \nwhere φ Poincare 0 ( ω, k ) is the Fourier transform of the boundary field. We now use the two integrals \n∫ 2 π 0 dθ e -irω sin θ -kr cos θ = 2 πJ 0 ( r √ ω 2 -k 2 ) (12) \n∫ 1 0 rdr (1 -r 2 ) ν -1 J 0 ( br ) = 2 ν -1 Γ( ν ) b -ν J ν ( b ) (13) \nJ ν ( √ ω 2 -k 2 Z ) ( ω 2 -k 2 ) ν/ 2 = 1 π (2 Z ) ν Γ( ν ) ∫ T ' 2 + Y ' 2 <Z 2 dT ' dY ' ( Z 2 -T ' 2 -Y ' 2 ) ν -1 e -iωT ' e -kY ' . (14) \nInserting this into (11) one gets \nφ ( T, X, Z ) = ν π ∫ T ' 2 + Y ' 2 <Z 2 dT ' dY ' ( Z 2 -T ' 2 -Y ' 2 Z ) ν -1 ∫ dωdk e -iω ( T + T ' ) e ik ( X + iY ' ) φ Poincare 0 ( ω, k ) (15) \nφ ( T, X, Z ) = ∆ -1 π ∫ T ' 2 + Y ' 2 <Z 2 dT ' dY ' ( Z 2 -T ' 2 -Y ' 2 Z ) ∆ -2 φ Poincare 0 ( T + T ' , X + iY ' ) . (16) \nWe identify the second line of (15) as φ Poincare 0 ( T + T ' , X + iY ' ), so we can write (recall ν = ∆ -1) \nThat is, we've succeeded in expressing the bulk field in terms of an integral over a disk of radius Z in the (real T , imaginary X ) plane. We can express the result in terms of the invariant distance (5), \nφ ( T, X, Z ) = ∆ -1 π ∫ T ' 2 + Y ' 2 <Z 2 dT ' dY ' lim Z ' → 0 ( 2 Z ' σ ( T, X, Z | T + T ' , X + iY ' , Z ' ) ) ∆ -2 φ Poincare 0 ( T + T ' , X + iY ' ) (17) \nWe'll obtain the generalization of this result to higher-dimensional AdS in the next subsection.", '2.3 de Sitter continuation': "Having seen that we need to analytically continue the boundary spatial coordinates to complex values in order to obtain a smearing function with compact support, we will now begin by Wick rotating the Poincar'e longitudinal spatial coordinates, setting X = iY . This turns the AdS metric (4) into \n( \nds 2 = R 2 Z 2 dZ 2 -dT 2 -| dY | 2 ) . \n) This is nothing but de Sitter space written in flat Friedmann-RobertsonWalker (FRW) coordinates, with Z playing the role of conformal time (note the flip in signature). The AdS boundary becomes the past boundary of de Sitter space. Up to a divergent conformal factor the induced metric on the past boundary is \nds 2 bdy = dT 2 + | dY | 2 \ni.e. a plane R d which should be thought of as compactified to a sphere S d by adding a point at infinity. The Penrose diagram is shown in Fig. 1. \nIn de Sitter space it's clear that the field at any point inside the Poincar'e patch can be expressed in terms of data on a compact region of the past boundary. 4 With this motivation we will construct a retarded Green's function in de Sitter space and use it to reproduce and generalize the smearing function (16) that we previously obtained from a Poincar'e mode sum. \nS \nFigure 1: The Penrose diagram for de Sitter space. Flat FRW coordinates cover the lower triangle. Horizontal slices are spheres. Each point on the diagram represents an S d -1 which shrinks to a point at the north and south poles (the right and left edges of the diagram). \n<!-- image --> \nThe de Sitter invariant distance function is \nσ ( T, Y, Z | T ' , Y ' , Z ' ) = 1 2 ZZ ' Z 2 + Z ' 2 -( T -T ' ) 2 -| Y -Y ' | 2 ) . \n( \n) We consider a scalar field of mass m in de Sitter space. For now we take m 2 R 2 > 1, however later we will analytically continue m 2 → -m 2 . The analytically continued mass can be identified with the mass of a field in AdS (note that the Wick rotation flips the signature of the metric). \nThe field at some bulk point can be written in terms of the retarded Green's function. The retarded Green's function coincides with the imaginary part of the commutator inside the past light-cone of the future point and vanishes outside this region. The field at some bulk point is therefore \nφ ( T, Y, Z ) = ∫ dT ' d d -1 Y ' ( R Z ' ) d -1 G ret ( T, Y, Z | T ' , Y ' , Z ' ) ←→ ∂ Z ' φ ( T ' , Y ' , Z ' ) (18) \nwhere the region of integration is over a spacelike surface of fixed Z ' inside the past light-cone of the bulk point. In the Z ' → 0 limit this becomes the disk \n( T -T ' ) 2 + | Y -Y ' | 2 < Z 2 . (19) \nAs Z ' → 0 (with other coordinates held fixed) the retarded Green's function takes the form [24] \nG ret ∼ iR -d +1 ( c ( -σ -i/epsilon1 ) -d/ 2+ i √ m 2 R 2 -( d 2 ) 2 + c ∗ ( -σ -i/epsilon1 ) -d/ 2 -i √ m 2 R 2 -( d 2 ) 2 -c.c. ) \nwhere we take branch cuts along the positive real σ axis and where \nc = Γ ( 2 i √ m 2 R 2 -( d 2 ) 2 ) Γ ( d 2 -i √ m 2 R 2 -( d 2 ) 2 ) 2 -d 2 + i √ m 2 R 2 -( d 2 ) 2 (4 π ) d +1 2 Γ ( 1 2 + i √ m 2 R 2 -( d 2 ) 2 ) . \nThe boundary field is defined as usual using (7). Choosing normalizable modes from the AdS viewpoint corresponds to taking only positive frequencies in the Z direction, which have a Z d/ 2+ i √ m 2 R 2 -( d 2 ) 2 Z -dependence. \nEvaluating (18) as Z ' → 0 we obtain the smearing function 5 \nK ( T ' , Y ' | T, Y, Z ) = Γ ( ∆ -d 2 +1 ) π d/ 2 Γ(∆ -d +1) ( Z 2 -( T -T ' ) 2 -| Y -Y ' | 2 Z ) ∆ -d θ ( Z 2 -( T -T ' ) 2 -| Y -Y ' | 2 ) . (20) \nHowever at this point we still have ∆ = d 2 + i √ m 2 R 2 -( d 2 ) 2 . By analytically continuing m 2 →-m 2 we can take ∆ to coincide with the conformal dimension in AdS. Since σ > 0 in the domain of integration this analytic continuation is straightforward. Furthermore we can shift iY → X + iY and iY ' → X + iY ' , assuming φ Poincare 0 is analytic everywhere in the strip | Y | < Z ; this is true for any given Poincar'e mode function (6). Thus we wind up with the integral representation \nφ ( T, X, Z ) = Γ ( ∆ -d 2 +1 ) π d/ 2 Γ(∆ -d +1) ∫ T ' 2 + | Y ' | 2 <Z 2 dT ' d d -1 Y ' ( Z 2 -T ' 2 -| Y ' | 2 Z ) ∆ -d φ Poincare 0 ( T + T ' , X + iY ' ) (21) \nThis matches (16) for d = 2. As a further check we can examine the limit Z → 0 where we should recover (7). In this limit the region of integration becomes very small so we can bring the boundary field out of the integral and we indeed recover (7).", '2.4 Recovering bulk correlators': "In this section we show that the smearing functions we have constructed can be used to reproduce bulk correlation functions. As a corollary, this shows that the operators we have defined will commute with each other at bulk spacelike separation. For simplicity we will only treat the case of a massless field in AdS 3 . \nThe Wightman function for a massless scalar in AdS 3 is \nG ( x | x ' ) = 〈 0 | φ ( x ) φ ( x ' ) | 0 〉 SUGRA = 1 4 πR 1 √ σ 2 -1 ( σ + √ σ 2 -1 ) (22) \nwhere σ is defined in (5), and where branch cuts are handled with a T → T -i/epsilon1 prescription. 6 We'll consider the correlation function between an arbitrary bulk point ( T, X, Z ) and a point near the boundary with coordinates ( T ' = 0 , X ' = 0 , Z ' → 0). Taking the appropriate limit of (22) we have \n〈 φ ( T, X, Z ) φ Poincare 0 (0 , 0) 〉 SUGRA = 1 2 πR Z 2 ( T 2 -X 2 -Z 2 ) 2 . (23) \nWe'd like to reproduce this from the CFT. To do this note that from (16) we have \nφ ( T, X, Z ) = 1 π ∫ T ' 2 + Y ' 2 <Z 2 dT ' dY ' φ Poincare 0 ( T + T ' , X + iY ' ) . (24) \nAlso by sending both points to the boundary in (22) we have the boundary correlator 7 \n〈 φ Poincare 0 ( T, X ) φ Poincare 0 (0 , 0) 〉 CFT = 1 2 πR 1 ( T 2 -X 2 ) 2 . (25) \nThus our claim is that we can reproduce (23) by computing \n1 π ∫ T ' 2 + Y ' 2 <Z 2 dT ' dY ' 〈 φ Poincare 0 ( T + T ' , X + iY ' ) φ Poincare 0 (0 , 0) 〉 = 1 2 π 2 R ∫ T ' 2 + Y ' 2 <Z 2 dT ' dY ' 1 ( T + T ' ) 2 -( X + iY ' ) 2 ) 2 (26) \n( \nLet's begin by studying this in the regime \n| T + X | > Z and | T -X | > Z . (27) \nIn this regime there are no poles in the range of integration, so (26) is welldefined without having to give a prescription for dealing with light-cone singularities in the CFT. It's convenient to work in polar coordinates, setting T ' = r cos θ and Y ' = r sin θ . Defining z = e iθ we have \n1 πR ∫ Z 0 rdr ∮ | z | =1 dz 2 πi z ( T + X + rz ) 2 ( z ( T -X ) + r ) 2 (28) \nEvaluating the contour integral gives \n1 πR ∫ Z 0 rdr T 2 -X 2 + r 2 ( T 2 -X 2 -r 2 ) 3 = 1 2 πR Z 2 ( T 2 -X 2 -Z 2 ) 2 (29) \nas promised. \nNow let's return to the question of dealing with light-cone singularities in the CFT. That is, let's ask how we can analytically continue this result outside the range (27). In general the integrand in (28) has two double poles, located at \nIn the range (27) we see that the contour always encircles the pole at z 1 and never encircles the pole at z 2 . When we try to go outside this range one of the poles crosses the integration contour | z | = 1. So to analytically continue the calculation outside the range (27) we merely have to deform the z contour of integration so that it continues to encircle the pole at z 1 and exclude the pole at z 2 . \nz = z 1 = -r T -X and z = z 2 = -T + X r . (30) \nOne might ask how one can distinguish the two poles in general. Recall that the boundary CFT correlator is defined with a T → T -i/epsilon1 prescription. \nThis means the poles are displaced to 8 \nz 1 = -r T -X -i/epsilon1 z 2 = -T + X r + i/epsilon1 . (31) \nWe see that z 1 is always in the lower half plane while z 2 is always in the upper half plane. So the general prescription is to only encircle the pole in the lower half plane. The i/epsilon1 prescription makes the z contour integral well-defined, since the poles never collide. It also makes the integral over r well-defined, since the poles in r are displaced off the real axis. \nThis lets us see how the bulk light-cone singularity emerges from the CFT. Let's perform the z integral in (28) first, followed by the r integral. The two poles pinch the z contour of integration when r 2 = r 2 0 ≡ ( T -i/epsilon1 ) 2 -X 2 . Thus the integral over z has a pole when r = ± r 0 . When one of these singularities in the complex r plane hits the r = Z endpoint of the contour for integrating over r , the integral over r diverges. This reproduces the bulk light-cone singularity at T 2 -X 2 = Z 2 , regulated by the appropriate i/epsilon1 prescription. \nSince our smeared operators have the correct 2-point functions, it follows that at infinite N they commute as operators in the CFT whenever the bulk points are spacelike separated. This relies on the fact that at infinite N the commutator is a c-number, and one can check that it vanishes at bulk spacelike separation by computing a correlator 〈 ψ |O 1 O 2 -O 2 O 1 | ψ 〉 in any state of the CFT. However at finite N the commutator becomes an operator. The delicate cancellations which occurred at infinite N become state-dependent and are no longer possible in general. Thus we do not necessarily expect the commutator to vanish at bulk spacelike separation. We discuss this point further in section 6.", '3 AdS 3 in Rindler coordinates': "We now specialize to AdS 3 . This is a particularly interesting example, since the BTZ black hole can be constructed as a quotient of AdS 3 [25]. After some preliminaries we discuss AdS 3 in accelerating Rindler-like coordinates. We show that our Poincar'e results can be translated into accelerating coordinates \nFigure 2: A slice of constant φ in AdS 3 , drawn as an AdS 2 Penrose diagram. The four Rindler wedges are separated by horizons at r = r + . \n<!-- image --> \nand, with the help of an antipodal map, can be used to describe local bulk operators inside the Rindler horizon.", '3.1 Preliminaries': "AdS 3 can be realized as the universal cover of a hyperboloid \n-U 2 -V 2 + X 2 + Y 2 = -R 2 (32) \ninside R 2 , 2 with metric ds 2 = -dU 2 -dV 2 + dX 2 + dY 2 . To describe this in \nRindler coordinates we set \nU = Rr r + cosh r + φ R V = R √ r 2 r 2 + -1 sinh r + t R 2 (33) X = R √ r 2 r 2 + -1 cosh r + t R 2 Y = Rr r + sinh r + φ R \nso that the induced metric is \nds 2 = -r 2 -r 2 + R 2 dt 2 + R 2 r 2 -r 2 + dr 2 + r 2 dφ 2 . (34) \nHere -∞ < t, φ < ∞ and r + < r < ∞ . The Rindler horizon is located at r = r + . These coordinates cover the right Rindler wedge of AdS 3 as shown in Fig. 2. One can continue into the future wedge by setting \nU = Rr r + cosh r + φ R V = R √ 1 -r 2 r 2 + cosh r + t R 2 X = R √ 1 -r 2 r 2 + sinh r + t R 2 Y = Rr r + sinh r + φ R 2 \nwith 0 < r < r + . One can extend these coordinates to the (left, past) wedges by starting from the (right, future) definitions and changing the signs of V and X . It will frequently be convenient to work with rescaled coordinates \nˆ t = r + t/R 2 ˆ φ = r + φ/R. \nAn AdS-invariant distance function is provided by \nσ ( x | x ' ) = -1 R 2 X µ X ' µ (35) \nin terms of the embedding coordinates X µ = X µ ( x ). For two points in the right Rindler wedge we have \nσ = rr ' r 2 + cosh( ˆ φ -ˆ φ ' ) -( r 2 r 2 + -1 ) 1 / 2 ( r ' 2 r 2 + -1 ) 1 / 2 cosh( ˆ t -ˆ t ' ) (36) \nwhile for a point ( ˆ t, r, ˆ φ ) inside the future horizon and a point ( ˆ t ' , r ' , ˆ φ ' ) in the { right left Rindler wedge we have \n} \nσ = rr ' r 2 + cosh( ˆ φ -ˆ φ ' ) ∓ ( 1 -r 2 r 2 + ) 1 / 2 ( r ' 2 r 2 + -1 ) 1 / 2 sinh( ˆ t -ˆ t ' ) (37)", '3.2 Rindler smearing functions': "We could set about constructing a smearing function starting from a Rindler mode sum. For points outside the Rindler horizon this was carried out in [7], while for points inside the horizon we set up but do not evaluate the mode sum in appendix A. However the Rindler mode sum is divergent and must be defined by analytic continuation in ˆ t and/or ˆ φ . The divergence means there is no smearing function with support on real values of the Rindler boundary coordinates. \nA simpler approach to constructing the Rindler smearing function is to begin with our Poincar'e result (16) and translate it into Rindler coordinates. The translation is easiest to understand in de Sitter space. Wick rotating ˆ φ = iy turns the AdS metric (34) into \nds 2 = R 2 r 2 + [ r 2 + r 2 -r 2 + dr 2 -( r 2 -r 2 + ) d ˆ t 2 -r 2 dy 2 ] . \nThis is de Sitter space in static coordinates. To avoid a conical singularity at r = 0 we must periodically identify y ∼ y +2 π . The right Rindler wedge becomes the past wedge of de Sitter space, as shown in Fig. 3. The induced metric on the past boundary is, up to a divergent conformal factor, \nds 2 bdy = d ˆ t 2 + dy 2 -∞ < ˆ t < ∞ , y ∼ y +2 π \ni.e. an infinite cylinder which can be compactified to a sphere by adding the north and south poles. This sphere can be identified with the past boundary that we identified working in Poincar'e coordinates. However note that \nr = \n∞ \nFigure 3: de Sitter space in static coordinates. \n<!-- image --> \nany observer inside the past wedge of de Sitter space can at most see one hemisphere of the past boundary, namely the region characterized by \n-∞ < ˆ t < ∞ -π/ 2 < y < π/ 2 . \nFor a point inside the past wedge of de Sitter we can construct a retarded Green's function that lets us express the value of the field in terms of data on the past boundary. In AdS this means we can express the value of the field anywhere in the right Rindler wedge in terms of a data on the right Rindler boundary. In fact the result is a simple translation of our Poincar'e result (16). We define the right boundary field in Rindler coordinates by \n∣ \nφ Rindler , R 0 ( ˆ t, ˆ φ ) = lim r →∞ r ∆ φ ( ˆ t, r, ˆ φ ) ∣ ∣ right boundary (38) \n∣ This is related to the Poincar'e boundary field by \nφ Rindler , R 0 ( ˆ t, ˆ φ ) = lim r →∞ ( rZ ) ∆ φ Poincare 0 ( T, X ) . (39) \nWe also have the boundary change of coordinates \ndTdX Z 2 = r 2 d ˆ td ˆ φ r 2 + . (40) \nMaking these substitutions in (16), the value of the field at a bulk point inside the right Rindler wedge of AdS 3 is \nφ ( ˆ t, r, ˆ φ ) = (∆ -1)2 ∆ -2 πr 2 + ∫ spacelike dxdy lim r ' →∞ ( σ/r ' ) ∆ -2 φ Rindler , R 0 ( ˆ t + x, ˆ φ + iy ) (41) \nwhere as r ' →∞ the AdS invariant distance (36) becomes \nσ ( ˆ t, r, ˆ φ | ˆ t + x, r ' , ˆ φ + iy ) = rr ' r 2 + [ cos y -( 1 -r 2 + r 2 ) 1 / 2 cosh x ] (42) \nand the integration is over 'spacelike separated' points on the Wick rotated boundary, that is, over real values of ( x, y ) such that σ > 0. \nThe result (41) for bulk points in the right Rindler wedge was obtained in [7], starting from a Rindler mode sum and defining it via an analytic continuation, or alternatively from a de Sitter Green's function. Now let's ask what happens for bulk points inside the Rindler horizon. It's clear from Fig. 2 that, if we were willing to work in Poincar'e coordinates, there would be no problem: we could use (16) to obtain a smearing function with compact support on the Poincar'e boundary. However if we wish to work in Rindler coordinates there is a problem: the smearing function extends outside the Rindler wedge, and covers points on the (real slice of) the boundary which are to the future of the right Rindler patch. 9 \nTo fix this we can use the antipodal map. 10 The antipodal map acts on the embedding coordinates of section 3.1 by \nA : X µ →-X µ . (43) \nIn terms of Rindler coordinates this can be realized by \nA : ˆ t → ˆ t + iπ, ˆ φ → ˆ φ + iπ . (44) \nNote that σ ( x | Ax ' ) = -σ ( x | x ' ). Fields with integer conformal dimension transform simply under the antipodal map, \nφ ( Ax ) = ( -1) ∆ φ ( x ) . (45) \nThis is discussed in appendix B, where we also treat the slightly more involved case of non-integer ∆. \nIn Rindler coordinates the antipodal map can be used to move the part of the smearing function which extends outside the right Rindler wedge over to the left boundary. To see this one starts with the Poincar'e result (16) and breaks the integration region up into two pieces. One piece gives a smearing function in the right Rindler wedge, while under the antipodal map the other piece becomes a smearing function in the left Rindler wedge. Thus for a bulk point inside the Rindler horizon we have \nφ ( ˆ t, r, ˆ φ ) = (∆ -1)2 ∆ -2 πr 2 + [∫ σ> 0 dxdy lim r ' →∞ ( σ/r ' ) ∆ -2 φ Rindler , R 0 ( ˆ t + x, ˆ φ + iy ) + ∫ σ< 0 dxdy lim r ' →∞ ( -σ/r ' ) ∆ -2 ( -1) ∆ φ Rindler , L 0 ( ˆ t + x, ˆ φ + iy ) ] . (46) \nHere as r ' →∞ the AdS invariant distance (35) becomes \nσ ( ˆ t, r, ˆ φ | ˆ t + x, r ' , ˆ φ + iy ) = rr ' r 2 + [ cos y ∓ ( r 2 + r 2 -1 ) 1 / 2 sinh x ] (47) \nwhen the boundary point is in the { right left } Rindler wedge. The integration is over points with σ > 0 on the right boundary and points with σ < 0 on the left boundary, and we define \nφ Rindler , L 0 ( ˆ t, ˆ φ ) = lim r →∞ r ∆ φ ( ˆ t, r, ˆ φ ) ∣ ∣ ∣ left boundary . (48)", '3.3 Reproducing bulk correlators': "It is instructive to check that the Rindler smearing functions we have constructed let us recover the correct bulk two-point functions from the CFT, 11 especially for points inside the Rindler horizon. Clearly of special importance is the point r = 0, where the Rindler coordinates become singular. So in this section we show how this works for a point located at r = 0 and a point near the right boundary. \nThe AdS Wightman function is \nG AdS ( x | x ' ) = 〈 0 | φ ( x ) φ ( x ' ) | 0 〉 SUGRA = 1 4 πR 1 √ σ 2 -1 1 ( σ + √ σ 2 -1) ∆ -1 . (49) \nHere | 0 〉 is the global or AdS-invariant vacuum state. Branch cuts are handled with a τ → τ -i/epsilon1 prescription, or equivalently σ → σ + i/epsilon1 sin( τ -τ ' ), where τ is the global time coordinate defined in appendix B. 12 We consider a point near the origin of Rindler coordinates ( t = 0 , r = r 0 , φ = 0), and a point near the right boundary with coordinates ( t, r, φ ). As r 0 → 0 and r → ∞ the invariant distance (37) is \nσ ≈ r r + ( r 0 r + cosh ˆ φ +sinh ˆ t ) . \nThus the AdS correlator approaches a finite, ˆ φ -independent value as r 0 → 0 \nG AdS (0 , 0 , 0 | ˆ t, r, ˆ φ ) ≈ 1 2 πR ( r + 2 r sinh ˆ t + i/epsilon1 ) ∆ . (50) \nThe fact that the correlator is independent of ˆ φ reflects the fact that r = 0 is a fixed point of the isometry ˆ φ → ˆ φ +const. \nNow let's see how this behavior is reproduced by the CFT. We'll work with a field of integer conformal dimension. At ˆ t = r = ˆ φ = 0 the smearing function (46) reduces to \nφ (0 , 0 , 0) = (∆ -1)2 ∆ -2 πr ∆ + ∫ ∞ 0 dx sinh ∆ -2 x ∫ π/ 2 -π/ 2 dy ( φ Rindler , R 0 ( x, iy ) + ( -1) ∆ φ Rindler , L 0 ( x, iy ) ) (51) \nwhile as r →∞ the smearing function (41) reduces to \nφ ( ˆ t, r, ˆ φ ) ≈ r -∆ φ Rindler , R 0 ( ˆ t, ˆ φ ) . (52) \nThis means we should be able to recover (50) by computing \n(∆ -1)2 ∆ -2 π ( rr + ) ∆ ∫ ∞ 0 dx sinh ∆ -2 x ∫ π/ 2 -π/ 2 dy (53) 〈( φ Rindler , R 0 ( x, iy ) + ( -1) ∆ φ Rindler , L 0 ( x, iy ) ) φ Rindler , R 0 ( ˆ t, ˆ φ ) 〉 CFT . \nFor convenience we'll work in the regime \nˆ t < 0 , ˆ t < ˆ φ < -ˆ t . (54) \nIn this regime the smeared CFT operators are never lightlike separated, so (53) is well defined without a prescription for dealing with lightcone singularities in the CFT. The appropriate CFT correlators can be obtained from (49) by sending the bulk points to the appropriate boundary 13 \n〈 φ Rindler , R 0 ( ˆ t, ˆ φ ) φ Rindler , R 0 ( ˆ t ' , ˆ φ ' ) 〉 CFT = ( r 2 + / 2) ∆ 2 πR ( cosh( ˆ φ -ˆ φ ' ) -cosh( ˆ t -ˆ t ' -i/epsilon1 ) ) ∆ 〈 φ Rindler , R 0 ( ˆ t, ˆ φ ) φ Rindler , L 0 ( ˆ t ' , ˆ φ ' ) 〉 CFT = ( r 2 + / 2) ∆ 2 πR ( cosh( ˆ φ -ˆ φ ' ) + cosh( ˆ t -ˆ t ' ) ) ∆ . (55) \nThen we have \n(∆ -1) r ∆ + 8 π 2 Rr ∆ ∫ ∞ 0 dx sinh ∆ -2 x ∫ π/ 2 -π/ 2 dy [ ( cosh( ˆ φ -iy ) -cosh( ˆ t -x ) ) -∆ +( -1) ∆ ( cosh( ˆ φ -iy ) + cosh( ˆ t -x ) ) -∆ ] = (∆ -1) r ∆ + 8 π 2 Rr ∆ ∫ ∞ 0 dx sinh ∆ -2 x ∫ π -π dy ( cosh( ˆ φ -iy ) -cosh( ˆ t -x ) ) -∆ = (∆ -1)2 ∆ -3 r ∆ + iπ 2 Rr ∆ ∫ ∞ 0 dx sinh ∆ -2 x ∮ | z | = e φ z ∆ -1 dz ( z -e ˆ t -x ) ∆ ( z -e -( ˆ t -x ) ) ∆ (56) \nIn the last line we set z = e ˆ φ -iy . In the regime (54) the z contour of integration always encircles the pole at e ˆ t -x and never encircles the pole at e -( ˆ t -x ) . To analytically continue outside (54) we proceed as in section 2.4 and deform the contour of integration so that it continues to encircle the appropriate pole. This continuation gives an integral that is independent of ˆ φ , and in this way the smearing function (51) captures the fact that r = 0 is a fixed point of the isometry φ → φ +const. It's entertaining to push the \ncalculation a little further and show that the CFT exactly reproduces the bulk correlator. Just to be concrete, let's set ∆ = 2. Then evaluating the contour integral in (56) gives \n-r 2 + πRr 2 ∫ ∞ 0 dx 2 cosh( ˆ t -x ) 2 sinh( ˆ t -x ) ) 3 = 1 2 πR ( r + 2 r sinh ˆ t ) 2 (57) \n( \n) in agreement with (50) for ∆ = 2. The result is also valid outside the range (54) using the analytic continuation described above.", '4 BTZ black hole': "To make a BTZ black hole starting from AdS 3 all we have to do is periodically identify the φ coordinate, φ ∼ φ + 2 π [25, 26]. Scalar fields on AdS 3 will descend to scalar fields on BTZ provided they satisfy φ ( t, r, φ ) = φ ( t, r, φ + 2 π ). The global AdS vacuum descends to the Hartle-Hawking vacuum state in BTZ. \nIn this construction we are identifying points separated by real values of the φ coordinate. Since the Rindler smearing functions we have constructed are translation invariant in φ , and since they only involve integration over the imaginary part of φ , we can apply our Rindler results to a BTZ black hole without modification. That is, (41) and (46) can be used to represent bulk fields in a BTZ spacetime; if the boundary field has the correct periodicity then so will the bulk field. This shows quite explicitly that we can recover local physics outside the BTZ horizon using operators that act on a single copy of the CFT, while to describe the region inside the horizon we must use operators that act on both the CFT and its thermofield double [6, 16, 17, 18, 27]. \nThe BTZ black hole has a spacelike singularity at r = 0, which has been studied from the CFT point of view in [17, 18, 21, 23]. 14 In the semiclassical limit that we are considering this singularity should be encoded in the CFT. We are now in a position to see this directly, by studying bulk correlators with one point close to the singularity. \nThe BTZ Wightman function is given by an image sum [28, 29] \nG BTZ ( x | x ' ) = ∞ ∑ n = -∞ G AdS ( t, r, φ | t ' , r ' , φ ' +2 πn ) . (58) \nThis diverges when r = 0, just because r = 0 is a fixed point of the isometry of shifting φ by a constant: when r = 0 the invariant distance (37) is independent of φ ' and the image sum diverges. To estimate the divergence, note that for small r 0 the BTZ image sum is cut off at | n | ≈ 1 2 π log( r + /r 0 ). This means the BTZ Wightman function diverges logarithmically near the singularity \nG BTZ (0 , r 0 , 0 | ˆ t, r, ˆ φ ) ∼ 1 2 π 2 R ( r + 2 r sinh ˆ t ) ∆ log r + r 0 as r 0 → 0 . \nHow does this divergence arise from the CFT viewpoint? A priori there are a number of possibilities: \n- · The CFT itself could be incomplete in the same sense as classical gravity.\n- · The mapping between CFT operators and local bulk fields could become singular at this point.\n- · The mapping could remain smooth, but the CFT operator moves outside the class of physically reasonable observables. \nThe boundary S-matrix in the gravity theory appears to be well-defined around the BTZ background by virtue of cosmic censorship, provided one avoids processes that produce naked singularities. Hence the same will be true of the CFT correlators, so in that sense the CFT gives a complete well-defined theory at large N . Thus the first possibility is ruled out. The mapping is non-singular, as can be seen explicitly in (51), which rules out the second possibility. It is the third possibility which is realized. \nBefore discussing this in more detail, let us follow through with our calculation of the bulk two point function using the CFT. As in section 3.3 we place one point near the singularity and the other near the right boundary. \nThen all we have to do is replace the AdS boundary correlators with BTZ boundary correlators in (53). 15 Boundary correlators in the BTZ geometry can be obtained from (55) by performing an image sum to make them 2 π periodic in φ [30]. However as we have seen (53) gives a result that is independent of φ . Therefore substituting BTZ boundary correlators in (53) leads to a divergent image sum. So the divergence is present in the CFT computation of the correlator, for the same reason it was present in the bulk. \nNow let's make some comments on the interpretation of this divergence. In AdS 3 as two bulk points coincide their correlator exhibits the expected Hadamard short-distance singularity \nG AdS ∼ 1 4 πR 2( σ -1) as σ → 1 . (59) \nGenerically as two points coincide in BTZ their correlator diverges in exactly the same way, because only one term in the image sum (58) will have a singularity. However if we place one point at the BTZ singularity then G BTZ diverges no matter where the other point is located. This is because r = 0 is a fixed point of the orbifold symmetry and the symmetry operation is of infinite order. \n√ \nWecan use the coefficient of the singularity (59) as a definition of the norm of these operators. For generic points the norm is finite, however the norm diverges for the operator at the fixed point. One way to see this is by using a point splitting regularization and considering lim /epsilon1 → 0 G BTZ (0 , 0 , 0 | /epsilon1 r , /epsilon1 φ , /epsilon1 t ). The invariant distance is independent of the coordinate separation in the φ direction if one point lies at r = 0, so G BTZ (0 , 0 , 0 | /epsilon1 r , /epsilon1 φ , /epsilon1 t ) diverges even at finite /epsilon1 . Thus the operator φ | r =0 has infinite norm. \nIn the CFT we interpret the operator (51) dual to φ | r =0 exactly as in the bulk. It is a non-normalizable operator which has divergent correlators with all operators of interest. This is how a well-behaved conformal field theory gives rise to a divergent correlation function: through the introduction of a non-normalizable operator. We will comment further in section 7 on how this picture generalizes when back-reaction and finite N are taken into account. \nFigure 4: An AdS black hole formed by collapse. The left edge of the diagram is the origin of AdS, the right edge is the AdS boundary. The dashed line is the black hole horizon while the solid diagonal line represents the infalling shell. \n<!-- image -->", '5 Collapse geometries': "As we have seen, it is possible to probe the region inside the horizon of a BTZ black hole using operators that act on both the left and right copies of the CFT. A similar result should hold for a general eternal AdS-Schwarzschild black hole. However in the more physical case of a black hole formed in collapse there is only a single asymptotic AdS region, and one might ask: can the region inside the horizon be described using the single copy of the CFT? \nFor simplicity let's work in AdS 3 and consider a large (stable) black hole formed by sending in a null shell from the boundary. The Penrose diagram is shown in Fig. 4. Consider a bulk point P inside the horizon and to the future of the shell. Can an operator inserted at that point be described in the CFT? \nThe answer is yes, and for fields with integer conformal dimension the explicit construction is quite simple. As can be seen from the global mode \nexpansion given in appendix B, fields with integer conformal dimension are single-valued on the AdS hyperboloid (periodic in global time with period 2 π ). Note from (33) that continuing ˆ t → ˆ t + iπ has the effect of changing the sign of two of the embedding coordinates, namely \nV →-V X →-X. \nThus for integer conformal dimension the boundary fields in the left and right Rindler wedges are related by \nφ Rindler , L 0 ( ˆ t, ˆ φ ) = φ Rindler , R 0 ( ˆ t + iπ, ˆ φ ) . (60) \n(This relation was also used in [16]). The collapse geometry can be made by taking the right and future regions of an eternal BTZ black hole and joining them across the shell to a piece of AdS 3 . For points to the future of the shell we can, by analytic continuation, pretend that we are in an eternal BTZ geometry. We can therefore use the relation (60) in our BTZ smearing function (46) to represent bulk operators that are located inside the horizon. \nThis shows that we can represent a bulk point inside the horizon in terms of a single CFT, provided we analytically continue in both the ˆ t and ˆ φ coordinates. Our explicit construction works for points that are to the future of the infalling shell. One could also ask about representing bulk points inside the shell, such as the point Q in Fig. 4. This is indeed possible, although the construction is more complicated since one must propagate modes across the shell [31].", '6 Comments on finite N': "We have seen that in the semiclassical limit one can construct local operators anywhere in the bulk of AdS. However at finite N , when the Planck length is finite, holography demands that the number of independent degrees of freedom inside a volume is finite, bounded by the area of the region in Planck units. In this section we attempt to understand how this comes about. \nThe smeared operators we have constructed in the CFT are still welldefined at finite N . For example in N = 4 Yang-Mills we can define the \noperator \nΦ( T, X, Z ) = ∫ dT ' d 3 X ' K ( T ' , X ' | T, X, Z ) Tr F 2 ( T ' , X ' ) (61) \nat any N . At finite N it does not obey the correct bulk dilaton equation of motion [10]. However it is a perfectly good operator in the gauge theory, and it has the right behavior in the largeN limit to be associated with a particular point in the bulk. So as a first step, it seems reasonable to associate Φ( T, X, Z ) with a point in the bulk, even at finite N . Since the bulk point was arbitrary, at first sight this means we can associate an infinite number of local operators with any given region in the bulk. \nThis might seem like a surprising conclusion, so let us give supporting evidence for our approach. Consider pure AdS D , dual to a CFT D -1 in its ground state. The conformal symmetry of the CFT is valid at any N . This means that, even when the Planck length is finite, AdS quantum gravity has an exact SO ( D -1 , 2) symmetry. Purely formally, we can realize this symmetry as acting on a set of coordinates ( T, X, Z ). The smearing functions we have constructed transform covariantly under SO ( D -1 , 2) [7] - a property which suffices to determine them up to an overall coefficient. 16 This means that at any N , the smearing functions we have defined are the unique way to start with a primary operator in the CFT and build a representation of SO ( D -1 , 2) which transforms as a scalar field in AdS. Since the construction we have outlined is fixed by the symmetries, the operators (61) are singled out even at finite N . \nHow can this continuum of operators be compatible with holography? We believe the answer is that only a few of these operators will commute with each other at finite N . At infinite N we managed to construct smeared operators in the CFT which commute with each other even though the smearing functions overlap on the boundary. We discussed this in section 2.4. But at finite N it is implausible that all the overlapping operators will commute. 17 Let's estimate how many commuting operators we do expect. If we take a local CFT operator and smear it, it will trivially commute with another smeared operator provided the two smearings are 'spacelike' to each other: \nthat is, provided the two smearing functions have supports which only involve points on the boundary that are at spacelike separation. In this case the two smeared operators will commute with each other by locality of the boundary theory. The condition for spacelike separation was studied in [6] for AdS 3 and is easily extended to any dimension. In Poincar'e coordinates, working on a hypersurface of fixed time, it boils down to the requirement that the separation between any two bulk operators satisfies | ∆ X | > 2 Z . Since the necessary separation gets larger as Z increases, the maximum number of commuting operators in a given region is achieved by placing them all at the boundary of the region. For example, inside a bulk region \n0 < X i < L Z 0 < Z < ∞ i = 1 · · · d -1 \nthe maximum number of trivially commuting operators is given when they are evenly spaced along the boundary of the region, at Z = Z 0 , with a characteristic coordinate spacing of order Z 0 . Thus according to this prescription there are at most ∼ ( L/Z 0 ) d -1 trivially commuting operators one can build in this region by smearing a single local operator in the CFT 18 . This corresponds to one commuting operator per AdS area (in units of the AdS radius of curvature R ). This is far too few degrees of freedom to describe a local bulk field. \nTurning back to the infinite number of operators described above, we note that - although they do not all commute - their correlation functions nonetheless look local up to 1 /N corrections that involve mixing with other operators. The infinite set of operators can therefore be used to describe bulk physics which is approximately local, at least as far as correlation functions are concerned, as long as the 1 /N corrections can be ignored. However note that if one tries to place operators too close together or in a state with large energy, their commutator may get a large contribution from smeared operators corresponding to bulk excitations which are outside the given spacetime volume. We should not associate such operators with independent degrees of freedom within the volume. Presumably there is a finite maximal set of operators that mutually commute up to terms that vanish as N → ∞ and remain inside the given volume. Bekenstein-style arguments [33] (made on the supergravity side) support this idea. It is this set of operators which we argue counts the independent degrees of freedom inside a volume. \nWe obtain a natural proposal for a basis of these operators by generalizing the above construction of trivially commuting operators. Let us consider all possible degrees of freedom within a given bulk volume, rather than those associated with a particular supergravity field. For concreteness, we will consider AdS 5 . We expect of order N 2 independent local operators in the boundary theory. Therefore we should be able to construct a basis of N 2 mutually commuting bulk operators as we did above for the trivially commuting operators. This implies a total of N 2 degrees of freedom per area in AdS units. This matches perfectly with the relation l 3 Planck = R 3 /N 2 and saturates the holographic bound.", '7 Conclusions': 'In this paper we developed the representation of local operators in the bulk of AdS in terms of non-local operators on the complexified boundary. We showed that these non-local operators reproduce the correct bulk-to-bulk correlation functions. In particular they reproduce the divergent correlators of an operator inserted at the BTZ singularity. We commented on black holes formed by collapse, and discussed the way in which bulk locality arises in the largeN limit but breaks down at finite N . \nLocal bulk operators thus provide a powerful tool for understanding the AdS/CFT correspondence. They give new insights into the way in which light-cone singularities and spacelike commutativity arise in the bulk. They enable us to probe non-trivial geometries, including regions inside horizons which are naively hidden from the boundary, and they show very explicitly how a bulk singularity can manifest itself in a well-behaved CFT. Our results were all obtained in the infinite N limit. However we argued that in some contexts (2-point functions in pure AdS) our results carry over exactly to any value of N . And based on consideration of the operators at infinite N we were able to give a qualitative picture of the independent bulk degrees of freedom at finite N . \nThere are a number of directions for future work. We begin with a few further remarks on the nature of the BTZ singularity from the CFT viewpoint. At leading order in a large N expansion, we found that a bulk field probe of the singularity is represented by a non-normalizable operator in the CFT. \nNote that back-reaction/finite N effects play a crucial role in understanding the physics near the singularity, even in the case of BTZ, as discussed in [28, 34, 16] (and references therein). Therefore we certainly expect large corrections to the smearing function within a Planck length of the singularity. It would be interesting to know whether these corrections render operators at the singularity normalizable, or whether one should simply abandon a bulk spacetime description of the physics in this region. Nevertheless it seems the operators defined by (46) have smooth analytic continuations through complex values of r from region 2 ++ (in the notation of [16]) to the past of the singularity to region 2 -+ to the future of the singularity, avoiding the Planck scale region near the singularity. This raises the question of whether the CFT also gives a smooth description of regions to the future of the singularity. An important criterion in deciding whether certain combinations of CFT correlators reproduce sensible bulk spacetime physics, is whether the set of amplitudes can be reproduced by a unitary local bulk Lorentzian spacetime effective action. This seems to be true for regions outside the horizon, and regions to the past of the singularity, but it is unlikely this will be true if one also includes operators to the future of the singularity. It would be very interesting show this explicitly. Moreover the resolution of the black hole information problem via AdS/CFT suggests [35, 36] that non-local terms appear in a bulk effective action connecting the region near the singularity with the region outside the horizon. The local operators constructed in the present work are an important first step in trying to reconstruct these new quantum gravity features of the bulk effective action. \nIn section 6 we commented on the way in which the number of commuting degrees of freedom is reduced at finite N . We also showed how bulk locality is recovered in correlation functions in the largeN limit, despite the seemingly low number of degrees of freedom (corresponding to a theory with a cutoff ∆ X > Z ): namely, through the presence of a continuum of bulk operators whose commutators are O (1 /N ). Constructing a precise analog of smearing functions at finite N and better understanding the analog of bulk spacetime geometry is an important open problem. \nFor eternal black holes we found that local operators inside the horizon are dual to operators which act on both copies of the CFT. In section 5 we showed that, at least in some cases, one could represent an operator inside the horizon of a black hole formed by collapse in terms of a single CFT, by using an operator which is analytically continued both in the spatial and \ntemporal coordinates of the CFT. These ideas will be further explored in [31]. \nThis leads to an interesting question, namely, whether there is an algorithm for constructing smearing functions with compact support in a general asymptotically AdS background. The smearing functions we have constructed in this paper can all be thought of as arising from a Wick rotation of the boundary spatial coordinates. This should certainly be a well-defined operation on the analytic correlators that arise from the CFT. However a general bulk geometry will typically not have an interpretation with a real metric after performing such a continuation. One could still try to represent the smearing function as a mode sum, but it is not clear that the smearing function will have compact support on the (complexified) boundary. One way to address this issue would be to attempt to find a procedure, purely within the CFT, for identifying a set of well-behaved smearing functions. The only obvious condition to impose is that in the semiclassical limit the smeared operators should commute at bulk spacelike separation. Is that enough to uniquely determine the smearing functions?', 'Acknowledgements': 'We thank Raphael Bousso, Asad Naqvi and Shubho Roy for valuable discussions. DK and DL are grateful to the 2006 Simons Workshop for hospitality. The work of AH and DK is supported by US Department of Energy grant DE-FG02-92ER40699. AH is supported in part by the Columbia University Initiatives in Science and Engineering. GL is supported in part by the Israeli science foundation, grant number 568/05. The research of DL is supported in part by DOE grant DE-FG02-91ER40688-Task A.', 'A Rindler mode sum': "In this appendix we set up the Rindler mode sum for a bulk point inside the horizon. It's convenient to introduce Kruskal coordinates on AdS 3 in which \nds 2 = -4 R 2 (1 + uv ) 2 dudv + r 2 dφ 2 . \nThese coordinates are defined by \nu = ( r -r + r + r + ) 1 / 2 e ˆ t v = -( r -r + r + r + ) 1 / 2 e -ˆ t \nin the right Rindler wedge and \nu = ( r + -r r + + r ) 1 / 2 e ˆ t u = ( r + -r r + + r ) 1 / 2 e -ˆ t \nin the future Rindler wedge; to cover the left and past wedges just change the signs of both u and v . A complete set of normalizable modes in the right Rindler wedge is given by \nφ R ( t, r, φ ) = e -iωt e ikφ r -∆ ( 1 -r 2 + r 2 ) -i ˆ ω/ 2 F ( ∆ 2 -i ˆ ω + , ∆ 2 -i ˆ ω -, ∆ , r 2 + r 2 ) (62) \nwhere ω, k ∈ R , ˆ ω ± = 1 2 (ˆ ω ± ˆ k ), ˆ ω = ωR 2 /r + , ˆ k = kR/r + . We can extend this mode to the entire Kruskal diagram by analytically continuing across the Rindler horizons. If we continue through the lower half of the complex u and v planes we get a mode which is positive frequency with respect to Kruskal time, while continuing through the upper half of the complex u and v planes gives a negative frequency Kruskal mode. 19 The analytic continuation is straightforward, with the help of a z → 1 -z transformation of the hypergeometric function. Define \nf ωk ( r ) = 1 r ∆ ( 1 -r 2 + r 2 ) -i ˆ ω/ 2 F ( ∆ 2 -i ˆ ω + , ∆ 2 -i ˆ ω -, ∆ , r 2 + r 2 ) g ωk ( r ) = 1 r ∆ ( r 2 + r 2 -1 ) -i ˆ ω/ 2 Γ(∆)Γ( i ˆ ω ) Γ((∆ / 2) + i ˆ ω + )Γ((∆ / 2) + i ˆ ω -) F ( ∆ 2 -i ˆ ω + , ∆ 2 -i ˆ ω -, 1 -i ˆ ω, 1 -r 2 + r 2 ) . \nThen a complete set of { positive negative } frequency Kruskal modes is given by \nφ ± R ( t, r, φ ) = e -iωt e ikφ f ωk ( r ) (63) φ ± F ( t, r, φ ) = e -iωt e ikφ ( g ωk ( r ) + e ∓ π ˆ ω g -ω,k ( r ) ) φ ± L ( t, r, φ ) = e ∓ π ˆ ω e -iωt e ikφ f ωk ( r ) \nin the (right, future, left) Rindler wedges. This means we can express the value of the field in the future wedge in terms of data on the right and left boundaries, via \nφ F ( t, r, φ ) = ∫ dωdk 1 4 π 2 g ωk ( r ) [ ∫ dt ' dφ ' ( e -iω ( t -t ' ) e ik ( φ -φ ' ) φ Rindler , R 0 ( t ' , φ ' ) + e -iω ( -t + t ' ) e ik ( φ -φ ' ) φ Rindler , L 0 ( t ' , φ ' ) ) ] . \n(Recall that time is oriented oppositely on the two boundaries, so for t = 0 this expression is in fact symmetric between the right and left boundaries.) Switching the order of integration and performing the ω and k integrals first gives a formal representation of the Rindler smearing function, essentially as the Fourier transform of g ωk . However it is easy to check that g ωk grows exponentially as k →±∞ . So we are not justified in switching the order of integration and the Fourier transform does not exist. One can presumably make sense of the Rindler smearing function in this approach by deforming the contours of integration as in [7]. For points inside the horizon this should reproduce the result (46) we obtained from Poincar'e coordinates.", 'B Non-integer ∆': "In this appendix we work out the generalization of the Rindler smearing function (46) appropriate for arbitrary conformal dimension. \nWe first need to discuss the generalization of the antipodal map. This is easiest to understand in global coordinates, where the embedding coordinates \nof section 3.1 are given by \nU = R cos τ/ cos ρ V = R sin τ/ cos ρ X = R cos θ tan ρ Y = R sin θ tan ρ \nfor -∞ < τ < ∞ , 0 ≤ ρ < π/ 2, θ ∼ θ +2 π . The induced metric is \nds 2 = R 2 cos 2 ρ ( -dτ 2 + dρ 2 +sin 2 ρdθ 2 ) . \nThe antipodal map acts by \nA : ( τ, ρ, θ ) → ( τ -π, ρ, θ + π ) . \nThe global mode expansion is \nφ ( τ, ρ, θ ) = ∞ ∑ n =0 ∞ ∑ l = -∞ a nl e -iω nl τ e ilθ sin | l | ρ cos ∆ ρ P ( | l | , ∆ -1) n (cos 2 ρ ) + c.c. \nwhere ω nl = 2 n + | l | +∆ and P n is a Jacobi polynomial. So fields which are { positive negative } frequency with respect to global time satisfy \nφ ± ( x ) = e ∓ iπ ∆ φ ± ( Ax ) . \nThis means the generalization of (46) to arbitrary conformal dimension is \nφ = (∆ -1)2 ∆ -2 πr 2 + [∫ σ> 0 dxdy lim r ' →∞ ( σ/r ' ) ∆ -2 φ Rindler , R 0 ( ˆ t + x, ˆ φ + iy ) (64) \n+ ∫ σ< 0 dxdy lim r ' →∞ ( -σ/r ' ) ∆ -2 ( e -iπ ∆ φ Rindler , L 0+ ( ˆ t + x, ˆ φ + iy ) + e iπ ∆ φ Rindler , L 0 -( ˆ t + x, ˆ φ + iy ) ) ] \nwhere we've decomposed the left boundary field into pieces φ Rindler , L 0 ± that are { positive negative } frequency with respect to global (equivalently, Kruskal) time. These may in turn be expressed in terms of integrals involving φ Rindler , L 0 and φ Rindler , R 0 over all time, as in appendix B of [6].", 'References': "- [1] J. M. Maldacena, 'The large N limit of superconformal field theories and supergravity,' Adv. Theor. Math. Phys. 2 (1998) 231-252, hep-th/9711200.\n- [2] S. S. Gubser, I. R. Klebanov, and A. M. 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1997NuPhB.486...77D

BPS spectrum of the five-brane and black hole entropy

1997-01-01

['-']

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We propose a formulation of ]]-dimensional M-theory in terms of five-branes with closed strings on their world-volume. We use this description to construct the complete spectrum of BPS states in compactifications to six and five dimensions. We compute the degeneracy for fixed charge and find it to be in accordance with U-duality (which in our formulation is manifest in six dimensions) and the statistical entropy formula of the corresponding black hole. We also briefly comment on the compactification to four dimensions.

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https://arxiv.org/pdf/hep-th/9603126.pdf

{'No Header': 'March 1996 cern-th /96-74 pupt -1604', 'Robbert Dijkgraaf': 'Department of Mathematics University of Amsterdam, 1018 TE Amsterdam', 'Erik Verlinde': 'TH-Division, CERN, CH-1211 Geneva 23 and Institute for Theoretical Physics Universtity of Utrecht, 3508 TA Utrecht \nand', 'Herman Verlinde': 'Institute for Theoretical Physics University of Amsterdam, 1018 XE Amsterdam and Joseph Henry Laboratories Princeton University, Princeton, NJ 08544', 'Abstract': 'We propose a formulation of 11-dimensional M-theory in terms of five-branes with closed strings on their world-volume. We use this description to construct the complete spectrum of BPS states in compactifications to six and five dimensions. We compute the degeneracy for fixed charge and find it to be in accordance with U-duality (which in our formulation is manifest in six dimensions) and the statistical entropy formula of the corresponding black hole. We also briefly comment on the compactification to four dimensions.', 'Introduction.': 'One of the outstanding challenges that one faces in building a successful theory of quantum gravity is to provide a microscopic description of black holes that explains and reproduces the Hawking-Bekenstein formula for the entropy [1]. This question has been hard to answer within string theory because, until recently, black holes only arose as particular classical solitons of the low energy effective field theory and hence could only be studied in string perturbation theory. This situation has changed quite dramatically through the recent developments related to string duality [2, 3]. It has been discovered that black holes and other RR-solitons have an exact description in string theory in terms of D-branes [4], and that their quantum states are related by duality to elementary strings excitations. These facts have been used recently in [5] to give a microscopic description in terms of D-branes of certain five-dimensional extremal black holes and to check that it reproduces the expected entropy formula (see also [6], and for extensions to 4 dimension [7]). The central idea of this computation is a comparison of the asymptotic growth of the number of BPS states in string theory with a given charge and the area or volume of the horizon of the corresponding black hole geometry. Specifically, the statistical entropy S ( Z ) as a function of the charge Z is expected to behave as \nS ( Z ) = 2 π | Z | α (1) \nwith α = 2 , 3 2 , 1 for dimension d = 4 , 5 , 6 respectively. The precise form of the right-hand side is further restricted by the required invariance under the U-duality symmetry group. \nU-duality and other string dualities are still rather mysterious in the D-brane description, because it necessarily makes a distinction between charges that correspond to RR-fields and those of the NS-sector of the theory. On the other hand, there are many indications that elementary strings and all other p -branes have a unified description in 11 dimensions [8]. One therefore expects that the U-duality invariance of the BPS spectrum can be explained by extending the D-brane analysis to this eleven-dimensional M-theory [3, 9, 10]. It is known that M-theory contains membranes and five-branes, which are charged relative to the three-form gauge field C 3 and its six-form dual ˜ C 6 respectively. However, by forming bound states the five-brane can also carry C 3 charges and it therefore seems the natural starting point for a unified treatment of all BPS states in string theory. \nIt has been suggested that the five-brane can be viewed as a D-object on which membranes can have boundaries [11]. In this picture the world-volume theory of the five-brane is induced by the boundary states of the membrane and thus is naturally described by a closed string theory. In this letter we propose to take this description seriously. We will present evidence that it indeed gives rise to a complete, U-duality invariant counting of BPS states. In our presentation we will focus on the central ideas of the construction, since the details of the calculations will be published elsewhere [13].', 'Strings on the Five-Brane': "For definiteness and simplicity, we will concentrate on toroidal compactifications of Mtheory, because this will allow us to make maximal use of the space-time supersymmetries. Also, we consider only flat five-branes with the topology of T 5 . The five-brane represents a soliton configuration that breaks half of the 32 space-time supersymmetries of M-theory. On the 5 + 1 dimensional world-volume the unbroken supercharges combine into 4 fourcomponent chiral spinors that generate a N = (4 , 0) supersymmetry. It is known [14] that the effective world-brane theory is, after appropriate gauge-fixing, described by a tensor multiplet containing 5 scalars, an anti-symmetric tensor b + with self-dual three-form field strength db + , and 4 chiral fermions ψ . These fields must represent the massless states of the closed strings that live on the world-brane. One may think about these strings as solitons of the effective world brane theory that are charged with respect to the tensor b + . The self-duality of db + implies that we are dealing with a 6-dimensional self-dual string theory (cf. [15]). \nThe string configurations break half of the world-brane supersymmetries, and hence the world sheet formulation of this string must contain 8 supercharges: 4 left-moving and 4 right-moving. The 8 remaining supercharges, those that are broken by the string, give rise to 4 left-moving and 4 right-moving fermionic Goldstone modes λ and λ . We assume that the world-sheet theory can be formulated in a light-cone gauge, and so one expects to have 4 bosonic fields x that describe the transversal directions inside the 5+1 dimensional world-volume of the five-brane. We find it convenient to label these fields using chiral spinor indices a and ˙ a of the transversal SO (4) rotation group. In this notation the left-moving fields on the string world-sheet are \nx a ˙ a ( z ) , λ a α ( z ) , (2) \nand the right-moving fields are \nx a ˙ a ( z ) , λ a ˙ α ( z ) , (3) \nwhere we used the fact that the fermions λ and λ have the same chirality with respect to the SO (4). The indices α ( ˙ α ) are (anti-)chiral spinor indices of another SO (4) which, as will become clear, becomes identified with part of the space-time rotations. On the five-brane it will be realized as an R-symmetry. \nWe propose to take the fields ( x a ˙ a , λ a α , λ a ˙ a ) as the complete field content of the worldsheet theory in the light-cone gauge, without worrying about a possible covariant formulation. In fact, we have essentially half of the world-sheet fields of the type IIB string in the Green-Schwarz formulation ∗ and we can use this analogy to check that we get indeed the right massless fields. The ground states must form a multiplet of the left-moving \nzero-mode algebra { λ α a , λ β b } = /epsilon1 ab /epsilon1 αβ . This gives 2 left-moving bosonic ground states | α 〉 and 2 fermionic states | a 〉 . By taking the tensor product with the right-moving vacua one obtains in total 16 ground states \n( | α, k L 〉 ⊕ | a, k L 〉 ) ⊗ ( | ˙ β, k R 〉 ⊕ | b, k R 〉 ) . (4) \nHere we also took into account the momenta ( k L , k R ), which form a Γ 5 , 5 lattice, since we have assumed that the five-brane has the topology of T 5 . Level matching implies that for the ground states one should have that | k L | = | k R | . Notice that these ground states are stable as long as the pair ( k L , k R ) is a primitive vector on Γ 5 , 5 . \nIgnoring for a moment the string winding numbers, we find that the ground states indeed represent the Fourier modes of the massless tensor multiplet on the five-brane. Specifically, the states | α ˙ β 〉 describe four scalars X α ˙ β = σ α ˙ β i X i that transform as a vector of the SO (4) R-symmetry, while | αb 〉 and | a ˙ β 〉 describe the eight helicity states of the 4 world-brane fermions ψ α and ψ ˙ α . Finally, the RR-like states | ab 〉 decompose into a fifth scalar, which we call Y , and the 3 helicity states of the tensor field b + . These are indeed the fields that parametrize the collective excitations of the five-brane soliton. In our description, however, these are just the low energy modes. The complete set of fluctuations of the five-brane are parametrized by the quantum string states. We also note that from the point of view of the string world-sheet the 5 scalars on the five-brane that describe the transverse oscillations naturally split up into four X i 's plus one 'RR'-scalar Y . The interpretation of these fields in terms of the five-brane geometry will be discussed in detail in [13]. From this point of view it can be shown that Y is naturally compactified. \nJust as in the type II superstring one can combine the 4 left-moving and 4 right-moving supercharges G ˙ aα = ∮ ∂x ˙ ab λ α b and G ˙ a ˙ α = ∮ ∂x ˙ ab λ ˙ α b together with the eight fermion zeromodes of λ aα and λ a ˙ β to construct the N = (4 , 0) supercharges Q on the world-volume of the five-brane. The fact that the string lives on a 5-torus, however, has important consequences for the supersymmetry algebra. Namely, the anti-commutator of the leftmoving supercharges produces the left-moving momentum k L , while the right-movers give k R . Hence the string states form representations of the N = (4 , 0) supersymmetry algebra † \n{Q aα , Q bβ } = /epsilon1 αβ ( P 0 1 ab + P m L Γ ab m ) , a ˙ α , Q b ˙ β } = /epsilon1 ˙ α ˙ β ( P 0 1 ab + m Γ ab m ) , (5) \n{Q \nP R \ncanceled by adding a term of the form ∫ b + tr R 2 to the world volume action of the five brane and by accompanying the world-sheet reparametrizations by a gauge transformation of the b + field. A similar mechanism is described in the last reference of [10]. \n† We use a Hamiltonian notation with only the spatial rotation group SO (5) manifest. Hence from now on the indices a and ˙ a are combined into a four-valued SO (5) spinor index a . The index m corresponds to a SO (5) vector. \nwhere Γ ab m are SO (5) gamma matrices. The operators P 0 , P m L and P m R act on multistring states that form the Hilbert space of the five-brane. The five-brane Hamiltonian P 0 measures the energy of the collection of strings, while 1 2 ( P m L + P m R ) measure the total momentum. But we see that the algebra also contains a vector central charge P m L -P m R , which measures the sum of the string winding numbers around the 5 independent onecycles on the T 5 of the world-brane.", 'Counting Multiple BPS Strings': "The self-dual string is an interacting theory. It has no weak coupling limit, since its coupling is fixed by the self-duality relation. In the following, however, we will assume that for the purpose of counting the number of BPS-states, it will be an allowed procedure to treat it as a theory of non-interacting strings. The BPS-restriction should indeed limit the possible interactions that can take place. Furthermore, we will find that our degeneracy formulas will be consistent with previous results obtained from D-brane technology [18, 19, 5], as well as with U-duality. \nOur aim is to count BPS states of the space-time theory that respect either 1 / 4 or 1 / 8 of the supersymmetries. On the world-brane this translates to a condition that either 8 or 4 of the 16 supercharges annihilate the states. The strongest BPS condition is obtained by demanding that \nε aα Q aα | BPS 〉 = 0 , (6) \nfor four independent spinors /epsilon1 aα , and at the same time imposing four similar relations for the supercharges Q a ˙ β . By dropping these latter four relations one gets a weaker BPS condition that, as we will see, allows many more states. In fact, we can easily treat both cases in parallel. \nWithout loss of generality we can assume that | BPS 〉 is an eigenstate of P m L and P m R with eigenvalue P m L and P m R . Since the condition (6) holds for all the states | BPS 〉 in the same multiplet, we can use the algebra (5) to deduce that \nε αa ( P 0 1 ab + P m L Γ ab m ) = 0 . (7) \nThe equation (7) only has solutions when P 0 = ±| P L | . To count how many states we have for a given value of P m L we have to consider the multiple string states that have a total left-moving momentum P m L ∼ ∑ k m L and total energy | P L | . Since the energy of a single string is bounded from below by | k L | , we deduce that in fact all momenta k m L must be in the same direction, namely that of P m L . For generic Γ 5 , 5 lattice this implies that ( k L , k R ) must be a multiple /lscript of the primitive vector ( ˆ P L , ˆ P R ) ∈ Γ 5 , 5 in the direction of ( P L , P R ). Thus we have \n( P L , P R ) = N P ( ˆ P L , ˆ P R ) , (8) \nwhere N P is an integer which is defined by this equation. Thus we see that in a BPS state of the five-brane, all world-brane momenta and windings of the strings are in the same direction. In other words, the BPS restriction implies that the five-brane behaves effectively as a 1+1-dimensional string-like object, as will become more evident in the following. \nNow let H /lscript denote the space of single string BPS states with momentum k = /lscript ˆ P that are in the left moving ground state. Via the level matching condition, its dimension is given by (here ˆ P 2 = | ˆ P L | 2 -| ˆ P R | 2 ) \ndim H /lscript = d ( 1 2 /lscript 2 ˆ P 2 ) , (9) \nwhich is the coefficient of the term q 1 2 /lscript 2 ˆ P 2 in the elliptic genus of T 4 \n∑ N d ( N ) q N = 16 ∏ n ( 1 + q n 1 -q n ) 4 . (10) \nWe will now make the assumption that we can treat the second quantized theory on the five-brane as a theory of non-interacting strings. The space of all multiple string states that satisfy the BPS conditions, and with a given total energy-momentum P , then takes the form \n∑ \nH P = ⊕ /lscriptN /lscript = N P ⊗ /lscript Sym N /lscript H /lscript , (11) \nwhere Sym N indicates that N -th symmetric tensor product. At first one may think that only the states with N P 'primitive' strings with /lscript = 1 should contribute, because only these are stable. This would imply that only the first term Sym N P H 1 must be kept. However, we propose that the correct interpretation of the Hilbert spaces H /lscript with /lscript > 1 is that they represent the contributions of the various fixed points of the permutation group, and hence describe the bound states of /lscript different primitive strings. These therefore also represent stable states, and should be counted as well. We thus conjecture that the exact dimension of H P is determined by the character expansion ‡ \n∑ N P q N P dim H P = (16) 2 ∏ /lscript ( 1 + q /lscript 1 -q /lscript ) 1 2 d ( 1 2 /lscript 2 ˆ P 2 ) . (12) \nThis formula somewhat resembles the expressions of Borcherds [16] for the denominator formula of a generalized Kac-Moody algebra. Presumably, by a similar calculation as in \n[17] one can relate its logarithm to the one-loop amplitude of the self-dual string. We further note that for ˆ P 2 = 0 (12) reduces to the standard (chiral) superstring partition function (since 1 2 d (0) = 8). This correspondence will be explained below. \nAn alternative but presumably equivalent description of the second quantized BPS string states on the five-brane is obtained by considering the sigma model on the 'target space' ∑ N Sym N T 4 . The intuitive picture behind this representation is that the lightcone description of an N -string state may also be thought of as that of a single string state on the N -fold symmetric product of its target space. § This description should be further supported by the correspondence between the above formula for the dimension of H P and the term at order q 1 2 N P ˆ P 2 in the expansion of the elliptic genus of the orbifold Sym N P T 4 . In this way of looking at it one finds that the asymptotic growth equals that of states at level h = 1 2 N P ˆ P 2 in a unitary conformal field theory with central charge c = 6 N P . The standard degeneracy formula gives \ndim H P ∼ exp ( 2 π √ 1 2 ( P 2 L -P 2 R ) ) . (13) \nThis same result can be obtained directly from (12). We will now turn to the discussion of the space-time meaning of the momentum vector P .", 'U-Duality Invariant BPS Spectrum in D = 6 .': "Let us now focus on the space-time interpretation of these BPS states in toroidal compactifications of M-theory. We consider situations in which at least 5 of the coordinates are compact so that the five-brane can wrap completely around an internal T 5 . We will also assume that at least 4 coordinates are uncompactified because these will be identified with the fields X i on the five-brane. This leaves us with two cases: d = 6 and d = 5. \nWe start with d = 6. The space-time effective action for M theory compactified on T 5 is given by N = (4 , 4) six-dimensional supergravity that contains as bosonic fields, besides the metric, 5 anti-symmetric tensors, 16 gauge fields and 25 scalars. The scalars parametrize a SO (5 , 5) /SO (5) × SO (5) manifold and hence the expected U-duality group is SO (5 , 5 , Z ). The N = (4 , 4) supersymmetry algebra is \n{ Q a α , Q b β } = ω ab p/ αβ , { Q a α , Q b β } = δ αβ Z ab , (14) \nwhere a, b = 1 , . . . , 4 are now SO (5) ∼ = Usp (4) spinor indices and ω ab is an anti-symmetric matrix, that will be used to raise and lower indices. The algebra contains 16 central \ncharges that are combined in the 4 × 4 matrix Z ab , where a, b = 1 , . . . , 4 are again SO (5) spinor indices. The central charge Z ab forms a 16 component spinor of the SO (5 , 5 , Z ) U-duality group, and takes its values on an integral lattice whose shape is determined by the expectation values of the scalars. \nNow let us look at the BPS states in the six-dimensional space-time theory that respect 1 / 4 of the space-time supersymmetries. The mass of these BPS state can be determined from Z ab by solving the eigenvalue equations \n( Z † Z ) a b ε b L = m 2 BPS ε a L , ( ZZ † ) a b ε b R = m 2 BPS ε a R . (15) \nEach of these equations has two independent solutions, as can be seen for example from the fact that the matrices Z † Z and ZZ † must be of the form \n( Z † Z ) ab = ( m 2 BPS -2 | K L | ) 1 ab +2 K m L Γ ab m , ( ZZ † ) ab = ( m 2 BPS -2 | K R | ) 1 ab +2 K m R Γ ab m . (16) \nSince | K L | = | K R | the combination ( K L , K R ) forms a null vector of SO (5 , 5 , Z ). As we will show below, BPS states with fixed ( K L , K R ) correspond to those multi-string states on the five-brane for which the total momentum and winding number vector ( P L , P R ) is equal to minus ( K L , K R ). Since this implies that P 2 L -P 2 R = 0, the corresponding BPS states are necessarily made up from the string ground states with k L | = k R | . \n| \n| | To derive the relations (16) we must understand how the space-time supersymmetry algebra, including its central charge, is realized on the world-volume of the five-brane. We assume that the world-brane theory is formulated in a light-cone gauge, so that the SO (5 , 1) space-time Lorentz group is broken to the SO (4) subgroup of transverse rotations. On the world-brane this group becomes identified with the R-symmetry, and so the previously introduced spinor indices α, ˙ α indeed correspond to the two spin representations of the space-time rotations. Notice that the space-time chirality is thus linked with the chirality on the string world-sheet. To construct the N = (4 , 4) space-time supersymmetry algebra on the world-brane we need to use the zero-mode algebra of the various world-brane fields. The transversal momentum p α ˙ β is as usual identified with the zero-modes of the canonical conjugate of the fields X α ˙ β . Similarly, there are fermion zeromodes S α b and S ˙ β a that represent the broken part of the space-time supercharges. They can be normalized such that they satisfy the algebra \n| \n{ S α a , S β b } = /epsilon1 αβ ω ab . (17) \nMore surprisingly, all 16 central charges Z ab appear also as zero modes, namely as the 10 fluxes of the self-dual three-form db + through the 3-cycles of T 5 , the 5 winding numbers \nof the scalar Y , and the 5-flux of its dual ∗ dY . Thus these charges are in one-to-one correspondence with the odd homology of T 5 , which naturally forms a spinor representation of SO (5 , 5 , Z ). To complete the space-time supersymmetry algebra, one also has to use the world-brane supercharges Q aα , which act on the zero-modes as \n{Q α a , S β b } = /epsilon1 αβ Z ab , {Q α a , S ˙ β b } = ω ab p/ α ˙ β . (18) \nFrom these equations it follows that Q α a contains a zero-mode contribution Z ab S bα + p/ α ˙ β S a ˙ β in addition to the non-zero-modes that we have been considering up to now. Then, from the world-brane supersymmetry algebra (5) one deduces that P 0 1 ab + P m L Γ ab m also contains a zero mode contribution 1 2 ( p 2 i 1 ab +( Z † Z ) ab ). \nWe can now use this to re-analyse the BPS conditions on the five brane. In d = 6 we have the condition that on physical states the total space-like momentum on the five-brane vanishes, P m L = 0. ¶ This implies that the oscillator contribution P m L and the zero-mode contribution K m L = 1 8 tr(Γ m Z † Z ) cancel, so that we derive that P m L = -K m L . Similarly one finds that P m R = -K m R . The BPS mass formula now follows by imposing the five-brane mass-shell condition P 0 = p + p -. This gives m 2 BPS = 1 4 tr( Z † Z ) + 2 | K | , where | K | = | K L | = | K R | . Thus the SO (5 , 5) vector ( P L , P R ) satisfies P 2 L -P 2 R = 0, and hence is indeed a null-vector. \nThe degeneracy formula follows now immediately by specializing the general formula (12) to the case ˆ P 2 = 0. So the number of BPS states is given by D ( N K ), where N K is the number of times that the primitive vector ˆ K fits in K , and D ( N ) is the degeneracy at level N of the standard superstring partition function \n∑ N D ( N ) q N = (16) 2 ∏ n ( 1 + q n 1 -q n ) 8 . (19) \nThis degeneracy formula is the unique U-duality invariant extension of the results obtained in [19, 18] from the counting of string and D-brane BPS states. Note that pure string or Dbrane configurations carry at most 8 charges that transform as an SO (4 , 4) vector ( q L , q R ), and their degeneracy is D ( 1 2 ( q 2 L -q 2 R )). The above result shows that for more general bound states between string and D-branes, the degeneracy is obtained by generalizing the T-duality invariant 1 2 ( q 2 L -q 2 R ) to the greatest common divisor of the ten components of the SO (5 , 5) vector ( K L , K R ) defined in (16). \nBPS Spectrum and Black Hole Entropy in D = 5 . \nIt is straightforward to extend the six-dimensional results to d = 5 and the BPS states that respect only 1 / 8 of the space-time supersymmetries. \nThe 5 anti-symmetric tensor fields in 6 dimensions can be decomposed into 5 fields B m L with self dual field strength and 5 fields B m R with anti-self dual field strength. Together ( B m L , B m R ) form a vector of SO (5 , 5 , Z ). After further compactification to d = 5 these antisymmetric tensor fields produce 10 additional gauge fields, with 10 corresponding charges ( W m L , W m R ). Together with the Kaluza-Klein momentum p and the 16 charges contained in Z ab this gives a total of 27 charges. These charges combine into one irreducible representation of the E 6(6) ( Z ) U-duality group: an 8 × 8 pseudo-real, anti-symmetric and traceless matrix Z [20]. This matrix can be expressed in terms of the SO (5 , 5) vector ( W L , W R ), scalar p and spinor Z as \nZ = p + W L · Γ Z Z † -p + W R · Γ . (20) \nBy dimensional arguments, the extremal black holes that carry these charges are expected to have a non-zero entropy that is the square root of a cubic expression in Z . There is one unique cubic E 6(6) invariant, namely tr Z 3 . In the above normalisation, this leads to the following prediction for the U-duality invariant entropy formula \nS ( Z ) = 2 π [ 1 2 p ( W 2 L -W 2 R ) + 1 8 W L · tr(Γ Z † Z ) -1 8 W R · tr(Γ ZZ † ) ] 1 / 2 = 2 π [ 1 2 p ( | pW L -K L | 2 -| pW R -K R | 2 ) ] 1 / 2 (21) \nwith K as defined in (16). \nHow can such a result be derived from the five-brane? Because we are considering 1/8 BPS states in space-time, in this case we will only have to impose the left-moving BPS conditions. So in terms of the self-dual string one expects to get multiple string states on the five-brane with right-moving oscillators. Note however, that this is only possible when the momentum ( P m L , P m R ) is no longer a null vector, but satisfies P 2 > 0. Also, we need to represent the additional charges ( W m L , W m R ) and the Kaluza-Klein momentum p on the Hilbert states of the five-brane. The interpretation of the vector charges ( W m L , W m R ) is roughly that they indicate the winding number of the five-brane around the extra compactification circle. In this way the Kaluza-Klein momentum p is fed into the fivebrane and this modifies the level matching relations on the five-brane to P m L = pW m L and similarly for the right-movers. ‖ Now following the same steps as before one finds that the \ntotal left-moving momentum of the multiple string states is \nP m L = pW m L -K m L (22) \nand similarly for P m R . \nEven though we are able to represent all 27 charges, only the SO (5 , 5 , Z ) subgroup of the U-duality group E 6(6) ( Z ) is a manifest symmetry of the five-brane. The full U-duality is hidden in the light-cone construction, which has extra subtleties because we work on a 'light-cone cylinder.' Because of this, our construction is in fact most straightforward for p = 1. In this case the counting of the states as explained above equation (13) goes through without change, and leads for the statistical entropy S = log(dim H P ) to the following result \nS = 2 π √ 1 2 ( W -K ) 2 . (23) \nComparing to the expected E 6(6) -invariant result (21) we see that for p /negationslash = 1 an extra factor 1 /p enters. In the light-cone gauge constructions that we have been using, this factor naturally follows from the relative normalisation of the time-coordinates on the world-sheet of the self-dual string and space-time. In this way we find that the five-brane description of the BPS states exhibits a maximal symmetry under the five-dimensional U-duality group. The fact that we furthermore reproduce the expected black hole entropy is a strong indication that the five-brane also gives an exhaustive description of the BPS spectrum in d = 5. It would be interesting to find a general covariant derivation of this result, starting from a formulation as in [21], since this could make the full E 6(6) -invariance manifest from the start.", 'Concluding remarks': "We have shown that at least down to 5 dimensions the string formulation of the fivebrane theory gives a unified description of all BPS states that is invariant under the U-duality group. The general quantisation of five-dimensional extended objects is still an unsolved problem, but we see convincing evidence that the 'BPS quantisation' of the five-brane reduces it effectively to a type II string, since the momenta k m on the worldbrane are aligned with the central charge K m . U-duality permutes these various strings, similar to the action of SL (2 , Z ) on the ( p, q ) strings in type IIB string theory in ten dimensions. The correspondence with the type II string also shows immediately that our description for the spectrum of 1/4 BPS states is fully Lorentz invariant. This is however somewhat less obvious for the 1/8 BPS states. \nWe can also consider compactifications of M-theory on other manifolds than tori, such as orbifolds [10]. Particular cases that we have analyzed (see [13]) are K 3 × S 1 and \nT 4 × ( S 1 / Z 2 ). In both of these compactifications the five-brane can be shown to behave as a heterotic string. In the latter case, the geometric explanation is that the worldbrane geometry of the five-brane must also be of the form of K 3 × S 1 . This result can be used to give a new explanation from M-theory of the various heterotic string/string dualities [22]. Furthermore, the counting of BPS states can be done in similar manner as in this letter [13], and is in agreement with the black hole entropy formula derived in [5]. \nFinally, let us comment on the generalization to four dimensions. Here one gets one extra electric charge p ' from the Kaluza-Klein momentum, which gives a total of 28 charges. But a general state in d = 4 can carry 28 magnetic and 28 electric charges that combine into the 56 of the E 7(7) ( Z ) U-duality group. In this case the entropy is expected to scale with the square root of a quartic expression in the central charge, which is now a general complex anti-symmetric 8 × 8 matrix. The unique quartic invariant is the so-called diamond function ♦ . In [23] this was used to conjecture that the macroscopic entropy in four dimensions equals S = 2 π √ ♦ . For the states obtained from the five-brane, this entropy formula can be expressed in terms of SO (5 , 5 , Z ) representations and takes the form \nS = 2 π √ p ' 2 p ( pW -K ) 2 . (24) \nWe believe that via a straightforward generalization of the above procedure one should be able to reproduce this result. This would give a complementary derivation of the recently obtained results [7]. It seems, however, that to obtain a fully E 7(7) -invariant description from the five-brane, new ingredients may be needed. Possibilities are the inclusion of D-brane configurations on the world-volume [24], or of bound states of several five-branes [11, 25], which would lead to non-abelian extensions of the self-dual string theory considered here.", 'Acknowledgements': 'We would like to thank S. Ferrara, C. Kounnas, W. Lerche, R. Minasian, and C. Vafa for interesting discussions and helpful comments. This research is partly supported by a Pionier Fellowship of NWO, a Fellowship of the Royal Dutch Academy of Sciences (K.N.A.W.), the Packard Foundation and the A.P. Sloan Foundation.', 'References': "- [1] J. Bekenstein, Phys. Rev. D7 (1973) 2333; Phys. Rev. D9 (1974) 3292. S. W. Hawking, Phys. Rev. D13 (1976) 191.\n- [2] C. Hull and P. Townsend, Nucl. Phys. B 438 (1995) 109.\n- [3] E. Witten, 'String Theory in Various Dimensions,' Nucl. Phys. B 443 (1995) 85.\n- [4] J. Polchinski, ' Dirichlet-Branes and Ramond-Ramond Charges,' hep-th/9510017. \n- [5] A. Strominger, C. 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Townsend, 'P-Brane Democracy,' hep-th/9507048.\n- [9] J. Schwarz, 'The Power of M-theory,' Phys. Lett. 367B (1996) 97, hep-th/9510086.\n- [10] P. Horava and E. Witten, 'Heterotic and Type I String Dynamics from 11 Dimensions,' hep-th/9510209; K. Dasgupta and S. Mukhi, 'Orbifolds of M-Theory,' hepth/9512196; E. Witten, 'Five-branes and M-theory on an Orbifold,' hep-th/9512219.\n- [11] P. Townsend, 'D-branes from M-branes,' hep-th/95012062; A. Strominger, 'Open P-Branes,' hep-th/9512059; K. Becker and M. Becker, 'Boundaries in M-theory,' hep-th/9602071.\n- [12] D. Kutasov and E. Martinec, 'New Principles for String/Membrane Unification,' hep-th/9602049; David Kutasov , Emil Martinec, and Martin O'Loughlin, 'Vacua of M-theory and N=2 Strings,' hep-th/9603116\n- [13] R. Dijkgraaf, E. Verlinde and H. Verlinde, in preparation.\n- [14] C. Callan, J. Harvey, and A. Strominger, Nucl. Phys. B 367 (1991) 60; D. M. Kaplan and J. Michelson, 'Zero-modes for the 11-dimensional membrane and five-brane,' hep-th/9601053\n- [15] M. J. Duff and J. X. Lu, 'Black and Super p -Branes in Diverse Dimensions,' Nucl. Phys. B416 (1994) 301; E. Witten, 'Some Comments on String Dynamics,' hepth/9510135.\n- [16] R. E. Borcherds, 'Automorphic Forms on O s +2 , 2 ( R ) and Infinite Products,' Invent. Math. 120 (1995) 161.\n- [17] J. Harvey and G. Moore, 'Algebras, BPS States, and Strings,' hep-th/9510182. \n- [18] C. Vafa,'Gas of D-Branes and Hagedorn Density of BPS States,' hep-th/9511026, 'Instantons on D-branes,' hep-th/9512078; M. Bershadsky, V. Sadov, and C. Vafa, 'D-Branes and Topological Field Theories,' hep-th/9511222.\n- [19] A. Sen, 'A Note on Marginally Stable Bound States in Type II String Theory' hepth/9510229; 'U Duality and Intersecting D-Branes,' hep-th/9511026; 'T-Duality of p-Branes,' hep-th/9512062.\n- [20] E. Cremmer, 'Supergravities in 5 Dimensions,' in Superspace and Supergravities, Eds. S.W. Hawking and M. Rocek (Cambridge Univ. Press, 1981) 267.\n- [21] E. Bergshoeff, E. Sezgin, and P. Townsend, 'Supermembranes and Eleven Dimensional Supergravity,' Phys. Lett. B189 (1987) 75; K. Becker, M. Becker, and A. Strominger, 'Five-branes, Membranes, and Non-perturbative String Theory,' hepth/9507158.\n- [22] M. Duff, R. Minasian, and E. Witten, 'Evidence for Heterotic/Heterotic Duality,' hep-th/9601036.\n- [23] R. Kallosh and B. Kol, 'E(7) Symmetric Area of the Black Hole Horizon,' hepth/9602014; S. Ferrara and R. Kallosh, 'Universality of Supersymmetric Attractors,' hep-th/9603090.\n- [24] M.R. Douglas, 'Branes within Branes,' hep-th/9512077\n- [25] G. Papadopoulos, P.K. Townsend, 'Intersecting M-branes,' hep-th/9603087."}

2005ApJ...629..362H

The Orbital Statistics of Stellar Inspiral and Relaxation near a Massive Black Hole: Characterizing Gravitational Wave Sources

2005-01-01

['black hole physics', 'gravitational waves', 'stars kinematics and dynamics', 'astrophysics', '-']

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We study the orbital parameter distribution of stars that are scattered into nearly radial orbits and then spiral into a massive black hole (MBH) due to dissipation, in particular by emission of gravitational waves (GWs). This is important for GW detection, e.g., by the Laser Interferometer Space Antenna (LISA). Signal identification requires knowledge of the waveforms, which depend on the orbital parameters. We use analytical and Monte Carlo methods to analyze the interplay between GW dissipation and scattering in the presence of a mass sink during the transition from the initial scattering-dominated phase to the final dissipation-dominated phase of the inspiral. Our main results are as follows. (1) Stars typically enter the GW-emitting phase with high eccentricities. (2) The GW event rate per galaxy is afew×10<SUP>-9</SUP> yr<SUP>-1</SUP> for typical central stellar cusps, almost independently of the relaxation time or the MBH mass. (3) For intermediate-mass black holes of ~10<SUP>3</SUP> M<SUB>solar</SUB> such as may exist in dense stellar clusters, the orbits are very eccentric and the inspiral is rapid, so the sources are very short-lived.

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https://arxiv.org/pdf/astro-ph/0503672.pdf

{'THE ORBITAL STATISTICS OF STELLAR INSPIRAL AND RELAXATION NEAR A MASSIVE BLACK HOLE: CHARACTERIZING GRAVITATIONAL WAVE SOURCES': 'CLOVIS HOPMAN 1 AND TAL ALEXANDER 1,2 ApJ accepted', 'ABSTRACT': 'We study the orbital parameters distribution of stars that are scattered into nearly radial orbits and then spiral into a massive black hole (MBH) due to dissipation, in particular by emission of gravitational waves (GW). This is important for GW detection, e.g. by the Laser Interferometer Space Antenna ( LISA ). Signal identification requires knowledge of the waveforms, which depend on the orbital parameters. We use analytical and Monte Carlo methods to analyze the interplay between GW dissipation and scattering in the presence of a mass sink during the transition from the initial scattering-dominated phase to the final dissipation-dominated phase of the inspiral. Our main results are (1) Stars typically enter the GW-emitting phase with high eccentricities. (2) The GW event rate per galaxy is few × 10 -9 yr -1 for typical central stellar cusps, almost independently of the relaxation time or the MBH mass. (3) For intermediate mass black holes (IBHs) of ∼ 10 3 M /circledot such as may exist in dense stellar clusters, the orbits are very eccentric and the inspiral is rapid, so the sources are very short-lived. \nSubject headings: black hole physics - stellar dynamics - gravitational waves', '1. INTRODUCTION': 'Dissipative interactions between stars and massive black holes (MBHs; M · /greaterorsimilar 10 6 M /circledot ) in galactic nuclei (e.g. Gebhardt et al. 2000, 2003), or intermediate mass black holes (IBHs; 10 2 <M · /lessorsimilar 10 4 M /circledot ), which may exist in dense stellar clusters, have been in the focus of several recent studies. The interest is mainly motivated by the possibility of using the dissipated power to detect the BH, or to probe General Relativity. Examples of such processes are tidal heating (e.g. Alexander &Morris 2003; Hopman, Portegies Zwart & Alexander 2004) and gravitational wave (GW) radiation (Hils & Bender 1995; Sigurdsson & Rees 1997; Ivanov 2002; Freitag 2001, 2003). \nA statistical characterization of inspiral orbits is of interest in anticipation of GW observations by the Laser Interferometer Space Antenna ( LISA ). LISA will be able to observe GW from stars at cosmological distances during the final, highly relativistic phase of inspiral into a ∼ 10 6 M /circledot MBH, thereby opening a new non-electromagnetic astronomical window. GW from inspiraling compact objects (COs) is one of the three major targets of the LISA mission (Barack & Cutler 2003, 2004; Gair et al. 2004), together with cosmological MBH-MBH mergers and Galactic CO-CO mergers. \nLISA can detect GW emission from stars with orbital period shorter than P L ∼ 10 4 s . In order for the shortest possible period to be small enough to be detectable by LISA , the MBH has to be of moderate mass, M · /lessorsimilar 5 × 10 6 M /circledot (Sigurdsson & Rees 1997). LISA is expected to be able to detect inspiral into MBHs of 10 6 M /circledot to distances as far as /greaterorsimilar 1 Gpc. \nThe detailed time-evolution of the GW depends on the eccentricity of the stellar orbit, and therefore probes both General Relativity and the statistical predictions of stellar dynamics theory. Due to the low signal to noise ratio, knowledge of the wave forms is required in advance. For this purpose, it is necessary to estimate the orbital characteristics of the \nGW-emitting stars, and in particular the distribution function (DF) of their eccentricities (Pierro et al. 2001; Glampedakis, Hughes & Kennefick 2002), as the wave forms are strong functions of the eccentricity (e.g. Barack & Cutler 2003; Wen & Gair 2005). This study focuses on inspiral by GW emission. However, it should be emphasized that inspiral is a general consequence of dissipation, and the formalism presented below can be extended in a straight-forward way to other dissipation processes, such as tidal heating. \nThe prompt infall of a star into a MBH and its destruction have been studied extensively ( § 2.1). Here we analyze a different process, the slow inspiral of stars (Alexander & Hopman 2003). A star on a highly eccentric orbit with small periapse r p , repeatedly loses some energy ∆ E every periapse passage due to GW emission, and its orbit gradually decays. At a distance r 0 from the MBH, where the orbital period is P 0 , the time-scale t 0 for completing the inspiral (i.e. decaying to a P → 0 orbit) is much longer than the time-scale P 0 needed to reach the MBH directly on a nearly radial orbit. While the orbit decays, two-body scatterings by other stars continually perturb it, changing its orbital angular momentum J by order unity on a timescale t J . Because t 0 /greatermuch P 0 , inspiraling stars are much more susceptible to scattering than those on infall orbits. If t 0 > t J , either because r 0 is large or r p is large (small ∆ E ), then the orbit will not have time to decay and reach an observationally interesting short period. Before that can happen, the star will either be scattered to a wider orbit where energy dissipation is no longer efficient, or conversely, plunge into the MBH. Inspiral is thus much rarer than direct infall. The stellar consumption rate, and hence the properties of the stellar distribution function (DF) at low J , are dominated by prompt infall, with inspiral contributing only a small correction. This DF describes the parent population of the inspiraling stars. \nWe show below that the DF of the small subset of stars on lowJ orbits that complete the inspiral and are GW sources is very different from that of the parent population (Fig. 2). This results from the interplay between GW dissipation and scattering in the presence of a mass sink during the transition ( t 0 ∼ t J ) from the initial scattering-dominated phase to the \nfinal dissipation-dominated phase of the inspiral. \nThis paper is organized as follows. In § 2 we recapitulate some of the results of loss cone theory for the prompt infall, and extend it to slow inspiral. In § 3 and § 4 we present a detailed analytical discussion of the main effects that determine the rates of GW events and their statistical properties. In § 4 we describe three different approaches for studying the problem: by Monte Carlo simulations, by solving numerically the 2D diffusion / dissipation equation in E and J , and by a simplified analytical model, which mimics the behavior of a typical star. In § 5 we apply the MC simulation to a MBH in a galactic nucleus, and an IBH in a stellar cluster. We summarize our results in § 6.', '2. THE LOSS-CONE': "The rate at which stars are consumed by an MBH and the effect this has on the stellar DF near it have been studied extensively (Peebles 1972; Frank & Rees 1976; Bahcall & Wolf 1976, 1977; Lightman and Shapiro 1977; Cohn & Kulsrud 1978; Syer & Ulmer 1999; Magorrian & Tremaine 1999; Miralda-Escud'e & Gould 2000; Freitag & Benz 2001; Alexander & Hopman 2003; Wang & Merritt 2004; see Sigurdsson 2003 for a comparative review). Self-consistent N-body simulations with stellar captures were recently performed by Baumgardt, Makino, & Ebisuzaki (2004a, 2004b) and by Preto, Merritt & Spurzem (2004). \nWe begin by summarizing these results, neglecting dissipative processes. We then extend the formalism to include dissipative processes.", '2.1. Prompt infall': "The stellar orbits are defined by a specific angular momentum J and relative specific energy ε = ψ ( r ) -v 2 / 2 (hereafter 'angular momentum' and 'energy'), where ψ is the relative gravitational potential, and v is the velocity of a star with respect to the MBH. A spherical mass distribution and a nearly spherical velocity distribution are assumed. \nOrbits in the Schwarzschild metric (unlike Keplerian orbits) can escape the MBH only if their angular momentum is high enough, J >J lc ( ε ) . The phase space volume J <J lc is known as the 'loss-cone'. As argued below, stars that are scattered to lowJ orbits are typically on nearly zero-energy orbits. For such orbits \nJ lc ( ε =0) = 4 GM · c , (1) \nThe size of the loss-cone J lc is nearly constant over the relevant range of ε . Only during the very last in-spiral phase the energy of the star becomes non-negligible compared to its rest-mass, in which case the loss-cone is slightly modified (see section [4.1]). Deviations from geodetic motion due to tidal interactions are neglected here. This assumption is justified for COs orbiting ∼ 10 6 M /circledot MBHs, where the tidal radius is much smaller than the event horizon. For main-sequence (MS) stars where the tidal radius lies outside the event horizon, the loss-cone is similarly defined as the minimal J required to avoid tidal disruption. \nStars that are initially on orbits with J <J lc will promptly fall into the MBH on an orbital timescale. Subsequently, the infall flow in J -space, F ( ε ; J ) , is set by the rate at which relaxation processes (here assumed to be multiple two-body scattering events) re-populate the loss-cone orbits. \nDiffusion in ε -space occurs on the relaxation timescale, \nt r ∼ ε/ ˙ ε , whereas diffusion in J -space occurs on the angular momentum relaxation timescale, \nt J ∼ J 2 / ˙ ( J 2 ) ∼ [ J/J m ( ε )] 2 t r , (2) \nwhere J m ( ε ) is the maximal (circular orbit) angular momentum for specific energy ε .The square root dependence of J on t J reflects the random walk nature of the process. Typically, J lc /lessmuch J m . In principle, stars can enter the loss-cone, J < J lc ( ε ) either by a decrease in J , or by an increase in ε (up to the last stable orbit). In practice, diffusion in J -space is much more efficient: the energy of a star must increase by many orders of magnitude in order for it to reach the losscone, which takes many relaxation times. The angular momentum of the star, on the other hand, needs only to change by order unity in order for the star to be captured, which happens on a much shorter time t J t r . \nThe ratio between J lc and the mean change in angular momentumperorbit, ∆ J , defines two dynamical regimes of losscone re-population (Lightman & Shapiro 1977). In the 'Diffusive regime' of stars with large ε (tight orbits), ∆ J /lessmuch J lc and so the stars slowly diffuse in J -space. The loss-cone remains nearly empty at all times since any star inside it is promptly swallowed. At J /greatermuch J lc the DF is nearly isotropic, but it falls logarithmically to zero at J /greaterorsimilar J lc (Eq. 10). In the 'full loss-cone regime' (sometimes also called the 'pinhole' or 'kick' regime) of stars with small ε (wide orbits), ∆ J /greatermuch J lc and so the stars can enter and exit the loss-cone many times before reaching periapse. As a result, the DF is nearly isotropic at all J /greaterorsimilar J lc . We argue below that only the diffusive regime is relevant for inspiral. \n≤ \nThe DF in the diffusive regime is described by the FokkerPlanck equation. We follow Lightman & Shapiro (1977), who neglect the small contribution of energy diffusion to F , and write the Fokker-Planck equation for the number density of stars N ( ε, J ; t ) as 1 \n∂N ( ε, J ; t ) ∂t = -∂ F ( ε ; J ) ∂J , (3) \nwhere \nF ( ε ; J ) = N ( ε, J ) 〈 ∆ J 〉 -1 2 ∂ ∂J N ( ε, J ) 〈 ∆ J 2 〉 . (4) \nThe diffusion coefficients 〈 ∆ J 〉 and 〈 ∆ J 2 〉 obey the relation \n〈 ∆ J 2 〉 = 2 J 〈 ∆ J 〉 ( J /lessmuch J m ) , (5) \n(Lightman & Shapiro 1977; Magorrian & Tremaine 1999). Much of the difficulty in obtaining an exact solution for the Fokker-Planck equation stems from the dependence of the diffusion coefficients on the DF; self-consistency requires solving a set of coupled equations. For many practical applications the diffusion coefficients are estimated in a non-selfconsistent way, for example by assuming local homogeneity and isotropy (e.g. Binney & Tremaine 1987). Irrespective of its exact form, 〈 ∆ J 2 〉 describes a random-walk process, and is therefore closely related to the relaxation time. In anticipation of the eventual necessity of introducing such approximations, we forgo from the outset the attempt to write down explicit expressions for the diffusion coefficients. Instead, we \nuse them to define the relaxation time t r as the time required to diffuse in J 2 by J 2 m , \nt r ( ε ) = J 2 m ( ε ) 〈 ∆ J 2 〉 . (6) \nWe then treat the relaxation time as a free parameter that characterizes the system's typical timescale for the evolution of the DF, and whose value can be estimated by Eqs. (25, 26) below. For simplicity, the relaxation time t r is assumed to be independent of angular momentum 2 , but can generally be a function of energy. \nAt steady state, the stellar current F ( ε ) is independent of J , \nF ( ε ) = 1 2 J 2 m ( ε ) t r [ N ( ε, J ) J -∂N ( ε, J ) ∂J ] . (7) \nSolving this equation yields \nN ( ε, J ) = -2 F t r J m ( ε ) 2 J ln J + CJ . (8) \nThe integration constants C and F that are determined by the boundary conditions N ( ε, J lc ) = 0 and N ( ε, J m ) = N iso ( ε, J m ) . The isotropic DF is separable in ε and J , \nN iso ( ε, J )d ε d J = 2 N iso ( ε ) J J 2 m ( ε ) d ε d J . (9) \nApplying these boundary conditions 3 to equation (8), the DF is given by \nN ( ε, J ) = 2 N iso ( ε ) J J 2 m ( ε ) ln( J/J lc ) ln( J m /J lc ) , (10) \nand the stellar current into the MBH per energy interval is \nF ( ε ) = N iso ( ε ) ln( J m /J lc ) t r ( ε ) . (11) \nNote that the capture rate in the diffusive regime depends only logarithmically on the size of the loss cone. \nThe prompt infall rate Γ p in the diffusive regime is then given by \nΓ p = ∫ ∞ ε p d εN iso ( ε ) ln( J m /J lc ) t r ( ε ) , (12) \nwhere the energy ε p separates the diffusive and full loss-cone regimes. A star samples all angular momenta J lc <J < J m in a relaxation time, and it is promptly captured once J <J lc . The total rate is therefore of order Γ p ∼ N iso ( <a p ) /t r ( a p ) , where a p is the typical radius associated with orbits of energy ε p ( a p is the semi-major axis for Keplerian orbits, see § 2.2) and N iso ( <a p ) is the number of stars within a p . The rate is logarithmically suppressed because of the diluted occupation of phase space near the loss cone.", '2.2. Keplerian orbits in a power-law cusp': 'The MBH dominates the stellar potential within the radius of influence, \nr h = GM · σ 2 , (13) \nwhere σ 2 is the 1D stellar velocity dispersion far from the MBH. The mass enclosed within r h is roughly equal to M · . Various formation scenarios predict that the spatial stellar number density at r < r h should be approximately a power law (e.g. Bahcall & Wolf 1976; Young 1980) \nn /star ( r ) = (3 / 2 -p ) N h 4 πr 3 h ( r r h ) -3 / 2 -p , (14) \nwhere N h is the number of stars inside r h . This corresponds to an energy distribution N ( ε )d ε ∝ ε p -5 / 2 d ε . A stellar cusp with p ∼ 0 has been observed in the Galactic Center (Alexander 1999; Genzel et al. 2003). For a single mass population, it was shown by analytical considerations that p = 1 / 4 (Bahcall & Wolf 1976). This has been confirmed recently by N-body simulations (Baumgardt et al. 2004a; Preto et al. 2004). \nMass segregation drives the heavy stars in the population to the center. The radial distribution of the different mass componentscan be then approximated as average power-laws, steeper ( p > 0 ) for the heavier masses and flatter ( p /lessorsimilar 0 ) for the lower masses (Bahcall & Wolf 1977; Baumgardt et al. 2004b). \nTypically, the diffusive regime is within the radius of influence. We therefore assume from this point on that the stars move on Keplerian orbits ( ψ = GM · /r ) in a power-law density cusp. The stellar orbits are characterized by a semi-major axis a , eccentricity e , periapse r p and period P , \na = GM · 2 ε , e 2 =1 -J 2 GM · a , r p = a (1 -e ) , P = 2 πa 3 / 2 √ GM · . (15) \nDuring most of the inspiral 1 -e /lessmuch 1 , and the periapse can be approximated by r p ≈ J 2 / 2 GM · . This remains valid until the last phases of the inspiral. \nThe prompt infall rate (Eq. 12) can be expressed in terms of the maximal semi-major axis, \nΓ p = ∫ a p 0 d aN iso ( a ) ln( J m /J lc ) t r ( a ) . (16) \nWe will assume Keplerian orbits throughout most of this paper except for section (4.1), where we employ the general relativistic potential of the MBH.', '2.3. Slow inspiral': 'The derivations of the conditions necessary for slow inspiral and of the inspiral rate follow closely those of prompt infall, but with two important differences. (1) The time to complete the inspiral is not the infall time P 0 , but rather t 0 /greatermuch P 0 . (2) There is no contribution from the full loss-cone regime, where stars are scattered multiple times each orbit. This is because inspiral in this regime would require that the very same star that was initially deflected into an eccentric orbit, be rescattered back into it multiple times. The probability for this \nhappening is effectively zero 4 . \nIn analogy to the radial scale a p of prompt infall, which delimits the volume where stars can avoid scattering for a time P 0 <t J , and thus maintain their infall orbit until they reach the MBH, the inspiral criterion t 0 <t J defines a critical radius a c /lessmuch a p (or equivalently, a critical energy ε c /greatermuch ε p ). Stars starting the inspiral from orbits with a 0 < a c ( ε 0 > ε c ) will complete it with high probability, whereas stars starting with a 0 >a c ( ε 0 <ε c ), will sample all J values before they spiral in significantly, regardless of J 0 and ultimately either (1) fall in the MBH, (2) diffuse in energy to much wider orbits or (3) into the much tighter orbits of the diffusive regime. Since we assume a steady state DF, outcomes (2) and (3) represent a trivial, DF-preserving redistribution of stars in phase space, which does not affect the statistical properties of the system. Whether stars spiral in or fall in depends, statistically, only on a 0 ( ε 0 ). We use below Monte Carlo simulations ( § 4.1, Fig. 3) to estimate the inspiral probability function, S ( a 0 ) , which describes the probability of completing inspiral when starting from an orbit with semi-major axis a 0 ( S → 1 for a 0 /lessmuch a c , S → 0 for a 0 /greatermuch a c ). The inspiral rate for stars of type s with number fraction f s is then \nΓ i = f s ∫ ∞ 0 d aN ( a ) S ( a ) ln( J m /J lc ) t r ( a ) /similarequal f s ∫ a c 0 d aN ( a ) ln( J m /J lc ) t r ( a ) , (17) \nwhere roughly S ( a c ) ∼ 0 . 5 .', '3. PARAMETER DEPENDENCE OF THE INSPIRAL RATE': "In this section we derive some analytical results for the inspiral rate. In order to keep the arguments transparent, we neglect relativistic deviations from Keplerian motion. Relativistic orbits are discussed in section (4.1). \nConsider a star of mass M /star orbiting a MBH of mass M · on a bound Keplerian orbit with semi-major axis a and angular momentum J . When the star arrives at periapse, it loses some orbital energy ∆ E by GW emission. As a result, the orbit shrinks and its energy increases. For highly eccentric orbits the periapse of the star is approximately constant during inspiral in absence of scattering. We define the inspiral time t 0 as the time it takes the initial energy ε 0 to grow formally to infinity. If the energy loss per orbit is constant, then for e → 1 \nt 0 = ∫ ∞ ε 0 d ε (d ε/ d t ) ≈ 1 ∆ E ∫ ∞ ε 0 d εP ( ε ) = 2 ε 0 P 0 ∆ E , (18) \nor \nt 0 ( r p , a ) = 2 π √ GM · a ∆ E . (19) \nFor GW, ∆ E is given by (Peters 1964) \n∆ E GW = 8 π 5 √ 2 f ( e ) M /star c 2 M · ( r p r S ) -7 / 2 , (20) \nwhere \nf ( e ) = 1 + 73 24 e 2 + 37 96 e 4 (1 + e ) 7 / 2 , (21) \nand r S = 2 GM · /c 2 is the Schwarzschild radius. \nDuring all but the last stages of the inspiral e ∼ 1 , in which case r p /r S = 4( J/J lc ) 2 and \n∆ E GW = E 1 ( J/J lc ) -7 , E 1 ≡ 85 π 3 × 2 13 M /star c 2 M · . (22) \nGravity waves also carry angular momentum, \n∆ J GW = -16 π 5 g ( e ) GM /star c ( r p r S ) -2 , (23) \ng ( e ) = 1 + 7 8 e 2 (1 + e ) 2 . (24) \nGenerally, the change in J in the course of inspiral is dominated by two-body scattering, and ∆ J GW can be neglected until a becomes very small. \nIt is convenient to refer the timescales in the system to the relaxation time at the MBH radius of influence, \nt h = A p ( M · M /star ) 2 P ( r h ) N h logΛ 1 , (25) \nwhere Λ 1 = M · /M /star ( r S /r h ) 1 / 4 (Miralda-Escud'e & Gould 2000), and A p /similarequal 0 . 2 for p =0 (Alexander & Hopman 2003). The relaxation time at any radius is then \nt r ( a ) = t h ( a r h ) p , (26) \nwhere we associate the typical relaxation time on an orbit with that at its semi-major axis. This is a good approximation, since theoretical arguments (Bahcall & Wolf 1976; 1977) and simulations (Freitag & Benz 2002; Baumgardt et al. 2004a 2004b; Preto et al. 2004) indicate that 0 /lessorsimilar p /lessorsimilar 0 . 25 , and so t r is roughly independent of radius. The angular momentum relaxation time is \nt J = [ J J m ( a ) ] 2 ( a r h ) p t h . (27) \nDissipational inspiral takes place in the presence of twobody scattering. When t 0 ∼ t J , both effects have to be taken into account. It is useful to parametrize the relative importance of dissipation and scattering by the dimensionless quantity \ns ( J, a ) ≡ t 0 ( J, a ) t J ( J, a ) = ( a d c ) 3 / 2 ( a r h ) -p ( J J lc ) 5 , (28) \nwhere we introduce the ( p -independent) length scale \nd c ≡ ( 8 √ GM · E 1 t h πc 2 ) 2 / 3 , (29) \nwhich is of the same order as a c (Eq. 30) and is /lessmuch r h . We define some critical value s crit /lessmuch 1 such that the inspiral is so rapid that the orbit is effectively decoupled from the perturbations. \nThe three phases of inspiral can be classified by the value of s . In the 'scattering phase' the star is far from the region in phase space where GW emission is efficient and s /greatermuch 1 . With time it may scatter to a lowerJ orbit, enter the 'transition phase', where s ∼ 1 , and start to spiral in. If it is not scattered into the MBH or to a wide orbit, it will eventually \nreach the stage where s<s crit . It will then enter the 'dissipation phase' where it spirals-in deterministically according to equations (20-24). Note that eventually the e → 1 approximation is no longer valid. \nHere we are mainly interested in understanding how the interplay between two-body scattering and energy dissipation in the first two phases sets the initial conditions for the GW emission in the final phase. It should be emphasized that the onset of the dissipation phase does not necessarily coincide with the emission of detectable GW. For example, while a star is well into the dissipation phase by the time s < s crit ∼ 10 -3 , the orbit has still to decay substantially before the GW frequency becomes high enough to be detected by LISA . \nWe derive an analytical order of magnitude estimate for the critical semi-major axis a c by associating it with s =1 orbits in the transition phase. Since s falls steeply with J , we set J = J lc and solve s ( J lc , a c )=1 for a c , obtaining \na c r h = ( d c r h ) 3 / (3 -2 p ) . (30) \nThe MC simulations below ( § 4.1) confirm that this analytical estimate corresponds within a factor of order unity to the semi-major axis where the inspiral probability S ( a c ) ∼ 0 . 5 , for a wide range of masses (see table [2]). Expression (17) for the rate can then be approximated by \nΓ i ∼ f s N h t h ln[ J m ( a c ) /J lc ] ( a c r h ) 3 / 2 -2 p , (31) \nwhere N h is the number of stars within r h . \nThe dependence of the inspiral rate on p (at fixed d c /r h and neglecting the logarithmic terms) can be examined by writing Γ i /similarequal g ( p ) f s ( N h /t h ) , where \ng ( p ) = ( d c r h ) 3(3 -4 p )/2(3 -2 p ) . (32) \nThe pre-factor g ( p ) grows with p over the relevant range (see also Ivanov 2002). This reflects the fact that the inspiral rate is determined by the number of stars within a c , rather than the total number of stars within r h . The concentration of the cusp increases with p , so that there are more stars within a c . This result suggests qualitatively that in a mass-segregated population, the heavier stars (higher p ) will have an enhanced GWevent rate compared to the light stars (lower p ). \n¿From equations (29-32) it follows that \nΓ i t -2 p/ (3 -2 p ) h . (33) \nSince p ∼ 0 for typical stellar cusps around MBHs, this means that the inspiral rate is nearly independent of the relaxation time for such cusps. This counter-intuitive result reflects the near balance between two competing effects. When scattering is more efficient, stars are supplied to inspiral orbits at a higher rate, but are also scattered off them prematurely at a higher rate, so the volume of the diffusive regime, which contributes stars to the inspiral ( ∼ a 3 c ), decreases. This is in contrast to the prompt disruption rate Γ p , which increases as the relaxation time becomes shorter 5 . It then follows that enhanced scattering, such as by massive perturbers (e.g. clusters, giant molecular clouds; Zhao, Haehnelt & Rees 2002) \n∝ \nincreases the prompt disruption rate, but will not enhance the rate of inspiral events. \nThe dependence of the GW event rate on the mass of the MBH can be estimated from Eq. (31) and the empirical M · -σ relation \nM · = 1 . 3 × 10 8 M /circledot ( σ 200 kms -1 ) β , (34) \nwhere β ∼ 4 (Ferrarese & Merritt 2000; Gebhardt et al. 2000; Tremaine et al. 2002). Note that the M · -σ relation implies that the stellar number density at the radius of influence, n h is larger for lighter MBHs: for example, for β =4 , n h ∝ N h r -3 h ∝ M -1 / 2 · , where we assumed that N h ∝ M · ; the consequences of this for the dependence of the rate for prompt tidal disruptions on the MBH mass were discussed by Wang &Merritt (2004). \nThe GW event rate depends on M · as \nΓ i ∝ M 3 /β -1 · . (35) \nThis dependence is weak, e.g. Γ i ∝ M -1 / 4 · for β = 4 . Thus, the rate becomes higher for lower mass MBHs. If M · ∼ 10 3 M /circledot IBHs indeed exist in stellar clusters and the M · -σ relation can be extrapolated to these masses, then they may be more likely to capture stars than MBHs. This, however, does not necessarily translate into more GW sources. The strain h of GW decreases with the mass of the MBH, so that these sources have to be closer by in order to observe their GW emission. Another restriction is that for IBHs the tidal force is so strong that white dwarfs are tidally disrupted well outside the event horizon, which precludes them from being LISA sources. These issues are further discussed in § 5.", '4. ORBITAL EVOLUTION WITH DISSIPATION AND SCATTERING': "We present three different methods for analyzing inspiral in the presence of scattering. The first approach is based on Monte Carlo (MC) simulations, which follow a star on a relativistic orbit, described by ε and J , and add small perturbations to simulate energy dissipation and random two-body scattering. The second approach consists of direct numerical integration of the time dependent diffusion-dissipation equation. The third approach is a heuristic semi-analytical effective model that can describe the 'effective' trajectory of a star through phase space, as well as the statistical properties of an ensemble of such trajectories. \nThe three approaches are complementary. The MC simulations allow a direct realization of the micro-physics of the system, since they follow the perturbed orbits of individual stars. They also offer much flexibility in setting the initial conditions of the numerical experiments, but the results are subject to statistical noise and are not easy to generalize. The diffusion-dissipation equation on the other hand deals directly with the DF, and allows an analytical formulation of the problem in terms of partial differential equations, which are solved numerically. However, computational limitations do not allow covering as large a dynamical range as in the MC simulations. Finally, the heuristic effective model has the advantage of its intuitive directness and relative simplicity of use. We find that all three methods give the same results for the same underlying assumptions (Fig. 2). This inspires confidence in the robustness of the analysis. \nTo compare the three methods, we stop the simulation at the point where s = s crit = 10 -3 , and we plot the DF of the angular momenta of the stars. From that point on the stars \nare effectively decoupled from the cluster and can spiral in undisturbed. We then use the MC method, which can be easily extended to follow the stars in the dissipation phase (section 5), to find the DF of the eccentricities of stars which enter the LISA band.", '4.1. Monte Carlo simulations': 'The MC simulations generally follow the scheme used by Hils & Bender (1995) to study the event rate of GW. The star starts on an initial orbit with ( ε 0 , J 0 ) such that dissipation by GW emission is negligible. The initial value of J is not of importance as the angular momentum is quickly randomized in the first few steps of the simulation. Every orbital period P ( ε ) , the energy and angular momentum are modified by δε and δJ , and the orbital period and periapse are recalculated (this diffusion approach is justified as long as P /lessmuch t J ). The simulation stops when the star decays to an orbit with s < s crit =10 -3 or when J <J lc and it falls in the MBH (escape to a less bound orbit is not an option here since energy relaxation is neglected and only dissipation is considered; see 2.3). \nWhen stars reach high eccentricities as a result of scattering, the periapse approaches the Schwarzschild radius to the point where the Newtonian approximation breaks down.The MC simulations take this into account by integrating the orbits in the relativistic potential of a Schwarzschild MBH. The periapse of the star moving in a Schwarzschild spacetime is related to its angular momentum by one of the three roots of the equation \n§ \nE 2 GR = ( c 2 -2 GM · r )( c 2 + J 2 r 2 ) ≡ V GR ( r, J ) . (36) \nThe term on left hand side of this equation is the squared specific relativistic energy of the star (including its rest mass), while the right hand side is the effective GR potential. A star on a bound orbit is not captured by the MBH as long as equation (36) has three real roots for r . The smallest root is irrelevant for our purposes. The intermediate root (the turningpoint) is the periapse r p of the orbit, and the largest root is the apo-apse r a . The semi-major axis a and eccentricity e are defined by a = ( r p + r a ) / 2 and e = 1 -r p /a (see Eq. 15). For bound orbits the loss cone J lc is a very weak function of energy and is well approximated by Eq. (1). \nFor given values of E and J , the GR periapse is smaller than the Newtonian one, and therefore the orbits can be more eccentric, and the dissipation can be stronger than implied by the Newtonian approximation. The Keplerian relations between energy, angular momentum and the orbital parameters (Eqs 15) are replaced by \nE 2 GR = ( q -2 -2 e )( q -2 + 2 e ) q ( q -3 -e 2 ) ; (37) \nJ 2 = q 2 q -3 -e 2 ( GM · c ) 2 , (38) \nwhere q = 2(1 -e 2 ) a/r S (Cutler, Kennefick & Poisson 1994). The condition for a non-plunging orbit can also be expressed in terms of the eccentricity and the semi-major axis, 2(1 -e 2 ) a = (6 + 2 e ) r S (Cutler et al. 1994). This corresponds to a maximal eccentricity for a star on a non-plunging orbit \ne max ( a ) = -r S 2 a + √ ( r S / 2 a ) 2 -3 r S /a +1 , (39) \nwhich increases with a . If a L = ( P 2 L / 4 π 2 GM · ) 1 / 3 is the maximal semi-major axis a star may have in order to be detectable by LISA , the maximal eccentricity of a star detectable by LISA is e max ( a L ) . \nThe step in J -space per orbit is the sum of three terms \nδJ ( ε, J ) = ∆ 1 J scat ( ε, J ) + χ ∆ 2 J scat ( ε, J ) -∆ J GW ( ε, J ) . (40) \nThe first and second terms represent two-body scattering (Eqs. 2,5,6), with 6 ∆ 1 J scat = 〈 ∆ J 〉 P = J 2 m P ( ε ) / (2 t r J ) and ∆ 2 J scat = √ 〈 ∆ J 2 〉 P = √ P ( ε ) /t r J m ( ε ) = √ P ( ε ) /t J J . The random variable χ takes the values ± 1 with equal probabilities. The third term is the deterministic angular momentum loss due to GW emission (Eq. 23).The energy step per orbit is deterministic (diffusion in energy is neglected) \nδE ( ε, J ) = ∆ E GW ( ε, J ) . (41) \nIn order to increase the speed of the simulation we use an adaptive time-step. We checked for some cases that taking a smaller time-step does not affect the results. \nThe DF of inspiraling stars is generated by running many simulations (typically 3 × 10 4 ) of stellar trajectories through phase space with the same initial value for J 0 for given initial semi-major axis a 0 , but with different random perturbations. We verified that the initial value of J 0 is irrelevant, as long as J 0 /greatermuch J lc . We record the fraction of stars that avoid falling in the MBH, S ( a 0 ) , and the value of J at s crit for those stars that reach the dissipation-dominated phase, thereby obtaining the DF W ( J ; a 0 ) . This is repeated for a range of a 0 values. The integrated DF over all cusp stars, W ( J ) , is then obtained by taking the average of all DFs, weighted by N ( a 0 ) S ( a 0 )d a 0 (cf Eq. 17).', '4.2. Diffusion equation with GW dissipation': "The DF of the inspiraling stars at the onset of the dissipation dominated phase can be obtained directly by solving the diffusion-dissipation equation. This approach was taken by Ivanov (2002), who included a GW dissipation term and obtained analytic expressions for the GW event rate for J /greatermuch J lc under various simplifying assumptions. Here we are interested DF of stars very near the loss-cone, and so we integrate the diffusion-dissipation equation numerically with two simplifying assumptions. (1) We neglect the drift term in the diffusion-dissipation equation. This can be justified by noting that the drift grows linearly with time, δJ 1 ( t ) = J 2 m t/ (2 t r J ) , while the change due to the random walk grows as δJ 2 ( t ) = ( t/t r ) 1 / 2 J m . For a star with initial angular momentum J the drift becomes important only for t > t drift = 4( J/J m ) 2 t r = 4 t J . Since we are interested in the timescales t ∼ t 0 ≤ t J , the drift is only a small correction. (2) We assume Keplerian orbits. These approximations are validated by the very good agreement with the MC simulation results (section 4.4). \nWith these two assumptions, the diffusion-dissipation is (cf Eqs. 3, 7) \n∂N ( ε, J ; t ) ∂t = J 2 m ( ε ) 2 t r ∂ 2 ∂J 2 N ( ε, J ; t ) \n6 The drift term ∆ 1 J scat represents a bias to scatter away from the MBH due to the 2D character of the direction of the velocity vector v . This can be expressed geometrically by considering a circle of radius ∆ v scat centered on v ( v +∆ v scat is the change per orbit due to scattering). This circle is intersected by a second circle of radius v that passes through v and is centered on the radius vector to the MBH. The section of the first circle that leads away from the MBH is slightly larger than the section leading toward it. \nFIG. 1.- Examples of tracks of stars from the MC simulation (solid lines), and a track from the effective model (dashed line) with ξ = 1 . The effective track represents the stochastic tracks well. The tracks extend to the point where s =10 -3 . The relevant parameters are given in the first line of table (2), i.e. inspiral of a white dwarf into a MBH. \n<!-- image --> \n+ ∂ ∂ε [ ˙ E GW ( J ) N ( ε, J ; t ) ] , (42) \nwhere ˙ E GW = ∆ E GW /P is the rate at which energy is lost to GW. The first term accounts for diffusion in J -space and the second represents the energy dissipated by GW emission. As with the MC simulations, the diffusion coefficient 〈 ∆ J 2 〉 = J 2 m /t r is an input parameter rather than resulting from a selfconsistent calculation. \nThe initial conditions consist of an isotropic cusp DF, N ( ε, J ) ∝ ε p -5 / 2 J and the boundary condition that the DF vanish on the loci s = s crit =10 -3 and J = J lc in the ( ε , J )-grid. The initial DF is evolved in time over the ( ε, J ) -grid until t ∼ t J , when a relaxed steady state is achieved. The integration is done using a Forward Time Centered Space (FTCS) representation of the diffusion term, and an 'upwind' differential scheme for the dissipation term (see e.g. Press et al. 1992). After each time step (which are chosen small enough to obey the Courant condition) the DF is re-normalized so that captured stars are replenished. \nAfter relaxation, the DF is non-zero up to the boundary s = s crit , as stars are redistributed in phase space by diffusion at large J and by dissipation at small J . The DF at s crit is then extracted to construct the angular momentum DF, W ( J ) . Although the stars continue their trajectory in phase space beyond s crit all the way to the last stable orbit, this does not change W ( J ) because the rapid energy dissipation does not allow much J -diffusion.", '4.3. Analytical model of effective orbit': "Stars follow complicated stochastic tracks in ( J, a ) space; due to scattering they move back and forth in J , while they always drift to smaller a due to GW dissipation (energy diffusion by scattering is neglected). The drift rate depends strongly on J . Figure (1) shows some examples of stellar tracks in phase space, taken from the MC simulations. \nIn this section we propose a heuristic analytical model, which captures some essential results of the MC simulations, while providing a more intuitive understanding. \nIn this model we define the equations of motion of an 'effective track' of a star, which is deterministic and can be solved analytically. With this method we follow stars that \nstart at some given initial semi-major axis a 0 and angular momentum J 0 , and follow the star during its first two stages of inspiral, i.e., until the moment that s<s crit . \nThe DF of stars which reach s = s crit is determined by the evolution of the orbital parameters ( J, a ) in the region where energy dissipation is efficient. Because of the strong dependence of the energy ∆ E which is dissipated per orbit on angular momentum (or, equivalently, periapse), this is a small region ∆ J in J -space, the size of which is of order of the loss-cone, ∆ J ∼ J lc . The time ∆ t it typically takes a star with semi-major axis a 0 performing a random walk in J to cross this region is ∆ t = [∆ J/J m ( a 0 )] 2 t h ( t r = t h assumed). This can be used to introduce an effective J -'velocity' ∆ J/ ∆ t = J 2 m ( a 0 ) /J lc t h . \nThe effective velocity of the semi-major axis is given by \n- \nd a ( J, a ) d t = da dE ˙ E = -2 a 2 GM · ∆ E GW ( J ) P ( a ) . (43) \nThese two equation define a deterministic time evolution in ( J, a ) space from given initial values ( J (0) = J 0 , a (0) = a 0 ), to a final point ( J f , a f ), where s = s crit . To recover W ( J ; a 0 ) at s crit , we introduce some scatter in the effective velocities d J/ d t , \nd j d t = -ξy j 2 m ( a 0 ) t h , (44) \nwhere j denotes angular momentum in terms of J lc , y is drawn from the positive wing of the normalized Gaussian distribution, G 1 ( y ) , and ξ ∼ 1 (a free parameter) is the width of the distribution, which can vary depending on the system's parameters. Equations (43,44) can be solved analytically, \nj ( t )= j 0 -yξj 2 m ( a 0 ) t t h , a ( t )= [ 1 -( d c /a 0 ) 3 / 2 6 ξyj 6 ] 2 a 0 . (45) \nThe stopping condition s ( j, a ) = s crit gives an additional relation between j and a , so that the final values are related by a f = s 2 / 3 crit j -10 / 3 f d c . This ties the initial and final values through the equations of motions \ny ( j f ) = (6 ξ ) -1 ( d c /a 0 ) 3 / 2 j -6 f 1 -s 1 / 3 crit ( d c /a 0 ) 1 / 2 j -5 / 3 f . (46) \nThe relation between the initial distribution of j -velocities and the final j distribution at is W ( j f ; a 0 ) = G 1 ( y )[d y ( j f ) / d j f ] . The integrated DF over all the cusp stars is \nW ( j f )d j f = ∫ a c N ( a ) W ( j f ; a ) d a ∫ a c N ( a ) d a . (47)", '4.4. Comparison of the different methods': "Wecomparethe results of the MC simulation, the diffusiondissipation equation and an integration of equations (44) and (43). In all cases the calculation is stopped when s =10 -3 ; at that point the eccentricity is still approximately unity, but scattering becomes entirely negligible so that even in the MC simulation the quantities evolve essentially deterministically according to equations (20-24). The calculations assume a cusp with slope p = 0 , M · = 3 × 10 6 M /circledot and M /star = 0 . 6 (corresponding to WDs). \nFIG. 2.- A comparison of the DFs of angular momenta for inspiraling stars in a p =0 stellar cusp, derived from (1) MC simulations, (2) direct integration of the diffusion-dissipation equation (Eq. 42) and (3) the effective model (Eq. 47, with ξ =0 . 6 ). The DFs are normalized over the displayed J -range. The DFs are shown at s =10 -3 , when the dynamics are dominated by dissipation and scattering no longer plays a role, but the orbits are still well outside the LISA band, P /greatermuch P L and e → 1 . This DF sets the initial conditions for the dissipation phase. The DF of the isotropic parent population (no sink, no dissipation; Eq. 9) is also shown for contrast. Unlike the DF of the inspiraling stars, it is dominated by highJ stars. \n<!-- image --> \nTABLE 1 COMPARISON OF ASSUMPTIONS IN THE 3 METHODS \n| Method | GR potential | ∆ 1 J scat | ∆ J GW | Stop at s crit |\n|-----------------------|----------------|--------------|----------|------------------|\n| Monte Carlo | yes | yes | yes | no a |\n| Diffusion/Dissipation | no | no | no | yes |\n| Effective track | no | no | no | yes | \na For the purpose of comparison to the other two methods, \nthe MC simulation was terminated at s crit . \nFigure (2) shows the very good correspondence between the three approaches, whose underlying assumptions and approximations are summarized in table (1). One important conclusion is that all stars enter the dissipation phase with very small angular momenta, 1 /lessorsimilar J/J lc /lessorsimilar 2 . For a suitable choice of ξ ∼ 1 , the actual complicated stochastic tracks are well mimicked by the tracks of the 'effective stars'. The semianalytical approach not only identifies the correct scale of angular momentum of inspiraling stars, but reproduces the DF. Its practical value lies in the relative ease of its application compared to the time consuming MC simulations or the integration of the diffusion-dissipation equation.", '5. INSPIRAL RATES AND DISTRIBUTION OF ORBITAL PARAMETERS': 'We now proceed to apply the MC simulation to different types of COs inspiraling into a 3 × 10 6 M /circledot MBHin a galactic center, and into a 10 3 M /circledot IBH in a stellar cluster. We find from the simulations the critical semi-major axis a c and calculate the DF of the eccentricities of LISA sources. We stop the simulations when the orbital period falls below the longest period detectable by LISA , P L =10 4 s , which corresponds to a semi-major axis of \na L = 23 . 5 M -2 / 3 6 r S , (48) \nTABLE 2 PARAMETERS FOR STELLAR POPULATIONS AND INSPIRAL RATES \n| Star | M /star | t h | r h | p | f s | a c c | a d c | Γ i - |\n|--------|----------------|---------|----------|-----|---------|---------------|---------|---------|\n| | [ M /circledot | [Gyr] | [pc] | | | [mpc] [mpc] [ | | Gyr 1 ] |\n| WD a | 0.6 | 10 | 2 | 0 | 0 . 1 | 30 | 20 | 7.8 |\n| NS a | 1.4 | 5 | 2 | 0 | 0.01 | 40 | 20 | 1.8 |\n| BH a | 10 | 1 | 2 | 1/4 | 1( - 3) | 20 | 10 | 4.7 |\n| NS b | 1.4 | 1( - 3) | 0.05 | 0 | 0.01 | 2 | 1 | 4.3 |\n| BH b | 10 | 1( - 3) | 0.05 1/4 | | 1(-3) | 5 | 2 | 6.7 | \na MBH with M · =3 × 10 6 M /circledot b IBH with M · =10 3 M /circledot . \nc From equation (30) \nd From the MC simulations \nwhere M · = 10 6 M 6 M /circledot .We record the eccentricity at that point and construct the DF of the eccentricity at P L . It is straightforward to integrate the orbits in the eccentricity histograms backward (forward) in time to larger (smaller) values of P , see e.g. Barack & Cutler (2003). \nThe stars are distributed according to a powerlaw distribution with different values for p . The total number of stars within r h was assumed to be N h = 2 M · /M /circledot , with different number fractions for the respective species. See table (2) for the assumed parameters of the stellar populations.', '5.1. Massive black holes in galactic centers': 'We assume a M · = 3 × 10 6 M /circledot MBH as a representative example and make the simplifying assumption that the MBH is not spinning. Real MBHs probably have non-zero angular momentum (e.g. see for evidence of spin of the MBH in our Galactic Center Genzel et al. 2003). An important qualitative difference is that in the Schwarzschild case the eccentricity of a GW-emitting star decreases monotonically with time, which this is not so for non-zero angular momentum (e.g. Glampedakis et al. 2002). We conducted several MC simulations with Kerr metric orbits and verified that in spite of changes in details, our overall results hold. \nThe tidal field of MBHs in this mass range disrupts main sequence stars before they can become detectable LISA sources. A possible exception is our own Galactic Center, where the weak GW emission from very low mass main sequence stars (which are the densest and so the most robust against tidal disruption) may be detected because of their proximity (Freitag 2003). However, here we only consider GW from COs. Table (2) summarizes the parameters assumed for the properties of white dwarfs (WDs), neutron stars (NSs) and stellar mass black holes (BHs). The BHs are assumed to be strongly segregated in a steep cusp because of their much heavier mass (Bahcall & Wolf 1977). The total (dark) mass in COs near MBHsis not known, but in future it may be constrained by deviations from Keplerian motions of luminous stars very near the MBH in the Galactic center (Mouawad et al. 2004). \nFig. (3) shows the normalized inspiral probability function S s ( a 0 ) , where s stands for WD, NS, and BH. The critical semi-major axis is similar for WDs and NSs, but much smaller for BHs, because of the smaller relaxation time and the higher central concentration due to mass segregation. The functions S s ( a ) are used to calculate the inspiral rates (Eq. 17) that are listed in table (2). The rate for WDs is highest, Γ WD = 7 . 8 Gyr -1 . We find that because of mass segrega- \nFIG. 3.- Dependence of the normalized fraction of inspiral events on initial semi-major axis, for the case of a MBH of M · =3 × 10 6 M /circledot (asterisks) and an IBH (triangles). The solid line is for white dwarfs, the dashed lines for neutron stars, and the dotted lines for stellar mass black holes. For the parameters of these cases see table (2). \n<!-- image --> \nion, the rate for stellar BHs is of the same order of magnitude as that for WDs, Γ BH = 4 . 7 Gyr -1 although their number fraction is lower by two orders of magnitude. The hierarchy of rates in table (2) is not, of course, necessarily the same as cosmic rates that LISA will observe; NSs and BHs are more massive than WDs, and can be observed at larger distances. \nFig. (4) shows the eccentricity DFs of stars on orbits with P = P L ; note that since a = a L is fixed, the orbits are fully determined by e . The DFs show a strong bias to large eccentricities. The maximal eccentricity possible for a star orbiting a 3 10 6 M /circledot MBHin the LISA band is e max =0 . 81 (Eq. 39). \nIt should be emphasized that the histogram in figure (4) can be obtained deterministically from the DF W ( J ; s crit =10 -3 ) in figure (2), because the stars have already reached the point where scattering is negligible. This would not be the case had the MC simulation been terminated at s crit = 1 (e.g. Freitag 2003), since then significant scatterings would continue to redistribute the orbital parameters. We find however that the final DF (at P = P L ), which is obtained by integrating W ( J ; s crit =1) forward in time according to the GW dissipation equations (Eqs. 20,23) without scattering, is not much different from that shown in Fig. (4). This coincidence is due to the fact that the stars with the largest eccentricities, which typically drop into the MBH, are replenished by the stars with slightly lower eccentricities. The main consequence of choosing s crit too large is an overestimate of the total event rate; stars which actually fall into the MBH are erroneously counted as contributing to the GW event rate. Incidentally, even though stars that do not complete the inspiral are not individually resolvable, they will contribute to the background noise in the LISA band (Barack & Cutler 2004). \n× \nA premature neglect of the effects of scattering in previous studies (e.g. Freitag 2001) probably explains in part why our derived rates are significantly lower. Those studies usually assumed that stars will spiral in without further perturbations once s =1 . We ran a simulation that was stopped at s crit =1 instead of s crit =10 -3 . The event rate in that unrealistic case is about ∼ 6 times higher. This does not explain all of the discrepancies, which are hard to track as different methods are used. The different stopping criterion may also explain whyour rates are lower than those estimated by Hils & Bender (1995), who used a method similar to ours, without specifying \nFIG. 4.- The probability DF of a CO entering the LISA band with eccentricity e for a MBH of mass M · =3 × 10 6 M /circledot . The histograms represent WDs, NSs and BH (from broad to narrow). Note that the maximal possible eccentricity for which P < P L is e max =0 . 81 . See table (2) for the cusp parameters. \n<!-- image --> \nthe criterion for inspiral.', '5.2. Intermediate mass black holes in stellar clusters': "Unlike MBHs with masses M · /greaterorsimilar 10 6 M /circledot , there is little firm observational evidence at this time for the existence of IBHs with masses M · ∼ 10 3 M /circledot (see review by Miller & Colbert 2004). However, there are plausible arguments arguing in favor of their existence. From a theoretical point of view, these objects are thought to form naturally, such as in population III remnants (Madau & Rees 2001), or in a runaway merger of young stars in dense stellar clusters (Portegies Zwart & McMillan 2001; Portegies Zwart et al. 2004). From an observational perspective, IBHs may power some of the ultraluminous X-ray sources (e.g. Miller, Fabian, & Miller 2004), for example by tidal capture of a main sequence companion star (Hopman et al. 2004). \nFor the purpose of estimating the orbital parameters of GW emitting stars, we assume that IBHs lie at the center of dense stellar clusters (see model parameters listed in table 2). White dwarfs are disrupted by an IBH before entering the LISA band, so that only the most compact sources, neutron stars and stellar mass BHs can emit GW in the LISA frequency band. The same values for the stellar population fractions f s where taken here as for MBHs. We note that this is not necessarily the case. For example, N-body simulations indicate that a large fraction of BHs may be ejected in an early stage of the cluster's life if a massive stellar object forms a tight binary with the IBH (Baumgardt et el. 2004b). \nFigure (3) shows the inspiral probability functions S s ( a ) , and Fig. (5) shows the eccentricity histograms of stars on orbits with P = P L . The maximum eccentricity still observable by LISA is nearly unity, and the IBH case shows even more clearly the strong tendency towards large eccentricities. In general, this effect becomes more prominent for lighter BHs. The high eccentricity makes the GW signal highly nonmonochromatic. The star spends most time at apo-apse, emitting a relatively weak, low frequency signal. At periapse short pulses of high frequency GW is emitted. For IBHs of M · ∼ 10 3 M /circledot , the frequency ν p of these short bursts at periapse is of the order ν p /lessorsimilar 100 Hz. This is too high to be measurable by LISA , but may be measurable by ground based detectors such as LIGO or VIRGO . \nFIG. 5.- The probability DF of a compact remnant entering the LISA band with eccentricity e for an IBH of mass M · =10 3 M /circledot . Only NSs (broad) and BHs (narrow) are considered; WDs are probably disrupted by the tidal field. For an IBH, the maximal possible eccentricity for which P < P L is nearly unity, e max = 0 . 998 , all inspiraling stars are likely to have eccentricities close to the maximum value. See table (2) for the cusp parameters. \n<!-- image --> \nAs anticipated (Eq. 35), the rate of inspiraling stars is comparable to that of a MBH (Table 2). However, due to the extremely high eccentricities and small semi-major axes of the orbits, the GW emission is very efficient and they spiral in on a very short time scale (on the order of a year).", '6. SUMMARY AND DISCUSSION': "Stars spiraling into MBH due to the emission of GW are an important potential source for future GW detectors, such as LISA . The detection of the signal against the noise will be challenging, and requires pre-calculated wave-forms. The waveforms depend on the orbital parameters of the inspiraling stars. Our main goal in this study was to derive the distribution of eccentricities of inspiraling stars. The orbital statistics of such stars reflect a competition between orbital decay through dissipation by the emission of GW, and scattering, which deflects stars into eccentric orbits but can also deflect them back to wider orbits or straight into the MBH. \nInspiral is a slow process, and unless the stars start close enough to the MBH, they will be scattered off their orbit. We identified a critical length scale a c which demarcates the volume from which inspiral is possible. We obtained an analytical expression for the inspiral rate and showed that it is of the order of a few per Gyr per galaxy, much smaller than the rate for direct capture (Alexander & Hopman 2003), that it is nearly independent of the relaxation time for typical stellar cusps, and that it grows slowly with decreasing MBH mass (assuming the M · -σ relation). Throughout we assumed a single powerlaw DF. Generalization to different profiles is straightforward. Qualitatively, the inspiral rate is determined by the stellar density near a c , and is not very sensitive to the exact profile far away at radii much smaller or much larger than a c . The rate of GW events depends on the number of COs inside a c (which is much smaller than the MBH radius of influence), and so mass-segregation can play an important role in enhancing the event rate by leading to a centrally concentrated distribution of COs. \nWe obtained a relatively low rate. One important reason for this is our stringent criterion for inspiral. This may also explain why our results deviate from Hils & Bender 1995, although they do not specify their precise criterion for inspiral. Comparison with other works are more complicated. Possi- \nble differences may stem from different normalizations of the central density, different CO fractions, different criteria for inspiral, and the way mass segregation is treated. An essential step in the future will be to analyze inspiral processes by N-body simulations, which are feasible already for small systems (Baumgardt et al. 2004a, 2004b; Preto et al. 2004). \nThe detection rate depends on the inspiral rates, but also on the mass, relaxation time, the orbital parameters (especially the eccentricity) and the details of the detection algorithm (Barack & Cutler 2003). Here we provide a simple recipe to estimate the number of detectable sources. We assume that the various dependencies above can be expressed by an effective strain ˆ h . \nThe strain of GW resulting from a star of mass M /star = mM /circledot orbiting a MBH of mass M · = 10 6 M /circledot M 6 /greatermuch M /star at a distance d = Gpc d Gpc , on a circular orbit of orbital frequency ν = 10 -3 s -1 ν -3 , is given by \nh = 1 . 7 × 10 -23 ν 2 / 3 -3 d -1 Gpc M 2 / 3 6 m. (49) \n(e.g., Sigurdsson & Rees 1997). \nThe cosmic rate of LISA events is given by \nΓ tot = ∫ M max M min dM · dn · dM · Γ i ( M · ) V ( M · ) , (50) \nwhere dn · /dM · is the number of MBHs per unit mass, per unit volume, and V ( M · ) is the volume in which stars can be observed by LISA . Aller & Richstone (2002) used the M · -σ relation to 'weigh' the MBHs. The spectrum is roughly approximated by \ndn · dM · = 10 7 ( 1 10 6 M /circledot ) M -1 6 Gpc -3 . (51) \nThe LISA sensitivity curve at ν = 10 -3 is h ∼ 10 -23 for one year of observation with signal to noise ratio S/N=1 (see e.g. http://www.srl.caltech.edu/lisa ). We adopt this value as a representative detection threshold for ˆ h . If the effective strain has to be be at least 10 -23 ˆ h -23 for the star to be observable, then, for a Euclidean Universe, \nV ( M · ) = 20 . 6 Gpc 3 ˆ h -3 -23 ν 2 -3 m 3 M 2 6 . (52) \nFinally, the rate per MBH is \nΓ i ( M · ) = Γ i (3 × 10 6 M /circledot )(3 M 6 ) -1 / 4 , (53) \nwhere we used the expression for the dependence of the inspiral rate on the MBH mass, equation (35), with β = 4 . The expression can be calibrated with the MC results for a 3 10 6 M /circledot MBH. \nIntegrating (50) from M 6 = 0 . 5 to M 6 = 5 gives \n× \nΓ tot = 1 . 5 yr -1 [ Γ i (3 × 10 6 M /circledot ) Gyr -1 ] m 3 ˆ h -3 -23 ν 2 -3 . (54) \nThe number of sources which LISA can observe at any moment is estimated by N = Γ tot × t L (¯ e, a L ) , where t L is the time a star with eccentricity ¯ e spends in the LISA band before being swallowed; here ¯ e is the average eccentricity of stars entering the LISA band. \nFor example, our calculations for WD inspiral indicate that Γ i (3 × 10 6 M /circledot ) = 7 . 8 Gyr -1 . For a circular orbit with a period of P = 10 3 s and M 6 = 1 , the inspiral time is t L = \nN ∼ \n54 yr , in which case the number of detectable sources would be 130 ˆ h -3 -23 ν 2 -3 . \nWe would like to emphasize that this estimate is to be treated with caution. The number of detectable sources depends strongly on the assumptions. In particular, GWs from eccentric sources are not monochromatic and may be harder to detect. The analysis in this paper can be used for a more detailed analysis of the number of detectable LISA sources. \nWe used three complementary methods to model the inspiral process, MC simulations, numerical solution of the diffusion/dissipation equation and an analytic effective orbit model. We followed the evolution of the orbital properties of the inspiraling stars to the point where they decoupled from the scattering. All three methods were in excellent agreement. We find that the distribution of orbital angular momenta is strongly peaked near J /greaterorsimilar J lc , with the detailed form of the distribution depending somewhat on the parameters of the \nstellar system. We demonstrated that estimates that 'freeze' the scattering prematurely may lead to erroneously high rates by counting stars that actually plunge into the MBH. We then used the most versatile method, the MC simulations, to continue evolving the orbits in a GR Schwarzschild potential, taking into account energy and angular momentum loss due to GW emission (and residual perturbations by scattering). We derived the distribution of eccentricities of the inspiraling stars as they enter the detection window (orbital period of P L /lessorsimilar 10 4 s for LISA ). \nOur main result is that the eccentricities of stars entering the LISA band are strongly skewed toward high values. \nTAis supported by ISF grant 295/02-1, Minerva grant 8484, and a New Faculty grant by Sir H. Djangoly, CBE, of London, UK.", 'REFERENCES': "Hopman, C., Portegies Zwart, S.F., & Alexander, T., 2004, ApJ, 604, L101 Ivanov, P. B., 2002, MNRAS, 336, 373, 2002 Lightman, A. P., & Shapiro, S. L., 1977, ApJ, 211, 244 \nMadau, P., & Rees, M. J., 2001, ApJ551, L27 \nMagorrian, J., & Tremaine, S., 1999, MNRAS, 309, 447 \nMiller, M. C., & Colbert, J. M., 2004, Int.J.Mod.Phys. 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2002PhRvD..65d4021G

Binary black holes in circular orbits. II. Numerical methods and first results

2002-01-01

['-', '-', '-', '-', 'methods numerical', '-', 'black hole physics', '-', '-', 'astrophysics']

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We present the first results from a new method for computing spacetimes representing corotating binary black holes in circular orbits. The method is based on the assumption of exact equilibrium. It uses the standard 3+1 decomposition of Einstein equations and conformal flatness approximation for the 3-metric. Contrary to previous numerical approaches to this problem, we do not solve only the constraint equations but rather a set of five equations for the lapse function, the conformal factor and the shift vector. The orbital velocity is unambiguously determined by imposing that, at infinity, the metric behaves like the Schwarzschild one, a requirement which is equivalent to the virial theorem. The numerical scheme has been implemented using multi-domain spectral methods and passed numerous tests. A sequence of corotating black holes of equal mass is calculated. Defining the sequence by requiring that the ADM mass decrease is equal to the angular momentum decrease multiplied by the orbital angular velocity, it is found that the area of the apparent horizons is constant along the sequence. We also find a turning point in the ADM mass and angular momentum curves, which may be interpreted as an innermost stable circular orbit (ISCO). The values of the global quantities at the ISCO, especially the orbital velocity, are in much better agreement with those from third post-Newtonian calculations than with those resulting from previous numerical approaches.

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https://arxiv.org/pdf/gr-qc/0106016.pdf

{'Binary black holes in circular orbits. II. Numerical methods and first results': "Philippe Grandcl'ement ∗ , Eric Gourgoulhon † and Silvano Bonazzola ‡ D'epartement d'Astrophysique Relativiste et de Cosmologie UMR 8629 du C.N.R.S., Observatoire de Paris, F-92195 Meudon Cedex, France (11 October 2001) \nWe present the first results from a new method for computing spacetimes representing corotating binary black holes in circular orbits. The method is based on the assumption of exact equilibrium. It uses the standard 3+1 decomposition of Einstein equations and conformal flatness approximation for the 3-metric. Contrary to previous numerical approaches to this problem, we do not solve only the constraint equations but rather a set of five equations for the lapse function, the conformal factor and the shift vector. The orbital velocity is unambiguously determined by imposing that, at infinity, the metric behaves like the Schwarzschild one, a requirement which is equivalent to the virial theorem. The numerical scheme has been implemented using multi-domain spectral methods and passed numerous tests. A sequence of corotating black holes of equal mass is calculated. Defining the sequence by requiring that the ADM mass decrease is equal to the angular momentum decrease multiplied by the orbital angular velocity, it is found that the area of the apparent horizons is constant along the sequence. We also find a turning point in the ADM mass and angular momentum curves, which may be interpreted as an innermost stable circular orbit (ISCO). The values of the global quantities at the ISCO, especially the orbital velocity, are in much better agreement with those from third post-Newtonian calculations than with those resulting from previous numerical approaches. \nPACS number(s): 04.25.Dm, 04.70.Bw, 97.60.Lf, 97.80.-d \nWe dedicate this work to the memory of our friend and collaborator Jean-Alain Marck.", 'I. INTRODUCTION': "Motivated by the construction of several gravitational wave detectors (LIGO, GEO600, TAMA300 and VIRGO) great efforts have been conducted in the past years to compute the waves generated by binary black holes. We presented in Ref. [1] (hereafter Paper I) a new method for getting quasi-stationary spacetimes representing binary black holes in circular orbits. See also Paper I for a review on issues and previous works in this field. \nThe basic approximation is to assume the existence of an helical Killing vector : \n/lscript = ∂ ∂t 0 +Ω ∂ ∂ϕ 0 , (1) \nwhere ∂/∂t 0 (resp. ∂/∂ϕ 0 ) is a timelike (resp. spacelike) vector which coincides asymptotically with the time coordinate (resp. azimuthal coordinate) vector of an asymptotically inertial observer. Basically, it means that the two black holes are on circular orbits with orbital velocity Ω [2]. This is of course not exact because the emission of gravitational waves will cause the two holes to spiral toward each other. But this is a valid approximation as long as the time-scale of the gravitational radiation is much longer than the orbital period, which should be true, at least for large separations. The existence of /lscript enables us to get rid of any time evolution. \nWe use the standard 3+1 decomposition of the Einstein equations [3]. We restrict ourselves to a space metric that is conformally flat, i.e. of the form : \nγ = Ψ 4 f , (2) \nwhere Ψ is a scalar field and f denotes the flat 3-metric [4]. Let us mention that the exact spacetime should differ from conformal flatness and that this assumption is only introduced for simplification and should be removed from later works. However it is important to note that it is consistent with the existence of the helical Killing vector and the assumption of asymptotic flatness. The ten Einstein equations then reduce to five equations, one for the lapse function N , one for the conformal factor Ψ and three for the shift vector /vector β (see Paper I for derivation) : \n∆ N = N Ψ 4 ˆ A ij ˆ A ij -2 ¯ D j ln Ψ ¯ D j N (3) \n∆ β i + 1 3 ¯ D i ¯ D j β j = 2 ˆ A ij ¯ D j N -6 N ¯ D j ln Ψ ) (4) \n( \n) ∆Ψ = -Ψ 5 8 ˆ A ij ˆ A ij (5) \nwhere ¯ D i denotes covariant derivative associated with f and ∆ := ¯ D k ¯ D k the ordinary Laplace operator. ˆ A ij is the reduced extrinsic curvature tensor related to K ij by ˆ A ij := Ψ 4 K ij and given by \nˆ A ij = 1 2 N ( Lβ ) ij , (6) \n( Lβ ) ij denoting the conformal Killing operator applied to the shift vector \n( Lβ ) ij := ¯ D i β j + ¯ D j β i -2 3 ¯ D k β k f ij . (7) \nEquations (3), (4) and (5) are a set of five strongly elliptic equations that are coupled. To solve such a system, we must impose boundary conditions. To recover the Minkowski spacetime at spatial infinity, i.e. asymptotical flatness, the fields must have the following behaviors : \nN → 1 when r →∞ (8) \n/vector β → Ω ∂ ∂ϕ 0 when r →∞ (9) \nΨ → 1 when r →∞ . (10) \nAs we wish to obtain solutions representing two black holes and not Minkowski spacetime, we must impose a nontrivial spacetime topology. In Paper I, we define the topology to be that of the real line R times the 3-dimensional Misner-Lindquist manifold [5,6] ; this defines two throats, being two disjointed spheres S 1 and S 2 of radii a 1 and a 2 , centered on points ( x 1 , 0 , 0) and ( x 2 , 0 , 0) (such that | x 1 -x 2 | > a 1 + a 2 ). Following Misner [5], Lindquist [6], Kulkarny et al. [7], Cook et al. [8-10] and others [11,12], we demand that the two sheets of the Misner-Lindquist manifold are isometric. Moreover we choose the lapse function N to be antisymmetric with respect to this isometry. We solve the Einstein equations only for the 'upper' sheet, i.e. only for the space exterior to the throats, with boundary conditions given by : \nN | S 1 = 0 and N | S 2 = 0 (11) \n∣ \n/vector β ∣ S 1 = 0 and /vector β ∣ ∣ S 2 = 0 (12) \n∣ ∣ where r 1 and r 2 are the radial coordinates associated with spheres S 1 and S 2 . Equations (11) reflect the antisymmetry of the lapse function N . The boundary conditions for the shift vector, given by Eqs. (12), represent two black holes in corotation (rotation synchronized with the orbital motion), which is the only case studied in this paper. Those boundary conditions should be easily changed to represent other states of rotation (like irrotation ). Equations (13) come from the isometry solely. \n∣ ∣ ∣ \n∣ ∣ ∣ ( ∂ Ψ ∂r 1 + Ψ 2 r 1 )∣ ∣ ∣ S 1 = 0 and ( ∂ Ψ ∂r 2 + Ψ 2 r 2 )∣ ∣ ∣ S 2 = 0 , (13) \n∣ \nThe orbital velocity Ω only appears in the boundary condition for the shift (see Eq. (9)). Equations (3), (4) and (5) can be solved for any value of Ω. So we need an extra condition to fix the right value for Ω. This is done by imposing that, at spatial infinity, the metric behaves like a Schwarzschild metric, i.e. by imposing that Ψ 2 N has no monopolar term in 1 /r : \nΨ 2 N ∼ 1 + α r 2 when r →∞ . (14) \nIn other words, Ω is chosen so that the ADM and the 'Komar-like' masses coincides, those masses being given by \nM ADM = -1 2 π ∮ ∞ ¯ D i Ψ dS i (15) \nM Komar = 1 4 π ∮ ∞ ¯ D i NdS i . (16) \nAs shown in [13] and in Paper I this is closely linked to the virial theorem for stationary spacetimes. We will see later that this uniquely determines the orbital velocity, and that this velocity tends to the Keplerian one at large separation. \nThis paper is organized as follows. Sec. II is dedicated to the presentation of the numerical scheme, that is based on multi-domain spectral methods. In Sec. III we present some tests passed by the code, which encompass comparison with the Schwarzschild and Kerr black hole and the Misner-Lindquist solution [5,6]. In Sec. IV we present results about a sequence of binary black holes in circular orbits. In particular we locate the innermost stable circular orbit and compare its location with other works. Sec. V is concerned with extension of this work, for getting more complicated and more realistic results.", 'A. Multi-domain spectral methods': "The numerical treatments used to solve the elliptic equations presented above is based on the same methods that we already successfully applied to binary neutron stars [14]. The sources of the equations being mainly concentrated around each hole we use two sets of polar coordinates centered around each throat (see Sec. I). Note however that the tensorial basis of decomposition is a Cartesian one. For example, a vector field /vector V will be given by its components on the Cartesian basis ( V x , V y , V z ) but each component is a function of the polar coordinates ( r, θ, ϕ ) with respect to the center of one hole or the other. \nWe use spectral methods to solve the elliptic equations presented in Sec. I ; the fields are given by their expansion onto some basis functions. Mainly, we use expansion on spherical harmonics with respect to the angles ( θ, ϕ ) and Chebyshev polynomials for the radial coordinate. Let us mention that there exists two equivalent descriptions : a function can be given in the coefficient space , i.e. by the coefficients of its spectral expansion, or in the configuration space by specifying its value at some collocation points [15]. \nThe sources of the elliptic equations being non-compactly supported, we must use a computational domain extending to infinity. This is done by dividing space into several types of domains : \n- · a kernel , a sphere containing the origin of the polar coordinates centered on one of the throats.\n- · several spherical shells extending to finite radius.\n- · a compactified domain extending to infinity by the use of the computational coordinate u = 1 r . \nThis technique enables us to choose the basis function so that the fields are regular everywhere, especially on the rotation axis and to impose exact boundary conditions at infinity. This has been presented with more details elsewhere [14,16-18]. Note that since the last domain extends to spacelike infinity, the surface integrals defining global quantities, such as (15) and (16), can be computed without any approximation. This contrasts with other numerical methods based on finite domains (cf. e.g. Ref. [19] and Fig. 1 therein). As two different sets of coordinates are used, one centered on each hole, we are left with two computational domains of this type, each describing all space and so overlapping. \nThe sources of the equations being concentrated around the two throats, we wish to split the total equations (5), (4) and (3) into two parts, each being centered mainly around each hole and solved using the associated polar coordinates set. So an equation of the type ∆ F = G will be split into \n∆ F 1 = G 1 (17) \n∆ F 2 = G 2 , (18) \nwith F = F 1 + F 2 and G = G 1 + G 2 . G a is constructed to be mainly concentrated around hole a , and so well described by polar coordinates around this hole. Therefore, the solved equations are : \n∆ N a = N Ψ 4 ˆ A ij ˆ A ij a -2 Ψ ¯ D j Ψ a ¯ D j N (19) \n∆ β i a + 1 3 ¯ D i ¯ D j β j a = 2 ˆ A ij ( ¯ D j N a -6 N Ψ ¯ D j Ψ a ) (20) \n∆Ψ a = -Ψ 5 8 ˆ A ij ˆ A ij a , (21) \nwhere the values with no index represent the total values and the values with index a represent the values 'mostly' generated by hole a ( a = 1 or 2). For example, we have ¯ D i N = ¯ D i N 1 + ¯ D i N 2 , ¯ D i N a being concentrated around hole a . Doing so, the physical equations and sources are given by the sum of equations (21), (20) and (19) for a = 1 and a = 2. For more details about such a splitting of the equations into two parts we refer to [14].", '1. Scalar Poisson equation solver with boundary condition on a single throat': 'Using spectral methods with spherical harmonics, the resolution of the scalar Poisson equation reduces to the inversion of banded matrices. We already presented in details in [17,18] the methods to solve such equations in all space, imposing regularity at the origin and exact boundary condition at infinity. In the case of black holes we wish to replace the regularity at the origin by boundary conditions on the spheres S 1 and S 2 and to solve only for the part of space exterior to those spheres. In Ref. [18] we have shown that, for each couple of indices ( l, m ) of a particular spherical harmonic, we can calculate one particular solution in each domain, two homogeneous solutions in the shells and only one in the kernel (due to regularity) and one in the external domain (due to boundary condition at infinity). The next step was to determine the coefficients of the homogeneous solutions by imposing that the global solution is C 1 at the boundaries between the different domains. \nIn the case of a single throat S , the boundary condition is given by a function of the angles solely, i.e. B ( θ, ϕ ). One wishes to impose that the solution or its radial derivative is equal to B on the sphere which corresponds respectively to a Dirichlet or a Neumann problem. We choose the kernel so that its spherical boundary coincides with the throat. So we do not solve in the kernel with represents the interior of the sphere. B is expanded in spherical harmonics and for each couple ( l, m ), we use one of the homogeneous solution in the first shell to fulfill the Dirichlet or Neumann boundary condition. After that we are left with one particular solution in every domain, one homogeneous solution in the innermost shell and in the external domain and two in the other shells. The situation is exactly the same as when a solution was sought in all space and the coefficients of the remaining homogeneous solutions are chosen to maintain continuity of the solution and of its first derivative. So the generalization of the scheme presented in [17,18] is straightforward and enables us to solve either the Dirichlet or Neumann problem, with any boundary condition imposed on the throat.', '2. Vectorial Poisson equation solver with boundary condition on a single throat': 'We presented extensively two different schemes to solve the vectorial Poisson equation (4) in all space in [18] (the Oohara-Nakamura [20] and Shibata [21] schemes). We present here an extension of the so-called Oohara-Nakamura scheme to impose boundary condition a throat and to solve only for the exterior part of space. The Shibata scheme has not been chosen because, the solution being constructed from auxiliary quantities, it is not obvious at all to impose boundary conditions on it. This is not the case with the Oohara-Nakamura scheme where the final solution is calculated directly as the solution of three scalar Poisson equations. More precisely the solution of (cf. Eq. 20) \n∆ β i + λ ¯ D i ¯ D j β j = V i ( λ /negationslash = -1) (22) \nis found by solving the set of three scalar Poisson equations \n∆ β i = V i -λ ¯ D i χ, (23) \nwhere χ is solution of \n∆ χ = 1 λ +1 ¯ D i V i . (24) \nLet us mention that this scheme should only be used with a source /vector V that is continuous. We use the scalar Poisson equation solvers with boundary condition previously described to solve for each Cartesian component of (23) with the appropriate boundary conditions. But let us recall (see [18]) that the Oohara-Nakamura scheme is only applicable if \nχ = ¯ D i β i (25) \nand that it only ensures that \n∆ χ -¯ D i β i ) = 0 . (26) \n( \nOne can easily show that (26) implies (25) if and only if \nχ | S = ¯ D i β i ∣ S , (27) \n∣ which is the boundary condition we must impose during the resolution of (24) to use this scheme. Let us mention that χ being calculated before /vector β , we must use some iterative procedure. We first solve (24) with an initial guess of the boundary condition and then determine /vector β by solving (23). Using that value, we can determine a new boundary condition for χ , using (27), and so a new /vector β . This procedure is repeated until it has sufficiently converged. The obtained /vector β is then solution of the vectorial Poisson equation with either a Dirichlet or Neumann type boundary condition on the sphere S . \n∣', '3. Elliptic solvers with boundary conditions on two throats': "In order to illustrate how boundary conditions are put on the two spheres S 1 and S 2 , let us concentrate on the Dirichlet problem for the scalar Poisson equation. One wishes to solve \n∆ F = G, (28) \nwith the boundary conditions \nF | S 1 = B 1 ( θ 1 , ϕ 1 ) (29) \nF | S 2 = B 2 ( θ 2 , ϕ 2 ) , (30) \nwhere B 1 and B 2 are arbitrary functions. As explained in Sec. II A, the total equation is split into two parts \n∆ F 1 = G 1 (31) \n∆ F 2 = G 2 , (32) \nthe equation labeled a = 1 or 2, being solved on the grid centered around hole a so that the sphere S a coincides with the innermost boundary of the first shell. \nDuring the first step we solve Eqs. (31) and (32) with the boundary conditions \nF 1 | S 1 = B 1 (33) \nF 2 | S 2 = B 2 (34) \nby means of the scalar Poisson equation solver described in Sec. II B 1. Doing so, the total solution F = F 1 + F 2 does not fulfill the boundary conditions (29)-(30). So we calculate the values of F 1 on the sphere S 2 and modify the boundary condition (34) by B ' 2 = B 2 -F 1 | S 2 . The same modification is done with the boundary condition (33). Then we solve once again for F 1 and F 2 . The whole procedure is repeated until a sufficient convergence is achieved. So we are left with a function F which is solution of the Poisson equation (28) and which fulfills a given Dirichlet-type boundary condition on two spheres (29)-(30). \nThe same thing can be done for the Neumann problem by modifying the boundary conditions using the radial derivatives of the functions F a . The same technique is applied for the vectorial Poisson equation. Let us mention that the iteration on the boundary conditions for /vector β , resulting from the presence of the two spheres, is done at the same time than the one on the quantity χ resulting from to the Oohara-Nakamura scheme (see Sec. II B 2). \nAs seen in the previous section, we can solve elliptic equations with various boundary conditions in all the space exterior to two non-intersecting spheres S 1 and S 2 . But a problem arises from the iterative nature of the total numerical procedure. Suppose that after a particular step the lapse N = N 1 + N 2 has been calculated by means of the two Poisson equations (19). From the very procedure of the elliptic solvers, N 1 (resp. N 2 ) is known everywhere outside sphere S 1 (resp. S 2 ). If the next equation to be solved is the one for the shift vector split like (20), N appears in the source term. We need to know the source everywhere outside the associated sphere S a ( a = 1 , 2) which includes the interior of the other sphere. So we must construct fields that are known in the all space. After each resolution, the fields are filled as smoothly as possible inside the associated sphere. In our example, after the resolution of (19), N 1 and N 2 are filled inside the spheres, so that the total function N is known everywhere. \nThe filling is performed, for each spherical harmonic ( l, m ), by the following radial function : \n- · 3 r 4 -2 r 6 ) ( α + βr 2 if l is even,\n- ) ( · ( 3 r 4 -2 r 6 αr + βr 3 ) if l is odd, \n( ) where the coefficients α and β are calculated so that the function is C 1 across the sphere S a . The multiplication by the polynomial ( 3 r 4 -2 r 6 ) ensures that the function is rather regular at the origin. Of course this choice of filling is not unique and the final result should be independent of the filling procedure, the fields outside the spheres depending only on the boundary conditions on those spheres. The choice of filling may only change the convergence of the numerical scheme. Let us stress that even if the fields are known, regular and C 1 everywhere, they have a physical meaning only outside the throats. The filling is only introduced for numerical purposes.", '1. Regularization of the shift': 'When one imposes corotation for the two black holes, that is a vanishing shift vector on the throats, isometry conditions (59), (60) and (61) of Paper I are trivially fulfilled. Unfortunately this is not the case for (62) and (63) of Paper I. We must find a way to impose that \nin order to get a truly isometric solution. \n∂β θ ∂r ∣ ∣ ∣ ∣ S i = 0 and ∂β ϕ ∂r ∣ ∣ ∣ ∣ S i = 0 (35) \nAnother problem comes from the computation of the reduced extrinsic curvature tensor ˆ A ij by means of (6). Because of division by N = 0 on S 1 and S 2 , we must impose that \n( Lβ ) ij ∣ ∣ S 1 = 0 and ( Lβ ) ij ∣ ∣ ∣ S 2 = 0 , (36) \n∣ \n∣ so that the extrinsic curvature tensor is regular everywhere. Because of the rigidity conditions (12) and for a truly isometric solution verifying (35), the regularity conditions (36) are fulfilled if and only if \n∂β r 1 ∂r 1 ∣ ∣ ∣ S 1 = 0 and ∂β r 2 ∂r 2 ∣ ∣ ∣ ∣ S 2 = 0 . (37) \n∣ \n∣ So, to get a truly isometric and regular solution, both the value and the radial derivative of /vector β must be zero on the throats : \n/vector β ∣ ∣ ∣ S i = 0 and ∂ /vector β ∂r ∣ ∣ ∣ ∣ S i = 0 . (38) \nBut when solving Eq. (4), one can only impose the value at infinity and one of those two conditions, i.e. we can only solve for the Dirichlet or Neumann problem, not for both. We choose to solve the equation (4) for the Dirichlet boundary condition : /vector β = 0 on both spheres. Doing so, the regularity conditions (37), as well as the remaining \n∣ \n( \n) \n) ( \nisometry conditions (35), are not necessarily fulfilled. After each step we must modify the obtained shift vector to enforce (37) and (35). The part of the shift generated by the hole 1 is modified by \n∣ \nβ i 1 ∣ new = β i 1 ∣ old + β i cor , 1 (39) \n∣ \n∣ where r 1 is the radial coordinate associated with hole 1, a 1 the radius of the throat and R an arbitrary radius, typically R = 2 a 1 . The correction procedure is only applied for a 1 ≤ r 1 ≤ R . Let us mention that the function of r 1 in front of ∂ β i ∣ ∣ old /∂r 1 in Eq. (40) has been chosen so that it maintains the value of the shift vector on the sphere 1 and its continuity ( C 1 function). The same operation is done for the other hole. After regularization, the shift vector satisfies (i) the rigidity condition (12), (ii) the isometry conditions (35), and (iii) the condition (37) ensuring the regularity of the extrinsic curvature, but it violates slightly the momentum constraint (4). \n∣ ∣ β i cor , 1 := -( R -r 1 ) 3 ( r 1 -a 1 ) ( R -a 1 ) 3 ∂ β i ∣ ∣ old ∂r 1 ∣ ∣ ∣ ∣ S 1 , (40) \n∣ \nAs seen in Paper I, the regularity is a consequence of the equation \n¯ D i β i = -6 β i ¯ D i ln Ψ . (41) \nBecause this equation is not part of the system we choose to solve, we do not expect that the correction function is exactly zero at the end of a computation. But we will verify in Sec. IV B 1 that it is only a small fraction of the shift vector (less than 10 -3 ), fraction which represents the deviation from Eq. (41) (see also Ref. [22] for an extended discussion). Moreover, we will see in Sec. III B that /vector β cor converges to zero for a single rotating black hole.', '2. Computation of the extrinsic curvature tensor': "Using the regularized shift vector presented above, we can compute the tensor ( Lβ ) ij , which is zero on both throats. To calculate the tensor ˆ A ij one must divide it by the lapse function which also vanishes on both throats. Near the throat 1, N has the following behavior \nN | r 1 → a 1 = ( r 1 -a 1 ) n 1 , (42) \nwhere n 1 is non zero on throat 1 (this supposes that r 1 = a 1 is only a single pole of N , which turns out to be true, ∂N/∂r 1 representing the 'surface gravity' of black hole 1). We can compute n 1 , using an operator that acts in the coefficient space of N and divides it by ( r 1 -a 1 ). The same operation is done with \n∣ \n( Lβ ) ij ∣ ∣ r 1 → a 1 = ( r 1 -a 1 ) l ij 1 . (43) \n∣ The divisions are also done on the second throat. To compute the extrinsic curvature tensor in all space we use \n- · ˆ A ij = l ij 1 / (2 n 1 ) in the first shell around throat 1\n- · ˆ A ij = l ij 2 / (2 n 2 ) in the first shell around throat 2\n- · ˆ A ij = ( Lβ ) ij / (2 N ) in all other regions. \nThis procedure enables us to compute the extrinsic curvature tensor everywhere, without any problem that could arise from a division by zero.", '3. Splitting of the extrinsic curvature tensor': 'In the split equations (19) and (21), the term ˆ A ij 1 appears. This term represents the part of the total extrinsic curvature tensor generated mostly by hole 1 so that the total tensor is given by \nˆ A ij = ˆ A ij 1 + ˆ A ij 2 . (44) \nFor the binary neutron stars treated in [14], those split quantities were constructed by setting ˆ A ij 1 = ( Lβ 1 ) ij / (2 N ). Such a construction is not applicable in the case of black holes. Indeed, only the total shift vector is such that \n( Lβ ) ij = 0 on the throats and not the split shifts /vector β 1 and /vector β 2 . If such a construction were applied the quantity ˆ A ij 1 would be divergent due to division by N = 0 on the throats. The computation presented in the previous section gets rid of such divergences but enables us to calculate only the total ˆ A ij . \nThe construction of ˆ A ij 1 and ˆ A ij 2 is then obtained by \nˆ A ij 1 = ˆ A ij H 1 (45) \nˆ A ij 2 = ˆ A ij H 2 , (46) \nwhere H 1 and H 2 are smooth functions such that H 1 + H 2 = 1 everywhere. We also want H 1 (resp. H 2 ) to be close to one near hole 1 (resp. 2) and close to zero near hole 2 (resp. 1), so that ˆ A ij 1 (resp. ˆ A ij 2 ) is mostly concentrated around hole 1 (resp. 2). So, we define H 1 by \n- · 1 if r 1 ≤ R int\n- · 1 2 cos 2 ( π/ 2 ( r 1 -R int ) / ( R ext -R int )) + 1 ] if R int ≤ r 1 ≤ R ext \n[ \n- · 0 if r 2 ≤ R int\n- · 1 2 sin 2 ( π/ 2 ( r 1 -R int ) / ( R ext -R int )) if R int ≤ r 2 ≤ R ext\n- · 1 2 if r 1 ≥ R ext and r 2 ≥ R ext , \nwhere r 1 (resp. r 2 ) is the radial coordinate associated with throat 1 (resp. 2). The radii R int and R ext are computational parameters, chosen so that the different cases presented above are exclusive. Typically, we choose R int = d/ 6 and R ext = d/ 2, where d is the coordinate distance between the centers of the throats. H 2 is obtained by permutation of indices 1 and 2.', 'D. Numerical implementation': "The numerical code implementing the method described above is written in LORENE (Langage Objet pour la RElativit'e Num'eriquE), which is a C++ based language for numerical relativity developed by our group. A typical run uses 12 domains (6 centered on each black hole) and N r × N θ × N ϕ = 33 × 21 × 20 (resp. N r × N θ × N ϕ = 21 × 17 × 16) coefficients in each domain in high resolution (resp. low resolution). For each value of Ω, a typical calculation takes 50 steps. To determine the right value of the angular velocity, by means of a secant method, it takes usually 5 different calculations with different values of Ω. The associated time to calculate one configuration is approximatively 72 hours (resp. 36 hours) for the high resolution (resp. for the low resolution) on one CPU of a SGI Origin200 computer (MIPS R10000 processor at 180 MHz). The corresponding memory requirement is 700 MB (resp. 300 MB) for the high resolution (resp. low resolution).", 'A. Schwarzschild black hole': 'In this section we solve Eqs. (3) and (5), with boundary conditions (11) and (13) on a single throat S . The behaviors at infinity are given by Eqs. (8) and (10). In this particular case, the shift vector /vector β is set to zero, so that ˆ A ij vanishes. This represents a single, static black hole, and we expect to recover the Schwarzschild solution in isotropic coordinates. \nThe computation has been conducted with a initial guess far from the expected result. More precisely, we began the computation by setting N = 1 and Ψ = 1 everywhere. Equations (3) and (5) are then solved by iteration. Let us mention that the boundary condition on the conformal factor, given by (13), is obtained by iteration. At each step we impose \n∂ Ψ J ∂r ∣ ∣ ∣ S = Ψ J -1 2 r ∣ ∣ ∣ S , (47) \n∣ ∣ where Ψ J is the conformal factor at the current step and Ψ J -1 at the previous one. Before beginning a new step, some relaxation is performed on the fields by \n∣ \n∣ \nQ J ← λQ J +(1 -λ ) Q J -1 , (48) \nwhere 0 < λ ≤ 1 is the relaxation parameter, typically λ = 0 . 5. Q stands for any of the fields for which we solve an equation ( N and Ψ solely for the static case). \nThe iteration is stopped when the relative difference between the lapse obtained at two consecutive steps is smaller than the threshold δN = 10 -13 . The computation has been performed with various number of collocation points and with two shells. All the errors are estimated by the infinite norm of the difference. \nFIG. 1. Relative difference between the calculated and the analytical lapse N with respect to the number of radial spectral coefficients for the Schwarzschild black hole. The circles denote the error in the innermost shell, the squares that in the other shell and the diamonds that in the external domain. \n<!-- image --> \nFIG. 2. Same as Fig. 1 for the conformal factor Ψ. \n<!-- image --> \nFigures 1 and 2 show a extremely good agreement with the exact analytical solution. The saturation level of approximatively 10 -13 is due to the finite number of digits (15) used in the calculations (round-off errors). Before the saturation, the error is evanescent (exponential decay with the number of collocation points), which is typical of spectral methods.', 'B. Kerr black hole': 'In this section we consider a single rotating black hole by setting /vector β /negationslash = 0. Let us mention that, since the Kerr solution is known to diverge from conformal flatness (see e.g. [23]), we will no be able to recover exactly the Kerr metric. In other words the obtained solution is expected to violate some of the 5 Einstein equations we decided to ignore. \nSo we solve Eqs. (3), (4) and (5) with boundary conditions (11) (12) and (13) on one single sphere. The values at infinity are chosen to maintain asymptotical flatness by using Eq. (8), (9) and (10). The two parameters of our rotating black hole are the radius of the throat S and the rotation velocity Ω. The total mass M and and angular momentum J are computed at the end of the iteration. \nInitialy the values of N and Ψ are set to those of a Schwarzschild black hole and the shift is set to zero. Relaxation is used for all the fields with a parameter λ = 0 . 5. As for the Schwarzschild computation, we use two shells with the same number N r × N θ × N ϕ of collocation points in the two shells and in the external compactified domain. The iteration is stopped when the relative difference between the shifts obtained at two consecutive steps is smaller than δβ = 10 -10 . \nBefore comparing the obtained solution to the Kerr metric we perform some self-consistency checks, by varying the number of coefficients of the spectral expansion. First of all, we need to verify that the regularization function applied to the shift by means of Eq. (39) has gone to zero at the end of the computation. Figure 3 shows that, for various values of the Kerr parameter J/M 2 , the relative norm of the regularization function decreases very fast, as the number of coefficients increases. The saturation value of 10 -11 is due to the criterium we choose to stop the computation δβ = 10 -10 . Had it been conducted for a greater number of steps, the saturation level of the double precision would have been reached. Figure 3 enables us to say that the shift solution of (4) fulfills the regularity conditions (12) for the extrinsic curvature tensor. Let us mention, that the fact that the conformal approximation is not valid, does not prevent the correction function /vector β cor from going to zero. \nFIG. 3. Relative norm of the regularization function given by Eq. (39) with respect to the Kerr parameter J/M 2 , for various numbers N r × N θ × N ϕ of collocation points. \n<!-- image --> \nAs seen in Paper I, the total angular momentum can be calculated in two different ways, using a surface integral at infinity: \nJ = 1 8 π ∮ ∞ ˆ A i j m j dS i , (49) \n(where m := ∂/∂ϕ ) or an integral on the throat: \nJ = -1 8 π ∮ r = a Ψ 6 ˆ A ij f jk m k d ¯ S i , (50) \nwhere d ¯ S i denotes the surface element with respect to the flat metric f . \nThe two results will coincide if and only if the momentum constraint \n¯ D i ( Ψ 6 ˆ A ij ) = 0 (51) \nhas been accurately solved in all the space. This is a rather strong test for the obtained value of ˆ A ij . Figure 4 shows that the relative difference between the two results rapidly tends to zero, as the number of coefficients increases. The same saturation level as in Fig. 3 is observed. \nThe last self-consistency check is to verify the virial theorem considered in Sec. I. In other words we wish to check if the ADM and Komar masses are identical, which should be the case for a Kerr black hole. We plotted the relative difference between these two masses, for various numbers of collocation points and rotation velocities in Fig. 5. Once more this difference rapidly tends to zero as the number of coefficient increases. Contrary to the case of two black holes, the angular velocity Ω is not constrained by the virial theorem, reflecting the fact that an isolated black hole can rotate at any velocity (smaller than the one of an extreme Kerr black hole). \nFIG. 4. Same as Fig. 3 for the relative difference between the angular momentum calculated by means of Eq. (49) and that by means of Eq. (50). \n<!-- image --> \n<!-- image --> \nwhere \nTo end with a single rotating black hole, we check how far the numerical solution is from an exact, analytically given, Kerr black hole. Given the ADM mass M and the reduced angular momentum a = J/M , an exact Kerr metric in quasi-isotropic coordinates would take the form \nds 2 = -N 2 Kerr dt 2 + B 2 Kerr r 2 sin 2 θ ( dϕ -N ϕ Kerr dt ) 2 + A 2 Kerr dr 2 + r 2 dθ 2 ) , (52) \n( \n) with N Kerr , N ϕ Kerr , A Kerr and B Kerr known functions. It obviously differs from asymptotical flatness because A /negationslash = B for a /negationslash = 0. So we define a pseudo-Kerr metric by setting B = A , which gives \nds 2 = -N 2 Kerr dt 2 +Ψ 4 Kerr [ r 2 sin 2 θ ( dϕ -N ϕ Kerr dt ) 2 + dr 2 + r 2 dθ 2 ) ] , (53) \n) where Ψ 4 Kerr = A 2 Kerr . After a numerical calculation, we compute the global parameters M and a , calculate the functions N Kerr , N ϕ Kerr and Ψ Kerr and compare them to the ones that have been calculated numerically. Note that N ϕ := β ϕ -Ω. The coefficients of the pseudo-Kerr metric are given by \nN 2 Kerr := 1 -2 MR Σ + 4 a 2 M 2 R 2 sin 2 θ Σ 2 ( R 2 + a 2 ) + 2 a 2 Σ MR sin 2 θ \n(54) \nΨ 4 Kerr := 1 + 2 M r + 3 M 2 + a 2 cos 2 θ 2 r 2 + ( M 2 -a 2 ) M 2 r 3 + ( M 2 -a 2 ) 2 16 r 4 (55) \nN ϕ Kerr := 2 aMR Σ( R 2 + a 2 ) + 2 a 2 MR sin 2 θ , (56) \nR := r + M 2 -a 2 4 r + M (57) \nΣ := R 2 + a 2 cos 2 θ. (58) \nThose analytical functions are then compared with that obtained numerically (see Fig. 6). As expected the difference between the fields is not zero and it increases with Ω, reflecting the fact that a Kerr black hole deviates more and more from conformal flatness as J increases. \nTo summarize the results about a single rotating throat, we are confident in the fact that the Eqs. (3), (4) and (5) have been successfully and accurately solved, with the appropriate boundary conditions. On the other hand we do not claim to recover the exact Kerr metric, for this latter differs from conformal flatness. \nFIG. 6. Relative difference between the pseudo-Kerr quantities defined by Eqs. (54)-(56) and the numerically calculated ones with respect to the angular velocity. The computation has been performed with N r × N θ × N ϕ = 25 × 17 × 16. The circles denote the differences on N , the squares on Ψ and the diamonds on N ϕ := β ϕ -Ω. \n<!-- image --> \n(', 'C. The Misner-Lindquist solution': 'Misner [5] and Lindquist [6] have found the conformal factor Ψ of two black holes in the static case, i.e. when /vector β = 0 (see also Ref. [24] and Appendices A and B of Ref. [25]). In such a case the equation for Ψ is only \n∆Ψ = 0 , (59) \nwhich was to be solved using boundary conditions (10) and (13). In the case of identical black holes, that is for two throats having the same radius a , the solution is analytical and does only depend on the separation parameter \nD := d a , (60) \nd being the coordinate distance between the centers of the throats. To check if our scheme enables us to recover such a solution, we solve Eq. (59) with the boundary conditions (10) and (13). We then compute the ADM mass by means of the formula (see Paper I) \nM = -1 2 π ∮ ∞ ¯ D i Ψ dS i (61) \nand compare the result to the analytical value given by a series in Lindquist article [6]. \nLet us mention that, even if Eq. (59) is a linear equation (the source is zero), the problem has to be solved by iteration because of our method for setting the boundary condition (13). The computation has been conducted with a relaxation parameter λ = 0 . 5 and until a convergence of δ Ψ = 10 -10 has been attained. The comparison between the analytical and calculated ADM masses is plotted on Fig. 7 for various values of the separation parameter D and various numbers of coefficients. The agreement is very good for every value of D . As for the Kerr black hole, when the number of coefficients increases, we attain the saturation level of a few 10 -10 is due to the threshold chosen for stopping the calculation. This test makes us confident about the iterative scheme used to impose boundary conditions onto the two throats S 1 and S 2 . \nFIG. 7. Relative difference between the calculated and analytical ADM mass for the Misner-Lindquist solution. The computation has been performed with various number of coefficients N r × N θ × N ϕ . \n<!-- image --> \nTo go a a bit further and check the decomposition of the sources into two parts, presented in Sec. II A, we wish to consider a test problem with a source different from zero. To do so we consider the Misner-Lindquist problem but decide to solve for the logarithm of Ψ, Φ = ln Ψ. The equation for Φ is \n∆Φ = -¯ D k Φ ¯ D k Φ (62) \nand it must be solved with the following boundary conditions \nΦ → 0 when r →∞ (63) \n∣ \n∂ Φ ∂r 1 ∣ ∣ ∣ S 1 = -1 2 a 1 and ∂ Φ ∂r 2 ∣ ∣ ∣ S 2 = -1 2 a 2 . (64) \n∣ ∣ The source of the equation for Φ containing Φ itself, it is split as described in Sec. II A by \n∣ \n∆Φ a = -¯ D k Φ ¯ D k Φ a , (65) \na being 1 or 2. At the end of a computation, we compute the ADM mass by using \nM = -1 2 π ∮ ∞ ¯ D i Φ dS i (66) \nand compare it with the analytical value. The computation used a relaxation parameter λ = 0 . 5 and has been stopped when the threshold δ Φ = 10 -7 has been reached. \nFIG. 8. Relative difference between the calculated and analytical ADM mass for the Misner-Lindquist solution calculated using Φ = ln Ψ, with respect to the number of coefficients in ϕ ( N θ = N ϕ +1 and N r = 2 N ϕ +1). The separation parameter is D = 10. \n<!-- image --> \nFigure 8 shows the resulting relative error estimated by means of the ADM mass for D = 10 and various numbers of coefficients. The convergence is evanescent , i.e. it is exponential as the number of coefficients increases. Unfortunately, this convergence is much slower than when the solution was computed using Ψ and Eq. (59). This comes from the very nature of the source of Eq. (65). The part of the equation split on the coordinates centered around throat 1 is the sum of two terms. The first one -¯ D k Φ 1 ¯ D k Φ 1 is centered around hole 1 and well described by spherical coordinates associated with this hole. We do not expect any problems with this term. The other term is -¯ D k Φ 1 ¯ D k Φ 2 and contains a part that is centered around hole 2. Describing this part using spherical coordinates around hole 1 is much more tricky and a great number of coefficients, especially in ϕ , is necessary to do it accurately. It is the presence of such a component at the location of the other hole that makes the convergence of the calculation much slower in this case. Of course, we expect to recover this effect in the calculation of orbiting black holes.', 'A. Numerical procedure': 'In this section we concentrate on equal mass black holes. The only parameter is the ratio D between the distance of the centers of the holes and the radius of the throats [see Eq. (60)]. We solve Eqs. (3), (4) and (5), with values at \ninfinity given by (8), (9) and (10) and with boundary conditions on the horizons by (11), (12) and (13). We solve for various values of Ω and choose for solution the only value that fulfills the condition (14). It turns out that this process uniquely determines the angular velocity. Let us call Ω true the only value that equals the ADM and the Komar-like masses. It happens that \n- · if Ω < Ω true then M Komar < M ADM\n- · if Ω > Ω true then M Komar > M ADM . \nThe fact that Ω -Ω true has always the same sign than M Komar -M ADM enables us to implement a very efficient procedure to determine the orbital velocity. It is found as the zero of the function M Komar (Ω) -M ADM (Ω) by means of a secant method. This is illustrated by Fig. 9, which shows the value ( M ADM -M Komar ) /M Komar for various values of Ω, with respect to the step of the iterative procedure. Those calculations have been performed for D = 16. The solid line denotes Ω true , the only value of Ω for which ( M ADM -M Komar ) /M Komar converges to 0. \nFIG. 9. Value of ( M ADM -M Komar ) /M Komar with respect to the step of the iteration, for D = 16 and for various values of Ω. The solid line denotes Ω true , the short-dashed line Ω = 0 . 95 Ω true and the long-dashed line Ω = 1 . 08 Ω true . \n<!-- image --> \nThe computations have been done either in low resolution with N r × N θ × N ϕ = 21 × 17 × 16 coefficients in each domain or in high resolution with N r × N θ × N ϕ = 33 × 21 × 20 coefficients in each domain. All the computation used a relaxation parameter λ = 0 . 5. We solve first for the static case Ω = 0 and use that solution as initial guess. For each value of Ω, the computation is stopped for a relative change on the shift vector as small as δβ = 10 -8 (resp. δβ = 10 -7 ) for the high resolution (resp. low resolution) between two consecutive steps. The secant procedure for the determination of the angular velocity has been conducted until | ( M ADM -M Komar ) /M Komar | < 10 -5 (resp. < 10 -4 ) for the high resolution (resp. low resolution), which gives a precision on Ω true of the order of 10 -4 (resp. 10 -3 ).', '1. Check of the momentum constraint': 'As discussed in Sec. II C 1, we have to slightly modify the shift vector to ensure both the regularity of the extrinsic curvature on the throats and the invariance of the shift under the inversion isometry. This modification of the shift, via the addition of the correction function /vector β cor , results in a slight violation of the momentum constraint Eq. (4). A good way to measure the magnitude of this violation is to check whether the total angular momentum J has the same value when calculated by surface integrals at infinity or on the throats. Indeed, as for the Kerr black hole, it has been shown in Paper I that J can be given either by one of the two following integrals: \nJ = 1 8 π ∮ ∞ ˆ A i j m j d ¯ S i , (67) \nJ = -1 8 π ∮ S 1 Ψ 6 ˆ A ij f jk m k d ¯ S i -1 8 π ∮ S 2 Ψ 6 ˆ A ij f jk m k d ¯ S i . (68) \nAny difference between those two formulas would reflect the fact that the momentum constraint (51) is not exactly fulfilled. \nFIG. 10. Relative difference between the regularized shift and the exact solution of (4) (circles) and relative difference between J calculated by means of (67) and (68) (squares). The filled symbols and the solid line denote the high resolution and the empty symbols and the dashed line the low resolution. \n<!-- image --> \nWe have plotted the relative difference δJ/J between the two integrals (67) and (68) in Fig. 10 as a function on the separation between the two holes. Also shown on the same figure is the relative norm of the correction function | /vector β cor | / | /vector β | . The correlation between the two curves shows that the error on the momentum constraint arises from the introduction of the correction function on the shift. It is also clear from Fig. 10 that increasing the number of coefficients in the spectral method does not make the correction function tend to zero. This means that the error in the momentum constraint come rather from the method (necessity to regularize the shift vector) than from some lack of numerical precision. \nAs discussed in Paper I, we had to regularize the shift vector because Eq. (41) is not enforced in our scheme. It has been argued recently by Cook [22] that if one reformulates the problem by assuming that the helical vector /lscript is not an exact Killing vector, but only an approximate one - as it is in reality - then the only freely specifiable part of the extrinsic curvature, as initial data, is (6), not (41). This means that the relation (41) between the extrinsic curvature and the shift is not as robust as the relation (6). \nHowever, we see from Fig. 10 that at the innermost stable circular orbit, which is located at D = 17 (cf. Sec. IV C), the error are very small: \nδJ/J = 2 10 -2 (69) \n| /vector β cor | = 8 10 -4 | /vector β | . (70) \nThe δJ/J error estimator maximizes the error on the momentum constraint because it integrates it in all space. Thus we conclude that momentum constraint is satisfied in our numerical results with a precision of the order 1%.', '2. Check of the Smarr formula': 'A good check of the global error in the numerical solution is the generalized Smarr formula derived in Paper I : \nM -2Ω J = -1 4 π ∮ S 1 Ψ 2 ¯ D i Nd ¯ S i -1 4 π ∮ S 2 Ψ 2 ¯ D i Nd ¯ S i . (71) \nFor any computation, one gets M , Ω and can compute the r.h.s. of Eq. (71) and use that equation to derive the value of J that fulfills the Smarr formula. That value is then compared to the ones calculated using Eqs. (67) and (68). The comparison is plotted in Fig. 11 for the two different resolutions. It turns out that the angular momentum calculated at infinity is better in fulfilling the Smarr formula than the one calculated on the throats by an order of magnitude and that the precision is better than 5 × 10 -3 . So, for all following purposes, we will use the value of J given by Eq. (67). \nFIG. 11. Relative error on the generalized Smarr formula (71). The circles denote the error obtained using J calculated at infinity [Eq. (67)] and the squares that obtained when evaluating J on the throats [Eq. (68)]. The filled symbols and the solid line denote the high resolution and the empty symbols and the dashed line the low resolution. \n<!-- image -->', '3. Check of Kepler law at large separation': "The next thing one wishes to test is the value of Ω, obtained from the virial criterium (14). In Newtonian gravity, two points particles on circular orbits obey the following relation, which is equivalent to Kepler's third law: \n4 J Ω 1 / 3 M 5 / 3 = 1 , (72) \nwhere M is the total mass, J the total angular momentum and Ω the orbital velocity. For large separations of the two throats we expect to recover this relation. Therefore, for every value of D , we evaluate \nI := 4 J Ω 1 / 3 M 5 / 3 (73) \nand check if I tends to 1 when D →∞ . \nFIG. 12. Value of I = 4 J ( Ω /M 5 ) 1 / 3 (low resolution runs) with respect to the separation parameter D . The horizontal dashed line corresponds to the value predicted by Kepler's third law. \n<!-- image --> \nThe value of I is plotted in Fig. 12 with respect to the distance parameter D . As expected, for large values of D , it tends to 1, implying that for large separations the system behaves like two point particles in Keplerian motion.", 'C. Evolutionary sequence': "Let us first present some figures about the metric fields. Figure 13 shows the total lapse function N , conformal factor Ψ and the shift vector /vector β and Fig. 14 the components ˆ A xx , ˆ A xy and ˆ A yy of the extrinsic curvature tensor. All those plots are cross-section in the orbital plane z = 0 and the coordinate system is a Cartesian one centered at the middle of the centers of the throats. The computation has been done using the high resolution. The separation parameter is D = 17. As it will be seen later, this separation corresponds to the turning point in the energy and angular momentum curves. \nFIG. 13. Isocontour of the lapse function N and of the conformal factor Ψ and plot of the shift vector /vector β , for D = 17, in the orbital plane z = 0. The computation has been done using the high resolution. The thick solid lines denote the surfaces of the throats. The kilometer scale of the axis corresponds to an ADM mass of 31 . 8 M /circledot . \n<!-- image --> \nFIG. 14. Isocontour of the extrinsic curvature tensor for D = 17 in the orbital plane z = 0. The solid (dashed) lines denote positive (negative) values. The thick solid lines denote the surfaces of the throats. The computation has been done using the high resolution. The kilometer scale of the axis corresponds to an ADM mass of 31 . 8 M /circledot . \n<!-- image --> \nIn the previous section, the only parameter we considered was the dimensionless separation parameter D . But there also exists a scaling factor. Suppose that all the distances in the computation are multiplied by some factor α . Another solution with the same value of D will be obtained, the global quantities being rescaled as \nM α = αM 1 (74) \nJ α = α 2 J 1 (75) \nΩ α = Ω 1 α , (76) \nwhere M 1 , J 1 and Ω 1 are the values before rescaling and M α , J α and Ω α the values after the rescaling. \nConsider a physical configuration corresponding to a value D ( n ) of the separation parameter, with global quantities M ( n ), J ( n ) and Ω ( n ). This system will evolve due to the emission of gravitational radiation. A subsequent configuration n + 1 will have D ( n +1) < D ( n ). But what scaling factor α should be applied to the configuration calculated for D ( n +1) to ensure that it represents the same physical system as before ? In other word, a physical sequence is a one parameter (the separation) family of configurations and we have to impose another condition to determine the scaling factor associated with each value of D . In the case of binary neutron stars the condition is obtained by imposing that the number of baryons is conserved (see e.g. Ref. [14]). This cannot be extended to the black holes case since no matter is present. We chose instead to define a sequence by requiring that the loss of energy (ADM mass) dM and angular momentum dJ due to gravitational wave emission are related by \ndM dJ ∣ ∣ ∣ sequence = Ω . (77) \n∣ \nThis relation is exact at least when one considers only the quadrupole formula (see e.g. page 478 of Ref. [26]). It turns out that it is also well verified for sequences of binary neutron stars [27,28]. So Eq. (77) should hold rather well for corotating black holes. \nThe scaling factor α associated with the separation parameter D ( n +1) can be computed from the global values at separation D ( n ) and the unscaled values at separation D ( n +1) as the solution the third degree equation \nM ( n ) -αM 1 ( n +1) J ( n ) -α 2 J 1 ( n +1) = 1 2 ( Ω( n ) + Ω 1 ( n +1) α ) , (78) \nwhich is a first-order translation of Eq. (77). To present the results, we define the following dimensionless quantities \n¯ M := M M 0 (79) \n¯ J := J M 2 0 (80) \n¯ Ω := M 0 Ω (81) \n¯ l := l M 0 , (82) \nwhere l is the proper separation of the holes, defined as the geometrical distance between the throats along the axis joining their centers. M 0 is some arbitrary mass used for normalization purpose. It is often convenient to choose M 0 to be the total mass of the system when the two holes are infinitely separated, i.e. the ADM mass when D → ∞ . Unlike other methods, this value is not an input parameter of our calculation. It can only be obtained by constructing a sequence until very large values of D , which would impose to calculate a great number of configurations. However, as will been seen further, the system will exhibit turning point in the total energy and angular momentum, thereafter assumed to be the signature of an innermost stable circular orbit (thereafter ISCO). We chose M 0 to be the total ADM mass of the system at that point : \nM 0 := M ADM | ISCO , (83) \nso that ¯ M is 1 at the location of the ISCO. \nThe values of the dimensionless quantities ¯ Ω, ¯ J , ¯ M and ¯ l along the sequence are given by Table I, for the high resolution. \nTABLE I. Values of dimensionless quantities along a sequence of corotating black holes obtained using the high resolution. The bold line denotes the values at the location of the ISCO. \n| D | ¯ Ω | ¯ J | ¯ M | ¯ l |\n|-----|-------------|------------|-----------|-----------|\n| 40 | 0 . 0296159 | 0 . 995862 | 1 . 00597 | 13 . 3502 |\n| 39 | 0 . 0307026 | 0 . 987981 | 1 . 00573 | 13 . 0699 |\n| 38 | 0 . 0318074 | 0 . 978701 | 1 . 00544 | 12 . 7892 |\n| 37 | 0 . 0330632 | 0 . 971834 | 1 . 00522 | 12 . 5075 |\n| 36 | 0 . 0344638 | 0 . 966535 | 1 . 00504 | 12 . 2247 |\n| 35 | 0 . 0358923 | 0 . 959407 | 1 . 00479 | 11 . 9414 |\n| 34 | 0 . 0374279 | 0 . 952494 | 1 . 00453 | 11 . 6572 |\n| 33 | 0 . 0390612 | 0 . 945264 | 1 . 00426 | 11 . 3721 |\n| 32 | 0 . 0408436 | 0 . 938754 | 1 . 004 | 11 . 086 |\n| 31 | 0 . 0427491 | 0 . 931913 | 1 . 00371 | 10 . 7988 |\n| 30 | 0 . 0448273 | 0 . 925596 | 1 . 00343 | 10 . 5104 |\n| 29 | 0 . 0470335 | 0 . 918635 | 1 . 00311 | 10 . 2211 |\n| 28 | 0 . 0494625 | 0 . 911941 | 1 . 00279 | 9 . 93012 |\n| 27 | 0 . 0521747 | 0 . 907183 | 1 . 00255 | 9 . 6379 |\n| 26 | 0 . 0550996 | 0 . 901696 | 1 . 00225 | 9 . 34411 |\n| 25 | 0 . 0582842 | 0 . 896173 | 1 . 00194 | 9 . 04881 |\n| 24 | 0 . 0617501 | 0 . 89047 | 1 . 0016 | 8 . 75188 |\n| 23 | 0 . 0656222 | 0 . 885511 | 1 . 00128 | 8 . 45286 |\n| 22 | 0 . 0699629 | 0 . 88151 | 1 . 00101 | 8 . 15164 |\n| 21 | 0 . 0747426 | 0 . 877312 | 1 . 00071 | 7 . 84823 |\n| 20 | 0 . 0801137 | 0 . 874079 | 1 . 00046 | 7 . 54243 |\n| 19 | 0 . 086184 | 0 . 871511 | 1 . 00024 | 7 . 23364 |\n| 18 | 0 . 0930453 | 0 . 869573 | 1 . 00007 | 6 . 9218 |\n| 17 | 0 . 100897 | 0 . 86885 | 1 | 6 . 60644 |\n| 16 | 0 . 109958 | 0 . 869769 | 1 . 0001 | 6 . 28724 |\n| 15 | 0 . 120329 | 0 . 870729 | 1 . 00021 | 5 . 96358 |\n| 14 | 0 . 132657 | 0 . 874838 | 1 . 00073 | 5 . 63471 |\n| 13 | 0 . 14712 | 0 . 880134 | 1 . 00147 | 5 . 30006 |\n| 12 | 0 . 164512 | 0 . 888448 | 1 . 00276 | 4 . 95863 | \nFIG. 15. ¯ M with respect to ¯ J along a sequence. The filled symbols and solid line denote the high resolution and the empty symbols and the dashed line the low one. \n<!-- image --> \nFIG. 16. ¯ M with respect to ¯ Ω along a sequence. The filled symbols and solid line denote the high resolution and the empty symbols and the dashed line the low one. \n<!-- image --> \nFIG. 17. ¯ J with respect to ¯ Ω along a sequence. The filled symbols and solid line denote the high resolution and the empty symbols and the dashed line the low one. 6.60644 \n<!-- image --> \nFIG. 18. ¯ l with respect to ¯ Ω along a sequence. The filled symbols and solid line denote the high resolution and the empty symbols and the dashed line the low one. \n<!-- image --> \nFigures 15, 16, 17 and 18 show the values of the dimensionless quantities along a sequence. The calculation has been performed with the high and low resolutions and for values of the parameter D ranging from 40 to 11. As previously mentioned, the sequence exhibits a minimum of ¯ J and ¯ M as the throats become closer, thereafter interpreted as the signature of an innermost stable circular orbit (ISCO) [29]. But at this point, we have to be cautious. Indeed, the relative variation of ¯ M and ¯ J along a sequence is rather small, and comparable to the precision estimated by means of the Smarr formula (see Sec. IV B). The exact location of the ISCO being very dependent on those small effects, we do not claim to have very precisely determined it. The following results should be confirmed with more precise calculations. \nAnother important quantity is the area of the black hole horizons which relates to the irreducible mass [30] (see also Box 33.4 of [31]). As discussed in Sec. II.B.6 of Paper I, in our case the apparent horizons coincide with the two throats. We therefore define the dimensionless irreducible mass by \n¯ M ir := 1 M 0 ( √ A 1 16 π + √ A 2 16 π ) , (84) \nwhere A a ( a = 1 , 2) denotes the area of the throat a , calculated according to the formula \nA a = ∮ S a Ψ 4 d ¯ S. (85) \nFIG. 19. Relative change of ¯ M ir along the sequence, with respect to the orbital velocity ¯ Ω. The filled symbols and the solid line denote the high resolution and the empty symbols and the dashed line the low resolution. \n<!-- image --> \nFigure 19 shows the relative change of ¯ M ir along the sequence. It exhibits a slight increase, but its variation is very small. It appears that, along the overall sequence, the variation is smaller than 10 -3 . The precision of our code being of that order, this result is compatible with the fact that ¯ M ir is constant. In other word it shows that imposing the condition (77) is equivalent to imposing that the irreducible mass is constant along a sequence. This constitutes in fact a very good test of our procedure. Indeed Friedman, Uryu & Shibata [32] have recently established the first law of binary black hole thermodynamics: \ndM = Ω dJ + κ 1 dA 1 + κ 2 dA 2 , (86) \nwhere κ 1 and κ 2 are two constants, representing the black holes surface gravity. For identical black holes ( κ 1 = κ 2 and dA 1 = dA 2 ), the above relation implies \ndM = Ω dJ ⇐⇒ dA a = 0 ( a = 1 , 2) . (87) \nHence the area of each black hole must be conserved during the evolution. In future works, this last criterium could be used to define a sequence, instead of the relation (77). \nWe choose an average value of the irreducible mass ¯ M ir = 1 . 0173 and we define then the binding energy of the system at the ISCO by \n∣ \n¯ E b ∣ ISCO = 1 -¯ M ir , (88) \n∣ the dimensionless total energy being equal to 1 at the location of the turning point. \nThe values of the dimensionless quantities ¯ Ω, ¯ J , ¯ E b and ¯ l at the ISCO are given in Table II and compared with the results from other approaches (see [29] for a review). 3-PN EOB stands for the third order post-Newtonian Effective One Body method for non-spinning black holes [33], with two values of the 3-PN parameter ω s : ω s = 0 and ω s = -9 . 34. 3-PN j-method stands for third order post-Newtonian j-method of [33]. Puncture denotes the results from the puncture method in the case of non-spinning black holes [34] and Conformal imag. the conformal imaging approach with various values of the individual spins for rotating black holes [11]. By definition ¯ M = 1 at the location of the ISCO for all the methods. The results from the different methods are also plotted in Fig. 20. \nFIG. 20. Values of ¯ E b and ¯ J with respect to ¯ Ω at the ISCO for different methods, including ours with high and low resolution. The references to previous studies are as follows: Damour et al. 2000: [33], Pfeiffer et al. 2000: [11] and Baumgarte 2000: [34]. S denotes the (fixed) spin of the black holes used in various methods. \n<!-- image --> \nTABLE II. Values of dimensionless quantities at the location of the ISCO. Comparison with other works. \n| Method | ¯ Ω | ¯ J | ¯ E b | ¯ l |\n|-----------------------------------------|----------|---------|------------|-----------|\n| 3-PN EOB ω s = 0 S = 0 [33] | 0 . 0868 | 0 . 847 | - 0 . 0170 | not given |\n| 3-PN EOB ω s = - 9 . 34 S = 0 [33] | 0 . 0722 | 0 . 877 | - 0 . 0152 | not given |\n| 3-PN j-method ω s = - 9 . 34 S = 0 [33] | 0 . 0731 | 0 . 877 | - 0 . 0153 | not given |\n| Puncture S = 0 [34] | 0 . 176 | 0 . 773 | - 0 . 0235 | 4 . 913 |\n| Conformal imag. S = 0 [11] | 0 . 162 | 0 . 779 | - 0 . 0230 | 5 . 054 |\n| Conformal imag. S = 0 . 08 [11] | 0 . 182 | 0 . 799 | - 0 . 0250 | 4 . 705 |\n| Conformal imag. S = 0 . 17 [11] | 0 . 229 | 0 . 820 | - 0 . 0279 | 4 . 040 |\n| This work (high res.) | 0 . 101 | 0 . 869 | - 0 . 0173 | 6 . 606 |\n| This work (low res.) | 0 . 105 | 0 . 867 | - 0 . 0173 | 6 . 450 | \nFigure 20 shows explicitly that the present results are in much better agreement with post-Newtonian calculations than with other numerical works. Note that the post-Newtonian point plotted on Fig. 20 corresponds to the value 0 of the (previously ambiguous) 3-PN 'static' parameter ω s . This is indeed the value recently determined by Damour et al. [35]. \nBut let us point out that it is rather difficult to compare precisely our results with the other works. The main problem comes from the fact that all those methods use individual spins of the black holes as input parameters. In the present paper we impose corotation, that is that the throats are spinning at the orbital velocity. The only value that can be computed is the total angular momentum J and, in general relativity, it cannot be split into orbital and spins parts in a invariant way. However, from the results of Pfeiffer et al. [11] one can see that increasing the spins of the black holes make the values ¯ Ω, ¯ J and -¯ E b at the ISCO greater. Taking rotation into account in the post-Newtonian methods [36] will probably make the orbital velocity and the binding energy at the ISCO match even better with our values. Work is under progress to compare with corotating post-Newtonian results [37]. \nSo, it appears that our results match pretty well with post-Newtonian methods. This is the most striking conclusion from our study. The difference between numerical and post-Newtonian results have often been imputed mostly to the conformal flatness approximation (see [29]). The fact that our result, using conformal flatness , is in much better agreement with PN calculations than other numerical works, makes us believe that the main worry of both conformal imaging and puncture methods lies elsewhere, possibly in the determination of Ω. Indeed, it is very unlikely that the orbits and so orbital velocity can be properly computed by solving only for the four constraint equations. Time should be involved at some level and one should take other Einstein equations into account, as we have done here.", 'V. CONCLUSIONS': 'The present work should be seen as a first step in trying to give some new insight to the binary black holes problem. The basic idea is to extend the numerical treatment beyond the resolution of the four constraint equations within a 3-dimensional spacelike surface. This is achieved by reintroducing time in the problem to deal with a 4-dimensional spacetime. The orbits are well defined by imposing the existence of a helical Killing vector and the orbital velocity is found as the only value that equals the ADM and the Komar-like masses, a requirement which is equivalent to the virial theorem. According to us those are the two most important features of our method. The approximation of conformal flatness for the 3-metric has only been used for simplicity. Sooner or later this problem will have to be solved using a general spatial metric and outgoing waves boundary conditions at large distances. The use of the inversion isometry to derive boundary conditions on the throats is also a weak assumption. In the future, it would be interesting to change the boundary conditions on the fields in order to investigate their influence on the results (see e.g. [22] for an alternative choice). Besides, changing the boundary conditions on the shift vector should enable us to describe other states of rotation of the black holes, as has been recently proposed by Cook [22]. \nThe numerical schemes are basically the same as those which have been previously successfully applied to binary neutron stars configurations [14]. They have been extended to solve elliptic equations with non-trivial boundary conditions imposed on two throats and exact boundary conditions at infinity. Those techniques passed numerous tests and recover the Schwarzschild and Kerr solutions as well as the Misner-Lindquist one for two static black holes [5,6]. A technical problem lies in the great number of coefficients needed to accurately describe the part of the sources located around the companion hole. This effect causes some lack of precision. But we can estimate the error it generates by varying the number of coefficients, and comparing the results. This is what we have done here, using 21 × 17 × 16 coefficients in each of the 12 domains for the low resolution computations and 33 × 21 × 20 coefficients for the high resolution ones. The accuracy, estimated from the generalized Smarr formula, is below 1%. \nAnother issue is the slight violation of the momentum constraint which arises from the necessity to regularize the shift vector. We have found that the modification of the shift vector with respect to the vector which satisfies the momentum constraint (4) is below 10 -3 , and that the error it induces in the momentum constraint equation is of the order 1%. In view of the other approximations performed in this work, especially the conformal flatness of the 3-metric, we find this to be very satisfactory. \nIn this article, we have defined a sequence of binary black holes by requiring that the ADM mass decrease is related to the angular momentum decrease via dM = Ω dJ . This relation is true for the loss due to gravitational radiation, at least when one considers only the quadrupole formula. We have then found that the area of the apparent horizons (irreducible mass) is constant along the sequence, in agreement with the first law of binary black holes thermodynamics recently derived by Friedman et al. [32]. \nThe location of the ISCO has been obtained and compared with the results from other methods [33,34,11]. It turns out that our results match the 3-PN methods much better than previous numerical works. The differences between numerical studies and 3-PN approximations have often been explained by the use of the conformal flatness \napproximation in the numerical calculations [33]. It seems to us that this is not the main explanation, for we are using this approximation. It certainly arises instead from the way Ω is determined. \nAnother natural extension of this work is to use the obtained configurations as initial data for binary black holes evolution codes (see [38] for a review and Refs. [39-41] for recent results). Initial data files containing the result of the present work are publically available on the CVS repository of the European Union Network on Sources of Gravitational Radiation [42]. Extraction of the wave-forms from a sequence would also be an interesting application [43,44].', 'ACKNOWLEDGMENTS': "This work has benefited from numerous discussions with Luc Blanchet, Brandon Carter, Thibault Damour, David Hobill, J'erˆome Novak and Keisuke Taniguchi. We warmly thank all of them. The code development and the numerical computations have been performed on SGI workstations purchased thanks to a special grant from the C.N.R.S. 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2009ApJ...701.1175S

X-ray Polarization from Accreting Black Holes: The Thermal State

2009-01-01

['black hole physics', 'polarization', 'radiative transfer', 'astronomy x rays', '-']

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We present new calculations of X-ray polarization from black hole (BH) accretion disks in the thermally dominated state, using a Monte Carlo ray-tracing code in full general relativity. In contrast to many previously published studies, our approach allows us to include returning radiation that is deflected by the strong-field gravity of the BH and scatters off of the disk before reaching a distant observer. Although carrying a relatively small fraction of the total observed flux, the scattered radiation tends to be highly polarized and in a direction perpendicular to the direct radiation Agol &amp; Krolik. For moderately large spin parameters (a/M gsim 0.9), this scattered returning radiation dominates the polarization signal at energies above the thermal peak, giving a net rotation in the polarization angle of 90°. We show how these new features of the polarization spectra from BHs in the thermal state may be developed into a powerful tool for measuring BH spin and probing the gas flow in the innermost disk. In addition to determining the emission profile, polarization observations can be used to constrain other properties of the system such as BH mass, inclination, and distance. New instruments currently under development should be able to exploit this tool in the near future.

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https://arxiv.org/pdf/0902.3982.pdf

{'X-ray Polarization from Accreting Black Holes: II. The Thermal State': 'Jeremy D. Schnittman \nDepartment of Physics and Astronomy, Johns Hopkins University Baltimore, MD 21218 \[email protected] \nand \nJulian H. Krolik \nDepartment of Physics and Astronomy, Johns Hopkins University Baltimore, MD 21218 \[email protected]', 'ABSTRACT': 'We present new calculations of X-ray polarization from black hole (BH) accretion disks in the thermally-dominated state, using a Monte-Carlo ray-tracing code in full general relativity. In contrast to many previously published studies, our approach allows us to include returning radiation that is deflected by the strong-field gravity of the BH and scatters off of the disk before reaching a distant observer. Although carrying a relatively small fraction of the total observed flux, the scattered radiation tends to be highly polarized and in a direction perpendicular to the direct radiation. For moderately large spin parameters ( a/M /greaterorsimilar 0 . 9), this scattered returning radiation dominates the polarization signal at energies above the thermal peak, giving a net rotation in the polarization angle of 90 · . We show how these new features of the polarization spectra from BHs in the thermal state may be developed into a powerful tool for measuring BH spin and probing the gas flow in the innermost disk. In addition to determining the emission profile, polarization observations can be used to constrain other properties of the system such as BH mass, inclination, and distance. New instruments currently under development should be able to exploit this tool in the near future. \nSubject headings: black hole physics - accretion disks - X-rays:binaries', '1. INTRODUCTION': "A recent flurry of new mission proposals has renewed interest in X-ray polarization from a variety of astrophysical sources. The Gravity and Extreme Magnetism SMEX ( GEMS ) mission 1 , which has recently been approved for phase A study in the latest round of NASA Small Explorer proposals, should be able to detect a degree of polarization δ /lessorsimilar 1% for a flux of a few mCrab (Black et al. 2003; Bellazzini et al. 2006). A similar detector for the International X-ray Observatory ( IXO ) could achieve sensitivity roughly 10 × greater ( /lessorsimilar 0 . 1% degree of polarization; Jahoda et al. 2007, Costa et al. 2008). A large number of galactic and extra-galactic sources are expected to be polarized at the δ /greaterorsimilar 1% level, including black holes, magnetars, pulsar wind nebulae, and active galactic nuclei. In this paper, we focus on accreting stellar-mass black holes (BHs) in the thermal state, which are characterized by a broad-band spectrum peaking around 1 keV. The typical level of polarization from these sources should be a few percent in the 1 -10 keV range, depending on BH spin and the inclination angle of the accretion disk. \nSymmetry arguments demand that in the flat-space (Newtonian) limit, the observed polarization from the disk must be either parallel or perpendicular to the BH/disk rotation axis. However, the effects of relativistic beaming, gravitational lensing, and gravito-magnetic frame-dragging can combine to give a non-trivial net rotation to the integrated polarization vector. Early work exploring these effects (Stark & Connors 1977; Connors & Stark 1977; Connors et al. 1980) showed that they create changes in the angle and degree of polarization that are strongest for higher photon energy. The reason for this trend is that the temperature in a standard accretion disk increases as the BH is approached, and that is, of course, where the relativistic effects are strongest. Since those first efforts, there have been studies of the predicted polarization from more general accretion geometries, including UV and X-ray emission from AGN disks (Laor et al. 1990; Matt et al. 1993) and 'lamp post' models for irradiating the accretion disk with a non-thermal source above the plane (Dovciak et al. 2004). Quite recently, Dovciak et al. (2008) investigated the effect of atmospheric optical depth on the polarization signal, and Li et al. (2008) applied the original calculations of thermal X-ray polarization to the problem of measuring the inclination of the inner accretion disk. \nDespite this relatively large body of literature dedicated to X-ray polarization from BH accretion disks, to the best of our knowledge, all previous work has modeled the relativistic effects by calculating the transfer function along geodesics between the observer and emitter. By its very nature, this method precludes the possibility of including the effects of returning \nradiation-photons emitted from one part of the disk and bent by gravitational lensing so that they are absorbed or reflected from another part of the disk (Cunningham 1976). Using new methodology, in this paper we show the contribution from this returning radiation is very important to the observed polarization. The details of our method are described in Schnittman (2009) (hereafter 'Paper I'). The most important feature of our approach is that the photons are traced from the emitting region in all directions, either returning to the disk, scattering through a corona, getting captured by the BH, or eventually reaching a distant observer. It should be noted that a sort of hybrid technique was used in Agol & Krolik (2000) to estimate the effects of returning radiation on polarization by situating an observer in the disk plane and shooting rays backwards to see how much flux was incident from other regions of the disk, but this approach cannot account for multiple scattering events, where the returning radiation passes over the BH more than once. \nThe inclusion of returning radiation leads to the most important of the new results presented in this paper, namely a transition between horizontal- and vertical-oriented polarization as the photon energy increases. At low energies we reproduce the 'Newtonian' result of a semi-infinite scattering atmosphere emitting radiation weakly polarized in a direction parallel to the emission surface, an orientation we call horizontal polarization (Chandrasekhar 1960). At higher energy, corresponding to the higher temperature of the inner disk, a greater fraction of the emitted photons returns to the disk and is then scattered to the observer. These scattered photons have a high degree of polarization and are aligned parallel to the disk rotation axis ( vertical ), as projected onto the image plane. At the transition point between horizontal and vertical polarization, the relative contributions of direct and reflected photons are nearly equal, and no net polarization is observed. \nSince the effects of returning radiation are greatest for photons coming from the innermost regions of the disk, the predicted polarization signature is strongly dependent on the behavior of gas near and inside the inner-most stable circular orbit (ISCO). Thus, polarization observations could be used to measure the spin of the black hole and also constrain the dissipation profile of the inner disk. For systems where the BH mass, distance, and disk inclination are known, a single polarization observation can constrain the emission profile at least well as fitting the thermal continuum spectrum (Gierli'nski et al. 2001; Davis et al. 2006; Shafee et al. 2006). When these system parameters are not known a priori , polarization can be used in many cases to measure them simultaneously with the emission profile (albeit with lower confidence than for the sources with known priors). \nThe structure of this paper is as follows: after briefly describing our methods in Section 2, we review in Section 3 some of the classical results of earlier polarization calculations from thermal disks, including only the radiation that reaches the observer directly from the disk. \nIn Section 4, we include the effects of returning radiation, focusing on new features in the polarization spectrum. Sections 5 and 6 contain a detailed discussion of our results and how they depend on model parameters, and in Section 7 we conclude.", '2. METHODOLOGY': "The inner disk of a stellar-mass BH binary can be well described locally as a planeparallel, electron scattering-dominated atmosphere (Shakura & Sunyaev 1973; Novikov & Thorne 1973). In the local fluid frame, the classical result of Chandrasekhar (1960) applies: the polarization vector of the emitted radiation is oriented parallel to the disk plane and normal to the direction of propagation. The degree of polarization varies from zero for photons emitted normal to the disk surface up to ∼ 12% for an inclination angle of 90 · . In addition to the polarization effects, the scattering of the outgoing flux causes limb darkening, effectively focusing the emitted radiation in the direction normal to the disk surface. Both the degree of polarization and the limb darkening/brightening factors are tabulated as a function of emission angle in Table XXIV in Chandrasekhar (1960). \nFor the local emission spectrum, we assume a diluted black-body spectrum with hardening factor f = 1 . 8 (Shimura & Takahara 1995). As demonstrated by Davis et al. (2005), this is a reasonable approximation for disks around stellar-mass black holes because the temperature is so high that there is little opacity due to anything but Compton scattering and free-free absorption. For a locally emitted flux F , the effective temperature is defined as \nT eff ≡ ( F σ ) 1 / 4 , (1) \nwhere σ is the Stefan-Boltzmann constant. The spectral intensity of the diluted emission is then given by \nI ν = 1 f 4 B ν ( f T eff ) , (2) \nwith B ν the black-body function. For the emitted flux function F ( R ), we consider two different models: that given by Novikov & Thorne (1973) for a zero-stress inner boundary condition at the ISCO, and a quasi-Newtonian expression F ( R ) ∼ R -3 that extends the classical scaling at large radius all the way down to the horizon. Recent global magnetohydrodynamic (MHD) simulations suggest that reality lies somewhere in between these two models (Noble et al. 2008). Outside of the ISCO, the gas follows circular, planar geodesic orbits as in Novikov & Thorne (1973). Inside the ISCO, the fluid continues along plunging trajectories with constant energy and angular momentum, resulting in a rapid decline of surface density with decreasing radius. \nand \nAs described in Paper I, at each point in the disk, multi-wavelength photon packets are emitted in all directions, with appropriate intensity weights and polarizations as a function of emission angle. We use a Monte Carlo algorithm with typically 10 5 -10 6 rays traced per radius, with 500 radial bins spaced logarithmically between the horizon and R = 100 M . Since the photon trajectories originate at the disk, the observer's inclination angle is not specified; rather, all photons are followed until they get captured by the BH, return to the disk plane, or reach 'infinity,' in practice a spherical hypersurface at large radius. At that point, the photons are binned by observer inclination, effectively simulating all inclinations simultaneously. By calculating the angle that each photon makes with the detector sphere, we are also able to create an image of the source, much like a classical pin-hole camera. \nEach photon packet contains a complete broad-band spectrum initially described by equation (2). Since Thomson scattering in the disk atmosphere is frequency-independent over the range of relevant X-ray energies ( ∼ 0 . 1 -10 keV for thermal emission from a solar-mass BH), the photon packet can be described by a single polarization amplitude and direction. \nFollowing Connors et al. (1980), we characterize the polarization of each photon packet by the normalized Stokes parameters: \nX = Q/I, (3a) \nY = U/I, (3b) \nδ = ( X 2 + Y 2 ) 1 / 2 , (4a) \nψ = 1 2 tan -1 ( Y/X ) , (4b) \nwhere ψ is the angle of polarization, ranging from -90 · to +90 · , and δ ≤ 1 is the degree of polarization, invariant along a geodesic. \nThe polarization vector f is a normalized space-like 4-vector perpendicular to the photon propagation direction: k · f = 0 and f · f = 1. It is parallel-transported along the photon's null geodesic, conserving the complex-valued Walker-Penrose integral of motion κ (Walker & Penrose 1970). Analogous to the way that Carter's constant (Carter 1968) can be used to constrain four-velocity components, κ can be used to reconstruct the polarization vector at any point along the geodesic. The two orthonormality conditions stated above, along with the real and imaginary parts of κ , give a total of four equations for the four components of the polarization vector f . When a ray reaches an observer at infinity, the polarization vector is projected onto the detector plane defined by the basis vectors e 1 and e 2 to \ngive the polarization angle: f = [0 , cos ψ e 1 +sin ψ e 2 ] (in our convention, e 2 is parallel to the projected rotation axis of the BH and disk). From δ and ψ , as measured by the detector, we can reconstruct the normalized Stokes parameters X and Y via equation (4). The observed spectrum I ν is propagated along the geodesic path as well, containing within it the emitted spectrum, modified by all relativistic effects as well as any energy change that may occur when returning radiation scatters off disk electrons. I ν provides the absolute normalization that, when applied to X and Y allows Q ν and U ν to be recovered for each photon bundle reaching the detector. Integrating over all photon bundles gives the polarization degree δ ν and angle ψ ν as a function of energy (see Paper I for details). These are ultimately the quantities that we wish to be able to measure and use to probe the accretion geometry of the BH.", '3. DIRECT RADIATION': "In Figure 1, we show a simulated image of a Novikov-Thorne accretion disk around a black hole with spin parameter a/M = 0 . 9 and luminosity 0 . 1 L E , corresponding to a disk whose X-ray spectrum peaks around 1 keV for a BH mass of 10 M /circledot . With the observer at an inclination of i = 75 · , significant relativistic effects are clearly apparent. The increased intensity on the left side of the disk is due to special relativistic beaming of the gas moving towards the observer, and the general relativistic light bending makes the far side of the disk appear warped and bent up towards the observer. Superposed on top of the intensity map is the polarization signature, represented by small black vectors whose lengths are proportional to the degree of polarization observed from that local patch of the disk. Far from the black hole, the polarization is essentially given by the classical result of Chandrasekhar (1960) for a scattering-dominated atmosphere: horizontal orientation with δ ≈ 4% when i = 75 · ; nearer to the black hole, a variety of relativistic effects alter the polarization. \nWhile the fundamental calculation of polarization from a thermal BH accretion disk is not at all new (Connors et al. 1980), to the best of our knowledge this is the first published polarization map as seen projected in the image plane. The usefulness of such a map is primarily as a tool to understand how relativistic effects determine the integrated polarization at different energies. \nThe two most prominent relativistic effects are gravitational lensing and special relativistic beaming, both lowering the net level of polarization seen by the observer. Gravitational lensing causes the far side of the disk to appear warped up towards the observer, and thus have a smaller effective inclination. Relativistic beaming causes photons emitted normal to the disk plane in the fluid frame to travel forward in the direction of the local orbital motion \nwhen seen by a distant observer; the result is a smaller effective inclination and thus degree of polarization. On the other hand, photons emitted at high inclination angle in the local fluid frame but against the direction of orbital motion are ultimately seen by observers located at lower inclination, and therefore raise the polarization in that direction. Naturally, these relativistic effects are most important close to the black hole, where the gas is also hottest and the photons have the highest energies. All these effects are clearly visible in Figure 1, which shows a smaller degree of polarization where the beaming is greatest (yellow region of high intensity in the left of the image) and the lensing is strongest (just above the center of the image). At the same time, the gas moving away from the observer on the right of the image has an enhanced level of polarization because the observer sees photons emitted at a larger inclination in the fluid frame. \nIntegrating over the entire disk, we can calculate the angle and degree of polarization as a function of energy. Plotted in Figure 2, these results essentially reproduce those of Connors et al. (1980), who performed a similar calculation using semi-analytic transfer functions from the observer to the disk. Again we see that at low energies (dominated by emission from large disk radii), the polarization is given by the Chandrasekhar result, and is a function only of the disk inclination angle. At higher energies, we begin to probe the inner regions of the accretion disk and see the relativistic effects described above that lower the total degree of polarization. In addition to reducing the polarization amplitude, the strong gravity near the black hole also rotates the angle of polarization due to the parallel transport: since the polarization vector f must remain perpendicular to the photon momentum k , as the geodesic path bends around the BH, the polarization angle must also rotate. For example, the high-energy photons beamed towards the observer in Figure 1 are also bent by the BH's gravity, initially moving to the left before curving back to the observer, thus producing the rotated polarization vectors seen just to the left of the image center (the same lensing effect rotates the red-shifted photons from the right side of the disk, but in an opposite direction). \nThe angle of polarization as a function of energy is plotted in the lower panel of Figure 2. For an inclined accretion disk with the black hole axis projected onto the vertical, or y-axis, ψ = 0 corresponds to horizontal polarization parallel to the disk surface. From equation (4), we see there is a point symmetry in the definition of ψ , giving ψ = ψ ± 180 · . As can be seen in the simulated image of Figure 1, the individual polarization vectors are rotated in the inner disk, due largely to gravitational lensing and, to a lesser degree, frame-dragging around a spinning black hole. While individual photons can experience significant rotation, the net result is a modest rotation of the total observed polarization in the clockwise direction, giving ψ < 0 at higher energies. From Figure 2, it is clear this effect is more pronounced for small inclinations, where the low amplitude of polarization makes it easier to 'overcome' the classical result with additional relativistic effects. Similarly, the polarization rotation \nis greater for more rapidly spinning black holes, where the accretion disk extends closer in towards the horizon and probes a stronger gravitational field. In this way, it has previously been proposed that the polarization degree and angle as a function of energy could be used to infer the spin of the black hole in the thermal state (Connors et al. 1980; Laor et al. 1990; Dovciak et al. 2008).", '4. RETURNING RADIATION: QUALITATIVE DISCUSSION': "When returning radiation is included, although little changes in terms of the total observed spectrum, the polarization picture (Fig. 3) changes significantly-in much of the inner disk, the integrated polarization rotates by 90 · , even though none of the model's physical parameters has been changed at all! \nThis effect can be understood qualitatively in very simple fashion (see also Agol & Krolik (2000)). For most reasonable stellar-mass BH accretion disk models, the opacity in the inner disk is dominated by electron scattering (Shakura & Sunyaev 1973; Novikov & Thorne 1973), so returning radiation in the ∼ 1 -10 keV energy range should scatter off the disk with negligible absorption. Detailed atmosphere calculations (S. Davis, private communication) show that the photospheric region is strongly scattering dominated for R /lessorsimilar 50 M whenever L/L Edd /greaterorsimilar 0 . 01. If these calculations were altered to take into account magnetic contributions to vertical support, which should be substantial at these altitudes in this regime (Hirose et al. 2009), absoroption opacity would likely be even weaker (however, when the central mass is much larger, as in an AGN, the inner disk temperature is much lower, giving significantly greater absorptive opacity in the X-ray band). Moreover, the typical photon scatters only once or twice before permanently departing the disk. Its outgoing polarization is therefore very sensitive to the electron scattering cross section's polarization-dependence: \n( dσ d Ω ) pol = r 2 0 | f i · f f | 2 , (5) \nwhere r 0 is the classical electron radius. In other words, scattering is strong only when the polarization direction changes little. This is possible for both senses of linear polarization when the change in photon direction is small, but when the scattering is nearly perpendicular, the new photon wave-vector becomes nearly parallel to one of the initial polarization directions. Thus, for nearly-perpendicular scattering, the outgoing light can be close to 100% polarized. \nFor observers at high inclination angles, such as in Figure 3, returning radiation photons initially emitted from the far side of the disk (top of the image) are reflected off the near \n(bottom) side with a relatively small scattering angle, maintaining a moderate horizontal polarization as in Figure 1. On the other hand, photons emitted from the left side of the disk can be bent back to the right side (or vice versa ), and then scatter at roughly 90 · to reach the observer, thereby aquiring a large vertical polarization component. Although relatively small in total flux, this latter contribution can dominate the polarization because it is so strongly polarized. A flux component of only 10%, when 100% polarized, contributes more to the polarized flux than does a 90% contribution that is only 5% polarized. \nBecause high-energy photons from the hotter inner parts of the disk experience stronger gravitational deflection, they are more likely to return to the disk than the low-energy photons emitted at larger radii. We find that, for each value of the spin parameter, there is a characteristic 'transition radius,' within which the returning radiation dominates and produces net vertical polarization. Outside this point, the direct radiation dominates and produces horizontal polarization. This transition radius is in fact only weakly dependent on a/M , ranging from R trans ≈ 7 M for a/M = 0 to R trans ≈ 5 M for a/M = 0 . 998. The location and shape of the observed polarization swing can be used to infer the radial temperature profile near the transition radius (see Sec. 6 below for more details). While the majority of the returning obviously originates from the inner-most disk, we find that most of it also scatters inside R /lessorsimilar 10 M as well (where the atmosphere opacity is completely dominated by electron scattering), due to the additional focusing effects of gravitational lensing.", '5. RETURNING RADIATION: QUANTITATIVE RESULTS': "Polarization maps like that shown in Figure 3 provide a useful picture for understanding the physical origin of the local polarization features, but are less useful in providing quantitative predictions for future observations. Since vertically- and horizontally-polarized photons add up to give zero net polarization, it is not clear from Figure 3 alone what the integrated polarization signal should be. Furthermore, Figure 3 shows only the energy-integrated polarization at each point in the disk, masking the important energy dependence of polarization seen in Figure 2. \nIn Figures 4-6 we show the spectral intensity, degree, and angle of polarization as a function of photon energy for a range of BH spins and inclinations. In each frame, we plot the total flux (solid curves), as well as the direct flux from the disk (dotted curves) and the returning radiation (dashed curves). Beginning with the spectral intensity, we see the standard broad thermal peak characteristic of BHs in the 'high soft' state. For BH masses around 10 M /circledot and luminosities of ∼ 0 . 1 L Edd , the thermal spectrum peaks around 1 keV. For a Novikov-Thorne (NT) emission profile, the direct radiation from a non-spinning BH (peak \nemissivity at radius R ∼ 9 M ) dominates over the returning radiation by a factor of roughly 100. As the spin increases and the peak emission region moves closer to the horizon, the relative fraction of observed flux that comes from returning radiation increases to ∼ 5% for a/M = 0 . 9 and ∼ 20% for a/M = 0 . 998. Naturally, the relative contribution from returning radiation increases with energy because the highest energy photons come from the innermost regions of the disk. In fact, for a/M = 0 . 998, the returning radiation actually dominates the spectrum above ∼ 10 keV. \nFor all spins, the fraction of returning radiation increases with inclination for two reasons: forward/backward scattering has a larger differential cross-section than right-angle scattering, so returning photons, which characteristically have large angles of incidence, are reflected at large angles from the axis and preferentially reach observers with high inclination; secondly, the direct radiation suffers limb-darkening effects at high inclination due both to the scattering atmosphere and also the diminished solid angle of the disk, while both effects affect the reflected radiation to a lesser degree (see Paper I for a detailed calculation). For the examples shown in Figures 4-6, the fraction of observed flux coming from returning radiation is larger for i = 75 · than i = 45 · by a factor of about two. \nAt low energies (below the thermal peak), the flux comes predominantly from the outer regions of the disk, where relativistic effects are negligible, so the classic results of Chandrasekhar (1960) are reproduced: horizontal polarization with amplitude of a few percent. The degree of polarization at low energies increases with observer inclination, ranging from δ = 1% for i = 45 · to δ = 3 . 5% for i = 75 · . Although the free-free absorption opacity in the disk increases below ∼ 1 keV, which should reduce the net polarization, we have ignored this effect for two reasons. First, large uncertainties in the vertical structure of the disk make it difficult to arrive at a quantitative prediction for the absorption profile, which is strongly dependent on local temperature and density. Second, most proposed X-ray polarization missions will be sensitive only to photons above 1 keV (Bellazzini et al. 2006), so we choose to focus on this energy range. \nAbove the thermal peak, we see the relativistic effects described in the previous Sections begin to dominate the polarization signal. The direct radiation (dotted curves) undergoes lensing and beaming, thereby reducing the polarization amplitude and rotating the angle. The polarization properties of the returning radiation (dashed curves) are roughly constant in the 1 -10 keV range, but as the relative flux from returning radiation increases, we see the total polarization makes a transition from following the direct behavior to following the returning behavior. In the process of making this transition from horizontal polarization at low energies to vertical polarization at high energies, the degree of polarization goes through a minimum as the two contributions cancel each other. This transition is sharpest for lower \ninclinations, where there is a smaller degree of horizontal polarization to overcome. Yet even for large inclination angles, when the BH spin is high enough, we still see a dramatic transition from horizontal to vertical polarization close to the thermal peak, caused by the enhanced level of returning radiation from the innermost disk. \nDue to this dependence on the relative contribution from returning radiation, if the emissivity profile is NT, the energy at which the polarization transition takes place may be a powerful diagnostic of BH spin. In Figure 7 we plot the polarization degree and angle as a function of energy for a range of spin parameters. In all cases, the BH mass is 10 M /circledot , i = 75 · , and the luminosity is 0 . 1 L Edd (due to differences in accretion efficiency, we use a fixed luminosity, not mass accretion rate, when comparing different spins). For inclinations of i = 45 · or 60 · , this same dependence on spin can be seen by comparing the top and middle rows of Figures 4-6. With even moderate energy resolution and polarization sensitivity of /lessorsimilar 1%, observations of BHs in the thermal state should be able to distinguish between Schwarzschild and extremal Kerr with high significance-if one can safely assume the NT surface brightness profile. \nHowever, recent relativistic MHD simulations suggest that, contrary to the NT assumption of zero stress and zero dissipation inside the ISCO, there are in fact significant stress and dissipation in the plunging region, leading to steadily increasing surface brightness (as measured in the local fluid frame) with decreasing radius (Noble et al. 2008). For a simplistic treatment of this flux, we apply a quasi-Newtonian emissivity profile with F ( R ) ∼ R -3 all the way to the horizon, regardless of the BH spin or ISCO location. The local spectral distribution is still given by equations (1-2). The resulting polarization signals are plotted in Figure 8, showing a clear reduction in the sensitivity to spin parameter. While there is still some variation between the different curves, it would likely be impossible to measure a/M with a first-generation X-ray polarization mission (see below, Sec. 6). \nThe transition energy in the polarization angle is a strong function of the luminosity, which in turn is a function of the temperature of the inner disk. Since the initial polarization and subsequent scattering-induced polarization are independent of frequency at these photon energies, we can think of the luminosity dependence simply as a rescaling of the entire polarization spectrum with inner disk temperature. This scaling is illustrated in Figure 9, which plots the polarization signature for a range of luminosities, in all cases using a NT flux profile with a diluted thermal spectrum and BH mass 10 M /circledot , for a/M = 0 (solid curves) and a/M = 0 . 9 (dashed curves). Since most galactic BHs are observed over a wide range in luminosities, even within the thermal state, multiple polarization observations could give even better constraints on the value of the spin parameter. By reproducing the same polarization at a shifted energy, such a measurement would also help confirm that the features in the \npolarization spectrum are in fact coming from relativistic effects in the inner disk. Again, if the underlying emission profile increases continuously down to the horizon (see Fig. 10), we lose some sensitivity in determining the spin, but gain sensitivity to L/L Edd because the dependence on that variable sharpens.", '6. PARAMETER ESTIMATION': "As shown in the previous section, spectropolarimetry measurements of BHs in the thermal state will give a powerful new way of measuring BH spin, if the underlying emission follows a Novikov-Thorne profile with zero dissipation inside of the ISCO. More generally, polarization observations will probe the emissivity profile directly, offering a new way to measure the temperature profile as a function of the geometrized radius ( R/M ). Well outside the ISCO, conservation laws constrain all models of time-steady disks to follow the NT form; however, nearer the ISCO and throughout the plunging region inside it, more variation is possible. Motivated by global MHD simulations such as those presented in Noble et al. (2008) and Beckwith et al. (2008), we describe the range of possible dissipation profiles by power-laws in radius in the plunging region matched smoothly to the NT profile just outside the ISCO. Figure 11 shows a sample of emission profiles for a/M = 0 . 9 and a range of power-law index α . In the limit α →-∞ , we reproduce NT; α = 3 is the pseudo-Keplerian case shown above in Figures 8 and 10. Also shown for comparison is the dissipation profile calculated by HARM3D for the case of a/M = 0 . 9 and a disk aspect ratio H/R /similarequal 0 . 1, which corresponds to α ≈ 0 . 5 -1 (Noble et al. 2008). \nTo estimate the ability of polarization observations to accurately determine a/M and α for a typical galactic BH binary, we consider two theoretical X-ray polarimeters: a 'firstgeneration' instrument with modest energy range 1 -10 keV, resolution ∆ E/E ≈ 1, and minimum polarization sensitivity of δ /greaterorsimilar 1% with 1 σ confidence. A 'next-generation' detector is characterized by a broader energy range 0 . 1 -100 keV, better resolution ∆ E/E ≈ 0 . 1, and ten times the collecting area, giving δ /greaterorsimilar 0 . 3% for the same source. We then select a few sample 'target' models with different emission profiles and generate simulated polarization data for these cases using the ray-tracing code described above. Scanning over a range of model parameters, we calculate a large number of 'template' polarization spectra to be matched against the targets. \nFor each point in parameter space, we calculate the overlap between a template spectrum and the target spectrum by calculating χ 2 for the specified detector sensitivity and source luminosity. Rather than attempting to combine the absolute luminosity, distance, detector response function and collecting area into an error estimate for a generic spectro- \nolarimeter, we simply describe the instrument and source together in terms of the minimum error achievable in a measurement of δ for a single energy bin. For a broad-band detector with constant sensitivity at all energies, the minimum error will correspond to the peak of the observed intensity spectrum, i.e. the bin with the most photon counts. The error in any other bin simply scales like the square root of the counts in that bin. We use for our observables the Stokes parameters Q ν and U ν , both normalized to the total observed flux I tot = ∫ I ν dν . For a peak sensitivity (minimum error) of δ min , the error on each value of Q ν and U ν is given by \n∆ Q ν = ∆ U ν = δ min I ν √ I peak I ν . (6) \nWith the target values of Q ν and U ν and their associated measurement errors at each energy, it is straightforward to calculate the χ 2 fit for the match between the target 'observation' and any given template spectrum. By generating multiple templates to cover the parameter space, we can estimate the significance with which the model parameters can be determined from future data. \nIn practice, this search through parameter space would be prohibitively expensive computationally if each template were constructed using the full Monte Carlo calculation (typically ∼ 10 8 rays traced per run). Fortunately, we are able to utilize transfer functions similar to those used in calculating the relativistic effects on broad-band continuum spectra (e.g. Davis et al. (2005)). The full details are described in Paper I, but a short summary of the method is as follows: for each value of the spin parameter (roughly 20 points spaced evenly in log(1 -a/M )), a single complete Monte Carlo simulation is carried out as above, with the emission spectrum at each radius given by a delta-function in energy. The initial polarization is still given by a scattering-dominated atmosphere, and the returning radiation is treated as before, typically adding a strong vertical component to the observed polarization. When each ray reaches an observer at infinity, the Stokes parameters I ν , Q ν , and U ν are recorded, each looking essentially like a delta function shifted in energy and amplitude by the relativistic effects of the ray-tracing. \nSorting all the rays by observer inclination angle, we construct a transfer function g ( a/M ) that takes the input emission profile I em ν ( R ) and generates the observed polarization in terms of I obs ν ( i ), Q obs ν ( i ), and U obs ν ( i ). For a given spin, a single transfer function can be used to generate polarization spectra for any set of model parameters M , L/L Edd , and α , all of which combine to give I em ν ( R ). The transfer function can also be used to calculate the transition radius R trans described above in Section 4. For emission profiles with steep gradients around R trans (as would be produced by large values of a/M or α ; see Figs. 8 and 10), the observed polarization spectrum displays a sharp swing around the photon energy corresponding to the disk temperature at that radius: E trans ≈ 3 T ( R trans ). On the other \nhand, moderate spin systems with NT emission profiles have a shallow temperature gradient around R trans , giving a broad swing in the polarization angle (Figs. 7 and 9), or none at all in low-spin cases where there is simply not enough emission from inside R trans to overcome the horizontal contribution from emission outside of R trans . \nFor reasonably high resolution spectra, we find the transfer function method gives a speed-up factor of better than 10,000 compared to the direct Monte Carlo calculation. This method is particularly straightforward in the thermal state, where the scattering cross sections and polarization transport are essentially independent of energy. However, even when including inverse-Compton coronal scattering, the transfer function method is still applicable, with the added dimension that the transfer function is dependent on the seed photon energy and accretion geometry (again, see Paper I for details). \nIn Figure 12, we show quality-of-fit contours for matching the target parameters of three different models: two NT radial profiles with a/M = 0 and a/M = 0 . 998, and one case with additional emissivity in the plunging region with α = 1 and a/M = 0 . 97. All three cases have M = 10 M /circledot , L/L Edd = 0 . 1, and i = 75 · . The top row of figures corresponds to cases where the disk inclination and Eddington-scaled luminosity (i.e. the BH mass and distance) are known a priori , and the target data is fit by scanning over a and α . The middle row repeats this calculation, but assumes prior knowledge only of the disk inclination, and then for each value of a and α minimizes χ 2 over L/L Edd . Finally, the bottom row assumes no prior knowledge about the system and minimizes χ 2 with respect to L/L Edd and inclination i . In each frame, the target parameters are marked with an 'X' and the contours show confidence intervals in significance units: ≤ 1 σ (white), 2 σ (blue), 3 σ (purple), 4 σ (red), 5 σ (orange), and > 5 σ (yellow). Here σ is given by a standard χ 2 distribution: σ = √ 2 ν , with ν the number of data points minus the number of free parameters. For large values of ν , these confidence intervals correspond very closely to those of a normal distribution: (68%, 95.5%, 99.7%, etc.). \nAs seen in Figure 9, the Schwarzschild case closely resembles the Chandrasekhar limit of horizontal polarization with amplitude of a few percent. When the emission cuts off at the ISCO, relativistic effects play a much smaller role in rotating the polarization of the direct radiation. Furthermore, the fraction of returning radiation is much smaller, minimizing the contribution from vertically polarized photons. The only way to generate such a polarization spectum is by producing the observed flux at large radii, from where the photons can propagate through nearly flat space to the distant observer, arriving with their polarization essentially unchanged. Thus if the observed polarization is relatively strong ( /greaterorsimilar 1%), horizontally oriented, and constant with energy, then the emission likely comes from large radii, in turn constraining the spin to be relatively small and strongly limiting the amount \nof dissipation inside the ISCO. This is evident from the upper-left frame in Figure 12, where a polarization observation of a Schwarzschild BH with a NT dissipation profile rules out the possibility of a high value for a/M or α . \nOn the other hand, when matching the polarization from a near-maximal spin BH with a/M = 0 . 998 (upper-right frame of Fig. 12), any system with a NT emission profile and spin a/M /lessorsimilar 0 . 97 is ruled out at the 3 σ confidence level. Yet as described above in Section 5, a non-spinning BH can appear to be rapidly spinning if there is significant emission inside the ISCO. Additionally, since the plunging region is so small for large values of a/M , templates with a range of α (i.e. the emission profile inside the ISCO) are nearly degenerate, as seen in Figure 8. These systems are characterized by a sharp transition in the polarization: it changes from strong, horizontal polarization at low energies to strong, vertical polarization above the thermal peak. \nFor intermediate cases with moderately high spin and α [(0 . 9 /lessorsimilar a/M /lessorsimilar 0 . 99); ( -2 /lessorsimilar α /lessorsimilar 2)], there is a broad minimum in the polarization amplitude between 1 and 10 keV (see, e.g., the red and orange curves in Fig. 7). If the inclination is known to be relatively high, as in our test cases, then this broad minimum can still constrain a/M and α (or at least some combination of the two parameters) reasonably well, as seen in the upper-middle plot of Figure 12. However, if we have no prior knowledge of the disk inclination, we could reproduce such a feature by assuming a low-inclination system, whose light would have little or no polarization near the thermal peak regardless of the emission profile. Therefore, as shown in the lower-middle panel of Figure 12, when minimizing χ 2 over all model parameters, it is always possible to get a decent fit to an unpolarized source, and the emission profile is poorly constrained. At the same time, if a non-zero polarization is detected, it can give a measure not only of the emission profile, but also of the disk inclination. \nThere are also a number of BH binary systems for which the inclination is known relatively well, but the BH mass M and/or distance D are not known. This is the situation illustrated in the middle row of Figure 12. As those panels show, substantial constraining power is retained even without knowledge of L/L Edd (and as the bottom panels show, even when the inclination angle is also unknown, spectropolarimetry still provides some limit on the possible range of α and a/M ). The reason is that with a single spectropolarimetry observation, we can measure, using the shape and energy of the polarization transition, the temperature profile T ( r ) in the inner disk as a function of radius in geometric units r ≡ R/M . From this temperature profile, we can constrain the system's ratio of bolometric luminosity to mass through the relation: \nL ∝ M 2 ∫ r T ( r ) 4 dr . (7) \nUsing the net flux gives L/D 2 , so that we have two constraint equations for three unknowns. \nIf either the BH mass or the distance can be measured by some other means, all three variables ( L , M , and D ) will be known. \nViewed in this way, spectropolarimetry measurements can be seen to have significant advantages relative to pure spectral measurements. When the inclination, mass, and distance are all known, detailed continuum-fitting can be used to measure the temperature profile in the inner disk, giving constraints on α and a/M that are comparable to what is shown in the top row of Figure 12. However, when one or more of these quantities (inclination, distance, or mass) is not known, the continuum-fitting method becomes less able to constrain the radial emission profile and thus the BH spin. \nAt the qualitative level, one can understand the greater power of spectropolarimetry as simply the result of having more observables: δ ( E obs ) and ψ ( E obs ) in addition to F ( E obs ), giving more constraints on the model parameters. However, there are more specific reasons as well. On the one hand, inferring parameters from continuum spectra can be very ambiguous because there are multiple ways to reproduce an observed spectrum by varying M , D , i , and L/L Edd . On the other hand, especially for high-inclination systems, the polarization tends to constrain individual parameters more directly: for example, its magnitude at low energies (where relativistic effects are weak) directly relates to the inclination. At higher energies, the distinctive swing in polarization angle is a direct probe of extreme gravitational lensing and returning radiation, giving a sensitive indicator of strongly relativistic effects in the inner disk. \nNot surprisingly, a next-generation polarimeter with roughly a factor of ten improvement in energy bandwidth, resolution, and collecting area can do a much better job at measuring all these system parameters. Figure 13 shows the same confidence contours as Figure 12, now for the next-generation detector. Almost all degeneracy has been removed, and even when we have no prior knowledge of the binary parameters, a/M , α , i , L/M 2 , and M/D can be determined with high precision. For the three cases considered in the bottom row of Figure 13, the inclination is recovered within ∼ 5 · and the accretion rate within ∼ 20%: \n70 · /lessorsimilar i /lessorsimilar 80 · (8a) \nand \n0 . 08 /lessorsimilar ( L L Edd )( 10 M /circledot M ) /lessorsimilar 0 . 12 . (8b) \nIn fact, at these levels of polarization sensitivity, the ability to measure the intrinsic properties of the system likely becomes more dependent on the accuracy of the underlying emission model, rather than on the signal-to-noise of the observation. For example, the simplified form of the emission spectrum may be modified by a more careful treatment of \nthe disk atmosphere (Davis et al. 2005). Additionally, the low-energy polarization may be strongly modified or reduced by absorption in the disk (Laor et al. 1990), Faraday rotation as the photons pass through turbulent magnetic fields (S. Davis, private communication), and even a small amount of inverse-Compton scattering that is sometimes seen in the thermal state.", '7. DISCUSSION': "We have presented here the first results of a new Monte Carlo ray-tracing code for calculating the X-ray polarization from black holes. This code, described in detail in Paper I, is most notable for its emitter-to-observer paradigm of radiation transport, which allows for the inclusion of returning radiation and electron scattering in a completely general accretion geometry. In this paper we focus on polarization signatures of the thermal state in stellarmass black holes, a condition in which the disk is optically thick and geometrically thin, and its opacity is scattering-dominated. The emitted radiation has a diluted thermal spectrum and is weakly polarized parallel to the disk surface. The integrated polarization spectrum seen by a distant observer contains distinct energy-dependent features, as the high-energy photons from the inner disk are modified by relativistic effects such as Doppler boosting and gravitational lensing near the BH. \nFor radiation originating very close to the BH, the most important relativistic effect is the strong gravitational lensing that causes the photons to get bent back onto the disk and scatter towards the observer. This scattering can induce very high levels of polarization, especially at large inclination angles, and leads to a distinct transition in the polarization angle from horizontal at low energies to vertical above the thermal peak. Observing such a swing in the polarization angle would give the most direct evidence to date for the extreme relativistic light bending predicted around black holes. Furthermore, by measuring the location and shape of this transition, we will be able to constrain the temperature profile of the inner disk. If we assume a NT disk with zero emission from inside the ISCO, the polarization transition energy gives a direct measurement of the BH spin (see Fig. 7). Alternatively, if we relax this assumption, the polarization can constrain models for the dissipation profile and provide compelling evidence for strong-field gravitational effects. \nFor BH systems where the mass, inclination, and distance are known from other observations, polarization measurements would be comparable in power to the continuum fitting method (Gierli'nski et al. 2001; Davis et al. 2006; Shafee et al. 2006) for the purpose of inferring the emissivity profile of the inner disk. However, when we lack knowledge of any one of these priors, polarization provides a significantly stronger tool than continuum fitting for \nconstraining both the shape of the emissivity profile and the unknown parameter(s). \nIn this paper, we have presented the first results from our new black hole X-ray polarization analysis code, corresponding to thermal emission from a geometrically thin accretion disk. In future work, we will extend its application to cases in which a hot corona partially covers a cooler disk (as may be the case in AGN) and to other accretion geometries, appropriate to other spectral states of Galactic black hole binaries.", 'REFERENCES': "Agol, E., & Krolik, J. H. 2000, ApJ, 528, 161 \nBeckwith, K., Hawley, J.F. & Krolik, J.H. 2008, MNRAS 390, 21 \nBellazzini, R., et al. 2006, Nucl. Instr. Meth., A560, 425 \nBlack, J. K., et al. 2003, Nucl. Instr. Meth., 513, 639 \nCarter, B. 1968, Phys. Rev., 174, 1559 \nChandrasekhar, S. 1960. Radiative Transfer , Dover, New York \nConnors, P. A., & Stark, R. F. 1980, Nature, 269, 128 \nConnors, P. A., Piran, T., & Stark, R. F. 1980, ApJ, 235, 224 \nCosta, E., et al. 2008, Proc. SPIE, vol. 7011-15, [arXiv:0810.2700] \nCunningham, C. T. 1976, ApJ, 208, 534 \nDavis, S.W., Blaes, O.M., Hubeny, I. & Turner, N.J. 2005, ApJ, 621, 372 \nDavis, S. W., Done, C., & Blaes, O. M. 2006, ApJ, 647, 525 \nDovciak, M., Karas, V., & Matt, G. 2004, MNRAS, 355, 1005 \nDovciak, M., Muleri, F., Goodmann, R. W., Karas, V., & Matt, G. 2008, MNRAS submitted, [arXiv:0809.0418] \nGierli'nski, M., Macio/suppresslek-Nied'zwiecki, A. & Ebisawa, K. 2001, MNRAS 325, 1253 \nHirose, S., Krolik, J. H., & Blaes, O. 2009, ApJ, 691, 16 \nJahoda, K., Black, K., Deines-Jones, P., Hill, J. E., Kallman, T., Strohmayer, T., & Swank, J. 2007, [arXiv:0701090] \nLaor, A., Netzer, H., & Piran, T. 1990, MNRAS, 242, 560 \nLi, L.-X., Narayan, R., & McClintock, J. E. 2008, ApJ submitted, [arXiv:0809.0866] \nMatt, G., Fabian, A. C., & Ross, R. R. 1993, MNRAS, 264, 839 \nNoble, S. C., Krolik, J. H., & Hawley, J. F. 2008, ApJ submitted, [arXiv:0808.3140] \nNovikov, I. D., & Thorne, K. S. 1973, in Black Holes , ed. C. DeWitt & B. S. DeWitt (New York: Gordon and Breach) \nRemillard, R. A., & McClintock, J. E. 2006, ARA& A, 44, 49 \nRybicki, G. B., & Lightman, A. P. 1979, Radiative Processes in Astrophysics (New York: Wiley-Interscience) \nSchnittman, J. D. 2009, in preparation (Paper I) \nShafee, R., McClintock, J. E., Narayan, R., Davis, S. W., Li, L.-X., Remillard, R. A. 2006, ApJ, 636, L113 \nShakura, N. I., & Sunyaev, R. A. 1973, A&A, 24, 337 \nShimura, T., & Takahara, F. 1995, ApJ, 445, 780 \nStark, R. F., & Connors, P. A. 1977, Nature, 266, 429 \nWalker, M., & Penrose, R. 1970, Commun. Math. Phys., 18, 265 \nFig. 1.- Ray-traced image of direct radiation from a thermal disk. The observer is located at an inclination of 75 · relative to the BH and disk rotation axis, with the gas on the left side of the disk moving towards the observer, which causes the characteristic increase in intensity due to relativistic beaming. The black hole has spin a/M = 0 . 9, mass M = 10 M /circledot , and is accreting at 10% of the Eddington limit with a Novikov-Thorne zero-stress emissivity profile, giving peak temperatures around 1 keV. The observed intensity is color-coded on a logarithmic scale and the energy-integrated polarization vectors are projected onto the image plane with lengths proportional to the degree of polarization. \n<!-- image --> \nFig. 2.- Polarization degree and angle from a thermal disk as a function of observed photon energy, including only direct radiation. The BH parameters are as in Fig. 1 (dashed curves) and also include identical calculations corresponding to a non-spinning BH (solid curves). The angle of polarization is measured with respect to the horizontal axis in the image plane with point symmetry through the origin: ψ = ψ -180 · . \n<!-- image --> \n<!-- image --> \nFig. 3.- Ray-traced image of radiation from a thermal disk, as in Fig. 1, but here including returning radiation. Photons emitted from the inner disk get bent by the BH and scatter off the opposite side of the disk towards the distant observer. \n<!-- image --> \nFig. 4.- Intensity spectrum and polarization degree and angle from a thermal disk, including direct and returning radiation, for a BH with a/M = 0, M = 10 M /circledot , L/L Edd = 0 . 1, and a Novikov-Thorne emission profile. In the left column we plot the observed flux, in the middle the degree of polarization, and on the right the angle of polarization. In each plot, the flux is divided into contributions from the direct (dotted curves), reflected (dashed curves), and total (solid curves). From top to bottom, the observer inclination is 45 · , 60 · , and 75 · . \n<!-- image --> \nFig. 5.- Intensity spectrum and polarization degree and angle from a thermal disk, as in Figure 4, but for a Kerr BH with a/M = 0 . 9. \n<!-- image --> \nFig. 6.- Intensity spectrum and polarization degree and angle from a thermal disk, as in Figure 4, but for a Kerr BH with a/M = 0 . 998. \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFig. 7.- Polarization degree and angle for a range of BH spin parameters. All systems have inclination i = 75 · , BH mass 10 M /circledot , luminosity L/L Edd = 0 . 1, and Novikov-Thorne radial emission profiles. \n<!-- image --> \n<!-- image --> \nFig. 8.- Same as Figure 7 but for a power-law emissivity profile F ( R ) ∼ R -3 all the way to the horizon. \n<!-- image --> \n<!-- image --> \nFig. 9.- Polarization degree and angle for a range of luminosities for a/M = 0 (solid curves) and a/M = 0 . 9 (dashed curves). All systems have inclination i = 75 · , BH mass 10 M /circledot , and Novikov-Thorne radial emission profiles. \n<!-- image --> \nFig. 10.- Same as Figure 9 but for a power-law emissivity profile F ( R ) ∼ R -3 all the way to the horizon. \n<!-- image --> \n<!-- image --> \nFig. 11.- Local flux (as measured in the fluid frame) versus radius for a/M = 0 . 9. In the plunging region, the flux is parameterized by a power law with index α , normalized to match the NT profile ( solid curve ) just outside the ISCO. Also shown is the emissivity calculated by HARM3D ( dotted curve ), a 3-dimensional global MHD code in full relativity (Noble et al. 2008). \n<!-- image --> \nFig. 12.- Contour plots of fit quality in matching emissivity profiles to simulated data for a first-generation X-ray polarimeter. The left column corresponds to simulated data from a non-spinning Schwarzschild BH with a NT emissivity profile ( α = -∞ ), the center column has a/M = 0 . 97 and α = 1, and the right column has a/M = 0 . 998 and α = -∞ . The top row shows the fit quality regions assuming the luminosity in Eddington units and the disk inclination are both known; the middle row assumes we know the disk inclination but not the accretion rate, and the bottom row assumes we know neither. The color coding is ≤ 1 σ (white), 2 σ (blue), 3 σ (purple), 4 σ (red), 5 σ (orange), and > 5 σ (yellow), where σ is the variance of the χ 2 ( ν ) distribution, as defined in the text. \n<!-- image --> \nFig. 13.- Contour plots of fit quality in matching emissivity profiles to simulated data for a next-generation X-ray polarimeter with broader energy range and larger collecting area. The different frames and their color codes are the same as in Figure 12. \n<!-- image -->"}

2016MNRAS.457.2122G

Masses and scaling relations for nuclear star clusters, and their co-existence with central black holes

2016-01-01

['-']

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Galactic nuclei typically host either a Nuclear Star Cluster (NSC, prevalent in galaxies with masses $\lesssim 10^{10}M_\odot$) or a Massive Black Hole (MBH, common in galaxies with masses $\gtrsim 10^{12}M_\odot$). In the intermediate mass range, some nuclei host both a NSC and a MBH. In this paper, we explore scaling relations between NSC mass (${\cal M}_{\rm NSC}$) and host galaxy total stellar mass (${\cal M}_{\star,\rm gal}$) using a large sample of NSCs in late- and early-type galaxies, including a number of NSCs harboring a MBH. Such scaling relations reflect the underlying physical mechanisms driving the formation and (co)evolution of these central massive objects. We find $\sim\!1.5\sigma$ significant differences between NSCs in late- and early-type galaxies in the slopes and offsets of the relations $r_{\rm eff,NSC}$--${\cal M}_{\rm NSC}$, $r_{\rm eff, NSC}$--${\cal M}_{\star,\rm gal}$ and ${\cal M}_{\rm NSC}$--${\cal M}_{\star,\rm gal}$, in the sense that $i)$ NSCs in late-types are more compact at fixed ${\cal M}_{\rm NSC}$ and ${\cal M}_{\star,\rm gal}$; and $ii)$ the ${\cal M}_{\rm NSC}$--${\cal M}_{\star,\rm gal}$ relation is shallower for NSCs in late-types than in early-types, similar to the ${\cal M}_{\rm BH}$--${\cal M}_{\star,\rm bulge}$ relation. We discuss these results in the context of the (possibly ongoing) evolution of NSCs, depending on host galaxy type. For NSCs with a MBH, we illustrate the possible influence of a MBH on its host NSC, by considering the ratio between the radius of the MBH sphere of influence and $r_{\rm eff, NSC}$. NSCs harbouring a sufficiently massive black hole are likely to exhibit surface brightness profile deviating from a typical King profile.

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https://arxiv.org/pdf/1601.02613.pdf

{'and Nadine Neumayer 1': '1 Max-Planck Instiut für Astronomie, Königstuhl 17, 69117 Heidelberg \n- 2 Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA\n- 3 Department of Astrophysics, American Museum of Natural History, Central Park West and 79th Street, New York, NY 10024\n- 4 Department of Astronomy, University of Alberta, 116 St and 85 Ave, Edmonton, AB T6G 2R3, Canada \nAccepted 2015 mm dd. Received 2015 mm dd', 'ABSTRACT': 'Galactic nuclei typically host either a Nuclear Star Cluster (NSC, prevalent in galaxies with masses glyph[lessorsimilar] 10 10 M glyph[circledot] ) or a Massive Black Hole (MBH, common in galaxies with masses glyph[greaterorsimilar] 10 12 M glyph[circledot] ). In the intermediate mass range, some nuclei host both a NSC and a MBH. In this paper, we explore scaling relations between NSC mass ( M NSC ) and host galaxy total stellar mass ( M glyph[star], gal ) using a large sample of NSCs in late- and earlytype galaxies, including a number of NSCs harboring a MBH. Such scaling relations reflect the underlying physical mechanisms driving the formation and (co)evolution of these central massive objects. We find ∼ 1 . 5 σ significant differences between NSCs in late- and early-type galaxies in the slopes and offsets of the relations r eff , NSC -M NSC , r eff , NSC -M glyph[star], gal and M NSC -M glyph[star], gal , in the sense that i ) NSCs in late-types are more compact at fixed M NSC and M glyph[star], gal ; and ii ) the M NSC -M glyph[star], gal relation is shallower for NSCs in late-types than in early-types, similar to the M BH -M glyph[star], bulge relation. We discuss these results in the context of the (possibly ongoing) evolution of NSCs, depending on host galaxy type. For NSCs with a MBH, we illustrate the possible influence of a MBH on its host NSC, by considering the ratio between the radius of the MBH sphere of influence and r eff , NSC . NSCs harbouring a sufficiently massive black hole are likely to exhibit surface brightness profile deviating from a typical King profile. \nKey words: galaxies: nuclei - galaxies: star clusters - galaxies: quasars: supermassive black holes', '1 INTRODUCTION': "A growing body of observational evidence indicates that the nuclear regions of galaxies are often occupied by a nuclear star cluster (NSC) and/or a super massive black hole (SMBH), with NSCs being identified in more than > 60 -70% of early- (e.g. Durrell 1997; Carollo et al. 1998; Geha et al. 2002; Lotz et al. 2004; Côté et al. 2006; Turner et al. 2012; den Brok et al. 2014) and late-type galaxies (e.g. Böker et al. 2002, 2004; Balcells et al. 2007b,a; Seth et al. 2006; Georgiev et al. 2009a; Georgiev & Böker 2014; Carson et al. 2015). Driven by apparent similarities in the scaling relations of SMBHs and NSCs with host galaxy properties, Ferrarese et al. (2006a) introduced the term Central Massive \nObject 1 (CMO), suggesting that the formation and evolution of both types of central mass concentration may be linked by similar physical processes. \nIndeed, the mass range of the two components of CMOs overlap, with SMBHs having M BH glyph[greaterorsimilar] 10 6 M glyph[circledot] (e.g. Gültekin et al. 2009; Rusli et al. 2013; McConnell & Ma 2013), and NSC masses falling in the range 10 4 glyph[lessorsimilar] M NSC glyph[lessorsimilar] 10 8 M glyph[circledot] . Both M BH and M NSC have repeatedly been found to correlate with a range of host galaxy properties including galaxy luminosity, mass, stellar velocity dispersion ( σ ), AGN activity etc. (e.g. Gültekin et al. 2009; Seth et al. 2008a; Kormendy & Ho 2013, and references therein). These correlations followed the earlier discoveries that the mass of the SMBH scales with the host galaxy B -band bulge luminosity \n(Kormendy & Richstone 1995), dynamical mass (e.g. Magorrian et al. 1998; Häring & Rix 2004), stellar velocity dispersion (Ferrarese & Merritt 2000; Gebhardt et al. 2000) and central light concentration (e.g. Graham et al. 2001). \nSimilar scaling relations are also found to hold between the mass of NSCs and their host galaxy bulge luminosity, mass (e.g. Ferrarese et al. 2006a; Wehner & Harris 2006) as well as morphological type (e.g. Rossa et al. 2006; Erwin & Gadotti 2012). The detailed shape of these relations possibly depends on host galaxy morphology, as suggested by Erwin & Gadotti (2012) who report a systematic difference in the NSC mass fraction between early- and late-type hosts. It is, however, still hotly debated which is the fundamental physical mechanism setting these scaling relations (e.g. gas accretion, cluster and/or galaxy mergers Silk & Rees 1998; McLaughlin et al. 2006; Li et al. 2007; Leigh et al. 2012; Antonini 2013), or any combination of these (see reviews by Kormendy & Ho 2013; Cole & Debattista 2015). Proper understanding of these issues is crucial for gaining insight into the formation and growth of CMOs, and, in turn, how a CMO might impact the evolution of the host galaxy. \nPerhaps one of the most intriguing observations is the coexistence of NSC and SMBH in galaxies with masses around M gal glyph[similarequal] 10 10 M glyph[circledot] (Filippenko & Ho 2003; Seth et al. 2008a, 2010; Graham & Spitler 2009; Neumayer & Walcher 2012), with the best-studied example being the center of the Milky Way (Schödel et al. 2007; Ghez et al. 2008; Gillessen et al. 2009; Genzel et al. 2010; Feldmeier et al. 2014; Schödel et al. 2014). Throughout this paper, we will use the term 'coexisting' whenever describing an NSC that contains a MBH. Finding coexisting NSCs and MBHs has triggered numerous studies to understand the nature of this co-existence and the processes involved in their formation, growth, mutual influence, and co-evolution (e.g. McLaughlin et al. 2006; Li et al. 2007; Nayakshin et al. 2009; Bekki & Graham 2010). \nFor example, Neumayer & Walcher (2012) discuss the possibility that NSCs are susceptible to destruction by BHs when M BH / M NSC >> 1 , or when the MBH sphere of influence becomes comparable to the size of the NSC (Merritt 2006). Recent N -body simulations have demonstrated that the capture and accretion of stars migrating within the BH sphere of influence can significantly contribute to the mass growth of black holes as well as the central core density of the host galaxy (Brockamp et al. 2011, 2014). From theoretical arguments, the growth rate of MBHs is expected to increase with MBH mass and likely requires a seed BH with M > 100 M glyph[circledot] , unless the BH host cluster is very dense (Baumgardt et al. 2004a,b, 2005, 2006). These simulations also show that a significant fraction of stars can escape from the cluster due to close encounters with the MBH (Baumgardt et al. 2004a, 2006). \nThe presence of a MBH can inhibit the onset of corecollapse in the NSC, and cause the NSC to expand, and ultimately to be disrupted (e.g. Merritt 2006, 2009; Tremaine 1995). Depending on M BH and the cluster core density (concentration and core velocity dispersion), the impact of tidal stress forces from the MBH on the NSC will become significant at a radius comparable to that of the MBH sphere of influence, r infl , BH which scales linearly with M BH . Therefore, the effect of a MBH on the stellar orbits in the NSC is likely to be more pronounced in massive host galaxies (because more massive galaxies host more massive MBHs). This is \ntrue even in formation scenarios that involve the merging of systems: regardless of whether a NSC spirals into a nucleus that already contains a MBH, or whether a MBH falls into a nucleus occupied by a NSC, the structure and integrity of the NSC will be impacted if the MBH mass is a sufficiently high fraction of the bound NSC mass (e.g. Antonini et al. 2012, 2015; Antonini 2013, see also refs. in § 3.3) \nIn massive globular clusters (GCs) and ultra compact dwarf galaxies (UCDs), the presence of MBHs is hotly debated. If confirmed, this would add support to the notion that some GCs may be the former nuclei of galaxies which lost significant amounts of mass in galaxy interactions/merging (e.g. Lützgendorf et al. 2013; Mieske et al. 2013; Seth et al. 2014, and refs therein). The (non-)presence of MBHs is therefore an extremely important factor for the study of the various types of compact stellar systems (NSCs, UCDs, and massive GCs), and possible evolutionary connections between them (e.g. Gregg et al. 2009; Price et al. 2009; Georgiev et al. 2009b,a, 2012; Taylor et al. 2010; Misgeld & Hilker 2011; Chiboucas et al. 2011; Brüns et al. 2011; Norris & Kannappan 2011; Foster et al. 2011; Pfeffer & Baumgardt 2013; Puzia et al. 2014; Georgiev & Böker 2014; Frank 2014; Norris et al. 2014; Seth et al. 2014). \nHere, we explore scaling relations between the size/mass of NSCs and the stellar mass of their host galaxies, M glyph[star] , sorted by host morphology. In this context, it is reasonable to consider the total mass of the host galaxy (rather than just bulge mass). This is because the bulge mass of an earlytype, elliptical galaxy is effectively equal to its total stellar mass, while in late-type galaxies, the bulge - if it exists at all - is negligible compared to the disk component. Therefore, the main mass reservoir for NSC and/or SMBH formation in late-type disks would be ignored in studies that only consider the bulge mass of the host. While a few previous studies (Carollo et al. 1998; Erwin & Gadotti 2012) have taken the approach of considering the total host galaxy mass, our work significantly improves on the number of objects and the galaxy mass range, taking advantage of our recent catalogue of NSCs in disk galaxies (Georgiev & Böker 2014). \nIn § 2, we describe the galaxy sample and the calculation of photometric masses for NSC and host galaxy. In § 3, we present the analysis and comparison between late- and early-type galaxies using relations between NSC mass and size, M NSC -r eff , NSC as well as between NSC properties and host galaxy stellar mass, r eff , NSC -M glyph[star] (§ 3.1), and M NSC -M glyph[star] (§ 3.2). For nuclei with co-existing NSC and SMBH, we show in § 3.3 the corresponding relations for the combined CMO mass, M BH+NSC -M glyph[star] , and the ratio between the radius of the BH sphere of influence and the NSC effective radius, r infl , BH / r eff , NSC . The results are discussed in § 4 and our conclusions are summarized in § 5.", '2.1 Morphological Sample Definitions': "One of our main goals for this study is to check for differences in the CMO properties between early- and late-type galaxies, i.e. in nuclei of dynamically 'hot' (bulge-dominated) and 'cold', (disk-dominated) host galaxies. Such a morphology- \nbased separation could indicate two different modes of evolution, e.g. active and inactive. Any observed differences between late- and early-type NSCs could therefore reflect the underlying environmental conditions for CMO formation (e.g. Leigh et al. 2015). \nTo define our NSC sample, We use the t -type galaxy morphological parameter defined by de Vaucouleurs et al. (1991). The overall sample is comprised of NSCs in spheroiddominated galaxies from Côté et al. (2006) and Turner et al. (2012), as well as in disk-dominated galaxies from Georgiev & Böker (2014) and Georgiev et al. (2009a). We divide this master sample into sub-samples of NSCs in early- and latetype hosts using the following criteria: the early-type subsample is comprised of all galaxies with t< 0 (i.e. bulge dominated Es-S0s), while the late-type sub-samples contains all galaxies with t > 3 (i.e. disk dominated, Sb and later). To demonstrate the clear separation of the two sub-samples, we show in Figure 1 histograms of the t-type distribution within each subsample. This approach enables us to identify general trends and differences between the properties of NSCs in bulge and disk-dominated host galaxies, such as those reported by Erwin & Gadotti (2012) who find a change in the the mass ratio M NSC / M glyph[star], gal occuring around t glyph[similarequal] 3 (their Fig. 4), i.e. close to the morphological separation between our late- and early-type sub-samples. \nWe also note that our two sub-samples are dominated by galaxies in different environments. While virtually all NSC hosts in the early-type sample are located in a cluster environment (Virgo or Fornax), the late-type galaxies are found mainly in a lower density (group) environment, except for 12 galaxies ( < 10% ) that are members of Virgo or Fornax according to catalogues of Binggeli et al. (1985) and Ferguson & Sandage (1990). We highlight these 12 objects with a solid histogram in Figure 1. \nRelevant only for the NSC-MBH discussion in § 3.3.1 and 3.3.2, we also use data for galaxies hosting both a NSC and a MBH from Neumayer & Walcher (2012) and for SMBH host galaxies from McConnell & Ma (2013). Both studies contain galaxies with a wide range of morphological types, and in order to separate these objects into early- and late-types, we adopt a galaxy morphology dividing line at t =3 .", '2.2 NSC photometry and mass': 'Deriving accurate photometric masses relies on properly accounting for foreground Galactic extinction to a given galaxy and precise knowledge of its distance. For this purpose, we retrieved the foreground Galactic extinction E ( B -V ) and the (median value of the) distance modulus for all sample galaxies from NED 2 . The NED extinction values are based on the Schlafly & Finkbeiner (2011) recalibration of the Schlegel et al. (1998) extinction map, for which we calculate filter-specific values assuming the Fitzpatrick (1999) reddening law with R V =3 . 1 . The values for E ( B -V ) and m -M used for the computation of photometric stellar masses are listed in Table A1. We emphasize that possible NSC reddening due to host galaxy self-absorption is not accounted for. A correction for an A V glyph[similarequal] 0 . 4 mag would increase the NSC \nFigure 1. Morphological distribution of the early- and late-type sub-samples. The t-type values are from HyperLEDA (based on de Vaucouleurs et al. 1991). \n<!-- image --> \nmass by a factor of two (at fixed age and metallicity), which needs to be considered when estimating the systematic uncertainties (see also § 2.2 and discussion in § 3.2). The values for M NSC and M glyph[star] derived as described in the next sections are tabulated in Table A1 3 . \nThe sample of NSCs in late-type galaxies in this work comes from the recently published catalogue of 228 NSCs in nearby ( glyph[lessorsimilar] 40 Mpc), moderately inclined spiral galaxies with t glyph[greaterorequalslant] 3 (Georgiev & Böker 2014). These selection criteria ensure that the effects of any light contamination from the host galaxy disk and (pseudo-)bulge on the derived NSC properties are minimized. The catalogue contains luminosities calculated from the flux within the best fitting King model of a given concentration index. This provides the most accurate photometry in a nuclear environment because is less affected by nearby contaminating sources. \nThe Bell et al. (2003) mass-to-light ratio ( M/L )-colour relations are available for the SDSS, 2MASS, and Johnson/Cousins magnitude systems. However, as discussed in Georgiev & Böker (2014), we prefer to work in the native WFPC2 magnitudes to avoid propagating uncertainties from transformations between the various photometric systems. For each NSC, we obtain the M/L -ratio using the NSCs magnitudes in Georgiev & Böker (2014) and the Bruzual & Charlot (2003) SSP models for solar metallicity and a Kroupa (2001) IMF. As shown by spectroscopic studies of NSCs in late-type galaxies, the assumption of solar metallicity is a reasonable one for these objects (e.g. Rossa et al. 2006; Walcher et al. 2006; Seth et al. 2006). To calculate the luminosity weighted photometric mass of the NSCs, M NSC , we used the available colour information in the various combinations of the most reliably calibrated WFPC2 filters ( F 300 W,F 450 W,F 555 W,F 606 W,F 814 W ). We obtain the SSP model M/L by matching the NSC colours to the model colours. If more than one colour is available, we calculate the error weighted mean of the different M/L -values to obtain M NSC , which helps to minimize systematic uncer- \nFigure 2. Ratio between M NSC measurements from spectroscopy (from Rossa et al. 2006) to those from photometric colours (this study). The error-weighted straight line fit (solid horizontal line) is plotted, together with the rms scatter of the data (shaded area). The dashed horizontal lines show the ± 50% range around a mass ratio of 1. The plotted errors are only from our study, as there are no error bars provided by Rossa et al. (2006). \n<!-- image --> \ninties (cf McGaugh & Schombert 2014). For NSCs with photometry in only one band (i.e. without colour information), we used the sample median colours containing that filter to calculate the error weighted mean of the M/L from the possible colour combinations containing that filter, e.g. for a NSC with only F 814 W magnitude, we used the median F 300 W -F 814 W, F 450 W -F 814 W and F 606 W -F 814 W colours of the NSC sample to calculate the error weighted SSP model M/L F 814 W . We checked, and expectedly, the calculated M NSC using the sample median colours showed no systematic difference between those NSCs with mass calculated from measured colour(s). \nAlthough these colours are representative for the entire sample of NSCs, we caution that there may still be a small bias in the NSC masses derived from different filters. We checked for this using NSCs observed in multiple filters, but did not find any systematic differences. We also note that NSCs with uncertainties larger than > 100% in M NSC (shown with gray symbols in subsequent figures) are excluded from the various fits 4 . \nThe formal errors of the measured colours are small due to the generally very high S/N of the NSC, and thus introduce only negligible uncertainties in the resulting M/L -ratios. However, they may still be affected by the possibility that a small mass fraction of the NSC ( δ M ≈ 10% ) is composed by a younger stellar population ( ∆ t > 5 Gyr) which will outshine the more massive older stellar component. This will cause a bias towards bluer integrated colours, younger SSP ages, and a lower M/L values, i.e. towards lower total M NSC mass (by up to a factor of 5, see also discussion in § 3.1 and 3.2). Our approach to derive M NSC in latetype hosts is similar to that in Seth et al. (2008a) who find good agreement between photometric and dynamical mass estimates to within a factor of two, which is comparable to the mass uncertainties calculated here. Nevertheless, in Figure 2 we illustrate how well our colour-based photometric NSC masses compare to those obtained from spectroscopic \n- 4 We found no significant differences in the best-fit parameters when including these NSCs in the fits, using weights that are inversely proportional to their uncertainty. \nanalysis (from stellar population fitting, Rossa et al. 2006, or line widths, Walcher et al. 2006). For those objects with both types of measurements, Figure 2 plots their ratio as a function of NSC mass. The shaded area is the rms scatter of the data around the best fit ( σ =0 . 42 dex). For reference, dashed horizontal lines show the ± 50% range around a mass ratio of 1. The plot shows that within the uncertainties, both estimates are generally in good agreement. The photometric estimates appear to be higher by about 20%, but the significance of this difference is only < 1 . 5 σ . Nevertheless, a slight overestimation of the photometric mass could be expected if the assumed NSC metallicity is too high. For example, the M/L s would differ by about 20% between solar and sub-solar metallicity in the Bruzual & Charlot (2003) SSP models. Thus, the systematic uncertainty in the photometric and spectroscopic mass estimates due to our choice of metallicity is much smaller than that caused by the degeneracy with stellar age, as discussed below. \nThe NSC photometry in early-type galaxies is collected from two galaxy cluster surveys conducted with with HST/ACS - the Virgo Cluster Survey (ACSVCS, Côté et al. 2004) with photometry for 56 NSCs (Côté et al. 2006) and the Fornax Cluster Survey (ACSFCS, Jordán et al. 2007) with measurements of 31 NSCs (Turner et al. 2012). The photometric mass of the NSCs of these samples are calculated in a similar way as for the late-types by using the g F 475 W -z F 850 LP colour and z -band magnitude in the tables of Côté et al. (2006) and Turner et al. (2012). We note that our approach in deriving M NSC (from colours at fixed solar metallicity) differs from that adopted in those studies ( M NSC at fixed age of 5 Gyr). According to the Bruzual & Charlot (2003) SSP models, at a fixed age of 5 Gyr, the M/L can vary by a factor of 2 with metallicity. For solar metallicity, the M/L increases again by a factor of 2 between ages of 5 and 14 Gyr. Therefore, either method carries an equal amount of uncertainty, roughly a factor of two. Our method therefore should yield M NSC values that are consistent with other studies to within a factor of two.', '2.3 Properties of NSCs with massive BHs': 'For the subsample of NSCs with MBHs, we use the measurements of Neumayer & Walcher (2012) who provide upper limits for M BH based on velocity dispersions and dynamical mass modelling from VLT/UVES spectra. \nTwelve NSCs (one late- and 11 early-type galaxies) in their sample are not in Georgiev & Böker (2014). For those NSCs, we use the luminosities obtained by Neumayer & Walcher from a Multi-Gaussian Expansion fitting technique (MGE, Emsellem et al. 1994; Cappellari 2002). The remaining seven NSCs in the Neumayer & Walcher (2012) sample are present in our HST sample, and we therefore use our photometry to calculate the NSCs masses, as explained in § 2.2. To check for systematic differences between the two studies, we calculate the ratio between our and the Neumayer & Walcher NSC sizes and masses. We find good agreement, with a mean ratios of 0 . 81 ± 0 . 18 for the NSC sizes, and 0 . 98 ± 0 . 16 for the NSC masses. The small apparent difference in derived sizes can likely be attributed to the different fitting techniques used, elliptical King profiles in Georgiev & Böker vs. MGE technique in Neumayer & Walcher. \nWe also include in our sample the NSC and MBH \nmasses of the Milky Way (MW) and Andromeda (M31). Their proximity makes these two nuclei the best-studied examples of systems with reliable mass measurements of both NSC and MBH. For the MW NSC, we use mass and size estimates from Schödel et al. (2014), M NSC = (2 . 5 ± 0 . 4) × 10 7 M glyph[circledot] , r eff , NSC = 4 . 2 ± 0 . 4 pc. The mass of the MBH in the MW, M BH = 4 . 26 × 10 6 M glyph[circledot] , is from Chatzopoulos et al. (2014) based on stellar kinematics. For the total stellar mass of the MW, we use the value derived by Licquia & Newman (2014), M glyph[star] = (6 . 08 ± 1 . 14) × 10 10 M glyph[circledot] , which is based on an improved Bayesian statistical analysis accounting for uncertainties in literature measurements. The MW gas mass is M MWgas = 1 . 25 × 10 10 M glyph[circledot] (about 17% of its stellar mass), of which atomic Hydrogen constitutes M HI = 8 × 10 9 M glyph[circledot] , warm ionized medium M H+ = 2 × 10 9 M glyph[circledot] , and molecular gas M H2 = 2 × 10 9 M glyph[circledot] (Kalberla & Kerp 2009). \nThe M31 nucleus is a rather complex system (Lauer et al. 1993). It is composed of a cluster that clearly stands out above the surrounding bulge within the central 10 pc and is dominated by light from old stellar populations (Kormendy & Bender 1999). Its inner 1.8 pc core features a bimodal component (Lauer et al. 1993, 1998, 2012), which is interpreted as a projection of the Keplerian orbits of stars in a central eccentric disc around the MBH (Tremaine 1995; Peiris & Tremaine 2003). The mass of the MBH is M BH = 1 . 4 × 10 8 M glyph[circledot] (Bender et al. 2005) and that of the NSC M NSC = 3 . 5 ± 0 . 8 × 10 7 M glyph[circledot] (Lauer et al. 1998; Kormendy & Ho 2013). We calculate the total stellar mass of M31 to be M glyph[star] = 7 . 88 ± 4 . 23 × 10 10 M glyph[circledot] , using B,V,I photometry from HyperLEDA 5 and using Bell et al. (2003) M/L -colour relations (see also § 2.5). Our value for the M 31 mass is consistent with other stellar population based estimates in the literature, e.g. 10-1 5 × 10 10 M glyph[circledot] (Tamm et al. 2012).', '2.4 Sample of massive black holes': 'Masses of 72 SMBHs and their host galaxies are taken from McConnell & Ma (2013). They collect literature data from various sources of the most up to date M BH measurements. This sample is used in § 3.3.2 for calculating the SMBH sphere of influence radius, r infl , BH .', '2.5 Photometric stellar mass of NSC host galaxies': 'We also calculate the total galaxy stellar mass (i.e. the sum of their bulge and disk components) for all NSC host galaxies in our sample. The total galaxy mass is an important quantity for the discussion of formation scenarios for both NSCs and SMBHs. This is especially true for late-type galaxies without prominent bulges, where material for the growth of either CMO must come predominantly from the disk. \nCalculating galaxy photometric mass, M glyph[star], gal , from integrated colours in the optical is a challenging task, mainly due to the age-metallicity degeneracy and assumptions of galaxy star formation history used by synthetic models. However, it has been demonstrated that the B -V colour (including for disk galaxies) offers a good representation of their stellar population (e.g. McGaugh & Schombert 2014). \nWe have therefore calculate M glyph[star], gal using the empirically calibrated M/L -galaxy colour relations of Bell et al. (2003). These relations were obtained by comparing galaxy SEDs at optical and NIR wavelengths, and by using composite stellar evolutionary models for a range of metallicities and star formation histories. \nFor the majority of the galaxies in our samples, we collect photometry ( B,B -V, I magnitudes) from HyperLEDA, i.e. for the galaxies in Georgiev & Böker (2014), Turner et al. (2012), Neumayer & Walcher (2012) and McConnell & Ma (2013). The only exception is the ACSVCS sample, for which we use the photometry derived from a dedicated isophotal analysis (Ferrarese et al. 2006b). \nThe McConnell & Ma (2013) catalogue provides host galaxy bulge stellar mass (either from spherical Jeans modelling of the bulge stellar dynamics, or from M/L -modeling based on galaxy colours, the latter approach being identical to ours). However, McConnell & Ma (2013) do not provide stellar mass estimates for those galaxies in their sample that contain a significant disk component, i.e. S0 and later types. For a consistent comparison to the NSC sample, we calculate the total (bulge+disk) stellar masses of their entire sample using magnitudes and colours obtained from HyperLEDA and the Bell et al. (2003) M/L -color relations. To check the consistency of our results, we compared our galaxy masses to those of McConnell & Ma (2013) for the early-type galaxies in their sample, i.e. for cases where M bulge glyph[similarequal] M glyph[star] , and find a very good agreement (to within 10%). Uncertainties of the photometric masses have been calculated by propagation of the photometric uncertainties and the uncertainties associated to the coefficients of the M/L -color relation. To avoid over-crowding in the figures, galaxies with uncertainties larger than 100% are shown with grey symbols.', '2.6 HI and X-ray gas masses': "A significant baryonic mass component in late-type galaxies is in the form of atomic HI gas. We therefore calculate the HI mass from the HyperLEDA 21-cm line magnitudes, m 21 , converted to flux ( F HI = 10 -0 . 4 × (17 . 40 -m 21) ) using the relation between the M HI and F HI , i.e. M HI = 2 . 36 × 10 5 × D 2 × F HI , where D is the distance in Mpc, calculated from the same distance modulus in NED used for calculating galaxy mass from its luminosity. We note that we did not correct the HI mass for He fraction or molecular gas. \nEarly-type galaxies are known to contain a hot gas component, detected as an X-ray halo resulting from thermal Bremsstrahlung emission, which is known to trace well the total gravitating mass (e.g. Forman et al. 1985; Fukazawa et al. 2006). Typically, the hot gas mass is no more than a few times 10 9 M glyph[circledot] for a range of galaxy morphologies, environments, and luminosities ( L K glyph[similarequal] 3 -15 × 10 10 L glyph[circledot] ) (e.g. Bogdán et al. 2013b,a; Anderson et al. 2013). This is only 6-7% of the galaxy stellar mass (e.g. O'Sullivan et al. 2003; Su & Irwin 2013) and is therefore not a significant component of the baryon mass budget. Nevertheless, for the earlytype massive galaxies in the McConnell & Ma (2013) SMBH sample we collect the hot gas mass measured by Su & Irwin (2013), based on Chandra and XMM data. Unfortunately, no X-ray measurements exist for most of our late-type sample, and we therefore do not list the hot gas mass fraction \nin Table A1. Given that the hot gas mass fraction is smaller in late-type galaxies than in ellipticals Li et al. (2011), and in any case accounts for only a small fraction of the total galaxy mass, this does not significantly affect our analysis or conclusions.", '3 RESULTS AND ANALYSIS': 'Quantifying the relations between NSCs and their host galaxy, and possible dependence on galaxy morphology, bears important constraints for models of NSC formation and evolution, as well as for possible evolutionary connections to the various incarnations of compact stellar systems (e.g. massive GCs, UCDs). In addition, they provide insights into mechanisms that may transform galaxies from late- to early-type morphologies. With our large sample of NSCs in late-type hosts, we significantly increase the number of well studied nuclei in disk-dominated galaxies. This enables a more statistically meaningful comparison of r eff , NSC and M NSC between early- and late-type galaxies, and extends the range of host galaxy masses to less massive systems.', '3.1 Relations between NSC size and NSC and host-galaxy masses': "The relations between the NSC effective radius and its mass ( r eff , NSC -M NSC ) as well as the stellar mass of its host galaxy ( r eff , NSC -M glyph[star], gal ) are shown in Figures 3 and 4, respectively. In both figures, late- and early-type host galaxies are shown with different symbol and line types, as indicated in the figure legend. We do not plot unresolved NSCs, i.e. those with only an upper limit to r eff , NSC (see Fig. 4 in Georgiev & Böker 2014). In both figures, the bottom panel shows the two-dimensional probability density distribution function (2D-PDF). The uncertainty weighted 2D-PDFs are estimated within a running box of size 0.3 dex within R 6 . The 2D-PDFs provide a first order quantification of the observed r eff , NSC -M NSC distribution. The thick dashed and solid contour lines in Figures 3 and 4 indicate the 1 σ dispersion of the data. \nTo more robustly quantify any differences in the r eff , NSC -M NSC and r eff , NSC -M glyph[star], gal relations between the two subsamples, we perform a maximum likelihood, linear (in log-log space) regression analysis by bootstrapping the data to account for the finite data sample and construct the posterior PDFs. Our fitting also accounts for the nonsymmetric measurement uncertainties, which are treated as a combination of two Gaussians, i.e. a split normal distribution. The fitted linear regression is of the form: \nlog 10 ( r eff , NSC /c 1) = α × log 10 ( M NSC /c 2) + β, (1) \nwhere the normalization constants ( c 1 , c 2 ) and the best fit values for the slope ( α ) and intercept ( β ) for the different subsamples are tabulated in Table 1. Description of the fitting technique and results for each relation are provided in the Appendix § A. \n6 R is a free software environment for statistical computing. The R-project is an official part of the Free Software Foundation's GNU project (http://www.r-project.org/). \n/circledot /circledot Figure 3. Top: Nuclear star cluster size - mass relation, r eff , NSC vs. M NSC for late- and early-type host galaxies. Small grey (light) symbols are for NSCs with uncertainties > 100% . Bottom: a contour plot of the two-dimensional probability density distribution (2D PDF) for the two subsamples. The different symbols and line types are for the different samples, as indicated in the legend. Thick contour lines mark the 1 σ of the 2D PDFs. The fit to the data is shown with lines, where the narrower darker (colour) shaded region indicates the uncertainties range of the fit slope and intercept. The wider and lighter (colour) shaded region is the 1 σ dispersion of the data. \n<!-- image --> \nThe values of the normalization constants are the highest probability density value of the center peaks in the contour plot in Fig. 3. We find that these normalization constants are important for minimizing the correlation between the slope and intercept, which provides a more realistic uncertainty estimate of the fits. The posterior probability density distributions of the slope and intercept for each fitted relation are shown in Figure A1 in § A. We find that log r eff , NSC scales with log M NSC with a slope of α = 0 . 321 +0 . 047 -0 . 038 for late-types and 0 . 347 +0 . 024 -0 . 024 for early-types. The log r eff , NSC is also observed to scale with host galaxy stellar mass with a slope of α = 0 . 356 +0 . 056 -0 . 057 for late-types and 0 . 326 +0 . 055 -0 . 051 for early-types. \nThe comparison between the 2D-PDFs of NSCs in the \n/circledot Figure 4. Top: Nuclear star cluster size versus total stellar mass of the host galaxy for NSCs in late- and early-type galaxies. Bottom: A contour plot of the two-dimensional probability density distribution of the two subsamples. The symbol, line types and shaded areas are the same as in Fig. 3. \n<!-- image --> \n/circledot \nlate- and early-type subsamples suggests that the r eff , NSC -M NSC distributions are consistent with each other to within 1 σ of the dispersion of the data (cf. the solid 1 σ contour lines in Fig. 3, bottom). Within the uncertainties, the relations for late- and early-type hosts also have very similar slopes, however, the zeropoint of the fitted relations differ beyond their 1 σ dispersion (cf. the broader shaded region around the highest density peaks in Fig. 3). This suggests that at fixed cluster mass, NSCs in late-type hosts are smaller by about a factor of 2 than their counterparts in early-type hosts. \nA similar difference in r eff , NSC between late- and earlytype galaxies is also observed as a function of galaxy stellar mass, ( r eff , NSC -M glyph[star], gal ), shown in Figure 4. The r eff , NSC increases with M glyph[star], gal with an identical slope for both samples, however, at fixed M glyph[star], gal , NSCs in late-type hosts are more compact by about a factor of 2. In this case, however, the statistical significance that both distributions differ is less than 1 σ , for the offset, slope and the 2D density distributions of the data (cf fits' shaded regions and solid density contours in Fig. 4). These differences are discussed in § 4.2. \nTable 1. Parameters of the fitted relations for late- and early-type NSC host galaxies. \n| Host type (1) | c1 (2) | c2 (3) | α (4) | β (5) | σ (6) |\n|----------------------------------------------------------------|----------------------------------------------------------------|----------------------------------------------------------------|----------------------------------------------------------------|----------------------------------------------------------------|----------------------------------------------------------------|\n| log( r eff , NSC /c 1) = α × log( NSC /c 2) + β | log( r eff , NSC /c 1) = α × log( NSC /c 2) + β | log( r eff , NSC /c 1) = α × log( NSC /c 2) + β | log( r eff , NSC /c 1) = α × log( NSC /c 2) + β | log( r eff , NSC /c 1) = α × log( NSC /c 2) + β | log( r eff , NSC /c 1) = α × log( NSC /c 2) + β |\n| Late Early | 3.31 6.27 | M 3.60e6 1.95e6 | 0 . 321 +0 . 047 - 0 . 038 0 . 347 +0 . 024 - 0 . 024 | - 0 . 011 +0 . 014 - 0 . 031 - 0 . 024 +0 . 022 - 0 . 021 | 0.133 0.131 |\n| log( r eff , NSC /c 1) = α × log( M glyph[star], gal /c 2) + β | log( r eff , NSC /c 1) = α × log( M glyph[star], gal /c 2) + β | log( r eff , NSC /c 1) = α × log( M glyph[star], gal /c 2) + β | log( r eff , NSC /c 1) = α × log( M glyph[star], gal /c 2) + β | log( r eff , NSC /c 1) = α × log( M glyph[star], gal /c 2) + β | log( r eff , NSC /c 1) = α × log( M glyph[star], gal /c 2) + β |\n| Late Early | 3.44 6.11 | 5.61e9 2.09e9 | 0 . 356 +0 . 056 - 0 . 057 0 . 326 +0 . 055 - 0 . 051 | - 0 . 012 +0 . 026 - 0 . 024 - 0 . 011 +0 . 015 - 0 . 040 | 0.139 0.143 |\n| log( M NSC /c 1) = α × log( M glyph[star], gal /c 2) + β | log( M NSC /c 1) = α × log( M glyph[star], gal /c 2) + β | log( M NSC /c 1) = α × log( M glyph[star], gal /c 2) + β | log( M NSC /c 1) = α × log( M glyph[star], gal /c 2) + β | log( M NSC /c 1) = α × log( M glyph[star], gal /c 2) + β | log( M NSC /c 1) = α × log( M glyph[star], gal /c 2) + β |\n| Late Early | 2.78e6 2.24e6 | 3.94e9 1.75e9 | 1 . 001 +0 . 054 - 0 . 067 1 . 363 +0 . 129 - 0 . 071 | 0 . 016 +0 . 023 - 0 . 061 0 . 010 +0 . 047 - 0 . 060 | 0.127 0.157 |\n| log( M NSC /c 1) = α × log( M glyph[star] +HI , gal /c 2) + β | log( M NSC /c 1) = α × log( M glyph[star] +HI , gal /c 2) + β | log( M NSC /c 1) = α × log( M glyph[star] +HI , gal /c 2) + β | log( M NSC /c 1) = α × log( M glyph[star] +HI , gal /c 2) + β | log( M NSC /c 1) = α × log( M glyph[star] +HI , gal /c 2) + β | log( M NSC /c 1) = α × log( M glyph[star] +HI , gal /c 2) + β |\n| Late+HI | 2.87e6 | 1.17e10 | 1 . 867 +0 . 158 - 0 . 133 | 0 . 041 +0 . 041 - 0 . 042 | 0.126 |\n| MBH+NSC | | 2.76e10 | 1 . 491 - 0 . 097 | - 0 . 019 - 0 . 054 | 0.233 |\n| log( M NSC+MBH /c 1) = α × log( M glyph[star], gal /c 2) + β | log( M NSC+MBH /c 1) = α × log( M glyph[star], gal /c 2) + β | log( M NSC+MBH /c 1) = α × log( M glyph[star], gal /c 2) + β | log( M NSC+MBH /c 1) = α × log( M glyph[star], gal /c 2) + β | log( M NSC+MBH /c 1) = α × log( M glyph[star], gal /c 2) + β | log( M NSC+MBH /c 1) = α × log( M glyph[star], gal /c 2) + β |\n| | 5.03e7 | | +0 . 149 | +0 . 111 | | \nNote. The fitted scaling relations are of the form log 10 ( y/c 1) = α ∗ log 10 ( x/c 2) + β , where in column (1) is the NSC host morphological type, columns (2) and (3) are the normalization constants obtained from the 2D PDFs (see § 3.1), in columns (4) and (5) are the slope and intercept and in (6) is the fit rms dispersion of the data, σ .", '3.2 NSC mass - host galaxy stellar mass relation': 'In Figure 5, we explore the relation between the NSC mass and host galaxy stellar mass, again separately for late(Fig. 5 a) and early-type galaxies (Fig. 5 b). We fit the two subsamples with the same technique as described in § 3.1. The best-fit relations are shown with solid lines, while the shaded regions represent the uncertainties of the fit coefficients (the narrower, darker region) and the 1 σ dispersion of the data (the broader, lighter region). The direct comparison between the relations for late- and early-type NSC host galaxies in Figure 5 c shows that within 1 σ , their 2D-PDFs (thick contour lines) are indistinguishable from each other. On the other hand, the comparison also shows that the fitted slopes (solid and dashed lines in Fig. 5 c) are different between the two sub-samples beyond the 1 σ level (i.e. the darker shaded region in Fig. 5 c do not overlap). This implies that at higher galaxy mass, early-types have more massive NSCs than late-types. A similar difference as a function of galaxy morphology has been also reported earlier (Rossa et al. 2006; Seth et al. 2008a; Erwin & Gadotti 2012). \nThe values of the best-fit coefficients are summarized in Table 1. Our result that the mass of the NSC scales with host galaxy stellar mass with a slope near unity for late-types, \n8 \nI. Y. Georgiev, T. Böker, N. Leigh, N. Lützgendorf, N. Neumayer \n<!-- image --> \n<!-- image --> \nFigure 5. Relation between nuclear star cluster mass, M NSC , and host galaxy stellar mass, M glyph[star], gal . Panels a) and b) show separately the relations for late- and early-type galaxies. The histograms on the y2-axes show the NSCs mass distributions for the different samples. In panel c) we compare the fitted relations from panels a) and b) and their 2D PDF distribution (E for early- and S for late-types). Thick contour lines indicate the 1 σ of the data PDF. Symbols, line types and shaded areas are the same as in Fig. 3. \n<!-- image --> \nα = 1 . 001 +0 . 054 -0 . 067 , is in good agreement with the literature, e.g. Erwin & Gadotti (2012), who derive a slope of α = 0 . 90 ± 0 . 21 between M NSC and total (bulge plus disk) stellar galaxy mass in a smaller sample of massive late-type spirals. \nWe note at this point that the slope of the relation for the late-types is similar to the slope defined by the MBH mass and host spheroid mass, M BH -M glyph[star], sph , which is α = 1 . 05 (McConnell & Ma 2013). However, the M BH -M glyph[star], sph has a zeropoint of 8.46, which is 0.6 dex higher compared to the 7 . 86 ± 0 . 1 for our late-types relation. In § 4.2 we discuss first whether the differences between the relations for late- and early-types are due to measurement biases or evolutionary differences, and then discuss the implications for the M CMO -M glyph[star], gal relation. \nIt is perhaps equally interesting to see how the M NSC -M glyph[star], gal relation changes when including the HI mass to the total host galaxy mass. Naturally, the effect on the M NSC -M glyph[star], gal relation will be larger for gas-rich late-type galaxies. To gauge the magnitude of this effect, we show in Figure 6 a the NSC mass distribution for both early- and late-types against host galaxy stellar mass, M glyph[star], gal , and in Figure 6 b against the total galaxy mass, M glyph[star] +HI . To guide the eye, we overplot again the best-fit relations from Fig. 5 in Figure 6 a. It is evident from Figure 6 b that when the HI mass is included, the relation for late-types steepens significantly. This is mostly because the low mass, late-type, galaxies have the highest HI mass fraction, and thus move noticeably to the right (i.e. toward higher total mass) in Figure 6 b, causing the relation to steepens. Due to the purely illustrative purposes of this comparison, we did not attempt to include He or molecular mass corrections. Those will only further strengthen the differences. As mentioned in § 2.6, ignoring the small fraction ( < 5% ) of the total galaxy mass contained in X-ray emitting hot gas mass should not significantly affect our results. \nIn Figure 6 c we plot M NSC / M glyph[star] gal , i.e. the fraction of galaxy mass contained in the NSC. Overall, both for lateand early-type hosts, the mass of the NSC is about 0.1% of the galaxy stellar mass ( M NSC / M glyph[star] gal glyph[similarequal] 10 -3 , cf the histogram in Fig. 6 c with a dispersion of about a factor of three. The slope of the M NSC -M gal relation adds to the broadening of the histogram projections. The observed NSC mass fractions in this late-type host galaxy sample are consistent \nwith the values (0.1-0.2%) reported by earlier studies (e.g. Rossa et al. 2006; Seth et al. 2008a; Graham & Spitler 2009; Erwin & Gadotti 2012; Kormendy & Ho 2013). We note that Erwin & Gadotti (2012) reported M NSC / M glyph[star] gal ∼ 0 . 2% for Hubble types earlier than Sbc (consistent with our results), but ∼ 0 . 03% for later Hubble types, which is lower than the peak of the distribution in this study. However, partly due to the slope of the M NSC -M glyph[star] gal relation the M NSC / M glyph[star] gal distribution shows a large dispersion in Fig. 6 c with a 1 σ range between 6 × 10 -6 (0.006%) and 3 × 10 -3 (0.1%).', '3.3 Relations for coexisting NSCs and MBHs': 'The identification and study of systems in which NSC and MBH coexist is important in order to make progress on a number of open questions. For example, it is not clear whether this coexistence is possible only in the nuclei of intermediate mass galaxies (few × 10 10 M glyph[circledot] ), or what the physical reason is for the dominance of one or the other at low and high galaxy mass. Understanding whether there is a common scaling relation for NSC and MBHs with host M glyph[star] promises to shed light on the processes that govern their growth, i.e. the processes funnelling matter (gas, stars, star clusters) towards the deepest point of the host galaxy potential (e.g. Li et al. 2007; Mayer et al. 2010; Hartmann et al. 2011; Antonini et al. 2012, 2015), and the feedback processes affecting the growth by either NSC or MBH. \nObservationally, however, it is extremely challenging to populate the respective scaling relations, mostly because in the absence of accretion activity, the dynamical effects of a low-mass MBH on the surrounding NSC are below the detection threshold of current instruments. On the high-mass end, one could ask why NSC are not observed around SMBHs with M BH > 10 8 M glyph[circledot] ? To address these questions, we first look at the combined mass of the NSC and MBH in coexisting systems (Neumayer & Walcher 2012), with an eye on the impact of a MBH that is massive enough to affect more than 50% of the NSC.', '3.3.1 M NSC+BH - host galaxy stellar mass relation': 'In Figure 7, we show the combined mass of the CMO (i.e. M NSC + M BH ) against host galaxy stellar mass for late- \n<!-- image --> \nFigure 6. Panel a) M NSC -M glyph[star], gal relation and in panel b) with added HI mass, M glyph[star] +HI , gal . NSCs in late- and early-type galaxies are shown with different symbol types and colours as indicated in the figure legend. For reference, with lines in panel a) we show the same fits as in Fig. 5, whereas in panel b) are the fits including the HI mass. The fit values for the different samples are given in Table 1. Panel c) shows the mass of NSC compared to host galaxy stellar mass. Symbols, line types and shaded areas are the same as in Fig. 3. \n<!-- image --> \nand early-type galaxies, plotted with light and dark symbols, respectively. The values for M NSC are calculated from luminosities in Georgiev & Böker (2014) and Neumayer & Walcher (2012) as described in § 2.2 and 2.3. \nThe maximum likelihood, bootstrapped, nonsymmetric error weighted fit is shown with a solid line in Figure 7 a. As before, the shaded regions indicate the uncertainty of the fit and the 1 σ dispersion of the data. The fit values are listed in Table 1. For comparison, we overplot the M BH -M bulge relation from McConnell & Ma (2013) with a dashed line, and with a dash-dotted line the M NSC -M glyph[star] gal relation for late-type galaxies obtained in § 3.2 (cf Fig. 5). We find that the sum of the NSC and MBH masses also defines a relation with host galaxy stellar mass, with a slope of α = 1 . 491 +0 . 149 -0 . 097 . This slope is similar to that of the early-type M NSC -M glyph[star] gal relation ( α = 1 . 363 +0 . 129 -0 . 071 ), but significantly steeper than the one for late-type hosts. We note that the M NSC+MBH -M glyph[star], gal is steeper than the M BH -M bulge relation (e.g. α = 1 . 05 ± 0 . 11 , 1 . 12 ± 0 . 06 , respectively McConnell & Ma 2013; Häring & Rix 2004). In their sample of massive late-type spirals, Erwin & Gadotti (2012) find a slope of α = 1 . 27 ± 0 . 26 for the M BH -M glyph[star] bulge relation, but unfortunately, they do not provide a fit against total galaxy mass. \nThe fact that the M NSC -M glyph[star] gal and M BH -M bulge relations have similar zeropoint and slope (cf. dash-dotted and dashed lines in Fig. 7 a) is perhaps not surprising, given that the bulge mass of early-type galaxies is a good approximation for the total galaxy stellar mass. In § 4.3, we further discuss these relations in the context of the coexistence of NSC and MBH and the transition from one to the other. \nIn Figure 7 b, we plot the mass ratio between the NSC and MBH, M BH / M NSC , against host galaxy stellar mass. The plot shows that at total stellar host masses around 5 × 10 10 M glyph[circledot] , the BH mass begins to dominate over the NSC mass, while for lower galaxy masses, the NSC outweighs the MBH. Mass ratios of glyph[lessmuch] 1 in late-type galaxies were first pointed by Seth et al. (2008a). Subsequently, Graham & Spitler (2009) also included data for coexisting NSCs and MBHs in early-types, but they considered the fractional mass ratio M BH /( M NSC + M BH ) against host spheroid mass. Neumayer & Walcher (2012), whose data we \n<!-- image --> \n<!-- image --> \nFigure 7. Sum of the mass of NSC with MBH ( panel a ) and their mass ratio ( panel b ) against host galaxy stellar mass (data from Neumayer & Walcher (2012)). The different symbol types indicate late- and early-type galaxies, as indicated in the legend. Labeled also are the Milky Way and M 31. The fit through the data in panel a) is shown with solid line and the shaded region indicates the uncertainty of the fit values and the rms of the data. For reference, dashed line is the McConnell & Ma (2013) M BH -M bulge relation and dash-dotted line is the M NSC -M glyph[star] relation for late-type galaxies obtained in § 3.2 (discussion in § 3.3 and 4.3). \n<!-- image --> \nuse here, plot directly M NSC vs. M BH . From the lack of correlation between the two, they conclude that NSCs and BHs do not correlate as strongly with each other as they do with their host galaxy.', '3.3.2 MBH to NSC size ratio': "There has been a large body of analytical and numerical work to understand the effect on the formation and evolution of a NSC due to the presence of a MBH (e.g. Tremaine 1995; Milosavljević & Merritt 2001; Peiris & Tremaine 2003; Merritt 2006, 2009; Baumgardt et al. 2005, 2006; Matsubayashi et al. 2007; Bekki & Graham 2010; Brockamp et al. 2011; Antonini 2013; Lupi et al. 2014; Mastrobuono-Battisti et al. 2014; Antonini et al. 2012, 2015, \nand many others). In this section, we attempt to explore this topic from the observational perspective by using the MBH sphere of influence ( r infl , BH ) and the effective (halfmass) radius of the NSC ( r eff , NSC ). These are observables that can be linked to theoretical expectations and provide observational information/expectation as to whether a significant fraction of the NSC stars/mass is influenced by the MBH. For an isotropic, virialized stellar cluster, the size ratio between NSC and MBH is effectively equivalent to their mass ratio, because in such an idealized system, r infl , BH = G M BH /σ 2 and r eff , NSC = G M NSC / σ 2 , and hence r infl , BH / r eff , NSC ≡M BH / M NSC . When r infl , BH / r eff , NSC = 1 , all stars within r eff , NSC are strongly bound to the MBH and have mostly Keplerian orbits. Thus, beyond the r eff , NSC , the cluster will have profile represented by a King model. Therefore, at this limit, the inner 50% of the NSC potential (i.e. the stellar orbits) are dominated by the MBH, and the outer 50% are dominated by the NSC potential/mass distribution. It follows that, for r infl , BH / r eff , NSC >> 1 , the 'classic' NSC SB profile should no longer exist, and the NSC potential should be entirely dominated and shaped by the MBH. In other words, it is reasonable to expect that when r infl , BH / r eff , NSC >> 1 , the NSC integrity may be compromised, to the point that the very definition of an NSC may change, both theoretically and observationally. \nIn what follows, we derive these two characteristic sizes for NSCs and MBHs, first for nuclei in which both are known to coexist. We calculate the radius of the BH sphere of influence as r infl , BH = G M BH /σ 2 , where M BH is from McConnell & Ma (2013) and Neumayer & Walcher (2012) and σ is the central velocity dispersion taken from HyperLEDA for the McConnell & Ma sample, or as measured by Neumayer & Walcher for their NSC-MBH sample. We note that the σ values from HyperLEDA may be biased toward higher values due to the often limited spatial resolution in measuring σ . Our calculated r infl , BH values for the McConnell & Ma sample may be affected by this bias. \nIn Figure 8 a we plot the ratio r infl , BH / r eff , NSC against host galaxy stellar mass. For reference, the data points for the Milky Way (MW) and M31 are labelled, and the unity ratio is indicated with a dashed horizontal line. As expected, the MW has a size ratio below one, while M 31 falls above the unity line. This is in line with the observed complex morphology of the M 31 nucleus, while the MW NSC structure and SB-profile are undisturbed by the presence of the MBHin its center. In other words, the large size ratio relates to the larger fraction of the M 31 NSC stars (mass) within the r infl , BH that are affected by the MBH (as also discussed in e.g. Peiris & Tremaine 2003). We note that the majority of the galaxies in the Neumayer & Walcher (2012) sample have a size ratio below one. It may be interesting to check whether those galaxies with size ratios similar or greater to M31 have similarly complex central morphologies. \nAnother question to ask from this observational perspective that can be related to theoretical expectations is to what extent galaxies with a SMBH could also harbor a 'classical' NSC with a radius in the range 2 - 5 pc? To address this question for the McConnell & Ma (2013) sample of 'pure' MBHs, we show in Figure 8 b the ratio between the derived r infl , BH and a 'nominal' NSC size of r eff , NSC =3pc, \nplotted against host galaxy mass 7 . The vast majority of the systems have a size ratio that falls significantly above unity, which is in line with theoretical expectations for the absence of a NSC. \nOn the other hand, one could also assume that a putative NSC in these galaxies has a mass corresponding to the extrapolation of the r eff , NSC -M glyph[star], gal relation shown in Figure 4. In this case, it implies that the NSC initially outgrew the MBH by a large amount, the theoretical size ratios are significantly smaller (see Figure 8 ), but still fall above unity, again favouring the strong impact by the SMBH on the NSC structure and its stellar velocity field. We further discuss these observations and their implications in § 4.3.", '4 DISCUSSION': 'We have found noticeable differences in the fitted relations between M NSC / r eff , NSC and host galaxy mass for different morphological types, as shown in Figures 3 and 4. We now investigate whether these differences could provide insight into the evolutionary path of NSCs in different hosts. In other words, can the properties of NSCs (size, mass) be traced to the various growth mechanism(s) that result from the internal secular evolution of the host, such as gas accretion and/or merging star clusters? For example, in late-type galaxies that are gas rich and harbor young stars and star clusters, NSC growth is more likely to be ongoing, while in early-type hosts, the only feasible mechanism today is the infall of old stellar populations.', '4.1 Possible measurement biases in r eff , NSC and M NSC': "We first discuss possible biases in the estimates for the sizes and photometric masses of NSCs. In late-type galaxies, the derived values for M NSC can be affected if the light (and hence the color) of the NSC is significantly influenced by a young stellar population (cf. § 2.5). This effect can cause the M/L ratio (and thus M NSC ) to be underestimated by up to a factor of 5, for example if 10% of the stellar mass is ≈ 5 Gyr younger than the rest (cf. § 2.2). However, this is opposite to what is observed in Figure 3, namely that at fixed r eff , NSC NSCs in late-type galaxies are more massive than those in early-type hosts. We conclude that the actual offset between the two populations in Figure 3 may well be more pronounced. \nAnother possible bias comes from underestimating r eff , NSC in late-type hosts if the NSC contains a significant fraction of young stars that are more centrally concentrated. We detected such an effect in Georgiev & Böker (2014) by measuring the ratio of NSC sizes in blue and red passbands (see also Kormendy & McClure 1993, Matthews et al. 1999 and Carson et al. 2015). However, as shown in Figure 10 of Georgiev & Böker (2014), this bias is < 5 %for our NSC sample, and is thus a negligible effect when interpreting Figure 3. We also do not expect a significant measurement bias caused by any contamination of NSC light from the underlying disk \nFigure 8. The ratio between the MBH sphere of influence radius, r infl , BH , and the NSC effective radius, r eff , NSC , against host galaxy stellar mass, M glyph[star] gal . Left: size ratios from directly measured quantities (Neumayer & Walcher 2012). Middle: shows the ratio r infl , BH / 3 pc for galaxies with SMBHs from McConnell & Ma (2013) (details in § 2.4 and 3.3). Right: ratio of the r infl , BH to a predicted r eff , NSC according to the r eff , NSC -M glyph[star], gal empirical relation derived in § 3.1, Figure 4. Dashed, grey line in all panels indicates a ratio of unity. Late- and early-type galaxies are shown with different symbols as indicated in the legend. For reference, the Milky Way, M 31 and M 32 are indicated with labels. \n<!-- image --> \nand/or bulge, because i) the galaxies in our late-type sample are selected to have a low inclination (see § 2.2) which minimizes this effect, and ii) our PSF-fitting methods implicitly account for any 'background' emission surrounding the NSC. Within 0.2 '' , only a very steep bulge would be of a concern, however, by construction of our catalogue of very late-types, we have no such cases. \nA last possible bias in measuring r eff , NSC and M NSC in early-type hosts may arise from an imperfect decomposition of the combined NSC-bulge surface brightness profile. This effect is more pronounced for luminous bulges with steeply rising surface brightness profiles. Côté et al. (2006) tested how well r eff , NSC and M V can be recovered by generating simulated data of NSCs with a range in size and luminosity. They find that irrespective of the input NSC size, r eff , NSC is recovered to better than 15%, with a bias toward underestimating r eff , NSC with increasing NSC magnitude. Accounting for such a bias would increase the offset between the earlyand late-type samples in Figures 3 and 4. As for the inferred luminosity (i.e. mass) of the NSC, Côté et al. (2006) estimate that it can be overestimated by < 0 . 1 mag for bright NSCs, and by as much as 0.5 mag for the faintest NSCs. This means that NSC masses in early-type hosts may be overestimated by up to a factor of three - again causing the separation of the two subsamples to become more pronounced. \nWethus conclude that the differences between the earlyand late-type samples seen in Figures 3 and 4 cannot be explained by observational biases in deriving r eff , NSC and M NSC , and in fact are likely to be more pronounced when observational biases are fully accounted for. This strengthens our finding that NSCs in late-type galaxies are more compact, both at fixed NSC mass and at fixed host galaxy mass. \n<!-- image --> \n/circledot /circledot", '4.2 Differences in NSC properties for different host morphologies': "As discussed in the last section, measurement biases are insufficient to explain the result that NSCs in late-type galaxies are more compact, both at fixed NSC mass and fixed host galaxy mass (Figures 3 and 4). An obvious question to ask therefore is which, if any, evolutionary effects could explain this difference? An increase in NSC size and mass over time has been demonstrated by a number of numerical simulations of merging clusters (Fellhauer & Kroupa 2002; Baumgardt et al. 2003; Bekki et al. 2004; Brüns et al. 2011; Antonini 2013; Gnedin et al. 2014; Arca-Sedda & Capuzzo-Dolcetta 2014) or/and mass build up from gas accretion (Hartmann et al. 2011). In particular, the slope of the r eff , NSC -M NSC relation found in this paper ( α glyph[similarequal] 0 . 34 +0 . 05 -0 . 04 , cf Table 1) is consistent with the slope of 0.4 found from cluster merger simulations (e.g. Bekki et al. 2004). This suggests that the smaller sizes of NSCs in late-types may well be explained by a scenario in which late-type nuclei have not (yet) experienced the infall of a large number of stellar clusters, i.e. that they are 'lagging behind' their counterparts in early-type hosts in the accumulation of stellar mass. Moreover, in late-type galaxies in-situ star formation may be the driving mechanism to grow the NSC, leading to a higher phase-space density and thus smaller sizes than what can be reached by cluster merging. \nWhat can the difference between late- and early-types in the M NSC -M glyph[star], gal relation in Figure 5, where early-type nuclei show a steeper slope, tell us in this context? As discussed above, M NSC in early-types may well be overestimated by about a factor of three in the most luminous host galaxies. While this effect certainly contributes to the steeper slope of early-type nuclei, there are also plausible evolutionary effects that may explain this. For example, the more eventful merger history of massive early-type hosts likely leads to an over-proportional growth of their NSCs caused by enhanced funneling of material to the center, both in the form of gas and star clusters. Late-type hosts, in contrast, have not ex- \nperienced significant mergers, and in this scenario, their nuclei would grow only proportionally to their host mass, resulting in a shallower slope compared to early-types. \nAs illustrated in Figure 6 b, disk-dominated NSC host galaxies contain significant amounts of (HI) gas. In a scenario in which late-type galaxies eventually turn into earlytype galaxies, one can ask how their NSCs move from the steeper late-type relation in Figure 6 b to the shallower earlytype relation. One possible path is to simply remove the gas, e.g. by ram pressure stripping, and galaxy-galaxy 'harassment' in a cluster environment. For a galaxy in isolation, the only plausible path to remove significant amounts of gas are stellar winds and/or supernovae from starburst regions (e.g. Mac Low & McCray 1988; Meurer et al. 1995). In fact, some low-mass late-type galaxies exhibit wind velocities above 1000 km/s, while their escape velocity is only 400 -500 km/s (e.g. Strickland & Heckman 2009). Such winds will naturally have the largest impact on the lowest mass galaxies due to their weaker potentials (e.g. Carraro 2014, and refs. therein). This could offer an evolutionary path for a NSC host galaxy from one relation to the other in Figure 6 b, even without involving interactions, leading to a shallower M NSC -M glyph[star] relation when the HI gas mass is removed. However, due to the galaxy density environments dichotomy between our samples, we can not exclude the possibility that environmental effects could make inapplicable one or the other discussed effects for cluster or isolated galaxies. \nWe conclude that the differences in r eff , NSC between late- and early-type galaxies are likely due to NSCs and their host galaxies being at different evolutionary stages. The differences in M NSC between the two morphological host types could possibly be explained by measurement biases, however, some plausible evolutionary effects can not be ruled out.", '4.3 Relations between NSCs and MBHs': "In this section we discuss what the relations between host galaxy stellar mass M glyph[star], gal and the parameters M MBH+NSC , M BH / M NSC and r infl , BH / r eff , NSC (§ 3.3) can tell us from observational point of view about the interplay between these two types of object. \nThe apparent lack of systems with MBH at galaxy masses below 10 9 M glyph[circledot] is noteworthy. The extrapolation of the M NSC+BH -M glyph[star] gal relation for coexisting NSCs and MBHs towards lower galaxy masses implies that in this range, a central MBH is expected to have a mass of M BH glyph[lessorsimilar] 10 4 -10 5 M glyph[circledot] . At typical galaxy distances of a few Mpc or more, this is below the detection limit of current instruments and analysis techniques. \nOn the other hand, there is evidence that low- to intermediate-mass BHs reside in (some) massive globular clusters (e.g. Lützgendorf et al. 2013, but see also Lanzoni et al. 2013). They appear to define a shallower scaling relation, which can potentially be explained if they are the remnant nuclei of stripped galaxies. Indeed, this is a popular formation scenario for dense stellar systems with MBHs, such as UCDs (Mieske et al. 2013; Seth et al. 2014). \nAt the high mass end, on the other hand, NSCs appear to become rare. This likely implies that as a galaxy grows in mass, there are processes at work that destroy NSCs, or prevent their formation on the first place. As discussed in \n§3.3.2, the 'classical' NSC surface brightness profile, which is normally well described by a King model, may no longer be a good representation if the radius of the MBH sphere of influence ( r infl , BH ) is significantly larger than the NSC effective (or half-mass) radius. In cases where there is only a MBHin the nucleus, the dissolution of an infalling NSC that passes through r infl , BH has been demonstrated in simulations (Antonini 2013; Mastrobuono-Battisti et al. 2014). A similar situation can also arise if a MBH spirals into the nucleus occupied by a NSC (e.g. in an Antennae-like merger where two galactic nuclei will coalesce; see Antonini et al. (2015) for the effect of mergers). \nIn the absence of infalling external objects, it is less clear how an NSC could be destroyed. As discussed in §3.3.2, in cases where NSC and MBH coexist, i.e. in the intermediate galaxy mass range, the NSC would be destroyed if it is 'outgrown' by the MBH. A potential example for this process is M 31 where the strong dynamical impact of the MBH on its surroundings are clearly present (e.g. Peiris & Tremaine 2003). The inner few pc of the M 31 nucleus is strongly axisymmetric and composed of three main central components - a central blue component at glyph[lessorsimilar] 0.2 pc along with a double-lobed redder component at ∼ 1-2 pc (i.e. each lobe is located on either side of the central blue component), which can be explained as the projection of an edgeon central disc of stars on Keplerian orbits around the MBH (Tremaine 1995; Peiris & Tremaine 2003; Brown & Magorrian 2013). \nOn the other hand, in the Milky Way nucleus, the other well-studied example of NSC-MBH system, the NSC has retained a normal star cluster profile that is well described by a standard King model (Schödel et al. 2014). This is expected due to the fact that, in contrast to M 31, the r infl , BH / r eff , NSC (or M BH / M NSC ) ratio in the Milky Way is less than 1 (cf. Figs. 7b and 8a). Next generation of large telescopes and instrumentation will help to extend this type of comparison to other nuclei with coexisting NSC and MBH. As pointed out by Georgiev & Böker (2014), there are a number of NSCs that are poorly described by a King model, and constraining their M BH / M NSC (or r infl , BH / r eff , NSC ) ratios would allow to check whether internal evolution due to the presence of a MBHis a viable explanation for their complex morphologies. \nOf course, it cannot be ruled out that the mass ratio M BH / M NSC is not affected by evolutionary effects at all, but is instead governed by the inability to form either object in the first place due to destructive feedback from the formation of the other. This 'competitive feedback' scenario has been discussed by Nayakshin et al. (2009).", '5 CONCLUSIONS': "We presented an updated analysis of various scaling relations between Nuclear Star Cluster (NSC) properties (mass and size) and the stellar mass of their host galaxies. We compared these scaling relations between late- and earlytype host galaxies, aided by the recent compilation of NSC properties in a large sample of late-type galaxies (Georgiev & Böker 2014). We added literature estimates of NSC properties in a number of other late- and early-type galaxies. Of special relevance are data for NSCs that harbor a massive black hole (MBH). \nOur study expands on earlier works (Seth et al. 2008b; Erwin & Gadotti 2012) that consider the total galaxy stellar mass (bulge plus disk), instead of only the bulge mass. This is especially relevant for late-type hosts which have most of their mass in the disk and therefore provides a more complete picture of the potential supply of matter (e.g. gas and star clusters) to the nucleus. For comparative purposes we add to the baryonic mass budget in these galaxies, their HI and X-ray masses (§ 2.6) to illustrate the amount of available material to further supply the evolution of the CMO. \nWe summarize our main results and their implications as follows: \n- · We provide photometric masses for all NSCs as well as their host galaxies, calculated from color-dependent massto-light ratios. These masses are listed in Table A1 (full version is available in the online version).\n- · The NSCs have a typical mass of a few × 10 6 M glyph[circledot] and constitute M NSC / M glyph[star] gal glyph[similarequal] 0 . 1% ± 0 . 2% of the total galaxy stellar mass (§ 3.2, Fig.5, 6), consistent with earlier results.\n- · We derive empirical scaling relations between r eff , NSC and M NSC and host galaxy total stellar mass for NSCs in late- and early-type host galaxies. The fit values of these scaling relations are provided in Table 1.\n- · The mass-size relation for NSCs shows a ∼ 1 . 5 σ significant difference between late- and early-type galaxies (Fig. 3) that cannot be explained by plausible measurement biases. At a given M NSC , NSCs in late-type hosts are on average twice as compact as their counterparts in early-type hosts (§ 3.1). We interpret this as evidence that NSCs in late-type galaxies are still evolving, i.e. they still have growth potential via gas accretion and/or cluster merging in the nucleus.\n- · The M NSC -M glyph[star] gal scaling relation for NSCs in earlytype hosts has a steeper slope than that for NSCs in latetype galaxies. Specifically, NSCs in early-types become progressively more massive with increasing total galaxy mass, compared to NSCs in late-type galaxies (Fig. 5 c. We interpret this result as likely being due to measurement bias, which can reach a factor of three in M NSC in massive earlytype galaxies. However, we can not exclude the possibility that the difference in slopes is real, as a number of physical processes could contribute to this trend, such as i) depletion of the host galaxy mass via ram pressure stripping and/or galaxy 'harassment', or ii) accelerated NSC growth in massive hosts due to their enhanced merger history.\n- · Coexisting NSC-MBH systems define a M BH+NSC -M glyph[star] gal relation (§ 3.3, Fig. 7) with a slope consistent with that defined by NSCs without MBHs in early-types, but steeper than both the well-known M BH -M bulge relation and the relation defined by late-type NSCs without MBHs. To within the fit uncertainties, the slopes of the M NSC -M glyph[star] gal , M BH+NSC -M glyph[star] gal and M BH -M bulge relations are consistent with each other. This is probably suggesting similar physical mechanisms driving NSC or/and MBH growth as a function of galaxy mass.\n- · We looked at the size ratio between the MBH sphere of influence and the NSC effective (or half-mass) radius. It covers a wide range of values ( 0 . 01 glyph[lessorequalslant] r infl , BH / r eff , NSC glyph[lessorequalslant] 100 , § 3.3.2, Fig. 8), and because r infl , BH / r eff , NSC ≡ M BH / M NSC , the limit r infl , BH / r eff , NSC glyph[greaterorequalslant] 1 implies that more than 50% of the bound NSC stars are on Keplerian orbits around the MBH. The best example for this scenario is the nucleus of \nM31, which has a r infl , BH / r eff , NSC > 1 , thus illustrating the dynamical influence of the MBH on its surroundings. The NSC-MBH system in the Milky Way nucleus, in contrast, has a size ratio below 1. It has thus, unsurprisingly, a surface brightness profile that is well described by a King model.", 'ACKNOWLEDGMENTS': 'IG would like to thank the science department of ESAESTEC in Noordwijk and the ESO-Garching visitor programme for partial support during the preparation of this paper, as well as the Max-Plank-Institut für Astronomie (Heidelberg) for support during the final stages of this work. NL gratefully acknowledges the generous support of an NSERC PDF award. We also thank Morgan Fouesneau for numerous discussions and help with fitting techniques. This research has made use of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. We also acknowledge use of the HyperLeda database (http://leda.univ-lyon1.fr).', 'References': "Anderson M. E., Bregman J. 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The method adopted here for fitting the various subsamples with a straight line (in a log -log space) also treats non symmetric uncertainties. It consists of a maximum likelihood estimation (MLE) of the model parameters and data bootstrapping to account for the finite data samples and build the probability density distribution (posterior) of model parameters (Nelder & Wedderburn 1972; Hogg et al. 2010). This method is very similar to an MCMC, however, instead of sampling the prior space to derive posterior density distributions, we sample from the bootstrapped data. The functional form of the measurement uncertainties used for the bootstrapping as well as one part of the noise model in the MLE is expressed as a combination of two Gaussians joined by common mode value, which in our case is the location ( µ ) of the data point: \nf ( t ; µ, glyph[epsilon1] 1 , glyph[epsilon1] 2 ) = A exp( -( t -µ ) 2 2 glyph[epsilon1] 2 1 ) , if t < µ , and f ( t ; µ, glyph[epsilon1] 1 , glyph[epsilon1] 2 ) = A exp( -( t -µ ) 2 2 glyph[epsilon1] 2 2 ) otherwise, - \nwhere A = √ 2 /π ( glyph[epsilon1] 1 + glyph[epsilon1] 2 ) -1 , and glyph[epsilon1] 1 , glyph[epsilon1] 2 is each side of the uncertainty. This distribution is known as a split normal distribution (Gibbons & Mylroie 1973; John 1982). The product of these functions along x and y , F = f ( x, µ x , glyph[epsilon1] x 1 , glyph[epsilon1] x 2 ) × f ( y, µ y , glyph[epsilon1] y 1 , glyph[epsilon1] y 2 ) allows to treat each data point with a probability density space defined by a split normal distribution, glyph[epsilon1] xy = { glyph[epsilon1] x 1 , glyph[epsilon1] x 2 , glyph[epsilon1] y 1 , glyph[epsilon1] y 2 } . \nOur dataset can be defined as, D of k independent observations, d k = { x k , y k } with uncertainties glyph[epsilon1] = { glyph[epsilon1] 2 k } = { glyph[epsilon1] 2 x 1 , glyph[epsilon1] 2 x 2 , glyph[epsilon1] 2 y 1 , glyph[epsilon1] 2 y 2 } described by the split normal distribution. The model, M (straight line with parameters, { α, β } ) is 'predicting' how the data should be distributed. We therefore do not test for other than linear (in log-log space) relation between the fitted quantities. The model dispersion is also described by a Gaussian model ( Σ ) with a variance ( σ 2 ) orthogonal to the linear regression. The 'noise' we can also note as E ( { glyph[epsilon1], Σ } ). Thus, the model can be written as: \ny k = α × x k + β + glyph[epsilon1] +Σ , \nwhere α and β are the slope and intercept model parameters. Thus, the density distribution of the data given the model can be expressed as: \np ( d k | M,E ) = 1 √ 2 πσ 2 exp {-1 2 σ 2 ( y k -α × x k -β ) 2 }× × f ( x k , µ x k , glyph[epsilon1] x k ) × f ( y k , µ y k , glyph[epsilon1] y k ) . \nFollowing the Bayes rule, which states that the posterior probability distribution of 'observing' a model M (the linear regression) given the distribution of the data ( D ) and its uncertainties ( E , incl. model noise) is: \nP ( M | D,E ) = N ∏ k =1 p ( d k | M,E ) × P ( M,E ) P ( D,E ) , \nwhere P ( D,E ) is the evidence, i.e. the probability of the data averaged over all parameters (it also assures that the posterior distribution integrates to unity), p ( d k | M,E ) is the above likelihood of the k -th data point given the model M . Since the denominator (the data) does not depend on the model parameters, { α, β } , the Bayesian estimator is obtained by maximizing the likelihood p ( d k | M,E ) P ( M,E ) \nwith respect to the model (straight line) parameters. Assuming that the model parameters α and β are uniformly distributed, then the Bayesian estimator is obtained by maximizing the likelihood function p ( d k | M,E ) . Or for convenience, to convert products into to sums, the natural logarithm of it: \nln p ( d k | M,E ) = -N ln 2 π 2 -1 2 N ∑ k =1 [ ln σ 2 -( y k -αx k -β ) 2 σ 2 --ln F ( x k , y k , µ x k , µ y k , glyph[epsilon1] x k , glyph[epsilon1] y k ) ] \nIn general, this treatment, that followed from discussion in Bailer-Jones (2015) will be described in details in Fouesneau et al. (in prep.). For normal distributions the likelihood function has a closed form expression for the estimator, however, if the estimator lacks a closed form, a solution can be obtained by MCMC. The estimator in our case has a closed form, however, we build our posterior distributions not by sampling from a wide model prior space (with MCMC technique, e.g. emcee Foreman-Mackey et al. 2013), but by sampling from a smaller data 'prior' space restricted by the data uncertainties. This data space is generated by bootstrap from the split normal distribution of the measurement uncertainties. We note that the classical bootstrapping ( N new data points) makes significant difference only for small or/and data with large scatter, σ 2 . In other words, the final posterior distribution describing the posterior distributions from j -number of model estimators is defined by: \nP ( M | D,E ) = n ∑ j =1 N ∏ k =1 P ( d k,j | M j , E j ) × P ( M j ) , \nwhere n is the number of new data samples (bootstraps). This number in our case was chosen to be 1500 to sample well the posterior distributions. The linear regression model fitted to our data is of the form: \nlog 10 ( y/c 1) = α × log 10 ( x/c 2) + β, (A1) \nwhere c 1 , c 2 are normalization constants, α and β are the slope and intercept, respectively. As also discussed in § 3.1, the right choice of normalization constants is important to minimize the correlation between the slope and intercept. This provides a more realistic estimation of the uncertainties for α, β and the dispersion, σ , of the data, because it determines the shape of the posterior probability density functions (PDFs). Those distributions are shown by histograms in Figure A1. The c 1 , c 2 constants are estimated a priori from the highest probability density value of the 2D-PDFs of the data. Figure A1 illustrates the results from fitting each parameter pair. Each line in the top-right panel represents one of the possible solutions (one bootstrap realisation of the data described by the uncertainty of each data point) fitted with linear regression, which coefficients are shown with dots in the lower-left panel. For illustration, contour lines show the density distribution of the probable solutions for the slope and intercept. The PDFs of α, β are shown with histograms, where with red lines is indicated their highest probability modal value. The solid red line in the top-right panel shows the final solution for the respective subsample. The two red, parallel lines indicate the 1 σ dispersion of the data with respect the solid line of the final solution. \nFigure A1. Results from the bootstrapped, non-symmetric error weighted, maximum likelihood fitting technique. Left and Right figure columns are for early- and late-type galaxies, respectively. Top row: NSC effective radius versus NSC mass (Fig. 3); middle row NSC effective radius versus host galaxy stellar mass (Fig. 4); bottom row: NSC mass - host galaxy stellar mass (Fig. 5). For each figure, the top-left and bottom-right panels show the projected distributions for each realization of the relation slope ( α ) and intercept ( β ), which are shown with dots in the bottom-left panel and some of those are shown with lines in the top-right panel . Their best value is estimated from the distribution mode value as indicated with solid lines (and density contours, in bottom-left). The thick (red) line plotted with the data ( top-right panel ) shows the best solution, where the two parallel lines are the 1 σ dispersion to that best fit. \n<!-- image --> \n18 I. Y. Georgiev, T. Böker, N. Leigh, N. Lützgendorf, N. NeumayerFigure A1. ( Cont'd ) Left: Fit for the NSC mass vs. host galaxy stellar plus HI mass, M NSC -M glyph[star] +HI , gal (Fig. 6 b); Right: Fit for the NSC plus MBH mass vs. host galaxy stellar mass, M BH+NSC -M glyph[star], gal (Fig. 7). \n<!-- image --> \nTable A1. Photometric masses of Nuclear Star Clusters and their host galaxies. (Full table available online only) \n| Object (1) | m - M [mag] (2) | t (3) | r eff , NSC [pc] (4) | M NSC [ 10 4 M glyph[circledot] ] (5) | M gal ,glyph[star] [ 10 7 M glyph[circledot] ] (6) | M gal , HI [ 10 7 M glyph[circledot] ] (7) | NSC filter (8) | M/L NSC [ M glyph[circledot] /L glyph[circledot] ] (9) | Galaxy filter (10) | M/L gal [ M glyph[circledot] /L glyph[circledot] ] (11) |\n|-----------------------|-------------------|---------|--------------------------------------|---------------------------------------------------|------------------------------------------------------|----------------------------------------------|------------------|----------------------------------------------------------|----------------------|-----------------------------------------------------------|\n| | | | +0 . 1 | 19 . 21 +0 . 48 | 1 . 22 ± 0 . 91 | | | +0 . 02 | | 1.16 |\n| DDO078 | 27.82 | 10. | 3 . 7 - 0 . 00 | - 0 . 48 | | 0 . 00 ± 0 . 00 | V I | 0 . 92 - 0 . 02 | B | |\n| ESO138-G10 | 30.84 | 7.9 | 7 . 7 +0 . 1 - 7 . 7 +0 . 7 | 2427 . 8 +890 . 19 - 60 . 69 8 . 34 +3 . 06 | 1824 . 23 ± 743 . 8 66 . 58 23 . 56 | 3274 . 47 ± 1432 . 55 | I | 0 . 45 +0 . 17 - 0 . 01 +0 . 17 | B BI | 1.16 |\n| ESO187-G51 ESO202-G41 | 31.13 | 9. | 1 . 8 - 0 . 8 +0 . 3 | - 0 . 21 +2 . 62 | ± 15 . 91 9 . 83 | 282 . 66 ± 130 . 17 176 . 86 ± 65 . 16 | I | 0 . 45 - 0 . 01 +0 . 17 | BI | 1.64 0.75 |\n| ESO241-G06 | 31. | 8.9 | 1 . 8 - 1 . 8 +0 . 00 | 7 . 13 - 0 . 18 +16 . 81 | ± 108 . 34 39 . 53 | 103 . 72 ± 54 . 93 | I I | 0 . 45 - 0 . 01 +0 . 17 | BI | 1.43 |\n| ESO271-G05 | 31.44 32.57 | 9. 9. | 2 . 2 - 0 . 4 3 . 1 +0 . 6 | 45 . 84 - 1 . 15 +32 . 62 | ± | 114 . 43 50 . 06 | V | 0 . 45 - 0 . 01 0 . 63 +0 . 21 | B | 1.16 |\n| | | | - 0 . 6 | 96 . 83 - 29 . 18 | 104 . 11 ± 105 . 00 | ± | | - 0 . 19 0 . 45 +0 . 17 | BI | |\n| ESO290-G39 | 31.33 | 9. | 4 . 6 +1 . 00 - 4 . 5 | 9 . 67 +3 . 54 - 0 . 24 | 16 . 72 ± 5 . 7 | 140 . 87 ± 74 . 61 | I | - 0 . 01 | | 0.63 |\n| ESO301-IG11 | 31.23 | 9.9 | 3 . 7 +0 . 3 - 0 . 4 | 61 . 09 +20 . 58 - 18 . 41 | 410 . 68 ± 151 . 75 | 0 . 00 ± 0 . 00 | V | 0 . 63 +0 . 21 - 0 . 19 | BI | 2.66 |\n| ESO357-G12 | 31.76 | 7. | 4 . 3 +0 . 3 - 4 . 3 | 186 . 21 +62 . 73 - 56 . 12 | 579 . 22 ± 320 . 78 | 409 . 9 ± 66 . 07 | V | 0 . 63 +0 . 21 - 0 . 19 | BI | 2.76 |\n| ESO358-G05 | 31.59 | 9. | 2 . 7 +0 . 1 - 2 . 7 | 88 . 15 +32 . 32 - 2 . 2 | 342 . 09 ± 107 . 02 | 100 . 93 ± 39 . 51 | I | 0 . 45 +0 . 17 - 0 . 01 | BI | 3.4 |\n| ESO359-G29 | 30.03 | 9.9 | 0 . 5 +0 . 00 - 0 . 5 | 14 . 84 +19 . 82 - 2 . 39 | 22 . 48 ± 13 . 03 | 60 . 82 ± 18 . 2 | B | 0 . 7 +0 . 93 - 0 . 11 | B | 1.16 |\n| ESO404-G03 | 32.58 | 3.9 | 2 . 1 +0 . 1 - 0 . 3 | 121 . 9 +41 . 07 - 36 . 74 | 2356 . 09 ± 737 . 05 | 165 . 51 ± 45 . 73 | V | 0 . 63 +0 . 21 - 0 . 19 | BI | 4.48 |\n| ESO418-G08 | 31.77 | 7.7 | 8 . 3 +0 . 2 - 8 . 3 | 55 . 11 +1 . 38 - 1 . 38 | 264 . 05 ± 77 . 07 | 190 . 96 ± 35 . 18 | UVI | 0 . 45 +0 . 01 - 0 . 01 | BI | 1.53 |\n| ESO498-G05 | 32.52 | 4.3 | 6 . 2 +0 . 00 - 6 . 2 | 805 . 36 +271 . 33 - 242 . 74 | 1316 . 96 ± 423 . 92 | 0 . 00 ± 0 . 00 | V | 0 . 63 +0 . 21 - 0 . 19 | BV | 2.25 |\n| ESO502-G20 | 31.07 | 5. | 1 . 6 +0 . 00 - 0 . 2 | 23 . 23 +7 . 83 - 7 . 00 | 257 . 7 ± 106 . 37 | 381 . 59 ± 149 . 37 | V | 0 . 63 +0 . 21 - 0 . 19 | BI | 2.83 |\n| ESO504-G17 | 31.84 | 4.1 | 2 . 3 +1 . 7 - 1 . 1 | 124 . 76 +386 . 77 - 3 . 12 | 232 . 88 ± 188 . 92 | 255 . 19 ± 82 . 26 | U | 0 . 15 +0 . 47 | BI | 1.48 |\n| ESO504-G30 | 31.71 | 7.7 | 2 . 6 +0 . 3 - 2 . 6 | 109 . 95 +40 . 32 - 2 . 75 | 104 . 81 ± 44 . 79 | 0 . 00 ± 0 . 00 | I | - 0 . 00 0 . 45 +0 . 17 - 0 . 01 | BI | 1.35 |\n| ESO508-G34 | 32.43 32.51 | 8.1 | 1 . 5 +0 . 00 - 1 . 5 38 . 9 +0 . 00 | 51 . 76 +17 . 44 - 15 . 6 +245 . 19 | 327 . 72 ± 108 . 61 | 345 . 78 ± 111 . 47 | V | 0 . 63 +0 . 21 - 0 . 19 +0 . 21 | BI | 1.77 |\n| ESO508-G36 | 32.3 | 4.2 4.8 | - 0 . 00 4 . 8 +0 . 00 | 727 . 77 - 219 . 35 206 . 06 +69 . 42 | 77 . 75 ± 33 . 37 5985 . 88 2672 . 97 | 0 . 00 ± 0 . 00 | V V | 0 . 63 - 0 . 19 0 . 63 +0 . 21 | B BI | 1.16 6.78 |\n| ESO549-G18 ESO572-G22 | 32.1 | 6.7 | - 4 . 8 3 . 8 +0 . 2 | - 62 . 11 82 . 79 +27 . 89 | ± 420 . 85 204 . 45 | 46 . 49 ± 9 . 63 138 . 65 63 . 85 | V | - 0 . 19 0 . 63 +0 . 21 | BI | 2.98 |\n| IC0239 | 30.76 | 6. | - 0 . 2 2 . 6 +0 . 1 | - 24 . 95 123 . 08 +6 . 44 | ± 969 . 32 104 . 01 | ± 1516 . 65 453 . 99 | BVI | - 0 . 19 1 . 13 +0 . 06 | BV | 1.45 |\n| IC0396 | 30.79 | 4. | - 0 . 1 16 . 3 +0 . 00 | - 5 . 25 4033 . 94 +13863 . 8 | ± 422 . 55 ± 272 . 03 | ± 66 . 95 ± 13 . 87 | BVI | - 0 . 05 0 . 65 +2 . 22 | B | 1.16 |\n| IC0769 | 32.88 | 4. | - 16 . 3 3 . 5 +0 . 3 - | - 1291 . 628 . 05 +211 . 59 - 189 . 3 | 628 . 83 ± 419 . 53 | 303 . 93 ± 97 . 98 | V | - 0 . 21 0 . 63 +0 . 21 | BI | 0.96 |\n| IC1933 | 31.41 | 6.2 | 3 . 5 16 . 4 +1 . 3 - 3 . 6 | 22 . 18 +7 . 47 - 6 . 69 | 169 . 43 ± 39 . 99 | 495 . 5 ± 193 . 96 | V | - 0 . 19 0 . 63 +0 . 21 - 0 . 19 | BV | 0.56 |\n| IC2056 | 31.93 | 3.8 | 16 . 1 +0 . 1 - 0 . 00 | 3477 . 06 +86 . 93 - 86 . 93 | 994 . 53 ± 298 . 79 | 144 . 98 ± 76 . 78 | BVI | 0 . 45 +0 . 01 - 0 . 01 | BV | 1.07 |\n| IC2129 | 31.72 | 7.6 | 1 . 9 +1 . 5 - 1 . 00 | 47 . 00 +145 . 7 - 1 . 17 | 1005 . 9 ± 366 . 97 | 141 . 77 ± 29 . 38 | U | 0 . 15 +0 . 47 | BI | 3.56 |\n| IC3021 | 32.65 | 9. | 7 . 2 +0 . 6 - 7 . 2 | 250 . 03 +84 . 24 - 75 . 36 | 8 . 08 ± 4 . 49 | 30 . 53 ± 9 . 14 | V | - 0 . 00 0 . 63 +0 . 21 - | BI | 0.15 |\n| IC3298 | 32.59 | 5.2 | 5 . 4 +0 . 5 - 1 . 00 | 228 . 03 +76 . 82 - 68 . 73 | 59 . 35 ± 14 . 99 | 50 . 94 ± 29 . 32 | V | 0 . 19 0 . 63 +0 . 21 | BI | 0.68 |\n| IC4710 | 29.75 | 8.9 | 0 . 8 +0 . 00 - 0 . 00 | 248 . 81 +22 . 57 - 177 . 33 | 125 . 12 ± 45 . 65 | 642 . 52 ± 251 . 51 | BVI | - 0 . 19 1 . 53 +0 . 14 | BV | 0.9 |\n| IC5256 | 33.52 | 8. | 5 . 2 +0 . 00 - 5 . 2 | 95 . 94 +32 . 32 - 28 . 92 | 1458 . 69 ± 476 . 79 | 0 . 00 ± 0 . 00 | V | - 1 . 09 0 . 63 +0 . 21 - 0 . 19 | BI | 2.52 |\n| IC5332 | 29.62 | 6.8 | 2 . 2 +0 . 1 - 0 . 2 | 18 . 00 +0 . 45 - 0 . 45 | 393 . 26 ± 253 . 18 | 2168 . 11 ± 698 . 92 | BI | 0 . 6 +0 . 02 - 0 . 02 | B | 1.16 |\n| M063 | 29.5 | 4. | 4 . 2 +0 . 00 - 4 . 2 | 972 . 95 +3016 . 15 - 24 . 32 | 3583 . 08 ± 307 . 57 | 4370 . 75 ± 1308 . 32 | U | 0 . 15 +0 . 47 | BV | 1.8 |\n| M074 | 29.93 | 5.2 | 3 . 6 +0 . 00 - 0 . 00 | 1128 . 37 +533 . 13 - 71 . 87 | 1897 . 02 1058 . 45 | 4625 . 14 958 . 48 | BVI | - 0 . 00 1 . 69 +0 . 8 | BV | 0.92 |\n| M083 | 28.41 | 5. | 0 . 9 +0 . 00 | 76 . 58 +237 . 38 | ± 4008 . 34 1806 . 37 | ± 32199 . 89 9638 . 59 | U | - 0 . 11 0 . 15 +0 . 47 | BV | 1.29 |\n| M100 | 31. | 4.1 | - 0 . 00 2 . 5 +0 . 00 - | - 1 . 91 1304 . 63 +1742 . 1 | ± | ± | | - 0 . 00 +0 . 93 | BV | 1.65 |\n| M101 | 29.13 | 6. | 2 . 5 0 . 4 +0 . 2 | - 210 . 28 169 . 08 +524 . 15 | 6871 . 13 ± 1916 . 88 | 1105 . 69 ± 254 . 6 | B | 0 . 7 - 0 . 11 0 . 15 +0 . 47 | BV | 0.77 |\n| M106 | 29.39 | 4. | - 0 . 1 4 . 4 +0 . 4 | - 4 . 23 93 . 78 +290 . 71 | 2015 . 99 ± 821 . 99 3439 . 25 959 . 47 | 12519 . 67 ± 4324 . 14 5879 . 1 2030 . 57 | U U | - 0 . 00 0 . 15 +0 . 47 | BV | 1.64 |\n| M108 | 30.6 | 6. | - 0 . 1 2 . 5 +0 . 2 - 2 . 5 | - 2 . 34 97 . 72 +32 . 92 | ± 2158 . 64 231 . 62 | ± | V | - 0 . 00 0 . 63 +0 . 21 | BV | 1.48 |\n| MCG-1-03-85 | 30.28 | 7. | 2 . 9 +0 . 2 | - 29 . 45 +83 . 72 | ± | 2400 . 79 ± 663 . 36 | | - 0 . 19 +0 . 86 | | |\n| MCG-3-13-63 | 33.22 | 3.9 | - 2 . 9 6 . 8 +0 . 5 | 250 . 53 - 44 . 86 571 . 76 +14 . 29 | 456 . 89 ± 137 . 27 462 . 85 356 . 81 | 998 . 21 ± 321 . 79 564 . 56 246 . 99 | BVI | 2 . 58 - 0 . 46 +0 . 01 | BV | 1.43 |\n| | 30.54 | 9.9 | - 1 . 5 +0 . 7 | - 14 . 29 45 . 3 +140 . 43 | ± 169 . 5 ± 50 . 92 | ± 306 . 65 63 . 55 | V I | 0 . 45 - 0 . 01 +0 . 47 | BI | 1.19 |\n| NGC0014 NGC0247 | 27.81 | 6.9 | 5 . 4 - 0 . 7 1 . 00 +0 . 00 | - 1 . 13 138 . 67 +46 . 72 - 41 . 8 | 281 . 59 ± 60 . 43 | ± 7894 . 3 2544 . 82 | U V | 0 . 15 - 0 . 00 0 . 63 +0 . 21 | BV | 0.97 1.07 |\n| NGC0275 | 31.7 | 6. | - 1 . 00 1 . 3 +0 . 00 - 0 . 1 | 242 . 78 +89 . 02 - 6 . 07 | 636 . 69 ± 233 . 68 | ± 468 . 85 ± 129 . 55 | I | - 0 . 19 0 . 45 +0 . 17 - 0 . | BV BI | 1.37 |\n| NGC0300 | 26.48 | 6.9 | 1 . 9 +0 . 00 - 0 . 00 | 75 . 37 +27 . 64 - 1 . 88 | 217 . 05 ± 97 . 81 | 33323 . 29 ± 16113 . 24 | I | 01 0 . 45 +0 . 17 - 0 . 01 | BV | 1.2 |\n| NGC0337A | 30.68 | 8. | 5 . 00 +1 . 00 - 0 . 2 | 17 . 27 +6 . 33 - 0 . 43 | 173 . 46 ± 74 . 45 | 1427 . 65 ± 394 . 48 | I | 0 . 45 +0 . 17 - 0 . 01 | BV | 0.76 |\n| NGC0406 | 31.57 | 4.9 | 1 . 3 +0 . 00 | 66 . 99 +22 . 57 | 451 . 77 ± 77 . 56 | 945 . 06 ± 304 . 65 | V | 0 . 63 +0 . 21 | BV | 1.16 |\n| NGC0428 | 30.86 | 8.6 | - 0 . 1 1 . 2 +0 . 00 | - 20 . 19 258 . 23 +237 . 9 | 346 . 47 66 . 92 | 1168 . 7 269 . 1 | UBVI | - 0 . 19 0 . 5 +0 . 46 | | |\n| NGC0600 | 31.8 | 7. | - 0 . 00 1 . 3 +0 . 1 | - 32 . 18 194 . 63 +71 . 36 | ± 488 . 23 94 . 3 | ± 471 . 81 119 . 5 | I | - 0 . 06 0 . 45 +0 . 17 | BV | 0.71 |\n| NGC0672 | 29.44 | 6. | - 0 . 1 4 . 1 +0 . 2 - 4 . 1 | - 4 . 87 65 . 65 +47 . 84 - | ± 271 . 02 ± 58 . 16 | ± 2696 . 42 745 . 05 | BVI | - 0 . 01 0 . 48 +0 . 35 | BV BV | 0.98 0.96 |\n| NGC0853 NGC0864 | 31.61 | 8.6 | 2 . 1 +0 . 00 - 2 . 1 | 5 . 64 201 . 94 +74 . 04 - 5 . 05 | 0 . 93 ± 5 . 6 | ± 134 . 45 ± 71 . 2 | I | - 0 . 04 0 . 45 +0 . 17 - 0 . 01 | BI | 0.12 |\n| | 31.44 | 5.1 | 2 . 7 +0 . 00 - 0 . 00 | 7246 . 31 +181 . 16 - 181 . 16 | 1256 . 46 ± 512 . 3 | 1471 . 9 ± 271 . 13 | V I | 5 . 92 +0 . 15 - 0 . 15 | BV | 0.95 |\n| | 30.47 | 7.9 | +0 . 1 | 215 . 38 +5 . 38 | 166 . 72 25 . 04 | | | +0 . 01 | | 0.98 |\n| NGC0959 | | | 7 . 4 - 7 . 4 | - 5 . 38 | | 196 . 18 49 . 69 | BVI | 0 . 45 01 | BV | |\n| NGC1003 NGC1042 | 30.16 | 6. | 4 . 7 +0 . 2 - 0 . 1 +0 . 1 | 671 . 37 +209 . 04 - 112 . 58 +844 . 18 | ± 249 . 00 ± 74 . 81 | ± 2310 . 86 ± 638 . 52 | BVI | - 0 . 3 . 76 +1 . 17 - 0 . 63 | BV | 0.9 |\n| NGC1058 | 29.56 29.85 | 6. 5.1 | 1 . 3 - 0 . 00 1 . 7 +0 . 00 - 1 . 6 | 212 . 82 - 77 . 86 5157 . 47 +128 . 94 - 128 . 94 | 210 . 81 ± 108 . 58 282 . 79 ± 84 . 96 | 776 . 82 ± 214 . 64 1124 . 09 ± 258 . 83 | BVI BV | 0 . 69 +2 . 74 - 0 . 25 4 . 33 +0 . 11 - 0 . 11 | BV BV | 0.97 1.15 | \nI. Y. Georgiev, T. Böker, N. Leigh, N. Lützgendorf, N. Neumayer \nTable A1 (cont'd) \n| Object (1) | m - M [mag] (2) | t (3) | r eff , NSC [pc] (4) | M NSC [ 10 4 M glyph[circledot] ] (5) | M gal ,glyph[star] [ 10 7 M glyph[circledot] ] (6) | M gal , HI [ 10 7 M glyph[circledot] ] (7) | NSC filter (8) | M/L NSC [ M glyph[circledot] /L glyph[circledot] ] (9) | Galaxy filter (10) | M/L gal [ M glyph[circledot] /L glyph[circledot] ] (11) |\n|-----------------|-------------------|---------|----------------------------------------|----------------------------------------------|------------------------------------------------------|----------------------------------------------|------------------|-----------------------------------------------------------------|----------------------|-----------------------------------------------------------|\n| NGC1073 | 30.7 | 5.3 | 0 . 5 +0 . 1 | 338 . 23 +8 . 46 | 490 . 07 ± 220 . 85 | | | 0 . 45 +0 . 01 | | 0.83 |\n| NGC1156 | 29.39 | 9.8 | - 0 . 00 +0 . 3 | - 8 . 46 70 . 48 +218 . 5 | 145 . 91 ± 46 . 97 | 1094 . 43 ± 252 . 00 | BVI U | - 0 . 01 0 . 15 +0 . 47 | BV BV | 0.68 |\n| NGC1249 | 30.95 | 6. | 1 . 3 - 1 . 3 1 . 9 +0 . 2 - 1 . 9 | - 1 . 76 798 . 07 +191 . 01 - 330 . 59 | 293 . 26 ± 31 . 47 | 940 . 4 ± 173 . 23 1325 . 05 ± 518 . 68 | BVI | - 0 . 00 3 . 24 +0 . 77 - 1 . 34 | BV | 0.71 |\n| NGC1258 | 32.28 | 5.7 | 3 . 3 +0 . 00 - 0 . 1 | 593 . 07 +291 . 72 - 529 . 06 | 1583 . 53 ± 652 . 77 | 105 . 39 ± 41 . 25 | BVI | 4 . 07 +2 . 00 - 3 . 63 | BI | 3.71 |\n| NGC1310 | 31.7 | 5. | 13 . 9 +1 . 8 - 1 . 2 | 187 . 93 +63 . 31 - 56 . 64 | 481 . 5 ± 216 . 99 | 0 . 00 ± 0 . 00 | V | 0 . 63 +0 . 21 - 0 . 19 | BV | 0.79 |\n| NGC1325 | 31.24 | 4. | 3 . 9 +0 . 1 - 0 . 00 | 2417 . 36 +230 . 17 - 1836 . 4 | 1054 . 76 ± 181 . 08 | 442 . 93 ± 122 . 39 | BVI | 5 . 54 +0 . 53 - 4 . 21 | BV | 1.6 |\n| NGC1325A | 31.7 | 6.9 | 3 . 5 +0 . 00 | +113 . 11 | 339 . 75 51 . 04 | 97 . 06 37 . 99 | V | 0 . 63 +0 . 21 - 0 . 19 | BV | 1.1 |\n| | | | - 0 . 00 | 335 . 73 - 101 . 19 | ± | ± | | +0 . 01 | | |\n| NGC1326A | 31.09 | 8.8 | 4 . 3 +0 . 4 - 0 . 8 | 13 . 47 +0 . 34 - 0 . 34 | 99 . 31 ± 100 . 19 | 348 . 46 ± 144 . 42 | V I | 0 . 45 - 0 . 01 | BI | 1.33 |\n| NGC1385 | 30.62 | 5.9 | 0 . 4 +0 . 1 - 0 . 00 | 207 . 59 +5 . 19 - 5 . 19 | 584 . 91 ± 100 . 42 | 488 . 56 ± 123 . 75 | BVI | 0 . 45 +0 . 01 - 0 . 01 | BV | 0.91 |\n| NGC1406 | 31.29 | 4.4 | 5 . 8 +0 . 2 - 0 . 00 | 3020 . 77 +75 . 52 - 75 . 52 | 652 . 77 ± 364 . 21 | 585 . 76 ± 121 . 39 | BVI | 5 . 92 +0 . 15 - 0 . 15 | BV | 1.24 |\n| NGC1483 | 31.87 | 4. | 1 . 6 +0 . 00 - 0 . 00 | 56 . 75 +19 . 12 - 17 . 11 | 270 . 85 ± 34 . 87 | 412 . 74 ± 180 . 57 | V | 0 . 63 +0 . 21 - 0 . 19 | BV | 0.73 |\n| NGC1487 | 29.74 | 7.2 | 0 . 6 +0 . 2 - 0 . 00 | 103 . 09 +2 . 58 - 2 . 58 | 89 . 97 ± 21 . 24 | 1190 . 13 ± 520 . 67 | BVI | 0 . 45 +0 . 01 - 0 . 01 | BV | 0.74 |\n| NGC1493 | 30.27 | 6. | 3 . 6 +0 . 5 - 0 . 00 | 354 . 17 +8 . 85 - 8 . 85 | 302 . 39 ± 84 . 36 | 595 . 58 ± 219 . 42 | V I | 0 . 45 +0 . 01 - 0 . 01 | BV | 0.93 |\n| NGC1494 | 30.92 | 7. | 4 . 6 +0 . 2 - 4 . 5 | 48 . 99 +1 . 22 - 1 . 22 | 541 . 61 ± 206 . 94 | 621 . 43 ± 271 . 87 | BI | 0 . 6 +0 . 02 - 0 . 01 | BI | 1.56 |\n| NGC1507 | 30.21 | 8.5 | 0 . 1 +2 . 3 - 0 . 1 +0 . 2 | 1644 . 72 +5098 . 63 - 41 . 12 +7 . 54 | 140 . 01 ± 51 . 08 | 615 . 48 ± 99 . 2 | U | 0 . 15 +0 . 47 - 0 . 00 | BV | 0.72 |\n| NGC1518 | 29.9 | 8.2 | 3 . 4 - 3 . 4 | 22 . 39 - 6 . 75 | 125 . 55 ± 29 . 64 | 1473 . 17 ± 542 . 74 | V | 0 . 63 +0 . 21 - 0 . 19 | BV | 0.73 |\n| NGC1559 | 30.91 | 5.9 | 1 . 5 +0 . 00 - 0 . 00 +0 . 00 | 456 . 74 +69 . 27 - 92 . 65 +1622 . 2 | 587 . 67 ± 176 . 56 | 833 . 89 ± 249 . 61 | BVI | 0 . 99 +0 . 15 - 0 . 2 +0 . 5 | BV | 0.53 |\n| NGC1566 NGC1569 | 29.99 27.21 | 4. 9.6 | 2 . 3 - 0 . 00 1 . 00 +0 . 00 - 1 . 00 | 14879 . 65 - 13453 . 3 1683 . 26 +42 . 08 - | 1524 . 64 ± 392 . 62 56 . 34 ± 19 . 34 | 1831 . 74 ± 590 . 48 1242 . 71 ± 658 . 13 | UVI UI | 4 . 58 - 4 . 14 0 . 6 +0 . 02 | BV BV | 1.33 0.37 |\n| NGC1744 | 30.18 | 6.7 | 3 . 3 +0 . 00 - | 42 . 08 113 . 24 +38 . 15 | 271 . 65 ± 110 . 76 | 1960 . 43 ± 541 . 69 | V | - 0 . 01 0 . 63 +0 . 21 | BV | 0.78 |\n| NGC1796 | 30.12 | 5.2 | 3 . 3 2 . 6 +0 . 00 | - 34 . 13 666 . 95 +2067 . 54 | 116 . 07 ± 14 . 94 | 166 . 96 ± 88 . 42 | U | - 0 . 19 0 . 15 +0 . 47 | BV | 0.99 |\n| NGC1809 | 30.83 | 5. | - 2 . 6 4 . 4 +0 . 00 | - 16 . 67 201 . 94 +5 . 05 | 600 . 64 ± 850 . 71 | 952 . 49 ± 394 . 77 | BVI | - 0 . 00 0 . 45 +0 . 01 | BV | 2.4 |\n| NGC1892 | 31.14 | 5.4 | - 0 . 1 1 . 1 +0 . 00 - 0 . 00 | - 5 . 05 738 . 1 +18 . 45 - | 369 . 43 ± 47 . 57 | 1095 . 34 ± 378 . 32 | BVI | - 0 . 01 5 . 92 +0 . 15 | BV | 1.00 |\n| NGC2090 | 30.45 | 4.5 | 11 . 9 +1 . 3 - 0 . 2 | 18 . 45 418 . 79 +141 . 09 - 126 . 22 | 1560 . 47 ± 770 . 21 | 1556 . 28 ± 394 . 18 | V | - 0 . 15 0 . 63 +0 . 21 - 0 . 19 | BV | 2.11 |\n| NGC2139 | 32.41 | 5.9 | 14 . 8 +0 . 4 - 14 . 8 | 2742 . 53 +68 . 56 - 68 . 56 | 1005 . 82 ± 172 . 68 | 942 . 48 ± 282 . 12 | UI | 0 . 6 +0 . 02 - 0 . 01 | BV | 0.54 |\n| NGC2207 | 31.88 | 4.6 | 42 . 4 +0 . 00 - 1 . 6 | 11046 . 82 +404 . 1 - 5719 . 5 | 1977 . 39 ± 1273 . 02 | 1096 . 35 ± 454 . 4 | UBVI | 5 . 85 +0 . 21 - 3 . 03 | BV | 1.3 |\n| NGC2283 | 29.97 | 5.9 | 5 . 1 +0 . 2 - 5 . 1 | 90 . 54 +105 . 44 - 6 . 94 | 1284 . 37 ± 1739 . 8 | 2389 . 38 ± 1045 . 33 | BVI | 0 . 48 +0 . 55 - 0 . 04 | BI | 3.6 |\n| NGC2344 | 31.01 | 4.3 | 20 . 5 +1 . 1 - 20 . 5 | 15095 . 95 +1470 . 7 - 11816 . 5 | 759 . 13 ± 146 . 62 | 359 . 68 ± 91 . 1 | BVI | 3 . 65 +0 . 36 - 2 . 86 | BV | 1.89 |\n| NGC2427 | 30.34 | 7.8 | 3 . 2 +0 . 1 - 0 . 00 | 403 . 64 +135 . 99 - 121 . 66 | 776 . 9 ± 166 . 72 | 1396 . 19 ± 546 . 52 | V | 0 . 63 +0 . 21 | BV | 1.51 |\n| NGC2500 | 30.01 | 7. | 4 . 9 +0 . 2 - 0 . 1 | 136 . 19 +3 . 4 - 3 . 4 | 220 . 08 ± 18 . 89 | 509 . 85 ± 70 . 44 | BI | - 0 . 19 0 . 6 +0 . 02 - | BV | 1.07 |\n| NGC2541 | 30.39 | 6. | 1 . 00 +0 . 1 - 0 . 1 | 47 . 64 +16 . 05 - 14 . 36 | 183 . 92 ± 82 . 88 | 1933 . 63 ± 311 . 66 | V | 0 . 01 0 . 63 +0 . 21 - 0 . 19 | BV | 0.71 |\n| NGC2552 | 30.48 | 9. | 1 . 00 +0 . 00 - 1 . 00 | 266 . 2 +97 . 61 - 6 . 66 | 121 . 67 ± 26 . 11 | 437 . 46 ± 70 . 51 | I | 0 . 45 +0 . 17 - 0 . 01 | BV | 0.67 |\n| NGC2748 | 31.62 | 4. | 2 . 1 +0 . 00 - 2 . 1 | 172 . 98 +58 . 28 - 52 . 14 | 1538 . 1 ± 363 . 08 | 655 . 9 ± 151 . 03 | V | 0 . 63 +0 . 21 - 0 . 19 | BV | 1.81 |\n| NGC2758 | 32.36 | 3.9 | 4 . 9 +0 . 2 - 0 . 00 | 368 . 12 +124 . 02 - 110 . 95 | 228 . 02 ± 90 . 82 | 668 . 23 ± 153 . 87 | V | 0 . 63 +0 . 21 - 0 . 19 | BI | 0.84 |\n| NGC2763 | 32.22 | 5.7 | 2 . 5 +0 . 2 - 2 . 5 | 381 . 25 +139 . 79 - 9 . 53 | 1324 . 29 ± 85 . 26 | 320 . 01 ± 81 . 05 | I | 0 . 45 +0 . 17 | BV | 1.11 |\n| NGC2805 | 32.24 | 6.9 | 1 . 9 +0 . 2 - 1 . 9 | 713 . 2 +261 . 51 - 17 . 83 | 1732 . 13 ± 743 . 42 | 1398 . 63 354 . 25 | I | - 0 . 01 0 . 45 +0 . 17 | BV | 0.78 |\n| NGC2835 | 30.18 | 5. | 3 . 4 +0 . 1 - 0 . 00 | 407 . 00 +926 . 03 - 89 . 37 | 505 . 08 ± 151 . 75 | ± 1855 . 03 ± 555 . 28 | BVI | - 0 . 01 1 . 48 +3 . 36 - | BV | 0.67 |\n| NGC2903 | 29.8 | 4. | 4 . 3 +0 . 00 - 0 . 00 | 966 . 52 +24 . 16 - 24 . 16 | 3148 . 39 ± 675 . 63 | 3057 . 34 ± 492 . 79 | V R | 0 . 32 0 . 45 +0 . 01 | BV | 1.47 |\n| NGC2997 | 30.43 | 5.1 | 9 . 4 +0 . 00 - 0 . 00 | 14179 . 87 +369 . 4 - 510 . 28 | 3623 . 1 ± 699 . 75 | 3369 . 11 ± 853 . 34 | BVI | - 0 . 01 5 . 91 +0 . 15 - 0 . 21 | B | 1.16 |\n| NGC3041 | 32.17 | 5.4 | 10 . 2 +2 . 00 - 10 . 2 | 2130 . 22 +1313 . 1 - | 3297 . 31 ± 424 . 56 | 390 . 67 62 . 97 | BVI | 2 . 03 +1 . 25 - | BV | 2.02 |\n| NGC3184 | 30.49 | 5.9 | 11 . 3 +1 . 3 - 0 . 00 | 1251 . 6 361 . 4 +121 . 76 | 1828 . 2 ± 588 . 49 | ± 1397 . 1 321 . 69 | V | 1 . 2 0 . 63 +0 . 21 - 0 . 19 | BV | 1.15 |\n| NGC3259 | 32.61 | 3.7 | 8 . 4 +0 . 00 - 0 . 00 | - 108 . 93 1621 . 78 +546 . 39 - 488 . 82 | 756 . 54 ± 669 . 39 | ± 854 . 31 ± 354 . 08 | V | 0 . 63 +0 . 21 - 0 . 19 | BI | 1.17 |\n| NGC3274 | 29.51 | 6.6 5.9 | 1 . 5 +0 . 1 - 1 . 5 +0 . 1 | 42 . 87 +12 . 8 - 8 . 97 +4 . 08 | 25 . 48 ± 1 . 64 | 788 . 59 ± 290 . 53 | UBVI | 0 . 55 +0 . 17 - 0 . 12 +0 . 01 | BV | 0.61 |\n| NGC3319 | 30.7 | | 4 . 7 - 0 . 1 | 163 . 39 - 4 . 08 6 | 312 . 97 ± 100 . 74 | 1200 . 02 ± 193 . 42 | V I | 0 . 45 - 0 . 01 +0 . 42 | BV | 0.67 |\n| NGC3338 NGC3344 | 31.98 31.4 | 5.1 4. | 14 . 4 +0 . 00 - 0 . 00 +0 . 00 | 24440 . 74 +2546 . - 778 . 4 +153 . 87 | 2895 . 8 ± 1367 . 14 4032 . 78 ± 692 . 34 | 1989 . 16 ± 503 . 82 2797 . 51 ± 772 . 98 | BVI BV | 4 . 02 - 0 . 13 0 . 45 +0 . 01 - 0 . 01 | BV BV | 1.14 1.13 |\n| NGC3346 | 31.75 | | 16 . 5 - 16 . 5 +0 . 1 | 6154 . 78 - 153 . 87 +85 . 01 | | | | +0 . 17 | | |\n| NGC3359 | | 5.9 | 1 . 1 - 0 . 00 | 231 . 85 - 5 . 8 | 281 . 5 ± 184 . 42 | 325 . 39 ± 89 . 91 | I | 0 . 45 - 0 . 01 | BI | 0.7 |\n| | 31.42 | 5.2 | 6 . 9 +0 . 1 - 0 . 2 | 879 . 26 +21 . 98 - 21 . 98 | 1427 . 88 ± 337 . 06 | 2626 . 18 ± 423 . 29 | BVI | 5 . 92 +0 . 15 - 0 . 15 | BV | 0.8 |\n| NGC3423 | 30.26 | 6. | 2 . 2 +0 . 1 - 0 . 1 | 194 . 11 +14 . 04 - 9 . 43 | 284 . 14 ± 60 . 98 | 763 . 22 ± 246 . 03 | BVI | 1 . 43 +0 . 1 - 0 . 07 | BV | 0.73 |\n| NGC3445 NGC3455 | 31.36 32.55 | 8.9 | 1 . 4 +0 . 2 - 0 . 00 +0 . 1 | 375 . 1 +9 . 38 - 9 . 38 | 160 . 8 ± 179 . 44 | 374 . 68 ± 94 . 9 | UI | 0 . 6 +0 . 02 - 0 . 01 | BV | 0.58 |\n| NGC3621 | 29.14 | 3.7 | 2 . 2 - 0 . 1 +0 . 00 | 639 . 72 +215 . 53 - 192 . 82 | 3717 . 11 ± 2075 . 5 | 329 . 62 ± 75 . 9 | V | 0 . 63 +0 . 21 - 0 . 19 | BI | 7.24 |\n| | 29.81 | 6.9 | 1 . 8 - 1 . 8 +0 . 1 | 657 . 65 +221 . 56 - 198 . 22 | 810 . 66 ± 400 . 12 | 7832 . 33 ± 3246 . 23 | V | 0 . 63 +0 . 21 - 0 . 19 | BV | 1.09 |\n| NGC3631 | | 5.2 | | 2515 . 14 +67 . 92 | 741 . 07 508 . 9 | 720 . 5 165 . 9 | BVI | 1 . 39 +0 . 04 | | 1.15 |\n| | | | 28 . 7 - 0 . 00 +0 . 00 | - 115 . 59 | ± | ± | | - 0 . 06 +0 . 39 | BV | |\n| NGC3666 NGC3756 | 31.43 31.73 | 5.2 4. | 9 . 6 - 0 . 00 2 . 6 +0 . 2 - 0 . 00 | 6423 . 24 +613 . 63 - 4901 . 0 +337 . 34 | 876 . 31 ± 529 . 89 | 815 . 83 ± 131 . 5 | BVI | | BI | 1.98 |\n| NGC3782 | 30.78 | 6.5 | 6 . 00 +0 . 7 - 0 . 5 | 3867 . 77 - 2614 . 6 38 . 41 +7 . 9 - 5 . 43 | 1454 . 92 ± 811 . 78 133 . 36 ± 31 . 67 | 456 . 93 ± 126 . 26 471 . 46 ± 162 . 84 | BVI BVI | 4 . 13 - 3 . 15 3 . 76 +0 . 33 - 2 . 54 0 . 51 +0 . 11 - 0 . 07 | BV BI | 1.33 1.08 | \n| Object (1) | m - M [mag] (2) | t (3) | r eff , NSC [pc] (4) | M NSC [ 10 4 M glyph[circledot] ] (5) | M gal ,glyph[star] [ 10 7 M glyph[circledot] ] (6) | M gal , HI [ 10 7 M glyph[circledot] ] (7) | NSC filter (8) | M/L NSC [ M glyph[circledot] /L glyph[circledot] ] (9) | Galaxy filter (10) | M/L gal [ M glyph[circledot] /L glyph[circledot] ] (11) |\n|-----------------|-------------------|---------|----------------------------------------------|-----------------------------------------------------------|------------------------------------------------------|----------------------------------------------|------------------|-----------------------------------------------------------------|----------------------|-----------------------------------------------------------|\n| NGC3906 | 30.71 | 6.7 | 1 . 5 +0 . 2 - 0 . 1 | 30 . 28 +11 . 1 - 0 . 76 | 29 . 42 ± 22 . 74 | 80 . 81 ± 29 . 77 | I | 0 . 45 +0 . 17 - 0 . 01 | BI | 0.52 |\n| NGC3913 | 31.15 | 6.6 | 24 . 8 +0 . 8 - 1 . 5 | 128 . 95 +35 . 42 - 33 . 85 | 188 . 09 ± 108 . 98 | 239 . 79 ± 60 . 73 | BVI | 0 . 59 +0 . 16 - 0 . 16 | BV | 1.02 |\n| NGC3949 | 31.35 | 4. | 1 . 6 +0 . 00 - 0 . 00 | 619 . 32 +371 . 1 - 46 . 24 | 969 . 8 ± 208 . 12 | 694 . 04 ± 143 . 83 | BVI | 0 . 47 +0 . 28 - 0 . 04 | BV | 0.74 |\n| NGC4030 | 31.91 | 4. | 8 . 5 +0 . 00 | 26624 . 37 +1098 . 7 | 35793 . 98 ± 11341 . 25 | 913 . 62 ± 210 . 37 | BVI | 5 . 82 +0 . 24 | BI | 7.43 |\n| NGC4041 | 31.78 | 4. | - 8 . 5 19 . 2 +0 . 00 | - 5196 . 9 +7422 . 96 | 1913 . 64 492 . 79 | | | - 1 . 14 +3 . 04 | BV | 1.15 |\n| NGC4062 | | 5.3 | - 0 . 00 2 . 5 +0 . 1 | 4618 . 78 - 747 . 4 254 . 68 +85 . 8 | ± 1712 . 36 367 . 47 | 760 . 71 ± 175 . 16 | BVI | 1 . 89 - 0 . 31 0 . 63 +0 . 21 | BV | 2.01 |\n| | 31.1 | 5.3 | - 0 . 00 2 . 8 +0 . 00 | - 76 . 76 110 . 15 +37 . 11 | ± 1076 . 21 438 . 81 | 430 . 96 ± 109 . 15 986 . 48 272 . 57 | V V | - 0 . 19 0 . 63 +0 . 21 - 0 . 19 | BV | 1.34 |\n| NGC4096 | 30.38 | | - 2 . 8 +0 . 1 | - 33 . 2 +5 . 27 | ± | ± | | +0 . 17 | | |\n| NGC4204 | 29.48 | 8. | 0 . 9 - 0 . 00 | 14 . 36 - 0 . 36 | 49 . 2 ± 26 . 92 | 483 . 02 ± 166 . 83 | I | 0 . 45 - 0 . 01 | BI | 2.06 |\n| NGC4208 | 31.19 | 4.9 | 12 . 3 +0 . 00 - 12 . 3 | 497 . 98 +12 . 45 - 12 . 45 | 1425 . 36 ± 336 . 46 | 201 . 81 ± 74 . 35 | UV | 0 . 45 +0 . 01 - 0 . 01 | BV | 1.47 |\n| NGC4214 | 27.43 | 9.8 | 1 . 5 +0 . 00 - 1 . 5 | 102 . 33 +34 . 47 - 30 . 84 | 79 . 57 ± 8 . 54 | 2590 . 38 ± 715 . 75 | V | 0 . 63 +0 . 21 - 0 . 19 | BV | 0.77 |\n| NGC4237 | 31.67 | 4. | 6 . 5 +0 . 00 - 0 . 00 | 1324 . 25 +291 . 43 - 55 . 44 | 2931 . 25 ± 1446 . 78 | 76 . 24 ± 19 . 31 | BVI | 0 . 98 +0 . 22 - 0 . 04 | BV | 2.82 |\n| NGC4242 | 29.1 | 7.9 | 2 . 2 +0 . 00 - 0 . 00 | 50 . 35 +16 . 96 - 15 . 18 | 145 . 33 ± 53 . 02 | 637 . 82 ± 176 . 24 | V | 0 . 63 +0 . 21 - 0 . 19 | BV | 1.02 |\n| NGC4254 | 30.97 | 5.2 | 7 . 6 +0 . 00 - 0 . 00 | 2916 . 02 +72 . 9 - 72 . 9 | 2707 . 32 ± 348 . 59 | 1566 . 01 ± 432 . 7 | UV | 1 . 45 +0 . 04 - 0 . 04 +0 . 21 | BV | 1.04 |\n| NGC4276 | 32.71 | 7.6 | 1 . 7 +0 . 1 - 0 . 3 | 306 . 19 +103 . 16 - 92 . 29 | 355 . 6 ± 264 . 63 | 109 . 21 ± 35 . 21 | V | 0 . 63 - 0 . 19 | BI | 0.78 |\n| NGC4299 | 31.13 | 8.5 | 0 . 9 +0 . 1 - 0 . 00 | 179 . 15 +65 . 69 - 4 . 48 | 196 . 28 ± 126 . 37 | 287 . 92 ± 72 . 93 | I | 0 . 45 +0 . 17 - 0 . 01 | BV | 0.63 |\n| NGC4393 | 30.3 | 6.7 | 2 . 4 +0 . 2 - 2 . 4 | 47 . 47 +9 . 68 - 13 . 12 | 19 . 17 ± 11 . 87 | 549 . 28 ± 215 . 01 | BVI | 0 . 95 +0 . 19 - 0 . 26 | BI | 0.63 |\n| NGC4395 | 28.15 | 8.9 | 1 . 5 +0 . 00 - 1 . 5 | 226 . 72 +302 . 74 - 36 . 54 | 86 . 57 ± 48 . 3 | 3871 . 41 ± 802 . 28 | B | 0 . 7 +0 . 93 - 0 . 11 | BV | 0.78 |\n| NGC4396 | 30.6 | 6.9 | 4 . 6 +0 . 5 - 0 . 4 | 30 . 62 +10 . 32 - 9 . 23 | 112 . 76 ± 24 . 2 | 246 . 81 ± 73 . 88 | V | 0 . 63 +0 . 21 - 0 . 19 | BV | 0.81 |\n| NGC4411b | 31.13 | 6.3 | 4 . 8 +0 . 00 - 0 . 00 +0 . 00 | 357 . 45 +131 . 06 - 8 . 94 +16484 . | 303 . 34 ± 97 . 64 | 282 . 66 ± 117 . 15 | I | 0 . 45 +0 . 17 - 0 . 01 +0 . 93 | BV | 1.14 |\n| NGC4414 NGC4449 | 31.3 27.95 | 5.2 9.8 | 26 . 5 - 26 . 5 5 . 5 +0 . 2 | 12344 . 88 - 1989 . 7 783 . 42 +263 . 94 | 6646 . 18 ± 1141 . 0 192 . 67 152 . 98 | 1037 . 53 ± 286 . 68 1877 . 91 1081 . 01 | B | 0 . 7 - 0 . 11 0 . 63 +0 . 21 | BV | 2.64 0.64 |\n| NGC4487 | 31.46 | 6. | - 0 . 00 1 . 1 +0 . 1 | - 236 . 13 841 . 81 +21 . 05 | ± 2391 . 55 ± 1668 . 16 | ± 649 . 28 179 . 4 | V | - 0 . 19 0 . 45 +0 . 01 | BV | 2.28 |\n| NGC4490 | 29.36 | 7. | - 0 . 1 4 . 3 +0 . 00 - 0 . 2 | - 21 . 05 99 . 54 +33 . 54 - 30 . 00 | 655 . 61 ± 239 . 18 | ± 4172 . 77 ± 1152 . 98 | BVI V | - 0 . 01 0 . 63 +0 . 21 | BI BV | 0.72 |\n| NGC4496A | 30.95 | 7.6 | 0 . 7 +1 . 2 - 0 . 7 | 168 . 31 +4 . 21 - 4 . 21 | 449 . 23 ± 192 . 81 | 721 . 49 ± 232 . 58 | V I | - 0 . 19 1 . 8 +0 . 04 - 0 . 04 | BV | 0.92 |\n| NGC4498 | 31.11 | 6.4 | 3 . 2 +0 . 00 - 3 . 2 | 136 . 14 +45 . 87 - 41 . 03 | 112 . 62 ± 119 . 99 | 206 . 4 ± 71 . 29 | V | 0 . 63 +0 . 21 - 0 . 19 | BI | 0.63 |\n| NGC4504 | 31.64 | 6. | 3 . 9 +0 . 00 - 3 . 8 | 709 . 76 +819 . 46 - 417 . 2 | 3458 . 9 ± 1861 . 6 | 1450 . 06 ± 534 . 22 | BVI | 1 . 06 +1 . 23 - 0 . 63 | BI | 3.23 |\n| NGC4517 | 30.14 | 6. | 2 . 3 +0 . 2 - 2 . 3 | 70 . 14 +23 . 63 - 21 . 14 | 1200 . 59 ± 334 . 94 | 1799 . 69 ± 497 . 27 | V | 0 . 63 +0 . 21 - 0 . 19 | BV | 1.71 |\n| NGC4522 | 31.22 | 6. | 11 . 8 +0 . 2 - 11 . 8 | 132 . 43 +44 . 62 - 39 . 92 | 421 . 92 ± 197 . 64 | 182 . 72 ± 88 . 35 | V | 0 . 63 +0 . 21 - 0 . 19 | BI | 1.67 |\n| NGC4525 | 31.11 | 5.9 | 0 . 9 +0 . 00 - 0 . 00 | 132 . 65 +88 . 61 - 10 . 69 | 323 . 63 ± 330 . 05 | 148 . 15 ± 34 . 11 | BVI | 0 . 48 +0 . 32 | BI | 1.82 |\n| NGC4534 | 30.93 | 7.8 | 4 . 1 +0 . 2 - 4 . 1 | 96 . 4 +33 . 68 - 32 . 15 | 123 . 77 ± 99 . 61 | 976 . 5 ± 157 . 39 | BVI | - 0 . 04 0 . 7 +0 . 24 - | BI | 0.99 |\n| NGC4540 | 31.13 | 6.2 | 3 . 2 +0 . 00 - 3 . 2 | 151 . 78 +55 . 65 - 3 . 79 | 843 . 45 ± 181 . 00 | 102 . 63 ± 51 . 99 | I | 0 . 23 0 . 45 +0 . 17 - 0 . 01 | BI | 2.01 |\n| NGC4559 | 29.63 | 6. | 1 . 3 +0 . 00 - 0 . 00 | 179 . 15 +4 . 48 - 4 . 48 | 543 . 47 ± 163 . 28 | 3600 . 6 ± 994 . 88 | BVI | 0 . 45 +0 . 01 - 0 . 01 | BV | 0.75 |\n| NGC4567 | 31.9 | 4. | 2 . 1 +0 . 00 - 2 . 00 | 4385 . 23 +1477 . 4 - 1321 . 7 | 2819 . 97 ± 1210 . 31 | 380 . 62 ± 210 . 34 | V | 0 . 63 +0 . 21 - 0 . 19 | BV | 1.9 |\n| NGC4571 | 31.09 | 6.5 | 2 . 3 +0 . 00 - 0 . 1 | 824 . 36 +20 . 61 - 20 . 61 | 558 . 88 ± 323 . 82 | 215 . 85 ± 64 . 61 | V I | 5 . 92 +0 . 15 - 0 . 15 | BV | 0.84 |\n| NGC4592 | 30.13 | 8. | 1 . 1 +0 . 1 - 0 . 1 | 63 . 39 +21 . 36 | 104 . 06 82 . 65 | 1900 . 68 ± 656 . 47 | V | 0 . 63 +0 . 21 | BI | 0.98 |\n| NGC4595 | 31.12 | 3.8 | 0 . 3 +1 . 4 - 0 . 2 | - 19 . 1 227 . 62 +5 . 69 | ± 207 . 69 107 . 01 | | BVI | - 0 . 19 0 . 45 +0 . 01 | BI | 1.03 |\n| NGC4597 | 31.15 | 8.7 | 3 . 4 +0 . 1 - 0 . 2 | - 5 . 69 176 . 19 +59 . 36 - | ± 349 . 52 ± 127 . 51 | 135 . 2 ± 34 . 25 801 . 36 313 . 68 | V | - 0 . 01 0 . 63 +0 . 21 | BI | 1.42 |\n| NGC4618 | 29.31 | 8.7 | 4 . 2 +0 . 00 - 0 . 2 | 53 . 11 268 . 26 +6 . 71 - 6 . 71 | 163 . 2 ± 52 . 53 | ± 1083 . 79 ± 349 . 37 | BI | - 0 . 19 10 . 58 +0 . 26 - 26 | BV | 0.77 |\n| NGC4625 | 29.57 | 8.8 | 8 . 9 +0 . 00 - 8 . 9 | 265 . 6 +6 . 64 - 6 . 64 | 67 . 24 ± 14 . 43 | 423 . 27 ± 116 . 95 | V I | 0 . 3 . 87 +0 . 1 - 0 . 1 | BV | 1.11 |\n| NGC4631 | 29.19 | 6.6 | 5 . 6 +0 . 2 - 0 . 1 | 63 . 39 +21 . 36 - 19 . 1 | 1207 . 2 ± 647 . 65 | | | 0 . 63 +0 . 21 | BV | 1.07 |\n| NGC4634 | 27.73 | 5.9 | 1 . 5 +0 . 1 - 1 . 5 | 2 . 45 +0 . 83 - 0 . 74 | 25 . 06 ± 9 . 14 | 7036 . 88 ± 2268 . 42 119 . 9 30 . 37 | V | - 0 . 19 0 . 63 +0 . 21 - | BV | 2.05 |\n| NGC4635 | 31.86 | 6.6 | 4 . 3 +0 . 4 - 0 . 3 | 617 . 53 +271 . 62 - 375 . 16 | 103 . 00 ± 82 . 84 | ± 141 . 71 ± 39 . 16 | V BVI | 0 . 19 2 . 28 +1 . 00 - 1 . 39 | BI | 0.56 |\n| NGC4639 | 31.7 | 3.5 | 10 . 3 +1 . 5 - 0 . 00 | 1122 . 41 +28 . 06 - 28 . 06 +7189 . 5 | 1849 . 56 ± 238 . 15 | 339 . 65 ± 109 . 49 | UB | 0 . 6 +0 . 02 - 0 . 01 | BV | 1.7 |\n| NGC4651 | 32.22 | 5.2 | 29 . 4 +0 . 3 - 0 . 00 | 65060 . 9 - 59834 . 4 | 3518 . 39 ± 679 . 53 | 1099 . 42 ± 278 . 47 | BVI | 5 . 46 +0 . 6 - 5 . 02 | BV | 1.08 |\n| NGC4656 NGC4700 | 28.99 | 9. | 2 . 00 +0 . 1 - 0 . 1 +0 . 00 | 144 . 94 +81 . 44 - 3 . 62 +90 . 68 | 442 . 54 ± 617 . 29 | 3779 . 27 ± 1131 . 27 | V I | 3 . 88 +2 . 18 - 0 . 1 | BV | 0.74 |\n| NGC4701 | 30.6 | 4.9 | 1 . 00 - 0 . 00 +0 . 00 | 269 . 15 - 81 . 12 +1463 . 81 | 194 . 67 ± 93 . 88 | 417 . 21 ± 124 . 89 | V | 0 . 63 +0 . 21 - 0 . 19 | BI | 0.76 |\n| | 32. | 5.9 | 2 . 4 - 0 . 00 | 3992 . 2 - 99 . 81 | 408 . 71 ± 26 . 31 | 807 . 63 ± 334 . 73 | I | 0 . 45 +0 . 17 - 0 . 01 | BV | 0.66 |\n| NGC4771 | 31.45 | 6.2 | 1 . 9 +0 . 1 | 428 . 64 +1328 . 78 - 10 . 72 | 2070 . 37 ± 1059 . 81 | 291 . 17 ± 73 . 75 | U | 0 . 15 +0 . 47 - 0 . 00 | BI | 4.17 |\n| NGC4775 | 32.12 | 6.9 | - 0 . 4 2 . 2 +0 . 00 - 0 . 2 | 1679 . 63 +615 . 86 | 2650 . 88 ± 2400 . 74 | 557 . 78 ± 179 . 81 | I | 0 . 45 +0 . 17 | BI | 1.34 |\n| NGC4781 | 30.83 | 7. | 4 . 9 +0 . 2 - 0 . 2 | - 41 . 99 149 . 28 +50 . 29 - 44 . 99 | 829 . 41 ± 374 . 2 | 652 . 92 ± 255 . 58 | V | - 0 . 01 0 . 63 +0 . 21 | BI | 1.18 |\n| NGC4790 | 31.87 | 4.8 | 5 . 00 +0 . 2 - 0 . 00 | 10142 . 65 +253 . 6 - 253 . 57 | 1683 . 52 ± 1749 . 13 | 343 . 3 118 . 57 | BI | - 0 . 19 0 . 6 +0 . 02 - 0 . 01 | BI | 2.37 |\n| NGC4806 | 32.45 | 4.9 | 2 . 5 +0 . 5 - 0 . 00 | 289 . 73 +97 . 61 - 87 . 33 | 1416 . 21 ± 537 . 25 | ± 173 . 52 95 . 89 | V | 0 . 63 +0 . 21 - 0 . 19 | BI | 2.04 |\n| NGC4861 | | 8.9 | 2 . 7 +0 . 2 - 0 . 4 | 13 . 74 +4 . 63 - | 176 . 15 189 . 00 | ± 597 . 54 178 . 87 | V | +0 . | B | |\n| | 30.32 | | | | | | | 0 . 63 21 - 0 . 19 | | 1.16 |\n| NGC4900 | 31.86 | 5.2 | 3 . 1 +0 . 1 - 3 . 00 | 4 . 14 1726 . 68 +43 . 17 - 43 . 17 | ± 1314 . 24 282 . 03 | ± 349 . 46 ± 72 . 42 | | BVI +0 . 01 | BV | 0.96 |\n| NGC4904 NGC4942 | 31.68 32.68 | 5.8 6.9 | 2 . 5 +0 . 00 - 0 . 2 12 . 1 +0 . 3 - 0 . 00 | 171 . 09 +62 . 73 - 4 . 28 1374 . 02 +462 . 91 - 414 . 14 | ± 837 . 04 ± 341 . 29 2422 . 53 ± 1427 . 51 | 257 . 33 ± 106 . 65 207 . 72 ± 100 . 44 | I V | 0 . 45 - 0 . 01 0 . 45 +0 . 17 - 0 . 01 0 . 63 +0 . 21 - 0 . 19 | BV BI | 1.31 3.2 | \n22 I. Y. Georgiev, T. Böker, N. Leigh, N. Lützgendorf, N. Neumayer \nTable A1 (cont'd) \n| Object | m - M [mag] | t | r eff , NSC [pc] | M NSC [ 10 4 M glyph[circledot] ] (5) | M gal ,glyph[star] [ 10 7 M glyph[circledot] ] (6) | M gal , HI [ 10 7 M glyph[circledot] ] | NSC filter | M/L NSC [ M glyph[circledot] /L glyph[circledot] ] | Galaxy filter | M/L gal [ M glyph[circledot] /L glyph[circledot] ] (11) |\n|-------------------|---------------|---------|------------------------------------|---------------------------------------------|------------------------------------------------------|------------------------------------------|--------------|------------------------------------------------------|-----------------|-----------------------------------------------------------|\n| (1) | (2) 31.35 | (3) | (4) 8 . 3 +0 . 8 | 1042 . 3 +351 . 16 | 2685 . 77 ± 749 . 26 482 . 79 165 . 77 | (7) | (8) | (9) 0 . 63 +0 . 21 | (10) | 1.7 |\n| NGC5054 | | 4.2 | - 0 . 00 +0 . 1 | - 314 . 16 +1 . 33 | | 433 . 9 ± 99 . 91 | V | - 0 . 19 +0 . 01 | BV | |\n| NGC5068 NGC5112 | 28.9 32.15 | 6. 5.8 | 4 . 9 - 0 . 2 6 . 6 +0 . 3 - 1 . 3 | 53 . 11 - 1 . 33 130 . 98 +3 . 27 - 3 . 27 | ± 664 . 3 ± 171 . 07 | 2045 . 07 ± 612 . 16 793 . 00 ± 200 . 85 | BVI BVI | 0 . 45 - 0 . 01 0 . 45 +0 . 01 - 0 . 01 | BV BV | 1.2 0.79 |\n| NGC5204 | 28.38 | 8.9 | 0 . 4 +0 . 00 - | 22 . 7 +3 . 39 - | 38 . 71 ± 4 . 15 | 1455 . 25 ± 301 . 57 | BVI | 0 . 8 +0 . 12 - 0 . 03 | BV | 0.67 |\n| NGC5264 | 28.27 | 9.7 | 0 . 00 0 . 4 +0 . 00 | 0 . 81 58 . 74 +1 . 47 | 34 . 12 ± 4 . 39 | 172 . 01 ± 43 . 57 | V I | 1 . 18 +0 . 03 | BV | 1.11 |\n| NGC5300 | 31.82 | 5.2 | - 0 . 00 20 . 6 +0 . 00 - 6 | - 1 . 47 1632 . 47 +46 . 78 | 924 . 83 ± 1480 . 2 | 295 . 33 61 . 2 | BVI | - 0 . 03 0 . 96 +0 . 03 - 0 . 06 | BI | 2.06 |\n| | | | 20 . +0 . 4 | - 103 . 23 +174 . 51 | | ± | | 0 . 76 +0 . 19 | BI | |\n| NGC5334 | 32.78 | 5.2 | 11 . 9 - 0 . 7 | 708 . 96 - 297 . 61 | 615 . 6 ± 894 . 66 | 520 . 15 ± 143 . 72 | BVRI | - 0 . 32 +0 . | | 0.76 |\n| NGC5398 | 29.6 | 7.9 | 1 . 1 +0 . 1 - 1 . 1 | 107 . 78 +16 . 27 - 75 . 1 | 2636 . 58 ± 2400 . 82 | 383 . 26 ± 105 . 9 | UBVI | 1 . 45 22 - 1 . 01 | BI | 2.96 |\n| NGC5427 | 32.62 | 5. | 8 . 6 +0 . 00 - 8 . 6 | 4897 . 71 +1650 . 1 - 1476 . 2 | 2979 . 75 ± 767 . 33 | 981 . 48 ± 203 . 39 | V | 0 . 63 +0 . 21 - 0 . 19 | BV | 1.07 |\n| NGC5474 | 28.9 | 6.1 | 3 . 9 +0 . 2 - 0 . 3 | 13 . 75 +0 . 34 - 0 . 34 | 118 . 19 ± 63 . 41 | 1732 . 64 ± 518 . 64 | BI | 0 . 6 +0 . 02 - 0 . 01 | BV | 0.93 |\n| NGC5530 | 30.35 | 4.2 | 2 . 6 +0 . 00 - 0 . 00 | 511 . 44 +30 . 05 - 193 . 4 | 2254 . 01 ± 581 . 45 | 782 . 04 ± 252 . 1 | BVI | 1 . 57 +0 . 09 - 0 . 59 | BI | 3.59 |\n| NGC5556 | 31.36 | 6.7 | 2 . 3 +0 . 5 - 0 . 1 | 159 . 38 +43 . 77 - 41 . 83 | 581 . 34 ± 237 . 03 | 720 . 55 ± 232 . 28 | BVI | 0 . 59 +0 . 16 - 0 . 16 | BV | 1.23 |\n| NGC5584 | 32.01 | 6. | 9 . 1 +0 . 4 - 9 . 1 | 177 . 89 +4 . 45 - 4 . 45 | 342 . 15 ± 466 . 96 | 548 . 89 ± 126 . 39 | UI | 0 . 6 +0 . 02 - 0 . 01 | BI | 0.72 |\n| NGC5585 | 29.7 | 6.9 | 1 . 5 +0 . 1 - 0 . 00 | 77 . 48 +1 . 94 - 1 . 94 | 217 . 54 ± 65 . 36 | 1747 . 53 ± 402 . 38 | V I | 0 . 45 +0 . 01 - 0 . 01 | BV | 0.78 |\n| NGC5668 | 32.15 | 6.9 | 2 . 9 +0 . 1 - 0 . 00 | 377 . 76 +9 . 44 - 9 . 44 | 1357 . 65 ± 262 . 21 | 697 . 07 ± 224 . 71 | V I | 0 . 45 +0 . 01 - 0 . 01 | BV | 0.98 |\n| NGC5669 | 31.99 | 6. | 4 . 5 +0 . 6 - 4 . 5 | 45 . 42 +16 . 65 - 1 . 14 | 974 . 93 ± 1642 . 38 | 716 . 05 ± 164 . 88 | I | 0 . 45 +0 . 17 - 0 . 01 | BI | 1.79 |\n| NGC5774 | 32.41 | 6.9 | 5 . 6 +0 . 9 - 1 . 00 | 91 . 46 +33 . 53 - 2 . 29 | 969 . 02 ± 1060 . 54 | 658 . 07 ± 212 . 14 | I | 0 . 45 +0 . 17 - 0 . 01 | BV | 1.14 |\n| NGC5789 | 32.07 32.18 | 7.8 6.9 | 2 . 8 +0 . 3 - 2 . 8 2 . 4 +0 . 4 | 77 . 48 +28 . 41 - 1 . 94 363 . 3 +183 . 73 | 299 . 97 ± 296 . 12 44397 . 45 40623 . 54 | 150 . 35 ± 45 . 01 | I | 0 . 45 +0 . 17 - 0 . 01 0 . 54 +0 . 27 | B | 1.16 2.68 |\n| NGC5964 NGC5970 | 32.36 | 5. | - 0 . 1 15 . 6 +0 . 00 | - 68 . 03 645 . 64 +217 . 52 | ± 4062 . 3 3399 . 85 | 704 . 83 ± 113 . 61 | BVI V | - 0 . 1 0 . 63 +0 . 21 | BI BV | 1.63 |\n| NGC6384 | 32.03 | 3.6 | - 0 . 00 15 . 9 +0 . 5 | - 194 . 6 1264 . 71 +426 . 09 19 | ± 4275 . 67 2293 . 86 | 360 . 52 ± 99 . 62 1247 . 46 258 . 51 | V | - 0 . 19 0 . 63 +0 . 21 | BV | 1.33 |\n| NGC6412 | 31.86 | 5.2 | - 0 . 00 2 . 00 +0 . 1 | - 381 . 399 . 22 +146 . 38 | ± 847 . 97 127 . 38 | ± 439 . 95 81 . 04 | I | - 0 . 19 0 . 45 +0 . 17 | BV | 0.91 |\n| NGC6503 | 28.66 | 5.8 | - 0 . 00 3 . 3 +0 . 00 - | - 9 . 98 97 . 72 +32 . 92 | ± 366 . 61 ± 70 . 81 | ± 1430 . 42 ± 395 . 24 | V | - 0 . 01 0 . 63 +0 . 21 | BV | 1.62 |\n| NGC6509 | 32.25 | 6.5 | 3 . 3 1 . 3 +1 . 4 - 0 . 4 | - 29 . 45 725 . 9 +18 . 15 - 18 . 15 | 1265 . 55 ± 869 . 07 | 512 . 83 ± 129 . 89 | V I | - 0 . 19 0 . 94 +0 . 02 - 0 . 02 | B | 1.16 |\n| NGC7090 | 29.7 | 5. | 1 . 1 +0 . 00 - 1 . 1 | 302 . 52 +126 . 25 - 223 . 1 | 435 . 67 ± 84 . 14 | 1536 . 12 ± 636 . 67 | BVI | 2 . 61 +1 . 09 - 1 . 93 | BV | 1.24 |\n| NGC7162 | 32.79 | 4.8 | 13 . 6 +0 . 00 | 695 . 01 +234 . 15 | 1138 . 66 219 . 92 | 377 . 05 147 . 59 | | 0 . 63 +0 . 21 | BV | 1.2 |\n| NGC7188 | 32.01 | 3.5 | - 13 . 6 4 . 2 +0 . 2 | - 209 . 48 512 . 85 +172 . 78 | ± 1330 . 59 453 . 73 | ± 36 . 26 7 . 52 | V | - 0 . 19 +0 . 21 | BI | 4.07 |\n| NGC7421 | 31.95 | 3.7 | - 0 . 00 13 . 6 +0 . 00 - 13 . 5 | - 154 . 58 1745 . 79 +588 . 17 - 526 . 19 | ± 1128 . 63 ± 193 . 76 | ± 137 . 36 ± 63 . 26 | V V | 0 . 63 - 0 . 19 0 . 63 +0 . 21 - 0 . 19 | BV | 1.36 |\n| NGC7424 | 30.3 | 6. | 6 . 8 +0 . 3 | 100 . 28 +2 . 51 - 2 . 51 | 567 . 91 ± 280 . 31 | 3249 . 35 897 . 83 | BVI | 0 . 45 +0 . 01 | BV | 0.84 |\n| NGC7462 | 30.57 | 3.6 | - 0 . 00 3 . 6 +0 . 2 | 454 . 98 +153 . 29 - 137 . 13 | 1330 . 71 ± 549 . 3 | ± 524 . 21 156 . 91 | V | - 0 . 01 0 . 63 +0 . 21 | BI | 3.49 |\n| NGC7689 | 32.05 | 5.9 | - 0 . 00 5 . 9 +0 . 2 - 0 . 00 | 1130 . 35 +414 . 46 - 28 . 26 | 3886 . 05 ± 719 . 8 | ± 738 . 87 ± 289 . 22 | I | - 0 . 19 0 . 45 +0 . 17 - 0 . 01 | BI | 2.72 |\n| NGC7713 | 30.07 | 6.7 | 1 . 4 +0 . 00 - 1 . 4 | 115 . 14 +2 . 88 | 155 . 89 73 . 6 | 789 . 18 ± 254 . 4 | UBVI | +0 . 01 | BV | 0.49 |\n| NGC7741 | 30.89 | 5.9 | 3 . 1 +0 . 7 - 3 . 1 | - 2 . 88 128 . 59 +3 . 21 | ± 574 . 29 271 . 13 | 832 . 81 ± 172 . 59 | BV | 0 . 45 - 0 . 01 0 . 45 +0 . 01 | BV | 0.82 |\n| UGC02302 | 30.73 | 8.8 | 1 . 8 +0 . 1 - 1 . 8 | - 3 . 21 51 . 77 +1 . 29 - 1 . 29 | ± 112 . 17 ± 98 . 69 | 912 . 09 ± 189 . 01 | V I | - 0 . 01 1 . 8 +0 . 04 - 0 . 04 | B | 1.16 |\n| UGC03574 | 31.47 | 5.8 | 3 . 9 +0 . 2 - 0 . 2 | 410 . 27 +382 . 53 - | 40089 . 42 ± 32447 . 31 | 680 . 31 ± 109 . 65 | BVI | 2 . 3 +2 . 14 | BI | 1.60 |\n| UGC03698 | 29.3 | 9.8 | 1 . 8 +0 . 1 - 0 . 00 | 250 . 14 3 . 23 +1 . 18 | | | I | - 1 . 4 0 . 45 +0 . 17 | B | 1.16 |\n| UGC03755 | 28.49 | 9.9 | 0 . 4 +0 . 3 - | - 0 . 08 5 . 99 +0 . 15 | 9 . 57 ± 5 . 34 5 . 62 5 . 91 | 109 . 31 ± 55 . 37 | V I | - 0 . 01 0 . 45 +0 . 01 | BV | 0.85 |\n| UGC03826 | 32.17 | 6.6 | 0 . 00 5 . 5 +0 . 3 - | - 0 . 15 109 . 95 +40 . 32 | ± 204 . 87 ± 180 . 25 | 159 . 33 ± 55 . 03 452 . 7 ± 83 . 39 | I | - 0 . 01 0 . 45 +0 . 17 | B | 1.16 |\n| UGC03860 | 29.3 | 9.8 | 5 . 5 1 . 4 +0 . 00 - 0 . 1 | - 2 . 75 4 . 18 +1 . 53 - 0 . 10 | 4 . 78 ± 0 . 72 | 183 . 08 ± 42 . 16 | I | - 0 . 01 0 . 45 +0 . 17 - 0 . | BV | 0.69 |\n| UGC04499 | 30.54 | 8. | 3 . 00 +0 . 4 | 13 . 34 +4 . 89 | 0 . 03 0 . 17 | | I | 01 0 . 45 +0 . 17 | | |\n| UGC04904 | 31.8 | 4.3 | - 3 . 00 2 . 4 +0 . 6 | - 0 . 33 112 . 57 +38 . 38 | ± 94 . 51 70 . 99 | 419 . 42 ± 106 . 23 | | - 0 . 01 4 . 52 +1 . 54 | BI | 0.02 |\n| UGC04988 | 31.63 | 9. | - 2 . 4 1 . 4 +0 . 2 - 0 . 1 | - 71 . 09 81 . 89 +30 . 03 - 2 . 05 | ± 42 . 3 ± 18 . 48 | 0 . 00 ± 0 . 00 46 . 68 ± 32 . 24 | BVRI I | - 2 . 86 0 . 45 +0 . 17 | B BI | 1.16 1.27 |\n| UGC05015 | 31.73 | 7.9 | 9 . 9 +0 . 4 - 9 . 9 +2 . 3 | 87 . 34 +32 . 02 - 2 . 18 +12 . 59 | 22 . 16 ± 14 . 3 | 135 . 47 ± 81 . 1 | I | - 0 . 01 0 . 45 +0 . 17 | BI | 0.87 |\n| UGC05189 | 33.13 1 | 0.0 | 39 . 6 - 3 . 6 | 503 . 68 - 12 . 59 | 378 . 12 ± 405 . 71 | 650 . 66 ± 494 . 41 | UI | - 0 . 01 0 . 6 +0 . 02 - 0 . 01 | BV | 0.6 |\n| UGC05428 UGC05692 | 27.86 28. | 9.8 | 2 . 5 +0 . 00 - 0 . 1 +0 . 00 | 6 . 84 +0 . 17 - 0 . 17 +0 . 53 | 1 . 64 ± 1 . 13 32 . 92 14 . 13 | 50 . 45 ± 54 . 6 0 . 00 0 . 00 | V I V I | 1 . 29 +0 . 03 - 0 . 03 0 . 94 +0 . 02 | B | 1.16 |\n| UGC05889 | | 8.8 | 0 . 3 - 0 . 00 +0 . 00 | 21 . 32 - 0 . 53 +3 . 06 | ± | ± | | - 0 . 02 +0 . 15 | BV | 2.56 |\n| | 29.19 | 8.9 | 1 . 00 - 0 . 00 | 122 . 49 - 3 . 06 | 15 . 75 ± 3 . 72 | 103 . 6 ± 28 . 63 | V I | 5 . 92 - 0 . 15 | BV | 1.2 |\n| UGC06192 | 32.3 | 3.5 | 2 . 9 +0 . 1 - | 147 . 64 +54 . 14 - | 29 . 64 ± 12 . 37 | 0 . 00 ± 0 . 00 | I | 0 . 45 +0 . 17 - 0 . 01 | BI | 0.83 |\n| UGC06931 | 31.23 | 9. | 2 . 9 1 . 9 +0 . 7 - 0 . 4 | 3 . 69 11 . 3 +4 . 14 - | 47 . 59 ± 31 . 35 | 106 . 18 ± 46 . 45 | I | 0 . 45 +0 . 17 - 01 | BI | 1.09 |\n| UGC06983 | 31.64 | 5.9 | 1 . 5 +4 . 3 - 0 . 2 | 0 . 28 60 . 24 +4 . 54 - 3 . 46 | 244 . 09 ± 254 . 09 | 598 . 94 ± 165 . 49 | BVI | 0 . 0 . 47 +0 . 04 - | BI | 1.07 |\n| UGC07943 UGC08041 | 31.49 | 6. | 6 . 6 +0 . 3 - 6 . 6 | 43 . 45 +14 . 64 - 13 . 1 | 217 . 31 ± 55 . 96 | 174 . 28 ± 72 . 23 | V | 0 . 03 0 . 63 +0 . 21 - 0 . 19 | B | 1.16 |\n| | 31.17 | 6.9 | 10 . 3 +0 . 4 - 0 . 1 | 546 . 03 +13 . 65 - 13 . 65 | 67 . 79 ± 105 . 77 | 373 . 58 ± 120 . 43 | BVI | 0 . 45 +0 . 01 - 0 . 01 | BI | 0.7 |\n| UGC08085 | 32.35 | 5.8 | 9 . 9 +2 . 7 - 1 . 4 | 106 . 17 +35 . 77 - 32 . 00 | 929 . 76 619 . 81 | 347 . 26 ± 119 . 94 | V | 0 . 63 +0 . - 0 . 19 | BI | 3.92 |\n| | | | | | | | | 21 | | |\n| UGC08516 | 31.83 | 5.9 | 11 . 2 +0 . 3 - 0 . 00 +2 . 3 | 236 . 16 +86 . 59 - 5 . 9 | ± 145 . 77 78 . 55 | 90 . 07 ± 20 . 74 | | I +0 . 17 | BI | 1.02 |\n| UGC09215 | 32.06 | 6.3 | 0 . 8 - 0 . 00 +0 . 1 | 1738 . 62 +911 . 3 - 1262 . 07 +6 . 4 | ± 595 . 19 ± 702 . 9 ± | 395 . 22 ± 145 . 6 ± | BVI | 0 . 45 - 0 . 01 2 . 63 +1 . 38 - 1 . 91 +0 . 1 | BI | 1.98 | \nTable A1 (cont'd) \n| Object (1) | m - M [mag] (2) | t (3) | r eff , NSC [pc] (4) | M NSC [ 10 4 M glyph[circledot] ] (5) | M gal ,glyph[star] [ 10 7 M glyph[circledot] ] (6) | M gal , HI [ 10 7 M glyph[circledot] ] (7) | NSC filter (8) | M/L NSC [ M glyph[circledot] /L glyph[circledot] ] (9) | Galaxy filter (10) | M/L gal [ M glyph[circledot] /L glyph[circledot] ] (11) |\n|--------------|-------------------|---------|-------------------------|-----------------------------------------|------------------------------------------------------|----------------------------------------------|------------------|----------------------------------------------------------|----------------------|-----------------------------------------------------------|\n| UGC12732 | 30.46 | 8.7 | 3 . 2 +0 . 1 - 0 . 1 | 76 . 77 +28 . 15 - 1 . 92 | 70 . 19 ± 61 . 76 | 1028 . 9 ± 213 . 22 | I | 0 . 45 +0 . 17 - 0 . 01 | B | 1.16 |\n| UGCA086 | 27.12 | 10 | 5 . 6 +0 . 00 - 5 . 6 | 74 . 68 +1 . 87 - 1 . 87 | 113 . 31 ± 121 . 58 | 3072 . 5 ± 1131 . 95 | V I | 0 . 45 +0 . 01 - 0 . 01 | B | 1.16 |\n| UGCA196 | 31.44 | 6.3 | 34 . 5 +0 . 00 - 34 . 5 | 392 . 64 +132 . 28 - 118 . 34 | 265 . 69 ± 83 . 02 | 751 . 41 ± 224 . 92 | V | 0 . 63 +0 . 21 - 0 . 19 | BI | 1.2 |\n| UGCA200 | 29.01 | 10 | 1 . 4 +0 . 00 - 1 . 4 | 8 . 98 +0 . 22 - 0 . 22 | 2 . 61 ± 2 . 07 | 0 . 00 ± 0 . 00 | V I | 0 . 45 +0 . 01 - 0 . 01 | B | 1.16 | \nNote. - Columns: (1) NSC host galaxy ID, (2) Adopted distance modulus and (3) galaxy morphological code ((1-2-3) are the same as in Georgiev & Böker 2014); (4) NSC effective radius measured in F 606 W or F 814 W or F 450 W or F 300 W , as available in this order. We note that differences in sizes in the different bands are < 10% (see Georgiev & Böker 2014, for discussion); (5) NSC mass calculated using the M/L given in (9) for the F 606 W (V) or F 814 W (I) or F 450 W (B) or F 300 W (U) filter in this priority order, e.g. if Colour is UBVI , the value is the M/L V error weighted average from the M/L V s from the various colour combinations containing V -band. NSC host galaxy stellar and HI mass (6 and 7), where the used colour and M/L B are in (10) and (11)."}

2005MNRAS.359..801K

Observations of the Blandford-Znajek process and the magnetohydrodynamic Penrose process in computer simulations of black hole magnetospheres

2005-01-01

['black hole physics', 'stars magnetic fields', 'methods numerical', 'astrophysics']

[]

In this paper we report the results of axisymmetric relativistic magnetohydrodynamic (MHD) simulations for the problem of a Kerr black hole immersed in a rarefied plasma with `uniform' magnetic field. The long-term solution shows properties that are significantly different from those of the initial transient phase studied recently by Koide. The topology of magnetic field lines within the ergosphere is similar to that of the split-monopole model with a strong current sheet in the equatorial plane. Closer inspection reveals a system of isolated magnetic islands inside the sheet and ongoing magnetic reconnection. No regions of negative hydrodynamic `energy at infinity' are seen inside the ergosphere and the so-called MHD Penrose process does not operate. However, the rotational energy of the black hole continues to be extracted via the purely electromagnetic Blandford-Znajek mechanism. In spite of this, no strong relativistic outflows from the black hole are seen to be developing. Combined with results of other recent simulations, our results signal a potential problem for the standard MHD model of relativistic astrophysical jets should they be found at distances as small as a few tens of gravitational radii from the central black hole.

[]

https://arxiv.org/pdf/astro-ph/0501599.pdf

{'S.S.Komissarov': 'Department of Applied Mathematics, The University of Leeds, Leeds LS2 9JT \n9 March 2022', 'ABSTRACT': "In this paper we report the results of axisymmetric relativistic MHD simulations for the problem of Kerr black hole immersed into a rarefied plasma with 'uniform' magnetic field. The long term solution shows properties which are significantly different from those of the initial transient phase studied recently by Koide(2003). The topology of magnetic field lines within the ergosphere is similar to that of the splitmonopole model with a strong current sheet in the equatorial plane. Closer inspection reveals a system of isolated magnetic islands inside the sheet and ongoing magnetic reconnection. No regions of negative hydrodynamic 'energy at infinity' are seen inside the ergosphere and the so-called MHD Penrose process does not operate. Yet, the rotational energy of the black hole continues to be extracted via purely electromagnetic mechanism of Blandford and Znajek(1977). However, this is not followed by development of strong relativistic outflows from the black hole. Combined with results of other recent simulations this signals a potential problem for the standard MHD model of relativistic astrophysical jets should they still be observed at distances as small as few tens of gravitational radii from the central black hole. \nKey words: black hole physics - magnetic fields - methods:numerical.", '1 INTRODUCTION': "Observations of various astrophysical phenomena such as active galactic nuclei and galactic microquasars often reveal powerful relativistic jets streaming away from a massive central object, most likely a black hole. The exact mechanism of generating powerful relativistic jets from black holes system is not yet known, although a number of interesting models have been put forward and studied with various degree of detail during the last few decades. \nBy now there has emerged a general consensus that the central engine should involve a rotating black hole linked with the jets by means of strong magnetic field and an accretion disc supporting this magnetic field by its electric currents. This magnetic field is believed to serve a number of key functions: 1) to power the jets via extracting the rotational energy of the black hole; 2) to provide the jet collimation via magnetic hoop stress; and 3) to suppress mixing of the rarefied jet plasma with the relatively dense surrounding medium such as the coronal plasma of the accretion disc, which is needed to ensure that the jet plasma remains magnetically dominated and, thus, can be accelerated to the ultra-relativistic speeds inferred from the observations. \nThe most simplified mathematical frameworks for \nhighly magnetised relativistic plasma are an ideal relativistic magnetohydrodynamics (MHD) and magnetodynamics (MD, which can be described as MHD in the limit of zero particle inertia, e.g Komissarov 2002). Both frameworks, however, involve rather complex equations particularly in the case of curved spacetime. This explains a relatively slow progress in the theory - only a limited number of analytical solutions have been found so far and only for the limiting case of a slowly rotating black hole where one can employ the perturbation method. The most important of them is the MD solution for a monopole magnetosphere by Blandford and Znajek (1977). Indeed, this solution describes an outgoing Poynting flux from a Kerr black hole as if the hole was a magnetized rotating conductor whose rotational energy can be efficiently extracted by means of magnetic torque. This most important feature of the Blandford-Znajek solution (BZ) is also the most intriguing and puzzling as such conductor does not really exist. \nOften, this missing element of the BZ model is artificially introduced in the form of the so-called 'membrane', or the 'stretched horizon', located somewhat above the real event horizon (Macdonald & Thorne 1982; Thorne et al.1986). This 'physical' interpretation of the BZ \nsolution provides simple means of communicating the results to wide astrophysical community and for this reason it has been almost universally accepted. However, its artificial nature and the fact that no proper physical interpretation had been pushed forward also made the result vulnerable to criticism on theoretical grounds and stimulated attempts to find alternative ways of magnetic extraction of rotational energy of black holes (Punsly & Coroniti 1990a; Punsly 2001; Takahashi et al.1990). These attempts seem to draw the inspiration from a completely non-magnetic mechanism proposed much earlier by Penrose (1969). The key role in this mechanism is played by the ergosphere of a rotating black hole. Within the ergosphere particles can acquire negative energy (or rather 'energy at infinity') and if such particles are swallowed by the black hole its energy can decrease. In the original Penrose process such particles are created via close range interaction (collisions, decay) with other particles which gain positive energy and carry it away. However, electrically charged particles can also be pushed onto orbits with negative energy by the Lorentz force and this is what makes possible the so-called 'MHD Penrose process' (Takahashi et al.1990; Koide et al.2002; Koide 2003) which is the common element of all alternative magnetic mechanisms. \nRecently there has been a renewed interest to this problem. The main reason for this is the arrival of robust numerical methods for relativistic astrophysics, e.g. (Pons et al.1998; Komissarov 1999; Koide et al.1999; Koldoba et al.2002; Gammie et al.2003; Del Zanna et al.2003; De Villiers & Hawley 2003). Timedependent MD and MHD simulations of monopole magnetospheres of black holes have demonstrated the asymptotic stability of the BZ solution (Komissarov 2001) and provided additional arguments in favour of the MD approximation in this particular case (Komissarov 2004a; Komissarov 2004b). Moreover, simulations of magnetically driven accretion disks have shown the development of low density axial regions, 'funnels', closely described by the BZ solution (De Villiers et al. 2003; McKinney & Gammie 2004). \nOn the other hand, the MHD simulations of a black hole immersed into a rarefied plasma with uniform magnetic field seemed to provide support for the MHD Penrose model (Koide et al.2002; Koide 2003). At least, it was found that by the end of the simulations the inflow of particles with negative energy at infinity accounted for about one half of the extracted energy. Unfortunately, the termination time of these simulations was surprisingly short, approximately one half of the rotational period of the black hole (presumably due to some computational problems.) As the result one could not tell whether the MHD Penrose mechanism would remain effective on a long-term basis. In this paper we present the results of new simulations which were carried out for much longer period of time and which bring new light on this important issue.", '2 BASIC EQUATIONS AND NUMERICAL METHOD': "In these simulations we solve numerically the equations of ideal MHD in the space-time of a Kerr black hole. This space-time is described using the foliation approach where \nthe time coordinate t parametrises a suitable filling of spacetime with space-like hypersurfaces described by the 3dimensional metric tensor γ ij . These hypersurfaces may be regarded as the 'absolute space' at different instances of time t . If { x i } are the spatial coordinates of the absolute space then the metric form can be written as \nds 2 = ( β 2 -α 2 ) dt 2 +2 β i dx i dt + γ ij dx i dx j (1) \nwhere α > 0 is called the 'lapse function' and β is the 'shift vector'. In this study we employ the Kerr-Schild coordinates, t, φ, r, θ , which like the well known Boyer-Lindquist coordinates ensure that none of the components of the metric form depend on t and φ . At spatial infinity these coordinate systems do not differ but the Kerr-Schild system does not have a coordinate singularity on the event horizon. Other details can be found in e.g. (Komissarov 2004a; McKinney & Gammie 2004). \nThe evolution equations of ideal MHD include the continuity equation, \n∂ t ( α √ γρu t ) + ∂ i ( α √ γρu i ) = 0 , (2) \nthe energy-momentum equations, \n∂ t ( α √ γT t ν ) + ∂ i ( α √ γT i ν ) = 1 2 ∂ ν ( g αβ ) T αβ α √ γ, (3) \nand the induction equation, \n∂ t ( B i ) + e ijk ∂ j ( E k ) = 0 . (4) \nHere g αβ is the metric tensor of spacetime, γ = det( γ ij ), e ijk is the Levi-Civita pseudo-tensor of space, ρ is the proper mass density of plasma, u ν is its four-velocity vector. The total stress-energy-momentum tensor, T µν , is a sum of the stress-energy momentum tensor of matter, \nT µν ( m ) = wu µ u ν -pg µν , (5) \nwhere p is the thermodynamic pressure and w is the enthalpy per unit volume, and the stress-energy momentum tensor of electromagnetic field, \nT µν ( e ) = F µγ F ν γ -1 4 ( F αβ F αβ ) g µν , (6) \nwhere F νµ is the Maxwell tensor of the electromagnetic field. The electric field, E , and the magnetic field, B , are defined via \nE i = α 2 e ijk ∗ F jk , (7) \nand \nB i = α ∗ F it , (8) \nwhere ∗ F µν is the Faraday tensor of the electromagnetic field, which is simply dual to the Maxwell tensor. In the limit of ideal MHD \nE = -v × B or E i = e ijk v j B k , (9) \nwhere v i = u i /u t is the usual 3-velocity of plasma. In addition, the magnetic field must satisfy the divergence free condition \n∂ i ( √ γB i ) = 0 . (10) \nNote, that 1) all the components of vectors and tensors appearing in these equations are the those measured in the coordinate basis, { ∂ ν } , of the Kerr-Schild coordinates; 2) \nFigure 1. The large scale structure of the numerical solution during the initial phase, t = 6. The colour image shows the distribution of rest mass density ,log 10 ρ . The contours show the magnetic flux surfaces (the lines of the poloidal component of magnetic field.) The arrows show the poloidal component of velocity relative to the coordinate grid. \n<!-- image --> \nthroughout the paper we employ such units that the speed of light c = 1, the gravitational constant G = 1, the black hole mass M = 1, and the factor 4 π does not appear in Maxwell's equations. \nOur numerical scheme is a 2D Godunov-type upwind scheme which utilises special relativistic Riemann solver described in Komissarov(1999) and uses the method of constraint transport (Evans & Hawley 1988) to preserve the magnetic field divergence free. Other details are outlined in (Komissarov 2004b).", '3.1 Setup': "In these simulations the rotational parameter a of the Kerr metric is a = 0 . 9 which gives the event horizon radius r + /similarequal 1 . 44. The axisymmetric computational domain covers 0 < θ < π and 1 . 35 < r < 53 . 1. The computational grid has 401 cells in the θ -direction, where it is uniform, ∆ θ = const , and 400 cells in the r -direction. The cell size in the r -direction, ∆ r , is such that the corresponding physical lengths in both directions are equal in the equatorial plane. \nThe usual axisymmetric boundary conditions are imposed at θ = 0 and θ = π boundaries. At the outer boundary, r = 53 . 1, the initial values of all variables are imposed throughout the whole run, the termination time for the simulations was set to t = 60 so no waves emitted from the dynamically active region near the black hole had a chance to get reflected of the boundary and effect the inner solution. \nThe inner boundary r = 1 . 35 is well inside the event horizon, which justifies use of 'radiative boundary conditions'. \nThe initial velocity field is set to be the same as the one of the fiducial observers of the Kerr-Schild foliation, who spiral towards the black hole (e.g. Komissarov 2004a). \nThe initial electromagnetic field has the same 'uniform' magnetic component, aligned with the rotational axis of the black hole, as in the vacuum solution of Wald: \nF µν = ( m [ µ,ν ] +2 ak [ µ,ν ] ) , (11) \nwhere k ν = ∂ t and m ν = ∂ φ are the Killing vectors of the Kerr spacetime (Wald 1974). However, the initial electric component is different as it has to satisfy the condition of perfect conductivity (9). \nThe initial thermodynamic pressure and the rest mass density of plasma are set to be p = 0 . 01 p m and ρ = 0 . 05 p m , where p m = ( B 2 -E 2 ) / 2 is the magnetic pressure. Finally, the equation of state employed in the simulations describes polytropic gas with the ratio of specific heats, Γ = 4 / 3. \nSummarising, although the setup of our simulations is not exactly the same as in Koide(2003) it is quite similar. In both cases we are dealing with magnetically dominated plasma of similar magnetisation. In both cases the initial magnetic field is described by the Wald vacuum solution (Wald 1974). Thus one would expect to obtain at least qualitatively similar solution in the common region of computational domains. Because of different space-time foliations, Koide(2003) used the Boyer-Lindquist coordinates, the difference in computational domains is not simply reduced to the difference in the range of the radial coordinate, r . Different definitions of the global time coordinate, t , should also be taken into account. This, however, is only significant in the spacial regions very close to the event horizon where the Boyer-Lindquist system becomes singular. \nAll conservative schemes for the relativistic magnetohydrodynamics have an upper limit on plasma magnetization above which they fail. At this limit, which somewhat varies from problem to problem and also depends on the resolution, the numerical error for the total energy density becomes comparable with the energy density of matter. This forces us to pump in fresh plasma in regions where its magnetization becomes dangerously high. Since in such regions the dynamical role of particles is rather insignificant this measure should not have a strong effect in most respects. The critical condition we set in these simulations is \nwW 2 -p = 0 . 01 B 2 , (12) \nwhere W is the Lorentz factor of the flow and B is the magnetic field strength as measured by the local FIDO. Should the energy density of matter drop below 0 . 01 B 2 , both ρ and p are artificially increased by the same factor. To minimise the effect of the mass injection on the flow the velocity of the injected matter is set to be equal to the local velocity of the flow. In fact, new particles must be constantly created in real magnetospheres of black holes but the details of this process can be rather different (Beskin et al.1991; Hirotani & Okamoto 1998; Phinney 1982). An additional lower limit was set on the value of the thermodynamic pressure, which was not allowed to drop below 0 . 01 ρ . \nFigure 2. Left panel: The u t component of plasma four-velocity and magnetic flux surfaces at t = 6. Right panel: the same as in the left panel but at t = 60. In both panels there is shown the same set of magnetic flux surfaces. The thick solid line shows the ergosphere. Note that -u t gives the specific 'energy at infinity' for a free particle and that the region of negative hydrodynamic energy at infinity is even somewhat narrower than the region of positive u t because of the pressure contribution. \n<!-- image -->", '3.2 Results and Discussion': "We have found that numerical solution exhibits two phases with rather different properties: 1) the rather short initial phase which is dominated by a rapid evolution in the neighbourhood of the black hole and 2) the final phase where solution settles to an approximate steady state in this region. \nIn the initial phase our numerical solution is indeed very similar to the one described in Koide (2003). Plasma mainly slides along the magnetic field lines towards the equatorial plane where it passes through an accretion shock and forms an equatorial disc (see fig.1). The magnetic field lines are also pulled towards the black hole. Further away from the hole this seems to be mainly due the velocity field of the initial solution as the subsequent increase of the magnetic pressure in the central region quickly halts this motion and the magnetic field lines of these distant regions eventually straighten up (see fig.4). However, near the horizon this pulling is certainly caused by the hole as the field lines never straighten up there. Since similar effect is also observed in simulations of inertia-free magnetospheres (Komissarov 2004a) it cannot not be explained simply by dragging of the magnetic field lines along with the accreted plasma and has to have a more general cause. In any case, these findings are in strong contrast with the exclusion of magnetic flux by rotating black holes found in axisymmetric vacuum solutions (Wald 1974; Bicak & Janis 1985). Those vacuum solutions were used in the past to argue low efficiency of the BZ mechanism in the case of rapidly rotating \nblack holes. Our results shows that this argument is incorrect and vacuum solutions have to be used with more caution. \nThe left panel of figure 2 shows the inner region of our solution at t = 6 in more detail. The colour image describes the distribution of the covariant time component of plasma's four-velocity, u t , whereas the solid lines show the magnetic flux surfaces (the lines of poloidal magnetic field). For a free particle -u t gives its specific energy at infinity. Since the hydrodynamic part of energy at infinity has the volume density \n-αT t t ( m ) = α ( -wu t u t + p ) (13) \nit can only be negative if u t > 0; because of the pressure contribution the region of negative hydrodynamic energy at infinity is somewhat narrower than the region of positive u t . What is even more important is that the flux of hydrodynamic energy at infinity \n-αT i t ( m ) = -αwu t u i (14) \nhas the opposite direction to the flow velocity v i = u i /u t , as required in the MHD Penrose process, if and only if u t is positive. Figure 2 shows that well in agreement with findings of (Koide et al.2002; Koide 2003) the inner part of the ergospheric disc does indeed have positive u t during the initial phase. Moreover, the magnetic field has a very similar topology too - all magnetic field lines threading the ergospheric disc have a turning point in the equatorial plane and do not cross the event horizon. \nThe reason for developing positive u t looks attractively simple. Due to the inertial frames dragging, all plasma enter- \nFigure 4. The large-scale structure of the solution at t = 60. Top left panel: the rest mass density, ρ ; Top right panel: the gas temperature, log 10 ( P/ρ ); Bottom left panel: H φ =(the total electric current through the circular loop r =const) / 2 π ; Bottom left panel: the angular velocity of magnetic field lines Ω / Ω h , where Ω h is the angular velocity of the black hole. The solid lines show the magnetic flux surfaces and the arrows show the poloidal velocity relative to the coordinate grid of the Kerr-Schild coordinates. \n<!-- image --> \ning the black hole ergosphere is forced to rotate in the same sense as the black hole (to be more precise this is what is observed by a distant observer. In Kerr-Schild coordinates dφ/dt may have both signs.) As the result a differential rotation inevitably develops along the magnetic field lines penetrating the ergosphere which causes twisting of these lines. The Alfven waves generated in this manner propagate away \nfrom the ergosphere to infinity and establish an outflow of energy in the form of Poynting flux. Because of the energy conservation the ergospheric plasma reacts by moving onto orbits of lower energy. Provided this plasma remains in the space between the event horizon and the ergosphere for long enough it may indeed end up having negative energy. \nIn the problem under consideration the strong magnetic \nFigure 3. The hydrodynamic (dashed line) and the electromagnetic (solid line) components of the energy flux ( -T r t ) through the event horizon at t = 60. \n<!-- image --> \nfield keeps the plasma of the ergospheric disc from falling into the black hole and allows it to gain negative energy (see fig.2). However, such configuration cannot be sustain forever within ideal MHD. Since, the energy is constantly extracted along the field lines of the equatorial disc the energy of the disc is constantly going down and no steady-state can be reached. At some point the magnetic configuration would have to change so that all magnetic field lines entering the ergosphere also penetrate the event horizon (Another possible option could be a non-steady behaviour with magnetic field lines pulled in and out of the event horizon all the time.) As one can see in the right panel of figure 2, where we present the solution at t = 60, this is more or less what is observed in our simulations. At around t = 20 the ergospheric disc is fully swallowed by the hole and a strong current sheet develops in its place. As the result, the structure of magnetic field becomes similar to that of the split-monopole model (Blandford & Znajek 1977). This configuration persists till the very end of the simulations when the whole solution in the inner part of the spacial domain seems to reach a state of an approximate equilibrium. \nIn fact, one can see in fig.2 at least two magnetic islands in the equatorial current sheet. Such structure is known be quite typical for the tearing mode instability (e.g. Priest & Forbes, 2000) and was suggested in the context of black hole magnetospheres in Beskin (2003). Here, the islands are formed during slow reconnection events resulting in the gradual escape of magnetic flux from the event horizon, the inner islands disappear into the hole whereas the outer ones move away from it and supply hot plasma for a thin outflow sheath clearly seen in the temperature plot in fig.4. This evolution is indeed very slow and might be related to \na somewhat excessive capture of magnetic field line during the early stages. \nThe most important feature of the solution after the restructuring of magnetic field is the total disappearance of the regions with positive u t (see the right panel of fig.2). Thus, the MHD Penrose process does no longer operate in the black hole ergosphere. This is confirmed in figure 3 which shows that the hydrodynamic flux of energy at infinity through the event horizon, -αT r t ( m ) , is everywhere negative. However, the electromagnetic flux of energy at infinity through the event horizon is positive almost everywhere with the exception of a very thin equatorial belt where it can be slightly negative (fig.3) . Thus, the pure electromagnetic Blandford-Znajek mechanism continues to operate. Moreover, the total flux of energy at infinity through the event horizon is positive and, in spite of the fact that the plasma magnetization is many orders of magnitude lower than that expected in typical astrophysical conditions, the BZ mechanism allows to extract the rotational energy of the black hole. \nOne important feature of the initial phase which has been found in the previous simulations (Koide 2003) and which persists in the longer run is the lack of any noticeable plasma outflow from the black hole. The only exception is a thin sheath of radius r /similarequal 3, most clearly seen in the top right panel of fig.4,) which is supplied with relativistically hot plasma ( p/ρ /similarequal 10 2 ) via the reconnection process in the ergospheric current sheet. The outflow in the sheath is rather slow with poloidal speed relative to the local FIDO never exceeding v p /similarequal 0 . 7 c ; it is most likely driven by the gas pressure. This finding is in striking contrast with the results of MHD simulations of monopole magnetospheres of black holes which show the development of a powerful ultrarelativistic wind predominantly in the equatorial direction (Komissarov 2004b). Being taken together, these two results suggest that divergence of magnetic field lines is a necessary condition for generating strong ultra-relativistic plasma outflows from black holes. Moreover, given the fact that the Lorentz factor of the monopole wind reaches the value of W = 3 in the equatorial direction only at the distance of around r=20 (or 20 r g , where r g = GM/c 2 , if one prefers this dimensional form) and much further away in the polar direction, one would not expected to find an MHD-driven collimated ultra-relativistic outflow at distances smaller than several tens of r g . Here there may be present only relativistic beams of particles accelerated by other mechanisms, e.g. electromagnetically. This makes the ongoing projects of studying the very bases (up to few tens of r g ) of astrophysical jets via short-wavelength VLBI observations particularly interesting (Krichbaum et al. 2004). \nThe bottom right panel of fig.4 shows the angular velocity of magnetic field lines, Ω, normalised to the angular velocity of the black hole Ω h = a/ ( r 2 + + a 2 ), where r + is the radius of the event horizon. In a steady state this parameter must be constant along magnetic field lines (e.g. Camenzind 1986) which is what is seen in the plot. Thus, at t = 60 our solution is very close to a steady state at least for r < 20. Another interesting result seen in this plot is that Ω /similarequal 0 . 5Ω h , the value obtained in (Blandford & Znajek 1977) for a slowly rotating black hole with monopole magnetic field. Finally, there is a sharp transition between the 'rotating column' of magnetic field lines attached to the black hole \nObservations of the Blandford-Znajek and the MHD Penrose processes in computer simulations of black hole magn \nFigure 5. Ω / Ω h (top panel) and H φ (bottom panel) at r=5 and t=60. \n<!-- image --> \nand the nonrotating 'soup' of magnetic field lines which fail to enter the ergosphere (see also fig.5). This is most likely a discontinuity somewhat smeared by numerical diffusion and perhaps by the process of reconnection in the equatorial current sheet described above. \nThe bottom left panel of fig.4 shows the distribution of H φ , the covariant azimuthal component of vector H introduced via \nH i = ∗ F ti . \nIn a steady-state \n∇ × H = J, \nwhere J is the electric current density (Komissarov 2004a). Thus, H φ at the point ( r, θ ) gives us the total electric current flowing through the loop r =const originated from this point. In force-free steady-state solutions H φ is also constant along magnetic field lines (Blandford & Znajek 1977; Komissarov 2004a). This is exactly what is seen in figure 4, thus confirming that at t = 60 the solution is almost forcefree and very close to a steady state for r < 20. The distribution of H φ also exhibits a discontinuity at the boundary of the 'rotating column' indicating a thin current sheet of return current. \nThe split-monopole structure of magnetic field found in these simulations within the ergosphere is in conflict with the electrodynamic simulations where the field lines exhibit a sharp turning point in the equatorial plane (Komissarov 2004a). There seem to be only two possible reasons for this difference. First of all, neither the inertia nor the pressure of plasma particles are accounted for in the electrodynamic model. At first glance, this does not seem to be important as even in the current MHD simulations both are small compared to the mass-energy density and pressure of magnetic field almost everywhere. But not exactly everywhere. In the equatorial current sheet the gas pressure dominates and plays a stabilizing role against otherwise quick reconnection of magnetic field lines. The second factor is \nthe resistivity model. While the electrodynamic simulations (Komissarov 2004a) utilize the anisotropic resistivity based on the inverse Compton scattering of background photons, the only resistivity present in the MHD model is numerical. At this point we cannot exclude that explicit inclusion of physical resistivity will have a strong effect on the outcome of MHD simulations. This matter will have to be addressed in future studies.", '4 SUMMARY AND CONCLUSIONS': "We have carried out axisymmetric ideal MHD simulations of plasma flows in the case of a rotating black hole immersed into an initially uniform magnetic field described by the vacuum solution of Wald (1974). Our main intention was to verify the results of previous simulations (Koide et al.2002; Koide 2003) claiming a rather significant role for the socalled MHD Penrose process of extracting the rotational energy of black holes, at least in this particular case. Our simulations show that this is true only for a short initial period during which a large fraction of magnetic field lines entering the black hole ergosphere exhibit a turning point in the equatorial plane. Eventually, those field lines are pulled into the black hole and within the ergosphere the magnetic field acquires the split-monopole structure. After this transient phase the regions of negative hydrodynamic energy at infinity are no longer present in the ergosphere and the the MHD Penrose process ceases to operate altogether. \nThe rotational energy yet continues to be extracted via purely electromagnetic Blandford-Znajek process (Blandford & Znajek 1977; Komissarov 2004a). This energy extraction, however, is not followed by development of large-scale outflow from the black hole in sharp contrast with the results of previous MHD simulations for the monopole configuration of magnetic field lines (Komissarov 2004b). The only exception is a thin current sheet located at the interface between the 'rotating column' of magnetic field lines attached to the black hole and the 'nonrotating soup' of field lines failing to enter the black hole ergosphere. The relatively slow outflow in this 'sheath' is likely to be driven by gas pressure and is fed by hot plasma produced during reconnection events in the equatorial current sheet. Given the results of these and other recent MHD simulations (Koide 2003; Komissarov 2004b) it seems impossible to generate via the standard MHD mechanism an outflow which becomes both ultrarelativistic and collimated already within the few first decades of the gravitational radius from a black hole. Should relativistic astrophysical jets continue to be seen at such small distances (Krichbaum et al. 2004), one will have to look for other explanations of their origin. \nOne of the main shortcoming of these simulations is the approximation of perfect conductivity. The only source of resistivity governing the process of magnetic reconnection in the developing current sheets is purely numerical. Future numerical models will have to include physical resistivity.", 'REFERENCES': "Beskin V.S., Istomin Y.N., Pariev V.I., 1991, Sov.Astron., 36(6) , 642. \nBeskin V.S., 2003, Phys.Uspekhi, 173 , 1247. Bicak J. and Janis V., 1985, MNRAS, 212 , 899. Blandford R.D. and R.L. Znajek R.L., 1977, MNRAS, 179 , 433. Camenzind, M., 1986, A&A, 162 , 32. Del Zanna L., Bucciantini N., Londrillo P., 2003, A&A, 400 , 397. De Villiers J.-P., Hawley J.F., 2003, ApJ, 589 , 458. De Villiers J.-P., Hawley J.F., Krolik J.H. 2003, ApJ, 599 , 1238. Evans C.R., Hawley J.F., 1988, ApJ, 332 , 659. Gammie C.F., McKinney J.C., T'oth G., 2003, ApJ, 589 , 444. Hirotani K., Okamoto I., 1998, ApJ, 497 , 563. 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2013PhRvD..87f3519K

Primordial black hole formation from an axionlike curvaton model

2013-01-01

['-', '-', '-', 'particles', '-', 'cosmology dark matter', '-', '-']

[]

We argue that the existence of the cold dark matter is explained by primordial black holes. We show that a significant number of primordial black holes can be formed in an axionlike curvaton model, in which the highly blue-tilted power spectrum of primordial curvature perturbations is achieved. It is found that the produced black holes with masses ∼10<SUP>20</SUP>-10<SUP>38</SUP>g account for the present cold dark matter. We also argue the possibility of forming the primordial black holes with mass ∼10<SUP>5</SUP>M<SUB>⊙</SUB> as seeds of the supermassive black holes.

[]

https://arxiv.org/pdf/1207.2550.pdf

{'No Header': 'ICRR-Report-616-2012-5 IPMU 12-0116', 'Primordial black hole formation from an axion-like curvaton model': 'Masahiro Kawasaki ( a,b ) , Naoya Kitajima ( a ) and Tsutomu T. Yanagida ( b ) \n- a Institute for Cosmic Ray Research, University of Tokyo, Kashiwa, Chiba 277-8582, Japan\n- b Kavli Institute for the Physics and Mathematics of the Universe, University of Tokyo, Kashiwa, Chiba 277-8568, Japan', 'Abstract': 'We argue that the existence of the cold dark matter is explained by primordial black holes. We show that a significant number of primordial black holes can be formed in an axion-like curvaton model, in which the highly blue-tilted power spectrum of primordial curvature perturbations is achieved. It is found that the produced black holes with masses ∼ 10 20 -10 38 g account for the present cold dark matter. We also argue the possibility of forming the primordial black holes with mass ∼ 10 5 M glyph[circledot] as seeds of the supermassive black holes.', '1 Introduction': 'The current cosmic microwave back ground (CMB) observations have revealed that the our present universe is filled with the unknown matter called dark matter, which cannot be explained within the framework of the well-established standard model of particle physics. The observed density parameter for the cold dark matter (CDM) is found by the WMAP [1] to be \nΩ CDM h 2 = 0 . 1126 ± 0 . 0036 , (1) \nwhere h is the dimensionless Hubble parameter defined via the present Hubble parameter: H 0 = 100 h km sec -1 Mpc -1 . In order to detect the dark matter, many experiments have been performed by now, but we have not found any meaningful signature yet. Therefore it is one of the most important problems of modern cosmology and particle physics to answer what the dark matter is. \nIt is usually assumed that the dark matter is the weakly-interacting massive particles (WIMPs). The supersymmetric (SUSY) model [2], which is one of the most promising model beyond the standard model, naturally provides such WIMPs as the lightest supersymmetric particle (LSP). Another promising candidate of dark matter is the axion, which is originally introduced to solve the strong CP problem in the standard model [3]. However, even if the SUSY particles or axion exist in the present universe, it may not be enough to explain the observed dark matter abundance depending on the model parameters. In such a case, we are forced to demand another candidate for dark matter. It is known that the primordial black holes (PBHs), the black holes formed in the early universe [4], can behave like CDM. In this paper, therefore, we argue the scenario in which the currently observed abundance of CDM is explained by PBHs. \nPBHs are expected to be formed through the collapses of the high density regions caused by the large primordial density perturbations [5]. Light PBHs with mass smaller than 10 15 g are evaporated by now through the Hawking radiation [6], implying that only the PBHs with masses M BH > 10 15 g can survive and contribute to the CDM. Furthermore, various cosmological and astrophysical constraints show that only the PBHs with mass 10 17 g < M BH < 10 27 g can be the dominant component of the current CDM [7]. Although it is not easy to build the model in which a significant number of PBHs are formed, various models were proposed in the literature. Focusing on the inflation models, for examples, PBH formation was proposed in double inflation models [8, 9, 10, 11] or running mass inflation models [12, 13]. \nAnother motivation to consider PBHs is the existence of supermassive black holes (SMBHs) at the center of galaxies [14, 15]. The observation of quasars (QSO) reveals that the SMBHs with mass M BH ≈ 10 9 M glyph[circledot] exist at the redshift z ≈ 6 [16]. These black holes cannot be explained within the purely astrophysical mechanism, so we must rely on the primordial origin. If PBHs with sufficiently large mass, M BH glyph[greaterorsimilar] 10 3 M glyph[circledot] , can be formed in the early universe, they can play roles of the seeds of SMBHs [17, 18]. \nIn order for PBHs to form through the primordial density perturbations, we need the strongly blue-tilted power spectrum of the curvature perturbations, which gives the large density perturbations at small scales while the large scale density perturbations are \nconsistent with the CMB observation. However, the observation indicates that the scale dependence of the power spectrum is slightly red-tilted at large scales. This inconsistency is solved by employing a curvaton. The curvaton was originally introduced to generate the primordial large scale curvature perturbations instead of the inflaton [19]. In the curvaton model, a scalar field (called curvaton) acquires fluctuations during inflation and after inflation it decays into the standard model particles producing the adiabatic perturbation in the radiation dominated universe. \nIn this paper, we consider that the curvaton is responsible for generating only the small-scale curvature perturbations while the large-scale perturbations are generated by an inflaton. After the decay of the curvaton, a significant number of PBHs can be formed through large density perturbations due to the curvaton. A specific model for the PBH formation with curvaton was proposed in [20], where three scalar fields (including inflaton and curvaton) with ad hoc couplings among them evolve non-trivially during inflation and leads to large density perturbations at small scales. Our mechanism for the PBH formation is completely different from that in Ref. [20]. We consider an axion-like curvaton field whose nature is very crucial for the PBH formation. Furthermore, axion-like fields often appear in various particle physics theories. We consider that one of such axionlike (curvaton) fields may play an important role for the PBH formation as studied in this paper. Ref. [12] also discussed PBH formation in curvaton model without concrete models. \nThe remainder of the paper is organized as follows. In section 2, we show an axion-like curvaton model and we see the largely blue-tilted spectrum for the curvature perturbations is achieved. In section 3, we consider the PBH formation within the axion-like curvaton model. Section 4 is devoted to the conclusion.', '2.1 The potential of the curvaton': 'In this section, we describe the axion-like curvaton model which was originally introduced in [21] (see also [22]) as an axion model with extremely blue-tilted spectrum of the isocurvature perturbations. The model is built in the framework of supersymmetry and has the following superpotential: \nW = hS (Φ ¯ Φ -f 2 ) , (2) \nwhere Φ, ¯ Φ and S are chiral superfields whose R -charges are +1, -1 and +2 respectively, f is some energy scale and h is a dimensionless coupling constant. Here we assume that the model has a global U (1) symmetry and Φ, ¯ Φ and S have charges +1, -1 and 0, respectively. In the limit of the global SUSY, the scalar potential is derived from (2) as \nV = h 2 | Φ ¯ Φ -f 2 | 2 + h 2 | S | 2 ( | Φ | 2 + | ¯ Φ | 2 ) , (3) \nwhere, the scalar components are denoted by the same symbols as the superfields. Provided that | S | < f is satisfied, S tends to the origin and Φ and ¯ Φ are settled on the flat \ndirection satisfying \nΦ ¯ Φ = f 2 with S = 0 , (4) \nwhich makes the scalar potential (3) vanish. Hereafter, we assume that the flat condition (4) is always satisfied. Including the supergravity effects, the Hubble-induced mass terms are added to the scalar potential [23] as \nV H = c 1 H 2 | Φ | 2 + c 2 H 2 | ¯ Φ | 2 + c S H 2 | S | 2 , (5) \nwhere c 1 , c 2 and c S are numerical constants assumed to be real, positive and of order unity. In addition, there also exist the low energy SUSY breaking terms, \nV m = m 2 1 | Φ | 2 + m 2 2 | ¯ Φ | 2 + m 2 S | S | 2 , (6) \nwhere m 1 , m 2 and m S are soft masses of order of the gravitino mass. Here, because we are interested in the inflationary epoch, we neglect the low energy SUSY breaking mass terms. Thus, the flat direction is lifted by only the Hubble-induced mass terms (5) and the minimums of Φ and ¯ Φ are determined as \n| Φ | min glyph[similarequal] ( c 2 c 1 ) 1 / 4 f, | ¯ Φ | min glyph[similarequal] ( c 1 c 2 ) 1 / 4 f. (7) \nNow, we decompose the complex scalar fields into the radial and the phase components as \nΦ = 1 √ 2 ϕ exp( iθ + ) , ¯ Φ = 1 √ 2 ¯ ϕ exp( iθ -) . (8) \nThen, along the flat direction, the massless direction is found as a linear combination of the phases, θ = ( θ + -θ -) / 2. Without loss of generality, we can take ϕ glyph[greatermuch] ¯ ϕ as the initial condition and neglect the dynamics of ¯ ϕ in the early epoch [22], so we follow the dynamics of only the complex scalar field Φ = ϕe iθ / √ 2 whose potential is given by \nV ϕ = 1 2 cH 2 ϕ 2 . (9) \nNote that since the mass is comparable to the Hubble parameter, ϕ rolls down the potential somewhat rapidly during inflation. \nIn our model, the curvaton is defined as the phase component of Φ. Note that the curvaton is well-defined only after ϕ reaches the minimum ( ϕ min ) and denoted as σ = ϕ min θ ∼ fθ . Here, we assume that the U (1) symmetry is broken by some non-perturbative effect and σ has the following potential in the low energy universe like the axion: \nV σ = Λ 4 [ 1 -cos ( σ f )] glyph[similarequal] 1 2 m 2 σ σ 2 , (10) \nwhere the second equality holds near the minimum σ min = 0 and the curvaton mass is defined as m σ = Λ 2 /f . After the Hubble parameter becomes smaller than the curvaton \nmass, the curvaton field starts to oscillate coherently with the initial amplitude σ i ∼ fθ and behaves as matter. \nLet us derive the ratio r of the curvaton energy density to that of the radiation at the time of the curvaton decay. In order to estimate this, we consider the two cases : (i) the reheating occurs after the curvaton starts to oscillate and (ii) the curvaton starts to oscillate after the reheating. In case (i), which is denoted as m σ > Γ I (Γ I : decay rate of the inflaton), we estimate r as \nr = ρ σ ( t dec ) ρ r ( t dec ) = ρ σ ( t R ) ρ r ( t R ) T R T dec glyph[similarequal] 1 6 ( fθ M P ) 2 T R T dec for m σ glyph[greaterorsimilar] T 2 R M P , (11) \nand, on the other hand, in case (ii), m σ < Γ I , we obtain \nr = ρ σ ( t osc ) ρ r ( t osc ) T osc T dec glyph[similarequal] 1 6 ( fθ M P ) 2 T osc T dec for m σ glyph[lessorsimilar] T 2 R M P , (12) \nwhere the subscripts dec, osc and R correspond to the epochs of the curvaton decay, the curvaton oscillation and the reheating, respectively. Hereafter, we consider only the case of r 1. \nThe curvaton decays when the Hubble parameter becomes equal to the decay rate of the curvaton and the decay temperature of the curvaton is determined from the decay rate. Here we assume that the interaction of the curvaton with its decay product is suppressed by f like an axion, so we denote the decay rate of the curvaton as \n≤ \nΓ σ = κ 2 16 π m 3 σ f 2 , (13) \nwhere κ is a dimensionless numerical constant assumed to be real, positive and smaller than 1. Then, the decay temperature of the curvaton is given by \nT dec = 0 . 5 ( g ∗ 100 ) -1 / 4 (Γ σ M P ) 1 / 2 , (14) \nwhere g ∗ is the relativistic degrees of freedom.', '2.2 Generating the curvature perturbation': 'To create a significant number of PBHs from the primordial density perturbations, we require the extremely blue spectrum with spectral index n s glyph[greaterorsimilar] 2 as we will see later. However, such a large spectral index is already ruled out by the CMB observation. In order not to contradict the observation, then, we build the model in which the almost scale-invariant large-scale curvature perturbations are generated by an inflaton and the small-scale curvature perturbations which are free from the CMB constraint are generated by the curvaton. Here, we investigate the possibility of PBH formation in the axion-like curvaton model introduced above. \nThe power spectrum of curvature perturbations is the sum of the contributions from the inflaton and the curvaton, which is written as \nP ζ ( k ) = P ζ, inf ( k ) + P ζ, curv ( k ) . (15) \nAs mentioned above, the power spectrum is dominated by the first term in rhs. of (15) for small k and by the second term for large k . It is quite reasonable that the contribution from the inflaton is dominant until the perturbation scale at least k ∼ 1 Mpc -1 leaves the horizon. For later convenience, we define k c as \nP ζ, curv ( k c ) = P ζ, inf ( k c ) glyph[similarequal] 2 × 10 -9 (16) \nwhere we have used the CMB normalization [1] for the spectrum of the large-scale curvature perturbations. Using this definition, our requirement is denoted as \nP ζ, curv ( k ) < P ζ, inf ( k ) ∼ 2 × 10 -9 for k glyph[lessorsimilar] k c and k c glyph[greaterorsimilar] 1 Mpc -1 . (17) \nLet us consider the power spectrum from the fluctuation of the curvaton field. From the definition of the curvaton, σ = ϕ min θ , the density perturbation of the curvaton is given by \nδρ σ ρ σ, 0 glyph[similarequal] 2 δσ σ 0 = 2 δθ θ 0 , (18) \nwhere we decompose each field into the homogeneous part and the small perturbation: X = X 0 ( t ) + δX ( t, glyph[vector]x ). Focusing on the super-horizon Fourier mode of δθ , δθ/θ 0 is conserved because the masses of both θ 0 and δθ are much smaller than the expansion rate of the universe [24]. This means that the resultant power spectrum remembers the fluctuation of θ at the horizon exit, written as P 1 / 2 δθ ( k ) glyph[similarequal] H inf / (2 πϕ 0 ( k )), where the argument k entering in ϕ 0 denotes the value when the scale k leaves the horizon. Thus, the power spectrum of the density perturbation for the curvaton is expressed as \nP 1 / 2 δ, curv ( k ) = 2 P 1 / 2 δθ ( k ) θ glyph[similarequal] H inf πϕ ( k ) θ . (19) \nHere and hereafter, we drop the subscript 0 to express the homogeneous value. From the above, the spectrum of the curvature perturbation from the curvaton is calculated as [19] \nP ζ, curv ( k ) = ( r 4 + 3 r ) 2 P δ, curv = ( 2 r 4 + 3 r ) 2 ( H inf 2 πϕ ( k ) θ ) 2 (20) \nNote that, after ϕ reaches the minimum, the power spectrum takes the constant value given by \nP ζ, curv ( k ) = P ζ, curv ( k ∗ ) ≈ ( 2 r 4 + 3 r ) 2 ( H inf 2 πfθ ) 2 for k > k ∗ , (21) \nwhere k ∗ is defined as the scale leaving the horizon at the time ϕ reaches the minimum ∼ f . With use of the curvaton spectral index n σ , the scale dependence of the power spectrum of the curvature perturbation is expressed as \nP ζ, curv ( k ) = P ζ, curv ( k c ) ( k k c ) n σ -1 for k ≤ k ∗ . (22) \nCombining this and the ϕ dependence, \nP ζ, curv ( k ) = P ζ, curv ( k c ) ( ϕ ( k c ) ϕ ( k ) ) 2 , (23) \nwe obtain the relation \nk = k c ( ϕ ( k c ) ϕ ( k ) ) 2 / ( n σ -1) for k ≤ k ∗ . (24) \nThe spectral index of the curvaton is calculated by solving the equation of motion of ϕ with potential (9), \n¨ ϕ +3 H ˙ ϕ + cH 2 ϕ = 0 , (25) \nwhose solution during inflation ( H glyph[similarequal] const.) is given by \nϕ ∝ e -λHt ∝ k -λ with λ = 3 2 -3 2 √ 1 -4 9 c. (26) \nTogether with (22) and (23) the spectral index is given by \nn σ -1 = 3 -3 √ 1 -4 9 c, (27) \nso we can obtain the extremely blue spectrum such as n σ ∼ 2 - 4 with appropriate choice of c [21].', '3 The PBH formation': 'In this section, we consider the formation of PBHs in our model. It is well-known that PBHs can be formed by collapse of overdensity regions in the radiation-dominated universe and their mass is as large as the horizon mass at the formation time [25, 26, 27], which is given by \nM BH = 4 π 3 ρ r H -3 glyph[similarequal] 0 . 05 M glyph[circledot] ( g ∗ 100 ) -1 / 2 ( T f GeV ) -2 glyph[similarequal] 1 × 10 13 M glyph[circledot] ( g ∗ 100 ) -1 / 6 ( k f Mpc -1 ) -2 , (28) \nwhere ρ r represents the energy density of the radiation and the subscript f represents the time of the PBH formation. Here we assume r ≤ 1 which means that PBHs are formed in radiation dominated universe after the curvaton decay (see also footnote 2). PBHs with M BH > 10 15 g do not evaporate through the Hawking radiation [6] until now and their abundance can contribute to the present CDM density. The current density parameter for such PBHs is calculated as \nΩ PBH h 2 = ρ PBH , eq ρ tot , eq Ω m h 2 glyph[similarequal] 5 × 10 7 β ( M glyph[circledot] M BH ) 1 / 2 , (29) \nwhere the subscript eq corresponds the time of the matter-radiation equality and Ω m glyph[similarequal] 0 . 13 h -2 is the density parameter for matter today. β is defined as the energy density fraction of the PBHs at the PBH formation, which is denoted as β ≡ ρ PBH ( t f ) /ρ tot ( t f ). Various cosmological and astrophysical constraints are imposed on β [7], from which only the PBHs with mass ∼ 10 17 g - 10 27 g can be the dominant component of the dark matter. For such PBHs, the constraint on β comes from the current observational value of the dark matter density (1), which implies \nβ < 3 × 10 -11 ( M BH M glyph[circledot] ) 1 / 2 . (30) \nAssuming that the PBHs are created by the collapse of overdensity regions with primordial gaussian density perturbations, β is estimated as [28] \nβ ≈ √ 〈 δ 2 〉 exp ( -1 18 〈 δ 2 〉 ) , (31) \nwhere δ = δρ/ρ is the density contrast and 〈 δ 2 〉 is its variance. The variance of the density perturbations is related to the power spectrum of curvature perturbations, and, in the comoving gauge in which the curvature perturbation is expressed as R , which coincides with ζ well outside the horizon, the following relation is known: \nP δ ( k ) = 4(1 + w ) 2 (5 + 3 w ) 2 P R ( k ) , (32) \nat the time the scale k leaves the horizon [29]. w is determined from the relation between the pressure and the energy density of the cosmic fluid P = wρ and it takes 1/3 in radiation dominated universe. The variance of the density perturbation smoothed over the scale R is estimated as \n〈 δ 2 ( R ) 〉 = ∫ ∞ 0 W 2 ( kR ) P δ ( k ) dk k , (33) \nwhere W ( kR ) is the window function in Fourier space. Assuming the Gaussian window function W ( kR ) = exp( -k 2 R 2 / 2) and taking into account P ζ ≈ P R and (21) and (22), we can approximate the variance as \n〈 δ 2 ( R ) 〉 = 8 81 P ζ, curv ( k ∗ ) [ ( k ∗ R ) -( n σ -1) γ (( n σ -1) / 2 , k 2 ∗ R 2 ) + E 1 ( k 2 ∗ R 2 ) ] , (34) \nFigure 1: The ratio of the smoothed variance of density perturbation to the power spectrum of curvature perturbation are shown. The horizontal axis is the wave number corresponding to the smoothing scale as k = R -1 divided by k ∗ . This curve is independent of n σ . \n<!-- image --> \nwhere γ ( a, x ) and E 1 are defined as \nγ ( a, x ) = ∫ x 0 t a -1 e -t dt, and E 1 ( x ) = ∫ ∞ x e -t t dt. (35) \nIf we wrire \n〈 δ 2 ( k -1 ) 〉 = α P ζ, curv ( k ) , (36) \nthe numerical coefficient α is taken to be 0 . 1-4 as shown in Fig. 1. \nWe show the energy density fraction of PBH in terms of P ζ, curv in Fig. 2(a) in the case of α = 1 (solid red line) and α = 0 . 1 (dashed green line). The dotted blue line (the smalldotted magenta line) corresponds to the upper limit in the case of M BH = 10 27 (10 17 ) g, which comes from the current observation of the CDM density. In order for PBHs to be the dominant component of dark matter, the required value of curvature perturbation is P ζ, curv ∼ 2 × 10 -3 (2 × 10 -2 ) for α = 1 (0 . 1). Substituting (20) and (36) into (31) and taking ϕ ( k ) ∼ f , the constraint (30) is rewritten in terms of H inf / ( fθ ) shown in Fig. 2(b). In this figure, the thick (thin) solid red line corresponds to r = 1 and α = 1 (0 . 1) and the thick dashed green line corresponds to r = 0 . 1 and α = 1. The breaking point of each line corresponds to the point at whch the quantum fluctuation of the curvaton, δσ = H inf / 2 π , becomes f . If H inf / 2 π > f , the amplitude of the quantum fluctuations of S overtakes the critical value f , which invalidates our underlying assumption (4). From Fig. 2(b) we need r ∼ 1 and fθ ∼ H inf to account for the present dark matter abundance. 1 \n<!-- image --> \nFigure 2: The energy density fraction of the PBH at the formation is shown. The horizontal axis correspond to P ζ, curv in Fig. 2(a) and H inf /fθ in Fig. 2(b). In Fig. 2(a), the solid red line and the dashed green line correspond to α = 1 and α = 0 . 1 respectively. In Fig. 2(b), the thick (thin) solid red line corresponds to r = 1 and α = 1 (0 . 1) and the thick dashed green line corresponds to r = 0 . 1 and α = 1. Breaking point of each line in Fig. 2(b) corresponds to δσ/σ = 1. The dotted blue line (the small-dotted magenta line) corresponds to the upper limit in the case of M BH = 10 27 (10 17 ) g, which comes from the current observational value of the CDM density parameter : Ω CDM = 0 . 23. \n<!-- image --> \nNow let us estimate the mass spectrum of PBHs in the present model. This is especially impotent for SMBHs since, taking into account the merging and accretion events prior to the formation of SMBHs, the mass spectrum of primordial seeds of SMBHs is required to have a sharply peaked shape [32]. With R = k -1 and Eq.(28) we rewrite the smoothed variance (34) in terms of PBH masses as \n〈 δ 2 ( M BH ) 〉 = 8 81 P ζ, curv ( k ∗ ) [( M ∗ M BH ) ( n σ -1) / 2 γ ( n σ -1 2 , M BH M ∗ ) + E 1 ( M BH M ∗ )] , (37) \nwhere M ∗ is the mass of PBH formed when the scale k ∗ enters the horizon. Using (37), we can calculate the mass function, which is defined as the number of PBHs per comoving volume whose mass range is M BH ∼ M BH + dM BH , as [33] : \ndn PBH dM BH = √ 1 18 π ¯ ρ r ( t ∗ ) M 2 BH ( M ∗ M BH ) 1 / 2 ∣ ∣ ∣ ∣ d ln 〈 δ 2 ( M BH ) 〉 d ln M BH ∣ ∣ ∣ ∣ β ( M BH ) 〈 δ 2 ( M BH ) 〉 , (38) \nwhere ¯ ρ r ( t ∗ ) is the radiation energy per comoving volume when the scale k ∗ enters the horizon and β ( M BH ) is the density fraction of PBH whose mass is M BH . Since we assume that PBHs are formed after the curvaton decays, the mass spectrum has the lower cutoff M min , which corresponds to the mass of PBH formed just after the curvaton decays. 2 The \nFigure 3: The mass spectrum of PBH, dn PBH /dM BH , is shown. The solid red line and dashed green line corresponds to M min /M ∗ = 10 -8 and M min /M ∗ = 10 -3 respectively and they are normalized by the their own peak values. They are independent of n σ . \n<!-- image --> \nmass spectrum of PBHs is shown in Fig. 3. The solid red and dashed green lines correspond to M min /M ∗ = 10 -8 and M min /M ∗ = 10 -3 respectively and they are normalized by the their own peak values. The mass spectrum depends only on M min and it is independent of n σ . It is clear that the dominant contribution to the energy density of PBHs comes from the smaller mass PBHs, so the constraint on the initial PBH abundance should be applied to the PBHs with M min . In particular, for M min /M ∗ = 10 -8 , it is seen that the number of PBHs with mass larger than ∼ 10 -4 M ∗ decreases drastically. This is due to the sudden decreasing of α , which implies the sudden decreasing of 〈 δ 2 〉 , for k glyph[lessorsimilar] 10 2 k ∗ (see Fig. 1). Thus we can obtain a very narrow mass spectrum by tuning M min /M ∗ as 10 -3 -10 -2 , which is required to explain SMBHs. \nNow, we investigate the parameters allowing the formation of PBHs which eventually becomes the dominant component of the CDM. We also impose several conditions to build the viable scenario, which are listed below. \n- · Going back to the time when the pivot scale k p = 0 . 002 Mpc -1 leaves the horizon, ϕ should be smaller than the Planck scale. Using (24) and taking into account f ≈ H inf /θ and ϕ ( k c ) ≈ 10 3 f , we get \nH inf < 2 ( n σ -1) / 2 10 -3( n σ +1) / 2 θM P ( k c Mpc -1 ) -( n σ -1) / 2 . (39) \nsuch a case, however, P δ is suppressed through the factor ( ρ σ /ρ r ) 2 at the formation. Since the number of produced PBHs is very sensitive to P δ and exponentially suppressed for small P δ , the number of those PBHs produced before the curvaton decay may be negligibly small. Thus, even if we include the above effect, the mass spectrum (Fig. 3) may slightly spread around the cutoff and nothing is affected in our discussion. \nCombining this with the constraint from the tensor-to-scalar ratio [1], H inf < 5 × 10 -5 M P , we get \nH inf < min [ 2 ( n σ -1) / 2 10 -3( n σ +1) / 2 θM P , 5 × 10 -5 M P ] , (40) \nwhere we set k c = 1 Mpc -1 . \n- · The mass of PBHs formed when the scale k ∗ reenter the horizon, M ∗ , is calculated with use of (16),(22) and (28). Since M ∗ is larger than the minimum mass of PBHs, we obtain the following condition: \nM min < M ∗ = 2 × 10 46 -12 / ( n σ -1) g ( g ∗ 100 ) -1 / 6 ( k c Mpc -1 ) -2 ( P ζ, curv ( k ∗ ) 2 × 10 -3 ) -2 / ( n σ -1) (41) \n- · The curvaton should decay before the Big Bang nucleosynthesis (BBN), that is T dec > 1 MeV. From (28), the minimum mass of PBHs is related to T dec as \nT dec glyph[similarequal] 1 × 10 3 GeV ( g ∗ 100 ) -1 / 4 ( 10 26 g M min ) 1 / 2 . (42) \nHence the minimum mass of PBHs is constrained as \nM min glyph[lessorsimilar] 1 × 10 38 g ( g ∗ 100 ) -1 / 2 , (43) \nwhich is combined with (41) and we get \nM min glyph[lessorsimilar] min [ 2 × 10 46 -12 / ( n σ -1) g , 1 × 10 38 g ] , (44) \nwhere we set g ∗ ≈ 100, k c = 1 Mpc -1 and P ζ, curv ( k ∗ ) = 2 × 10 -3 . \n- · The reheating temperature is also constrained. In the case of Γ I < m σ , from (11) and (42) we obtain \nT R ≈ 6 × 10 3 GeV ( 10 26 g M min ) 1 / 2 ( M P H inf ) 2 , (45) \nwhere we take fθ ≈ H inf , r ≈ 1 and g ∗ ≈ 100. In the case of Γ I > m σ , on the other hand, we get the similar relation by simply replacing T R with T osc . The reheating temperature or the curvaton oscillation temperature is constrained from the inequality (40). \nWe show the parameter space allowing for the PBH to take a role of the dominant component of the CDM in Fig. 4 in the case of Γ I < m σ and k c = 1 Mpc -1 . The allowed region is inside the respective contours. The dashed-and-dotted-cyan line is the lower limit on the PBH mass, which comes from the current upper limit on the tensor-to-scalar \nFigure 4: The allowed parameter region in which PBHs become the dominant component of the CDM in our model is shown. The allowed region is inside the respective contours. The dotted-blue line and the solid-red line correspond to the boundary in the case of n σ = 3 and 2 respectively. The dashed-and-dotted-cyan line is the lower limit on the PBH mass coming from the upper limit on the tensor-to-scalar ratio. We have taken r = 1 and θ = 1. \n<!-- image --> \nratio. For the PBH to take a role of the dominant component of the CDM, we need the somewhat high reheating temperature T R glyph[greaterorsimilar] 10 12 GeV. \nThen, we investigate the allowed region of our model parameters, f and m σ , in which the current dark matter density can be explained by the PBH. The allowed parameter region becomes much narrower than Fig. 4 if we take into account the decay rate formula (13). The relation between the decay temperature of the curvaton and PBH mass given by (42) is rewritten as \nf glyph[similarequal] 1 × 10 14 GeV κ ( M min 10 26 g ) 1 / 2 ( m σ 10 6 GeV ) 3 / 2 , (46) \nwhich constrains the allowed region in f -m σ plane when 10 17 g < M min < 10 27 g is imposed. In the case of m σ > Γ I , (11) is translated into the following inequality : \nf glyph[greaterorsimilar] 1 × 10 14 GeV κ 1 / 3 ( m σ 10 6 GeV ) 1 / 3 , (47) \nwhere we set θ = 1 and g ∗ ≈ 100. In the case of m σ < Γ I , on the other hand, the relation from (12) becomes approximately same as (47) but replacing glyph[greaterorsimilar] with glyph[similarequal] . Moreover, the constraints (40) is trivially translated into upper bound on f by simply replacing H inf with fθ . The constraint (44) is rewritten as \nf glyph[lessorsimilar] 1 × 10 18 GeV κ ( M c 10 34 g ) 1 / 2 ( m σ 10 6 GeV ) 3 / 2 , (48) \n<!-- image --> \nFigure 5: The allowed regions for the PBH to be the dominant dark matter in f -m σ plane and f - Λ plane are shown. Inside the thick solid-red (dashed-green) lines, the conditions (47) and (48) are satisfied for κ = 1 (0 . 01). The thick (thin) dashed-anddotted-cyan lines corresponds to the upper limit which comes from the maximum mass of PBH dark matter: M BH = 10 27 g for κ = 1 (0 . 01), so the allowed parameters are inside the yellow shaded regions. The thin small-dotted magenta lines correspond to M BH = 10 25 g for κ = 0 . 01. We have taken n σ = 2 and θ = 1 and assumed m σ > Γ I in both figures. \n<!-- image --> \nwhere M c is defined as the right-hand-side of (44). \nWe summarize the above constraints in Fig. 5(a) and Fig. 5(b) for n σ = 2. In the case of m σ > Γ I , the conditions (47) and (48) correspond to the region inside the thick solid-red (dashed-green) lines for κ = 1 (0 . 01). In addition, it must be below the thick (thin) dashed-and-dotted-cyan lines corresponding to the upper bound of the PBH mass, 10 27 g for κ = 1 (0 . 01). Thus, the allowed parameters are inside the yellow shaded regions. In the opposite case, m σ < Γ I , allowed parameters are on the lower boundary of these regions. From these, it is found that f and m σ must be f ∼ 5 × 10 13 - 10 14 GeV, m σ 5 × 10 5 - 10 8 GeV and Λ 10 10 - 10 11 GeV to explain the current CDM abundance. 3 \n∼ \n× Another outcome of our model is the possibility of explaining the seeds of SMBHs. The initial mass fraction of PBHs as seeds of SMBHs is constrained by the observed comoving number density of QSOs: a 3 n QSO glyph[similarequal] (6 ± 2) × 10 -10 Mpc -3 [16]. The comoving number density of PBHs is given by \n∼ \na 3 n PBH glyph[similarequal] 6 × 10 18 β Mpc -3 ( g ∗ 10 ) -1 / 4 ( M glyph[circledot] M BH ) 3 / 2 , (49) \nso, compared with a 3 n QSO , β is estimated as \nβ ∼ 2 × 10 -21 ( g ∗ 10 ) 1 / 4 ( M BH 10 5 M glyph[circledot] ) 3 / 2 . (50) \nSince a quite large mass and a narrow mass spectrum of the PBH are needed to explain the SMBH, we set M min ∼ M ∗ which leads α glyph[similarequal] 0 . 1, H inf /fθ ∼ 2 and P ζ, curv ( k ∗ ) ∼ 1 × 10 -2 . Then the parameter space is constrained by the same way as those of the PBH dark matter case and we summarize it in Fig. 6. \nIn Fig. 6(a), the allowed region is inside the solid-red line (the dotted-blue line) for n σ = 2 . 5 ( n σ = 2 . 75). The dashed-and-dotted-cyan line (the small-dotted-magenta line) corresponds to M BH = 10 5 M glyph[circledot] (10 4 M glyph[circledot] ), on which the SMBH is explained by the PBH. In Fig. 6(b), the allowed region is inside the solid-red line (the dashed-green line) for κ = 1 (0 . 01). The thick (thin) dashed-and-dotted-cyan line and small-dotted-magenta line correspond to M BH = 10 5 M glyph[circledot] and M BH = 10 4 M glyph[circledot] respectively for κ = 1 (0 . 01). We found that our model can provide the seeds of SMBHs for T R glyph[greaterorsimilar] 10 9 GeV, f ∼ 10 12 GeV, m σ 0 . 5 - 100 GeV and Λ ∼ 10 6 - 10 7 GeV. \n∼ \n∼ However, the SMBHs cannot be a significant part of the dark matter density of the universe. Fortunately, various axion-like particles often appears in particle physics theories. One of them may be the curvaton which is responsible for SMBHs as discussed above. Another axion field can play a role of the usual QCD axion which solves the strong CP problem. If the Peccei-Quinn scale f a is ∼ 10 12 GeV, the QCD axion can account for the dark matter of the universe. The coincidence of two independent scales f glyph[similarequal] f a ∼ 10 12 GeV may be very interesting. Furthermore, it is pointed out that the axion dark matter is a good candidate consistent with the presence of the primordial SMBHs [34]. The required scale Λ ∼ 10 6 - 10 7 GeV is coincide with the SUSY breaking scale when it is mediated by gauge interactions, which suggests that the dynamics generates the curvaton mass may be related to physics of SUSY breaking.', '4 Conclusion': 'We considered the axion-like curvaton model based on the SUSY, in which the curvaton is identified as the phase direction contained in some complex scalar field. Because of the Hubble-induced mass of the radial part of the complex scalar field, the power spectrum of the curvature perturbations from the curvaton becomes extremely blue such as n σ = 2 4. In order not to contradict with the WMAP observation of the spectral index, in our model, the large scale perturbations ( k glyph[lessorsimilar] 1 Mpc -1 ) are generated by the inflaton giving the almost scale-invariant power spectrum and the contribution from the curvaton becomes significant at sufficiently small scales. We showed that, by use of such a extremely blue spectrum, the PBHs are formed from the collapse of the overdensity regions and the produced PBHs have a peaked mass spectrum. It is found that, in a certain parameter region, PBHs with mass 10 17 -10 27 g can eventually become the dominant component \n<!-- image --> \nFigure 6: The allowed regions for the PBH to be the seed of the SMBHs in M BH -T R plane and f -m σ plane are shown in Fig. 6(a) and Fig. 6(b) respectively. In Fig. 6(a), the allowed region is inside the solid-red line (the dotted-blue line) for n σ = 2 . 5 ( n σ = 2 . 75) and the dashed-and-dotted-cyan line (small-dotted-magenta line) corresponds to M BH = 10 5 M glyph[circledot] (10 4 M glyph[circledot] ). In Fig. 6(b), the allowed region is inside the solid-red line (the dashed-green) and the thick (thin) dashed-and-dotted-cyan line and small-dotted-magenta line correspond to M BH = 10 5 M glyph[circledot] and M BH = 10 4 M glyph[circledot] respectively for κ = 1 (0 . 01). We have taken r = 1 and θ = 1 in both figures and n σ = 2 . 75 in Fig. 6(b). \n<!-- image --> \nof the CDM. Furthermore, it is found that the PBHs with quite large masses ( ∼ 10 5 M glyph[circledot] ) and very narrow mass spectrum can be formed and these can be the seeds of SMBHs. \nIn this paper we have derived the scalar potential in the frame work of supergravity. However, we can build the model without supersymmetry if we start with the potential (3). The Hubble induced mass terms (5) which are necessary for generating the blue-tilted power spectrum can be obtained through couplings with the inflaton field. For example, suppose that a scalar ϕ causes chaotic inflation and its potential is given by V ( ϕ ) = λϕ 4 . Then the term like gϕ 2 | Φ | 2 ( g : small coupling) lead to the Hubble induced mass term for Φ if we take appropriate g .', 'Acknowledgment': 'We thank Fuminobu Takahashi for useful discussions. This work is supported by Grantin-Aid for Scientific research from the Ministry of Education, Science, Sports, and Culture (MEXT), Japan, No. 14102004 (M.K.), No. 21111006 (M.K.) and also by World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan. N.K. is supported by the Japan Society for the Promotion of Science (JSPS).', 'References': "- [1] E. Komatsu et al. [WMAP Collaboration], Astrophys. J. Suppl. 192 , 18 (2011) [arXiv:1001.4538 [astro-ph.CO]].\n- [2] For a review, see S. P. 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Lyth, 'Cosmological Inflation and Large-scale Structure,' (Cambridge University Press, Cambridge, England, 2000).\n- [30] M. Y. .Khlopov and A. G. Polnarev, Phys. Lett. B 97 , 383 (1980); A. G. Polnarev and M. Y. .Khlopov, Sov. Phys. Usp. 28 , 213 (1985) [Usp. Fiz. Nauk 145 , 369 (1985)].\n- [31] C. T. Byrnes, E. J. Copeland and A. M. Green, arXiv:1206.4188 [astro-ph.CO].\n- [32] R. Bean and J. Magueijo, Phys. Rev. D 66 , 063505 (2002) [astro-ph/0204486].\n- [33] H. I. Kim and C. H. Lee, Phys. Rev. D 54 , 6001 (1996).\n- [34] T. Bringmann, P. Scott and Y. Akrami, Phys. Rev. D 85 , 125027 (2012) [arXiv:1110.2484 [astro-ph.CO]]."}

2014CQGra..31a5002J

Distinguishing black holes from naked singularities through their accretion disc properties

2014-01-01

['-', '-', '-', 'gravitation', 'black hole physics', '-', '-', '-', '-']

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We show that, in principle, a slowly evolving gravitationally collapsing perfect fluid cloud can asymptotically settle to a static spherically symmetric equilibrium configuration with a naked singularity at the center. We consider one such asymptotic final configuration with a finite outer radius, and construct a toy model in which it is matched to a Schwarzschild exterior geometry. We examine the properties of circular orbits in this model. We then investigate the observational signatures of a thermal accretion disc in this spacetime, comparing them with the signatures expected for a disc around a black hole of the same mass. Several notable differences emerge. A disc around the naked singularity is much more luminous than one around an equivalent black hole. Also, the disc around the naked singularity has a spectrum with a high frequency power law segment that carries a major fraction of the total luminosity. Thus, at least some naked singularities can, in principle, be distinguished observationally from the black holes of the same mass. We discuss the possible implications of these results.

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https://arxiv.org/pdf/1304.7331.pdf

{'Distinguishing black holes from naked singularities through their accretion disk properties': 'Pankaj S. Joshi, 1, ∗ Daniele Malafarina, 2, 1, † and Ramesh Narayan 3, ‡ \n1 Tata Institute of Fundamental Research, Homi Bhabha Road, Colaba, Mumbai 400005, India 2 Department of Physics, Fudan University, 220 Handan Road, Shanghai 200433, China 3 Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA \nWe show that, in principle, a slowly evolving gravitationally collapsing perfect fluid cloud can asymptotically settle to a static spherically symmetric equilibrium configuration with a naked singularity at the center. We consider one such asymptotic final configuration with a finite outer radius, and construct a toy model in which it is matched to a Schwarzschild exterior geometry. We examine the properties of circular orbits in this model. We then investigate observational signatures of a thermal accretion disk in this spacetime, comparing them with the signatures expected for a disk around a black hole of the same mass. Several notable differences emerge. A disk around the naked singularity is much more luminous than one around an equivalent black hole. Also, the disk around the naked singularity has a spectrum with a high frequency power law segment that carries a major fraction of the total luminosity. Thus, at least some naked singularities can, in principle, be distinguished observationally from black holes of the same mass. We discuss possible implications of these results.', 'I. INTRODUCTION': 'The no hair theorems in black hole physics state that a black hole is completely characterized by three basic parameters, namely, its mass, angular momentum, and charge. Because of this extreme simplicity, the observational properties of a black hole are determined uniquely by these three intrinsic parameters plus a few details of the surrounding environment. \nThere is compelling observational evidence that many compact objects exist in the universe, and there are indications that some of these might have horizons [1]. However, there is as yet no direct proof that any of these objects is necessarily a black hole. The dynamical equations describing collapse in general relativity do not imply that the final endstate of gravitational collapse of a massive matter cloud has to be a black hole (see [2]); other possibilities are also allowed. In the case of a continual collapse, general relativity predicts that a spacetime singularity must form as the endstate of collapse. However, recent collapse studies show that, depending on the initial conditions from which the collapse evolves, trapped surfaces form either early or late during the collapse. Correspondingly, the final singularity may either be covered, giving a black hole, or may be visible as a naked singularity. If astrophysical objects of the latter kind hypothetically form in nature, it is important to be able to distinguish them from their black hole counterparts through observational signatures. \nGeneral relativity has never been tested in the very strong field regime, and very little is known about how matter behaves towards the end of the gravitational collapse of a massive star, when extremely large densities are reached and quantum effects possibly become relevant. Over the years, it has often been suggested that some exotic stable state of matter might occur below the neutron degeneracy threshold, allowing for the existence of quark stars, or boson stars, or even more exotic astrophysical objects (see for example [3]). \nAt present we cannot rule out the possibility that such compact objects do exist. On the other hand, the densities and sizes of compact objects in the universe vary enormously, depending on their mass. While a stellar mass quark star could have a density greater than that of a neutron star, a supermassive compact object at the center of a galaxy might have a density comparable to that of ordinary terrestrial matter. This means that it is difficult to come up with a single paradigm for all compact objects by simply modifying the equation of state of matter. Also, the processes that lead to the formation of stellar mass compact objects are different from those leading to supermassive objects. For one thing, time scales are very different. Stellar mass compact objects form in a matter of seconds, when the core of a massive dying star implodes under its own gravity. We know little about the physical processes that lead to the formation of supermassive compact objects, but whatever it is, it doubtless operates far more slowly than in the stellar-mass case. \nIn our view it is important to study viable theoretical models that, under reasonable physical conditions, lead to the formation of different kinds of compact objects, and to investigate the properties of these different end states. In recent times, much attention has been devoted to the observational properties of spacetimes that describe very compact objects or singularities where no horizon is present [4]. Many valid solutions of Einstein field equations exist which describe spacetimes that are not black holes. These are either vacuum solutions with naked singularities or collapse solutions in the presence of matter. Naked \nsingularity models include the Reissner-Nordstrom, Kerr and Kerr-Newman geometries with a range of parameter values that differentiate the black hole and naked singularity regimes. Collapse models include dust collapse, models with perfect fluids and those with other equations of state [2]. \nNote further that the classical event horizon structure of the Kerr metric can be altered in many ways. One way is by overspinning the Kerr black hole in order to obtain a naked singularity [5]. Another is by introducing deformations or scalar fields to alter the spacetime and thus expose a naked singularity [6]. Although the physical viability of some of these examples is not clear, the fact remains that classical general relativity allows for the formation of naked singularities in a variety of situations. If singular solutions represent a breakdown of the theory in the regime of strong gravity, then a study of some of these models might provide clues to new physics. For example, string theory or quantum gravity corrections can remove the Kerr singularity, leaving open the possibility of non singular superspinning solutions [7]. \nIn a recent paper [8], the present authors showed that equilibrium configurations describing an extended compact object can in principle be obtained from gravitational collapse. The models we described correspond to a slowly evolving collapsing cloud which settles asymptotically to a static final configuration that is either regular or has a naked singularity at the center. In this context, a naked singularity must be understood as a region of arbitrarily large density that is approached as comoving time goes to arbitrarily large values. We briefly examined the properties of accretion disks around the naked singularity solutions. There were significant physical differences compared to disks around black holes, and it followed that there could be astrophysical signatures that could distinguish black holes from naked singularities. \nIn the above previous work, we considered the case of purely tangential pressure and vanishing radial pressure. Are the solutions we obtained a consequence of this extreme simplification, or are they representative of a generic class of solutions that survive even for more reasonable equations of state? We answer this question here at least partly by exploring gravitational collapse of perfect fluid objects with an isotropic pressure. Even in this more realistic case, we find that objects with either regular interiors or naked singularities form readily as a result of gravitational collapse. We explore the similarities and differences between these new solutions and the earlier solutions which had purely tangential pressure. \nOur main purpose in the present paper is to investigate whether the naked singularity models derived here and in our previous work can be observationally distinguished from black holes of the same mass. We therefore take a further step in this direction by calculating the spectral energy distributions of putative accretion disks. We show that important differences exist in the physical properties of accretion disks around naked singularities compared to those around black holes, which may help us distinguish black holes from naked singularities through observations of astrophysical objects. Although the toy models considered here are unlikely to be realized physically, some general features of these objects are revealed by our analysis and show that naked singularities could be observationally distinguished from black holes. \nThe structure of the paper is as follows: In section II we describe a procedure by which a static perfect fluid object with a Schwarzschild exterior metric can be obtained via gravitational collapse from regular initial conditions. In section III, we use the above procedure to obtain a toy model of a static final configuration with a naked singularity at the center. We then describe the properties of accretion disks in this toy spacetime and compare these models to disks around a Schwarzschild black hole of the same mass. We also briefly discuss other density profiles of astrophysical interest that could be studied within the same framework. In the final section IV, we discuss possible applications to astrophysical observations.', 'II. GRAVITATIONAL COLLAPSE': "Spherical collapse in general relativity can be described by a dynamical spacetime metric of the form \nds 2 = -e 2 ν dt 2 + R ' 2 G dr 2 + R 2 d Ω 2 , (1) \nwhere ν , R and G are functions of the comoving coordinates t and r . For the perfect fluid case, the energy-momentum tensor is given by T 0 0 = /epsilon1, T 1 1 = T 2 2 = T 3 3 = p . The Einstein equations then take the form \np = -˙ F R 2 ˙ R , (2) ' \n/epsilon1 = R 2 R ' , (3) \nF \nν ' = -p ' /epsilon1 + p , (4) \n˙ G = 2 ν ' R ' ˙ RG , (5) \nwhere ( ' ) denotes a derivative with respect to comoving radius r and (˙) denotes a derivative with respect to time t in the ( r, t ) representation. The function F , called the Misner-Sharp mass, describes the amount of matter enclosed within the shell labeled \nF = R (1 -G + e -2 ν ˙ R 2 ) . (6) \nEquations (2)-(6) give five relations among the six unknown functions p ( r, t ) , /epsilon1 ( r, t ) , ν ( r, t ) , G ( r, t ) , F ( r, t ) and R ( r, t ) . We thus have the freedom to specify one free function. An assumed equation of state relating pressure to energy density during the evolution of the system would fix this remaining freedom and give a closed set of equations. It could happen, however, that this approach leads to an analytically intractable problem. Moreover, we do not know if the collapsing matter will have the same unchanged equation of state as it evolves to higher and higher densities as the collapse progresses. There is in fact no astrophysical or mathematical requirement that the equation of state must be fixed as the collapse evolves in time. We therefore prefer here instead to choose the functional form of F ( r, t ) , which corresponds implicitly to fixing the equation of state, which could however change with time and space coordinates. We choose a scenario such that we approximate some standard equation of state at early times, switching to some other, possibly exotic, kind of matter at later times. \nWe use the scaling degree of freedom in the definition of R to fix the initial condition R ( r, t i ) = r , where t i is the initial time. To describe collapse we require ˙ R < 0 , which guarantees that R decreases monotonically with respect to t . Hence, we may change coordinates from ( r, t ) to ( r, R ) , thus in effect considering t = t ( r, R ) . Correspondingly, we can view the original function F ( r, t ) as a function of r and R and write F = F ( r, R ) . In the following, we use ( , r ) to denote a derivative with respect to r in the ( r, R ) representation, i.e., \nF ' = F ,r + F ,R R ' . (7) \nThe total mass of the system is not conserved during collapse, unless one requires the further condition that F ( r b , t ) = const , where r b corresponds to the boundary of the system. Therefore, we cannot in general match a collapse solution to an exterior Schwarzschild metric. However, matching to a generalized Vaidya spacetime at the boundary R b ( t ) = R ( r b , t ) is always possible [9]. \nThe procedure to solve the above system of Einstein equations is the following. We choose the free function F ( r, R ) globally and use equations (2) and (3) to obtain p ( r, R ( r, t )) and /epsilon1 ( r, R ( r, t )) as functions of r , R and its derivatives. We then integrate equation (4) to obtain ν , \nν ( r, R ) = -∫ r 0 p ' /epsilon1 + p d ˜ r, (8) \nand integrate equation (5) to obtain G , \nG ( r, R ) = b ( r ) e 2 ∫ R r ν ' R ' d ˜ R . (9) \nThe free function b ( r ) results from the integral in equation (9); it is related to the velocity profile in the collapsing cloud. The integral in equation (8) again gives a free function of t , but this can be absorbed via a redefinition of the time coordinate. Once we have ν ( r, R ) and G ( r, R ) as functions of r , R and its derivatives, we can integrate the Misner-Sharp mass equation (6) that becomes a differential equation involving R and its derivatives. We can write it in the form \nt ,R = -e -ν √ F R + G -1 , (10) \nand its integration gives t ( r, R ) , or equivalently R ( r, t ) thus solving the system. \nSubstituting equations (8) and (9) in equation (1), the metric of the collapsing spacetime takes the form \nds 2 = -e -2 ∫ r 0 p ' /epsilon1 + p d ˜ r dt 2 + R ' 2 b ( r ) e 2 ∫ R r ν ' R ' d ˜ R dr 2 + R 2 d Ω 2 . (11) \nIt may not always be possible to fully integrate the system of Einstein equations globally. However, this is not always needed, because by considering the behaviour of the functions involved, it is often possible to extract useful information about collapse and to integrate the solution at least in a neighborhood close to the center. \nTypically some restrictions are required in order for the collapse model to be considered physically viable. They are: \n- 1. Absence of shell crossing singularities, which arise from a breakdown of the coordinate system at locations where collapsing shells intersect. This requirement implies R ' > 0 . We note in this connection that there is always a neighborhood of the central line r = 0 of the collapsing cloud which contains no shell-crossing singularities throughout the collapse evolution [10]. \n- 2. Energy conditions, of which the usual minimal requirement is the weak energy condition, viz., positivity of the energy density ( /epsilon1 > 0 ) and of the sum of density and pressure ( /epsilon1 + p > 0 ). This requires F ' ( r, R ) > 0 and F ,r ( r, R ) > 0 .\n- 3. Regularity at the center during collapse before the formation of the singularity. This includes the requirements that forces and pressure gradients vanish at the center and that the energy density has no cusps at r = 0 . The corresponding requirements are F ( r, R ) /similarequal r 3 m ( r, R ) near r = 0 , m ' (0 , R ) = 0 and p ' (0 , R ) = 0 .\n- 4. Absence of trapped surfaces at the initial time. This last requirement is given by positivity of [1 -( F/R )] t = t i and translates to ˙ R 2 t = t i < ( e 2 ν G ) t = t i . \nOur aim now is to construct a dynamical collapse evolution such that the pressure eventually balances gravitational attraction and the collapsing object settles into a static configuration. Such a scenario is of course not always possible in gravitational collapse of a massive matter cloud, and there are matter configurations that can only collapse indefinitely without achieveing any possible equilibrium. Such is the case of collapse of a pressureless dust cloud [11], or similar systems where pressure is very insignificant. Another possibility in collapse is that the cloud bounces back after reaching a minimum radius. Typically, since we begin with a collapsing configuration, for each shell labeled by the comoving radius r , three different behaviours are possible: \n- 1. Collapse: ˙ R < 0 . If ˙ R is negative at all times the shell will collapses to a central singularity.\n- 2. Bounce: ˙ R = 0 at a certain time, with R > 0 at this time. The shell bounces back and re-expands.\n- 3. Equilibrium configuration: ˙ R = R = 0 . Here the collapse slows down and the shell achieves a static configuration. \nShells that achieve an equilibrium configuration in a certain sense mark the separation between the region that collapses indefinitely and the region that eventually bounces back [12]. Both collapse and bounce would occur typically very rapidly, on a dynamical time scale which is proportional to the mass. Even in the case of supermassive compact objects, this time is much shorter than other typical time scales. \nIt would appear that all three possibilities above represent generic behavior during gravitational collapse, depending on the masses and velocities of the collapsing shells and the physical scenario involved in collapse. For example, for a very massive star, an indefinite collapse would seem inevitable if the star cannot shed away enough of its mass in a very short collapse time scale to achieve any possible equilibrium. On the other hand, for a much larger mass scale such as galactic or yet larger scales, the collapse could proceed much more slowly and even arrive at an equilibrium to form a stable massive object. It is well known of course that gravitational collapse plays a key role in the formation of large scale structures in the universe. In many such cases, an evolving collapse would slow down, eventually to form a stable massive object. In such cases, scenarios such as the one considered here could be relevant. \nWe note that in order to achieve an equilibrium object as the endstate of collapse or in order to have a useful quasi-equilibrium configuration, it is necessary that each collapsing shell in the cloud must individually achieve the condition ˙ R = R = 0 . Such an object can be well approximated by a static configuration. The equation of motion (6) can be written in terms of R , for any fixed comoving radius r , in the form of an effective potential: \nV ( r, R ) = -˙ R 2 = -e 2 ν ( F R + G -1 ) . (12) \nWealready know that no static configuration is possible for the pressureless (dust) case, where V is negative at all times. When there is pressure, it is still possible for V to be negative at all times, giving continued collapse. However, other possibilities are also allowed since at any given time t each shell r can be either collapsing, expanding or still giving rise to a wide array of scenarios. We shall consider here the simple but very common case where V , as a function of R for fixed r , is a polynomial of second order in R . We see that one generic possibility in this case is that V has two distinct zeroes. In this case, the shell will bounce at a finite radius and re-expand. Another possibility is that the two zeroes of V coincide (namely V has one root of double multiplicity) corresponding to an extremum at V = 0 . In this case, the shell will coast ever more slowly towards the radius corresponding to V = 0 and will halt without bouncing having reached such radius with zero velocity and zero acceleration as t goes to infinity. A global static configuration for this form of the potential is then achieved only if each shell satisfies the condition that it has one double root. It is not difficult to see that the velocity of the cloud during collapse is always non-zero and therefore such an equilibrium configuration can be achieved only in a limiting sense, as the comoving time t goes to infinity (for a more detailed discussion on the condition leading to equilibrium see [8]). For more general forms of the potential the allowed regions of dynamics for the shell and its behaviour are decided by the multiplicity of the roots of V . \nThe condition that the spacetime evolves towards an equilibrium configuration is thus \n˙ R = R = 0 , (13) \nwhich is equivalent to \nFrom equation (12) we find \nV ,R = e 2 ν ( F R 2 -F ,R R + G ,R ) -2 ν ,R e 2 ν ( F R + G -1 ) . (15) \nThe system achieves an equilibrium configuration if the solution of the equation of motion (10), given by R ( r, t ) , tends asymptotically to an equilibrium solution R e ( r ) such that the conditions given in equations (13) or (14) are satisfied. Therefore, in order to have \nR ( r, t ) ---→ t →∞ R e ( r ) , (16) \nwe must choose the free function F ( r, R ) in the dynamical collapse scenario such that the quantities F , ν , G tend to their respective equilibrium limits: \nF ( r, R ) F e ( r ) = F ( r, R e ( r )) , (17) \nν ( r, R ) → ν e ( r ) = ν ( r, R e ( r )) , (18) \n→ \n→ G ( r, R ) → G e ( r ) = G ( r, R e ( r )) . (19) \nImposing the conditions (14), we thus obtain two equations which fix the behaviour of G and G ,R at equilibrium: \nG e ( r ) = 1 -F e R e , (20) \n( G ,R ) e = G ,R ( r, R e ( r )) = F e R 2 e -( F ,R ) e R e , (21) \nwhere the velocity profile b ( r ) in equation (9) has been absorbed into G e ( r ) . \nNote that at equilibrium the area radius R becomes a monotonic increasing function of r . Therefore if we define the new radial coordinate ρ at equilibrium as \nρ ≡ R e ( r ) , (22) \nwe can rewrite the functions at equilibrium as \nV = V ,R = 0 . (14) \nF e ( r ) = F( ρ ) , (23) \nν e ( r ) \n= \nφ ( ρ ) , \n(24) \nG e ( r ) = G( ρ ) = 1 -F ρ . (25) \nThen, from equations (3) and (4), two of the Einstein equations for a static source become \n/epsilon1 ( ρ ) = F ,ρ ρ 2 , (26) \np ,ρ \n= \n-( /epsilon1 + p ) φ ,ρ , \n(27) \nwhere ( , ρ ) denotes a derivative with respect to the new static radial coordinate ρ . The second equation is the well known Tolman-Oppenheimer-Volkoff (TOV) equation. The third static Einstein equation, namely \np = 2 φ ,ρ ρ G( ρ ) -F( ρ ) ρ 3 , (28) \nis obtained from equation (2) by imposing the equilibrium condition and making use of equation (5) at equilibrium. The metric (11) at equilibrium then becomes the familiar static spherically symmetric spacetime, \nds 2 = -e 2 φ dt 2 + dρ 2 G + ρ 2 d Ω 2 , ρ ≤ ρ b ≡ R e ( r b ) . (29) \nThis interior metric is matched to a Schwarzschild vacuum exterior at the boundary ρ b = R e ( r b ) . By matching g ρρ at the matching radius ρ b and making use of equation (25), we see that the total gravitational mass M T of the interior is given by \n1 -2 M T ρ b = G ( ρ b ) = 1 -F ( ρ b ) ρ b . (30) \nNote that, in principle, the interior metric in equation (29) need not be regular at the center, as the eventual singularity is achieved as the result of collapse from regular initial data. A singularity at the center of the static final configuration is then interpreted as a region of arbitrarily high density that is achieved asymptotically as the comoving time t goes to arbitrarily large values. Therefore when considering static interiors with a singularity we are in fact approximating a slowly evolving configuration, where shells have typically very 'small' velocities and where the central region can reach very high densities. \nTypically, for a static perfect fluid source of the Schwarzschild geometry, various conditions could be required in order to ensure physical reasonability [13]. Some of these are: the matter satisfies an energy condition, matching conditions with the exterior Schwarzschild geometry, vanishing of the pressure at the boundary, monotonic decrease of the energy density and pressure with increasing radius, sound speed within the cloud should be smaller than the speed of light. If the energy conditions are satisfied during collapse, they will be satisfied by the equilibrium configuration as well, i.e., the positivity of the energy density and sum of density and pressure at the origin follows from the same condition during collapse. Also, requiring only the weak energy condition allows the possibility of negative pressures either during the collapse phase or in the final equilibrium configuration. \nSince we have required here the static configuration to evolve from regular initial data, we may omit the condition that the final state must be necessarily regular at the center. This leaves open the possibility that a central singularity might develop as the equilibrium is reached, where the singularity has to be understood in the sense explained above. Absence of trapped surfaces at the initial time ensures that, if a singularity develops, it will not be covered by a horizon [14]. In fact it can be shown that if trapped surfaces do form at a certain time as the cloud evolves, then the collapse cannot be halted and the whole cloud must collapse to a black hole or to a singularity where the first point of singularity is visible but the later portions of the singularity become covered in a black hole. We note that some of the above conditions, although desirable, may be neglected in special cases. Many interior solutions are available in the literature describing a static sphere of perfect fluid matching smoothly to a Schwarzschild exterior geometry (see [15] for a list of solutions or [16] for an algorithm to construct interior solutions). As discussed above, we can construct dynamically evolving collapse scenarios that lead asymptotically to the formation of such static configurations. If we are able to achieve a static configuration from collapse from regular initial data, and if we view a singularity as a region where density increases to arbitrarily high values, signaling a breakdown of the ability of general relativity to model the spacetime, we may then neglect the requirement of regularity at the center.", 'III. A TOY MODEL OF A STATIC SPHERICALLY SYMMETRIC PERFECT FLUID INTERIOR': 'In the following, we look for static interiors with a singularity at the origin. We wish to investigate the properties of such solutions and to establish whether such hypothetical naked singularity objects could in principle be distinguished in terms of observational signatures from black hole counter-parts of the same mass. \nWe start with the most general static spherically symmetric metric written in the form given in equation (29). As we have seen, the perfect fluid source Einstein equations give \n/epsilon1 = F ,ρ ρ 2 , (31) \np = 2 φ ,ρ ρ [ 1 -F( ρ ) ρ ] -F( ρ ) ρ 3 , (32) \np ,ρ = -( /epsilon1 + p ) φ ,ρ , (33) \nwhere we have absorbed the factor 4 π into the definition of /epsilon1 and p , and defined F ,ρ = d F /dρ . The third (Tolman-OppenheimerVolkoff) equation, combined with the second equation, gives \np ,ρ = -( /epsilon1 + p ) [ pρ 3 +F( ρ ) ] 2 ρ [ ρ -F( ρ )] . (34) \nSince the above system of Einstein equations consists of three equations with four unknowns, in order to close the system we can either specify an equation of state that relates p to /epsilon1 or use the freedom to choose arbitrarily one of the other functions [17]. As discussed earlier, we opt for the latter approach and specify the form of the mass profile F( ρ ) , which describes the final mass distribution obtained from collapse. Once we specify F , equation (34) reduces to a first-order ordinary differential equation, which we can solve. \nWe follow the same procedure that we used in the pure tangential pressure case [8] and obtain a solution of the form \nF( ρ ) ρ = M 0 = const , ρ ≤ ρ b . (35) \nBy equation (30), this solution corresponds to a total mass M T given by \n2 M T = F( ρ b ) = M 0 ρ b . (36) \nHence, to avoid a horizon, we require M 0 < 1 . Solving the first Einstein equation (31), the energy density becomes \n/epsilon1 = M 0 ρ 2 , (37) \nwhich clearly diverges as ρ → 0 , indicating the presence of a strong curvature singularity at the center. The TOV equation (34) now becomes \np ,ρ = -( /epsilon1 + p ) 2 ρ 2(1 -M 0 ) = -( M 0 + pρ 2 ) 2 2 ρ 3 (1 -M 0 ) , (38) \nwhich can be integrated by defining the auxiliary parameter \nλ = √ 1 -2 M 0 1 -M 0 , M 0 = 1 -λ 2 2 -λ 2 . (39) \nClearly, we require λ ∈ [0 , 1) , or equivalently M 0 < 1 / 2 (for values of M 0 > 1 / 2 similar considerations apply with λ = √ -(1 -2 M 0 ) / (1 -M 0 ) but we do not consider this case here). In terms of λ , the solution of equation (38) can be written as \np = 1 2 -λ 2 1 ρ 2 [ (1 -λ ) 2 A -(1 + λ ) 2 Bρ 2 λ A -Bρ 2 λ ] , (40) \nwhere A and B are arbitrary integration constants. Then, solving the remaining Einstein equation gives \ne 2 φ = ( Aρ 1 -λ -Bρ 1+ λ ) 2 . (41) \nThis solution, among other similar interior static solutions, was first investigated by Tolman in 1939 for a specific choice of λ [18]. The energy density of the solution may be rewritten as \n/epsilon1 = ( 1 -λ 2 2 -λ 2 ) 1 ρ 2 . (42) \nThe existence of a strong curvature singularity at the center can be confirmed by an analysis of the Kretschmann scalar K near ρ = 0 , which gives \nK = 16 Q 4 (1 -λ 2 ) 2 + ρ 4 ( Q 2 ,ρ -2 QQ ,ρρ ) 2 +8 ρ 2 Q 2 Q 2 ,ρ 4 ρ 4 Q 4 (2 -λ 2 ) 2 , Q ( ρ ) = e 2 φ ( ρ ) , (43) \nwhich is clearly singular in the limit ρ 0 . \nThe static metric of the above solution takes the form \n→ \nds 2 = -( Aρ 1 -λ -Bρ 1+ λ ) 2 dt 2 +(2 -λ 2 ) dρ 2 + ρ 2 d Ω 2 , ρ ≤ ρ b . (44) \nIt is matched at the boundary ρ b to a vacuum Schwarzschild spacetime with total mass M T . Since g ρρ in the interior is a constant, the matching will not in general be smooth, though it will be continuous. From the matching conditions for g ρρ and g tt at the boundary we obtain, \nρ b = 2(2 -λ 2 ) (1 -λ 2 ) M T , (45) \n-Bρ 2 λ b = A -ρ λ -1 b √ 2 -λ 2 . (46) \nFor all values of λ ∈ [0 , 1) , the pressure is maximum at the center and decreases outwards, becoming zero at a finite radius ρ p given by \nρ 2 λ p = A B (1 -λ ) 2 (1 + λ ) 2 . (47) \nSince we require the pressure to vanish at the boundary of the cloud, we obtain the further condition ρ b = ρ p , which, together with equation (46), fixes the two integration constants A and B : \nA = (1 + λ ) 2 ρ λ -1 b 4 λ √ 2 λ 2 , (48) \nB = (1 -λ ) 2 ρ -λ -1 b 4 λ √ 2 -λ 2 . (49) \n- \nThe sound speed inside the cloud is given by c 2 s = ∂p/∂/epsilon1 . We find \nc 2 s = p /epsilon1 + 4 λ 2 ABr 2 λ (1 -λ 2 )( A -Br 2 λ ) 2 , p /epsilon1 = A 1 -λ 1+ λ -B 1+ λ 1 -λ ρ 2 λ A -Bρ 2 λ , (50) \nfrom which we see that c s (0) = (1 -λ )(1 + λ ) < 1 , as required, and c s decreases as the radius ρ goes from 0 to ρ b . We may also rewrite the above relations in the form of an equation of state, p = p ( /epsilon1 ) , \np ( /epsilon1 ) = /epsilon1 A ( 1 -λ 1+ λ )( 2 -λ 2 1 -λ 2 ) λ /epsilon1 λ -B ( 1+ λ 1 -λ ) A ( 2 -λ 2 1 -λ 2 ) λ /epsilon1 λ -B . (51) \nIn the limit λ → 0 , corresponding to M 0 → 1 / 2 , we obtain the most compact member of the above family of solutions. It has an equation of state p ≈ /epsilon1 and hence sound speed c s → 1 . Note, however, that at the outer boundary, p → 0 but /epsilon1 does not vanish, so the equation of state is not strictly isothermal. For this solution, the matching radius with the Schwarzschild exterior is at ρ b = 4 M T , i.e., twice the Schwarzschild radius. In the case of pure tangential pressure, which we considered in our previous paper [8], we found physically meaningful solutions down to ρ b = 3 M T , and even more compact solutions with exotic properties. This is one respect in which the perfect fluid model differs from the tangential pressure case. \nFor λ = 1 / 2 , corresponding to M 0 = 3 / 7 , we obtain an equation of state \np = /epsilon1 3 1 -8 B A √ 7 3 /epsilon1 -B , (52) \nwhich approaches the radiation equation of state p = /epsilon1/ 3 as ρ → 0 . For this model, ρ b = (14 / 3) M T , i.e., the object is a little less compact than the model with λ → 0 . \n→ Finally, the case λ → 1 , M 0 → 0 , corresponds to ρ b /M T →∞ , and hence an infinitely large object. In this limit, our solution reduces to the classical Newtonian singular isothermal sphere solution. Correspondingly, A → 1 , B → 0 , and the metric in equation (44) reduces to the metric of flat space. \nConsidering next the energy conditions, it is easy to see that M 0 > 0 implies positivity of the energy density and pressure. The pressure decreases from its maximum value at the center to zero at the boundary. From \n/epsilon1 + p = (1 -λ )(1 + λ ) 2 -λ 2 1 ρ 2 ( 1 + p /epsilon1 ) , (53) \nwe see that the weak energy condition is satisfied throughout the interior of the cloud.', 'A. Properties of circular orbits': "We wish to investigate basic observational properties of accretion disks orbiting in the above family of spacetimes. To this end, we study circular geodesics. Our assumption is that a test particle orbiting inside the 'cloud' does not interact with the material of the cloud but merely feels its gravitational influence [34]. \nFigure 1 shows the variation of the energy per unit mass E with radius ρ for a selection of models corresponding to λ = 1 / 100 , 1 / 2 , 1 / √ 2 , 2 / √ 5 . The boundaries of the four models are at (see eq. 45) ρ b = 4 . 0002 , 14 / 3 , 6 and 12. In all the models, E goes to zero as ρ → 0 , with a power-law dependence: E ∼ ρ 1 -λ . At ρ = ρ b , each model is matched to an exterior Schwarzschild spacetime with mass M T = 1 . Panel (b) in Figure 1 shows the matching region. \n<!-- image --> \nFIG. 1: (a) Energy per unit mass E of circular orbits as a function of radius ρ for four perfect fluid models of unit mass ( M T = 1 ). The parameter λ and the radius of the boundary ρ b of the four models are, respectively: ( λ, ρ b ) = (1 / 100 , 4 . 0002) ; (1 / 2 , 14 / 3) ; (1 / √ 2 , 6) ; (2 / √ 5 , 12) . The curves are labeled by their values of λ . Each solution is matched to a Schwarzschild exterior at ρ = ρ b . (b) Close up of the matching region between the perfect fluid interior and the Schwarzschild exterior. The models with λ = 1 / 100 , 1/2, have ρ b lying inside the innermost stable circular orbit ρ ISCO = 6 of the Schwarzschild spacetime. For these two solutions, E decreases between ρ b and ρ ISCO , indicating a zone of unstable circular orbits. \n<!-- image --> \nOur perfect fluid model has five parameters, M T , λ , ρ b , A , B , and there are three matching conditions at the boundary ρ = ρ b , viz., matching of g tt and g ρρ with the exterior Schwarzschild metric and the condition p ( ρ b ) = 0 . Thus, we are free to choose two of the five parameters. For convenience, in the following we choose the total gravitational mass of the cloud M T to be equal to unity. This still leaves one free parameter, which we choose to be λ . Once we pick a value of λ , the boundary radius ρ b is given by equation (45), and the coefficients A and B are given by equations (48) and (49). \nGiven the spherical symmetry of the metric we can always choose the coordinate θ such that the geodesic under consideration lies in the equatorial plane ( θ = π/ 2 ). Time-like geodesics in this plane satisfy \n-H ( ρ ) ( dt dτ ) 2 +(2 -λ 2 ) ( dρ dτ ) 2 + ρ 2 ( dϕ dτ ) 2 = -1 , (54) \nwhere H ( ρ ) = ( Aρ 1 -λ Bρ 1+ λ ) 2 . b \n- \nFrom the two killing vectors, ζ b , η a , associated with time-translational symmetry and rotational symmetry, we can calculate two conserved quantities, the energy per unit mass, E = g ab ζ a u b = e 2 φ dt/dτ = Hdt/dτ , and the angular momentum per unit mass, L = g ab η a u b = ρ 2 dϕ/dτ . For circular geodesics we must have dρ/dτ = 0 . Therefore, we obtain \nE 2 = 2 H 2 2 H H ,ρ ρ , (55) \nL 2 ρ 2 = H ,ρ ρ 2 H -H ,ρ ρ . (56) \n- \nBy construction, E is continuous across the matching boundary, but dE/dρ is not. Note in particular that the vacuum Schwarzschild spacetime has an innermost stable circular orbit (ISCO) at ρ ISCO = 6 . For ρ < ρ ISCO , E increases with decreasing ρ , which is one of the consequences of the absence of stable orbits. The model with λ = 2 / √ 5 has its boundary at ρ b = 12 and hence has stable circular orbits all the way from large radii down to ρ → 0 . So too does the model with λ = 1 / √ 2 \nFIG. 2: (a) Angular momentum per unit mass L of circular orbits as a function of radius ρ for the same four perfect fluid models shown in Figure 1. The values of λ are: 1 / 100 , 1 / 2 , 1 / √ 2 , 2 / √ 5 (the curves corresponding to the first and last solutions are labeled). (b) Close up of the matching region between the perfect fluid interior and the Schwarzschild exterior. Note, as in Figure 1, that the solutions with λ = 1 / 100 , 1 / 2 have L decreasing between ρ b and ρ ISCO = 6 . Circular orbits are unstable over this range of radius. \n<!-- image --> \nwhich has its matching radius at the ISCO, ρ b = ρ ISCO . However, the other two models ( λ = 1 / 2 , 1 / 100 ) have ρ b < ρ ISCO . Hence, these models have stable circular orbits inside ρ b and outside ρ ISCO = 6 , but no stable orbits in between. \nFigure 2 shows analogous results for the angular momentum per unit mass L . Here, at small radii, L ∼ ρ for all the models. As in the case of E , the angular momentum matches continuously across the boundary ρ = ρ b to the external Schwarzschild solution. As panel (b) shows, the two models with λ = 1 / 100 and 1 / 2 have dL/dρ < 0 for a range of radii between ρ b and ρ ISCO . This corresponds to the region with unstable circular orbits. \nThe angular velocity ω = dφ/dt of particles on circular orbits is given by \nω 2 = -g tt,ρ g φφ,ρ = H ,ρ 2 ρ . (57) \nThis quantity scales as ω ∼ ρ -λ as ρ → 0 . Figure 3 shows a plot of ω vs ρ for the same four models as before.", 'B. Properties of accretion disks': "Gas in an accretion disk loses angular momentum as a result of viscosity and moves steadily inwards along a sequence of nearly circular orbits. Using just the properties of circular geodesics, and without needing to know the detailed properties of the viscous stress, it is possible to calculate the radiative flux emitted at each radius in the disk [20]. \nSince in our model both E and L tend to zero as ρ goes to zero, no energy or angular momentum is added to the central singularity by the gas in the accretion disk. The central singularity may be considered to be 'stable' in this sense. Indeed, since E → 0 , all the mass energy of the accreting gas is converted to radiation and returned to infinity, i.e., the net radiative luminosity as measured at infinity satisfies L ∞ = ˙ mc 2 , where ˙ m is the rate at which rest mass is accreted. Accretion disks around our model naked singularity solutions are thus perfect engines that convert mass into energy with 100% efficiency. \nFrom the behaviour of E and L of circular geodesics (Figs. 1 and 2), we can distinguish two different regimes of accretion, depending on the value of λ : \n- · For λ ∈ [1 / √ 2 , 1) , corresponding to ρ b ∈ [6 M T , + ∞ ) , particles in the accretion disk follow circular geodesics of the Schwarzschild exterior until they reach the matching radius ρ b at the outer edge of the cloud. Inside the cloud, the particles switch smoothly and continuously to the circular geodesics of the interior solution. Thus, the accretion disk extends without any break from arbitrarily large radii down to the singularity ρ → 0 . \n<!-- image --> \nFIG. 3: (a) Angular velocity ω of circular orbits as a function of radius ρ for the same four perfect fluid models shown in Figure 1. The values of λ are: 1 / 100 , 1 / 2 , 1 / √ 2 , 2 / √ 5 (the curves corresponding to the first and last solutions are labeled). (b) Close up of the matching region between the perfect fluid interior and the Schwarzschild exterior. \n<!-- image --> \n- · For λ ∈ (0 , 1 / √ 2) , corresponding to ρ b ∈ (4 M T , 6 M T ) , particles reach the ISCO of the exterior Schwarzschild spacetime at ρ ISCO = 6 . Inside this radius no stable circular orbits are allowed, so the gas in the disk plunges with constant E = E ISCO and L = L ISCO until it crosses the boundary of the cloud at ρ b . Inside the cloud, circular geodesics are allowed again. The gas penetrates into the cloud until it reaches a radius ρ L at which the specific angular momentum of a local circular geodesic L is equal to L ISCO . At this radius, the gas settles into a stable circular orbit and radiates away any excess energy. Further evolution than proceeds in the standard fashion, with the gas steadily moving to smaller radii until ρ → 0 . The accretion disk is thus divided into two parts, one in the vacuum exterior over radii ρ ≥ ρ ISCO and the other inside the cloud over radii ρ ≤ ρ L . \nIn the following, we focus on models belonging to the first regime, where we have a continuous disk extending with no break from large radius down to the singularity. Specifically, we consider models with λ = 1 / √ 2 , 2 / √ 5 , which have matching boundaries at ρ b = 6 , 12 , respectively. The radiative properties of accretion disks in these spacetimes may be calculated using the relations given in [20]. In the local frame of the accreting fluid, the radiative flux emitted by the disk (which is the energy per unit area per unit time) is given by \nF ( ρ ) = -˙ m 4 π √ -g ω ,ρ ( E -ωL ) 2 ∫ ρ ρ in ( E -ωL ) L , ˜ ρ d ˜ ρ , (58) \nwhere ˙ m is the rest mass accretion rate, assumed to be constant, ρ in is the radius of the inner edge of the accretion disk, which is zero for our singular cloud models, and g is the determinant of the metric of the three-sub-space ( t, ρ, φ ) , \ng ( ρ ) = -(2 -λ 2 ) ρ 2 H ( ρ ) . (59) \nThe solid curves in Figure 4(a) show the variation of F ( ρ ) vs ρ for the two chosen models. The flux diverges steeply as the gas approaches the center: F ∼ ρ -(3 -λ ) . This is not surprising, considering that the cloud is singular in this limit. Perhaps more surprising is the discontinuity in the flux at the boundary between the cloud and the external vacuum metric, as seen clearly in Figure 4(b). While all the quantities E , L , ω , g which are present in equation (58) are continuous across the boundary, the derivative ω ,ρ is not. The discontinuity in ω ,ρ causes the jump in F as ρ crosses ρ b . The dotted lines in the two panels correspond to an accretion disk around a Schwarzschild black hole. In this case, the inner edge of the disk is at ρ in = ρ ISCO = 6 , and the flux cuts off at this radius. \nWe note of course that the flux F is a local quantity measured in the frame of the fluid and is not directly observable. A more useful quantity is the luminosity L ∞ (energy per unit time) that reaches an observer at infinity. The differential of L ∞ with \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFIG. 4: (a) The two solid lines show the variation of the radiative flux F of an accretion disk, as measured by a local observer comoving with the fluid, for two perfect fluid singular models with λ = 1 / √ 2 ( ρ b = 6 ) and λ = 2 / √ 5 ( ρ b = 12 ). The dotted line corresponds to a disk around a Schwarzschild black hole, while the two dashed lines are for accretion disks in models with purely tangential pressure [8], which are discussed in section III C. (b) Close-up of the flux profiles near the matching radius. Note the discontinuity in the flux, which is caused by the discontinuous behavior of ω ,ρ in equation (58). (c) Profile of the differential luminosity reaching an observer at infinity, d L ∞ /d ln ρ , for the same models. (d) Close-up of the region near the matching radius. \n<!-- image --> \nrespect to the radius ρ can be computed from F by the following relation [20]: \nd L ∞ d ln ρ = 4 πρ √ -gE F . (60) \nPanels (c) and (d) in Figure 4 show d L ∞ /d ln ρ for the two models under consideration. We see that the luminosity behaves in a perfectly convergent fashion as ρ → 0 : d L ∞ /d ln ρ ∼ ρ 1 -λ . By integrating this quantity over ln ρ , we can calculate the net luminosity L ∞ observed at infinity. We have confirmed that this is equal to ˙ mc 2 for the two singular models. That is the disk has 100% efficiency - it converts the entire rest mass energy of the accreting gas into radiation. The dotted lines in the panels indicate the very different behavior of a disk around a Schwarzschild black hole. Since such a disk is truncated at the ISCO, the \nluminosity is much less. In this case, we recover the standard result, L ∞ = (1 -E ISCO ) ˙ mc 2 = 0 . 05719 ˙ mc 2 , i.e., an efficiency of 5.719%. \nIs it possible to distinguish observationally whether a given accretion system is a black hole or one of the toy singular objects described in this paper? We have seen above that the accretion efficiencies are very different. However, the efficiency is not easily determined via observations since there is no way to obtain an indepedent estimate of the mass accretion rate ˙ m . A more promising avenue is the spectral energy distribution of the disk radiation. \nFollowing standard practice, we assume that each local patch of the disk radiates as a blackbody. Defining a characteristic temperature T ∗ as follows (where we have included physical units), \nσT 4 ∗ ≡ ˙ mc 2 4 π ( GM T /c 2 ) 2 , (61) \nwhere σ is the Stefan-Boltzmann constant, the local blackbody temperature of the radiation emitted at any radius ρ with flux F (eq. 58) is given by \nT BB ( ρ ) = [ F ( ρ )] 1 / 4 T ∗ . (62) \nThis radiation is transformed by gravitational and Doppler redshifts by the time it reaches an observer at infinity and hence appears to have a different temperature. The transformation depends on the orientation of the observer with respect to the disk axis. To avoid getting into too much detail, we simply use a single characteristic redshift z , corresponding to an observer along the disk axis, \n1 + z ( ρ ) = [ -( g tt + ω 2 g φφ )] -1 / 2 , (63) \nand assume that the radiation emitted at radius ρ has a temperature at infinity, independent of direction, given by \nT ∞ ( ρ ) = T BB ( ρ ) / (1 + z ) . (64) \nIt is then straightforward to convert the differential luminosity calculated in equation (60) into the spectral luminosity distribution L ν, ∞ observed at infinity. The result is \nν L ν, ∞ = 15 π 4 ∫ ∞ ρ in ( d L ∞ d ln ρ ) (1 + z ) 4 ( hν/kT ∗ ) 4 / F exp[(1 + z ) hν/kT ∗ F 1 / 4 ] -1 d ln ρ. (65) \nAs a check, we have verified numerically that the integral over frequency of the spectral luminosity L ν, ∞ obtained via the above relation is equal to ˙ mc 2 (100% efficiency) for the singular models and equal to 0 . 05719 ˙ mc 2 for the Schwarzschild case (5.719% efficiency). \nFigure 5 shows spectra corresponding to the same models considered in Figure 4. For a given accretion rate ˙ m , the Schwarzschild black hole model gives a much lower luminosity than the naked singular models. However, as we discussed earlier, this is not observationally testable. More interesting is the fact that the spectra have noticeably different shapes. At low frequencies, all the models have the same spectral shape νL ν ∼ ν 4 / 3 , which is the standard result for disk emission from large non-relativistic radii [21]. However, there are dramatic differences at high frequencies. \nThe emission from a disk around a Schwarzschild black hole cuts off at the ISCO radius ρ ISCO = 6 . Correspondingly, there is a certain maximum temperature for the emitted radiation, which causes the spectrum to cutoff abruptly at a frequency hν/kT ∗ somewhat below unity. The two perfect fluid singular models, on the other hand, behave very differently. In these models, the disk extends all the way down to ρ → 0 . Consequently, the temperature T BB of the emitted radiation rises steadily and diverges as ρ → 0 . The emission from all the inner radii combines to produce a power-law spectrum at high frequencies [35]: ν ν, ∞ ∼ ν -8(1 -λ ) / (6 λ -2) . \nL \n∼ As Figure 5 shows, the high-frequency power-law segment of the spectrum carries a substantial fraction (more than half) of the total emission from disks around our naked singularity models. The presence of this strong power-law spectrum is thus a characteristic feature of these models which can be used to distinguish them qualitatively from disks around black holes. Several well-known astrophysical black hole candidates are known to have spectra with a strong thermal cutoff, similar to that seen in the dotted line in Figure 5 [22]. These systems show very little power-law emission at high frequencies. In these cases at least we can state with some confidence that the central mass does not have a naked singularity of the sort discussed in this paper.", 'C. Comparison of the perfect fluid and tangential pressure models': 'In our previous paper [8], we considered a relatively restricted model in which the fluid in the cloud has non-zero pressure only in the tangential direction. Because of vanishing radial pressure, matching pressure across the boundary radius ρ b between \nFIG. 5: Spectral luminosity distribution of radiation from the accretion disk models in Figure 4. The dotted line corresponds to a disk around a Schwarzschild black hole, the solid lines correspond to disks around two perfect fluid naked singularity models discussed in this paper, and the dashed lines are for disks around two tangential pressure naked singularity models discussed in [8]. \n<!-- image --> \nthe cloud and the external Schwarzschild spacetime was trivial, and hence the solution was more easily determined. For radii inside the cloud, the metric of the pure tangential pressure model takes the form, \nds 2 = -H ( ρ ) dt 2 + dρ 2 (1 -M 0 ) + ρ 2 d Ω 2 , ρ ≤ ρ b , (66) \nwith H ( ρ ) = (1 -M 0 ) ( ρ/ρ b ) M 0 / (1 -M 0 ) and M 0 = 2 M T /ρ b , where M T is the total gravitational mass of the cloud as measured in the vacuum exterior. \nThe close similarity of the tangential pressure model and the perfect fluid model described in the present paper is obvious, e.g., compare the above expressions with equations (44) and (36). The constant M 0 in the tangential pressure model plays the role of λ for the perfect fluid case, and in both cases this parameter determines the compactness of the cloud as measured by its dimensionless radius ρ b /M T . The limits M 0 → 0 and λ → 1 in the two models correspond to infinitely large and dilute clouds that are fully non-relativistic: ρ b /M T →∞ . In this limit, the two models are essentially identical. One minor difference between the models occurs in the opposite limit. For the perfect fluid cloud, the most compact configuration we find has λ = 0 , which corresponds to ρ b = 4 M T . In contrast, for the tangential pressure model, the most compact physically valid configuration has M 0 = 2 / 3 , which corresponds to ρ b = 3 M T , i.e., a more compact object. \nAnother difference between the two models is that the tangential pressure cloud has a particularly straightforward metric in which g tt = H ( ρ ) varies with radius as a simple power-law. Therefore, all quantities behave as power-laws and the analysis is easy. In the case of the perfect fluid model, the extra matching condition on the pressure at ρ = ρ b results in the metric coefficient g tt involving two power-laws. The term involving the coefficient A in equation (44) dominates as ρ → 0 and behaves just like the lone power-law term in the tangential pressure cloud case. The second term involving B is required in order to satisfy the pressure boundary condition and plays a role only as ρ approaches ρ b . This term causes various quantities like E , L , ω , etc. to deviate from perfect power-law behavior as ρ ρ b (see Figs. 1-4). \nAs far as observables are concerned the two models behave quite similarly. In Figs. 4 and 5, the two perfect fluid models with λ = 1 / √ 2 and 2 / √ 5 have boundaries at ρ b = 6 and 12 , respectively, and their properties are shown by the solid lines. For \n→ \ncomparison, the dashed lines show results for two tangential pressure models with M 0 = 1 / 3 and 1 / 6 which have boundaries at the same radii, ρ b = 6 , 12 . While the agreement between the two sets of models is not perfect, as is to be expected since the models are different, we see very good qualitative agreement. In particular, note that, just as in the perfect fluid models, the tangential pressure models too produce a strong power-law high energy tail in the spectra of their accretion disks (Fig. 5), which may be used to distinguish these models from disks around a Schwarzschild black hole (dotted line).', 'D. Comparison with the Newtonian singular isothermal sphere': 'A simple and commonly used model in astrophysics is the singular isothermal sphere. This is a spherically symmetric selfgravitating object with an equation of state p = /epsilon1c 2 s , where p is the pressure, /epsilon1 is the density (this is usually called ρ but we use /epsilon1 here since we have already defined ρ to be the radius), and c s is the isothermal sound speed. The model satisfies the condition of hydrostatic equilibrium under the action of Newtonian gravity. The solution is \n/epsilon1 ( ρ ) = c 2 s 2 πGρ 2 , p ( ρ ) = c 4 s 2 πGρ 2 , M ( ρ ) = 2 c 2 s ρ G , (67) \nwhere M ( ρ ) is the mass interior to radius ρ . The variations of /epsilon1 , p and M with ρ are very reminiscent of the variation of /epsilon1 , p and F in the perfect fluid relativistic cloud model described earlier. \nOne deficiency of the basic singular isothermal sphere model is that it extends to infinite radius, where the mass is infinite. In order to obtain an object with a finite radius, one needs to change the equation of state such that the pressure goes to zero at a finite density. This is easily arranged as follows: \n/epsilon1 ( ρ ) = c 2 s 2 πGρ 2 = ρ 2 b ρ 2 /epsilon1 b , /epsilon1 b ≡ c 2 s 2 πGρ 2 b , p ( ρ ) = c 2 s ( /epsilon1 -/epsilon1 b ) , M ( ρ ) = 4 π/epsilon1 b r 2 b ρ, ρ ≤ ρ b , (68) \nwhere ρ b is the radius of the boundary and /epsilon1 b is the density at that radius. \nCircular orbits inside a singular isothermal sphere behave very simply, with velocity and angular momentum given by \nv circ = √ c s ρ = const , L = √ 2 c s ρ, ρ ≤ ρ b . (69) \nThe constancy of v circ is the chief attraction of the singular isothermal sphere. It provides a simple way of reproducing the observed flat rotation curves of galaxies. The scaling of L with ρ in the singular isothermal sphere is exactly the same as the variation of L with ρ in the perfect fluid relativistic cloud. At first sight it appears that the flat rotation curve property is not reproduced in the relativistic model. For example, ωρ scales as ρ 1 -λ . However, note that ω is defined as dφ/dt , where t is the time measured at infinity. The appropriate quantity to consider is dφ/dt loc , where t loc is time measured by a local ZAMO (Zero Angular Momentum Observer) at radius ρ . The two times are related by a factor of [ H ( ρ )] 1 / 2 which varies as Aρ 1 -λ in the limit ρ → 0 . Thus, for ZAMOs, the rotation velocity is indeed independent of radius in the deep interior of the cloud. (Close to the boundary, there are small deviations because of the term involving B in H ( ρ ) ). \nConsider next the energy of circular orbits in the singular isothermal sphere model. If we include the rest mass energy and add to it the orbital kinetic energy and the potential energy, then a Newtonian calculation gives for the energy of a particle of unit mass \nE circ = 1 -c 2 s c 2 ( 1 + 2 ln ρ b ρ ) , ρ ≤ ρ b . (70) \nThe weak logarithmic divergence at small radii is rarely a problem since we generally have c s /lessmuch c . Nevertheless, the presence of the logarithm implies that, in principle, at a sufficiently small radius, the Newtonian model predicts a negative total energy for the particle. This is clearly unphysical. \nIt is reassuring that the logarithmic divergence is not present in our relativistic cloud models. Both the pefect fluid model and the tangential pressure model have E varying as a power-law with radius: E ∼ ρ 1 -λ for the perfect fluid case and E ∼ ρ M 0 / 2(1 -M 0 ) for the tangential pressure case. In both cases, E asymptotes precisely to zero as ρ → 0 and does not go negative anywhere. Effectively, the relativistic models, being more self-consistent, regularize the logarithm of the Newtonian model by replacing it with a power-law. The index of the power-law is nearly 0 in the Newtonian limit but becomes as large as unity for the maximally compact configuration, viz., λ = 1 and M 0 = 2 / 3 for the perfect fluid and tangential pressure models, respectively.', 'E. Other models': 'As we have seen above, the perfect fluid relativistic model described in this paper is equivalent to the Newtonian singular isothermal sphere model used in astrophysics. The relativistic generalization was obtained by assuming that the energy density \n/epsilon1 ( ρ ) in the relativistic model has the same functional form as /epsilon1 ( ρ ) in the Newtonian model (compare eqs. 37 and 68), and then solving for the pressure p and the spacetime metric. This procedure can be followed with any other trial model of /epsilon1 ( ρ ) of interest. \nOne simple example is to consider a superposition of the toy perfect fluid model described in this paper with the well known constant density Schwarzschild interior. The density profile then takes the form \n/epsilon1 = M 0 ρ 2 + M 1 , (71) \nwhich corresponds to \nF ( ρ ) = M 0 ρ + M 1 ρ 3 / 3 . (72) \nThe density in this model approaches the singular interior of our perfect fluid model as ρ → 0 but resembles a constant density interior with /epsilon1 = M 1 as ρ → ρ b . It turns out that the TOV equation can be explicitly integrated in this case, though the expression for p is fairly complicated and involves hypergeometric functions. \nOther examples of more interest to astrophysics could be similarly considered. One natural generalization of the singular isothermal sphere is the Jaffe density profile [23], \n/epsilon1 = M 0 r 0 ρ 2 ( ρ 0 + ρ ) 2 , (73) \nwhere M 0 is a constant and ρ 0 describes the characteristic radius of the object. This density profile corresponds to \nF ( ρ ) = M 0 ρ ( ρ 0 + ρ ) . (74) \nThe Jaffe model behaves just like the singular isothermal sphere as ρ → 0 , yet it has a finite total mass given by M 0 and hence does not need to be artifically truncated as we had to in the case of the singular isothermal sphere. The Jaffe model is a special case of a more general class of models, the Dehnen density profile [24], \n/epsilon1 = (3 -γ ) M 0 ρ 0 ρ γ ( ρ + ρ 0 ) 4 -γ . (75) \nThe Jaffe profile corresponds to γ = 2 , while the case with γ = 1 is known as the Hernquist model [25]. The Dehnen profile implies a mass function \nF ( ρ ) = M 0 ( ρ ρ 0 + ρ ) 3 -γ . (76) \nFinally, we could also consider the Navarro-Frenk-White (NFW) profile [26], which is given by \n/epsilon1 = M 0 ρ ( ρ 0 + ρ ) 2 , (77) \nwhere again M 0 is a constant. This model has a logarithmically diverging mass as ρ , and is thus a little less attractive. \nIt is not easy to solve the TOV equations for the pressure p and the spacetime metric of any of the above models analytically. However, numerical solutions are easily obtained. Note that all these popular models technically have naked singularities at their centers. Obtaining relativistic generalizations along the lines followed in this paper would be worthwhile. \n→∞', 'IV. CONCLUDING REMARKS': "In the present paper we considered a self-bound spherical cloud of perfect fluid and derived static non-vacuum solutions of the Einstein equations which posseses a central naked singularity. We showed that these solutions could be obtained asymptotically as the final result of the slow collapse of a massive matter cloud. The solutions described here closely parallel those we obtained in [8] for a fluid with pure tangential pressure. \nWe studied the properties of steady thermal accretion disks in our naked singularity spacetimes. Focusing on those models that have a disk extending continuously from a large radius ρ down to ρ → 0 , i.e., models with λ ≥ 1 / √ 2 for the perfect fluid case and M 0 ≤ 1 / 3 for the tangential pressure case, we showed that accretion disk spectra would consist of a multi-temperature blackbody at low frequencies joining smoothly to a power-law at high frequencies. Notably, the disk luminosity would be dominated by the high-energy tail, which is a characteristic feature of these models. The spectrum of an equivalent accretion \ndisk around a black hole would have only the low-frequency multi-temperature blackbody component and would be missing the high frequency tail (or at best have a weak tail). \nAccretion disks around astrophysical black hole candidates do show power-law tails, but these are usually interpreted as coronal emission from hot gas above the disk. In those cases where the disk emission is definitely thermal, e.g., in the so-called Thermal State of accreting stellar-mass black holes [27], the spectrum is invariably dominated by the low-frequency multitemperature emission and the power-law tail tends to be quite weak. A number of stellar-mass black holes have been observed in the Thermal State [28]. In the case of these systems at least we may conclude that the central compact objects are not naked singularities of the type discussed in the present paper; the objects are presumably true black holes. However, it is not possible to state anything with certainty in the case of other black hole candidates that have not been observed in the Thermal State. \nThe existence of horizons in stellar mass astrophysical sources has been a matter of discussion in recent years (see for example [1], [29] and references therein). Some observations suggest that the departure of such objects from black holes must be very small [30]. All the same, when it comes to more massive sources, excess luminosity from Ultra Luminous X-ray Sources cannot at present be explained fully by the usual black hole accretion disk models and seem to require the existence of as yet undiscovered intermediate mass black holes [31]. \nSince there is a strong thermal cutoff at high frequencies present in the spectrum of the black hole models while it is absent in the perfect fluid naked singularity models studied here, this strenghtens the argument that the astrophysical sources that exhibit the same behaviour would be black holes. On the other hand, ultra luminous sources exist in the universe and at present it is not clear if it will be possible to fit all the observations within the black hole paradigm. It is possible that different kinds of sources exist in nature, besides black holes, stars and neutron stars. These could be of the singular type discussed here or could be regular objects composed of ordinary or exotic matter. All these could possibly form from collapse processes as described here and they would have their distinctive observational features (see for example [33]). In such a scenario, the comparison and models such as the ones discussed here may be useful to understand better and analyze future observations. \nFor the toy models presented here we found that the radiant energy flux and the spectral energy distribution are much greater as compared to a black hole of the same mass, and therefore we obtained indications that if some sources of similar kind do exist in the universe it might be possible to distinguish them observationally from black holes. \nWe also found here that the qualitative features described in [8] are preserved when we consider perfect fluid sources, instead of sources sustained only by tangential stresses. We can therefore conjecture that the increased flux and luminosity may be a generic feature of any source with a singularity at the center where the accretion disk can in principle extend until r = 0 . As it is known, perfect fluids are important in the context of astrophysics as they can be used to model many sources and objects of astrophysical relevance. Of course, different matter models will imply different luminosity spectrums that might or might not eventually be distinguished from one another. \nFinally, we note that these models require a certain fine tuning since all the collapsing shells must have the right velocity so that the effective potential leads asymptotically to ˙ R = R = 0 . Nevertheless the class of static final configurations can be understood in the sense of an idealized approximation for a slowly evolving cloud with small velocities. In this sense these scenarios appear to be generic as they approximate a wide variety of slowly evolving matter clouds. \n- [1] R. Narayan and J. E. McClintock, New Astron. Rev. 51 , 733-751 (2008); and references therein.\n- [2] P. S. Joshi and D. Malafarina, Int. J. Mod. Phys. D 20 , 2641 (2011); and references therein.\n- [3] B. Freedman and L. D. McLerran, Phys. Rev. D 17 , 11091122 (1978); M. Colpi, S. L. Shapiro and I. Wasserman, Phys. Rev. Lett. 57 , 24852488 (1986); D. F. Torres, S. Capozziello and G. Lambiase, Phys. Rev. D 62 (10), 104012 (2000); P. O. Mazur and E. Mottola, arXiv:0109035 [gr-qc]; D. F. Torres, Nucl. Phys. B 626 , 377 (2002); P. Jaikumar, S. Reddy and A. Steiner, Phys. Rev. Lett. 96 , 041101 (2006); M. Visser, C. Barcelo, S. Liberati and S. Sonego, arXiv:0902.0346 [gr-qc]; F. E. Schunck and E. W. Mielke, Class. Quantum Grav. 20 , 301 (2003).\n- [4] A. N. Chowdhury, M. Patil, D. Malafarina and P. S. Joshi, Phys. Rev. D 85 , 104031 (2012); D. Pugliese, H. 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Lake, Phys. Rev. D 38 , 1315 (1988); R. P. A. C. Newman, Class. Quantum Grav. 3 , 527 (1986); D. Christodoulou, Commun. Math. Phys. 93 , 171 (1984); D. M. Eardley and L. Smarr, Phys. Rev. D 19 , 2239 (1979). \n- [12] T. A. Madhav, R. Goswami and P. S. Joshi, Phys Rev D 72 , 084029 (2005); A. Mahajan, T. Harada, P. S. Joshi and K. I. Nakao, Prog. Theor. Phys. 118 , 865-878 (2007).\n- [13] K. Lake, Phys. Rev. D 67 , 104015 (2003).\n- [14] P. S. Joshi and I. H. Dwivedi, Class. Quantum Grav. 16 , 41 (1999).\n- [15] M. S. R. Delgaty and K. Lake, Comput. Phys. Commun. 115 , 395 (1998).\n- [16] D. Martin and M. Visser, Phys. Rev. D 69 , 104028 (2004).\n- [17] R. Wald, General Relativity , University Of Chicago Press, p. 125 (1984).\n- [18] R. C. Tolman, Phys. Rev. 55 , 364 (1939).\n- [19] D. Bini, D. Gregoris, K. Rosquist and S. Succi, Gen. Rel. Grav. 44 (10), 2669 (2012).\n- [20] D. Novikov and K. S. Thorne in Black Holes , ed. C. DeWitt and B. S. DeWitt, New York: Gordon and Breach, p. 343 (1973); D. N. Page and K. S. Thorne, Astrophys. J. 191 , 499 (1974).\n- [21] J. Frank, A. King and D. J. Raine, Accretion Power in Astrophysics , Third edition, Cambridge Univ. Press (2002).\n- [22] S. W. Davis, C. Done and O. Blaes, Astrophys. J. 647 , 525 (2006); L. Gou et al., Astrophys. J. 718 , L122 (2010); J. F. Steiner et al., Mon. Not. Royal Astr. Soc. 416 , 941 (2011).\n- [23] W. Jaffe, Mon. Not. Royal Astr. Soc. 202 , 995 (1983).\n- [24] W. Dehnen, Mon. Not. Royal Astr. Soc. 265 , 250 (1993).\n- [25] L. Hernquist, Astrophys. J. 356 , 359 (1990).\n- [26] J. F. Navarro, C. S. Frenk and S. D. M. White, Astrophys. J. 463 , 563 (1996).\n- [27] R. A. Remillard and J. E. McClintock, Ann. Rev. Astron. Astrophys. 44 , 49 (2006).\n- [28] J. E. McClintock, R. Narayan and J. F. Steiner, arXiv astro-ph:1303.1583.\n- [29] E. Maoz, Astrophys. J. 494 , L181 (1998); A. E. Broderick, A. Loeb and R. Narayan, Astrophys. J. 701 , 1357 (2009); M. A. Abramowicz, W. Kluzniak and J. P. Lasota, Astron. Astrophys. 396 , L31 (2002); C. Bambi, arXiv:1205.4640 [gr-qc].\n- [30] S. S. Doeleman et.al. Nature, 455 , 78 (2008).\n- [31] J. M. Miller, A. C. Fabian and M. C. Miller, Astrophys. J. 614 , L117 (2004).\n- [32] Critical Collapse of Einstein Cluster Ashutosh Mahajan, Tomohiro Harada, Pankaj S. Joshi, Ken-ichi Nakao, Prog. Theor. Phys. 118 , 865-878 (2007).\n- [33] C. Bambi and D. Malafarina, arXiv:1307.2106; P. S. Joshi and D. Malafarina, Compact objects from gravitational collapse: an analytical toy model (in progress).\n- [34] Of course as the particles approach the center friction and viscosity will play an increasingly important role thus deviating the behaviour from that of a classical accretion disk [19]. Nevertheless we can assume that until a certain radius the approximations made here are satisfied.\n- [35] Note that models with λ < 1 / 3 do not have a high energy power-law tail. These models belong to the class of models with λ < 1 / 2 which we have decided to ignore in this paper since their disks have a gap between the ISCO of the external Schwarzschild spacetime and the surface of the cloud. \n√"}

2013CQGra..30x4004R

The spin of supermassive black holes

2013-01-01

['-', '-']

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Black hole spin is a quantity of great interest to both physicists and astrophysicists. We review the current status of spin measurements in supermassive black holes (SMBH). To date, every robust SMBH spin measurement uses x-ray reflection spectroscopy, so we focus almost exclusively on this technique as applied to moderately-luminous active galactic nuclei (AGN). After describing the foundations and uncertainties of the method, we summarize the current status of the field. At the time of writing, observations by XMM-Newton, Suzaku and NuSTAR have given robust spin constraints on 22 SMBHs. We find a significant number of rapidly rotating SMBHs (with dimensionless spin parameters a &gt; 0.9) although, especially at the higher masses (M &gt; 4 × 10<SUP>7</SUP>M<SUB>⊙</SUB>), there are also some SMBHs with intermediate spin parameters. This may be giving us our first hint at a mass-dependent spin distribution which would, in turn, provide interesting constraints on models for SMBH growth. We also discuss the recent discovery of relativistic x-ray reverberation which we can use to ‘echo map’ the innermost regions of the accretion disc. The ultimate development of these reverberation techniques, when applied to data from future high-throughput x-ray observatories such as LOFT, ATHENA+, and AXSIO, will permit the measurement of black hole spin by a characterization of strong-field Shapiro delays. We conclude with a brief discussion of other electromagnetic methods that have been attempted or are being developed to constrain SMBH spin.

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https://arxiv.org/pdf/1307.3246.pdf

{'Christopher S. Reynolds': "Dept. of Astronomy and the Joint Space Science Institute, University of Maryland, College Park, MD 20901, USA. \nE-mail: \nAbstract. Black hole spin is a quantity of great interest to both physicists and astrophysicists. We review the current status of spin measurements in supermassive black holes (SMBH). To date, every robust SMBH spin measurement uses X-ray reflection spectroscopy, so we focus almost exclusively on this technique as applied to moderately-luminous active galactic nuclei (AGN). After describing the foundations and uncertainties of the method, we summarize the current status of the field. At the time of writing, observations by XMM-Newton , Suzaku and NuSTAR have given robust spin constraints on 22 SMBHs. We find a significant number of rapidly-rotating SMBHs (with dimensionless spin parameters a > 0 . 9) although, especially at the higher masses ( M > 4 × 10 7 M /circledot ), there are also some SMBHs with intermediate spin parameters. This may be giving us our first hint at a mass-dependent spin distribution which would, in turn, provide interesting constraints on models for SMBH growth. We also discuss the recent discovery of relativistic X-ray reverberation which we can use to 'echo map' the innermost regions of the accretion disk. The ultimate development of these reverberation techniques, when applied to data from future high-throughput X-ray observatories such as LOFT , ATHENA+ , and AXSIO , will permit the measurement of black hole spin by a characterization of strong-field Shapiro delays. We conclude with a brief discussion of other electromagnetic methods that have been attempted or are being developed to constrain SMBH spin.", '1. Introduction': "Supermassive black holes (SMBHs) are commonplace in today's Universe, being found at the center of essentially every galaxy [1]. As such, they have attracted the attention of astrophysicists and physicists alike. Astrophysicists have long been fascinated by the extremely energetic phenomena powered by the accretion of matter onto black holes [2]. Beyond the natural draw of studying 'cosmic fireworks', it is now believed that the energy released by growing SMBHs is important in shaping the properties of today's galaxies and galaxy clusters [3, 4]. Thus, SMBHs are important players in the larger story of galactic-scale structure formation. To the gravitational physicist, SMBHs provide the ultimate laboratory in which to test the prediction of General Relativity (GR) and it's extensions [5]. \nThis article discusses recent progress in observational studies of strong gravitational physics close to SMBHs and, in particular, the characterization of black hole (BH) spin. In order to bring focus to this discussion, we shall address exclusively studies of SMBHs. A large body of parallel work exists directed towards stellar-mass BHs; we direct the interested reader to some excellent reviews [6, 7]. Before beginning our discussion, it is worth reiterating why SMBH spin is an interesting quantity to pursue. Assuming that standard GR describes the macroscopic physics, spin provides a powerful way to probe the growth history of SMBHs and, ultimately, their formation pathways (an issue that remains mysterious). In essence, scenarios in which SMBH growth is dominated by BH-BH mergers predict a population of modestly spinning SMBHs, whereas growth via gas accretion can lead to a rapidly-spinning or a very slowly-spinning population depending upon whether the accreting matter maintains a coherent angular momentum vector over the time it takes to double the BH mass [8, 9]. SMBH spin can also be a potent energy source, and may well drive the powerful relativistic jets that are seen from many BH systems [10, 11]. On more fundamental issues, generic deviations from or extensions to GR tend to show up at the same order as spin effects (e.g. [12]) thus observational probes capable of diagnosing spin are also sensitive to physics beyond GR. In addition, spinning BHs have recently been shown to be unstable in extensions of GR with massive gravitons; for sufficiently massive gravitons, SMBHs should spin down on astrophysically-relevant timescales [5, 13] (although see [14]). Thus, observations of spinning SMBHs can set upper limits on the graviton mass. \nWe structure this paper as follows. In Section 2, we briefly review the basic physics relevant to the discussion of accreting SMBHs. In Section 3, we describe the basic anatomy of an accreting SMBH (Section 3.1), and then present the main tool employed to measure BH spin, namely X-ray reflection spectroscopy (Sections 3.2 and 3.3). We illustrate the method with a detailed discussion of the SMBH in the galaxy NGC 3783 (Section 3.4), discuss the uncertainties and caveats that should be attached to these spin measurements (Section 3.5), and then summarize recent attempts to map out the distribution of BH spins (Section 3.6). While the long-term future of precision SMBH spin measurements will likely be dominated by space-based \ngravitational wave astronomy [15](GWA), Section 4 concludes with a brief discussion of other electromagnetic probes of spin that are complementary to X-ray spectroscopy and can be developed in the pre-GWA era. \nIn what follows, the mass of the SMBH shall be denoted M . Following the usual convection, we shall mostly use units in which G = c = 1; thus, length and time are measured in units of M with 1 M /circledot ≡ 1 . 48 km ≡ 4 . 93 µ s", '2. Review of the basic physics': 'We begin by stating two basic but important facts about a SMBH at the center of a galaxy. Firstly, the BH is believed to dominate the gravitational potential out to at least 10 5 M . Secondly, the ubiquitous presence of plasma in the galactic environment precludes any significant charge build-up on the BH. Assuming standard GR (as we shall do henceforth), these two conditions imply that the spacetime is characterized by the Kerr metric [16], parameterized solely by the mass M and dimensionless spin parameter a ≡ Jc/GM 2 , where J is the angular momentum of the BH. Following the usual astrophysical convention, we shall work in Boyer-Lindquist coordinates [17] in which the Kerr line element is given by \nds 2 = -( 1 -2 Mr Σ ) dt 2 -4 aM 2 r sin 2 θ Σ dt dφ + Σ ∆ dr 2 (1) + Σ dθ 2 + ( r 2 + a 2 M 2 + 2 a 2 M 3 r sin 2 θ Σ ) sin 2 θ dφ 2 , \nwhere ∆ = r 2 -2 Mr + a 2 M 2 , Σ = r 2 + a 2 M 2 cos 2 θ . The event horizon is given by the outer route of ∆ = 0, i.e., r evt = M (1 + √ 1 a 2 ). \nThere are two other locations of special astrophysical importance within the Kerr metric. The surface defined by Σ = 0 (which completely encompasses the event horizon) is the static limit ; within the static limit, any (time-like or light-like) particle trajectory is forced to rotate in the same sense as the BH as seen by a distant observer (i.e. dφ/dt > 0). The stationary nature of the spacetime (i.e. the existence of a time-like Killing vector, ∂ t ) allows us to define the total conserved energy of any test-particle orbit around the BH. A curious property of the Kerr metric is the existence of orbits within the static limit (but exterior to the event horizon) for which this conserved energy is negative. Penrose [18, 19] showed that the existence of these orbits permits, in principle, the complete extraction of the rotational energy of a spinning BH, resulting in this region being called the ergosphere . Blandford & Znajek [10] demonstrated that magnetic fields supported by currents in the surrounding plasma can interact with a spinning BH to realize a Penrose process; this is the basis of spin-powered models for BH jets. \n- \nA second astrophysically important location is the innermost stable circular orbit (ISCO), i.e. the radius inside of which circular test-mass (time-like) orbits in the θ = π/ 2 plane become unstable to small perturbations. This is given by the expression, \nr isco = M ( 3 + Z 2 ∓ [(3 -Z 1 )(3 + Z 1 +2 Z 2 )] 1 / 2 ) , (2) \nwhere the ∓ sign is for test particles in prograde and retrograde orbits, respectively, relative to the spin of the BH and we have defined, \nZ 1 = 1 + 1 -a 2 ) 1 / 3 (1 + a ) 1 / 3 +(1 -a ) 1 / 3 ] , (3) \n( \n[ \nFor a = 1 (maximal spin in the prograde sense relative to the orbiting particle), we have r isco = M . This is the same coordinate value as possessed by the event horizon but, in fact, the coordinate system is singular at this location and there exists finite proper distance between the two locations. As a decreases, r isco monotonically increases through r isco = 6 M when a = 0 to reach a maximum of r = 9 M when a = -1 (maximal spin retrograde to the orbiting particle). As we discuss below, the ISCO sets an effective inner edge to the accretion disk (at least for the disk configurations that we shall be considering here). Thus, the spin dependence of the ISCO directly translates into spin-dependent observables; as spin increases and the radius of the ISCO decreases, the disk becomes more efficient at extracting/radiating the gravitational binding energy of the accreting matter, the disk becomes hotter, temporal frequencies associated with the inner disk are increased, and the gravitational redshifts of the disk emission are increased. \n) ] Z 2 = ( 3 a 2 + Z 2 1 ) 1 / 2 . (4)', '3. X-ray spectroscopic measurements of SMBH spin': 'We now turn to a discussion of the main observational technique that has been used to date to obtain robust measurements of SMBH spin - X-ray reflection spectroscopy.', '3.1. Basic structure of a SMBH accretion disk': "In order to characterize the rather subtle effects of BH spin, we must select astrophysical systems that we believe possess particularly simple and well-defined accretion geometries. In our discussion, we only consider sources that have 'moderately high' accretion rates, i.e., those with luminosities in the range ∼ 10 -2 L Edd to ∼ 0 . 3 L Edd where L Edd ≈ 1 . 3 × 10 31 ( M/M /circledot ) W is the usual Eddington limit (the luminosity above which fully ionized hydrogen is blown away from the BH by radiation pressure). This means that we are studying bonafide active galactic nuclei (AGN). For such systems, the accretion geometry is described by the cartoon shown in Fig. 1. Outside of the ISCO, the accretion disk is geometrically-thin and consists of gas following almost testparticle circular orbits (hereafter, we shall use the term 'keplerian motion' to mean test-particle circular orbits). The gradual re-distribution of angular momentum by magnetohydrodynamic (MHD) turbulence within the disk [21, 22] leads to a gradual inwards radial drift of matter superposed on this background of circular motion. Inside of the ISCO, material spirals rapidly into the BH approximately conserving energy and angular momentum. The radial velocities become relativistic (becoming c at the horizon itself) and so, by conservation of baryon number, the density of the flow plummets \nFigure 1. Schematic sketch of the accretion disk geometry discussed in Section 3.1. The accretion disk is geometrically-thin and flat (in the symmetry plane of the Kerr BH). Within the ISCO, the flow plunges into the BH, effectively truncating the disk signatures at the ISCO. In SMBH systems, the disk has a temperature of ∼ 10 5 K and the X-rays arise from a geometrically extended structure (the 'corona') elevated off the disk plane. The X-ray source irradiates the accretion disk's surface, producing observable reflection signatures in the X-ray spectrum. Figure courtesy of Lijun Gou, closely following similar figure in [20]. \n<!-- image --> \nwithin this region, and the matter becomes fully ionized. Indeed, it may even become optically-thin as indicated in the cartoon. The transition of the accretion flow from approximately circular motion to inward plunging at the ISCO is particularly important for our discussion - it sets an inner radius to the region of the disk that produced any observable effects.. This transition is seen in MHD simulations of accretion disks [23, 24], although a full characterization of the inner disk structure must await the development of global, radiation-dominated GR-MHD simulations. Noting that accretion disk theory is still an active and on-going field of research, we shall proceed under the assumption that this transition occurs at the ISCO (although, as we shall point out when appropriate, small shifts in the transition away from the ISCO due to finite disk thickness effects may well be an important source of systematic error for BH spin). \nIn general, of course, the angular momentum vector of the incoming material about the BH (and hence the orientation of the accretion disk at large radius) may be misaligned with the spin angular momentum of the BH. In what is now the standard picture, Bareen & Peterson [25] argue that the effects of differential Lens-Thirring precession slave the inner accretion disk to the BH symmetric plane ( θ = π/ 2), producing a disk that gradually twists from alignment with the mass reservoir at the outside, to alignment with the black hole in its inner region. The Bardeen-Petterson argument \nsuggests that the inner r ∼ 100 M of the accretion disk will be essentially flat in the θ = π/ 2 plane. However, the details of this process, including the value of the characteristic warp radius, depends upon the transport of the out-of-plane angular momentum in an MHD turbulent disk; only now has it become tractable to examine this problem with MHD simulations and we eagerly await results. For the rest of the discussion, the effects of disk warping shall be neglected. \nThe temperature of the inner accretion disk around a SMBH for these accretion rates is generally in the range 10 5 -10 6 K ; the resulting thermal (quasi-blackbody) emission produces a strong bump peaking in the UV/EUV. Indeed, this UV bump can often dominate the observed energy output of an AGN. However, essentially every AGN also shows significant emission in the X-ray band. This has the form of a power-law component with flux-density F ( ν ) ∝ ν -α , α ≈ 1, and can carry up to 10-20% of the total power radiated by the AGN. This 'hard component' emission must arise from much hotter, optically-thin material associated with either a magnetically heated 'corona' above the disk or dissipation in the base regions of a jet (the yellow cloud in Fig. 1). On the basis of detailed spectral studies, the most likely physical process producing the X-ray emission is the inverse Compton scattering of optical/UV photons from the accretion disk by ∼ 10 9 K electrons in the corona.", "3.2. X-ray 'reflection' from the SMBH accretion disk": 'This configuration of a flat, thin, cold, keplerian disk below a strong continuum X-ray source proves to be fortuitous. The observed X-ray emission is (usually) dominated by radiation that comes straight from the corona to us. However, the X-rays from the corona also irradiate the accretion disk. With energy-dependent ratios, the incident photons are either photoelectrically-absorbed by ions in the surface layers of the disk or Compton scattered back out of the disk. The de-excitation of the resulting excited ions produces a rich X-ray spectrum of radiative-recombination features and fluorescent emission lines. Collectively, the Compton scattered continuum and these emission features make up (what is poorly termed) the X-ray reflection spectrum [26, 27, 28, 29]. \nA representative calculation of the rest-frame X-ray reflection (using the xillver code of [29]) is shown in Fig. 2 (left). Particularly important is the iron-K α line at 6.4 keV [30]. This line is strong due to the high astrophysical abundance and high fluorescence yield of iron. It is also relatively isolated in the spectrum, making it easy to identify and characterize even if it is strongly broadened and distorted (as we will discuss in Section 3.3). However, there are other characteristic features of the reflection spectrum. At low X-ray energies, the spectrum is dominated by a dense forrest of emission lines from low atomic number elements (with N, O, Ne, Mg, Si, and S being particularly important). At high X-ray energies, the absorption cross-sections become small and Compton-scattering dominates, producing a broad hump in the spectrum that peaks at 20-30 keV. Of course, the precise form of the spectrum depends upon several things, most notable (i) the ionization state of the disk surface (characterized \n<!-- image --> \nFigure 2. Left panel : Incident power-law continuum (red dashed line) and rest-frame reflection spectrum produced by the accretion disk (solid blue line) computed using the xillver model of [29] assuming an ionization parameter of ξ = 5ergcms -1 and cosmic iron abundance. Right panel : Effects of relativistic Doppler and gravitational redshift effects on this reflection spectrum assuming a viewing inclination of i = 30 · , irradiation index of q = 3, and a rapidly spinning BH ( a = 0 . 99; black line) or nonspinning ( a = 0; blue line) BH. \n<!-- image --> \nby the ionization parameter ξ = 4 πF ion /n where F ion is the ionizing flux impinging on the disk surface and n is the electron number density at the X-ray photosphere), (ii) the elemental composition of the disk matter (with the iron fraction being of special importance), and (iii) the shape of the irradiating continuum.', '3.3. Relativistic effects and spin-effects in the X-ray spectrum': 'The geometry outlined above is a relativists dream - the accretion disk is a thin structure following almost test-particle orbits close to the BH and emitting well-defined, sharp spectral features. The observed spectrum will thus be strongly modified by relativistic Doppler shifts and gravitational redshift (see Fig. 2 [right]). By identifying these spectral features and modeling the broadening effects we can determine the properties of the accretion disk itself (e.g., the ionization state of the surface layers and the elemental abundances of the disk) as well as the inclination of the disk and the location of the ISCO (hence the BH spin). \nOperationally, the analysis of real data is facilitated by making the approximation that the rest-frame disk reflection spectrum is invariant across the region of interest. We then model the observer-frame spectrum by convolving the reflection spectrum with a kernel describing the GR photon transfer effects (Doppler and gravitational shifts). This kernel is constructed by ray-tracing from the the disk surface ( θ = π/ 2 plane) to the observer, taking full advantage of the conserved quantities in the Kerr metric (see [31] for a review of this calculation). The kernel depends upon the viewing inclination of the accretion disk and the BH spin. It also depends upon the radial distribution of the irradiating X-ray flux (and hence the weight that should be given to each radius of the disk in the blurring kernel); this is an unknown function and something we would like to \nFigure 3. Th 2009 Suzaku spectrum of NGC3783 ratioed against a power-law continuum. Figure from [39]. \n<!-- image --> \ndetermine from the data. Thus, these kernels are calculated and tabulated as a function of emission radius, and the full kernel is constructed (for a given viewing inclination and spin) by integrating over radii weighted by the irradiation profile [32, 30, 33, 34]. We commonly parameterize the irradiation profile with a power-law form, F ion ∝ r -q , where q is known as the irradiation index . High-quality datasets sometimes require a broken power-law form in order to fully capture the shape of the broadened reflection spectrum in which case we have two irradiation indices ( q 1 , q 2 ) and a break radius [35, 36, 37]. A non-parametric treatment of the irradiation profile [38] confirms the appropriateness of a broken power-law form.', '3.4. A worked example : the spin of the SMBH in NGC3783': "In principle, the procedure to determine SMBH spin is now straightforward. We take a high signal-to-noise X-ray spectrum of an AGN with a high throughput observatory such as XMM-Newton , Suzaku or NuSTAR . We then compare model spectra (such as illustrated in Fig. 2 [right]) with the observed spectrum and, using standard techniques (e.g. direct χ 2 -minimization or Monte Carlo Markov Chain [MCMC] methods), determine the best-fitting values and error ranges on all model parameters, including spin. \nHowever, AGN possess complex structures in addition to the accretion disk, including winds of photoionized plasma that are blown off the disk as well as large-scale cold structures that may act as the reservoir for the accretion disk. These structures imprint their own patterns of absorption, emission and reflection on the X-ray spectrum \nFigure 4. Spin constraints on the SMBH in NGC3783 from the 2009 Suzaku observation. Left panel : ∆ χ 2 as a function of the spin parameter a . Different lines show the effects of different data analysis assumptions; a fiducial analysis (black), an analysis in which the warm absorber parameters are frozen at their best values (red), an analysis in which the cross normalization of the two Suzaku instruments (the XIS and HXD/PIN) are allowed to float (blue), and an analysis that ignores all data below 3 keV (green). Figure from [36]. Right panel : Probability distribution for spin parameter a from a Monte Carlo Markov Chain (MCMC) analysis using the fiducial spectral model. Figure from [45]. \n<!-- image --> \nand these must be included in our spectral model. We illustrate this with a discussion of the AGN in the spiral galaxy NGC3783, one of the targets of the Suzaku Key Project on BH spin. Figure 3 shows the ratio of the Suzaku data to a reference power-law model. The soft X-ray spectrum is dominated by an absorption feature from a highlysubrelativistic wind of photoionized plasma flowing away from the BH (the so-called warm absorber , [40, 41, 42, 43]). This absorption can be accurately modeled by spectral synthesis codes such as xstar [44]. Shifting attention to the iron-K band around 6 keV, we see in Fig. 3 a strong narrow iron line due to the fluorescence of cold gas that is distant from the SMBH. This can be accurately described by a low-ionization X-ray reflection spectrum. \nOur final spectral model must include all of these additional components (with their associated parameters), and the spin constraints must be derived marginalizing over all other parameters. Figure 4 shows the constraints for NGC3783 derived by both a simple χ 2 minimization procedure and an MCMC approach. Both approaches constrain the spin to be a > 0 . 89 at the 90% confidence level [36, 45]. Figure 5 illustrates precisely how the models are constraining the spin by showing the residuals between the model prediction and the data when the 'wrong' spin is forced into the fit. We see that, when a non-spinning BH is imposed in the fit to the NGC3783 data, the broad iron line is too narrow and over-predicts the flux in the 5-6 keV band. For further detailed discussion of the statistical robustness of these fits, including a treatment of the subtle statistical correlation between BH spin and iron abundance, see [45]. \nFigure 5. Left panel : Suzaku /XIS spectra overlaid with the best fitting model (top) and the corresponding data/model ratio (bottom). Right panel : Same, except that the spin parameter has been frozen at a = 0 and (for physical consistency) the irradiation indices have been frozen at q 1 = q 2 = 3. All other parameters have been allowed to fit freely. For both panels, the model components are colored as follows: absorbed power-law continuum (red), distant reflection (green), and relativistically smeared disk reflection (blue). Figure from [45]. \n<!-- image -->", '3.5. Health warnings': "As in any field of experimental physics, we must be aware of the systematic errors in our measurements or, worse, the possible ways that our methodology may completely fail to capture the reality of a given astrophysical source. Here, we walk through some of the main concerns relevant for SMBH spin measurements. \nThe single biggest concern is that, in some sources, we may be fundamentally mischaracterizing the spectrum. In [46], it is shown that multiple absorbers that partially cover the X-ray source can mimic the relativistically-blurred reflection spectrum, at least in the 0.5-10 keV band. A significant issue faced by these models is their geometric plausibility - the absorbers are postulated to be some significant distance from the SMBH ( r /greatermuch 100 M ) whereas the X-ray source is very compact, lying within 10 M of the SMBH (the compactness of the X-ray source has recently been confirmed by gravitational microlensing studies of multiply-lensed quasars, e.g. [47]). Thus, partial covering of the X-ray source requires either a very special viewing angle (so that our line of sight just clips the edge of the absorber) or a finely clumped wind with tuned clumping factors (such that, at any given time, ∼ 50% of our sight-lines to the compact source are covered by clumps). Still, it is important to address this model from a hard-headed data point of view. This was recently achieved using a joint XMMNewton/NuSTAR observation of the AGN in the spiral galaxy NGC 1365 that, for the first time, allowed a high-quality X-ray spectrum of an AGN to be measured in the 0.5-60 keV band. As reported in [48], the canonical partial-covering absorption model can fit the 0.5-10 keV spectrum but fails when extrapolated up to the higher-energy \nportion of the spectrum. Adding in an extra component partial covering component that is marginally optically-thick to Compton scattering can produce agreement with the observed spectrum, but then the inferred absorption-corrected luminosity of the AGN would exceed the Eddington limit and hence this solution would be unphysical. It was concluded by [48] that the data strongly favor the relativistic disk reflection model over partial-covering absorption. \nWithin the disk reflection paradigm, there are some important issues to note. Firstly, our method assumes that the accretion disk observables truncate at the ISCO; the spin constraint is almost entirely driven by this fact. While there is support for this assumption from MHD simulations of accretion disks [23, 24], our knowledge of disk physics is incomplete. In particular, the inner accretion disk may be subject to visco-thermal instabilities that could, sporadically, truncate disk outside of the ISCO. Although there is no evidence for this transitory disk truncation in most AGN, there is an important class of AGN that do appear to show this phenomenon - the broadline radio galaxies (BLRGs). BLRGs are in our target range of accretion rates, display the strong optical/UV/X-ray emission characteristic of a geometrically-thin accretion disks and corona, but also possess powerful relativistic jets. The jet experiences abrupt and sporadic injections of energy (i.e., geyser-like eruptions) and, during one of these injection-events, the inner regions of the thin-disk appear to be destroyed before refilling back down to the ISCO[49, 50, 51, 52]. Thus, some monitoring of the object is required to know whether the spin measurement can be considered robust or not [52]. \nSecondly, there is the question of which aspect of the data is driving the fit. In the most robust cases, such as NGC3783 discussed in Section 3.4, the constraint is clearly driven by the profile of the broad iron line. However, in addition to the broad iron line, the ionized reflection spectrum also contributes to the soft part of the spectrum ( < 1 keV); the forrest of emission features in the soft reflection spectrum are broadened into a pseudo-continuum that produces a rather smooth 'soft excess' in the spectrum. Many sources do indeed display a soft excess and, in some cases, the high signal-to-noise in the soft part of the spectrum can drive the constraints on the spin parameter. If the observed soft excess is truly due to disk reflection, this is a valid and powerful use of the data. However, there are other possible origins for the soft excess, most notably inverse Compton scattering of UV photons from the cold disk by 'warm' material (possibly a 'chromosphere' lying on the disk surface). Thus, in some sources, the origin of the soft excess and the value of the spin parameter are coupled questions that are still under examination (see detailed discussion of this issue for the luminous AGN Fairall 9 in [53]).", '3.6. Survey of current results': "Without further ado, we present the current state of the field of SMBH spin measurements. Over the past few years there have been an explosion in the number of published SMBH spin measurements, including samples by Walton et al. [54] and \nFigure 6. Masses M and spin parameters a of the 19 SMBHs for which both parameters are constrained. Following the conventions of the primary literature, the spin measurements are shown with 90% error ranges, whereas the masses are shown with 1 σ . This is an updated version of a similar figure appearing in [56]. \n<!-- image --> \nPatrick et al. [55]. Spins have been published by multiple groups and, in the same timeframe, the spectral models and methodologies have been improved. Thus, with some hindsight, one cannot simply harvest SMBH spin measurements from the literature without exercising caution. Following [56], we propose filtering the published values with quality-control criteria; (i) the spin measurement is based on spectral models employing full disk reflection spectra and not just isolated 'pure iron line' models, (ii) the iron abundance is treated as a free parameter (required since iron abundance and spin are correlated variables), (iii) the inclination is treated as a free parameter and can be constrained (this helps protect against fits that are driven purely by the soft excess), (iv) the irradiation index can be constrained and fits to give q > 2 (required for a well-posed model in which the reflection spectrum is not unphysically dominated by the outer disk radius). These filter criteria eliminate 13 sources from the Walton [54] and Patrick [55] samples; the excluded sources span the range of AGN type and luminosity. Taking the compilation described in Table 1 of [56] and updating with the XMM-Newton/NuSTAR result on NGC1365 [48] and the multi-epoch study of 3C120 [52], we now have spin constraints on 22 AGN that satisfy the quality control criteria. \nIn Fig. 6 we take the 19 of these objects that also have mass estimates (from various techniques; see [56]) and place them on the ( M,a )-plane. There are several interesting points to note about this plot. Firstly, there is clearly a population of rapidly spinning BHs ( a > 0 . 9), especially below masses of 4 × 10 7 M /circledot . This is a strong indication that these SMBHs grew (at least in their final mass doubling) by the accretion of gas with a \nFigure 7. The 2-10keV Rossi X-ray Timing Explorer light curve of the AGN MCG6-30-15 from a long observation in August 1997. \n<!-- image --> \ncoherent angular momentum. Secondly, there are some SMBHs for which intermediate spins (0 . 4 < a < 0 . 8) are inferred, and these tend to be the highest mass systems ( M > 4 × 10 7 M /circledot ). While the small number statistics and ill-defined selection effects prevent firm conclusions from being drawn, this may be the first hints for a massdependence to the SMBH spin distribution, with a more slowly spinning population (corresponding to growth via BH-BH mergers or incoherent accretion [57]) emerging at the highest masses. Lastly, there are no retrograde spins measured even though our technique is capable of finding them. A single epoch analysis of the BLRG 3C120 suggested an accretion disk truncated at r ∼ 10 M , possibly indicating a rapid retrograde spin [58], but a multi-epoch analysis revealed that this was a rapidly-rotating prograde BH with a disk that undergoes transitory truncation related to jet activity [52].", '3.7. The emerging field of broad iron line reverberation': 'An important characteristic of accretion onto BHs that we have not yet addressed is the time-variability. Fundamentally, the variability is driven by a combination of local instabilities (such as the magnetorotational instability that drives MHD turbulence [22]) and more poorly understood global instabilities (that may drive limit-cycle or explosive behavior). In AGN, this variability is particularly dramatic in the X-ray band since this emission comes from the central regions of accretion disk (where all of the characteristic timescales are shortest) and originates from a corona which may be unstable to, for example, reconnection-driven eruptions. An example X-ray light curve from the Rossi \nFigure 8. Relativistic transfer function showing the response of a 6.4 keV iron line from the disk surface to a δ -function flare in the driving X-ray continuum source. For the purposes of this illustration, the continuum source is assume to be a point source on the spin axis at r = 10 M , the BH is rapidly spinning a = 0 . 998 and the accretion disk has an inclination of i = 30 · . \n<!-- image -->', 'X-ray Timing Explorer (RXTE) is shown in Fig. 7.': "Since the X-ray source is elevated off the disk surface, we can use this rapid variability to 'echo map' the reflection from the inner accretion disk. In essence, by characterizing the time-delays between flares in the direct X-ray continuum and the reflection spectrum, we can map out the geometry of the SMBH/disk/corona system. This was examined theoretically by the author some years ago [59] who computed transfer functions relating the observed response of the iron-K α emission line from the disk to a δ -function continuum flare. An example of such a transfer function for a rapidly spinning BH ( a = 0 . 998) is shown in Fig. 8. Beyond the obvious fact that the timedelay gives information on the actual location of the X-ray source (something which is hard to obtain via a pure spectral analysis), it is interesting to note that the transfer function bifurcates into two branches. The upper branch corresponds to the normal echo sweeping out to larger radii (and irradiating more slowly moving material) giving an ever narrowing line. The lower branch corresponds to strongly Shapiro-delayed [60] reflection from the very innermost regions of the disk close to the event horizon. The slope of this branch is a function of the spin parameter a , especially when the BH is close to extremal. \nWhile our observatories do not yet have the collecting area to allow us to see this \nFigure 9. Right panel : Time-lags as a function of energy (i.e. the lag-spectrum) for the XMM-Newton data of the bright AGN NGC4151. Show here is the lag-spectrum for temporal frequencies < 2 × 10 -5 Hz (blue triangles) and (5 -50) × 10 -5 Hz (red diamonds). Left panel : Simple Gaussian fits to these lack spectra, demonstrating that the iron-line feature in the lag spectrum has a centroid energy that decreases and (marginally) a width that increases as one probes high temporal frequencies. Figure from [61]. \n<!-- image --> \nresponse of the iron line to a single flare, reverberation has recently been discovered via statistical analyzes of long XMM-Newton datasets of bright AGN. The analysis is subtle and must account for the fact that (i) there is no single time-delay that characterizes the response of the reflection continuum and, in fact, the measured time delays are both a function of energy and the temporal-frequency being probes; and (ii) the direct continuum source itself has intrinsic energy- and temporal-frequency dependent time lags. Most analyses use Fourier methods. For two given (non-overlapping) energy bands, time-sequences of X-ray flux ( a ( t ) and b ( t )) are extracted and their Fourier transforms (˜ a ( ω ) and ˜ b ( ω ) )are computed. We then form the cross-spectrum C ( ω ) ≡ ˜ a ∗ ˜ b , which we can write in terms of a frequency-dependent amplitude and phase, C ( ω ) = | C ( ω ) | e iφ ( ω ) . The frequency-dependent time-delay between the two lightcurves is then given by ∆ t = φ ( ω ) /ω . \nThe first detections of reverberation from the inner accretion disk were made by probing delays between the continuum and the soft excess [62, 63, 64]. However, recently, there are clear detections of reverberation associated with the broad iron line [61, 65, 66]. Figure 9 shows the first such detection of broad iron line reverberation in the bright AGN NGC4151. The magnitude of the lags suggests that the X-ray source is displaced ∼ 4 M above the center of the disk; this is not necessarily a natural location for strong coronal emission and may suggest that the X-ray source is actually associated \nwith the dissipative base of a jet or some other dissipative structure associated with a BH magnetosphere. The exact shape of the lag-spectrum depends upon the temporal frequency being probes, with the line appearing broader and more highly redshifted at higher temporal-frequencies. This is expected since the higher temporal-frequencies are probing smaller radii in the accretion disk, and suggests that we are gaining our first hints at the nature of the transfer function. \nOne important point to note (in the spirit of a sanity check) is that the size/location of the X-ray source as inferred from X-ray reverberation is broadly consistent with recent estimates derived from a completely different technique, studies of microlensing in multiply lensed quasars [67, 68].", '4. Conclusions': 'In this review, we have summarized the methodology by which we can measure SMBH spins using X-ray reflection spectroscopy and have described the current status of such measurements. Today, we have spins for approximately two dozen SMBHs; we find a population of rapidly rotating BHs ( a > 0 . 9) but hints of a more slowly rotating population emerging at higher SMBH masses. The main limitation to applying this technique more widely are the long integration times needed with current observatories in order to obtain spectra with sufficient signal-to-noise. The next generation of proposed high throughput X-ray observatories such as LOFT , ATHENA+ and AXSIO will permit us to push these studies to fainter and more distant objects. This, in turn, will allow well-defined samples to be obtained and permit investigations of the SMBH spin distribution as a function of cosmic time. These high-throughput observatories will also unlock the full power of reverberation studies, allowing us to detect strong-field Shapiro delays in the reverberation transfer function. \nThe long term future of SMBH spin measurements will likely be dominated by space-based GWA, once such observatories are deployed. GWA will permit SMBH spin measurements with a precision and accuracy probably unattainable with X-ray spectroscopy. However, given the uncertain timeframe for the deployment of space-based GW observatories, it is interesting to pursue electromagnetic probes of SMBH spin as vigorously as possible. We end this review by mentioning a few other electromagnetic techniques that have been discussed or are actively being developed. \nIf the accretion disk is efficient at radiating away the gravitational potential energy of the accreting matter, the overall efficiency of the accretion disk (defined as η ≡ L/ ˙ Mc 2 , where L is the total radiative luminosity of the system and ˙ M ≡ dM/dt ) is a function of spin. With some work, the total L can be estimated straightforwardly. The mass accretion rate ˙ M is more challenging to constrain, but it can be estimated through the detailed temperature/shape of the thermal emission from the accretion disk. Applying this method to a sample of 80 quasars, [69] found tentative evidence for efficiencies (and hence spin parameters) that increase with increasing SMBH mass. Of course, this is discrepant with the result presented in Fig. 6 - future explorations \nof this discrepancy must properly account for selection effects as well as the systematic errors in the efficiency-based measurements. \nOne interesting technique focuses on powerful jetted systems. If we assume that the jets are powered by the Blandford-Znajek (i.e. magnetic Penrose) process, and if we can make an estimate of the magnetic field in the immediate vicinity of the BH (from accretion theory), then the BH spin parameter can be determined once the jet power is measured. Practical difficulties include the significant uncertainty in deriving jet powers from observables (which may be strongly influenced by an unknown relativistic beaming factor). However, it is encouraging that a careful application of this method [70] produces results that are sensible; given the analytic approximation used to relate spin and jet power, it would be possible to infer a > 1, but the observed population of AGN seem to obey the constraint a | < 1. \n| \n| Of course, a tremendous amount of observational firepower (including a recent 3 Ms campaign by Chandra ) has been directed at the SMBH at the center of our own Galaxy (coincident with the radio source Sgr A ∗ ). Unfortunately, the X-ray reflection technique cannot be applied here - the accretion rate is extremely small, and the disk is believed to be in a hot, geometrically-thick (quasi-spherical) state that is optically-thin to Xray emission and hence does not produce any X-ray reflection features [71, 72]. Instead, calculations of photon production/propagation through GR-MHD simulations of the hot flow have been used to make predictions of the radio spectrum and frequency-dependent polarization as a function of system parameters. In principle, comparison with the observations can then constrain the spin. However, due to the inherent uncertainties in the calculation of this complex radio continuum, the constraints to date have not yet proved conclusive, with some groups obtaining high inferred spin [73, 74] and others obtaining rather lower values [75]. \nAnother particularly exciting development, again aimed squarely at Sgr A ∗ and the SMBH in the nearby elliptical galaxy M87, are the attempts to directly image the horizon-scale structures using mm- or submm-wave Very Long Baseline Interferometry (VLBI). At these short radio wavelengths, the accretion flow is optically-thin to synchrotron self-absorption and hence we might hope to see the shadowing effect of the event horizon against the background radio emission [76]. The properties of the shadow depend upon the spin (and spin-orientation) of the SMBH [77, 12]. Great strides have already been made towards the goal of detecting these structures, with current interferometry being able to detect horizon scale structure but with an insufficient number of baselines to enable true image reconstruction [78]. In M87, current data have also resolved a compact (10 M ) structure possibly related to the jet-launching region [79] which may, itself, probe SMBH spin as well as jet launching [74]. It will be exciting to watch these developments in the next few years. \nThroughout this review, we have assumed the validity of standard GR (and hence the Kerr metric) and have focused on attempted to extract the spin parameter. A more ambitious problem is to use (electromagnetic) observations of accreting black holes to explore deviations from and/or extensions to GR [80, 81, 82, 83, 84]. A generic difficulty \nencountered in these studies is that, to lowest order, the deviations from GR are present in the quadrupole moment of the potential and hence are degenerate with the spin parameter. Still, the implications are profound, and increasingly detailed studies of BH systems with electromagnetic and, eventually, gravitational wave probes remain our best hope for breaking GR. \nThe author thanks Laura Brenneman, Philip Cowperthwaite, Andrew Fabian, Anne Lohfink, Jon Miller, Rubens Reis, and Abdu Zoghbi for stimulating conservations that have helped formulate the discussion presented here. The author also gratefully acknowledges the hospitality of the Institute of Astronomy at the National Central University (Jhongli, Taiwan) during which much of this review was written, as well as funding from NASA under grants NNX10AE41G (Astrophysics Theory), NNX10AR31G (Suzaku Guest Observer Program) and NNX12AE13G (Astrophysics Data Analysis).', 'References': "- [1] J. Kormendy and D. Richstone. Inward Bound-The Search For Supermassive Black Holes In Galactic Nuclei. Ann. Rev. Astron. & Astrophys. , 33:581, 1995.\n- [2] M. J. Rees. Black Hole Models for Active Galactic Nuclei. Ann. Rev. Astron. & Astrophys. , 22:471-506, 1984.\n- [3] A. J. Benson. Galaxy formation theory. Physics Reports , 495:33-86, October 2010.\n- [4] A. C. Fabian. Observational Evidence of Active Galactic Nuclei Feedback. Ann. Rev. Astron. & Astrophys. , 50:455-489, September 2012.\n- [5] E. Berti. Astrophysical Black Holes as Natural Laboratories for Fundamental Physics and StrongField Gravity. 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2014PhRvD..90b4026F

Neutron star-black hole mergers with a nuclear equation of state and neutrino cooling: Dependence in the binary parameters

2014-01-01

['-', '-', '-', '-', 'methods numerical', 'relativity', '-', '-', '-', '-']

[]

We present a first exploration of the results of neutron star-black hole mergers using black hole masses in the most likely range of 7M⊙-10M<SUB>⊙</SUB>, a neutrino leakage scheme, and a modeling of the neutron star material through a finite-temperature nuclear-theory based equation of state. In the range of black hole spins in which the neutron star is tidally disrupted (χ<SUB>BH</SUB>≳0.7), we show that the merger consistently produces large amounts of cool (T ≲1 MeV), unbound, neutron-rich material (M<SUB>ej</SUB>∼0.05M<SUB>⊙</SUB>-0.20M⊙). A comparable amount of bound matter is initially divided between a hot disk (T<SUB>max</SUB>∼15 MeV) with typical neutrino luminosity of L<SUB>ν</SUB>∼<SUP>1053</SUP> erg /s, and a cooler tidal tail. After a short period of rapid protonization of the disk lasting ∼10 ms, the accretion disk cools down under the combined effects of the fall-back of cool material from the tail, continued accretion of the hottest material onto the black hole, and neutrino emission. As the temperature decreases, the disk progressively becomes more neutron rich, with dimmer neutrino emission. This cooling process should stop once the viscous heating in the disk (not included in our simulations) balances the cooling. These mergers of neutron star-black hole binaries with black hole masses of M<SUB>BH</SUB>∼7M⊙-10M⊙, and black hole spins high enough for the neutron star to disrupt provide promising candidates for the production of short gamma-ray bursts, of bright infrared postmerger signals due to the radioactive decay of unbound material, and of large amounts of r-process nuclei.

[]

https://arxiv.org/pdf/1405.1121.pdf

{'Neutron star-black hole mergers with a nuclear equation of state and neutrino cooling: Dependence in the binary parameters': "Francois Foucart, 1 M. Brett Deaton, 2 Matthew D. Duez, 2 Evan O'Connor, 1 Christian D. Ott, 3 Roland Haas, 3 Lawrence E. Kidder, 4 Harald P. Pfeiffer, 1, 5 Mark A. Scheel, 3 and Bela Szilagyi 3 \n1 Canadian Institute for Theoretical Astrophysics, University of Toronto, Toronto, Ontario M5S 3H8, Canada 2 Department of Physics & Astronomy, Washington State University, Pullman, Washington 99164, USA 3 Theoretical Astrophysics 350-17, California Institute of Technology, Pasadena, California 91125, USA 4 Center for Radiophysics and Space Research, Cornell University, Ithaca, New York, 14853, USA \n- 5 Canadian Institute for Advanced Research, 180 Dundas St. West, Toronto, ON M5G 1Z8, Canada \nWe present a first exploration of the results of neutron star-black hole mergers using black hole masses in the most likely range of 7 M /circledot -10 M /circledot , a neutrino leakage scheme, and a modeling of the neutron star material through a finite-temperature nuclear-theory based equation of state. In the range of black hole spins in which the neutron star is tidally disrupted ( χ BH > ∼ 0 . 7 ), we show that the merger consistently produces large amounts of cool ( T < ∼ 1 MeV ), unbound, neutron-rich material ( M ej ∼ 0 . 05 M /circledot -0 . 20 M /circledot ). A comparable amount of bound matter is initially divided between a hot disk ( T max ∼ 15 MeV ) with typical neutrino luminosity L ν ∼ 10 53 erg / s , and a cooler tidal tail. After a short period of rapid protonization of the disk lasting ∼ 10 ms , the accretion disk cools down under the combined effects of the fall-back of cool material from the tail, continued accretion of the hottest material onto the black hole, and neutrino emission. As the temperature decreases, the disk progressively becomes more neutron-rich, with dimmer neutrino emission. This cooling process should stop once the viscous heating in the disk (not included in our simulations) balances the cooling. These mergers of neutron star-black hole binaries with black hole masses M BH ∼ 7 M /circledot -10 M /circledot and black hole spins high enough for the neutron star to disrupt provide promising candidates for the production of short gamma-ray bursts, of bright infrared post-merger signals due to the radioactive decay of unbound material, and of large amounts of r-process nuclei. \nPACS numbers: 04.25.dg, 04.40.Dg, 26.30.Hj, 98.70.-f", 'I. INTRODUCTION': "The coalescence and merger of black holes and neutron stars in binary systems is one of the main sources of gravitational waves that is expected to be detected by the next generation of ground-based detectors (Advanced LIGO [1] and VIRGO [2], KAGRA [3]). If at least one of the members of the binary is a neutron star, bright post-merger electromagnetic signals could also be observable. For example, the formation of hot accretion disks around remnant black holes provides a promising setup for the generation of short gamma-ray bursts (SGRB), while the radioactive decay of unbound neutron-rich material could power an infrared transient days after the merger ('kilonova') [4]. Numerical simulations of these mergers in a general relativistic framework are required both to model the gravitational wave signal around the time of merger, and to determine which binaries can produce detectable electromagnetic signals and how the properties of these signals are related to the physical parameters of the source. \nThese two objectives demand very different types of simulations. To model the gravitational wave signal, long and very accurate simulations of the last tens of orbits before merger are required. But during this phase, the important physical effects can be recovered using simple models for the neutron star matter (e.g. Gamma-law, piecewise polytrope) [5, 6]. On the other hand, to assess whether a given binary can power detectable electromagnetic signals and to predict nucleosynthesis yields, shorter inspiral simulations are acceptable but a detailed description of a wider array of physical effects is re- \nquired: magnetic fields, neutrino radiation, nuclear reactions and the composition and temperature dependence of the properties of neutron-rich, high-density material all play an important role in the post-merger evolution of the system. Any ejected material also has to be tracked far from the merger site, thus requiring accurate evolutions of the fluid in a much larger region than during inspirals. The simulations presented here focus on the second issue in the context of neutron star-black hole (NSBH) mergers. \nGeneral relativistic simulations have only recently begun to include the effects of neutrino radiation on the post-merger evolution of the remnant of binary neutron star (BNS) [79] and NSBH [10] mergers through the use of 'leakage' schemes providing a simple prescription for the cooling of the fluid through neutrino emission. These schemes are largely based on methods already used in the simulation of postmerger remnants by codes with approximate treatments of gravity [11-14]. More advanced (and computationally expensive) methods based on a moment expansion of the radiation fields [15] have also been developed for general relativistic simulations of binary mergers [16]. An energy-integrated version of the moment formalism was recently used for the first time to study BNS mergers [17]. Both general relativistic and non-relativistic simulations have shown that neutrino cooling can play an important role in the evolution of the disk. Emission and absorption of neutrinos can also affect the composition of the ejecta, and the mass of heavy elements produced as a result of r-process nucleosynthesis in the unbound material [17-19]. Finally, energy deposition by neutrino-antineutrino annihilation in the baryon-poor region \nabove the disk and near the poles of the black hole could play a role (either positive or negative) in the production of short gamma-ray bursts [20]. \nMagnetic fields are also expected to play a critical role in the evolution of NSBH remnants. Simulations show magnetic effects to be unimportant before merger for realistic field strengths [21, 22]. However, if the merger produces an accretion disk, the disk will be subject to the magneto-rotational instability (MRI) [23], which will induce turbulence, leading to angular momentum transport and energy dissipation that drives the subsequent accretion. Most NSBH simulations with magnetic fields fail to resolve MRI growth, but it is seen if a sufficiently strong poloidal seed field is inserted [24]. \nFinally, the equation of state of the fluid used to model the neutron star plays an important role before, during and after a NSBH merger. Before merger, it determines the response of the neutron star to the tidal field of the black hole, which, for low mass black holes, could cause measurable differences in the gravitational wave signal [6, 25]. During merger, it determines whether the neutron star is disrupted by the black hole (mostly by setting the radius of the neutron star), allowing the formation of an accretion disk and the ejection of unbound material, or whether it just falls directly into the black hole. Different equations of state can also lead to different qualitative features for the evolution of the tidally disrupted material. After merger, knowing the temperature and composition dependence of the equation of state is necessary to properly evolve the forming accretion disk. Until recently, most general relativistic simulations used Gamma-law or piecewise-polytropic equations of state, which can only provide us with accurate results up to the disruption of the neutron star. For post-merger evolution, compositionand temperature-dependent nuclear-theory based equations of state are required both to properly model the properties of the fluid and to be able to compute its composition and its interaction with neutrinos. Only two general relativistic simulations of NSBH mergers using such equations of state have been presented so far. First, a single low-mass case without neutrino radiation [26] showed only moderate differences with an otherwise identical simulation using a simpler Gammalaw equation of state. More recently, a first simulation with our neutrino leakage scheme for a relatively low mass, highly spinning black hole [10] indicated that the effects of neutrino cooling were more important than the details of the equation of state. Simulations using zero-temperature, nuclear-theory based equations of state were also reported as part of a couple of studies of the radioactive emission coming from unbound neutron-rich material [27, 28], but only a few general properties of the ejecta have been provided at this point. Accordingly, our best estimates for the dependence of the merger and post-merger evolution of NSBH binaries on the equation of state of nuclear matter come from a set of studies using simpler models for the nuclear matter [26, 29, 30], which need to be complemented by simulations using temperaturedependent, nuclear-theory based equations of state. \nIn this paper, we study with the SpEC code [31] the merger and post-merger evolution of NSBH binaries for black holes of mass M BH = 7 M /circledot -10 M /circledot (around the current es- \ntimates of the peak of the mass distribution of stellar mass black holes [32, 33]). We use a nuclear-theory based equation of state with temperature and composition dependence of the fluid properties, and a leakage scheme to approximate the effects of neutrino cooling. Given the computational cost of the numerical simulations, we limit ourselves to a single equation of state (LS220 equation of state from Lattimer & Swesty [34]) and only consider relatively low mass neutron stars ( M NS = 1 . 2 M /circledot -1 . 4 M /circledot ). We also use black holes with spins high enough for the neutron star to be tidally disrupted before plunging into the black hole (dimensionless spins χ BH = 0 . 7 -0 . 9 , as estimated from simulations using simpler equations of state [35]). Indeed, for lower spin black holes, the post-merger evolution is trivial and we do not expect the production of appreciable nucleosynthesis output or detectable electromagnetic signals. For the same black hole masses M BH = 7 M /circledot -10 M /circledot but higher black hole spins, on the other hand, the neutron star is tidally disrupted during the merger. We show that large amounts of neutronrich, low entropy material are ejected, which will undergo robust r-process nucleosynthesis. Bound material which escapes rapid accretion onto the black hole forms an accretion disk, albeit generally of slightly lower mass than for binaries of more symmetric mass ratio and comparable black hole spins (we find M disk ∼ 0 . 05 M /circledot -0 . 15 M /circledot ). The disk is initially hot ( T max ∼ 15 MeV ), with a high neutrino luminosity ( L ν ∼ 10 53 erg / s ). But about 10 ms after merger, the combined effect of the accretion of hot material by the black hole, the fall-back of cold material from the tidal tail, the emission of neutrinos, and the expansion of the disk causes a rapid decline of both the temperature and the luminosity. In a realistic disk, in the inner regions where neutrino cooling is efficient, this decline would presumably stop when the neutrino cooling and the viscous heating due to MRI turbulence roughly balance each other. However, our simulations at this point do not include magnetic effects, and would not resolve the MRI even if they did. And we do not include any parametrized viscosity either. Accordingly, here the disk continues to cool until we end the simulation, about 40 ms after merger. With both a massive, hot accretion disk and a significant amount of ejected unbound material, NSBH mergers in this part of the parameter space would thus be prime candidates for the emission of both prompt (e.g. short gamma-ray bursts) and delayed (e.g. 'kilonovae') electromagnetic signals. \nWe find that the impact of using the LS220 equation of state appears, for the lower mass black holes, fairly weak before merger. Most of the difference in post-merger evolution compared to previous simulations using neutron stars of similar radii but simpler equations of state can be attributed to the effect of neutrino cooling. For higher mass black holes, on the other hand, the disruption occurs very close to the black hole horizon, and the detail of the tidal response of the neutron star can cause more significant differences. The disruption occurs later, faster, and creates a much narrower tidal tail for the LS220 equation of state than for the commonly used Gammalaw equation of state with Γ = 2 . These differences also cause the disruption of the neutron star to be extremely hard to resolve numerically, and require the use of a much finer grid \nFIG. 1: Mass-radius relationship for the LS220 equation of state, the 3 equations of state presented in Hebeler et al. [36], and a Γ = 2 polytrope of similar radius for a 1 . 4 M /circledot star. We consider the ADM mass of isolated neutron stars, and their areal radius. Dashed lines indicate the masses used in this work. The H-Soft and H-Stiff equations of state bracket the ensemble of mass-radius relationships obtained for the family of equations of state presented in [36]. These equations of state match both neutron star mass constraints and nuclear physics constraints from chiral effective field theory. H-Inter is a representative intermediate case. \n<!-- image --> \nclose to the black hole horizon than what is typically used in general relativistic simulations of NSBH mergers (see Sec. II). \nWe will begin with a discussion of the numerical methods used, the chosen initial configurations and an estimate of the errors in the simulations (Sec. II). We then discuss the general properties of the disruption and the evolution of the accretion disk (Sec. III), before a more detailed presentation of the neutrino emission from the disk, and of the evolution of its composition (Sec. IV), and a summary of the expected evolution of the disk over timescales longer than our simulations (Sec. V). Finally, Sec. VI briefly discusses the gravitational wave signal. \nUnless units are explicitly given, we use the convention G = c = 1 , where G is the gravitational constant and c the speed of light.", 'A. Equation of state': "As in our first study of NSBH mergers including neutrino cooling [10], we model the nuclear matter using the Lattimer & Swesty equation of state [34] with nuclear incompressibility parameter K 0 = 220 MeV and symmetry energy S ν = 29 . 3 MeV (hereafter LS220), using the table available on http://www.stellarcollapse.org and described in O'Connor & Ott 2010 [37]. This equation of state lies within the allowed range of neutron star radii, as determined by Hebeler et al. [36] from nuclear theory constraints and the existence of neutron stars of mass ∼ 2 M /circledot [38, 39]. Although it is \nFIG. 2: Density profile as a function of radius (in isotropic coordinates) for the LS220 equation of state, the 3 equations of state presented in Hebeler et al. [36], and a Γ = 2 polytrope of similar radius for a 1 . 4 M /circledot star. Note that in isotropic coordinates, the radius of the surface is not equal to the circumferential radius. \n<!-- image --> \nTABLE I: Properties of the neutron stars before tidal effects become strong. M b NS is the baryonic mass, M NS the ADM mass of the star, if isolated, R NS the circumferential radius, and C NS ≡ M NS /R NS the compactness. \n| M NS | R NS | C NS | M b NS |\n|---------------------|-------------|-----------|-------------------|\n| 1 . 20 M /circledot | 12 . 8 km 0 | . 139 1 . | 31 M /circledot |\n| 1 . 40 M /circledot | 12 . 7 km 0 | . 163 1 | . 55 M /circledot | \nnot fully consistent with the most recent constraints from nuclear experiments (in particular, measurements of the Giant Dipole Resonance [40, 41]), the LS220 equation of state produces neutron stars over a range of masses with structural properties within the limits set by chiral effective field theory (see Figs. 1-2). Using the LS220 equation of state is, at the very least, a significant step forward from Gamma-law equations of state, or even from the temperature-dependent nuclear-theory based equation of state first used in NSBH simulations [26, 42], which do not meet these constraints. Tabulated equations of state which do satisfy all known constraints have recently been developed (see e.g. [41]) and are available in tabulated form on stellarcollapse.org. Those equations of state will be the subject of upcoming numerical studies. \nTable I summarizes the properties of the two neutron stars that we use in this work (with masses M NS = 1 . 2 M /circledot and M NS = 1 . 4 M /circledot ). With radii of ∼ 12 . 7 km , they lie within the range of sizes allowed by chiral effective field theory constraints (see Fig. 1) [36]. The most recent measurements of neutron star radii in X-ray binaries [43] and the millisecond pulsar PSR J0437-4715 [44] are also consistent with R NS ∼ 12 . 7 km (but see [45] for predictions of more compact neutron stars).", 'B. Initial configurations': 'The initial conditions for our simulations are chosen so that the merger leads to the disruption of the neutron star, and thus \npotentially to the formation of an accretion disk and the ejection of unbound material. We additionally require the black hole mass to be within the range currently favored by observations of galactic black holes [32, 33] and by population synthesis models [46, 47], M BH ∼ 7 M /circledot -10 M /circledot . In this mass range, the neutron star will only be disrupted for rapidly spinning black holes and large neutron stars. An approximate threshold for the disruption of the neutron star to occur is indeed [35] \nC NS < ∼ ( 2 + 2 . 14 q 2 / 3 R ISCO 6 M BH ) -1 , (1) \nwhere C NS = M NS /R NS is the compactness of the star, q = M BH /M NS is the mass ratio, and R ISCO is the radius of the innermost stable circular orbit for an isolated black hole of the same mass and spin. For a canonical neutron star of mass 1 . 4 M /circledot described by the LS220 equation of state, this requires a dimensionless spin χ BH > ∼ 0 . 75 (resp. χ BH > ∼ 0 . 55 ) for a black hole of mass 10 M /circledot (resp. 7 M /circledot ). For the LS220 equation of state, the critical spin separating disrupting from non-disrupting neutron stars increases with the neutron star mass. Accordingly, we will consider black hole spins in the range χ BH = 0 . 7 -0 . 9 and neutron star masses in the range M NS = 1 . 2 M /circledot -1 . 4 M /circledot . \nThe simulations presented in this paper are fairly costly: at our standard resolution, a single run might require 100 , 000 -150 , 000 CPU-hours (more than 3 months using 48 processors). As we are considering a 3-dimensional parameter space (neutron star and black hole mass, and black hole spin magnitude), we can only afford a very coarse coverage of each parameter. In this first parameter space study using the LS220 equation of state, we consider 9 simulations covering 2 black hole masses ( 7 M /circledot , 10 M /circledot ), 2 neutron star masses ( 1 . 2 M /circledot , 1 . 4 M /circledot ), and 3 spins ( χ BH = 0 . 7 , 0 . 8 , 0 . 9 , the lower spin being only used for M NS = 1 . 4 M /circledot , M BH = 7 M /circledot ). The initial parameters for each configuration are summarized in Table II, while the properties of the neutron star for each choice of M NS are given in Table I. For reference, our first simulation using the LS220 equation of state and a leakage scheme for the neutrino radiation used a M NS = 1 . 4 M /circledot neutron star, with a less massive ( M BH = 5 . 6 M /circledot ), rapidly rotating ( χ BH = 0 . 9 ) black hole [10]. In the following sections, we will refer to the various simulations by their names listed in Table II, in which the first two numbers refer to the mass of the neutron star and the mass of the black hole, and the third number to the spin of the black hole, e.g. M12-10-S9 correspond to a binary with M NS = 1 . 2 M /circledot , M BH = 10 M /circledot and χ BH = 0 . 9 . \nWe obtain constraint satisfying initial data using the spectral elliptic solver Spells [48, 49], at a separation chosen to provide 5 -8 orbits before merger. The initial data use the quasi-circular approximation [49, 50], which causes the binary to be on slightly eccentric orbits ( e ∼ 0 . 03 -0 . 04 , see Table II). \nTABLE II: Initial configurations studied. All cases use the LS220 equation of state. M BH and χ BH are the mass and dimensionless spin of the black hole, M NS the mass of the neutron star, M Ω 0 orbit the initial orbital frequency multiplied by the total mass M = M BH + M NS , and e is the initial eccentricity. \n| Name | M BH | χ BH | M NS | M Ω 0 orbit | e |\n|-----------|-----------------|--------|--------------------|---------------|-------|\n| M12-7-S8 | 7 M /circledot | 0.8 | 1 . 2 M /circledot | 0.0364 | 0.027 |\n| M12-7-S9 | 7 M /circledot | 0.9 | 1 . 2 M /circledot | 0.0364 | 0.026 |\n| M12-10-S8 | 10 M /circledot | 0.8 | 1 . 2 M /circledot | 0.0396 | 0.031 |\n| M12-10-S9 | 10 M /circledot | 0.9 | 1 . 2 M /circledot | 0.0396 | 0.033 |\n| M14-7-S7 | 7 M /circledot | 0.7 | 1 . 4 M /circledot | 0.0438 | 0.039 |\n| M14-7-S8 | 7 M /circledot | 0.8 | 1 . 4 M /circledot | 0.0437 | 0.037 |\n| M14-7-S9 | 7 M /circledot | 0.9 | 1 . 4 M /circledot | 0.0437 | 0.037 |\n| M14-10-S8 | 10 M /circledot | 0.8 | 1 . 4 M /circledot | 0.0441 | 0.042 |\n| M14-10-S9 | 10 M /circledot | 0.9 | 1 . 4 M /circledot | 0.044 | 0.043 |', 'C. Summary of the neutrino leakage scheme': "The neutrino leakage scheme used in this work is a first attempt at including the effects of neutrino radiation on the evolution of the remnant of NSBH mergers and at providing estimates of the properties of the emitted neutrinos (luminosity, species, average energy). A leakage scheme is a local prescription for the number and energy of neutrinos emitted from a given point in the disk, based on the local properties of the fluid and on an estimate of the optical depth. A detailed description of our leakage implementation can be found in Deaton et al. (2013) [10]. It is essentially a generalization to 3 dimensions of the leakage code implemented in GR1D (O'Connor & Ott 2010 [37]), which is itself inspired by the work of Ruffert et al. (1996) [11] and Rosswog & Liebendorfer (2003) [12]. Our leakage scheme aims at modeling the effects of neutrino cooling, while we do not consider the effects of neutrino heating. \nAs a brief summary, the main components of the leakage code are: \n- · A local prescription for the free emission of neutrinos as a function of the fluid properties (i.e. emission in the optically thin regime). We include β -capture processes, e + -e -pair annihilation, plasmon decay [11] and nucleon-nucleon Bremsstrahlung [51]. We compute the luminosity and number emission for ν e , ¯ ν e and ν x , where ν x stands for all other neutrino species ( ν µ,τ , ¯ ν µ,τ ), which are assumed to all have the same interactions with the fluid.\n- · Alocal prescription to compute the opacity to neutrinos as a function of the fluid properties and of the chemical potentials. We take into account scattering on nucleons and heavy nuclei and absorptions on nucleons [37].\n- · A prescription for the computation of the optical depth between the current point and the boundaries of the domain. We estimate this by integrating the opacities along the coordinate axes, and along the two most promising diagonal directions (i.e. those corresponding to the coordinate axes along which the optical depth is \nminimum). The minimum optical depth along the selected path is taken as the optical depth at the current point (see Sec. II D for an estimate of the influence of this choice). \n- · A local prescription for the rate at which neutrinos can escape through diffusion (i.e. in the optically thick regime), for which we follow Rosswog & Liebendorfer (2003) [12].\n- · A choice of an effective emission rate interpolating between the optically thin and optically thick limit. If Q free is the emission rate in the free emission regime and Q diff the emission rate in the diffusive regime, we use \nQ eff = Q diff Q free Q free + Q diff (2) \n(for both the energy and number emission rates). \nCompared to Deaton et al. 2013 [10], we have implemented two modifications to the leakage scheme. First, we compute the neutrino chemical potentials used in the determination of the opacities \nµ ν = µ eq ν (1 -e -〈 τ ν 〉 ) , (3) \nwhere µ eq ν is the β -equilibrium value of the potential and 〈 τ ν 〉 the energy-averaged optical depth, by iteratively solving for ( µ ν , τ ν ). This is required because τ ν is itself a function of the opacities. In [10], we instead used an analytical approximation for 〈 τ ν 〉 as a function of the local properties of the fluid when computing µ ν . Second, instead of treating the black hole as an optically thick region, we let neutrinos freely escape through the excision boundary. In [10], the black hole was treated as an optically thick region to avoid spuriously including the emission from hot points at the inner edge of the disk in the total neutrino luminosity. In this paper, when computing the optical depth along a direction crossing the excision boundary, we terminate the integration at that boundary. But points for which the direction of smallest optical depth is towards the black hole are excluded when computing the total neutrino luminosity of the disk. The cooling and composition evolution of the fluid are thus properly computed at the inner edge of the disk, but without affecting estimates of the neutrino luminosity. These changes neither appear to significantly modify the results of the simulations at late times, nor the properties or evolution of the post-merger accretion disk.", 'D. Error estimates': "To estimate the errors in the various observables discussed in this paper for simulations with M BH = 7 M /circledot , we perform a convergence test on simulation M14-7-S9, as well as a series of tests checking that the treatment of the low-density regions, the boundary condition on the region excised from the numerical domain inside of the apparent horizon of the black hole, and the method used to compute the neutrino optical depth do \nnot affect the results within the expected numerical errors. For the simulations with M BH = 10 M /circledot , which are significantly harder to resolve, we also simulate the late disruption phase using a fixed mesh refinement and a much finer grid and find that the standard grid choices used for the lower mass ratio cases are no longer appropriate in this regime. Issues specific to the high mass ratio cases are discussed at the end of this section. \nFor the convergence test with M BH = 7 M /circledot , we perform additional simulations at a lower and a higher resolutions. The SpEC code [31] uses two separate numerical grids [52]: a finite difference grid on which we evolve the general relativistic fluid equations, and a pseudospectral grid on which we evolve Einstein's equations. The finite difference grid is regularly regenerated so that it covers the region in which matter is present, but ignores the rest of the volume covered by the pseudospectral grid. The coupling between the two sets of equations then requires interpolation from one grid to the other, which we perform at the end of each time step. During the plunge and merger, the three resolutions have N FD = (120 3 , 140 3 , 160 3 ) points on the finite difference grid, which only covers the region in which matter is present. This leads to variation of the physical grid spacing over time, from ∆ x ∼ 200 m at the beginning of the plunge to ∆ x ∼ 2 km -3 km while we follow the ejected material as it escapes the grid, to finally ∆ x ∼ 1 km at the end of the simulation. The resolution on the spectral grid on which we evolve Einstein's equations is chosen adaptively, so that the relative truncation error of the spectral expansion of the metric and of its spatial derivatives is kept below /epsilon1 sp = (1 . 9 × 10 -4 , 1 × 10 -4 , 5 . 9 × 10 -5 ) at all times (as measured from the amplitude of the coefficients of the spectral expansion within each subdomain of the numerical grid). During the short inspiral, the finite difference grid covers a much smaller region (a box of size 2 R NS ∼ 26 km ) and has slightly less points ( N FD = (100 3 , 120 3 , 140 3 ) ). The truncation error on the spectral grid is kept at about the same level, except close to the black hole and neutron star where we require the truncation error to be a factor of 10 smaller. At these resolutions, we find that for many quantities the convergence is faster than second order, which is the expected convergence rate in the high resolution limit when finite difference errors dominate the error budget. This is presumably due to contributions to the error of terms with a higher convergence rate, such as grid-to-grid interpolation (third order), the evolution of Einstein's equations (exponentially convergent everywhere but in regions in which a discontinuity is present), or the timestepping algorithm (third order). When we observe a convergence rate faster than second-order, we assume second order convergence at the two highest resolutions to get an upper bound on the error. In previous simulations with the SpEC code, we have generally found this error estimate (which is about 4 times the difference between the results of the high and medium resolutions) to be much higher than the actual error in the results. Using this estimate, we find the following upper bounds for the errors in our medium resolution runs: \n· ∼ 25% relative error in the final disk mass \n- · ∼ 60% relative error in the mass and kinetic energy of the ejected material;\n- · 1% relative error in the mass and spin of the black hole at a given time;\n- · ∼ 50% relative error in the neutrino luminosity obtained from the leakage scheme - smaller than the error due to the use of the leakage algorithm itself, which might be a factor of a few (see e.g. [12]).\n- · ∼ 10 km / s of error in the recoil velocity due to gravitational wave emission ( v GW kick in Table III), due mostly to the extrapolation of the waveform to infinity. \nBeyond the finite resolution of the numerical grids, our choice of numerical methods cause additional errors which are important to quantify. First, we usually stop evolving the metric variables ∼ 15 ms after merger. By that point, the mass of the post-merger disk is at most ∼ 1% of the mass of the black hole. Previous tests on configurations with a relatively higher disk mass ( M )disk ∼ 0 . 03 M BH ) have however shown that, at the level of accuracy currently reached by our simulations, this has no observable effect on the results [53]. Similary, ∼ 1% -2% of the mass of the system leaves the grid as unbound or marginally bound material. By performing simulations in which the matter evolution was followed up to different radii, we have confirmed that the error thus induced (both in the evolution of the post-merger remnant and the evolution of Einstein's equation) is not measurable at our current accuracy, up to the point at which fall-back of the neglected bound material would affect the evolution of the disk. But this will only occur after multiple orbital timescale of the disk, by which point the fact that we neglect the effective viscosity induced by the growth of the magnetorotational instability is expected to be a much larger source of error. \nIn the leakage scheme, one potential source of error is the choice of specific directions over which the optical depth is computed (in our case, the coordinate axes and a subset of the diagonals). To assess the effect of that choice, we tried two alternative methods. In the first test, we computed the optical depth along every coordinate diagonal. In the second, we computed the optical depth by finding the path of smallest optical depth among all paths leading from the current point to a domain boundary (approximated as all paths formed of cellto-cell piecewise segments and connecting the current point to the outer boundary). We do this by computing the optical depth at a cell center τ c from the optical depth at the center of neighboring cells τ i and the optical depth along the line connecting the center of the cell to the center of the neighboring cells τ c -i , i.e. \nτ c = min i ( τ i + τ c -i ) . (4) \nIndeed, if the τ c -i are the optical depths along the paths through which neutrinos can escaped the most easily from the neighboring cells ' i ', then Eq. (4) returns the optical depth along the path through which neutrinos can the most easily escape from the current cell ' c '. We thus only need to iterate Eq. (4) until τ c converges at all points - which occurs rapidly \nif τ c is initialized to its value at the previous time step. This method is inspired by the recent work of Perego et al. [54] and Neilsen et al. [9], and closely follows the algorithm presented in [9]. Both tests lead to changes of ∼ 20% in the neutrino luminosities, thus showing that the exact method used to compute the optical depth is a fairly minor source of error compared to the systematic error due to the use of a leakage scheme. \nAs a last test of our code for simulations with M BH = 7 M /circledot , we consider the effect of the treatment of the lowdensity region. This region is dynamically unimportant, but could conceivably affect the neutrino luminosities if it develops a very hot atmosphere. We also study the impact of the black hole excision by using linear extrapolation of the density and velocities to the faces of our grid located at the border of the excised region instead of just copying the value of the fields from the nearest cell center, as well as by limiting the temperature of the fluid on the excision boundary to T = 10MeV . The resulting changes in the post-merger properties of the system are well below our estimates of the numerical error due to the finite resolution of our grid. \nWe now turn our attention to the higher mass ratio configurations. These simulations are significantly more difficult to resolve due to a number of issues: \n- · The disruption of the neutron star occurs very close to the black hole: except for simulation M12-10-S9, the tidal tail begins to form as high-density material has already begun to cross the apparent horizon of the black hole. But the accuracy of our code is lower close to the apparent horizon, due to the use of one-sided stencils close to the excised region inside of the horizon. Additionally, stricter control of the velocities and temperature within the low-density regions is necessary for stability in that excised region. Hence, properly resolving the disruption of the neutron star requires higher resolution in such cases.\n- · Compared to previous simulations using a Gamma-law ( Γ = 2 ) equation of state [55], simulations using the LS220 equation of state show important differences in the qualitative features of the tidal disruption. Initially, the neutron star remains more compact with the LS220 equation of state, and most of the expansion of the neutron star occurs later in the plunge (see Fig. 3), but more rapidly. This makes the disruption of the neutron star more difficult to resolve with the nuclear-theory based equation of state. This is ultimately a consequence of the large difference between the stiffness of the LS220 equation of state at high and low densities (with respect to nuclear saturation density).\n- · Once a tidal tail forms, tidal compression is very efficient and forces the tail to be much thinner than for a Γ = 2 equation of state (see Fig. 3). This is, again, a consequence of the properties of the equation of state at low densities. Because disruption occurs as the neutron star is already plunging into the black hole, the effects of tidal compression can be significant. We find \n<!-- image --> \nFIG. 3: Comparison of the distribution of matter during disruption for the LS220 equation of state used in this work ( left , simulation M1410-S8), and a Γ = 2 equation of state ( right , simulation Q7S9-R12i0 from [55]). Snapshots taken in the equatorial plane at a time at which 0 . 25 M /circledot of material remains outside of the black hole. Both simulations lead to ∼ 0 . 1 M /circledot of material remaining outside of the black hole at late times - yet the distribution of that material between bound and unbound material and the properties of the tidal tail are very different. In each case, the thick black line shows the location of the apparent horizon of the black hole. \n<!-- image --> \nthat the tail has a typical coordinate width (and vertical height) of only a few kilometers (note that Fig. 3 uses a logarithmic scale for the density). This should be compared with the radius of the apparent horizon of the black hole, which in our gauge is ∼ 30 km . Accordingly, the typically resolutions used in the simulation of NSBH mergers are insufficient in this case. Indeed, simulations using fixed mesh refinement typically use 20 -30 grid points across the radius of the black hole (see e.g. [29]), while our finite difference grid generally has a higher resolution at the beginning of the disruption ( ∼ R NS / 50 ∼ 0 . 2 km ), but rapidly becomes coarser as we progressively expand the grid in order to follow the expansion of the tidal tail. Additionally, our simulations suffer from lower accuracy in the immediate neighborhood of the excised region, as opposed to codes which do not excise the inside of the black hole and can use high-order finite difference stencils crossing the horizon. In SpEC, and for large black hole spins, the excised region can be very close to the horizon of the black hole [56], and thus some regions outside of the horizon have to use low-order stencils including points on the boundary of the evolved computational domain where errors in the fluid properties are larger. \nAccordingly, as soon as the resolution of our finite difference grid drops below ∆ x ≈ 0 . 5 km -1 km , simulations with M BH = 10 M /circledot and the LS220 equation of state suffer great losses of accuracy. This typically takes the form of a rapid spurious acceleration of the tail material within the unresolved region, which then becomes unbound with large asymptotic velocities ( v/c ∼ 0 . 6 -0 . 9 ). To resolve the tidal tail during \nthis phase requires resolution that our code cannot maintain while following the unbound ejecta and bound tidal tail far away from the black hole. One solution is to use mesh refinement for the finite difference grid. Although we have implemented a fixed mesh refinement algorithm for our finite difference grid, the cost of interpolating between the different evolution grids currently makes it impractical to use more than 2-3 levels of refinement, and thus to resolve accurately the material close to the black hole while following the ejecta for a few milliseconds, especially for high mass ratio simulations which require a small grid spacing on both our finite difference and pseudospectral grids. Instead. we choose to continue simulations M12-10-S8 and M12-10-S9 through disruption while maintaining a resolution in the region close to the black hole of ∆ x < 0 . 7 km (and vertically ∆ x = 0 . 2 km ) for M12-10-S9, with 2 levels of refinement, and ∆ x < 0 . 4 km (and vertically, ∆ x = 0 . 1 km ) for the more challenging M1210-S8 , with 3 level of refinements. Each refinement level is a box using 140 3 grid points, and the grid spacing is multiplied by a factor of 2 between levels. Both of those simulations confirmed that the rapid acceleration observed at lower resolution is indeed a numerical artifact. The higher resolution simulations eject ∼ 0 . 1 M /circledot -0 . 15 M /circledot of material at velocities v/c < ∼ 0 . 5 . Using a fixed mesh refinement with a resolution similar to the one used in simulation M12-10-S9 proved to be insufficient to resolve the configurations for which the disruption occurs as the neutron star crosses the apparent horizon of the black hole (M12-10-S8, M14-10-S8, M14-10-S9). Clearly, with these grids covering such a small region around the black hole (up to ∼ 5 M BH ∼ 75 km ), we can only measure the ejected mass and the mass remaining outside of the \nblack hole (and the latter will only be a rough estimate). We cannot follow the evolution of the disk as it expands and is impacted by fall-back from the tidal tail. Hence, for the high mass ratio simulations with fixed mesh refinement (M12-10S8, M12-10-S9), we stop the evolution when an accretion disk forms. Similarly, for the high mass ratio simulations without mesh refinement (M14-10-S8, M14-10-S9), we only obtain estimates of the ejected mass and of the mass remaining outside of the black hole: we cannot reliably continue the evolution once the resolution drops below ∆ x ≈ 0 . 5 km -1 km .", 'E. Tracer particles': 'A major motivation for the simulations presented here is to characterize the unbound outflow: its mass, asymptotic velocity distribution, composition, and entropy. In a stationary spacetime and in the absence of pressure forces, a fluid element with 4-velocity u α is unbound if it has positive specific energy E = -u t -1 (under those conditions, E is constant). In our simulations, we consider a fluid element as unbound if E > 0 as it leaves the grid - even though it has a finite pressure and the metric is not yet completely stationary. Thus, another potential source of error in our conclusions comes from the finite outer boundary of our fluid evolution grid. \nTo test that the outflow reaches its asymptotic state before exiting the simulation, we evolve one hundred Lagrangian tracer particles in the weakly bound and unbound flow. That is, the position of tracer particle A is evolved according to the local 3-velocity: dx i A /dt = v i ( x i A ) , and fluid quantities at tracer positions are monitored by interpolating from the fluid evolution grid. \nIn Fig. 4, we plot some fluid quantities for four representative randomly chosen unbound particles in the ejecta of case M12-7-S9. We see that the energies stabilize after only about two milliseconds of post-merger evolution. The electron fraction Y e becomes stationary the soonest because the tail is too cold for strong charged-currentinteractions that would change Y e . Entropy also levels off, albeit more slowly. Over the 4 ms of evolution, the density decreases by about three orders of magnitude. \nWe are thus confident that near-merger gravitational fluctuations, pressure forces, and neutrino emission do not significantly contaminate our predictions about the outflows. It is important to note that the entropy and energy may change significantly outside the evolved region because of recombination and nucleosynthesis in the decompressing material. Charged current heating from the background neutrino field, which we do not include in our simulations, can also change the composition of the ejecta, particularly for material unbound by disk winds (as opposed to the material ejected during the disruption of the neutron star). The late-time, largedistance evolution of the ejecta is a matter of ongoing study. \n8 \nFIG. 4: The evolution of four representative tracer particles representing unbound fluid elements. We plot the baryon density ρ , the specific entropy S , the electron fraction Y e , and the specific energy E ≡ -u t -1 . The small glitches in the entropy evolution are artifacts of stencil changes with the low-order interpolator used for Lagrangian output. Note, the tracer particle denoted by the solid black line begins closest to the black hole. \n<!-- image -->', 'A. Disruption of the neutron star': 'Let us now consider the results of these simulations, starting with the disruption and merger dynamics. At the time of tidal disruption, the results are largely unaffected by neutrinos (which act over timescales ∼ 10 ms , while the merger takes ∼ 1 ms ). Instead, the main components required to properly model the system are the size and structure of the neutron star, and general relativity. \nWe choose the initial parameters of these simulations so that the neutron star is disrupted, thus allowing the formation of an accretion disk and/or the ejection of material. This is indeed what we observe. The mass remaining outside of the black hole after disruption (including unbound material) and the final black hole mass and spin are in agreement with previous numerical studies using simpler equations of state, either in the same range of parameters [55], or for slightly lower black hole spins [29]. They are also largely consistent with semi-analytical models mostly based on lower mass simulations with simpler equations of state [35, 57]. However, going beyond these global properties of the post-merger remnant, we observe differences in the qualitative features of the disruption phase, modifying the distribution of the material remaining outside of the black hole between unbound material, bound tail and accretion disk (summarized in Table III). \nThe systems with the more symmetric mass ratio ( q = 5 , \nTABLE III: Properties of the final remnant. M f BH is the mass of the black hole at the end of the simulation, χ f BH is its dimensionless spin, v kick , GW is the predicted recoil velocity due to gravitational wave emission (i.e. not taking into account the larger recoil from the asymmetric ejection of material), while v BH recoil is the measured coordinate velocity of the black hole after merger. M 5ms out is the baryon mass outside of the black hole 5 ms after merger (including unbound material). M 20ms disk , M 20ms tail , bound and M ej are the baryon mass of the disk, bound tail material, and unbound material 20 ms after merger. By conservation of baryon number, M 5ms out -M 20ms disk -M 20ms tail , bound -M ej is the mass accreted onto the black hole in the 15 ms between the two sets of measurements. E ej and 〈 v ej 〉 are the kinetic energy and average asymptotic velocity of the unbound material. Fluid quantities for the higher mass ratio simulations (second half of the table) generally have larger uncertainties, as discussed in Sec. II D, and are only provided when they can be reliably extracted from the simulation. \n| Name | M f BH χ f BH v | GW kick (km/s) | v BH recoil (km/s) | M 5ms out | M 20ms disk | M 20ms tail , bound | M ej | E ej (10 51 erg) | 〈 v ej 〉 /c |\n|-----------------|---------------------------|------------------|----------------------|-----------------------|---------------------|-----------------------|----------------------------------|--------------------|------------------|\n| M12-7-S8 | 7 . 79 M /circledot 0.85 | 60 | 675 | 0 . 31 M /circledot | 0 . 09 M /circledot | 0 . 04 M /circledot | 0 . 14 M /circledot | 8.5 | 0.24 |\n| M12-7-S9 | 7 . 74 M /circledot 0.91 | 69 | 874 | 0 . 37 M /circledot | 0 . 13 M /circledot | 0 . 03 M /circledot | 0 . 16 M /circledot | 11 | 0.25 |\n| M14-7-S7 8 . | 07 M /circledot 0.81 | 79 | 151 | 0 . 15 M /circledot | 0 . 04 M /circledot | 0 . 05 M /circledot | 0 . 04 M /circledot | 1.7 | 0.20 |\n| M14-7-S8 8 . | 00 M /circledot 0.87 | 76 | 185 | 0 . 25 M /circledot | 0 . 08 M /circledot | 0 . 05 M /circledot | 0 . 06 M /circledot | 2.0 | 0.18 |\n| M14-7-S9 7 | . 95 M /circledot 0.92 | 94 | 345 | 0 . 35 M /circledot | 0 . 14 M /circledot | 0 . 04 M /circledot | 0 . 07 M /circledot | 2.5 | 0.18 |\n| M12-10-S8 | 10 . 75 M /circledot 0.83 | 45 | - | ∼ 0 . 25 M /circledot | - | - | ∼ (0 . 10 - 0 . 15) M /circledot | - | - |\n| M12-10-S9 | 10 . 58 M /circledot 0.89 | 49 | - | ∼ 0 . 40 M /circledot | - | - | ∼ (0 . 10 - 0 . 15) M /circledot | ∼ 10 - 20 | ∼ 0 . 3 - 0 . 35 |\n| M14-10-S8 | 11 . 00 M /circledot 0.85 | 22 | - | ∼ 0 . 10 M /circledot | - | - | ∼ (0 . 05 - 0 . 10) M /circledot | - | - |\n| M14-10-S9 10 70 | . M /circledot 0.90 | 54 | - | ∼ 0 . 30 M /circledot | - | - | ∼ (0 . 10 - 0 . 15) M /circledot | - | - | \nwith M NS = 1 . 4 M /circledot and M BH = 7 M /circledot ) provide the most traditional results: between 0 . 15 M /circledot and 0 . 35 M /circledot of material remains outside of the black hole 5 ms after merger. The higher black hole spins naturally lead to more massive remnants, as expected from the fact that the innermost stable circular orbit is at a lower radius for prograde orbits around a more rapidly spinning black hole. We find that the system ejects 0 . 04 M /circledot -0 . 07 M /circledot of material, which appears compatible with the range of ejected mass listed for NSBH binaries at χ BH = 0 . 75 by Hotokezaka et al. [27]: although [27] does not list values for individual systems, they find a range M ej ∼ 0 . 02 M /circledot -0 . 04 M /circledot for the neutron star closest in size to the 1 . 4 M /circledot neutron star used in this work and mass ratios q = 3 -7 . The higher masses found here can easily be attributed to larger black hole spins. The average velocity of the ejecta is 〈 v 〉 ∼ 0 . 2 c , where the average 〈 X 〉 of a quantity X on a spatial slice Σ is here defined as \n〈 X 〉 = ∫ Σ ρXW √ gdV Σ ρW √ gdV , (5) \nρ is the baryon density of the fluid, W its Lorentz factor, and g is the determinant of the 3-metric on slice Σ . This is slightly lower than in [27] ( 〈 v 〉 ∼ 0 . 25 c -0 . 29 c ). As for the bound material, 20 ms after merger 0 . 04 M /circledot -0 . 14 M /circledot has formed a hot accretion disk cooled by neutrinos (discussed in Sections III C-IV), 0 . 04 M /circledot -0 . 05 M /circledot has left the grid on highly eccentric bound trajectories, while the rest ( 0 . 02 M /circledot -0 . 10 M /circledot ) has accreted onto the black hole. Extrapolating the remnant mass to lower spins indicate that the neutron star will disrupt for χ BH > ∼ 0 . 55 , as predicted from simulations with simpler equations of state [35]. \n∫ \nAt higher mass ratios, we generally find that a larger amount of material is ejected. Our two simulations with M NS = 1 . 2 M /circledot and M BH = 7 M /circledot find about the same amount of material in the disk and bound tail as for the q = 5 cases ( M disk ∼ 0 . 1 M /circledot , M tail ∼ 0 . 05 M /circledot ), but about 0 . 15 M /circledot is ejected at speeds 〈 v 〉 ∼ 0 . 25 c (see Fig. 5). The closest results to compare to are again those of Hotokezaka et \nal. [27], for the H4 equation of state, which has a compactness C NS = 0 . 147 for a mass M NS = 1 . 35 . Hotokezaka et al. [27] find M ej = 0 . 04 M /circledot -0 . 05 M /circledot for q = 3 -7 and χ BH = 0 . 75 , and similar average velocities. Even considering the different spins used and the expected error bars, the more massive ejecta found here appear to be an indication of a dependence of the ejected mass on the internal structure of the neutron star: from our results, we would predict M ej ∼ 0 . 13 M /circledot for q = 5 . 8 and χ BH = 0 . 75 . The difference with the results of Hotokezaka et al. [27] is slightly out of the 60% relative error that we consider to be a strict upper bound on the error in our measurement of M ej , and which is most likely a significant overestimate of that error. The most likely reason for an equation of state dependence of the ejected mass at higher mass ratios is that the disruption of the neutron star then occurs as the neutron star is plunging into the black hole. The mass and properties of the ejected material then depend not only on the separation at which disruption occurs, but also on the timedependent response of the neutron star to the tidal disruption. For the same reason, the properties of the post-merger remnant also have a steeper dependence in the mass and spin of the black hole for more massive black holes. \nAs discussed in Sec. II D, for the simulations with M BH = 10 M /circledot we can only resolve the disruption of the neutron star and rapid accretion onto the black hole if we use a very high resolution grid covering only a small area around the black hole. This is due largely to qualitative differences in the disruption process: the neutron star disrupts as high-density material has already begun to cross the apparent horizon, and the entire disruption and tail formation process occurs within a distance of about 3 M BH ∼ 45 km of the black hole center. All material surviving the merger is on highly eccentric orbits, and the tidal tail experiences strong tidal compression in the directions in which the tidal field of the black hole causes trajectories to converge. The tidal tail is reduced to a thin stream of matter only a few kilometers wide. Not surprisingly, in this regime the result of the merger is very sensitive to the parameters of the binary: for example, a small change in the spin of \nFIG. 5: Matter distribution during the disruption of the neutron star for case M12-7-S8. About half of the remnant material is unbound, while a relatively low mass hot disk forms. \n<!-- image --> \nthe black hole (e.g. δχ BH ∼ 0 . 1 , as in our simulations) drastically changes the amount of material which remains outside of the black hole after disruption. Changes in the stiffness of the equation of state, which affect the distribution of matter during disruption, also become even more important than in the previously discussed configurations (see Fig. 3 for a comparison with a similar simulation with a Γ = 2 equation of state). Surprisingly, even in this case, the total amount of material surviving disruption remains similar for both the Γ = 2 and LS220 equations of state. We can for example compare simulation M14-10-S9 of this work, for which about 0 . 3 M /circledot remains outside the black hole after merger, and simulation R13i0 of Foucart et al. 2013 [55], with the same black hole parameters and a similar neutron star radius R NS = 13 . 3 km , in which 0 . 31 M /circledot of material remains. But more material is unbound during merger for the LS220 equation of state: M ej > ∼ 0 . 1 M /circledot here, while M ej ∼ 0 . 05 M /circledot in Foucart et al. 2013 [55]. And the disk is initially less massive here: half of the material promptly formed a disk in [55], nearly all of the material is on highly eccentric orbits here. The long term evolution of the disk will thus be more significantly affected by the fall-back of tail material than expected from the simulations with a Γ = 2 equation of state. Compared to the analytical prediction of [35], there is an excess of material surviving the disruption, which is what we usually find for high spin black holes and high mass ratio. But the estimate that disruption will only occur for χ BH > ∼ 0 . 65 (resp. χ BH > ∼ 0 . 75 ) for M NS = 1 . 4 M /circledot (resp. 1 . 2 M /circledot ) appears to be accurate although it is of course dangerous to draw such a conclusion by extrapolating from only 2 simulations with inaccurate estimates of the remnant mass. \nAll of the simulations presented here thus appear to have post-merger remnants with large amounts of both bound and unbound material, providing promising setups for potential \nFIG. 6: Baryon mass left on the numerical grid as a function of time, for all simulations with M BH = 7 M /circledot . \n<!-- image --> \nelectromagnetic signals. The typical timescales for the merger can be observed on Fig. 6, which shows the baryon mass remaining on the grid at any given time for all simulations with M BH = 7 M /circledot . The disruption of the neutron star and rapid accretion of material onto the black hole occur over ∼ 1 ms . Then accretion slows down and the disk evolves over timescales of ∼ 0 . 1s . The rapid variation in the mass present on the grid around 5 ms -10 ms after merger is due to the unbound material leaving the grid (and discrete jumps are due to modifications of the location of the outer boundary of the grid as the algorithm in charge of deciding which regions of the spacetime to cover abandons low-density regions containing unbound or marginally bound material). Not shown are the simulations for M BH = 10 M /circledot , for which most of the material escaping early accretion onto the black hole rapidly leaves the grid. In the following sections, we will now look in more detail into the properties of the unbound material and the evolution of the disk.', 'B. Properties of the outflows': 'Given the large mass of material ejected by these mergers, NSBH binaries with M BH ∼ 7 M /circledot -10 M /circledot appear to be prime candidates to produce post-merger electromagnetic counterparts, and large quantities of r-process elements (see Table III). Although the mass and velocity of these outflows vary from case to case, other properties of the ejecta are more consistent. Since the material is ejected before neutrino emission has had a chance to modify its composition, it has a low electron fraction ( Y e < 0 . 1 ). The neutron star matter initially has a low entropy per baryon, and is not significantly heated during the tidal compression of the tail and the ejection of material: the entropy per baryon of the ejecta for the simulations with M BH = 7 M /circledot is 〈 S 〉 /k B ∼ 4 -5 , and it is only slightly higher for simulations with M BH = 10 M /circledot ( 〈 S 〉 /k B ∼ 5 -7 ). The outflow is expected to robustly undergo r-process nucleosynthesis, with the production of material with A > 120 \nFIG. 7: Distribution of asymptotic velocities for the unbound material in simulation M14-7-S9. Other simulations with M BH = 7 M /circledot have similar distributions, with a slightly higher peak value when M NS = 1 . 2 M /circledot . \n<!-- image --> \ninsensitive to variations in the properties of the outflow within the range of values observed here [19]. Accordingly, all of the simulations considered here are promising setups for the production of infrared transients [4, 19] and heavy r-process elements. \nThe asymptotic velocity distribution of the ejecta (its expected velocity at infinite distance from the black hole) is shown in Fig. 7 for simulation M14-7-S9. For all configurations with M BH = 7 M /circledot , we find similar distributions spanning 0 < v < 0 . 5 c , with slight variations in the location of the peak. A more asymmetric mass ratio generally results in higher asymptotic velocities for the ejecta. This additional acceleration is presumably due to the fact that, since the disruption occurs on an eccentric orbit passing close to the black hole, the fluid can be efficiently accelerated by the rapid compression and decompression of the tidal tail under the influence of the tidal field of the black hole (see also Sec. III A). This effect is specific to high mass ratio binaries, for which the disruption occurs as the neutron star plunges into the black hole and the properties of the disrupted material are sensitive to the equation of state (see Fig. 3). \nAs pointed out in Kyutoku et al. 2013 [58], the ejection of unbound material is not isotropic. In our simulations, the ejecta is confined within ∼ 20 · of the equatorial plane, and covers an azimuthal angle ∆ φ ∼ π (see Fig. 5). An important consequence of this anisotropy is that a kick is imparted onto the black hole with v kick , ej ∼ M ej v ej /M BH . For simulation M12-7-S9, for which we follow tracer particles, we measure the linear momentum carried away by the ejecta and find \nv kick , ej ∼ 0 . 5 M ej M BH 〈 v 〉 ej ∼ 770 km / s . (6) \nThe factor of 0 . 5 comes from the fact that not all of the ejecta is ejected in the same direction. We estimate it from the direction of v i as the tracer particles cross spheres of constant radius. Even considering the uncertainties in our measurements of M ej and 〈 v 〉 ej , this is clearly larger than the kick velocity due to gravitational wave emission (see Table III): our simulations predict v kick , ej ∼ 150 km / s -800 km / s but \nFIG. 8: Average entropy per baryon on the computational grid as a function of time, for all simulations with M BH = 7 M /circledot . \n<!-- image --> \nv kick , GW ∼ 20 km / s -100 km / s . These velocities play an important role when assessing whether globular clusters can retain a black hole after it merges with a neutron star, and in the determination of the rate of NSBH mergers occurring in those clusters [59]. At the high end of this velocity range, the kick could even be above the escape velocity of small galaxies. \nThe amplitude of this kick is qualitatively confirmed by measurements of the coordinate velocity of the black hole, defined as its average velocity on the grid between 5 ms and 15 ms after merger. Although coordinate velocities are gauge dependent and should thus be taken with a certain skepticism, the values measured here are compatible with the predictions from Eq. (6) while being much larger than the predicted recoil velocities due to asymmetric emission of gravitational waves (see Table III).', 'C. Disk evolution': "Constraining the formation and the long-term evolution of the accretion disks resulting from NSBH mergers is an important step towards assessing their potential as short gamma-ray burst progenitors. Those disks are also likely to power outflows (e.g. via neutrino-driven or magnetically-driven winds, or recombination of α -particles), causing optical and radio signals. \nIn the simulations with M BH = 7 M /circledot , we follow the postmerger evolution of the system. We define the time of merger t merge as the time at which 50% of the neutron star has been accreted onto the black hole, and the disk mass M disk as the mass remaining in the accretion disk 20 ms after t merge . We evolve the disks for about 40 ms after t merge , switching to a fixed metric evolution after about 15 ms . We find a range of disk masses M disk = 0 . 04 M /circledot -0 . 14 M /circledot . At late times, the accretion timescale in these disks is t acc ∼ 150 ms for the M NS = 1 . 2 M /circledot simulations and t acc ∼ 75 ms for the M NS = 1 . 4 M /circledot simulations. \nThe evolution of the disk can be subdivided into two main periods. For the first 5 ms -10 ms , shocks in the forming disk heat the material and raise the entropy of the fluid (see Fig. 8). \nFIG. 9: Distribution of matter around the black hole measured 5 ms , 10 ms and 40 ms after merger, for simulation M14-7-S8. \n<!-- image --> \nFIG. 10: Distribution of the average temperature and electron fraction around the remnant black hole for simulation M14-7-S8, at the same times as in Fig. 9. The average is taken after subdividing all material into radial bins identical to those shown in Fig. 9. \n<!-- image --> \nThe resulting disk is compact and hot (see below). However, average temperatures rapidly decrease through a combination of effects: the expansion of the hot compact disk, the emission of neutrinos, and the fall-back of cool material from the tail while hot material is accreted onto the black hole. An example of this evolution is shown in Figs. 9 and 10, where we look in more detail at simulation M14-7-S8. 5 ms after merger, about a third of the remaining material is in a compact hot disk, with most of the mass at r ∼ 5 M BH ∼ 60 km and temperature T ∼ 5 MeV -15 MeV . The rest is in a cool tidal tail, part of which is unbound. 5 ms later, the disk has spread and slightly cooled down ( T ∼ 3 MeV -10 MeV ), while the fall-back rate from the tidal tail begins to decrease. At later times, and given that we do not include the effects of viscosity, we no longer have any significant source of heating in the disk. As hotter material accretes onto the black hole, cooler material falls onto the disk, and neutrinos extract energy from the system, the disk cools down. At the end of the simulation, most of the disk is at T ∼ 2 MeV (except for the small amount \nof hotter material at the inner edge of the disk). It is also confined to a small radial extent of 50 km < r < 100 km . As discussed in Deaton et al. 2013 [10], this is a direct consequence of the loss of energy due to emitted neutrinos. Visualizations of the disk for the same simulation are shown in Fig. 11, at the same times. The evolution of the composition of the fluid will be discussed in the next section, as it is tightly linked to the properties of the neutrino emission. \nAfter the first 10 ms, matter in the disks largely settles into circular orbits, although they remain highly non-axisymmetric throughout the post-merger evolution. In Fig. 12, we show vertically- and azimuthally-averaged profiles of the density, temperature, entropy, and electron fraction at three times, separated by 8 ms, for the case M12-7-S8. Other systems that produce large disks show very similar profiles. The disk evolves under the influence of accretion and neutrino emission, causing the disk to cool and the inner regions to evacuate. The S and Y e profiles approach the 'U' shape seen in our previous simulation [10]. Fig. 13 shows the angular velocity and angular momentum profiles 31 ms after merger. The inner disk is almost entirely rotation-supported, as can be seen from the agreement between the actual rotation and the curves for a circular geodesic orbit. Beyond 70 km, the actual rotation is significantly slower than geodesic, a sign that the outer disk has significant pressure support. This is confirmed by the agreement between the actual rotation rate and what is required for a hydrodynamic equilibrium circular orbit, an agreement that only breaks down in the very outer regions where the disk has not equilibrated. It is also observed in the fact that the sound speed c s decreases much more slowly with radius than the angular velocity. The small change in c s is a consequence of the moderate variation in temperature across the disk and the very weak dependence of c s on density for the \ndisk's range of densities and temperatures. The low \n∂c s ∂ρ \n∣ ∣ ∣ T,Y e comes from passing through a minimum of c s ; at lower densities, the pressure is dominated by relativistic particles, while at higher densities the degeneracy pressure is stronger. We observe that the angular momentum of a freely falling particle on a circular orbit actually has a minimum at around 34km, which is also where the radial epicyclic frequency of perturbed circular orbits (measured as in [60]) vanishes. This is the innermost stable circular orbit (ISCO). Inside the ISCO, the fluid picks up a significant radially ingoing velocity component, and, as would be expected, the angular momentum profile is relatively flat. It is worth pointing out the difference in this inner-disk behavior from what was found in our previous paper [10], which involved a less-massive black hole and a more-massive accretion disk. In those simulations, we did not resolve a geodesic ISCO, but we did see a sharp increase in the orbital energy in the inner disk, connected to strong pressure forces. This was associated with an apparent inner disk instability. No such effects are seen for these systems with more massive black holes and less massive disks (see Sec. IV). \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \nt -t merge \n= 5 ms \n<!-- image --> \nt -t merge \n= 10 ms \n<!-- image --> \nt -t merge \n= 40 ms \nFIG. 11: Disk evolution for case M14-7-S8. The 3 snapshots are taken 5 ms after merger (top, unbound material leaves the grid), 10 ms after merger (middle, end of rapid protonization of the disk), and 40 ms after merger (bottom, end of the simulation). Left: 3D distribution of baryonic matter. Center: Electron fraction in the vertical slice perpendicular to the orbital plane of the disk which passes through the initial location of the black hole and neutron star. The white line is the density contour ρ = 10 10 g / cm 3 . Right: Temperature in the same vertical slice. In the first two snapshots, the disk is hot and rapidly protonizing. At later times, the disk cools down to T ∼ 2 -3 MeV and becomes very neutron rich again ( Y e ∼ 0 . 05 ). \n<!-- image -->", 'IV. NEUTRINO COOLING AND DISK COMPOSITION': "Neutrino emission plays an important role in the postmerger evolution of NSBH remnants. It is indeed the main source of energy loss for the resulting accretion disk, and may also deposit energy in the low-density regions above the disk through ν ¯ ν annihilation, or drive outflows through neutrino absorption in the upper regions of the disk. The simple leakage scheme used in our simulations can only address the cooling of the disk, as we do not take into account the effects of neutrino heating. A first guess at the importance of ν ¯ ν anni- \non and neutrino absorptions can however be made from the intensity of the neutrino emission coming from the disk. \nTo understand the evolution of our disks under neutrino emission, it is helpful to compare our results to previous simulations of accretion tori using a pseudo-Newtonian potential and with more controlled initial conditions (Setiawan et al. 2006 [13]). There are a few important differences between this paper's simulations and those presented in [13]. The approximate treatment of gravity used in [13] might significantly affect the behavior of the inner disk, especially for high-spin cases. Also, the initial conditions in [13] were axisymmet- \nFIG. 12: The late-time post-merger evolution of the disk formed in case M12-7-S8. We plot density-weighted average values along cylindrical shells, with the shell radius defined from the proper circumference of the cylinder on the equator. Data at three times are shown: 23, 31, and 39 ms after the merger time. \n<!-- image --> \nFIG. 13: The equatorial velocity in the φ -direction, v φ ≡ u φ r/u t , and the specific angular momentum L ≡ -u φ /u t in the disk for case M12-7-S8 measured 31 ms after merger. For comparison, the sound speed, c s , is also included in the top panel, as are the v φ and L values for geodesic circular orbit and hydrodynamic equilibrium circular orbit for the given metric and pressure profile in both panels. \n<!-- image --> \nric (taking the azimuthal average of the final result of an earlier NSBH merger simulation performed with a Newtonian code [61]), were colder than the initial post-merger state of our simulations ( T ∼ 1 MeV -2 MeV in most of the disk), and used lower black hole spin parameters (as the post-Newtonian potential performs worse with higher spins). Some of their simulations also include an explicit viscosity, which in the most extreme cases ( α = 0 . 1 ) causes the disk to heat to temperatures similar to our initial conditions ( T ∼ 10 MeV ). The typical density ( ρ 0 ∼ 10 10 -10 11 g / cm 3 ) and electron fraction ( Y e < 0 . 05 ) are fairly similar to those found in our disks resulting from general relativistic mergers. \nFor disks with T ∼ 10 MeV , Setiawan et al. [13] find to- \nFIG. 14: Total neutrino luminosity (all species, at infinity) as a function of time, for all simulations with M BH = 7 M /circledot . \n<!-- image --> \nFIG. 15: Luminosity of the electron neutrinos, electron antineutrinos, and for each of the other 4 species ( ν µ,τ , ˜ ν µ,τ ), for simulation M147-S8. \n<!-- image --> \nl neutrino luminosities L ν ∼ 10 53 erg / s , dominated by the emission of electron antineutrinos. This is expected because the disk is both extremely neutron-rich and at a low enough density that, at this temperature, beta-equilibrium would require a more balanced distribution of neutrons and protons. As a consequence, the disk rapidly protonizes, with the electron fraction rising to Y e ∼ 0 . 1 -0 . 3 on a timescale of 20 ms . This is fairly similar to the conditions observed in our disk at early times ( ∼ 10 ms after merger), except that our initial conditions are asymmetric and that the high temperatures come from shock heating during disk formation rather than viscosity. Accordingly, the early evolution of the disk and its neutrino emission follow a similar pattern. In Fig. 14, we show the total neutrino luminosity for all simulations with M BH = 7 M /circledot . At early times, we find L ν ∼ (2 -5) × 10 53 erg / s , which appears consistent with Setiawan et al. [13], given the slightly higher temperatures present in our disks. As in [13], the luminosity also increases with the disk mass - although the high variability of our disks makes it difficult to derive an actual scaling (it was L ν ∝ M 2 disk in [13]). The emission is dominated by electron antineutrinos, with τ and µ neutrinos having fairly negligible contributions to the total luminosity \nTABLE IV: Neutrino luminosities L ν and average energy 〈 /epsilon1 ν 〉 10 ms after merger. All luminosities are in unites of 10 52 erg / s , and energies are in MeV. L ν x is the luminosity for each of the 4 types of neutrinos ν µ,τ , ¯ ν µ,τ individually. \n| Name | L 10ms ν e | L 10ms ¯ ν e | L 10ms ν x | 〈 /epsilon1 〉 10ms ν e | 〈 /epsilon1 〉 10ms ¯ ν e | 〈 /epsilon1 〉 10ms ν x |\n|----------|--------------|----------------|--------------|----------------------------|------------------------------|----------------------------|\n| M12-7-S8 | 9 | 13 | 0.2 | 11 | 14 | 16 |\n| M12-7-S9 | 10 | 14 | 0.2 | 10 | 14 | 14 |\n| M14-7-S7 | 6 | 11 | 0.2 | 12 | 16 | 18 |\n| M14-7-S8 | 10 | 14 | 0.2 | 11 | 15 | 16 |\n| M14-7-S9 | 13 | 23 | 1.5 | 11 | 14 | 16 | \nFIG. 16: Neutrino number emission as a function of time, for all simulations with M BH = 7 M /circledot . A short period of rapid protonization occurs during the first 10 ms -15 ms after merger. Then, the emission of electron neutrinos and antineutrinos become more balanced. \n<!-- image --> \nFIG. 17: Average electron fraction of the material on the grid, for all simulations with M BH = 7 M /circledot . The early rise in 〈 Y e 〉 is due to preferential emission of electron antineutrinos, while the late time evolution is due to the accretion of the higher Y e material onto the black hole and, for M NS = 1 . 2 M /circledot , the emission of a slight excess of electron neutrinos. \n<!-- image --> \n(see Fig. 15 for simulation M14-7-S8, and the summary of the properties of the neutrino emission for all simulations in Table IV). The net lepton number emission, shown in Fig. 16, is highly variable but clearly causes a protonization of the disk - an effect confirmed by the evolution of the average electron fraction in the fluid (Fig. 17). Finally, the energy of the \nFIG. 18: Equilibrium electron fraction Y e, eq in the free streaming regime, defined as the value of Y e at which the rate of positron and electron captures are equal for optically thin material, i.e. ignoring the effect of neutrino absorption on the matter. The black (thick) set of curves result from the balancing of the electron and positron capture rates used in our leakage code. There are corrections to these capture rates that make the energy dependence not ∝ E 2 . These cannot be included in our leakage scheme, but would adjust the value of Y e, eq to the red set of curves. These corrections include those due to the neutron-proton rest mass difference, weak magnetism, and finite electron mass. The largest difference between these rates occurs either at low densities, where neutrino emission is low, or high temperatures, where the free streaming assumption is not valid. The various lines of each color show Y e, eq ( T ) for a range of baryon densities (solid lines, labeled by log ρ = 9 , 10 , 11 , 12 in g / cm 3 ). The dotted lines are separated by ∆log ρ = 1 / 3 . We only show the range 0 . 035 < Y e < 0 . 55 covered by our equation of state table. \n<!-- image --> \nemitted neutrinos is very consistent across all disks (see Table IV), with the electron neutrinos having the lowest energy ( 〈 /epsilon1 ν e 〉 ∼ 10 MeV -12 MeV ), and the electron antineutrinos being slightly more energetic ( 〈 /epsilon1 ¯ ν e 〉 ∼ 14 MeV -16 MeV ). The other species of neutrinos have energies comparable to 〈 /epsilon1 ¯ ν e 〉 , are more variable, and are emitted at much lower rates. This is slightly lower than the neutrino energies observed for the hottest disks in [13] ( 〈 /epsilon1 〉 ∼ 20 MeV ). The large variations observed in the neutrino luminosity during the first ∼ 10 ms following the merger (see Fig. 14) are due to the chaotic nature of the disk formation process, and are particularly large for the lower mass disks. For example, simulation M14-7-S7 has a period ∼ 5 ms after merger in which there is very little material in the hot inner disk (most of the material shocked at early times has been accreted by the black hole), and the neutrino luminosity is very low. As the remaining material from the cold tidal tail falls onto the forming disk and heats \nup, the luminosity rises again. And once a massive and more symmetric accretion disk forms, the variability in the neutrino luminosity decreases. Simulations with more massive postmerger remnants, on the other hand, rapidly form massive, hot inner disks and show less variability during these first 10 ms . \nAt later times, interesting changes in the properties of the disk arise due to the cooling of the disk and the fall-back of tail material. As hotter, more proton-rich material close to the black hole is accreted, and cooler, more neutron-rich material ( Y e < 0 . 05 ) from the tidal tail is added to the disk, we observe a decrease in the temperature of the disk (see previous section), and in its average electron fraction. Additionally, at the lower temperature observed in the disk after 20 ms -40 ms the matter is mainly optically thin. In this regime, and under the assumption of no neutrino heating, the lepton number emission (and therefore the Y e evolution) is determined by the electron and positron capture rates alone. The Y e where these rates are balanced, denoted as Y e, eq , as a function of density and temperature is shown in Fig. 18. For the conditions of our disks at late times ( T ∼ 2 -3 MeV, ρ ∼ 10 11 g/cm 3 ), the equilibrium value of Y e is lower than the instantaneous value of Y e ∼ 0 . 1 . Therefore, at that time, the neutrino emission from the optically thin parts of the disks is predominantly of positive lepton number and can lead to net deleptonization. This is particularly visible in simulations M12-7-S8 and M12-7S9, with a net excess of electron neutrinos over antineutrinos towards the end of the simulation (see Fig. 16). In general, the neutronization of the disk is due to a combination of the accretion of neutron rich material and the preferential emission of electron neutrinos. Note that, even at late times, the luminosity of the electron antineutrinos remains larger than the luminosity of the electron neutrinos, since their average energy is larger. Due to the steep dependence on temperature of the charged-current reactions n + e + → p + ¯ ν e and p + e -→ n + ν e , which are the main contributors to the neutrino luminosity, the disk also becomes significantly dimmer as it cools down. \nIn nature, this evolution of the disk towards a lower electron fraction will eventually stop. Indeed, a physical disk will have a non-zero effective viscosity (presumably provided by the magneto-rotational instability). When the heating due to viscous effects compensates the cooling due to neutrino emission, the disk should reach a more steady temperature profile. In the range of black hole masses and spins studied here, we thus expect the neutrino emission to be composed of a short burst of mostly electron antineutrinos, with L ν ∼ 10 53 erg / s , lasting ∼ 10 ms and followed by a period of more constant luminosity at a level set by the viscosity of the disk, decaying on a timescale comparable to the lifetime of the disk. The emission should cause rapid protonization of the high-density regions of the disk in the first 10 ms , but can then contribute to re-neutronization as the disk cools down. \nFor such a configuration, the evolution of Y e in the lowdensity regions above the disk (from which a wind might be launched) is expected, at early times, to be affected by the asymmetry in the number of electron neutrinos and antineutrinos emitted, and can slow down the protonization of the wind material resulting from neutrino captures [18]. The effect of \nthe neutrino emission on a disk wind over longer timescales is more uncertain, particularly when general relativistic effects on the neutrino trajectories are taken into account. This is mainly because the antineutrinos can begin free-streaming much closer to the black hole than the neutrinos, for which the optical depth is generally higher. Surman et al. [62] have recently shown, based on neutrino fluxes derived from Newtonian simulations, that disk winds could then be proton-rich, with an evolution resulting in the production of large amounts of Ni 56 but no heavy elements ( A > 120 ), as opposed to the material unbound during tidal disruption (see also [63]). Given the sensitivity of the process to the geometry of the neutrino radiation, which we do not directly compute, and the geometry of the wind, which we could only obtain by including neutrino heating, further studies are required to determine in which category the winds emitted by the disks produced in our simulations fall. \nIt is also worth noting some differences with the disk obtained for a lower mass black hole in Deaton et al. 2013 [10]. In [10], the combination of a high black hole spin ( χ BH = 0 . 9 ) and a low black hole mass ( M BH = 5 . 6 M /circledot ) led to the formation of a massive disk ( M disk ∼ 0 . 3 M /circledot ). The disks presented here initially have optical depths of a few ( τ ν e ∼ 3 -5 > τ ¯ ν e ). Towards the end of the simulations, they have τ ν e ∼ 1 and are optically thin to electron antineutrinos. On the contrary, the more massive disk in [10] remained optically thick until the end of the simulation, 40 ms after merger (with τ ν e ∼ 15 , τ ¯ ν e ∼ 5 ). The inner region of that disk was also susceptible to a convective instability, due to the negative gradient dL/dr < 0 of the angular momentum L of the disk at radii r < ∼ 27 km , which prevented the disk to evolve towards an axisymmetric configuration at late times. The less massive disks observed in this work, on the other hand, satisfy the SolbergHøiland criteria for convective stability of an axisymmetr ic equilibrium fluid [64] everywhere outside of their ISCO. More generally, the massive accretion disk in Deaton et al. [10] was hotter, denser, and had a higher electron fraction than any of the disks observed here. However, unless astrophysical black holes are less massive than expected, or are very rapidly spinning ( χ BH > 0 . 9 ), we expect the less optically thick disks obtained in this paper to be more representative of the postmerger remnants of NSBH binaries.", 'V. LONG-TERM EVOLUTION OF THE DISK': 'Since our code does not include the effects of magnetic fields, or any ad-hoc prescription for the viscosity in the disk, important physical effects will be missing if we attempt to evolve the post-merger disks over timescales comparable to the disk lifetime. From existing simulations using approximate treatments of gravity [13, 65, 66], we can however obtain reasonable estimates of what should happen. As mentioned in the previous section, the cooling of the disk will stop when neutrino cooling and viscous heating balance each other. For α ∼ 0 . 01 -0 . 1 , this is expected to lead to maximumtemperatures in the disk of T max ∼ 5 MeV -10 MeV and Y e ∼ 0 . 1 -0 . 3 [13]. Over longer timescales, viscous transport \nof angular momentum will drive mass accretion onto the black hole, and the viscous spreading of the disk. Over timescales of a few seconds, nuclear recombination and viscous heating can then unbind ∼ 10% of the matter in the disk [65] (see also [66]). Although this ejecta is more proton-rich than the material ejected during the disruption of the neutron star, Fernandez & Metzger [65] estimate that its electron fraction ( Y e ∼ 0 . 2 ) and specific entropy ( S ∼ 8 k B ) are low enough to allow the production of 2nd and 3rd peak r-process elements. In NSBH mergers, this process would probably be of less importance than the prompt ejection of material: 10% of the disk mass is much less than the mass ejected at the time of merger, and likely difficult to detect if its properties are similar to those of the dynamical ejecta. This is quite different from the results from BNS mergers, in which only 10 -4 M /circledot -10 -2 M /circledot of material is promptly ejected, and the ejected material is strongly heated by shocks and significantly affected by the neutrino radiation field [17]: the different components of the outflow can then be of similar importance, and have very different properties. \nThe exact mechanism leading to the production of relativistic jets from the post-merger remnants of NSBH binaries is more uncertain. Mildly relativistic outflows along the spin-axis of the black hole have been obtained by Etienne et al. [24], but only after seeding a coherent poloidal field B θ ∼ 10 14 G in the disk resulting from a NSBH merger. For black hole spins initially aligned with the orbital angular momentum of the binary (and thus aligned with the postmerger accretion disk), the magnetic flux in the post-merger remnant of a NSBH binary is likely to be too low to generate the highly efficient relativistic jets which have been observed in numerical simulations of magnetically choked accretion flows [67], at least until the accretion rate onto the black hole decreases by several orders of magnitude, or unless the MRI can somehow lead to the growth of a coherent poloidal field. But given the relatively high spins obtained after merger ( χ BH ∼ 0 . 9 ), if such a flow can be established it would have an efficiency η = E jet / ( ˙ Mc 2 ) ∼ 100% [67], or a jet power E jet ∼ (10 53 erg / s) × ( ˙ M ) / (0 . 1 M /circledot / s) . Even if relativistic jets can only be launched when the accretion rate drops to ˙ M /lessmuch 1 M /circledot / s , this remains sufficient to explain the energy output of short gamma-ray bursts. For black hole spins misaligned with the orbital angular momentum of the binary, the angular momentum of the post-merger accretion disk can be misaligned by 5 · -15 · with respect to the final black hole spin [53, 68]. Etienne et al. [24] suggest that a large scale, coherent poloidal field could be obtained from the mixing between poloidal and toroidal fields in a tilted accretion disk, thus allowing the formation of a jet at earlier times. Unfortunately, the small number of existing general relativistic simulations of NSBH binaries with misaligned black hole spins [53, 55] do not include the effects of magnetic fields. \nFinally, a few percents of the energy radiated in neutrinos is expected to be deposited in the region along the spin axis of the black hole [13], through ν ˜ ν annihilations. This energy deposition, which at early times in our disks would be E ν ˜ ν ∼ 10 51 erg / s , might also be sufficient to power short gammaray bursts. Whether jets can be launched prompty or only \nafter the accretion rate drops, the disks produced by disrupting NSBH mergers for M BH ∼ 7 M /circledot -10 M /circledot thus appear to be promising candidates for the production of short gamma-ray bursts.', 'VI. GRAVITATIONAL WAVES': 'Given the relatively small number of orbits simulated in this work (5 to 8) and the lower phase accuracy of simulations with a tabulated equation of state compared to simulations with a Γ -law equation of state, our simulations are not competitive with longer, more accurate recent results (e.g. [55, 69]) as far as the gravitational wave modeling of the inspiral phase is concerned. There are however two effects which can be studied even with these lower accuracy simulations: the influence of the tidal disruption of the neutron star on the gravitational wave signal, and the kick velocity given to the black hole as a result of an asymmetric emission of gravitational waves around the time of merger.', 'A. Neutron star disruption': 'The effect of the disruption of the neutron star on the gravitational wave signal generally falls into one out of three categories, as shown in Kyutoku et al. 2011 [29] (see also Pannarale et al. 2013 [70] for a phenomenological model of the waveform amplitude). When tidal disruption occurs far out of the innermost stable circular orbit (i.e., for low mass black holes), the spectrum of the gravitational wave signal shows a sharp cutoff at the disruption frequency. At the opposite end, when the neutron star does not disrupt (that is, for higher mass black holes of low to moderate spins), the gravitational wave spectrum is similar to what is seen during the mergerringdown of a binary black hole system (i.e. a bump in the spectrum at the ringdown frequency, and an exponential cutoff at higher frequencies). Finally, for neutron stars disrupting close to the innermost stable circular orbit or during their plunge into the black hole (for higher mass black holes with large spin), the spectra show a shallower cutoff of the signal at frequencies above the disruption frequency (in effect, a smaller contribution of the ringdown of the black hole to the signal). \nMost of the simulations presented here fall into the third category, as should be expected from the dynamics of the merger. A simple approximation for the frequency of tidal disruption can be obtained by equating the tidal forces acting on the neutron star with the gravitational forces on its surface, i.e. in Newtonian theory \nM BH d 3 tide R NS ∝ M NS R 2 NS , (7) \nwhere d tide is the binary separation at which tidal disruption occurs. We then obtain a gravitational wave frequency at tidal \nFIG. 19: Gravitational wave spectra for all simulations. Solid lines are rescaled in amplitude by the chirp mass M chirp = ( M NS M BH ) 0 . 6 / ( M NS + M BH ) 0 . 2 and in frequency by the expected correction to the tidal disruption frequency. Dashed curves show the spectra before rescaling in amplitude and frequency. The amplitude is chosen arbitrarily (as it scales with the distance between the binary and the observer). Once rescaled, all spectra have roughly the same cut-off frequency. \n<!-- image --> \ndisruption which scales as \nf GW , tide ∝ √ GM NS R 3 NS . (8) \nWethus expect very similar high frequency signals for all simulations, with the cutoff occurring at a frequency ∼ 9% larger for simulations using a neutron star of mass M NS = 1 . 4 M /circledot than for those using a neutron star of mass M NS = 1 . 2 M /circledot , and the amplitude scaling with the chirp mass M chirp = ( M NS M BH ) 0 . 6 / ( M NS + M BH ) 0 . 2 . Fig. 19 shows that this is indeed the case - the mass and spin of the black hole only contribute to the disruption signal by determining how sharp the cutoff is, and not where the disruption starts. For the LS220 equation of state, we get \nf GW , tide ∼ 1 . 2 kHz ( M NS 1 . 2 M /circledot ) 1 / 2 ( 12 . 7 km R NS ) 3 / 2 . (9) \nThe shape of the merger-ringdown waveform shows more variations for the heavier neutron stars ( M NS = 1 . 4 M /circledot ), as simulations with M BH = 10 M /circledot are close to the transition between disrupting and non-disrupting systems.', 'B. Kick velocity': 'The second type of information that we can extract from the measured gravitational wave signal is the kick velocity given to the black hole at merger as a result of asymmetric emission of gravitational waves. We find kicks spanning the range v kick ∼ 25 km / s -95 km / s , with the expected approximate \nscaling of v kick ∝ q -2 . These kick values are only slightly below what we would expect for binary black hole mergers with the same masses and spins (less than 20% smaller, see [71] for the binary black hole predictions). The kicks are however increasing with the black hole spin, while the opposite trend is observed in the fitting formula for binary black holes when the spins are aligned with the orbital angular momentum and q > ∼ 4 . In contrast, the difference with binary black hole results was shown to be much larger for lower mass black holes [53], which showed a stronger relative suppression of the gravitational recoil. \nFor disrupting NSBH binaries of large mass ratio and rapidly spinning black holes, however, those gravitationalwave induced kicks appear to be significantly weaker than the recoil velocities due to the asymmetric ejection of unbound material (see Sec. III B).', 'VII. CONCLUSIONS': "We performed a first numerical study of NSBH mergers in the most likely range of black hole masses, M BH = 7 M /circledot -10 M /circledot , using a general relativistic code, a leakage scheme to estimate neutrino emission, and a hot nucleartheory based equation of state (LS220). We consider relatively high black hole spins χ BH = 0 . 7 -0 . 9 and low mass neutron stars ( M NS = 1 . 2 M /circledot -1 . 4 M /circledot ), so that the neutron star is disrupted by the tidal field of the black hole before merger. Under those conditions, our simulations show that NSBH mergers reliably eject large amounts of neutron-rich material, with M ej = 0 . 04 M /circledot -0 . 20 M /circledot . The ejected material has a low entropy per baryon 〈 S 〉 < 10 k B , and is ejected early enough that its composition is largely unaffected by neutrino emission and absorption. Accordingly, it should consistently produce heavy r-process elements - as opposed to BNS mergers, which show a wider variety of outflow compositions [17]. \nThe mergers produce accretion disks of masses M disk ∼ 0 . 05 M /circledot -0 . 15 M /circledot , similar to the ejected masses. Due to shock heating, these disks are initially hot ( T ∼ 5 MeV -15 MeV ), and are luminous in neutrinos ( L ν ∼ 10 53 erg / s ). Preferential emission of electron antineutrinos causes the disks to rapidly protonize during the first 10 ms after merger, from Y e ∼ 0 . 06 to Y e ∼ 0 . 1 -0 . 4 . However, at later times, the cooling of the disk combined with the fallback of cold, neutron rich material from the tidal tail reverses this process. About 30 ms after merger, the denser areas of the disks are already much colder ( T ∼ 2 MeV -3 MeV ) and more neutron rich ( Y e < 0 . 1 ) than at early times. The disk is then expected to evolve mainly under the influence of MRIdriven turbulence, and spread from its initial dense and compact state ( ρ ∼ 10 11 g / cm 3 , r ∼ 5 M BH ∼ 50 km ) over a viscous timescale of t ν ∼ 0 . 1s . These effects are however not modeled in our code, and we thus stop the simulations 40 ms after merger. Outflows from these accretion disks are likely to be subdominant compared to the material dynamically ejected during the disruption of the neutron star. On the other hand, the disks are promising configurations for the production of short gamma-ray bursts. \nOur simulations confirm that, for disrupting NSBH binaries, large recoil velocities v kick , ej > ∼ 100 km / s can be imparted to the black hole. These recoil velocities are not due to the gravitational wave emission, but rather to the asymmetry in the ejection of unbound material, as proposed by Kyutoku et al. [29]. \nFinally, we showed that the disruption of a neutron star with a nuclear-theory based equation of state has, for the larger black hole masses considered here, very different properties from those observed for lower mass black holes or simpler equations of state. The tidal tail is strongly compressed, to width of only a few kilometers, and the entire disruption process occurs within a distance r < ∼ 3 M BH of the black hole (in the coordinates of our simulation). In such cases, the properties of the post-merger remnant are more sensitive to the initial parameters of the binary, and to the details of the equation of state. Unfortunately, due to the differences of scale between the width of the tail, the radius of the black hole, and the much larger distance to which we need to follow the ejecta, this also makes these configurations significantly harder to resolve numerically. In the current version of our code, we can only evolve these mergers by using a very fine grid close to the black hole, and letting the disrupted material exit the grid early. Improvements to the adaptivity of our grid will be required in order for such binaries to be reliably evolved over the same timescales as lower mass configurations. \nThis first study involved a fairly small number of cases, so many questions have yet to be addressed. Perhaps most interesting is the variation in merger outcomes from different plausible assumptions about the equation of state. We have found that LS220 and Γ = 2 results differ, but it would be much more interesting to know if viably 'realistic' equations of state can be distinguished by their merger outcomes. Also, a more complete understanding of any of these NSBH post-merger evolutions will require simulations that include the remaining crucial physical effects. As ejecta continues to expand far from the merger location, its density and temperature drop below the thresholds for maintaining nuclear statistical equilibrium, and the subsequent compositional and thermal evolution can only be modeled using nuclear reaction networks. Additionally, tracking the disk evolution will require including the effects of MRI turbulence, either directly through MHD evolutions (the optimal solution), or at least in- \n- [1] G. M. Harry (for the LIGO Scientific Collaboration), Class. Quantum Grav. 27 , 084006 (2010).\n- [2] The Virgo Collaboration, Advanced Virgo Baseline Design (2009), [VIR-0027A-09], URL https://tds.ego-gw.it/ql/?c=6589 .\n- [3] K. Somiya (KAGRA Collaboration), Class. Quantum Grav. 29 , 124007 (2012), 1111.7185.\n- [4] B. D. Metzger and E. Berger, Astrophys. J. 746 , 48 (2012), 1108.6056.\n- [5] J. S. Read, B. D. Lackey, B. J. Owen, and J. L. Friedman, Phys. Rev. D79 , 124032 (2009), 0812.2163.\n- [6] B. D. Lackey, K. Kyutoku, M. Shibata, P. R. Brady, and J. L. \ndirectly through some sort of effective viscosity prescription. Finally, using radiation transport instead of a leakage scheme to evolve the neutrino radiation field is needed to capture the production of neutrino-driven winds, to estimate the deposition of energy and the production of e + e -pairs in the lowdensity region along the black hole spin axis, and to generally improve the accuracy of our predictions for the neutrino emission from the post-merger accretion disk.", 'Acknowledgments': "The authors wish to thank Luke Roberts, Curran Muhlberger, Nick Stone, Carlos Palenzuela, Albino Perego, Zach Etienne and Alexander Tchekhovskoy for useful discussions over the course of this project, and the members of the SXS collaboration and the participants and organizers of the MICRA 2013 workshop for their suggestions and support. F.F. gratefully acknowledges support from the Vincent and Beatrice Tremaine Postdoctoral Fellowship. The authors at CITA gratefully acknowledge support from the NSERC Canada, from the Canada Research Chairs Program, and from the Canadian Institute for Advanced Research. M.D.D. and M.B.D. acknowledge support through NASA Grant No. NNX11AC37G and NSF Grant PHY-1068243. L.K. gratefully acknowledges support from National Science Foundation (NSF) Grants No. PHY-1306125 and No. AST-1333129, while the authors at Caltech acknowledge support from NSF Grants No. PHY-1068881 and No. AST-1333520 and NSF CAREER Award No. PHY-1151197. Authors at both Caltech and Cornell also thank the Sherman Fairchild Foundation for their support. Computations were performed on the supercomputer Briar'ee from the Universit'e de Montr'eal, managed by Calcul Qu'ebec and Compute Canada. 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2019PhRvD..99j3531W

Prospective constraints on the primordial black hole abundance from the stochastic gravitational-wave backgrounds produced by coalescing events and curvature perturbations

2019-01-01

['-', '-', '-']

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For a variety of ongoing and planned gravitational-wave (GW) experiments, we study expected constraints on the fraction (f<SUB>PBH</SUB>) of primordial black holes (PBHs) in dark matter by evaluating the energy-density spectra of two kinds of stochastic GW backgrounds. The first one is produced from an incoherent superposition of GWs emitted from coalescences of all of the binary PBHs. The second one is induced through nonlinear mode couplings of large primordial curvature perturbations inevitably associated with the generation of PBHs in the early Universe. In this paper, we focus on PBHs with masses 10<SUP>-8</SUP> M<SUB>⊙</SUB>≤M<SUB>PBH</SUB>&lt;1 M<SUB>⊙</SUB> , since they are not expected to be of stellar origin. In almost all mass ranges, we show that the experiments are sensitive enough to constrain the fraction for 10<SUP>-5</SUP>≲f<SUB>PBH</SUB>≲1 by considering the GWs from coalescing events and 10<SUP>-10</SUP>≲f<SUB>PBH</SUB>≲1 by considering the GWs from curvature perturbations. Exceptionally, the fraction cannot be constrained for f<SUB>PBH</SUB>≲3 ×10<SUP>-3</SUP> by these two GW backgrounds only in the narrow mass range 2 ×10<SUP>-5</SUP> M<SUB>⊙</SUB>≲M<SUB>PBH</SUB>≲4 ×10<SUP>-5</SUP> M<SUB>⊙</SUB> .

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https://arxiv.org/pdf/1903.05924.pdf

{'Prospective constraints on the primordial black hole abundance from the stochastic gravitational-wave backgrounds produced by coalescing events and curvature perturbations': 'Sai Wang, 1, ∗ Takahiro Terada, 1, † and Kazunori Kohri 1, 2, ‡ \n1 Institute of Particle and Nuclear Studies, KEK, 1-1 Oho, Tsukuba 305-0801, Japan 2 The Graduate University for Advanced Studies (SOKENDAI), 1-1 Oho, Tsukuba 305-0801, Japan \nFor a variety of on-going and planned gravitational-wave (GW) experiments, we study expected constraints on the fraction ( f PBH ) of primordial black holes (PBHs) in dark matter by evaluating the energy-density spectra of two kinds of stochastic GW backgrounds. The first one is produced from an incoherent superposition of GWs emitted from coalescences of all the binary PBHs. The second one is induced through non-linear mode couplings of large primordial curvature perturbations inevitably associated with the generation of PBHs in the early Universe. In this paper, we focus on the PBHs with their masses of 10 -8 M glyph[circledot] ≤ M PBH < 1 M glyph[circledot] , since they are not expected to be of a stellar origin. In almost all ranges of the masses, we show that the experiments are sensitive to constrain the fraction for 10 -5 glyph[lessorsimilar] f PBH glyph[lessorsimilar] 1 by considering the GWs from coalescing events and 10 -13 glyph[lessorsimilar] f PBH glyph[lessorsimilar] 1 by considering the GWs from curvature perturbations. Exceptionally, only in a narrow range of masses for M PBH glyph[similarequal] 10 -7 M glyph[circledot] , the fraction cannot be constrained for f PBH glyph[lessorsimilar] 10 -13 by those two GW backgrounds.', 'I. INTRODUCTION': 'The first detection of gravitational waves (GWs) from a binary black hole (BBH) merger by the first Advanced LIGO (aLIGO) observing run [1] has revived extensive interests in primordial black holes (PBHs) [2, 3], which are produced directly from the gravitational collapses of the enhanced inhomogeneities in the primordial Universe. Until now, the origin of these black holes (BHs) and the formation mechanism of BBHs are still under debate. Besides an astrophysical origin [4-6], the possibility that these BHs are of a primordial origin is also considered [7-20]. Recently, it has been proposed that the PBHs are capable of accounting for the event rate of BBH mergers observed by aLIGO [7, 8], although the formation mechanisms of PBH binaries bring about uncertainties of a couple of orders of magnitude (see e.g. Ref. [20] and references therein). The PBHs can be one of the most promising candidates for the cold dark matter (CDM) [11]. Currently, the nature of CDM is still uncertain [21]. There is not definitive evidence for the weakly interacting massive particles (WIMPs) which is a prime candidate for CDM [22-26]. Conventionally, one defines the abundance of PBHs in CDM as a dimensionless fraction of the form f PBH = Ω PBH / Ω CDM , where Ω PBH and Ω CDM denote the present energy-density fractions of PBHs and CDM, respectively. This quantity has been constrained in a variety of mass ranges by a variety of observations (see e.g. Refs. [20, 27] and references therein), for example, the microlensing events caused by massive astrophysical compact halo objects [28-31], the gas accretion effect of PBHs on the cosmic microwave background (CMB) [32-34], the null detection of a third-order Shapiro time delay using a pulsar timing array [35], and the claimed event rate of BBH mergers from aLIGO [7, 8, 36], etc. \nThe PBHs can be also a useful probe to the primordial curvature perturbations [37], since the former are formed via directly gravitational collapses of the latter [2, 3]. Contrary to the astrophysical processes for which only BHs heavier than O (1) solar mass can be produced [38], the small-mass BHs could be also produced by the strong gravity inside the highly compressed overdensities in the early Universe [39]. The PBH mass depends on the PBH formation redshift z f , namely M glyph[similarequal] 30 M glyph[circledot] [4 × 10 11 / (1 + z f )] 2 [8], where M glyph[circledot] is the solar mass (= 2 × 10 33 g). Since inflation models [40-46] predict the properties of the primordial curvature perturbations, which determine the mass and abundance of PBHs (see e.g. Refs. [20, 27, 47-49] and references therein), our observational knowledge of the PBHs is important to learn about the physics of the inflationary Universe. \nRecently, it has been proposed that the energy-density fraction of PBHs can be constrained by measuring the energydensity spectrum of the stochastic gravitational-wave background (SGWB). First, the SGWB can be produced from an incoherent superposition of GWs emitted from all the coalescing PBH binaries. The null detection of such a SGWB by the first aLIGO observing run [50] has been used to independently constrain f PBH [51-54]. For example, Ref. [51] \n∗ Electronic address: [email protected] † Electronic address: [email protected] \nobtained the tightest observational constraint on f PBH in the mass range 1 -10 2 M glyph[circledot] , pushing the existing observational constraints tighter by one order of magnitude. The possibility to detect the SGWB from PBHs, in particular from subsolar-mass PBHs, by upcoming aLIGO observing runs was also predicted [51]. Second, the SGWB is induced from the enhanced primordial curvature perturbations [55-58] 1 . By making use of the semi-analytic calculation of the induced GW spectrum [77, 78], the null detection of such a SGWB by a variety of GW detectors has been used to obtain constraints on the spectral amplitude of primordial curvature perturbations [79-81]. The constraints on the induced SGWB can be recast as the constraints on the abundance of PBHs, and vice versa [70, 82-85]. \nIn this paper, we focus on the small-mass PBHs for 10 -8 M glyph[circledot] ≤ M PBH ≤ 1 M glyph[circledot] . Correspondingly, we calculate the energy-density fraction of the above two kinds of SGWBs, and report the expected constraints on the energy-density fraction of PBHs from the null detection of the SGWBs by several on-going and planned GW experiments (see details in Ref. [86]), which include Square Kilometre Array (SKA) [87], Laser Interferometer Space Antenna (LISA) [88, 89], DECi-hertz Interferometer Gravitational wave Observatory (DECIGO) [90] and B-DECIGO [91], Big Bang Observer (BBO) [92], Einstein Telescope (ET) [93], and aLIGO design sensitivity [94]. Although we focus on the mass range 10 -8 M glyph[circledot] ≤ M PBH ≤ 1 M glyph[circledot] , the method of our analysis is equally applicable to the PBS masses outside of this range. In this context, the authors of Refs. [95, 96] studied the SGWB induced by the curvature perturbations associated with the PBHs of masses around 10 -12 M glyph[circledot] as it may explain the whole abundance of the dark matter. The authors of Ref. [80] obtained the constraints on the primordial curvature perturbations by studying the detectability of the curvature-induced SGWB in a wide frequency range corresponding to a wide PBH mass range. \nFirst, following Ref. [51], we evaluate the energy-density fraction of the SGWB from binary PBH coalescence, by assuming a monochromatic mass distribution of PBHs. This choice of the delta function is reasonable since the mass distribution of PBHs is insensitive to the details of the spectral shape of primordial curvature perturbations especially after taking into account coarse graining within the Hubble horizon and the effects of critical collapse [48]. In addition, the inflation scenario does not favor a significantly extended PBH mass distribution [11]. Second, following Ref. [77], we evaluate the energy-density fraction of the induced SGWB, by assuming a delta function for the power spectrum of primordial curvature perturbations. In principle, the spectrum of the induced SGWB depends on the details of the spectral shape of primordial curvature perturbations. Recently, Ref. [80] found a spread of the SGWB spectrum by studying a log-normal distribution for the power spectrum of primordial curvature perturbations. So the results obtained by this work can be regarded as the conservative one. 2 See also Ref. [97] which discusses the effects of a broad spectrum. \nThe rest of the paper is arranged as follows. In Sec. II, we briefly review the formation of PBHs in the early Universe, given the power spectrum of primordial curvature perturbations. In Sec. III, we evaluate the energy-density fraction of the SGWB from binary PBH coalescence, and use it to obtain expected constraints on f PBH from a variety of on-going and planned GW detectors. In Sec. IV, we evaluate the induced SGWB from the enhanced primordial curvature perturbations, and obtain the expected constraints on the energy-density fraction of PBHs from SKA and LISA. The conclusions and discussions are given in Sec. V.', 'II. FORMATION OF PRIMORDIAL BLACK HOLES': 'Given the power spectrum of the primordial curvature perturbations, we can evaluate a probability of the PBH production, the mass function of PBHs and the PBH abundance [20, 27]. In this work we assume that the PBHs are formed in the early Universe which is radiation dominated (RD). First of all, we need to estimate the wavenumber scale k which is related with a given mass scale M H within the Hubble horizon at the time of horizon re-entry. According to Appendix A, it is represented by \nk k ∗ = 7 . 49 × 10 7 ( M glyph[circledot] M H ) 1 / 2 ( g ∗ ,ρ ( T ( M H )) 106 . 75 ) 1 / 4 ( g ∗ ,s ( T ( M H )) 106 . 75 ) -1 / 3 , (1) \nwhere k ∗ = 0 . 05Mpc -1 . Here we can numerically obtain the temperature at the formation T ( M H ) by using Eq. (A6). The effective degrees of freedom of relativistic particles, i.e. g ∗ ρ and g ∗ s , are precisely calculated for the Standard \nModel in Ref. [98]. Here we interpolate the tabulated data provided by the associated website 3 to this reference. The phenomena of critical collapse [11, 99] could describe the formation of PBHs with mass M in the early Universe, depending on the horizon mass M H and the amplitude of density fluctuation δ . We have the following relation \nM = KM H ( δ -δ c ) γ , (2) \nwhere K = 3 . 3, γ = 0 . 36 and δ c = 0 . 45 are numerical constants 4 . The above equation can be inverted to express δ in terms of M/M H , namely, δ = ( M/ ( KM H )) 1 /γ + δ c , which is useful in the following calculations. \nIn the RD Universe, the coarse grained density perturbation is given by \nσ 2 ( k ) = ∫ + ∞ -∞ d ln q w 2 ( q/k ) ( 4 9 ) 2 ( q k ) 4 T 2 ( q, τ = 1 /k ) P ζ ( q ) , (3) \nwhere w ( q/k ) = exp( -q 2 / (2 k 2 )) is a Gaussian window function, and T ( q, τ ) = 3(sin y -y cos y ) /y 3 ( y ≡ qτ/ √ 3) is a transfer function (see e.g. Refs. [102, 103] for details). We consider the power spectrum of primordial curvature perturbations P ζ ( k ) to be a delta function of ln k , i.e., \nP ζ ( k ) = Aδ (ln k -ln k 0 ) , (4) \nwhere k 0 is a given constant wavenumber, and A is a dimensionless amplitude. By substituting Eq. (4) into Eq. (3), we obtain \nσ 2 ( k ) = 16 Ae -1 /x 2 [ cos 2 ( 1 √ 3 x ) + x ( 3 x sin 2 ( 1 √ 3 x ) -√ 3 sin ( 2 √ 3 x ))] , (5) \nwhere x ≡ k/k 0 is a dimensionless quantity. We show σ 2 ( k ) /A versus k/k 0 in a figure at the end of Appendix A. To convert σ ( k ) to the mass function of PBHs, by making use of the Press-Schechter formalism [104], we calculate the probability of the PBH production, i.e., \nβ M H = ∫ ∞ δ c M M H P M H ( δ ( M )) dδ ( M ) = ∫ ∞ -∞ M M H P M H ( δ ( M )) dδ ( M ) d ln M d ln M ≡ ∫ ∞ -∞ ˜ β M H ( M ) d ln M , (6) \nwhich accounts for the fraction of the Hubble volumes that collapse into PBHs when the horizon mass is M H . ˜ β M H ( M ) is the distribution of the (logarithmic) masses of PBHs resulting after the critical collapse. Here P M H ( δ ) denotes a Gaussian probability distribution of primordial density perturbations at the given horizon scale corresponding to M H . It is represented by \nP M H ( δ ( M )) = 1 √ 2 πσ 2 ( k ( M H )) exp ( -δ 2 ( M ) 2 σ 2 ( k ( M H )) ) , (7) \nwhere σ ( k ( M H )) is computed by making use of Eq. (5), and k ( M H ) is given by Eq. (1). The explicit form of ˜ β M H ( M ) is written to be [105] \n˜ β M H ( M ) = K √ 2 πγσ ( k ( M H )) ( M KM H ) 1+ 1 γ exp -1 2 σ 2 ( k ( M H )) ( δ c + ( M KM H ) 1 γ ) 2 . (8) \nThe mass function of PBHs is defined as f ( M ) = 1 Ω CDM d Ω PBH d ln M , and the abundance of PBHs in CDM is given by f PBH = ∫ f ( M )d ln( M/M glyph[circledot] ). We obtain the mass function of PBHs as follows (see e.g. Ref. [48]) \nf ( M ) = Ω m Ω CDM ∫ ∞ -∞ ( g ∗ ,ρ ( T ( M H )) g ∗ ,ρ ( T eq ) g ∗ ,s ( T eq ) g ∗ ,s ( T ( M H )) T ( M H ) T eq ) ˜ β M H ( M ) d ln M H , (9) \nwhere T eq is the temperature of the Universe at the epoch of matter-radiation equality. In Fig. 1, we depict several \nFIG. 1: Mass functions f ( M ) originated from the delta function power spectrum P ζ ( k ) (colored solid). From right to left, based on Eq. (1), the value of k 0 is chosen so that the corresponding M H is log 10 ( M H /M glyph[circledot] ) = 0 , -1 , -2 , -3 , -4 , -5 , -6 , -7 , -8. The normalization A is chosen so that f PBH is equal to the upper bound on f ( M ) with the aforementioned values of M H . To be specific, we take 10 2 A as 5.8898, 5.4658, 5.1116, 4.8011, 4.6799, 4.6490, 4.3002, 3.8740 and 3.6813, respectively. The existing observational constraints (HSC [106] (green dotted), OGLE [107] (blue dotted), EROS/MACHO [108, 109] (cyan dotted), caustic crossing [110] (purple dotted) and their combination (red dashed)) are plotted for comparison. \n<!-- image --> \nexamples (colored solid) for the mass function f ( M ) which is originated from P ζ ( k ) in Eq. (4). To be specific, we choose a horizon mass to be M H = 10 -i M glyph[circledot] ( i = 0 , 1 , 2 , ..., 8), each of which determines the value of its own k 0 = k 0 ( M H ). Here (10 2 A ) is 5.8898, 5.4658, 5.1116, 4.8011, 4.6799, 4.6490, 4.3002, 3.8740 and 3.6813, respectively, so that f PBH equals the upper limit on f ( M ) with the aforementioned values of M H . For comparison, we plot the existing observational constraint (red dashed) on the PBH mass function. The constraint used here arises from the microlensing observations of Subaru/HSC [106], OGLE [107], EROS-2 [108], MACHO [109], and the caustic crossing [110].', 'III. STOCHASTIC GRAVITATIONAL-WAVE BACKGROUND DUE TO BINARY PRIMORDIAL BLACK HOLE MERGERS': 'Two different mechanisms have been proposed to form binaries from the PBHs. One scenario assumes that two PBHs could form a binary due to the energy loss via gravitational radiation when they pass by each other accidentally in the late Universe [7, 9]. The other one assumes that two nearby PBHs form a binary due to the tidal force from a third neighboring PBH in the early Universe [8, 111, 112]. Both scenarios are capable of explaining the merger rates of BBHs reported by aLIGO. However, the first one requires the PBHs to contribute most of the CDM, which is disfavored by various observational constraints in the relevant mass range. On the other hand, the second one is still allowed. In this work, we thus adopt the formation scenario 5 of PBH binaries proposed in Ref. [111] and revisited by Refs. [8, 10, 36, 112-119]. In Appendix B we show a brief summary of the formalism for such a scenario. \nWe calculate the SGWB spectrum produced from the coalescing PBH binaries. In general, the dimensionless energy-density spectrum of the SGWB is defined as Ω GW = ρ -1 c dρ GW /d ln ν , where ρ GW is the GW energy density, and ν is the GW frequency [120]. Knowing the merger rate of PBH binaries in Eq. (B2), according to Ref. [51], we can compute the SGWB energy-density spectrum within the frequency interval ( ν, ν + dν ). It is given by \nΩ GW ( ν ) = ν ρ c ∫ ν cut ν -1 0 R PBH ( z ) (1 + z ) H ( z ) dE GW dν s ( ν s ) dz , (10) \nwhere dE GW dν s ( ν s ) is the GW energy spectrum of a BBH coalescence (see details in Refs. [121, 122], or a brief summary in Appendix C), ν s is the frequency in the source frame and is related to the observed frequency ν through ν s = (1+ z ) ν , and ν cut is the cutoff frequency for a given BBH system. \nFIG. 2: Energy-density spectrum (colored solid) of the SGWB due to binary PBH coalescence which is just allowed by the existing observational constraints on the PBH abundance. The SGWB spectra with the cutoff frequencies from right to left correspond to the peaks from left to right in Fig. 1 (same colors). The sensitivity curves (colored dashed/dotted) of the GW detectors are also plotted for comparison. \n<!-- image --> \nFIG. 3: Expected constraints on the PBH abundance from the null detection of the SGWB by LISA (orange solid), B-DECIGO (green dashed), DECIGO (green solid), BBO (blue solid), ET (cyan solid), and aLIGO (purple dashed). The present existing constraint (red dashed) is plotted for comparison. \n<!-- image --> \nFor the PBH binaries for the component masses 10 -i M glyph[circledot] ( i = 0 , 1 , 2 , ..., 8), which correspond to the examples of the PBH mass function in Fig. 1, we plot the corresponding energy-density fractions of the SGWB due to binary PBH coalescence at the existing observational constraints on the PBH abundance in Fig. 2. The color coding is the same as \nthat in Fig. 1. For comparison, we depict the sensitivity curves 6 of several GW experiments (colored dashed/dotted curves), which include pulsar timing array (SKA [87]), space-based GW interferometers (LISA [89], DECIGO [90] and B-DECIGO [91], BBO [92]), third-generation ground-based GW interferometer (ET [93]) and second-generation ground-based GW interferometer (aLIGO [94]). If the spectrum predicted in a model intersects the sensitivity curve of a given experiment, the expected signal-to-noise ratio (SNR) is equal to or greater than unity, which means a possible detection of such a spectrum by this experiment. \nNull detection of the SGWB by the given future or on-going GW experiment can place an upper bound on the magnitude of the energy-density fraction of the SGWB at a given frequency band, and can be further recast to constrain the maximum PBH abundance. From Fig. 2, all the GW experiments have possible contributions to improve the existing observational constraints on the PBH abundance, since their sensitivity curves intersect some spectra. Therefore, by regarding the sensitivity curves of all these experiments as upper bounds on the SGWB spectrum, we evaluate the expected upper limits on the PBH abundance from these experiments. We depict our results in Fig. 3. \nOur results are as follows. SKA, LISA and aLIGO will give us relatively weak constraints in future. It is notable that this expected limit from aLIGO is surely stronger than the current one which was reported recently by Ref. [36]. Both ET and B-DECIGO also have similar constraints on the abundance. All the above four experiments are expected to improve the existing observational constraints on the subsolar-mass PBHs. However, both DECIGO and BBO are expected to significantly improve the existing constraints over the mass range O (10 -6 ) ≤ M/M glyph[circledot] ≤ O (10 0 ).', 'IV. STOCHASTIC GRAVITATIONAL-WAVE BACKGROUND INDUCED BY PRIMORDIAL CURVATURE PERTURBATIONS': 'The SGWB can be also induced by the enhanced primordial curvature perturbations via the scalar-tensor mode coupling in the second-order perturbation theory [56]. In Appendix D, we show a brief summary of the evaluations of the induced SGWB spectrum. For details, see Ref. [77] and references therein. In the following, we will use Appendix D to calculate the energy-density spectrum of the induced SGWB, given the form of P ζ ( k ) in Eq. (4). We consider the minimal case in which the statistics of the curvature perturbations is Gaussian 7 and neglect the time evolution of the mass function of PBHs due to accretion, but generalizations can be found in Ref. [127]. \nAccording to Eqs. (D1)-(D3), we obtain the dimensionless energy-density spectrum of the induced SGWB as \nΩ IGW ( ν = k 2 π ) = Ω r , 0 ( g ∗ ( T ( k )) g ∗ ( T eq ) )( g ∗ ,s ( T ( k )) g ∗ ,s ( T eq ) ) -4 / 3 × 3 A 2 64 ( 4 -˜ k 2 4 ) 2 ˜ k 2 ( 3 ˜ k 2 -2 ) 2 × [ π 2 ( 3 ˜ k 2 -2 ) 2 Θ(2 -√ 3 ˜ k ) + ( 4 + ( 3 ˜ k 2 -2 ) ln ∣ ∣ ∣ ∣ 1 -4 3 ˜ k 2 ∣ ∣ ∣ ∣ ) 2 ] Θ(2 -˜ k ) , (11) \nwhere ν = k/ 2 π denotes the frequency of GW, and the dimensionless wavenumber ˜ k = k/k 0 is introduced for simplicity. Based on Appendix A, cosmic temperature T can be numerically related with M H and k , and then with ν . Here Θ( x ) denotes the Heaviside theta function with variable x . \nSimilarly to Fig. 2, we plot the energy-density fractions of the induced SGWB due to the enhanced primordial curvature perturbations in Fig. 4. Both A and k 0 are chosen as those in Sec. II. The same color coding is used as in Fig. 1. The double-peak structures arise from the property of the delta function for P ζ ( k ) in Eq. (4). For a broader distribution for P ζ ( k ), e.g., a log-normal distribution in Ref. [80], one could find a spread of the SGWB spectrum. Therefore, our discussions in the next two paragraphs could be regarded as conservative. \nIn Fig. 5, besides the energy-density spectra of the induced SGWB (colored solid, same as Fig. 4), we depict the sensitivity curves of SKA [87] (red dashed) and LISA [89] (orange dashed) for comparison. For a given spectrum of the induced SGWB, we conservatively drop the right-handed peak since such a spiky structure exists only for source spectra with a tiny width. Similarly to the discussions in last section, if a model-predicting spectrum intersects the sensitivity curve of a given experiment, it is possible to measure such a spectrum by this experiment. In such a \nFIG. 4: Energy-density spectrum of the SGWB nonlinearly induced by the primordial curvature perturbations. The SGWB spectra (colored solid) with the peaks from right to left correspond to the mass functions with the peaks from left to right in Fig. 1 (same colors). \n<!-- image --> \nFIG. 5: Similarly to Fig. 4, we plot the energy-density spectrum of the induced SGWB (colored solid), but the right-handed peak is conservatively dropped. We depict the sensitivity curves of SKA (red dashed) and LISA (orange dashed) for comparison. \n<!-- image --> \ncase, both SKA and LISA are expected to exclude most of the parameter space, or equivalently improve the existing observational constraints on the PBH abundance significantly. \nAssuming the null detection of the induced SGWB from the enhanced primordial curvature perturbations, similarly to Fig. 3, we plot the expected constraints on the PBH abundance from SKA (red shaded) and LISA (orange shaded) in Fig. 6. The shaded regions mean the excluded parts of the parameter space by these experiments. In fact, here we first obtain the constraints on A from the induced SGWB, and then recast them as the upper limits on f PBH according to the formulae in Sec. II. \nFinally, we can combine the results in Fig. 3 with Fig. 6 to obtain Fig. 7. Generally speaking, the slopes of the upper bounds (i.e., boundaries of shaded regions) from the induced SGWB are significantly sharper than those (i.e., colored curves) from the SGWB due to the coalescing PBH binaries. This property can be easily understood as follows. On the one hand, we directly constrained the magnitude of f PBH by calculating the SGWB from the coalescing PBH binaries. \nFIG. 6: Expected constraints on the PBH abundance versus the PBH mass from the null detection of the induced SGWB by SKA (red shaded) and LISA (orange shaded). The existing observational constraint (red dashed) is also plotted for comparison. \n<!-- image --> \nFIG. 7: Expected constraints on the PBH abundance versus the PBH mass from the null detection of the two kinds of SGWBs. The existing observational constraint (red dashed) is plotted for comparison. \n<!-- image --> \nThe detection of such a SGWB requires a significant amount of PBH binaries in the Universe. This implies that the GW detectors can probe the enhanced primordial curvature perturbations only if A ∼ O (0 . 1). When A glyph[lessmuch] O (0 . 1), there would be so few PBHs in the Universe that the thresholds of GW detectors are not triggered. On the other hand, by detecting the induced SGWB, we directly obtained the constraints on A , which were recast as the indirect constraints on f PBH . In fact, by detecting the induced SGWB, the GW detectors can probe the primordial curvature perturbations of arbitrary amplitudes within their sensitivities. Since the induced SGWB spectrum is proportional to A 2 while f PBH is exponentially sensitive to A , we obtained the sharper slopes for the upper bounds from the induced SGWB than those from the SGWB due to coalescing PBH binaries in Fig. 7. Thus, the constraints on f PBH from the SGWB induced by the curvature perturbations are stronger than those from the SGWB whose origin is merger events except for a narrow gap around 10 -7 M glyph[circledot] corresponding to the relatively weak observational sensitivity around about 10 -6 Hz. Nevertheless, both types of the SGWB are complementary and useful for the consistency check of the PBH hypothesis since those two types of the SGWB have their own individual features in the spectra and are probed by different observations which are supposed to measure GWs at different frequency bands.', 'V. CONCLUSIONS': 'It has been known that PBHs can form binaries in the early Universe, and a PBH binary can merge to a new heavier BH due to the energy loss via gravitational radiation. Based on Ref. [8], the merger rate of PBH binaries depends on the abundance and mass of PBHs. Given the existing constraints on the mass function of PBHs, following Ref. [51], we have evaluated the energy-density spectra of SGWBs which arise from coalescences of PBH binaries with component masses 10 -i M glyph[circledot] ( i = 0 , 1 , 2 , ..., 8). From Fig. 2, we found that some of them intersect the sensitivity curves of several future GW experiments. This means that the existing limits can be improved by these experiments in the future if these experiments do not detect the SGWB. By making use of these sensitivity curves as upper limits on the SGWB energy-density fraction, we have evaluated the expected upper limits on the abundance of PBHs, and shown our results in Fig. 3. In particular, both DECIGO and BBO are expected to significantly improve the existing limits over the mass range 10 -6 M glyph[circledot] -10 0 M glyph[circledot] . \nThe generation of PBHs in the early Universe requires large amplitudes of the primordial curvature perturbations, which can always induce the SGWB. By taking into account the existing constraints on the mass function of PBHs and making use of the semi-analytic formula in Ref. [77], we have calculated the energy-density spectrum of the induced SGWB, and shown our results in Fig. 4. We find several intersections between the induced SGWB spectra and the sensitivity curves of SKA and LISA in Fig. 5. This implies that these experiments can improve the existing upper limits on the mass function of PBHs in the future if they claim the null detection of the induced SGWB energy-density fraction. In this case, the shaded regions in Fig. 6 will be excluded by SKA and LISA, respectively. \nFinally, by combining Fig. 3 with Fig. 6 to obtain Fig. 7, we found stronger constraints on f PBH from the SGWB induced by curvature perturbations than those from the SGWB due to coalescing events, except for a narrow gap around 10 -7 M glyph[circledot] . However, both types of the SGWB are complementary and useful for the consistency check of the PBH hypothesis.', 'Acknowledgments': 'This work is supported in part by the JSPS Research Fellowship for Young Scientists (TT) and JSPS KAKENHI Grants No. JP17H01131 (SW and KK) and No. JP17J00731 (TT), and MEXT KAKENHI Grants No. JP15H05889 (KK), and No. JP18H04594 (KK).', 'Appendix A: Relation between k and M H in the radiation dominated Universe': 'During the radiation dominated (RD) era of the Universe, the relation between the wavenumber k and the horizon mass M H is obtained as follows. By definition, we have \nk = a ( M H ) H ( M H ) . (A1) \nThe value of the scale factor a ( M H ), when the mode corresponding to M H re-enters the Hubble horizon, is obtained by using the entropy conservation to be \na ( M H ) a 0 (= 1) = ( g ∗ ,s ( T 0 ) g ∗ ,s ( T ( M H )) ) 1 / 3 T 0 T ( M H ) , (A2) \nwhere T 0 = 2 . 725K denotes the present temperature of the CMB, and the temperature T ( M H ) is given by the Friedmann equation, i.e., \n3 H 2 ( M H ) M 2 G = ρ ≈ ρ rad = π 2 g ∗ ,ρ ( T ( M H )) 30 T 4 ( M H ) , (A3) \nwhere M G = M P / √ 8 π is the reduced Planck mass. The relation between the horizon mass M H and the Hubble radius H -1 is given by \nM H = 4 π 3 ( H ( M H )) -3 ρ . (A4) \nCombining Eq. (A4) with the left equality of Eq. (A3), we have the following formula between H and M H , i.e., \nH ( M H ) =4 π M 2 G M H . (A5) \nBy combining Eq. (A5) with the right equality of Eq. (A3), we thus obtain a relation between M H and T , i.e., \nM H = 12 ( 10 g ∗ ,ρ ( T ) ) 1 / 2 M 3 G T 2 . (A6) \nCombining Eqs. (A1), (A2), (A5) and (A6) together, we obtain \nk k ∗ = 7 . 49 × 10 7 ( M glyph[circledot] M H ) 1 / 2 ( g ∗ ,ρ ( T ( M H )) 106 . 75 ) 1 / 4 ( g ∗ ,s ( T ( M H )) 106 . 75 ) -1 / 3 , (A7) \nwhere M glyph[circledot] denotes the solar mass and k ∗ = 0 . 05Mpc -1 . \nIn the following, we depict Fig. 8 to show σ 2 ( k ) /A versus k/k 0 . In the wavenumber space, the peak of the coarse grained perturbations shifts from the original peak k 0 . Numerically, the shifted peak is obtained as k = 0 . 730715 k 0 . \nk (peak of PBHs) k (peak of primordial curvature perturbations) = 0 . 730715 . (A8) \nIn our example by assuming P ζ ( k ) to be the delta function, the wavenumber in the denominator is nothing but k 0 . When a PBH forms, the shorter scales had already experienced the radiation pressure and have been smoothened. Therefore, the PBH mass scale corresponds to the coarse grained perturbation scale. In other words, k in Eqs. (1) and (A7) should be the one appearing in the numerator of the left-hand side of Eq. (A8). \nFIG. 8: Coarse grained density function calculated by assuming P ζ ( k ) to be a delta function of ln k . \n<!-- image -->', 'Appendix B: Formalism for the merger rate of PBH binaries': "Given the fraction of PBHs in CDM, namely f PBH 8 , for a fixed PBH mass M , the probability that a PBH binary coalesces within the cosmic time interval ( t, t + dt ) is given by (see e.g. Refs. [8, 20] for details) \ndP t = 3 58 [ -( t t 0 ) 3 8 + ( t t 0 ) 3 37 ] dt t for t < t c 3 58 ( t t 0 ) 3 8 [ -1 + ( t t c ) -29 56 ( 4 π 3 f PBH ) -29 8 ] dt t for t ≥ t c , (B1) \nwhere we define t 0 = (3 / 170) { ¯ x 4 / [( GM ) 3 (4 πf PBH / 3) 4 ] } and t c = t 0 (4 πf PBH / 3) 37 / 3 , and ¯ x = [3 M/ (4 πρ PBH , eq )] 1 / 3 is the physical mean separation of PBHs at the epoch of matter-radiation equality. Here ρ PBH , eq is the energy density of the PBHs at the epoch of matter-radiation equality. Multiplying dP t /dt by the present average number density of PBHs, one can obtain the merger rate of PBH binaries as \nR PBH ( z ) = ( f PBH Ω CDM ρ c M ) dP t dt . (B2) \nThe redshift z is related to the cosmic time t through t = ∫ ∞ z dz ' / [(1 + z ' ) H ( z ' )], where H ( z ) = H 0 [Ω r, 0 (1 + z ) 4 + Ω m, 0 (1 + z ) 3 +Ω Λ ] 1 / 2 is Hubble parameter at redshift z . The quantities Ω r, 0 and Ω m, 0 denote the present energydensity fractions of radiations and non-relativistic matter, respectively. The present energy-density fraction of dark energy is derived as Ω Λ = 1 -Ω r, 0 -Ω m, 0 . Here ρ c = 3 H 2 0 M 2 G is the critical energy density of the Universe. Throughout this paper, we adopt the ΛCDM model with cosmological parameters measured by Planck satellite [128].", 'Appendix C: Energy spectrum of gravitational waves': 'In the non-spinning limit, the inspiral-merger-ringdown energy spectrum for a BBH coalescence takes the following form [121, 122] \ndE GW dν s ( ν s ) = ( Gπ ) 2 / 3 M 5 / 3 c 3 ν -1 / 3 s for ν s < ν 1 w 1 ν 2 / 3 s for ν 1 ≤ ν s < ν 2 w 2 σ 4 ν 2 s ( σ 2 +4( ν s -ν 2 ) 2 ) 2 for ν 2 ≤ ν s ≤ ν 3 0 for ν 3 < ν s (C1) \nwhere ν s is a GW frequency in the source frame, w 1 and w 2 are two normalization constants that make the spectrum to be continuous. The parameters ν i ( i = 1 , 2 , 3) and σ can be expressed in terms of M t and η as follows \nπM t ν 1 = (1 -4 . 455 + 3 . 521) + 0 . 6437 η -0 . 05822 η 2 -7 . 092 η 3 (C2) \nπM t ν 2 = (1 -0 . 63) / 2 + 0 . 1469 η -0 . 0249 η 2 +2 . 325 η 3 (C3) \nπM t σ = (1 -0 . 63) / 4 -0 . 4098 η +1 . 829 η 2 -2 . 87 η 3 (C4) \nπM t ν 3 = 0 . 3236 -0 . 1331 η -0 . 2714 η 2 +4 . 922 η 3 (C5) \nwhich can be found in Table 1 of Ref. [122]. Here M c is the chirp mass, i.e., M 5 / 3 c = m 1 m 2 ( m 1 + m 2 ) -1 / 3 , and M t = m 1 + m 2 is the total mass. The symmetric mass ratio is defined by η = m 1 m 2 ( m 1 + m 2 ) -2 , which gives 0 . 25 in this work, since we assume the monochromatic mass of PBHs. The cutoff frequency is given to be ν cut = ν 3 .', 'Appendix D: Curvature-induced gravitational waves in a nutshell': "We briefly summarize the semi-analytic calculation of the SGWB spectrum induced in the RD era from the non-linear (tensor-scalar-scalar) mode coupling. The details are described in Ref. [77] and references therein. The energy-density fraction of the induced SGWB is given by \nΩ GW ( k ) | today = Ω r , 0 24 ( g ∗ ,ρ ( T ) g ∗ ,ρ ( T eq ) )( g ∗ ,s ( T ) g ∗ ,s ( T eq ) ) -4 / 3 ( k aH ) 2 P h ( τ, k ) , (D1) \nwhere cosmic temperature T = T ( M H ( k )) with the horizon mass M H ( k ), aH and conformal time τ are to be evaluated at (a time somewhat after) the horizon entry of the relevant mode (when the Ω GW has reached a temporary asymptotic value). In fact, T ( M H ( k )) can be numerically evaluated by combining Eq. (A6) with Eq. (A7). The last two factors in the above formula are given by \n( k aH ) 2 P h ( τ, k ) = 4 ∫ ∞ 0 dv ∫ 1+ v -| 1 -v | du [ 4 v 2 -(1 + v 2 -u 2 ) 2 4 uv ] 2 ( kτ ) 2 I 2 ( v, u, kτ glyph[greatermuch] 1) P ζ ( kv ) P ζ ( ku ) . 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2019PhRvD..99f4011S

Stability of scalarized black hole solutions in scalar-Gauss-Bonnet gravity

2019-01-01

['-', '-', '-']

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Scalar-tensor theories of gravity where a new scalar degree of freedom couples to the Gauss-Bonnet invariant can exhibit the phenomenon of spontaneous black hole scalarization. These theories admit both the classic black hole solutions predicted by general relativity as well as novel hairy black hole solutions. The stability of hairy black holes is strongly dependent on the precise form of the scalar-gravity coupling. A radial stability investigation revealed that all scalarized black hole solutions are unstable when the coupling between the scalar field and the Gauss-Bonnet invariant is quadratic in the scalar, whereas stable solutions exist for exponential couplings. Here, we elucidate this behavior. We demonstrate that, while the quadratic term controls the onset of the tachyonic instability that gives rise to the black hole hair, the higher-order coupling terms control the nonlinearities that quench that instability and, hence, also control the stability of the hairy black hole solutions.

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https://arxiv.org/pdf/1812.05590.pdf

{'On the stability of scalarized black hole solutions in scalar-Gauss-Bonnet gravity': "Hector O. Silva, 1, ∗ Caio F. B. Macedo, 2, † Thomas P. Sotiriou, 3, 4, ‡ Leonardo Gualtieri, 5, § Jeremy Sakstein, 6, ¶ and Emanuele Berti 7, ∗∗ \n1 eXtreme Gravity Institute, Department of Physics, Montana State University, Bozeman, MT 59717 USA \n2 Campus Salinópolis, Universidade Federal do Pará, Salinópolis, Pará, 68721-000, Brazil \n3 School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, UK \n4 School of Physics and Astronomy, University of Nottingham, University Park, Nottingham, NG7 2RD, UK \n5 Dipartimento di Fisica 'Sapienza' Università di Roma & Sezione INFN Roma1, Piazzale Aldo Moro 5, 00185, Roma, Italy \n6 Center for Particle Cosmology, Department of Physics and Astronomy, \nUniversity of Pennsylvania, 209 S. 33rd St., Philadelphia, PA 19104, USA \n7 Department of Physics and Astronomy, Johns Hopkins University, 3400 N. Charles Street, Baltimore, MD 21218, USA \n(Dated: March 27, 2019) \nScalar-tensor theories of gravity where a new scalar degree of freedom couples to the Gauss-Bonnet invariant can exhibit the phenomenon of spontaneous black hole scalarization . These theories admit both the classic black hole solutions predicted by general relativity as well as novel hairy black hole solutions. The stability of hairy black holes is strongly dependent on the precise form of the scalar-gravity coupling. A radial stability investigation revealed that all scalarized black hole solutions are unstable when the coupling between the scalar field and the Gauss-Bonnet invariant is quadratic in the scalar, whereas stable solutions exist for exponential couplings. Here we elucidate this behavior. We demonstrate that, while the quadratic term controls the onset of the tachyonic instability that gives rise to the black hole hair, the higher-order coupling terms control the nonlinearities that quench that instability, and hence also control the stability of the hairy black hole solutions.", 'I. INTRODUCTION': 'A century after the inception of general relativity (GR), we have entered the era of gravitational wave astronomy. The LIGO/Virgo Collaboration has already observed ten binary black hole (BH) mergers and one binary neutron star merger [1], and this number is only expected to grow as the sensitivity is increased and future detectors (such as LIGOIndia and KAGRA) come online. This new observational window offers us the unprecedented opportunity to test gravity on new distance and energy scales using some of the most extreme objects in the Universe [2-6]. Indeed, the events already observed have provided new bounds on modified gravity theories and important consistency tests [7-15]. \nAstrophysical BHs in GR are simple objects characterized by two numbers: their mass and spin. Because of their simplicity, they are attractive probes for testing GR. Any observational test of gravity must compare the predictions of GR with those coming from competing theories. Without alternative predictions, one cannot verify whether GR is correct, or even quantify the amount by which GR is the preferred theory. This is where the challenges typically arise: the vast majority of well-motivated modifications of GR are subject to no-hair theorems that preclude the existence of new BH charges. In particular, many of these models contain an additional scalar degree of freedom ϕ [2, 16], and the no-hair theorems preclude the existence of BHs with some new scalar \ncharge Q [17-23]. A notable exception to this rule are theories that include a coupling of the scalar field to the Gauss-Bonnet invariant G = R 2 -4 R ab R ab + R abcd R abcd , where R , R ab and R abcd are the Ricci scalar, the Ricci tensor and the Riemann tensor, respectively. Such scalar-Gauss-Bonnet (sGB) couplings arise in the low-energy effective field theory (EFT) derived from string theory, and it has long been known that they give rise to non-Schwarzschild and non-Kerr BHs (see e.g. [24, 25]). A linear coupling between a scalar field and the Gauss-Bonnet invariant features prominently in the EFT of shift-symmetric scalars [21], since the Gauss-Bonnet term is a topological invariant and a total divergence in four dimensions. Again, studies have found new hairy BH solutions in these theories [21, 26-30]. The fact that LIGO/Virgo has observed BH merger events consistent with GR implies that such couplings are necessarily small [31]. \nRecently, a new possibility was pointed out and named \'spontaneous BH scalarization": certain sGB theories admit both the BH solutions of GR and hairy BH solutions [32, 33]. This phenomenon can occur in theories where a scalar field is coupled to the Gauss-Bonnet invariant and the coupling function respects Z 2 symmetry and vanishes for some constant ϕ 0. The last condition guarantees that GR BHs are admissible solutions. The coupling with the Gauss-Bonnet invariant acts as an effective mass term for the scalar perturbations around these solutions. When the BH mass lies within a certain interval, this effective mass term is negative in parts of the BH exterior, triggering a tachyonic instability and producing a nonzero scalar charge. The scalarized solutions exist in BH mass bands whose onset coincides with the tachyonic instability, and whose termination is due to regularity conditions on the horizon that arise from nonlinear effects [32, 33]. Similar scalarization phenomena have been studied also for neutron stars [33, 34], Reissner-Nordström and Kerr BHs [35-39], and scalar-tensor gravity coupled with Born-Infeld electrodynamics [40, 41]. \nThere is no a priori guidance for the functional dependence of the coupling function f ( ϕ ) . Reference [33] focused on the quadratic coupling f ( ϕ ) ∼ ϕ 2 , as this is the simplest case where the tachyonic instability should be present and the leading-order term is expected to control the onset of the instability. Reference [37] focused on the exponential coupling f ( ϕ ) ∼ exp ( βϕ 2 ) instead. Recent studies suggest that scalarized BH solutions are unstable under radial perturbations for the quadratic coupling, while solutions within the exponential coupling model have better stability properties [42]. \nThis paper is concerned with understanding the nature of the instability of the quadratic coupling function. In particular, we will show that the radial instability of the quadratic model is directly linked to the fact that, in this model, the scalar field equation is linear in the scalar. This implies that gravitational backreaction is crucial for quenching the tachyonic instability that leads to scalarization, and that backreaction determines the properties of the scalarized solution in this model. Here we find that introducing nonlinearity in the scalar provides a different quenching mechanism for the tachyonic instability, changes the properties of the scalarized solutions, and removes the radial instability. The simplest setup to demonstrate these points is a theory where the quadratic coupling is augmented by a quartic term 1 . \nThe plan of the paper is as follows. In Sec. II we briefly review sGB gravity, the necessary conditions for the existence of scalarized BH solutions, and our chosen coupling functions. In Sec. III we study quartic sGB gravity in the decoupling limit, computescalar field bound states and investigate their stability. In Sec. IV we obtain full nonlinear BH solutions in this theory and discuss their stability under radial perturbations. In Sec. V we summarize our findings.', 'II. SCALAR-GAUSS-BONNET GRAVITY': "In sGB gravity, a real, massless scalar field is coupled to gravity through the Gauss-Bonnet invariant G . The action of sGB gravity is \nS = 1 2 /uni222B.dsp d 4 x √ -g [ R -1 2 g ab ϕ ; a ϕ ; b + f ( ϕ )G ] , (1) \nwhere ϕ is the scalar field and g ab is the spacetime metric. We use geometrical units, such that 8 π G = c = 1. The field equation for the scalar field in sGB gravity is \n/square ϕ = -f ,ϕ ( ϕ )G , (2) \nwhile the equation for the spacetime metric is \nR ab -1 2 g ab R = T ab (3) \nwhere T ab is the sum of the matter stress-energy tensor (which is vanishing for BH solutions) and an effective stressenergy tensor which depends on f ,ϕ ( ϕ ) [29]. Different choices of the function f ( ϕ ) correspond to different sGB gravity theories. In particular, f ( ϕ ) ∼ exp ( αϕ ) (where α can be different depending on the specific stringy scenario) corresponds to Einstein-dilaton Gauss-Bonnet (EdGB) gravity [24, 25, 27, 44-53], which can arise in the low-energy effective actions of some string theories [54, 55]; f ( ϕ ) ∼ ϕ corresponds to shift-symmetric sGB gravity, which is invariant under ϕ → ϕ + constant, and is so far the only known shift-symmetric scalar-tensor theory with second-order field equations to allow for asymptotically flat, hairy BH solutions [21, 26, 28]. \nRemarkably, EdGB and shift-symmetric sGB gravity do not admit Schwarzschild BH solutions: all static, spherically symmetric BH solutions in this theory have nontrivial scalar field configurations. This feature, however, is not shared by all sGB gravity theories. As shown in [32, 33], sGB gravity admits the BH solutions of GR (with a constant scalar field) if f ,ϕ ( ϕ 0 ) = 0 for some constant ϕ 0, and it also admits BH solutions with nontrivial scalar field configurations if f ,ϕϕ G < 0. In these theories, the f ,ϕϕ G term in the field equations acts as a negative mass term, triggering a tachyonic instability 2 , which can in principle lead to the development of 'scalar hair'. This process of spontaneous scalarization is analogous to that studied in compact stars in scalar-tensor gravity [57, 58], but, crucially , it does not rely on any coupling with matter, and therefore it could potentially be tested through the observation of gravitational waves from binary BH mergers. \nTwo examples of sGB gravity theories satisfying the conditions for spontaneous scalarization have been studied: quadratic sGB gravity [33], in which f ( ϕ ) ∼ ϕ 2 , and exponential sGB gravity [32], where f ϕ ) ∼ 1 exp (-3 ϕ 2 2 ) 3 . \n( \n) ∼ (-) In this work we will study quartic sGB gravity , with a coupling term of the form \n- \n/ \nf ( ϕ ) ≡ 1 8 ¯ ηϕ 2 + 1 16 ¯ ζϕ 4 , (4) \nwhere ¯ η , ¯ ζ are coupling constants with dimensions of [length] 2 , and the numerical factors are chosen for convenience. The scalar field is dimensionless, while G has units of [length] -4 . When ¯ ζ = 0, we obtain the quadratic sGB gravity theory considered in [33], which allows for spontaneous scalarization when ¯ η > 0. For small values of the scalar field, the exponential sGB gravity studied in [32] \nf ( ϕ ) = λ 2 12 1 -exp (-3 ϕ 2 / 2 ) ] (5) \n[ \nreduces to quartic sGB gravity with \n¯ η = λ 2 , ¯ ζ = -3 2 λ 2 , (6) \nplus ϕ 6 ) terms. \nO( \n) Since the field equations (3) reduce to Einstein's equations when f ,ϕ ( ϕ ) = 0, quartic sGB gravity admits the GR solutions provided that \nf ,ϕ = ϕ 4 ( ¯ η + ¯ ζϕ 2 ) = 0 . (7) \nϕ ± = ± √ | ¯ η / ¯ ζ | . (8) \nTherefore, it always admits the GR solutions with ϕ ≡ 0, and if ¯ η ¯ ζ < 0 it admits two additional, real-valued, constant scalar field solutions ϕ ≡ ϕ + and ϕ ≡ ϕ -, where", 'III. DECOUPLING LIMIT OF QUARTIC SCALAR-GAUSS-BONNET GRAVITY': 'Our goal in this section is to understand whether a quartic term in the coupling function f ( ϕ ) i.e., a nonzero value of ¯ ζ can stabilize static, spherically symmetric BH solutions, which are known to be unstable in the quadratic case. Since quadratic sGB gravity only admits scalarized solutions if ¯ η ∼ f ,ϕϕ > 0, in the following we shall assume ¯ η > 0, as in [33, 42]. \nTo begin, we consider the decoupling limit in which we neglect the backreaction from the metric. Thus, we study the scalar field equation (2) \n/square ϕ = -f ,ϕ ( ϕ )G = -ϕ 4 ( ¯ η + ¯ ζϕ 2 ) G , (9) \non a fixed Schwarzschild background with mass M . As discussed in [32, 33], this class of theories admits two kinds of solutions: the GR solutions, i.e. a constant scalar field with ϕ ≡ 0 , ϕ ± ; and the scalarized solutions, in which the scalar field has a nontrivial configuration.', 'A. Static bound-state solutions': "As a first step, we look for static, bound-state solutions of quartic sGB gravity in the decoupling limit. As in Ref. [33], we consider a time-independent scalar field and we expand it in spherical harmonics in standard Schwarzschild coordinates: \nϕ = 1 r /summationdisplay.1 /lscript m ¯ σ /lscript m ( r ) Y /lscript m ( θ, φ ) . (10) \nDue to the nonlinearity introduced by the quartic term in Eq. (4), the wave equation is only separable for spherically symmetric configurations ( /lscript = 0). Defining ¯ σ = ¯ σ 00 and introducing the dimensionless variables \nσ ≡ ¯ σ /( 2 M ) , ρ ≡ r /( 2 M ) , η ≡ ¯ η /( 2 M ) 2 , ζ ≡ ¯ ζ /( 2 M ) 2 , \nwe obtain the following nonlinear differential equation: \nσ '' + 1 ( ρ -1 ) [ σ ' ρ -σ ρ 2 + 3 ησ ρ 5 + 3 ζσ 3 ρ 7 ] = 0 , (11) \nwhere primes denote derivatives with respect to ρ . \nIntroducing the tortoise coordinate ρ ∗ = ρ + log ( ρ -1 ) , this equation becomes a Schrödinger-like equation with a nonlinear potential. We cannot straightforwardly apply results from quantummechanicsto determine when Eq. (11) admits boundstate solutions, as done e.g. in Refs. [36, 59, 60] using criteria derived in [61], and therefore we must study Eq. (11) numerically. We start the integrations close to the event horizon ρ = 1 (typically at ρ = 1 + 10 -5 ) using a power series solution for σ valid in the near-horizon region. To leading order we have \nσ = σ h + ( σ h -3 ησ h -3 ζσ 3 h )( ρ -1 ) . (12) \nWe then integrate out to a large ρ (typically ≈ 10 5 ) for given values of ( η, ζ ) and an arbitrary value σ h ≡ σ ( 1 ) . For each choice of the coupling constants η , ζ , and for each choice of σ h, Eq. (11) admits a unique solution with σ ' h ≡ σ ' ( 1 ) = σ h ( 1 -3 η -3 ζσ 2 h ) , ϕ h = σ h and ϕ ' h = -3 σ h ( η + ζσ 2 h ) . We find that the scalar field diverges as ρ → ∞ if and only if ϕ ' h / ϕ h > 0, i.e. if η + ζσ 2 h > 0. Since we assume that η > 0, this condition is always satisfied if ζ > 0, while if ζ < 0 the condition is satisfied for ϕ -< ϕ h < ϕ + [see Eq. (8)]. \nSince we are interested in scalarized solutions with the same asymptotic behavior as in GR, we require that the scalar field vanishes at infinity. We find that this condition can only be enforced for η larger than a threshold value η thr = 0 . 726; for each η > η thr, there is a discrete set of values of σ h which correspond to the solutions satisfying the boundary condition at infinity. This value, i.e. ¯ η thr / M 2 = 4 η thr = 2 . 904, coincides with the first (zero-node) eigenvalue in the quadratic theory of Ref. [33]. We focus on the nodeless solution because previous work[42] showed that this is the only stable scalarized solution in the exponential theory. Therefore it is natural to ask whether the nodeless solution is stable in the (simpler) quartic theory. \nThe results of the integration of Eq. (11) are shown in Fig. 1, where we plot the scalar field at the horizon, ϕ h, compatible with the boundary conditions, in the case of ζ = -( 3 / 2 ) η [corresponding to the choice in Eq. (6)], for the solution with no nodes. Note that quartic sGB gravity is symmetric under ϕ → -ϕ , so we only show solutions with ϕ > 0. For η < η thr the only solution is ϕ h = 0, and the scalar field is zero everywhere. A scalarized solution appears for η > η thr. As η increases, the value of ϕ h for the scalarized solution also increases, and it tends to the limit ϕ + as η → ∞ . The same qualitative behavior occurs for different (negative) values of ζ . \nIn the same figure we show the corresponding curve for the quadratic theory [33]. In this case the scalar field equation in the decoupling limit is linear, therefore only a discrete set of values of η fulfills the boundary conditions, and the zero-node solution corresponds to η = η thr. For this value all choices of ϕ h are equivalent, since the solution of a linear equation is defined modulo an overall multiplicative constant. \nFIG. 1. Bound scalar field solutions (with no nodes) in the quartic theory with ζ = -( 3 / 2 ) η . Solutions for which the effective potential V eff defined in Eq. (15) is (is not) positive definite correspond to the solid (dashed) line. The horizontal line (marked by a circle) corresponds to the constant ϕ + = √ 3 / 2 solution, whereas the vertical line corresponds to the solutions of the quadratic sGB. For η = 0, the theory reduces to that of scalar field minimally coupled to gravity. No-hair theorems force ϕ to have (any) constant value. See text for a detailed description. \n<!-- image -->", 'B. Linear stability analysis in the decoupling limit': 'Let us now analyze the stability of the static, spherically symmetric solutions discussed in Sec. III A. Let \nσ = σ 0 + δσ, (13) \nwere σ 0 is a solution of Eq. (11) (e.g. found numerically as in Sec. III A). Substituting this perturbation in Eq. (11) and neglecting O( δσ 2 ) terms, we find the linear equation \nd 2 δσ d ρ 2 ∗ -V eff ( ρ ) δσ = 0 , (14) \nwith effective potential \nV eff ≡ ( 1 -1 ρ ) [ /lscript ( /lscript + 1 ) ρ 2 + 1 ρ 3 -( 3 η ρ 6 + 9 ζ ρ 8 σ 2 0 )] . (15) \nFollowing [61], a sufficient (but not necessary) condition for instability is \n/uni222B.dsp ∞ 1 V eff 1 -1 / ρ d ρ = /lscript ( /lscript + 1 ) + 1 2 -3 η 5 -9 ζ /uni222B.dsp ∞ 1 ϕ 2 0 ρ 6 d ρ < 0 , (16) \nwhere ϕ 0 = δσ 0 / ρ and the integral must be computed numerically. \nIn the quadratic sGB theory ( ζ = 0) the solution is unstable for η > 5 / 6 /similarequal 0 . 83. Remarkably, in this case the effective potential does not depend on ϕ 0. Therefore, when \nthe Schwarzschild solution is unstable - the instability eventually leading to spontaneous scalarization - the bound state solution is also unstable. This simple qualitative reasoning suggests that, as shown in [42], scalarized solutions in the quadratic sGB theory are always unstable. \nEq. (16) also shows that when ζ < 0 the contribution of the integral is positive, and therefore it tends to stabilize BH solutions. Moreover the integral term vanishes as ϕ 0 → 0. This suggests that in the quartic theory, for certain values of the coupling constants ( η, ζ ) the Schwarzschild solution is unstable, while the scalarized solution is not. This is consistent with the results found in [42] for the exponential theory, which is equivalent to the quartic theory with ζ = -( 3 / 2 ) η if we ignore terms of order O( ϕ 6 ) .', 'C. Numerical results': 'In Fig. 2 we plot the scalar charge-mass ( Q -M ) diagram corresponding to nodeless ( n = 0) scalarized BH solutions in the quartic theory for fixed values of | ζ / η | = 0 , 0 . 5 , 1 , 1 . 5 and 2 . 0. For completeness we also show the corresponding diagram for the exponential coupling from Ref. [32]. The vertical line represents the threshold η = η thr. \nNumerically we found that solutions with ζ / η /lessorsimilar -0 . 8 bend to the left in the Q -M diagram (and so do solutions corresponding to the exponential coupling), while solutions with \nFIG. 3. Eigenfrequencies of the unstable modes ω I as function of the coupling η for different ζ / η and for the Schwarzschild BH in sGB gravity. The Schwarzschild BH is unstable for η /greaterorsimilar 0 . 726, as indicated by the linear analysis outlined in Sec. III. For η /lessorsimilar 0 . 726 we can see the instability time scale for the nodeless solutions for the cases ζ / η > -0 . 5. The inset zooms-in into these frequencies. \n<!-- image --> \nζ / η larger than this critical value bend to the right. Solutions illustrating this behavior near this critical value are shown in the inset of Fig. 2. This different behavior corresponds to different radial stability properties. When ζ / η = -0 . 5 there is a gap in the parameter space in which BH solutions do not exist: the first derivative of the scalar field at the horizon is complex in this region. This happens because Eq. (23) cannot be satisfied for parameter choices that lie in the gap. We verified that the gap exists in the parameter range -0 . 64 < ζ / η < 0, and it is indeed related to the polynomial form of the existence condition given by Eq. (22). \nThetwodifferentbranchesof solutions that exist (e.g.) when ζ / η = -0 . 5present different stability properties,one being stable and the other unstable. We can understand this behavior qualitatively using intuition built from Sec. III. From (16), we know that not only the magnitude of the coupling parameter ζ , but also the amplitude of the background solution ϕ 0 contributes to quenching the instability. In the unstable segment of the ζ / η = -0 . 5 solutions, the scalar field has small amplitude (i.e., small scalar charge) and therefore it cannot quench the instability. This is not the case for more charged solutions, which are stable. \nTo analyze the stability, in Fig. 3 we plot the unstable mode frequencies for the same set of theories with ( η, ζ ) shown in Fig. 2, and also for Schwarzschild BH solutions in the same theory. Note that for Schwarzschild BH solutions ϕ 0 = 0, and therefore the frequency does not depend on ζ . Hence, all unstable scalarized BHs branch out of the same threshold value of η ≈ 0 . 726, in agreement with the value obtained in Sec. III. All BH solutions in quartic theories with ζ / η > 0 shown in Fig. 3 are unstable to radial perturbations, again in agreement with the analysis from the decoupling limit, and the instability time scale τ = | ω -1 I | of these modes decreases as ζ / η increases (as long as ζ / η > 0). \nAs noted when discussing Fig. 2, there is a gap in the parameter space of BH solutions with ζ / η = -0 . 5. The two branches have different behavior in the Q -M plane: the branch with small values of Q / ¯ η 1 / 2 is more similar to solutions with ζ / η > -0 . 5, and the branch with large values of Q / ¯ η 1 / 2 is more similar to solutions with ζ / η < -0 . 5. We performed a radial stability analysis searching for unstable modes in the two branches. We found no unstable modes in the large Q / ¯ η 1 / 2 branch, but we found unstable modes in the small Q / ¯ η 1 / 2 branch, and these are the ζ η = -0 . 5 modes shown in Fig. 3. \n/ \n-The case ζ / η = -0 . 5, presenting stable and unstable solutions, is similar to the ones shown in the inset of Fig. 2. Note that the unstable modes for ζ / η = -0 . 5 have small M ω I (see inset of Fig. 3). For the solutions with values of ζ / η presented in the inset in Fig. 2 the mode frequency is even smaller, being challenging to find numerically. Our numerical findings suggest that in quartic sGB with ζ / η < -0 . 8 scalarized BHs with n = 0 are stable.', 'IV. NONLINEAR BLACK HOLE SOLUTIONS AND THEIR RADIAL (IN)STABILITY': "In GR, radial perturbations describe nonradiative fields. The perturbation equations can be solved analytically and correspond to a change in mass of the Schwarzschild BH solution, as expected from Birkhoff's theorem [62]. In modified theories of gravity radial perturbations can be radiative, with important consequences for the stability of the spacetime. \nAs mentioned in the introduction, Ref. [42] studied radial perturbations of a static, spherically symmetric BH for a generic coupling function f ( ϕ ) in sGB gravity. We mostly follow their treatment. For brevity, we will only outline the \nprocedure to obtain the perturbation equations and our numerical calculation of the radial oscillation modes. \nThe spherically symmetric, radially perturbed spacetime up to first order in the perturbations has line element \nd s 2 = -exp [ 2 Φ ( r ) + ε F t ( t , r )] d t 2 + exp [ 2 Λ ( r ) + ε F r ( t , r )] d r 2 + r 2 d Ω 2 , (17) \nwhere ε is a small bookkeeping parameter and d Ω 2 = d θ 2 + sin 2 θ d φ 2 is the line element of the unit-sphere. To the same order, the scalar field is given by \nϕ = ϕ 0 ( r ) + ε ϕ 1 ( t , r ) r . (18) \nBy inserting Eqs. (17) and (18) into the field equations (2)-(3) and expanding in powers of ε we get equations for the background metric functions ( Φ , Λ ) at zeroth order in ε , and for the radial perturbations ( F t , F r , ϕ 1 ) at first order in ε . Let us first discuss the background equations and boundary conditions.", 'A. Black hole scalarization in quartic sGB gravity': "The zeroth-order equations for ( Φ , Λ , ϕ 0 ) can be cast as a coupled system of two first-order equations for Φ and Λ and a second-order equation for ϕ 0 [25, 32, 42]. BH solutions are obtained by imposing that the metric functions exp ( 2 Φ ) and exp (-2 Λ ) vanish and that the scalar field ϕ 0 be regular at the horizon: \nexp ( 2 Φ ) ∼ ( r -r h ) + O[( r -r h ) 2 ] , (19) \nϕ 0 ∼ ϕ 0 , h + ϕ ' 0 , h ( r -r h ) + O[( r -r h ) 2 ] , (21) \nexp (-2 Λ ) ∼ ( r -r h ) + O[( r -r h ) 2 ] , (20) \nwhere r h is the horizon radius, while ϕ 0 , h and ϕ ' 0 , h denote the scalar field and its first derivative at the horizon. Using a near-horizon expansion of the field equations, we find that BH solutions correspond to the condition (cf. [29, 32, 33] or more details) \nϕ ' 0 , h = -r h ϕ 0 , h ( η + ζϕ 2 0 , h ) 1 -√ √ 1 -6 ϕ 2 0 , h r 4 h ( η + ζϕ 2 0 , h ) 2 . (22) \nWhen Eq. (22) is not satisfied, ϕ '' 0 diverges at the horizon. By requiring that the first derivative of the scalar field at the horizon be real and using Eq. (22) we find a condition for the existence of the solutions: \n6 ϕ 2 0 , h ( η + ζϕ 2 0 , h ) 2 < r 4 h . (23) \nAt large distances, an expansion of the background equations in powers of r -1 leads to \nexp ( 2 Φ ) ∼ 1 -2 M r + O( r -2 ) , (24) \nexp (-2 Λ ) ∼ 1 -2 M r + O( r -2 ) , (25) \nϕ 0 ∼ ϕ 0 , ∞ + Q r + O( r -2 ) , (26) \nwhere M is the total (ADM) mass and Q is the scalar charge. Since we are not interested in cosmological effects and we require that the hairy solution has the same asymptotic properties as the GR solution, we will also impose ϕ 0 , = 0. \nTo obtain scalarized BH solutions we proceed as follows. Weintegrate the differential equations for the background from the horizon outwards imposing the boundary conditions (19)(21) with the constraint (23), using a guess value for the scalar field at the horizon ϕ 0 , h. Numericalsolutions in the far-horizon region ( r /greatermuch r h) are then compared with the boundary conditions (24)-(25). Not all sets of ( ϕ 0 , h , η, ζ ) allow for BH solutions satisfying both boundary conditions. This generates a boundary value problem that can be solved by a shooting method. In practice, we fix the values of ( η, ζ ) , and we find the values of ϕ 0 , h by shooting and requiring that the scalar field vanishes in the far region. \n∞", 'B. Radial perturbations of scalarized black holes': 'Let us now consider the radial stability of the BH solutions foundin the preceding section. By manipulating the first-order equations we can show that the functions ( F t , F r , ϕ 1 ) are not independent: F t and F r can be written in terms of ϕ 1, where ϕ 1 obeys the differential equation [42, 45] \nh ( r ) ∂ 2 ϕ 1 ∂ t 2 -∂ 2 ϕ 1 ∂ r 2 + k ( r ) ∂ϕ 1 ∂ r + p ( r ) ϕ 1 = 0 . (27) \nHere h , k and p are functions of r that depend on the background metric functions: cf. Eq. (14) of [42]. \nBy a suitable redefinition of the functions ( h , k , p ) and using a harmonic-time decomposition ϕ 1 ( t , r ) = ϕ 1 ( r ) e -i ω t , we can write the above equation in a Schrödinger-like form. This is useful for analyzing the effective potential felt by the perturbations [42]. However here we will deal directly with the differential equation in the form (27), mainly because it is simpler to solve it numerically. We introduce a compactified dimensionless coordinate \nx ≡ 1 -r h / r , (28) \nsuch that the horizon and spatial infinity are mapped to x = 0 and x = 1, respectively. To integrate Eq. (27) numerically we impose the standard boundary conditions at the horizon and at spatial infinity: \nϕ 1 ( x ) = { e i ω r ∗ r ∗ →∞ ( x → 1 ) e -i ω r ∗ r ∗ →-∞ ( x → 0 ) , (29) \nwhere r ∗ is the tortoise coordinate. \n∗ The differential equation (27) together with boundary conditions (29) yields a boundary-value problem for the complex eigenvalue ω = ω R + i ω I . Stable modes have ω I < 0, while unstable modes have ω I > 0. Therefore, to study the radial stability of the solutions we can search for purely imaginary modes with ω I > 0. To obtain these modes, we use again a shooting method. We perform two integrations starting at x = 0 and at x = 1. At each boundary we impose that the \nFIG. 2. Q -M diagram for scalarized solutions in quartic sGB gravity with n = 0, considering different values for ζ / η . For comparison, we also show the solutions for exponential sGB gravity [cf. Eq. (5)]: for small Q / ¯ η 1 / 2 (i.e. small scalar field amplitudes), the curve overlaps with the case | ζ / η | = 1 . 5, as it should. The vertical line represents the scalarization threshold η = η thr . Solutions to the left (right) of the vertical dotted line are stable (unstable). The inset shows additional illustrative curves showing the behavior near the scalarization threshold η thr . Curves with ζ / η /lessorsimilar -0 . 8 are always to the left of the scalarization threshold. \n<!-- image --> \nscalar field is zero and that its first derivative is constant. We can fix the scalar field amplitude to unity because Eq. (27) is linear. The integration of Eq. (27) from the horizon yields a first solution ϕ (-) 1 , and the integration from infinity yields a second solution ϕ ( + ) 1 . We match the two solutions at an intermediate point x m . The eigenvalue ω = i ω I corresponds to the frequency at which the Wronskian \nW = [ ϕ (-) 1 d ϕ ( + ) 1 d x -ϕ ( + ) 1 d ϕ (-) 1 d x ] x = x m (30) \nvanishes. We checked that the modes are stable under variations of the numerically chosen values of the near-horizon radius, of the large radius representing spatial infinity, and of the matching point x m . Additionally, we checked that our results reproduce those for the exponential and quadratic couplings presented in Ref. [42].', 'V. CONCLUSIONS': "In this work we have investigated the radial stability of scalarized BH solutions in sGB gravity. Motivated by the radial instability of quadratic sGB solutions found in [42], we have shown that adding higher-order (quartic) corrections to the original quadratic sGB model of [33] can stabilize the solutions. \nOur analysis provided a clear picture for the physical interpretation of this results. At the linearized level, scalarization manifests as a tachyonic instability that triggers the growth of the scalar field. For the end-point of the instability to be a hairy solution, the tachyonic instability needs to be quenched by some nonlinear effects. In quadratic sGB gravity however, the field equation for the scalar field is linear in the scalar and hence the only quenching mechanism would be backreaction. This is nicely demonstrated by our decoupling limit analysis, where backreaction is entirely ignored and the tachyonic instability is always present. 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This suggests that scalarized BH solutions are stable in this region of the theory's parameter space. \nMore generally, our results clearly demonstrate that the quadratic coupling between the scalar and the Gauss-Bonnet invariant controls the onset of the scalarization, whereas the higher-ordercorrections (in the scalar) control the end-point of the tachyonic instability that triggers scalarization, and hence they are crucial for the properties of the hairy black holes solutions. \nNote Added - While this work was being completed, a preprint with similar conclusions appeared as an e-print [63]. Where our works overlap, our conclusions agree with theirs.", 'ACKNOWLEDGMENTS': "This work was supported by the H2020-MSCA-RISE-2015 Grant No. StronGrHEP-690904 and by the COST action CA16104 'GWverse'. H.O.S was supported by NASA Grants No. NNX16AB98G and No. 80NSSC17M0041. T.P.S. acknowledgespartial support from the STFC Consolidated Grant No. ST/P000703/1. 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D84 , 087501 (2011), arXiv:1109.3996 [gr-qc].\n- [49] K. Yagi, L. C. Stein, N. Yunes, and T. Tanaka, Phys. Rev. D85 , 064022 (2012), [Erratum: Phys. Rev.D93,no.2,029902(2016)], arXiv:1110.5950 [gr-qc].\n- [50] K. Yagi, Phys. Rev. D86 , 081504 (2012), arXiv:1204.4524 [gr-qc].\n- [51] A. Maselli, P. Pani, L. Gualtieri, and V. Ferrari, Phys. Rev. D92 , 083014 (2015), arXiv:1507.00680 [gr-qc].\n- [52] B. Kleihaus, J. Kunz, S. Mojica, and E. Radu, Phys. Rev. D93 , 044047 (2016), arXiv:1511.05513 [gr-qc].\n- [53] J. L. Blázquez-Salcedo, C. F. B. Macedo, V. Cardoso, V. Ferrari, L. Gualtieri, F. S. Khoo, J. Kunz, and P. Pani, Phys. Rev. D94 , 104024 (2016), arXiv:1609.01286 [gr-qc].\n- [54] D. J. Gross and J. H. Sloan, Nucl. Phys. B291 , 41 (1987).\n- [55] R. R. Metsaev and A. A. Tseytlin, Nucl. Phys. B293 , 385 (1987).\n- [56] Y. S. Myung and D.-C. Zou, Phys. Rev. D98 , 024030 (2018), arXiv:1805.05023 [gr-qc].\n- [57] T. Damour and G. Esposito-Farèse, Phys. Rev. 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2014PhLB..730...14B

Regular black hole metrics and the weak energy condition

2014-01-01

['-']

[]

In this work we construct a family of spherically symmetric, static, charged regular black hole metrics in the context of Einstein-nonlinear electrodynamics theory. The construction of the charged regular black hole metrics is based on three requirements: (a) the weak energy condition should be satisfied, (b) the energy-momentum tensor should have the symmetry T00=T11, and (c) these metrics have to asymptotically behave as the Reissner-Nordström black hole metric. In addition, these charged regular black hole metrics depend on two parameters which for specific values yield regular black hole metrics that already exist in the literature. Furthermore, by relaxing the third requirement, we construct more general regular black hole metrics which do not behave asymptotically as a Reissner-Nordström black hole metric.

[]

https://arxiv.org/pdf/1401.2136.pdf

{'Regular black hole metrics and the weak energy condition': "Leonardo Balart 1 , 2 ∗ and Elias C. Vagenas 3 † \n- 1 I.C.B. - Institut Carnot de Bourgogne UMR 5209 CNRS, \nFacult ' e des Sciences Mirande, Universit ' e de Bourgogne, \n- 9 Avenue Alain Savary, BP 47870, 21078 Dijon Cedex, France\n- 2 Departamento de Ciencias F ' ı sicas, Facultad de Ingenier ' ı a y Ciencias, Universidad de La Frontera, Casilla 54-D, Temuco, Chile and\n- 3 Theoretical Physics Group, Department of Physics, \nKuwait University, P.O. Box 5969, Safat 13060, Kuwait", 'Abstract': 'In this work we construct a family of spherically symmetric, static, charged regular black hole metrics in the context of Einstein-nonlinear electrodynamics theory. The construction of the charged regular black hole metrics is based on three requirements: (a) the weak energy condition should be satisfied, (b) the energy-momentum tensor should have the symmetry T 0 0 = T 1 1 , and (c) these metrics have to asymptotically behave as the Reissner-Nordstrom black hole metric. In addition, these charged regular black hole metrics depend on two parameters which for specific values yield regular black hole metrics that already exist in the literature. Furthermore, by relaxing the third requirement, we construct more general regular black hole metrics which do not behave asymptotically as a Reissner-Nordstrom black hole metric.', 'I. INTRODUCTION': "Charged regular black hole solutions exist in the framework of Einstein-nonlinear electrodynamics theory and are obtained as solutions of Einstein equations that are characterized by the fact that the metric as well as the curvature invariants R , R µν R µν , R κλµν R κλµν do not present singularities anywhere 1 . \nThe Bardeen black hole is the first of a series of regular black hole solutions obtained [2]. If we write the most general form of a static line element with spherical symmetry \nds 2 = -f ( r ) dt 2 + f ( r ) -1 dr 2 + r 2 ( dθ 2 +sin 2 θdφ 2 ) , (1) \nwhere the metric function can be written as \nf ( r ) = 1 -2 m ( r ) r , (2) \nthen, we can write the mass function of the Bardeen black hole as [2] 2 \nm ( r ) = Mr 3 ( r 2 + g 2 ) 3 / 2 . (3) \nSuch a metric has event horizons located at r ± if g 2 ≤ (16 / 27) M 2 , where g can be interpreted as the monopole charge of a self-gravitating magnetic field described by nonlinear electrodynamics [3]. Furthermore, if r →∞ , we get that the metric behaves as \nf ( r ) → 1 -2 M r + 3 Mg 2 r 3 . (4) \nSimilarly, when r → 0 the Bardeen metric function behaves as the de Sitter black hole, that is as \nf ( r ) → 1 -2 M g 3 r 2 . (5) \nLater on, other solutions [4]-[8] were obtained, using the F-P dual formalism, by considering the action of general relativity coupled to nonlinear electrodynamics, namely \nS = ∫ d 4 x √ -g ( 1 16 π R -1 4 π L ( F ) ) . (6) \nHere, the Lagrangian L ( F ) is a nonlinear function of the Lorentz invariant F = 1 4 F µν F µν which, for weak fields, describes the Maxwell theory, and the corresponding regular black hole solution asymptotically behaves as the Reissner-Nordstrom black hole. \nThe regular black hole may also be characterized by energy conditions [9, 10] that the corresponding energy-momentum tensor should satisfy. In the context of regular black holes, three energy conditions have been utilized by several authors. In particular: \n- · The strong energy condition (SEC) states that T µν t µ t ν ≥ 1 2 T µ µ t ν t ν for all timelike vector t µ . This condition implies that gravitational force is attractive.\n- · The dominant energy condition (DEC) states that T µν t µ t ν ≥ 0 and that T µν t µ must be non-spacelike for all timelike vector t µ or, equivalently, T 00 ≥ | T ij | for each i, j = 1 , 2 , 3. This means that the local energy density measured by a given observer must be nonnegative, and that the speed of the energy density flow associated with this observer cannot exceed the speed of light.\n- · The weak energy condition (WEC) states that the energy density of matter measured by an observer, whose 4-velocity is t µ , satisfies T µν t µ t ν ≥ 0 for all timelike vector t µ , that is, the local energy density cannot be negative for all observers. The DEC implies the WEC. \nWhen a black hole is regular, the SEC is necessarily violated somewhere inside the horizon [11]. However, a regular black hole could satisfy the WEC or the DEC everywhere [8]. Considering the line element (1) and the metric function (2), we can write the components of the respective energy-momentum tensor as \nT 0 0 = T 1 1 = 2 8 πr 2 dm ( r ) dr , T 2 2 = T 3 3 = 1 8 πr d 2 m ( r ) dr 2 , (7) \nand hence the WEC can be expressed equivalently in terms of the mass function by the following inequalities \n1 r 2 dm ( r ) dr ≥ 0 , (8) \n2 r dm ( r ) dr ≥ d 2 m ( r ) dr 2 . (9) \nFrom these relations, it is obvious that the Bardeen black hole satisfies the WEC everywhere [2]. The same applies to regular black hole solutions reported in Refs. [4], [7], [8], [12], [13], [14], and [15]. However, there are other regular black hole solutions reported in Refs. [5], [6], [7], [16], and [17], which do not satisfy the WEC. \nand \nThe regular black hole solutions that satisfy the WEC and their energy-momentum tensor has the symmetry T 0 0 = T 1 1 , necessarily have de Sitter behavior at r → 0 as was shown in Ref. [13], and illustrated here in Eq. (5) for the Bardeen black hole solution. As we will see later, this leaves an extra condition which allows the WEC to be better exploited compared to the DEC when one is building solutions. However, it should be noted that a de Sitter behavior at the center of a regular black hole is not sufficient by itself to ensure that the solution satisfies the WEC. In addition, it has been shown that for a regular black hole in nonlinear electrodynamics which satisfies the WEC, the non-existence theorems [16] can be circumvented by removing the condition of the Maxwell weak field limit imposed at the center of the black hole. In this way, regular black hole solutions with electric charge do exist [8]. \nFurthermore, there are other features that characterize regular black holes which are due to the nonlinearities of the field equations. For example, the thermodynamic quantities of the regular black holes do not satisfy the Smarr formula [18], the identity of Bose-Dadhich [19] which refers to the relation between the Brown-York energy and the Komar charge, is not satisfied by regular black holes [20]. \nIn all the above-mentioned solutions, one asymptotically recovers the Schwarzschild black hole metric, and if the condition r 2 m ' ( r ) /negationslash = 0 as r → ∞ is satisfied, then one recovers the Reissner-Nordstrom black hole metric. \nIn this Letter, in the context of Einstein-nonlinear electrodynamics theory, we will construct a family of spherically symmetric, static, charged regular black hole metrics without utilizing the aforesaid methods but by imposing three conditions: (a) the weak energy condition should be satisfied, (b) the energy-momentum tensor should have the symmetry T 0 0 = T 1 1 , and (c) these metrics have to asymptotically behave as the Reissner-Nordstrom black hole metric. In addition, by relaxing the third condition, i.e., condition (c), we construct more general regular black hole metrics which do not behave asymptotically as a Reissner-Nordstrom black hole metric. The Letter is organized as follows. In Sec. II, we present the equations that we will use in the construction of our metrics. In Sec. III, we obtain a general metric for charged regular black holes that satisfies the WEC and asymptotically behaves as the Reissner-Nordstrom black hole metric. In addition, we discuss a specific case of the aforesaid general metric function. In Sec. IV, we extend the analysis to obtain metric functions that do not necessarily asymptotically behave as the Reissner-Nordstrom \nsolution. Finally, in Sec. V, we briefly summarize our results.", 'II. WEC EQUATIONS': 'Up to now, regular black holes solutions have been constructed by searching for, or postulating, the Lagrangian function L ( F ) in the framework of F-P dual formalism [21] 3 . As already mentioned in the Introduction, here we will construct a family of spherically symmetric, static, charged regular black hole metrics by imposing three conditions: (a) the weak energy condition should be satisfied, (b) the energy-momentum tensor should have the symmetry T 0 0 = T 1 1 , and (c) these metrics have to asymptotically behave as the ReissnerNordstrom black hole metric. Therefore, in this section we will derive the equations which will be used in this construction, from the WEC. \nFor simplicity and future convenience, we replace the variable r that appears in Eqs. (8) and (9) with a new variable x which is defined as r = 1 /x and obviously x ∈ [0 , ∞ ). It is evident that by employing the new variable x , the WEC inequalities, namely Eqs. (8) and', '(9), now read': 'x 4 dm ( x ) dx ≤ 0 (10) \n4 x 3 dm ( x ) dx + x 4 d 2 m ( x ) dx 2 ≤ 0 . (11) \nIt should be noted that these conditions plus regularity imply that the regular black hole metric function must satisfy the following limit \n-dm ( x ) dx x 4 → b when x →∞ (12) \nwhere b is a positive constant. \nAt this point, we should stress that if we want the regular black hole metric to behave asymptotically as the Schwarzschild black hole, then we have to demand our metric function to satisfy the following condition \nm ( x ) = 0 when x → 0 . (13) \n/negationslash \nand \nHowever, if we want the regular black hole metric to behave asymptotically as the ReissnerNordstrom black hole, then we must also require our metric function to satisfy the condition \n-dm ( x ) dx /negationslash = 0 when x → 0 . (14) \nIt is easily seen that two mass functions which satisfy the aforesaid conditions, i.e. Eqs. (10)(14), are defined by the following WEC equations \n-dm ( x ) dx = c 1 (1 + c 2 x α ) 4 /α (15) \nand \n-dm ( x ) dx = c 3 (1 + c 4 x 1 /β ) 4 β (16) \nwhere α is a positive integer, β is a positive constant, c 1 , c 2 , c 3 , and c 4 are arbitrary but also positive constants related by c 1 = ( c 2 ) 4 /α b and c 3 = ( c 4 ) 4 /β b . However, the latter mass function has to be discarded since its expansion leaves fractional powers (with the exception of the case with β = 1).', 'III. REGULAR BLACK HOLE METRICS': 'In this section, we will construct a family of spherically symmetric, static, charged regular black hole metrics by imposing the following three conditions on them: (a) the weak energy condition should be satisfied, (b) the energy-momentum tensor should have the symmetry T 0 0 = T 1 1 , and (c) the metrics have to asymptotically behave as the Reissner-Nordstrom black hole metric. \nFirst we transform the WEC equation given by Eq. (15) into the integral form \nm ( x ) = ∫ ∞ x c 1 (1 + c 2 y α ) 4 /α dy (17) \nand then we compute the above integral. The mass function is, thus, given by the expression \nm ( x ) = c 1 3( c 2 ) 4 /α 1 x 3 2 F 1 ( 3 α , 4 α ; 3 + α α ; -1 c 2 x α ) (18) \nwhere 2 F 1 ( a, b ; c ; z ) is the Gauss hypergeometric function . \nAt this point, we demand the metric function given in Eq. (2), i.e., \nf ( r ) = 1 -2 m ( r ) r , (19) \nto behave asymptotically as the Reissner-Nordstrom black hole metric, i.e., \nf ( r ) = 1 -2 M r + q 2 r 2 . (20) \nNow, we substitute Eq. (18) in Eq. (19) and Taylor expand it around r = 0. By comparing the coefficients of the series expansion at the asymptotic limit, i.e., r → ∞ , with the corresponding ones in Eq. (20), we define the constants c 1 and c 2 as follows \nc 1 = q 2 2 (21) \nc 2 = [ q 2 Γ( 1 α )Γ( α +3 α ) 6 M Γ( 4 α ) ] α . (22) \nTherefore, the mass function given by Eq. (18) becomes \nm ( r ) = r 3 q 2 6 ( 6 Γ( 4 α ) Γ( 1 α )Γ( α +3 α ) M q 2 ) 4 2 F 1 ( 3 α , 4 α ; 3 + α α ; -( 6 Γ( 4 α ) Γ( 1 α )Γ( α +3 α ) M q 2 r ) α ) . (23) \nThis is the mass function of a charged regular black hole metric given by Eq. (19) which asymptotically behaves as the Reissner-Nordstrom black hole if α is a positive constant. It is noteworthy that for α = 2, we retrieve the regular black hole metric given in Ref. [8]. \nFurthermore, in the context of Einstein-nonlinear electrodynamics theory, the electric field associated with the above regular black hole metric is given as [22] \nE = q r 2 ( 1 + ( Γ( 1 α )Γ( 3+ α α ) 6 Γ( 4 α ) q 2 Mr ) α ) -(4+ α ) /α , (24) \nwhich behaves as E = q r 2 when r →∞ . \nFinally, as an example, we choose α = 3 in which case the metric function is of the form \nf ( r ) = 1 -2 M r 1 -1 ( 1 + ( 2 Mr q 2 ) 3 ) 1 / 3 . (25) \nThis regular black hole metric has event horizons if the electric charge satisfies the condition q ≤ 1 . 0257 M . Moreover, in the context of Einstein-nonlinear electrodynamics theory, the associated electric field is given as [22] \nE = q r 2 ( 1 + ( q 2 2 Mr ) 3 ) -7 / 3 . (26)', 'IV. MORE GENERAL METRICS': 'In this section, we will construct more general regular black hole metrics. The same analysis as in the previous section will be adopted here but we will not demand the metric to behave asymptotically as the Reissner-Nordstrom black hole metric. For this reason, we will relax the condition given in Eq. (14). The mass function m ( x ) will now satisfy the following WEC equation \n-dm ( x ) dx = c 1 x µ -4 (1 + c 2 x α ) µ/α (27) \nwhere α and µ are integers and also α ≥ 1 and µ ≥ 4 and, thus, it is given by the expression \nm ( x ) = c 1 3( c 2 ) µ/α 1 x 3 2 F 1 ( 3 α , µ α ; 3 + α α ; -1 c 2 x α ) . (28) \nThe coefficients c 1 and c 2 are now given by \nc 1 = q 2 2 Γ( 1 α )Γ( µ α ) Γ( 4 α )Γ( µ -3 α ) ( 6Γ( 4 α ) Γ( 1 α )Γ( α +3 α ) M q 2 ) 4 -µ (29) \nc 2 = [ 6Γ( 4 α ) Γ( 1 α )Γ( α +3 α ) ) M q 2 ] -α . (30) \nThus, the mass function of the regular black hole metric becomes \nm ( r ) = r 3 q 2 6 Γ( 1 α )Γ( µ α ) Γ( 4 α )Γ( µ -3 α ) ( 6 Γ( 4 α ) Γ( 1 α )Γ( α +3 α ) M q 2 ) 4 2 F 1 ( 3 α , µ α ; 3 + α α ; -( 6 Γ( 4 α ) Γ( 1 α )Γ( α +3 α ) M q 2 r ) α ) . (31) \nThis is the mass function of a charged regular black metric given by Eq. (19), but which does not asymptotically behave as the Reissner-Nordstrom black hole metric except for the case with µ = 4. \nIt is worthy of notice that, with an appropriate choice of parameters, we can derive from Eq. (31) several regular black hole metrics which already exist in the literature. For instance, if we choose µ = 5 and α = 2, we obtain \nm ( r ) = Mr 3 ( r 2 + π 2 q 4 64 M 2 ) 3 / 2 . (32) \nBy replacing π 2 q 4 / (64 M 2 ) with g 2 in Eq. (32) , we recover the Bardeen metric whose mass function is given by Eq. (3). \nNow, if we take µ = 6 and α = 3, and we replace the factor q 6 8 M 3 with 2 l 2 M , we obtain the regular black hole metric given in Ref. [15] with mass function \nm ( r ) = Mr 3 r 3 +2 l 2 M . (33) \nFinally, if we choose µ = 3 and let α be arbitrary, then we obtain the following mass function \nm ( r ) = M 1 -1 ( 1 + ( 2 M q 2 r ) 3 ) ( α -3) / 3 . (34)', 'V. CONCLUSIONS': 'In this Letter, we have constructed a family of spherically symmetric, static, charged regular black hole metrics in the context of Einstein-nonlinear electrodynamics theory. Our analysis is based on the fact that we impose three conditions on the black hole metrics: (a) the weak energy condition should be satisfied, (b) the energy-momentum tensor should have the symmetry T 0 0 = T 1 1 , and (c) these metrics have to asymptotically behave as the ReissnerNordstrom black hole metric. Moreover, by relaxing the third requirement, we construct more general regular black hole metrics which do not behave asymptotically as a ReissnerNordstrom black hole metric. In addition, we discuss as examples several special cases of the more general regular black hole metrics. These special cases have been obtained by choosing specific values for the parameters that characterize the mass function of the more general regular black hole metric. Some of these regular black hole metrics already exist in the literature but they are obtained in the context of F-P dual formalism. All the above regular black hole metrics also satisfy the DEC, although it was not imposed as a condition.', 'Acknowledgments': "L.B. would like to thank Direcci'on de Investigaci'on y Postgrado de la Universidad de La Frontera (DIUFRO) for the financial support.", 'Appendix A: Dual P Formalism': "An equivalent method for deriving the regular black hole metrics obtained here is the F-P dual formalism. For this reason, we briefly present here the description based on the F-P \ndual representation of nonlinear electrodynamics obtained by a Legendre transformation [21] and reproduce the results of Sec. III. \nThe regular black hole metrics can be described by the metric function and its corresponding electromagnetic field which arise as a solution of Einstein field equations coupled to a nonlinear electrodynamics, that is of the action given by Eq. (6). One can also describe the considered system in terms of an auxiliary field defined by P µν = ( dL/dF ) F µν . The dual representation is obtained by means of a Legendre transformation \nH = 2 F dL dF -L (35) \nwhich is a function of the invariant P = 1 4 P µν P µν . Thus, we can express the Lagrangian L depending on P µν as \nL = 2 P dH dP -H , (36) \nand the electromagnetic field as \nF µν = dH dP P µν . (37) \nThe energy-momentum tensor in the F-P dual representation is given by \nT µν = 1 4 π dH dP P µα P α ν -1 4 π g µν ( 2 P dH dP -H ) . (38) \nIt follows from the components of T µν that M ' ( r ) = -r 2 H ( P ). Hence, we can obtain the corresponding mass function. \nAs an example, we give the function H ( P ) for the regular black hole metrics of Sec. III \nH ( P ) = P (1 + Ω ( -P ) α/ 4 ) 4 /α (39) \nwhere \nand Ω is defined as \nP = -q 2 2 r 4 (40) \nΩ = ( q 3 / 2 6 M Γ( 1 α )Γ( 3+ α α ) Γ( 4 α ) ) α . (41) \n- [3] E. Ayon-Beato and A. Garcia, Phys. Lett. B 493 , 149 (2000).\n- [4] E. Ayon-Beato and A. Garcia, Phys. Rev. Lett. 80 , 5056 (1998).\n- [5] E. Ayon-Beato and A. Garcia, Phys. Lett. B 464 , 25 (1999).\n- [6] E. Ayon-Beato and A. Garcia, Gen. Rel. Grav. 31 , 629 (1999).\n- [7] E. Ayon-Beato and A. Garcia, Gen. Rel. Grav. 37 , 635 (2005).\n- [8] I. Dymnikova, Class. Quant. Grav. 21 , 4417 (2004).\n- [9] S. W. Hawking and G. F. R. Ellis, 'The Large scale structure of space-time,' Cambridge University Press, Cambridge, 1973.\n- [10] S. M. Carroll, 'Spacetime and geometry: An introduction to general relativity,' San Francisco, USA: Addison-Wesley (2004).\n- [11] O. B. Zaslavskii, Phys. Lett. B 688 , 278 (2010).\n- [12] S. Ansoldi, P. Nicolini, A. Smailagic and E. Spallucci, Phys. Lett. B 645 , 261 (2007).\n- [13] I. Dymnikova, Gen. Rel. Grav. 24 , 235 (1992).\n- [14] P. Nicolini, A. Smailagic and E. Spallucci, Phys. Lett. B 632 , 547 (2006).\n- [15] S. A. Hayward, Phys. Rev. Lett. 96 , 031103 (2006).\n- [16] K. A. Bronnikov, Phys. Rev. D 63 , 044005 (2001).\n- [17] A. Burinskii and S. R. Hildebrandt, Phys. Rev. D 65 , 104017 (2002).\n- [18] N. Breton, Gen. Rel. Grav. 37 , 643 (2005).\n- [19] S. Bose and N. Dadhich, Phys. Rev. D 60 , 064010 (1999).\n- [20] L. Balart, Phys. Lett. B 687 280 (2010).\n- [21] I. H. Salazar, A. Garcia and J. Plebanski, J. Math. Phys. 28 2171 (1987).\n- [22] Leonardo Balart and Elias C. Vagenas, paper in preparation."}

2013A&A...558A..59S

Massive black hole factories: Supermassive and quasi-star formation in primordial halos

2013-01-01

['cosmology theory', '-', '-', '-', '-', 'stars fundamental parameters', '-']

[]

Context. Supermassive stars and quasi-stars (massive stars with a central black hole) are both considered as potential progenitors for the formation of supermassive black holes. They are expected to form from rapidly accreting protostars in massive primordial halos. <BR /> Aims: We explore how long rapidly accreting protostars remain on the Hayashi track, implying large protostellar radii and weak accretion luminosity feedback. We assess the potential role of energy production in the nuclear core, and determine what regulates the evolution of such protostars into quasi-stars or supermassive stars. <BR /> Methods: We followed the contraction of characteristic mass shells in rapidly accreting protostars, and inferred the timescales for them to reach nuclear densities. We compared the characteristic timescales for nuclear burning with those for which the extended protostellar envelope can be maintained. <BR /> Results: We find that the extended envelope can be maintained up to protostellar masses of 3.6 × 10<SUP>8</SUP> ṁ<SUP>3</SUP> M<SUB>⊙</SUB>, where ṁ denotes the accretion rate in solar masses per year. We expect the nuclear core to exhaust its hydrogen content in 7 × 10<SUP>6</SUP> yr. If accretion rates ṁ ≫ 0.14 can still be maintained at this point, a black hole may form within the accreting envelope, leading to a quasi-star. Alternatively, the accreting object will gravitationally contract to become a main-sequence supermassive star. <BR /> Conclusions: Due to the limited gas reservoir in typical 10<SUP>7</SUP> M<SUB>⊙</SUB> dark matter halos, the accretion rate onto the central object may drop at late times, implying the formation of supermassive stars as the typical outcome of direct collapse. However, if high accretion rates are maintained, a quasi-star with an interior black hole may form.

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https://arxiv.org/pdf/1305.5923.pdf

{'Massive black hole factories: Supermassive and quasi-star formation in primordial halos': 'Dominik R.G. Schleicher 1 , Francesco Palla 2 , Andrea Ferrara 3 , 4 , Daniele Galli 2 , and Muhammad Latif 1 \n- 1 Institut für Astrophysik, Georg-August-Universität Göttingen, Friedrich-Hund-Platz 1, D-37077 Göttingen, Germany e-mail: [email protected], [email protected]\n- 2 INAF-Osservatorio Astrofisico di Arcetri, Largo Enrico Fermi 5, I - 50125 Firenze, Italy\n- e-mail: [email protected], [email protected]\n- 3 Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy\n- e-mail: [email protected]\n- 4 Kavli IPMU (WPI), the University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, 277-8583, Japan \nSeptember 13, 2018', 'ABSTRACT': 'Context. Supermassive stars and quasi-stars (massive stars with a central black hole) are both considered as potential progenitors for the formation of supermassive black holes. They are expected to form from rapidly accreting protostars in massive primordial halos. \nAims. We explore how long rapidly accreting protostars remain on the Hayashi track, implying large protostellar radii and weak accretion luminosity feedback. We assess the potential role of energy production in the nuclear core, and determine what regulates the evolution of such protostars into quasi-stars or supermassive stars. \nMethods. We follow the contraction of characteristic mass scales in rapidly accreting protostars, and infer the timescales for them to reach nuclear densities. We compare the characteristic timescales for nuclear burning with those for which the extended protostellar envelope can be maintained. \nResults. We find that the extended envelope can be maintained up to protostellar masses of 3 . 6 × 10 8 ˙ m 3 M /circledot , where ˙ m denotes the accretion rate in solar masses per year. We expect the nuclear core to exhaust its hydrogen content in 7 × 10 6 yr. If accretion rates ˙ m /greatermuch 0 . 14 can still be maintained at this point, a black hole may form within the accreting envelope, leading to a quasi-star. Alternatively, the accreting object will gravitationally contract to become a main-sequence supermassive star. \nConclusions. Due to the limited gas reservoir in typical 10 7 M /circledot dark matter halos, the accretion rate onto the central object may drop at late times, implying the formation of supermassive stars as the typical outcome of direct collapse. However, if high accretion rates are maintained, a quasi-star with an interior black hole may form.', '1. Introduction': "Supermassive black holes (SMBHs) with more than 10 9 M /circledot have been observed at z > 6 (Fan et al. 2001; Fan et al. 2003, 2004, 2006), and recently even at z = 7 . 085 (Mortlock et al. 2011). While theoretical considerations indicate a variety of potential pathways, including direct collapse to a black hole, collapse to a supermassive star, collapse of stellar clusters or clusters of black holes (Rees 1984; Haiman 2006; Volonteri & Bellovary 2012), substantial efforts have been made to assess the validity of such scenarios. As the most straightforward possibility, one could imagine these SMBHs to originate from remnants of the first stars, which were proposed to be very massive with 100 -1000 M /circledot (Abel et al. 2002; Bromm & Larson 2004). Stellar evolution calculations by Heger & Woosley (2002) indeed show that black holes may form for stellar masses between 10 -30 M /circledot , or 100 -300 M /circledot , thus yielding the seeds for further growth to SMBHs. \nNew calculations following the evolution of primordial star formation beyond the first peak however show that fragmentation can occur efficiently (Stacy et al. 2010; Turk et al. 2009; Clark et al. 2011; Greif et al. 2011; Smith et al. 2011; Stacy et al. 2012). While the effect of the accretion luminosity on the protostellar core seems to \nbe minor (Smith et al. 2011, 2012), UV feedback during the main sequence phase appears to set the upper limit to the stellar mass to 50 -100 M /circledot (Hosokawa et al. 2011, 2012b; Susa 2013). Even if very massive stars could form, the resulting HII region is likely to inhibit accretion for at least 10 8 yrs (Milosavljević et al. 2009a,b). An additional problem has been identified by Whalen & Fryer (2012), who showed that black holes resulting from such stars would be born with strong dynamical kicks, implying that they should readily be expelled from the halo. \nOn the other hand, dense stellar clusters may form at high redshift in the presence of trace amounts of dust (Schneider et al. 2003; Omukai et al. 2005; Clark et al. 2008; Omukai et al. 2008; Dopcke et al. 2011; Schneider et al. 2011; Dopcke et al. 2012; Klessen et al. 2012; Omukai 2012; Schneider et al. 2012). The relativistic instability of such clusters has been explored in a semianalytic framework by Devecchi & Volonteri (2009) and Devecchi et al. (2010, 2012), finding characteristic black hole masses of up to ∼ 3000 M /circledot . The latter is already substantially higher than the masses of the first stars, though it may still be difficult to obtain masses as high as 10 9 M /circledot at z = 7 . In the presence of a significant black hole spin, which will be rapidly obtained in the presence of efficient accretion, one typically expects an additional growth by \nabout four orders of magnitude, thus requiring seed masses of the order 10 5 M /circledot (Shapiro 2005). \nThe formation of such massive seeds has been considered in the context of the direct collapse model, where gas in a 10 7 -10 8 M /circledot halo is expected to collapse without fragmentation into a single central object (Koushiappas et al. 2004; Begelman et al. 2006; Lodato & Natarajan 2006; Spaans & Silk 2006; Volonteri et al. 2008; Begelman & Shlosman 2009). In order to avoid fragmentation, molecular hydrogen needs to be efficiently dissociated, requiring a strong ambient UV field (Omukai 2001; Bromm & Loeb 2003; Shang et al. 2009; Schleicher et al. 2010; Latif et al. 2011; Van Borm & Spaans 2013). Such radiation backgrounds may have been provided by nearby starburst galaxies, which may indeed occur frequently enough to explain the observed abundance of SMBHs at z ∼ 6 (Dijkstra et al. 2008; Agarwal et al. 2012). \nUsing numerical simulations, Wise et al. (2008) have modeled the gravitational collapse of massive primordial halos cooling via atomic hydrogen, reporting an isothermal density profile and angular momentum transport via bar-like instabilities during the formation of the first peak. Regan & Haehnelt (2009) presented the first study following the evolution beyond the first peak, and modeling the formation of self-gravitating disks on parsec-scales. While these studies employed a typical resolution of 16 cells per Jeans length, it was recently demonstrated that turbulent structures can only be resolved with a resolution of at least 32 cells per Jeans length, preferably more (Sur et al. 2010; Federrath et al. 2011; Turk et al. 2012). In a set of highresolution simulations following the formation of the first peak, Latif et al. (2013a,b) found extended turbulent structures in the center of massive primordial halos, but no signs of simple disks or bar-like instabilities. Disks do however form at later stages of the evolution, with masses of ∼ 1000 M /circledot on scales of 30 AU, and characteristic accretion rates of ∼ 1 M /circledot yr -1 (Latif et al. 2013c). As a result, very massive central objects may indeed form in rather short cosmic times. The dynamics at early times have also been explored by Choi et al. (2013), finding a somewhat different result including the formation of toroidal structures on scales of several parsec. It is however not fully clear to which extent the latter is a result of the simplified initial conditions employed in their calculation. In order to provide more quantitative predictions regarding the conditions where black holes may form, Prieto et al. (2013) explored the correlations of the baryon spin with the spin of the dark matter, showing that the resulting correlation is however weak, and that the baryon properties cannot be naively extrapolated from the dark matter. \nWhile the accumulation of high masses seems feasible from a hydrodynamical point of view, the resulting object is still considerably more uncertain. As a first step, we expect the formation of a massive protostars, as the gas becomes optically thick at high densities (Omukai & Palla 2001, 2003; Hosokawa et al. 2012a). It is however unclear how these objects are going to evolve, and whether they will form a supermassive star or a quasi-star. While a supermassive star denotes a conventional star with very high masses of 10 3 -10 6 M /circledot (Shapiro & Teukolsky 1986), a quasi-star refers to an object with similar mass, but where the central core has collapsed into a low-mass black hole (Begelman et al. 2006; Begelman 2010). The latter then \naccretes mass from the stellar envelope, while the quasi-star as a whole may accrete at a rate larger than the Eddington rate of the black hole. A central question however concerns the conditions under which we expect the formation of a quasi-star as opposed to a supermassive star, and which of these objects should be expected as the generic outcome in the context of the direct collapse model. Also the properties of these objects are of high interest, as the amount of stellar feedback may influence the final accretion rates. \nIn order to explore that question, Hosokawa et al. (2012a) recently followed the evolution of rapidly accreting protostars, showing that they expand as cool supergiants, thus inhibiting the feedback from accretion luminosity and potentially allowing accretion to proceed for very long times. Hosokawa et al. also found that nuclear burning starts at stellar masses of about ∼ 50 M /circledot , and is maintained until the end of their calculation at 10 3 M /circledot . Therefore, their model indicates no transition towards a quasi-star at least during the evolutionary phase considered. It is however important to assess how long this phase of efficient accretion can be maintained, and under which conditions a transition to a supermassive main-sequence star can be expected, which can potentially influence the accretion flow via radiative feedback (Omukai & Inutsuka 2002; Johnson et al. 2011, 2012). While the majority of such supermassive stars directly collapses into a black hole (Fryer & Heger 2011), a small mass window exists around 55000 M /circledot where violent supernova explosions of up to 10 55 erg may occur (Johnson et al. 2013). \nFollowing a different approach, Begelman et al. (2006) and Begelman (2010) have proposed the formation of a black hole in the interior of rapidly accreting objects, leading to quasi-stars as the progenitors of SMBHs. Begelman et al. (2006) argued that for quasi-stars with 10 4 -10 5 M /circledot , the typical accretion timescale is considerably shorter than the timescale for nuclear burning, and as a result, the latter may not be able to stop the collapse of the central core. A potential advantage of such a configuration is that the accretion onto the central object is not limited by the Eddington accretion rate of the black hole, but by the Eddington accretion rate of the more massive quasi-star (Begelman et al. 2008). Begelman (2010) considers objects with more than 10 6 M /circledot , which first evolve towards the main sequence as a supermassive star, but then form a black hole after an extended phase of nuclear burning. Ball et al. (2011) followed the evolution of black holes in a quasi-star employing the Cambridge STARS stellar evolution package (Eggleton 1971; Pols et al. 1995), finding that the black hole may efficiently accrete 10% of the stellar mass before hydrostatic equilibrium breaks down. They also report that the results are sensitive to the boundary condition in the interior. The contraction of an embedded isothermal core in a stellar envelope was further explored by Ball et al. (2012) in the framework of the Schönberg-Chandrasekhar limit. \nWhile the models for the quasi-stars typically employed a Thompson-scattering opacity, Hosokawa et al. (2012a) reported that H -rather than Thomson scattering dominates the opacity in the protostellar atmosphere. They find luminosities close to the Eddington luminosity, \nL Edd = 4 πGMm p c σ T = 3 . 8 × 10 4 L /circledot ( M M /circledot ) , (1) \nwhere G is the gravitational constant, m p the proton mass, c the speed of light, σ T the Thomson scattering cross sec- \nion and M the mass of the star. The temperatures in the atmosphere are rather cool, ∼ 5000 K, as the protostars are on the Hayashi track. As a result, the characteristic radii evolve as \nR ini = 2 . 6 × 10 2 R /circledot ( M M /circledot ) 1 / 2 . (2) \nThe protostars are thus considerably more extended and show an increasing stellar radius as a function of mass, while the models of Begelman et al. (2006) and Begelman (2010) indicate a constant radius as a function of stellar mass. The difference is crucial, as the stellar radius regulates the temperature on the surface, and thus the strength of protostellar feedback. In fact, the behavior reported by Hosokawa et al. (2012a) is well-known in cases where the accretion timescale, \nt acc = M ˙ M , (3) \nis much smaller than the Kelvin-Helmholtz timescale, \nt KH = GM 2 RL . (4) \nHowever, Hosokawa et al. (2012a) report that this behavior extends into the regime where t KH < t acc , i.e. where Kelvin-Helmholtz contraction is faster than mass growth via accretion. The physical mechanism which allows this phase to continue will be discussed in detail in section 2. Taking the maximum luminosity of the star \nL max ∼ 0 . 6 L /circledot ( M M /circledot ) 11 / 2 ( R R /circledot ) -1 / 2 , (5) \nwhich assumes Kramer's opacity κ ∝ ρT -3 / 5 (Hayashi et al. 1962) and an initial radius as given in Eq. (2), they show that both timescales are equal at protostellar masses of \nM eq = 14 . 9 M /circledot ( ˙ m 0 . 01 ) 0 . 26 , (6) \nwhere we parametrized the accretion rate as ˙ M ≡ ˙ m M /circledot yr -1 . On the other hand, the accreting star remains on the Hayashi track up to stellar masses of 1000 M /circledot , the highest mass reached in the calculation of Hosokawa et al. (2012a). The extended envelope locked at T eff ∼ 5000 K allows efficient accretion with only moderate feedback for a longer period. \nIn this paper, we aim to assess how long this efficient accretion phase can be maintained beyond M ∼ 1000 M /circledot without strong feedback from the protostar. In addition, we will discuss the potential impact of nuclear burning for the evolution of the accreting objects, and the conditions under which a quasi-star as opposed to a supermassive star may form. For this purpose, we explore the interplay of mass accretion and Kelvin-Helmholtz contraction in section 2. We calculate the impact of nuclear burning in section 3. A final discussion of our results is provided in section 4. The impact of additional processes such as deuterium shell burning and hydrogen burning via the pp-chain is assessed in the appendix. \nFig. 1. The enclosed mass as a function of radius in the protostar calculated from Eq. (10), assuming a total stellar mass of 10 5 M /circledot and mass accretion rates ˙ m = 0 . 1 , 1 ., 10 . The current time t is calculated as t = ( M/M /circledot ) / ˙ m . For a given mass shell, larger accretion rates imply larger radii, as the star had less time to contract. \n<!-- image -->", '2. The interplay of mass accretion and Kelvin-Helmholtz contraction': 'In the following, we will sketch why the protostar may maintain large envelopes even if the Kelvin-Helmholtz timescale becomes shorter than the timescale for accretion. For this purpose, we consider mass shells of enclosed mass M , located at radii R ( M,t ) . A mass shell M forms at the time M/ ˙ M , with an initial radius as given by Eq. (2). The radii of these mass shells evolve on their Kelvin-Helmholtz timescale, given as \nt KH ( M,R ) = GM 2 R ( M,t ) L Edd . (7) \nAs a result, we have \ndR dt = -R ( M,t ) t KH ( M,R ) . (8) \nThe equation is integrated between the initial radius R ini when the mass shell forms, given in (2), corresponding to the time t ini ( M ) = M/ ˙ M , to the current radius R at time t . The integration yields \n1 R = 1 R ini + 4 πm p c σ T M ( t -t ini ( M )) . (9) \nInserting the initial radius as given in Eq. (2) and expressing the result in astrophysical units, we obtain \n1000 R /circledot R = 1000 R /circledot 2 . 6 × 10 2 ( M/M /circledot ) 1 / 2 + 1 . 04 yr -1 ( t -t ini ( M )) M/M /circledot . (10) \nThe first term on the right-hand side thus dominates right after the formation of a mass shell and determines its initial radius, while the second term describes its evolution due to Kelvin-Helmholtz contraction. It is remarkable that at late \ntimes, the second term dominates and the evolution of the radii appears to be independent of their initial position. This can be understood, as our expression for the KelvinHelmholtz timescale, Eq. (7), scales with the inverse radius of the mass shell. As a result, the evolution slows down during the contraction, implying that the Kelvin-Helmholtz timescale may increase significantly in the interior. An example for the resulting structure in a 10 5 M /circledot star is given in Fig. 1, including both the regime where the first and the second term of the equation dominate. \nWe now aim to estimate when a given mass shell reaches the densities of nuclear burning, which we take as ρ nuc ∼ 1 g cm -3 (Hosokawa et al. 2012a). We obtain a radius of nuclear burning, which is given as \nR nuc = ( 3 M 4 πρ nuc ) 1 / 3 ∼ 1 . 2 R /circledot ( M M /circledot ) 1 / 3 . (11) \nA comparison with Eq. (2) shows that this radius always remains smaller than the radius of the star by at least a factor of 1000 , and in fact increases more gradually with stellar mass. A given mass shell M will only reach nuclear densities on timescales much longer than the initial KelvinHelmholtz timescale t KH , which increases as R -1 . We can thus determine the time when nuclear densities are reached by equating (11) with the second term in (10). As a result, we obtain \n∆ t = t -t ini ( M ) = 710 yrs ( M M /circledot ) 2 / 3 . (12) \nDuring the time interval ∆ t , the star will accrete an additional mass ∆ M = ˙ M ∆ t . Considering a given mass shell, we can calculate the ratio \n∆ M M = ˙ M ∆ t M ∼ 710 ˙ m ( M M /circledot ) -1 / 3 , (13) \nwhich describes how much additional mass is accreted before the shell reaches nuclear densities. The ratio drops below 1 only for protostellar masses of \nM ≥ 3 . 6 × 10 8 ˙ m 3 M /circledot . (14) \nWhile for typical accretion rates of ˙ m ∼ 10 -3 , this happens already at mass scales of order unity, this transition occurs only at substantially larger masses above 1000 M /circledot for ˙ m > 10 -2 . While a given shell evolves towards nuclear densities, the amount of additional mass that is accreted is thus substantially higher than the mass in that shell. From Eq. (12), we can further derive the mass in the nuclear core in the limit that t /greatermuch t ini ( M ) . We obtain \nM nuc = ( t 710 yr ) 3 / 2 M /circledot = t 3 / 2 710 M /circledot , (15) \nwhere we defined t 710 = t/ (710 yr ) . As we neglected the term M/ ˙ M in Eq. (12), we note that the derivation here implicitly assumes a sufficient mass supply to the protostar in order to feed the core. Clearly, this approximation will break down at the mass scale derived in (14). At that point, both the protostellar mass as well as the mass in the nuclear core may become approximately constant, and we expect a transition towards supermassive main sequence stars. \nFig. 2. The ratio of nuclear luminosity provided via the CNO-cycle over the Eddington luminosity of the star (A.5) as a function of protostellar mass for different mass accretion rates. The CNO luminosity is calculated here as the minimum of the estimate in Eq. (24 and the Eddington luminosity of the nuclear core (25). At high protostellar masses, one can clearly recognize the transition to the regime where nuclear burning is limited by the Eddington luminosity of the core. Larger accretion times imply that less time was available for contraction, thus reduced nuclear luminosities. \n<!-- image --> \nFrom this model, we already obtain a set of relevant conclusions concerning the evolution of the protostar. In particular, if the accretion rate is high, a substantial amount of additional matter is accreted before a given shell reaches nuclear densities. As a result, the evolution of the protostar is dominated not by the interior shells, but by the additional matter which is accreted during the contraction. This behavior will only change when the timescale of accretion becomes comparable to the timescale on which the outer mass shell is able to reach nuclear densities. As we show here, the latter is significantly larger than the Kelvin-Helmholtz timescale of the star, and scales with the R -1 nuc . The transition occurs when a critical mass scale of 3 . 6 × 10 8 ˙ m 3 M /circledot is reached, implying a transition towards a supermassive main sequence star. We note that the transition may even occur earlier if ˙ m decreases with time.', '3. The nuclear evolution': 'In this section, we will assess the potential role of nuclear burning. For this purpose, we show that the transition to the CNO-cycle rapidly occurs and regulates the production of helium in the nuclear core. We will then calculate under which conditions the fuel in the nuclear core will be exhausted while the protostar maintains its extended envelope. The latter implies the potential formation of a black hole in the interior, thus a transition into a quasi-star.', '3.1. Importance of the CNO-cycle': 'While the pp-cycle will dominate in the very beginning (see appendix), we show here that the transition to the CNOcycle rapidly occurs as a result of helium burning via the tripleα process. Following Padmanabhan (2000), the en- \nrgy production rate via tripleα is given as \n/epsilon1 3 α = 5 . 1 × 10 8 ρ 2 Y 3 T 3 9 e -4 . 4027 /T 9 erg g -1 s -1 , (16) \nwhere Y ∼ 0 . 25 denotes the mass fraction of helium and T 9 denotes the gas temperature in units of 10 9 K. It is straightforward to check that the energy production rate changes by several orders of magnitude for temperature changes of the order 10% . As shown by Hosokawa et al. (2012a), the highest temperatures with T 9 ∼ 0 . 15 are expected in the center of the nuclear core, which has an enhanced nuclear density of 10 g cm -3 . With a volume filling factor /epsilon1 fill ∼ 0 . 01 -0 . 1 , the contribution of that region to helium burning is significant due to the steep temperature dependence. For the following estimates, we will assume that efficient mixing occurs throughout the nuclear core, implying that the heavy elements produced in the central region will be available throughout the core. On the other hand, if mixing is inefficient, nuclear burning via the CNO cycle will dominate even more in the central region due to its higher metallicity. \nNoting that the energy released by a single tripleα reaction corresponds to 1 . 166 × 10 -5 erg, the heavy element production rate per unit volume is given as \nn 3 α = /epsilon1 3 α 1 . 166 × 10 -5 erg . (17) \nThe CNO mass in the core thus evolves as \n˙ M CNO = ˙ n 3 α /epsilon1 fill M nuc × 12 m p , (18) \nand an integration yields \nM CNO = 7 . 1 × 10 -17 /epsilon1 fill t 5 / 2 710 M /circledot . (19) \nAssuming efficient mixing, the resulting metallicity in the nuclear core is then \nZ = M CNO M nuc = 6 . 3 × 10 -9 t 710 /epsilon1 fill . (20) \nWe recall that the energy production rate in the CNO cycle is given as (Padmanabhan 2000) \n/epsilon1 CNO = 4 . 4 × 10 25 ρXZ T 2 / 3 9 e -15 . 228 /T 1 / 3 9 erg g -1 s -1 . (21) \nSince the energy production rate depends sensitively on the temperature, we expect its contribution in the innermost core to be dominant. For the temperature T 9 = 0 . 15 , it is straightforward to show that energy production via the CNO cycle becomes comparable to the pp cycle for a metallicity of Z c = 2 × 10 -12 . A comparison with Eq. (20) shows that the CNO cycle thus dominates after a short time of \nt CNO = 2 . 25 /epsilon1 -1 fill , -1 yr , (22) \nwhere we introduced /epsilon1 fill = 0 . 1 /epsilon1 fill , -1 . We note here that this timescale is probably not accurate, as our assumptions in section 2 (in particular concerning the Eddington luminosity) only become valid at later times. Nevertheless, this result implies that the CNO cycle can be expected to be relevant early on, and thus needs to be considered. As the production of elements becomes increasingly efficient at late times, we expect that the expression (20) will yield a reasonable estimate in the period of interest. \nWith the above assumptions, the CNO luminosity due to nuclear burning is given as \nL CNO = /epsilon1 CNO ,Z c ( Z Z c ) M nuc /epsilon1 fill , (23) \nwhere /epsilon1 CNO ,Z c denotes the energy production rate via the CNO cycle evaluated at the critical metallicity Z c . Inserting (20), we obtain \nL CNO = 1 . 3 × 10 5 /epsilon1 2 fill t 5 / 2 710 L /circledot . (24) \nThe Eddington luminosity of the nuclear core is given as \nL Edd , core = 3 . 8 × 10 4 ˙ mt 3 / 2 710 L /circledot . (25) \nThe ratio of these luminosities to the Eddington luminosity of the star is given in Fig. (2). Even here, the luminosity produced by nuclear burning approaches the Eddington luminosity of the star only around stellar masses of 10 8 M /circledot , implying no relevant impact on the stellar evolution during the earlier stages. A comparison of these expressions yields the timescale \nt c = 2 . 1 × 10 4 /epsilon1 -2 fill , -1 yr . (26) \nFor /epsilon1 fill , -1 ∼ 1 , as indicated by Hosokawa et al. (2012a) for ˙ m = 0 . 1 , the CNO luminosity would exceed the Eddington luminosity of the core after ∼ 2 × 10 4 yr, shortly after the end of their simulations. However, when the Eddington luminosity is reached, one would expect an expansion of the nuclear core, implying lower densities and an adiabatically decreasing temperature. Due to this thermostat, it is likely that the core will adjust to a state maintaining its Eddington luminosity. \nWe will now demonstrate that the nuclear core is not converted to helium before the critical timescale t c after which the core radiates at its Eddington luminosity. We consider the production of helium by the CNO process, given as \n˙ N He , CNO = L CNO 4 . 3 × 10 -5 erg , (27) \nand the helium mass production rate \n˙ M He , CNO = 12 m p × ˙ N He , CNO . (28) \nAn integration yields \nM He , CNO = 1 . 0 × 10 -3 /epsilon1 2 fill t 7 / 2 710 M /circledot . (29) \nA comparison with the mass of the core indicates that they become comparable at \nt He , CNO = 2 . 25 × 10 5 /epsilon1 -1 fill , -1 yr , (30) \nimplying that a pure helium core will only be formed once the nuclear luminosity is equal to the Eddington luminosity of the core. \nFig. 3. The fraction of helium in the nuclear core, corresponding to the amount of exhausted fuel for nuclear burning, as a function of protostellar mass for different accretion rates. Higher accretion rates imply less time for protostellar contraction. As a result, nuclear densities are reached later, implying less time for nuclear burning. \n<!-- image -->', '3.2. Formation of a helium core': 'We will now calculate the helium production rate under the assumption that the formation of a helium core occurs once the nuclear luminosity equals the Eddington luminosity of the core. Thus, \n˙ N He = L Edd , core 4 . 3 × 10 -5 erg . (31) \nThe helium mass production rate is thus given as \n˙ M He = 4 m p × ˙ N He (32) \nand an integration yields \nM He = 1 . 05 × 10 -4 t 5 / 2 710 M /circledot . (33) \nUnder this assumption, the nuclear core becomes a pure helium core after t f /similarequal 6 . 8 × 10 6 yr. We note that at this point, the mass of the nuclear core is given as M f /similarequal 9 . 4 × 10 5 M /circledot . In order to illustrate how the latter translate into total stellar masses, the ratio of helium mass to total mass in the core is displayed in Fig. 3 for different accretion rates, showing that the hydrogen fuel is indeed exhausted for stellar masses of 3 × 10 5 -3 × 10 7 M /circledot depending on the accretion rate. After that point, the burning of heavier elements will occur, but only last for a short period. As the mass of the core is considerably larger than the Chandrasekhar mass scale and the Tolman-Oppenheimer-Volkoff mass scale, the core may collapse and form a black hole within the star, a so-called quasi-star, as proposed by Begelman et al. (2006) and Begelman (2010). The interior black hole may then continue accreting from the stellar envelope, while the accretion rate of the quasi-star will only be limited by the Eddington rate corresponding to the total mass of the configuration. The derivation of t f and M f however implicitly assumes that enough mass has been accreted in order to build up the core, requiring that \nM f /lessmuch t f ˙ M. (34) \nIn order to form a quasi-star, we thus require an accretion rate of ˙ M /greatermuch M f /t f , i.e. \n˙ m /greatermuch 0 . 14 . (35) \nFor lower accretion rates, we expect that the extended envelope of the star cannot be maintained at late times, implying that the star will contract and form a supermassive main sequence star. During this period, both the stellar mass and the mass in the nuclear core will be approximately constant, implying a characteristic hydrogen burning timescale of 2 × 10 6 yr. For stellar masses above 300 M /circledot , these are expected to collapse to massive black holes (Fryer et al. 2001; Fryer & Heger 2011).', '4. Discussion and outlook': "To investigate the interplay of accretion and KelvinHelmholtz-contraction of the protostar, we have presented an analytical model following the evolution of the mass shells from the extended envelope down to nuclear densities. Based on this model, we are able to explain the fundamental result by Hosokawa et al. (2012a) that the accreting protostar keeps its extended envelope beyond the adiabatic accretion phase. The reason is that the timescale on which a given mass shell may reach nuclear densities is much longer than the Kelvin-Helmholtz timescale of the star, as the protostellar radius is considerably larger than the radius of the nuclear core. Instead of the Kelvin-Helmholtz timescale evaluated at the protostellar radius, the relevant timescale is thus the Kelvin-Helmholtz timescale at the expected core radius after contraction, providing an appropriate estimate of the evolutionary timescale. As a consequence, rapidly accreting protostars are expected to keep their extended envelopes until stellar masses of 3 . 6 × 10 8 ˙ m 3 M /circledot . At this point, the accretion timescale becomes longer than the Kelvin-Helmholtz timescale in the stellar interior, implying that the star will subsequently contract towards the main sequence. Then, we expect it to follow the typical evolution of a supermassive (proto)star (Shapiro & Teukolsky 1986). \nWe further assessed the impact of nuclear burning for the evolution of these stars. While deuterium burning will never be relevant during the protostellar evolution, hydrogen burning via the pp-cycle could potentially affect the stellar evolution after 10 15 years, which is however practically irrelevant. However, due to the efficient metal production via the tripleα process, a transition to the CNO cycle is expected early on. The energy production by the CNO cycle is considerably more efficient, implying that the core can radiate at its Eddington luminosity. As a result, a helium core forms after 7 × 10 6 yr. The expected mass of the core is then 10 6 M /circledot , which provides an upper limit on the initial mass of the interior black hole. Therefore, if the available gas reservoir is sufficient to maintain accretion rates ˙ m /greatermuch 0 . 14 until this point, the central core may collapse during the protostellar evolution phase, and the object becomes a quasi-star. Alternatively, the protostar will evolve towards the main sequence to become a supermassive star. Such an object is then expected to collapse into a supermassive black hole at the end of its lifetime (Fryer & Heger 2011). \nOur results have been derived based on a comparison of the mass accumulated by accretion with the characteristic mass scale where we expect a pure helium core. This mass \nscale is based on the reasonable assumption that the luminosity of the core is given by its Eddington luminosity, in agreement with the numerical results by Hosokawa et al. (2012a). Uncertainties are however present, as our calculation for the evolution of the core mass assumes that the luminosity of each mass shell is given by the Eddington luminosity. Following Hosokawa et al. (2012a), the latter is a good but not precise approximation. Further corrections can be expected in particular when approaching nuclear densities, when feedback from nuclear burning becomes significant. Additional effects could be introduced as a result of rotation, which was not included in this model. We thus expect our results to provide an order of magnitude estimate for the critical accretion rate, which may be determined more accurately employing stellar evolution calculations, but also a more realistic model for the time-dependent accretion rate. We also note here that Hosokawa et al. (2012a) reported a slight dependence of their results on the employed boundary conditions for the protostar. While their main calculations adopted shock boundary conditions, they also explored the effect of lowerentropy accretion provided by photospheric boundary conditions. In that case, a slightly higher accretion rate of 0 . 3 M /circledot yr -1 appears to be required to maintain the extended envelopes, but the overall evolution remains very similar. \nFor comparison, we note that recent simulations by Latif et al. (2013c) reported accretion rates of ∼ 1 M /circledot yr -1 in halos with ∼ 10 7 M /circledot . Adopting a constant accretion rate, the transition to a supermassive main sequence star would occur at a stellar mass of 3 . 6 × 10 8 M /circledot . As the nuclear fuel is however exhausted earlier, a central black hole may form before this transition, giving rise to an evolution as sketched by Ball et al. (2011). However, due to the limited gas reservoir, and as here the halo mass is in fact comparable to the mass scale of the corresponding quasistar, it seems likely that the accretion rate will substantially decrease at late times, implying that the transition towards the supermassive star should occur at an earlier stage which is then determined by the time evolution of the accretion rate, and that supermassive stars with 10 4 -10 5 M /circledot might be the most generic outcome of the collapse. More massive objects appear to be possible at least in principle, if a larger gas reservoir is available in more massive dark matter halos. \nIn order to assess the potential influence of rotation, we estimated the amount of rotational support in the central 10 3 M /circledot clumps reported by Latif et al. (2013c), which varied between 5 -20% in different simulations. With densities of ∼ 10 -10 g cm -3 , these are still orders of magnitudes below the characteristic densities within the protostar, such that we cannot yet draw strong conclusions regarding the final amount of rotational energy. However, we note that Stacy et al. (2012) reported a significant amount of rotation in primordial protostars based on numerical simulations of Greif et al. (2012), and a quite similar case can be expected here. As recently shown by Reisswig et al. (2013), the latter may have a substantial impact on the collapse of supermassive stars, implying the potential formation of a black hole binary and a subsequent merger, accompanied with efficient emission of gravitational waves. The latter provides a potential pathway of probing black hole formation scenarios with LISA 1 . In a narrow mass range around ∼ 55000 M /circledot , \none may further expect the occurence of highly energetic supernovae with energies up to 10 44 erg, which can be potentially detected with JWST 2 (Johnson et al. 2013). \nReturning to the fate of supermassive stars, their evolution in the presence of UV feedback was assessed by Omukai & Inutsuka (2002) and Johnson et al. (2012) in the case of spherical symmetry. These authors find that UV feedback is unable to stop accretion for rates above ∼ 0 . 1 M /circledot yr -1 . As a result, the expected stellar mass is \nM UV ∼ 10 3 M /circledot ( ˙ m 10 -3 ) 8 / 7 . (36) \nFor accretion rates of ∼ 0 . 1 M /circledot yr -1 , our results exceed this value, as the protostars remain on the Hayashi track up to a mass of ∼ 3 × 10 5 M /circledot , making feedback inefficient. Therefore, taking protostellar evolution into account favours the formation of more massive objects. In all cases, a critical question concerns the time evolution of the accretion rate, since the transition to supermassive stars is regulated by the accretion rate at late times. While this paper provides a first assessment for the case of constant accretion rates and spherical symmetry, the potential implications of timedependent accretion rates need to be addressed in more detail in the future, along with the implications of rotation during protostellar evolution. \nAcknowledgements. DRGS and ML thank for funding from the Deutsche Forschungsgemeinschaft (DFG) via the SFB 963/1 'Astrophysical flow instabilities and turbulence' (project A12). DRGS further acknowledges financial support via the Schwerpunktprogramm SPP 1573 'Physics of the Interstellar Medium' under grant SCHL 1964/1-1. DG and FP acknowledge the financial support of PRININAF 2010 'The Formation of Stars'.", 'References': "Abel, T., Bryan, G. L., & Norman, M. 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L. 2012, ApJ, 745, 154 \nVan Borm, C. & Spaans, M. 2013, ArXiv e-prints 1304.4057 Volonteri, M. & Bellovary, J. 2012, Reports on Progress in Physics, \n- 75, 124901 Volonteri, M., Lodato, G., & Natarajan, P. 2008, MNRAS, 383, 1079 Whalen, D. J. & Fryer, C. L. 2012, ApJ, 756, L19 Wise, J. H., Turk, M. J., & Abel, T. 2008, ApJ, 682, 745", 'Appendix A: Additional nuclear processes': 'In section 3.1, we have shown that the transition to the CNO cycle rapidly occurs, implying that the latter will regulate the formation of a helium core as discussed in section 3.2. However, additional nuclear processes are expected to simultaneously occur, which we discuss here for completeness. In the two sub-sections below, we assess the role of deuterium shell burning as well as the impact of hydrogen burning via the pp-change during the stellar evolution, showing that these processes will only have a minor impact on the stellar evolution.', 'Appendix A.1: Deuterium burning': 'As reported by Hosokawa & Omukai (2009), deuterium shell burning may occur even before nuclear burning starts in the core. The luminosity from deuterium shell burning is given as \nL D = 1 . 5 × 10 5 L /circledot ( ˙ m 0 . 1 )( [D / H] 2 . 5 × 10 -5 ) , (A.1) \nwhere [D/H] denotes the deuterium abundance relative to hydrogen. For an accretion rate of ˙ m = 0 . 1 , Hosokawa et al. (2012a) demonstrated that deuterium shell burning starts only at protostellar masses of ∼ 80 M /circledot , and presumably even higher masses in case of higher accretion rates. A comparison with the Eddington accretion rate in Eq. (1) thus shows that the luminosity from deuterium shell burning will never become dominant in this mass range, but in fact becomes increasingly less relevant for larger masses. \nIt is also clear that deuterium shell burning is not going to occur close to the outer surface. While the characteristic temperature for a given mass shell scales as T ∝ GM/R , we note that, from Eq. (10), GM/R ∝ t = M/ ˙ M at times t /greatermuch M/ ˙ M . In this regime, the central temperature is almost spatially constant, consistent with the results of Hosokawa et al. (2012a), but already considerably larger than the deuterium burning temperature of ∼ 10 6 K. Deuterium burning will thus occur when the first term in Eq. (10) is still relevant. From Fig. 1, it is evident that the radius of the shells changes considerably in that regime, while the mass in the shells is almost unchanged. Adopting a scaling relation of T ∝ GM/R with almost constant M implies that the radius has to change by 2-3 orders of magnitude for the atmospheric temperature of 5000 K to increase to a value above 10 6 K. We therefore expect that the radius of deuterium burning will correspond to a fixed fraction of the protostellar radius as long as the protostar maintains its bloated envelope. From a comparison with Hosokawa et al. (2012a) at a stellar mass of 1000 M /circledot , we obtain the normalization of this relation to be \nR D = 10 R /circledot ( M M /circledot ) 1 / 3 . (A.2) \nAs long as the protostar remains on the Hayashi track, the deuterium burning shell will not be able to move towards the atmosphere, and, therefore, it does not have an impact on the evolution of the protostar.', 'Appendix A.2: The pp-cycle': 'We consider first the energy production in the nuclear core via the pp-cycle, as the composition of the star is initially \nFig. A.1. The ratio of nuclear luminosity provided via the pp-cycle (A.4) over the Eddington luminosity of the star (A.5) as a function of protostellar mass for different mass accretion rates. Larger accretion times imply that less time was available for contraction, thus reduced nuclear luminosities. \n<!-- image --> \nprimordial. As shown by Hosokawa et al. (2012a), the typical temperature in the core is ∼ 10 8 K, and the nuclear density ∼ 1 g cm -3 . We adopt here the expression of Padmanabhan (2000) for the energy production rate, noting that \n/epsilon1 pp = 2 . 4 × 10 4 ρX 2 T 2 / 3 9 e -3 . 38 /T 1 / 3 9 erg s -1 g -1 , (A.3) \nwhere T 9 denotes the temperature in units of 10 9 K and X ∼ 0 . 75 the mass fraction of hydrogen. While the ppcycle considered here has only a moderate temperature dependence, we note that the CNO cycle scales as T 20 around temperatures of 10 6 K, and the tripleα process as T 40 around temperatures of 10 8 K. For the latter cases, we will thus need to take into account the higher temperatures within the center of the core, as they substantially contribute to the energy production via nuclear burning. Now, with ρ nuc ∼ 1 g cm -3 and T 9 = 0 . 1 , we have /epsilon1 pp = 43 erg g -1 s -1 . The luminosity provided by the ppcycle is then given as \nL pp = /epsilon1 pp M nuc = 21 . 5 t 3 / 2 710 L /circledot . (A.4) \nFor comparison, the Eddington luminosity of the protostar is given as \nL Edd = 3 . 8 × 10 4 ˙ MtL /circledot = 2 . 7 × 10 7 ˙ mt 710 L /circledot . (A.5) \nThe ratio of these luminosities is given in Fig. (A.1), showing that it remains considerably smaller than unity for stellar masses up to 10 8 M /circledot . Equating the two expressions, we find that the luminosity resulting from the pp-burning is relevant only at very late times t pp = 1 . 1 × 10 15 yr . \nFor practical accretion rates of ˙ m = 10 -3 -10 and protostellar masses between 10 3 -10 8 M /circledot , the contribution from the pp-chain will never be relevant, and can thus be neglected for the overall evolution of the stars. \nWe now estimate the amount of helium produced in the core. During one fusion event, an average energy of \n4 . 3 × 10 -5 erg is released, implying a helium production rate of \n˙ N He , pp = L pp 4 . 3 × 10 -5 erg . (A.6) \nThe helium mass in the core thus evolves as \n˙ M He , pp = 4 m p × ˙ N He,pp . (A.7) \nAn integration yields \nM He , pp = 6 . 0 × 10 -8 t 5 / 2 710 M /circledot . (A.8) \nEquating the resulting mass with the total mass in the core, a helium core can be expected after a time of 1 . 2 × 10 10 yr. In summary, when only the pp-cycle is considered, nuclear burning is rather inefficient, and never going to have a significant impact on the evolution of the star. However, we will show in the next subsection that a transition to the CNO-cycle should be expected, and the implications of nuclear burning will then become more relevant.'}

2014ApJ...780...86E

Nuclear Star Formation Activity and Black Hole Accretion in Nearby Seyfert Galaxies

2014-01-01

['galaxies nuclei', 'galaxies seyfert', 'astronomy infrared', '-']

[]

Recent theoretical and observational works indicate the presence of a correlation between the star-formation rate (SFR) and active galactic nucleus (AGN) luminosity (and, therefore, the black hole accretion rate, \dot{M}_BH) of Seyfert galaxies. This suggests a physical connection between the gas-forming stars on kpc scales and the gas on sub-pc scales that is feeding the black hole. We compiled the largest sample of Seyfert galaxies to date with high angular resolution (~0.''4-0.''8) mid-infrared (8-13 μm) spectroscopy. The sample includes 29 Seyfert galaxies drawn from the AGN Revised Shapley-Ames catalog. At a median distance of 33 Mpc, our data allow us to probe nuclear regions on scales of ~65 pc (median value). We found no general evidence of suppression of the 11.3 μm polycyclic aromatic hydrocarbon (PAH) emission in the vicinity of these AGN, and we used this feature as a proxy for the SFR. We detected the 11.3 μm PAH feature in the nuclear spectra of 45% of our sample. The derived nuclear SFRs are, on average, five times lower than those measured in circumnuclear regions of 600 pc in size (median value). However, the projected nuclear SFR densities (median value of 22 M <SUB>⊙</SUB> yr<SUP>-1</SUP> kpc<SUP>-2</SUP>) are a factor of 20 higher than those measured on circumnuclear scales. This indicates that the SF activity per unit area in the central ~65 pc region of Seyfert galaxies is much higher than at larger distances from their nuclei. We studied the connection between the nuclear SFR and \dot{M}_BH and showed that numerical simulations reproduce our observed relation fairly well.

[]

https://arxiv.org/pdf/1311.0703.pdf

{'NUCLEAR STAR FORMATION ACTIVITY AND BLACK HOLE ACCRETION IN NEARBY SEYFERT GALAXIES': "Pilar Esquej 1,2,3 , Almudena Alonso-Herrero 2,4 , Omaira Gonz'alez-Mart'ın 5,6 , Sebastian F. Honig 7,8 , Antonio Hern'an-Caballero 2,3 , Patrick F. Roche 9 , Cristina Ramos Almeida 5,6 , Rachel E. Mason 10 , Tanio D'ıaz-Santos 11 , Nancy A. Levenson 12 , Itziar Aretxaga 13 , Jos'e Miguel Rodr'ıguez Espinosa 5,6 , Christopher Packham 14 \nDraft version June 5, 2018", 'ABSTRACT': "Recent theoretical and observational works indicate the presence of a correlation between the star formation rate (SFR) and the active galactic nuclei (AGN) luminosity (and, therefore, the black hole accretion rate, ˙ M BH ) of Seyfert galaxies. This suggests a physical connection between the gas forming stars on kpc scales and the gas on sub-pc scales that is feeding the black hole. We compiled the largest sample of Seyfert galaxies to date with high angular resolution ( ∼ 0 . 4 -0 . 8 '' ) mid-infrared (8-13 µ m) spectroscopy. The sample includes 29 Seyfert galaxies drawn from the AGN Revised Shapley-Ames catalogue. At a median distance of 33 Mpc, our data allow us to probe nuclear regions on scales of ∼ 65pc (median value). We found no general evidence of suppression of the 11.3 µ m polycyclic aromatic hydrocarbon (PAH) emission in the vicinity of these AGN, and used this feature as a proxy for the SFR. We detected the 11.3 µ m PAH feature in the nuclear spectra of 45% of our sample. The derived nuclear SFRs are, on average, five times lower than those measured in circumnuclear regions of 600 pc in size (median value). However, the projected nuclear SFR densities (median value of 22 M /circledot yr -1 kpc -2 ) are a factor of 20 higher than those measured on circumnuclear scales. This indicates that the SF activity per unit area in the central ∼ 65pc of Seyfert galaxies is much higher than at larger distances from their nuclei. We studied the connection between the nuclear SFR and ˙ M BH and showed that numerical simulations reproduce fairly well our observed relation. Subject headings: \nSubject headings: galaxies: nuclei - galaxies: Seyfert - infrared: galaxies", '1. INTRODUCTION': "One of the most important challenges in modern cosmology is to disentangle the physics behind the processes underlying galaxy formation and evolution. Observations over the past decades have revealed that supermassive black holes (SMBHs) likely reside at the centers of all galaxies with a bulge and that the properties of these black holes and their host galaxies are tightly correlated (e.g. Magorrian et al. 1998; Ferrarese & Merritt \n- 1 Centro de Astrobiolog'ıa, INTA-CSIC, Villafranca del Castillo, 28850, Madrid, Spain 2\n- 3 Departamento de F'ısica Moderna, Universidad de Cantabria, Avda. de Los Castros s/n, 39005 Santander, Spain\n- 2 Instituto de F'ısica de Cantabria, CSIC-Universidad de Cantabria, 39005 Santander, Spain\n- 4 Augusto G. Linares Senior Research Fellow\n- 5 Instituto de Astrof'ısica de Canarias (IAC), C/V'ıa L'actea, 38205, La Laguna, Spain\n- 6 Departamento de Astrof'ısica, Universidad de La Laguna (ULL), 38205, La Laguna, Spain\n- 7 UCSB Department of Physics, Broida Hall 2015H, Santa Barbara, CA, USA\n- 8 Institut fur Theoretische Physik und Astrophysik, ChristianAlbrechts-Universitat zu Kiel, Leibnizstr. 15, 24098 Kiel, Germany\n- 9 Department of Physics, University of Oxford, Oxford OX1 3RH, UK\n- 10 Gemini Observatory, Northern Operations Center, 670 N. A'ohoku Place, HI 96720, USA\n- 11 Spitzer Science Center, 1200 East California Boulevard, Pasadena, CA 91125, USA\n- 12 Gemini Observatory, Casilla 603, La Serena, Chile\n- 13 Instituto Nacional de Astrof'ısica, ' Optica y Electr'onica (INAOE), Aptdo. Postal 51 y 216, 72000 Puebla, Mexico\n- 14 Department of Physics and Astronomy, University of Texas at San Antonio, One UTSA Circle, San Antonio, TX 78249, USA \n2000; Gebhardt 2000; Kormendy & Ho 2013). The coevolution of galaxies and their corresponding SMBHs depends on some physical mechanism, referred to as feedback, that links accretion and ejection of gas residing on a sub-pc scale in galactic nuclei to the rest of the galaxy (Silk & Rees 1998; King 2010; Nayakshin & Zubovas 2012). The connection between star formation (SF) activity on different physical scales in a galaxy and the presence of an active galactic nucleus (AGN) has been a long discussed topic. However, there still are many uncertainties under consideration to disentangle the processes behind such a relation (see e.g. Hopkins & Quataert 2010, and references therein). \nIn the standard unification model, the powering mechanism of AGN is gas accretion onto a central SMBH. However, the physics of angular momentum transfer to the vicinity of the black hole is still unclear (see Alexander & Hickox 2012, for a recent review). Given that the angular momentum of inflowing gas produced by galaxy mergers or other large scale structures (e.g., bars) cannot be removed instantaneously, many studies proposed that the inflowing gas could form a circumnuclear disk where SF can take place. Kawakatu & Wada (2008, and references therein) put forward a model for such a circumnuclear disk, which might be coincident with the putative torus of the unification model of AGN (Antonucci 1993). This model predicts that SF would mostly take place in the outer parts of a 100 pc-size torus (Wada & Norman 2002). Cid Fernandes & Terlevich (1995) proposed the presence of a starburst in the obscuring torus as a solution for the absence of conspicuous broad lines in Seyfert 2s. \nThe starburst disk model of Thompson et al. (2005) estimates that most of the gas is supplied from outside the inner 200pc, but this is better suited for ultraluminous infrared galaxies due to the high star formation rates (SFRs) considered. Ballantyne (2008) presented an update of the Thompson et al. (2005) model with typical maximum SFRs of ∼ 1 M /circledot yr -1 , that could also potentially obscure the AGN. These nuclear pcsized starbursts will mostly be associated with low luminosity AGN (i.e. Seyferts and low-ionization nuclear emission-line regions - LINERs). From an observational point of view, nuclear starbursts have been detected in Seyfert 2 galaxies and LINERs using UV images obtained with the Hubble Space Telescope (Heckman et al. 1995; Gonz'alez Delgado et al. 1998; Colina et al. 2002). \nThe numerical simulations of Hopkins & Quataert (2010) predict a relation with some scatter between the SFR of the galaxy on different scales - going from 10kpc-scale to the central parsec - and the black hole accretion rate ( ˙ M BH ). This correlation appeared to be more prominent on smaller physical scales. However, simulations also indicate dynamical delays between the peaks of the SF and the BH growth (Hopkins 2012), in agreement with results from observational works (e.g., Davies et al. 2007; Wild et al. 2010; Ramos Almeida et al. 2013). \nMid-infrared (mid-IR) spectroscopy is a powerful tool to explore the nature of AGN and SF activity in galaxies. Among the most remarkable characteristics of the mid-IR spectra of galaxies is the presence of polycyclic aromatic hydrocarbons (PAH) emission, with the most prominent features being at 6.2, 7.7, 8.6, 11.3 and 17 µ m. They are due to the stretching and bending vibrations of aromatic hydrocarbon materials, where the shortest wavelength features are dominated by the smallest PAHs (e.g. Tielens 2010). This type of emission mostly originates in photo-dissociation regions where aromatic molecules are heated by the radiation field produced by young massive stars (Roche & Aitken 1985; Roche et al. 1991). Therefore, PAHs are often used as indicators of the current SFR of galaxies. Note that they can also be excited by UV emission from B stars and thus PAH emission probes SF over a few tens of million years (e.g., Peeters et al. 2004; D'ıaz-Santos et al. 2010). \nPAH features are detected in AGN, although they generally appear weak when compared with those of star forming galaxies (Roche et al. 1991). It has been proposed that the PAH molecules might be destroyed in the vicinity of an active nucleus due to the presence of a hard radiation field (Voit 1992a). There is also evidence that different PAHs might behave differently. Diamond-Stanic & Rieke (2010) showed that the 11.3 µ m PAH feature emission is a reliable indicator of the SFR in AGN, at least for Seyfert-like AGN luminosities and kpc scales, while the 6.2, 7.7, and 8.6 µ mfeatures appear suppressed. Sings of variations between the different features have been reported by many authors (e.g. Peeters et al. 2004; Galliano et al. 2008). For instance, Smith et al. (2007) found that the ratio of the PAH emission at 7.7 and 11.3 µ m is relatively constant among pure starbursts, while it decreases by up to factor of 5 for galaxies hosting a weak AGN. They interpreted this as a selective destruction of the smallest PAH carriers by the \nhard radiation arising from the accretion disk, ruling out the explanation in terms of ionization of the molecules (see also Siebenmorgen et al. 2004). \nA number of works have studied the SF activity using PAH emission and its relation to the AGN activity. Shi et al. (2007) demonstrated that the SF contribution increases from Palomar-Green QSO, to 2MASS QSO, and radio galaxies. Using measurements of the 7.7 and 11.3 µ m PAH using Spitzer /IRS data, they found higher SFRs for more intense nuclear activity, which indicates that the AGN selection technique influences the level of SF activity detected in the corresponding host galaxies. Watabe et al. (2008) investigated the nuclear vs. circumnuclear SF for a sample of Seyfert galaxies using ground-based observations of the 3.3 µ m PAH feature. Assuming that the this PAH traces the SF activity, they found that both SF and AGN activity are correlated (see also, Imanishi 2003; Imanishi & Wada 2004). Such a relation implied that SF in the inner region of the AGN (within a few hundred parsecs from the center) might have a greater influence on ˙ M BH . On the other hand, Mason et al. (2007) found weak or absent PAH emission in the central 20 pc of the Seyfert 1 galaxy NGC1097, whilst in the circumnuclear region, strong 3.3 and 11.3 µ m PAH bands were detected. In the case of NGC 1097, the absence of PAH emission may be related to destruction/ionization of PAH molecules by hard photons from the nuclear star cluster. \nDiamond-Stanic & Rieke (2012) recently found a strong correlation between the kpc-scale SF derived using the 11.3 µ mPAH feature and 24 µ mobservations for Seyfert galaxies. However, the limited angular resolution of their Spitzer data ( ∼ 4 -5 '' ) did not allow them to resolve nuclear ( ∼ 100 pc) scales, and it is unclear if the measured PAH feature is associated with the galaxy or to the nuclear environment. \nIn this work we compile a sample of 29 Seyfert galaxies from the revised Shapley-Ames (RSA) galaxy catalogue (Sandage & Tammann 1987) with published groundbased mid-IR high angular resolution spectroscopy obtained on 8m-class telescopes. At a median distance of 33Mpc, this sample allows us to study the nuclear SF activity around AGN on scales of ∼ 65pc. We also use mid-IR spectra taken with the Infrared Spectrograph (IRS, Houck et al. 2004) on board Spitzer for all objects in our sample to investigate the extended ( ∼ 600 pc) SF in the host galaxy. This enables us to study the relation SFR-˙ M BH at different scales in the local Universe. \nThis paper is structured as follows: Section 2 describes the sample selection and data analysis. In Section 3 we study the nuclear 11.3 µ m PAH feature emission. Section 4 compares the circumnuclear and nuclear SF activity and its relation with ˙ M BH . Finally, our conclusions are summarized in Section 5. Throughout this work we assumed a ΛCDM cosmology with (Ω M , Ω Λ ) = (0.3, 0.7) and H 0 = 70 km s -1 Mpc -1 .", '2.1. Sample': "Our sample (see Table 1) is drawn from the galaxymagnitude-limited RSA Seyfert sample, which includes the 89 Seyfert galaxies brighter than B T =13mag from Maiolino & Rieke (1995) and Ho et al. (1997). We \nTABLE 1 Sample properties \n| Object | D L (a) (Mpc) | b/a | Type | log L 2 - 10keV (b) (erg s - 1 ) | log L agn (c) (erg s - 1 ) | log( M BH ) ( M /circledot ) | Refs. |\n|--------------|-----------------|-------|--------|------------------------------------|------------------------------|--------------------------------|----------|\n| Circinus | 4.2 | 0.4 | Sy2 | 42.6 ( ∗ ) | 43.8 | 6.42 | (1) |\n| ESO 323-G077 | 65 | 0.7 | Sy1 | 42.7 | 43.9 | 7.40 | (2) |\n| IC 5063 | 49 | 0.7 | Sy2 | 42.8 | 44 | 7.74 | (1) |\n| Mrk 509 | 151.2 | 0.8 | Sy1 | 43.9 | 45.4 | 7.86 | (3,4,5) |\n| NGC 1068 | 16.3 | 0.9 | Sy2 | 43.0 ( ∗ ) | 44.3 | 7.59 | (1) |\n| NGC 1365 | 23.5 | 0.5 | Sy1 | 42.1 (d) | 43.1 | 8.20 | (6) |\n| NGC 1386 | 12.4 | 0.4 | Sy2 | 41.6 ( ∗ ) | 42.6 | 7.42 | (1) |\n| NGC 1808 | 14.3 | 0.6 | Sy2 | 40.4 | 41.2 | . . . | (7) |\n| NGC 2110 | 33.6 | 0.7 | Sy1 | 42.6 | 43.7 | 8.30 | (3,4,8) |\n| NGC 2992 | 33.2 | 0.3 | Sy1 | 43.1 | 44.4 | 7.72 | (8) |\n| NGC 3081 | 34.4 | 0.9 | Sy2 | 42.5 | 43.6 | 7.13 | (1) |\n| NGC 3227 | 16.6 | 0.7 | Sy1 | 42.4 | 43.5 | 7.62 | (9,4,10) |\n| NGC 3281 | 46.1 | 0.5 | Sy2 | 42.6 | 43.8 | 7.91 | (1) |\n| NGC 3783 | 42 | 0.9 | Sy1 | 43.2 | 44.5 | 7.48 | (3,4,10) |\n| NGC 4151 | 14.3 | 0.7 | Sy1 | 42.1 | 43.2 | 7.66 | (11,12) |\n| NGC 4388 | 36.3 | 0.2 | Sy2 | 42.9 | 44.1 | 7.23 | (1) |\n| NGC 4507 | 51 | 0.8 | Sy2 | 43.1 | 44.4 | 7.65 | (1) |\n| NGC 4945 | 3.6 | 0.2 | Sy2 | 42.3 ( ∗ ) | 43.4 | 6.15 | (13,14) |\n| NGC 5128 | 3.7 | 0.8 | Sy2 | 41.9 | 42.9 | 7.84 | (4) |\n| NGC 5135 | 59.3 | 0.7 | Sy2 | 43.1 ( ∗ ) | 44.4 | 7.29 | (1) |\n| NGC 5347 | 33.6 | 0.8 | Sy2 | 42.4 ( ∗ ) | 43.5 | 6.97 | (1,15) |\n| NGC 5506 | 26.6 | 0.3 | Sy1 | 43.0 | 44.3 | 7.95 | (1) |\n| NGC 5643 | 17.2 | 0.9 | Sy2 | 41.4 | 42.3 | 7.40 | (16,15) |\n| NGC 7130 | 70 | 0.9 | Sy2 | 43.1 ( ∗ ) | 44.4 | 7.59 | (1) |\n| NGC 7172 | 37.4 | 0.6 | Sy2 | 42.2 | 43.3 | 7.67 | (17) |\n| NGC 7213 | 25.1 | 0.9 | Sy1 | 42.1 | 43.1 | 7.74 | (5) |\n| NGC 7469 | 70.8 | 0.7 | Sy1 | 43.3 | 44.7 | 7.08 | (3,4,6) |\n| NGC 7479 | 34.2 | 0.7 | Sy1 | 42.0 | 43 | 7.68 | (7,10) |\n| NGC 7582 | 22.6 | 0.4 | Sy1 | 41.9 (d) | 42.9 | 7.13 | (1,18) | \nNotes. -(a) Distances from NED. (b) Compton-thick sources according to Marinucci et al. (2012). Hard X-ray luminosities are corrected by a factor 70. (c) AGN bolometric luminosities calculated from X-ray luminosities after applying the bolometric corrections of Marconi et al. (2004). (d) Changing-look AGN (e.g. Bianchi et al. 2005). Data from an intermediate state. ( ∗ ) Compton-thick sources. \nReferences. (1) Marinucci et al. (2012), (2) Malizia et al. (2007), (3) Dadina et al. (2007), (4) Tueller et al. (2008), (5) Asmus et al. (2011), (6) Risaliti et al. (2005), (7) Brightman et al. (2011), (8) Woo & Urry (2002), (9) Honig et al. (2010), (10) Diamond-Stanic & Rieke (2012), (11) Wang et al. (2011), (12) Beckmann et al. (2006), (13) Guainazzi et al. (2000), (14) Muller et al. (2003), (15) Beifiori et al. (2009), (16) Guainazzi et al. (2004), (17) Akylas et al. (2001), (18) Piconcelli et al. (2007). \nselected galaxies with existing high angular resolution ( ∼ 0 . 4 -0 . 8 '' ) mid-IR spectra observed on 8m-class telescopes. The sample contains a total of 29 Seyfert galaxies, of which 16 (55%) are Type 2 and 13 (45%) are Type 1 AGN. We included in the Seyfert 1 category those galaxies classified as Seyfert 1.5, 1.8, and 1.9, as well as those with broad near-IR lines. \nWe used the hard 2 -10keV X-ray luminosity (see Table 1 for references) as a proxy for the AGN bolometric luminosities after correcting for absorption and applying the bolometric corrections of Marconi et al. (2004). The high column density in Compton-thick objects (defined as those having N H > 10 24 cm -2 , see Table 1) prevents us from measuring the intrinsic nuclear luminosity below 10 keV. Instead, one can only derive the reflection component from model fitting. Assuming that the [O iii ] forbidden line is a tracer of the AGN intrinsic luminosity \nand comparing it with the observed hard X-ray emission of Compton-thick AGN, Marinucci et al. (2012) derived a correction factor of 70, that we used to correct the observed 2 -10 keV luminosities of these objects 1 This large correction factor is also theoretically justified by the torus model proposed by Ghisellini et al. (1994). This is the method commonly used for Compton-thick sources and applied in other several works, as in e.g. Bassani et al. (1999); Panessa et al. (2006). We expect the nuclear mid-IR and X-ray luminosities to be well correlated (e.g. Levenson et al. 2009; Asmus et al. 2011), which is fulfilled for sources in our sample (Honig et al. 2010; Gonz'alez-Mart'ın et al. 2013). The only significant \n1 The X-ray luminosity of NGC 7479 is not corrected by such a factor because, despite its high N H , its Compton-thick nature is not confirmed. Therefore, its luminosity could be up to a factor of 60-70 times higher (Panessa et al. 2006). \nFig. 1.Distribution of the AGN bolometric luminosities for Sy1 (black histogram, horizontal filling lines) and Sy2 (red histogram, vertical lines) galaxies in our sample. \n<!-- image --> \noutlier, NGC1808, indicates that for this source the AGN does not dominate the continuum mid-IR emission (see figure 5 in Gonz'alez-Mart'ın et al. 2013). The uncertainties in L agn are driven by the scatter on the relationship for the bolometric correction which, in general, is significantly larger than the error on the X-ray luminosities. Based on the L agn determination, Young et al. (2010) derived typical uncertainties of 0.4 dex (see also Marinucci et al. 2012). \nOur sample spans AGN bolometric luminosities in the range log L agn = 41 . 2 -45 . 5 erg s -1 , with a median value of 43.7 erg s -1 . This is a fair representation of the full RSA Seyfert sample (see figure 1 of Diamond-Stanic & Rieke 2012) 2 . As can be seen from Figure 1, Type 1 and Type 2 sources have similar distributions of L agn , with median values (in logarithm scale) of 43.7 and 43.8 erg s -1 for Sy1 and Sy2s, respectively. We also list in Table 1 the BH masses of the galaxies in our sample and corresponding references. There is no available black hole mass measurement for NGC 1808. The median value for our sample is 3.9 × 10 7 M /circledot , which is similar to the 3.2 × 10 7 M /circledot median value for the complete RSA Seyfert sample (Diamond-Stanic & Rieke 2012). In terms of the Eddington ratio, we sample values of L agn /L edd = 10 -4 -0 . 3.", '2.2. Observations': "Ground-based mid-IR spectroscopic observations of the 29 Seyfert galaxies were taken with three different instruments. They operate on 8m-class telescopes and cover the N -band, ∼ 8 -13 µ m. Table 2 summarizes details of the mid-IR spectroscopic observations, along with references where the data were originally published. Observations taken with the Thermal-Region Camera Spectrograph (T-ReCS Telesco et al. 1998) on the 8.1 m Gemini-South Telescope used the low resolution mode, which provides a spectral resolution of R = ∆ λ/λ ∼ 100, and slit widths between 0.31 and 0.70 '' . Observations \nTABLE 2 Ground-based high angular resolution mid-IR spectroscopy \n| Galaxy | Instrument | slit ( '' ) | Refs |\n|----------------|--------------|---------------|--------|\n| ESO 323 - G077 | VISIR | 0.75 | (4) |\n| IC 5063 | T-ReCS | 0.67 | (5,2) |\n| | VISIR | 1 | (4) |\n| Mrk 509 | VISIR | 0.75 | (4) |\n| NGC 1068 | Michelle | 0.36 | (6) |\n| | VISIR | 0.4 | (4) |\n| NGC 1365 | T-ReCS | 0.35 | (7,2) |\n| NGC 1386 | T-ReCS | 0.31 | (2) |\n| NGC 1808 | T-ReCS | 0.35 | (8,2) |\n| NGC 2110 | Michelle | 0.36 | (9) |\n| | VISIR | 0.75 | (4) |\n| NGC 2992 | Michelle | 0.4 | (10) |\n| NGC 3081 | T-ReCS | 0.65 | (2) |\n| NGC 3281 | T-ReCS | 0.35 | (2,11) |\n| NGC 3227 | VISIR | 0.75 | (4) |\n| NGC 3783 | VISIR | 0.75 | (4) |\n| NGC 4151 | Michelle | 0.36 | (12) |\n| NGC 4388 | Michelle | 0.4 | (10) |\n| NGC 4507 | VISIR | 1 | (4) |\n| NGC 5135 | T-ReCS | 0.7 | (13,2) |\n| NGC 5347 | Michelle | 0.4 | (10) |\n| NGC 5506 | T-ReCS | 0.36 | (14,2) |\n| NGC 5643 | T-ReCS | 0.36 | (2) |\n| | VISIR | 0.75 | (4) |\n| NGC 7130 | T-ReCS | 0.7 | (13,2) |\n| NGC 7172 | T-ReCS | 0.36 | (14,2) |\n| NGC 7213 | VISIR | 0.75 | (4) |\n| NGC 7469 | VISIR | 0.75 | (4) |\n| NGC 7479 | T-ReCS | 0.35 | (2) |\n| NGC 7582 | T-ReCS | 0.7 | (2) |\n| | VISIR | 0.75 | (3) | \nReferences. (1) Roche et al. (2006), (2) Gonz'alez-Mart'ın et al. (2013), (3) Honig et al. (2008), (4) Honig et al. (2010), (5) Young et al. (2007), (6) Mason et al. (2006), (7) Alonso Herrero et al. (2012), (8) Sales et al. (2013), (9) Mason et al. (2009), (10) Colling (2011), (11) Sales et al. (2011), (12) Alonso Herrero et al. (2011), (13) D'ıaz Santos et al. (2010), (14) Roche et al. (2007). \nperformed by Michelle (Glasse et al. 1997) on the 8.1 m Gemini-North telescope, which has a higher spectral resolution by a factor of two ( R ∼ 200), were obtained with slit widths of ∼ 0 . 4 '' . Finally, observations with the VLT spectrometer and imager for the mid-infrared (VISIR, Lagage et al. 2004) instrument mounted on the 8.2m VLT UT3 telescope at the ESO/Paranal observatory were obtained with the low spectral resolution mode ( R ∼ 300) and a slit width of 0 '' .75 or 1 '' (and 0.4 '' for NGC1068). For the typical distances of our sample the ground-based slit widths probe typical physical scales of ∼ 65pc. These range from ∼ 7-255pc for all objects except for Mrk 509 (545 pc), which is by far the most distant galaxy in the sample. Sixteen sources were observed with Gemini /T-ReCS (Gonz'alez-Mart'ın et al. 2013, and references therein). \nFig. 2.Spitzer /IRS SL spectrum (thick line) from CASSIS compared with the ground-based nuclear T-ReCS spectrum (thin line) from Gonz'alez-Mart'ın et al. (2013) of NGC 7130, one of the galaxies in our sample. We show the location of the 11.3 µ mPAH feature, with the blue shaded area indicating the spectral range used for obtaining the integrated flux. The red lines are the fitted local continua. The rms of the spectrum is shown in yellow. We note that the [Ne ii ] 12.81 µ m emission line is contaminated by the 12.7 µ mPAH feature, which cannot be resolved. The complete sample is shown in Figure A.1 of the Appendix. \n<!-- image --> \nThirteen sources have VLT/VISIR observations (see Honig et al. 2010, for details), with 4 overlapping with the T-ReCS sample. Finally, 6 Seyfert galaxies were observed with Gemini/Michelle (Mason et al. 2006; Alonso-Herrero et al. 2011; Colling 2011), of which two sources also have VISIR observations. We refer the reader to the original papers for details on the observations and the data reduction. \nWe retrieved mid-IR Spitzer /IRS spectra (for all sources except for NGC 1068) from the Cornell Atlas of Spitzer /IRS Source (CASSIS v4, Lebouteiller et al. 2011). We used staring mode observations taken with the short-low (SL) module covering the spectral range ∼ 5 -15 µ m. The spectral resolution is R ∼ 60 -120. The CASSIS database provides spectra with optimal extraction regions to ensure the best signal-to-noise ratio and are fully reduced. We only needed to apply a small offset to stitch together the two short-wavelength modules SL1 and SL2 (but note that this does not affect the PAH measurement, see Section 2.3). NGC1068 is part of the GOALS programme (Armus et al. 2009) and IRS SH data have been obtained from the NASA/IPAC Infrared Science Archive (IRSA). Assuming a typical spatial resolution of 3.7 '' for the SL module of IRS given by the slit width, this corresponds to a physical scale of about ∼ 600pc for our sample, i.e. a factor of 10 less resolved than for the ground-based data. Figure 2 shows a comparison of Spitzer against ground-based data of NGC 7130 for illustration. We present the spectra of the full sample in the Appendix (see Figure A.1).", '2.3. Measuring the 11 . 3 µ m PAH feature': "A number of methods have been developed to provide accurate measurements of the PAH feature fluxes, specially for the relatively large spectral range covered by IRS. These include, among others, PAHFIT (Smith et al. \n2007), DecompIR (Mullaney et al. 2011), and spline fit (e.g. Uchida et al. 2000; Peeters et al. 2002). They are useful for decomposing IR spectra, especially when the AGN emission is contaminated by extra-nuclear emission. These techniques, however, might not be appropriate for the limited wavelength coverage of ground-based data and/or weak PAH features (see Smith et al. 2007). \nWe measured the flux of the 11.3 µ m PAH feature following the method described in Hern'an-Caballero & Hatziminaoglou (2011). We fitted a local continuum by linear interpolation of the average flux in two narrow bands on both sides of the PAH. We then subtracted this fitted local continuum and integrated the residual data in a spectral range centered on 11.3 µ m ( λ rest =11.05 -11.55 µ m), to obtain the PAH flux. Figure 2 illustrates the method. As can be seen from this figure, this procedure slightly underestimates the PAH feature flux due to losses at the wings of the line profiles and overlaps between adjacent PAH bands. We corrected for these effects by assuming that the line has a Lorentzian profile of known width and applying a multiplicative factor (see Hern'an-Caballero & Hatziminaoglou 2011, for details). We measured the equivalent width (EW) of the 11.3 µ m PAH feature by dividing the integrated PAH flux by the interpolated continuum at the center of the feature. We derived the uncertainties in the measurements by performing Monte Carlo simulations. This was done by calculating the dispersion around the measured fluxes and EWs in a hundred simulations of the original spectrum with random noise distributed as the rms. \nAdditionally, PAH fluxes measured using a local continuum tend to be smaller than those using a continuum fitted over a large spectral range. To scale up our flux values to the total emission in the PAH features, we used the multiplicative factor of 2. This value was derived by Smith et al. (2007) for the 11.3 µ mPAH, after comparing results obtained by the spline fitting and the PAHFIT full decomposition. For consistency in all measurements, we used the same method described above for both the ground-based and IRS spectra. We detected the 11.3 µ m PAH feature at the 2 σ level or higher significance in all objects of our sample observed with IRS, except for NGC 3783 (see Figs. 2 and A.1). Our measurements of the 11 . 3 µ m PAH fluxes agree well with those of Diamond-Stanic & Rieke (2012) using PAHFIT , even though they used their own spectral extraction from IRS data.", '3. NUCLEAR 11.3 µ m PAH FEATURE EMISSION': "In this section we investigate the nuclear 11.3 µ m PAH feature emission in our sample of galaxies. Hereinafter, nuclear scales generally refer to the physical regions observed with the T-ReCS/Michelle/VISIR instruments, whereas circumnuclear scales are those probed with the IRS spectroscopy. The only exceptions are the most nearby (distances of ∼ 4 Mpc) galaxies Circinus, NGC 4945, and NGC 5128. To explore similar physical regions in comparison to the rest of the sample we used the IRS observations as our nuclear spectra for the three galaxies. For these sources the circumnuclear data are from Siebenmorgen et al. (2004) and Galliano et al. \nTABLE 3 Nuclear measurements of the sample. \n| Name | Ins | Slit/size ( '' /pc) | L 11 . 3 µ mPAH (10 40 erg s - 1 ) | EW 11 . 3 µ mPAH (10 - 3 µ m) | SFR nuclear ( M /circledot yr - 1 ) | SFR nuclear SFR circ | f 11 . 3 µ mPAH f [NeII] |\n|--------------|----------|-----------------------|--------------------------------------|---------------------------------|---------------------------------------|------------------------|----------------------------|\n| Circinus (a) | IRS | 3.70/75 | 5.1 ± 0.3 | 61 ± 1 | 0.13 | 0.23 | 0.83 |\n| ESO323-G077 | VISIR | 0.75/235 | < 9.3 | < 9 | < 0.23 | < 0.07 | < 0.56 |\n| IC5063 | T-ReCS | 0.65/153 | < 8.4 | < 12 | < 0.21 | < 0.53 | . . . |\n| Mrk509 | VISIR | 0.75/545 | 44.5 ± 17.6 | 10 ± 4 | 1.11 | 0.18 | . . . |\n| NGC1068 | Michelle | 0.36/28 | 14.4 ± 2.7 | 9 ± 1 | 0.36 | 0.36 | . . . |\n| NGC1365 | T-ReCS | 0.35/40 | < 1.9 | < 18 | < 0.05 | < 0.06 | < 0.97 |\n| NGC1386 | T-ReCS | 0.31/19 | < 0.5 | < 31 | < 0.01 | < 0.26 | < 0.91 |\n| NGC1808 | T-ReCS | 0.35/24 | 8.2 ± 0.5 | 365 ± 22 | 0.21 | 0.17 | 0.74 |\n| NGC2110 | VISIR | 0.75/121 | < 1.3 | < 7 | < 0.03 | < 0.12 | . . . |\n| NGC2992 | Michelle | 0.40/64 | < 3.8 | < 30 | < 0.09 | < 0.14 | < 0.33 |\n| NGC3081 | T-ReCS | 0.65/107 | < 1.9 | < 18 | < 0.05 | < 0.25 | < 0.34 |\n| NGC3227 | VISIR | 0.75/60 | 2.9 ± 0.5 | 63 ± 11 | 0.07 | 0.18 | 0.58 |\n| NGC3281 | T-ReCS | 0.35/78 | < 6.2 | < 9 | < 0.16 | < 0.98 | < 0.40 |\n| NGC3783 | VISIR | 0.75/151 | < 3.3 | < 6 | < 0.08 | < 0.79 | < 0.22 |\n| NGC4151 | Michelle | 0.36/25 | < 2.4 | < 17 | < 0.06 | < 1.03 | < 0.36 |\n| NGC4388 | Michelle | 0.40/70 | < 7.7 | < 68 | < 0.19 | < 0.38 | < 0.18 |\n| NGC4507 | VISIR | 1.00/245 | < 4.5 | < 5 | < 0.11 | < 0.12 | < 0.10 |\n| NGC4945 (b) | IRS | 3.70/64 | 0.4 ± 0.1 | 358 ± 17 | 0.01 | 0.13 | 0.13 |\n| NGC5128 (a) | IRS | 3.70/66 | 0.6 ± 0.1 | 65 ± 2 | 0.01 | 0.18 | 0.43 |\n| NGC5135 (c) | T-ReCS | 0.70/200 | 5.9 ± 2.2 | 34 ± 12 | 0.15 | 0.04 | 0.63 |\n| NGC5347 | Michelle | 0.40/65 | < 6.5 | < 56 | < 0.16 | < 0.76 | < 0.64 |\n| NGC5506 | T-ReCS | 0.35/45 | 6.4 ± 2.8 | 15 ± 6 | 0.16 | 0.32 | 0.40 |\n| NGC5643 | VISIR | 0.75/62 | 1.6 ± 0.2 | 49 ± 4 | 0.04 | 0.24 | 0.29 |\n| NGC7130 (c) | T-ReCS | 0.70/236 | 47.1 ± 3.3 | 166 ± 11 | 1.18 | 0.22 | 1.11 |\n| NGC7172 | T-ReCS | 0.35/63 | < 2.8 | < 43 | < 0.07 | < 0.16 | < 0.27 |\n| NGC7213 | VISIR | 0.75/91 | < 0.7 | < 8 | < 0.02 | < 0.14 | < 0.06 |\n| NGC7469 | VISIR | 0.75/255 | 47.7 ± 4.7 | 31 ± 3 | 1.19 | 0.10 | 0.82 |\n| NGC7479 | T-ReCS | 0.35/58 | < 5.6 | < 37 | < 0.14 | < 0.82 | . . . |\n| NGC7582 | T-ReCS | 0.70/76 | 3.9 ± 0.7 | 50 ± 9 | 0.10 | 0.30 | 0.41 |", 'Notes. -': 'The values of the EW and f 11 . 3 µ mPAH /f [NeII] are derived from measurements using the fitted local continuum. Upper limits at a 2 σ significance are included for non detections with the < symbol.', '(2008) (see Table 3 for more information).': 'The median value of the nuclear physical sizes probed with our data is 65 pc (see Table 3). If two nuclear spectra exist for the same galaxy, we used the one sampling a physical scale closest to the median value, for consistency with the rest of the galaxies. This information along with the physical scales probed with the nuclear spectra are given in Table 3. Note that this is approximately a factor of 10 improvement in physical resolution with respect to the circumnuclear median value of 600 pc. Hereinafter, we will use the term size as referring to the physical scale probed, which is determined by the slit widths of the observations.', '3.1. Detection of the 11 . 3 µ m PAH feature': "The 11.3 µ mPAH feature is weak or it might not even be present in a large fraction of galaxies in our sample, as \ncan be seen from the nuclear spectra in Figs. 2 and A.1. We deemed the feature as detected if the integrated flux is, at least, two times above the corresponding measured error. This is equivalent to having the PAH feature detected with a significance of 2 σ or higher. The nondetections are given as upper limits at a 2 σ level, that is, with a 95% probability that the real flux is below the quoted value. \nTable 3 gives the nuclear luminosities and EWs of the 11.3 µ m PAH detections, as well as 2 σ upper limits for the remaining objects. Note that the flux of the 11.3 µ mPAH feature is not corrected for extinction. Thus, its proper characterisation might be hampered in cases of high extinction, i.e. when the PAH molecules are embedded behind the silicate grains and the feature is buried within the silicate absorption at ∼ 9.7 µ m. This also depends on the location of the material \ncausing the extinction relative to the PAH emitting region. Another additional complication is the presence of crystalline silicate absorption at around 11 µ m, which has been detected in local ultraluminous infrared galaxies (Spoon et al. 2006) and in local Seyferts (Roche et al. 2007). In particular, Colling (2011) detected crystalline silicate absorption that could be blended with the 11.3 µ mPAH feature in some of the galaxies in our sample, namely, NGC 4388, NGC 5506, NGC 7172, and NGC 7479 (see also Section 3.2). \nUsing T-ReCS/VISIR/Michelle data we detected nuclear 11.3 µ m PAH emission in 10galaxies. For the three most nearby Seyferts, the 11.3 µ m PAH feature is detected in the IRS observations, while in the corresponding T-ReCS/VISIR spectra the feature is below the detection limits. Taking this into account, we detected nuclear 11.3 µ m PAH feature emission in 13 out of 29 galaxies (45% of the sample). The detection rate is similar for Seyfert 1s and Seyfert 2s (40% and 50%, respectively). The observed EWs of the feature (using the fitted local continuum) are between ∼ 0 . 01 -0 . 4 µ m. These values are much lower than those typical of high metallicity star forming galaxies (see Hern'an-Caballero & Hatziminaoglou 2011). This is expected given that we are probing smaller regions around the nucleus, and probably the continuum emission is mostly arising from dust heated by the AGN. \nTo study a possible extra-nuclear origin of the PAH feature we investigated the morphology of the galaxies in our sample. We compiled the b/a axial ratio (measurements from NED, RC3 D 25 /R 25 isophotal B -band diameters) to determine the inclination of the host galaxy, where b and a are the minor and mayor axis, respectively. Axial ratios b/a < 0.5 are considered as edge-on galaxies, whereas face-on galaxies have b/a > 0.5 (see Table 1). With this definition we find 11 edge-on and 18 face-on galaxies. Out of the 13 sources with detection of the nuclear 11.3 µ mPAH feature, we find 5 (44%) edge-on and 8 (45%) face-on galaxies. We do not find that a positive detection predominates in edgeon galaxies, where material of the host galaxy along our line of sight could be misinterpreted as nuclear SF. However, we cannot rule out a dominant contribution from extra-nuclear SF in the most edge-on galaxies in our sample. Gonz'alez-Mart'ın et al. (2013) found that the host galaxies could significantly contribute to the nuclear component for sources with the deepest silicate absorption features. \nThe majority of the nuclear 11.3 µ m PAH detections in our sample are galaxies with well-documented nuclear starbursts and/or recent SF activity based on UV and optical observations (Gonz'alez Delgado et al. 1998, NGC 5135, NGC 7130), modelling of the optical spectra (Storchi-Bergmann et al. 2000, NGC 5135, NGC 5643, NGC 7130, NGC 7582), and near-IR integral field spectroscopy (Davies et al. 2007; Tacconi-Garman & Sturm 2013, Circinus, NGC 1068, NGC 3227, NGC 3783, NGC 5128).", '3.2. Stacking nuclear spectra with undetected 11 . 3 µ m PAH emission': 'In this section we further investigate those galaxies with weak or no detected 11.3 µ m PAH feature emission. We stacked the individual spectra deemed to have \nFig. 3.Result from the stacking of nuclear spectra (from T-ReCS, Michelle, and VISIR) without 11.3 µ m PAH detections. The stacked spectrum for sources with weak silicate features (six galaxies, thick line) shows a 2 σ detection of the 11.3 µ m PAH feature. In the stacked spectrum (thin line) of the seven galaxies with deep silicate features the PAH feature remains undetected. The flux density units are arbitrary. The individual spectra were normalized at 12 µ m. See Section 3.2 for details. \n<!-- image --> \nundetected 11.3 µ m PAH features according to our 2 σ criterion (see Table 3 and Section 3.1). We divided them in two groups. The first includes galaxies with a weak silicate feature: ESO 323-G077, NGC 1365, NGC 3081, NGC 3783, NGC 4151, and NGC 4507. The second group contains galaxies with relatively deep silicate features: NGC 1386, NGC 2992, NGC 3281, NGC 4388, IC 5063, NGC 7172 and NGC 7479. We excluded from the stacking NGC 2110 and NGC 7213 because the silicate feature is strongly in emission, and NGC 5347 because the spectrum is very noisy. We normalized the spectra at 12 µ m and then used the IRAF task scombine with the average option to combine the different observations. \nFigure 3 shows the stacked spectra for the two groups. We applied the same method as in Section 3.1 to determine if the PAH feature is detected. We found that the 11.3 µ mPAHappears detected in the stacked nuclear spectrum of the galaxies with weak silicate features at a 2 σ level. The derived EW of the 11.3 µ m PAH is 8 × 10 -3 µ m. The feature remains undetected in the stacked nuclear spectrum of sources with deep silicate features. This could be explained in part as due to extinction effects, given that the silicate absorption in these galaxies likely comes from cold foreground material (Goulding et al. 2012). We also note that the minimum around 11 µ m in the stacked spectrum of galaxies with deep silicate features could be from crystalline silicates. Indeed, Colling (2011) found that inclusion of crystalline silicates improved the fit of the silicate features in NGC 7172, NGC 7479, and NGC 4388. Therefore, it would be expected to also appear in the stacked spectrum.', '3.3. Is the 11 . 3 µ m PAH feature suppressed in the vicinity of AGN?': "It has been known for more than 20 years now that PAH emission is weaker in local AGN than in high metallicity star forming galaxies, although some AGN do \nalso show strong PAH features on circumnuclear scales (Roche et al. 1991). It is not clear, however, if the decreased detection of PAH emission and the smaller EWs of the PAH features in AGN are due to 1) an increased mid-IR continuum arising from the AGN, 2) destruction of the PAH carriers in the harsh environment near the AGN (Roche et al. 1991; Voit 1992a) or 3) decreased SF in the nuclear region (Honig et al. 2010). Additionally, there is a prediction that smaller PAH molecules would be destroyed more easily in strong radiation fields (see e.g. Siebenmorgen et al. 2004), also indicating that different PAHs may behave differently. \nThe effects of an increasing continuum produced by the AGN is clearly seen in the 3.3 µ m PAH map of the central region of NGC 5128. The ratio of the feature-tocontinuum (i.e., the EW of the feature) decreases towards the AGN, whereas the feature peaks in the center (see Tacconi-Garman & Sturm 2013, for more details). This implies that the PAH molecules are not destroyed in the harsh environment around the AGN of this galaxy (see also Section 3.4). Similarly, D'ıaz-Santos et al. (2010) showed that at least the molecules responsible for the 11.3 µ m PAH feature can survive within < 100pc from the AGN. \nSome recent observational works reached apparently opposing conclusions on the PAH emission of AGN on physical scales within a few kpc from the nucleus, but note that these are for much larger physical scales than those probed here. Diamond-Stanic & Rieke (2010) demonstrated for the RSA Seyferts that the 11.3 µ mPAH feature is not suppressed, whereas other mid-IR PAH features are. LaMassa et al. (2012), on the other hand, combined optical and mid-IR spectroscopy of a large sample of AGN and star forming galaxies and concluded that in AGN-dominated systems (higher luminosity AGN) the 11.3 µ mPAH feature does get suppressed. \nTo investigate this issue, we plotted in Figure 4 the observed nuclear EW of the 11.3 µ mPAHfeature against the AGN bolometric luminosity for our sample, with smaller symbols indicating regions closer to the AGN. This figure does not show any clear trend. If the decreased nuclear EW of the 11.3 µ mPAH feature were due to the AGN mid-IR continuum in more luminous AGN, we would expect a trend of decreasing EW for increasing AGN bolometric luminosities. Alternatively, we would expect the same trend if the PAH molecules responsible for the 11.3 µ m feature were suppressed/destroyed more easily in luminous AGN. From this figure we can see that at a given AGN luminosity we sometimes detect nuclear 11.3 µ m PAH emission, whereas in other cases we do not. In other words, we do not see clear evidence in our sample for the 11.3 µ m PAH feature to be suppressed in more luminous AGN, at least for the AGN bolometric luminosities covered in our sample of Seyfert galaxies. \nAs can be seen from Figure 4, there is no clear influence of the probed physical region sizes on the observed EWs. Hence, we do not see a tendency for the EW of the PAH feature to decrease for smaller physical regions. This would be the case if we were to expect a higher AGN continuum contribution and/or PAH destruction as we get closer to the AGN. No trend is either present when plotting the observed EWs with respect to luminosity densities. We note, however, that for the three closest Seyferts (Circinus, NGC 4945, \nFig. 4.Nuclear EW of the 11.3 µ m PAH feature versus the AGN bolometric luminosity. Filled star symbols are detections whereas open squares are upper limits. The sizes/colors of the symbols (see figure legend) indicate the different physical sizes probed, which are determined by the slit widths of the observations. Hereinafter, we have marked Compton-thick objects in all plots using a double-star or a double-square for detections or upper limits, respectively. \n<!-- image --> \nNGC 5128) the 11.3 µ m PAH feature is not detected in ground-based high-resolution T-ReCS spectra, which probe scales of ∼ 7 -15 pc for these sources (Roche et al. 2006; Gonz'alez-Mart'ın et al. 2013).", '3.4. Nuclear PAH molecules shielded by the dusty torus?': "As we have shown in the previous sections and as presented by others (e.g., Miles et al. 1994; Marco & Brooks 2003; D'ıaz-Santos et al. 2010; Honig et al. 2010; Gonz'alez-Mart'ın et al. 2013; Sales et al. 2013; Tacconi-Garman & Sturm 2013) PAH emission is detected in the vicinity (from tens to a few hundreds parsecs) of the harsh environments of some AGN. Therefore, at least in some galaxies, the PAH molecules are not destroyed (or at least not completely) near the AGN. They must be shielded from the AGN by molecular material with sufficient X-ray absorbing column densities (Voit 1992a; Miles et al. 1994; Watabe et al. 2008). As pointed out by Voit (1992b), for PAH features to be absent due to destruction, they have to be fragmented more quickly than they can be rebuilt. In other words, PAHs will exist if the rate of reaccretion of carbon onto the PAHs is higher than the evaporation rate caused by the AGN. Using the parameters of the Voit (1991) model, Miles et al. (1994) estimated the column density of hydrogen required to keep the evaporation rate of PAHs below the rate of reaccretion of carbon onto the PAHs. As derived in Miles et al. (1994), the time scale for X-ray absorption in terms of the hydrogen column density of the intervening material N H (tot), the distance from the AGN fixed in our case by the slit width D agn , and the X-ray luminosity of the AGN, can be written as \n<!-- image --> \nFig. 5.Flux ratio of the 11.3 µ m PAH and the [Ne ii ] lines plotted against the AGN luminosity for the total (AGN+SF) [Ne ii ] emission (left) and corrected when possible for the AGN contribution (right panel), using values from Mel'endez et al. (2008) (these are shown with name tags on the figure, with the corresponding percentage of star formation). See text for details. \n<!-- image --> \ncolumns of the order of 10 23 cm -2 or even higher. \nτ ≈ 700 yr ( N H (tot) 10 22 cm -2 ) 1 . 5 ( D agn kpc ) 2 ( 10 44 erg s -1 L X ) . (1) \nVoit (1992b) estimated that the time scale needed for reaccretion of a carbon atom on to a fractured PAH should be at least 3000 years for the typical conditions of the interstellar medium. \nThe protecting material, which has to be located between the nuclear sites of SF and the AGN, is likely to be that in the dusty torus postulated by the unified model. In a number of works (Ramos Almeida et al. 2009, 2011; Alonso-Herrero et al. 2011, 2012) we have demonstrated that the clumpy torus models of Nenkova et al. (2008a,b) accurately reproduce the nuclear infrared emission of local Seyfert galaxies. These models are defined by six parameters describing the torus geometry and the properties of the dusty clouds. These are, the viewing angle ( i ) and radial extent ( Y ) of the torus, the angular ( σ ) and radial distributions ( q ) of the clouds along with its optical depth ( τ V ), and the number of clouds along the equatorial direction ( N 0 ). The optical extinction of the torus along the line of sight is computed from the model parameters as A LOS V = 1 . 086 N 0 τ V e ( -( i -90) 2 /σ 2 ) mag. According to Bohlin et al. (1978), the absorbing hydrogen column density is then calculated following N LOS H /A LOS V = 1 . 9 × 10 21 cm -2 mag -1 . \nWith the derived torus model parameters, we estimated the hydrogen column density of the torus material in our line of sight for a sample of Seyferts. This typically ranges from N LOS H /similarequal 10 23 to a few times 10 24 cm -2 (see Ramos Almeida et al. 2009, 2011; Alonso-Herrero et al. 2011, 2012, for further details). Using Equation 1 for the hard X-ray luminosities (Table 1) and distances from the AGN probed by our spectroscopy (see Table 3), and setting τ =3000yr as proposed by Voit (1992a), we require hydrogen column densities of at least a few 10 23 cm -2 to protect the PAHs from the AGN radiation. Evidence for such values for the N H are found for our sample, as we derived absorbing \nIn Ramos Almeida et al. (2011), we also demonstrated that Seyfert 2s are more likely to have higher covering factors than Seyfert 1s. Assuming that the nuclear SF occurs inside the torus, the PAH molecules may be more shielded in the nuclear region of Seyfert 2s. However, even the N H values along our line of sight from the torus model fits of Seyfert 1s (which would be a lower limit to the total N H in the torus) with 11.3 µ m PAH detections are sufficient to protect the PAH carriers. This is the case for four Seyfert 1s (NGC3227, NGC5506, NGC7469, and NGC7582), as can be seen from the modelling by Alonso-Herrero et al. (2011). Note that the column densities that we refer to are not only those absorbing the X-rays but also including material much farther away from the accreting BH. \nAnother interesting aspect to keep in mind from Equation 1 is that the column densities needed to protect the PAH molecules from the AGN X-ray emission become higher for more luminous AGN as well as for distances closer to the AGN. However, we emphasize that for the AGN luminosities of the RSA Seyferts and distances from the AGN probed by the observations presented here, the PAH molecules are likely to be shielded from the AGN by material in the torus residing on smaller scales. Also, part of the obscuring material even on these nuclear scales can reside in the host galaxy as shown by Gonz'alez-Mart'ın et al. (2013). Thus, another source of opacity that might prevent the PAHs from being destroyed are dust lanes in galaxies or dust in the nuclear regions of merger systems. This might be the case for five sources in our sample, namely NGC 4945, NGC5128, NGC5506, NGC7130 and NGC7582.", '4. NUCLEAR STAR FORMATION RATES IN SEYFERT GALAXIES': "4.1. Relation between the 11 . 3 µ m PAH feature and the [Ne ii ] 12 . 81 µ m emission line on nuclear scales \nIn star forming galaxies, the luminosity of the [Ne ii ] 12.81 µ m emission line is a good indicator of the SFR (Roche et al. 1991; Ho & Keto 2007; D'ıaz-Santos et al. 2010). We note that these measurements are contaminated by the 12.7 µ m PAH feature, \nwhich is not easily resolvable. \nIn Seyfert galaxies the situation is more complicated because this line can be excited by both SF and AGN activity. The AGN contribution to the [Ne ii ] varies from galaxy to galaxy in local Seyfert galaxies and other AGN (see e.g. Mel'endez et al. 2008; Pereira-Santaella et al. 2010). For the RSA Seyfert sample, Diamond-Stanic & Rieke (2010) used IRS spectroscopy to compare the circumnuclear SFRs computed with the [Ne ii ] line and the 11.3 µ m PAH feature as a function of EW of the PAH. They found that the ratio of the two circumnuclear SFRs (on a kpc scale) is on average unity, with some scatter for galaxies with large PAH EWs > 0.3 µ m. The most discrepant measurements were for those galaxies with elevated [O iv ]/[Ne ii ] and low EW of the PAH, that is, AGN dominated galaxies. \nIn Figure 5 (left panel) we show the observed nuclear PAH/[Ne ii ] ratio as a function of the AGN luminosity. To correct for the 12.7 µ mPAH contamination, we have used a median ratio of the 11.3 versus the 12.7 µ mPAH features of 1.8, derived in Smith et al. (2007), and subtracted it from the [Ne ii ] measurement. NGC 1365, NGC 1386 and NGC 5347 are not included in the plot because both lines are undetected and, therefore, the value in the Y axis is completely unconstrained. We derived the [Ne ii ] fluxes using the same technique as explained for the PAH in Section 2.3, integrating the line between 12.6 and 12.9 µ m. For seven galaxies, namely IC5063, Mrk509, NGC1068, NGC2110, NGC5135, NGC7130 and NGC7479, the [Ne ii ] line falls outside the wavelength range covered by our observations. They are not included in the plot except for NGC5135 and NGC7130, whose values have been extracted from D'ıaz-Santos et al. (2010). Most of the galaxies with detections of the nuclear 11.3 µ m PAH feature and [Ne ii ] show ratios similar to those of high metallicity star forming galaxies ( ∼ 0 . 7 -2, see e.g., Roche et al. 1991; D'ıaz-Santos et al. 2010), even if the [Ne ii ] fluxes are not corrected for AGN emission. On the other hand, most of the nuclear spectra with non detections show upper limits to the PAH/[Ne ii ] ratio below 0.5. \nTo derive the nuclear [Ne ii ] flux solely due to star formation, we can use the fractional SF contribution to the [Ne ii ] line within the IRS aperture estimated by Mel'endez et al. (2008). For sources with strong AGN contribution (higher than 50%), we estimated the [Ne ii ] flux coming from the AGN, which can be subtracted from the observed nuclear [Ne ii ] flux. This is shown in Figure 5 (right panel). We note that for NGC 2992, NGC 3227, NGC 4151, NGC 5506 and NGC 7172, the estimated AGN contribution to the total [Ne ii ] is higher than the nuclear [Ne ii ] value, indicating that the AGN [NeII] contributions were overestimated. For these galaxies, we did not apply any correction. Figure 5 shows that for those Seyferts with a nuclear 11.3 µ mPAH detection the PAH/[Ne ii ] SF ratio does not decrease with the AGN bolometric luminosity. This would be expected if the PAH emission was to be suppressed. Therefore, given that the PAH/[Ne ii ] SF ratio does not show a dependence on L agn , we conclude that the 11.3 µ m PAH feature emission can be used to estimate the nuclear SFRs (see next section).", '4.2. Circumnuclear ( ∼ 600 pc) vs Nuclear ( ∼ 65 pc) scales': "In the vicinity of an AGN, we expect the chemistry and/or the heating as dominated by X-rays from the socalled X-ray dominated regions (XDRs). In principle, XDRs could also contribute to PAH heating through the photodissociation and photoionization by FUV photons produced via excitation of H and H 2 in collisions with secondary electrons. However, as we derived in Section 3.4, the torus appears to provide the appropriate environment to shield the PAH molecules from the AGN emission, on typical physical scales of a few parsecs up to a few tens of parsecs. We thus expect little or no contribution of UV AGN produced photons to the PAH heating in the nuclear scales of Seyfert galaxies. Hereinafter, we will assume that the aromatic molecules are heated by the radiation field produced by young massive stars and that the 11.3 µ m PAH luminosity can be used to estimate the SFR. \nWe derived nuclear and circumnuclear SFRs using the PAH 11.3 µ m feature luminosities and applying the relation derived in Diamond-Stanic & Rieke (2012) \nSFR ( M /circledot yr -1 ) = 9 . 6 × 10 -9 L (PAH 11 . 3 µm , L /circledot ) (2) \nusing PAHFIT measurements of galaxies with IR (81000 µ m) luminosities L IR < 10 11 L /circledot , using the Rieke et al. (2009) templates and a Kroupa IMF. This is appropriate for our sample, as the median value of the IR luminosity of the individual galaxies is 5 × 10 10 L /circledot 3 . The uncertainties in the derived SFRs using Equation 2 are typically 0.28 dex (see Diamond-Stanic & Rieke 2012, for full details). \nFor the 13 galaxies with nuclear 11.3 µ m PAH detections, the nuclear SFRs span two orders of magnitude between ∼ 0.01-1.2 M /circledot yr -1 for regions of typically ∼ 65pc in size. The non-detections indicate that the nuclear SFR of RSA Seyfert galaxies from the high angular resolution spectroscopy can be ∼ 0 . 01 -0 . 2 M /circledot yr -1 , or even lower. \nThe projected nuclear SFR densities are between 2 and 93 M /circledot yr -1 kpc -2 with a median value 22 M /circledot yr -1 kpc -2 . These are consistent with the simulations of Hopkins et al. (2012) for similar physical scales. The two galaxies with the largest SFR densities are NGC1068 and NGC1808 with values of 414 and 329 M /circledot yr -1 kpc -2 , respectively. We notice that for those galaxies in common with Davies et al. (2007) (namely Circinus, NGC 1068, NGC 3783, NGC 7469, NGC 3227) we find quite discrepant values for the SFR density, with ours lying below those in Davies et al. (2007) except for NGC 1068. It might be due to the use of different SF histories and SFR indicators. We note that with the 11.3 µ mPAH feature we cannot explore age effects (see D'ıaz-Santos et al. 2008, Figure 8) as this feature can be excited by both O and B stars, and thus it integrates over ages of up to a few tens of millions of years (Peeters et al. 2004), unlike the measurements \nFig. 7.Observed nuclear SFR vs. ˙ M BH relation. Predictions from Hopkins & Quataert (2010) are shown as dashed lines. We show the ˙ M BH ≈ 0.1 × SFR relation, which is expected for r < 100 pc, and the 1:1 relation which, is expected for the smallest physical scales ( r < 10 pc). The solid line represents the fit to our detections of the nuclear 11.3 µ mPAH feature (see text for details). \n<!-- image --> \nFig. 6.Comparison between the SFR on different scales, where SFR circumnuclear implies typical physical scales of ∼ 0.6 kpc and SFR nuclear is for ∼ 65 pc scales. Non-detections are plotted at a 2 σ level. Symbols as in Figure 4. The dashed line shows the median value of the nuclear/circumnuclear SFR ratio for the detections of the nuclear PAH feature (see text and Table 3). The dotted line indicates a nuclear/circumnuclear ratio of one. \n<!-- image --> \nin Davies et al. (2007) that sample younger populations. In addition, we detect neither nuclear nor circumnuclear SF in NGC 3783 based on the PAH measurements. On the other hand, PAHs can also be found in the interstellar medium (ISM) as being excited in lessUV rich environments, such as reflection nebulae (e.g. Li & Draine 2002). However, the decreased strength of the IR emission features in these objects seems to indicate the low efficiency of softer near-UV or optical photons in exciting PAHs in comparison to SF (Tielens 2008). \nThe circumnuclear SFRs in our sample are between 0.2 and 18.4 M /circledot yr -1 (see also Diamond-Stanic & Rieke 2012), and the median circumnuclear SFR densities are 1.2 M /circledot yr -1 kpc -2 . These are similar to those of the CfA (Huchra & Burg 1992) and 12 µ m(Rush et al. 1993) samples, derived using the 3.3 µ m PAH feature (see Imanishi 2003; Imanishi & Wada 2004). \nThe comparison between the nuclear and circumnuclear SFRs for our sample clearly shows that, in absolute terms, the nuclear SFRs are much lower (see Table 3). This is in good agreement with previous works based on smaller samples of local AGN (e.g. Siebenmorgen et al. 2004; Watabe et al. 2008; Honig et al. 2010; Gonz'alez-Mart'ın et al. 2013). The median value of the ratio between the nuclear and circumnuclear SFRs for the detections of the 11.3 µ mfeature is ∼ 0.18 (see also Table 3), with no significant difference for type 1 and type 2 Seyferts ( ∼ 0 . 18 and ∼ 0 . 21, respectively). \nIn Figure 6 we plot the nuclear and circumnuclear SFRs probing typical physical scales of ∼ 65pc and ∼ 600pc, respectively. Again, non-detections are plotted as upper limits at the 2 σ level. Overall, for our detections, the fraction of the SFR accounted for by the central ∼ 65pc region of our Seyferts ranges between ∼ 535% of that enclosed within the aperture corresponding to the circumnuclear data. \nWhile the nuclear SFRs are lower than the cir- \ncumnuclear SFRs, the median nuclear projected SFR densities are approximately a factor of 20 higher than the circumnuclear ones in our sample (median values of 22 and 1 M /circledot yr -1 kpc -2 , respectively). This shows that the SF is not uniformly distributed. Conversely, it is more highly concentrated in the nuclear regions of the RSA Seyferts studied here. This is in agreement with simulations of Hopkins et al. (2012). The molecular gas needed to maintain these nuclear SFR densities appears to have higher densities in Seyfert galaxies than those of quiescent (non Seyferts) galaxies (Hicks et al. 2013).", '4.3. Nuclear star formation rate vs. black hole accretion rate': 'Hopkins & Quataert (2010) performed smoothed particle hydrodynamic simulations to study the inflow of gas from galactic scales ( ∼ 10kpc) down to /lessorsimilar 0.1 pc, where key ingredients are gas, stars, black holes (BHs), self-gravity, SF and stellar feedback. These numerical simulations indicate a relation (with significant scatter) between the SFR and ˙ M BH that holds for all scales, and that is more tightly coupled for the smaller physical scales. The model of Kawakatu & Wada (2008) predicts that the AGN luminosity should also be tightly correlated with the luminosity of the nuclear (100 pc) SF in Seyferts and QSOs, and also that L nuclear , SB / L AGN is larger for more luminous AGN. \nAccording to Alexander & Hickox (2012, and references therein), ˙ M BH and AGN luminosities follow the relation \n˙ M BH ( M /circledot yr -1 ) = 0 . 15(0 . 1 //epsilon1 )( L agn / 10 45 erg s -1 ) (3) \nwhere we used /epsilon1 = 0 . 1 as the typical value for the mass-energy conversion efficiency in the local Universe (Marconi et al. 2004). We obtained ˙ M BH ranging between 5 × 10 -6 and 0 . 5 M /circledot yr -1 for our sample. \nUncertainties in the ˙ M BH estimations are dominated by those in L agn , i.e. 0.4 dex, as mentioned in Section 2.1. Figure 7 shows the observed nuclear SFR against ˙ M BH for the Seyferts in our sample. The different sizes of the symbols indicate different physical sizes of the probed regions. The prominent outlier in this figure is NGC 1808 4 . We also show in Figure 7 as dashed lines predictions from the Hopkins & Quataert (2010) simulations for r < 100 pc ( ˙ M BH ≈ 0.1 × SFR) and r < 10 pc ( ˙ M BH ≈ SFR). These radii encompass approximately the physical scales probed by our nuclear SFR. The prediction from the Kawakatu & Wada (2008) disk model for Seyfert luminosities and BH masses similar to those of our sample falls between the two dashed lines ( ∼ 1:0.4 relation). \nTo derive a possible correlation between the nuclear SFR and ˙ M BH , we applied a simple fit to the nuclear SFR detections (excluding NGC 1808) and obtained a nearly linear relation (slope of 1.01, and uncertainties of 0.4 dex in both parameters, Equation 4), which is close to the 1:0.1 relation (see Figure 7 and below). \nlog ˙ M BH = 1 . 01 × log SFR nuclear -1 . 11 (4) \nAlso including the upper limits in the fit we obtained a very similar result (slope of 0.95). In contrast, Diamond-Stanic & Rieke (2012) obtained a slightly superlinear relation ( ˙ M BH ∝ SFR 1 . 6 ), when the SFRs are measured in regions of 1 kpc radius. This behaviour (i.e., the relations becoming linear on smaller scales) is nevertheless predicted by the Hopkins & Quataert (2010) simulations. \nAs can be seen in Figure 7, the Hopkins & Quataert (2010) predictions for r < 100 pc reproduce fairly well the observed relation for our sample. We do not find a tendency for galaxies with SFRs measured in regions closer to the AGN (slit sizes of less than 100pc) to have larger ˙ M BH to SFR ratios (i.e., to lie closer to the 1:1 relation) than the rest, as predicted by Hopkins & Quataert (2010). It is worth noting that these authors caution that their work do not include the appropriate physics for low accretion rates ( << 0 . 1 M /circledot yr -1 ). The scatter in the theoretical estimations and the limited size of our sample of Seyfert galaxies prevent us from further exploring this issue. Future planned observations with the mid-IR CanariCam (Telesco et al. 2003) instrument on the 10.4m Gran Telescopio de Canarias (GTC) will allow a similar study for larger samples of Seyfert galaxies.', '5. SUMMARY AND CONCLUSIONS': "We have presented the largest compilation to date of high angular resolution (0 . 4 -0 . 8 '' ) mid-IR spectroscopy of nearby Seyfert galaxies obtained with the T-ReCS, VISIR, and Michelle instruments. We used the 11.3 µ mPAHfeature to study the nuclear SF activity and \n- 4 NGC 1808 is a low-luminosity AGN and shows the lowest AGN bolometric luminosity in our sample. It seems plausible that SF, unlike the AGN as for the rest of the sample, is the dominant mechanism contributing to the nuclear mid-IR emission (see Mason et al. 2012), as this source is completely off the midIR vs hard X-ray correlation for AGN (see Gonz'alez-Mart'ın et al. 2013). \nits relation to the circumnuclear SF, as well as with ˙ M BH . The sample includes 29 Seyfert galaxies (13 Seyfert 1 and 16 Seyfert 2 galaxies) belonging to the nearby RSA AGN sample (Maiolino & Rieke 1995; Ho et al. 1997). It covers more than two orders of magnitude in AGN bolometric luminosity, with the galaxies located at a median distance of 33 Mpc. Our data allow us to probe typical nuclear physical scales (given by the slit widths) of ∼ 65pc. We used the hard X-ray luminosity as a proxy for the AGN bolometric luminosity and ˙ M BH . We used mid-infrared Spitzer/IRS spectroscopy to study the SF taking place in the circumnuclear regions (a factor of 510 larger scales). \nThe main results can be summarized as follows: \n- 1. The detection rate of the nuclear 11.3 µ m PAH feature in our sample of Seyferts is 45 % (13 out of 29 sources), at a significance of 2 σ or higher. Additionally, the stacked spectra of six galaxies without a detection of the 11.3 µ m PAH feature and weak silicate features resulted into a positive detection of the 11.3 µ m PAH feature above 2 σ .\n- 2. There is no evidence of strong suppression of the nuclear 11.3 µ m PAH feature in the vicinity of the AGN, at least for the Seyfert-like AGN luminosities and physical nuclear regions (65 pc median value) sampled here. In particular, we do not see a tendency for the EW of the PAH to decrease for more luminous AGN. The hydrogen column densities predicted from clumpy torus model fitting (a few 10 23 cm -2 up to a few 10 24 cm -2 ) would be, in principle, sufficient to shield the PAH molecules from AGN X-ray photons in our Seyfert galaxies.\n- 3. The nuclear SFRs in our sample derived from the 11.3 µ m PAH feature luminosities are between 0.01 and 1.2 M /circledot yr -1 , where we assumed no XDRcontribution to the PAH heating. There is a significant reduction of the 11.3 µ mPAH flux from circumnuclear (median size of 600pc) to nuclear regions (median size of 65 pc), with a typical ratio of ∼ 5. Although this indicates that the SFRs are lower near the AGN in absolute terms, the projected SFR rate density in the nuclear regions (median value of 22 M /circledot yr -1 kpc -2 ) is approximately 20 higher than in the circumnuclear regions. This indicates that the SF activity is highly concentrated in the nuclear regions in our sample of Seyfert galaxies.\n- 4. Predictions from numerical simulations for the appropriate physical regions are broadly consistent with the observed relation between the nuclear SFR and ˙ M BH in our sample (slope of 1.01 ± 0 . 4). Although limited by the relatively small number of sources in our sample, we do not find decreased nuclear SFR-to-˙ M BH ratios for regions closer to the AGN, as predicted by the Hopkins & Quataert (2010) simulations. \nWe thank the anonymous referee for comments that helped us to improve the paper, and E. Piconcelli for useful discussion. PE and AAH acknowledge support from the Spanish Plan Nacional de Astronom'ıa y Astrof'ısica under grant AYA2009-05705-E. PE is partially funded by Spanish MINECO under grant AYA2012-39362-C02-01. AAH and AHC acknowledge support from the Augusto G. Linares Program through the Universidad de Cantabria. CRA ackowledges financial support from the Instituto de Astrof'ısica de Canarias and the Spanish Plan Nacional de Astronom'ıa y Astrof'ısica under grant AYA2010-21887C04.04 (Estallidos). OGM and JMRE acknowledge support from the Spanish MICINN through the grant AYA201239168-C03-01. SFH acknowledges support by Deutsche Forschungsgemeinschaft (DFG) in the framework of a research fellowship (Auslandsstipendium). \nThis research has made use of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. 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 Inova¸c˜ao (Brazil) and Ministerio de Ciencia, Tecnolog'ıa e Innovaci'on Productiva (Argentina). The Cornell Atlas of Spitzer/IRS Sources (CASSIS) is a product of the Infrared Science Center at Cornell University, supported by NASA and JPL.", 'REFERENCES': "- Akylas, A., Georgantopoulos, I., & Comastri, A. 2001, MNRAS, 324, 521 \nAlexander, D. M., & Hickox, R. C. 2012, New AR, 56, 93 Alonso-Herrero, A., Pereira-Santaella, M., Rieke, G. 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In a few cases, the nuclear data lie above the circumnuclear data, this can be due to calibration uncertainties. \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFig. A.1.Spectra of the sample. Continued. \n<!-- image --> \n<!-- image --> \n<!-- image -->'}

2021PhLB..81336040K

Solar-mass primordial black holes explain NANOGrav hint of gravitational waves

2021-01-01

['-', '-', '-', '-']

[]

The NANOGrav collaboration for the pulsar timing array (PTA) observation recently announced evidence of an isotropic stochastic process, which may be the first detection of the stochastic gravitational-wave (GW) background. We discuss the possibility that the signal is caused by the second-order GWs associated with the formation of solar-mass primordial black holes (PBHs). This possibility can be tested by future interferometer-type GW observations targeting the stochastic GWs from merger events of solar-mass PBHs as well as by updates of PTA observations.

[]

https://arxiv.org/pdf/2009.11853.pdf

{'No Header': 'KEK-TH-2260 KEK-Cosmo-0263 CTPU-PTC-20-22', 'Solar-Mass Primordial Black Holes Explain NANOGrav Hint of Gravitational Waves': 'Kazunori Kohri a,b,c and Takahiro Terada d \n- a Institute of Particle and Nuclear Studies, KEK, 1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan b The Graduate University for Advanced Studies (SOKENDAI), \n1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan \n- c Kavli Institute for the Physics and Mathematics of the Universe (WPI), \nUniversity of Tokyo, Kashiwa 277-8583, Japan \n- d Center for Theoretical Physics of the Universe, \nInstitute for Basic Science (IBS), Daejeon, 34126, Korea', 'Abstract': 'The NANOGrav collaboration for the pulsar timing array (PTA) observation recently announced evidence of an isotropic stochastic process, which may be the first detection of the stochastic gravitational-wave (GW) background. We discuss the possibility that the signal is caused by the second-order GWs associated with the formation of solar-mass primordial black holes (PBHs). This possibility can be tested by future interferometer-type GW observations targeting the stochastic GWs from merger events of solar-mass PBHs as well as by updates of PTA observations.', '1 Introduction': 'Gravitational-wave (GW) astronomy started with the successful observations of GWs from merger events of binary black holes by LIGO/Virgo collaborations [1]. GWs are also a valuable probe for the early Universe cosmology and particle physics. In particular, interests in primordial black holes (PBHs) [2-4] were reactivated after the first detection of GWs [57]. In the PBH scenario, GWs can be emitted not only from the merger of binary PBHs but also from the enhanced curvature perturbations that form PBHs [8-10]. This is due to the scalar-tensor mode couplings appearing at the second-order of the cosmological perturbation theory [11-16]. It is interesting that we can indirectly probe physics of inflation by probing the primordial scalar (curvature/density) perturbations inferred from the second-order GWs and PBH abundances [17-23]. \nRecently, the North American Nanohertz Observatory for Gravitational Waves (NANOGrav) released its 12.5-year pulsar timing array (PTA) data [24]. They search for an isotropic stochastic GW background by analyzing the cross-power spectrum of pulsar timing residuals. They reported evidence of a stochastic process, parametrized as a power-law, whose amplitude and slope are common among pulsars. The significance of the quadrupole nature in the overlap reduction function is not conclusive, whereas the monopole and dipole are relatively disfavored. This implies that the NANOGrav collaboration might have detected an astrophysical or cosmological stochastic GW background. \nIt should be noted that the NANOGrav 12.5-yr signal strength is greater than the upper bound derived in their previous 11-yr result [25] as well as that in Parkes PTA (PPTA) [26] (see Ref. [27] for the NANOGrav 11-yr constraints on PBHs and also Ref. [28] related particularly to European PTA (EPTA) constraints [29]). This apparent tension is explained primarily by the different choices of the Bayesian priors [24, 30], so all analyses can be correct given their assumptions including the priors. Specifically, the most relevant prior is on the amplitude of the red noise component associated with each pulsar. Previous PTA analyses used the uniform prior in the linear scale, whereas the NANOGrav 12.5-yr analyses used the uniform prior in the log scale. The effect of the difference is studied in detail in Ref. [30], and they found that the injected GW signal in their simulations tends to be absorbed by the red noise component more easily in the case of the (linearly) uniform prior. Moreover, the 95% confidence-level upper bound on the amplitude of the GW becomes smaller than the injected GW signal in about 50% of their simulations. This implies that the previous analyses are conservative for GW detection, but it can be regarded as aggressive in terms of upper limits. In this way, the putative GW signal and existing constraints can be consistent with each other once we take into account the differences of the priors on the pulsar red \nnoise. To claim the detection of the GW signals, however, it is also crucial to establish the quadrupole (Hellings-Downs [31]) nature of the GWs. \nAssuming the observed stochastic process is due to the detection of stochastic GW background, the NANOGrav paper [24] studied the possibility that the GWs are produced from supermassive black hole merger events (e.g., see Ref. [32]). Other possibilities for the sources of GWs include cosmic strings [33-35], the PBH formation [36, 37], and a phase transition of a dark (hidden) sector [38, 39]. \nIn this paper, we discuss the possibility that the putative GW signal is the secondorder GWs induced by the curvature perturbations that produced solar-mass PBHs. The main difference from Refs. [36, 37] is the mass range of the dominant PBH component. Ref. [36] concluded that the solar-mass PBHs abundance must be negligible and also that the supermassive black holes may be responsible for the NANOGrav signal. Ref. [37] considered a wide spectrum of the curvature perturbations and studied the possibility that the dark matter abundance is explained by O (10 -14 ) solar mass PBHs and a subdominant abundance of the solar-mass PBHs explain the NANOGrav signal. Further comparisons with Refs. [36, 37] are made in Section 5. We compare the second-order GWs and the NANOGrav result in Section 2 and interpret it in terms of PBH parameters in Section 3. Then, we discuss future tests of the scenario by measuring the stochastic GW background from mergers of solar-mass PBHs in Section 4. After the discussion in Section 5, we conclude in Section 6. We adopt the natural unit glyph[planckover2pi1] = c = 8 πG = 1.', '2 NANOGrav signals and second-order GWs': 'NANOGrav measures the strain of the GWs which is assumed to be of the power-law type in the relevant range of the analysis, \nh ( f ) = A GWB ( f f yr ) α , (1) \nwhere f is the frequency, f yr = 3 . 1 × 10 -8 Hz, A GWB is the amplitude, and α is the slope. More directly, they measure the timing-residual cross-power spectral density, whose slope is parametrized as -γ = 2 α -3. They report preferred ranges of the parameter space spanned by A GWB and γ . \nThese parameters are related to the energy-density fraction parameter Ω GW ( f ) = ρ GW ( f ) /ρ total in the following way, where ρ total is the total energy density of the Universe and the GW \nenergy density is given by ρ GW = ∫ dln f ρ GW ( f ): [25] \nΩ GW ( f ) = 2 π 2 f 2 yr 3 H 2 0 A 2 GWB ( f f yr ) 5 -γ , (2) \nwhere H 0 ≡ 100 h km/s/Mpc is the current Hubble parameter. \nIn this paper, we discuss the possibility to explain the putative signal by the secondary, curvature-induced GWs produced at the formation of O (1) M glyph[circledot] PBHs. For such PBHs, it turns out that f glyph[greaterorsimilar] f yr does not contribute significantly, and so we consider the frequency range 2 . 5 × 10 -9 Hz ≤ f ≤ 1 . 2 × 10 -8 Hz [24, 33], which corresponds to the orange contour of figure 1 of Ref. [24]. \nThe current strength of the second-order, curvature-induced GWs is given by Ω GW ( f ) = D Ω GW,c ( f ), where D = ( g ∗ ( T ) /g ∗ , 0 )( g ∗ ,s, 0 /g ∗ ,s ( T )) 4 / 3 Ω r is the dilution factor after the matterradiation equality time with Ω r being the radiation fraction 1 , and Ω GW,c ( f ) is the asymptotic value of Ω GW ( f ) well after the production of the GWs but before the equality time. This is given by \nΩ GW,c ( f ) = 1 12 ( 2 πf aH ) 2 ∫ ∞ 0 d t ∫ 1 -1 d s [ t ( t +2)( s 2 -1) ( t + s +1)( t -s +1) ] 2 × I 2 ( t, s, kη c ) P ζ ( π ( t + s +1) f ) P ζ ( π ( t -s +1) f ) , (3) \nwhere aH is the conformal Hubble parameter evaluated at the conformal time η c , P ζ ( k ) is the dimensionless power spectrum of the primordial curvature perturbations, and I 2 ( t, s, kη c ) is the oscillation average of the kernel function, whose analytic formula has been derived in Refs. [41, 42]. For the recent discussions on gauge (in)dependence, see Refs. [43-51]. \nFor the primordial curvature perturbations, we assume that there is a smooth local peak on top of the quasi-scale-invariant power spectrum measured at the cosmic-microwavebackground (CMB) scale. Such a peak can be approximated by the log-normal power spectrum \nP ζ ( k ) = A s √ 2 πσ 2 exp ( -(ln k/k ∗ ) 2 2 σ 2 ) , (4) \nwhere k = 2 πf is the wave number, A s is the amplitude, σ 2 is the variance, and ln k ∗ is \nthe average. (One can match the position of the peak, its height, and its width by the Taylor series expansion. Note that the tail parts do not need to be precisely approximated as the log-normal function.) We take σ = 1 throughout the paper as a simple representative value. An O (1) value of σ can be expected, e.g., if one assumes that the local feature of P ζ ( k ) originates from a local feature of the inflaton potential, which can be, e.g., a locally flat part (an approximate inflection point) [52], a bump, or a dip [53] in the single-field case, corresponding to some physical phenomenon occurring in O (1) e-folding time of the Hubble expansion. 2 We treat A s and k ∗ as free parameters. These can be translated to the GWparameters A GWB and γ and to the PBH parameters f PBH and M PBH , which are defined below. In the case of the log-normal power spectrum, the full (approximate) analytic formula of Ω GW,c ( f ) is available [58] although we compute it numerically with the aid of extrapolation into the IR tail using the formula of Ref. [60]. \nAn example of the spectrum of the second-order GWs is shown as the thick black line in Fig. 1. Also shown are power-law lines whose amplitude and slope correspond to points on the contours of the NANOGrav favored region on the ( A GWB , γ )-plane (the green contours in Fig. 2). The blue and cyan lines correspond to points on the upper half of 1 σ and 2 σ contours, while the orange and yellow lines correspond to points on the lower half of 1 σ and 2 σ contours, respectively. The shaded regions are the constraints from the previous PTA observations: EPTA [29], NANOGrav 11-yr [25], and PPTA [26]. The pink line at the bottom right is the prospective constraint of SKA [61]. \nIn the figure, there seems an apparent tension between the NANOGrav 12.5-yr result and the existing PTA constraints. As mentioned in the introduction, this does not necessarily mean contradiction, but it reflects the intrinsic uncertainties of Bayesian analyses. The uniform prior on the red noise for each pulsar (adopted in the existing constraints) tends to pre-assign and overestimate the power in red noise components [30], and the reweighting of the samples of the previous data in accordance with the log-uniform prior indeed weaken the previous constraints [24, 30]. An ongoing joint investigation among the PTA datasets implies a similar tendency to the results of Ref. [24] also for data other than those of NANOGrav 11-yr [24] (namely, EPTA and PPTA). Therefore, we do not worry too much about the apparent tension between these preexisting PTA constraints and our explanation for the NANOGrav 12.5-yr hint of the GWs in the following analyses. \nFigure 1: Example of the spectrum of the second-order GWs induced by the curvature perturbations that produced PBHs of M PBH = 1 M glyph[circledot] and f PBH = 1 × 10 -4 (thick black line). The power-law lines in the interval 2 . 5 × 10 -9 Hz ≤ f ≤ 1 . 2 × 10 -8 Hz are also shown that correspond to a rough visual guide of the NANOGrav signal range. The amplitudes and slopes of blue (cyan) and orange (yellow) lines are on the upper and lower 1 σ (2 σ ) contours of the NANOGrav signal, respectively. The previous PTA constraints are shown by shaded regions: EPTA [29], NANOGrav 11-yr [25], and PPTA [26]. The pink line at the bottom right is the prospective constraint of SKA [61]. \n<!-- image -->', '3 Implications for the PBH mass and its abundance': 'The relations between the second-order GWs and the properties of PBHs are as follows. The GWs are induced by the enhanced curvature perturbations, which also produce PBHs. The energy density fraction β of the PBHs at the formation time, which also has the meaning of the formation probability of a PBH in a given Hubble patch, is calculated in the PressSchechter formalism [62] 3 as \nβ = ∫ ∞ δ c d δ 1 √ 2 πσ 2 2 exp ( -δ 2 2 σ 2 2 ) glyph[similarequal] 1 2 Erfc ( δ c √ 2 σ 2 2 ) , (5) \nwhere we have assumed that the primordial curvature perturbations have the Gaussian statistics, δ c is the critical value of the coarse-grained density perturbations that produces \na PBH [68-74], for which we take δ c = 0 . 42 [74, 75] 4 , Erfc is the complementary error function, and the variance σ 2 2 of the coarse-grained density perturbations is defined as \nσ 2 2 ( k ) = 16 81 ∫ ∞ -∞ dln xw 2 ( x ) x 4 P ζ ( xk ) , (6) \nwhere w ( x ) is the window function, which we take as the modified Gaussian function w ( x ) = exp( -x 2 / 4). This window function was introduced in Ref. [76] and used as one of the two benchmark choices for the window function in Ref. [59]. Note that the choice of the window function significantly affects the abundance of the PBHs [77] (see also Ref. [78]) unless compensating parameters for the critical collapse are taken [59]. We will come back to this point in the discussion section. \nThe present energy density fraction of PBHs relative to cold dark matter is denoted by f PBH = ρ PBH /ρ CDM . This is related to β as follows, \nf PBH = ∫ dln M Ω m Ω CDM g ∗ ( T ) g ∗ ( T eq ) g ∗ ,s ( T eq ) g ∗ ,s ( T ) T T eq glyph[epsilon1]β , (7) \nwhere the subscript m and eq denote the non-relativistic matter and the equality time, the temperature T is evaluated at the horizon entry of the corresponding mode k , and glyph[epsilon1] denotes the fraction of the horizon mass that goes into the PBH, which we take glyph[epsilon1] = 3 -3 / 2 [4]. More detailed explanation for PBH formation and parameter dependencies can be found, e.g., in Refs. [79, 80] and in reviews [81-86]. \nWe relate k and the horizon mass in the standard way, i.e., using the Friedmann equation. Note, however, that there is a discrepancy between the average PBH mass M PBH and a naive horizon mass corresponding to k ∗ because of two reasons: the peak position of σ 2 2 ( k ) is smaller than k ∗ , and each PBH mass is glyph[epsilon1] times smaller than the corresponding horizon mass. These shifts of peak positions were discussed, e.g., in Ref. [87] and recently emphasized again [59]. \nConcretely, the relation among the wave number k ∗ , the corresponding frequency f ∗ = k ∗ / (2 π ), the corresponding horizon mass M , and the average PBH mass M PBH is as follows: \nM PBH 1 . 0 M glyph[circledot] glyph[similarequal] M 0 . 31 M glyph[circledot] glyph[similarequal] ( k ∗ 3 . 3 × 10 6 Mpc -1 ) -2 glyph[similarequal] ( f ∗ 5 . 0 × 10 -9 Hz ) -2 . (8) \nWe vary the scalar amplitude in the range 0 . 015 ≤ A s ≤ 0 . 040 and the average PBH mass in the range 0 . 2 ≤ M PBH /M glyph[circledot] ≤ 5. The resultant Ω GW h 2 is fitted by a power-law line in the aforementioned range 2 . 5 × 10 -9 Hz ≤ f ≤ 1 . 2 × 10 -8 Hz to extract the amplitude of the GW strain A GWB and the slope γ . Note that A GWB ∝ A s , but it also depends on k * (or M PBH ) since the pivot scale is fixed to f yr (see eq. (1)). The result is shown in Fig. 2. From the figure, we see that a large fraction of the scanned parameter space can explain the NANOGrav signal. \nFigure 2: Parameter scan in the range 0 . 015 ≤ A s ≤ 0 . 040 and 0 . 2 ≤ M PBH /M glyph[circledot] ≤ 5 shown as the red shaded region. A larger A s corresponds to a larger A GWB , and a larger M PBH corresponds to a larger γ . The thin red lines correspond to f PBH = 10 -1 , 10 -4 , 10 -7 , and 10 -10 from top to bottom. The 1 σ and 2 σ NANOGrav contours are also shown. \n<!-- image --> \nThe scanned parameter range for A s corresponds to that of the PBH abundance f PBH as shown in Fig. 3. The upper and lower ends correspond to M PBH = 0 . 2 M glyph[circledot] and 5 M glyph[circledot] , respectively. \nCombining the information in Figs. 2 and 3, one can map the NANOGrav contours onto the PBH parameter space ( M PBH , f PBH ), which are shown as the green contours in Fig. 4. The non-smoothness of the contours largely originates from the non-smoothness of the original NANOGrav contours. The uncertainty of extracting the data from the original contours is magnified in this figure compared to Fig. 2. Therefore, the 1 σ and 2 σ boundary has an uncertainty of very roughly an order of magnitude. \nFig. 4 shows that the PBH mass should be around a solar mass to explain the NANOGrav \nFigure 3: Relation between the scalar amplitude A s and the PBH abundance f PBH for M PBH /M glyph[circledot] = 0 . 2 (top, solid), 1 (middle, dashed), and 5 (bottom, dotted). \n<!-- image --> \nsignal. Also, it shows that f PBH close to unity is disfavored, but f PBH ∼ 0 . 1 is within the 2 σ contour depending on the value of M PBH . \nA part of such regions is excluded by existing constraints shown by shaded regions at the top of the figure. These include the microlensing constraints by EROS/MACHO collaborations [88, 89], the caustic crossing constraint [90], Advanced LIGO constraints on the subsolar mass range (individual events [91] and superposition of events [92, 93]), and the constraints due to photo-emission during gas accretion onto PBHs [94-96]. There are many subdominant but independent and complementary constraints around this mass range (see Ref. [85]). There is also the LIGO/Virgo constraints on supersolar mass range [97, 98]. Ref. [98] implies a substantial dependence on the width of the mass function, so we do not include it in Fig. 4.', '4 Testing the scenario with the GWs from mergers': "The solar-mass PBH possibility for NANOGrav can be tested by the detection of stochastic GW background from the superposition of binary solar-mass PBH merger events. The GW \nFigure 4: NANOGrav contours (green) on the plane of the average PBH mass M PBH and the PBH abundance f PBH . The dark shaded regions at the top are constraints from EROS-2 [88] and MACHO [89] (brown), caustic crossing [90] (purple), Advanced LIGO O2 (subsolar mass range) [91] (gray), Advanced LIGO non-detection of the stochastic GW background [92, 93] (cyan), and the E -mode polarization of the CMB due to the disk-shaped gas accretion [94] (blue). \n<!-- image --> \nspectrum is obtained as \nΩ merger GW ( f ) = f 3 H 2 0 ∫ f cut f -1 0 d z R ( z ) (1 + z ) H ( z ) d E GW d f s , (9) \nwhere f cut (= O (1 /M PBH )) is the UV cutoff frequency at the source frame (i.e., without the redshift factor) (see Refs. [99, 100] for the IR 'cutoff' frequency), f s is the frequency at the source frame, z is the redshift, R is the comoving merger rate, and E GW is the energy of the GWs at the source frame. The expressions of f cut , R , and d E GW / d f s are found in Appendices B and C of Ref. [87]. See also Refs. [7, 84, 92, 101, 102] for more details. The frequency f cut is just the maximal cutoff appearing around the end of the merger process. \nThe result is shown in Fig. 5 as the black lines where M PBH = 1 M glyph[circledot] and f PBH = 10 -2 (solid), 10 -3 (dashed), 10 -4 (dotted), and 10 -5 (dot-dashed). Various prospective constraints (see the caption) 5 as well as the lines in Fig. 1 are also shown. We do not show the M PBH \nFigure 5: GW spectrum from the superposition of binary PBH merger events (thin black) with M PBH = 1 M glyph[circledot] and f PBH = 10 -2 (solid), 10 -3 (dashed), 10 -4 (dotted), and 10 -5 (dotdashed). Future prospects of various GW observations are also shown: SKA [61], LISA [103], TianQin [104, 105], BBO [106], DECIGO [107], AION [108], AEDGE [109], Advanced LIGO Hanford and Livingston [110] combined with Advanced Virgo [111] as well as LIGO India [112, 113] and KAGRA [114, 115] (HLVIK), and Einstein Telescope [116] and two thirdgeneration Cosmic Explorers [117] (ET+2CE). The shaded red region is the Advanced LIGO O2 constraint [118]. Sensitivity curves have been read from Refs. [33, 108, 119, 120]. The top side of the figure is the upper bound Ω GW h 2 < 1 . 8 × 10 -6 from the (non-)adiabatic N eff bound of big-bang nucleosynthesis [42]. The existing PTA constraints and NANOGrav power-law guides are also shown as in Fig. 1. \n<!-- image --> \ndependence in the figure, but the spectra shift to the left as M PBH increases. Eq. (8) clearly shows that the characteristic frequency f ∗ of the second-order GWs scales as M -1 / 2 PBH , whereas the counterpart for the GWs from mergers scales as f cut ∼ M -1 PBH (see the text below eq. (9)) as demonstrated in Ref. [87]. Note that the thick black line corresponds to the second-order GWs for M PBH = 1 M glyph[circledot] and f PBH = 10 -4 , but the f PBH dependence is weak (see Fig. 3). The top end of the figure is the upper bound Ω GW h 2 < 1 . 8 × 10 -6 [42] from the fact that the GWs contribute to the effective number of neutrinos N eff and affect the big-bang nucleosynthesis. We can see from the figure that a large part of the parameter space can be probed by the future GW observations.", '5 Discussion': "Our results depend on various assumptions. Some of them have been already stated, but we emphasize them again. First, we do not consider the effect of the critical collapse [83, 128130] since it occurs only when the spherical symmetry is precisely respected. It is clear that the rare high-peak has approximately the spherical shape [131], but the spherical symmetry must be realized to high precision for the critical collapse to happen [68]. On the other hand, Refs. [36, 37] include the effect of the critical collapse. It will be interesting to compare our results with an analysis including the critical collapse effect using a consistent parameter set [59]. In our preliminary study, we found a qualitatively similar feature that f PBH tends to become larger than those reported in Refs. [36, 37]. \nSecond, we have chosen the modified Gaussian window function, whose width is twice as large as the standard Gaussian window function. This boosts the value of f PBH for a given value of A s . This may be the largest difference compared to Refs. [36, 37] in which much smaller f PBH 's were reported. \nThird, we have not taken into account the nonlinear relation between the primordial curvature perturbations and the density perturbations (see Refs. [63, 132]). This inevitably leads to non-Gaussianity of the density perturbations [132]. Also, the inclusion of the intrinsic non-Gaussianity of the primordial curvature perturbations significantly affects f PBH [23, 133, 134]. It also affects the second-order GWs [23, 135-139]. \nFourth, we have not included the transfer function of the curvature perturbations in the definition of σ 2 2 . This is preferred in Ref. [76]. If we include the transfer function, however, σ 2 2 will reduce by 'several' percent. This reduces f PBH non-negligibly. \nIt is also worth mentioning that we have not taken into account the softening of the equation-of-state during the phase transition/crossover of quantum chromodynamics (QCD). See Refs. [37, 140, 141] for its enhancement effect on the PBH abundance f PBH for a given scalar amplitude A s . Depending on the boost factor, this may realize a better fit for the NANOGrav signal simultaneously with stronger and more easily detectable GWs from mergers of the solar-mass binary PBHs. The softening also slightly affects the spectrum of the second-order GWs [142]. \nWe discussed a possible detection of the PBHs with the masses of O (1) M glyph[circledot] only by a future interferometer-type GW observations in Section 4. Complementarily, however, we can also measure such PBHs by the future optical/IR telescopes through microlensing events, e.g., Subaru HSC towards M31 for 10 year observations [143] or by the future precise CMB observations of E - and B -mode polarization due to photon emission from an accretion disk around a PBH, e.g., by LiteBIRD [144] or CMB-S4 [145].", '6 Conclusion': 'In this paper, we have interpreted the recently reported NANOGrav 12.5-yr excess of the timing-residual cross-power spectral density in the low-frequency part as a stochastic GW background. We conclude that, under our assumptions, the second-order GWs induced by the curvature perturbations that produced a substantial amount of O (1) solar-mass PBHs can explain the NANOGrav stochastic GW signal. In particular, the abundance of the PBHs can be sufficiently large so that future GW observations can test this possibility by measuring the stochastic GW background produced by mergers of the solar-mass PBHs. This is nontrivial since the suitable scalar amplitude A s could a priori produce too many PBHs that are excluded by existing observational constraints or too few PBHs that do not lead to the detectable stochastic GW background from merger events. Similarly, for a given f PBH , the second-order GWs could be too strong or weak. Since the relation between A s and f PBH depends crucially on the ambiguity for the choice of the windows function as discussed in the previous section, a further study to refine the PBH formation criterion is necessary.', 'Note Added': 'Taking into account the uncertainties of PBH abundance calculations, i.e., the different choices of the window function, the value of δ c (see footnote 4), etc., our results are largely consistent with those of Ref. [37] [146]. The difference from Ref. [36] is also discussed in the note added in Ref. [36]. 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2000ApJ...544..993S

Complete RXTE Spectral Observations of the Black Hole X-ray Nova XTE J1550-564

2000-01-01

['black hole physics', '-', 'astronomy x rays', 'astrophysics']

[]

We report on the X-ray spectral behavior of XTE J1550-564 during its 1998-1999 outburst. XTE J1550-564 is an exceptionally bright X-ray nova and is also the third Galactic black hole candidate known to exhibit quasi-periodic X-ray oscillations above 50 Hz. Our study is based on 209 pointed observations using the PCA and HEXTE instruments on board the Rossi X-Ray Timing Explorer (RXTE) spanning 250 days and covering the entire double-peaked eruption that occurred from 1998 September until 1999 May. The spectra are fitted to a model including multicolor blackbody disk and power-law components. The spectra from the first half of the outburst are dominated by the power-law component, whereas the spectra from the second half are dominated by the disk component. The source is observed in the very high and high/soft outburst states of black hole X-ray novae. During the very high state, when the power-law component dominated the spectrum, the inner disk radius is observed to vary by more than an order of magnitude; the radius decreased by a factor of 16 in one day during a 6.8 crab flare. If the larger of these observed radii is taken to be the last stable orbit, then the smaller observed radius would imply that the inner edge of the disk is inside the event horizon! However, we conclude that the apparent variations of the inner disk radius observed during periods of increased power-law emission are probably caused by the failure of the multicolor disk/power-law model; the actual physical radius of the inner disk may remain fairly constant. This interpretation is supported by the fact that the observed inner disk radius remains approximately constant over 120 days in the high state, when the power-law component is weak, even though the disk flux and total flux vary by an order of magnitude. The mass of the black hole inferred by equating the approximately constant inner disk radius observed in the high/soft state with the last stable orbit for a Schwarzschild black hole is M<SUB>BH</SUB>=7.4 M<SUB>solar</SUB>(D/6 kpc)(cosi)<SUP>-1/2</SUP>.

[]

https://arxiv.org/pdf/astro-ph/0005599.pdf

{'Complete RXTE Spectral Observations of the Black Hole X-ray Nova XTE J1550-564': 'Gregory J. Sobczak 1 , Jeffrey E. McClintock 2 , Ronald A. Remillard 3 , Wei Cui 3 , Alan M. Levine 3 , Edward H. Morgan 3 , Jerome A. Orosz 4 , and Charles D. Bailyn 5', 'ABSTRACT': 'We report on the X-ray spectral behavior of XTE J1550-564 during its 1998-99 outburst. XTE J1550-564 is an exceptionally bright X-ray nova and is also the third Galactic black hole candidate known to exhibit quasiperiodic X-ray oscillations above 50 Hz. Our study is based on 209 pointed observations using the PCA and HEXTE instruments onboard the Rossi X-ray Timing Explorer spanning 250 days and covering the entire double-peaked eruption that occurred from 1998 September until 1999 May. The spectra are fit to a model including multicolor blackbody disk and power-law components. The spectra from the first half of the outburst are dominated by the power-law component, whereas the spectra from the second half are dominated by the disk component. The source is observed in the very high and high/soft outburst states of black hole X-ray novae. During the very high state, when the power-law component dominated the spectrum, the inner disk radius is observed to vary by more than an order of magnitude; the radius decreased by a factor of 16 in one day during a 6.8 Crab flare. If the larger of these observed radii is taken to be the last stable orbit, then the smaller observed radius would imply that the inner edge of the disk is inside the event horizon! However, we conclude that the apparent variations of the inner disk radius observed during periods of increased power-law emission are probably caused by the failure of the \nmulticolor disk/power-law model; the actual physical radius of the inner disk may remain fairly constant. This interpretation is supported by the fact that the observed inner disk radius remains approximately constant over 120 days in the high state, when the power-law component is weak, even though the disk flux and total flux vary by an order of magnitude. The mass of the black hole inferred by equating the approximately constant inner disk radius observed in the high/soft state with the last stable orbit for a Schwarzschild black hole is M BH = 7 . 4 M /circledot ( D/ 6 kpc )(cos i ) -1 / 2 . \nSubject headings: black hole physics - stars: individual (XTE J1550-564) X-rays: stars', '1. Introduction': "The X-ray nova and black hole candidate XTE J1550-564 was discovered with the All Sky Monitor (ASM; Levine et al. 1996) onboard the Rossi X-ray Timing Explorer (RXTE) just after the outburst began on 1998 September 6 (Smith et al. 1998). The source exhibited a flare on 1998 September 19-20 that reached 6.8 Crab (or 1.6 × 10 -7 erg s -1 cm -2 ) at 2-10 keV. The discovery of XTE J1550-564 prompted a series of almost daily pointed RXTE observations with the Proportional Counter Array (PCA; Jahoda et al. 1996) and the High-Energy X-ray Timing Experiment (HEXTE; Rothschild et al. 1998) instruments. The first 14 RXTE observations were part of a guest observer program with results reported by Cui et al. (1999). They found that during the initial X-ray rise (0.7-2.4 Crab at 2-10 keV), the source exhibited very strong quasiperiodic X-ray oscillations (QPOs) in the range 0.08-8 Hz. The spectral and timing analysis of 60 additional RXTE observations, reported in Sobczak et al. (1999a) and Remillard et al. (1999a), revealed the presence of canonical outburst states characteristic of black hole X-ray novae (see Tanaka & Lewin 1995) and X-ray QPOs at a few Hz and ∼ 200 Hz. XTE J1550-564 is one of only a few Galactic black hole candidate known to exhibit QPOs above 50 Hz (Remillard et al. 1999a; Homan, Wijnands, & van der Klis 1999); the others are 4U 1630-47 (Remillard & Morgan 1999) and the microquasars GRS 1915+105 (Morgan, Remillard, & Greiner 1997) and GRO J1655-40 (Remillard et al. 1999b). \nThe X-ray light curve of XTE J1550-564 from the ASM aboard RXTE is shown in Figure 1. The outburst exhibits a 'double-peaked' profile, with the first half generally dominated by power-law emission, and the second half generally dominated by emission from the accretion disk (see § 3). The double-peaked profile of the outburst is different from \nthe outbursts of classical X-ray novae like A0620-00 (see Chen, Shrader, & Livio 1997), but is similar to the outburst behavior of the microquasar GRO J1655-40 (Sobczak et al. 1999b). \nThe optical (Orosz, Bailyn, & Jain 1998) and radio (Campbell-Wilson et al. 1998) counterparts of XTE J1550-564 were identified shortly after the source was discovered. The presence of an optical counterpart, with B ∼ 22 mag in quiescence (Jain et al. 1999), is especially important since this will allow radial velocity studies of the companion star during quiescence that could confirm the black hole nature of the primary. \nHerein we present spectral results for 209 X-ray observations spanning the entire 250 days of the 1998-99 outburst of XTE J1550-564. These observations include the rising phase observations with results first reported by Cui et al. (1999) (RXTE program P30188-06), all of our RXTE guest observer program (P30191), and all of the public observations of this source (P30435 & P40401). The spectral analysis of PCA observations 15-75 (see Table 1) was first reported by Sobczak et al. (1999a). A timing study based on those same RXTE observations is presented in Remillard et al. (1999a) and observations of the optical counterpart are presented in Jain et al. (1999). Low frequency QPOs (0.08-18 Hz) were observed during 74 of the 209 observations reported in the present paper. The frequencies, amplitudes, and coherence factors (Q) of these QPOs can be found in Table 1 of Sobczak et al. (2000).", '2. Observations and Analysis': 'We present spectral results for 209 observations of XTE J1550-564 (see Fig. 1) obtained using the PCA and HEXTE instruments onboard RXTE. The PCA consists of five xenon-filled detector units (PCUs) with a total effective area of ∼ 6200 cm -2 at 5 keV. The PCA is sensitive in the range 2-60 keV, the energy resolution is ∼ 17% at 5 keV, and the time resolution capability is 1 µ sec. The HEXTE consists of two clusters of 4 NaI/CsI phoswich scintillation detectors, each with an effective area of 800 cm 2 , covering the energy range 15 to 250 keV with an energy resolution of 9 keV at 60 keV. A journal of the PCA/HEXTE observations of XTE J1550-564 is given in Table 1, including exposure times, count rates, etc. \nThe PCA and HEXTE pulse height spectra were not fit simultaneously because of uncertainty in the cross-calibration of the instruments. This uncertainty is apparent when fitting the spectrum of the Crab. Fitting the Crab spectrum (e.g. 1997 December 15 and 1999 March 24) to a power-law using either PCA or HEXTE data alone yields χ 2 ν ∼ 1 for \neach detector; however, fitting PCA and HEXTE data simultaneously, even while floating the relative normalization of the detectors, results in χ 2 ν ∼ 4. For this reason, we present separate fits to the PCA and HEXTE spectra.', '2.1. PCA': "The PCA data were taken in the 'Standard 2' format, which consists of 129 channel spectra accumulated for each PCU every 16 s. The data span PCA gain epochs 3 & 4 (epoch 4, which covers observations 170-209, began when the PCA gain was lowered on 1999 March 22 at 16:30 UT). The epoch 3 response matrix for each PCU was obtained from the 1998 January distribution of response files, and the response matrix for epoch 4 was obtained from the RXTE GOF website and dated 1999 March 31. We tested the latest epoch 4 response matrices released with FTOOLS v5.0 and found that there are no significant changes in the fits. The pulse height spectrum from each PCU was fit over the energy range 2.5-20 keV for epoch 3, using a systematic error in the count rates of 1%. For epoch 4, the pulse height spectrum from each PCU was fit over the energy range 3-20 keV, using a systematic error in the count rates of 2%. The lower limit of the energy range was raised to 3 keV for epoch 4 because the sensitivity of the detectors at low energy decreased when the gain was lowered, and a systematic error of 2% was adopted in order to maintain χ 2 ν ∼ 1 for the Crab nebula. All of the spectra were corrected for background using the standard bright source background models appropriate for each epoch. We began using the epoch 4 faint source background model beginning on 1999 April 27, when the source count rate dropped below 40 c/s/PCU. Only PCUs 0 & 1 were used for the spectral fitting reported here and the spectra from both PCUs were fit simultaneously using XSPEC (Sobczak et al. 1999a,b). \nThe PCA spectral data were fit to the widely used model consisting of a multicolor blackbody accretion disk plus power-law (Tanaka & Lewin 1995; Mitsuda et al. 1984; Makishima et al. 1986). The fits were significantly improved by including a smeared Fe absorption edge near 8 keV (Ebisawa et al. 1994; Inoue 1991) and a weak Fe emission line; the best-fit line had a central energy around 6.5 keV, a width held fixed at 1.2 keV (FWHM), and an equivalent width ∼ < 100 eV. Interstellar absorption was modeled using the Wisconsin cross-sections (Morrison & McCammon 1983). All of the observations were first fit with a floating hydrogen column density ( N H ), which was generally in the range 1.5-2.5 × 10 22 cm -2 (Jain et al. 1999 estimated N H = 0 . 9 × 10 22 cm -2 from optical reddening). The fits were insensitive to N H differences in this small range, and in the final analysis presented here, N H was fixed at 2 . 0 × 10 22 cm -2 . There are a total of eight free \nparameters: the apparent temperature ( T in ) and radius ( R in ) of the inner accretion disk, the power-law photon index (Γ) and normalization ( K ), the edge energy and optical depth ( τ Fe ) of the Fe absorption feature, and the central energy and normalization of the Fe emission line. \nThe addition of the Fe emission and absorption components is motivated in Figures 2a & 2b, which show the ratio of a typical spectrum to the best fit model without and with the Fe emission and absorption. The addition of the Fe emission and absorption components reduces the χ 2 ν from 7.9 to 0.9 in this example. The Fe line and edge energies also agree with the relation in Figure 17 of Nagase (1989) and indicate variations in the ionization state of Fe during the outburst. \nThe fitted temperature and radius of the inner accretion disk presented here ( T in and R in ) are actually the color temperature and radius of the inner disk, which may be affected by spectral hardening due to electron scattering (Shakura & Sunyaev 1973; Shimura & Takahara 1995). The physical interpretation of these parameters remains uncertain and is discussed below. The reader should also note that the inner disk radius is obtained from the normalization of the multicolor disk model, R in (cos i ) 1 / 2 / ( D/ 6 kpc ), which is a function of the distance D and inclination i of the system. We use i = 0 and D = 6 kpc for XTE J1550-564, but the actual distance and inclination are unknown. Six representative spectra are shown in Figures 3a-3f. The model parameters and component fluxes (see Tables 2 & 3) are plotted in Figures 4a-4f. All uncertainties are given at the 1 σ confidence level. Unless otherwise noted, the spectral parameters discussed in this paper are those derived using the PCA as opposed to the HEXTE spectra.", '2.2. HEXTE': "The standard HEXTE reduction software was used for the extraction of the HEXTE archive mode data. The HEXTE modules were alternatingly pointed every 32 s at source and background positions, allowing background subtraction with high sensitivity to time variations in the particle flux at different positions in the spacecraft orbit. Clusters A and B were fit simultaneously and the normalization of each cluster was allowed to float independently, since there is a small systematic difference between the normalizations of the two clusters. We used the HEXTE response matrices released 1997 March 20. Only the data above 20 keV was used because of uncertainty in the response at lower energies. \nThe source was not detected in the HEXTE during all observations. For those observations in which the source was detected, the HEXTE spectra were fit to a power-law \nmodel from 20 keV to the maximum energy at which the source was detected, which ranged from 50 to 200 keV. In a number of cases we found that the HEXTE spectra could not be adequately fit using a pure power-law model in the observed energy range (Fig. 5a-5d). In these instances, we used a power-law with a high energy cutoff of the form (cf. Grove et al. 1998): \nN ( E ) = KE -Γ for E ≤ E cut (1a) \n= KE -Γ exp (( E cut -E ) /E fold ) for E ≥ E cut (1b) \nThe addition of the high energy cutoff improved the value of χ 2 ν from 4.9 to 0.7 for the observation on 1998 Sept. 7 (Fig. 5) and gave similarly dramatic improvements for many other observations. A three parameter cutoff model (the equivalent of E cut = 0) can also fit the data in some cases. Similarly, the 'comptt' model in XSPEC v.10 (Titarchuk 1994) can be used to fit most of these data, but it tends to cutoff more rapidly than the data at high energy. These data can also be adequately fit by a model including a pure power-law component plus a broad gaussian or reflection component. The presence of a reflection component is possible, given the inclusion of Fe absorption when fitting the PCA spectra (see above). If a reflection component is present, the fit parameters indicate that it would contribute as much as half of the 2-20 keV or 2-100 keV flux for the 1998 Sept. 7 observation. Observations at higher energies are necessary to determine whether the high energy cutoff seen in the 20-200 keV range is due to a physical cutoff in the power-law component or an underlying reflection feature. \nThe HEXTE model parameters are given in Table 4. The parameters of the high energy cutoff are given only for those cases where the addition of the cutoff improved the value of χ 2 ν by more than 30%. The value of χ 2 ν for the pure power-law model is also given for comparison in these cases. Upper limits for the 20-100 keV HEXTE flux were determined by assuming a fixed power-law photon index of 2.5. The reader should also note that the normalization of the HEXTE instrument is systematically ∼ 20-30% lower than the PCA; we have not adjusted our spectral fits of the HEXTE data to correct for this discrepancy.", '3. Spectral Results': 'Six representative spectra are shown in Figures 3a-3f. These spectra illustrate the range of X-ray spectra for XTE J1550-564, in which (a) a strong power-law with high energy cutoff dominates a warm disk (rising phase), (b) an intense power-law component dominates a hot disk (very high state flare), (c,d) the disk dominates a weak power-law \n(high/soft state), (e) a strong power-law dominates a warm disk, and (f) a weak power-law dominates a cool disk. \nAs discussed in the introduction, the outburst of XTE J1550-564 can be divided into two halves, which we now discuss in turn. We define the first half as extending from the start of the outburst on MJD 51062 (1998 September 6) to approximately MJD 51150 (1998 December 3). During the initial rise (MJD 51063-51072), the spectra are dominated by the power-law component with a photon index that gradually steepens from Γ = 1 . 5 to 2.5 (Fig. 4c, Table 2), and the source displays strong 0.08-8 Hz QPOs (Cui et al. 1999). A high energy cutoff is observed in the HEXTE spectrum during the initial rise with E fold ∼ 70-150 (Fig. 3a, Table 4). Following the initial rise, the spectra from MJD 51074 to 51115 remain dominated by the power-law component (Fig. 3b) which has photon index Γ ∼ 2.4-2.9 (Fig. 4c). The source displays strong 3-13 Hz QPOs during this time, whenever the power-law contributes more than 60% of the observed X-ray flux (Remillard et al. 1999a; Sobczak et al. 1999a). This behavior is consistent with the very high state of Black Hole X-ray Novae (BHXN) (see Tanaka & Lewin (1995) and references therein for details on the spectral states of BHXN). A high energy cutoff in the power-law tail is observed during a number of very high state observations (Table 4). The peak luminosity (bolometric disk luminosity plus 2-100 keV power-law luminosity) during the flare on MJD 51075 is L = 1 . 2 × 10 39 ( D/ 6 kpc ) 2 erg s -1 , which corresponds to the Eddington luminosity for M = 9 . 6 M /circledot at 6 kpc (Sobczak et al. 1999a). \nAfter MJD 51115 (1998 October 29), the source fades rapidly, the power-law component decays and hardens (Γ = 2 . 0 -2 . 4), and the disk component begins to dominate the spectrum (see Fig. 3c). The source generally shows little temporal variability during this time (Remillard et al. 1999a; Sobczak et al. 1999a). We identify this behavior with the high/soft state . However, during this time the source occasionally exhibits QPOs at ∼ 5 Hz and power density spectra that have properties intermediate between the very high and high/soft states or the high/soft and low/hard states (Remillard et al. 1999a; Sobczak et al. 1999a). The low/hard state was not observed and the intensity increased dramatically after MJD 51150, marking the start of the second half of the outburst. \nDuring the second half of the outburst (MJD 51150-51230), most of the observed spectra are dominated by the disk component (Fig. 3d), which contributes > 85% of the 2-20 keV flux (Fig. 4e), and no QPOs are observed. Figure 4c shows that Γ increased sharply at the onset of the second half of the outburst from Γ ∼ 2 to 4. These features are typical of the high/soft state of BHXN. This dichotomy between the power-law-dominated first half and disk-dominated second half of the outburst is shared by GRO J1655-40, and cannot be easily explained by the standard disk instability model. \nAlso during the second half of the outburst, there are a few instances lasting two or three days when the power-law hardens from Γ ∼ 4 to 2.5 (Fig. 4c). During one of these instances (MJD 51201), the power-law flux increases by almost a factor of two and the source is also detected in the HEXTE (see Table 4). After MJD 51230 (1999 February 21), the power-law hardens considerably (Fig. 4c) and there is an intense power-law flare, which begins on MJD 51241 (1999 March 4; see Fig. 4d), accompanied by a sharp decline in the disk flux (Fig.4e). QPOs from 5-18 Hz also reappear during this power-law flare. A sample spectrum from this flare is shown in Figure 3e. A high energy cutoff in the power-law component is marginally detected in two HEXTE observations during the second half of the outburst (Table 4). The total flux decreases steadily as the power-law flare fades after MJD 51260 (1999 March 23; see Fig. 4f). The spectrum of one of the last few observations resembles the low/hard state and is shown in Figure 3f. \nA comparison of the 2-12 keV (soft) PCA flux and the 20-100 keV (hard) HEXTE flux is shown in Figure 6. During the first few observations, the hard flux exceeds or is approximately equal to the soft flux. Following the initial rise, during the very high state in the first half of the outburst, the soft flux exceeds the hard flux by almost an order of magnitude. During the high/soft state in the second half of the outburst, the source is usually undetectable in the HEXTE, and the upper limits on the hard flux show that the soft flux exceeds the hard flux by more than 2.5 orders of magnitude. This strong dominance of the soft flux over the hard flux is characteristic of the high/soft state. The hard flux increases again during the power-law flare and subsequent decline that mark the end of the outburst. Similar variations in the relative strength of the soft and hard flux between outburst states were also observed for the BHXN Nova Muscae 1991 (Esin, McClintock, & Narayan 1997) and are a good means of differentiating the very high and high/soft states. \nFrom Figure 4b, it appears that the inner radius of the disk does not appear to be constant throughout the outburst cycle. From Figure 4b and Table 2, we see that the intense 6.8 Crab flare on MJD 51075 (1999 September 19) is accompanied by a dramatic decrease of the inner disk radius from 33 to 2 km over one day. Similar behavior was observed for GRO J1655-40 during its 1996-97 outburst: The observed inner disk radius decreased by almost a factor of four during periods of increased power-law emission in the very high state, and it was generally larger in the high/soft state (Sobczak et al. 1999b). Below we discuss both the problems and possible interpretations that are relevant to our spectral results, focussing on the observed variation of the inner disk radius.', '4.1. The Inner Disk Radius': 'The physical radius of the inner disk may vary in XTE J1550-564 and GRO J1655-40 - by as much as a factor of 16 in one day in the case of XTE J1550-564. Another possibility, however, is that the apparent decrease of the inner disk radius observed during intense flares is caused by the failure of the multicolor disk/power-law model at these times. This failure may be caused by one or both of the following effects: (1) Variations in spectral hardening, which occurs because electron scattering as opposed to free-free absorption dominates the opacity in the inner disk (Shakura & Sunyaev 1973). This causes the observed (color) temperature of the disk to appear higher than the effective temperature, which decreases the normalization of the multicolor disk model from which the radius is inferred (Ebisawa et al. 1994; Shimura & Takahara 1995). The spectral hardening correction is likely not to be constant in extreme situations, such as the 6.8 Crab flare, and any increase in spectral hardening will appear as a decrease in the observed inner disk radius. (2) The Compton upscattering of soft disk photons, which is likely the origin of the power-law component in BHXN. In this case, an increase in power-law emission naturally implies an increase in the Comptonization of soft disk photons. Therefore, the measured normalization of the multicolor disk (from which the radius is derived) represents only a fraction of the intrinsic X-ray emission from the disk because the photons that are upscattered to produce the power-law component are missing. This causes the inferred radius to decrease (Zhang et al. 1999), although a corresponding increase in the mass accretion rate is necessary to explain the associated increase in the apparent disk temperature. Thus, intense flares may cause an apparent decrease in the radius of the inner disk due to increased spectral hardening and/or Compton upscattering of soft disk photons, while the actual physical radius may remain fairly constant. \nThe above interpretation is bolstered by the constancy of the inner disk radius of XTE J1550-564 when the power-law is weak: The observed inner disk radius remains approximately constant at ∼ 40 km (assuming i = 0 · , D = 6 kpc) over 120 days from MJD 51120-51240 when the power-law contributes less than 20% of the 2-20 keV flux, even though the disk flux and total flux vary by an order of magnitude. The observed inner disk radius in GRO J1655-40 also remains approximately constant over more than 150 days during the high state, when the power-law contributes less than 10% of the flux (Sobczak et al. 1999b). Similar behavior has been observed for several other BHXN and Galactic black hole candidates, where the observed stable inner disk radius was plausibly identified with the last stable orbit (Tanaka & Lewin 1995). If we hypothesize that the stable value of the radius for XTE J1550-564 is a reasonable measure of the radius of the last stable orbit \n(6 r g where r g = GM/c 2 ), then a drop by a factor of 20 (observed between the peak flare and the high/soft state) would imply a decrease from 6 r g to 0 . 3 r g , which is well within the event horizon of the black hole. This result is independent of the assumed distance and inclination. We thus conclude that the small inner disk radius observed during the 6.8 Crab flare is unphysical and that the inner disk radius during the most intense power-law activity cannot be reliably determined from the multicolor disk model. This interpretation is supported by the work of Merloni, Fabian, & Ross (1999) who used a self-consistent model for radiative transfer and vertical temperature structure in a Shakura-Sunyaev disk and found that (1) the multicolor disk model systematically underestimates the inner disk radius when most of the gravitational energy is dissipated in the corona and (2) the multicolor disk model gives stable, acceptable results for high accretion rates ( ∼ > 10% of the Eddington accretion rate) and/or when a lower fraction of the gravitational energy is dissipated in the corona. \nIn contrast to the apparent decrease of the inner disk radius observed during the 6.8 Crab flare, the radius appears to increase at times near the beginning and end of the outburst cycle (Fig. 4b). It is not clear whether these observed increases of the inner disk radius are physical or not; in the following we discuss two possible causes. \nThe Advection-Dominated Accretion Flow (ADAF) model predicts changes in the inner disk radius during the initial rise and the final decline of the outburst. According to the ADAF model, the inner edge of the disk should move inward (from hundreds of gravitational radii) to the last stable orbit during the initial rise, and it should move outward again at the end of the outburst during the transition from the high/soft state to the low/hard state (Esin, McClintock, & Narayan 1997). However, the ADAF model predicts that the radius should move inward on a viscous timescale of several days (Hameury et al. 1998) during the initial rise; instead, about 30 days elapsed between the initial rise and the time when the observed inner disk radius reached an approximately constant value (Fig. 4b). On the other hand, the observed increase of the inner disk radius near the end of the outburst from MJD 51279-51293 is in better agreement with the predictions of the ADAF model. The critical mass accretion rate at which the soft-to-hard transition takes place in the ADAF model is approximately ˙ m crit ∼ 1 . 3 α 2 (in Eddington units), where α is the standard viscosity coefficient (Esin et al. 1997). The ADAF model predicts that the inner edge of the disk should begin to move outward from the last stable orbit when ˙ m ∼ < ˙ m crit . We can estimate the mass accretion rate ( ˙ m ) of XTE J1550-564 in Eddington units by assuming that the 6.8 Crab flare corresponds to the Eddington accretion rate and scaling the inferred accretion rate by the total observed flux. The ratio of the total observed flux on MJD 51279, when the observed radius begins to increase, to the total flux during the 6.8 Crab flare corresponds to a mass accretion rate ˙ m ∼ 0 . 02 (Table 3). The mass accretion \nrate at this time corresponds to the critical accretion rate for α = 0 . 12, which is smaller than the value α = 0 . 25 used by Esin et al. (1997) to model the outburst of Nova Muscae 1991, but is still in reasonable agreement with the predictions of the ADAF model. \nIt is also possible that the observed increase of the inner disk radius at early times is not physical, but rather due to the flattening of the radial temperature profile from the T ∼ r 3 / 4 assumed in the multicolor disk model. This effect would cause the radius inferred from the multicolor disk model to increase. The flattening of the radial temperature profile in the disk could be caused by irradiation of the disk during the initial rise and following the intense 6.8 Crab flare.', '4.2. Black Hole Mass': 'We can estimate the mass of the black hole in XTE J1550-564 by equating the observed inner disk radius with the last stable orbit predicted by general relativity. Merloni et al. (1999) find that the multicolor disk model, with appropriate corrections for spectral hardening and relativistic effects, gives stable, acceptable results for high accretion rates ( ∼ > 10% of the Eddington accretion rate) and/or when a lower fraction of the gravitational energy is dissipated in the corona. In this case, the actual inner disk radius ( R in ) can be determined from the stable observed (color) radius ( r in ) during the high/soft state according to the formula: \nR in = ηg ( i ) f 2 r in , (2) \nwhere η is the ratio of the inner disk radius to the radius at which the disk emissivity peaks, g ( i ) is the correction for relativistic effects as a function of inclination, f is the correction for constant spectral hardening ( f = 1 . 7; Shimura & Takahara 1995), and r in = 40 km ( D/ 6 kpc )(cos i ) -1 / 2 for XTE J1550-564 (see above). Both η and g ( i ) are functions of black hole angular momentum. Assuming a Schwarzschild black hole with 0 · ≤ i ≤ 70 · , for which η = 0 . 63 and g ( i ) ∼ 0 . 9 (Zhang, Cui, & Chen 1997; Sobczak et al. 1999a), eq. (2) becomes R in = 66 km ( D/ 6 kpc )(cos i ) -1 / 2 . Equating R in with the last stable orbit for a Schwarzschild black hole (6 GM/c 2 ), we find: \nM BH = 7 . 4 M /circledot ( D/ 6 kpc )(cos i ) -1 / 2 . (3) \nFor example, for i = 60 · and D = 6 kpc, M BH = 10 M /circledot .', '5. Conclusion': "We have analyzed RXTE data acquired during observations of the X-ray nova XTE J1550-564. Satisfactory fits to all the PCA data were obtained with a model consisting of a multicolor disk, a power-law, and Fe emission and absorption components. XTE J1550-564 is observed in the very high and high/soft canonical outburst states of BHXN. We distinguished these two states based on the relative flux contribution of the disk and power-law spectral components, the value of the power-law photon index, the presence or absence of QPOs, and the relative strength of the 2-12 keV and 20- 100 keV fluxes. \nThe source exhibited an intense (6.8 Crab) flare on MJD 51075, during which the inner disk radius appears to have decreased dramatically from 33 to 2 km (for i = 0 and D = 6 kpc). However, the apparent decrease of the inner disk radius observed during periods of increased power-law emission may be caused by the failure of the multicolor disk model; the actual physical radius of the inner disk may remain fairly constant. This interpretation is supported by the fact that the observed inner disk radius remains approximately constant over 120 days from MJD 51120-51240, when the power-law is weak, even though the disk flux and total flux vary by an order of magnitude. The mass of the black hole inferred by equating the approximately constant inner disk radius observed in the high/soft state with the last stable orbit for a Schwarzschild black hole (see § 4.2) is M BH = 7 . 4 M /circledot ( D/ 6 kpc )(cos i ) -1 / 2 . \nThe outburst of XTE J1550-564 has many features in common with the most recent outburst of the microquasar GRO J1655-40 (Sobczak et al.1999b). During the first half of their outbursts, the X-ray spectra of both sources are dominated by the power-law component and both the observed inner disk radius and flux exhibit extreme variability. Following their initial outbursts, the flux from both sources declined, only to be followed by a second outburst. During the second half of their outbursts, the X-ray spectra of both sources are primarily disk-dominated with an approximately constant inner disk radius and slowly varying intensity. Neither of these sources can be described as a 'canonical' BHXN. \nThis work was supported, in part, by NASA grants NAG5-3680 and NAS5-30612. Partial support for J.M. and G.S. was provided by the Smithsonian Institution Scholarly Studies Program. W.C. would like to thank Shuang Nan Zhang and Wan Chen for extensive discussions on spectral modeling and interpretation of the results.", 'REFERENCES': "| Campbell-Wilson, D., McIntyre, V., Hunstead, R., & Green, A. 1998, IAU Circ. 7010 |\n|------------------------------------------------------------------------------------------------------------------------------------------------------|\n| Chen, W., Shrader, C. R., & Livio, M. 1997, ApJ, 491, 312 |\n| Cui, W., Zhang, S. N., Chen, W., & Morgan, E. H. 1999, ApJ, 512, L43 |\n| 375 |\n| Grove, J. E., Johnson, W. N., Kroeger, R. A., McNaron-Brown, K., Skibo, J. G., Hulburt, E. O., & Philips, B. 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E., Remillard, R. A., Bailyn, C. D., & Orosz, J. A. 1999b, ApJ, 520, 776 \nSobczak, G. J., McClintock, J. E., Remillard, R. A., Cui, W., Levine, A. M., Morgan, E. H., Orosz, J. A., & Bailyn, C. D. 2000, ApJ, 531, 537 \nTanaka, Y. & Lewin, W. H. G. 1995, in X-ray Binaries, ed. W. H. G. Lewin, J. van Paradijs, & E. P. J. van den Heuvel (Cambridge: Cambridge Univ. Press), p. 126 \nTitarchuk, L. 1994, ApJ, 434, 570 \nZhang, S. N., Cui, W., & Chen, W. 1997, ApJ, 482, L155 \nZhang, S. N., et al. 1999, in preparation \nFig. 1.- (Upper Panel) The 2-12 keV ASM lightcurve and (Lower Panel) the ratio of the ASM count rates (5-12 keV)/(3-5 keV) for XTE J1550-564. The small, solid vertical lines in the top panel indicate the times of pointed RXTE observations; the downward arrows indicate the observations during which high frequency 161-283 Hz QPOs are present. \n<!-- image --> \nFig. 2.- The ratio data/model for (a) the best fit multicolor disk plus power-law model and (b) the multicolor disk plus power-law plus Fe emission & absorption model for a representative high/soft state spectrum (MJD 51126, 1998 Nov. 9). The addition of the Fe emission & absorption components improves the χ 2 ν from (a) 7.9 to (b) 0.9 in this example. \n<!-- image --> \nFig. 3.- Sample PCA spectra from (a) the rising phase on MJD 51063 (1998 Sept. 7), (b) the flare on MJD 51075 (1998 Sept. 19), (c,d) the high/soft state on MJD 51121 & 51211 (1998 Nov. 4 & 1999 Feb. 2), (e) the power-law flare during the decline of the outburst (MJD 51249; 1999 March 12), and (f) one of the last observations on MJD 51307 (1999 May 9) during gain epoch 4. The individual components of the model are also shown (dashed lines). Although error bars are plotted for all the data, they are usually not large enough to be visible. \n<!-- image --> \nFig. 4.- Spectral parameters and fluxes for PCA observations of XTE J1550-564. See the text for details on the spectral models and fitting. The quantities plotted here are (a) the color temperature of the accretion disk T in in keV, (b) the inner disk radius R ∗ = R in (cos i ) 1 / 2 / ( D/ 6 kpc ) in km, where i is the inclination angle and D is the distance to the source in kpc, (c) the power-law photon index Γ, the unabsorbed 2-20 keV flux in units of 10 -7 erg s -1 cm -2 for (d) the power-law, (e) the disk, and (f) the total. Data points from gain epoch 4, beginning on MJD 51260 (MJD = JD-2,400,000.5), are plotted using an open square. When error bars are not visible, it is because they are comparable to or smaller than the plotting symbol. The dots plotted without error bars in (d) correspond to the right axis and are shown to highlight the behavior of the faint power-law component during the second half of the outburst. \n<!-- image --> \nFig. 5.- The ratio data/model for (a) power-law fit to the Crab on 1997 December 15, (b) later power-law fit to the Crab on 1999 March 24, (c) power-law fit to the first observation of XTE J1550-564 on 1998 September 7, and (d) the same observation of XTE J1550-564 fit to a model consisting of a power-law with a high energy cutoff (see Eq. 1). \n<!-- image --> \nFig. 6.- Plot of the 2-12 keV PCA flux and the 20-100 keV HEXTE flux vs. time (MJD = JD-2,400,000.5). Representative error bars are plotted for every 5th point. Upper limits for the HEXTE data are plotted at the 3 σ level of confidence (recall that the normalization of the HEXTE instrument is systematically ∼ 20-30% lower than the PCA). \n<!-- image --> \nT ABLE /1 PCA Obser v a tions of XTE J/1/5/5/0/{/5/6/4 \n| Obs /# | Date /(UT/) | MJD a | Exp osure /(s/) | Coun t Rate b | HR c |\n|-----------|----------------------------------------|----------------------------------------------------------|-------------------|----------------------------|----------------------------------|\n| | | | | /(c//s//PCU/) | |\n| /1 | /9/8/0/9/0/7 | | /1/8/0/8 | /9/5/0 | /0/./6/2/2 |\n| | /9/8/0/9/0/8 | /5/1/0/6/3/./6/9/7 /5/1/0/6/4/./0/0/6 | /5/7/4/4 | /9/9/5 | /0/./5/5/3 |\n| /2 /3 | /9/8/0/9/0/9 | | /3/5/6/8 | | /0/./4/8/5 |\n| /4 | /9/8/0/9/0/9 | /5/1/0/6/5/./0/6/8 | /2/1/4/4 | /1/7/8/8 /1/7/6/0 | /0/./4/7/1 |\n| /5 | /9/8/0/9/1/0 | /5/1/0/6/5/./3/4/2 /5/1/0/6/6/./0/6/7 | /2/8/8/0 | /2/1/4/3 | /0/./4/1/9 |\n| /6 | /9/8/0/9/1/0 | | /2/6/5/6 | /2/2/9/1 | /0/./3/9/8 |\n| | | /5/1/0/6/6/./3/4/5 | | | |\n| /7 | /9/8/0/9/1/1 | /5/1/0/6/7/./2/7/1 | /3/2/0/0 | /2/7/9/7 | /0/./3/6/5 |\n| /8 /9 | /9/8/0/9/1/2 | /5/1/0/6/8/./3/4/5 | /2/5/6/0 | /3/0/9/6 /3/5/0/4 | /0/./3/1/7 /0/./2/8/8 |\n| | /9/8/0/9/1/3 | /5/1/0/6/9/./2/7/5 | /2/8/8/0 | | |\n| /1/0 | /9/8/0/9/1/4 | /5/1/0/7/0/./1/3/1 | /3/7/6/0 | /3/4/8/2 | /0/./2/9/1 |\n| /1/1 | /9/8/0/9/1/4 | /5/1/0/7/0/./2/7/4 | /2/8/9/6 | /3/5/1/3 | /0/./2/9/0 |\n| /1/2 | /9/8/0/9/1/5 | /5/1/0/7/1/./2/0/0 | /3/5/5/2 | /3/9/2/8 | /0/./2/7/9 |\n| /1/3 | /9/8/0/9/1/5 | | /1/1/3/6 | /3/2/7/7 | /0/./3/0/9 |\n| /1/4 | /9/8/0/9/1/6 | /5/1/0/7/1/./9/9/6 /5/1/0/7/2/./3/4/5 | /2/6/2/4 | /4/0/2/2 | |\n| | | | | | /0/./2/6/9 |\n| /1/5 | /9/8/0/9/1/8 | /5/1/0/7/4/./1/4/0 | /2/0/2/0 | /5/1/5/1 | /0/./2/4/2 |\n| /1/6 | /9/8/0/9/1/9 | /5/1/0/7/5/./9/8/7 | /2/9/7/5 | /1/3/1/9/5 | /0/./2/2/0 |\n| /1/7 | /9/8/0/9/2/0 | /5/1/0/7/6/./8/0/2 | /5/0/0/0 | /6/3/6/0 | /0/./2/3/2 |\n| /1/8 | /9/8/0/9/2/0 | /5/1/0/7/6/./9/5/3 | /4/9/3/0 /3/6/8/0 | /7/1/5/3 | /0/./2/0/0 |\n| /1/9 | /9/8/0/9/2/1 /9/8/0/9/2/1 | /5/1/0/7/7/./1/4/3 /5/1/0/7/7/./2/1/1 | /9/7/1/0 | /7/9/7/0 /6/8/6/6 | /0/./2/1/3 /0/./2/2/3 |\n| /2/0 | | /5/1/0/7/7/./8/6/9 | /1/0/4/5/0 | | |\n| /2/1 | /9/8/0/9/2/1 /9/8/0/9/2/2 | | | /5/5/5/7 | /0/./2/3/6 |\n| /2/2 | /9/8/0/9/2/3 | /5/1/0/7/8/./1/3/2 | /4/1/0/0 | /4/7/7/9 | /0/./2/4/5 |\n| /2/3 /2/4 | /9/8/0/9/2/4 | /5/1/0/7/9/./7/9/4 | /2/7/0/0 /3/1/3/0 | /3/9/6/3 /3/8/0/7 | /0/./2/5/0 /0/./2/5/2 |\n| /2/5 | /9/8/0/9/2/5 | /5/1/0/8/0/./0/7/8 | /4/4/3/0 | /3/3/4/5 | /0/./2/9/7 |\n| /2/6 | | /5/1/0/8/1/./0/6/2 | | | |\n| | /9/8/0/9/2/6 | /5/1/0/8/2/./0/0/2 | /7/1/7/0 | /3/2/6/6 | /0/./3/0/0 |\n| /2/7 /2/8 | /9/8/0/9/2/7 /9/8/0/9/2/8 | /5/1/0/8/3/./0/0/2 | /3/8/6/0 /4/7/5/0 | /3/1/7/2 /3/1/2/0 | /0/./3/0/6 /0/./3/1/0 |\n| /2/9 | /9/8/0/9/2/9 /9/8/0/9/2/9 | /5/1/0/8/4/./3/4/2 /5/1/0/8/5/./2/7/0 /5/1/0/8/5/./9/2/0 | /4/5/0/0 | /3/6/0/9 | /0/./2/7/6 /0/./3/0/3 |\n| /3/0 | | /5/1/0/8/5/./9/9/1 | /1/8/0/0 | /3/0/7/6 | |\n| | | | /5/0/6/0 | /3/1/0/0 | /0/./3/0/0 |\n| /3/1 /3/2 | /9/8/0/9/2/9 /9/8/0/9/3/0 | /5/1/0/8/6/./8/8/9 | /5/6/3/0 | /3/2/3/8 | /0/./2/8/9 |\n| /3/3 /3/4 | /9/8/1/0/0/1 /9/8/1/0/0/2 | /5/1/0/8/7/./7/2/3 /5/1/0/8/8/./0/0/7 /5/1/0/8/9/./0/0/8 | /8/3/9/0 /6/0/5/0 | /3/1/4/6 /3/0/4/5 | /0/./2/9/2 /0/./3/0/0 |\n| /3/5 | /9/8/1/0/0/3 | | | /2/9/0/7 | /0/./3/0/6 |\n| /3/6 | /9/8/1/0/0/4 | /5/1/0/9/0/./1/4/3 | /3/0/3/0 /3/1/6/0 | /3/1/4/3 | /0/./2/8/9 |\n| /3/7 | /9/8/1/0/0/4 | /5/1/0/9/0/./7/0/4 | /3/5/0/0 | /2/9/9/0 | /0/./2/9/2 |\n| /3/8 | /9/8/1/0/0/5 | /5/1/0/9/1/./7/4/3 | /2/8/3/0 | /3/7/4/1 | /0/./2/6/2 |\n| /3/9 | | | /3/0/3/0 | /4/0/2/1 | /0/./2/5/2 |\n| /4/0 | /9/8/1/0/0/7 /9/8/1/0/0/8 | /5/1/0/9/3/./1/4/3 /5/1/0/9/4/./1/4/3 /5/1/0/9/4/./5/7/2 | /2/9/7/0 | /3/0/1/8 | /0/./2/9/0 |\n| /4/1 /4/2 | /9/8/1/0/0/8 /9/8/1/0/0/9 /9/8/1/0/1/0 | /5/1/0/9/5/./6/0/9 /5/1/0/9/6/./5/7/2 | /3/8/5/0 /1/3/0/0 | /3/1/6/1 /2/8/7/1 /3/0/8/7 | /0/./2/7/1 /0/./2/8/0 /0/./2/7/0 |\n| /4/3 | | | /4/0/5/0 | | |\n| /4/4 | /9/8/1/0/1/1 | /5/1/0/9/7/./5/7/2 | /1/4/8/0 | /2/7/5/1 | /0/./2/8/4 |\n| /4/5 | /9/8/1/0/1/1 /9/8/1/0/1/2 | /5/1/0/9/7/./8/0/9 /5/1/0/9/8/./2/7/5 | /1/0/5/0 | /2/5/7/5 | /0/./2/9/0 |\n| /4/6 | | | /1/7/1/0 | /2/7/8/5 | /0/./2/7/8 |\n| /4/7 | /9/8/1/0/1/3 | /5/1/0/9/9/./2/1/5 | /1/4/8/0 | /2/6/7/9 | /0/./2/8/1 |\n| | /9/8/1/0/1/3 | | /2/6/2/0 | /2/6/4/0 | /0/./2/8/3 |\n| /4/8 | | /5/1/0/9/9/./6/0/8 | | | |\n| /4/9 /5/0 | /9/8/1/0/1/4 /9/8/1/0/1/5 | /5/1/1/0/0/./2/8/7 | /2/7/0/0 /1/6/3/0 | /3/1/4/2 | /0/./2/6/2 |\n| | /9/8/1/0/1/5 | | /2/0/8/0 | /3/0/7/1 | /0/./2/5/6 /0/./2/6/0 |\n| /5/1 | | /5/1/1/0/1/./6/0/7 /5/1/1/0/1/./9/4/1 | | /3/0/2/1 | |\n| /5/2 | /9/8/1/0/2/0 | /5/1/1/0/6/./9/5/3 | /9/3/0 | /3/7/5/5 | /0/./2/1/9 |\n| /5/3 | /9/8/1/0/2/2 /9/8/1/0/2/3 | /5/1/1/0/8/./0/7/6 | /9/8/7/0 | /3/5/4/8 | /0/./2/2/4 |\n| /5/4 | | /5/1/1/0/9/./7/3/7 | /1/2/3/0 | /3/2/7/2 | /0/./2/2/3 |\n| /5/5 | /9/8/1/0/2/4 /9/8/1/0/2/5 | /5/1/1/1/0/./2/7/0 | /3/5/6/0 /8/5/0 | /3/0/1/3 /2/8/6/2 | /0/./2/0/0 /0/./2/3/0 |\n| /5/6 | /9/8/1/0/2/6 | /5/1/1/1/1/./6/0/2 | | | |\n| | | /5/1/1/1/2/./8/0/2 | /1/7/2/0 | /2/6/9/5 | /0/./2/3/1 |\n| /5/7 /5/8 | /9/8/1/0/2/7 /9/8/1/0/2/9 | /5/1/1/1/3/./6/6/8 | /1/0/5/0 /2/0/7/0 | /2/3/9/6 /1/8/0/2 | /0/./2/2/1 /0/./2/1/3 |\n| /5/9 | | | | | |\n| | /9/8/1/0/3/1 | /5/1/1/1/5/./2/8/0 | /1/9/5/0 | /1/2/4/3 | /0/./1/6/0 |\n| /6/0 /6/1 | /9/8/1/1/0/2 | /5/1/1/1/7/./3/5/1 /5/1/1/1/9/./0/0/3 | /3/6/8/0 | /1/0/2/5 /7/6/0 | /0/./1/7/7 /0/./0/7/4 |\n| /6/2 | /9/8/1/1/0/4 | | /3/7/5/0 | /6/1/1 | |\n| | /9/8/1/1/0/7 | /5/1/1/2/1/./0/0/3 | /2/1/9/0 | | /0/./1/5/1 |\n| /6/3 | | /5/1/1/2/4/./7/2/7 | /4/6/8/0 | /6/4/9 | /0/./2/0/4 |\n| | /9/8/1/1/0/9 /9/8/1/1/1/1 | /5/1/1/2/6/./5/9/5 | /2/5/2/0 | /5/0/4 | /0/./2/0/3 |\n| /6/4 /6/5 | | | | | |\n| | /9/8/1/1/1/3 | /5/1/1/2/8/./5/6/4 | | | |\n| /6/6 | /9/8/1/1/1/5 | /5/1/1/3/0/./4/5/7 | /4/8/0/0 /2/9/8/0 | /4/5/4 /3/8/6 | /0/./2/0/9 /0/./1/8/0 |\n| /6/7 /6/8 | /9/8/1/1/1/7 | | | /3/1/1 | |\n| /6/9 | /9/8/1/1/1/9 | /5/1/1/3/2/./4/7/9 /5/1/1/3/4/./4/3/6 /5/1/1/3/6/./7/7/7 | /5/0/5/0 /6/5/0/0 | /2/6/1 | /0/./1/1/4 /0/./1/7/2 |\n| /7/0 /7/1 | /9/8/1/1/2/0 /9/8/1/1/2/2 | /5/1/1/3/7/./9/2/3 /5/1/1/3/9/./9/9/1 | /4/2/2/0 /1/6/0/0 | /3/3/1 /3/2/2 | /0/./2/4/6 /0/./3/5/8 |\n| | /9/8/1/1/2/3 | /5/1/1/4/0/./7/0/6 | /2/5/0/0 | | /0/./2/4/5 |\n| /7/2 | | | | /2/8/6 | | \nT ABLE /1/| Continue d \n| Obs | Date /(UT/) | MJD a | Exp osure /(s/) | Coun t Rate b | HR c |\n|----------------------|----------------------------------------|----------------------------------------------------------|-------------------|----------------------------|----------------------------------|\n| /# | /(c//s//PCU/) | /(c//s//PCU/) | /(c//s//PCU/) | /(c//s//PCU/) | /(c//s//PCU/) |\n| /7/3 | /9/8/1/1/2/6 | /5/1/1/4/3/./8/0/0 | /1/8/8/0 | /2/5/8 | /0/./2/2/4 |\n| /7/4 | /9/8/1/1/2/8 | | /5/6/6/0 | /1/6/8 | /0/./1/9/0 |\n| /7/5 | /9/8/1/1/3/0 | /5/1/1/4/5/./4/7/6 /5/1/1/4/7/./3/3/5 | /4/6/0/0 | /2/2/0 | /0/./2/2/1 |\n| /7/6 | | | | | /0/./2/1/0 |\n| | /9/8/1/2/0/3 | /5/1/1/5/0/./0/7/3 /5/1/1/5/2/./0/8/0 | /2/2/2/0 /2/3/6/0 | /2/0/6 /2/3/0 | /0/./0/3/4 |\n| /7/7 /7/8 | /9/8/1/2/0/5 /9/8/1/2/0/5 | /5/1/1/5/2/./8/6/6 | /4/8/3/0 | /2/5/8 | /0/./0/9/0 |\n| /7/9 | /9/8/1/2/0/7 | /5/1/1/5/4/./0/1/6 | /1/6/5/0 | /3/1/4 | /0/./0/0/0 |\n| /8/0 /8/1 | /9/8/1/2/0/8 | /5/1/1/5/5/./0/6/7 | /3/0/3/0 | /3/8/5 | /0/./0/0/0 /0/./0/0/0 |\n| | /9/8/1/2/1/0 | /5/1/1/5/7/./4/9/3 | /2/4/1/0 | /6/3/5 | |\n| /8/2 | /9/8/1/2/1/3 | /5/1/1/6/0/./2/7/6 | /2/2/5/0 | | /0/./0/0/0 |\n| | | | /2/4/3/0 | /1/0/3/1 | |\n| /8/3 /8/4 | /9/8/1/2/1/5 /9/8/1/2/1/6 | /5/1/1/6/2/./2/0/2 /5/1/1/6/3/./2/0/5 | /8/7/0 | /1/4/2/2 /1/6/0/9 | /0/./0/0/0 /0/./0/0/0 |\n| /8/5 | | | | /1/9/0/7 | |\n| | /9/8/1/2/1/7 | /5/1/1/6/4/./2/0/5 | /9/3/0 | | /0/./0/0/3 |\n| /8/6 | /9/8/1/2/1/8 | /5/1/1/6/5/./5/9/0 | /3/1/8/0 | /2/3/2/7 | /0/./0/0/3 |\n| /8/7 | /9/8/1/2/1/9 | /5/1/1/6/6/./0/1/1 | /1/4/0/0 | /2/4/2/3 | /0/./0/0/0 |\n| /8/8 | /9/8/1/2/2/0 | /5/1/1/6/7/./4/1/8 | /1/3/9/0 | /2/8/1/1 | /0/./0/3/2 |\n| /8/9 | /9/8/1/2/2/1 | /5/1/1/6/8/./0/1/0 | /1/7/1/0 | /2/8/6/1 | /0/./0/0/0 |\n| /9/0 | /9/8/1/2/2/2 | /5/1/1/6/9/./0/0/9 | /4/1/5/0 | /3/1/2/7 | /0/./0/0/0 |\n| /9/1 | /9/8/1/2/2/3 | /5/1/1/7/0/./5/2/0 | /1/2/3/0 | /3/3/5/9 | /0/./0/0/6 |\n| /9/2 | /9/8/1/2/2/5 | /5/1/1/7/2/./0/6/5 | /2/5/8/3 | /3/5/0/6 | /0/./0/0/0 |\n| /9/3 | /9/8/1/2/2/6 | /5/1/1/7/2/./9/9/0 | /3/2/5/0 | /3/5/6/8 | /0/./0/0/2 |\n| /9/4 | /9/8/1/2/2/7 | /5/1/1/7/3/./9/9/0 | /3/1/8/0 | /3/6/4/1 | /0/./0/0/5 |\n| /9/5 | /9/8/1/2/2/7 | /5/1/1/7/4/./7/2/2 | /3/1/1/0 | /3/7/6/9 | /0/./0/0/6 |\n| /9/6 | /9/8/1/2/2/8 | /5/1/1/7/5/./7/9/6 | /5/4/0/0 | /3/8/5/2 | /0/./0/1/0 |\n| /9/7 | /9/8/1/2/2/9 | /5/1/1/7/6/./8/4/7 | /3/9/0/0 | /3/8/7/1 | /0/./0/1/2 |\n| /9/8 | /9/8/1/2/3/0 | /5/1/1/7/7/./8/4/6 | /4/0/0/0 | /3/9/6/5 | /0/./0/1/0 |\n| /9/9 | /9/8/1/2/3/1 /9/9/0/1/0/1 | /5/1/1/7/8/./8/4/6 /5/1/1/7/9/./9/1/9 | /4/0/0/0 | /3/9/1/1 | /0/./0/1/1 |\n| | | | /3/4/9/0 | /4/1/8/3 | |\n| /1/0/0 /1/0/1 | /9/9/0/1/0/2 /9/9/0/1/0/4 | /5/1/1/8/0/./8/4/5 /5/1/1/8/2/./3/3/7 | /4/1/1/0 /4/1/2/0 | /4/5/4/6 | /0/./0/3/0 /0/./0/7/3 /0/./0/0/9 |\n| /1/0/2 | | /5/1/1/8/3/./5/4/0 | | /4/1/6/3 | |\n| /1/0/3 | /9/9/0/1/0/5 | /5/1/1/8/4/./7/1/1 | /2/6/0/0 | /4/2/5/8 | /0/./0/0/7 /0/./0/0/8 |\n| /1/0/4 | /9/9/0/1/0/6 | | /5/5/5/0 | /4/2/5/0 | |\n| /1/0/5 | /9/9/0/1/0/7 | /5/1/1/8/5/./1/2/6 | /1/7/6/0 | /4/2/6/6 | /0/./0/2/2 |\n| /1/0/6 /1/0/7 | /9/9/0/1/0/7 /9/9/0/1/0/9 | /5/1/1/8/5/./8/4/6 /5/1/1/8/7/./1/2/6 | /8/6/4/0 /4/1/2/0 | /4/2/3/3 /4/2/5/9 | /0/./0/0/7 /0/./0/0/9 |\n| /1/0/8 | /9/9/0/1/1/0 | /5/1/1/8/8/./1/9/4 | /4/4/7/0 | /4/2/6/3 | /0/./0/0/6 /0/./0/0/3 |\n| /1/0/9 | /9/9/0/1/1/0 | /5/1/1/8/8/./7/3/1 | /1/6/5/0 | /4/2/4/0 | |\n| /1/1/0 | /9/9/0/1/1/2 /9/9/0/1/1/3 | /5/1/1/9/0/./0/5/4 /5/1/1/9/1/./4/8/5 | /1/2/1/0 /4/2/3/0 | /4/3/0/7 | /0/./0/0/7 |\n| /1/1/1 | | /5/1/1/9/2/./1/8/7 | | /4/3/8/2 | /0/./0/0/7 |\n| /1/1/2 /1/1/3 | /9/9/0/1/1/4 /9/9/0/1/1/4 /9/9/0/1/1/6 | /5/1/1/9/2/./5/9/3 | /4/4/2/0 | | |\n| /1/1/4 | /9/9/0/1/1/8 | /5/1/1/9/4/./5/0/0 | /1/5/0/0 /2/7/2/0 | /4/4/9/1 /4/3/4/1 /4/5/1/3 | /0/./0/0/6 /0/./0/0/9 /0/./0/0/8 |\n| /1/1/5 /1/1/6 | /9/9/0/1/1/8 | /5/1/1/9/6/./0/5/1 /5/1/1/9/6/./5/2/2 | /4/3/5/0 /2/0/3/0 | /4/6/0/7 /4/4/4/3 | /0/./0/0/7 /0/./0/0/8 |\n| | | | | | /0/./0/0/8 |\n| | /9/9/0/1/2/0 | /5/1/1/9/8/./0/5/1 | | | |\n| /1/1/7 | /9/9/0/1/2/2 | | /4/1/0/0 | /4/6/3/8 | /0/./0/7/0 |\n| /1/1/8 /1/1/9 | /9/9/0/1/2/3 | /5/1/2/0/0/./3/8/3 /5/1/2/0/1/./7/8/3 | /3/0/7/0 /2/6/7/0 | /5/0/3/1 /5/4/4/3 | /0/./1/0/6 |\n| /1/2/0 | /9/9/0/1/2/5 | /5/1/2/0/3/./2/4/9 | /4/6/3/0 | /4/7/8/0 | /0/./0/1/5 |\n| /1/2/1 | /9/9/0/1/2/6 | /5/1/2/0/4/./2/4/8 /5/1/2/0/5/./8/5/0 | /1/5/0/0 | | |\n| /1/2/2 | /9/9/0/1/2/7 /9/9/0/1/2/8 | | /2/9/7/0 | /4/7/7/1 /4/8/0/9 | /0/./0/1/0 /0/./0/1/1 /0/./0/0/9 |\n| | | /5/1/2/0/6/./7/8/1 | /2/6/9/0 | | |\n| /1/2/3 /1/2/4 | /9/9/0/1/3/0 | /5/1/2/0/8/./1/7/9 | | /4/7/1/4 | /0/./0/2/9 |\n| | /9/9/0/1/3/0 | /5/1/2/0/8/./8/4/5 | /1/1/5/0 | /4/8/8/4 | /0/./0/1/3 |\n| /1/2/5 | /9/9/0/1/3/1 | /5/1/2/0/9/./7/1/1 | /3/3/3/0 | /4/7/5/1 /4/7/0/7 | /0/./0/1/4 |\n| /1/2/6 /1/2/7 | /9/9/0/2/0/2 | /5/1/2/1/1/./7/0/9 | /2/4/9/0 /2/6/2/0 | /4/5/8/4 | /0/./0/1/1 |\n| | /9/9/0/2/0/3 | /5/1/2/1/2/./7/7/5 | | /4/5/7/3 | /0/./0/1/2 |\n| /1/2/8 | /9/9/0/2/0/4 | | /3/0/9/0 | | |\n| /1/2/9 /1/3/0 | /9/9/0/2/0/5 /9/9/0/2/0/6 | /5/1/2/1/3/./7/7/4 /5/1/2/1/4/./9/7/5 | /3/1/5/0 /3/4/6/0 | /4/5/5/6 /4/7/3/4 /4/6/2/4 | /0/./0/1/3 /0/./0/4/9 /0/./0/4/6 |\n| /1/3/1 | | /5/1/2/1/5/./8/3/9 | /7/1/4/0 | /4/4/6/8 | |\n| | /9/9/0/2/0/7 | /5/1/2/1/6/./8/3/8 | | | /0/./0/1/9 |\n| /1/3/2 | | | /3/6/1/0 | | |\n| /1/3/3 | /9/9/0/2/0/8 /9/9/0/2/1/0 | /5/1/2/1/7/./7/0/4 /5/1/2/1/9/./0/5/3 | /3/1/0/0 | /4/4/3/8 /4/3/5/2 | /0/./0/0/9 /0/./0/0/9 |\n| /1/3/4 | /9/9/0/2/1/1 | | /3/9/2/0 | | |\n| | | /5/1/2/2/0/./5/0/8 | | /4/3/0/8 | |\n| /1/3/5 | /9/9/0/2/1/1 | /5/1/2/2/0/./8/3/6 | /1/8/9/0 /5/2/0/0 | /4/3/0/0 | /0/./0/1/3 /0/./0/1/4 |\n| /1/3/6 | /9/9/0/2/1/2 | /5/1/2/2/1/./5/6/7 | /2/3/0/0 | /4/2/4/5 | /0/./0/0/1 |\n| /1/3/7 | | | | | |\n| | /9/9/0/2/1/4 | /5/1/2/2/3/./7/6/6 | | | |\n| /1/3/8 | /9/9/0/2/1/5 | /5/1/2/2/4/./6/9/9 | /4/5/0/0 /3/5/3/0 | /4/1/4/2 /4/0/5/2 | /0/./0/0/7 /0/./0/0/6 |\n| /1/3/9 /1/4/0 | /9/9/0/2/1/7 /9/9/0/2/1/8 | /5/1/2/2/6/./2/9/6 | /2/3/2/0 /3/0/2/0 | /4/0/6/8 /3/9/7/0 | /0/./0/0/8 /0/./0/0/3 |\n| /1/4/1 /1/4/2 /1/4/3 | /9/9/0/2/1/9 | /5/1/2/2/7/./8/3/7 /5/1/2/2/8/./6/9/6 /5/1/2/3/0/./4/2/4 | /3/6/1/5 | | /0/./0/0/6 /0/./0/0/5 |\n| | /9/9/0/2/2/1 | | /3/4/8/0 | /3/9/1/0 /3/8/6/9 | |\n| | /9/9/0/2/2/2 | /5/1/2/3/1/./2/9/3 | /1/3/0/0 | /3/9/9/7 | /0/./0/2/4 |\n| /1/4/4 | | | | | | \nb \nc \n/1 \nCrab \n/= \n/2/5/0/0 \nc//s//PCU \nT ABLE /1/| Continue d \n| Obs /# | Date /(UT/) | MJD a | Exp osure /(s/) | Coun t Rate b | HR c |\n|----------------------|----------------------------------------|----------------------------------------------------------|----------------------------|------------------------|-----------------------|\n| | /9/9/0/2/2/2 | | /3/1/5/0 | /(c//s//PCU/) /3/9/4/0 | /0/./0/2/1 |\n| /1/4/5 | /9/9/0/2/2/3 | /5/1/2/3/1/./4/2/2 /5/1/2/3/2/./2/5/2 | /1/5/4/0 | | /0/./1/0/1 |\n| /1/4/6 | | | | /4/3/6/8 | |\n| /1/4/7 | /9/9/0/2/2/3 | /5/1/2/3/2/./8/6/0 | /8/6/0 | | /0/./1/1/4 |\n| /1/4/8 | /9/9/0/2/2/4 | /5/1/2/3/3/./8/3/5 | /3/1/4/0 | /4/4/4/2 /4/3/6/2 | /0/./1/1/0 |\n| | /9/9/0/2/2/6 | /5/1/2/3/5/./1/9/9 | /5/8/3/0 | /4/3/9/0 | /0/./1/2/5 |\n| /1/4/9 | /9/9/0/2/2/8 | | /4/0/8/0 | /3/6/4/2 | /0/./0/3/4 |\n| /1/5/0 /1/5/1 | /9/9/0/3/0/2 | /5/1/2/3/7/./6/1/9 /5/1/2/3/9/./0/8/1 | /5/9/8/0 | /3/7/8/8 | /0/./0/8/7 |\n| /1/5/2 | /9/9/0/3/0/3 | | /2/1/6/0 | /3/9/1/2 | /0/./1/0/1 |\n| /1/5/3 | | /5/1/2/4/0/./0/6/3 | | | |\n| | /9/9/0/3/0/4 | /5/1/2/4/1/./8/2/8 | /3/1/3/0 | /4/1/7/1 | /0/./1/6/6 |\n| /1/5/4 | /9/9/0/3/0/5 | /5/1/2/4/2/./5/0/7 | /1/5/2/5 | /4/0/2/3 | /0/./1/7/4 |\n| /1/5/5 | /9/9/0/3/0/7 | /5/1/2/4/4/./4/9/6 | /6/8/5 | /4/8/9/4 | /0/./1/9/2 |\n| /1/5/6 | /9/9/0/3/0/8 | /5/1/2/4/5/./3/5/3 | /2/7/9/0 | /4/5/8/3 | /0/./2/1/7 |\n| /1/5/7 | /9/9/0/3/0/9 | /5/1/2/4/6/./4/1/4 | /2/8/2/0 | /4/4/8/8 | /0/./1/8/3 |\n| /1/5/8 | /9/9/0/3/1/0 | /5/1/2/4/7/./9/7/9 | /2/8/3/0 | /4/3/9/2 | /0/./2/1/2 |\n| | /9/9/0/3/1/1 | | | | |\n| /1/5/9 /1/6/0 | /9/9/0/3/1/2 | /5/1/2/4/8/./0/9/1 /5/1/2/4/9/./4/0/0 | /1/3/6/5 /1/2/7/5 | /4/4/0/8 | /0/./2/0/6 |\n| /1/6/1 | /9/9/0/3/1/3 | /5/1/2/5/0/./6/9/3 | /2/9/2/0 | /4/0/9/5 | /0/./2/2/9 /0/./2/6/1 |\n| /1/6/2 | /9/9/0/3/1/6 | /5/1/2/5/3/./2/2/5 | /1/1/7/0 | /2/5/7/1 /3/5/1/8 | /0/./2/2/5 |\n| | | /5/1/2/5/4/./0/9/2 | | | |\n| /1/6/3a | /9/9/0/3/1/7a /9/9/0/3/1/7b | /5/1/2/5/4/./1/0/7 | /1/0/9/0 | /2/9/1/0 | /0/./2/7/9 |\n| /1/6/3b /1/6/4 | /9/9/0/3/1/8 | /5/1/2/5/5/./0/9/1 | /1/8/9/0 /5/1/7/0 | /3/3/3/5 /2/3/4/6 | /0/./2/6/7 /0/./1/8/2 |\n| | /9/9/0/3/2/0 | | | | |\n| /1/6/5 | | /5/1/2/5/7/./3/6/2 | /6/0/0 | /1/8/9/2 | /0/./1/6/7 |\n| /1/6/6 | /9/9/0/3/2/1 | /5/1/2/5/8/./0/8/8 | /9/0/0 | /1/8/7/4 | /0/./1/7/6 |\n| /1/6/7 | /9/9/0/3/2/1 | /5/1/2/5/8/./4/9/6 | /1/0/9/0 | /1/8/7/4 | /0/./1/7/6 |\n| /1/6/8 | /9/9/0/3/2/1 | /5/1/2/5/8/./9/7/5 | /1/8/3/5 | /1/7/8/4 | /0/./2/0/5 |\n| /1/6/9 | /9/9/0/3/2/2 | /5/1/2/5/9/./2/5/2 | /9/0/0 | /1/7/4/5 | /0/./2/0/4 |\n| /1/7/0 y | /9/9/0/3/2/3 | /5/1/2/6/0/./5/5/2 | /2/3/1/0 | /1/4/3/0 | /0/./1/7/6 |\n| /1/7/1 | /9/9/0/3/2/4 | /5/1/2/6/1/./7/6/6 | /1/9/6/0 | /1/3/0/5 | /0/./1/6/9 |\n| /1/7/2 /1/7/3 | /9/9/0/3/2/6 /9/9/0/3/2/7 | /5/1/2/6/3/./1/0/8 /5/1/2/6/4/./7/4/8 | /1/4/4/5 /3/5/6/0 | /1/1/7/7 /9/4/4 | /0/./1/5/8 /0/./1/6/6 |\n| /1/7/4 | /9/9/0/3/2/8 | /5/1/2/6/5/./6/1/3 | /2/6/1/0 | /9/1/0 | /0/./1/8/5 |\n| /1/7/5 /1/7/6 | /9/9/0/3/2/9 /9/9/0/3/3/0 | /5/1/2/6/6/./8/8/0 /5/1/2/6/7/./6/1/2 | /3/2/2/0 /2/7/2/5 | /7/2/5 /7/5/6 | /0/./1/2/2 /0/./1/6/9 |\n| /1/7/7 | /9/9/0/4/0/1 | /5/1/2/6/9/./6/7/7 | /3/5/8/0 | /6/8/3 | /0/./1/7/0 |\n| /1/7/8 | /9/9/0/4/0/2 | /5/1/2/7/0/./7/4/2 | /3/6/7/0 | /7/0/3 | /0/./1/9/4 |\n| | | | /2/0/0/0 | | /0/./2/0/6 |\n| /1/7/9 | /9/9/0/4/0/3 | /5/1/2/7/1/./4/0/8 | | /6/7/6 | |\n| /1/8/0 /1/8/1 | /9/9/0/4/0/5 /9/9/0/4/0/6 | /5/1/2/7/3/./5/4/1 /5/1/2/7/4/./4/7/1 | /5/2/2/5 /2/6/1/0 | /4/7/2 /4/1/3 | /0/./1/6/7 /0/./1/5/4 |\n| /1/8/2 | /9/9/0/4/0/8 | /5/1/2/7/6/./2/7/8 | /1/2/5/5 | /3/0/8 | /0/./1/3 |\n| /1/8/3 | /9/9/0/4/0/9 | /5/1/2/7/7/./4/0/1 | /2/3/7/0 | /2/9/0 | /0/./1/1 |\n| /1/8/4 /1/8/5 | /9/9/0/4/1/0 /9/9/0/4/1/1 | /5/1/2/7/8/./6/8/6 | /1/5/7/0 /3/8/0/0 | /2/5/0 /2/2/3 | / / / / / |\n| | | /5/1/2/7/9/./5/3/1 | | | / / / |\n| /1/8/6 | /9/9/0/4/1/2 /9/9/0/4/1/5 | /5/1/2/8/0/./5/4/4 | /2/6/0/0 /1/3/0/0 | /1/9/1 /2/0/5 | / |\n| /1/8/7 | /9/9/0/4/1/7 | /5/1/2/8/3/./2/2/4 /5/1/2/8/5/./1/9/7 | | /1/1/8 | / / / /0/./1/1 |\n| /1/8/8 | | | /2/8/9/0 | | /0/./2/0 |\n| /1/8/9 | /9/9/0/4/1/8 | /5/1/2/8/6/./0/5/8 | /3/0/5/5 | /1/1/4 | |\n| /1/9/0 | /9/9/0/4/1/9 | /5/1/2/8/7/./2/5/7 | /1/2/5/5 | /1/0/1 | /0/./1/5 |\n| /1/9/1 | /9/9/0/4/2/0 | /5/1/2/8/8/./3/8/2 | /9/4/0 | /8/0/./0 | /0/./1/1 |\n| /1/9/2 | /9/9/0/4/2/1 | /5/1/2/8/9/./1/4/6 | /1/3/6/0 | /1/0/9 | / / / |\n| /1/9/3 /1/9/4 | /9/9/0/4/2/2 /9/9/0/4/2/3 | /5/1/2/9/0/./9/8/4 /5/1/2/9/1/./1/8/3 | /6/8/5 /5/3/0/0 | /1/0/8 /9/1/./0 | /0/./1/1 /0/./1/3 |\n| | | | | /4/1/./4 | |\n| /1/9/5 | /9/9/0/4/2/4 | /5/1/2/9/2/./4/5/1 | /3/9/2/5 | | / / / |\n| /1/9/6 | /9/9/0/4/2/5 /9/9/0/4/2/7 | /5/1/2/9/3/./4/5/0 | /3/8/7/0 | /5/4/./1 /2/8/./2 | / / / |\n| /1/9/7 | | /5/1/2/9/5/./5/2/0 | /4/2/8/0 | | / / / |\n| /1/9/8 | /9/9/0/4/2/9 | /5/1/2/9/7/./3/7/6 | /3/8/2/0 | /1/4/./1 | / / / |\n| /1/9/9 | /9/9/0/4/3/0 /9/9/0/5/0/1 | /5/1/2/9/8/./0/5/2 | /5/4/8/0 /4/8/4/0 | /1/0/./9 /8/./4 | / / / |\n| /2/0/0 | | /5/1/2/9/9/./3/4/4 | | | / / / |\n| /2/0/1 | /9/9/0/5/0/2 /9/9/0/5/0/4 | /5/1/3/0/0/./4/5/8 | /2/6/2/0 /2/5/1/0 | /7/./0 /5/./3 | / / / / / |\n| | /9/9/0/5/0/5 | /5/1/3/0/2/./4/5/9 | | | / |\n| /2/0/2 /2/0/3 | | | /2/0/3/0 | /4/./1 | |\n| /2/0/4 | /9/9/0/5/0/7 | /5/1/3/0/3/./3/9/1 /5/1/3/0/5/./1/2/4 | | /9/./1 | / / / / |\n| /2/0/5 | /9/9/0/5/0/9 | | /3/5/3/0 /3/6/2/0 | | / / / / |\n| | | /5/1/3/0/7/./3/2/2 | | /2/6/./7 | / |\n| /2/0/6 /2/0/7 /2/0/8 | /9/9/0/5/1/1 /9/9/0/5/1/3 /9/9/0/5/1/5 | /5/1/3/0/9/./7/2/0 /5/1/3/1/1/./5/1/9 /5/1/3/1/3/./3/1/7 | /3/7/4/0 /1/7/8/0 /1/7/7/0 | /1/1/./9 /5/./9 | / / / |\n| /2/0/9 | /9/9/0/5/2/0 | /5/1/3/1/8/./7/6/8 | | /6/./0 | / / / |\n| | | | /2/9/7/0 | /0/./0 | / / / / / / | \na Start of Observ ation/. M J D /= J D /BnZr /2 /; /4/0/0 /; /0/0/0 /: /5 \nrate/: \n/1/3/-/3/0 \nk eV \n// \n/6/-/1/3 \nk eV/. \nT ABLE /2 Spectral P arameters f or XTE J/1/5/5/0/{/5/6/4 / \n| Obs /# | Date /(UT/) | MJD a | T col /(k eV/) | R / in b d /(km/) | Photon Index | P o w er/- c la w | F e Edge /(k eV/) | / F e d | E line e /(k eV/) | N line e /; f / /1/0/0 | Line Eq/. Width | / /2 / /(dof/) |\n|------------------------------------------------------------------------------------------------|------------------------------------------|---------------------------------------|------------------------------------------------------------------------------|--------------------------------------------------------------------------|----------------------------------------------------------------------------|---------------------------------------------------------|-----------------------------------------------------------|----------------------------------------------------------------------------------------|----------------------------------------------------------------------|------------------------------------------------------------------|------------------------------------|-------------------------------------|\n| | /9/8/0/9/0/7 /1 /: /4/7 | /5/1/0/6/3/./6/9/7 | /0 /: /8/9/2 /+/0 /: /0/5/7 | /7 /: /4/6 /+/1 /: /1/6 | /+/0 /: /0/2 | /1 /: /0/0 /+/0 /: /0/5 | /8 /: /2/8 /+/0 /: /3/3 | /0 /: /4/0 /+/0 /: /0/8 | /6 /: /5/2 /+/0 /: /1/0 | /0 /: /5/6 /+/0 /: /1/1 | /8/5 /+/1/6 | /0/./8/4/( /8/7/) |\n| /1 | /9/8/0/9/0/8 | /5/1/0/6/4/./0/0/6 | /BnZr /0 /: /0/6/5 /0 /: /8/0/3 /+/0 /: /0/5/7 | /BnZr /0 /: /8/2 /1/0 /: /7/0 /+/2 /: /2/3 | /BnZr /0 /: /0/2 /1 /: /5/3 /+/0 /: /0/1 | /BnZr /0 /: /0/5 /1 /: /4/7 /+/0 /: /0/7 | /BnZr /0 /: /3/3 /8 /: /0/2 /+/0 /: /2/7 | /BnZr /0 /: /0/7 /0 /: /4/6 /+/0 /: /0/8 | /BnZr /0 /: /1/1 /6 /: /4/9 /+/0 /: /0/9 | /BnZr /0 /: /1/3 /0 /: /7/4 /+/0 /: /1/5 | /BnZr /1/9 /8/7 /+/1/7 | /0/./3/3/( /8/7/) |\n| /2 /3/0 /: /7/9 | /9/8/0/9/0/9 | /5/1/0/6/5/./0/6/8 | /BnZr /0 /: /0/6/9 /0 /: /5/5/2 /+/0 /: /0/7/2 | /BnZr /1 /: /4/1 /+/1/5 /: /8/9 | /BnZr /0 /: /0/1 /1 /: /7/2 /+/0 /: /0/1 | /BnZr /0 /: /0/6 /3 /: /9/9 /+/0 /: /0/8 | /BnZr /0 /: /2/7 /7 /: /8/5 /+/0 /: /1/8 | /BnZr /0 /: /0/7 /0 /: /4/0 /+/0 /: /0/5 | /BnZr /0 /: /0/9 /6 /: /6/1 /+/0 /: /0/9 | /BnZr /0 /: /1/7 /1 /: /3/0 /+/0 /: /2/4 | /BnZr /1/9 /8/3 /+/1/5 | /0/./4/9/( /8/7/) |\n| /3 /4 /9/8/0/9/0/9 | | /5/1/0/6/5/./3/4/2 | /BnZr /0 /: /0/6/8 /0 /: /5/1/7 /+/0 /: /0/7/3 | /BnZr /9 /: /1/5 /3/6 /: /8/2 /+/2/2 /: /1/2 | /BnZr /0 /: /0/1 /1 /: /7/6 /+/0 /: /0/1 | /BnZr /0 /: /0/8 /4 /: /3/2 /+/0 /: /0/8 | /BnZr /0 /: /1/7 /7 /: /8/7 /+/0 /: /1/9 | /BnZr /0 /: /0/5 /0 /: /3/7 /+/0 /: /0/4 | /BnZr /0 /: /0/8 /6 /: /5/5 /+/0 /: /0/7 | /BnZr /0 /: /2/3 /1 /: /5/7 /+/0 /: /1/2 | /BnZr /1/4 /9/8 /+/7 | /0/./5/9/( /8/7/) |\n| /5 /9/8/0/9/1/0 | | /5/1/0/6/6/./0/6/7 | /BnZr /0 /: /0/6/7 /0 /: /3/5/2 /+/0 /: /0/6/0 | /BnZr /1/2 /: /1/5 /1/6/0 /: /3/3 /+/2/2/8 /: /7/9 | /BnZr /0 /: /0/1 /1 /: /9/1 /+/0 /: /0/0 | /BnZr /0 /: /0/9 /7 /: /3/2 /+/0 /: /0/8 | /BnZr /0 /: /1/8 /7 /: /9/8 /+/0 /: /0/9 | /BnZr /0 /: /0/5 /0 /: /2/5 /+/0 /: /0/4 | /BnZr /0 /: /0/7 /6 /: /6/1 /+/0 /: /0/6 | /BnZr /0 /: /2/3 /2 /: /3/1 /+/0 /: /2/2 | /BnZr /1/4 /1/1/6 /+/1/0 | /0/./6/6/( /8/7/) |\n| /6 | /9/8/0/9/1/0 | /5/1/0/6/6/./3/4/5 | /BnZr /0 /: /0/5/9 /0 /: /3/2/7 /+/0 /: /0/5/5 | /BnZr /9 /: /5/3 /2/4/1 /: /0/5 /+/3/3/8 /: /8/0 | /BnZr /0 /: /0/0 /1 /: /9/8 /+/0 /: /0/0 | /BnZr /0 /: /0/8 /8 /: /8/8 /+/0 /: /0/9 | /BnZr /0 /: /1/9 /8 /: /1/8 /+/0 /: /1/9 | /BnZr /0 /: /0/4 /0 /: /2/1 /+/0 /: /0/4 | /BnZr /0 /: /0/6 /6 /: /7/0 /+/0 /: /0/6 | /BnZr /0 /: /2/2 /2 /: /4/6 /+/0 /: /2/1 | /BnZr /1/1 /1/1/8 /+/1/0 | /0/./6/9/( /8/7/) |\n| /7 | /9/8/0/9/1/1 | /5/1/0/6/7/./2/7/1 | /BnZr /0 /: /0/3/9 /0 /: /3/4/5 /+/0 /: /0/4/1 | /BnZr /1/2/1 /: /2/4 /2/6/5 /: /6/4 /+/1/5/9 /: /8/8 | /BnZr /0 /: /0/0 /2 /: /1/2 /+/0 /: /0/0 | /BnZr /0 /: /0/9 /1/3 /: /9/7 /+/0 /: /1/3 | /BnZr /0 /: /2/0 /8 /: /2/4 /+/0 /: /2/2 | /BnZr /0 /: /0/4 /0 /: /1/7 /+/0 /: /0/4 | /BnZr /0 /: /0/6 /6 /: /6/6 /+/0 /: /0/6 | /BnZr /0 /: /2/1 /3 /: /1/9 /+/0 /: /2/5 | /BnZr /1/0 /1/2/6 /+/1/0 | /0/./6/3/( /8/7/) |\n| /8 /1/4/6 /: /2/0 /+/5/2 /: /3/8 | /9/8/0/9/1/2 /2 /: /3/1 /+/0 /: /0/1 | /5/1/0/6/8/./3/4/5 | /BnZr /0 /: /0/1/9 /0 /: /4/2/9 /+/0 /: /0/3/3 /2/1 /: /9/8 /+/0 /: /2/9 | 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/: /9/3 /+/0 /: /1/5 /BnZr /0 /: /1/4 /+/0 /: /1/2 | /1 /: /0/0 /+/0 /: /0/6 | /6 /: /9/8 /BnZr /0 /: /1/2 | /1 /: /2/6 /+/0 /: /2/2 /BnZr /0 /: /2/3 | /BnZr /1/4 /7/7 /+/1/3 /BnZr /1/3 | /0/./8/0/( /8/7/) |\n| | | | | | /2 /: /5/1 | /1/9 /: /9/4 | /8 /: /9/0 /BnZr /0 /: /1/1 | /BnZr /0 /: /0/6 | | | /8/4 | |\n| | | | | | | | | | | /+/0 /: /2/0 | | |\n| | | | | | | | | | | | /+/1/4 | |\n| /5/6 | | | | | | | | | | | | |\n| | | | | /8/7 /+/1/3 | | | | | | | | |\n| | | | | | | | | | | | | /0/./5/6/( /8/7/) |\n| /5/8 /9/8/1/0/2/7 /0 /: /8/7/0 /3/6 /: /9/2 | /2 /: /5/4 | /5/1/1/1/3/./6/6/8 | /1/5 /: /6/0 | /1 /: | /1/8 | /8 /: /8/5 | | /6 /: /8/7 | /1 /: /2/8 | | | /0/./6/4/( /8/7/) |\n| | /BnZr /0 /: /0/2 | | /BnZr /0 /: /8/6 | | | | | | | | | |\n| /+/0 /: /0/0/8 /+/0 /: /7/3 | /+/0 /: /0/2 | | /+/0 /: /9/1 | /: /1/1 | /+/0 /: /0/7 | | /+/0 | /+/0 | /: /1/1 | | | | \nT ABLE /2/| Continue d \n| Obs /# | Date /(UT/) | MJD a | T col /(k eV/) | R / in b /(km/) | Photon Index | P o w er/- c la w | F e Edge d /(k eV/) | / F e d /(k eV/) | E line e | N line e /; f / /1/0/0 | Line Eq/. Width | / /2 / /(dof/) |\n|-------------------------------------------------------------------------------------------------------------------------------------------|---------------------------|---------------------------------------|----------------------------------------------------------------------------------|---------------------------------------------------------------------------------------------------------------------------------------------------------------------|------------------------------------------------------------------------|----------------------------------------------------------------------|------------------------------------------------------------------------|---------------------------------------------------------------------------------------|--------------------------------------------------------------------|----------------------------------------------------------------------|-------------------------------------------|-------------------------------------|\n| /0 /: /8/3/7 /3/9 /: /1/7 | /9/8/1/0/2/9 /2 /: | /5/1/1/1/5/./2/8/0 | /+/0 /: /0/0/6 | /+/0 /: /6/7 | /4/7 /+/0 /: /0/2 | /8 /: /3/8 /+/0 /: /5/0 | /8 /: /8/4 /+/0 /: /1/0 | /1 /: /3/8 /+/0 /: /0/7 /6 /: /8/3 | /+/0 /: /1/0 | /0 /: /9/1 /+/0 /: /1/3 | /9/3 /+/1/3 | /0/./8/3/( /8/7/) |\n| /5/9 | /9/8/1/0/3/1 | /5/1/1/1/7/./3/5/1 | /BnZr /0 /: /0/0/6 /0 /: /8/1/4 /+/0 /: /0/0/4 | /BnZr /0 /: /6/5 /4/5 /: /2/6 /+/0 /: /6/0 | /BnZr /0 /: /0/2 /2 /: /2/2 /+/0 /: /0/4 | /BnZr /0 /: /4/8 /1 /: /5/5 /+/0 /: /1/7 | /BnZr /0 /: /1/0 /8 /: /6/1 /+/0 /: /1/2 | /BnZr /0 /: /0/7 /1 /: /8/1 /+/0 /: /1/2 | /BnZr /0 /: /1/1 /6 /: /6/6 /+/0 /: /1/3 | /BnZr /0 /: /1/4 /0 /: /4/8 /+/0 /: /0/8 | /BnZr /1/4 /9/1 /+/1/4 | /1/./5/0/( /8/7/) |\n| /6/0 /4/6 /: /9/5 | /9/8/1/1/0/2 | /5/1/1/1/9/./0/0/3 | /BnZr /0 /: /0/0/4 /0 /: /7/7/8 /+/0 /: /0/0/4 | /BnZr /0 /: /5/7 /+/0 /: /6/7 | /BnZr /0 /: /0/4 /2 /: /2/7 /+/0 /: /0/4 | /BnZr /0 /: /1/5 /1 /: /5/0 /+/0 /: /1/7 | /BnZr /0 /: /1/2 /8 /: /3/8 /+/0 /: /1/2 | /BnZr /0 /: /1/2 /1 /: /9/5 /+/0 /: /1/4 | /BnZr /0 /: /1/4 /6 /: /6/3 /+/0 /: /1/4 | /BnZr /0 /: /0/8 /0 /: /3/7 /+/0 /: /0/6 | /BnZr /1/4 /8/6 /+/1/5 | |\n| /6/1 /9/8/1/1/0/4 | | /5/1/1/2/1/./0/0/3 | /BnZr /0 /: /0/0/4 /0 /: /7/6/0 /+/0 /: /0/0/3 | /BnZr /0 /: /6/5 /4/6 /: /4/1 /+/0 /: /5/7 /8 /: /4/8 | /BnZr /0 /: /0/4 /2 /: /3/5 /+/0 /: /0/6 | /BnZr /0 /: /1/5 /0 /: /6/2 /+/0 /: /1/1 | /BnZr /0 /: /1/2 /+/0 /: /1/2 /2 /: | /BnZr /0 /: /1/4 /8/0 /+/0 /: /1/8 | /BnZr /0 /: /1/6 /6 /: /4/7 /+/0 /: /1/4 | /BnZr /0 /: /0/7 /0 /: /2/6 /+/0 /: /0/5 | /BnZr /1/7 /9/0 /+/1/6 | /1/./0/1/( /8/7/) /1/./2/5/( /8/7/) |\n| /6/2 /6/3 /9/8/1/1/0/7 | /2 /: | /5/1/1/2/4/./7/2/7 | /BnZr /0 /: /0/0/3 /0 /: /7/2/1 /+/0 /: /0/0/4 | /BnZr /0 /: /5/5 /4/7 /: /3/4 /+/0 /: /6/7 /8 /: /4/2 | /BnZr /0 /: /0/6 /2/0 /+/0 /: /0/5 | /BnZr /0 /: /0/9 /0 /: /6/3 /+/0 /: /0/9 | /BnZr /0 /: /1/2 /+/0 /: /1/3 /2 /: | /BnZr /0 /: /1/9 /2/0 /+/0 /: /1/7 | /BnZr /0 /: /1/4 /6 /: /6/3 /+/0 /: /1/1 | /BnZr /0 /: /0/4 /0 /: /2/4 /+/0 /: /0/4 | /BnZr /1/5 /1/1/5 /+/2/0 | /0/./9/4/( /8/7/) |\n| /BnZr /0 /: /0/0/4 /BnZr /0 /: /6/5 /6/4 | /9/8/1/1/0/9 | /5/1/1/2/6/./5/9/5 | /0 /: /7/0/6 /+/0 /: /0/0/4 | /4/6 /: /9/0 /+/0 /: /7/3 | /BnZr /0 /: /0/5 /2 /: /3/1 /+/0 /: /0/2 | /BnZr /0 /: /0/8 /1 /: /5/9 /+/0 /: /1/0 | /BnZr /0 /: /1/3 /8 /: /6/3 /+/0 /: /0/8 | /BnZr /0 /: /1/7 /1 /: /8/1 /+/0 /: /0/7 | /BnZr /0 /: /1/3 /BnZr /0 /6 /: /6/2 /+/0 /: /1/1 | /: /0/4 /0 /: /2/7 /+/0 /: /0/4 /BnZr /0 /: /0/4 | /BnZr /2/1 /9/5 /+/1/4 /BnZr /1/5 | /0/./9/1/( /8/7/) |\n| /6/5 | /9/8/1/1/1/1 | /5/1/1/2/8/./5/6/4 | /BnZr /0 /: /0/0/4 /0 /: /6/8/2 /+/0 /: /0/0/4 | /BnZr /0 /: /7/1 /4/8 /: /7/4 /+/0 /: /8/7 | /BnZr /0 /: /0/2 /2 /: /1/9 /+/0 /: /0/4 | /BnZr /0 /: /0/9 /0 /: /7/9 /+/0 /: /0/8 | /BnZr /0 /: /0/8 /8 /: /3/5 /+/0 /: /1/3 | /BnZr /0 /: /0/7 /1 /: /8/7 /+/0 /: /1/1 | /BnZr /0 /: /1/2 /6 /: /3/4 /+/0 /: /1/8 | /0 /: /1/8 /+/0 /: /0/4 | /7/6 /+/1/8 | /0/./9/8/( /8/7/) |\n| | | | /BnZr /0 /: /0/0/4 /+/0 /: /0/0/4 | /BnZr /0 /: /8/2 /+/0 /: /7/6 | /BnZr /0 /: /0/4 /+/0 /: /0/3 | /BnZr /0 /: /0/7 /+/0 /: /0/5 | /BnZr /0 /: /1/2 /+/0 /: /1/0 | /BnZr /0 /: /1/2 /+/0 /: /0/9 | /BnZr /0 /: /2/0 /+/0 /: /1/2 | /BnZr /0 /: /0/4 /+/0 /: /0/3 | /BnZr /1/8 /+/1/6 | |\n| /6/6 /9/8/1/1/1/3 /0 /: /6/7/1 /4/8 /: /2/7 | /9/8/1/1/1/5 | /5/1/1/3/0/./4/5/7 /5/1/1/3/2/./4/7/9 | /BnZr /0 /: /0/0/4 /0 /: /6/5/9 /+/0 /: /0/0/5 | /BnZr /0 /: /7/5 /4/7 /: /5/3 /+/0 /: /9/2 | /2 /: /1/6 /BnZr /0 /: /0/3 /+/0 /: /0/5 | /0 /: /7/2 /BnZr /0 /: /0/5 /0 /: /6/3 /+/0 /: /0/8 | /8 /: /4/5 /BnZr /0 /: /1/0 /+/0 /: /1/9 | /1 /: /8/2 /BnZr /0 /: /0/9 /6 /: /4/5 /1 /: /6/4 /+/0 /: /1/4 | /BnZr /0 /: /1/3 /6 /: /6/0 /+/0 /: /1/3 /BnZr /0 /: /1/6 | /0 /: /1/8 /BnZr /0 /: /0/3 /0 /: /1/5 /+/0 /: /0/4 | /9/4 /BnZr /1/7 /1/0/6 /+/2/7 /BnZr /2/8 | /1/./2/0/( /8/7/) /2/./8/9/( /8/7/) |\n| /6/7 /6/8 | /9/8/1/1/1/7 | /5/1/1/3/4/./4/3/6 | /BnZr /0 /: /0/0/5 /0 /: /6/4/3 /+/0 /: /0/0/3 | /BnZr /0 /: /9/6 /4/8 /: /2/7 /+/0 /: /8/1 | /2 /: /1/8 /BnZr /0 /: /0/5 /+/0 /: /0/4 | /BnZr /0 /: /0/7 /+/0 /: /0/4 | /8 /: /4/0 /BnZr /0 /: /1/8 /8 /: /3/5 /+/0 /: /1/2 | /BnZr /0 /: /1/4 /2 /: /0/8 /+/0 /: /1/2 | /6 /: /4/2 /+/0 /: /1/4 /0 /: /1/2 | /BnZr /0 /: /0/4 /+/0 /: /0/2 | /1/0/4 /+/1/9 | /1/./9/4/( /8/7/) |\n| /6/9 | /9/8/1/1/1/9 | /5/1/1/3/6/./7/7/7 | /BnZr /0 /: /0/0/4 /0 /: /6/1/3 /+/0 /: /0/0/3 | /BnZr /0 /: /7/6 /5/1 /: /1/2 /+/0 /: /7/4 | /2 /: /1/7 /BnZr /0 /: /0/4 /+/0 /: /0/4 | /0 /: /3/8 /BnZr /0 /: /0/4 /+/0 /: /0/4 | /BnZr /0 /: /1/2 /+/0 /: /1/0 | /BnZr /0 /: /1/2 /+/0 /: /1/0 | /BnZr /0 /: /1/6 /5 /: /9/6 /+/0 /: /1/8 | /BnZr /0 /: /0/2 /0 /: /0/9 /+/0 /: /0/2 | /BnZr /2/0 /6/5 /+/1/4 /BnZr /1/5 | /1/./3/3/( /8/7/) |\n| | | /5/1/1/3/7/./9/2/3 | /BnZr /0 /: /0/0/3 /+/0 /: /0/0/5 | /BnZr /0 /: /7/3 /+/1 /: /1/0 | /2 /: /2/3 /BnZr /0 /: /0/3 /+/0 /: /0/2 | /0 /: /4/2 /BnZr /0 /: /0/3 /+/0 /: /0/6 | /8 /: /1/1 /BnZr /0 /: /1/0 | /2 /: /0/8 /BnZr /0 /: /1/1 /+/0 /: /0/8 | /BnZr /0 /: /1/0 /+/0 /: /1/3 | /BnZr /0 /: /0/2 /+/0 /: /0/3 | /+/1/9 | |\n| /7/0 /0 /: /6/2/0 /7/1 | /9/8/1/1/2/0 | | /BnZr /0 /: /0/0/5 /0 /: /6/1/4 /+/0 /: /0/0/6 | /4/6 /: /7/7 /BnZr /1 /: /0/5 /+/1 /: /2/2 | /2 /: /2/4 /BnZr /0 /: /0/3 /+/0 /: /0/3 | /0 /: /9/8 /BnZr /0 /: /0/6 | /8 /: /5/4 /+/0 /: /0/9 /BnZr /0 /: /1/0 | /1 /: /6/4 /BnZr /0 /: /0/8 /BnZr /0 /: /+/0 /: /0/8 /+/0 /: | /6 /: /6/1 /1/4 /BnZr /0 /1/6 /+/0 | /0 /: /1/3 /: /0/3 /: /0/4 | /7/7 /BnZr /2/0 /+/1/8 | /1/./5/6/( /8/7/) |\n| | /9/8/1/1/2/2 | /5/1/1/3/9/./9/9/1 | /BnZr /0 /: /0/0/6 /+/0 /: /0/0/5 | /4/3 /: /7/2 /BnZr /1 /: /1/7 /+/1 /: /1/7 | /2 /: /1/9 /BnZr /0 /: /0/3 /+/0 /: /0/2 | /1 /: /0/0 /+/0 /: /0/6 /BnZr /0 /: /0/6 | /8 /: /5/6 /+/0 /: /0/9 /BnZr /0 /: /0/9 | /1 /: /6/6 /BnZr /0 /: /0/9 /+/0 /: /0/4 | /6 /: /3/3 /BnZr /0 /: /1/9 /+/0 /: /1/7 | /0 /: /1/4 /BnZr /0 /: /0/4 /+/0 /: /0/3 | /6/8 /BnZr /1/8 /+/1/8 | /0/./9/9/( /8/7/) |\n| /7/2 /BnZr /0 /: /0/0/5 /BnZr /1 /: /1/3 | /9/8/1/1/2/3 | /5/1/1/4/0/./7/0/6 | /0 /: /6/0/3 | /4/5 /: /0/0 | /2 /: /2/4 /BnZr 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/8/5 /+/0 /: /2/0 /BnZr /0 /: /2/2 /0 /: /9/6 | /BnZr /0 /: /2/7 /1 /: /5/7 /+/0 /: /3/1 | /4/3 /BnZr /1/2 /6/3 /+/1/2 /BnZr /1/2 | /1/./7/2/( /8/7/) |\n| /1/0/1 | | | | | /2 /: /7/2 | | /8 /: /7/6 | /2 /: /4/3 | /6 /: /8/7 /+/0 /: /1/4 | | | /1/./0/0/( /8/7/) |\n| /BnZr /0 /: /0/0/4 /BnZr /0 /: /3/6 /1/0/2 /9/9/0/1/0/4 /1 /: /0/8/0 /+/0 /: /0/0/3 /3/8 /: /3/4 /+/0 /: /3/6 | /3 /: | /5/1/1/8/2/./3/3/7 | | /8 /: /6/6 | /BnZr /0 /: /0/4 /8/7 /+/0 /: /0/5 | /BnZr /1 /: /6/1 /4/8 /: /2/1 /+/6 /: /3/0 | /BnZr /0 /: /0/9 /+/0 /: /1/1 /4 /: | /BnZr /0 /: /1/6 /6/1 /+/0 /: /2/6 /6 /: /5/( f ixed | /BnZr /0 /: /1/5 /) /0 /: /2/3 /+/0 | /BnZr /0 /: /3/1 /: /3/0 | /7 /+/1/0 | /1/./5/5/( /8/7/) |\n| /BnZr /0 /: /0/0/2 /1/0/3 | /9/9/0/1/0/5 | /5/1/1/8/3/./5/4/0 | /1 /: /0/8/3 /+/0 /: /0/0/3 /BnZr /0 /: /0/0/2 | /BnZr /0 /: /3/7 /3/8 /: /4/8 /+/0 /: /3/7 | /BnZr /0 /: /0/5 /3 /: /8/6 /+/0 /: /0/5 /BnZr /0 /: /0/5 | /BnZr /6 /: /0/7 /4/8 /: /0/6 /+/6 /: /5/4 | /BnZr /0 /: /1/1 /8 /: /6/7 /+/0 /: /1/3 | /BnZr /0 /: /2/6 /4 /: /0/2 /+/0 /: /2/7 | /6 /: /5/( f ixed /) | /BnZr /0 /: /2/3 /0 /: /1/6 /+/0 /: /3/0 /BnZr /0 /: /1/6 | /BnZr /7 /5 /+/1/0 /BnZr /5 /4 /+/6 | /1/./2/0/( /8/8/) |\n| /1/0/4 /1 /: /0/8/8 /BnZr /0 /: /0/0/2 /3/8 /: /0/6 /BnZr /0 /: /2/8 | /9/9/0/1/0/6 /3 /: | /5/1/1/8/4/./7/1/1 | /+/0 /: /0/0/2 | /BnZr /0 /: /3/7 /+/0 /: /2/7 | /8/9 /+/0 /: /0/4 /BnZr /0 /: /0/4 | /BnZr /6 /: /1/2 /5/1 /: /0/7 /+/5 /: /0/8 /BnZr /4 /: /7/1 | /BnZr /0 /: /1/3 /8 /: /6/9 /+/0 /: /0/8 /BnZr /0 /: /0/9 /4 /: | /BnZr /0 /: /2/6 /5/2 /+/0 /: /2/0 | /6 /: /5/( f ixed /) | /0 /: /1/3 /+/0 /: /2/1 /BnZr /0 /: /1/3 | /BnZr /4 | /1/./7/0/( /7/9/) |\n| /1/0/5 /+/0 /: /0/0/3 /+/0 /: /5/3 | /9/9/0/1/0/7 | /5/1/1/8/5/./1/2/6 | /1 /: /0/6/6 /BnZr /0 /: /0/0/3 /+/0 /: /0/0/2 | /4/0 /: /6/9 /BnZr /0 /: /5/8 /+/0 /: /3/3 | /3 /: /5/1 /+/0 /: /0/8 /BnZr /0 /: /1/0 /+/0 /: /0/4 | /2/3 /: /7/6 /+/6 /: /3/6 /BnZr /5 /: /6/7 | /9 /: /0/8 /+/0 /: /1/4 /BnZr /0 /: /1/5 | /BnZr /0 /: /2/0 /4 /: /3/2 /+/0 /: /3/2 | /7 /: /3/3 /+/0 /: /2/0 | /0 /: /5/8 /+/0 /: /2/2 | /3/7 /+/1/4 /BnZr /1/5 | /1/./4/4/( /8/6/) |\n| /1/0/6 /BnZr /0 /: /0/0/2 /BnZr /0 /: /3/5 | /9/9/0/1/0/7 /4 /: | /5/1/1/8/5/./8/4/6 | /1 /: /0/9/7 | /3/7 /: /2/4 /8 /: /6/5 | /0/3 /BnZr /0 /: /0/5 | /5/7 /: /1/2 /+/7 /: /3/7 /BnZr /6 /: /8/5 | /+/0 /: /1/1 /BnZr /0 /: /1/1 | /BnZr /0 /: /3/3 /4 /: /2/5 /+/0 /: /2/9 /BnZr /0 /: /2/8 | /BnZr /0 /: /2/6 /6 /: /5/( f ixed /) /BnZr /0 | /BnZr /0 /: /2/4 /0 /: /0/0 /+/0 /: /2/0 /: /0/0 | /0 /+/6 /BnZr /0 | /1/./9/6/( /7/4/) |\n| /1/0/7 | /9/9/0/1/0/9 | /5/1/1/8/7/./1/2/6 | /1 /: /0/9/6 /+/0 /: /0/0/3 /BnZr /0 /: /0/0/3 /+/0 /: /0/0/3 | /3/7 /: /5/6 /+/0 /: /3/7 /BnZr /0 /: /3/8 | /3 /: /9/6 /+/0 /: /0/7 /BnZr /0 /: /0/7 | /4/9 /: /5/8 /+/7 /: /8/4 /BnZr /7 /: /2/8 | /8 /: /9/1 /+/0 /: /1/1 /BnZr /0 /: /1/1 | /4 /: /5/2 /+/0 /: /3/0 /BnZr /0 /: /3/1 | /6 /: /5/( f ixed /) | /0 /: /0/0 /+/0 /: /2/3 /BnZr /0 /: /0/0 /+/0 /: /1/7 | /0 /+/7 /BnZr /0 /0 /+/5 | /1/./9/3/( /6/2/) |\n| /1/0/8 | /9/9/0/1/1/0 | /5/1/1/8/8/./1/9/4 | /1 /: /0/9/7 | /3/7 /: /3/5 /+/0 /: /3/7 | /4 /: /1/3 /+/0 /: /0/6 | /6/6 /: /7/4 /+/8 /: /9/3 | /8 /: /8/1 /+/0 /: /1/2 | /3 /: /9/1 /+/0 /: /3/2 | /6 /: /5/( f ixed /) | /0 /: /0/0 /BnZr /0 /: /0/0 | /BnZr /0 | /1/./3/1/( /6/2/) |\n| /BnZr /0 /: /0/0/3 /BnZr /0 /: /3/7 /1/0/9 /+/0 /: /3/2 /1/1/0 | /9/9/0/1/1/0 /9/9/0/1/1/2 | /5/1/1/8/8/./7/3/1 | /1 /: /1/0/6 /+/0 /: /0/0/2 /BnZr /0 /: /0/0/3 /+/0 /: /0/0/2 | /3/6 /: /5/9 /BnZr /0 /: /3/3 /+/0 /: /2/8 | /BnZr /0 /: /0/6 /4 /: /5/2 /+/0 /: /0/5 /BnZr /0 /: /0/5 /+/0 /: /0/5 | /BnZr /8 /: /4/9 /1/1/0 /: /5/0 /+/1/1 /: /7/3 /BnZr /1/1 /: /1/6 | /BnZr /0 /: /1/3 /8 /: /7/5 /+/0 /: /2/2 /BnZr /0 /: /2/2 /+/0 /: /1/3 | /BnZr /0 /: /3/4 /2 /: /6/5 /+/0 /: /4/8 /BnZr /0 /: /5/0 /3 /: /8/6 /+/0 /: /3/9 | /6 /: /5/( f ixed /) | /+/0 /: /0/8 | /+/2 | /1/./0/0/( /8/8/) |\n| /1 /: /1/0/5 /BnZr /0 /: /0/0/3 /BnZr /0 /: /2/8 /1/1/1 | /4 /: | /5/1/1/9/0/./0/5/4 | /0 /: /0/0 /BnZr /0 /: /0/0 /0 /: /0/0 /+/0 /: /0/6 | /3/7 /: /0/1 | /2/7 /BnZr /0 /: /0/6 /4 /: /4/6 /+/0 /: /0/4 | /7/6 /: /1/0 /+/7 /: /9/4 /BnZr /7 /: /7/2 /+/1/0 /: /2/6 | /8 /: /8/8 /BnZr /0 /: /1/4 | /BnZr /0 /: /4/1 | /6 /: /5/( f ixed /) | /BnZr /0 /: /0/0 | /0 /BnZr /0 /0 /+/1 /BnZr /0 /+/2 | /1/./5/3/( /8/8/) |\n| /9/9/0/1/1/3 /1 /: /1/2/0 /+/0 /: /0/0/2 /BnZr /0 /: /0/0/2 /3/6 /: /0/6 /+/0 /: /3/1 /BnZr /0 /: /3/2 /1/1/2 /+/0 /: /0/0/2 /+/0 /: /3/4 | /9/9/0/1/1/4 | /5/1/1/9/1/./4/8/5 /5/1/1/9/2/./1/8/7 | /BnZr /0 /: /0/0 /0 /: /0/0 /+/0 /: /1/3 | /8 /: /5/4 /BnZr /0 /0 /+/4 /BnZr /0 /+/2 | /BnZr /0 /: /0/4 /4 /: /1/2 /+/0 /: /0/4 /BnZr /0 /: /0/4 | /1/0/1 /: /3/0 /BnZr /9 /: /9/8 /7/3 /: /2/5 /+/7 /: /8/9 | /+/0 /: /2/2 /BnZr /0 /: /2/0 /2 /: /8 /: /7/2 /+/0 /: /1/2 | /2/3 /+/0 /: /4/4 /BnZr /0 /: /4/5 /3 /: /7/8 /+/0 /: /2/9 | /6 /: /5/( f ixed /) /+/0 /6 /: /5/( f ixed /) | /0 /: /0/0 /: /0/8 /BnZr /0 /: /0/0 | /0 | /1/./4/9/( /8/8/) |\n| /1 /: /1/1/5 /1/1/3 | /9/9/0/1/1/4 | /5/1/1/9/2/./5/9/3 | /+/0 /: /0/9 | /0 /BnZr /0 /+/3 | /4 /: /3/3 /+/0 /: /0/5 | /BnZr /8 /: /0/4 /: /4/8 /+/1/0 /: /1/4 | /BnZr /0 /: /1/2 /8 /: /8/9 /+/0 /: /1/9 | /BnZr /0 /: /3/1 /2 /: /7/2 /+/0 /: /3/8 /BnZr /0 /: /4/1 | /6 /: /5/( f ixed /) | /0 /: /0/0 /BnZr /0 /: /0/0 | | /1/./5/3/( /7/5/) /0/./8/6/( /8/8/) |\n| /1/1/4 | /9/9/0/1/1/6 /9/9/0/1/1/8 | /5/1/1/9/4/./5/0/0 | /BnZr /0 /: /0/0/3 /1 /: /1/1/2 /+/0 /: /0/0/2 /BnZr /0 /: /0/0/3 /+/0 /: /0/0/4 | /3/6 /: /5/7 /BnZr /0 /: /3/3 /3/6 /: /3/5 /+/0 /: /3/4 /BnZr /0 /: /3/3 | /BnZr /0 /: /0/5 /4 /: /2/5 /+/0 /: /0/6 /BnZr /0 /: /0/3 | /BnZr /9 /: /9/4 /+/1/4 /: /4/0 | /BnZr /0 /: /2/0 /8 /: /7/4 /+/0 /: /1/3 /BnZr /0 /: /2/1 | /3 /: /0/1 /+/0 /: /5/0 /BnZr /0 /: /3/0 /3 /: /4/4 /+/0 /: /3/0 | /6 /: /5/( f ixed /) | /+/0 /: /1/2 | | |\n| /1/1/5 | | /5/1/1/9/6/./0/5/1 | /1 /: /1/2/5 /BnZr /0 /: /0/0/2 /1 /: /1/2/1 /+/0 /: /0/0/2 /BnZr /0 /: /0/0/3 | /3/6 /: /0/3 /BnZr /0 /: /5/3 /3/6 /: /4/6 /+/0 /: /3/3 /BnZr /0 /: /3/4 | /4 /: /1/5 /+/0 /: /0/4 | /8/6 /: /4/7 /BnZr /5 /: /8/0 /8/0 /: /6/3 /+/8 /: /5/3 | /8 /: /6/9 /+/0 /: /1/2 | /BnZr /0 /: /2/9 | /6 /: /5/( f ixed /) | /0 /: /0/0 /BnZr /0 /: /0/0 /0 /: /0/0 /+/0 /: /1/4 /BnZr /0 /: /0/0 | /0 /BnZr /0 /0 /+/4 /BnZr /0 | /1/./1/1/( /8/8/) /2/./3/3/( /8/7/) |\n| /1/1/6 | /9/9/0/1/1/8 | | | | | | | | | /+/0 /: /0/7 | | |\n| | | | /+/0 /: /0/0/2 | | /BnZr /0 /: /0/4 | | | | | | | |\n| | | | | | | | | | /6 /: /5/( f ixed /) | /0 /: /0/0 /BnZr /0 /: /0/0 /0 /: /0/0 /+/0 /: /2/5 | /+/2 | |\n| | | | | /+/0 /: /3/5 | /4 /: /5/2 /+/0 /: /0/5 /BnZr /0 /: /0/5 | /BnZr /7 /: /9/3 | | | | | | |\n| | | | /1 /: /1/1/9 /BnZr /0 /: /0/0/2 | /3/6 /: /2/9 | | | /BnZr /0 /: /1/3 /8 /: /6/9 /+/0 /: /2/5 | /2 /: /1/6 /+/0 /: /4/6 /BnZr /0 /: /5/1 | /6 /: /5/( f ixed /) | | | |\n| | | /5/1/1/9/6/./5/2/2 | /+/0 /: /0/0/2 | /BnZr /0 /: /3/0 | | | | | | | | |\n| /1/1/7 | /9/9/0/1/2/0 | /5/1/1/9/8/./0/5/1 | /1 /: /1/1/5 /BnZr /0 /: /0/0/2 | /3/7 /: /0/5 /+/0 /: /3/5 /BnZr /0 /: /3/5 | /3 /: /9/8 /+/0 /: /0/4 | /+/1/0 /: /8/1 | | /+/0 /: /2/5 | | | /0 /BnZr /0 | |\n| | | | /BnZr /0 /: /0/0 | /9/5 /+/0 /: /2/0 | | /: /9/0 /BnZr /1/2 /: /3/7 | /BnZr /0 /: /2/4 /8 /: /7/1 /+/0 /: /1/1 | /3 /: /8/0 /BnZr /0 /: /2/6 | | | | |\n| | | | | | /BnZr /0 /: /0/4 | /1/1/9 | | | | | /0 /+/7 | |\n| | | | | | | | | | | | | /1/./4/0/( /8/8/) |\n| | | | | | | /+/7 /: /3/4 | | | | | | /1/./7/6/( /7/9/) |\n| | | | | | | | | | | /BnZr /0 | | |\n| | | | | | | /6/5 /: /5/2 /+/1/5 | /6/5 /: /5/2 /+/1/5 | /6/5 /: /5/2 /+/1/5 | /6/5 /: /5/2 /+/1/5 | /6/5 /: /5/2 /+/1/5 | /6/5 /: /5/2 /+/1/5 | /6/5 /: /5/2 /+/1/5 |\n| /1/1/8 /9/9/0/1/2/2 /1 /: /0/6/5 /+/0 /: /0/0/5 /BnZr /0 /: /0/0/4 /4/2 /: /1/5 /BnZr /0 /: /4/6 | /2 /: | /5/1/2/0/0/./3/8/3 | /+/0 /: /4/4 /+/0 /: | /0 /: /1/1 /8 /: /8/9 /: /1/1 /0 /: /1/0 /2 /: /3/6 /+/0 /: /1/7 /BnZr /0 /: /1/8 /6 /: /9/3 /+/0 /: /1/9 /BnZr /0 /: /2/0 /1 /: /6/9 /+/0 /: /4/2 /BnZr /0 /: /4/2 | /8/3 /0/5 /BnZr /0 /: /0/5 | /BnZr /7 /: /3/7 /1/9 /: /0/1 /+/2 /: /9/9 /BnZr /2 /: /7/9 | /BnZr /+/0 /BnZr | | | | /6/1 /BnZr /1/5 | /0/./5/6/( /8/7/) | \nT ABLE /2/| Continue d \n| Obs /# | Date /(UT/) | MJD a | T col /(k eV/) | R / in b /(km/) | Photon Index | P o w er/- c la w | F e Edge d /(k eV/) | / F e d /(k eV/) | E line e | N line e /; f / /1/0/0 | Line Eq/. Width | / /2 / /(dof/) |\n|------------------------------------------------------------------------------------------------------------------------|------------------------------------|---------------------------------------|----------------------------------------------------------------------------------|--------------------------------------------------------------------------|--------------------------------------------------------------------------|--------------------------------------------------------------------------------|----------------------------------------------------------------|--------------------------------------------------------------------------------------------|-----------------------------------------------------------------------------|----------------------------------------------------------------------|------------------------------------------|---------------------------------------|\n| /4/1 /: /9/2 | /9/9/0/1/2/3 /2 /: /6/5 | /5/1/2/0/1/./7/8/3 | /+/0 /: /0/0/4 | /+/0 /: /3/8 | /+/0 /: /0/3 /1/9 | /: /4/5 /+/2 /: /0/6 | /8 /: /8/6 /+/0 /: /0/9 | /1 /: /8/2 /+/0 /: /1/2 | /6 /: /8/4 /+/0 /: /1/4 | /2 /: /2/9 /+/0 /: /4/0 | /6/9 /+/1/2 | /1/./1/3/( /8/7/) |\n| /1/1/9 /1 /: /1/1/0 | /9/9/0/1/2/5 | /5/1/2/0/3/./2/4/9 | /1 /: /0/6/6 /BnZr /0 /: /0/0/5 /+/0 /: /0/0/4 | /BnZr /0 /: /3/9 /3/8 /: /1/1 /+/0 /: /5/1 | /BnZr /0 /: /0/4 /3 /: /7/9 /+/0 /: /0/4 | /BnZr /1 /: /9/5 /5/4 /: /1/1 /+/8 /: /0/8 | /BnZr /0 /: /0/9 /8 /: /8/9 /+/0 /: /1/6 | /BnZr /0 /: /1/2 /2 /: /9/0 /+/0 /: /3/7 | /BnZr /0 /: /1/4 /7 /: /4/5 /+/0 /: /2/5 | /BnZr /0 /: /4/1 /0 /: /5/4 /+/0 /: /2/4 | /BnZr /1/2 /3/1 /+/1/3 | /0/./9/9/( /8/7/) |\n| /1/2/0 | /9/9/0/1/2/6 | /5/1/2/0/4/./2/4/8 | /BnZr /0 /: /0/0/4 /1 /: /1/2/6 /+/0 /: /0/0/3 | /BnZr /0 /: /5/7 /+/0 /: /4/5 | /BnZr /0 /: /0/5 /4 /: /0/2 /+/0 /: /0/5 | /BnZr /7 /: /3/7 /7/7 /: /3/5 /+/9 /: /9/6 | /BnZr /0 /: /1/7 /8 /: /5/8 /+/0 /: /1/6 | /BnZr /0 /: /3/4 /3 /: /6/2 /+/0 /: /3/4 | /BnZr /0 /: /3/1 /6 /: /5/( f ixed /) | /BnZr /0 /: /2/6 /+/0 /: /3/2 | /BnZr /1/5 /+/9 | |\n| /1/2/1 /3/6 /: /4/7 | | /5/1/2/0/5/./8/5/0 | /BnZr /0 /: /0/0/3 /1 /: /1/2/2 /+/0 /: /0/0/2 | /BnZr /0 /: /4/6 /3/6 /: /8/7 /+/0 /: /3/6 /+/0 /: | /BnZr /0 /: /0/5 /3 /: /9/7 /+/0 /: /0/4 | /BnZr /9 /: /5/2 /+/7 /: /9/5 | /BnZr /0 /: /1/5 /1/2 | /BnZr /0 /: /3/5 /+/0 /: /2/5 | /6 /: /5/( f ixed /) | /0 /: /0/0 /BnZr /0 /: /0/0 /+/0 /: /3/2 | /0 /BnZr /0 /+/9 | /0/./8/4/( /8/8/) |\n| /1/2/2 /9/9/0/1/2/7 /1/2/3 /9/9/0/1/2/8 | /4 /: /2/1 | /5/1/2/0/6/./7/8/1 | /BnZr /0 /: /0/0/3 /+/0 /: /0/0/3 | /BnZr /0 /: /3/7 /+/0 /: /3/3 | /BnZr /0 /: /0/4 /+/0 /: /0/4 /8/9 | /7/2 /: /6/8 /BnZr /7 /: /2/3 /: /9/8 /+/9 /: /2/5 | /8 /: /6/5 /BnZr /0 /: /1/0 /+/0 /: /1/4 /3 /: /0/8 | /3 /: /7/9 /BnZr /0 /: /2/5 /+/0 /: /3/4 /6 /: /5/( f ixed | /) /0 /: /0/0 | /0 /: /0/0 /BnZr /0 /: /0/0 /+/0 /: /1/0 | /0 /BnZr /0 /0 /+/3 | /1/./6/8/( /8/7/) /1/./6/8/( /8/7/) |\n| /1/2/4 | /9/9/0/1/3/0 | /5/1/2/0/8/./1/7/9 | /1 /: /1/2/8 /BnZr /0 /: /0/0/2 /+/0 /: /0/0/3 | /3/6 /: /1/7 /BnZr /0 /: /3/5 /+/0 /: /4/2 | /BnZr /0 /: /0/4 /+/0 /: /0/5 | /BnZr /8 /: /5/6 /+/4 /: /1/1 | /8 /: /6/5 /BnZr /0 /: /1/5 /+/0 /: /1/1 | /BnZr /0 /: /3/3 /2 /: /8/8 /+/0 /: /1/9 | /7 /: /1/5 /+/0 /: /1/4 | /BnZr /0 /: /0/0 /0 /: /9/7 /+/0 /: /2/4 | /BnZr /0 /4/3 /+/1/0 | /1/./2/5/( /8/7/) |\n| /1/2/5 | /9/9/0/1/3/0 | /5/1/2/0/8/./8/4/5 | /1 /: /0/8/8 /BnZr /0 /: /0/0/3 /1 /: /1/0/1 /+/0 /: /0/0/4 | /4/0 /: /1/2 /BnZr /0 /: /4/5 /3/8 /: /7/8 /+/0 /: /4/9 | /3 /: /2/8 /BnZr /0 /: /0/5 /+/0 /: /0/5 | /2/8 /: /8/8 /BnZr /3 /: /8/1 /+/7 /: /2/1 | /8 /: /8/5 /BnZr /0 /: /1/1 /9 /: /0/5 /+/0 /: /1/5 | /BnZr /0 /: /2/0 /3 /: /0/9 /+/0 /: /3/1 | /BnZr /0 /: /1/7 /7 /: /4/7 /+/0 /: /1/9 | /BnZr /0 /: /2/5 /0 /: /6/3 /+/0 /: /2/2 | /BnZr /1/1 /3/8 /+/1/3 | /8/6/) |\n| | | | /BnZr /0 /: /0/0/3 /+/0 /: /0/0/2 | /BnZr /0 /: /5/3 /+/0 /: /3/5 | /3 /: /7/2 /BnZr /0 /: /0/5 /+/0 /: /0/4 | /4/5 /: /8/6 /BnZr /6 /: /6/6 /+/7 /: /0/2 | /BnZr /0 /: /1/6 /+/0 /: /1/1 | /BnZr /0 /: /3/1 /+/0 /: /2/3 | /BnZr /0 /: /2/2 | /BnZr /0 /: /2/4 /+/0 /: /2/9 | /BnZr /1/4 /+/8 | /1/./3/2/( |\n| /1/2/6 /9/9/0/1/3/1 /1/2/7 | /3 /9/9/0/2/0/2 | /5/1/2/0/9/./7/1/1 /5/1/2/1/1/./7/0/9 | /1 /: /1/1/2 /BnZr /0 /: /0/0/2 /1 /: /1/0/7 /+/0 /: /0/0/2 | /3/7 /: /4/9 /BnZr /0 /: /3/6 /+/0 /: /3/6 | /: /8/7 /BnZr /0 /: /0/4 /+/0 /: /0/4 | /5/7 /: /4/1 /BnZr /6 /: /4/9 /+/7 /: /1/1 | /8 /: /7/3 /BnZr /0 /: /1/0 /3 /: /8/0 /8 /: /7/3 /+/0 /: /1/2 | /BnZr /0 /: /2/4 /6 /: /5/( f ixed /4 /: /1/5 /+/0 /: /2/4 | /) /6 /: /5/( f ixed /) | /0 /: /0/0 /BnZr /0 /: /0/0 /0 /: /0/0 /+/0 /: /3/1 | /0 /BnZr /0 /0 /+/9 | /1/./3/3/( /8/7/) /1/./2/3/( /8/8/) |\n| /3/7 /: /4/8 /1/2/8 /1 /: /1/0/5 | /3 /: /9/1 /9/9/0/2/0/3 /3 /: /8/7 | /5/1/2/1/2/./7/7/5 | /BnZr /0 /: /0/0/3 /+/0 /: /0/0/2 | /BnZr /0 /: /3/6 /3/7 /: /5/8 /+/0 /: /3/6 | /BnZr /0 /: /0/5 /+/0 /: /0/4 /5/7 | /5/9 /: /0/8 /BnZr /6 /: /8/1 /: /1/6 /+/6 /: /6/6 | /BnZr /0 /: /1/0 /8 /: /6/6 /+/0 /: /1/0 | /BnZr /0 /: /1/3 /3 /: /8/1 /+/0 /: /2/2 | /6 /: /5/( f ixed /) | /BnZr /0 /: /0/0 /0 /: /0/0 /+/0 /: /1/6 | /BnZr /0 /0 /+/4 | /1/./0/7/( /8/8/) |\n| /1/2/9 /9/9/0/2/0/4 | | /5/1/2/1/3/./7/7/4 | /BnZr /0 /: /0/0/3 /1 /: /0/9/0 /+/0 /: /0/0/3 | /BnZr /0 /: /3/6 /3/9 /: /1/9 /+/0 /: /5/1 | /BnZr /0 /: /0/4 /+/0 /: /0/5 | /BnZr /6 /: /5/0 /4/0 /: /4/2 /+/7 /: /0/0 | /BnZr /0 /: /1/0 /+/0 /: /1/5 | /BnZr /0 /: /2/3 /3 /: /3/1 /+/0 /: /3/5 | /7 /: /4/4 /+/0 /: /2/1 | /BnZr /0 /: /0/0 /0 /: /5/8 /+/0 /: /2/2 | /BnZr /0 /3/6 /+/1/3 | /1/./1/6/( /8/7/) |\n| /1/3/0 | | /5/1/2/1/4/./9/7/5 | /BnZr /0 /: /0/0/3 /+/0 /: /0/0/3 | /BnZr /0 /: /5/5 /+/0 /: /3/7 | /3 /: /7/0 /BnZr /0 /: /0/6 /+/0 /: /0/5 | /BnZr /6 /: /5/5 /+/2 /: /8/1 | /8 /: /9/4 /BnZr /0 /: /1/5 /+/0 /: /1/0 | /BnZr /0 /: /3/4 /+/0 /: /1/7 | /BnZr /0 /: /2/5 /+/0 /: /1/8 | /BnZr /0 /: /2/4 /+/0 /: /3/0 | /BnZr /1/5 /+/1/2 | |\n| /1/3/1 /+/0 /: /0/0/3 /+/0 /: /3/4 | /9/9/0/2/0/5 | | /1 /: /0/6/6 /BnZr /0 /: /0/0/3 | /4/1 /: /5/1 /BnZr /0 /: /3/9 /BnZr /0 /: /+/0 /: | /2 /: /9/5 /BnZr /0 /: /0/5 /+/0 /: /0/4 | /1/7 /: /9/5 /BnZr /2 /: /6/5 /+/2 /: /5/3 | /8 /: /8/7 /0/9 /0/9 | /2 /: /7/3 /BnZr /0 /: /1/8 /BnZr /0 /: /+/0 /: /1/6 /+/0 /: | /7 /: /0/1 /1/9 /BnZr /0 /1/7 /+/0 | /1 /: /2/2 /: /3/0 /: /2/8 | /5/1 /BnZr /1/2 /+/1/2 | /0/./8/1/( /8/7/) |\n| /9/9/0/2/0/6 /1 /: /0/5/9 /4/1 /: /9/4 | | /5/1/2/1/5/./8/3/9 /5/1/2/1/6/./8/3/8 | /BnZr /0 /: /0/0/3 /1 /: /0/7/7 /+/0 /: /0/0/4 | /BnZr /0 /: /3/5 /3/9 /: /6/0 /+/0 /: /5/3 | /2 /: /9/7 /BnZr /0 /: /0/5 /+/0 /: /0/5 | /1/7 /: /1/9 /BnZr /2 /: /3/8 /+/6 /: /3/0 | /8 /: /9/1 /BnZr /0 /: /0/9 /2 /: /7/6 /+/0 /: /1/2 | /BnZr /0 /: /1/7 /6 /: /9/9 /+/0 /: /2/7 | /BnZr /0 /: /1/9 /1 /: /1/9 /7 /: /1/1 /+/0 /: /2/5 | /BnZr /0 /: /2/8 /0 /: /6/5 /+/0 /: /2/6 | /5/1 /BnZr /1/2 /+/1/2 | /1/./0/0/( /8/7/) |\n| /1/3/2 | /9/9/0/2/0/7 | | | | /3 /: /5/8 | /3/8 /: /7/4 | /8 /: /8/0 | /3 /: /5/2 | | | /3/2 | /1/./0/7/( /8/7/) |\n| /1/3/3 /9/9/0/2/0/8 /1 /: /0/9/8 /3/7 /: /7/5 | | /5/1/2/1/7/./7/0/4 /5/1/2/1/9/./0/5/3 | /BnZr /0 /: /0/0/3 /1 /: /0/8/3 /+/0 /: /0/0/3 | /BnZr /0 /: /3/6 /3/9 /: /1/5 /+/0 /: /3/6 | /3 /: /9/2 /BnZr /0 /: /0/4 /+/0 /: /0/4 | /5/7 /: /6/8 /BnZr /6 /: /6/3 /+/5 /: /1/3 | /8 /: /6/6 /BnZr /0 /: /1/2 /4 /: /2/1 /+/0 /: /1/2 | /BnZr /0 /: /2/7 /6 /: /5/( f ixed /3 /: /5/6 /+/0 /: /2/7 | /) /0 /: /1/6 /7 /: /4/2 /+/0 /: /1/4 | /BnZr /0 /: /1/6 /0 /: /5/5 /+/0 /: /1/5 | /4 /BnZr /4 /3/7 /+/9 | /1/./2/9/( /8/8/) /2/./5/3/( /8/1/) |\n| /1/3/4 | /9/9/0/2/1/0 | | /BnZr /0 /: /0/0/2 /+/0 /: /0/0/5 | /BnZr /0 /: /3/8 /+/0 /: /6/4 | /3 /: /7/8 /BnZr /0 /: /0/5 /+/0 /: /0/7 | /3/9 /: /3/5 /BnZr /4 /: /9/2 /+/8 /: /6/2 | /9 /: /0/4 /BnZr /0 /: /1/2 /+/0 /: /1/7 | /BnZr /0 /: /2/8 /+/0 /: /4/2 | /BnZr /0 /: /1/6 /+/0 /: /2/5 | /BnZr /0 /: /1/5 /+/0 /: /2/7 | /BnZr /1/0 /+/1/7 | |\n| /1/3/5 /1 /: /0/7/1 | /9/9/0/2/1/1 | /5/1/2/2/0/./5/0/8 /5/1/2/2/0/./8/3/6 | /BnZr /0 /: /0/0/4 /+/0 /: /0/0/4 | /4/0 /: /0/6 /BnZr /0 /: /7/2 /+/0 /: /6/1 | /3 /: /6/6 /BnZr /0 /: /0/8 /+/0 /: /0/7 | /3/5 /: /2/9 /BnZr /7 /: /6/6 /+/7 /: /8/5 | /8 /: /9/8 /BnZr /0 /: /1/8 | /3 /: /8/6 /BnZr /0 /: /4/1 /7 /: /3/4 /+/0 /: /3/9 | /BnZr /0 /: /3/4 /+/0 /: /2/6 | /0 /: /6/0 /BnZr /0 /: /3/0 /+/0 /: /2/7 | /3/8 /BnZr /1/9 /+/1/6 | /0/./6/1/( /8/6/) |\n| /1/3/6 | /9/9/0/2/1/1 | | /1 /: /0/7/8 | /3/9 /: /3/3 | /3 /: /6/2 | /3/3 /: /3/5 | /8 /: /9/4 /+/0 /: /1/7 | /3 /: /6/0 | /7 /: /3/0 | /0 /: /5/6 | /3/4 | /0/./7/5/( /8/6/) |\n| /1/3/7 /BnZr /0 /: /0/0/2 /BnZr /0 /: /3/7 | /9/9/0/2/1/2 | /5/1/2/2/1/./5/6/7 | /BnZr /0 /: /0/0/4 /1 /: /0/9/3 /+/0 /: /0/0/3 | /BnZr /0 /: /6/7 /3/7 /: /4/5 /+/0 /: /3/6 /BnZr /0 /: | /BnZr /0 /: /0/8 /4 /: /0/0 /+/0 /: /0/5 /BnZr /0 /: /0/5 | /BnZr /7 /: /0/3 /5/8 /: /6/3 /+/7 /: /5/3 /BnZr /7 /: /1/0 | /BnZr /0 /: /1/8 /8 /: /7/2 /+/0 /: /1/2 /1/0 | /BnZr /0 /: /3/8 /4 /: /7/0 /+/0 /: /2/8 /BnZr /0 /: /2/8 | /BnZr /0 /: /3/8 /6 /: /5/( f ixed /) /BnZr /0 | /BnZr /0 /: /3/0 /0 /: /0/0 /+/0 /: /2/6 /: /0/0 | /BnZr /1/8 /0 /+/8 /BnZr /0 | 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/: /3/0 /+/0 /: /2/0 | /1/0/5 /BnZr /2/3 /+/2/3 | /0/./3/4/( /3/9/) g g |\n| /1/6/5 | /9/9/0/3/2/0 | /5/1/2/5/7/./3/6/2 | /0 /: /8/4/8 /8 /: | /4/6 /: /7/6 /6/7 | /2 /: /4/0 | /4 /: /2/8 | | /1 /: /8/1 | /6 /: /7/6 | /0 /: /9/4 | /1/0/9 | /0/./5/0/( /3/9/) |\n| /1/6/6 /9/9/0/3/2/1 /0 /: /8/4/6 /+/0 /: /0/0/7 /4/4 /: /2/9 /+/0 /: /9/0 | /9/9/0/3/2/1 | /5/1/2/5/8/./0/8/8 /5/1/2/5/8/./4/9/6 | /BnZr /0 /: /0/0/7 /0 /: /8/3/4 /+/0 /: /0/0/7 | /BnZr /0 /: /8/7 /4/4 /: /6/6 /+/0 /: /9/5 | /2 /: /5/2 /+/0 /: /0/4 /BnZr /0 /: /0/4 | /7 /: /0/7 /+/0 /: /8/2 /BnZr /0 /: /7/3 /+/0 /: /7/9 | /8 /: /7/5 /+/0 /: /1/6 /BnZr /0 /: /1/6 | /1 /: /4/5 /+/0 /: /1/4 /BnZr /0 /: /1/4 /+/0 /: /+/0 /: /1/3 | /6 /: /8/1 /1/6 /BnZr /0 /: /1/7 /+/0 /+/0 /: /1/6 | /0 /: /8/5 /: /1/9 /BnZr /0 /: /2/0 /+/0 /: /1/9 | /9/2 /+/2/0 /BnZr /2/1 /+/2/0 | /0/./5/2/( /3/9/) g g |\n| /1/6/7 | | | | | /2 /: /5/3 /+/0 /: /0/4 | /7 /: /4/6 | /8 /: /6/1 /+/0 /: /1/4 | /1 /: /5/4 | /6 /: /7/3 | /0 /: /8/4 | /8/8 | /0/./5/6/( /3/9/) |\n| /1/6/8 /0 /: /8/1/3 | /9/9/0/3/2/1 /2 /: /5/0 | /5/1/2/5/8/./9/7/5 | /BnZr /0 /: /0/0/8 /+/0 /: /0/0/9 /8 /: | /BnZr /0 /: /9/1 /4/3 /: /6/1 /+/1 /: /1/1 /5/7 /+/0 /: | /BnZr /0 /: /0/4 /+/0 /: /0/3 /8 | /BnZr /0 /: /7/1 /: /6/4 /+/0 /: /7/5 | /BnZr /0 /: /1/4 /1/3 /1 /: /5/3 | /BnZr /0 /: /1/3 /+/0 /: /1/1 /6 /: /7/4 /+/0 /: | /BnZr /0 /: /1/7 /1/4 /0 /: /9/1 /+/0 | /BnZr /0 /: /2/0 /: /2/1 | /BnZr /2/1 /9/1 /+/2/0 | /0/./6/4/( /3/9/) g |\n| /BnZr /0 /: /0/0/9 /BnZr /1 /: /0/5 /1/6/9 /0 /: /8/2/3 /+/0 /: /0/0/9 | /9/9/0/3/2/2 | /5/1/2/5/9/./2/5/2 | /BnZr /0 /: /2/2 /1 /: /0/4 /+/0 /: /1/9 /BnZr /0 /: /2/0 | /4/2 /: /3/0 /+/1 /: /0/5 /BnZr /1 /: /0/0 | /BnZr /0 /: /0/3 /2 /: /4/6 /+/0 /: /0/4 /BnZr /0 /: /0/4 | /BnZr /0 /: /6/9 /7 /: /4/7 /+/0 /: /7/3 /BnZr /0 /: /6/7 | /BnZr /0 /: /1/2 /8 /: /8/7 /+/0 /: /1/5 /BnZr /0 /: /1/5 /1 | /BnZr /0 /: /1/1 /: /3/7 /+/0 /: /1/1 /BnZr /0 /: /1/1 | /BnZr /0 /: /1/6 /6 /: /8/0 /+/0 /: /1/4 /BnZr /0 /: /1/4 | | /BnZr /2/1 /1/1/2 /+/2/0 /BnZr /2/1 | /1/./0/8/( /3/9/) g |\n| /1/7/0 y /9/9/0/3/2/3 /0 /: /8/2/9 /+/0 /: /0/1/1 /BnZr /0 /: /0/1/1 /4/7 /: /8/3 /+/1 /: /9/8 /BnZr /1 /: /8/3 /1/7/1 | /2 /: /3/1 /9/9/0/3/2/4 | /5/1/2/6/0/./5/5/2 | /0 /: /6/2 /+/0 /: /2/4 /BnZr /0 /: /2/3 | /+/0 /: /8/2 /BnZr /3/1 /1/0/7 /+/3/0 /BnZr /3/1 | /+/0 /: /0/5 /BnZr /0 /: /0/5 /2 /2 /: /3/1 /+/0 /: /0/5 | /: /1/1 /+/0 /: /3/4 /BnZr /0 /: /2/8 /2 /: /0/1 /+/0 /: /3/1 | /8 /: /2/9 /1/8 /BnZr /0 /: /1/8 /8 /: /3/4 /+/0 /: /1/7 | /1 /: /7/2 /+/0 /: /2/2 /BnZr /0 /: /2/2 /1 /: /7/6 /+/0 /: /2/0 | /6 /: /5/( f ixed /) | | /+/3/1 | /1/./1/8/( /7/2/) |\n| /0 /: /8/0/7 /+/0 /: /0/1/1 /+/2 /: /0/5 | | /5/1/2/6/1/./7/6/6 | /0 /: /7/2 /+/0 /: /2/1 /BnZr /0 /: /2/1 | /6 /: /5/( f ixed /) /6 /: /5/( f ixed /) /0 /6 /: /5/( f ixed /) /0 | /BnZr /0 /: /0/5 /+/0 /: /0/6 | /BnZr /0 /: /2/6 /+/0 /: /2/5 | /BnZr /0 /: /1/8 /+/0 /: /2/2 | /BnZr /0 /: /2/0 /1 /: /6/5 /+/0 /: /2/3 | | /: /7/6 /+/0 /: /2/5 | /+/4/1 | /1/./0/2/( /7/2/) |\n| /1/7/2 /9/9/0/3/2/6 /1/7/3 | /9/9/0/3/2/7 | /5/1/2/6/3/./1/0/8 /5/1/2/6/4/./7/4/8 | /BnZr /0 /: /0/1/1 /0 /: /8/0/6 /+/0 /: /0/1/5 /BnZr /0 /: /0/1/5 /+/0 /: /0/1/0 | /4/9 /: /3/3 /BnZr /1 /: /9/2 /4/7 /: /9/7 /+/2 /: /8/8 /BnZr /2 /: /7/0 | /2 /: /1/6 /BnZr /0 /: /0/6 /2 /: /2/7 /+/0 /: /0/4 | /1 /: /3/8 /BnZr /0 /: /2/1 /+/0 /: /1/7 | /8 /: /6/5 /BnZr /0 /: /2/3 /8 /: /3/5 /+/0 /: /1/6 | /BnZr /0 /: /2/3 /1 /: /7/5 /+/0 /: /1/7 /BnZr /0 /: /1/7 | | /BnZr /0 /: /2/6 /+/0 /: /1/3 | /1/2/4 /BnZr /4/2 | /0/./9/8/( /3/2/) g |\n| | /9/9/0/3/2/8 | /5/1/2/6/5/./6/1/3 | /0 /: /7/6/8 /BnZr /0 /: /0/0/9 /0 /: /7/5/9 /+/0 /: /0/1/0 | /4/8 /: /8/4 /+/1 /: /8/8 /BnZr /1 /: /9/6 /+/2 /: /0/2 | /BnZr /0 /: /0/4 /+/0 /: /0/4 | /1 /: /3/8 /BnZr /0 /: /1/5 /+/0 /: /1/6 | /BnZr /0 /: /1/6 /+/0 /: /1/4 | /1 /: /8/0 /+/0 /: /1/6 | /6 /: /5/( f ixed /) | /: /5/1 /BnZr /0 /: /1/4 /0 /: /4/8 /+/0 /: /1/3 | /1/1/4 /+/3/0 /BnZr /3/1 /+/3/0 | /0/./9/3/( /7/2/) |\n| /1/7/4 | | | | /4/8 /: /6/9 | | | | | | /BnZr /0 /: /1/3 | /1/1/2 /BnZr /3/1 | /2/./0/7/( /7/2/) |\n| | /9/9/0/3/2/9 | | /BnZr /0 /: /0/0/9 | | /2 /: /2/4 /BnZr /0 /: /0/4 | | /8 /: /3/8 | /BnZr /0 /: /1/5 | | | | |\n| /1/7/5 | | | | /BnZr /1 /: /9/2 | | | | | /6 /: /5/( f ixed /) | /0 /: /3/6 /+/0 /: /1/4 /BnZr /0 /: /1/4 /0 /: /3/3 /+/0 /: /1/7 | /+/4/3 | |\n| | | | | /4/7 /: /1/0 /+/2 /: /6/8 | /2 /: /2/9 /+/0 /: /0/7 /BnZr /0 /: /0/6 /+/0 /: /0/6 | /1 /: /4/1 /BnZr /0 /: /1/4 /+/0 /: /1/8 | | | | | | |\n| | | | /0 /: /7/5/4 /+/0 /: /0/1/3 | | | /0 /: /9/1 | /BnZr /0 /: /1/5 /8 /: /4/8 /+/0 /: /2/3 /BnZr /0 /: /2/4 | /BnZr /0 /: /2/4 /1 /: /9/4 /+/0 /: /2/8 | /6 /: /5/( f ixed /) | | | |\n| | | /5/1/2/6/6/./8/8/0 | /BnZr /0 /: /0/1/2 /0 /: /7/3/4 /+/0 /: /0/1/4 | /BnZr /2 /: /4/8 /4/9 /: /3/1 /+/3 /: /1/9 | | | | /1 /: /7/7 /+/0 /: /2/5 | | | /1/1/3 /BnZr /4/3 | |\n| /1/7/6 | | /5/1/2/6/7/./6/1/2 | /BnZr /0 /: /0/1/4 | | /2 /: /3/5 | /BnZr /0 /: /1/5 /+/0 /: /3/5 | | | | /BnZr /0 /: /2/0 | /8/7 /+/4/6 | /0/./9/4/( /3/2/) g g |\n| /+/0 /: /0/1/0 /+/1 /: /1/7 /BnZr /0 /: /0/0/7 /BnZr /0 /: /8/8 /BnZr /0 /: /0/0/9 | | | | /: /0/1/6 /+/1 /: /3/7 /0 /: /0/1/7 /BnZr /1 /: /1/1 /+/0 /: /BnZr /0 /: | /+/0 /: /0/3 /BnZr /0 /: /0/6 /1 | /+/1 /: /0/9 /BnZr /0 /: /6/1 /: /7/1 | /1/7 /1/8 /8 /: /1/3 /+/0 /: /2/1 | /+/0 /: /1/3 /+/0 /: /BnZr /0 /: /1/9 /BnZr /0 /: | /1/7 /+/0 /1/7 /BnZr /0 | /: /2/9 /: /2/1 | /+/2/3 /BnZr /2/4 | |\n| | | | | | | | | | | | /BnZr /5/4 | |\n| | /9/9/0/3/3/0 | | | /BnZr /3 /: /0/2 | | | | | | | | |\n| | | | | | | | | /BnZr /0 /: /2/4 | | | | /0/./6/9/( /3/2/) |\n| /1/7/7 /9/9/0/4/0/1 /0 /: /7/1/7 /+/0 /: /0/0/9 /BnZr /0 /: /0/1/0 /4/8 /: /6/2 /+/2 /: /2/6 /BnZr /2 /: /0/8 | /2 /: /3/8 | /5/1/2/6/9/./6/7/7 | /8 /: | /BnZr /0 /: /4/1 /+/0 /: /BnZr /0 /: f ixed /) /0 /: /3/8 /+/0 /BnZr | /BnZr /0 /: /0/5 /+/0 /: /0/4 /BnZr /0 /: /0/3 /1 | /BnZr /0 /: /2/5 /: /8/2 /+/0 /: /1/8 /BnZr /0 /: /1/6 | /2/3 /1/4 /1/4 /1 /: /7/8 | /+/0 /: /1/3 /BnZr /0 /: /1/3 /6 /: /5/( | /0 | /: /1/0 /: /1/0 | /1/1/4 /+/3/0 /BnZr /3/0 | /1/./2/2/( /7/2/) | \n/ \nFixed N H /= /2 /: /0 /-/1/0 cm /. a Start of Observ ation/. M J D /= J D /BnZr /2 /; /4/0/0 /; /0/0/0 /: /5 \nb R / in /= R in /(cos i /) /1 /= /2 /= /( D /= /6 k pc /)/, where i is the inclination angle and D is the distance to the source in kp c/. \nc Photons d Fixed \ne Gaussian \nf Photons g \nPCU /0 only /. y Start of PCA Gain Ep o c h /4 \nwidth \nof \n/7 \nk eV \nusing \n/'smedge/' \nmo del \nin \nXSPEC/. \ns /BnZr \ns /BnZr \nwith \nfixed \nFWHM \n/= \n/1/./2 \nk eV \n/( / \n/= \n/0 /: /5 \nk eV/)/. \n/1 \n/1 \ncm \ncm \n/BnZr \n/BnZr \n/2 \n/2 /. \nk eV \n/2/2 \n/BnZr \n/2 \n/BnZr \n/1 \nat \n/1 \nk eV/. \nT ABLE /2/| Continue d \n| Obs /# | Date /(UT/) | MJD a | T col /(k eV/) | R / in b /(km/) | Photon Index | P o w er/- c la w Norm/. | F e Edge d /(k eV/) | / F e d | E line e /(k eV/) | N line e /; f / /1/0/0 | Line Eq/. Width /(eV/) | / /2 / /(dof/) |\n|---------------|---------------------------|---------------------------------------|-------------------------------------------------------------|-------------------------------------------------------------------|-----------------------------------------------------|-----------------------------------------------------|-------------------------------------------------------|-----------------------------------------------------------|-------------------------------------------|-----------------------------------------------------|--------------------------------|---------------------------------------|\n| /1/7/8 | /9/9/0/4/0/2 | /5/1/2/7/0/./7/4/2 | /0 /: /7/2/0 /+/0 /: /0/1/4 | /4/1 /: /6/0 /+/2 /: /6/4 | /2 /: /4/1 /+/0 /: /0/3 | /3 /: /1/4 /+/0 /: /2/7 | /8 /: /5/1 /+/0 /: /1/4 | /1 /: /5/0 /+/0 /: /1/1 | /6 /: /5/( f ixed /) | /0 /: /5/5 /+/0 /: /1/3 | /1/2/4 /+/2/8 | /1/./1/6/( /7/2/) |\n| /1/7/9 | /9/9/0/4/0/3 | /5/1/2/7/1/./4/0/8 | /BnZr /0 /: /0/1/4 /: /7/1/7 /+/0 /: /0/1/5 | /BnZr /2 /: /4/9 /4/0 /: /4/0 /+/2 /: /7/9 | /BnZr /0 /: /0/3 /2 /: /3/8 /+/0 /: /0/3 | /BnZr /0 /: /2/3 | /BnZr /0 /: /1/5 | /BnZr /0 /: /1/1 | | /BnZr /0 /: /1/4 | /BnZr /3/0 /1/0/5 /+/3/0 | |\n| | | | /0 /BnZr /0 /: /0/1/5 | /BnZr /2 /: /5/1 | /BnZr /0 /: /0/3 | /2 /: /9/7 /+/0 /: /2/6 /BnZr /0 /: /2/3 | /8 /: /4/4 /+/0 /: /1/4 /BnZr /0 /: /1/4 | /1 /: /6/2 /+/0 /: /1/2 /BnZr /0 /: /1/1 | /6 /: /5/( f ixed /) | /0 /: /4/6 /+/0 /: /1/3 /BnZr /0 /: /1/3 | /BnZr /3/1 | /1/./2/6/( /7/2/) |\n| /1/8/0 | /9/9/0/4/0/5 | /5/1/2/7/3/./5/4/1 | /0 /: /6/7/4 /+/0 /: /0/0/8 /BnZr /0 /: /0/0/9 | /4/9 /: /9/6 /+/2 /: /3/2 /BnZr /2 /: /1/1 | /2 /: /3/0 /+/0 /: /0/4 /BnZr /0 /: /0/4 | /1 /: /0/5 /+/0 /: /1/1 /BnZr /0 /: /0/9 | /8 /: /4/6 /+/0 /: /1/3 /BnZr /0 /: /1/3 | /1 /: /9/0 /+/0 /: /1/3 /BnZr /0 /: /1/2 | /6 /: /5/( f ixed /) | /0 /: /2/6 /+/0 /: /0/7 /BnZr /0 /: /0/7 | /1/2/3 /+/3/0 /BnZr /3/2 | /1/./3/0/( /7/2/) |\n| /1/8/1 | /9/9/0/4/0/6 | /5/1/2/7/4/./4/7/1 | /0 /: /6/6/4 /+/0 /: /0/0/8 /BnZr /0 /: /0/0/8 | /4/9 /: /7/8 /+/2 /: /2/3 /BnZr /2 /: /1/1 | /2 /: /2/4 /+/0 /: /0/5 /BnZr /0 /: /0/4 | /0 /: /7/5 /+/0 /: /0/9 /BnZr /0 /: /0/8 | /8 /: /2/8 /+/0 /: /1/6 /BnZr /0 /: /1/7 /+/0 /: /2/9 | /1 /: /7/8 /+/0 /: /1/6 /BnZr /0 /: /1/5 /+/0 /: /4/0 | /6 /: /5/( f ixed /) | /0 /: /2/2 /+/0 /: /0/6 /BnZr /0 /: /0/7 | /1/2/8 /+/3/4 /BnZr /3/8 | /1/./2/9/( /7/2/) |\n| /1/8/2 | /9/9/0/4/0/8 | /5/1/2/7/6/./2/7/8 | /0 /: /6/4/4 /+/0 /: /0/1/2 /BnZr /0 /: /0/1/2 | /5/0 /: /5/7 /+/3 /: /1/8 /BnZr /3 /: /1/4 | /2 /: /2/7 /+/0 /: /1/1 /BnZr /0 /: /0/9 | /0 /: /4/6 /+/0 /: /1/6 /BnZr /0 /: /1/0 | /8 /: /0/6 /BnZr /0 /: /3/2 | /2 /: /2/1 /BnZr /0 /: /3/2 | /6 /: /5/( f ixed /) | /0 /: /1/3 /+/0 /: /0/7 /BnZr /0 /: /0/9 | /1/2/4 /+/6/6 /BnZr /8/5 | /0/./7/2/( /3/2/) g |\n| /1/8/3 | /9/9/0/4/0/9 | /5/1/2/7/7/./4/0/1 | /0 /: /6/2/7 /+/0 /: /0/0/8 /BnZr /0 /: /0/0/8 | /5/1 /: /2/8 /+/2 /: /4/2 /BnZr /2 /: /1/8 | /2 /: /3/1 /+/0 /: /0/5 /BnZr /0 /: /0/5 | /0 /: 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/0/1 | /7 /: /4/6 /+/0 /: /7/1 | /1 /: /9/5 /+/1 /: /1/1 | /6 /: /5/( f ixed /) | /0 /: /0/1 /+/0 /: /0/2 | /1/5/0 /+/2/2/9 /BnZr /1/5/0 | /0/./4/8/( /3/2/) g |\n| /2/0/8 | /9/9/0/5/1/5 | /5/1/3/1/3/./3/1/7 | /BnZr /0 /: /1/9/2 / / / | /BnZr /3 /: /5/8 / / / | /BnZr /0 /: /1/8 /+/0 /: /0/5 | /BnZr /0 /: /0/1 | /BnZr /0 /: /4/4 / / / | /BnZr /1 /: /1/3 / / / | / / / | /BnZr /0 /: /0/1 / / / | / / / | /0/./9/1/( /7/7/) |\n| /2/0/9 | /9/9/0/5/2/0 | /5/1/3/1/8/./7/6/8 | / / / | / / / | /2 /: /2/7 /BnZr /0 /: /0/5 | /0 /: /0/5 /+/0 /: /0/0 /BnZr /0 /: /0/0 | | | | | | |\n| | | | | | / / / | / / / | / / / | / / / | / / / | / / / | / / / | / / / | \nT ABLE /3 Unabsorbed /2/{/2/0 keV Flux f or XTE J/1/5/5/0/{/5/6/4 \n| Obs | Date | MJD a | T otal b /7 /BnZr /1 /BnZr /2 | Disk b /BnZr /7 /BnZr /1 /BnZr /2 | P o w er/-La w b /BnZr /7 /BnZr /1 /BnZr /2 | Disk//T 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/: /0/5 /+/0 /: /0/5 |\n| /1/5/1 | /9/9/0/3/0/2 | /5/1/2/3/9/./0/8/1 | /0 /: /6/2/6 | /0 /: /5/2/8 | /0 /: /0/9/7 /+/0 /: /0/2/0 | /0 /: /8/4 |\n| /1/5/2 | /9/9/0/3/0/3 | /5/1/2/4/0/./0/6/3 | /0 /: /6/4/6 /+/0 /: /0/2/6 /BnZr /0 /: /0/2/3 /+/0 /: /0/3/1 | /0 /: /5/3/1 /+/0 /: /0/1/4 /BnZr /0 /: /0/1/4 /+/0 /: /0/1/4 | /0 /: /1/1/3 /+/0 /: /0/2/2 /BnZr /0 /: /0/1/9 /+/0 /: /0/2/8 | /0 /: /8/2 /+/0 /: /0/5 /BnZr /0 /: /0/5 /+/0 /: /0/5 |\n| /1/5/4 | /9/9/3/0/0/5 | /5/1/2/4/2/./5/0/7 | /BnZr /0 /: /0/2/9 /0 /: /6/7/0 /+/0 /: /0/3/4 | /BnZr /0 /: /0/1/3 /0 /: /3/6/7 /+/0 /: /0/1/4 | /BnZr /0 /: /0/2/6 /+/0 /: /0/3/1 | /BnZr /0 /: /0/4 /0 /: /5/5 /+/0 /: /0/5 |\n| /1/5/5 | /9/9/0/3/0/7 | /5/1/2/4/4/./4/9/6 | /BnZr /0 /: /0/3/1 /+/0 /: /0/4/7 | /BnZr /0 /: /0/1/4 /+/0 /: /0/1/7 | /0 /: /3/0/1 /BnZr /0 /: /0/2/8 /+/0 /: /0/4/4 | /BnZr /0 /: /0/5 /+/0 /: /0/4 |\n| /1/5/6 | | | /0 /: /7/9/8 /BnZr /0 /: /0/4/4 | /0 /: /3/0/3 /BnZr /0 /: /0/1/6 | /0 /: /4/9/2 /BnZr /0 /: /0/4/0 | /0 /: /3/8 /BnZr /0 /: /0/4 |\n| | /9/9/0/3/0/8 | /5/1/2/4/5/./3/5/3 | /0 /: /7/4/3 /+/0 /: /0/4/4 /BnZr /0 /: /0/4/0 | /0 /: /2/1/8 /+/0 /: /0/1/5 /BnZr /0 /: /0/1/5 | /0 /: /5/2/4 /+/0 /: /0/4/1 /BnZr /0 /: /0/3/7 | /0 /: /2/9 /+/0 /: /0/4 /BnZr /0 /: /0/4 |\n| /1/5/7 | /9/9/0/3/0/9 | /5/1/2/4/6/./4/1/4 | /0 /: /7/4/2 /+/0 /: /0/5/4 /BnZr /0 /: /0/4/9 | /0 /: /2/9/5 /+/0 /: /0/2/1 /BnZr /0 /: /0/2/1 | /0 /: /4/4/5 /+/0 /: /0/5/0 /BnZr /0 /: /0/4/4 | /0 /: /4/0 /+/0 /: /0/6 |\n| /1/5/8 | /9/9/0/3/1/0 | /5/1/2/4/7/./9/7/9 | /0 /: /7/1/0 /+/0 /: /0/5/6 /BnZr /0 /: /0/5/0 | /0 /: /2/2/4 /+/0 /: /0/2/0 /BnZr /0 /: /0/2/0 | /0 /: /4/8/3 /+/0 /: /0/5/2 | /BnZr /0 /: /0/5 /0 /: /3/2 /+/0 /: /0/5 /BnZr /0 /: /0/5 |\n| /1/5/9 | /9/9/0/3/1/1 | /5/1/2/4/8/./0/9/1 | /0 /: /7/1/7 /+/0 /: /0/6/1 /BnZr /0 /: /0/5/5 | /0 /: /2/3/8 /+/0 /: /0/2/0 /BnZr /0 /: /0/2/1 | /BnZr /0 /: /0/4/6 /0 /: /4/7/7 /+/0 /: /0/5/7 /BnZr /0 /: /0/5/0 | /0 /: /3/3 /+/0 /: /0/6 |\n| /1/6/0 | /9/9/0/3/1/2 | /5/1/2/4/9/./4/0/0 | /0 /: 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/8/5 /+/0 /: /2/5 /BnZr /0 /: /2/0 |\n| /1/7/3 | /9/9/0/3/2/7 | /5/1/2/6/4/./7/4/8 | /0 /: /1/9/1 /BnZr /0 /: /0/1/6 /+/0 /: /0/1/6 | /0 /: /1/6/1 /BnZr /0 /: /0/1/5 | /0 /: /0/3/0 /+/0 /: /0/0/6 /BnZr /0 /: /0/0/5 /+/0 /: /0/0/6 | /0 /: /8/4 /+/0 /: /1/6 /BnZr /0 /: /1/4 /+/0 /: /1/6 |\n| /1/7/4 | /9/9/0/3/2/8 | /5/1/2/6/5/./6/1/3 | /0 /: /1/8/1 | /0 /: /1/4/9 /+/0 /: /0/1/5 | /0 /: /0/3/2 | /0 /: /8/2 |\n| /1/7/5 | /9/9/0/3/2/9 | /5/1/2/6/6/./8/8/0 | /BnZr /0 /: /0/1/5 /0 /: /1/5/4 /+/0 /: /0/1/9 /BnZr /0 /: /0/1/7 /+/0 /: /0/2/2 | /BnZr /0 /: /0/1/4 /0 /: /1/3/5 /+/0 /: /0/1/8 /BnZr /0 /: /0/1/6 | /BnZr /0 /: /0/0/5 /0 /: /0/1/9 /+/0 /: /0/0/7 /BnZr /0 /: /0/0/5 | /BnZr /0 /: /1/4 /0 /: /8/8 /+/0 /: /2/4 /BnZr /0 /: /1/9 |\n| /1/7/6 /1/7/7 | /9/9/0/3/3/0 | /5/1/2/6/7/./6/1/2 | /0 /: /1/6/0 /BnZr /0 /: /0/1/9 | /0 /: /1/2/7 /+/0 /: /0/2/0 /BnZr /0 /: /0/1/8 | /0 /: /0/3/2 /+/0 /: /0/1/1 /BnZr /0 /: /0/0/8 | /0 /: /8/0 /+/0 /: /2/5 /BnZr /0 /: /2/0 /+/0 /: /1/6 |\n| | /9/9/0/4/0/1 | /5/1/2/6/9/./6/7/7 | /0 /: /1/4/1 /+/0 /: /0/1/3 | /0 /: /1/0/8 /+/0 /: /0/1/2 | /0 /: /0/3/2 /+/0 /: /0/0/5 | /0 /: /7/7 | \nT ABLE /3/| Continue d \n| Obs /# | Date /(UT/) | MJD a | T otal b /(/1/0 /BnZr /7 erg s /BnZr /1 cm /BnZr /2 /) | Disk b /(/1/0 /BnZr /7 erg s /BnZr /1 cm /BnZr /2 | P o w er/-La w b /(/1/0 /BnZr /7 erg s /BnZr /1 cm /BnZr /2 /) | Disk//T otal |\n|----------|---------------|--------------------|-----------------------------------------------------------------------------------------------|-----------------------------------------------------------------------------------------------|-------------------------------------------------------------------|-----------------------------------------------------------|\n| /1/7/8 | /9/9/0/4/0/2 | /5/1/2/7/0/./7/4/2 | /0 /: /1/3/6 /+/0 /: /0/1/5 | /0 /: /0/8/1 /+/0 /: /0/1/2 | /0 /: /0/5/4 /+/0 /: /0/0/8 | /0 /: /6/0 /+/0 /: /1/7 |\n| /1/7/9 | /9/9/0/4/0/3 | /5/1/2/7/1/./4/0/8 | /0 /: /1/2/9 /+/0 /: /0/1/5 | /0 /: /0/7/5 /+/0 /: /0/1/2 | /0 /: /0/5/4 /+/0 /: /0/0/8 | /: /5/8 /+/0 /: /1/7 |\n| /1/8/0 | /9/9/0/4/0/5 | /5/1/2/7/3/./5/4/1 | /BnZr /0 /: /0/1/3 /+/0 /: /0/0/9 | /BnZr /0 /: /0/1/1 | /BnZr /0 /: /0/0/7 /+/0 /: /0/0/4 | /0 /BnZr /0 /: /1/4 |\n| | | | /0 /: /1/0/2 /BnZr /0 /: /0/0/8 | /0 /: /0/8/0 /+/0 /: /0/0/9 /BnZr /0 /: /0/0/8 | /0 /: /0/2/2 /BnZr /0 /: /0/0/3 | /0 /: /7/9 /+/0 /: /1/6 /BnZr /0 /: /1/4 |\n| /1/8/1 | /9/9/0/4/0/6 | /5/1/2/7/4/./4/7/1 | /0 /: /0/9/0 /+/0 /: /0/0/8 | /0 /: /0/7/3 /+/0 /: /0/0/8 | /0 /: /0/1/7 /+/0 /: /0/0/4 | /0 /: /8/1 /+/0 /: /1/7 |\n| /1/8/2 | /9/9/0/4/0/8 | /5/1/2/7/6/./2/7/8 | /BnZr /0 /: /0/0/8 /0 /: /0/7/2 /+/0 /: /0/1/1 | /BnZr /0 /: /0/0/7 /0 /: /0/6/3 /+/0 /: /0/0/9 | /BnZr /0 /: /0/0/3 /0 /: /0/1/0 /+/0 /: /0/0/6 | /BnZr /0 /: /1/4 /0 /: /8/7 /+/0 /: /2/8 |\n| /1/8/3 | /9/9/0/4/0/9 | /5/1/2/7/7/./4/0/1 | /BnZr /0 /: /0/1/0 /0 /: /0/6/7 /+/0 /: /0/0/7 | /BnZr /0 /: /0/0/9 /0 /: /0/5/5 /+/0 /: /0/0/6 | /BnZr /0 /: /0/0/3 /0 /: /0/1/1 /+/0 /: /0/0/3 | /BnZr /0 /: /2/2 /0 /: /8/3 /+/0 /: /1/8 |\n| /1/8/4 | /9/9/0/4/1/0 | /5/1/2/7/8/./6/8/6 | /0 /: /0/6/0 /+/0 /: /0/1/4 | /0 /: /0/4/9 /+/0 /: /0/1/1 | /0 /: /0/1/1 /+/0 /: /0/0/8 | /0 /: /8/1 /+/0 /: /3/8 |\n| /1/8/5 | /9/9/0/4/1/1 | /5/1/2/7/9/./5/3/1 | /BnZr /0 /: /0/1/0 /0 /: /0/5/8 /+/0 /: /0/1/7 | /BnZr /0 /: /0/0/9 /0 /: /0/4/6 /+/0 /: /0/1/6 | /BnZr /0 /: /0/0/4 /0 /: /0/1/3 /+/0 /: /0/0/5 | /BnZr /0 /: /2/7 /0 /: /7/9 /+/0 /: /5/6 |\n| /1/8/6 | /9/9/0/4/1/2 | /5/1/2/8/0/./5/4/4 | /BnZr /0 /: /0/1/3 /0 /: /0/4/9 /+/0 /: /0/1/5 | /BnZr /0 /: /0/1/2 /0 /: /0/3/9 /+/0 /: /0/1/3 | /BnZr /0 /: /0/0/3 /0 /: /0/0/9 /+/0 /: /0/0/7 | /BnZr /0 /: /3/4 /0 /: /8/1 /+/0 /: /7/0 |\n| /1/8/7 | /9/9/0/4/1/5 | /5/1/2/8/3/./2/2/4 | /BnZr /0 /: /0/1/4 /0 /: /0/4/1 /+/0 /: /0/1/2 | /BnZr /0 /: /0/1/3 /0 /: /0/3/2 /+/0 /: /0/1/1 | /BnZr /0 /: /0/0/5 /0 /: /0/0/9 /+/0 /: /0/0/3 | /BnZr /0 /: /4/0 /0 /: /7/8 /+/0 /: /5/5 |\n| /1/8/8 | /9/9/0/4/1/7 | /5/1/2/8/5/./1/9/7 | /BnZr /0 /: /0/0/9 /0 /: /0/3/5 /+/0 /: /0/1/1 | /BnZr /0 /: /0/0/8 /0 /: /0/2/6 /+/0 /: /0/1/1 | /BnZr /0 /: /0/0/2 /0 /: /0/0/8 /+/0 /: /0/0/2 | /BnZr /0 /: /3/3 /0 /: /7/6 /+/0 /: /6/4 |\n| /1/8/9 | /9/9/0/4/1/8 | /5/1/2/8/6/./0/5/8 | /BnZr /0 /: /0/0/8 /0 /: /0/3/2 /+/0 /: /0/1/2 | /BnZr /0 /: /0/0/8 /0 /: /0/2/3 /+/0 /: /0/1/1 | /BnZr /0 /: /0/0/2 /0 /: /0/0/9 /+/0 /: /0/0/2 | /BnZr /0 /: /3/5 /0 /: /7/2 /+/0 /: /7/5 |\n| /1/9/0 | /9/9/0/4/1/9 | /5/1/2/8/7/./2/5/7 | /0 /: /0/2/8 /+/0 /: /0/1/2 | /0 /: /0/2/0 /+/0 /: /0/1/1 | /0 /: /0/0/8 /+/0 /: /0/0/3 | /0 /: /7/1 /+/0 /: /7/9 |\n| /1/9/1 | /9/9/0/4/2/0 | /5/1/2/8/8/./3/8/2 | /BnZr /0 /: /0/0/7 /0 /: /0/2/6 /+/0 /: /0/1/0 | /BnZr /0 /: /0/0/7 /0 /: /0/2/0 /+/0 /: /0/1/0 | /BnZr /0 /: /0/0/2 /0 /: /0/0/6 /+/0 /: /0/0/2 | /BnZr /0 /: /3/8 /0 /: /7/6 /+/0 /: /7/6 |\n| /1/9/2 | /9/9/0/4/2/1 | /5/1/2/8/9/./1/4/6 | /BnZr /0 /: /0/0/7 /0 /: /0/2/9 /+/0 /: /0/1/8 | /BnZr /0 /: /0/0/7 /0 /: /0/1/6 /+/0 /: /0/1/8 | /BnZr /0 /: /0/0/1 /0 /: /0/1/3 /+/0 /: /0/0/3 | /BnZr /0 /: /3/9 /0 /: /5/4 /+/1 /: /1/8 |\n| /1/9/3 | /9/9/0/4/2/2 | /5/1/2/9/0/./9/8/4 | /BnZr /0 /: /0/0/9 /0 /: /0/3/1 /+/0 /: /0/9/6 | /BnZr /0 /: /0/0/9 /0 /: /0/1/6 /+/0 /: /0/9/6 | /BnZr /0 /: /0/0/2 /0 /: /0/1/5 /+/0 /: /0/0/3 | /BnZr /0 /: /4/1 /0 /: /5/1 /+/6 /: /5/7 |\n| /1/9/4 | /9/9/0/4/2/3 | /5/1/2/9/1/./1/8/3 | /BnZr /0 /: /0/1/5 /0 /: /0/2/5 /+/0 /: /0/2/1 /BnZr /0 /: /0/0/9 /0 /: /0/1/7 /+/0 /: /0/1/1 | /BnZr /0 /: /0/1/5 /0 /: /0/1/3 /+/0 /: /0/2/1 /BnZr /0 /: /0/0/9 /0 /: /0/1/3 /+/0 /: /0/1/1 | /BnZr /0 /: /0/0/2 /0 /: /0/1/1 /+/0 /: /0/0/2 /BnZr /0 /: /0/0/2 | /BnZr /0 /: /5/0 /0 /: /5/3 /+/1 /: /6/8 /BnZr /0 /: /4/5 |\n| /1/9/5 | /9/9/0/4/2/4 | /5/1/2/9/2/./4/5/1 | /BnZr /0 /: /0/0/6 /+/0 /: /0/1/8 | /BnZr /0 /: /0/0/6 | /0 /: /0/0/4 /+/0 /: /0/0/1 /BnZr /0 /: /0/0/1 | /0 /: /7/6 /+/1 /: /4/7 /BnZr /0 /: /5/2 |\n| /1/9/6 | /9/9/0/4/2/5 | /5/1/2/9/3/./4/5/0 | /0 /: /0/1/9 /BnZr /0 /: /0/0/9 | /0 /: /0/1/3 /+/0 /: /0/1/8 /BnZr /0 /: /0/0/9 | /0 /: /0/0/7 /+/0 /: /0/0/1 /BnZr /0 /: /0/0/1 | /0 /: /6/6 /+/2 /: /3/0 /BnZr /0 /: /5/6 |\n| /1/9/7 | /9/9/0/4/2/7 | /5/1/2/9/5/./5/2/0 | /0 /: /0/0/9 /+/0 /: /0/0/8 | /0 /: /0/0/6 /+/0 /: /0/0/8 | /0 /: /0/0/3 /+/0 /: /0/0/1 | /0 /: /7/1 /+/2 /: /0/4 |\n| /1/9/8 | /9/9/0/4/2/9 | /5/1/2/9/7/./3/7/6 | /BnZr /0 /: /0/0/4 /0 /: /0/0/7 /+/0 /: /0/0/6 | /BnZr /0 /: /0/0/4 /0 /: /0/0/5 /+/0 /: /0/0/6 | /BnZr /0 /: /0/0/1 /0 /: /0/0/1 /+/0 /: /0/0/1 | /BnZr /0 /: /5/5 /0 /: /7/8 /+/2 /: /1/8 |\n| /1/9/9 | /9/9/0/4/3/0 | /5/1/2/9/8/./0/5/2 | /BnZr /0 /: /0/0/3 /0 /: /0/0/5 /+/0 /: /0/0/4 | /BnZr /0 /: /0/0/3 /0 /: /0/0/4 /+/0 /: /0/0/3 | /BnZr /0 /: /0/0/1 /0 /: /0/0/1 /+/0 /: /0/0/1 | /BnZr /0 /: /6/0 /0 /: /8/2 /+/2 /: /0/3 |\n| /2/0/0 | /9/9/0/5/0/1 | /5/1/2/9/9/./3/4/4 | /BnZr /0 /: /0/0/3 /0 /: /0/0/5 /+/0 /: /0/0/4 | /BnZr /0 /: /0/0/3 /0 /: /0/0/5 /+/0 /: /0/0/4 | /BnZr /0 /: /0/0/0 /0 /: /0/0/1 /+/0 /: /0/0/1 | /BnZr /0 /: /6/2 /0 /: /8/4 /+/2 /: /1/3 |\n| /2/0/1 | /9/9/0/5/0/2 | /5/1/3/0/0/./4/5/8 | /BnZr /0 /: /0/0/2 / / / | /BnZr /0 /: /0/0/2 / / / | /BnZr /0 /: /0/0/0 / / / | /BnZr /0 /: /6/2 / / / |\n| /2/0/2 | /9/9/0/5/0/4 | /5/1/3/0/2/./4/5/9 | / / / | / / / | / / / | / / / |\n| /2/0/3 | /9/9/0/5/0/5 | /5/1/3/0/3/./3/9/1 | / / / | / / / | / / / | / / / |\n| /2/0/4 | /9/9/0/5/0/7 | /5/1/3/0/5/./1/2/4 | / / / | / / / | / / / | / / / |\n| /2/0/5 | /9/9/0/5/0/9 | /5/1/3/0/7/./3/2/2 | /0 /: /0/0/4 /+/0 /: /0/3/8 /BnZr /0 /: /0/0/1 | /0 /: /0/0/1 /+/0 /: /0/3/8 /BnZr /0 /: /0/0/1 | /0 /: /0/0/3 /+/0 /: /0/0/1 /BnZr /0 /: /0/0/1 | /0 /: /1/9 /+ / / / / /BnZr /0 /: /1/9 |\n| /2/0/6 | /9/9/0/5/1/1 | /5/1/3/0/9/./7/2/0 | /0 /: /0/0/4 /+/0 /: /0/2/9 /BnZr /0 /: /0/0/1 | /0 /: /0/0/2 /+/0 /: /0/2/9 /BnZr /0 /: /0/0/1 | /0 /: /0/0/2 /+/0 /: /0/0/0 /BnZr /0 /: /0/0/0 | /0 /: /4/2 /+/9 /: /6/8 |\n| /2/0/7 | /9/9/0/5/1/3 | /5/1/3/1/1/./5/1/9 | /0 /: /0/0/2 /+/0 /: /0/0/2 | /0 /: /0/0/0 /+/0 /: /0/0/2 | /0 /: /0/0/1 /+/0 /: /0/0/1 | /BnZr /0 /: /4/0 /0 /: /2/9 /+/3 /: /7/1 |\n| /2/0/8 | /9/9/0/5/1/5 | /5/1/3/1/3/./3/1/7 | /BnZr /0 /: /0/0/1 /0 /: /0/0/1 /+/0 /: /0/0/0 | /BnZr /0 /: /0/0/1 / / / | /BnZr /0 /: /0/0/1 /0 /: /0/0/1 /+/0 /: /0/0/0 | /BnZr /0 /: /3/5 / / / |\n| /2/0/9 | /9/9/0/5/2/0 | /5/1/3/1/8/./7/6/8 | /BnZr /0 /: /0/0/0 / / / | / / / | /BnZr /0 /: /0/0/0 / / / | / / / | \na Start of Observ ation/. M J D /= J D /BnZr /2 /; /4/0/0 /; /0/0/0 /: /5 \nStart of Observ ation/. M J D /= y Start of PCA Gain Ep o c h /4 \nT ABLE /4 HEXTE Spectral P arameters f or XTE J/1/5/5/0/{/5/6/4 / \n| | Date | MJD a | /BnZr H E X T E | E cut | E f old Best | / /2 / | PL / /2 / | E max | /2/0/{/1/0/0 k eV Flux b /; c /(/1/0 /BnZr /8 erg s /BnZr /1 cm /BnZr /2 /) | /2/0/{/1/0/0 k eV Flux b /; c /(/1/0 /BnZr /8 erg s /BnZr /1 cm /BnZr /2 /) |\n|-----------|---------------------------|---------------------------------------|-----------------------------------------------------------|-----------------------------------------------------|--------------------------------------------------|------------------|-------------------|---------------|-------------------------------------------------------------------------------|-------------------------------------------------------------------------------|\n| /1 | /9/8/0/9/0/7 | /5/1/0/6/3/./6/9/5 | /1 /: /5/6 /+/0 /: /0/2 | /4/4 /: /9 /+/2 /: /7 | /1/1/4 /+/7 | /0/./6/5/(/7/9/) | /4/./9/1 | /1/7/0 | /1 /: /1/7 /+/0 /: /0/9 | /1 /: /1/7 /+/0 /: /0/9 |\n| | | /5/1/0/6/4/./0/0/8 | /BnZr /0 /: /0/3 /1 /: /5/2 /+/0 /: /0/3 | /BnZr /2 /: /7 /: /1 /+/1 /: /5 | /BnZr /6 /8/0 /+/3 | /1/./2/9/(/8/5/) | /2/6/./9/2 | /2/0/0 | /BnZr /0 /: /0/9 /1 /: /2/2 /+/0 /: /1/4 | /BnZr /0 /: /0/9 /1 /: /2/2 /+/0 /: /1/4 |\n| /2 | /9/8/0/9/0/8 | | /BnZr /0 /: /0/4 | /3/1 /BnZr /1 /: /7 | | | | | /BnZr /0 /: /1/6 | /BnZr /0 /: /1/6 |\n| /3 | /9/8/0/9/0/9 | /5/1/0/6/5/./0/6/6 | /1 /: /8/3 /+/0 /: /0/5 | /2/8 /: /5 /+/1 /: /5 | /BnZr /4 /6/8 /+/3 | /1/./0/8/(/7/9/) | /2/0/./1/6 | /1/7/0 | /1 /: /4/3 /+/0 /: /2/2 | /1 /: /4/3 /+/0 /: /2/2 |\n| /4 | /9/8/0/9/0/9 | /5/1/0/6/5/./3/4/4 | /BnZr /0 /: /0/6 /2 /: /0/0 /+/0 /: /0/4 | /BnZr /1 /: /6 /3/1 /: /1 /+/1 /: /8 | /BnZr /4 /7/2 /+/4 | /0/./9/5/(/7/7/) | /1/1/./0/6 | /1/6/0 | /1 /: /2/9 /+/0 /: /1/9 | /1 /: /2/9 /+/0 /: /1/9 |\n| /5 | /9/8/0/9/1/0 | /5/1/0/6/6/./0/6/6 | /2 /: /1/6 /+/0 /: /0/4 /BnZr /0 /: /0/5 | /2/7 /: /8 /+/1 /: /1 /BnZr /1 /: /3 | /7/5 /+/4 /BnZr /5 | /1/./2/2/(/7/5/) | /9/./4/2 | /1/5/0 | /1 /: /1/8 /+/0 /: /1/4 /BnZr /0 /: /1/6 | /1 /: /1/8 /+/0 /: /1/4 /BnZr /0 /: /1/6 |\n| /6 | /9/8/0/9/1/0 | /5/1/0/6/6/./3/4/4 | /2 /: /2/2 /+/0 /: /0/6 /BnZr /0 /: /0/7 /+/0 /: /0/4 | /2/7 /: /5 /+/1 /: /9 /BnZr /1 /: /8 /+/1 /: /3 | /7/4 /+/6 /BnZr /6 /+/6 | /0/./9/7/(/7/7/) | /8/./6/6 | /1/6/0 | /1 /: /1/4 /+/0 /: /2/2 /BnZr /0 /: /2/1 | /+/0 /: /1/5 |\n| /7 | /9/8/0/9/1/1 | /5/1/0/6/7/./2/7/0 | /2 /: /3/5 /BnZr /0 /: /0/5 /+/0 /: /0/4 | /2/6 /: /0 /BnZr /1 /: /2 /+/1 /: /7 | /8/0 /BnZr /6 /1/./1/6/(/7/5/) /+/1/3 | | /6/./8/1 | /1/5/0 | /1 /BnZr /0 /: /1/5 /+/0 /: /1/1 | /: /0/7 |\n| /8 | /9/8/0/9/1/2 /9/8/0/9/1/3 | /5/1/0/6/8/./3/4/4 | /2 /: /6/1 /BnZr /0 /: /0/5 /+/0 /: /0/3 | /2/7 /: /6 /BnZr /1 /: /8 | /1/1/6 /BnZr /1/2 | /1/./0/3/(/7/1/) | /2/./9/8 | /1/4/0 | /0 /: /8/8 | /BnZr /0 /: /1/2 |\n| /9 | | /5/1/0/6/9/./2/7/3 | /2 /: /7/2 /BnZr /0 /: /0/5 | /3/0 /: /9 /+/2 /: /5 | /1/2/9 /+/1/4 | /0/./9/8/(/6/9/) | /2/./6/2 | /1/3/0 | /0 /: /7/8 /+/0 /: /0/9 | /0 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/: /4/0 /+/0 /: /0/2 | | | | | | /0 /: /4/9 /+/0 /: /0/3 | /0 /: /4/9 /+/0 /: /0/3 |\n| /5/7 /5/8 | /9/8/1/0/2/6 /9/8/1/0/2/7 | /5/1/1/1/3/./6/6/8 | /BnZr /0 /: /0/2 /2 /: /4/1 /+/0 /: /0/5 | / / / / / / | / / / /1/./2/2/(/5/5/) | | / / / | /1/0/0 /5/0 | /BnZr /0 /: /0/3 /0 /: /3/8 /+/0 /: /0/6 | |\n| | | | | | / / / | /1/./3/4/(/2/7/) | | | | | \nT ABLE /4/| Continue d \n| Obs /# | Date /(UT/) | MJD a | H E X T E E /(k eV/) | cut E /(k eV/) | f old Best /(dof/) | / /2 / PL | / /2 / E max | /(k eV/) | /2/0/{/1/0/0 k eV Flux b /; c /(/1/0 /BnZr /8 erg s /BnZr /1 cm /BnZr /2 /) | /2/0/{/1/0/0 k eV Flux b /; c /(/1/0 /BnZr /8 erg s /BnZr /1 cm /BnZr /2 /) |\n|-----------------------------|-----------------------------------------------------|----------------------------------------------------------|------------------------------------------------------------------------|---------------------------------------------|------------------------------------|-------------------------------------------|------------------------------|------------------------------------------------------------------------|-------------------------------------------------------------------------------|-------------------------------------------------------------------------------|\n| /5/9 | /9/8/1/0/2/9 | /5/1/1/1/5/./2/8/1 | /: /1/8 /+/0 /: /0/3 /BnZr /0 /: /0/3 | / | / / / | /0/./8/8/(/4/9/) | / / / | /8/0 | /0 /: /2/8 /+/0 /: /0/3 | /0 /: /2/8 /+/0 /: /0/3 |\n| | | | /2 | / / | | | | | /BnZr /0 /: /0/3 /+/0 /: /0/3 | /BnZr /0 /: /0/3 /+/0 /: /0/3 |\n| /6/0 | /9/8/1/0/3/1 | | /1 /: /9/2 /+/0 /: /0/6 | / / / | / / / | /1/./4/2/(/3/7/) | / / / | /6/0 | /0 /: /1/3 /BnZr /0 /: /0/2 | /0 /: /1/3 /BnZr /0 /: /0/2 |\n| /6/1 | /9/8/1/1/0/2 | /5/1/1/1/7/./3/5/2 /5/1/1/1/9/./0/0/4 | /BnZr /0 /: /0/6 /1 /: /9/0 /+/0 /: /0/6 | / / / | / / / /0/./8/6/(/4/3/) | | / / / | /7/0 | /0 /: /1/1 /+/0 /: /0/2 | /0 /: /1/1 /+/0 /: /0/2 |\n| /6/2 | /9/8/1/1/0/4 | /5/1/1/2/1/./0/0/4 | /BnZr /0 /: /0/6 /2 /: /5/( f ixed /) | / / / | / / / | / / / | / / / | /1/0/0 | /BnZr /0 /: /0/2 /< /0 /: /0/3/0 | /BnZr /0 /: /0/2 /< /0 /: /0/3/0 |\n| /6/3 /6/4 | /9/8/1/1/0/7 /9/8/1/1/0/9 | /5/1/1/2/4/./7/2/7 /5/1/1/2/6/./5/9/4 | /2 /: /5/( f ixed /) /1 /: /9/8 /+/0 /: /0/7 | / / / / / | / / / / / / /0/./6/3/(/3/1/) | / / / | / / / / / / | /1/0/0 /5/5 | /< /0 /: /0/5/0 /: /1/0 /+/0 /: /0/2 | /< /0 /: /0/5/0 /: /1/0 /+/0 /: /0/2 |\n| /6/5 | /9/8/1/1/1/1 | /5/1/1/2/8/./5/6/2 | /BnZr /0 /: /0/6 /1 /: /+/0 /: /1/3 | / / / / | / / / /0/./8/2/(/2/7/) | | / / / | /5/0 | /0 /BnZr /0 /: /0/2 /0 /: /0/9 /+/0 /: /0/5 | /0 /BnZr /0 /: /0/2 /0 /: /0/9 /+/0 /: /0/5 |\n| /6/6 | /9/8/1/1/1/3 | /5/1/1/3/0/./4/5/7 | /7/5 /BnZr /0 /: /1/2 /1 /: /7/4 /+/0 /: /0/9 | / / | / / | /0/./9/7/(/2/7/) | / / / | /5/0 | /BnZr /0 /: /0/3 /0 /: /0/9 /+/0 /: /0/3 | /BnZr /0 /: /0/3 /0 /: /0/9 /+/0 /: /0/3 |\n| /6/7 | /9/8/1/1/1/5 /9/8/1/1/1/7 | /5/1/1/3/2/./4/8/0 /5/1/1/3/4/./4/3/8 | /BnZr /0 /: /0/9 /2 /: /5/( f ixed /) | / / / / | / / / / | / / / | / / / | /1/0/0 | /BnZr /0 /: /0/2 /< /0 /: /0/5/5 | /BnZr /0 /: /0/2 /< /0 /: /0/5/5 |\n| /6/8 /6/9 | /9/8/1/1/1/9 | /5/1/1/3/6/./7/7/7 | /2 /: /5/( f ixed /) | / / / / | | | / / / | /1/0/0 | /< /0 | /: /0/3/3 |\n| /7/0 | /9/8/1/1/2/0 | /5/1/1/3/7/./9/2/2 | /2 /: /5/( f ixed /) f ixed /) | / / | / / / / | / / / / / / | / / / | /1/0/0 /1/0/0 | /< /0 /: /0/3/2 /< /0 /: /0/6/3 | /< /0 /: /0/3/2 /< /0 /: /0/6/3 |\n| | | | /2 /: /5/( | / / / / | / / / / | / / / | / / / | | | |\n| /7/1 /7/2 | /9/8/1/1/2/2 /9/8/1/1/2/3 | /5/1/1/3/9/./9/9/2 /5/1/1/4/0/./7/0/7 | /2 /: /5/( f ixed /) /+/0 /: /1/5 | / / / / | / / / / | / / / /0/./3/4/(/2/7/) | / / / | /1/0/0 /5/0 | /< /0 /: /0/7/6 /: /0/7 /+/0 /: /0/4 | /< /0 /: /0/7/6 /: /0/7 /+/0 /: /0/4 |\n| /7/3 | /9/8/1/1/2/6 | /5/1/1/4/3/./8/0/1 | /2 /: /2/3 /BnZr /0 /: /1/9 ixed /) | / / / / | / / / / | / / / | / / / / / / | /1/0/0 | /0 /BnZr /0 /: /0/3 /< /0 /: /0/9/0 | /0 /BnZr /0 /: /0/3 /< /0 /: /0/9/0 |\n| /7/4 | /9/8/1/1/2/8 | /5/1/1/4/5/./4/7/7 | /2 /: /5/( f /2 /: /5/( f ixed /) | / / / / / | / / / | / / / | / / / /1/0/0 | | | |\n| /7/5 /7/6 | /9/8/1/1/3/0 /9/8/1/2/0/3 | /5/1/1/4/7/./3/3/6 /5/1/1/5/0/./0/7/4 | /2 /: /5/( f ixed /) / / / /) / / / | / | / / / / | / / / / / / | / / / / / / / / / | /1/0/0 /1/0/0 | /< /0 /: /0/2/8 /< /0 /: /0/4/8 | /< /0 /: /0/2/8 /< /0 /: /0/4/8 |\n| | /9/8/1/2/0/5 | /5/1/1/5/2/./0/7/8 | /2 /: /5/( f ixed /) / / / | / | | / / / | | /1/0/0 | /< /0 /: /0/3/4 | /< /0 /: /0/3/4 |\n| /7/7 /7/8 | /9/8/1/2/0/5 | /5/1/1/5/2/./8/6/7 | /2 /: /5/( f ixed /2 /: /5/( f ixed /) / / / | / / / / | / / / / | / | / / / /1/0/0 | | /< /0 /: /0/1/8 /< /0 | /: /0/1/3 |\n| /7/9 /8/0 | /9/8/1/2/0/7 /9/8/1/2/0/8 | /5/1/1/5/4/./0/1/6 /5/1/1/5/5/./0/6/6 | /2 /: /5/( f ixed /) / / / /2 /: /5/( f ixed /) / / / | / | / / / / | / / / / / / | / / / / / / | /1/0/0 /1/0/0 | /< /0 /: /0/1/4 /< /0 /: /0/1/2 | /< /0 /: /0/1/4 /< /0 /: /0/1/2 |\n| /8/1 | /9/8/1/2/1/0 | /5/1/1/5/7/./4/9/2 | /2 /: /5/( f ixed /) / / | / | / / / / | / / / | / / / | /1/0/0 | /< /0 /: /0/1/0 | /< /0 /: /0/1/0 |\n| /8/2 /8/3 | /9/8/1/2/1/3 /9/8/1/2/1/5 | /5/1/1/6/0/./2/7/7 /5/1/1/6/2/./2/0/3 | /2 /: /5/( f ixed /) / / /2 /: /5/( f ixed /) / / | / / / / | / / / / | / / / / / / | / / / / / / | /1/0/0 /1/0/0 | /< /0 /: /0/0/8 /< /0 /: /0/0/6 | /< /0 /: /0/0/8 /< /0 /: /0/0/6 |\n| /8/4 | /9/8/1/2/1/6 | /5/1/1/6/3/./2/0/7 | /2 /: /5/( f ixed /) | / / / / | / / | / / / | / / / | /1/0/0 | /< /0 /: /0/3/0 | /< /0 /: /0/3/0 |\n| /8/5 /8/6 /8/7 | /9/8/1/2/1/7 /9/8/1/2/1/8 /9/8/1/2/1/9 | /5/1/1/6/4/./2/0/7 /5/1/1/6/5/./5/9/0 /5/1/1/6/6/./0/1/2 | / / / /2 /: /5/( f ixed /) /2 /: /5/( /) | / / / / / / / / / / / / / / / | | / / / / / / | / / / / / / | / / / /1/0/0 | / / / /< /0 /: /0/1/2 | / / / /< /0 /: /0/1/2 |\n| /8/8 /8/9 | /9/8/1/2/2/0 /9/8/1/2/2/1 | /5/1/1/6/7/./4/1/8 /5/1/1/6/8/./0/1/2 | f ixed / / / f ixed /) | / / / / / | / / / / | / / / / / / | / / / / / / | /1/0/0 / / / | /< /0 /: /0/1/3 / / / | /< /0 /: /0/1/3 / / / |\n| /9/0 | /9/8/1/2/2/2 | /5/1/1/6/9/./0/0/8 | /2 /: /5/( /2 /: /5/( f ixed /) f ixed /) | / / / / / / / / / / / / | | / / / / / / | / / / / / / / / / | /1/0/0 | /< /0 /: /0/1/4 /< /0 /: /0/1/3 | /< /0 /: /0/1/4 /< /0 /: /0/1/3 |\n| /9/1 /9/2 | /9/8/1/2/2/3 /9/8/1/2/2/5 /9/8/1/2/2/6 | /5/1/1/7/0/./5/2/0 /5/1/1/7/2/./0/6/6 /5/1/1/7/2/./9/8/8 | /2 /: /5/( /2 /: /5/( f ixed /) f ixed /) | / / / / / / / / / / / / | | / / / / / / | / / / / / / | /1/0/0 /1/0/0 /1/0/0 | /< /0 /: /0/1/5 /< /0 /: /0/1/6 | /< /0 /: /0/1/5 /< /0 /: /0/1/6 |\n| /9/3 | /9/8/1/2/2/7 | /5/1/1/7/3/./9/8/8 | /2 /: /5/( f ixed /) | / / / | / / / | | | | /< /0 /: /0/1/4 | /< /0 /: /0/1/4 |\n| /9/4 /9/5 /9/6 | /9/8/1/2/2/7 /9/8/1/2/2/8 | /5/1/1/7/4/./7/2/3 /5/1/1/7/5/./8/7/9 | /2 /: /5/( f ixed /) f ixed /) | / / / / / / / / / | / / / | / / / / / / | / / / / / / / / / | /1/0/0 /1/0/0 | /< /0 /: /0/1/8 /< /0 /: /0/1/6 | /< /0 /: /0/1/8 /< /0 /: /0/1/6 |\n| | | | /2 /: /5/( /2 /: /5/( | / / / / / / / / / | / / | / | /1/0/0 | /1/0/0 | /< /0 /: /0/2/3 | /< /0 /: /0/2/3 |\n| /9/7 /9/8 /9/9 | /9/8/1/2/2/9 /9/8/1/2/3/0 /9/8/1/2/3/1 | /5/1/1/7/6/./8/4/8 /5/1/1/7/7/./8/4/4 | /2 /: /5/( f ixed /) /2 /: /5/( f ixed /) | / / / / | / / | / / / | / / / / / / | /1/0/0 | /< /0 /: /0/2/5 /< /0 /: /0/2/3 | /< /0 /: /0/2/5 /< /0 /: /0/2/3 |\n| /1/0/0 | /9/9/0/1/0/1 | /5/1/1/7/8/./8/4/4 | /2 /: /5/( f ixed /) /+/0 /: /1/1 | / / / / / / / / / | / / / | / / / / / / | / / / | /1/0/0 | /< /0 /: /0/2/7 | /< /0 /: /0/2/7 |\n| | | /5/1/1/7/9/./9/1/8 | /2 /: /1/4 /BnZr /0 /: /1/0 | / / / | / / / / / / | | | /1/0/0 | /0 /: /0/7 /+/0 /: /0/3 | /0 /: /0/7 /+/0 /: /0/3 |\n| /1/0/1 /1/0/2 | /9/9/0/1/0/2 /9/9/0/1/0/4 | /5/1/1/8/0/./8/4/4 /5/1/1/8/2/./3/3/6 | /2 /: /1/2 /+/0 /: /0/3 /BnZr /0 /: /0/3 f ixed /) | / / | / / / | /0/./9/4/(/2/7/) /1/./2/5/(/5/7/) / / / | / / / / / / | /5/0 /1/0/0 | /BnZr /0 /: /0/2 /0 /: /2/0 /+/0 /: /0/2 /BnZr /0 /: /0/2 | /BnZr /0 /: /0/2 /0 /: /2/0 /+/0 /: /0/2 /BnZr /0 /: /0/2 |\n| /1/0/3 | /9/9/0/1/0/5 | /5/1/1/8/3/./5/3/9 | /2 /: /5/( f ixed /) | / / / / | / / / | / / / | / / / / / / / / / | /1/0/0 | /< /0 /: /0/1/8 /< /0 /: /0/2/2 | /< /0 /: /0/1/8 /< /0 /: /0/2/2 |\n| /1/0/4 | /9/9/0/1/0/6 | /5/1/1/8/4/./7/1/1 | /2 /: /5/( f ixed /) | / / | / / / | / / / | | /1/0/0 | | |\n| /1/0/5 | /9/9/0/1/0/7 | /5/1/1/8/5/./1/2/5 | /2 /: /5/( /2 /: /5/( f ixed /) | / / / / / / / / / / | | / / / | / / / /1/0/0 | /1/0/0 | /< /0 /: /0/1/7 /< /0 | /: /0/3/2 |\n| /1/0/6 /1/0/7 /1/0/8 /1/0/9 | /9/9/0/1/0/7 /9/9/0/1/0/9 /9/9/0/1/1/0 /9/9/0/1/1/0 | /5/1/1/8/5/./8/4/8 /5/1/1/8/7/./1/2/5 /5/1/1/8/8/./1/9/5 | /2 /: /5/( f ixed /) /2 /: /5/( f ixed /) | / / / / / / / | / / / / | / / / / / / / / / | / / / / / / / / / | /1/0/0 /1/0/0 | /< /0 /: /0/1/2 /< /0 /: /0/0/8 | /< /0 /: /0/1/2 /< /0 /: /0/0/8 |\n| | | /5/1/1/8/8/./7/3/0 | /2 /: /5/( f ixed /) f ixed /) | / / / / | / / / / / / / / / / | / / / / / / | / / / / / / | /1/0/0 | /< /0 /: /0/1/0 /< /0 /: /0/0/9 | /< /0 /: /0/1/0 /< /0 /: /0/0/9 |\n| /1/1/0 /1/1/1 | /9/9/0/1/1/2 /9/9/0/1/1/3 | /5/1/1/9/0/./0/5/5 /5/1/1/9/1/./4/8/4 | /2 /: /5/( /2 /: /5/( f ixed /) f ixed /) | / / / / / | | | / / / | /1/0/0 | /< /0 /: /0/1/3 | /< /0 /: /0/1/3 |\n| /1/1/2 /1/1/3 | /9/9/0/1/1/4 /9/9/0/1/1/4 | /5/1/1/9/2/./1/8/8 /5/1/1/9/2/./5/9/4 | /2 /: /5/( /2 /: /5/( f ixed /) f ixed /) | / / / / / / / / | / / / / | / / / / / / / / / | / / / / / / | /1/0/0 /1/0/0 /1/0/0 | /< /0 /: /0/0/9 /< /0 /: /0/1/1 | /< /0 /: /0/0/9 /< /0 /: /0/1/1 |\n| | /9/9/0/1/1/6 | /5/1/1/9/4/./5/0/0 | /2 /: /5/( f ixed /) | / / / | / / | | / | /1/0/0 | | |\n| /1/1/4 /1/1/5 /1/1/6 | /9/9/0/1/1/8 /9/9/0/1/1/8 | /5/1/1/9/6/./0/5/1 /5/1/1/9/6/./5/2/3 | /2 /: /5/( /2 /: /5/( f ixed /) | / / / / / / / | / / | / / / | / / / / / / / / | /1/0/0 | /< /0 /: /0/1/8 /< /0 /: /0/1/3 | /< /0 /: /0/1/8 /< /0 /: /0/1/3 |\n| /1/1/7 | /9/9/0/1/2/0 | /5/1/1/9/8/./0/5/1 | / / / f ixed /) | / / / / | / / / / / / / / / | / / / / / / / / / | / / / | /1/0/0 / / / | /< /0 /: /0/1/5 / / / | /< /0 /: /0/1/5 / / / |\n| /1/1/8 | /9/9/0/1/2/2 | /5/1/2/0/0/./3/8/3 | /2 /: /5/( f ixed /) | / / / | | | / / / | /1/0/0 | /< /0 /: /0/1/9 | /< /0 /: /0/1/9 |\n| /1/1/9 /1/2/0 | /9/9/0/1/2/3 /9/9/0/1/2/5 | /5/1/2/0/1/./7/8/1 | /2 /: /5/( /2 /: /2/0 /+/0 /: /0/2 /BnZr /0 /: /0/2 | / / / / / / | / / / / / / | / / / /0/./9/1/(/5/1/) | / / / | /1/0/0 | /< /0 /: /1/8/9 /0 /: /3/8 /+/0 /: /0/3 | /< /0 /: /1/8/9 /0 /: /3/8 /+/0 /: /0/3 |\n| /1/2/1 | /9/9/0/1/2/6 | /5/1/2/0/3/./2/5/0 /5/1/2/0/4/./2/4/6 | /2 /: /5/( f ixed /) /5/( f ixed /) | / / / / | / / | / / / | / / / / / / | /1/0/0 /1/0/0 | /BnZr /0 /: /0/3 /< /0 /: /0/3/2 /< /0 /: /0/3/8 | /BnZr /0 /: /0/3 /< /0 /: /0/3/2 /< /0 /: /0/3/8 |\n| /1/2/2 | /9/9/0/1/2/7 /9/9/0/1/2/8 | /5/1/2/0/5/./8/5/2 /5/1/2/0/6/./7/8/1 | /2 /: /2 /: /5/( f ixed /) /: /5/( f ixed /) /: /5/( f ixed /) | / / / / / / / | / / / | / / / | / / / | /1/0/0 /1/0/0 | /< /0 /: /0/2/5 | /< /0 /: /0/2/5 |\n| /1/2/3 /1/2/4 | /9/9/0/1/3/0 | /5/1/2/0/8/./1/8/0 | /2 | / / | / / / / / / | / / / / / / | / / / / / / | /1/0/0 /1/0/0 | /< /0 /: /0/1/6 /< /0 /: /0/7/9 | /< /0 /: /0/1/6 /< /0 /: /0/7/9 |\n| /1/2/5 | /9/9/0/1/3/0 | /5/1/2/0/8/./8/4/4 | /2 /2 /: /5/( f ixed /) | / / / / / / / / / | | / / / | / / / | /1/0/0 | /< /0 /: /0/2/9 | /< /0 /: /0/2/9 |\n| /1/2/6 /1/2/7 /1/2/8 | /9/9/0/1/3/1 /9/9/0/2/0/2 /9/9/0/2/0/3 | /5/1/2/0/9/./7/1/1 /5/1/2/1/1/./7/1/1 /5/1/2/1/2/./7/7/3 | /2 /: /5/( f ixed /) /2 /: /5/( f ixed /) /: /5/( f ixed /) | / / / / / / / / / / / | / / / / | / / / / / / / / / | / / / / / / / / / | /1/0/0 /1/0/0 | /< /0 /: /0/2/7 /< /0 /: /0/2/1 /< /0 /: /0/2/6 | /< /0 /: /0/2/7 /< /0 /: /0/2/1 /< /0 /: /0/2/6 |\n| /1/2/9 | /9/9/0/2/0/4 /9/9/0/2/0/5 | /5/1/2/1/3/./7/7/3 /5/1/2/1/4/./9/7/3 | /2 /2 /: /5/( f ixed /) /: /0/9 /+/0 /: /0/4 | / / / / / / / / / | / / / | / / / / | / / | /1/0/0 /1/0/0 | /< /0 /: /0/2/8 /0 /: /1/4 /+/0 /: /0/2 | /< /0 /: /0/2/8 /0 /: /1/4 /+/0 /: /0/2 |\n| /1/3/0 | | | | | / | / | / | | /BnZr /0 /: /0/2 | /BnZr /0 /: /0/2 |\n| | | | /2 /BnZr /0 /: /0/4 | / / / / | / / /1/./2/2/(/5/7/) | | / / / | /1/0/0 | /+/0 /: /0/1 | /+/0 /: /0/1 |\n| /1/3/1 /1/3/2 /1/3/3 | /9/9/0/2/0/6 /9/9/0/2/0/7 /9/9/0/2/0/8 | /5/1/2/1/5/./8/4/0 /5/1/2/1/6/./8/3/6 /5/1/2/1/7/./7/0/3 | /2 /: /0/7 /+/0 /: /0/3 /BnZr /0 /: /0/3 /2 /: /5/( f ixed /) | / / / / / / | / / / / / / | /0/./7/2/(/6/3/) / / / | / / / / / / | /1/1/0 /1/0/0 | /0 /: /1/3 /BnZr /0 /: /0/1 /< /0 /: /0/3/8 /< /0 /: /0/2/0 | /0 /: /1/3 /BnZr /0 /: /0/1 /< /0 /: /0/3/8 /< /0 /: /0/2/0 |\n| /1/3/4 | /9/9/0/2/1/0 | /5/1/2/1/9/./0/5/5 | /2 /: /5/( f ixed /) /: /5/( f ixed /) | / / | / | / / / / / / | / / / / / / | /1/0/0 | /< /0 /: /0/2/4 | /< /0 /: /0/2/4 |\n| /1/3/5 | /9/9/0/2/1/1 | /5/1/2/2/0/./5/0/8 | /2 /: /5/( f ixed /) | / / / / / | / / / / / / / / | / / / / / / | / / | /1/0/0 /1/0/0 | /< /0 /: /0/2/1 | /< /0 /: /0/2/1 |\n| /1/3/6 /1/3/7 | /9/9/0/2/1/1 /9/9/0/2/1/2 | /5/1/2/2/0/./8/3/6 /5/1/2/2/1/./5/6/6 /5/1/2/2/3/./7/6/6 | /2 /2 /: /5/( f ixed /) /: /5/( f ixed /) /: /5/( f ixed /) | / / / / / / | / / / / / / | / / / | / / / / / / / / / / | /1/0/0 /1/0/0 | /< /0 /: /0/3/1 | /< /0 /: /0/3/1 |\n| /1/3/8 | /9/9/0/2/1/4 | | /2 /2 /: /5/( f ixed /) | / / / / / | / / / | | | | /< /0 /: /0/1/7 /< /0 /: /0/1/6 | /< /0 /: /0/1/7 /< /0 /: /0/1/6 |\n| | | | | | | | /1/0/0 | | | |\n| /1/3/9 /1/4/0 /1/4/1 | /9/9/0/2/1/5 /9/9/0/2/1/7 /9/9/0/2/1/8 | /5/1/2/2/4/./6/9/9 /5/1/2/2/6/./2/9/7 /5/1/2/2/7/./8/3/6 | /: /5/( f ixed /) /: /5/( f ixed /) | / / | / / / / / / / / / | / / / / / / / / / | / / / / / / / / / | /1/0/0 /1/0/0 | /< /0 /: /0/1/4 /< /0 /: /0/2/1 | /< /0 /: /0/1/4 /< /0 /: /0/2/1 |\n| | /9/9/0/2/1/9 | /5/1/2/2/8/./6/9/5 | /2 /2 /2 /2 /: /5/( f ixed /) | / / / / / / | / / / | | /1/0/0 | | /< /0 /: /0/1/3 /< /0 /: /0/1/3 | /< /0 /: /0/1/3 /< /0 /: /0/1/3 |\n| /1/4/2 /1/4/3 | | /5/1/2/3/0/./4/2/6 | /2 /: /5/( f ixed /) /2 /: /5/( f ixed /) | / / / / | / / / / | / / / | / / / | /1/0/0 | /< /0 /: /0/1/7 | /< /0 /: /0/1/7 |\n| /1/4/4 /1/4/5 | /9/9/0/2/2/1 /9/9/0/2/2/2 /9/9/0/2/2/2 | /5/1/2/3/1/./2/9/3 /5/1/2/3/1/./4/2/2 | /: /5/( f ixed /) | / / / / / / | / / / | / / / / / / | / / / / / / | /1/0/0 /1/0/0 | /< /0 /: /0/5/4 /< /0 /: /0/4/5 | /< /0 /: /0/5/4 /< /0 /: /0/4/5 |\n| | | | /2 /2 /: /0/7 /+/0 /: /0/4 | / / / / / / / / | /0/./9/9/(/3/7/) | / / / | / / / / / / | /1/0/0 | /+/0 /: /0/4 | /+/0 /: /0/4 |\n| /1/4/6 /1/4/7 | /9/9/0/2/2/3 /9/9/0/2/2/3 | /5/1/2/3/2/./2/5/0 /5/1/2/3/2/./8/5/9 /5/1/2/3/3/./8/3/6 | /BnZr /0 /: /0/4 /2 /: /1/3 /+/0 /: /0/3 /BnZr /0 /: /0/3 /+/0 /: /0/3 | / / / / / / | / / / / / / | | /6/0 /1/0/0 | /0 /: /2/9 | /0 /: /2/9 | /0 /: /2/9 |\n| | | | | | /0/./8/1/(/5/7/) | | | /BnZr /0 /: /0/4 /0 /: /3/4 /+/0 /: /0/4 /BnZr /0 /: /0/4 /+/0 /: /0/3 | /BnZr /0 /: /0/4 /0 /: /3/4 /+/0 /: /0/4 /BnZr /0 /: /0/4 /+/0 /: /0/3 | /BnZr /0 /: /0/4 /0 /: /3/4 /+/0 /: /0/4 /BnZr /0 /: /0/4 /+/0 /: /0/3 |\n| /1/4/8 | /9/9/0/2/2/4 | | /2 | | | | | | | |\n| | | | /: /1/9 | / / | /0/./9/1/(/7/1/) | / / / | | | /0 /: /3/0 | /BnZr /0 /: /0/3 |\n| | | | /BnZr /0 /: /0/3 | / | | | / / / | /1/3/0 | | |\n| | | | | | / / / | | | | | | \nT ABLE /4/| Continue d \n| Obs /# | Date /(UT/) | MJD a | /BnZr H E X T E | E cut /(k eV/) | E f old /(k eV/) | Best / /2 / /(dof/) | PL / /2 / | E max /(k eV/) | /2/0/{/1/0/0 k eV Flux b /; /(/1/0 /BnZr /8 erg s /BnZr /1 cm /BnZr /2 |\n|----------------------|----------------------------------------|----------------------------------------------------------|-----------------------------------------------------|---------------------------|---------------------------|---------------------------|---------------------------|--------------------------|--------------------------------------------------------------------------|\n| /1/4/9 | /9/9/0/2/2/6 | /5/1/2/3/5/./1/9/9 | /1 /: /9/4 /+/0 /: /1/4 | /: /4 /+/1/8 /: /1 | /1/8/2 /+/1/1/1 | /0/./9/0/(/7/5/) | /1/./3/4 | /1/5/0 | /0 /: /3/7 /+/0 /: /1/9 |\n| /1/5/0 | /9/9/0/2/2/8 | /5/1/2/3/7/./6/1/7 | /BnZr /0 /: /0/6 /1 /: /9/1 /+/0 /: /1/1 | /BnZr /2/2 /: /4 / / / | /BnZr /3/6 / / / | /0/./9/0/(/2/7/) | / / / | /5/0 | /BnZr /0 /: /0/6 /0 /: /0/8 /+/0 /: /0/4 |\n| | | | /BnZr /0 /: /1/0 /+/0 /: /0/2 | | | | | | /BnZr /0 /: /0/2 /+/0 /: /0/2 |\n| /1/5/1 /1/5/2 | /9/9/0/3/0/2 /9/9/0/3/0/3 | /5/1/2/3/9/./0/8/2 /5/1/2/4/0/./0/6/2 | /2 /: /1/3 /BnZr /0 /: /0/2 /2 /: /1/6 /+/0 /: /0/3 | / / / / / / | / / / / / / | /1/./0/3/(/5/7/) | / / / | /1/0/0 | /0 /: /2/0 /BnZr /0 /: /0/1 /0 /: /2/5 /+/0 /: /0/3 |\n| | | | /BnZr /0 /: /0/3 | | | /0/./9/7/(/5/3/) | / / / | /9/0 | /BnZr /0 /: /0/2 /+/0 /: /0/3 |\n| /1/5/3 | /9/9/0/3/0/4 | /5/1/2/4/1/./8/2/8 | /2 /: /3/9 /+/0 /: /0/2 /BnZr /0 /: /0/2 | / / / | / / / | /0/./9/6/(/6/7/) | / / / | /1/2/0 | /0 /: /4/2 /BnZr /0 /: /0/3 |\n| /1/5/4 | /9/9/3/0/0/5 | /5/1/2/4/2/./5/0/8 | /2 /: /3/6 /+/0 /: /0/3 /BnZr /0 /: /0/3 | / / / | / / / | /0/./8/4/(/6/3/) | / / / | /1/1/0 | /0 /: /4/5 /+/0 /: /0/4 /BnZr /0 /: /0/4 |\n| /1/5/5 | /9/9/0/3/0/7 | /5/1/2/4/4/./4/9/6 | /2 /: /5/6 /+/0 /: /0/2 | / / / | / / / | /0/./9/8/(/6/3/) | / / / | /1/1/0 | /0 /: /6/4 /+/0 /: /0/5 |\n| /1/5/6 | /9/9/0/3/0/8 | /5/1/2/4/5/./3/5/2 | /2 /: /5/9 /+/0 /: /0/1 | / / / | / / / | /1/./3/4/(/6/7/) | / / / | /1/2/0 | /0 /: /7/1 /+/0 /: /0/3 |\n| /1/5/7 | /9/9/0/3/0/9 | /5/1/2/4/6/./4/1/4 | /BnZr /0 /: /0/1 /2 /: /5/3 /+/0 /: /0/1 | / / / | / / / | /1/./4/2/(/7/7/) | / / / | /1/5/0 | /BnZr /0 /: /0/3 /0 /: /5/6 /+/0 /: /0/2 |\n| /1/5/8 | /9/9/0/3/1/0 | /5/1/2/4/7/./9/8/0 | /BnZr /0 /: /0/1 /2 /: /5/8 /+/0 /: /0/1 | / / / | / / / | /0/./9/2/(/7/3/) | / / / | /1/4/0 | /BnZr /0 /: /0/2 /0 /: /7/0 /+/0 /: /0/3 |\n| /1/5/9 | /9/9/0/3/1/1 | /5/1/2/4/8/./0/9/0 | /BnZr /0 /: /0/1 /2 /: /5/3 /+/0 /: /0/1 | / / / | / / / | /0/./9/1/(/6/7/) | / / / | /1/2/0 | /BnZr /0 /: /0/3 /0 /: /6/7 /+/0 /: /0/3 |\n| /1/6/0 | /9/9/0/3/1/2 | | /BnZr /0 /: /0/2 /2 /: /5/1 /+/0 /: /0/3 | / / / | / / / | /0/./7/9/(/5/3/) | / / / | /9/0 | /BnZr /0 /: /0/3 /0 /: /7/1 /+/0 /: /0/7 |\n| /1/6/1 | /9/9/0/3/1/3 | /5/1/2/4/9/./3/9/8 /5/1/2/5/0/./6/9/1 | /BnZr /0 /: /0/3 /2 /: /3/0 /+/0 /: /0/4 | /3/3 /: /7 /+/4 /: /1 | /1/5/3 /+/2/8 | /1/./1/9/(/6/1/) | /1/./8/5 | /1/1/0 | /BnZr /0 /: /0/6 /0 /: /6/1 /+/0 /: /0/8 |\n| /1/6/2 | /9/9/0/3/1/6 | /5/1/2/5/3/./2/2/7 | /BnZr /0 /: /0/8 /2 /: /4/1 /+/0 /: /0/3 | /BnZr /6 /: /7 / / / | /BnZr /1/2 / / / | /1/./5/6/(/5/3/) | / / / | /9/0 | /BnZr /0 /: /1/4 /0 /: /6/3 /+/0 /: /0/8 |\n| /1/6/3a | /9/9/0/3/1/7a | /5/1/2/5/4/./0/9/0 | /BnZr /0 /: /0/3 /2 /: /3/8 /+/0 /: /0/2 | / / / | / / / | /0/./9/3/(/5/7/) | / / / | /1/0/0 | /BnZr /0 /: /0/7 /0 /: /6/3 /+/0 /: /0/4 |\n| | | | /BnZr /0 /: /0/2 /+/0 /: /0/4 | /+/3 /: /5 | /+/2/3 | | | | /BnZr /0 /: /0/4 /+/0 /: /1/4 |\n| | | | /BnZr /0 /: /0/5 /+/0 /: /0/2 | /BnZr /3 /: /8 | /BnZr /1/8 | | | | /BnZr /0 /: /2/3 /+/0 /: /0/2 |\n| | | | /BnZr /0 /: /0/2 /+/0 /: /1/2 | | | | | | /BnZr /0 /: /0/2 /+/0 /: /1/0 |\n| /1/6/6 | /9/9/0/3/2/1 | /5/1/2/5/8/./0/8/6 | /BnZr /0 /: /1/1 /2 /: /1/5 /+/0 /: /0/7 | / / / | / / / | /0/./5/6/(/2/7/) | / / / | /5/0 | /BnZr /0 /: /0/7 /0 /: /2/1 /+/0 /: /0/6 |\n| /1/6/7 | /9/9/0/3/2/1 | /5/1/2/5/8/./4/9/6 | /BnZr /0 /: /0/7 /2 /: /0/6 /+/0 /: /0/5 | / / / | / / / | /1/./0/4/(/3/7/) | / / / | /6/0 | /BnZr /0 /: /0/4 /0 /: /2/4 /+/0 /: /0/5 |\n| /1/6/8 | /9/9/0/3/2/1 | /5/1/2/5/8/./9/7/7 | /BnZr /0 /: /0/5 /2 /: /2/3 /+/0 /: /0/3 | / / / | / / / | /0/./9/5/(/6/7/) | / / / | /1/2/0 | /BnZr /0 /: /0/4 /0 /: /2/6 /+/0 /: /0/3 |\n| /1/6/9 | /9/9/0/3/2/2 | /5/1/2/5/9/./2/5/4 | /BnZr /0 /: /0/3 /2 /: /1/3 /+/0 /: /0/9 | / / / | / / / | /0/./9/8/(/3/7/) | / / / | /6/0 | /BnZr /0 /: /0/2 /0 /: /2/8 /+/0 /: /1/0 |\n| /1/7/0 | /9/9/0/3/2/3 | /5/1/2/6/0/./5/5/1 | /BnZr /0 /: /0/9 /1 /: /8/4 /+/0 /: /0/7 | / / / | / / / | /1/./1/1/(/2/7/) | / / / | /5/0 | /BnZr /0 /: /0/7 /0 /: /1/6 /+/0 /: /0/4 |\n| /1/7/1 | /9/9/0/3/2/4 | /5/1/2/6/1/./7/6/6 | /BnZr /0 /: /0/7 /1 /: /8/1 /+/0 /: /0/7 | / / / | / / / | /0/./9/3/(/3/7/) | / / / | /6/0 | /BnZr /0 /: /0/3 /0 /: /1/5 /+/0 /: /0/4 |\n| /1/7/2 | /9/9/0/3/2/6 | /5/1/2/6/3/./1/0/9 | /BnZr /0 /: /0/7 /1 /: /7/8 /+/0 /: /1/1 | / / / | / / / | /1/./1/9/(/2/7/) | / / / | /5/0 | /BnZr /0 /: /0/3 /0 /: /1/5 /+/0 /: /0/6 |\n| /1/7/3 | /9/9/0/3/2/7 | /5/1/2/6/4/./7/4/6 | /BnZr /0 /: /1/1 /1 /: /9/5 /+/0 /: /0/6 | / / / | / / / | /0/./8/5/(/5/7/) | / / / | /1/0/0 | /BnZr /0 /: /0/4 /0 /: /1/1 /+/0 /: /0/2 |\n| /1/7/4 | /9/9/0/3/2/8 | /5/1/2/6/5/./6/1/3 | /BnZr /0 /: /0/6 /1 /: /9/6 /+/0 /: /0/8 | / / / | / / / | /0/./9/9/(/2/7/) | / / / | /5/0 | /BnZr /0 /: /0/2 /0 /: /1/2 /+/0 /: /0/4 |\n| /1/7/5 | /9/9/0/3/2/9 | /5/1/2/6/6/./8/7/9 | /BnZr /0 /: /0/8 /1 /: /9/4 /+/0 /: /1/3 | / / / | / / / | /0/./8/5/(/2/7/) | / / / | /5/0 | /BnZr /0 /: /0/3 /0 /: /0/7 /+/0 /: /0/4 |\n| /1/7/6 | /9/9/0/3/3/0 | /5/1/2/6/7/./6/1/3 | /BnZr /0 /: /1/3 /1 /: /7/7 /+/0 /: /0/8 | / / / | / / / | /1/./1/5/(/2/7/) | / / / | /5/0 | /BnZr /0 /: /0/2 /0 /: /1/2 /+/0 /: /0/3 |\n| /1/7/7 | /9/9/0/4/0/1 | /5/1/2/6/9/./6/7/6 | /BnZr /0 /: /0/7 /1 /: /8/3 /+/0 /: /0/8 | / / / | / / / | /1/./0/7/(/2/7/) | / / / | /5/0 | /BnZr /0 /: /0/3 /0 /: /1/1 /+/0 /: /0/3 |\n| /1/7/8 | /9/9/0/4/0/2 | /5/1/2/7/0/./7/4/2 | /BnZr /0 /: /0/8 /2 /: /0/7 /+/0 /: /0/4 | / / / | / / / | /1/./1/0/(/5/7/) | / / / | /1/0/0 | /BnZr /0 /: /0/3 /0 /: /1/5 /+/0 /: /0/2 |\n| /1/7/9 | /9/9/0/4/0/3 | /5/1/2/7/1/./4/0/6 | /BnZr /0 /: /0/4 /2 /: /0/1 /+/0 /: /0/6 | / / / | / / / | /0/./5/7/(/3/9/) | / / / | /7/0 | /BnZr /0 /: /0/2 /0 /: /1/5 /+/0 /: /0/3 |\n| /1/8/0 | /9/9/0/4/0/5 | /5/1/2/7/3/./5/3/9 | /BnZr /0 /: /0/6 /1 /: /9/7 /+/0 /: /0/7 | / / / | / / / | /0/./7/5/(/4/3/) | / / / | /7/0 | /BnZr /0 /: /0/3 /0 /: /0/8 /+/0 /: /0/2 |\n| | /9/9/0/4/0/6 | | /BnZr /0 /: /0/7 /+/0 /: /1/4 | | | | | | /BnZr /0 /: /0/2 |\n| /1/8/1 | /9/9/0/4/0/8 | /5/1/2/7/4/./4/7/3 | /1 /: /5/7 /BnZr /0 /: /1/4 | / / / | / / / | /0/./6/7/(/2/5/) | / / / | /4/9 | /0 /: /0/8 /+/0 /: /0/5 |\n| /1/8/2 | /9/9/0/4/0/9 | /5/1/2/7/6/./2/7/7 /5/1/2/7/7/./4/0/2 | /2 /: /5/( f ixed /) /2 | / / / / | / / / | / / / | / / / | /1/0/0 | /BnZr /0 /: /0/3 /< /0 /: /0/3/1 |\n| /1/8/3 /1/8/4 | /9/9/0/4/1/0 /9/9/0/4/1/1 | /5/1/2/7/8/./6/8/8 | /: /5/( f ixed /) /2 /: /5/( f ixed /) | / / / / / | / / / / / / | / / / / / / | / / / | /1/0/0 /1/0/0 | /< /0 /: /0/3/8 /< /0 /: /0/4/1 |\n| /1/8/5 | /9/9/0/4/1/2 | /5/1/2/7/9/./5/3/1 | /2 /: /5/( f ixed /) | / / / | / / / | / / / | / / / / / / | /1/0/0 | /< /0 /: /0/3/0 |\n| /1/8/6 /1/8/7 | /9/9/0/4/1/5 /9/9/0/4/1/7 | /5/1/2/8/0/./5/4/3 /5/1/2/8/3/./2/2/3 | /2 /: /5/( f ixed /) /2 /5/( f ixed /) | / / / / / / | / / / / / / | / / / / / / | / / / / / / | /1/0/0 /1/0/0 | /< /0 /: /0/2/3 /< /0 /: /0/3/0 |\n| /1/8/8 /1/8/9 | /9/9/0/4/1/8 | /5/1/2/8/5/./1/9/5 /5/1/2/8/6/./0/5/9 | /: /2 /: /5/( f ixed /) /2 /: /5/( f ixed /) | / / / / / / | / / / / / / | / / / / / / | / / / / / / | /1/0/0 /1/0/0 | /< /0 /: /0/2/5 /< /0 /: /0/2/4 |\n| /1/9/0 /1/9/1 | /9/9/0/4/1/9 /9/9/0/4/2/0 | /5/1/2/8/7/./2/5/8 /5/1/2/8/8/./3/8/3 | /2 /: /5/( f ixed /) /2 /5/( f ixed /) | / / / / / / | / / / / / / | / / / / / / | / / / / / / | /1/0/0 /1/0/0 | /< /0 /: /0/3/7 /< /0 /: /0/1/9 |\n| /1/9/2 | /9/9/0/4/2/1 | /5/1/2/8/9/./1/4/5 | /: /2 /: /5/( f ixed /) /: /5/( f ixed /) | / / / / | / / / / / / | / / / / / / | / / / | /1/0/0 | /< /0 /: /0/3/9 /< /0 /: /0/4/9 |\n| /1/9/3 /1/9/4 /1/9/5 | /9/9/0/4/2/2 /9/9/0/4/2/3 /9/9/0/4/2/4 | /5/1/2/9/0/./9/8/4 /5/1/2/9/1/./1/8/4 /5/1/2/9/2/./4/4/9 | /2 /2 /: /5/( f ixed /) /: /5/( f ixed /) | / / / / / / / / | / / / | / / / / / / | / / / / / / / / / | /1/0/0 /1/0/0 | /< /0 /: /0/3/4 /< /0 /: /0/1/7 |\n| /1/9/6 | /9/9/0/4/2/5 | | /2 /: /5/( f ixed /) | / / | / / / | | / / / | /1/0/0 | /< /0 /: /0/2/0 |\n| /1/9/7 /1/9/8 | /9/9/0/4/2/7 /9/9/0/4/2/9 | /5/1/2/9/3/./4/4/9 /5/1/2/9/5/./5/2/0 /5/1/2/9/7/./3/7/5 | /2 /: /5/( f ixed /) /: /5/( f ixed /) | / / / | / / / / / / / / / | / / / / / / / / / | / / / / / / | /1/0/0 /1/0/0 | /< /0 /: /0/1/3 /< /0 /: /0/0/7 |\n| /1/9/9 | /9/9/0/4/3/0 | /5/1/2/9/8/./0/5/1 | /2 /2 /2 /: /5/( f ixed /) | / / / / / / / | / / / | / / / | / / / | /1/0/0 | /< /0 /: /0/0/5 |\n| /2/0/0 /2/0/1 | /9/9/0/5/0/1 /9/9/0/5/0/2 | /5/1/2/9/9/./3/4/4 /5/1/3/0/0/./4/5/7 | /2 /: /5/( f ixed /) / / / | / / / / | / / / / / / | / / / / / / | / / / / / / | /1/0/0 /1/0/0 | /< /0 /: /0/0/3 / / / |\n| /2/0/2 /2/0/3 | /9/9/0/5/0/4 | /5/1/3/0/2/./4/6/1 | / / / | / / / / / | / / / | / / / | / / / | / / / / / / | / / / |\n| /2/0/4 /2/0/5 | /9/9/0/5/0/5 /9/9/0/5/0/7 /9/9/0/5/0/9 | /5/1/3/0/3/./3/9/1 /5/1/3/0/5/./1/2/5 /5/1/3/0/7/./3/2/0 | / / / / / / /2 /: /5/( f ixed /) | / / / / / / / / / | / / / / / / / / / | / / / / / / / / / | / / / / / / / / / | / / / / / / /1/0/0 | / / / / / / /< /0 /: /0/2/0 |\n| /2/0/6 /2/0/7 | /9/9/0/5/1/1 /9/9/0/5/1/3 /9/9/0/5/1/5 | /5/1/3/0/9/./7/1/9 /5/1/3/1/1/./5/2/0 /5/1/3/1/3/./3/1/6 | /2 /: /5/( f ixed /) /2 /: /5/( f ixed /) / / / | / / / / / | / / / / / / | / / / | / / / / / / | /1/0/0 /1/0/0 | /< /0 /: /0/1/5 /< /0 /: /0/0/9 |\n| | | | | | | / / / | / / / | | / / / |\n| | | | | | | / / / | | | |\n| /2/0/8 /2/0/9 | /9/9/0/5/2/0 | /5/1/3/1/8/./7/7/0 | / / / | / / / / / / / | / / / / / / | / / / | / / / | / / / / / / | / / / | \n- / HEXTE only from /2/0/{ E max k eV/. a\n- Start of Observ ation/. M J D /= J D /BnZr /2 /; /4/0/0 /; /0/0/0 /: /5 b Flux is normalized to the HEXTE /(Cluster A/)/, whic h is / /2/0/{/3/0/% lo w er than the PCA normalization/. c\n- HEXTE only from /2/0/{ E max k eV/. a Start of Observ ation/. M J D /= J D /BnZr /2 /; /4/0/0 /; /0/0/0 /: /5\n- Flux is normalized to the HEXTE /(Cluster A/)/, whic h is / /2/0/{/3/0/% lo w er than c Upp er limits for the /2/0/{/1/0/0 k eV flux are giv en at the /3 / lev el of confidence/."}

2009PhRvD..80j4039B

An analytic Lifshitz black hole

2009-01-01

['-', '-', '-', '-', '-']

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A Lifshitz point is described by a quantum field theory with anisotropic scale invariance (but not Galilean invariance). In , gravity duals were conjectured for such theories. We construct analytically a black hole that asymptotes to a vacuum Lifshitz solution; this black hole solves the equations of motion of some simple (but somewhat strange) extensions of the models of . We study its thermodynamics and scalar response functions. The scalar wave equation turns out to be exactly solvable. Interestingly, the Green’s functions do not exhibit the ultralocal behavior seen previously in the free Lifshitz scalar theory.

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https://arxiv.org/pdf/0909.0263.pdf

{'Koushik Balasubramanian and John McGreevy': 'Center for Theoretical Physics, MIT, Cambridge, Massachusetts 02139, USA', 'Abstract': "A Lifshitz point is described by a quantum field theory with anisotropic scale invariance (but not Galilean invariance). In 0808.1725, gravity duals were conjectured for such theories. We construct analytically a black hole which asymptotes to a vacuum Lifshitz solution; this black hole solves the equations of motion of some simple (but somewhat strange) extensions of the models of 0808.1725. We study its thermodynamics and scalar response functions. The scalar wave equation turns out to be exactly solvable. Interestingly, the Green's functions do not exhibit the ultralocal behavior seen previously in the free Lifshitz scalar theory.", '1 Introduction': "A great deal of progress has been made in the study of quantum field theories and their holographic duals. The possible scope of this enterprise is not yet clear; for example, the correspondence seems to extend to some systems without Lorentz invariance. Recently, attempts have been made to apply the holographic principle to study condensed matter systems near a critical point (for reviews, see [1, 2]). There are many scale-invariant field theories that are not Lorentz invariant, which are of interest in studying such critical points. In such a theory, time and space can scale differently i.e. , t → λ z t, /vectorx → λ/vectorx under dilatation. The relative scale dimension of time and space, z , is called the 'dynamical exponent'. Such a scale invariance is exhibited by a Lifshitz theory, which we will take to mean an anisotropic scale-invariant theory which is not Galilean-invariant. The following Gaussian action provides a simple example of a (free) Lifshitz theory in d space dimensions: \nS [ χ ] = ∫ d d xdt [ ( ∂ t χ ) 2 -K ( ∇ 2 χ ) 2 ] (1.1) \nThis action describes a fixed line parametrized by K , and the dynamical exponent is z = 2. This theory describes the critical behavior of e.g. quantum dimer models [3]. In many ways, the d = 2 , z = 2 version of the theory (1.1) is like a relativistic boson in 1+1 dimensions 1 . The scaling behavior of the ground-state entanglement entropy for this class of theories was studied recently in [4, 5]. This analysis also supports the similarity with 2d CFT, in that a universal leading singular behavior is found. \nIn the free theory, the boson has logarithmic correlators \n〈 χ ( x ) χ (0) 〉 ∼ ∫ dωd 2 k 1 ω 2 -k 4 e i /vector k · /vectorx -iωt ∼ ln x. (1.2) \nAs in the familiar d = z = 1 case, the operators of definite scaling dimension are not the canonical bose field itself, but rather its exponentials and derivatives. In the connection with quantum dimer models, the bose field is a height variable constructed from the dimer configuration, and the exponentials of the bose field are order parameters for various dimersolid orderings [3]. At zero temperature, the logarithmic behavior of the correlator of the bose fields implies that the two-point function of the order parameter decays as a power \nlaw. However, the equal-time correlators at finite temperature are ultra-local in the infinitevolume limit [7]: they vanish at any nonzero spatial separation. In [7], it was suggested that this might be a mechanism for the kind of local criticality (scaling in frequency, but not momentum) seen in the strange metal phase of the cuprates and in heavy fermion materials. One is led to wonder whether this property should is shared by interacting Lifshitz theories, and whether the Lifshitz scaling is sufficient to produce this behavior. In [7] the addition of perturbative interactions was shown to lead to a finite correlation length; these perturbations violate the Lifshitz scaling. Below we will show that interactions which preserve the Lifshitz scaling need not give ultralocal behavior. \nGravity solutions with Lifshitz-type scale invariance were found in [8]. They found that the following family of metrics, parametrized by z , provide a geometrical description of Lifshitz-like theories (with z as the dynamical exponent): \nds 2 = L 2 ( -dt 2 r 2 z + /vector dx 2 + dr 2 r 2 ) , (1.3) \nwhere /vectorx denotes a d -dimensional spatial vector. For d = 2, this metric extremizes the folowing action: \nS = 1 2 ∫ d 4 x ( R -2Λ) -1 2 ∫ ( F (2) ∧ /starF (2) + F (3) ∧ /starF (3) ) -c ∫ B (2) ∧ F (2) , (1.4) \nwhere F (2) = dA (1) , F (3) = dB (2) and Λ is the four dimensional cosmological constant. They computed the two-point function for the case when z = 2 and showed that it exhibits power law decay. They also studied the holographic renormalization group flow for this case and found that AdS 4 is the only other fixed point of the flow. Lifshitz vacuum solutions were shown to be stable under perturbations of the bulk action in [9]. \nIn this paper we shall study a black hole solution which asymptotes to the Lifshitz spacetime with d = 2 , z = 2. In section 2, an analytical solution for a black hole that asymptotes to the planar Lifshitz spacetime is written down. We present several actions whose equations of motion it solves; they all involve some matter sector additional to (1.4). Section 3 presents an analysis of the thermodynamics of this black hole. In section 4, we solve the wave equation for a massive scalar field in this background; surprisingly, this equation is exactly solvable. We use this solution to calculate the two-point functions of boundary operators in section 5. \nSince we found the solution described in this paper, some related work has appeared. [10] constructs a black hole solution in a related background with slightly different asymptotics. Danielsson and Thorlacius [11] found numerical solutions of black holes in global Lifshitz spacetime. Interestingly, these are solutions to precisely the system studied by [8], with no additional fields. Related solutions were found by [12, 27]. [13] found solutions of type IIB supergravity that are dual to Lifshitz-like theories with spatial anisotropy and z = 3 / 2; these solutions have a scalar field which breaks the scaling symmetry. To our knowledge, a string embedding of z = 2 Lifshitz spacetime is still not known; obstacles to finding such an embedding are described in [14].", '2.1 Vacuum solution': "The tensor fields in [8] can be rewritten as one massive gauge field. The Chern-Simons-like coupling is responsible for the mass. A familiar example is that of a 2-form field strength F and a 3-form field strength H in five dimensions with L = F ∧ /starF + H ∧ /starH + F ∧ H : this gives the same equation of motion as L = F ∧ /starF + A 2 . In the four dimensional case studied in [8], the dual of the 3-form field strength in four dimensions is a scalar field ϕ . Then \nB 2 ∧ F 2 = -F 3 ∧ A 1 +bdy terms = -/star dϕ ∧ A 1 = -√ g∂ µ ϕA µ . (2.1) \nThe action then reduces to \nF 2 ∧ /starF 2 +( ∂ϕ + A ) 2 , (2.2) \nand ϕ shifts under the A gauge symmetry, and we can fix it to zero, and this is just a massive gauge field 2 . Hence, the zero-temperature Lifshitz metric \nds 2 = -dt 2 r 2 z + d/vectorx 2 + dr 2 r 2 , (2.3) \nis a solution of gravity in the presence of cosmological constant and a massive gauge field, and the gauge field mass is m 2 = dz . The bulk curvature radius has been set to one here and throughout the paper; in these units, the cosmological constant is Λ = -z 2 +( d -1) z + d 2 2 . The gauge field profile is A = Ω r -z dt (in the r coordinate with the boundary at r = 0), and \nthe strength of the gauge field is (for d = 2) \nΩ 2 = 8 z 2 + z -2 z ( z +2) . \nWe note in passing that the Schrodinger spacetime is a solution of the same action with a different mass for the gauge field and a different cosmological constant [16, 17]. Therefore we find the perhaps-unfamiliar situation where the same gravitational action has solutions with very different asymptopia. Another recent example where this happens is 'chiral gravity' in three dimensions, which has asymptotically AdS solutions as well as various squashed and smushed and wipfed solutions [15]. \nGiven this fact, one might expect that the Lifshitz spacetime can be embedded into the same type IIB truncations as the Schrodinger spacetime (see [18, 19] and especially [20]). However, the scalar equation of motion is not satisfied by the Lifshitz background since F 2 is non-zero.", '2.2 Black hole solution': "We shall now study a black hole in four dimensions that asymptotically approaches the Lifshitz spacetime with z = 2. We first observe that there is such a black hole in a system with a strongly-coupled scalar ( i.e. a scalar without kinetic terms). The action is \nS 1 = 1 2 ∫ d 4 x ( R -2Λ) -∫ d 4 x ( e -2Φ 4 F 2 + m 2 2 A 2 + ( e -2Φ -1 ) ) . (2.4) \nA solution of this system is \nΦ = -1 2 log ( 1 + r 2 /r 2 H ) , A = f/r 2 dt \nds 2 = -f dt 2 r 2 z + d/vectorx 2 r 2 + dr 2 fr 2 , (2.5) \nf = 1 -r 2 r 2 H . \nwith \nNote that the metric has the same simple form as in the RG flow solution (eqn (4.1)) of [8]. \nWe can get the same contributions to the stress tensor as from the scalar without kinetic terms from several more-reasonable systems. One such system is obtained by adding a \nsecond massive gauge field B which will provide the same stress-energy as the scalar. It has a slightly unfamiliar action: \nS 2 = 1 2 ∫ d 4 x ( R -2Λ) -∫ d 4 x ( 1 4 B 2 dA 2 + m 2 A 2 A 2 + 1 4 dB 2 -m 2 B 2 (1 -B 2 ) ) (2.6) \nwhere A, B are one-forms, and m 2 A = 4 and m 2 B = 2 . The solution looks like B = B ( r ) dr, A = A ( r ) dt and the metric is same as (2.5). In the solution, the scalar functions take the form \nB ( r ) = √ g rr ( 1 + r 2 r 2 H ) , A ( r ) = Ω fr -z dt \nNote that B ( r ) isn't gauge-trivial (even though its field strength vanishes) because of the mass term. Since B ( r ) asymptotes to 1, the effective gauge coupling of the field A is not large at the boundary. \nThe system with a strongly-coupled scalar in (2.4) is not equivalent to the system (2.6) with two gauge fields. For example, there are solutions of (2.4) where the scalar has a profile that depends both on r and x ; such configurations do not correspond to solutions of (2.6). \nIt is not clear whether the solution written above is stable. We leave the analysis of the stability of such solutions to small perturbations to future work. As weak evidence for this stability, we show in the next section that these black holes are thermodynamically stable. \nAnother action with this Lifshitz black hole (2.5) as a solution is \nS 3 = 1 2 ∫ d 4 x ( R -2Λ -1 2 dB 2 -( ∂ Φ -B ) 2 -m A A 2 -1 2 e -2Φ F 2 -V (Φ) ) (2.7) \nwhere V (Φ) = 2 e -2Φ -2. In the solution, the metric and gauge field A take the same form as in (2.5). The other fields are \ne -2Φ = 1 + r 2 r 2 H , B = d Φ . \nNote that the action (2.7) is not invariant under the would-be gauge transformation \nB → B + d Λ , Φ → Φ+Λ , \nbecause of the coupling to F 2 -4 (the sum of the gauge kinetic term and the potential term) 3 . We are not bothered by this: it means that in quantizing the model, mass terms for the fluctuations B will be generated; however, such a mass term is already present. \nWe would also like to point out that in the three systems S 1 , 2 , 3 described above, the stressenergy tensor of the fields with local propagating degrees of freedom satisfy the dominant energy condition 4 , i.e. T (Φ ,A,B ) µν = R µν -( 1 2 R +Λ) g µν ) satisfies the following \n( \n) T tt T xx = T tt T yy > -1 and T tt T rr > -1 . \nHence, there are no superluminal effects in the bulk. This is basically a consequence of the fact that the squared-masses of the gauge fields are positive.", '3 Lifshitz black hole thermodynamics': "The Hawking temperature and entropy can be calculated using the near horizon geometry. The Hawking temperature is the periodicity of the Euclidean time direction in the near horizon metric (proportional to the surface gravity) i.e. , T = κ 2 π | r = r H , with \nκ 2 = -1 2 ∇ a v b ∇ b v a \nwhere v = ∂ t . Hence, \nT = 1 2 πr 2 H . (3.1) \nThe entropy of the black hole is \nS = Area of Horizon 4 G N = L x L y 4 G 4 r 2 H . (3.2) \nWe shall now evaluate the free energy, internal energy and pressure by calculating the on-shell action and boundary stress tensor. In order to renormalize the action, it is essential to add counterterms which are intrinsic invariants of the boundary (see [21]). \nConsider the following gravitational action: \nS = 1 2 ∫ M d 4 x √ g ( R -2Λ -e -2Φ 4 F 2 -m 2 A 2 A 2 -V (Φ) ) -∫ ∂M d 3 x √ γ ( K + c N e -2Φ n µ A ν F µν ) (3.3) + 1 2 ∫ ∂M d 3 x √ γ ( 2 c 0 -c 1 Φ -c 2 Φ 2 ) + 1 2 ∫ ∂M d 3 x √ γ ( ( c 3 + c 4 Φ) A 2 + c 5 A 4 ) . \nThe second line of (3.3) contains extrinsic boundary terms: the Gibbons-Hawking term, and a 'Neumannizing term' which changes the boundary conditions on the gauge field. The last line of (3.3) describes the intrinsic boundary counterterms 5 . In the above expression, we have set 8 πG = 1. We have written the analysis in terms of S 1 (2.4); the analysis can be adapted for S 2 (2.6) by simply replacing Φ in (3.3) by -1 2 log B 2 . If Neumann boundary conditions are imposed on the gauge field, then c N = 1 and c i = 0 for i ≥ 3. Similarly, c N = 0, if Dirichlet boundary condition is imposed on the gauge field. \nThe boundary stress tensor resulting from (3.3) is \nT µν = K µν -( K -c 0 + 1 2 c 1 Φ+ 1 2 c 2 Φ 2 ) γ µν + e -2Φ 2 ( n r A µ ∂ r A ν + n r A ν ∂ r A µ -n r A α ∂ r A α γ µν ) \n+ ( c 3 + c 4 Φ+2 c 5 A 2 ) A µ A ν -1 2 ( c 3 + c 4 Φ+ c 5 A 2 ) A 2 γ µν (3.4) \nThe values of c i are determined by demanding that the action is 'well-behaved'. The action is well-behaved if the variation of the action vanishes on-shell and if the residual gauge symmetries of the metric are not broken. The values of c i which makes the action well-defined also render finite the action and boundary stress tensor (please see appendix A). Implementing this procedure, we find for the energy density, pressure and free energy \nE = P = -F = 1 2 T S = L x L y 2 r 4 H (3.5) \nSatisfying the first law of thermodynamics (in the Gibbs-Duhem form E + P = T S ) is a nice check on the sensibility of our solution, since it is a relation between near-horizon ( T, S ) and near-boundary ( E , P ) quantities. \nRecently, [28] have described an alternative set of boundary terms for asymptotically Lifshitz theories. They do not include the Neumannizing term, but instead include an intrinsic but nonanalytic A µ A µ term. \n√ \n1 2 ∫ ∂M d 3 x √ γ ( 2 c ' 0 + c ' 1 Φ+ c ' 2 Φ 2 ) + 1 2 ∫ ∂M d 3 x √ γ ( ( c ' 3 + c ' 4 Φ)( A 2 -1) + c ' 5 ( A 2 -1) 2 ) \nwhich has the same form as (3.3).", '4 Scalar response': 'In this section, we study a probe scalar in the black hole background (2.5). The scalar can be considered a proxy for the mode of the metric coupling to T x y .', '4.1 Exact solution of scalar wave equation': "Consider a scalar field φ of mass m in the black hole background (2.5) 6 . \nLet u ≡ r 2 r 2 H . Fourier expand: \nφ = ∑ k φ k ( u ) e -iωt + i /vector k · /vectorx \nThe wave equation takes the form: \n0 = u ( -fk 2 + uω 2 ) + m 2 f 4 f 2 u 2 φ k ( u ) -1 fu φ ' k ( u ) + φ '' k ( u ) \nwhere k 2 ≡ /vector k 2 . Near the horizon, the incoming ( -) and outgoing (+) waves are \nφ k ∼ (1 -u ) ± iω/ 2 . \nThe solutions near the boundary at u = 0 are \nφ k ∼ u 1 ± 1 2 √ 4+ m 2 \nThe exact solution to the wave equation is φ k ( u ) = f -iω/ 2 u 1 -1 2 √ m 2 +4 G k ( u ) with \nG k ( u ) = A 1 2 F 1 ( a + , b + ; c + , u ) u √ m 2 +4 + A 2 2 F 1 ( a -, b -; c -, u ) (4.1) \nand \n( a ± , b ± ; c ± ) ≡ ( -iω 2 ± √ m 2 +4 2 -1 2 √ -k 2 -ω 2 +1+ 1 2 , -iω 2 ± √ m 2 +4 2 + 1 2 √ -k 2 -ω 2 +1+ 1 2 ; 1 ± √ m 2 +4; u ) \nWe emphasize that this is the exact solution to the scalar wave equation in this black hole; such a solution is unavailable for the AdS d> 3 black hole. The difference is that the equation \nhere has only three regular singular points, whereas the AdS 5 black hole wave equation has four. This is because in the AdS 5 black hole, the emblackening factor is f = 1 -u 2 which has two roots, whereas ours is just f = 1 -u . \nThe other example of a black hole with a solvable scalar wave equation is the BTZ black hole in AdS 3 [25] 7 . The origin of the solvability in that case is the fact that BTZ is an orbifold of the zero-temperature solution. This is not the origin of the solvability in our case - this black hole is not an orbifold of the zero-temperature solution. This may be seen by comparing curvature invariants: they are not locally diffeomorphic. More simply, if the black hole were an orbifold, it would solve the same equations of motion as the vacuum solution. The fact that we were forced to add an additional matter sector (such as Φ or B µ ) to find the black hole solution immediately shows that they are not locally diffeomorphic. \nNow we ask for the linear combination of (4.1) which is ingoing at the horizon. In terms of ν ≡ √ 4 + m 2 , γ ≡ √ 1 -ω 2 -k 2 , this is the combination with \nA 1 A 2 = -( -1) ν Γ( ν ) Γ( -ν ) Γ ( 1 2 (1 -iω -ν -γ ) ) Γ ( 1 2 (1 -iω + ν -γ ) ) Γ ( 1 2 (1 -iω -ν + γ ) ) Γ ( 1 2 (1 -iω + ν + γ ) ) . (4.2) \nIn the massless case, one of the hypergeometric functions in (4.1) specializes to a Meijer G-function, and the solution is φ k = u 2 f -iω/ 2 G k ( u ) with \nG k ( u ) = c 2 2 F 1 ( -iω 2 -1 2 √ -k 2 -ω 2 +1+ 3 2 , -iω 2 + 1 2 √ -k 2 -ω 2 +1+ 3 2 ; 3; u ) + \nIn this solution, the coefficient of c 1 (the MeierG function) is purely ingoing at the horizon. \nc 1 G 2 , 0 2 , 2 u ∣ ∣ ∣ ∣ ∣ ∣ ∣ 1 2 ( iω -√ -k 2 -ω 2 +1 -1 ) , 1 2 ( iω + √ -k 2 -ω 2 +1 -1 ) -2 , 0 \n∣", '4.2 Correlators of scalar operators': "In the previous section we wrote the solution for the wave equation in this black hole for a scalar field with an arbitrary mass. As mentioned earlier, the BTZ black hole also shares this property of having a scalar wave equation whose solutions are hypergeometric. Hence, \none might expect that the two-point function of scalar operators in a Lifshitz-like theory to have a form that is similar to that of 2D CFTs. \nThe momentum space correlator for a scalar operator of dimension ∆ = ∆ -is determined from the ratio of the non-normalizable and normalizable parts of the solution. The asymptotic behavior of the solution in (4.1) is \nφ ∼ u ∆ + 2 ( A 1 + O ( u )) + u ∆ -2 ( A 2 + O ( u )) (4.3) \nHence, the retarded Green's function (two-point function) is \nG ret ( ω, /vector k ) = -A 1 A 2 = ( -1) ν Γ( ν ) Γ( -ν ) Γ ( 1 2 (1 -iω -ν -γ ) ) Γ 1 2 (1 -iω + ν -γ ) ) Γ ( 1 2 (1 -iω -ν + γ ) ) Γ 1 2 (1 -iω + ν + γ ) ) (4.4) \n( \n) ) with ν and γ defined above equation (4.2). Note that the correlator has a form very similar to that of a 2D CFT. It would be nice to know the precise connection between z = 2 Lifshitzlike theories in 2 + 1 D with 2D CFTs that is responsible for this similarity. Note that the poles of the retarded Green's function do not lie on a straight line in the complex frequency plane, as they do for 2D CFTs. \n( \nNext, we would like to see whether the correlators exhibit ultra local behavior at finite temperature as observed in the free scalar Lifshitz theory [7]. We find that the Green's function is not ultra-local - this removes the possibility that Lifshitz-symmetric interactions require ultralocal behavior. \nWe will now calculate the two-point function of a scalar operator of dimension ∆ = 4 at finite temperature. In this case, the correlator is given by the coefficient of r 4 in the asymptotic expansion of the solution near r = 0. Kachru et. al. [8] showed that the correlator exhibits a power law decay at zero temperature. \nWe can evaluate this correlator by extracting the coefficient of the u 2 term (note that u ∝ r 2 ) in the asymptotic expansion of the solution of the massless scalar wave equation. The behavior of the solution near u = 0 is \nφ ( u, /vector k, ω ) = 1 -u 4 ( /vector k 2 +2 iω ) -u 2 64 ( ( /vector k 2 ) 2 +4 ω 2 ) [ -3+2 ψ ( 1 2 ( -1 + iω -√ 1 -/vector k 2 -ω 2 )) \n+2 ψ ( 1 2 ( -1 + iω + √ 1 -/vector k 2 -ω 2 )) +2 γ E +2ln u ] + O ( u 3 ) (4.5) \nwhere γ E is Euler's constant, ψ is the digamma function. The behavior of the solution in the Euclidean black hole can be obtained by replacing ω by -i | ω | . The choice of the negative sign \ngives the solution which is ingoing at the horizon, as appropriate to the retarded correlator [22]. Henceforth, we shall work with the solution for the Euclidean case. The correlator is the sum of the two digamma functions. All other terms in the coefficient of u 2 are contact terms. Hence, the correlator in momentum space is \n〈O ( -ω, -/vector k ) O ( ω, /vector k ) 〉 ∝ ( ( /vector k 2 ) 2 -4 ω 2 ) [ ψ ( 1 2 ( -1 + | ω | -√ 1 -/vector k 2 + ω 2 )) + ψ ( 1 2 ( -1 + | ω | + √ 1 -/vector k 2 + ω 2 )) ] (4.6) \nAfter dropping the contact terms, the above expression can be written as follows \n〈O ( -ω, -/vector k ) O ( ω, /vector k ) 〉 ∝ ∞ ∑ n =1 A n \nwhere \nWe can now calculate the correlators in coordinate space by performing the Fourier transform of the above expression 8 . This is given by \nA n = (2 n -3) + | ω | (2 n -3) 2 +2 | ω | (2 n -3) + k 2 -1 = a n + | ω | a 2 n +2 a n | ω | + k 2 -1 \nD ( | /vectorx | , t ) = [ (4 ∂ 2 t -( ∇ 2 ) 2 ) ] ∑ n F n (4.7) \nwhere, D ( | /vectorx | , t ) is the two-point function and \n∑ n F n = ∑ n ∫ kdkdωdθ a n + | ω | a 2 n +2 a n | ω | + k 2 -1 / 4 e ik | /vectorx | cos θ + iωt \nThe short distance ( r /lessmuch r H ) behavior of the equal time correlator is \nD ( | /vectorx | /lessmuch r H , 0) = [ (4 ∂ 2 t -( ∇ 2 ) 2 ) F ] t =0 , | /vectorx |→ 0 ∝ 1 | /vectorx | 8 . (4.8) \nAs a check, we note that, the short distance behavior of this expression reproduces the zero-temperature answer | /vectorx | -8 found in [8]. \nThe long distance ( | /vectorx | /greatermuch r H ) behavior is \nThe correlator is not ultra-local, unlike the thermal correlator in free scalar Lifshitz theory. \nD ( | /vectorx | /greatermuch r H , 0) = [ (4 ∂ 2 t -( ∇ 2 ) 2 ) F ] t =0 , | /vectorx |→∞ ∝ e -√ 2 | /vectorx | /r H | /vectorx | 3 / 2 . (4.9)", '5 Outlook': "An important defect of our work which cannot have avoided the reader's attention is the fact that the matter content which produces the stress-energy tensor for this black hole is unfamiliar and contrived. There is no physical reason why terms such as A 2 B 2 should not be added. In our defense, a perturbation analysis in the coefficient of such terms indicates that a corrected solution can be constructed. \nIt is not clear how to embed such solutions in a UV-complete gravity theory; a stringy description is not known yet even for the zero temperature case. Such a description would help in finding specific Lifshitz-like field theories with gravity duals. It would be nice to understand the connection (if it exists) between the Lifshitz spacetime and non-Abelian Lifshitz-like gauge theories [23, 24]. \nAcknowledgements We thank Shamit Kachru and Mike Mulligan for correspondence on attempts to construct more familiar actions with Lifshitz black hole solutions and for helpful advice on Fourier transforms. We also thank Allan Adams, Sean Hartnoll, Alex Maloney, Omid Saremi, T. Senthil and Larus Thorlacius for useful discussions and comments. This work was supported in part by funds provided by the U.S. Department of Energy (D.O.E.) under cooperative research agreement DE-FG0205ER41360 and by the National Science Foundation under Grant No. NSF PHY05-51164.", 'A Regularizing the action and boundary stress tensor': 'In this appendix, we will show that the on-shell action and boundary stress tensor can be rendered finite by making the action well-behaved, i.e. the action is stationary on-shell under an arbitrary normalizable variation of the bulk fields, and the boundary terms in the action must not break the residual gauge symmetries of the metric. \nWe will first find the constraints imposed by finiteness of the free energy, internal energy and pressure on c i . \nThe free energy of the boundary theory is \n-F = S onshell β = 1 2 L x L y [ 64 c N -8 c 0 +16 c 1 +8 c 2 +6 c 3 +16 c 4 -15 c 5 32 r 4 H -32 + 4 c 1 +8 c 0 +6 c 3 +2 c 4 -5 c 5 16 /epsilon1 2 r 2 H + 24 + 2 c 3 -c 5 -8 c N +8 c 0 8 /epsilon1 4 ] (A.1) \nwhere β is inverse temperature. We must set -c 5 + 24 -8 c N + 8 c 0 + 2 c 3 = 0 and -c 1 -8 -2 c 0 -3 / 2 c 3 -1 / 2 c 3 + 5 / 4 c 5 = 0 to get rid of the divergences in the on-shell action. Further, finiteness of the boundary stress tensor and conformal ward identities impose more constraints on the counterterms. \nThe internal energy of the boundary theory is \nE = -L x L y √ γT t t = -L x L y ( 16 + 8 c 0 -2 c 3 +3 c 5 +8 c N 8 /epsilon1 4 (A.2) \n-32 + 8 c 0 +4 c 1 -6 c 3 -2 c 4 +15 c 5 16 r 2 H /epsilon1 2 -8 c 0 -16 c 1 -8 c 2 +6 c 3 +16 c 4 -45 c 5 +64 c N 64 r 4 H ) (A.3) \nSimilarly, the expression for pressure is \nP = 1 2 L x L y √ γT i i = L x L y √ γT x x = 1 2 L x L y [ 64 c N -8 c 0 +16 c 1 +8 c 2 +6 c 3 +16 c 4 -15 c 5 32 r 4 H -32 + 4 c 1 +8 c 0 +6 c 3 +2 c 4 -5 c 5 16 /epsilon1 2 r 2 H + 24 + 2 c 3 -c 5 -8 c N +8 c 0 8 /epsilon1 4 ] (A.4) \nNote that F = -P , as expected in the grand canonical ensemble. Hence, the condition for the divergences in pressure to cancel is same as the condition for divergences in the on-shell action to cancel. However, finiteness of energy imposes additional constraints on the counterterms. In the case of Schrodinger black hole, it is not possible to get rid of the divergence in the energy without the Neumannizing term [19]. \nThe conformal Ward identity for conservation of the dilatation current requires z E = d P , and in our discussion d = z = 2. The residual gauge freedom of the metric is broken if this condition is not satisfied (see [16]). Note that making the boundary stress tensor finite does not ensure this condition. We must set c 2 = 7 / 2 for the conformal Ward identity to hold. After imposing these conditions, we find \nE = P = -F = L x L y 15 -2 c 1 -26 c N 16 r 4 H (A.5) \nIn order to have a well-defined variational principle, we must ensure that δS = 0 onshell. We shall now determine the value of c 1 using this condition 9 . The variation of the action is \nδS = ∫ bulk EOM+ 1 2 ∫ bdy d 3 x √ γ [ T µ ν δγ ν µ + ( ( c N -1) e -2Φ n ν F νµ + ( c 3 + c 4 Φ+2 c 5 A 2 ) A µ ) δA µ + \nThe first term vanishes onshell. Therefore, the boundary terms must also vanish onshell. Let us assume, for convenience that Dirichlet boundary condition is imposed on the gauge field ( c N = 0). Prescribing boundary conditions is equivalent to prescribing the coefficient of the non-normalizable mode of the solution. The allowed variations at the boundary fall faster than the non-normalizable part of the solution, i.e. , \nc N A µ δ ( n ν e -2Φ F νµ ) -1 2 ( c 1 +2 c 2 Φ -c 4 A 2 -4 c N A µ n ν F νµ e -2Φ ) δ Φ ] (A.6) \nδγ µ ν = δγ µ ν (1) r 2 + δγ µ ν (2) r 4 + . . . δA µ = r -2 ( δA µ (1) r 2 + δA µ (2) r 4 + . . . ) δ Φ = r 2 δ Φ (1) + r 4 δ Φ (2) + . . . (A.7) \nSubstituting these expressions in (A.6) and using the conditions on c i for energy and pressure to be finite 10 , we find \nδS = ∫ d 3 x ( √ γT µ ν r 2 δγ ν µ (1) + O ( r 2 ) δA µ (1) + ( c 2 -c 1 r 2 H ) ( δ Φ 1 + O ( r 2 )) ) (A.8) \nSince E and P are finite, the first term in the integrand vanishes at the boundary. Hence, c 1 = c 2 = 7 / 2 for the variation of the action to vanish on-shell. Using the values of c i found above in (A.5) we get \nE = P = -F = L x L y 2 r 4 H \nAfter restoring factors of 8 πG , \nE = P = -F = L x L y 16 πGr 4 H = -1 2 T ∂ F ∂T = 1 2 T S \nWe have shown that the stress tensor and on-shell action can be regularized by making the action well-behaved, i.e. δS must vanish on-shell and the counterterms should not break any residual gauge symmetry.', 'References': "- [1] S. Sachdev and M. Mueller, 'Quantum Criticality and Black Holes,' arXiv:0810.3005 [cond-mat.str-el].\n- [2] S. A. 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2010PhRvD..81h4030J

Uncertainty relation on a world crystal and its applications to micro black holes

2010-01-01

['-', '-', '-', 'quantum theory', '-', '-', '-']

[]

We formulate generalized uncertainty relations in a crystal-like universe—a “world crystal”—whose lattice spacing is of the order of Planck length. In the particular case when energies lie near the border of the Brillouin zone, i.e., for Planckian energies, the uncertainty relation for position and momenta does not pose any lower bound on involved uncertainties. We apply our results to micro black holes physics, where we derive a new mass-temperature relation for Schwarzschild micro black holes. In contrast to standard results based on Heisenberg and stringy uncertainty relations, our mass-temperature formula predicts both a finite Hawking’s temperature and a zero rest-mass remnant at the end of the micro black hole evaporation. We also briefly mention some connections of the world-crystal paradigm with ’t Hooft’s quantization and double special relativity.

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https://arxiv.org/pdf/0912.2253.pdf

{'Uncertainty relation on world crystal and its applications to micro black holes': "Petr Jizba, 1, 2, ∗ Hagen Kleinert, 1, † and Fabio Scardigli 3, ‡ \n1 ITP, Freie Universitat Berlin, Arnimallee 14 D-14195 Berlin, Germany 2 FNSPE, Czech Technical University in Prague, B˘rehov'a 7, 115 19 Praha 1, Czech Republic 3 Leung Center for Cosmology and Particle Astrophysics (LeCosPA), Department of Physics, National Taiwan University, Taipei 106, Taiwan \nWe formulate generalized uncertainty relations in a crystal-like universe whose lattice spacing is of the order of Planck length - 'world crystal'. In the particular case when energies lie near the border of the Brillouin zone, i.e., for Planckian energies, the uncertainty relation for position and momenta does not pose any lower bound on involved uncertainties. We apply our results to micro black holes physics, where we derive a new mass-temperature relation for Schwarzschild micro black holes. In contrast to standard results based on Heisenberg and stringy uncertainty relations, our mass-temperature formula predicts both a finite Hawking's temperature and a zero rest-mass remnant at the end of the micro black hole evaporation. We also briefly mention some connections of the world crystal paradigm with 't Hooft's quantization and double special relativity. \nPACS numbers: 04.70.Dy, 03.65.-w", 'I. INTRODUCTION': "Recent advances in gravitational and quantum physics indicate that in order to reconcile the two fields with each other, a dramatic conceptual shift is required in our understanding of spacetime. In particular, the notion of spacetime as a continuum may need revision at scales where gravitational and electro-weak interactions become comparable in strength [1]. For this reason there has been a recent revival of interest in approximating the spacetime with discrete coarse-grained structures at small, typically Planckian, length scales. Such structures are inherent in many models of quantum-gravity, such as spacetime foam [2], loop quantum gravity [3, 4, 5], noncommutative geometry [6, 7, 8, 9], black-hole physics [10] or cosmic cellular automata [11, 12, 13, 14, 15]. \nDespite a vast gap between the Planck length ( /lscript p ≈ 1 . 6 · 10 -35 m) and smallest length scales that can be probed with particle accelerators ( ≈ 10 -18 m), the issue of Planckian physics might not be so speculative as it seems. In fact, probes such as Planck Surveyor [16] or the related IceCube [17]- which just started or are planned to start in the near future, are supposed to set various important limits on prospective models of the Planckian world. \nOne of the simplest toy-model systems for Planckian physics is undoubtedly a discrete lattice. Discrete lattices are routinely used, for instance, in computational quantum field theory [18, 19, 20], but with a few notable exceptions [21, 22, 23], they mainly serve as numerical regulators of ultraviolet divergences. Indeed, a major point of renormalized theories is precisely to extract lattice-independent data from numerical computa- \ntions. One may, however, investigate the consequences of taking the lattice no longer as a mere computational device, but as a bona-fide discrete network, whose links define the only possible propagation directions for signals carrying the interactions between fields sitting on the nodes of the network. \nRecently one of us proposed a model of a discrete, crystal-like universe - 'world crystal' [20, 23, 24]. There, the geometry of Einstein and Einstein-Cartan spaces can be considered as being a manifestation of the defect structure of a crystal whose lattice spacing is of the order of /lscript p . Curvature is due to rotational defects, torsion due to translational defects. The elastic deformations do not alter the defect structure, i.e., the geometry is invariant under elastic deformations. If one assumes these to be controlled by a second-gradient elastic action, the forces between local rotational defects, i.e., between curvature singularities, are the same as in Einstein's theory [25]. Moreover, the elastic fluctuations of the displacement fields possess logarithmic correlation functions at long distances, so that the memory of the crystalline structure is lost over large distances. In other words, the Bragg peaks of the world crystal are not δ -function-like, but display the typical behavior of a quasi-long-range order, similar to the order in a Kosterlitz-Thousless transition in two-dimensional superfluids [23]. \nThe purpose of this note is to study the generalized uncertainty principle (GUP) associated with the quantum physics on the world crystal and to derive physical consequences related to micro black hole physics. In view of the fact that micro black holes might be formed at energies as low as the TeV range [26, 27, 28] - which will be shortly available in particle accelerators such as the LHC, it is hoped that the presented results may be more than of a mere academic interest. \nThe structure of our paper is as follows: In Section II we present some fundamentals of a differential calculus on a lattice that will be needed in the text. In Section III we construct position and momentum operators on a 1D \nlattice and compute their commutator. We then demonstrate that the usual Weyl-Heisenberg algebra W 1 for ˆ p and ˆ x operators is on a 1D lattice deformed to the Euclidean algebra E (2). By identifying the measure of uncertainty with a standard deviation we derive the related GUP on a lattice. This is done in Section IV. There we focus on two critical regimes: long-wave regime and the regime where momenta are at the border of the first Brillouin zone. Interestingly enough, our GUP implies that quantum physics of the world-crystal universe becomes 'deterministic' for energies near the border of the Brillouin zone. In view of applications to micro black hole physics, we derive in Section V the energy-position GUP for a photon. Implications for micro black holes physics are discussed in Section VI. There we derive a mass-temperature relation for Schwarzschild micro black holes. On the phenomenological side, the latter provides a nice resolution of a long-standing puzzle: the final Hawking temperature of a decaying micro black hole remains finite , in contrast to the infinite temperature of the standard result where Heisenberg's uncertainty principle operates. Besides, the final mass of the evaporation process is zero, thus avoiding the problems caused by the existence of massive black hole remnants. Entropy and heat capacity are discussed in Section VII. Finally, in Section VIII we outline a connection of our results with 't Hooft's approach to deterministic quantum mechanics and with deformed (or double) special relativity. Section IX is devoted to concluding remarks. For completeness, we present in Appendix an alternative derivation of the micro black hole mass-temperature formula.", 'II. DIFFERENTIAL CALCULUS ON A LATTICE': 'In this section we quickly review some features of a differential calculus on a 1D lattice. An overview discussing more aspects of such a calculus can be found, e.g., in Refs. [18, 19, 20]. Independent and very elegant derivation of these can be also done in the framework of a non-commutative geometry [29, 30, 31]. \nOn a lattice of spacing /epsilon1 in one dimension, the lattice sites lie at x n = n/epsilon1 where n runs through all integer numbers. There are two fundamental derivatives of a function f ( x ): \n( ∇ f )( x ) = 1 /epsilon1 [ f ( x + /epsilon1 ) -f ( x )] , ( ¯ ∇ f )( x ) = 1 /epsilon1 [ f ( x ) -f ( x -/epsilon1 )] . (1) \nThey obey the generalized Leibnitz rule \n( ∇ fg )( x ) = ( ∇ f )( x ) g ( x ) + f ( x + /epsilon1 )( g )( x ) , ( ¯ ∇ fg )( x ) ¯ f )( x ) g ( x ) + f ( x /epsilon1 ¯ g )( x ) . (2) \n∇ )( ∇ \n= ( ∇ \n- \nOn a lattice, integration is performed as a summation: \n∫ dxf ( x ) ≡ /epsilon1 ∑ x f ( x ) , (3) \nwhere x runs over all x n . \nFor periodic functions on the lattice or for functions vanishing at the boundary of the world crystal, the lattice derivatives can be subjected to the lattice version of integration by parts: \n∑ x f ( x ) ∇ g ( x ) = -x g ( x ) ¯ ∇ f ( x ) , (4) \n∑ ∑ x f ( x ) ¯ ∇ g ( x ) = -∑ x g ( x ) ∇ f ( x ) . (5) \n∑ \nOne can also define the lattice Laplacian as \n∇ ¯ ∇ f ( x ) = ¯ ∇∇ f ( x ) = 1 /epsilon1 2 [ f ( x + /epsilon1 ) -2 f ( x )+ f ( x -/epsilon1 )] , (6) \nwhich reduces in the continuum limit to an ordinary Laplace operator ∂ 2 x . Note that the lattice Laplacian can also be expressed in terms of the difference of the two lattice derivatives: \n∇ ¯ ∇ f ( x ) = 1 /epsilon1 [ ∇ f ( x ) -¯ ∇ f ( x ) ] . (7) \nThe above calculus can be easily extended to any number D of dimensions [18, 19, 23].', 'III. POSITION AND MOMENTUM OPERATORS ON A LATTICE': "Consider now the quantum mechanics (QM) on a 1D lattice in a Schrodinger-like picture. Wave function are square-integrable complex functions on the lattice, where 'integration' means here summation, and scalar products are defined by \n〈 f | g 〉 = /epsilon1 ∑ x f ∗ ( x ) g ( x ) . (8) \nIt follows from Eq. (4) that \n〈 f |∇ g 〉 = -〈 ¯ ∇ f | g 〉 , (9) \nso that ( i ∇ ) † = i ¯ ∇ , and neither i ∇ nor i ¯ ∇ are hermitian operators. The lattice Laplacian (6), however, is hermitian. \nThe position operator ˆ X /epsilon1 acting on wave functions of x is defined by a simple multiplication with x : \n( ˆ X /epsilon1 f )( x ) = xf ( x ) . (10) \nSimilarly we can define the lattice momentum operator ˆ P /epsilon1 . In order to ensure hermiticity we should relate it to the symmetric lattice derivative [19, 30, 32]. Using (9) we have \n( ˆ P /epsilon1 f )( x ) = /planckover2pi1 2 i [( ∇ f )( x ) + ( ¯ ∇ f )( x )] = /planckover2pi1 2 i/epsilon1 [ f ( x + /epsilon1 ) -f ( x -/epsilon1 )] . (11) \nFor small /epsilon1 , this reduces to the ordinary momentum operator ˆ p ≡ -i /planckover2pi1 ∂ x , or more precisely \nˆ P /epsilon1 = ˆ p + O ( /epsilon1 2 ) . (12) \nThe 'canonical' commutator between ˆ X /epsilon1 and ˆ P /epsilon1 on the lattice reads \n( [ ˆ X /epsilon1 , ˆ P /epsilon1 ] f ) ( x ) = i /planckover2pi1 2 [ f ( x + /epsilon1 ) + f ( x -/epsilon1 )] ≡ i /planckover2pi1 ( ˆ I /epsilon1 f )( x ) . (13) \nThe last line defines a lattice-version of the unit operator as the average over the two neighboring sites. Note that all three operators ˆ X /epsilon1 , ˆ P /epsilon1 , and ˆ I /epsilon1 are hermitian under the scalar product (8). \nIt was noted in [32] that the operators ˆ X /epsilon1 , ˆ P /epsilon1 and ˆ I /epsilon1 generate the Euclidean algebra E (2) in 2D. Indeed, setting ˆ M = /epsilon1 ˆ X /epsilon1 , ˆ P 1 = /epsilon1 ˆ P /epsilon1 / /planckover2pi1 and ˆ P 2 = ˆ I /epsilon1 we obtain \n[ ˆ M, ˆ P 1 ] = i ˆ P 2 , [ ˆ M, ˆ P 2 ] = -i ˆ P 1 , [ ˆ P 1 , ˆ P 2 ] = 0 . \nThe generator ˆ M corresponds to a rotation, while ˆ P 1 and ˆ P 2 represent two translations. In the limit /epsilon1 → 0, the Lie algebra of E (2) contracts to the standard WeylHeisenberg algebra W 1 : ˆ X /epsilon1 → ˆ x , ˆ P /epsilon1 → ˆ p , ˆ I /epsilon1 → ˆ 1 1. Thus ordinary QM is obtained from lattice QM by a contraction of the E (2) algebra, with the lattice spacing /epsilon1 playing the role of the deformation parameter . \nAll functions on the lattice can be Fourier-decomposed with wave numbers in the Brillouin zone: \nf ( x ) = ∫ π//epsilon1 -π//epsilon1 dk 2 π ˜ f ( k ) e ikx , (14) \nwith the coefficients \n˜ f ( k ) = /epsilon1 ∑ x f ( x ) e -ikx . (15) \nThis implies the good-old de Broglie relation \n(ˆ p ˜ f )( k ) = /planckover2pi1 k ˜ f ( k ) , (16) \nand its lattice version \n( -i ∇ ˜ f )( k ) = K ˜ f ( k ) , ( -i ¯ ∇ ˜ f )( k ) = ¯ K ˜ f ( k ) , (17) \nwith the eigenvalues \nK ≡ ( e ik/epsilon1 -1) /i/epsilon1 = ¯ K ∗ . (18) \nFrom (17) we find the Fourier transforms of the operators ˆ X /epsilon1 , ˆ P /epsilon1 , ˆ I /epsilon1 : \n( ˆ X /epsilon1 ˜ f )( k ) = i d dk ˜ f ( k ) , (19) \n( ˆ P /epsilon1 ˜ f )( k ) = /planckover2pi1 /epsilon1 sin( k/epsilon1 ) ˜ f ( k ) , (20) \n( ˆ I /epsilon1 ˜ f )( k ) = cos( k/epsilon1 ) ˜ f ( k ) . (21) \nWith the help of (21) we can rewrite the commutation relation (13) equivalently as \n( [ ˆ X /epsilon1 , ˆ P /epsilon1 ] f ) ( x ) = i /planckover2pi1 cos ( /epsilon1 ˆ p/ /planckover2pi1 ) f ( x ) . (22) \nThe latter allows to identify the lattice unit operator ˆ I /epsilon1 with cos ( /epsilon1 ˆ p/ /planckover2pi1 ). Indeed, ˆ I /epsilon1 = ˆ 1 1 on all lattice nodes.", 'IV. UNCERTAINTY RELATIONS ON LATTICE': "We are now prepared to derive the generalized uncertainty relation implied by the previous commutators. We shall define the uncertainty of an observable A in a state ψ by the standard deviation \n(∆ A ) ψ ≡ √ 〈 ψ | ( ˆ A -〈 ψ | ˆ A | ψ 〉 ) 2 | ψ 〉 . (23) \nFollowing the conventional Robertson-Schrodinger procedure (see, e.g., Ref. [33, 34, 35]), we derive on the spacetime lattice the inequality \n(∆ X /epsilon1 ) ψ (∆ P /epsilon1 ) ψ ≥ 1 2 ∣ ∣ ∣ 〈 ψ | [ ˆ X /epsilon1 , ˆ P /epsilon1 ] | ψ 〉 ∣ ∣ ∣ = /planckover2pi1 2 ∣ ∣ ∣ 〈 ψ | ˆ I /epsilon1 | ψ 〉 ∣ ∣ ∣ = /planckover2pi1 2 |〈 ψ | cos ( /epsilon1 ˆ p/ /planckover2pi1 ) | ψ 〉| . (24) \nFor brevity we will omit in the following the subscript ψ in (∆ A ) ψ and set ψ | · · · | ψ ψ . \n〈 \n| · · · | Let us now study two critical regimes of the GUP (24): the first is the long-wavelengths regime where 〈 ˆ p 〉 ψ → 0; the second regime is near the boundary of the Brillouin zone where 〈 ˆ p 〉 ψ → π /planckover2pi1 / 2 /epsilon1 . To this end we first rewrite 〈 cos ( /epsilon1 ˆ p/ /planckover2pi1 ) 〉 ψ as \n〉 ≡ 〈· · · 〉 \n〈 cos ( /epsilon1 ˆ p/ /planckover2pi1 ) 〉 ψ = ∞ ∑ n =0 ∫ ∞ 0 dp /rho1 ( p ) ( -1) n ( /epsilon1p/ /planckover2pi1 ) 2 n (2 n )! , (25) \nwhere /rho1 ( p ) ≡ | ψ ( p ) 2 . \n≡ | In the first case, /rho1 ( p ) is peaked around p /similarequal 0, so that the relation (25) becomes approximately \n| \n〈 cos ( /epsilon1 ˆ p/ /planckover2pi1 ) 〉 ψ = 1 -/epsilon1 2 p 2 2 /planckover2pi1 2 + O ( p 4 ) , (26) \nwhere p 2 ≡ 〈 ˆ p 2 〉 ψ . We should stress that expansion (26) is not an expansion in /epsilon1 but rather in /epsilon1p/ /planckover2pi1 . So if we speak of /rho1 ( p ) as being peaked around p /similarequal 0 we mean that p /lessmuch /planckover2pi1 //epsilon1 . \n/lessmuch Applying now the identity \n〈 ˆ A 2 〉 ψ = (∆ A ) 2 + 〈 ˆ A 〉 2 ψ , (27) \nwe obtain from (24) \n∆ X /epsilon1 ∆ P /epsilon1 /greaterorsimilar /planckover2pi1 2 ∣ ∣ ∣ ∣ 1 -/epsilon1 2 p 2 2 /planckover2pi1 2 ∣ ∣ ∣ ∣ = /planckover2pi1 2 ∣ ∣ ∣ ∣ 1 -/epsilon1 2 2 /planckover2pi1 2 [ (∆ p ) 2 + 〈 ˆ p 〉 2 ψ ] ∣ ∣ ∣ ∣ . (28) \nFor mirror-symmetric states where 〈 ˆ p 〉 ψ = 0 this implies \n∆ X /epsilon1 ∆ P /epsilon1 /greaterorsimilar /planckover2pi1 2 ( 1 -/epsilon1 2 2 /planckover2pi1 2 (∆ p ) 2 ) . (29) \nHere we have substituted | ... | by ( ... ) since we assume that /epsilon1 /similarequal /lscript p (Planckian lattice) and that ∆ p is close to zero (this is our original assumption). Therefore /epsilon1 2 (∆ p ) 2 / 2 /planckover2pi1 2 1. \nFor Planckian lattices with the relation (12), we can neglect higher powers of /epsilon1 in (29) and write \n/lessmuch \n∆ X /epsilon1 ∆ P /epsilon1 /greaterorsimilar /planckover2pi1 2 ( 1 -/epsilon1 2 2 /planckover2pi1 2 (∆ P /epsilon1 ) 2 ) . (30) \nIn the second case, where 〈 ˆ p 〉 ψ → /planckover2pi1 π/ 2 /epsilon1 , i.e. near the border of the Brillouin zone, we use the expansion: \n〈 cos[ π/ 2 + ( /epsilon1 ˆ p/ /planckover2pi1 -π/ 2)] 〉 ψ = 〈 sin( π/ 2 -/epsilon1 ˆ p/ /planckover2pi1 ) 〉 ψ = ∞ ∑ n =0 ∫ ∞ 0 dp /rho1 ( p ) ( -1) n ( π/ 2 -/epsilon1p/ /planckover2pi1 ) 2 n +1 (2 n +1)! . (31) \nUnder the assumption that /rho1 ( p ) is peaked near the border of the Brillouin zone, the first term in the expansion is dominant, and the uncertainty relation reduces to \n∆ X /epsilon1 ∆ P /epsilon1 ≥ /planckover2pi1 2 ∣ ∣ π 2 -/epsilon1 /planckover2pi1 〈 ˆ p 〉 ψ ∣ ∣ . (32) \n∣ ∣ Since k = p/ /planckover2pi1 lies always inside the Brillouin zone, we have 〈 ˆ p 〉 ψ ≤ π /planckover2pi1 / 2 /epsilon1 and can therefore in (32) substitute | ... | by ( ... ). Finally, using again (12), we can write for the GUP close to the boundary of the Brillouin zone \n∣ \n∣ \n∆ X /epsilon1 ∆ P /epsilon1 /greaterorsimilar /planckover2pi1 2 ( π 2 -/epsilon1 /planckover2pi1 〈 ˆ P /epsilon1 〉 ψ ) . (33) \nAs the momentum reaches the boundary of the Brillouin zone, the right-hand sides of (32)-(33) vanish, so that lattice quantum mechanics at short wavelengths is permitted to exhibit classical behavior - no irreducible lower bound for uncertainties of two complementary observables appears! \nIt is worth noting that the uncertainty relation (33) leads to the same physical conclusions as those found, on a different ground, by Magueijo and Smolin in Ref. [36]. In particular, the world-crystal universe can become 'deterministic' for energies near the border of the Brillouin zone, i.e., for Planckian energies. \nLet us remark that the scenario in which the universe at Planckian energies is deterministic rather than being dominated by tumultuous quantum fluctuations is a recurrent theme in 't Hooft's 'deterministic' quantum mechanics [37, 38, 39, 40, 41, 42, 43]. \nIt is straightforward to generalize the above formulas to higher dimensions. In this context, a useful inequality is \n∆ X i /epsilon1 ∆ | P /epsilon1 | ≥ /planckover2pi1 2 |〈 ψ | ( ˆ P i /epsilon1 / | ˆ P /epsilon1 | ) cos ( /epsilon1 i ˆ p i / /planckover2pi1 ) | ψ 〉| = /planckover2pi1 2 |〈 ψ | ε (ˆ p i ) cos ( /epsilon1 i ˆ p i / /planckover2pi1 ) | ψ 〉| , (34) \nwhich will be needed in the following. Here ε ( . . . ) is the sign function, and \n| ˆ P /epsilon1 | = /planckover2pi1 √ √ √ √ D ∑ j =1 [ sin( /epsilon1 j ˆ p j / /planckover2pi1 ) /epsilon1 j ] 2 . (35) \nInequality (34) should be contrasted with inequality (24) where the momentum is without an absolute value. \nIn a particular case when states ψ are a combination of only positive or only negative momentum eigenstates (e.g., incident or reflected particle states) we can simply write \n∆ X i /epsilon1 ∆ | P /epsilon1 | ≥ /planckover2pi1 2 |〈 ψ | cos ( /epsilon1 i ˆ p i / /planckover2pi1 ) | ψ 〉| = /planckover2pi1 2 [ 1 -2 〈 sin 2 ( /epsilon1 i ˆ p i / 2 /planckover2pi1 )〉 ψ ] . (36)", 'V. IMPLICATIONS FOR PHOTONS': "We may now use the inequality (36) to derive the GUP for photons. \nThe vector potential of a photon in the Lorentz gauge in 1 + 1 dimensions satisfies the wave equation \n1 c 2 ∂ 2 t A µ ( x, t ) = ∂ 2 x A µ ( x, t ) . (37) \nA plane wave solution A µ ( x ) = /epsilon1 µ exp[ i ( kx -ω ( k ) t )] possesses the well-known linear dispersion relation \nω ( k ) = c | k | , (38) \nwith /epsilon1 µ being a polarization vector. On a onedimensional lattice, the operator ∂ 2 x is replaced by the lattice Laplacian ¯ ∇∇ , and the spectrum becomes, on account of Eq. (6) and (17), \nω ( k ) c = √ K ¯ K = √ 2 [1 -cos( k/epsilon1 )] /epsilon1 = 2 /epsilon1 ∣ ∣ ∣ sin ( k/epsilon1 2 )∣ ∣ ∣ , (39) \n∣ ∣ which reduces to (38) for /epsilon1 → 0. Denoting the energy on the lattice /planckover2pi1 ω by E /epsilon1 , we obtain the dispersion relation \n∣ \n∣ \nE /epsilon1 /planckover2pi1 c = 2 /epsilon1 ∣ ∣ sin ( p /epsilon1 2 /planckover2pi1 )∣ ∣ . (40) \n∣ ∣ We can also define the associated energy operator ˆ E c by replacing p by ˆ p . \n∣ \n∣ \nFor states ψ with /rho1 ( p ) sharply peaked around small p , we can use a spectral expansion analog of (25) to obtain \n∆ E /epsilon1 /similarequal c ∆ | p | /similarequal c ∆ | P /epsilon1 | . (41) \nHere we have neglected higher powers of momentum and used the fact that we deal with a Planckian lattice. In deriving we have also applied the cumulant expansion: \n〈 ˆ E /epsilon1 〉 ψ = (2 /planckover2pi1 c//epsilon1 ) 〈| sin (ˆ p /epsilon1/ 2 /planckover2pi1 ) |〉 ψ = ∣ ∣ ∣ ∣ c p -c /epsilon1 2 p 3 24 /planckover2pi1 2 + O ( p 5 ) ∣ ∣ ∣ ∣ . (42) \nWith the help of (36), (40), and (41) we can write in the long-wavelength regime \n∆ X /epsilon1 ∆ E /epsilon1 ≥ /planckover2pi1 c 2 [ 1 -/epsilon1 2 2 /planckover2pi1 2 c 2 〈 E 2 /epsilon1 〉 ψ ] . (43) \nHere 〈 E 2 /epsilon1 〉 ψ is the average quadrat of the photon energy, and thus the square root of it can be formally identified with the energy change in the detector, i.e. ∆ E /epsilon1 . From this follows that if the uncertainty of a photon position in a state ψ is ∆ X /epsilon1 , then the energy of a detector changes at least by amount \n∆ E /epsilon1 /similarequal /planckover2pi1 c 2 [ 1 -/epsilon1 2 2 /planckover2pi1 2 c 2 〈 E 2 /epsilon1 〉 ψ ] 1 ∆ X /epsilon1 , (44) \nper particle. Remembering the Einstein relation ∆ E = 2 π /planckover2pi1 c/λ , we can interpret 4 π ∆ X /epsilon1 as being a lattice equivalent of photon's wavelength λ . \nIt is interesting to observe that in the short-wavelength case we can deduce from the exact GUP \n∆ X /epsilon1 ∆ | P /epsilon1 | ≥ /planckover2pi1 2 [ 1 -/epsilon1 2 2 /planckover2pi1 2 c 2 〈 E 2 /epsilon1 〉 ψ ] , (45) \nthat near the border of the Brillouin zone ∆ X /epsilon1 takes the approximate form (cf. Eq. (40)) \n∆ X /epsilon1 /similarequal /epsilon1 π [ 1 -/epsilon1 2 2 /planckover2pi1 2 c 2 〈 E 2 /epsilon1 〉 ψ ] /similarequal /epsilon1 π [ π 2 -/epsilon1 /planckover2pi1 〈 ˆ P /epsilon1 〉 ψ ] . (46) \nIn the derivation we have used that fact that \n∆ | P /epsilon1 | ≤ √ 〈 ˆ P 2 /epsilon1 〉 /similarequal π /planckover2pi1 2 /epsilon1 . (47) \nRelation (46) represents the smallest attainable positional uncertainty near the border of the Brillouin zone. It will be useful in the following two sections.", 'VI. APPLICATIONS TO MICRO BLACK HOLES': "An interesting playground where one can apply the above lattice GUP's is the hypothetical physics of micro black holes. Their mass-temperature relation depends sensitively on the actual form of the energy-position uncertainty relation. From this one can deduce non-trivial phenomenological consequences. The passage from the energy-position uncertainty relation to the micro black hole mass-temperature relation has been intensively studied in recent years. For definiteness we shall follow here the treatment of Refs. [44, 45, 46, 47, 48, 49, 50, 51]. An alternative derivation based on the so-called Landauer principle will be presented in Appendix. \nWe start with an assumption that the lattice spacing is roughly of order Planck length, i.e., /epsilon1 = a/lscript p , where a > 0 is of order of unity. Let us now imagine that we \nhave found a black hole on the lattice as a discretized version of a Schwarzschild solution. It is a pile up of disclinations. If the Schwarzschild radius is much larger than the lattice spacing /epsilon1 , this will not look much different from the well-known continuum solution. We must avoid too small black holes, for otherwise, completely new physics will set in near the center, due to the high concentration of defects. These will cause the 'melting' of the world crystal at a critical defect density [53], and the emerging trans-horizon general relativity would look completely different from Einstein's theory. \nFollowing the classical argument of the Heisenberg microscope [54], we know that the smallest resolvable detail δx of an object goes roughly as the wavelength of the employed photons. If E is the (average) energy of the photons used in the microscope, then \nδx /similarequal /planckover2pi1 c 2 E . (48) \nConversely, with the relation (48) one can compute the energy E of a photon with a given (average) wavelength λ /similarequal δx . As a consequence of Eq. (43), we can write the lattice version of this standard Heisenberg formula as \nδX /epsilon1 /similarequal /planckover2pi1 c 2 E /epsilon1 [ 1 -/epsilon1 2 2 /planckover2pi1 2 c 2 ( E /epsilon1 ) 2 ] , (49) \nwhich links the (average) wavelength of a photon to its energy E /epsilon1 . Since the lattice spacing is /epsilon1 = a/lscript p and the Planck energy E p = /planckover2pi1 c/ 2 /lscript p , Eq. (49) can be rewritten as \nδX /epsilon1 /similarequal /planckover2pi1 c 2 E /epsilon1 -a 2 /lscript p E /epsilon1 8 E p . (50) \nLet us now loosely follow the argument of Refs. [44, 45, 46, 47, 48, 49, 50, 51] and consider an ensemble of unpolarized photons of Hawking radiation just outside the event horizon. From a geometrical point of view, it's easy to see that the position uncertainty of such photons is of the order of the Schwarzschild radius R S of the hole. An equivalent argument comes from considering the average wavelength of the Hawking radiation, which is of the order of the geometrical size of the hole (see e.g. Ref. [49], chapter 5). By recalling that R S = /lscript p m , where m = M/M p is the black hole mass in Planck units ( M p = E p /c 2 ), we can estimate the photon positional uncertainty as \nδX /epsilon1 /similarequal 2 µR S = 2 µ/lscript p m. (51) \nThe proportionality constant µ is of order unity and will be fixed shortly. According to the above arguments, m must be assumed to be much larger than unity, in order to avoid the melting transition. With (51) we can rephrase Eq. (50) as \n2 µm /similarequal E p E /epsilon1 -a 2 8 E /epsilon1 E p . (52) \nAccording to the equipartition principle the average energy E /epsilon1 of unpolarized photons of the Hawking radiation is linked with their temperature T as \nE /epsilon1 = k B T . (53) \nIn order to fix µ , we go to the continuum lattice limit /epsilon1 → 0 ( a → 0), and require that formula (52) predicts the standard semiclassical Hawking temperature: \nT H = /planckover2pi1 c 3 8 πGk B M = /planckover2pi1 c 4 πk B R S . (54) \nThis fixes µ = π . \nDefining the Planck temperature T p so that E p = k B T p / 2 and measuring all temperatures in Planck units as Θ = T/T p , we can finally cast formula (52) in the form \n2 m = 1 2 π Θ -ζ 2 2 π Θ , (55) \nwhere we have defined the deformation parameter ζ = a/ (2 π √ 2). \nAs already mentioned, in the continuum limit both /epsilon1 and a tend to zero and (50) reduces to the ordinary Heisenberg uncertainty principle. In this case Eq. (55) boils down to \nm = 1 4 π Θ . (56) \nThis is the dimensionless version of Hawking's formula (54) for large black holes. \nHistorically, the validity of (54) was also postulated for micro black holes on the assumption that the black hole thermodynamics is universally valid for any black hole, be it formed via star collapse, or primordially via quantum fluctuations. Such an assumption is by no means warranted without some further input about mesoscopic and/or microscopic energy scales (much like in ordinary thermodynamics) and, in fact, we have seen that corrections should be expected at short world-crystal scales. \nIt is instructive to compare our mass-temperature relation (55) with the one suggested by the so-called stringy uncertainty relation [55, 56]. There the sign of the correction term in (55) is positive: \n2 m = 1 2 π Θ + ζ 2 2 π Θ . (57) \nThe phenomenological consequences of the relation (55) are quite different from those of the stringy result (57). In Fig. 1 we compare the two results, and add also the curve for the ordinary Hawking relation (56). Considering m and Θ as functions of time, we can follow the evolution of a micro black hole from the curves in Fig. 1. For the stringy GUP, the blue line predicts a maximum temperature \nΘ max = 1 2 πζ , (58a) \nFIG. 1: Diagrams for the three mass-temperature relations, ours (red), Hawking's (green), and stringy GUP result (blue), with ζ = √ 2, as an example. As a consequence of the lattice uncertainty principle the evaporation ends at a finite temperature with a zero rest-mass remnant. \n<!-- image --> \nand a minimum rest mass \nm min = ζ . (58b) \nThe end of the evaporation process is reached in a finite time, the final temperature is finite, and there is a remnant of a finite rest mass (cf. Refs. [44, 45, 46, 47, 48, 50, 51, 52]). \nFrom the standard Heisenberg uncertainty principle we find the green curve, representing the usual Hawking formula. Here the evaporation process ends, after a finite time, with a zero mass and a worrisome infinite temperature. In the literature, the undesired infinite final temperature predicted by Hawking's formula has so far been cured only with the help of the stringy GUP, which brings the final temperature to a finite value. This result is, however, also questionable since it implies the existence of finite-mass remnants in the universe. Though by some authors such remnants are greeted as relevant candidates for dark matter [57], others point out that their existence would create further complications such as the entropy/information problem [58], detectability issue, or their (excessive) production in the early universe [27, 59]. \nIn contrast to these results, our lattice GUP predicts the red curve. This yields a finite end temperature \nΘ max = 1 2 πζ , (59) \nwith a zero-mass remnant. The mass-temperature formula (55) thus solves at once several problems by predicting the end of the evaporation process at a finite final temperature with zero -mass remnants. \nIs should be stressed that since the photon GUP (43) and (49) holds only for states ψ where 〈 ˆ p 2 〉 ψ /lessmuch /planckover2pi1 2 //epsilon1 2 , our reasonings are warranted only for \nE /epsilon1 /lessmuch /planckover2pi1 c//epsilon1 /similarequal E p /a ⇒ 2 πζ /lessmuch 1 Θ . (60) \nThis implies, in particular, that when Θ is close to Θ max our long-wavelength approximation cannot be trusted. \nTo understand the behavior of the system close to Θ max we must turn to the short-wavelength limit, Eqs. (33) and (46). In this regime the momenta lie close to the border of the Brillouin zone 〈 ˆ p 〉 ψ /similarequal 〈 ˆ P /epsilon1 〉 ψ /similarequal π /planckover2pi1 / (2 /epsilon1 ), and Eq. (40) implies \nE /epsilon1 /similarequal √ 2 /epsilon1 /planckover2pi1 c , (61) \nwhich for the Planckian lattice, where /epsilon1 = a/lscript p , gives E /epsilon1 /similarequal E p / ( π ζ ). Considering again the uncertainty in the photon position as δX /epsilon1 /similarequal 2 π/lscript p m (cf. Eq. (51)), GUP (46) then predicts \nδX /epsilon1 ∝ m /similarequal a 2 π 2 ( 1 -a 2 /lscript 2 p 2 /planckover2pi1 2 c 2 E 2 p π 2 ζ 2 ) = 0 . (62) \nWe can thus conclude that the mass of the micro black hole must go to zero. This is also consistent with our previous long-wavelength considerations. The micro black hole therefore evaporates completely, without leaving remnants.", 'VII. ENTROPY AND HEAT CAPACITY': 'In this section we exhibit the modified thermodynamic entropy and heat capacity of a black hole implied by the new mass-temperature formula (55).', 'A. Entropy': "From the first law of black hole thermodynamics [60] we know that the differential of the thermodynamical entropy of a Schwarzschild black hole reads \ndS = dE T H , (63) \nwhere dE is the amount of energy swallowed by a black hole with Hawking temperature T H . In Eq. (63) the increase in the internal energy is equal to the added heat because a black hole makes no mechanical work when its entropy/surface changes (expanding surface does not exert any pressure). \nRewriting Eq. (63) with the dimensionless variables m and Θ we get \ndS = k B 2 dm Θ . (64) \nInserting here formula (55) we find \ndS = k B 2 dm Θ = -k B 4 ( 1 2 π Θ 3 + 2 πζ 2 Θ ) d Θ . (65) \nBy integrating dS we obtain S = S (Θ). Just as formula (55), the relation (65) can be trusted only for Θ /lessmuch Θ max = 1 / 2 πζ . Thus, when integrating (65), we should do this only up to a cutoff ˜ Θ max /lessmuch Θ max . The additive constant in S can be then be fixed by requiring that S = 0 when Θ → ˜ Θ max . This is equivalent to what is usually done when calculating the Hawking temperature for a Schwarzschild black hole. There one fixes the additive constant in the entropy integral to be zero for m = 0, so that S ( m = 0) = S (Θ → ∞ ) = 0 (the minimum mass attainable in the standard Hawking effect is m = 0). Thus we obtain \nS = k B 4 ∫ ˜ Θ max Θ ( 1 2 π Θ ' 3 + 2 πζ 2 Θ ' ) d Θ ' (66) \nwhere the sign was chosen in order to have a positive entropy. \nThe integral (66) yields \nS (Θ) = k B 16 π ( 1 Θ 2 -1 ˜ Θ 2 max +8 π 2 ζ 2 log ˜ Θ max Θ ) . (67) \nThe entropy is always positive, and S → 0 for Θ → ˜ Θ max .", 'B. Heat Capacity': 'With entropy formulae (65) and (67) at hand we can now compute the heat capacity of a (micro) black hole in the world-crystal. This will give us important insights on the final stage of the evaporation process. Again, we shall obtain formulae valid only for Θ Θ max = 1 / (2 πζ ). \nThe heat capacity C of a black hole is defined via the relation \n/lessmuch \ndQ = dE = CdT . (68) \nThe pressure exerted on the environment by the expanding black hole surface is zero. Hence we do not need to specify which C is meant. \nWith the help of (63) and (68) we obtain \nC = T ( dS dT ) = Θ ( dS d Θ ) , (69) \nwhich yields \nC = -πk B 2 [ ζ 2 + 1 (2 π Θ) 2 ] . (70) \nFrom this clearly follows that C is always negative. \nMost condensed-matter systems have C > 0. However, because of instabilities induced by gravity this is generally not the case in astrophysics [61, 62], especially in black hole physics. A Schwarzschild black hole has C < 0 which indicates that the black hole becomes hotter by radiating. The result (70) implies that this scenario holds also for micro black holes in the world-crystal. \nIn case of stringy GUP, we have to use Eq. (57) as the mass-temperature formula. The expression for the heat capacity then reads \nC = πk B 2 [ ζ 2 -1 (2 π Θ) 2 ] . (71) \nSince also here 0 < Θ < Θ max = 1 / 2 πζ , black holes have negative specific heat also according to the stringy GUP. However, stringy GUP displays a striking difference with respect to lattice GUP. In fact, since in principle we can trust Eq. (57) also when Θ /similarequal Θ max = 1 / 2 πζ , then from (71) we have, in such limit, C = 0. This means that for the stringy GUP the specific heat vanishes at the end point of the evaporation process in a finite time, so that the black hole at the end of its evolution cannot exchange energy with the surrounding space. In other words, the black hole stops to interact thermodynamically with the environment. The final stage of the Hawking evaporation, according to the stringy GUP scenario, contains a Planck-size remnant with a maximal temperature Θ = Θ max , but thermodynamically inert. The remnant behaves like an elementary particle - there are no internal degrees of freedom to excite in order to produce a heat absorption or emission. \nTo understand the heat exchange in the last live stage of the world-crystal black hole we cannot use the longwavelength formula (70). Instead, we must turn to Eq. (46). Since in our scenario the micro black hole disappears at the critical temperature Θ max , it is more appropriate to write the mass-temperature formula (46) in the form \nm /similarequal θ (Θ max -Θ) √ 2 π ζ ( 1 -a 2 8 Θ 2 ) , (72) \nwhere θ ( t ) is the Heaviside step function. For the specific heat this implies \nC /similarequal -θ (Θ max -Θ) √ 2 π k B ζ 3 Θ . (73) \nSo, in contrast to the stringy result, a world-crystal black hole exchanges heat with its environment by radiation until the last moment of its existence, and unlike the Schwarzschild black hole, the heat exchange with the environment increases in the final stage of its evaporation. In addition, because of the θ -function in (73), the transition from the universe with the world-crystal black hole to the one without it is of first order.', 'VIII. FURTHER APPLICATIONS': 'So far we have studied the consequence of the GUP on the micro black holes. Let us briefly mention two further applications.', "A. 't Hooft's proposal": "The first application relates to 't Hooft's proposal which purports to justify that our quantum world is merely a low-energy limit of a deterministic system operating at a deeper, perhaps Planckian, level of dynamics [37, 38]. As a deterministic substrate 't Hooft has proposed various cellular automata (CA) models. \nIn general, a CA is an array of cells forming a discrete lattice. All cells are typically equivalent and can take one of a finite number of possible discrete states . Like space, time is discrete as well. At each time step every cell updates its state according to a transition rule which takes into account the previous states of cells in the neighborhood, including its own state. In this sense the evolution is deterministic. \nOne of the simplest CA considered by 't Hooft is the 1dimensional periodic CA with 4-state cells, and with the nearest neighbor (N-N) transition rule, see Fig. 2. This \nFIG. 2: CA with discrete time evolution described by Eq. (74) and with the periodicity condition σ i = σ i +4 . The right-hand shows an equivalent graphical representation. \n<!-- image --> \n'clock like' CA can be generally described with 2 N +1 cells σ i ( i = -N,.. . , N ) each with two possible states { 0 , 1 } (cell is, e.g., white or black). The discrete time evolution with the elementary time step δt is described by the N-N transition rule \n( σ i -1 , σ i , σ i +1 ) → σ ' i = σ i ( t + δt ) : (0 , 0 , 0) → 0 , (0 , 0 , 1) → 0 , (0 , 1 , 0) → 0 , (1 , 0 , 0) → 1 , (1 , 1 , 0) → 0 , (0 , 1 , 1) → 0 , (1 , 0 , 1) → 0 , (1 , 1 , 1) → 0 , (74) \nwith the Born-von Karman periodicity condition σ i = σ i +2 N +1 . \nThe cells can be algebraically represented by orthonormal vectors \nσ -N = 0 0 . . . 1 ; σ -N +1 = 1 0 . . . 0 ; . . . σ N = 0 . . . 1 0 , \nσ -N = σ N +1 . On the basis spanned by σ i the \nelementary-time step evolution operator is: \nˆ U ( δt = τ ) = e -i ˆ Hδt = e -i π 2 N +1 0 1 1 0 . . . . . . 1 0 , (75) \nwhich, among others, defines the Hamiltonian ˆ H . The pre-factor e -i π 2 N +1 is known as 't Hooft's phase convention. Because ˆ U 2 N +1 = -ˆ 1 1 one can diagonalize ˆ U as \nˆ U diag = e -i π 2 N +1 diag ( e -i 2 πN 2 N +1 , . . . , 1 , . . . , e i 2 πN 2 N +1 ) . \nIf we denote the eigenstates of ˆ H as | n 〉 , we find that \n| n 〉 = 1 √ 2 N +1 N ∑ /lscript = -N exp [ -i 2 πn 2 N +1 /lscript ] σ /lscript , (76) \nwith n = -N,.. . , N and the ensuing 'energy' spectrum reads \nˆ H | n 〉 = ω N ( n + 1 2 ) | n 〉 , ω N ≡ 2 π (2 N +1) δt . (77) \nThe energy values E n resemble the spectrum of the harmonic oscillator, except that the n 's are bounded and can attain negative values. We shall be coming back to this issue shortly. \nPosition and momentum can be represented by operators with the matrix representations \n· · · \n , \nwhere /epsilon1 = 2 π/ (2 N +1). With these we obtain the commutator \nˆ X /epsilon1 = ( -N +1) /epsilon1 0 0 · · · 0 0 ( -N +2) /epsilon1 0 0 . . . . . . . . . 0 · · · N/epsilon1 0 0 · · · 0 -N/epsilon1 , ˆ P /epsilon1 = 2 0 1 0 0 · · · -1 1 0 1 0 · · · 0 0 -1 0 1 · · · 0 . . . . . . . . . 0 0 · · · 0 1 1 0 -1 0 (78) \n1 i/epsilon1 -. . 1 \n[ ˆ X /epsilon1 , ˆ P /epsilon1 ] = i 2 0 1 0 0 · · · 1 1 0 1 0 · · · 0 0 1 0 1 · · · 0 . . . . . . . . . 0 0 · · · 0 1 1 0 · · · 1 0 . (79) \nIn deriving (79) we have used the the periodicity condition σ -1 = σ 2 N . By comparing this with the evolution operator (75) we have \n[ ˆ X /epsilon1 , ˆ P /epsilon1 ] = i cos[( ˆ H -ω N / 2) δt ] , (80) \nwhich for small δt gives \n[ ˆ X /epsilon1 , ˆ P /epsilon1 ] /similarequal i [ 1 -δt 2 2 ( ˆ H -ω N / 2) 2 ] . (81) \nIn addition, we deduce from (75) and (78) that \nsin[( ˆ H -ω N / 2) δt ] = /epsilon1 ˆ P /epsilon1 , (82) \nwhich is compatible with the result (20). From (82) follows that ˆ H depends only on ˆ P /epsilon1 but not on ˆ X /epsilon1 . So ˆ P /epsilon1 and ˆ H are simultaneously diagonalizable. By defining the operators ˆ K ± as \nˆ K + = e -i ˆ X /epsilon1 ˆ P /epsilon1 , ˆ K -= ˆ P /epsilon1 e i ˆ X /epsilon1 , (83) \n( ˆ K -= ˆ K † + ), so that \nˆ K + | n 〉 = 2 N +1 2 π sin [ 2 π 2 N +1 n ] | n +1 〉 , ˆ K -| n 〉 = 2 N +1 2 π sin [ 2 π 2 N +1 ( n -1) ] | n -1 〉 , (84) \none can persuade itself that ˆ H and ˆ K ± close the deformed algebra which in the large N limit (i.e., in the small /epsilon1 or δt limit) reduces to \n[ ˆ H, ˆ K ± ] = ± ω ˆ K ± , [ ˆ K + , ˆ K -] = -2 ˆ H ω , (85) \nwith ω = ω ∞ . \nNote that for large N one can identify (77), (84), and (85) with the representation of SU (1 , 1) known as the discrete series D + 1 / 2 ⊕ D -1 / 2 (cf. Ref. [63]). Generally, the Lie algebra D + k ⊕ D -k is defined through the relations: \nˆ L 3 | k, m 〉 = ( m + k ) | k, m 〉 , ˆ L + | k, m 〉 = √ ( m +2 k )( m +1) | k, m +1 〉 , ˆ L -| k, m 〉 = √ ( n +2 k -1) m | k, m -1 〉 , [ ˆ L 3 , ˆ L ± ] = ± ˆ L ± , [ ˆ L + , ˆ L -] = -2 ˆ L 3 . (86) \nHere m + k = ± k, ± ( k + 1) , ± ( k + 2) , . . . and k = 1 2 , 1 , 3 2 , 2 , . . . is the so-called Bargmann index which labels the representations. From this we have that D + 1 / 2 ⊕ D -1 / 2 corresponds to \nˆ L 3 | 1 2 , m 〉 = ( m +1 / 2) | 1 2 , m 〉 , ˆ L + | 1 2 , m 〉 = ( m +1) | 1 2 , m +1 〉 , ˆ L -| 1 2 , m 〉 = m | 1 2 , m -1 〉 , (87) \nIdentification with the largeN limit of (77) and (84) is established when we identify ˆ H in (87) with ω ˆ L 3 , ˆ K ± with ˆ L ± , and set m = n . \nAt this stage one can invoke 't Hooft's loss of information condition [37, 38], and project out the negative part of the spectra. A plausible rationale for this step can \nbe found, e.g., in irreversibility of computational process due to a finite storage capacity [64]. \nAfter the negative energy spectrum is removed (erased), we obtain only the positive discrete series D + 1 / 2 (i.e., representation where m = 0 , 1 , 2 , . . . ), and the Hamiltonian morphs into a non-negative spectrum Hamiltonian H + . \nThe usual W (1)-algebra of the quantum harmonic oscillator emerges after we introduce the following mapping in the universal enveloping algebra of SU (1 , 1): \nˆ a = 1 √ ˆ L 3 +1 / 2 ˆ L -, ˆ a † = ˆ L + 1 √ ˆ L 3 +1 / 2 . (88) \nThe latter gives a one-to-one (non-linear) mapping between the deterministic cellular automaton system (with information loss) and the quantum harmonic oscillator. The reader will recognize the mapping Eq. (88) as the non-compact analog [65] of the well-known HolsteinPrimakoff representation for SU (2) spin systems [66]. \nOur operators ˆ X /epsilon1 , ˆ P /epsilon1 and ˆ I /epsilon1 may be viewed as the same cellular automaton E (2) algebra as discussed in Section III. Thus we can conclude that in the Planckian scale the system must behave deterministically - which is one of the defining property of cellular automata. It is only at low energies when the loss of information leads to the emergent degrees of freedom resulting in the usual quantum mechanical description", 'B. Double special relativity': "The second application relates to the idea of double (or doubly or deformed) special relativity (DSR) (see, e.g., Refs. [36, 67]). The general idea is that if the Planck length is a truly universal quantity, then it should look the same to any inertial observer. This demands a modification (deformation) of the Lorenz transformations, to accommodate an invariant length scale. In Ref. [36] the nonlinearity of the deformed Lorenz transformations lead the authors to novel commutators between spacetime coordinates and momenta, depending on the energy \n[ ˆ x i , ˆ p j ] = i /planckover2pi1 ( 1 -E E p ) δ i j , (89) \nwhere E is the energy scale of the particle to which the deformed Lorenz boost is to be applied, while E p is the Planck energy. This suggests that they have an energydependent Planck 'constant' /planckover2pi1 ( E ) = /planckover2pi1 (1 -E/ E p ). Their model also implies that /planckover2pi1 ( E ) → 0 for E → E p . For energies much below that Planck regime, the usual Heisenberg commutators are recovered, but when E /similarequal E p one has /planckover2pi1 ( E p ) /similarequal 0. So the Planck energy is not only an invariant in this model, but the world looks also apparently classical at the Planck scale, similarly as in 't Hooft's proposal. \nThe connections of the DSR model with our proposal are at this point self evident. Our GUP (22), (24) implies that, at the boundary of the Brillouin zone, when 〈 ˆ p 〉 ψ → /planckover2pi1 π/ 2 /epsilon1 , i.e. for Planck energies E /epsilon1 /similarequal (2 √ 2 /a ) E p , the fundamental commutator vanishes \n[ ˆ X /epsilon1 , ˆ P /epsilon1 ] /similarequal 0 , (90) \nand since \n∆ X /epsilon1 ∆ P /epsilon1 /greaterorsimilar 0 , (91) \nlattice quantum mechanics at short wavelengths allows for classical behavior, that is uncertainties of two complementary observables can be simultaneously zero. \nHowever, if we express the fundamental commutator (22) of our model in terms of energy, using the exact relation (40), we find (for /epsilon1 = a/lscript p ) \n[ ˆ X /epsilon1 , ˆ P /epsilon1 ] f ( x ) = i /planckover2pi1 ( 1 -a 2 8 ˆ E 2 E 2 p ) f ( x ) . (92) \nThis means that the deforming term in our model is quadratic in the energy, instead of the linear dependence in the energy of the DSR model (89).", 'IX. DISCUSSION AND SUMMARY': "It should be noted that the present lattice generalization of the uncertainty principle is not an approximate description, but it is an exact formula necessarily implied by our model of lattice space time. The great majority of the GUP research has always borrowed the deformed commutator [ˆ x, ˆ p ] = i /planckover2pi1 (1+ κ ˆ p 2 ) either from string theory, or from heuristic arguments about black holes [55, 56]. To be precise, even in string theory [55] the formula expressing the GUP is not derived from the basic features of the model, but instead it is deduced from high-energy gedanken experiments of string scatterings. In contrast to this we have derived all results from a simple lattice model of spacetime, and from the analytic structure of the basic commutator (22). \nWe have calculated the uncertainties on a crystal-like universe whose lattice spacing is of the order of Planck length - the so-called world crystal. When the energies lie near the border of the Brillouin zone, i.e., for Planckian energies, the uncertainty relations for position and momenta do not pose any lower bound on the associated uncertainties. Hence the world crystal universe can become'deterministic' at Planckian energies. In this high-energy regime, our lattice uncertainty relations resemble the double special relativity result of Magueijo and Smolin. \nThe scenario in which the universe at Planckian energies is deterministic rather than being dominated by quantum fluctuations is a starting point in 't Hooft's 'deterministic' quantum mechanics. \nWith the generalized uncertainty relation at hand we have been able to derive a new mass-temperature relation for Schwarzschild micro black holes. In contrast to standard results based on Heisenberg or stringy uncertainty relations, our mass-temperature formula predicts both finite Hawking's temperature and a zero rest-mass remnant at the end of the evaporation process. Especially the absence of remnants is a welcome bonus which allows to avoid such conceptual difficulties as entropy/information problem or why we do not experimentally observe the remnants that must have been prodigiously produced in the early universe. \nApart from the mass-temperature relation we have also computed two relevant thermodynamic characteristics, namely entropy and heat capacity. Particularly the heat capacity provided an important insight into the last life stage of the world-crystal micro black holes. In contrast to the stringy result, our result indicates that worldcrystal micro black hole exchanges heat with its environment (radiate) till the last moment of its existence, and unlike the Schwarzschild micro black hole, the heat exchange with the environment increases in the final stage of the evaporation. In addition, the transition from the universe with the world-crystal micro black hole to the one without is of the first order. \nSince the world crystal physics allows for deterministic description of the physics at Planckian energies, we have included in this paper a discussion of 't Hooft's periodic automaton model which gives at low energy scales rise to a genuine quantum harmonic oscillator. In addition, such an automaton has a close connection with our world crystal paradigm. Here we have re-derived 't Hooft's result in a new way. In contrast to 't Hooft derivation [38] we have matched the algebra of the automaton variables with the SU (1 , 1) algebra, and in contrast to Ref. [63] we have worked with a different set of dynamical variables. In the contraction limit (i.e., in the limit of many lattice spacings - low energy limit) we have recovered the canonical W (1)-algebra and were able to identify the 'emergent' harmonic oscillator variables. \nThere are several aspects of the double special relativity that are worth noting in the connection with our generalized uncertainty relation. Essentially, we have seen that the fundamental commutator in DSR as well as in our lattice GUP goes to zero at Planck energy. For both models, the world should therefore be manifestly 'classical' in the Planck regime, a feature very different from the common believe. This is a striking prediction supported by both models, although the lines of thought followed in the two research paths are completely different and independent. Moreover, this aspect presents also a strong resemblance with the results obtained in the research line of 'deterministic' quantum mechanics.", 'X. ACKNOWLEDGEMENTS': 'One of us (P.J.) is grateful to G. Vitiello for instigating discussions. This work was partially supported by the Ministry of Education of the Czech Republic (research plan MSM 6840770039), and by the Deutsche Forschungsgemeinschaft under grant Kl256/47. F.S. acknowledges financial support by the National Taiwan University under the contract NSC 98-2811-M-002-086, and thanks ITP Freie Universitat Berlin for warm hospitality.', 'Appendix: Landauer principle': "Here we wish to provide an alternative derivation of the mass-temperature formula (55) based on Landauer's principle. To this end we consider an ensemble of unpolarized photons that are going to deliver to a micro black hole one single bit of information per particle. In order to be sure that each photon delivers only one bit of information - namely the information that it is there, somewhere inside in the black hole, its position uncertainty must be 'maximal', i.e., it should not be smaller than Schwarzschild's radius R S as otherwise the photon would deliver to black hole also extra bits of information concerning its entry point (or better sector) on the horizon. At the same time its wavelength should not be bigger than R S , as otherwise the photon would bounced off the black hole without getting trapped. In this view, the position uncertainty of a photon in the ensemble must be of order of Schwarzschild's radius R S i.e., ∆ X /epsilon1 /similarequal µR S . Factor µ is Bekenstein's deficit coefficient which ensures a correct Hawking's formula in a continuum limit. \nAn extra bit of information added to the micro black hole will increase its energy at least by amount ∆ E /epsilon1 so that (cf. (43)) \n∆ X /epsilon1 ∆ E /epsilon1 /similarequal /planckover2pi1 c 2 [ 1 -/epsilon1 2 2 /planckover2pi1 2 c 2 (∆ E /epsilon1 ) 2 ] . (93) \nIn the following we denote ∆ E /epsilon1 simply as E /epsilon1 to stress that ∆ E /epsilon1 an energy increas due to one photon. With the explicit form for Planck's energy \nE p = /planckover2pi1 c 2 /lscript p ≈ 0 . 61 · 10 19 GeV , (94) \nthe relation (93) can be cast to \n∆ X /epsilon1 /similarequal /planckover2pi1 c 2 E /epsilon1 -a 2 /lscript p E /epsilon1 8 E p . (95) \nIf we use further the fact that, R S = /lscript p m , where m is the relative mass of the black hole in Planck units, i.e., m = M/M p ( M p = E p /c 2 ), we can rewrite (95) as \n2 mµ /similarequal E p E /epsilon1 -a 2 E /epsilon1 8 E p . 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1997NuPhB.488..236B

Classical and quantum N = 2 supersymmetric black holes

1997-01-01

['-']

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We use heterotic/type-II prepotentials to study quantum/classical black holes with half the N = 2, D = 4 supersymmetries unbroken. We show that, in the case of heterotic string compactifications, the perturbatively corrected entropy formula is given by the tree-level entropy formula with the tree-level coupling constant replaced by the perturbative coupling constant. In the case of type-II compactifications, we display a new entropy/area formula associated with axion-free black-hole solutions, which depends on the electric and magnetic charges as well as on certain topological data of Calabi-Yau three-folds, namely the intersection numbers, the second Chern class and the Euler number of the three-fold. We show that, for both heterotic and type-II theories, there is the possibility to relax the usual requirement of the non-vanishing of some of the charges and still have a finite entropy.

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https://arxiv.org/pdf/hep-th/9610105.pdf

{'No Header': 'CERN-TH/96-276 \nHUB-EP-96/53 \nSU-ITP-96-41 \nTHU-96/34 \nhep-th/9610105', 'CLASSICAL AND QUANTUM N = 2 SUPERSYMMETRIC BLACK HOLES': 'Klaus Behrndt a , Gabriel Lopes Cardoso b , Bernard de Wit c , Renata Kallosh d , Dieter Lust a , Thomas Mohaupt a 1 \na Humboldt-Universitat zu Berlin, Institut fur Physik, D-10115 Berlin, Germany b Theory Division, CERN, CH-1211 Geneva 23, Switzerland c Institute for Theoretical Physics, Utrecht University, 3508 TA Utrecht, Netherlands d Physics Department, Stanford University, Stanford, CA 94305-4060, USA', 'ABSTRACT': 'We use heterotic/type-II prepotentials to study quantum/classical black holes with half the N = 2 , D = 4 supersymmetries unbroken. We show that, in the case of heterotic string compactifications, the perturbatively corrected entropy formula is given by the tree-level entropy formula with the treelevel coupling constant replaced by the perturbative coupling constant. In the case of type-II compactifications, we display a new entropy/area formula associated with axion-free black-hole solutions, which depends on the electric and magnetic charges as well as on certain topological data of Calabi-Yau three-folds, namely the intersection numbers, the second Chern class and the Euler number of the three-fold. We show that, for both heterotic and type-II theories, there is the possibility to relax the usual requirement of the nonvanishing of some of the charges and still have a finite entropy. \nOctober 1996 \nCERN-TH/96-276', '1 Introduction': 'Recently there has been considerable progress in the understanding of microscopic and macroscopic properties of supersymmetric black holes in string theory. Using the Dirichlet-brane interpretation of type-II solitons, the microscopic entropy of certain stringy black holes could be explicitly calculated [1] in agreement with the macroscopic Bekenstein-Hawking entropy formula. In [2] it was shown that, while the values of the moduli at spatial infinity are more or less arbitrary parameters, their values at the horizon are entirely fixed in terms of the (quantized) magnetic and electric charges of the black hole. In contradistinction with the black-hole mass, which is governed by the value of the central charge at infinity and thus depends on the charges and the moduli values at infinity, the entropy-area formula is given in terms of the central charge at the horizon [3]. Because the moduli are fixed at the horizon in terms of the charges, irrespective of their possible values at spatial infinity, the entropy-area is thus expressible in terms of the charges. This result is natural from the point of view that the entropy should follow from a counting of independent quantum-mechanical states, which seems to preclude any dependence on continuous parameters such as the moduli values at spatial infinity. In [3] it was also shown that the central charge acquires a minimal value at the horizon and that the extremization of the central charge provides the specific moduli values at the horizon. \nThe restricted behaviour at the horizon is related to the enhancement to full N = 2 supersymmetry near the horizon, while globally the field configurations leave only half the supersymmetries unbroken (so that we are dealing with true BPS states). Thus the black holes can be regarded as solitonic solutions that interpolate between the maximally supersymmetric field configurations at spatial infinity and at the horizon. Particularly simple solutions, called double extreme black holes [4], are given by those configurations where the moduli take constant values from the horizon up to spatial infinity. \nUsing the extremization procedure of [3], the macroscopic entropy formulae for N = 4 and N = 8 extreme black hole solutions [5] were obtained in perfect agreement with the construction of explicit black-hole solutions. The N = 4 , 8 entropies are completely unique and they depend only on the quantized magnetic/electric charges and they are invariant under the perturbative and non-perturbative duality symmetries, such as T -duality, S -duality [6] and string/string duality [7, 8, 9]. \nIn four-dimensional N = 2 string theories new features of black-hole physics arise which destroy the uniqueness of the N = 2 entropy formula. In particular there exists a large number of different N = 2 string vacua so that the extreme black-hole solutions depend on \nthe specific details of the particular N = 2 string model. Consequently the same features are present for the N = 2 entropy formula. Nevertheless, the N = 2 entropy, being proportional to the extremized N = 2 central charge Z , still depends on the quantized magnetic/electric charges, although the nature of the dependence is governed by the particular string model. The N = 2 central charge Z and the N = 2 BPS spectrum can be directly calculated from the N = 2 holomorphic prepotential which describes the two-derivative couplings of the N = 2 vector multiplets in the effective N = 2 string action [10] (or, in a symplectic basis where the prepotential does not exist [11], from the symplectic sections). Therefore the parameters of the prepotential of a given N = 2 string model determine the black-hole entropy as well as the values of the scalar fields at the horizon. \nDepending on whether one is discussing heterotic or type-II N = 2 string vacua, the parameters of the prepotential have a rather different interpretation. To be more specific, let us first consider four-dimensional N = 2 heterotic string compactifications on K 3 × T 2 , where the number of vector multiplets N V (not counting the graviphoton), the number of hypermultiplets N H and the couplings are specified by a particular choice of the SU (2) instanton gauge bundle. The classical prepotential is completely universal and corresponds to a scalar non-linear σ -model based on the coset space SU (1 , 1) U (1) ⊗ SO (2 ,N V -1) SO (2) × SO ( N V -1) . Extremizing the corresponding central charge Z the classical N = 2 black hole entropy and the moduli on the horizon have been computed explicitly [4, 12, 13], and the result agrees with the truncated N = 4 formulae. \nSince in heterotic N = 2 string compactifications the dilaton field S can be described by a vector multiplet, the heterotic prepotential receives perturbative corrections only at the one-loop level [15, 16]; in addition there are non-perturbative contributions. The heterotic one-loop corrections to the prepotential, being independent of the dilaton S , split into a cubic polynomial, a constant term and an infinite series of terms which are exponentially suppressed in the decompactification limit of large moduli fields. It is an interesting observation that the coefficients of the exponential terms are given in terms of q -expansion coefficients of certain modular forms as explicitly shown for models with N V = 3 , 4 in [17, 18, 19]. Thus, the one-loop black hole solutions are determined by an infinite set of integer numbers; hence the extremization problem of the corresponding one-loop central charges is very involved and difficult. Nevertheless, we are able to derive a simple formula for the black-hole entropy in terms of the heterotic string-coupling and the target-space duality-invariant inner product of the charges, which holds to all orders in perturbation theory. This formula does not depend explicitly on the values of the moduli fields. At the horizon the values of the moduli can be determined explicitly in \ncertain cases when neglecting all exponential terms in the large moduli limit. Hence new quantum features of black holes already become important when considering only cubic corrections to the classical prepotential. \nIt is well established that the N = 2 heterotic string on K 3 × T 2 is dual to type-IIA (IIB) compactification on a suitably chosen Calabi-Yau three-fold [20, 21, 22]. In fact, it was shown [20, 23, 24, 25, 18, 19] for models with N V = 3 , 4 that the type-IIA and heterotic prepotential agree in heterotic weak-coupling limit. On the type-IIA side the N = 2 prepotential of the Kahler class moduli is completely classical because the type-II dilaton corresponds to a hypermultiplet and has no couplings to the vector fields. More specifically, the cubic couplings of the type-IIA prepotential are determined by the topological intersection numbers of the corresponding Calabi-Yau space; the coefficients of the exponential terms are given in terms of the rational Calabi-Yau instanton numbers. In this paper we focus on the limit of large Kahler-class moduli, i.e. we will discuss the influence of the classical intersection numbers C ABC , as well as terms constant and linear in the moduli, on the Calabi-Yau black-hole solutions. The linear ones are related to the second Chern class of the three-fold and the constant one is related to the Euler characteristic [26]. Hence we find new entropy formulae which depend only on the magnetic/electric charges and topological data on the Calabi-Yau manifold. \nOur paper is organized as follows. In the next section we will briefly introduce the N = 2 vector couplings and the N = 2 central charge in terms of the N = 2 prepotential. We will recall the structure of the prepotentials in four-dimensional N = 2 heterotic and type-IIA string vacua, and also their relations via heterotic/typeII string-string duality. In section 3 we show that there is a rather elegant and simple way to find the solutions of the extremization problem of the N = 2 central charge, which can be used to compute the values of the moduli on the black-hole horizon and the black-hole entropy as a function of the quantized electric/magnetic charges. While these solutions cannot be determined in full generality, we can generally prove a simple formula for the entropy for perturbative heterotic vacua, as a product of the inverse square of the perturbative string-coupling constant (which itself depends on the electric/magnetic charges) and the target-space duality-invariant inner product of the charges. A particular class of solutions that can generally be evaluated for cubic prepotentials, is the class of non-axionic black-holes. This result covers the type-IIA Calabi-Yau black-hole entropy in case of small contributions of the rational instanton configurations, i.e., in the limit of large Kahler-class moduli. We will also discuss the influence of linear terms in the prepotential on the black hole entropy. In the Calabi-Yau case these linear terms are related to the second Chern class of the three-fold [26]. In section four we discuss the relation of our solution to intersecting \nbranes in higher dimensions and suggest their M-theory interpretation. In the last section we summarize our results.', '2.1 General formulae': 'The vector couplings of N = 2 supersymmetric Yang-Mills theory are encoded in a holomorphic function F ( X ), where the X denote the complex scalar fields of the vector supermultiplets. With local supersymmetry this function depends on one extra field, in order to incorporate the graviphoton. The theory can then be encoded in terms of a holomorphic function F ( X ) which is homogeneous of second degree and depends on complex fields X I with I = 0 , 1 , . . . N V . Here N V counts the number of physical vector multiplets. \nThe resulting special geometry [10, 27] can be defined more abstractly in terms of a symplectic section V , also referred to as period vector: a (2 N V +2)-dimensional complex symplectic vector, expressed in terms of the holomorphic prepotential F according to \nV = ( X I F J ) , (2.1) \nwhere F I = ∂F/∂X I . The N V physical scalar fields of this system parametrize an N V -dimensional complex hypersurface, defined by the condition that the section satisfies a constraint \n〈 ¯ V , V 〉 ≡ ¯ V T Ω V = -i, (2.2) \nwith Ω the antisymmetric matrix \nΩ = ( 0 1 -1 0 ) . (2.3) \nThe embedding of this hypersurface can be described in terms of N V complex coordinates z A ( A = 1 , . . . , N V ) by letting the X I be proportional to some holomorphic sections X I ( z ) of the complex projective space. In terms of these sections the X I read \nX I = e 1 2 K ( z, ¯ z ) X I ( z ) , (2.4) \nwhere K ( z, ¯ z ) is the Kahler potential, to be introduced below. In order to distinguish the sections X I ( z ) from the original quantities X I , we will always explicitly indicate their z -dependence. The overall factor exp[ 1 2 K ] is chosen such that the constraint (2.2) \nis satisfied. Furthermore, by virtue of the homogeneity property of F ( X ), we can extract an overall factor exp[ 1 2 K ] from the symplectic sections (2.1), so that we are left with a holomorphic symplectic section. Clearly this holomorphic section is only defined projectively, i.e., modulo multiplication by an arbitrary holomorphic function. On the Kahler potential these projective transformations act as Kahler transformations, while on the sections V they act as phase transformations. \nThe resulting geometry for the space of physical scalar fields belonging to vector multiplets of an N = 2 supergravity theory is a special Kahler geometry, with a Kahler metric g A ¯ B = ∂ A ∂ ¯ B K ( z, ¯ z ) following from a Kahler potential of the special form \nK ( z, ¯ z ) = -log ( i ¯ X I (¯ z ) F I ( X I ( z )) -iX I ( z ) ¯ F I ( ¯ X I (¯ z )) ) . (2.5) \nA convenient choice of inhomogeneous coordinates z A are the special coordinates, defined by \nX 0 ( z ) = 1 , X A ( z ) = z A , A = 1 , . . . , N V . (2.6) \nIn this parameterization the Kahler potential can be written as [28] \nK ( z, ¯ z ) = -log ( 2( F + ¯ F ) -( z A -¯ z A )( F A -¯ F A ) ) , (2.7) \nwhere F ( z ) = i ( X 0 ) -2 F ( X ). \nWe should point out that it is possible to rotate the basis specified by (2.1) by an Sp (2 N V + 2 , Z ) transformation in such a way that it is no longer possible to associate them to a holomorphic function [11]. As long as all fundamental fields are electrically neutral (which is the case in the context of this paper), this is merely a technical problem, as one can always rotate back to the basis where a prepotential exists [29]. As shown in [11] the supergravity Lagrangian can be expressed entirely in terms of the symplectic section V , without restricting its parameterization so as to correspond to a prepotential F ( X ). \nThe Lagrangian terms containing the kinetic energies of the gauge fields are \n4 π L gauge = -i 8 ( N IJ F + I µν F + µνJ -¯ N IJ F -I µν F -µνJ ) , (2.8) \nwhere F ± I µν denote the selfdual and anti-selfdual field-strength components and N IJ ( z, ¯ z ) is the field-dependent tensor that comprises the inverse gauge couplings g -2 IJ = i 16 π ( N IJ -¯ N IJ ) and the generalized θ angles θ IJ = π 2 ( N IJ + ¯ N IJ ). \nNow we define the tensors G ± µνI as \nG + µνI = N IJ F + J µν , G -µνI = ¯ N IJ F -J µν , (2.9) \nwhich describe the (generalized) electric displacement and magnetic fields. The set of Bianchi identities and equation of motion for the Abelian gauge fields are invariant under the transformations \nF + I µν -→ ˜ F + I µν = U I J F + J µν + Z IJ G + µνJ . G + µνI -→ ˜ G + µνI = V I J G + µνJ + W IJ F + J µν , (2.10) \nwhere U , V , W and Z are constant, real, ( N V +1) × ( N V +1) matrices, which have to satisfy the symplectic constraint \nO -1 = Ω O T Ω -1 where O = ( U Z W V ) . (2.11) \nThe target-space duality group Γ is a certain subgroup of Sp (2 N V +2 , Z ). It follows that the magnetic/electric charge vector Q = ( p I , q J ), defined by ( ∮ F I , ∮ G J ) = (2 πp I , 2 πq J ), transforms as a symplectic vector, where we stress that the identification of magnetic and electric charges is linked to the symplectic basis. Since N = 2 supersymmetry relates the X I to the field strengths F + I µν , while the F I are related to the G + µν I , the period vector V also transforms as a symplectic vector: \n˜ X I = U I J X J + Z IJ F J , ˜ F I = V I J F J + W IJ X J . (2.12) \nFinally consider N = 2 BPS states, whose masses are equal to the central charge Z of the N = 2 supersymmetry algebra. In terms of the magnetic/electric charges Q and the period vector V the BPS masses take the following form [11]: \nM 2 BPS = | Z | 2 = |〈 Q,V 〉| 2 = e K | q I X I ( z ) -p I F I ( z ) | 2 = e K ( z, ¯ z ) |M ( z ) | 2 . (2.13) \nIt follows that M 2 BPS is invariant under symplectic transformations (2.12). \nAn example of a prepotential arising in string compactifications is given by the cubic prepotential \nF ( X ) = d ABC X A X B X C X 0 , (2.14) \nwhere d ABC are some real constants. The corresponding Kahler potential is given by \nK ( z, ¯ z ) = -log ( -id ABC ( z -¯ z ) A ( z -¯ z ) B ( z -¯ z ) C ) . (2.15) \nIn the case of heterotic string compactifications, both the classical prepotential as well as certain perturbative corrections to it are described by a cubic prepotential of the type (2.14). In the case of type IIA compactifications, the d ABC are just proportional to the classical intersection numbers: d ABC = -1 6 C ABC .', '2.2 The heterotic prepotential': 'In the following, we will discuss a class of heterotic N = 2 models, obtained by compactifying the E 8 × E 8 string on K 3 × T 2 . The moduli z A ( A = 1 , . . . , N V ) comprise the dilaton S , the two toroidal moduli T and U as well as Wilson lines V i ( i = 1 , . . . , N V -3): \nS = -iz 1 , T = -iz 2 , U = -iz 3 , V i = -iz i +3 . (2.16) \nWe will, in the following, collectively denote the moduli T, U and V i by T a , so that a = 2 , . . . , N V . The generic unbroken Abelian gauge group U (1) N V +1 depends on the specific choice of SU (2) bundles with instanton numbers ( d 1 , d 2 ) = (12 -n, 12 + n ) when compactifying to six dimensions on K 3 (see [20, 30, 31] for details). For example, for n = 0 , 1 , 2, a complete Higgsing is possible which leads to the three-parameter S -T -U models with no Wilson-line moduli ( N V = 3). It is, however, also possible to not completely Higgs away the six-dimensional gauge group E 7 × E 7 , and for n = 0 , 1 , 2 one obtains in this way heterotic models with one Wilson-line modulus V . Here we have four vector multiplets, so that we are dealing with a four-parameter S -T -U -V model ( N V = 4). \nFor this class of models, the heterotic prepotential has the form \nF het = -ST a η ab T b + h ( T a ) + f NP ( e -2 πS , T a ) , (2.17) \nwhere \nT a η ab T b = T 2 T 3 -N V ∑ I =4 ( T I ) 2 , a, b = 2 , . . . , N V . (2.18) \nThe dilaton S is related to the tree-level coupling constant and to the theta angle by S = 4 π/g 2 -iθ/ 2 π . The first term in (2.17) is the classical part of the heterotic prepotential, h ( T a ) denotes the one-loop contribution and f NP is the non-perturbative part, which is exponentially suppressed for small coupling. Note that the perturbative corrections are entirely due to one-loop effects, owing to nonrenormalization theorems. In the following we focus on the perturbative contributions. \nThe classical prepotential leads to the metric of the special Kahler manifold SU (1 , 1) U (1) ⊗ SO (2 ,N V -1) SO (2) × SO ( N V -1) with corresponding tree-level Kahler potential \nK = -log [ ( S + ¯ S ) ] -log [ ( T a + ¯ T a ) η ab ( T b + ¯ T b ) ] . (2.19) \nDue to the required embedding of the T -duality group into the N = 2 symplectic transformations, it follows [15, 16] that the heterotic one-loop prepotential h ( T a ) must obey \nwell-defined transformation rules under this group. The function h ( T a ) leads to the following modified Kahler potential [15], which represents the full perturbative contribution, \nK = -log[( S + ¯ S ) + V GS ( T a , ¯ T a )] -log [ ( T a + ¯ T a ) η ab ( T b + ¯ T b ) ] , (2.20) \nwhere \nV GS ( T a , ¯ T a ) = 2( h + ¯ h ) -( T a + ¯ T a )( ∂ T a h + ∂ ¯ T a ¯ h ) ( T a + ¯ T a ) η ab ( T b + ¯ T b ) (2.21) \nis the Green-Schwarz term [32] describing the mixing of the dilaton with the moduli T a . Note that the true perturbative coupling constant is given by \n4 π g 2 pert = 1 2 ( S + ¯ S + V GS ( T a , ¯ T a ) ) . (2.22) \nTo be more specific, let us recall the precise form of the one-loop prepotential h ( T a ) [17, 18, 19]. For simplicity we limit the discussion here to the models with N V = 4. Any of the S -T -U -V models considered can be simply truncated to the three-parameter S -T -U model upon setting V → 0. For the class of S -T -U -V models considered here, the one-loop prepotential is given by \nh ( T, U, V ) = p n ( T, U, V ) -c -1 4 π 3 ∑ k,l,b ∈ Z ( k,l,b ) > 0 c n (4 kl -b 2 ) Li 3 ( e [ ikT + ilU + ibV ]) , (2.23) \nwhere c = c n (0) ζ (3) 8 π 3 and e [ x ] = exp2 πix . The coefficients c n (4 kl -b 2 ) are the expansion coefficients of particular Jacobi modular forms [19]. p n is a cubic polynomial of the form [33, 34, 19] \np n ( T, U, V ) = -1 3 U 3 -( 4 3 + n ) V 3 +(1 + 1 2 n ) UV 2 + 1 2 nTV 2 . (2.24) \nIt is important to note that the expression (2.23) is valid in the specific Weyl chamber Re T > Re U > 2Re V . \nNow consider taking the limit S, T, U, V →∞ subject to Re S > Re T > Re U > 2Re V , in which all non-perturbative as well as perturbative exponential terms are suppressed. Then, the heterotic prepotential is simply given by the cubic polynomial F het = -S ( TU -V 2 ) + p n ( T, U, V ). In the limit V → 0, the perturbative prepotential is completely universal. In the large-moduli limit S, T, U → ∞ (Re S > Re T > Re U ), which is the decompactification limit to 5 dimensions, the prepotential of these three-parameter models takes the form \nF het = -STU -1 3 U 3 -c, (2.25) \nwhere c = c = c STU (0) ζ (3) 8 π 3 and c STU ( kl ) = ∑ b c n (4 kl -b 2 ) (for any n ). Using (2.21) it is straightforward to compute the one-loop term V GS which follows from the prepotential (2.25): \nV GS ( T, ¯ T, U, ¯ U ) = ( U + ¯ U ) 2 3( T + ¯ T ) -4 c ( T + T )( U + U ) . (2.26)', '2.3 The type-IIA prepotential': 'As already mentioned, the prepotential in type-IIA Calabi-Yau compactifications, which depends on the Kahler-class moduli t A ( A = 1 , . . . , N V = h 1 , 1 ), is of purely classical origin. Nevertheless it has the same structure as the heterotic prepotential. In fact, for dual heterotic/type-IIA pairs the prepotentials are identical upon a suitable identification of the Kahler-class moduli t A in terms of the heterotic fields S and T a . \nThe type-IIA prepotential has the following general structure [35]: \nF II = -1 6 C ABC t A t B t C -χζ (3) 2(2 π ) 3 + 1 (2 π ) 3 ∑ d 1 ,...,d h n r d 1 ,...,d h Li 3 ( e [ i ∑ A d A t A ]) , (2.27) \nwhere we work inside the Kahler cone σ ( K ) = { ∑ A t A J A | t A > 0 } . (The J A denote the (1,1)-forms of the Calabi-Yau three-fold M , which generate the cohomology group H 2 ( M, R )). The cubic part of the type-IIA prepotential is given in terms of the classical intersection numbers C ABC , whereas the coefficients n r d 1 ,...,d h of the exponential terms denote the rational instanton numbers of genus 0. Hence, in the limit of large Kahler class moduli, t A → ∞ , only the classical part, related to the intersection numbers, survives. Consider, for example, the four-parameter model based on the compactification on the Calabi-Yau three-fold P 1 , 1 , 2 , 6 , 10 (20) with h 1 , 1 = 4 and Euler number χ = -372 [33]. The cubic intersection-number part of the type-IIA prepotential for this model is given as \n-F II cubic = t 2 (( t 1 ) 2 + t 1 t 3 +4 t 1 t 4 +2 t 3 t 4 +3( t 4 ) 2 ) + 4 3 ( t 1 ) 3 +8( t 1 ) 2 t 4 + t 1 ( t 3 ) 2 +2( t 1 ) 2 t 3 +8 t 1 t 3 t 4 +2( t 3 ) 2 t 4 +12 t 1 ( t 4 ) 2 +6 t 3 ( t 4 ) 2 +6( t 4 ) 3 . (2.28) \nSome of the rational instanton numbers n r d 1 ,d 2 ,d 3 ,d 4 for this model are displayed in [33, 19]. This model is dual to the previously discussed heterotic string compactification with N V = 4 and n = 2 [19]. The necessary identification of heterotic and type-IIA moduli is given as \nt 1 = U -2 V, t 2 = S -T, t 3 = T -U, t 4 = V , (2.29) \nand one can explicitly check that for some instanton numbers and for the Euler number the relations \nn r l + k, 0 ,k, 2 l +2 k + b = -2 c 2 (4 kl -b 2 ) , χ = 2 c 2 (0) (2.30) \nare indeed satisfied. In addition, the cubic heterotic prepotential p 2 (cf. (2.24)) and the Calabi-Yau prepotential (cf. (2.28)) agree. \nFinally, let us mention that, in a particular symplectic basis, it is very convenient to add to the Calabi-Yau prepotential (2.27) a topological term which is determined by the second Chern class c 2 of the three-fold M , which gives rise to terms linear in the Kahler-class moduli fields: \nF II = -1 6 C ABC t A t B t C + h ∑ A c 2 · J A 24 t A + · · · . (2.31) \nThe real numbers c 2 · J A = ∫ M c 2 ∧ J A are the expansion coefficents of c 2 with respect to the basis J ∗ A of the cohomology group H 4 ( M, R ) which is dual to the basis J A of H 2 ( M, R ) (i.e. ∫ M J ∗ A ∧ J B = δ AB ). It is clear from eq.(2.12) that adding such a linear term to the prepotential is equivalent to performing a symplectic transformation with U = V = 1, Z = 0 and W 0 A = c 2 · J A 24 ; hence it has just the effect of a constant shift in the theta angles [36]. In the next section we will see that adding such a topological linear term to the prepotential may have interesting effects on the N = 2 black hole entropy as a function of the magnetic/electric charges. \nAs an example we consider the three parameter model based on the Calabi-Yau P 1 , 1 , 2 , 8 , 12 (24) with h 1 , 1 = 3 and χ = -480, which is dual to the heterotic string compactification with N V = 3 and n = 2. The corresponding prepotential can be simply obtained from (2.28) by setting V = 0. Here the linear topological term takes the form \n3 ∑ A c 2 · J A 24 t A = 23 6 t 1 + t 2 +2 t 3 = S + T + 11 6 U . (2.32)', '3 N = 2 Supersymmetric black holes': "In this section we consider extreme dyonic black holes in the context of N = 2 supergravity. The fields corresponding to these black holes spatially interpolate between two maximally supersymmetric field configurations. One is the trivial flat space at spatial infinity, which allows constant values for the moduli fields. The other is the BertottiRobinson metric near the horizon, where the fields are restricted to (covariantly) constant moduli and graviphoton field strength (the latter is directly related to the value of the central charge at the horizon). The interpolating fields leave only half the supersymmetries invariant, so that we are dealing with true BPS states. For these black holes, the mass is equal to the central charge taken at spatial infinity, so that \nM 2 ADM = | Z ∞ | 2 = e K ( z, ¯ z ) |M ( z ) | 2 ∣ ∣ ∣ ∞ , (3.1) \nwhere the moduli fields z are taken at spatial infinity. Hence the mass depends generically on the magnetic/electric charges and the asymptotic values of the moduli fields. \nNear the horizon the values of the moduli fields, and thus the value of the central charge, are strongly restricted by the presence of full N = 2 supersymmetry. In [3] it was proved that this implies that the central charge becomes extremal on the horizon. The result of this is that one can express the values of the moduli at the horizon in terms of the magnetic/electric charges p I and q I . The value of the central charge at the horizon is related to the Hawking-Bekenstein entropy, \nS π = | Z hor | 2 , (3.2) \nwhere we have conveniently adjusted the value of Newton's constant. The area of the black hole, which equals four times the entropy, has an interpretation as the mass of the Bertotti-Robinson universe. The crucial observation here is that the entropy and related quantities depend only on the quantized magnetic/electric charges (with N = 2 supersymmetry the nature of this dependence is governed by the particular string vacuum), while the mass of the black hole depends on the charges as well as on the asymptotic values of the moduli. The latter are, in principle, arbitrary parameters that do not depend on the charges and, when approaching the black hole, evolve according to a damped geodesic equation towards the fixed-point values at the horizon, which are given in terms of the charges. \nThere exist so-called double extreme black holes, introduced in [4], for which the moduli remain constant away from the horizon. In that case the central charge remains constant and thus the black hole mass is equal to the Bertotti-Robinson mass. The moduli at spatial infinity take the same values as near the horizon, so that the black-hole mass itself is now also a function of the magnetic/electric charges. Consequently, for double extreme black holes we find that M ADM is a function of the p I and q I . \nIn this section we study the extremization problem at the black-hole horizon to obtain the value of the moduli and the black-hole entropy as a function of the charges p I and q I . We cast this problem in a convenient form, which can be formulated in terms of a variational principle (cf. (3.12)). This allows us to construct a variety of explicit solutions. Then, in the second subsection, we consider the black-hole entropy for heterotic N = 2 supersymmetric string compactifications to all orders in string perturbation theory and derive a general formula for the entropy. An important feature of this formula is its invariance under target-space duality. In the next subsection we consider so-called nonaxionic black holes, where one can conveniently obtain explicit solutions. Finally in subsection 3.4 we consider the entropy for type-II compactifications.", '3.1 Extremization of the N = 2 central charge': "Let us start and exhibit some features of the double extreme black holes, for which the moduli remain constant. The metric of these black holes is of the extreme ReissnerNordstrom form with the mass equal to the square root of the area divided by 4 π . In isotropic coordinates the metric is \nds 2 = - 1 + √ A/ 4 π r -2 dt 2 + 1 + √ A/ 4 π r 2 d/vectorx 2 . (3.3) \nOne can also present the metric as (˜ r = r + √ A/ 4 π ) \nds 2 = - 1 -√ A/ 4 π ˜ r 2 dt 2 + 1 -√ A/ 4 π ˜ r -2 d ˜ r 2 + ˜ r 2 d 2 Ω . (3.4) \nThe mass is defined via the large-˜ r expansion \ng tt = ( 1 -2 M ADM ˜ r + · · · ) (3.5) \nand the metric shows that \nM ADM = √ A 4 π . (3.6) \nIn this form it is clear that the horizon is at g tt = 0 = ⇒ ˜ r = √ A/ 4 π . Therefore the area of the horizon is indeed given by \n4 π (˜ r 2 ) hor = A . (3.7) \nAs discussed above, to obtain the value of the moduli at the horizon for extreme N = 2 black holes, one can determine the extremal value of the central charge in moduli space. This implies that \n∂ A | Z | = 0 . (3.8) \nThese equations are difficult to solve in general. They are, however, equivalent to the following set of equations [3] \n¯ Z V -Z ¯ V = iQ, (3.9) \nwhere Q is the magnetic/electric charge vector Q = ( p I , q J ). The above relation is closely related to the fact that the field configurations are fully supersymmetric at the horizon. Here we note that these equations can be independently justified on the basis of symplectic covariance. Assuming that the moduli near the horizon depend exclusively on the magnetic/electric charges and satisfy equations of motions that transform in a \nwell-defined way under symplectic (duality) reparametrizations, the symplectic period vector must be proportional to the symplectic charge vector. As the period vector is complex and the charge vector is real, there is a complex proportionality factor which must be a symplectic invariant. Using (2.2) we derive that this factor is precisely the central charge Z and find the above result (3.9). \nFrom (3.9) one can determine the period vector, which is defined in terms of N V complex moduli. We do this by reformulating the equation and the corresponding expression of the black-hole entropy in terms of a variational principle. To do this, we first introduce a new symplectic vector Π by \nΠ = ( Y I F J ( Y ) ) where Y I ≡ ¯ Z X I . (3.10) \nObserve that Y I and thus the vector Π is U (1) invariant, so that it is not subject to Kahler transformations. In terms of Π, (3.9) and (3.2) turn into \nΠ -¯ Π = iQ, S π = | Z hor | 2 = i 〈 ¯ Π , Π 〉 . (3.11) \nThe equations (3.11) are governed by a variational principle associated with a 'potential' \nV Q ( Y, ¯ Y ) ≡ -i 〈 ¯ Π , Π 〉 - 〈 ¯ Π+Π , Q 〉 . (3.12) \nV Q takes an extremal value whenever Y and ¯ Y satisfy the first equation (3.11). This extremal value is given by the second expression (3.11) for the entropy. Using (2.5) the entropy can be also written as \nS π = | Y 0 | 2 exp [ -K ( z, ¯ z ) ] ∣ ∣ ∣ hor . (3.13) \n∣ \nwhere Y 0 and the special coordinates z A are evaluated at the horizon. \nLet us now consider the construction of solutions to the equations (3.11). Written in components they read \nY I -¯ Y I = ip I , F I ( Y ) -¯ F I ( ¯ Y ) = iq I . (3.14) \nTo solve these equations it does not help to go to a special symplectic basis (although the equations may take a more 'suggestive' form), as this only corresponds to taking linear combinations. Although we assumed the existence of the holomorphic prepotential, the above variational principle can also be formulated in a basis where such a prepotential does not exist, but for the purpose of this paper this feature is not important. The components of Π comprise 2 N V + 2 complex quantities, but only N V + 1 of them are independent (as the others are determined in terms of the prepotential). So generically, \nthe above equation fixes Π in terms of p I and q J . Before considering an explicit example, we note the following convenient relations, which follow from (3.14) for p 0 and p A , \n( z A -¯ z A ) Y 0 = i ( p A -p 0 ¯ z A ) . (3.15) \nAs an example, consider the following cubic prepotential \nF ( Y ) = -b Y 1 Y 2 Y 3 Y 0 + a ( Y 3 ) 3 Y 0 . (3.16) \nThe solution to (3.14) for a general magnetic/electric charge vector ( p I , q I ) where I = 0 , 1 , 2 , 3, reads \nY 0 \n= \np 3 + ip 0 ¯ U U + ¯ U \n, \nY 1 = -Y 0 p 3 + ip 0 ¯ U ( -ip 1 ¯ U + q 2 b ) , \nY 2 = -Y 0 p 3 + ip 0 ¯ U ( -ip 2 ¯ U + q 1 b ) , Y 3 = iU Y 0 , (3.17) \nwhere U is determined by the following equation \nq 0 -iq 3 ¯ U = b p 3 + ip 0 ¯ U ( -ip 1 ¯ U + q 2 b )( -ip 2 ¯ U + q 1 b ) + ap 3 ( U 2 +2 U ¯ U -2 ¯ U 2 ) + ia p 0 U ¯ U ( U +2 ¯ U ) . (3.18) \nThe entropy can be determined as a function of U , by making use of (3.13), \nS π = ∣ ∣ ∣ U + ¯ U p 3 -ip 0 U ∣ ∣ ∣ ∣ 2 { ( b p 1 p 3 + q 2 p 0 )( b p 2 p 3 + q 1 p 0 ) b -a } . (3.19) \n∣ \n∣ What remains is to solve (3.18). For the case a = 0, b = 1, this is a quadratic equation for U with solution \nU = i q 0 p 0 + q 1 p 1 + q 2 p 2 -q 3 p 3 2( q 3 p 0 + p 1 p 2 ) ± √ √ √ √ q 1 q 2 -q 0 p 3 q 3 p 0 + p 1 p 2 -( q 0 p 0 + q 1 p 1 + q 2 p 2 -q 3 p 3 ) 2 4( q 3 p 0 + p 1 p 2 ) 2 . (3.20) \nThese solutions with the corresponding value for the entropy can be compared to previous results [12, 4, 13] (for the results of the second and third work this comparison requires a conversion to the appropriate symplectic basis).", '3.2 Perturbative entropy formula for heterotic string compactifications': 'The classical entropy formula for N = 2 supersymmetric heterotic string compactifications has been derived in the perturbative string basis [4, 13]. It was shown to be invariant (as should be expected) under the target-space duality group, which, at the \nclassical level, is just equal to SO (2 , N V -1). The entropy was also constructed in the symplectic basis corresponding to the first term in (2.17) in [12]. \nIn this section we derive the entropy formula, but now to all orders of string perturbation theory. It reads \n∣ \nS π = 8 π g 2 pert ∣ ∣ ∣ hor ( p 0 q 1 + p a η ab p b ) . (3.21) \n∣ The perturbative string coupling depends on the values of the dilaton field and the moduli at the horizon. The charges p and q refer to the magnetic/electric charges as defined in the symplectic basis associated with (2.17). This is, however, not the basis defined by perturbative string theory, where the magnetic charges are equal to N I = ( p 0 , q 1 , p 2 , . . . ). These magnetic charges transform linearly under target-space duality transformations, \nN I → ˆ U I J N J , (3.22) \nwhere the matrix ˆ U belongs to a subgroup of SO (2 , N V -1 , Z ). In terms of the string basis we find that we are dealing with an invariant under these transformations [11, 15, 13] \np 0 q 1 + p a η ab p b = N 0 N 1 + N a η ab N b ≡ 〈 N,N 〉 . (3.23) \nThus, in the perturbative string basis, eq. (3.21) reads \nS π = 8 π g 2 pert ∣ ∣ ∣ hor 〈 N,N 〉 . (3.24) \n∣ \nDue to nonrenormalization theorems, this result is true to all orders in perturbation theory and takes precisely the same form as the classical entropy formula [13], with the tree-level coupling constant replaced by its full perturbative value. As the latter is invariant under target-space duality [15], the perturbative entropy formula is invariant under target-space duality. In fact, since 〈 N,N 〉 is invariant under target-space duality transformations, whereas the dilaton is not, at least not beyond the classical level, it was natural to expect that the corrected entropy formula should be given by the tree-level formula with the dilaton replaced by some one-loop target-space duality invariant object. It is gratifying to see that this object is precisely the true loop-counting parameter of heterotic string theory. \nLet us now show that (3.21) indeed holds. Inserting eqs. (2.20) into eq. (3.13) yields \nS π = ( S + ¯ S + V GS ) | Y 0 | 2 ( T a + ¯ T a ) η ab ( T b + ¯ T b ) . (3.25) \nUsing that T a = -iz a , and inserting (3.15) and its complex conjugate into (3.25) yields \n| Y 0 | 2 ( T a + ¯ T a ) η ab ( T b + ¯ T b ) = ( p a + ip 0 ¯ T a ) η ab ( p b -ip 0 T b ) . (3.26) \nOn the other hand, it follows from (3.14) and (2.17) that \nF 1 ( Y ) -¯ F 1 ( ¯ Y ) = -Y a η ab Y b Y 0 + ¯ Y a η ab ¯ Y b ¯ Y 0 = iq 1 . (3.27) \nUsing that ¯ Y a = Y a -ip a , we obtain \nF 1 ( Y ) -¯ F 1 ( ¯ Y ) = -Y a η ab Y b Y 0 + ¯ Y a η ab ( Y b -ip b ) ¯ Y 0 = -iT a η ab Y b -i ¯ T a η ab ( Y b -ip b ) = Y 0 ( T a + ¯ T a ) η ab T b -¯ T a η ab p b = iq 1 . (3.28) \nUsing once more (3.15), we establish \np 0 q 1 + p a η ab p b = ( p a + ip 0 ¯ T a ) η ab ( p b -ip 0 T b ) . (3.29) \nCombining (3.25), (3.26) and (3.29) and using the expression for the perturbative stringcoupling constant (2.22) yields the desired result (3.21).', '3.3 The axion-free case': "Axion-free solutions are solutions with Re z a = 0. For these solutions (3.15) takes the form \nz A (2 Y 0 -ip 0 ) = ip A . (3.30) \nFirst let us assume that 2 Y 0 -ip 0 = Y 0 + ¯ Y 0 = λ /negationslash = 0. In that case we easily derive the following result for the Y I , \nY 0 = 1 2 ( λ + ip 0 ) , Y A = ip A λ + ip 0 2 λ . (3.31) \nConsider the second set of equations (3.14) applied to an arbitrary prepotential of the heterotic/type-II form, \nF ( Y ) = d ABC Y A Y B Y C Y 0 + ic ( Y 0 ) 2 , (3.32) \nwhere c is a real constant. In principle we could allow additional quadratic terms, which would still be explicitly solvable, at least for axion-free solutions. Arbitrary quadratic terms with real coefficients can be easily incorporated by making use of suitable symplectic reparametrization. This will be discussed in the next subsection. For the case above (3.14) now yields the following equations, \nq 0 = d ABC p A p B p C λ 2 +2 cλ , q A = -3 p 0 λ 2 d ABC p B p C , (3.33) \nleading to the condition \n3 p 0 q 0 + p A q A = 6 cλ p 0 . (3.34) \nObserve that the first condition (3.33) can only be satisfied for ( q 0 -2 λ ) d ABC p A p B p C > 0. The entropy can be computed from (3.13) and reads \nS π = -2( q 0 -2 cλ ) [ λ + ( p 0 ) 2 λ ] . (3.35) \nFor cp 0 = 0 we can express λ in terms of the charges, \n/negationslash \nλ = 3 p 0 q 0 + p A q A 6 c p 0 , (3.36) \nOn the other hand, when cp 0 = 0 we have a constraint on the charges, \n3 p 0 q 0 + p A q A = 0 . (3.37) \nFor c = 0 and q 0 = 0 we can express λ as \n/negationslash \nλ = ± √ d ABC p A p B p C q 0 . (3.38) \nPlugging this into (3.33) one can express the charges q A in terms of the remaining ones, q 0 , p 0 , p A . Positivity of the entropy requires q 0 λ < 0. In the following we choose the moduli z A to live on the upper-half plane Im z A > 0 and for convenience we restrict ourselves to charges with q 0 < 0 and p A > 0. Then the moduli z A take the form \nz A = i p A √ q 0 d ABC p A p B p C . (3.39) \nAs a special case, consider the non-axionic solution (3.33) with c = p 0 = 0 and, consequently, q A = 0. This constitutes a solution with only N V +1 independent, non-vanishing charges, which we take to satisfy p A > 0, q 0 < 0, for definiteness. The entropy is given by \nand the moduli are given in (3.39). In particular, for the cubic prepotential (3.16) we find that \nas well as \nS π = 2 √ q 0 d ABC p A p B p C , (3.40) \nz 1 = p 1 z 3 p 3 , z 2 = p 2 z 3 p 3 , z 3 = i √ √ √ √ q 0 p 3 -b p 1 p 2 + a ( p 3 ) 2 , (3.41) \nS π = 2 √ -q 0 ( b p 1 p 2 p 3 -a ( p 3 ) 3 ) . (3.42) \nFor the values b = -3 a = 1, the cubic prepotential (3.16) describes the one-loop corrected heterotic prepotential (2.25) of the S -T -U model in the decompactification limit Re S > \nRe T > Re U → ∞ . Consistency of this limit requires the following ordering of the absolute values of the charges: -q 0 /greatermuch p 1 > p 2 > p 3 /greatermuch 0. It will be shown in section 4 that this hierarchy of charges also guarantees the suppression of α ' corrections. \nAlso note that the solution (3.38) we found for the case cp 0 = 0 is a good approximate solution for the general case c /negationslash = 0 (with general p 0 ). Recalling that c = χζ (3) 16 π 3 , which is of order 1 for typical Calabi-Yau Euler numbers χ with | χ | ≤ 1000, we expect that the constant term in the prepotential will only give a small contribution when the moduli are large. Comparing the exact solution for λ in the case c /negationslash = 0 to the solution (3.38) one can show that both differ by terms of order √ c 2 d ABC p A p B p C q 3 0 , which is small for | q 0 | /greatermuch | p A | , i. e. for large moduli. \nThe second class of solutions corresponds to Y 0 = 1 2 ip 0 , which implies (for finite z A ) that all the p A must vanish. Now the stabilization equations (3.14) imply that \nq A = 3 p 0 d ABC z B z C , q 0 = 0 . (3.43) \nHence the only nonzero charges are p 0 and (some of) the q A . The above N V quadratic equations for the N V purely imaginary parameters z A can usually be solved straightforwardly. Note that there is no dependence on the constant term c in this case, because Y 0 is purely imaginary. \nTo demonstrate this second solution we reconsider the prepotential corresponding to (3.16). The equations for q A take the form \nq 1 = b p 0 TU , q 2 = b p 0 SU , q 3 = b p 0 ST -3 ap 0 U 2 , (3.44) \nwith S , T , U real. These solutions can be solved for S , T and U , \n2 √ b p 0 q 1 q 2 S = 2 √ b p 0 q 2 q 1 T = √ √ √ √ q 3 +2 √ -3 aq 1 q 2 b + √ √ √ √ q 3 -2 √ -3 aq 1 q 2 b , 2 √ -3 ap 0 U = √ √ √ √ q 3 +2 √ -3 aq 1 q 2 b -√ √ √ √ q 3 -2 √ -3 aq 1 q 2 b . (3.45) \nThe charges and the coefficients a and b must be chosen such that S , T and U are positive.", '3.4 The entropy formula in type-II compactifications': "The entropy formula for extreme black holes in type-II compactifications will depend on electric and magnetic charges as well as on topological data of the Calabi-Yau manifold, on which one has compactified the type-II string theory. The topological data appearing \nin the prepotential are the classical intersection numbers C ABC as well as the expansion coefficients c 2 · J A of the second Chern class c 2 of the three-fold, which were defined in section 2.3. These data are related to the (real) coefficients d ABC , c and W 0 A of the associated prepotential, \nF ( Y ) = d ABC Y A Y B Y C Y 0 + W 0 A Y 0 Y A + ic ( Y 0 ) 2 , (3.46) \nby d ABC = -1 6 C ABC and W 0 A = c 2 · J A 24 . \nFor extreme black holes based on the prepotential (3.46), the entropy formula will generically be given by \nS π = | Z hor | 2 = 1 4 A ( ( p I , q I ) , C ABC , c 2 · J A , χ ) . (3.47) \nAs is well known, quadratic polynomials with real coefficients can be introduced into any N = 2 prepotential by a suitable symplectic reparametrization. So the above case (3.46) is covered by our previous analysis, provided we perform the corresponding symplectic rotation on the associated charges, \n( ˜ p I ˜ q I ) = ( 1 0 W 1 )( p I q I ) , (3.48) \nwhere W AB = W 00 = 0. Note that a non-vanishing W 00 would not allow us to eliminate the term ic ( Y 0 ) 2 in the prepotential, because W 00 must be real, wheras ic is imaginary. More general theta shifts with W AB /negationslash = 0 , W 00 /negationslash = 0 would generate quadratic and constant terms in Y 0 with real coefficents. We will discard these terms, because they don't have a topological interpretation. \nIn the following the electric and magnetic charges of the former solution are denoted by ˜ q I and ˜ p I , respectively, whereas the electric and magnetic charges of the latter are denoted by q I and p I . Note that this symplectic transformation induces a shift to the theta angles and thus a corresponding shift of the electric charges [36]. Thus, it follows that the entropy for the former solution can be computed from the entropy for the latter by performing the above substitution of the electric charges. \nConsider, for instance, the axion-free solution (3.40) discussed in the previous subsection, based on the cubic prepotential (2.14), with p 0 = q A = 0 and setting c = 0. Then we have for the symplectically transformed solution that \nq 0 = ˜ q 0 -W 0 A ˜ p A , (3.49) \nand for its entropy that \nS π = 2 √ (˜ q 0 -W 0 A ˜ p A ) d BCD ˜ p B ˜ p C ˜ p D . (3.50) \nThus, we can in particular set ˜ q 0 = 0, that is, we have a solution that is determined by magnetic charges ˜ p A only, which is non-singular and has non-vanishing entropy \nS π = 2 √ -( W 0 A ˜ p A ) d BCD ˜ p B ˜ p C ˜ p D . (3.51) \nIn the effective action, the term proportional to W in (3.46) manifests itself in the presence of the additional term in the action \nδS ∼ ∫ W A 0 F A ∧ F 0 . (3.52) \nSince F 0 is an electric gauge field and F A is a magnetic monopole field, this integral is non-vanishing.", '4 Relation to higher-dimensional geometries': "The black-hole solutions discussed so far appeared in the context of either a compactification of the heterotic string on K 3 × T 2 or of the type-II string on a Calabi-Yau three-fold. Type-II string theory, on the other hand, is dual [9] to M -theory compactified on CY × S 1 [39]. In this section we discuss how the black-hole geometries associated with (3.40) arise from a compactification of the higher-dimensional spacetime, that is, by a compactification of M -theory. We focus on those black-hole geometries that can either be obtained by a type-II string compactification on a Calabi-Yau three-fold with h 1 , 1 = 3, or that are associated with the S -T -U models on the heterotic side. \nOn the M -theory side, we can regard these black-hole solutions as arising from compactifications of certain 11-dimensional solutions describing three intersecting M -5-branes with a boost along the common string. Let us first consider the simplest such 11-dimensional solution, which can be compactified on a 6-dimensional torus, [37]: \nds 2 11 = 1 ( H 1 H 2 H 3 ) 1 3 [ dudv + H 0 du 2 + H 1 H 2 H 3 d/vectorx 2 + + H 1 ( dy 2 1 + dy 2 2 ) + H 2 ( dy 2 3 + dy 2 4 ) + H 3 ( dy 2 5 + dy 2 6 ) ] . (4.1) \nHere, the H 1 , H 2 and H 3 parametrize the three 5-branes and they are harmonic functions with respect to /vectorx . The internal space is spanned by the coordinates y . Each 5-brane wraps around a 4-cycle; e.g. the H 1 -5-brane around ( y 3 , y 4 , y 5 , y 6 ), and any two 4-cycles intersect each other in a 2-cycle. \nNext, let us look at more complicated 11-dimensional solutions which can be compactified on Calabi-Yau three-folds. For a generic Calabi-Yau three-fold, the intersection of three \nof the 4-cycles is determined by the classical intersection numbers C ABC . This leads us to make the following ansatz for the 11-dimensional metric, in analogy to (4.1), \nds 2 11 = 1 ( 1 6 C ABC H A H B H C ) 1 3 [ dudv + H 0 du 2 + 1 6 C ABC H A H B H C d/vectorx 2 + H A ω A ] , (4.2) \nwhere ω A ( A = 1 , 2 , 3) are the 2-dimensional line elements, which correspond to the intersection of two of the 4-cycles. Below, we will fix the harmonic functions H A for the solution (3.40) in the double extreme limit. \nAfter compactifying the internal coordinates in (4.2), we obtain a magnetic string solution in D = 5 dimensions. Similarly to the extreme Reissner-Nordstrom black hole in D = 4 dimensions, this magnetic solution has a non-singular horizon with the asymptotic geometry AdS 3 × S 2 [40]. In order to obtain a regular solution in D = 4 dimensions as well, we first have to perform a boost along this string (parameterized by H 0 ), which will keep the compactification radius G uu finite everywhere. This boost will induce momentum modes propagating along the magnetic string. Turning off these modes has the consequence that this radius shrinks to zero size on the horizon and that the solution becomes singular. Thus, performing the boost adds one electric charge to the three magnetic charges. Then, all the radii of the Calabi-Yau 2- and 4-cycles as well as of the string will also stay finite on the horizon. The resulting 4-dimensional metric defines an extreme Reissner-Nordstrom geometry given by \nds 2 4 = -1 √ -1 6 H 0 C ABC H A H B H C dt 2 + √ -1 6 H 0 C ABC H A H B H C d/vectorx 2 . (4.3) \nNext, let us consider the dual heterotic string solution with fields S , T and U . This will allow us to determine the harmonic functions H A . We will restrict ourselves to the classical solution, that is to (3.42) with b = 1 , a = 0. \nFirst, we will have to change the symplectic basis. That is, we will have to go from the basis corresponding to (2.17) to the perturbative basis preferred by the heterotic string. This requires a symplectic reparametrization, after which p 1 is no longer a magnetic, but an electric charge: p 1 → -q 1 . Hence in the heterotic string basis the solution is now characterised by 2 magnetic ( p 2 , p 3 ) and 2 electric ( q 0 , q 1 ) charges. The classical S -T -U black hole can then be obtained from the 6-dimensional solution [38] \nds 2 6 = 1 H 1 ( dudv + H 0 du 2 ) + H 2 ( 1 H 3 ( dx 4 + /vector V d/vectorx ) 2 + H 3 d/vectorx 2 ) (4.4) \n( /epsilon1 ijk ∂ j V k = ∂ i H 3 ). It describes a fundamental string lying in a solitonic 5-brane. Again, in order to keep the compactification radii finite, we need to perform a boost along the string and put a Taub-NUT soliton in the transversal space. From the resulting solution, \nwe can immediately read off the S, T and U fields. By compactifying over u and x 4 , we obtain for the internal metric that \nG rs = ( H 0 H 1 0 0 H 2 H 3 ) = H 2 H 3 ( (Re U ) 2 0 0 1 ) , (4.5) \nand thus we find for the scalar fields that \nS = e -2 φ = e -2 ˆ φ √ | G rs | = √ H 0 H 1 H 2 H 3 , T = √ | G rs | = √ H 0 H 2 H 1 H 3 , U = √ H 0 H 3 H 1 H 2 (4.6) \n( ˆ φ is the 6-dimensional dilaton). In the double extreme limit, we have to fix the values of the scalars at infinity so that they are constant everywhere. For the harmonic functions this means that \nH 0 = √ 2 q 0 ( c + 1 r ) , H 1 = √ 2 q 1 ( c + 1 r ) , (4.7) H 2 = √ 2 p 2 ( c + 1 r ) , H 3 = √ 2 p 3 ( c + 1 r ) , \nwith c -4 = 4 q 0 q 1 p 2 p 4 (in order to obtain asymptotic Minkowski geometry in D = 4). The limit of large q 0 now has the consequence that the boost or momentum along the string becomes large. Hence, this direction decompactifies and we obtain the 5-dimensional string solution. The metric (in the Einstein frame) is in this case again given by (4.3) with C 123 = 6 as the only non-vanishing element. \nWe can now insert the harmonic functions (4.7) into the metric (4.3) with general coefficients C ABC . In this way we precisely recover the metric (3.3). Note that in our notation -q 0 1 6 C ABC p A p B p C > 0. When approaching the horizon r → 0, we obtain the Bertotti-Robertson geometry which is non-singular ( AdS 2 × S 2 ) and restores all supersymmetries. The radius of the S 2 is given by the mass and so the area of the horizon is A = 4 πM 2 = 8 π √ -q 0 1 6 C ABC p C p B p C . This is the metric in the Einstein frame. The string, however, couples to the string-frame metric, which can easily be given on the heterotic side. Replacing -1 6 C ABC by d ABC and using the dilaton value S = -iz 1 with z 1 given by (3.39), we find that \nds 2 str = √ q 0 d ABC p A p B p C | q 0 p 1 | ds 2 (4.8) = -1 | q 0 p 1 | ( c + 1 r ) -2 dt 2 -d ABC p A p B p C p 1 ( c + 1 r ) 2 d/vectorx 2 \n(for q 0 < 0 and p A > 0). This again has a throat geometry for r → 0 \nds 2 str →-e 2 η/R dt 2 + dη 2 + R 2 d Ω 2 , R 2 = -d ABC p A p B p C p 1 (4.9) \n( r ∼ exp( η/R )). Since the curvature has its maximum inside the throat, we can keep higher curvature corrections ( ∼ O ( α ' )) under control if the radius of the throat is sufficiently large: -d ABC p A p B p C /greatermuch p 1 . This means that sufficiently large magnetic charges ensure that all higher curvature terms can be suppressed.", '5 Summary': 'Supersymmetric black holes provide us with a tool to probe the properties of the future fundamental theory which will describe non-perturbative quantum gravity. This theory is expected to explain in a quantum-mechanical context the existence of all non-perturbative states, or solitons, in string theory and in supergravity and also to control the interaction between these states. Meanwhile, in the absence of such a theory, it is important to study supersymmetric black holes and their properties as the most particular representatives of the non-perturbative states of quantum gravity. One of the remarkable property of all ( N = 2 , D = 4) supersymmetric black holes is the topological nature of the area of the black hole horizon in the sense that the area does not depend on the values of the moduli fields at spatial infinity [2]. The explanation of the entropy via the counting of string states [1] was thus established for a class of black holes for which the entropy depends only on electric and magnetic charges. In this paper we have found various new area formulae for a class of N = 2 , D = 4 supersymmetric theories. The choice of the prepotentials is motivated by various versions of string theory at the classical level as well as by string-loop corrections. \nOn the heterotic side we have studied the prepotentials which include the contributions from string-loop corrections. At the perturbative level, we could prove that the entropy takes precisely the same form as the tree-level entropy [4, 13], where the tree-level coupling constant S + ¯ S is replaced by the perturbative coupling constant, which originates entirely from one-loop effects and contains the Green-Schwarz modification: \nS π = ( S + ¯ S + V GS ) ∣ ∣ hor ( p 0 q 1 + p a η ab p b ) . (5.1) \n∣ \nTherefore, we confirmed the conjecture [13] that the string loops will affect the area formula only via a perturbative modification of the string coupling. In case that the oneloop heterotic prepotential can be approximated by a cubic polynomial, as it is true for large moduli values, the one-loop string coupling can be explicitly expressed in terms of the magnetic/electric charges for non-axionic black-hole solutions. For solutions that are not axion-free, the explicit expressions depend on solving some higher-order polynomial equation, as exhibited in section 3.1. \nOn the type-IIA side, our new area formulae imprint also the topological data of the Calabi-Yau manifold, in particular the intersection numbers C ABC . The fact that this symmetric tensor enters the area formula for the five-dimensional black holes was known before [3]. However, in five dimensions the area formula is implicit, as one still has to minimize it in the moduli space. Here, for the first time, we have found the area formulae of four-dimensional black holes which depend on charges and on arbitrary intersection numbers C ABC . In addition, we have found an interesting dependence of the area on the second Chern class of the three-fold c 2 . Here we deal with the Witten-type shift [36] of the electrical charge via magnetic charge in the presence of axions. Finally the entropy will in general depend on the Euler number χ of the Calabi-Yau three-fold, but this contribution is small compared to the other effects we studied. \nIn the simplest case, when all the moduli z A are imaginary (the axion-free solution with p 0 = q A = 0 and c = 0), the entropy is given by \nS = 2 π √ ( -q 0 + c 2 · J A 24 p A ) C BCD 6 p B p C p D . (5.2) \nThis formula reproduces some previously known solutions, in particular for the S -T -U black holes [12], where C 123 = 6 and c 2 = 0, and where the entropy of the simplest nonaxion solution was found to be S = 2 π √ | q 0 p 1 p 2 p 3 | . One can now address the following issue: which fundamental theory is capable of giving a microscoping interpretation to (5.2)? \nAn interesting feature of the new area formulae is the possibility to relax some of the electric charges due to the above mentioned shift effect via c 2 · J A terms. A simple example of such a relaxation is as follows. When applying the theta-angle shift to the S -T -U black hole [12] with one magnetic and three electric charges, which has the area formula S = 2 π √ | p 0 q 1 q 2 q 3 | , we obtain an entropy formula which is non-vanishing for q 1 = q 2 = q 3 = 0, even though there now is only one magnetic charge present: S /π = 2( p 0 ) 2 √ 1 24 3 | ( c 2 · J 1 )( c 2 · J 2 )( c 2 · J 3 ) | . \nIn conclusion, we have found various qualitatively new features of supersymmetric black holes in N = 2 , D = 4 supergravity theories motivated by string theory.', 'Acknowledgements': 'This work is supported by the European Commission TMR programme ERBFMRXCT96-0045. 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2010ApJ...708..427L

Type 2 Active Galactic Nuclei with Double-Peaked [O III] Lines: Narrow-Line Region Kinematics or Merging Supermassive Black Hole Pairs?

2010-01-01

['black hole physics', 'cosmology observations', 'galaxies active', 'galaxies quasars', 'surveys', '-']

[]

We present a sample of 167 type 2 active galactic nuclei (AGNs) with double-peaked [O III] λλ4959,5007 narrow emission lines, selected from the Seventh Data Release of the Sloan Digital Sky Survey. The double-peaked profiles can be well modeled by two velocity components, blueshifted and redshifted from the systemic velocity. Half of these objects have a more prominent redshifted component. In cases where the Hβ emission line is strong, it also shows two velocity components whose line-of-sight (LOS) velocity offsets are consistent with those of [O III]. The relative LOS velocity offset between the two components is typically a few hundred km s<SUP>-1</SUP>, larger by a factor of ~1.5 than the line full width at half maximum of each component. The offset correlates with the host stellar velocity dispersion σ<SUB>*</SUB>. The host galaxies of this sample show systematically larger σ<SUB>*</SUB>, stellar masses, and concentrations, and older luminosity-weighted mean stellar ages than a regular type 2 AGN sample matched in redshift, [O III] λ5007 equivalent width, and luminosity; they show no significant difference in radio properties. These double-peaked features could be due to narrow-line region kinematics, or binary black holes. The statistical properties do not show strong preference for or against either scenario, and spatially resolved optical imaging, spectroscopy, radio or X-ray follow-up are needed to draw firm conclusions.

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https://arxiv.org/pdf/0908.2426.pdf

{'TYPE 2 AGNS WITH DOUBLE-PEAKED [O III ] LINES: NARROW LINE REGION KINEMATICS OR MERGING SUPERMASSIVE BLACK HOLE PAIRS?': 'XIN LIU 1 , YUE SHEN 1 , MICHAEL A. STRAUSS 1 , AND JENNY E. GREENE 1,2 Accepted to ApJ, November 3, 2009', 'ABSTRACT': 'We present a sample of 167 type 2 AGNs with double-peaked [O III ] λλ 4959,5007 narrow emission lines, selected from the Seventh Data Release of the Sloan Digital Sky Survey. The double-peaked profiles can be well modeled by two velocity components, blueshifted and redshifted from the systemic velocity. Half of these objects have a more prominent redshifted component. In cases where the H β emission line is strong, it also shows two velocity components whose line-of-sight (LOS) velocity offsets are consistent with those of [O III ]. The relative LOS velocity offset between the two components is typically a few hundred kms -1 , larger by a factor of ∼ 1 . 5 than the line full width at half maximum of each component. The offset correlates with the host stellar velocity dispersion σ ∗ . The host galaxies of this sample show systematically larger σ ∗ , stellar masses, and concentrations, and older luminosity-weighted mean stellar ages than a regular type 2 AGN sample matched in redshift, [O III ] λ 5007 equivalent width and luminosity; they show no significant difference in radio properties. These double-peaked features could be due to narrow-line region kinematics, or binary black holes. The statistical properties do not show strong preference for or against either scenario, and spatially resolved optical imaging, spectroscopy, radio or X-ray followup are needed to draw firm conclusions. \nSubject headings: black hole physics - galaxies: active - cosmology: observations - quasars: general - surveys', '1. INTRODUCTION': "Binary 3 supermassive black holes (SMBHs) are possible outcomes of the hierarchical mergers of galaxies (e.g., Begelman et al. 1980; Milosavljevi'c & Merritt 2001; Yu 2002). In one of the leading hypotheses, major mergers between galaxies are responsible for triggering nuclear starbursts and quasar activity (e.g., Hernquist 1989). Despite the success of the merger scenario in explaining much of the observed phenomenology of AGN statistics (e.g., Kauffmann & Haehnelt 2000; Volonteri et al. 2003; Wyithe & Loeb 2003; Hopkins et al. 2008; Shen 2009) and the cores of massive elliptical galaxies (e.g., Faber et al. 1997; Kormendy & Bender 2009), direct observational evidence for binary SMBHs is surprisingly scarce. \nThe fraction of binary quasars at separations of tens to hundreds of kpc (halo) scales, is /lessorsimilar 0 . 1% at 1 /lessorsimilar z /lessorsimilar 5 (e.g., Hennawi et al. 2006; Myers et al. 2008; Hennawi et al. 2009). On kpc (galactic) scales there are only a handful of unambiguous low-redshift cases known, in which both active SMBHs are detected in X-rays (NGC 6240 and Mrk 463, Komossa et al. 2003; Bianchi et al. 2008), or in the radio (3C 75, Owen et al. 1985). On sub-kpc scales, only one case is known(0402+379,Rodriguez et al. 2006) of a pair of BHs detected by VLBI with a projected separation of ∼ 7 pc. There are several possible reasons that binary SMBHs are so rare compared to the expectations from the merger scenario: they spend most of their time at separations far below kpc scales and hence are difficult to resolve spatially; one or both of the SMBHs are heavily obscured; both BHs are rarely active at the same time, and dynamical differences between a single BH and a binary BH are not readily discernable at cosmolog- \n- 3 Hubble Fellow, Princeton-Carnegie Fellow \nical distances. \nRecently, Comerford et al. (2009a) conducted a systematic survey of type 2 AGNs in the DEEP2 galaxy sample at ¯ z ∼ 0 . 6 (Davis et al. 2003). They argue that 37 of the 107 AGNs in their sample are inspiralling binary SMBHs, based on significant velocity offsets between the narrow-line region (NLR) emission lines [O III ] λλ 4959,5007 and the systemic redshift measured from stellar absorption features. In particular, two of these objects (one reported by Gerke et al. 2007) show double-peaked [O III ] lines with velocity offsets of a few hundred kms -1 and spatial offsets of several kpc; they interpreted these as binary SMBHs when both BHs are active. Such double-peaked [O III ] emission line features have also been seen in a few optically-selected AGNs (e.g., Heckman et al. 1981; Zakamska et al. 2003; Zhou et al. 2004). One advantage of using spectroscopy to find binary SMBHs is that it is not limited by spatial resolution, as long as the separation is not so small that the NLRs are no longer distinct. On the other hand, such spectral features may be due to biconical outflows or disk rotation around a single SMBH (e.g., Axon et al. 1998; Veilleux et al. 2001; Crenshaw et al. 2009). Here we focus on double-peaked narrow line objects in which the associated line emitting gas is on scales of ∼ 100 pc to several kpc. Therefore in the binary SMBH scenario, our double-peaked sample would include objects whose two SMBHs along with their own NLR gas are still co-rotating in the galactic potential, well before forming a pc-scale binary under dynamical friction (e.g., Milosavljevi'c & Merritt 2001). For doublepeaked broad line objects which may involve outflows or disks on /lessorsimilar pc scales, see, e.g., Eracleous & Halpern (1994) and Strateva et al. (2003). \nWe have carried out a systematic search in the SDSS (York et al. 2000) spectroscopic database for emission line AGNs with double-peaked features or peculiar [O III ] line shifts relative to the systemic redshift as measured from stellar absorption features. Here we report our initial results on a sample of 167 type 2 AGNs with double-peaked [O III ] emis- \nTABLE 1 THE SAMPLE \n| SDSS Designation | Plate | Fiber | MJD | z | σ ∗ | FWHM[OIII] , 1 | FWHM[OIII] , 2 | V [O III] , 1 | V [O III] , 2 | V H β , 1 | V H β , 2 |\n|-------------------------------------------|---------|---------|-------|----------|-------|------------------|------------------|-----------------|-----------------|-------------|-------------|\n| J000249 . 07 + 004504 . 8 . . . . . . . . | 388 | 345 | 51793 | 0 . 0868 | 243 | 271 | 219 | - 361 | 168 | - 282 | 197 |\n| J000911 . 58 - 003654 . 7 . . . . . . . . | 388 | 148 | 51793 | 0 . 0733 | 203 | 231 | 220 | - 198 | 134 | - 196 | 129 |\n| J010750 . 48 - 005352 . 9 . . . . . . . . | 670 | 55 | 52520 | 0 . 5202 | 0 | 281 | 645 | - 546 | 37 | - 540 | - 113 |\n| J011659 . 59 - 102539 . 1 . . . . . . . . | 660 | 213 | 52177 | 0 . 1503 | 181 | 304 | 203 | - 159 | 145 | - 81 | 189 | \nNOTE. - The full table is available in the electronic version of the paper. The subscripts '1' and '2' denote blueshifted and redshifted components. V [O III] and V H β denote velocity offsets relative to the systemic redshift measured over [O III ] λλ 4959,5007 and H β . Velocities are in units of km s -1 . The entry reads zero if the quantity is unmeasurable. The typical uncertainty in σ ∗ is ∼ 15kms -1 . For [O III ], the typical statistical errors in the best-fit velocity offsets and FWHMs are ∼ 5 km s -1 and ∼ 10 km s -1 . For H β , the statistical errors are generally a few times larger because H β is weaker than [O III ]. \nFIG. 1.- Redshift distribution of the parent AGN and double-peaked samples. \n<!-- image --> \nsion lines 4 . We describe our sample selection in §2; the statistical properties of the sample are presented in §3, and we discuss our results and conclude in §4. A cosmology with Ω m = 0 . 3, Ω Λ = 0 . 7, and h = 0 . 7 is assumed throughout.", '2.1. The Sample': 'Our parent sample is the MPA-JHU SDSS DR7 galaxy sample 5 drawn from the SDSS DR7 spectroscopic database (Abazajian et al. 2009); those include objects spectrally classified as galaxies by the specBS pipeline (Adelman-McCarthy et al. 2008) or quasars that are targeted as galaxies (Strauss et al. 2002; Eisenstein et al. 2001) and have redshifts z < 0 . 7. We adopt redshifts z and stellar velocity dispersions σ ∗ from the specBS pipeline. When unavailable from the pipeline, we measure σ ∗ using the direct-fitting algorithm described in Greene & Ho (2006) and Ho et al. (2009); we have checked that this code gives results consistent with those of specBS . We have checked the accuracy of the redshift by refitting the stellar continuum \n4 After we submitted the paper, two similar statistical studies by Smith et al. (2009) and Wang et al. (2009) appeared on arXiv. Smith et al. (2009) focus on broad line AGNs with double-peaked [O III ] lines selected from SDSS quasar spectra and find a similar fraction ( ∼ 1%) of doublepeaked objects that we found. Because of pipeline misclassification, their double-peaked sample also contains narrow line objects, some of which overlap with our catalog. Wang et al. (2009) studied a sample of 87 doublepeaked narrow line AGNs of which 42 objects overlap with our sample. \nwith galaxy templates (Liu et al. 2009), as well as comparing multiple epoch spectra of the same objects. The quoted redshift faithfully traces the stellar absorption features (which we take as the systemic redshift), with typical statistical errors of a few kms -1 and systematic errors of ∼ 10 km s -1 . Additional spectral and photometric properties such as emission-line fluxes (determined from equivalent widths and continuum fluxes), spectral indices, stellar masses, and concentration indices are taken from the MPA-JHU SDSS DR7 data product. \nWe select our AGN sample from this parent sample by the following criteria: 1) the rest-frame wavelength ranges [4700 , 5100] Å and [4982 , 5035] Å centered on the [O III ] λ 5007 line have median signal-to-noise ratio (S/N) > 5 pixel -1 and bad pixel fraction < 30%; 2) the [O III ] λ 5007 line is detected at > 5 σ and has a rest-frame equivalent width (EW) > 4 Å; 3) the line flux ratio [O III ] λ 5007/H β > 3 if z > 0 . 33, or the diagnostic line ratios [O III ] λ 5007/H β and [N II ] λ 6584/H α lie above the theoretical upper limits for starformation excitation from Kewley et al. (2001) on the BPT diagram (Baldwin et al. 1981) if z < 0 . 33. This procedure yields ∼ 14,300 type 2 AGNs. We supplement this sample with ∼ 400 type 2 quasars from Reyes et al. (2008) which are not included in the MPA-JHU data products. These additional type 2 quasars extend to somewhat higher redshifts ( z < 0 . 83). Our final AGN sample includes 14,756 objects with high S/N and high spectral quality around the [O III ] lines, suitable for the analysis that follows. The redshift distribution of this parent AGN sample is shown in Figure 1. \nIn many AGNs and starbursts, the narrow forbidden emission lines [O III ] λλ 4959,5007 are known to have peculiarities such as extended blue wings, velocity offsets from systemic redshifts, and complex line profiles (e.g., Heckman et al. 1981; Whittle 1985a; Zakamska et al. 2003; Zhou et al. 2006; Komossa et al. 2008). We here focus on a more dramatic subset - those with double-peaked [O III ] emission lines. A complete analysis of the nature and statistical properties of other peculiar [O III ] line characteristics will be presented elsewhere. \nThe identification of such double-peaked objects requires that both [O III ] λ 4959 and [O III ] λ 5007 are better fit with two components rather than a single component. As the initial screening process we visually inspected the spectra of all the 14,756 AGNs and identified 167 objects with unambiguous double-peaked [O III ] lines. We only include objects which have well-detected double peaks in both [O III ] λ 4959 and [O III ] λ 5007 with similar profiles; we do not include those with complex line profiles such as lumpy, winged, or multicomponent features. We plot the redshift distribution of the double-peaked sample in Figure 1, and the SDSS images and \nFIG. 2.- Example double-peaked AGNs. Shown here are SDSS gri color-composite images and host-subtracted spectra (data points in red and smoothed curve in black) along with our best fits (model in green and individual components in magenta) for the H β -[O III ] region. The vertical lines are drawn at the systemic redshift of each galaxy as measured from stellar absorption features. \n<!-- image --> \nspectra of three such examples in Figure 2. We list all the 167 double-peaked AGNs and their line measurements in Table 1. We caution that there could be some double-peaked narrow-line objects in SDSS DR7 missed by our selection if the pipeline got the redshifts wrong because of the peculiar line profiles, although this fraction is likely to be no larger than a few percent. \nWe measured the line properties of these double-peaked objects as follows. The galaxy continuum was subtracted using the best-fit template constructed from a linear combination of instantaneous starburst models of Bruzual & Charlot (2003) with ten different ages, as described in Liu et al. (2009). In the 11 objects for which the stellar continuum is too weak for measuring σ ∗ and for template fitting, a simple powerlaw model was used. The continuum-subtracted [O III ] region ( λλ 4930 -5040Å) was fit by a pair of Lorentzian functions convolved with the measured instrumental resolution of the spectra ( σ ∼ 65 km s -1 ). The redshift and line width for each velocity component of [O III ] λ 4959 and [O III ] λ 5007 \nwere forced to be the same. The flux ratio of [O III ] λ 5007 to [O III ] λ 4959 was allowed to vary (but we found it was always close to 3). In cases where H β is measurable, we further fit a double-Lorentzian to the H β region ( λλ 4850 -4880 Å), where the positions of the centroids were allowed to vary, but the widths of the two H β components were fixed to the bestfit values for the two [O III ] components. We then repeated this fitting process with a double-Gaussian model, and took as the best fit the model which had the smaller reduced χ 2 for each object. We visually inspected all the fits and verified that the line profiles are generally well reproduced by the model. For [O III ], the typical statistical errors in the best-fit velocity difference between the two components v off, line fluxes and FWHMsare ∼ 5 km s -1 , ∼ 5%, and ∼ 10 km s -1 , but we caution that the actual uncertainties are likely to be larger. For H β , the statistical errors are generally a few times larger because H β is weaker than [O III ]. \nFIG. 3.- Selection completeness as a function of median S / N , v off, FWHM, total [O III ] λ 5007 EW and the flux ratio between the two components, as determined from our Monte Carlo simulations. In the middle panel, black and blue lines are for the two components respectively. \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFIG. 4.- Narrow-line region physical properties of the double-peaked components. Objects with stronger redshifted components are plotted as red open circles whereas those with stronger blueshifted components are plotted as blue plus signs. The Spearman correlation coefficient and the probability of null correlation are labelled in each plot for the whole ( black ), the stronger-blueshifted ( blue ), and the stronger-redshifted ( red ) samples, respectively. (a). Ionization parameter (indicated by the flux ratio between [O III ] λ 5007 and H β ) of the blueshifted component versus that of the redshifted component. (b). Line width ratio of the double-peaked components versus Balmer decrement (as a measure of reddening). (c). Electron density (inferred from the flux ratio between the [S II ] λλ 6717,6731 doublet) of the blueshifted component versus that of the redshifted component. \n<!-- image --> \nOur selection of double-peaked objects was done by eye and is by no means objective. To gain some sense of our completeness we performed Monte Carlo simulations. We generated mock spectra of double-peaked objects in the H β -[O III ] region with random distributions of continuum S/N, EW, velocity offset, and FWHM of the two components of [O III ]. We randomly assigned a Gaussian or Lorentzian profile to each line and broadened the spectrum using an instrumental Gaussian broadening of σ = 65 km s -1 . In this way we generated 2,000 mock spectra of double-peaked objects and mixed them with 18,000 mock spectra of single-peaked objects with random spectral properties. We then visually inspected the 20,000 mock spectra and identified double-peaked objects in the same way as we did for the real sample. We show in Figure 3 the selection completeness as a function of median S/N, v off, FWHM, total [O III ] λ 5007 EW, and flux (EW) ratio between the two components f EW. In general the completeness increases with increasing v off, [O III ] λ 5007 EW, and f EW, and decreases with increasing FWHM, as expected from the way our double-peaked objects are identified. As a result there is a selection bias against objects with small v off and large FWHM that we discuss later. In addition we are very incomplete for objects with v off /lessorsimilar 200kms -1 partly due to the spectral resolution limit 6 . On the other hand, we find that the false de- \nion rate is very low ( < 2% of the double-peaked sample) - probably an advantage of visual inspection over automated algorithms. We note that the completeness estimated this way is not the absolute completeness, which depends on the actual underlying distributions of all relevant properties. Nevertheless, these completeness estimates are useful to correct for selection biases. For example, when other properties are fixed, double-peaked objects with larger EWs are easier to identify, by a relative amount that can be determined from our simulation results. \nAfter accounting for the dependence of selection completeness on [O III ] EW, we find tentative evidence that the fraction of double-peaked objects in the underlying AGN sample depends on [O III ] λ 5007 luminosity ( L [OIII]): it increases by a factor of ∼ 2 from L [OIII] ∼ 10 7 . 0 L /circledot to L [OIII] ∼ 10 8 . 5 L /circledot , and has no clear dependence on redshift in the range probed. In most cases the two velocity components are redshifted and blueshifted with respect to the systemic redshift (except that in a few cases, one component is apparently coincident with the systemic velocity within the errors), and about half of them have more prominent redshifted components - a subset which may be more unusual since objects with outflows may tend to have more prominent blueshifted components (see below). In cases where H β is measurable, it also shows a double-peaked feature with a similar velocity offset as that of [O III ]. Hereafter we use v off measured from [O III ] in our analysis as it is more robust than that from H β . As illustrated in Figure 4(a), both velocity components have excitation diagnostic line ra- \nFIG. 5.- Host galaxy properties (stellar velocity dispersion σ ∗ , stellar mass M ∗ , spectral indices Dn (4000) and H δ A indicating the luminosity-weighted mean stellar age and the post-starburst fraction, and r -band concentration index R 90 , r / R 50 , r ) of the double-peaked sample ( red ) compared to the control sample ( black ). The control sample is randomly drawn from the parent AGN sample to have identical redshift and [O III ] λ 5007 EW distributions as the double-peaked sample. For each distribution the median is marked with a vertical line. \n<!-- image --> \n/[O III ] characteristic of AGNs 7 . The FWHM of each velocity component is typically smaller than v off by a factor of ∼ 1 . 5.', '3.1. Host Galaxy Properties': 'Are there any characteristics of the host galaxies of the double-peaked objects that are different from ordinary AGNs? The double-peaked sample is not selected uniformly as we demonstrated in §2.2. To account for selection biases, we construct a control sample from the underlying AGN sample with identical redshift, [O III ] λ 5007 EW and luminosity distributions as the double-peaked sample. We compare their host-galaxy properties, including stellar mass and velocity dispersion, SDSS colors, stellar population parameters Dn (4000) and H δ A (indicating the luminosity-weighted mean stellar age and the post-starburst fraction in the past ∼ 0.1-1 Gyr, respectively; e.g., Kauffmann et al. 2003), intrinsic extinction (estimated by the Balmer decrement), effective radius, concentration (characterized by, R 90 , r / R 50 , r , the ratio of radii that contain 90% and 50% of the r -band Petrosian flux), and inclination (estimated by the ratio between the major and minor isophotal axis). As shown in Figure 5, the double-peaked sample has systematically larger stellar velocity dispersions and masses, older mean stellar ages (and redder colors), smaller fractions of post-starburst populations in the past 0.1-1 Gyr, and higher concentrations than the control sample. The difference in the centroids of the two distributions are roughly half of the width of the distributions. Kolmogorov-Smirnov (KS) tests show that the probabilities that the two samples are drawn from the same distribution are P KS < 10 -3 . All other properties of the host galaxies we examined were indistinguishable with KS probabilities P KS > 10 -1 . We also match the double-peaked and the control samples with the FIRST (Becker et al. 1995; White et al. 1997) and ROSAT (Voges et al. 1999, 2000) surveys and find that they contain very similar fractions of matches with both surveys ( ∼ 35% with FIRST and ∼ 2% with ROSAT). The subset that has a more prominent redshifted component is indistinguishable from the rest of the double-peaked sample in terms of the properties studied above.', '3.2. Correlations between Line Properties': 'There are several apparent correlations among the dynamical properties and the NLR physical conditions of the \ndouble-peaked sample. Figures 4(a) and 4(c) illustrate that the two velocity components have similar ionization parameters (as indicated by the diagnostic line ratio H β /[O III ]) and electron densities (measured from the ratio of the doublet [S II ] λλ 6717,6731 for 107 objects in the sample which have low enough redshifts to have [S II ] coverage in the SDSS spectra and whose inferred electron densities of both components fall in the range of [10, 10 4 ] cm -3 ). Figures 6(a)-(d) show that v off is correlated with both σ ∗ and the line FWHMs of the two components, and the line FWHMs are correlated with each other. In all these four relations, the correlation for objects with more prominent redshifted components appears somewhat stronger than that for the rest. There is some selection bias at the low v off versus large FWHM corner, where the double-peaked feature is difficult to identify. However, our Monte-Carlo simulations show no selection bias against objects with high v off and small FWHMs (§2.2). In particular, the correlation between v off and σ ∗ suggests that the doublepeaked line-emitting regions are on galactic scales where the galactic potential dominates the bulk kinematics. Furthermore, Figures 6(e) and 6(f) show that both the [O III ] line flux ratio and the line width ratio of the two velocity components are anti-correlated with the ratio of their LOS velocity offsets relative to the systemic redshift (see also Wang et al. 2009), which most likely results from momentum conservation.', '4.1. NLR Kinematics': 'The double-peaked feature may result from particular NLR geometries such as biconical outflows 8 or rotating disks on kpc scales. In these scenarios, there is only one SMBH and the observed blueshifted and redshifted emission-line peaks arise from NLR gas moving towards and away from us. Certain nearby Seyfert galaxies known to have biconical outflows show double-peaked emission when spectra are taken over the whole galaxy. Such prototypical cases include NGC 1068 (e.g., Axon et al. 1998) and NGC 3079 (e.g., Duric & Seaquist 1988). It is very likely that there are some such sources in our sample. Several objects in our sample show clear galactic disks in their SDSS images. Here the correlation between line width and velocity shift naturally arises, as the bulge gravitational potential well influences the motion of NLR gas. The correlation between the diagnostic line ratio [O III ]/H β of the two components shown in Figure 4(a) may \n8 In principle, a NLR with inflowing gas and dust can also produce the observed features, but this scenario seems less favored considering the difficulty of removing high angular momentum given the large v off (e.g., Heckman et al. 1981). \nFIG. 6.- Dynamical properties of the double-peaked components. Objects with stronger redshifted components are plotted as red open circles whereas those with stronger blueshifted components are plotted as blue plus signs. The Spearman correlation coefficient and the probability of null correlation are labelled in each plot for the whole ( black ), the stronger-blueshifted ( blue ), and the stronger-redshifted ( red ) samples, respectively. (a)-(c). stellar velocity dispersion σ ∗ , FWHM of the blueshifted component, and FWHM of the redshifted component versus velocity offset between the double-peaked components. (d). FWHM of the blueshifted component versus that of the redshifted component. (e) Velocity-offset ratio of the double-peaked components versus the flux ratio. (f). Velocity-offset ratio of the double-peaked components versus the FWHM ratio. \n<!-- image --> \nbe explained if the NLR gas of both velocity components is ionized by a single SMBH. \nCan all the objects in our double-peaked sample be ascribed to NLR geometry? Outflows and inflows of NLR gas have long been invoked to explain several commonly observed features in NLR high ionization lines (such as [O III ]), including extended wings (usually blueshifted from the systemic redshift), line asymmetries, and broader line widths than those of low ionization lines (e.g., Heckman et al. 1981; Penston et al. 1984; Rodríguez-Ardila et al. 2006). In the cases of NGC 1068 (e.g., Axon et al. 1998) and Mrk 78 (e.g., Heckman et al. 1981; Pedlar et al. 1989; Whittle & Wilson 2004) known to have biconical outflows, the velocity splitting relative to the systemic redshift is larger on the blueshifted component, which is also brighter and has a larger width than the redshifted component. The statistical significance of these trends is unclear, however (e.g., Whittle et al. 1988). Unlike these, our double-peaked sample has comparable line widths in the blueshifted and redshifted components as shown in Figure 6(d); half of our double-peaked objects have a more prominent redshifted component than the blueshifted component. In addition, our sample has consistent v off in H β and in [O III ] whenever H β is measurable, with no significant ionization stratification as expected in NLR outflows. These possible discrepancies may arise because our sample has appreciably larger [O III ] luminosity than these previous low redshift comparison samples, and these properties may be a function of lu- \nosity or redshift or both; alternatively, they may suggest that not all objects in our double-peaked sample exhibit NLR outflows. \nFurther possible clues come from host galaxy and line properties. The FIRST detected fraction for the double-peaked objects is comparable to that of the control sample. The hostgalaxy inclination distribution shows no significant difference either, whereas our selection of a large LOS v off for the double components would bias towards either edge-on or face-on inclinations for the rotating disk or bipolar outflow populations, unless the outflow orientation is not closely linked to the angular momentum axis of the host galaxies. Radio outflows in nearby Seyfert galaxies and low-ionization AGNs indeed appear to orient randomly with respect to the host galaxy axes (e.g., Ulvestad & Wilson 1984; Gallimore et al. 2006). In addition, while a correlation between v off and line FWHM is observed in NLR outflows (e.g., Komossa et al. 2008), the ratio of velocity shift to line width is of order unity, unlike the ratio of ∼ 1.5 seen in our double-peaked sample. Finally, as shown in Figure 4(b), there is no obvious correlation in our sample between the Balmer decrement and the line-width ratio between the two components of the double-peaked feature (as a proxy for asymmetry). Such a correlation might exist if the blueshifted and redshifted NLR gas components are intrinsically symmetric and the apparent asymmetry results from extinction. Heckman et al. (1981) observed a correlation between the asymmetry of a single line profile (which, how- \ner we caution, is different from our asymmetry indicator - the line width ratio of the double-peaked components) and Balmer decrement in a sample of 36 nearby Seyferts thought to contain outflows, although later studies on nearby Seyferts (e.g., Whittle 1985b) found no strong correlations.', '4.2. Merging SMBH Pairs': "Another explanation for the double-peaked features is merging SMBH pairs. In this scenario, both BHs, along with their own NLR gas (with scales of order hundreds of pc), are co-rotating in the galactic potential on /greaterorsimilar kpc scales, well before dynamical friction causes their orbit to decay and form a gravitationally bounded compact binary (e.g., Milosavljevi'c & Merritt 2001). The v offσ ∗ and v off-FWHM correlations in this scenario would naturally arise from dynamics. The comparable numbers of objects with more prominent redshifted and blueshifted components and the null correlation between the Balmer decrement and the doublepeak asymmetry may both be accommodated if the two emission components are associated with two SMBHs. The similar distribution of host-galaxy inclination angles to the control sample could be explained, given that there is no preference for the rotation axis of galaxies in the merging SMBH-pair scenario. \nOn the other hand, the correlation between the diagnostic line ratio [O III ]/H β of the two components shown in Figure 4(a) may be difficult to explain if the NLR gas of each component is primarily ionized by its own BH. In addition, the higher galaxy concentration for objects with double peaks (Figure 5) seems inconsistent with these objects being in the early stage of mergers, although we may be biased to more compact objects by the necessity to get both merging objects within the 3-arcsec fiber aperture. The older mean stellar ages and the smaller post-starburst fraction in the past 0.1-1 Gyr also seem counterintuitive, if there is excess star formation in mergers. However, these differences might also result from the mass-age correlation (e.g., Kauffmann et al. 2003) given that the median σ ∗ of the double-peaked sample is larger than that of the control sample (Figure 5), or if the associated star formation is on a different time scale from what is probed by the Dn (4000) and H δ A indices adopted here, or it is suppressed by AGN activity. \nIn view of the above arguments, we conclude that our double-peaked sample may contain a population of NLR outflows or rotating disks, as well as objects that are merging SMBHs; The statistical properties of the sample do not show any strong preference for or against either scenario. In Figure 2 we show several intriguing cases. The first example (SDSSJ114642.5 + 511029.9) is a merging system of which the off-centered galaxy is partially covered by the 3-arcsec diameter fiber. We have checked that the two galaxies are both detected in the Ks band by 2MASS (Skrutskie et al. 2006), and have found evidence that both are AGNs using spatially resolved longslit spectroscopy, the results of which will be presented elsewhere. Only 13 of the 167 objects in our doublepeaked sample show possible evidence for two cores in the SDSS images within the 3-arcsec diameter of the spectro- \nscopic fibers. The second object (SDSSJ122709.8 + 124854.5) does not have a resolved double core in its SDSS image, and its emission lines have well separated peaks, symmetrically shifted from the systemic velocity, and the redshifted component is more prominent. High resolution HST imaging may help reveal potential double cores unresolved in SDSS images (e.g., Comerford et al. 2009b). For example, one of our double-peaked objects (SDSSJ130128.8 -005804.3, at z = 0 . 2455) shows no double cores in its SDSS image, but Zakamska et al. (2006) used HST imaging to show that it consists of two galaxies with a projected separation of 1.3 arcsec (corresponding to 5.0 kpc) which well fits into the SDSS 3arcsec diameter fiber. We will need spatially resolved spectroscopy to determine whether both of the galaxies are AGN. The last object in Figure 2 has a strong component coincident with the systemic redshift and a weaker blueshifted component and an even weaker redshifted component (not fitted), which could arise from a classic NLR region plus bipolar outflows. Spatially-resolved optical imaging, spectroscopy, radio and/or X-ray followup are still needed to help draw firm conclusions on the nature of these double-peaked narrow line objects. \nWe thank J. Krolik and W. Voges for helpful comments, and an anonymous referee for a careful and useful report that improves the paper. X.L., Y.S., and M.A.S. acknowledge the support of NSF grant AST-0707266. Support for J.E.G. was provided by NASA through Hubble Fellowship grant HF01196 awarded by the STSI, which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA, under contract NAS 5-26555. \nFunding 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/. \nThe SDSS is managed by the Astrophysical Research Consortium for the Participating Institutions. 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2001PhRvD..63f4015C

Charged brane-world black holes

2001-01-01

['-', '-', '-', '-', '-', '-', 'particles', '-', 'astrophysics', '-', '-']

[]

We study charged brane-world black holes in the model of Randall and Sundrum in which our universe is viewed as a domain wall in asymptotically anti-de Sitter space. Such black holes can carry two types of ``charge,'' one arising from the bulk Weyl tensor and one from a gauge field trapped on the wall. We use a combination of analytical and numerical techniques to study how these black holes behave in the bulk. It has been shown that a Reissner-Nordström geometry is induced on the wall when only Weyl charge is present. However, we show that such solutions exhibit pathological features in the bulk. For more general charged black holes, our results suggest that the extent of the horizon in the fifth dimension is usually less than for an uncharged black hole that has the same mass or the same horizon radius on the wall.

[]

https://arxiv.org/pdf/hep-th/0008177.pdf

{'Charged Brane-World Black Holes': 'Andrew Chamblin [email protected] \nCenter for Theoretical Physics, MIT, Bldg. 6-304, 77 Massachusetts Ave., Cambridge, MA 02139', 'Harvey S. Reall': '[email protected] \nDAMTP, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom', 'Hisa-aki Shinkai': '[email protected] \nCentre for Gravitational Physics and Geometry, 104 Davey Lab., Department of Physics, The Pennsylvania State University, University Park, Pennsylvania 16802-6300', 'Tetsuya Shiromizu': "[email protected] MPI fur Gravitationsphysik, Albert-Einstein Institut, D-14476 Golm, Germany Department of Physics, The University of Tokyo, Tokyo 113-0033, Japan and Research Centre for the Early Universe(RESCEU), The University of Tokyo, Tokyo 113-0033, Japan (November 2, 2000 (revised version) to appear in Phys. Rev. D. hep-th/0008177) \nWe study charged brane-world black holes in the model of Randall and Sundrum in which our universe is viewed as a domain wall in asymptotically anti-de Sitter space. Such black holes can carry two types of 'charge', one arising from the bulk Weyl tensor and one from a gauge field trapped on the wall. We use a combination of analytical and numerical techniques to study how these black holes behave in the bulk. It has been shown that a Reissner-Nordstrom geometry is induced on the wall when only Weyl charge is present. However, we show that such solutions exhibit pathological features in the bulk. For more general charged black holes, our results suggest that the extent of the horizon in the fifth dimension is usually less than for an uncharged black hole that has the same mass or the same horizon radius on the wall. \nPACS numbers: 04.50.+h;98.80.Cq;12.10.-g;11.25.Mj", 'I. INTRODUCTION': "In many of the brane-world scenarios, the matter fields which we observe are trapped on the brane [1-4] (see also [5] for older proposals). If matter trapped on a brane undergoes gravitational collapse then a black hole will form. Such a black hole will have a horizon that extends into the dimensions transverse to the brane: it will be a higher dimensional object. \nWithin the context of the second Randall-Sundrum (RS) scenario [4], it is important that the induced metric on the domain wall ∗ is, to a good approximation, the solution predicted by standard General Relativity in four dimensions. Otherwise the usual astrophysical properties of black holes and stars would not be recovered. \nIn a recent paper [6], the gravitational collapse of uncharged , non-rotating matter in the second model of RS \nwas investigated. There it was proposed that what would appear to be a four-dimensional black hole from the point of view of an observer in the brane-world, is really a fivedimensional 'black cigar', which extends into the extra fifth dimension. If this cigar extends all the way down to the anti-de Sitter (AdS) horizon, then we recover the metric for a black string in AdS. However, such a black string is unstable near the AdS horizon. This instability, known as the 'Gregory-Laflamme' instability [7], implies that the string will fragment in the region near the AdS horizon. However, the solution is stable far from the AdS horizon. Thus, one may conclude that the late time solution describes an object that looks like the black string far from the AdS horizon (so the metric on the domain wall is close to Schwarzschild) but has a horizon that closes off before reaching the AdS horizon. A similar effect occurs when there is more than one extra dimension transverse to the brane [8]. These conclusions are supported by an exact calculation for a three dimensional RS model [9]. \nIn this paper, we consider black holes charged under gauge fields which are trapped on the brane . The flux lines of such gauge fields can pierce the horizon only where it \nactually intersects the brane. The bulk theory is the same as for the uncharged case so one might expect that the black cigar solution would still describe the bulk metric of such a charged brane-world black hole. The effect of the charge might simply be to modify the position of the brane in the bulk spacetime. If this were the case, then we might be able to repeat the analysis of [6] by starting with the black string metric and solving the Israel equations appropriate for the presence of a gauge field on the brane. However, in the Appendix we prove that this is not possible. It is still conceivable that the bulk metric is the same as that of the black cigar , but unfortunately the form of the cigar metric is not known. We are therefore forced to study charged brane-world black holes numerically. \nA recent paper [10] has claimed to give a solution describing a non-charged black hole in the RS scenario. By using the brane-world Einstein equations derived in [11], it was shown that a Reissner-Nordstrom (RN) geometry could arise on the domain wall provided that the bulk Weyl tensor take a particular form at the wall. We regard this solution as unsatisfactory for two reasons. First, there is no Maxwell field on the domain wall so the black hole cannot be regarded as charged † . Secondly, only the induced metric on the domain wall was given - the bulk metric was not discussed. The solution is simply a solution to the Hamiltonian constraint of general relativity and gives appropriate initial data for evolution into the bulk. Until this evolution is performed and boundary conditions in the bulk are imposed, it is not clear what this solution represents. For example, it might give rise to some pathology such as a naked curvature singularity. We would then not regard it as a brane-world black hole, which should have a regular horizon [6,9]. One aim of the present paper is to evolve the initial data of [10] in order to understand what this 'solution' really describes. \nThe second aim of this paper is to study brane-world black holes that are charged with respect to a Maxwell field on the brane. We start by solving the Hamiltonian constraint on the brane to give an induced metric that is close to, but not exactly, Reissner-Nordstrom. The 'charge' of [10] arises as an integration constant in the metric. We then evolve this 'initial' data away from the domain wall in order to study the resulting bulk spacetime. Our solution to the Hamiltonian constraint is based on a metric ansatz that is almost certainly not obeyed by the true solution describing a charged brane-world black hole. However, we expect our ansatz to be sufficiently close to the true solution that useful results can be obtained without a knowledge of the exact metric, just as in [6]. \nOur results suggest that it is more natural to take the 'charge squared' parameter of [10] to be negative than positive since the latter gives an apparent horizon that grows relative to the black string as one moves away from the brane. For black holes charged with respect to a Maxwell field, we find that the horizon shrinks in the fifth dimension. In both cases (and for black holes carrying both charges), we obtain a numerical upper bound on the length of the horizon in the fifth dimension. We find that increasing either type of charge tends to decrease this length, even if the horizon radius on the brane is held fixed. \nIt is worth emphasizing that this paper is quite distinct from recent papers which have appeared on the subject of charged black holes in brane-world scenarios [13-15]. This is because these papers all study the effects of bulk charges on the brane-world geometry, whereas our analysis deals with gauge degrees of freedom that are truly localized on the brane. One consistent interpretation of the RN solution of [10] would be as the induced metric on the brane in the (bulk) charged black string solution of [14,15]. However, in this paper we will study whether sense can be made of this solution without introducing bulk gauge fields. \nRelated numerical work on uncharged brane-world black holes has recently appeared in [16]. The difference between that paper and the present work is that we will prescribe 'initial' data on the brane and evolve it in the spacelike direction transverse to the brane, whereas in [16], initial data was prescribed on a spacelike hypersurface and evolved in a timelike direction. \nThe outline of this paper is as follows. First, we set up the basic notation and formalism for a covariant treatment of the second brane-world scenario of Randall and Sundrum. Next we solve the Hamiltonian constraint for 'initial' data on the brane and obtain a RN solution with small corrections. We then numerically evolve the solution into the bulk subject to the constraint that the metric solve the vacuum Einstein equations with a negative cosmological constant. Finally we discuss the properties of the resulting bulk spacetime.", 'A. Covariant formulation of brane-world gravity': "We shall be discussing a thin domain wall in a five dimensional bulk spacetime. We shall assume that the spacetime is symmetric under reflections in the wall. The 5-dimensional Einstein equation is \n(5) R µν -1 2 (5) g µν (5) R = κ 2 5 (5) T µν , (2.1) \nwhere κ 2 5 = 8 πG 5 and G 5 is the five dimensional Newton constant. The energy-momentum tensor has the form \n(5) T µν = -Λ 5 (5) g µν + δ ( χ )[ -λh µν + T µν ] . (2.2) \nIn the above, the brane is assumed to be located at χ = 0 where χ is a Gaussian normal coordinate to the domain wall. χ = 0 is the fixed point of the Z 2 reflection symmetry. Λ and λ denote the bulk cosmological constant and the domain wall tension respectively. h µν is the induced metric on the wall, given by h µν = (5) g µν -n µ n ν where n µ is the unit normal to the wall. The effect of the singular source at χ = 0 is described by Israel's junction condition [17] \nK µν | χ =0 = -1 6 κ 2 5 λh µν -1 2 κ 2 5 ( T µν -1 3 h µν T ) . (2.3) \nHere, K µν denotes the extrinsic curvature of the domain wall, defined by K µν = h ρ µ h σ ν ∇ ρ n σ . In equation (2.3), we are calculating the extrinsic curvature on the side of the domain wall that the normal point into . This is because we want to evolve initial date prescribed on the wall in the direction of this normal. Using the Gauss equation and the junction condition, we recover the Einstein equation on the brane [11]: \n(4) G µν = -Λ 4 h µν +8 πG 4 T µν + κ 4 5 π µν -E µν , (2.4) \nwhere \nΛ 4 = 1 2 κ 2 5 Λ 5 + 1 6 κ 2 5 λ 2 ) (2.5) \n( 4 λ π \n) G 4 = κ 5 48 (2.6) π µν = 1 12 TT µν -1 4 T µα T α ν + 1 8 h µν T αβ T αβ -1 24 h µν T 2 (2.7) \nand E µν is the 'electric' part of the 5-dimensional Weyl tensor: \nE µν = (5) C µανβ n α n β (2.8) \nWe shall now specialize to the RS model. This has \nΛ 5 = -6 κ 2 5 /lscript 2 , λ = 6 κ 2 5 /lscript , (2.9) \nwhich implies \nΛ 4 = 0 , G 4 = G 5 /lscript . (2.10) \nThe matter on the domain wall will be assumed to be a Maxwell field. This implies T = 0, so we can rewrite the Einstein equation as \n(4) R µν = 8 πG 4 T µν -κ 4 5 4 T µρ T ρ ν -E µν . (2.11) \nThe Israel equation gives the extrinsic curvature of the wall: \nK µν | χ =0 = -1 /lscript h µν -κ 2 5 2 T µν . (2.12)", 'B. Strategy': 'We adopt the following procedure: We take a certain charged black hole geometry for the brane. When we solve for the bulk, we Wick rotate twice. This gives a Kaluza-Klein bubble spacetime [21,22] from which we obtain boundary conditions at the condition on the bubble surface. Wick rotating back gives boundary conditions at the bulk horizon for our problem. The Kaluza-Klein bubble spacetime is reviewed in Appendix B.', 'C. Metric and field equations': 'We assume that the induced metric on the brane takes the form \nds 2 = -U ( r ) dt 2 + dr 2 U ( r ) + r 2 d Ω 2 2 , (2.13) \nwhere d Ω 2 2 = dθ 2 +sin 2 θdϕ 2 . Note that this is a guess . It is unlikely that the exact metric describing a braneworld black hole would have precisely this form - in general one would expect the coefficients of dt 2 and dr 2 to be independent (when the coefficient of d Ω 2 2 is fixed as r 2 ). However, we know that the induced metric describing a charged black hole should be close to ReissnerNordstrom, which can be written in this form, so our ansatz is probably quite a good guess. We expect that deviations from the exact metric will give rise to pathologies when this initial data is evolved into the bulk. Even so, the analysis of [6] shows that it is possible to extract quite a lot of information from a pathological solution. The function U ( r ) will be determined from the Hamiltonian constraint equation below. The bulk metric is assumed to take the form \nds 2 = N ( χ, r ) 2 dχ 2 -e 2 a ( χ,r ) U ( r ) dt 2 + e 2 b ( χ,r ) dr 2 U ( r ) + e 2 c ( χ,r ) r 2 d Ω 2 2 , (2.14) \nN is the lapse function which describes the embedding geometry of the hypersurface spanned by ( t, r, θ, ϕ ) during the evolution in the bulk spacetime. \nThe extrinsic curvature of a hypersurface of constant χ (with unit normal n = Ndχ ) is given by \nK t t = ˙ a N , K r r = ˙ b N and K θ θ = K ϕ ϕ = ˙ c N , (2.15) \nwhere a dot denotes ∂ χ . The spacetime is described by the evolution equation, \n˙ K µ ν = N ( (4) R µ ν -KK µ ν + 4 /lscript 2 δ µ ν ) -D µ D ν N, (2.16) \nthe Hamiltonian constraint equation, \n(4) R -K 2 + K µν K µν = -12 /lscript 2 , (2.17) \nand the momentum constraint equation, \nD µ K µ ν -D ν K = 0 . (2.18) \nHere (4) R µ ν and (4) R are the Ricci tensor and Ricci scalar on hypersurfaces of constant χ .', "A. Brane Geometry : Charged black hole 'initial data'": "The action for the Maxwell field on the brane is taken to be \nS = -1 16 πG 4 ∫ d 4 x √ -hF µν F µν , (3.1) \ngiving energy-momentum tensor \nT µν = 1 4 πG 4 ( F µρ F ν ρ -1 4 h µν F ρσ F ρσ ) . (3.2) \nThe field strength F is related to a potential A by F = dA . The equations of motion are satisfied if we take A = -Φ( r ) dt with Φ( r ) = Q/r . This gives \nT µν = 1 8 πG 4 Q 2 r 4 diag ( U , -U -1 , r 2 , r 2 sin 2 θ ) . (3.3) \nThis can be substituted into the right hand side of the Israel equation (2.12) to give an expression for the extrinsic curvature. This can then be substituted into the Hamiltonian equation (2.17), along with our metric ansatz to give an equation for U ( r ). Solving this equation gives ‡ \nU ( r ) = 1 -2 G 4 M r + β + Q 2 r 2 + l 2 Q 4 20 r 6 , (3.4) \nwhere M and β are arbitrary constants of integration. Substituting into the Einstein equation on the domain wall gives \n-E µν = ( β r 4 + l 2 Q 4 2 r 8 ) diag ( U , -U -1 , r 2 , r 2 sin 2 θ ) . (3.5) \nIt is interesting to compare -E µν with 8 πG 4 T µν since these quantities appear on an equal footing in the effective Einstein equation (2.4). It is clear that the constant of integration β is in some sense analogous to Q 2 , which is why the authors of [10] obtained a RN solution. However, since their solution did not have a Maxwell field, it cannot really be regarded as a charged black hole in the usual sense. Rather it carries 'tidal' charge associated with the bulk Weyl tensor. β might be regarded as a five dimensional mass parameter. \nWe shall only consider initial data that corresponds to an object with an event horizon (in the four dimensional sense) on the domain wall. In some cases there may be more than one horizon. We shall use r + to denote the position of the outermost horizon, i.e., the largest solution of U ( r ) = 0. This has to be found numerically except when Q = 0. \nOur 'initial data' is given by \n∣ \n∣ a ( χ = 0 , r ) = b (0 , r ) = c (0 , r ) = 0 . (3.8) \n˙ a N ∣ ∣ ∣ χ =0 = ˙ b N ∣ ∣ ∣ ∣ χ =0 = -1 /lscript + /lscript 2 Q 2 r 4 , (3.6) \n∣ ∣ ∣ \n∣ ∣ ˙ c N ∣ ∣ χ =0 = -1 /lscript -/lscript 2 Q 2 r 4 , (3.7) \nWe shall study the following cases: \n- (i) No electromagnetic charge , i.e., Q = 0. In this case, the induced metric on the domain wall is exactly RN [10]. The horizon radius is \nr + = M + √ M 2 -β. (3.9) \n√ The induced metric has a regular horizon if β ≤ M 2 . Note that there is nothing to stop us choosing β to be negative , which emphasizes the difference between the solution of [10] and a charged black hole. If we take β to be negative then the induced metric has only one horizon, instead of the two horizons of a non-extreme RN black hole. \n- (ii) No tidal charge , i.e., β = 0. In this case, the induced metric on the domain wall is Reissner-Nordstrom with a correction term. Note that -E µν is non-zero but is of order 1 /r 8 , which suggests that the total 'tidal energy' on the wall is zero. \nWe shall also consider the general case (iii) Both charges non-zero , i.e., β = 0, Q = 0. \n/negationslash \n/negationslash", 'B. Bulk Geometry': "The bulk geometry is obtained by integrating (2.15) and (2.16) in the χ -direction numerically. We use the \nstandard 'free-evolution' method, that is we do not solve the constraint equations (2.17) and (2.18) during the evolution, but instead use them to monitor the accuracy of the simulation. \nWe obtain the solution numerically in the region r + < r < r e with r e ∼ 5 r + . Boundary conditions at r = r + are specified by first Wick rotating χ = iT , t = iτ , which takes the metric to a Kaluza-Klein bubble metric (see Appendix B). Therefore we can apply the numerical techniques that are used in the study of KaluzaKlein bubbles [24], although the physics of Kaluza-Klein bubbles is unrelated to the physics of black holes. It was shown in the Appendix of [24] that at the inner boundary r = r + , a and b evolve synchronously, that is, a ( T, r + ) = b ( T, r + ). Analytically continuing back to our original spacetime yields the boundary condition a ( χ, r + ) = b ( χ, r + ). The evolution equation for the trace of K µν and the momentum constraint are also used at r = r + . At the outer boundary r = r e , we assume the components of the extrinsic curvature [equation (2.15)] fall off like -1 //lscript + O r -4 ) [cf. (3.7)]. We apply the geodesic gauge condition (slicing condition), N = 1. \n( \n) We use the Crank-Nicholson integrating scheme with two iterations [25]. The numerical code passed convergence tests, and the results shown in this paper are all obtained to acceptable accuracy. \nWe were only able to solve numerically in a region near the domain wall with a maximum value for χ of O (1). This is because the volume element of surfaces of constant χ decreases exponentially as one moves away from the wall, just as in pure AdS. The evolution was stopped when √ g became too small to monitor accurately. \nWe are interested in how charge affects the shape of the horizon, in particular how far it extends into the fifth dimension. This will be measured by the ratio of the physical size of the apparent horizon r + e c ( χ,r + ) , to that of a black string [6] with the same horizon radius r + on the wall § . The size of the black string apparent horizon in the bulk is r + e -χ//lscript , so the ratio is \n- \nR ( χ ) = e c ( χ,r + )+ χ//lscript . (3.10) \nWe remark that the only apparent horizon that appears during the χ -evolution is at r = r + . Here we define apparent horizon as the outermost region of negative expansion of the outgoing null geodesic congruences, where we define the expansion rate, θ + , as \nθ + = (3) ∇ a s a + (3) K -s a s b (3) K ab , (3.11) \nwhere s a = (1 / √ g rr ) ∂ r is an outwards pointing unit vector in the 3-dimensional metric. We checked (3.11) during the evolution and confirmed its positivity for r > r + . \nOur initial conditions give the behaviour of the ratio R ( χ ) near the brane: \n˙ R ( χ ) | χ =0 = -/lscript 2 Q 2 r 4 + ≤ 0 , (3.12) \nand \nR ( χ ) | χ =0 = 3 Q 2 + β r 4 + -/lscript 2 Q 4 2 r 8 + . (3.13) \nFor model (i) ( Q = 0), ˙ R ( χ ) | χ =0 = 0, but R ( χ ) | χ =0 = β/r 4 + . This gives R ( χ ) | χ =0 < 0 for the case with β < 0, which indicates that the ratio decreases, while R ( χ ) | χ =0 > 0 for the case with β > 0, which indicates that the ratio increases. We have plotted the numerical results for this ratio in Fig.1 (a) and (b) (henceforth we shall set /lscript = G 4 = 1 and assume M /greatermuch 1, as appropriate for an astrophysical black hole.). Fig.1 (a) and (b) suggests that a negative value for β is the natural choice since the apparent horizon grows (relative to the black string) in the fifth dimension when β is positive. \nFor model (ii) ( β = 0), ˙ R ( χ ) | χ =0 < 0 and the ratio always decreases [see Fig.1 (c)]. Model (iii) ( Q /negationslash = 0 and β /negationslash = 0) is non-trivial. We present numerical results in Fig.2. The plot is for M = 5, Q = 3 and β = 0 , ± 5 , ± 10 , ± 15, where β = 15 is close to the extreme ∗∗ case for this choice of Q . The qualitative features are combinations of the plots in Fig.1. Note that β seems to have the greatest effect on the bulk evolution. Again, the case with negative β appears to be the natural choice since positive β gives a growing horizon.", 'C. Bulk Geometry: extent of the horizon': "In this section, we shall estimate how far the horizon extends into the fifth dimension by combining analytical and numerical work. Following the conjugate points theorem [26], we shall show that for a charged black hole, the trace of the extrinsic curvature diverges at a finite distance from the brane. The trace of the evolutional equation is given by \n˙ K = (4) R -K 2 + 16 /lscript 2 = -K µν K µν + 4 /lscript 2 , (3.14) \nwhere we used the Hamiltonian constraint in the second line. Now define k µν as \nso \n˙ k | brane < 0 (3.20) \nThis implies that there is a χ = χ 0 such that \nk = k 0 < 0 (3.21) \nFrom (3.16), one obtains \n1 + 8 /lscript | k | ≤ ( 1 + 8 /lscript | k 0 | ) e 2( χ 0 -χ ) //lscript , (3.22) \nfrom which it follows that k diverges before χ = χ crit , where \nχ crit = χ 0 + /lscript 2 log ( 1 + 8 /lscript | k 0 | ) . (3.23) \nThe divergence in k implies that K also diverges. Near χ = χ crit , | k | behaves like \nk ≤ -4 χ crit -χ . (3.24) \nThe case with Q = 0 is more difficult to analyze because ˙ k | brane = 0. We can use equation (2.15) (with N = 1) to give \nk | r = r + = 2 ∂ χ ( a + c + 2 χ /lscript ) , (3.25) \nwhere we have used the synchronous evolution boundary condition a = b at r = r + . In Fig.3, we have plotted a + c +2 χ//lscript at r = r + . It is clear from this plot that k becomes negative in the bulk when β < 0. In fact k also becomes negative when β > 0. Thus, even in the Q = 0 case, there exists a χ = χ 0 such that k = k 0 < 0. The above argument can then be used to show that when \nK µ ν =: -1 /lscript h µ ν + k µ ν . (3.15) \nThe trace part of k µν , k = k µ µ , is expected to measure the volume expansion relative to the AdS 'background' geometry. In term of k µν , (3.14) can be written as \n˙ k -2 /lscript k + 1 4 k 2 = -˜ k µν ˜ k µν ≤ 0 , (3.16) \nwhere ˜ k µν is the traceless part of k µν . On the brane the 'initial' condition is \nk µν | brane = ˜ k µν | brane = -4 πG 5 T µν , (3.17) \nwhich implies \nk | brane = 0 (3.18) \nFor the case with Q = 0, \n/negationslash \n˜ k µν ˜ k µν > 0 , (3.19) \nQ = 0 and β /negationslash = 0, K must diverge before χ = χ crit , where χ crit is given by equation (3.23). We have therefore proved that if Q /negationslash = 0 or β /negationslash = 0 then K diverges before χ = χ crit . \nIt follows from equations (3.24) and (3.25) that \n( a + c ) | r = r + ≤ 2 log ( χ crit -χ ) , (3.26) \n→ Conservatively, the divergence of K indicates that the geodesic slicing has broken down (when N = 1, ∂ χ is the tangent vector of spacelike geodesics), in other words a caustic has occurred. The numerical study therefore cannot be extended further using this slicing. This has, however, a physical meaning because the apparent horizon is located at constant r = r + in the bulk. The horizon will encounter the caustic before reaching the AdS Cauchy horizon. The caustic can therefore be viewed as the endpoint of the horizon, i.e., the tip of the black cigar. Our analysis has only shown that the geodesic slicing must break down at the caustic so, in principle, this point may be regular, i.e., there may exist a coordinate chart that covers a neighbourhood of this point †† . However, we do not regard this as very likely. Our guess for the induced metric on the domain wall is unlikely to be exactly correct, so in general we would expect some pathology such as a naked curvature singularity to appear in the bulk. We cannot check whether curvature invariants diverge at χ = χ crit since our numerical evolution cannot be extended as far as χ = χ crit . \nwhich implies that √ -g tends to zero at least as fast as ( χ crit χ ) 4 as χ → χ crit . \n- \nWhether the bulk solution is regular or not, equation (3.23) gives us an upper bound on the extent of the horizon in the direction transverse to the domain wall, i.e., the length of the black cigar. We have plotted this upper bound in Fig.4 taking the values for χ 0 and k 0 at the endpoint of our numerical evolution. The first graph shows how χ crit depends on Q and β when M is fixed. Note that when Q = β = 0, the numerical solution is simply the black string ‡‡ , which has χ crit = ∞ . Increasing Q clearly has the effect of decreasing χ crit , which is not surprising since increasing Q also shrinks the horizon radius on the domain wall r + . Perhaps more surprising is that making β more negative also appears to decrease χ crit even though this increases the horizon radius on the wall [see equation (3.9)]. The solid curve on this diagram has both M and r + fixed. It is clear that χ crit decreases along this curve as Q or β increases. \nThe second graph of Fig.4 plots the same curve (fixed M and fixed r + ) for different values of M . The trend seems to be the same in each case. \nThe final graph of Fig.4 is for fixed r + (rather than fixed M ). Increasing β appears to decrease χ crit when Q is small but has no significant effect when Q is large. When β is non-zero, increasing Q has the effect of initially slightly increasing χ crit , but ultimately decreases it substantially. The gross trend appears to be that increasing either type of charge leads to a decrease in the length of the horizon. \nIn most of these graphs, χ crit < r + , so the extent of the horizon in the fifth dimension is smaller than the horizon radius on the domain wall, just as for the uncharged black cigar.", 'IV. SUMMARY AND DISCUSSION': "In this paper we have studied charged black holes in the second RS model. We have seen that two types of charge can arise on the brane, one coming from the bulk Weyl tensor [10] and one from a Maxwell field trapped on the brane . Starting from an ansatz for the induced metric on the brane, we have solved the constraint equations of 4+1 dimensional gravity to find metrics describing charged brane-world black holes. In the absence of Maxwell charge, one can obtain a Reissner-Nordstrom solution [10]. If Maxwell charge is included then one can obtain a geometry that is Reissner-Nordstrom with small corrections. \nUsing these induced metrics as 'initial' data, we have solved the bulk field equations numerically. We have found that the RN solution of [10] has an apparent horizon that grows (relative to the black string apparent horizon) in the dimension transverse to the brane unless the 'charge squared' parameter β is taken to be negative §§ . It therefore seems unlikely that this solution really corresponds to a charged brane-world black hole. Of course, if a bulk gauge field is included then the work of [10] (with β > 0) has a natural interpretation as the induced metric on the brane arising from the charged black string solution of [14,15]. \nIf β < 0 and/or Q /negationslash = 0 then the horizon shrinks relative to the black string horizon. For all cases (including β > 0), we have found that the trace of the extrinsic curvature diverges at a finite distance from the brane, with the volume element of the spacetime tending to \nzero. For β ≤ 0, we have interpreted this as the end point of the horizon of the black hole. Our results suggest that increasing the charges of a brane-world black hole will decrease the length of its horizon in the fifth dimension, even when the horizon radius on the brane is kept fixed. This implies that, by adjusting Q , one can change the five dimensional horizon area while keeping the four dimensional horizon area fixed. One might think that this would lead to a difference between the entropies calculated from these horizon areas, which would be bad news for hopes of recovering General Relativity as the effective four dimensional theory of gravity on the brane. However, the exponential decrease in the volume element as one moves away from the brane implies that the dominant contribution to the five dimensional area comes from the region of the horizon that is closest to the brane [9]. Changes near the other end of the horizon give only subleading corrections to the five dimensional area, allowing the four and five dimensional entropies to agree at leading order. \nWe suspect that our solutions will generically have a curvature singularity at the point where the trace of the extrinsic curvature diverges. This is because it seems rather improbable that our ansatz for the induced metric on the brane should turn out to be exactly right. However, we expect that for each value of Q there will be some value of β for which a small change in our initial data would smooth out this singularity, leading to a regular geometry describing a brane-world black hole carrying Maxwell charge Q . This smoothing would probably not significantly affect the position of the 'tip' of the horizon, for which we have obtained an upper bound on the distance from the brane. This is to be contrasted with the uncharged case in which one takes the induced metric on the brane to be Schwarzschild. Evolving this into the bulk gives the black string metric, for which the singularity occurs at the AdS horizon, which is at infinite proper distance from the brane along spacelike geodesics. Asmall perturbation of the metric on the brane takes one from the black string to the black cigar, which has a regular AdS horizon and a black hole horizon with a tip at finite distance from the brane. \nFor the black string, the stability analysis of [6] shows that the horizon extends a distance of order d = /lscript log( G 4 M//lscript ) into the fifth dimension, so d /lessmuch r + . Our results give only an upper bound for d in the charged case. It would be nice if the stability analysis could be extended to the charged case. However, the instability only sets in when the proper radius of the horizon becomes smaller than the anti-de Sitter length scale and we were not able to extend our numerical evolution this far. Our upper bound seems rather on the large side, since it appears to give d ∼ r + for small Q and β . However, for large Q and/or β , figure 4(c) shows that d /lessmuch r + , so our upper bound is probably tighter in this case. \nThe main outstanding problem remains to find the exact bulk metric that describes a brane-world black hole. This was solved for uncharged black holes in the 3 di- \nmensional RS model by using the 4 dimensional AdS C-metric in the bulk [9]. Unfortunately, the higher dimensional generalization of this metric is not known. It would be interesting to see whether charged black holes in the 3 dimensional RS model could be constructed by using the same bulk as in [9] but simply slicing along a different hypersurface. It would also be interesting to use the methods of [18-20] to find linearized solutions describing spherical distributions of matter charged with respect to a brane-world gauge field.", 'ACKNOWLEDGMENTS': "AC thanks Dan Freedman, Andreas Karch, Philip Mannheim, Joe Minahan and Lisa Randall for useful conversations. AC is partially supported by the U.S. Dept. of Energy under cooperative research agreement DE-FC02-94ER40818. HSR thanks Stephen Hawking for useful conversations. AC and HSR thank the organizers of the Santa Fe 2000 Summer Workshop on 'Supersymmetry, Branes and Extra Dimensions' for hospitality while this work was being completed. HS appreciates the hospitality of the CGPG group, and was supported in part by NSF grants PHY-9800973, and the Everly research funds of Penn State. HS was supported by the Japan Society for the Promotion of Science as a research fellow abroad. TS thanks D. Langlois for discussions. His work is partially supported by the Yamada foundation. Numerical computations were performed using machines at CGPG.", 'APPENDIX A: BRANE BENDING AND THE BLACK STRING': "One candidate for a black hole formed by gravitational collapse of charged brane-world matter on a domain wall in AdS is the black string solution in AdS, which has the metric \nds 2 = /lscript 2 z 2 ( -U ( r ) dt 2 + U ( r ) -1 dr 2 + r 2 d Ω 2 2 + dz 2 ) (A1) \nwhere U ( r ) = 1 -2 G 4 M/r . As discussed in [6], surfaces of constant z trivially satisfy the Israel matching conditions provided that the tension satisfies λ = ± 6 /κ 5 2 /lscript . Thus, we may slice the spacetime along such a surface of constant z , and match to a mirror image, in order to obtain the Schwarzschild solution on the domain wall. \nWe now want to consider what happens when we allow the black hole to be electrically charged with respect to some U (1) gauge field living on the brane. Thus, we must add in an extra term to the brane-world stress energy tensor of the form \nT µν = 1 4 πG 4 ( F µρ F ρ ν -1 4 q µν F ρσ F ρσ ) (A2) \nwhere the electric gauge potential has the form \nA = -Φ( r ) dt (A3) \nso that \nF = Φ ' ( r ) dt ∧ dr (A4) \nwhere ' denotes differentiation with respect to r . \nNow, as a first guess we might try to support this stress-energy on the brane by allowing the brane to bend in the black string background in such a way that the extrinsic curvatures would still satisfy the Israel equations. \nIn other words, we allow the position z of the brane to depend on the radial direction r . Solving the Maxwell equations yields \nΦ ' ( r ) = -Q r 2 ( 1 + z ' 2 U ) 1 / 2 (A5) \nTo compute the extrinsic curvature of the timelike hypersurface swept out by z = z ( r ), we introduce an orthonormal basis which consists of a unit normal vector \nn = /epsilon1/lscript z 1 + Uz ' 2 ( dz -z ' dr ) , (A6) \nwhere /epsilon1 = ± 1, a unit timelike tangent \n√ \nu = z /lscript U -1 / 2 ∂ ∂t , (A7) \nand the spacelike tangents \nt = z /lscript √ U 1 + Uz ' 2 ( z ' ∂ ∂z + ∂ ∂r ) , (A8) \ne φ = z /lscriptr sin θ ∂ ∂φ , (A9) \ne θ = z /lscriptr ∂ ∂θ (A10) \nIt follows that the non-vanishing components of the extrinsic curvature in this basis are \nK uu = /epsilon1 /lscript √ 1 + Uz ' 2 (1 + 1 2 U ' zz ' ) , (A11) \nK θθ = K φφ = -/epsilon1 /lscript √ 1 + Uz ' 2 ( 1 + U r zz ' ) , (A12) \n(A13) \nK tt = -/epsilon1U /lscript ( 1 + Uz ' 2 ) 3 / 2 ( zz '' + z ' 2 + U -1 + U ' zz ' 2 U ) . \nUnder the assumption of Z 2 symmetry, the Israel equations reduce to (2.12). The three independent components of K µν give three independent equations: \nK tt = 1 /lscript -z 4 Q 2 2 /lscript 3 r 4 K uu = -1 /lscript + z 4 Q 2 2 /lscript 3 r 4 (A14) K θθ = 1 /lscript + z 4 Q 2 2 /lscript 3 r 4 . \nIt is straightforward to show that it is impossible to solve these three equations simultaneously unless one takes Q = 0 and z =constant, which is the uncharged solution of [6]. It is therefore not possible to support the stress-energy of a point charge by simply allowing the brane to bend in the black string background. It follows that the bulk has to change once the brane-world charge is introduced. In other words, brane-world charge will induce changes in the bulk Weyl tensor, and this is exactly what we have found in our numerical analysis.", 'APPENDIX B: KALUZA-KLEIN BUBBLE': "The double Wick rotation( χ → it, t → iτ ) of the metric of Eq. (2.13) gives us the Euclidean induced metric: \nds 2 = U ( r ) dτ 2 + dr 2 U ( r ) + r 2 d Ω 2 2 . (B1) \nThe largest r = r + such that U ( r + ) = 0 is interpreted as the position of the bubble surface. Around r = r + , the metric can be expanded \nds 2 /similarequal U ' ( r + )( r -r + ) dτ 2 + dr 2 U ' ( r + )( r -r + ) + r 2 + d Ω 2 2 . (B2) \nIn term of the new coordinate R := √ r -r + , \nds 2 /similarequal 4 U ' ( r + ) [ R 2 d ( U ' ( r + ) τ 2 ) 2 + dR 2 ] + r 2 + d Ω 2 2 . (B3) \nWe can see easily that the metric will be regular if we assume that the τ direction is periodic with period 4 π/U ' ( r + ). \nIn the case of U ( r ) = 1 -r 2 0 /r 2 with λ = Λ = 0, the exact five dimensional solution for time-symmetric initial data ( K µν = 0) is \nds 2 5 = -r 2 dt 2 + U ( r ) dτ 2 + U -1 ( r ) dr 2 + r 2 cosh 2 t d Ω 2 2 . (B4) \nThis is the Witten-bubble spacetime [21]. Another example of initial data for a Kaluza-Klein bubble spacetime was given in Ref. [22] and its classical time evolution has been investigated in Ref. [23,24]. \n- [1] N. Arkani-Hamed, S. Dimopoulos and G. 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Gubser, hep-th/9912001.\n- [13] N. Kaloper, E. Silverstein and L. Susskind, hepth/0006192.\n- [14] H. Lu and C.N. Pope, hep-th/0008050.\n- [15] I. Oda, hep-th/0008055.\n- [16] T. Shiromizu and M. Shibata, hep-th/0007203.\n- [17] W. Israel, Nuovo. Cimento. 44B , 1(1966); erratum: 48B , 463 (1967).\n- [18] J. Garriga and T. Tanaka, Phys. Rev. Lett. 84 , 2778 (2000).\n- [19] M. Sasaki, T. Shiromizu and K. Maeda, Phys. Rev. D62 , 024008 (2000).\n- [20] S. B. Giddings, E. Katz, and L. Randall, JHEP 0003 , 023 (2000).\n- [21] E. Witten, Nucl. Phys. B195 , 481 (1982).\n- [22] D. Brill and G. T. Horowitz, Phys. Lett. 262 , 437 (1991).\n- [23] S. Corley and T. Jacobson, Phys. Rev. D49 , 6261(1994).\n- [24] H. Shinkai and T. Shiromizu, Phys. Rev. D62 , 024010 (2000).\n- [25] e.g., S.A. Teukolsky, Phys. Rev. D61 , 087501 (2000).\n- [26] S. W. Hawking and G. F. R. Ellis, The large scale structure of space-time , (Cambridge Univ. Press, 1973); R. M. Wald, General Relativity , (Univ. of Chicago Press, 1984). \n<!-- image --> \n<!-- image --> \nFIG. 1. Ratio of the physical size of apparent horizon to size of black string apparent horizon, R ( χ ) [cf. eq. (3.10)], plotted as a function of χ . Fig.(a) is for model (i). Lines are of β = 0 , ± 0 . 2 M 2 , ± 0 . 5 M 2 , ± 0 . 9 M 2 and ± M 2 , where M = 10 . 0. We see that the qualitative behaviour of R ( χ ) depends on the sign of β . Fig.(b) is for the extremal case of model (i), β = M 2 with different values of M . Results are also plotted for β = -M 2 . Fig.(c) is for model (ii) for which R ( χ ) is monotonically decreasing. \n<!-- image --> \nFIG. 2. Ratio of physical size of apparent horizon to size of black string apparent horizon, R ( χ ) [cf. eq. (3.10)], for nonzero Q and β . We have set M = 5, Q = 3 and β = 0 , ± 5 , ± 10 , ± 15 for this plot. The main features are a combination of plots in Fig.1. \n<!-- image --> \nFIG. 3. The quantity a + c +2 χ//lscript at r = r + plotted for M = 5 , Q = 0 and β = 0 , -5 , -10 , -25 and -50. \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFIG. 4. Critical value χ crit [eq.(3.23)] plotted for non-zero Q and β ( ≤ 0) black holes. (a) [above left] We have set M = 5 for this plot. Note that for the uncharged case, χ crit = ∞ . The solid curve is for the special cases with r + = 10 . 0. (b) [above right] Critical value χ crit for combinations of parameters ( Q,β ) which produce a black hole with r + = 10 . 0. M = 5 . 0 , 6 . 25 , 7 . 5 , 8 . 75 and 10 . 0 are chosen for these plots. The black dots denote the ends of the lines at β = 0 and the other ends projected onto the χ crit = 3 plane. (c) [below] The same plot as for (b), but M is specified so as to fix r + = 10 for given ( Q,β ). \n<!-- image -->"}

2015MNRAS.447.2059M

Geometrical constraints on the origin of timing signals from black holes

2015-01-01

['stars binaries close', 'stars black holes', '-', 'stars low mass brown dwarfs', 'stars oscillations', 'astronomy x rays', '-']

[]

We present a systematic study of the orbital inclination effects on black hole transients fast time-variability properties. We have considered all the black hole binaries that have been densely monitored by the Rossi X-ray Timing Explorer satellite. We find that the amplitude of low-frequency quasi-periodic oscillations (QPOs) depends on the orbital inclination. type-C QPOs are stronger for nearly edge-on systems (high inclination), while type-B QPOs are stronger when the accretion disc is closer to face-on (low inclination). Our results also suggest that the noise associated with type-C QPOs is consistent with being stronger for low-inclination sources, while the noise associated with type-B QPOs seems inclination independent. These results are consistent with a geometric origin of the type-C QPOs - for instance arising from relativistic precession of the inner flow within a truncated disc - while the noise would correspond to intrinsic brightness variability from mass accretion rate fluctuations in the accretion flow. The opposite behaviour of type-B QPOs - stronger in low-inclinations sources - supports the hypothesis that type-B QPOs are related to the jet, the power of which is the most obvious measurable parameter expected to be stronger in nearly face-on sources.

[]

https://arxiv.org/pdf/1404.7293.pdf

{'S.E. Motta 1 , P. Casella 2 , M. Henze 1 , T. Mu˜noz-Darias 3 , A. Sanna 4 , R. Fender 3 , T. Belloni 5': "1 ESAC, European Space Astronomy Centre, Villanueva de la Ca˜nada, E-28692 Madrid, Spain \n2 INAF-Osservatorio Astronomico di Roma, Via Frascati 33, I-00040, Monteporzio Catone, Italy \n3 University of Oxford, Department of Physics, Astrophysics, Denys Wilkinson Building, Keble Road, Oxford OX1 3RH, UK \n4 Dipartimento di Fisica, Universit'a degli Studi di Cagliari, SP Monserrato-Sestu km 0.7, 09042 Monserrato, Italy \n5 INAF-Osservatorio Astronomico di Brera, Via E. Bianchi 46, I-23807 Merate, Italy \n16 June 2018", 'ABSTRACT': 'We present a systematic study of the orbital inclination effects on black-hole transients fast time-variability properties. We have considered all the black-hole binaries that have been densely monitored by the Rossi XTE satellite. We find that the amplitude of low-frequency quasi periodic oscillations (QPOs) depends on the orbital inclination. Type-C QPOs are stronger for nearly edge-on systems (high inclination), while type-B QPOs are stronger when the accretion disk is closer to face-on (low inclination). Our results also suggest that the noise associated with type-C QPOs is consistent with being stronger for low-inclination sources, while the noise associated to type-B QPOs seems inclination independent. These results are consistent with a geometric origin of the type-C QPOs - for instance arising from relativistic precession of the inner flow within a truncated disk - while the noise would correspond to intrinsic brightness variability from mass accretion rate fluctuations in the accretion flow. The opposite behavior of type-B QPOs - stronger in low inclinations sources - supports the hypothesis that type-B QPOs are related to the jet, the power of which is the most obvious measurable parameter expected to be stronger in nearly face-on sources. \nKey words: Black hole - binaries: close - X-rays', '1 INTRODUCTION': "Quasi-periodic oscillations (QPOs) were discovered several decades ago in the X-ray flux emitted from accreting black hole (BH) binaries and have been observed in many systems. In a Fourier power density spectrum (PDS) they take the form of relatively narrow peaks and appear together with different kinds of broad-band noise (e.g. Takizawa et al. 1997, Casella et al. 2005, Motta et al. 2012). It is now clear that QPOs are a common characteristic of accreting BHs and they have been observed also in neutron stars (NS) binaries (e.g. van der Klis 1989, Homan et al. 2002, Belloni et al. 2007), in cataclysmic variable (see e.g. Patterson et al. 1977) in the so-called ultra luminous X-ray sources (possibly hosting intermediate-mass BHs, e.g. Strohmayer & Mushotzky 2003, Strohmayer & Mushotzky 2003) and even in Active Galactic Nuclei (AGNs, e.g. Gierli'nski et al. 2008, Middleton & Done 2010). \nLow-frequency QPOs (LFQPOs), with frequencies ranging from a few mHz to ∼ 20 Hz were first observed with Ariel 6 in the BH binary GX 339-4 (Motch et al. 1983) and observations with Ginga provided the first indications for the existence of multiple types of LFQPOs (see e.g. Miyamoto & Kitamoto 1991 and \nTakizawa et al. 1997). Three main types of LFQPOs, type-A, typeB, and type-C 1 , originally identified in the PDS of XTE J1550-564 (see e.g. Wijnands et al. 1999; Homan et al. 2001; Remillard et al. 2002, Casella et al. 2005, Motta et al. 2011a), have been seen in several sources. These are distinct from the high-frequency QPOs, found at frequency up to ∼ 450Hz, which we do not consider in this work (but see Belloni et al. 2012 for a review). \nThe origin of LFQPOs is still unclear and there is no consensus about their physical nature, although their study provides a valuable way to explore accretion around accreting compact objects. Several models have been proposed to explain the origin and the evolution of LFQPO in X-ray binaries. Some of them invoke \n1 Type-C QPOs are by far the most common type of QPOs observed in BH binaries. Their amplitude is usually large and they are characterized by a variable frequency ranging the 0.1-30 Hz interval. Type-B QPOs are less common than the type-C QPOs, they are usually observed along the transition form hard to soft in transient BH binaries with frequencies around ∼ 6Hz. Among LFQPOs, type-A QPOs are the least common of all. They usually appear in the soft states of transient BH binaries as broad and weak peaks. \nthe effects of General Relativity (GR) (e.g. Stella & Vietri 1999, Ingram et al. 2009, Motta et al. 2014), while others are based on different kinds of instabilities (e.g. Titarchuk & Osherovich 1999, Tagger & Pellat 1999, Lamb & Miller 2001). On the other hand, several doubts still surround the origin of type-B QPOs, for which no comprehensive model has been proposed. However, it has been speculated that type-B QPOs are associated with the strong relativistic jets that occurs in BH binaries (BHB) during specific state transitions (Fender et al. 2009, Miller-Jones et al. 2012). \nIt has been known for a long time that inclination strongly affects the observed properties of AGNs (see e.g. Antonucci 1993, Urry & Padovani 1995, Risaliti et al. 2002, Bianchi et al. 2012). Over the last years, it has become increasingly clear that the same is true for Galactic accreting BHBs. Ponti et al. (2012) found strong evidences that the accretion disk winds observed in the radio quiet soft states of BHBs have an equatorial geometry with opening angles of a few degrees and therefore can only be observed in sources where the disc is inclined at a large angle i to the line of sight ( high-inclination sources , as opposed to low-inclination sources , where the orbital plane is closer to perpendicular to the line of sight). More recently, the results by Mu˜noz-Darias et al. (2013) supported the hypothesis that the inclination modifies the q-shaped tracks that BHB in outburst display in a hardness-intensity diagram (HID, Homan et al. 2001), which can be at least partially explained by considering inclination-dependent relativistic effects on the accretion disc. Finally, Corral-Santana et al. (2013) have found that obscuration effects similar to those observed in AGN, can be relevant in very high inclination BHBs. However, Gallo et al. (2003) and Soleri & Fender (2011) noted that there is no evidence that the hard state radio luminosity is a function of inclination. \nIn this work, we use data collected by the Rossi X-ray Timing Explorer (RXTE)/Proportional Counter Array (PCA) satellite to analyse the effects of inclination on the fast time-variability properties of BHBs.", '2 OBSERVATIONS AND DATA ANALYSIS': 'The main goal of this work is to perform a systematic study on the effect of the orbital inclination on the most common QPOs observed in BHs (i.e. type-C and type-B QPOs) and their associated noise. \nTo do so, we assume that there is no intrinsic difference between low inclination and high inclination systems. Therefore, any systematic difference observed between the QPO properties of systems viewed at different inclinations has to be mainly due to this parameter. We base our assumption on the fact that the outburst evolution and the macroscopic properties (e.g. orbital parameters, see Mu˜noz-Darias et al. 2013 where a similar source selection is performed) of these systems do not show any obvious difference.', '2.1 The Sample': 'For our analysis we considered only the sources that have shown LFQPOs. We only investigate those that have been densely monitored by RXTE, in order to maximize the chances of observing high-inclination features (i.e. X-ray absorption dips or eclipses), if present. This is because accurate inclination measurements are not generally available for most of the sources (see Casares & Jonker \n2014) 2 . We therefore used the presence/absence of absorption dips as the main discriminant to separate high and low inclination sources (see also Appendix A and B). Note that transient BHBs usually have relatively short orbital periods (from a few days down to a few hours, see e.g. Casares & Jonker 2014), therefore absorption dips can appear quite often and it is in principle reasonably easy to detect them with a good observation coverage. This simplistic approach was applied in two previous works where it has been shown that the sources of both groups present, depending on the orbital inclination, different behaviours when looking at the spectral properties (i.e. colours and disk temperatures, Mu˜noz-Darias et al. 2013) and at the presence/absence of winds (Ponti et al. 2012). \nOur sample includes the following 14 sources: Swift J1753.501, 4U 1543-47, XTE J1650-500, GX 339-4, XTE J1752-223, XTE J1817-330, 4U 1630-47, GRO J1655-40, H1743-322, MAXI J1659-15, XTE J1748-288, XTE J1550-564, XTE J1859+226, MAXI J1543-564 (see Tab. 1). XTE J1859+226 and MAXI J1543564 have been treated separately because they cannot be unambiguously placed in either the high-inclination or the low-inclination group (see Appendix B). We refer to them as intermediateinclination sources. \nWe excluded from this work XTE J1118+480, XTE J1652453 and XTE J1720-318, GS 1354-64 3 because they did not display any significant QPO during the outbursts covered by RXTE. We also excluded GRS 1915+105 (see e.g. Fender & Belloni 2004, Soleri et al. 2008) and IGR J17091-3624 (e.g. Altamirano et al. 2011) because of their unusual outburst evolution, which is different to that of the other sources (i.e. they did not display a clear q-shaped HID). \nThe classification above results in a grouping where high and low inclination roughly mean inclination angle larger and smaller than ∼ 65-70 · , respectively. This value, however, must be taken with caution as the exact threshold for high-inclination depends on the characteristic of each system, such as mass ratio between black hole and companion star, orbital separation, dimension of the accretion disk and of the bulge that forms were the material from the companion star hits the accretion disk. This topic is addressed in detail in Appendix A.', '2.2 Data Analysis': 'We examined all the RXTE archival observations of the sources in our sample. For each observation we computed power spectra from RXTE/PCA data using custom software under IDL 4 in the energy band 2-26 keV (absolute PCA channel 0 to 62). We used 128s-long intervals and a Nyquist frequency of 1024 Hz. 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 fractional rms (Belloni & Hasinger 1990). \nWe selected for our analysis only observations where a somewhat narrow (quality factor 5 Q > 2) low-frequency ( < 50 Hz) feature was identifiable on top of flat-top or power-law shaped noise \n- 3 Additionally, GS 1354-64, has been observed by RXTE only 7 times.\n- 4 GHATS,http://www.brera.inaf.it/utenti/belloni/GHATS Package/Home.html\n- 5 Q = ν centroid /FWHM , where ν centroid is the centroid frequency of the QPO fitted with a Lorentzian and FWHM its full width half maximum) \nTable 1. List of black hole transients and outbursts included in this work. In the column comments we report some informations about the behaviour of the sources relevant to distinguish between high and low inclination systems. With high inclination evolution (High-IE) , intermediate inclination evolution (Int.IE) and low-inclination evolution (Low-IE) we refer to high, intermediate and low disk temperatures, respectively, as reported in Mu˜noz-Darias et al. 2013. As discussed by these authors, the differences in the disk temperatures can be largely ascribed to the inclination of the disk to the line of sight.With the term spikes we refer to flux spikes visible in both the lightcurve and HIDs of most high-inclination systems (see Mu˜noz-Darias et al. 2013). The term winds or dipping in the comments column indicate that equatorial winds or absorption dips, respectively, have been reported for that source. \n| System | Outburst | Inclination e ( · ) | Comments | Ref. | Type-A | Type-B | Type-C |\n|------------------|--------------------------------------------------------------|-------------------------------|-----------------------------------|------------|----------|----------|----------|\n| Swift J1753.5-01 | 2005-2010 | ∼ 40-55 | Failed outburst ( d ) | 0 | | | 32 |\n| 4U 1543-47 | 2002 | 20 . 7 ± 1 . 5 | Low-IE; | 1 | 2 | 3 | 11 |\n| XTE J1650-500 | 2001 | > 47 | Low-IE; | 2 | | 1 | 25 |\n| GX 339-4 | 2002, 2004, 2007, 2010 | /greaterorequalslant 40 ( c ) | Low-IE; | 3 | 4 | 23 | 54 |\n| XTE J1752-223 | 2009 | /lessorequalslant 49 ( a ) | Low-IE; | 4 | | 2 | 4 |\n| XTE J1817-330 | 2006 | | Low-IE; | | 2 | 9 | 2 |\n| XTE J1859+226 | 1999 | /greaterorequalslant 60 ( b ) | Int.-IE (spikes) | 5 | 5 | 19 | 24 |\n| MAXI J1543-564 | 2011 | | Spikes; | | | | 5 |\n| XTE J1550-564 | 1998, 2000 | 74 . 7 ± 3 . 8 | high-IE (spikes); dipping; | 6,7 | 1 | 18 | 48 |\n| 4U1630-47 | 2002, 2003, 2004, 2005 | | high-IE (spikes); dipping; winds; | 8, 9 | | 6 | 19 |\n| GRO J1655-40 | 1996, 2005 | 70 . 2 ± 1 | high-IE (spikes); dipping; winds; | 8, 10 | | 4 | 50 |\n| H1743-322 | 2003, 2004, 2008 (Jan. and Oct.), 2009, 2010 (Jan. and Aug.) | 75 ± 3 ( a ) | high-IE (spikes); dipping; winds; | 11, 12, 13 | | 42 | 108 |\n| MAXI J1659-152 | 2010 | | Spikes; dipping; | 14 | | 8 | 40 |\n| XTE J1748-288 | 1998 | | Spikes; dipping; | | | | 7 | \n- (a) Assuming that the radio jet is perpendicular to the accretion disk.\n- (b) Inclination ∼ 60 degrees if accretion disc does not contribute to the optical luminosity in quiescence.\n- (c) The constrain is placed assuming that the mass of the BH should not be larger than 20 M /circledot . \n(d) Failed outburst (see Soleri et al. 2013). \n- (e) Inclination measurements come either form the X-rays or from multi-wavelength observations (moslty optical and radio).\n- REFERENCES:(0) Neustroev et al. (2014) ; (1) Orosz et al. (2002); (2) Orosz et al. (2004); (3) Mu˜noz-Darias et al. (2008); (4) Miller-Jones et al. (2011); (5) Corral-Santana et al. (2011); (6) Orosz et al. (2011); (7) Homan et al. (2001); (8) Kuulkers et al. (1998); (9) Tomsick et al. (1998); (10) Greene et al. (2001); (11) Corbel et al. (2005); (12) Steiner et al. (2012); (13) Homan et al. (2005b); (14) Kuulkers et al. (2013); \ncomponents in the PDS. PDS fitting was carried out with the XSPEC package by using a one-to-one energy-frequency conversion and a unit response. Following Belloni et al. (2002), we fitted the noise components with a number of broad Lorentzian shapes. LFQPOs are well-described by a variable number of narrow Lorentzians depending on the presence of harmonic peaks. When more than one peak was present, we identified the fundamental based on the QPO evolution along the outburst. Based on the results of the fitting, we excluded from the analysis non-significantly detected features (significance 6 /lessorequalslant 3 σ ). We measured the rms as the square root of the integral over all the available frequencies of each component in the fit to the PDS. This means that the rms values we reported are measured between 1/128 Hz and 1024 Hz. For each PDS, we measured the fractional rms of each component of the spectrum (i.e. each Lorentzian used to fit the spectrum). Whenever a QPO is formed by more than one harmonic peak, we measured the rms of the QPO adding in quadrature the rms of the harmonic peaks. The rms of the noise comes from all the remaining components. The total rms is measured adding in quadrature the rms of all the components of the PDS. We report in Table 2 the rms of the QPO, the rms of the noise and the total rms for each observation. \nWe classified the LFQPOs following the method outlined by Motta et al. 2012 (based on Casella et al. 2005), dividing them into type-A, type-B and type-C QPOs. We collected a total of 429 type- \nC QPOs, 135 type-B QPOs and 14 type-A QPOs (two from 4U 1543-47, two from XTE J1550-564, two from XTE J1817-330, four from XTE J1859+226 and four from GX 339-4). Due to the very low number of detections, we decided to exclude type-A QPOs from this work. In order to differentiate between the rms of the Type-B/C QPO, the rms of the noise associated with a Type-B/C QPO and the total rms of a PDS where we detected a type-B/C QPO, we will refer to type-B/C QPO rms, type-B/C noise rms and type-B/C total rms, respectively.', '3 RESULTS': "After classifying QPOs, we plotted the QPO rms and the noise rms as a function of the QPO centroid frequency, dividing the sources in high inclination and low inclination: we do not separate the different sources in order to give a better idea of the data general trend. The results are shown in Fig. 1 and 2. Blue and red points always correspond to high-inclination and low inclination sources, respectively. For clarity's sake, in each figure we display the data in the following way: in the top panel we show the data for high inclination sources in blue and the data from low-inclination sources in dark grey. In the middle panel we show the data from low inclination sources in red and the data from high-inclination sources in light grey. In the bottom panel the light and dark grey dots are the same as in the top and middle panes, while the blue circles and red squares are 'average' points, obtained applying a logarithmic rebin in frequency to the grey points. The blues circles and the red \n<!-- image --> \nFigure 1. QPO rms as a function of the QPO centroid frequency for Type-C QPOs (left figures) and Type-B QPOs (right figures). For each figure we plot as follows. In the top panel we show data from high-inclination (HI) sources (blue) and from low-inclination (LI) sources (dark grey). In the middle panel we plot the data from low-inclination sources (red) and from high inclination sources (light grey). In the bottom panel light grey and dark grey points are the same as in the top and middle panel, the blue circles correspond to QPO detected in high-inclination sources, the red squares correspond to QPOs detected in low-inclination sources. The red squares and blue circles are 'average' points and have been obtained by applying a logarithmic rebin in frequency to the grey points. The dashed blue and red lines are smooth fits to the blue and red points respectively, for visualising. Here we did not separate the sources, but we only distinguish between high- and low-inclination ones. Note that the axes are scaled differently for type-C and -B QPOs. \n<!-- image --> \nsquares represent data from high and low-inclination sources, respectively. \nIn Fig. 3 we compare the distributions of rms for type-C and and type-B QPOs. In Fig. 4 we do the same, but this time we consider the noise associated to type-C and type-B QPOs. In fig. 5 we compare the distributions of QPO centroid frequency for type-C and type-B QPOs. The plots in Fig. from 3 to 5 show the histograms and the corresponding empirical cumulative distribution functions (ECDF). Blue represents high inclination, red low inclination and an overlap of the two colors in a histogram is indicated by mixing them into purple. As in the previous plots, we do not separate the different sources.", '3.1 Statistical analysis': "From Fig. 1 it is clear that at least the Type-B and Type-C QPO rms is different in high and low-inclination sources in a quite large frequency range. To test the significance of these differences we carried out a two-sample Wilcoxon hypothesis test (U-test, also known as Mann-Whitney test, Mann & Whitney 1947). This is a non-parametric rank sum test designed to check for a difference in \nlocation shift. The idea is similar to testing the significance of the difference between two sample means in the popular Student's ttest. However, in our case many of the measured distributions are clearly not normal and therefore the t-test cannot be used. The only assumption of the U-test is that the shape of the two distributions is approximately similar. This is fulfilled here. \nFor the distributions in Fig. 3, where the similarity of shape might be doubted, we also performed a Kolmogorov-Smirnov test (KS-test), a different non-parametric test, which does not have this limitation. However, in order to interpret easily the results of the KS-test, the two ECDF should not intersect, a requirement that is clearly not fulfilled in the majority of cases in Figs. 3 to 5. Therefore, we based our statistical inference primarily on the U-test. This analysis was performed within the R software environment (R Development Core Team 2011). \nIn Table 3 we summarise the results of the U-test for the highand low-inclination ECDF in Figs. 3 to 5. If the plots suggested an obvious difference between two ECDF we used an one-sided test, i.e. we tested whether there is a location shift into a specific direction (e.g. red is right of blue). For cases in which there was no clear visible difference we tested for a non-zero location shift (i.e. \n<!-- image --> \nFigure 2. Rms of the noise associated with the QPOs in Fig. 1 as a function of the QPO centroid frequency. Color code is the same for Fig. 1. The average points have been produced as described in Fig. 1. \n<!-- image --> \n. \nTable 2. QPO, noise and total rms from all the observations on the sources of our sample. We separated type-C and type-B QPOs and high and low inclination sources. The full table is available online only. \n| # | ID | Frequency [ Hz ] | QPO rms % | Noise rms % | Total rms % |\n|-------------------------------------|-------------------------------------|-------------------------------------|-------------------------------------|-------------------------------------|-------------------------------------|\n| Type-C - Low inclination ( < 70 · ) | Type-C - Low inclination ( < 70 · ) | Type-C - Low inclination ( < 70 · ) | Type-C - Low inclination ( < 70 · ) | Type-C - Low inclination ( < 70 · ) | Type-C - Low inclination ( < 70 · ) |\n| | Swift J1753.5-01 | Swift J1753.5-01 | Swift J1753.5-01 | Swift J1753.5-01 | Swift J1753.5-01 |\n| 1 | 91094-01-01-00 | 0.639 +0 . 009 - 0 . 009 | 6.7 +0 . 6 - 0 . 6 | 33.5 +0 . 7 - 0 . 6 | 34.1 +0 . 7 - 0 . 6 |\n| 2 | 91094-01-01-01 | 0.823 +0 . 006 - 0 . 006 | 10.5 +0 . 6 - 0 . 6 | 30.7 +1 . 2 - 1 . 1 | 32.5 +1 . 1 - 1 . 0 |\n| 3 | 91094-01-01-02 | 0.842 +0 . 005 - 0 . 005 | 9.6 +0 . 5 - 0 . 5 | 31.9 +1 . 6 - 1 . | 33.3 +1 . 5 - 1 . 4 |\n| 4 | 91094-01-01-03 | 0.824 +0 . 005 - 0 . 005 | 12.3 +0 . 5 - 0 . 5 | 5 30.2 +3 . 8 - 3 . 3 | 32.6 +3 . 6 - 3 . |\n| 5 | 91094-01-01-04 | 0.832 +0 . 006 - 0 . 006 | 14.6 +0 . 8 - 0 . 7 | 29.0 +1 . 4 - 1 . 3 | 1 32.5 +1 . 3 - 1 . 2 |\n| ... | | | | | | \ntwo-sided test). Table 3 lists, for each test, the resulting p-value, which is the probability of obtaining a difference at least as large as observed if the underlying distributions were in fact identical.", '3.2 Results from the tests': "From the statistical tests described in Sec. 3.1, we find that the rms distributions of both type-C and type-B QPOs of high-inclination vs low-inclination sources are significantly different. \n- · Figure 3: here we compare the QPO rms distribution for typeC and type-B QPOs. For both QPOs the distributions of high and low-inclination sources are very different. The type-C QPO rms ap- \nFigure 3. Type-B and type-C QPO rms distributions and correspondent ECDF. The distributions are different with high significance. Histograms are normalized to unit area. A color version of the figure is available online. \n<!-- image --> \nFigure 4. Type-B and type-C QPO associated noise rms distributions and correspondent ECDF. There is a significant difference for the type-C QPO noise distributions of about 4 σ -equivalent confidence level. Histograms are normalized to unit area. A color version of the figure is available online. \n<!-- image --> \nFigure 5. Type-B and type-C QPO frequency distributions for high- and low-inclination sources. The high- and low-inclination distribution for both type-C and type-B QPOs are broadly comparable. Histograms are normalized to unit area. A color version of the figure is available online. \n<!-- image --> \nars to be, on average, higher for high-inclination sources, while the opposite is true for type-B QPOs that are stronger in lowinclination systems. In the case of type-C QPOs, the high and lowinclination distributions are different mainly at intermediate frequencies ( ∼ 0.5-10 Hz), while they overlap at lower and higher frequencies (see Fig. 1). Also in the case of type-B QPOs the distributions are significantly different and both a U and a KS-test 7 give similar results. \n- · Figure 4: this plot compares the rms of the noise associated with type-C and type-B QPOs. Again we separate type-B and -C QPOs. While the high- and low-inclination populations of typeB QPOs noise rms distributions are not significantly different, we detect a significant difference (about 4 σ ) in the case of type-C QPO noise, with the noise stronger for low-inclination sources (i.e. the noise does the opposite with respect to the QPOs). However, the difference is mainly due to the points between 1 and 8Hz (see Fig. 2).\n- · Figure 5: this plot compares the QPO frequencies for type-C and type-B QPOs separately. The high and low-inclination type-B QPOfrequency distributions are broadly comparable, while the two type-C QPO frequency distributions are marginally different. This is probably due to the high-frequency tail in the high-inclination type-C QPO frequency distribution. This feature decreases the similarity between the distributions and the formal test should therefore be interpreted with care.\n- · For completeness, we also compared the total rms distributions from the observations where a type-C or a type-B QPO was found. The distributions do not show significant differences. In the case of type-C QPOs, this is probably due to the fact that the QPO and the associated noise distributions show an opposite dependence on inclination that shifts the total rms distributions closer together. In the case of Type-B QPOs the total rms distribution is dominated by the noise (inclination independent) and the small difference between the type-B distributions is probably due to the presence of the (significantly different) QPO rms. \nIn order to study in greater detail the dependence of the amplitude (the rms) of the QPO on inclination, we produced two additional plots. The top and bottom left panels in Fig. 6 combine the ECDFs of the type-C QPO rms for all the individual sources. The right panels show the same for type-B QPOs. High-inclination sources (top panels) are plotted in cyan, light blue, blue, purple, dark blue and black and low-inclination sources (bottom panels) are represented in yellow, orange, dark orange, red, pink and brown. In the case of type-C QPOs there is some overlap, but high and lowinclination sources can clearly be distinguished. The situation is somewhat less clear for the type-B QPO rms distributions, because of the dominance of GX 339-4, and maybe the fact that a couple of sources have so few measurements. XTE J1650-500 and XTE J1752-223 only displayed one and two type-B QPOs, respectively, therefore they do not provide any valid information. \nSince GX 339-4 provides more type-B QPOs to the lowinclination group than any other source, we performed the tests described in Sec. 3.1 also excluding this source from the sample. For both the type-C and type-B noise distributions, as well as in the type-C QPOs distributions, we obtain similar results to what we report in Tab. 3 (low vs high inclination average QPO rms distributions without GX 339-4: p-value for type-C QPO is 2.033e06). However, in the case of type-B QPOs we found that the dif- \nTable 3. Results of the statistical tests performed on the parameter distributions for the type-C, type-B QPOs and their associated noise. 'a' and 'b' indicate that a one-sided or a two-sided test, respectively, has been performed. In the 'significance' column we indicate whether the distributions we compare in each line are significantly different. S: distributions are significantly different (p-value < 0.01), MS: distributions are marginally different (0.01 /lessorequalslant p-value /lessorequalslant 0.05), NS: distributions are not significantly different (p-value > 0.05). \n| low vs high inclination | p-value | Significance |\n|---------------------------|----------------------------------------------------|----------------|\n| Type-C QPOs rms | 3.421 × 10 - 8 (U-test) a 8.482 × 10 - 9 (KS-test) | S S |\n| Type-B QPOs rms | 1.047 × 10 - 6 (U-test) a | S |\n| | 5.857 × 10 - 8 (KS-test) | S |\n| Type-C QPOs noise rms | 1.108 × 10 - 5 (U-test) b | S |\n| Type-B QPOs noise rms | 0.1686 (U-test) b | NS |\n| Type-C QPOs frequency | 0.044 (U-test) b | MS |\n| Type-B QPOs frequency | 0.169 (U-test) b | NS | \nference in the high and low-inclination distributions becomes only marginally significant (low vs high inclination average QPO rms distributions without GX 339-4: p-value for type-B QPO is 0.023), which could be mainly due to the reduction in sample size (number of data points reduced from 38 to 15). Thus, it appears that the contribution from GX 339-4 dominates the low-inclination type-B QPOpopulation. We also tested for significant differences between the QPO rms ECDF of GX 339-4 and the average low-inclination QPOrms distributions and we found no significant differences (GX 339-4 vs low-inclination average QPO rms distributions: p-value for type-C QPO is 0.397; p-value for type-B QPO is 0.46), suggesting that GX 339-4 shows no obvious deviation from the average low-inclination sources behavior. Therefore, our results still hold despite the dominance of GX 339-4.", '3.3 The case of XTE J1859+226 and MAXI J1543-564': 'As noted in Sec. 2.1, we could not unambiguously classify XTE J1859+226 and MAXI J1543-564 as high or low-inclination systems. Therefore, we treat them separately here. In Fig. 7 we show a comparison between the QPO and noise rms ECDF for XTE J1859+226 and MAXI J1543-564 and the correspondent average ECDF for high and low-inclination systems (shown in Fig. 3). We see that in the relevant cases (Type-B/C QPO rms and type-C noise), despite some overlap with the average distributions, both systems qualitatively resemble the behavior of high-inclination sources rather than low-inclination sources. We note, however, that we only have 5 type-C QPOs from MAXI J1543-564, therefore the statistics is fairly low in this case.', '4 DISCUSSION': "We used a large archival data-set from the RXTE satellite to measure, with a 'population-statistics approach', the amplitude of the aperiodic variability (i.e. QPO amplitude and noise amplitude) in a number of BH X-ray transients. We collected a total of 564 QPOs: 128 type-C QPOs and 38 type-B QPOs from six low-inclination sources and 272 type-C QPOs and 78 type-B QPOs from six high inclination sources. In addition, we collected 29 type-C QPOs and 19 type-B QPOs from intermediate-inclination sources. \nFigure 7. Cumulative QPO rms distributions (for type-C and B QPOs, top left and top righ panels, respectively) and noise rms distributions (associated with type-C and -B QPOs, bottom left and bottom right panels, respectively) for XTE J1859+226 (dark green line) and MAXI J1543-564 (ligh green line). The red and blue line correspond to the average ECDF of low and high-inclination sources, respectively, as it is shown in Fig. 3. Note that in the case of Type-B QPOs, only XTE J1859+226 is shown since MAXI J1543-564 did not show any significant type-B QPOs during its outburst. A color version of the figure is available online. \n<!-- image --> \nThese are our main findings: \n- · We find a highly significant difference between the QPO rms distributions of high-inclination vs low-inclination sources. This is true for both type-C and type-B QPOs (even though the lowinclination type-B QPO population is somewhat dominated by GX 339-4). Our study confirms results obtained for a smaller sample of sources and observations by Schnittman et al. (2006) for the case of type-C QPOs. The QPO amplitude correlates significantly with the inclination of the binary system: type-C QPOs are generally stronger in high inclination sources (i.e. closer to edge on), while type-B QPOs are generally stronger in low-inclination sources (i.e. closer to face-on sources).\n- · We find a significant difference in the noise distributions associated with type-C QPOs, while in the case of type-B QPOs the associated noise seems to be inclination independent. On average, the noise is consistent with being stronger for low-inclination sources (at least in a certain frequency range), at variance with the behavior of QPOs. This constitute the first evidence that the broad band noise associated with the QPOs shows a different behavior with respect to the QPO themselves and, for this reason, could have a separate origin.\n- · Comparing these results with the properties of XTE J1859+226 and MAXI J1543-564, we conclude that the inclination of these sources is consistent with being intermediate to high rather than low inclination, in line with what the properties listed in Tab. 1 suggested. Of course, accurate inclination measurements for both sources are needed to confirm this result. \nRecently, with a grouping very similar to what we used, but through a different method, Heil et al. (2014) found the same inclination-dependence in the amplitude of type-C QPOs that we have found. Heil et al. (2014) also reported that the broad band noise associated with type-C QPOs is inclination independent. However, since the comparison between high and low-inclination sources performed by these authors is qualitative, their results are \nFigure 6. Cumulative QPO rms distributions (for type-C and B QPOs, left and righ panels, respectively) for the individual sources of our sample. Different colors correspond to diffferent sources (see legend). To facilitate the comparison between the two population of sources, we first show the high-inclination sources in different colors and the low inclination sources in grey (top panels) and then the low inclination sources in colours and the high inclination sources in grey. A color version of the figure is available online. \n<!-- image --> \nstill broadly consistent with ours.", '4.1.1 The sample': 'Our sample of sources is quite small and tests on a different, independent sample of sources are needed to confirm our findings. Furthermore, in this work we focused only on the general differences in the QPOs and noise distributions, without investigating the details of the distributions themselves. However, the different dependence of the QPO rms on the orbital inclination already suggest that type-B and type-C QPOs are intrinsically different and could come from two well-defined geometrical regions. Most likely, they are the effect of different physical processes and/or the result of different geometrical configuration/physical properties (e.g. change in plasma temperature, ionization state, viscosity or optical depth) of the accreting material in the region where they are produced. This is supported by the results obtained by Motta et al. (2012) who discovered simultaneous type-C and type-B QPO in the data of the black hole binary GRO J1655-40. This result essentially ruled out the possibility that type-B QPOs could arise from the same physical phenomenon originating type-C QPOs - regardless of what this phenomenon is - supporting what was already suggested by \nMotta et al. (2011a) based on the dependence of the different kind of QPOs on the hard emission.', '4.1.2 Source classification and grouping': "Since the inclination measurements are often not precise and affected by large uncertainties, our source classification mostly relies on the assumption that absorption dips are a strong indication of high orbital inclination. Even though this association is commonly adopted (see Appendix B), it represents a critical caveat for our results, as any new observation showing absorption dips might turn a low-inclination source into an high-inclination one (while it practically impossible for a high-inclination source to turn into a low-inclination one). Our choice of considering only wellmonitored sources is intended to minimize the possible missing of an high-inclination feature. Even so, the misclassification of one or more sources is still possible as it is known that absorption dips can disappear during an outburst (see e.g. Smale & Wachter 1999, Kuulkers et al. 2013) - making their detection harder - or can be detected only after being well-sampled by years of X-ray observations, as it happened in the case of H1743-322 (Homan et al. 2005b; see also D'A'ı et al. 2014). In order to investigate the effects of such misclassification, we performed again the U-tests described in Sec. 3.1 on the high and low-inclination distributions, but this time moving, one by one, the sources of the low-inclination group to the high-inclination group. We see that, predictably, the signif- \ncance of the difference between the high and low-inclination distributions tends to decrease when moving a low-inclination source to the high-inclination group. However, the difference remains always significant (p-value always < 0.001), with the exception of GX339-4 in the case of the type-B QPOs rms distributions (p-value = 0.082). This is probably mainly due to the fact that GX 339-4, as we already discussed in Sec. 3.2, dominates (without biasing it) the low-inclination type-B QPOs rms distribution. We note, however, that GX 339-4 8 is the best monitored BHB of our sample and therefore it is also the least likely source to be misclassified. \nAdditionally, we see a considerable scatter within the QPO rms distributions of the high- and low-inclination groups (see Fig. 3). While the average difference between the two groups is highly significant, there is also an overlap of the individual source ECDFs. However, in Fig.3 we clearly see that there are strong differences between the individual sources and that the QPO rms distributions cannot be considered similar. Evidently, it is in principle possible to group the sources differently from our categorization based on inclination. To test the goodness of our grouping, we randomly divided the twelve sources of our sample into two groups of six and used the same statistical test as above to compute the significance of the difference between the QPO rms distributions. For 10 6 of these random samplings we found that in about 5% of all cases the result is at least as significant as the one reported in Tab. 3 (after correcting for the large number of comparisons made). This is in agreement with the scatter visible in Fig.3. Nonetheless, unlike all those random groupings, our initial classification was made a priori based on physical arguments. Therefore, its statistical significance conveys a physically meaningful result.", '4.1.3 Inner-outer disk misalignment': 'It has been recently suggested (see Veledina et al. 2013) that a misalignment between orbital spin and BH spin (presumably aligned with the jet axis) might significantly influence the rms amplitude of type-C QPOs in BHBs. In this context, the relevant angle for the precession is the angle between the BH spin and the line of sight, therefore a misalignment between spin and orbital plane (not considered here, as we use the angle between the line of sight and the orbital spin) could in principle largely affect the QPO amplitude. It is also worth noticing that the outer disk inclination and the inner disk inclination might differ from one another as the inner disk can be strongly affected by the BH spin, i.e. the inner disk might tend to align with the plane normal to the BH spin, while the outer disk tends to stay aligned with the orbital plane. The difficulty in estimating the BH spin/orbital spin misalignment constitutes an additional source of uncertainty and, together with the fact that most inclination measurements are poorly constrained, (especially in the case of nearly face-on sources, see Appendix B) prevents us from studying the details of the effects of inclination on the ECDFs.', '4.2 About the origin of LFQPOs': "The existing models that attempt to explain the origin of LFQPOs are generally based on two different mechanisms: instabilities and geometrical effects. In the latter case, the physical process typically invoked is precession. \nTitarchuk & Fiorito (2004) proposed the so called transition layer model , where type-C QPOs are the result of viscous magnetoacoustic oscillations of a spherical bounded transition layer, formed by matter from the accretion disc adjusting to the sub-keplerian boundary conditions near the central compact object. This mechanism would not produce inclination dependent QPOs and is, therefore, unable to explain our results. \nCabanac et al. (2010) proposed a model to explain simultaneously type-C QPOs and the associated broad band noise. Magnetoacoustic waves propagating within the corona makes it oscillate, causing a modulation in the efficiency of the Comptonization process on the embedded photons. This should produce both the typeC QPOs (thanks to a resonance effect) and the noise that comes with them. This model predicts that both type-C QPOs and their associated noise would be inclination-independent. Therefore, also the predictions of this model do not match the observed properties of the LFQPOs reported here. \nTagger & Pellat 1999 proposed a model based on the accretion ejection instability (AEI), according to which a spiral density wave in the disc, driven by magnetic stresses, becomes unstable by exchanging angular momentum with a Rossby vortex. This instability forms low azimuthal wavenumbers, standing spiral patterns which would be the origin of LFQPOs. Varni'ere & Tagger (2002) and Varni'ere et al. (2012) suggested that all the tree types of QPOs (A, B and C) can be produced through the AEI in three different regimes: non-relativistic (type-C), relativistic (type-A, where the AEI coexist with the Rossby Wave Instability (RWI), see Tagger & Pellat 1999) and during the transition between the two regimes. Varni'ere & Blackman (2005) investigated the impact of a clean spiral density wave on the emission from an accreting BH and found that the rms of type-C QPOs is expected to be higher when the disk is close to edge on (Varni'ere & Blackman 2005), therefore the AEI model succeeds in partly reproduce our findings. However, as the effects of the interplay of the RWI and the AEI must be taken into account, further investigation is needed to fully understand the predicted behavior of type-B and type-A QPOs, as well of their associated noise (Varni'ere, private communication). \nIn summary, all the models based on instabilities fail in fully explaining the properties of the QPOs and of the noise. Therefore, we are left only with the models involving geometrical effects. The maximum radius at which a ∼ Hz modulation can be produced in a 10 M /circledot BHwould come from orbital motion at > 100 R g from the BH. Hence, in principle, type-C QPOs could be the result of modulation/occasional obscuration of the emission from the inner hot flow operated by structures in the disk (e.g. clumps in the accretion flow). However, in this case, a mechanism producing both obscuring structures - that would need to be unrealistically vertically extended to be able to partially obscure the emission from the inner flow - and a mainly equatorial emission from the inner flow is necessary to explain what we observed. Furthermore, Motta et al. (2014) and Motta et al. (2014a) have shown that the Lense-Thirring mechanism successfully explain the evolution of type-C QPOs (and other timing features) in two BHBs (GRO J1655-40 and XTE J1550-564). \nIngram et al. (2009) proposed a model based on the relativistic precession as predicted by GR that attempts to explain type-C QPOs and their associated noise. This model requires a cool optically thick, geometrically thin accretion disc (Shakura & Sunyaev 1973) truncated at some radius, filled by a hot, geometrically thick accretion flow. This geometry is known as truncated disc model (Esin et al. 1997, Poutanen et al. 1997). In this framework, type-C QPOs arise from the Lense-Thirring precession of a radi- \nally extended section of the hot inner flow that modulates the Xray flux through a combination of self-occultation, projected area and relativistic effects that become stronger with inclination (see Ingram et al. 2009). The broad-band noise associated with typeC QPOs, instead, would arise from variations in mass accretion rate from the outer regions of the accretion flow that propagate towards the central compact object, modulating the variations from the inner regions and, consequently, modulating also the radiation in an inclination-independent manner (see Ingram & van der Klis 2013). The predictions of this model matches our findings if the anisotropy of the hot-flow emission is considered. Veledina et al. (2013) simulated the X-ray and optical emission predicted by this model for variable source inclination angles, taking into account also the possible relativistic effects on the accretion flow. From Fig. 3 in Veledina et al. (2013), we infer that the angular dependence of the coronal emission varies with the distance from the BH. In particular, X-ray emission from the outer regions of the hotflow (tens to hundreds gravitational radii) is more visible in lowinclination sources, while the opposite is true for the inner regions (3-5 gravitational radii), whose emission is more visible in highinclination sources. Therefore, low-inclination sources should have a softer coronal emission than high-inclination sources. The broad band noise is also known to have a soft spectrum, at least in the hard states (Gierli'nski & Zdziarski 2005). Although little can be said about the intrinsic energy dependence of the noise in the intermediate states (because of the soft emission from the disk) it is reasonable to assume that the noise itself does not change much its properties from the hard to the intermediate state, thus remaining soft (i.e. coming predominantly from the outer regions of the hot flow). This implies that the noise associated to type-C QPOs is expected to be stronger in face on sources, as we do observe in our data (at least when the QPO is found in a certain frequency range). In conclusion, the predictions of the model proposed by Ingram et al. (2009); Ingram & Done (2011, 2012); Ingram & van der Klis (2013) are consistent with the observed properties of both type-C QPOs and noise. \nType-B QPOs show an opposite behaviour to type-Cs, i.e. their amplitude is larger for low-inclination sources. Only the model by Tagger & Pellat (1999) attempts to explain the origin of type-B QPOs, however our findings points to an intrinsic difference between type-C and type-B QPOs, which are still not fully explained by their model. \nType-B QPOs have been tentatively associated with the relativistic ejections usually observed along the hard-to-soft transitions in transient BHBs (Fender et al. 2004), even though a clear causeeffect relation between the two phenomena has not been identified (Fender et al. 2009, Miller-Jones et al. 2012). There are two main ways in which the observed dependence on inclination of type-B QPOs could be produced, assuming that the type-B QPOs come from the jet: (i) a lack of obscuration when the observer is able to look directly into the jet; (ii) relativistic aberration due to the jet flow. In case (i) the oscillator responsible for the production of type-B QPOs needs not be physically associated with the relativistic outflow, while in case (ii) it would actually be part of the jet. However, in case (i) we would expect abrupt changes in the amplitude of the type-B QPO (see Nespoli et al. 2003), while more continuous changes would be expected in case (ii). In any case, the currently available data and the poor inclination constraints do not allow to confirm either hypothesis. \nDespite the fact that the launch of the relativistic jet are expected to provoke significant changes in the structure of the accretion flow, only very small spectral changes are observed in the \nX-ray spectra in correspondence to the radio flares associated with the relativistic jets (a few % in the hard emission, see Motta et al. 2011b). The sole dramatic changes observed (almost) simultaneously to the launch of the jets are the transitions between type-C and type-B QPOs (see e.g. Miyamoto et al. 1991, Takizawa et al. 1997). We note, however, that it has been suggested that major radio flares seen across the hard to soft transition could be produced by internal shocks associated with varying/increasing jet velocities. If this is the case, only small changes are expected in the X-ray energy spectra and PDS (see e.g. Malzac 2013). \nOur results support the hypothesis that type-B QPOs are related to the relativistic jet (Fender et al. 2009), since there is no other obvious mechanism that would be stronger face-on. Therefore, our results support what has been already proposed by Fender et al. 2004 and Fender et al. 2009: the transitions between type-C and type-B QPOs track the dramatic changes in the configuration and/or in the physical properties of the accretion flow linked to the launch of relativistic jets (but see the case of GX 339-4, 2002 outburst, in Fender et al. 2009). As the mechanism of production of type-B QPOs and the launching mechanism of the jets is still unknown, further evidences are required to confirm this statement. Finally, the fact that the noise associated with Type-B QPOs seems to be inclination independent might indicate that the type-B QPOs noise is produced by yet another mechanism, that is not sensitive to inclination effects.", '5 SUMMARYAND CONCLUSIONS': "We have analyzed a large sample of archival RXTE observations were we detected low frequency QPOs. We assumed that there are no intrinsic differences between the sources of our sample and that the presence of absorption dips in the light-curve of a source corresponds to high orbital inclination. We have shown that inclination has a strong effect on the QPOs. We found that: \n- · Type-C QPOs appear stronger in high inclination sources.\n- · Type-B QPOs show the opposite behavior, being stronger for low-inclination sources.\n- · The noise associated with both type-C QPOs is consistent with being stronger for low-inclination sources, while the noise associated with type-B QPOs is consistent with being inclination independent. \nOur results suggest that: \n- · type-C QPOs, type-B QPOs and the broad band noise associated with type-C QPOs are geometrically/physically different phenomena.\n- · type-C QPOs are consistent with having a geometrical origin. In particular, we find that the relativistic precession is the only mechanism that satisfies all our observational constraints and therefore is favoured by our results.\n- · at variance with type-C QPOs, the associated broad band noise might, instead, correspond to intrinsic brightness variability induced by fluctuations in the mass accretion rate propagating in an hot flow that emits in a non-isotropic way.\n- · fast transition between type-C and type-B QPOs could be the best trackers in the X-rays of the relativistic ejections typical of most BH transients. \nSEMacknowledges Peggy Varniere, Lucy Heil, Erik Kuulkers and Jari Kajava for useful comments and discussion on this work. SEM and MH acknowledge support from the ESA research fellowship program and from \nthe ESAC Faculty. SEM also acknowledges the Observatory of Rome and Brera for hospitality. TMD and PC acknowledge support from the ESAC Faculty and ESA for hospitality. TMB and SEM acknowledge support from INAF PRIN 2012-6. TMD acknowledges funding via an EU Marie Curie Intra-European Fellowship under contract no. 2011-301355. PC acknowledges support by a Marie Curie FP7-Reintegration-Grants under contract no. 2012-322259. This research has made use of data obtained from the High Energy Astrophysics Science Archive Research Center (HEASARC), provided by NASA's Goddard Space Flight Center.", 'APPENDIX A: ON THE RELATION BETWEEN HIGH-INCLINATION AND ABSORPTION DIPS': "Absorption dips in the X-ray light-curves usually recur at the orbital period of a system and are thought to be caused by obscuration by material located in a thickened outer region ('bulge') of the accretion disk. The bulge forms as a consequence of the interaction of the accretion disk with the inflowing gas stream from the companion star (see e.g., White & Swank 1982,Walter et al. 1982). The presence of absorption dips is commonly considered a signature of high orbital inclination (but see Galloway 2012 9 ) and can allow measurement of the orbital period of a source (White & Swank 1982, White & Mason 1985, D'ıaz Trigo et al. 2009, Casares & Jonker 2014). Absorption dips in low-mass X-ray binaries (LMXBs) have been observed mainly around orbital phase 0.7-0.9, while eclipses are expected at phase zero when the accretion disk is seen at ∼ 90 · , i.e., when the companion star is closest to us and in front of the compact object (e.g., Parmar & White 1988). Occasionally, 'secondary absorption dips' are observed at a 0.5 phase difference with respect to the phase at which 'regular dips' occur are also observed. They are explained as being due to material migrating to the other side after the impact with the disk (see, e.g., Frank et al. 1987, Armitage & Livio 1998, and references therein). \nThe orbital inclination angle with respect to the line of sight necessary to produce absorption dips depends both on the geometry of the system and on the properties of the accretion flow (e.g. the size of the accretion disk and of the bulge) and therefore might be different for every source. However, it is still possible to estimate the minimum inclination angle that would provoke absorption dips in a standard BHB. This lower limit on the inclination is based on the size of the bulge and not on the disk opening angle, which has been generally estimated to be /similarequal 12 · (e.g., de Jong et al. 1996, Bayless et al. 2010). White & Holt (1982) estimated the size of the bulge responsible for absorption dips as 19 · ± 6 · for the LMXB 4U 1822 -37. Taking this value as typical for LMXBs, we can thus set a lower limit on the disk inclination of 65 · . We note, however, that if the accretion disk is tilted or warped, the lower limit for the inclination could be as low as 55 · , by considering that for a generic LMXBs, a disk tilt of about 10 · is expected (Foulkes et al. 2010). \nIn several LMXBs the absorption dips appear for only part of the outburst (see, e.g., in MAXI J1659 -152, Kuulkers et al. 2013, 4U1630 -47 and GROJ1655 -40, Kuulkers et al. 1998, Kuulkers et al. 2000, Tomsick et al. 1998, H1743-322, Homan et al. 2005). The explanation for this lies on the fact that for transient BHXBs, changes in the accretion mode cause the appearance or disappearance of dips. For instance, Kuulkers et al. (2000) interpreted the (deep) absorption dips during the rise and plateau phase of the outburst in GRO J1655 -40 as due to filaments in the stream of material coming from the companion star and splashing into the accretion disk, overflowing above and below the impact region. If the inclination is high enough, the impact region itself comes also into the line of sight (e.g., Frank et al. 1987). However, the presence of absorption features all around the orbit for neutron stars (e.g., Parmar et al. 2002) shows that at \n9 Galloway 2012 reported apparent dipping activity early in the 2011 outburst of the NS X-ray binary Aql X-1. Such features had not been previously reported in Aql X-1, but resembled the absorption dips observed in other X-ray binaries. Galloway (2012) therefore concluded that, since apparent dipping behaviour can occur at times, the system inclination is at the high end of its likely range (36 · -55 · ), Robinson et al. 2001 \nleast part of the photo-ionised plasma is distributed equatorially along the whole plane of the disk, indicating that absorption is due to a structure in the disk rather than by filaments. In that scenario, the cause for the disappearance of dips in BHXBs could be, e.g., a strong ionisation of the plasma in bright (but hard) states of the outburst, which renders the plasma transparent and therefore invisible to the observer. An alternative explanation could be that a change of the structure of the accretion flow could diminish the thickness of the bulge and cause the absorption dips to disappear.", 'APPENDIX B: ON THE INCLINATION OF THE SOURCES OF OUR SAMPLE': 'The inclination angle of a binary with respect to the line of sight can be obtained, in absence of eclipses, through indirect methods based on information from the lightcurve and the spectrum of the optical companion star. The binary inclination is commonly obtained through fitting optical/NIR light curves with synthetic ellipsoidal models, since the amplitude of the modulation normally observed in the OIR lightcurves strongly depends on the inclination angle. However, the vast majority of transient XRBs posses a faint K-M donor star and therefore their lightcurves can be seriously contaminated by other non-stellar sources of light (e.g. the outer accretion disk), affecting critically the inclination measurement.', 'B1 Swift J1753.5-0127': 'SWIFT J1753.50127 is an X-ray transient discovered by the Swift/BAT on 2005 as a bright variable X-ray source (Palmer et al. 2005). Although the mass of the primary has not been dynamically measured yet, the system displayed a number of characteristics that suggest that the binary hosts a BH (Soleri et al. 2013). \nNeustroev et al. (2014) reported results from optical and UV observations of Swift J1753.5-01. Despite the fact that very low values for inclination are to be excluded (e.g. no absorption lines - typical of high-inclination bright X-ray novae - and orbital-like modulation in the light-curve that suggest inclination larger than 40 · ), high orbital inclination values must be also excluded for two reasons. First, extensive photometry and spectroscopy of Swift J1753.5-01 rule out any significant absorption dip or eclipse, commonly considered a signature of high-inclination. Second, given the orbital parameters of Swift J1753.5-01, inclinations larger than ∼ 55 · are highly improbable, since it would require a physically unacceptable black-hole mass smaller than ∼ 2M /circledot .', 'B2 4U 1543-47': '4U 1543-47 is a bright soft X-ray transient, firstly observed in outburts in 1971 (Matilsky et al. 1972). Based on its spectral properties, the compact object has been classified as a BH candidate. \nThe presence of (small) ellipsoidal modulations in the optical light curve of the source allowed Orosz et al. (1998) to constrain the inclination between 24 · and 36 · (obtained fitting the V and I optical light-curves), with extreme hard lower and upper limits at 20 · and 40 · , respectively. The second set of limits takes into account possible systematics that the ellipsoidal modelling does not account for.', 'B3 XTE J1650-500': 'XTE J1650-500 was discovered by RXTE in 2001 (Remillard 2001) and classified as a strong BH candidate based on its X-ray spectrum and variability properties (see e.g. Wijnands et al. 2001). \nOrosz et al. (2004) performed a photometric and spectroscopic analysis of XTE J1650-500, and reported a lower limit to the orbital inclination of ∼ 47 · , assuming no disk contamination of the optical light-curve. Because of lack of eclipses, Orosz et al. (2004) concluded that the inclination must be lower than ∼ 70 · , even though the exact value depends on the mass ration on the binary. However, these authors noted that to obtain an inclination \nof ∼ 70 · the accretion disk should contribute 80% of the optical emission. Additionally, Orosz et al. (2004) showed that an inclination higher than 70 · would yield a BH mass lower than 4M /circledot .', 'B4 GX 339-4': 'GX 339-4 was discovered by OSO 7 in 1972 (Markert et al. 1973). Even when GX 339-4 was first discovered, its aperiodic X-ray variability on timescales from milliseconds to seconds suggested that the compact object in the system is a BH. This transient source is among the most studied by RXTE, having shown 4 outbursts in the last decade. \nCowley et al. 2002 reported the results from simultaneous optical photometric and spectroscopic observations. Only small amplitudes in the emission-line velocity were visible, suggesting that GX 339-4 is seen at a low orbital inclination angle. This is confirmed by the lack of eclipses and absorption dips in both the optical and X-ray light-curves (Mu˜noz-Darias et al. 2008), that allow to set a upper limit on the inclination of ∼ 70 · . However, a lower limit to the inclination can be set assuming that the BH mass should not exceed 20M /circledot : under this hypothesis, the large estimate of the mass function for GX 339-4 implies an inclination larger than 40 · .', 'B5 XTE J1752-223': 'XTE J1752-223 was discovered on 2009 October 23 by RXTE (Markwardt et al. 2009). The properties of the X-ray spectrum and the lack of pulsations in the X-ray band suggested that the source was a black-hole candidate (Markwardt et al. 2009), conclusion supported by Mu˜noz-Darias et al. (2010). \nSince XTE J1752-223 has not shown absorption dips nor eclipses, we can assume an upper limit on the inclination of ∼ 70 · . Miller-Jones et al. (2011) reported the detection of compact radio emission from the core of XTE J1752-223, which can be associated to the relativistic ejections from the source. Assuming the radio jet as perpendicular to the orbital plane, the inclination of XTE J1752-223 must be lower than ∼ 49 · .', 'B6 XTE J1859+226': 'XTE J1859+226 was discovered during its 1999 outburst (Wood et al. 1999) and its X-ray properties allowed to classify it as a BH candidate. Corral-Santana et al. (2011) performed photometry of the source and found a mass function significantly smaller than the ones previously estimated (see e.g. Filippenko & Chornock 2001), but still consistent with the presence of a black hole. \nThe lack of eclipses allowed Corral-Santana et al. (2011) to set an upper limit to the source inclination to 70 · . Assuming no contribution from the accretion disk to the optical emission, an inclination of at least 60 · is needed to reproduce the large modulations observed in the light-curve. A disk contribution to the optical light-curve equal to 25% would result in an inclination angle of ∼ 70 · . This suggest that XTE J1859+226 might be at intermediate inclination with respect to the line of sight. This is confirmed by the spectral properties of the source. Despite the lack of clear absorption dips, Mu˜noz-Darias et al. (2013) noted that XTE J1859+27 showed the characteristic flux spikes in its hardness intensity diagrams (HIDs) and lightcurves, typical of high-inclination sources.', 'B7 XTE J1550-564': 'XTE J1550-564 was discovered on 1998 (Smith 1998) by RXTE. Orosz et al. (2011) performed moderate-resolution optical spectroscopy and near-infrared photometry of the source and were able to determine the mass function for the binary system. Orosz et al. (2011) fitted the light-curves from different wavebands obtained over a 8 years of observations (20012009) with a number of models, combining different subsets of data and assuming variable disk-contributions to the optical emission and obtaining different values for the orbital inclination. Although the fit to the light-curve \nfrom a small subset of data (collected in 2006-2007) yielded an inclination of ∼ 57 · , the best fit to the overall optical data (considered more reliable by the authors) corresponds to an inclination of (74.69 ± 3.8) · .', 'B8 4U 1630-47': '4U 1630-47 has shown outbursts every ∼ 600 days since at least 1969 (Kuulkers et al. 1997). Even if the nature of the compact object is still to be confirmed, the X-ray spectral and timing properties of 4U 1630-47 point to a BH. \nKuulkers et al. (1998) discovered absorption dips of a typical duration of a few minutes in the X-ray light-curve of 4U 1630-47, but no eclipses, that allowed these authors to constrain the inclination of the source between 65 · and 75 · . The lack of other significant information does not allow to place any other constraints on the source inclination.', 'B9 GROJ1655-40': 'GRO J1655-40 was discovered in 1994 by the Compton Gamma-Ray Observatory (Zhang et al. 1994). Since this system has a relatively luminous F companion star, the quiescent light-curve is dominated by star light at an unusual extent and, as a result, an unusually precise determination of the orbital parameters is possible. \nOrosz et al. 1997 reported a first inclination measurement of 69.50 ± 0.08. Later works, based on larger datasets, reported refined inclination measurements, all within the 63 · -75 · range (van der Hooft et al. 1998, Shahbaz et al. 1999). Greene et al. (2001) performed multi-wavelenght photometry of GRO J1655-40, obtaining an inclination of (70.2 ± 1.9) · . The most recent fit to the multi-wavelength light-curve of GRO J1655-40 is reported in Beer & Podsiadlowski (2002), which obtained an inclination of (68.65 ± 1.5) · .', 'B10 H1743-322': 'First detected during an outburst in 1977 (Kaluzienski & Holt 1977), H1743-322 was rediscovered as a new source in 2003 by INTEGRAL (Revnivtsev et al. 2003) and only subsequently associated with the previously known source (Markwardt & Swank 2003). Since that time, several smaller outbursts of the source have been observed. From its similarities with the dynamically confirmed BH XTE J1550-564, H1743-322 was classified as a BH candidate (see McClintock et al. 2009), although no mass function has yet been reported. \nHoman et al. (2005b) and Miller et al. (2006) reported observations of absorption dips from this source, appeared only after years of monitoring of the source. This indicates that the orbital inclination is relatively high, possibly on the order of ∼ 80 · . Steiner et al. (2012) determined the inclination of the radio jet with respect to the line of sight to be (75 ± 3) · , which confirms the high inclination of the source if we assume that the disk and the jet axes are (almost) aligned.', 'B11 MAXI J1659-152': 'MAXI J1659152 was discovered in September 2010 by Swift /BAT. The source was initially designated as a gamma ray burst and subsequently classified as an X-ray transient thanks to MAXI data (Negoro et al. 2010). \nKennea et al. (2010) reported frequent intensity drops in the Xray lightcurve, possibly attributed to absorption dips or partial eclipses. Kuulkers et al. (2010) and Kuulkers et al. 2013 confirmed the detection of absorption disp and established an orbital period of ∼ 2.42h for the source. The presence of the absorption dips also allowed these authors to constrain the inclination of the orbital plane of MAXI J1659-152 between 65 and 80 · .', 'B12 MAXI J1543-564': 'MAXI J1543-564 was discovered by MAXI in 2011. The spectral and timing properties of the sources allowed to classify the compact object in the system as a BH candidate Stiele et al. 2011. \nNo mass function has been reported for this source and no information are available to allow to measure or constrain the inclination of this source. Hints pointing to intermediate inclination can be found in the typical spikes that can be observed in the HID and light-curve of the system, reported by Stiele et al. (2011) and discussed by Mu˜noz-Darias et al. (2013). However, since no absorption dips nor eclipses have been observed (yet), it is not possible to further constrain the inclination of the source.', 'B13 XTE J1748-288': 'XTE J1748-288 was discovered on 1998 by the RXTE/ASM (Smith et al. 1998). 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H., 1982, ApJ, 253, L61\n- Wijnands R., Homan J., van der Klis M., 1999, ApJ, 526, L33\n- Wijnands R., Miller J. M., Lewin W. H. G., 2001, IAU Circ., 7715, 2 \nWood A., Smith D. A., Marshall F. E., Swank J., 1999, IAU Circ., 7274, 1 \n- Zhang S. N., Wilson C. A., Harmon B. A., Fishman G. J., Wilson R. B., Paciesas W. S., Scott M., Rubin B. C., 1994, IAU Circ., 6046, 1\n- Zhang W., Jahoda K., Swank J. H., Morgan E. H., Giles A. B., 1995, ApJ, 449, 930 \nTable 2: QPO, noise and total rms from all the observations on the sources of pour sample. We separated type-C and type-B QPOs and high and low inclination sources. . \n| # | ID | Frequency [ Hz ] | QPO rms % | Noise rms % | Total rms % |\n|-------------------------------------------|-------------------------------------------|-------------------------------------------|-------------------------------------------|-------------------------------------------|-------------------------------------------|\n| Type-C - Low inclination | Type-C - Low inclination | Type-C - Low inclination | Type-C - Low inclination | Type-C - Low inclination | Type-C - Low inclination |\n| Swift J1753.5-01 | Swift J1753.5-01 | Swift J1753.5-01 | Swift J1753.5-01 | Swift J1753.5-01 | Swift J1753.5-01 |\n| 1 | 91094-01-01-00 | 0.639 +0 . 009 - 0 . 009 | 6.7 +0 . 6 - 0 6 | 33.5 +0 . 7 - 0 . 6 | 34.1 +0 . 7 - 0 . 6 |\n| 2 | 91094-01-01-01 | 0.823 +0 . 006 - 0 006 | . 10.5 +0 . 6 - 0 6 | 30.7 +1 . 2 - 1 1 | 32.5 +1 . 1 - 1 0 |\n| 3 | 91094-01-01-02 | . 0.842 +0 . 005 - 0 005 | . 9.6 +0 . 5 - 0 5 | . 31.9 +1 . 6 - 1 5 | . 33.3 +1 . 5 - 1 4 |\n| 4 | 91094-01-01-03 | . 0.824 +0 . 005 - 0 005 | . 12.3 +0 . 5 - 0 5 | . 30.2 +3 . 8 - 3 3 | . 32.6 +3 . 6 - 3 1 |\n| 5 | 91094-01-01-04 | . 0.832 +0 . 006 | . 14.6 +0 . 8 - 0 7 | . 29.0 +1 . 4 | . +1 . |\n| 6 | 91094-01-02-00 | - 0 . 006 0.714 +0 . 003 | . 11.9 +0 . 4 - 0 4 | - 1 . 3 +1 . 6 | 32.5 3 - 1 . 2 |\n| 7 | 91094-01-02-01 | - 0 . 003 +0 . 005 | . +0 . 5 | 29.1 - 1 . 5 +1 . 1 | 31.5 +1 . 5 - 1 . 4 |\n| | | 0.738 - 0 . 005 | 10.8 - 0 . 5 | 31.8 - 0 . 9 +1 3 | 33.5 +1 . 0 - 0 . 9 |\n| 8 | 91094-01-02-02 | 0.698 +0 . 004 - 0 . 004 | 11.4 +0 . 5 - 0 . 5 | 31.5 . - 1 . 3 . | 33.5 +1 . 2 - 1 . 2 |\n| 9 | 91094-01-02-03 | 0.630 +0 . 005 - 0 . 005 +0 . 003 | 12.8 +0 . 8 - 0 . 7 +0 . 8 | 32.2 +1 5 - 1 5 | 34.6 +1 . 5 - 1 4 |\n| 10 | 91423-01-01-00 | 0.671 - 0 . 003 | 14.7 - 0 7 | . 29.4 +1 . 6 - 1 . 8 | . 32.9 +1 . 5 - 1 6 |\n| 11 | 91423-01-01-04 | 0.895 +0 . 006 - 0 . 006 | . 12.1 +0 . 5 - 0 . 5 | 29.7 +0 . 9 - 0 . 8 | . 32.1 +0 . 9 - 0 8 |\n| 12 | 91423-01-02-00 | 0.614 +0 . 005 - 0 . 005 | 9.4 +0 . 4 - 0 . 4 | 35.5 +0 . 6 - 0 . 8 | . 36.7 +0 . 6 - 0 8 |\n| 13 | 91423-01-02-05 | 0.589 +0 . 005 - 0 005 | 13.8 +1 . 0 - 0 9 | 31.6 +2 . 4 - 2 6 | . 34.5 +2 . 3 - 2 4 |\n| 14 | 91423-01-02-06 | . 0.577 +0 . 005 - 0 005 | . 9.2 +0 . 4 - 0 4 | . 35.5 +0 . 4 - 0 | . 36.7 +0 . 4 |\n| 15 | 91423-01-03-00 | . 0.486 +0 . 004 - 0 003 | . 10.7 +0 . 5 - 0 | . 5 29.6 +1 . 5 - 1 4 | - 0 . 5 31.4 +1 . 4 - 1 3 |\n| 16 | 91423-01-03-02 | . 0.523 +0 . 005 - 0 005 | . 5 9.5 +0 . 5 - 0 4 | . 34.7 +0 . 5 - 0 5 | . 36.0 +0 . 5 |\n| 17 | 91423-01-03-03 | . 0.578 +0 . 006 - 0 006 | . 8.2 +0 . 6 - 0 6 | . 34.7 +0 . 9 - 0 | - 0 . 5 35.7 +0 . 9 |\n| 18 | 91423-01-03-04 | . 0.510 +0 . 005 | . 13.1 +0 . 6 - 0 6 | . 9 28.5 +1 . 7 - 1 7 | - 0 . 9 31.3 +1 . 6 1 5 |\n| 19 | 91423-01-03-05 | - 0 . 005 0.516 +0 . 006 | . 9.1 +0 . 5 - 0 5 | . 34.9 +0 . 3 0 | - . 36.1 +0 . 3 |\n| 20 | 91423-01-03-06 | - 0 . 006 0.478 +0 . 005 - 0 005 | . 11.8 +0 . 6 - 0 | - . 3 29.7 +1 . 0 | - 0 . 3 32.0 +1 . 0 |\n| 21 | 91423-01-03-07 | . 0.457 +0 . 003 - 0 003 | . 6 11.7 +0 . 4 | - 0 . 9 27.1 +1 . 0 - 0 9 | - 0 . 9 29.5 +0 . 9 - 0 8 |\n| 22 | 91423-01-04-00 | . 0.460 +0 . 005 - 0 005 | - 0 . 4 11.9 +0 . 6 - 0 6 | . 28.7 +1 . 7 - 1 6 | . 31.1 +1 . 6 - 1 5 |\n| 23 | 91423-01-04-01 | 0.463 +0 . 005 - 0 . 005 +0 005 | 10.1 +0 . 7 - 0 . 7 +0 5 | 39.7 +1 . 4 - 1 . 4 +0 3 | 41.0 +1 . 4 - 1 . 4 +0 3 |\n| 24 | 91423-01-04-02 | 0.442 . - 0 . 005 | 9.2 . - 0 . 5 | 34.7 . - 0 . 3 | 35.9 . - 0 . 3 . |\n| 25 | 91423-01-04-03 | 0.425 +0 . 006 - 0 . 006 | 11.3 +0 . 8 | 30.7 +1 . 9 - 1 . 7 | 32.8 +1 8 - 1 6 |\n| 26 | 91423-01-04-04 | 0.440 +0 . 006 - 0 006 | - 0 . 7 11.6 +0 . 8 | 29.7 +2 . 5 - 2 4 | . 31.9 +2 . 4 - 2 2 |\n| 27 | 91423-01-04-05 | . 0.468 +0 . 006 - 0 . 006 | - 0 . 8 10.0 +0 . 4 - 0 . 4 | . 29.3 +4 . 0 - 1 . 5 | . 31.0 +3 . 7 - 1 . 4 |\n| 28 | 91423-01-04-06 | 0.471 +0 . 006 - 0 . 006 | 8.1 +0 . 6 - 0 . 5 | 34.9 +0 . 3 - 0 . 3 | 35.9 +0 . 3 - 0 3 |\n| 29 | 91423-01-05-01 | 0.373 +0 . 005 - 0 005 | 9.0 +0 . 9 - 0 8 | 29.8 +1 . 5 - 1 3 | . 31.1 +1 . 4 - 1 2 |\n| 30 | 91423-01-05-02 | . 0.326 +0 . 006 - 0 006 | . 7.2 +0 . 7 - 0 6 | . 35.2 +0 . 4 - 0 4 | . 36.0 +0 . 4 - 0 4 |\n| 31 | 91423-01-06-00 | . 0.295 +0 . 004 | . 5.8 +0 . 6 - 0 6 | . 35.4 +0 . 3 0 3 | . 35.8 +0 . 4 |\n| 32 | 95105-01-11-00 | - 0 . 004 4.520 +0 . 118 - 0 119 | . 9.0 +0 . 9 - 0 8 | - . 21.1 +0 . 8 - 0 7 | - 0 . 4 31.6 +2 . 7 |\n| . . - 2 . 7 4U 1543-47 | . . - 2 . 7 4U 1543-47 | . . - 2 . 7 4U 1543-47 | . . - 2 . 7 4U 1543-47 | . . - 2 . 7 4U 1543-47 | . . - 2 . 7 4U 1543-47 |\n| 33 | 70124-02-01-00 | 9.009 +0 . 027 027 | 6.0 +0 . 3 3 | 20.7 +2 . 4 - 2 0 | 21.6 +2 . 3 - 2 0 |\n| 34 | 70124-02-02-00 | - 0 . 9.219 +0 . 066 - | - 0 . 5.7 +0 . 5 - 0 | . 19.3 +0 . 9 - 0 8 | . 20.1 +0 . 9 |\n| 35 | 70124-02-03-00 | 0 . 066 5.564 +0 . 149 - 0 | . 5 9.6 +2 . 6 - 1 7 | . 26.6 +7 . 4 - 4 6 | - 0 . 8 28.3 +7 . 0 |\n| 36 | 70128-01-01-00 | . 140 4.317 +0 . 038 038 | . 11.2 +1 . 0 - 0 9 | . 24.1 +1 . 6 - 1 6 | - 4 . 4 26.6 +1 . 5 - 1 5 |\n| | 70128-01-01-01 | - 0 . 5.383 +0 . 030 0 | . +0 . 5 | . +1 . 6 | . +1 . 6 |\n| 37 | 70133-01-01-00 | - . 029 5.973 +0 . 139 | 5.6 - 0 . 5 +0 . 1 | 27.0 - 1 . 5 +0 . 4 | 29.1 - 1 . 5 +0 . 3 |\n| 38 39 | 70133-01-06-00 | - 0 . 137 6.092 +0 . 103 - 0 103 | 2.6 - 0 . 2 2.4 +0 . 2 - 0 2 | 3.8 - 0 . 4 5.7 +0 . 2 - 0 2 | 4.6 - 0 . 3 6.2 +0 . 2 - 0 2 |\n| 40 | 70133-01-08-00 | . 6.646 +0 . 116 | . 1.6 +0 . 2 - 0 | . +0 . 2 | . 4.5 +0 . 2 |\n| | | - 0 . 118 | . 2 +0 . | 4.1 - 0 . 2 | - 0 . 2 . |\n| 41 | 70133-01-30-00 | 11.102 +0 . 097 - 0 092 | 4.8 7 - 0 7 | 15.8 +2 . 5 - 2 1 | 16.6 +2 4 - 2 0 |\n| 42 43 | 70133-01-31-00 70133-01-31-01 | 9.716 - 0 . 026 9.443 +0 . 028 - 0 | 6.4 - 0 . 2 6.1 +0 . 4 | 24.6 - 1 . 8 21.2 +1 . 2 | 25.4 - 1 . 8 22.1 +1 . 2 |\n| 028 - 0 . 3 - 1 . 2 - 1 . 2 XTE J1650-500 | 028 - 0 . 3 - 1 . 2 - 1 . 2 XTE J1650-500 | 028 - 0 . 3 - 1 . 2 - 1 . 2 XTE J1650-500 | 028 - 0 . 3 - 1 . 2 - 1 . 2 XTE J1650-500 | 028 - 0 . 3 - 1 . 2 - 1 . 2 XTE J1650-500 | 028 - 0 . 3 - 1 . 2 - 1 . 2 XTE J1650-500 |\n| 44 | 60113-01-03-00 | 1.992 +0 . 020 0 020 | 5.6 +1 . 3 - 0 9 | 29.3 +1 . 4 - 1 3 | 29.8 +1 . 4 - 1 3 |\n| 45 | 60113-01-04-00 | - . 2.230 +0 . 022 - 0 021 | . 5.9 +0 . 7 - 0 8 | . 30.5 +2 . 7 - 2 5 | . 31.0 +2 . 7 - 2 4 |\n| 46 | 60113-01-05-00 | . 1.305 +0 . 016 - 0 016 | . 8.7 +0 . 6 - 0 | . 31.1 +1 . 0 - 1 0 | . 32.3 +1 . 0 - 1 0 |\n| 47 | 60113-01-05-01 | . 2.745 +0 . 038 - 0 035 | . 6 4.4 +0 . 5 - 0 5 | . 33.5 +1 . 3 - 1 3 | . 33.8 +1 . 3 - 1 3 |\n| 48 | 60113-01-06-00 | . 1.540 +0 . 017 - 0 . 016 | . 7.4 +0 . 5 - 0 5 | . 31.4 +2 . 3 - 2 . 2 | . 32.2 +2 . 3 - 2 1 |\n| 49 | 60113-01-07-00 | 1.914 +0 . 016 - 0 . 016 | . 6.9 +0 . 5 - 0 . 4 | 30.4 +1 . 7 - 1 . 6 | . 31.2 +1 . 6 - 1 . 6 | \nContinued on next page", 'S. Motta et al.': 'Table 2 - continued from previous page \n| # | ID | Frequency [ Hz ] | QPO rms % | Noise rms % | Total rms % |\n|----------------------------------|----------------------------------|---------------------------------------|----------------------------------|----------------------------------|----------------------------------|\n| 482 | 40124-01-40-01 | 5.244 +0 . 066 | 3.6 +0 . 2 | 0.0 | 8.5 +1 . 0 - 1 0 |\n| 483 | | - 0 . 067 +0 . | - 0 . 2 +0 . 1 | +0 . | . +0 . |\n| | 40124-01-41-00 | 4.744 047 - 0 047 | 2.3 | 2.7 1 - 0 1 | 3.5 1 - 0 1 |\n| 484 | 40124-01-42-00 | . 11.632 +0 . 694 | - 0 . 1 2.5 +0 . 3 | . 4.3 +0 . 3 - 0 3 | . 4.3 +0 . 3 - 0 3 |\n| 485 | 40124-01-49-01 | - 0 . 694 5.202 +0 . 078 | - 0 . 3 3.2 +0 . 2 | . 3.3 +0 . 7 - 0 7 | . 4.6 +0 . 5 - 0 5 |\n| 486 | 40124-01-61-00 | - 0 . 080 4.585 +0 . 106 - 0 112 | - 0 . 2 2.9 +0 . 2 - 0 2 | . 1.8 +0 . 2 - 0 2 | . 3.4 +0 . 2 - 0 2 |\n| - High inclination ( > 70 · ) | - High inclination ( > 70 · ) | - High inclination ( > 70 · ) | - High inclination ( > 70 · ) | - High inclination ( > 70 · ) | - High inclination ( > 70 · ) |\n| XTE J1550-564 | XTE J1550-564 | XTE J1550-564 | XTE J1550-564 | XTE J1550-564 | XTE J1550-564 |\n| 487 | 30191-01-32-00 | 5.403 +0 . 034 - 0 034 | 4.5 +0 . 3 | 2.3 +0 . 3 - 0 2 | 9.0 +0 . 2 - 0 2 |\n| 488 | 30191-01-34-01 | . 4.913 +0 . 009 - 0 009 | - 0 . 2 4.3 +0 . 1 | . 1.7 +0 . 1 - 0 1 | . 8.4 +0 . 3 - 0 3 |\n| 489 | 30191-01-41-00 | . 10.291 +0 . 280 - 0 280 | - 0 . 1 1.5 +0 . 2 - 0 2 | . 3.8 +0 . 1 - 0 1 | . 4.1 +0 . 1 0 |\n| 490 | 40401-01-50-00 | . 5.870 +0 . 024 - 0 024 | . 4.2 +0 . 1 | . 3.0 +0 . 1 - 0 1 | - . 1 +0 . 1 |\n| | | . +0 . | - 0 . 1 +0 . 2 | . +0 . | 7.7 - 0 1 |\n| 491 | 40401-01-51-00 | 5.649 067 - 0 067 | 4.1 - 0 2 | 2.6 1 - 0 1 | . 8.1 +0 . 2 - 0 2 |\n| 492 | 40401-01-51-01 40401-01-53-00 | 5.741 - 0 . 020 6.333 +0 . 009 | 4.2 - 0 . 2 4.2 +0 . 1 | 2.8 - 0 . 2 3.9 +0 . 2 | 9.1 - 0 . 2 . |\n| 493 | 40401-01-55-00 | - 0 . 009 6.139 +0 . 009 | - 0 . 1 4.4 +0 . 0 | - 0 . 1 3.9 +0 . 2 | 10.0 +0 3 - 0 . 2 +0 . 1 |\n| 494 495 | 40401-01-56-00 | - 0 . 009 6.276 +0 . 035 034 | - 0 . 0 5.0 +0 . 1 | - 0 . 2 3.6 +0 . 2 2 | 9.6 - 0 . 1 11.2 +0 . 3 |\n| 496 | 40401-01-56-01 | - 0 . 6.277 +0 . 054 | - 0 . 1 4.9 +0 . 2 | - 0 . 0.0 | - 0 . 3 4.9 +0 . 2 |\n| | 40401-01-58-00 | - 0 . 054 5.673 +0 . 045 | - 0 . 2 5.2 +0 . 4 | 2.4 +0 . 1 | - 0 . 2 10.0 +0 . 4 |\n| 497 | 40401-01-58-01 | - 0 . 045 3.110 +0 . 020 | - 0 . 2 4.9 +0 . 1 | - 0 . 1 9.3 +0 . 3 | - 0 . 3 13.9 +0 . 2 |\n| 498 499 | 40401-01-59-01 | - 0 . 020 5.757 +0 . 108 | - 0 . 1 3.9 +0 . 3 | - 0 . 3 0.8 +0 . 2 1 | - 0 . 2 8.2 +0 . 4 |\n| 500 | 40401-01-61-00 | - 0 . 104 5.486 +0 . 097 | - 0 . 3 3.9 +0 . 2 | - 0 . 0.0 | - 0 . 4 7.5 +0 . 4 |\n| 501 | 40401-01-61-01 | - 0 . 099 9.142 +0 . 670 | - 0 . 2 3.5 +0 . 3 | 0.0 | - 0 . 4 8.4 +1 . 0 |\n| 502 | 50134-01-02-00 | - 0 . 670 3.580 +0 . 013 | - 0 . 3 8.9 +0 . 2 | 3.7 +0 . 2 | - 1 . 0 13.4 +0 . 7 |\n| 503 | 50134-02-02-00 | - 0 . 013 4.421 +0 . 012 | - 0 . 2 6.7 +0 . 2 | - 0 . 2 3.9 +0 . 1 | - 0 . 7 10.2 +0 . 3 |\n| 504 | 30191-01-02-00 | - 0 . 012 4.940 +0 . 028 | - 0 . 2 0.3 +0 . 0 - 0 0 | - 0 . 1 7.1 +0 . 0 | - 0 . 3 7.2 +0 . 0 - 0 0 |\n| - 0 . 028 . - 0 . 0 . 4U 1630-47 | - 0 . 028 . - 0 . 0 . 4U 1630-47 | - 0 . 028 . - 0 . 0 . 4U 1630-47 | - 0 . 028 . - 0 . 0 . 4U 1630-47 | - 0 . 028 . - 0 . 0 . 4U 1630-47 | - 0 . 028 . - 0 . 0 . 4U 1630-47 |\n| 505 | 70417-01-09-00 | 4.651 +0 . 160 - 0 160 | 1.0 +0 . 1 - 0 1 | 3.7 +0 . 1 - 0 | 9.4 +0 . 4 - 0 4 |\n| 506 | 80117-01-05-00 | . 4.663 +0 . 058 0 | . 2.2 +0 . 1 - 0 | . 1 11.1 +0 . 4 4 | . 13.4 +0 . 4 - 0 4 |\n| 507 | 80117-01-07-01 | - . 058 4.628 +0 . 028 - 0 027 | . 1 0.9 +0 . 1 - 0 1 | - 0 . 5.3 +0 . 2 | . 9.5 +0 . 4 |\n| 508 | 80117-01-07-02 | . 4.532 +0 . 021 0 | . 0.8 +0 . 1 - 0 | - 0 . 2 8.5 +0 . 3 | - 0 . 4 8.8 +0 . 2 |\n| 509 | 80117-01-10-00 | - . 021 4.547 +0 . 038 039 | . 1 0.9 +0 . 1 - 0 1 | - 0 . 3 5.1 +0 . 2 | - 0 . 2 9.6 +0 . 3 |\n| 510 | 80417-01-01-00 | - 0 . 4.835 +0 . 063 - 0 063 | . 1.7 +0 . 1 - 0 1 | - 0 . 2 11.4 +0 . 2 - 0 2 | - 0 . 3 13.6 +0 . 3 - 0 3 |\n| GRO J1655-40 | GRO J1655-40 | GRO J1655-40 | GRO J1655-40 | GRO J1655-40 | GRO J1655-40 |\n| 511 | 10255-01-05-00 | 10.635 +0 . 084 - 0 . 085 | 1.7 +0 . 1 - 0 . 1 | 6.2 +0 . 1 - 0 . 1 | 6.6 +0 . 1 - 0 1 |\n| 512 | 10255-01-08-00 | 9.221 +0 . 112 | 1.6 +0 . 1 - 0 | 4.6 +0 . 1 - 0 1 | . 8.1 +0 . 1 - 0 |\n| 513 | 91702-01-02-00G | - 0 . 114 6.442 +0 . 061 | . 1 7.1 +0 . 4 5 | . 8.3 +0 . 5 0 | . 1 11.0 +0 . 5 |\n| | | - 0 +0 | - 0 . +0 . | - . 4 +0 . | - 0 . 5 +0 . 4 |\n| 514 | 91702-01-58-00 | . 061 6.793 . 032 - 0 032 | 0.9 1 - 0 | 7.7 5 - 0 5 | 10.3 - 0 |\n| 1 3 H 1743-322 | 1 3 H 1743-322 | 1 3 H 1743-322 | 1 3 H 1743-322 | 1 3 H 1743-322 | 1 3 H 1743-322 |\n| 515 | 80135-02-02-00 | 5.349 +0 . 040 | 4.4 +0 . 2 - 0 4 | 8.8 +1 . 3 - 0 4 | 9.8 +1 . 1 - 0 |\n| 516 | 80135-02-02-000 | - 0 . 049 5.438 +0 . 012 - 0 012 | . 4.9 +0 . 1 - 0 1 | . 8.5 +0 . 3 - 0 4 | . 4 9.8 +0 . 2 - 0 |\n| 517 | 80135-02-02-01 | . 4.738 +0 . 021 - 0 021 | . 4.7 +0 . 1 - 0 | . 6.2 +0 . 2 - 0 2 | . 4 7.8 +0 . 2 |\n| 518 | 80144-01-01-00 | . 5.073 +0 . 071 - 0 070 | . 1 5.0 +0 . 5 0 4 | . 11.2 +0 . 5 0 4 | - 0 . 2 12.3 +0 . 5 0 4 |\n| 519 | 80144-01-01-01 | . 5.420 +0 . 039 0 | - . 4.4 +0 . 2 - 0 2 | - . 8.4 +0 . 3 - 0 3 | - . 9.5 +0 . 3 - 0 3 |\n| 520 | 80144-01-01-02 | - . 039 5.455 +0 . 051 | . 4.9 +0 . 3 | . 9.4 +0 . 4 | . 10.6 +0 . 4 0 4 |\n| 521 | 80144-01-02-00 | - 0 . 053 5.231 +0 . 027 - 0 027 | - 0 . 3 4.1 +0 . 1 - 0 1 | - 0 . 4 6.9 +0 . 2 - 0 2 | - . 8.0 +0 . 2 - 0 2 |\n| 522 | 80144-01-02-01 | . 5.570 +0 . 044 - 0 044 | . 3.6 +0 . 3 - 0 3 | . 9.4 +0 . 3 - 0 3 | . 10.1 +0 . 3 - 0 3 |\n| 523 | 80144-01-03-01 | . 4.963 +0 . 027 - 0 027 | . 4.5 +0 . 1 | . 6.9 +0 . 3 - 0 3 | . 8.2 +0 . 3 - 0 3 |\n| | | . 5.528 +0 . 032 | - 0 . 1 6.8 +0 . 1 | . +0 . 4 | . +0 . 4 |\n| 524 | 80146-01-11-00 | - 0 . 032 +0 . | - 0 . 1 +0 . | 13.9 - 0 . 4 | 15.5 - 0 . 4 |\n| 525 | 80146-01-12-00 | 5.787 038 - 0 . 038 | 5.5 3 - 0 . 3 | 13.5 +0 . 4 - 0 . 4 | 14.6 +0 . 4 - 0 . 4 |\n| 526 527 | 80146-01-15-00 80146-01-16-00 | 5.179 +0 068 - 0 . 068 4.492 +0 . 035 | 7.5 +0 2 - 0 . 2 2.3 +0 . 1 | 12.9 +0 9 - 0 . 8 13.4 +0 . 5 | 14.9 +0 8 - 0 . 7 13.6 +0 . 5 |\n| 528 | 80146-01-26-00 | - 0 . 036 5.106 +0 . 034 - 0 034 | - 0 . 1 4.2 +0 . 2 - 0 3 | - 0 . 5 9.6 +0 . 4 - 0 3 | - 0 . 5 10.4 +0 . 4 |\n| | 80146-01-27-00 | . 5.277 +0 . 017 - 0 017 | . 3.7 +0 . 1 - 0 1 | . 8.8 +0 . 2 | - 0 . 3 +0 . 2 |\n| 529 | 80146-01-48-00 | . 5.399 +0 . 027 - 0 027 | . 5.1 +0 . 2 - 0 2 | - 0 . 2 9.3 +0 . 2 | 9.5 - 0 . 2 +0 . 2 |\n| 530 531 | 80146-01-49-00 | . 5.253 +0 . 031 - 0 032 | . 5.1 +0 . 2 - 0 | - 0 . 2 9.9 +0 . 5 | 10.6 - 0 . 2 11.1 +0 . 5 |\n| | | . 4.557 +0 . 021 - 0 . 021 | . 3 2.9 +0 . 1 - 0 . 1 | - 0 . 5 +0 . 2 . 2 | - 0 . 5 9.5 +0 . 2 2 |\n| 532 | 80146-01-51-00 | | | 9.0 - 0 | - 0 . | \nContinued on next page', 'Inclination dependence of BH timing properties': 'Table 2 - continued from previous page \n| # | ID | Frequency [ Hz ] | QPO rms % | Noise rms % | Total rms % |\n|---------------|-------------------------------|---------------------------------------|---------------------------------|------------------------|---------------------------------|\n| 533 | 80146-01-51-01 | 4.221 +0 . 050 - 0 053 | 2.2 +0 . 1 - 0 1 | 8.6 +0 . 2 - 0 2 | 8.6 +0 . 2 - 0 2 |\n| 534 | 80146-01-52-00 | 5.137 +0 . 016 - 0 . 016 | 3.9 +0 . 1 - 0 . 1 | 7.4 +0 . 1 - 0 . 1 | 8.3 +0 . 1 - 0 . 1 |\n| 535 | 80146-01-52-01 | 5.077 +0 . 019 - 0 019 | 5.1 +0 . 1 - 0 1 | 6.4 +0 . 1 - 0 1 | 8.1 +0 . 1 - 0 1 |\n| 536 | 80146-01-53-00 | 4.931 +0 . 021 - 0 . 021 | 4.9 +0 . 1 - 0 . 1 | 7.0 +0 . 2 - 0 . 2 | 8.5 +0 . 2 - 0 . 2 |\n| 537 | 80146-01-53-01 | 5.082 +0 . 040 - 0 040 | 4.6 +0 . 2 - 0 2 | 6.3 +0 . 3 3 | 7.8 +0 . 3 - 0 3 |\n| | | . +0 . 016 016 | . +0 . | - 0 . +0 . 3 | . 9.9 +0 . - 0 3 |\n| 538 | 80146-01-54-00 | 4.853 - 0 . | 3.8 2 - 0 . 2 | 9.2 - 0 . 3 | 3 . |\n| 539 | 80146-01-55-00 | 5.173 +0 . 020 - 0 . 020 | 4.1 +0 . 1 - 0 . 1 | 6.8 +0 . 2 - 0 . 2 | 7.9 +0 . 2 - 0 2 |\n| 540 | 80146-01-56-00 | 4.861 +0 . 018 - 0 . 018 | 4.8 +0 . 1 - 0 . 1 | 5.9 +0 . 2 - 0 . 2 | . 7.6 +0 . 1 - 0 . 1 |\n| 541 | 80146-01-58-00 | 3.864 +0 . 044 - 0 . 044 | 2.1 +0 . 1 - 0 1 | 7.2 +0 . 1 - 0 1 | 7.5 +0 . 1 - 0 1 |\n| 542 | 80146-01-59-00 | 4.991 +0 . 040 - 0 040 | . 5.4 +0 . 3 2 | . 5.4 +0 . 7 - 0 5 | . 7.7 +0 . 5 4 |\n| 543 | 80146-01-60-00 | . 5.373 +0 . 014 - 0 014 | - 0 . 4.7 +0 . 1 - 0 1 | . 8.1 +0 . 1 - 0 1 | - 0 . 9.4 +0 . 1 - 0 1 |\n| 544 | 80146-01-62-00 | . 4.549 +0 . 022 - 0 . 022 +0 . | . 3.1 +0 . 1 - 0 . +0 . | . 6.7 +0 . 2 - 0 . 2 | . 7.4 +0 . 1 - 0 . 1 |\n| 545 | 80146-01-65-00 | 4.938 012 - 0 . 012 | 1 4.9 1 - 0 . 1 | 5.8 +0 . 1 - 0 . 1 | 7.6 +0 . 1 - 0 . 1 |\n| 547 | 80146-01-67-00 | 0 027 5.152 +0 . 030 0 | 0 3 4.7 +0 . 4 | 0 2 6.8 +0 . 3 | 8.3 +0 . - 0 3 |\n| 546 548 | 80146-01-66-00 | 5.160 - . - . 034 +0 . 029 | 4.9 - . - 0 . 4 +0 . 3 | 6.3 - . - 0 . 3 | 8.0 - 0 . 3 4 . |\n| | 80146-01-68-00 | 5.323 - 0 . 029 | 4.9 - 0 . 3 | 8.3 +0 . 7 - 0 . 7 | 9.6 +0 . 6 - 0 6 |\n| 549 | 80146-01-69-00 | 5.363 +0 . 020 - 0 . 020 030 | 4.9 +0 . 1 - 0 . 1 | 6.9 +0 . 3 - 0 . 4 | . 8.4 +0 . 2 - 0 . 3 |\n| 550 551 | 94413-01-03-02 94413-01-03-03 | 3.752 +0 . - 0 . 030 3.728 +0 . 022 - | 4.9 +0 . 2 - 0 . 2 5.1 +0 . 2 - | 0.0 5.1 +0 . 2 - | 4.9 +0 . 2 - 0 . 2 5.1 +0 . 2 - |\n| 552 | 95360-14-04-00 | 0 022 3.563 +0 . 014 - 0 014 | 0 2 5.5 +0 . 1 - 0 1 | 0 2 2.8 +0 . 1 - 0 . 1 | 0 2 6.2 +0 . 1 - 0 1 |\n| 553 | 95405-01-01-01 | . 1.608 +0 . 043 | . +0 . 3 3 | +0 . | . +0 . 4 |\n| | | - 0 . 043 | 3.8 - 0 . | 4.0 5 - 0 . 4 | 5.5 - 0 . 4 |\n| 554 | 95405-01-02-00 | 1.939 +0 . 035 - 0 . 035 | 6.9 +0 . 4 - 0 . | 5.8 +0 . 3 - 0 . 3 | 6.9 +0 . 4 - 0 . 4 |\n| 555 | 95405-01-02-04 | 2.346 +0 . 062 . 062 | 4 6.1 +0 . 5 - 0 . 5 | 0.0 | 6.1 +0 . 5 - 0 . 5 |\n| 556 | 95405-01-02-05 | - 0 1.832 +0 . 025 - 0 . 025 | 6.0 +0 . 5 - 0 . 5 | 5.8 +1 . 0 - 1 . 0 | 8.3 +0 . 8 - 0 . 8 |\n| MAXI J1659-15 | MAXI J1659-15 | MAXI J1659-15 | MAXI J1659-15 | MAXI J1659-15 | MAXI J1659-15 |\n| 557 | 95108-01-27-00 | 3.888 +0 . 013 - 0 . 013 | 4.7 +0 . 1 - 0 . 1 | 6.0 +0 . 4 - 0 . 4 | 7.7 +0 . 3 - 0 . 3 |\n| 558 | 95108-01-28-00 | 4.103 +0 . 020 - 0 020 | 5.5 +0 . 2 - 0 | 9.1 +0 . 4 - 0 6 | 15.6 +0 . 8 - 0 8 |\n| 559 | 95118-01-01-01 | 3.857 +0 . 016 - | 2 5.8 +0 . 2 - 0 2 | 6.1 +0 . 9 - 1 3 | 8.4 +0 . 7 - 0 9 |\n| 560 | 95118-01-02-00 | 0 . 016 3.469 +0 . 034 - 0 | . 4.5 +0 . 1 - 0 1 | . 3.5 +0 . 3 - 0 3 | . 5.7 +0 . 2 0 |\n| 561 | 95118-01-07-00 | . 035 3.287 +0 . 015 - 0 015 | . 6.4 +0 . 2 - 0 2 | . 7.3 +0 . 8 - 0 8 | - . 2 9.7 +0 . 6 - 0 6 |\n| 562 | 95118-01-09-00 | 5.080 +0 . 160 - 0 160 | 2.7 +0 . 7 - 0 6 | 6.9 +0 . 4 - 0 4 | 7.4 +0 . 4 - 0 4 |\n| 563 | 95118-01-10-00 | 3.564 +0 . 017 | 6.1 +0 . 3 | 9.5 +0 . 9 | 11.3 +0 . 8 7 |\n| | | - 0 . 017 +0 . | - 0 . 3 +0 . 4 | - 0 . 8 | - 0 . |\n| 564 | 95118-01-12-00 | 1.972 040 - 0 . 040 | 5.0 - 0 . 4 | 4.5 +0 . 8 - 0 . 6 | 6.7 +0 . 6 - 0 . 5 |'}

2020PhRvD.101b4056K

Including higher order multipoles in gravitational-wave models for precessing binary black holes

2020-01-01

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Estimates of the source parameters of gravitational-wave (GW) events produced by compact binary mergers rely on theoretical models for the GW signal. We present the first frequency-domain model for the inspiral, merger, and ringdown of the GW signal from precessing binary black hole systems that also includes multipoles beyond the leading-order quadrupole. Our model, PhenomPv3HM, is a combination of the higher-multipole nonprecessing model PhenomHM and the spin-precessing model PhenomPv3 that includes two-spin precession via a dynamical rotation of the GW multipoles. We validate the new model by comparing to a large set of precessing numerical-relativity simulations and find excellent agreement across the majority of the parameter space they cover. For mass ratios &lt;5 the mismatch improves, on average, from ∼6 % to ∼2 % compared to PhenomPv3 when we include higher multipoles in the model. However, we find mismatches ∼8 % for a mass-ratio-6 and highly spinning simulation. We quantify the statistical uncertainty in the recovery of binary parameters by applying standard Bayesian parameter estimation methods to simulated signals. We find that, while the primary black hole spin parameters should be measurable even at moderate signal-to-noise ratios (SNRs) ∼30 , the secondary spin requires much larger SNRs ∼200 . We also quantify the systematic uncertainty expected by recovering our simulated signals with different waveform models in which various physical effects—such as the inclusion of higher modes and/or precession—are omitted and find that even in the low-SNR case (∼17 ) the recovered parameters can be biased. Finally, as a first application of the new model we analyze the binary black hole event GW170729. We find larger values for the primary black hole mass of 58.25<SUB>-12.53</SUB><SUP>+11.73</SUP> M<SUB>⊙</SUB> (90% credible interval). The lower limit (∼46 M<SUB>⊙</SUB> ) is comparable to the proposed maximum black hole mass predicted by different stellar evolution models due to the pulsation pair-instability supernova (PPISN) mechanism. If we assume that the primary black hole in GW170729 formed through a PPISN, then out of the four PPISN models we consider only the model of Woosley [1] is consistent with our mass measurements at the 90% confidence level.

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https://arxiv.org/pdf/1911.06050.pdf

{'Including higher order multipoles in gravitational-wave models for precessing binary black holes': "Sebastian Khan, 1, 2, 3 Frank Ohme, 2, 3 Katerina Chatziioannou, 4 and Mark Hannam 1, 5 \n1 School of Physics and Astronomy, Cardi GLYPH<11> University, The Parade, Cardi GLYPH<11> , CF24 3AA, United Kingdom \n2 Max Planck Institute for Gravitational Physics (Albert Einstein Institute), Callinstr. 38, 30167 Hannover, Germany \n3 Leibniz Universitat Hannover, D-30167 Hannover, Germany \n4 Center for Computational Astrophysics, Flatiron Institute, 162 5th Ave, New York, NY 10010 \n5 Dipartimento di Fisica, Universit'a di Roma 'Sapienza', Piazzale A. Moro 5, I-00185 Roma, Italy \n(Dated: March 16, 2020) \nEstimates of the source parameters of gravitational-wave (GW) events produced by compact binary mergers rely on theoretical models for the GW signal. We present the first frequency-domain model for inspiral, merger and ringdown of the GW signal from precessing binary-black-hole systems that also includes multipoles beyond the leading-order quadrupole. Our model, PhenomPv3HM , is a combination of the higher-multipole nonprecessing model PhenomHM and the spin-precessing model PhenomPv3 that includes two-spin precession via a dynamical rotation of the GW multipoles. We validate the new model by comparing to a large set of precessing numerical-relativity simulations and find excellent agreement across the majority of the parameter space they cover. For mass ratios < 5 the mismatch improves, on average, from GLYPH<24> 6% to GLYPH<24> 2% compared to PhenomPv3 when we include higher multipoles in the model. However, we find mismatches GLYPH<24> 8% for a mass-ratio 6 and highly spinning simulation. We quantify the statistical uncertainty in the recovery of binary parameters by applying standard Bayesian parameter estimation methods to simulated signals. We find that, while the primary black hole spin parameters should be measurable even at moderate signal-to-noise ratios (SNRs) GLYPH<24> 30, the secondary spin requires much larger SNRs GLYPH<24> 200. We also quantify the systematic uncertainty expected by recovering our simulated signals with di GLYPH<11> erent waveform models in which various physical e GLYPH<11> ects, such as the inclusion of higher modes and / or precession, are omitted and find that even at the low SNR case ( GLYPH<24> 17) the recovered parameters can be biased. Finally, as a first application of the new model we have analysed the binary black hole event GW170729. We find larger values for the primary black hole mass of 58 : 25 + 11 : 73 GLYPH<0> 12 : 53 M GLYPH<12> (90% credible interval). The lower limit ( GLYPH<24> 46 M GLYPH<12> ) is comparable to the proposed maximum black hole mass predicted by di GLYPH<11> erent stellar evolution models due to the pulsation pair-instability supernova (PPISN) mechanism. If we assume that the primary Black Hole (BH) in GW170729 formed through a PPISN then out of the four PPISN models we considered only the model of Woosley [1] is consistent with our mass measurements at the 90% level. \nPACS numbers: 04.80.Nn, 04.25.dg, 95.85.Sz, 97.80.-d", 'I. INTRODUCTION': "The second generation Gravitational Wave (GW) detectors - Advanced LIGO [2] and Virgo [3] - have so far published observations of 11 compact binary mergers from the first two observing runs [4], including one binary neutron star merger that was also observed across the electromagnetic spectrum [5]. The third observing run is currently underway, with further sensitivity improvements planned in the coming years [6]. GW observations have already begun to constrain models of the formation and rates of stellar mass compact binary mergers [7], and to make strong-field tests of the general theory of relativity [8]. \nModels for the GW signal, parametrized in terms of the properties of the system (such as masses, spins, and orientation), are compared with detector data to infer the source properties of GW events. The GW signal is commonly expressed in a multipole expansion where we denote terms beyond the leading order quadrupole contribution as 'higher order multipoles'. These higher order multipoles are typically much weaker than the dominant quadrupolar multipole, but grow in relative strength for systems that are more asymmetric in mass. Past studies have shown that for events where the signal contains measurable power in the higher multipoles, parameter estimates can be biassed when using only a dominant- \nmultipole model. Conversely, it is also true that for some systems we are able to measure the source parameters more accurately using a higher multipole model [9-14]. \nAnother important physical e GLYPH<11> ect is spin precession, where couplings between the orbital and spin angular momenta can cause the orbital plane to precess and thus cause modulations of the observed GW [15, 16]. In terms of the GW multipoles, precession mixes together di GLYPH<11> erent orders ( m -multipoles) of the same degree ( ' -multipoles), complicating a simple description of the waveform [17-24]. By not taking into account precession and higher order multipoles in our waveform models we may not be able to confidently detect and accurately characterize signals where these e GLYPH<11> ects are important [25-30]. These events are also likely to be very interesting astrophysically, providing valuable information about Binary Black Hole (BBH) formation mechanisms and hence are events with high scientific gain that we wish to model and measure accurately. \nThe field of waveform modeling has seen sustained development over almost two decades and is currently thriving, with improvements to current and development of novel methods allowing for more accurate and e GLYPH<14> cient models to be applied in data analysis pipelines [10, 31-54]. In this work we take a step towards including as many important physical e GLYPH<11> ects as possible in waveform models, by constructing the \nFIG. 1. Comparison of the detector response strain h ( t ) viewed at an inclination angle of GLYPH<25>= 2. Solid grey: NR simulation (SXS:BBH:0058). A mass-ratio q GLYPH<17> m 1 = m 2 = 5, precessing BBH simulation with a dimensionless-spin magnitude of GLYPH<31> 1 = 0 : 5 generated with a total mass of 80 M GLYPH<12> . The NR signal contains all the '; m modes up to and including ' = 4. Top panel, blue: Precessing model SEOBNRv3 [57], with the ((2 ; GLYPH<6> 2) ; (2 ; GLYPH<6> 1)) modes in the co-precessing frame. Middle panel, orange: Precessing model PhenomPv3 [50] with only the 2 ; GLYPH<6> 2 modes in the co-precessing frame. Bottom panel, green: Precessing model presented here, PhenomPv3HM , with the ( '; j m j ) = ((2 ; 2) ; (2 ; 1) ; (3 ; 3) ; (3 ; 2) ; (4 ; 4) ; (4 ; 3)) modes in the coprecessing frame. The orientation-averaged mismatch 1 GLYPH<0> M (see Sec. III A) is 8 % for the top panel, 7 % for the middle and 1 % for the bottom. We only plot the last GLYPH<24> 7 GW cycles for clarity but the behaviour is qualitatively the same throughout the 29-orbit inspiral ( GLYPH<24> 60 GW cycles). \n<!-- image --> \nmost physically complete phenomenological model to date. Wepresent a frequency-domain model for the GW signal from the inspiral, merger and ringdown of a BBH system. The BHs are allowed to precess and we also model the contribution to the GW signal from higher order multipoles. This combines the progress made in two earlier models: a precessing-binary model that includes accurate two-spin precession e GLYPH<11> ects during the inspiral [55, 56] ( PhenomPv3 [50]), and an approximate higher-multipole aligned-spin model ( PhenomHM [10]). \nFigure 1 demonstrates the improved accuracy that is achievable by our new model, PhenomPv3HM , compared to other existing models that include the e GLYPH<11> ect of spin precession, but not higher order multipoles. We compare the observed GW signal predicted by our new model against a high-mass-ratio, \nprecessing NR simulation 1 (thick grey line). We plot the GW signal observed at an inclination angle 2 of GLYPH<25>= 2 rad to emphasise the e GLYPH<11> ect of precession. We use all multipoles in the range 2 GLYPH<20> ' GLYPH<20> 4 when computing the NR GW polarisations. \nWe compute the mismatch (defined in Section III A) between three di GLYPH<11> erent precessing waveform models and the NR waveform, and average over all possible orientations. The top panel shows the optimal waveform (in blue) when we use SEOBNRv3 [43] and the middle panel shows (in orange) the result when we use PhenomPv3 . In this context the optimal waveform maximises the overlap over coalescence time, template phase and polarisation angle and the intrinsic parameters are fixed to the values from the NR simulation. As shown in [50] SEOBNRv3 and PhenomPv3 have overlaps of GLYPH<24> 99 % and GLYPH<24> 98 % respectively to this NR waveform when only the ' = 2 multipoles are considered. When we include higher order multipoles in the NR waveform we find the overlap drops to only GLYPH<24> 92 % and GLYPH<24> 93 % respectively. This is an example where the exclusion of higher multipoles in template models can lead to unacceptable losses in signal-to-noise ratio (SNR). The bottom panel shows, in green, the best fitting PhenomPv3HM template. We find remarkable agreement, even through the inspiral, merger and ringdown stages. The overlap is now 99 % and the subtle modulation visible is accurately captured by our model. It is useful to point out here that, in PhenomPv3HM , the higher multipole and the precession elements of the model have not been calibrated to NR simulations, but when this is done we expect the accuracy to improve further. \nThe rest of the paper is organized as follows. In Sec. II we describe how our model is constructed. In Sec. III A we present results where we have compared our model against precessing NR simulations including higher order mutlipoles up to and including ' = 4 to demonstrate its accuracy across the parameter space where we have NR simulations. We have also performed a parameter estimation study to quantify the impact on parameter recovery when using a model that includes both higher multipoles and precession, the results of which are presented in section III B. \nFinally, in section III C we have analysed data for the GW170729 event, publicly available at the Gravitational Wave Open Science Center (GWOSC) [59], which has evidence for non-zero BH spin [4] and unequal masses [11].", 'II. METHOD': "Our method to build a model for the GW signal from precessing BBHs is based upon the novel ideas of Refs. [18, 23, 60], where the GW from precessing binaries can be modelled as a dynamic rotation of non-precessing systems. In \nRefs. [35, 37, 57] the authors used these ideas to build the first precessing Inspiral-Merger-Ringdown (IMR) models. \nOur goal is to derive frequency-domain expressions for the GW polarisations ˜ h + = GLYPH<2> ( f ) in terms of the multipoles ˜ h ' m ( f ). We start from the complex GW quantity, h = h + GLYPH<0> ih GLYPH<2> , in the time-domain and decompose this into spin weight GLYPH<0> 2 spherical harmonics, \nh ( t ; ~ GLYPH<21>; GLYPH<18>; GLYPH<30> ) = X ' > 2 X GLYPH<0> ' 6 m 6 ' h '; m ( t ; ~ GLYPH<21> ) GLYPH<0> 2 Y '; m ( GLYPH<18>; GLYPH<30> ) : (1) \nThis is a function of the time t , the intrinsic source parameters (masses and spin angular momenta of the bodies) denoted by ~ GLYPH<21> , and the polar angles GLYPH<18> and GLYPH<30> of a coordinate system whose z GLYPH<0> axis is aligned with the total angular momentum ~ J of the binary at some reference frequency. To approximate the precessing multipoles h prec '; m ( t ) we perform a dynamic rotation of the non-precessing multipoles h non GLYPH<0> prec '; m ( t ), \nh prec '; m ( t ) = X GLYPH<0> ' 6 m 0 6 ' h non GLYPH<0> prec '; m 0 ( t ) D ' m 0 ; m ( GLYPH<11> ( t ) ; GLYPH<12> ( t ) ; GLYPH<15> ( t )) : (2) \nWe define the Wigner D-matrix as D ' m 0 ; m ( GLYPH<11>; GLYPH<12>; GLYPH<15> ) = e im GLYPH<11> d ' m 0 ; m ( GLYPH<0> GLYPH<12> ) e GLYPH<0> im 0 GLYPH<15> and the Wigner d -matrix is given in Ref. [61]. \nNext we transform to the frequency domain using the stationary phase approximation [62] under the assumption that the precession angles modify the signal via a slowly varying amplitude, giving us an expression for the frequency-domain multipoles in terms of the co-precessing frame multipoles, \n˜ h prec '; m ( f ) = X GLYPH<0> 6 '< m 0 6 ' ˜ h non GLYPH<0> prec '; m 0 ( f ) D ' m 0 ; m ( GLYPH<11>; GLYPH<12>; GLYPH<15> ) : (3) \nFor brevity we omit the explicit dependence on frequency for the the precession angles ( GLYPH<11>; GLYPH<12>; GLYPH<15> ) but they are evaluated at the stationary points t ( f ) = 2 GLYPH<25> f = m 0 [39]. \nThe frequency-domain GW polarisations ˜ h + = GLYPH<2> ( f ) are defined as the Fourier transform (FT) of the real-valued GW polarisations h + = GLYPH<2> ( t ), which we write as, \n˜ h + ( f ) = FT[Re( h ( t ))] = 1 2 GLYPH<16> ˜ h ( f ) + ˜ h GLYPH<3> ( GLYPH<0> f ) GLYPH<17> ; (4) \n˜ h GLYPH<2> ( f ) = FT[Im( h ( t ))] = i 2 GLYPH<16> ˜ h ( f ) GLYPH<0> ˜ h GLYPH<3> ( GLYPH<0> f ) GLYPH<17> : (5) \nTo arrive at the final expression for the frequency-domain GW polarisations we substitute Eq. (3) into Eqs. (4) and (5), assuming f > 0 and symmetry through the orbital plane in the co-precessing frame 3 , leading to, \n˜ h prec + ( f ) = 1 2 X ' > 2 X m 0 > 0 ˜ h non GLYPH<0> prec '; m 0 ( f ) ' X m = GLYPH<0> ' GLYPH<16> A ' m 0 ; m + ( GLYPH<0> 1) ' A GLYPH<3> ' GLYPH<0> m 0 ; m GLYPH<17> ; (6) \n˜ h prec GLYPH<2> ( f ) = GLYPH<0> i 2 X ' > 2 X m 0 > 0 ˜ h non GLYPH<0> prec '; m 0 ( f ) ' X m = GLYPH<0> ' GLYPH<16> A ' m 0 ; m GLYPH<0> ( GLYPH<0> 1) ' A GLYPH<3> ' GLYPH<0> m 0 ; m GLYPH<17> : (7) \nTo shorten the expression we define the auxillary matrix A ' m 0 ; m GLYPH<17> GLYPH<0> 2 Y '; mD ' m 0 ; m and omit the explicit angular dependence of GLYPH<0> 2 Y '; m and the precession angles in D ' m 0 ; m . The summation over ' and m 0 are over the modes included in the co-precessing frame. Here we use the PhenomHM model [10], which contains the ( '; j m 0 j ) = ((2 ; 2) ; (2 ; 1) ; (3 ; 3) ; (3 ; 2) ; (4 ; 4) ; (4 ; 3)) modes. \nDue to precession the properties of the remnant BH in the precessing system are di GLYPH<11> erent to those in the equivalent nonprecessing system. We use the same prescription as described in Sec.III.C of Ref. [50] to include the in-plane-spin contribution to the spin of the remnant BH. This modified final spin vector changes the ringdown spectrum of the aligned-spin multipoles. \nLastly, we note that the models for the three ingredients (the non-precessing model, the precession angles, and the BH remnant model) are independent in our construction, and can therefore each be updated when any of them are improved.", 'A. Mismatch Computation': "The standard metric to assess the accuracy of GW signal models is to calculate the noise-weighted inner product between the template model and an accurate signal waveform. As our signal we use NR waveforms from the publicly available SXS catalogue [58, 63, 64] generated using the NR injection infrastructure in LALSuite [65]. From this catalogue we select the precessing configurations with the highest numerical resolution. This set contains 90 systems with q 2 [1 ; 6], however, the majority of cases have q 6 3. We have 2 cases at q = 5 and one case at q = 6. There are six cases that have at least one BH with a dimensionless spin magnitude j GLYPH<31> j > 0 : 5 whereas the majority of cases have j GLYPH<31> j 6 0 : 5 4 . For the exact list of NR configurations and specific details on how the mismatch calculations were performed we refer the reader to Ref. [50] where we presented an identical analysis but restricted the signals to contain the ' = 2 multipoles. \nTABLE I. Match results from Sec. III A. We quote the mean value of the match for each inclination angle considered ( GLYPH<19> 2 [0 ; GLYPH<25>= 3 ; GLYPH<25>= 2] rad) and averaged over all cases in the mass-ratio category for the Mtot = 100 M GLYPH<12> case. The subscript and superscript are the minimum and maximum values of the match for the mass-ratio category considered. \n| Waveform Model | PhenomPv3 | PhenomPv3 | PhenomPv3HM | PhenomPv3HM | PhenomPv3HM | PhenomPv3HM |\n|------------------|-------------------------|--------------|---------------|-------------------------|-------------------------|---------------|\n| Mass-Ratio (#) | 0 | GLYPH<25>= 3 | GLYPH<25>= 2 | 0 | GLYPH<25>= 3 | GLYPH<25>= 2 |\n| 1 6 q 6 2 (72) | 0 : 998 0 : 999 0 : 993 | 0 : 989 0 0 | 0 : 982 0 0 | 993 967 0 : 997 0 0 | : 999 : 993 0 : 993 0 0 | 0 : 987 0 0 |\n| q = 3 (15) | 0 : 989 0 : 999 0 : 974 | 0 : 959 0 0 | 0 : 941 0 0 | : 946 : 933 0 : 993 0 0 | : 997 : 985 0 : 987 0 0 | 0 : 984 0 0 |\n| q = 5 (2) | 0 : 973 0 : 978 0 : 968 | 0 : 941 0 0 | 0 : 911 0 0 | : 925 : 897 0 : 990 0 0 | : 990 : 989 0 : 971 0 0 | 0 : 978 0 0 |\n| q = | 0 : 863 | 0 : | 0 : 939 | 0 : | 0 : 914 | 0 : |\n| 6 (1) | | 919 | | 950 | | 898 | \nFIG. 2. The results of the comparison between PhenomPv3 (top), PhenomPv3HM (bottom) and the precessing NR simulations from the public SXS catalogue. The figure shows the mismatch average over a reference phase and polarisation angle (1 GLYPH<0> M ) as a function of the total mass for an inclination of GLYPH<19> = GLYPH<25>= 3. The worst case is SXS:BBH:0165, a short ( GLYPH<24> 6 orbits) signal with mass ratio 1:6 and high precession. \n<!-- image --> \nSince PhenomPv3 is constructed from PhenomD it only has the ' = j m j = 2 modes in the co-precessing frame and therefore we expect this model to perform poorly when the contribution to the signal due to higher modes is not negligible. As PhenomPv3HM is constructed from PhenomHM and contains the ( '; j m j ) = ((2 ; 2) ; (2 ; 1) ; (3 ; 3) ; (3 ; 2) ; (4 ; 4) ; (4 ; 3)) modes in the co-precessing frame we expect it to outperform PhenomPv3 . In the NR signal we include multipoles with ' 2 [2 ; 4] to be consistent with the highest modeled ' mode in PhenomPv3HM . \nWe use the expected noise curve for Advanced LIGO oper- \nating at design sensitivity [66] 5 with a low-frequency cuto GLYPH<11> of 10Hz. Due to the presence of higher modes the orbital phase of the binary is no longer degenerate with the phase of the observed waveform, which means the standard method to analytically maximise over the template phase is not applicable. It is possible, however, to analytically maximise over the template polarisation using the skymax-SNR derived in Ref. [28]. In our match calculation we analytically maximise over the template polarisation and relative time shift and numerically optimise over the template orbital reference phase and frequency. Finally we average the match, weighted by the optimal SNR, over the signal orbital reference phase and polarisation angle. See Sec. III A in Ref. [50] for details. \nFigure 2 shows the orientation-averaged-mismatch [50] as a function of the total mass of the binary for an inclination angle GLYPH<19> = GLYPH<25>= 3. Here GLYPH<19> is the angle between the Newtonian orbital angular momentum and the line of sight at the start frequency of the NR waveform. The first row uses the dominant multipoleonly model, PhenomPv3 , and the second row uses the new higher multipole model, PhenomPv3HM , presented here. We clearly see that for q > 3 that it is important to include higher modes in the template model. \nIn Table I we summarise the results of our validation study by tabulating the results as the match (as opposed to mismatch as in Fig. 2) for each model according to mass ratio and inclination angle. Next to the mass-ratio range in parentheses is the number of NR cases in that mass-ratio category. Each entry in the table is calculated as follows: for the Mtot = 100 M GLYPH<12> case we average the match over all cases in the mass-ratio category and write the minimum and maximum match as subscript and superscript respectively. \n- 1 6 q 6 2 : In this mass-ratio range both PhenomPv3 and PhenomPv3HM perform comparably, most likely due to the strength of higher multipoles scaling with mass ratio. \nq = 3 : Here we start to see the importance of the higher multipoles to accurately describe the NR signal. For GLYPH<19> = 0 PhenomPv3 has an average match of 0 : 989. However, as the inclination angle increases, thus emphasising more of the higher multipole content of the signal, the average match drops to 0 : 941 and can be as low as 0 : 933. On the other hand, PhenomPv3HM is able to describe the NR data to an average accuracy of 0 : 984 with a minimum value of 0 : 974 for inclined \nsystems. \nq = 5 : At this mass-ratio the loss in performance for PhenomPv3 is noticable even for low inclination values. At GLYPH<19> = 0 the average match is 0 : 973 dropping to 0 : 911 at GLYPH<19> = GLYPH<25>= 2. The match for PhenomPv3HM at GLYPH<19> = 0 remains high at 0 : 99, but reduces to 0 : 978 at GLYPH<19> = GLYPH<25>= 2. Note that we only have two NR simulations at q = 5 and are thus unable to rigorously test the model at this and similar mass ratios. \nq = 6 : When comparing to this NR simulation we find both models perform substantially worse than the q = 5 cases with even PhenomPv3 outperforming PhenomPv3HM with matches as low as 0 : 898. We have verified that we obtain matches of GLYPH<24> 0 : 97 when restricting the NR waveform to just the ' = 2 multipoles, consistent with our previous study [50]. We conclude that either our model is outside its range of validity or that this NR simulation is inaccurate for the higher multipoles, however, our results are robust against NR simulations of this configuration at multiple resolutions. This NR simulation, SXS:BBH:0165 is exceptional for a few reasons. First, it is a high mass-ratio system where higher multipoles are more important. Second, it is a strongly precessing system with primary GLYPH<31> 1 = (0 : 74 ; 0 : 19 ; GLYPH<0> 0 : 5) and secondary GLYPH<31> 2 = ( GLYPH<0> 0 : 19 ; 0 :; GLYPH<0> 0 : 23) spin vectors. Finally, it is also very short, only containing GLYPH<24> 6 : 5 orbits. We encourage more NR simulations in this region by di GLYPH<11> erent NR codes to (i) cross check the results and (ii) populate this region with more data with which to test and refine future models. \nWe conclude from our study that PhenomPv3HM greatly improves the accuracy towards precessing BBHs for systems with mass-ratio up to 5:1. We expect to be able to greatly improve the accuracy and extend towards higher mass-ratio by further calibrating the higher order multipoles and precession e GLYPH<11> ects to NR simulations.", 'B. Parameter Uncertainty': 'One of the main purposes of a waveform model is to estimate the source parameters of GW events. With models we can quantify the expected parameter uncertainty as a function of the parameter space [68-76]. Instead of a computationally intense systematic parameter estimation campaign we have chosen to focus on one configuration and study in detail the dependency of parameter recovery on SNR. We wish to study a system where both precession and higher modes are important and guided by previous studies [25, 26, 77] we chose to study a double precessing spin, mass-ratio 3 BBH signal with a total mass of 150 M GLYPH<12> in the detector frame. Starting at a frequency of 10 Hz this system produces a waveform with about 20 GW cycles and merges at a frequency of about 120 Hz. See Table II for specific injection values, where ( GLYPH<18> JN ; GLYPH<30> ) define the direction of propagation in the source frame, ( GLYPH<11>; GLYPH<14> ) are the right-ascension and declination of the source and is the polarisation angle. \nWe simulate this fiducial signal with PhenomPv3HM and recover its parameters using the parallel tempered MCMC algorithm implemented as LALInferenceMCMC in the publicly available LALInference software [78] with PhenomPv3HM as \nTABLE II. Injection parameters and results from parameter estimation of simulated signals. We quote the median and 90% credible interval. \n| Parameter | Injection Value | GLYPH<26> = 17 : 6 DL = 3000 | GLYPH<26> = 35 : 2 DL = 1500 | GLYPH<26> = 176 DL = 300 |\n|-----------------------------|-------------------|---------------------------------------------|------------------------------------|-----------------------------------------|\n| m det 1 = M GLYPH<12> | 112.500 | 102 : 98 + 13 : 38 GLYPH<0> 12 : 26 | 107 : 71 + 7 : 96 GLYPH<0> 7 : 43 | 112 : 38 + 1 : 73 GLYPH<0> 1 : 75 |\n| m det 2 = M GLYPH<12> | 37.500 | 40 : 62 + 5 : 92 GLYPH<0> 5 : 29 | 39 : 00 + 2 : 88 GLYPH<0> 2 : 66 | 37 : 55 + 0 : 54 GLYPH<0> 0 : 52 |\n| M det total = M GLYPH<12> | 150.000 | 143 : 64 + 11 : 28 GLYPH<0> 9 : 74 | 146 : 78 + 6 : 48 GLYPH<0> 5 : 94 | 149 : 93 + 1 : 49 GLYPH<0> 1 : 46 |\n| M det c = M GLYPH<12> | 54.940 | 55 : 08 + 3 : 37 GLYPH<0> 3 : 20 | 55 : 05 + 1 : 69 GLYPH<0> 1 : 69 | 54 : 95 + 0 : 36 GLYPH<0> 0 : 34 |\n| q | 0.333 | 0 : 39 + 0 : 11 GLYPH<0> 0 : 08 | 0 : 36 + 0 : 05 GLYPH<0> 0 : 04 | 0 : 33 + 0 : 01 GLYPH<0> 0 : 01 |\n| GLYPH<18> 1 / rad | 1.052 | 1 : 14 + 0 : 36 GLYPH<0> 0 : 37 | 1 : 10 + 0 : 27 GLYPH<0> 0 : 19 | 1 : 05 + 0 : 04 GLYPH<0> 0 : 04 |\n| GLYPH<18> 2 / rad | 2.090 | 1 : 73 + 1 : 01 GLYPH<0> 1 : 21 | 2 : 04 + 0 : 72 GLYPH<0> 1 : 09 | 2 : 09 + 0 : 14 GLYPH<0> 0 : 12 |\n| GLYPH<1> GLYPH<30> 12 / rad | 1.571 | 2 : 82 + 3 : 11 GLYPH<0> 2 : 49 | 1 : 79 + 3 : 42 GLYPH<0> 1 : 35 | 1 : 58 + 0 : 24 GLYPH<0> 0 : 24 |\n| GLYPH<18> JN / rad | 1.050 | 1 : 62 + 0 : 60 GLYPH<0> 0 : 73 | 1 : 21 + 0 : 92 GLYPH<0> 0 : 23 | 1 : 05 + 0 : 03 GLYPH<0> 0 : 03 |\n| cos( GLYPH<30> ) | 1.000 | GLYPH<0> 0 : 04 + 1 : 03 GLYPH<0> 0 : 96 | 0 : 46 + 0 : 54 GLYPH<0> 1 : 46 | 1 : 00 + 0 : 00 GLYPH<0> 0 : 01 |\n| GLYPH<11> / rad | 1.047 | 4 : 235 + 0 : 124 GLYPH<0> 3 213 | 1 : 070 + 3 : 277 GLYPH<0> 0 036 | 1 : 047 + 0 : 004 GLYPH<0> 0 004 |\n| GLYPH<14> / rad | 1.047 | GLYPH<0> 1 : 020 + 2 : 098 GLYPH<0> 0 : 125 | 1 : 025 + 0 : 037 GLYPH<0> 2 : 155 | 1 : 047 + 0 : 004 GLYPH<0> 0 : 004 |\n| / rad | 1.047 | 1 : 52 + 0 : 66 GLYPH<0> 0 : 76 | 1 : 28 + 0 : 80 GLYPH<0> 0 : 39 | 1 : 05 + 0 : 04 GLYPH<0> 0 : 04 |\n| GLYPH<31> e GLYPH<11> | 0.200 | 0 : 204 + 0 : 129 GLYPH<0> 0 : 136 | 0 : 201 + 0 : 070 GLYPH<0> 0 : 074 | 0 : 200 + 0 : 016 GLYPH<0> 0 : 017 |\n| GLYPH<31> p | 0.700 | 0 : 681 + 0 : 186 GLYPH<0> 0 : 285 | 0 : 705 + 0 : 098 GLYPH<0> 0 : 105 | 0 : 699 + 0 : 020 GLYPH<0> 0 : 024 |\n| j GLYPH<31> 1 j | 0.806 | 0 : 77 + 0 : 15 GLYPH<0> 0 : 27 | 0 : 80 + 0 : 07 GLYPH<0> 0 : 09 | 0 : 81 + 0 : 02 GLYPH<0> 0 : 02 |\n| j GLYPH<31> 2 j | 0.806 | 0 : 45 + 0 : 47 GLYPH<0> 0 : 40 | 0 : 59 + 0 : 35 GLYPH<0> 0 : 42 | 0 : 80 + 0 : 13 GLYPH<0> 0 : 11 |\n| DL / Mpc | see heading | 3086 : 50 + 739 : 44 GLYPH<0> 571 : 98 | 1465 : 76 + 177 : GLYPH<0> 157 : | 84 87 300 : 12 + 7 : 14 GLYPH<0> 6 : 94 | \nthe template model. We perform three separate, zero-noise, injections to investigate how our results depend on the injected SNR. Specifically we inject the signal at luminosity distances of 3000 Mpc, 1500 Mpc and 300 Mpc corresponding to a three-detector network SNR of 17, 35 and 176 respectively. We use the design sensitivity noise curves for the LIGO Hanford, LIGO Livingston and Virgo detectors [66]. \nWe present our results by tabulating the median and 90% credible interval on binary parameters in Tab. II and source frame parameters in Tab. III. We also plot the 90% credible interval as a function of the injected SNR for a few chosen parameters in Figs. 3, 4 and 5. In the high SNR limit the uncertainty on the parameters should decrease linearly wtih SNR i.e., as 1 =GLYPH<26> [79], which is shown as a dashed black line in these figures. \nIn the following discussion we change our convention for the mass-ratio to q GLYPH<17> m 2 = m 1 2 [0 ; 1] and will abbreviate the width of the 90% credible interval of parameter X at an SNR of GLYPH<26> as C GLYPH<26> 90% ( X ).', '1. Masses': 'Figure 3 shows the source frame mass parameters; primary mass m src 1 , secondary mass m src 2 , chirp mass M src c , total mass M src total and mass-ratio q . We find good scaling with respect to \nTABLE III. Source frame injection parameters and results from parameter estimation of simulated signals. We quote the median and 90% credible interval. \n| | GLYPH<26> = 17 | GLYPH<26> = 17 | GLYPH<26> = 34 | GLYPH<26> = 34 | GLYPH<26> = 176 | GLYPH<26> = 176 |\n|---------------------------|------------------|----------------------------------------|------------------|----------------------------------------|-------------------|------------------------------------|\n| Parameter | Inj. | Rec. | Inj. | Rec. | Inj. | Rec. |\n| m src 1 = M GLYPH<12> | 74.56 | 67 : 60 + 8 : 22 GLYPH<0> 8 : 35 | 87.789 | 84 : 28 + 5 : 36 GLYPH<0> 5 : 24 | 105.724 | 105 : 54 + 1 : 55 GLYPH<0> 1 : 58 |\n| m src 2 = M GLYPH<12> | 24.853 | 26 : 60 + 3 : 95 GLYPH<0> 3 : 55 | 29.263 | 30 : 48 + 2 : 47 GLYPH<0> 2 : 21 | 35.241 | 35 : 27 + 0 : 51 GLYPH<0> 0 : 50 |\n| M src total = M GLYPH<12> | 99.413 | 94 : 35 + 6 : 64 GLYPH<0> 7 : 16 | 117.052 | 114 : 82 + 3 : 97 GLYPH<0> 3 : 96 | 140.965 | 140 : 80 + 1 : 28 GLYPH<0> 1 : 27 |\n| M src c = M GLYPH<12> | 36.412 | 36 : 11 + 2 : 29 GLYPH<0> 2 : 34 | 42.872 | 43 : 04 + 1 : 31 GLYPH<0> 1 : 28 | 51.631 | 51 : 61 + 0 : 31 GLYPH<0> 0 : 30 |\n| DL = Mpc | 3000 | 3086 : 50 + 739 : 44 GLYPH<0> 571 : 98 | 1500 | 1465 : 76 + 177 : 84 GLYPH<0> 157 : 87 | 300 | 300 : 12 + 7 : 14 GLYPH<0> 6 : 94 |\n| z | 0.509 | 0 : 52 + 0 : 10 GLYPH<0> 0 : 08 | 0.281 | 0 : 28 + 0 : 03 GLYPH<0> 0 : 03 | 0.064 | 0 : 065 + 0 : 001 GLYPH<0> 0 : 001 | \nFIG. 3. 90% credible intervals for the source frame mass parameters as a function of injected SNR \n<!-- image --> \n1 =GLYPH<26> for all source frame mass parameters. \nTable III shows the injection and recovered values. Even at the high total masses we consider here we find that the chirp mass is still the best measured parameter with C 17 90% ( M src c ) = 4 : 63 M GLYPH<12> and C 176 90% ( M src c ) = 0 : 61 M GLYPH<12> . The total mass is the next best measured mass parameter with low and high SNR accuracies of C 17 90% ( M src total ) = 13 : 8 M GLYPH<12> and C 176 90% ( M src total ) = 2 : 55 M GLYPH<12> respectively. We find the primary mass can be measured to an accuracy of C 17 90% ( m src 1 ) = 16 : 57 M GLYPH<12> for low SNR and C 176 90% ( m src 1 ) = 3 : 13 M GLYPH<12> for high SNR. And for the secondary mass we find C 17 90% ( m src 2 ) = 7 : 5 M GLYPH<12> and C 176 90% ( m src 2 ) = 1 : 01 M GLYPH<12> for low and high SNR respectively. Finally, we are able to constrain the mass-ratio to C 17 90% ( q ) = 0 : 19 and C 176 90% ( q ) = 0 : 02.', '2. Spins': 'Figure 4 shows the primary and secondary spin magnitude j GLYPH<31> 1 j ; j GLYPH<31> 2 j , the e GLYPH<11> ective aligned-spin GLYPH<31> e GLYPH<11> and e GLYPH<11> ective \nFIG. 4. 90% credible intervals for the BH spin magnitudes and effective spin parameters as a function of injected SNR. \n<!-- image --> \nprecessing-spin GLYPH<31> p parameters. \nWith the exception of j GLYPH<31> 2 j we find good agreement with the 1 =GLYPH<26> scaling. This suggests that for j GLYPH<31> 2 j the two weaker injections do not have high enough SNR for the posterior distribution function for this parameter to be approximated by a Gaussian [79]. That being said, we do observe the 90% width decrease with SNR albeit at a slower rate. At SNR of 17 and 34 we find we are not able to place strong constraints on j GLYPH<31> 2 j with C 17 90% ( j GLYPH<31> 2 j ) = 0 : 87 and C 34 90% ( j GLYPH<31> 2 j ) = 0 : 77. However, at the high SNR of 176 we begin to constrain the spin magnitude at the level of C 176 90% ( j GLYPH<31> 2 j ) = 0 : 24, approximately the same level of uncertainty as GLYPH<31> e GLYPH<11> at a SNR of 17. This is consistent with the study of non-precessing binaries in Ref. [80], which concluded that the secondary spin will not be measurable for SNRs below GLYPH<24> 100, but our results suggest that this carries over to precessing systems. \nThe primary spin magnitude is measured with much higher precision than the secondary spin magnitude. However, constraining this parameter to a 90% width of less than 0.2 requires an SNR of GLYPH<24> 30. This parameter does follow the 1 =GLYPH<26> scaling very well and for high SNR cases we estimate the sta- \nFIG. 5. 90% credible intervals for the BH spin orientation parameters as a function of injected SNR. \n<!-- image --> \ntistical uncertainty to be C 176 90% ( j GLYPH<31> 1 j ) = 0 : 04. \nOf the e GLYPH<11> ective spin parameters the e GLYPH<11> ective aligned parameter GLYPH<31> e GLYPH<11> is the best measured quantity. This is closely related to the leading order spin e GLYPH<11> ect in Post Newtonian (PN) theory [81, 82] appearing at 1.5 PN order. For all three SNRs the median value is always within 10 GLYPH<0> 3 of the true value with the uncertainties ranging from C 17 90% ( GLYPH<31> e GLYPH<11> ) = 0 : 265 to C 176 90% ( GLYPH<31> e GLYPH<11> ) = 0 : 033. \nTurning towards the e GLYPH<11> ective precession spin parameter, GLYPH<31> p , at the lowest SNR we find the marginalised posterior for GLYPH<31> p has a median value of 0 : 681, close to the true value but with a wide uncertainty of C 17 90% ( GLYPH<31> p ) = 0 : 47, spanning almost half of the full range. The evolution of the median value does not change significantly with increasing SNR however, our measurement uncertainty does decrease with increasing SNR as expected and we find C 35 90% ( GLYPH<31> p ) = 0 : 203 for the medium SNR and C 176 90% ( GLYPH<31> p ) = 0 : 044 for the high SNR case. \nFigure 5 shows the spin orientation parameters. GLYPH<18> 1 and GLYPH<18> 2 are the polar angles of the primary and secondary spin vectors with respect to the orbital angular momentum at the reference frequency. The angle GLYPH<1> GLYPH<30> 12 is the angle between the primary and secondary spin vectors projected into the instantaneous orbital plane at the reference frequency. This angle is particularly useful when characterising precessing binaries as GLYPH<1> GLYPH<30> 12 = 0 or GLYPH<1> GLYPH<30> 12 = GLYPH<25> are resonant spin configurations (if other conditions on the mass-ratio and spin magnitudes are met) [83]. \nWe find GLYPH<18> 1 has good SNR scaling with C 17 90% ( GLYPH<18> 1) = 0 : 73 rad ( GLYPH<24> 42 deg) and C 176 90% ( GLYPH<18> 1) = 0 : 08 rad ( GLYPH<24> 5 deg). Furthermore, GLYPH<18> 2 and GLYPH<1> GLYPH<30> 12 are measured much less accurately and require SNRs of GLYPH<24> 60 and GLYPH<24> 100 to achieve statistical uncertainties of GLYPH<24> 1 rad ( GLYPH<24> 60 deg), respectively. However, in the event of a high SNR signal we find we are able to constrain GLYPH<18> 2 to C 176 90% ( GLYPH<18> 2) = 0 : 26 rad ( GLYPH<24> 15 deg) and GLYPH<1> GLYPH<30> 12 to C 176 90% ( GLYPH<1> GLYPH<30> 12) = 0 : 48 rad ( GLYPH<24> 28 deg). \nIn summary, we find that the primary spin magnitude j GLYPH<31> 1 j \nand polar angle GLYPH<18> 1 can be constrained at an SNR of GLYPH<24> 30, while the seconday spin magnitude j GLYPH<31> 2 j and polar angle GLYPH<18> 2, as well as the the information about the relative orientation of the spin vectors GLYPH<1> GLYPH<30> 12 are not constrained until we reach an SNR of GLYPH<24> 200.', '3. Waveform Systematics': 'Parameter estimation on a GW event with a waveform model that does not include relevant physics e GLYPH<11> ects could result in biased results. To quantify the size of the bias due to neglecting higher modes and / or precession for this signal we repeat our parameter estimation analsis with four additional models. \nThe waveform models we use are listed in Tab. IV, where we mark whether or not each model contains precession and / or higher modes. PhenomD is the baseline model upon which the other Phenom models used in this work are built. We include two di GLYPH<11> erent precessing models PhenomPv2 and PhenomPv3 to gauge systematics on precession. PhenomHM includes higher modes but is a non-precessing model and finally the precessing and higher mode model PhenomPv3HM presented in this article. \nOur results are presented in Fig. 6. From left to right the columns show the one dimensional marginalised posterior distribution for the; m det 1 , m det 2 , GLYPH<31> e GLYPH<11> and GLYPH<31> p. The rows from top to bottom show the results for the low ( GLYPH<26> = 17), medium ( GLYPH<26> = 34) and high ( GLYPH<26> = 176) SNR injections. The true value is shown as a vertical dashed black line. For all SNRs we find biases in the recovered masses for all models other than PhenomPv3HM i.e., the model that was used to produce the synthetic signal. This suggests that for real GW signals that are similar to this injection require analysis with models that contain both the e GLYPH<11> ects of precession and higher modes. For the high SNR case multi-mode posteriors are found for the PhenomD case. For GLYPH<31> e GLYPH<11> we find that for the low SNR injection the true value is within the 90% credible interval (CI) and therefore not considered biased however, as the SNR of the injection is increased we find that GLYPH<31> e GLYPH<11> can become heavily biased for the two precessing models but remains unbiased for the non-precessing models. For GLYPH<31> p we find that the precessing and non-higher mode models ( PhenomPv2 and PhenomPv3 ) consistently favour larger values of GLYPH<31> p as the SNR increases. Interestingly we also start to find large di GLYPH<11> erences between PhenomPv2 and PhenomPv3 at the high SNR cases.', 'C. GW170729 Analysis': 'Of the ten binary-black-hole observations reported by the LIGO-Virgo collaborations [4], GW170729 shows the strongest evidence for unequal masses, making it the most likely signal for which higher modes could impact parameter measurements. This motivated the study in Ref. [11], where the authors analysed GW170729 with two new aligned-spin and higher mode models ( SEOBNRv4HM [41] and PhenomHM [10]). They found that the models preferred to \nFIG. 6. One dimensional marginal posterior probability distributions for detector-frame primary and secondary masses (1st and 2nd columns respectively), e GLYPH<11> ective aligned-spin GLYPH<31> e GLYPH<11> and e GLYPH<11> ective precession spin GLYPH<31> p parameters (3rd and 4th columns respectively). Each row, from top to bottom, shows results for the low ( GLYPH<26> = 17), medium ( GLYPH<26> = 34) and high ( GLYPH<26> = 176) SNR injections. The true value is marked as a vertical black dashed line. The prior is shown as a black historgram. We show results for PhenomD (red), PhenomHM (purple), PhenomPv2 (green), PhenomPv3 (blue) and PhenomPv3HM (orange). Note the results for GLYPH<31> p do not show PhenomD or PhenomHM as they are aligned-spin models only. \n<!-- image --> \nTABLE IV. Waveform models that we use to analyse GW170729 and highlighting which physical e GLYPH<11> ects are included each model. \n<!-- image --> \ninterpret the data as the GW signal coming from a higher mass-ratio system with estimates for the mass-ratio changing from 0 : 62 + 0 : 36 GLYPH<0> 0 : 23 for PhenomPv2 to 0 : 52 + 0 : 26 GLYPH<0> 0 : 21 for PhenomHM (90% credible interval). This event also has evidence for a positive GLYPH<31> e GLYPH<11> , although when analysed with higher modes the 90% credible interval for GLYPH<31> e GLYPH<11> extended to include zero. This event has also been analysed in [86] with the aligned-spin and higher mode model NRHybSur3dq8 [47] where the the authors draw similar conclusions. Motivated by this we prioritise GW170729 to analyse first with PhenomPv3HM and compare to existing results. We use the posterior samples for PhenomHM from [11], and for PhenomPv2 from [87]. Results for PhenomD , PhenomPv3 and PhenomPv3HM were computed for this work using the LALInferenceMCMC code [88]. \nIn Fig. 7 we show the joint posterior for the source frame component masses ( m src 1 , m src 2 ) in the upper left; the aligned e GLYPH<11> ective spin and mass-ratio ( GLYPH<31> e GLYPH<11> , q ) in the upper right and finally the luminosity distance and inclination angle ( DL , GLYPH<19> ) in the bottom plot. The quantitative parameter estimates for the source properties are provided in Tab. V. Our posterior on the e GLYPH<11> ective precession parameter GLYPH<31> p is consistent with previous results and shows no significant di GLYPH<11> erences due to different choice of precession model (between PhenomPv2 and \nPhenomPv3 ) or including both precession and higher modes as in PhenomPv3HM . We find that the marginal posterior e GLYPH<11> ective aligned-spin parameter GLYPH<31> e GLYPH<11> and luminosity distance DL are remarkably similar to the results from PhenomHM . The posterior for the inclination angle GLYPH<19> for IMPhenomPv3HM has more support for more inclined viewing angles; however, the change is minor. \nInterestingly, not only do we find remarkably consistent results between PhenomD and PhenomPv2 as discussed in [11] but also with PhenomPv3 . This indicates that precession alone does not influence our inference for this event. However, including precession in addition to higher modes in the analysis does noticeably shift the posterior, albeit not very significantly in terms of the 90% CIs, which mostly overlap. \nWe find that the one-dimensional marginal posterior for the mass-ratio is pushed further towards lower mass-ratio values (more asymmetric) when using PhenomPv3HM where we find q = 0 : 47 + 0 : 28 GLYPH<0> 0 : 16 (90% level), implying that including precession and higher modes reinforces the findings of [11]. As more asymmetric masses are favoured the estimate for the primary mass (source frame) is shifted towards higher values and the secondary mass is shifted towards lower values where we find m src 1 = 58 : 25 + 11 : 73 GLYPH<0> 12 : 53 M GLYPH<12> and m src 2 = 28 : 18 + 9 : 83 GLYPH<0> 7 : 65 M GLYPH<12> . \n: : By favouring larger mass estimates for the primary BH we challenge formation models to describe this event through standard stellar evolution mechanisms. In particular our results inform the pulsational pair-instability supernova (PPISN) mechanism [1, 89]. The population synthesis analysis in [90] investigated the resulting distribution of BH masses subject to di GLYPH<11> erent PPISN models. They find that in three out of the four models that they explore, the maximum BH mass is GLYPH<24> 40 M GLYPH<12> [91-93], and in one of the models the maximum BH mass is GLYPH<24> 58 M GLYPH<12> [1]. In Fig. 8 we show the one-dimensional marginal posterior for the source-frame primary mass resulting from the analysis using PhenomPv2 (blue), PhenomHM (orange) and PhenomPv3HM (purple). The \n90% credible interval of each result is shown as the shaded area under their respective curves. The vertical black dashed lines denote the maximum BH mass from the four di GLYPH<11> erent PPISN models that were investigated in [90]. We do not show the posterior for PhenomD or PhenomPv3 as these are consistent with the PhenomPv2 posterior. \nWhen using PhenomPv2 to analyse the data we find that the maximum BH mass for all PPISN models are consistent with the posterior. When we include non-precessing, higher modes ( PhenomHM ) the PPISN models that predict maximum BH masses of GLYPH<24> 40 M GLYPH<12> [91-93] are excluded at the following level. In the posterior 1 : 3% of samples have a mass of 6 40 M GLYPH<12> . As noted previously when we include both precession and higher modes the primary mass shifts slightly higher resulting in 0 : 6% of samples having a mass of 6 40 M GLYPH<12> . If we assume that the primary BH in the GW170729 binary underwent a PPISN then the following PPISN models [91-93] are disfavoured at greater than 90% credibility and the maximum BH mass as predicted by [1] is consistent with our results. There are some caveats to these results however. In [90] the authors uses a linear fit to the PPISN model of [1] that systematically predicts larger remnant BH masses for pre-supernova helium (core) masses M He > 60 M GLYPH<12> than the model of [1] predicts. This in turn leads to larger maximum BH masses for this particular model. However, the size of this systematic uncertainty is unknown. Another caveat in the analysis of [90] is that the models [92, 93] have an uncertianty of GLYPH<24> 5 M GLYPH<12> to account for the di GLYPH<11> erence between the gravitational and baryonic mass [94].', 'IV. DISCUSSION AND FUTURE': "In this work we have presented the first, frequency-domain, phenomenological IMR model for spin-precessing BBHs that also includes the e GLYPH<11> ects of subdominant multipoles - beyond the quadrupole - in the co-precessing frame. By comparing to a large set of precessing NR simulations we find that our simple model is able to accurately reproduce the expected GW signal with an accuracy of 99% (97%) for small (high) inclinations, a significant improvement over models that do not include subdominant multipoles, which have accuracies of 97% (91%) for small (high) inclinations. \nPrecise measurements of BH spins from GW observations requires high SNR events in part due to the relatively high PN order that spin e GLYPH<11> ects appear at. We performed an idealized parameter estimation analysis to quantify the precision to which the BH spin magnitude and orientation can be measured, ignoring any e GLYPH<11> ects of systematic error on the waveform. We find, for this particular system (See Table II), that the primary spin parameters are more tightly constrained than the secondary spin, as expected for an unequal-mass system such as this. In the following discussion we remind the reader that the low, medium and high SNR cases have corresponding values of 17, 34 and 176 respectively. The primary spin magnitude can be constrained to a 90% CI of 0 : 42 for the low SNR case (about half the width of the physical range) and to a 90% CI of 0 : 04 for the high SNR case. The secondary spin \nmagnitude cannot be meaningfully constrained until the high SNR case with a 90% CI of 0 : 24. The primary spin polar angle shows reasonably good agreement with the expected SNR scaling and can be constrained to GLYPH<24> 42 deg (low SNR) and GLYPH<24> 5 deg (high SNR) at 90% CI. The secondary spin polar angle shows poor agreement with the expected SNR scaling and we find can only be meaningfully constrained ( GLYPH<24> 15 deg) for the high SNR case. The azimuthal angle between the spins ( GLYPH<1> GLYPH<30> 12) shows poor scaling with SNR. We find that only the highest SNR case was able to constrain GLYPH<1> GLYPH<30> 12 . 28 deg. Our parameter estimation study is only a point estimate for the size of the uncertainty on binary properties and a systematic study that explores the parameter space of precessing binaries is required to draw more general conclusions [68, 69]. However, recent work in understanding precession better may help make such a study tractable by focusing on regions where we expect precession to be measurable [95, 96]. \nWe have analysed the GW event GW170729 with the new precessing and higher mode model. We have shown that while the general interpretation of this event is unchanged we find that even small shifts in the posteriors due to using di GLYPH<11> erent waveform models, with di GLYPH<11> erent physical e GLYPH<11> ects incorporated, can be enough to inform astrophysical models such as the PPISN mechanism as we considered in this paper. If we assume that the primary BH in the GW170729 binary underwent a PPISN then we disfavour the PPISN models from [91-93] at greater than 90% credibility and our results are consistent with [1]. See [97] for a recent investigation into the location of the PPISN model mass-gap. \nOur model is analytic and natively in the frequencydomain, and as such it can be readily used in likelihood acceleration methods such as reduced order quadrature (ROQ) [98] or 'multibanding' techniques [99]. This model can be used to determine the impact on GW searches, event parameter estimation and population inference due to the e GLYPH<11> ects of precession and higher modes. \nWe expect to be able to greatly improve PhenomPv3HM , and similar models, by using models for the underlying higher multipole aligned-spin model that have been calibrated to NR waveforms [100]. Likewise, a model for the precession dynamics tuned to precessing NR simulations will improve its performance [101]. Although our model is a function of the 7 dimensional intrinsic parameter space of non-eccentric BBH mergers it is not 7 dimensional across the entire coalescence. It is true during the inspiral, but during the pre-merger and merger we use an e GLYPH<11> ective aligned-spin parametrization. Work is underway to develop an NR calibrated aligned-spin model with the e GLYPH<11> ects of two independent aligned-spins [102]. In addition, promising attempts to dynamically enhance incomplete models via singular-value-decomposition have recently been presented [49], and the model introduced here can easily be employed by such an automated tuning process. \nWith regards to higher modes we only include a subset of the complete list of modes, specifically ( '; j m j ) = ((2 ; 2) ; (2 ; 1) ; (3 ; 3) ; (3 ; 2) ; (4 ; 4) ; (4 ; 3)). We also ignore mode mixing [103] and the asymmetry between the + m and GLYPH<0> m modes, which are responsible for out of plane recoils [104]. \nWe plan to extend this model to include tidal e GLYPH<11> ects as \nFIG. 7. GW170729 parameter estimation results. Top left: component source-frame masses ( m src 1 , m src 2 ). Top right: e GLYPH<11> ective spin and mass-ratio ( GLYPH<31> e GLYPH<11> , q ). Bottom: inclination angle and luminosity distance ( GLYPH<19> , DL ). The contour lines correspond to a credible level of 90%. \n<!-- image --> \nTABLE V. Parameter estimation results for GW170729. Masses are quoted in the source frame. We quote the median and the 90% symmetric credible interval of the one-dimensional marginalised posterior distributions. \n| Parameter | PhenomD | PhenomHM | PhenomPv2 | PhenomPv3 | PhenomPv3HM |\n|--------------------------------------------------------|----------------------------------|----------------------------------|----------------------------------|----------------------------------|----------------------------------|\n| Primary Source Mass: m src 1 = M GLYPH<12> | 50 : 55 + 14 : 02 GLYPH<0> 10 64 | 56 : 36 + 11 : 08 GLYPH<0> 12 41 | 51 : 22 + 16 : 19 GLYPH<0> 10 99 | 51 : 39 + 16 : 35 GLYPH<0> 11 58 | 58 : 25 + 11 : 73 GLYPH<0> 12 53 |\n| Secondary Source Mass: m src 2 = M GLYPH<12> | 32 : 18 + 10 : 18 GLYPH<0> 8 84 | 29 : 45 + 9 : 72 GLYPH<0> 8 36 | 32 : 43 + 9 : 75 GLYPH<0> 9 46 | 31 : 67 + 10 : 43 GLYPH<0> 9 27 | 28 : 18 + 9 : 83 GLYPH<0> 7 65 |\n| Total Source Mass: M src total = M GLYPH<12> | 82 : 80 + 15 : 29 GLYPH<0> 10 82 | 85 : 16 + 14 : 00 GLYPH<0> 10 53 | 83 : 93 + 14 : 74 GLYPH<0> 10 91 | 83 : 52 + 14 : 94 GLYPH<0> 11 09 | 86 : 18 + 13 : 42 GLYPH<0> 10 77 |\n| Mass-Ratio: q | 0 : 64 + 0 : 31 GLYPH<0> 0 : 24 | 0 : 52 + 0 : 31 GLYPH<0> 0 : 18 | 0 : 63 + 0 : 32 GLYPH<0> 0 26 | 0 : 62 + 0 : 33 GLYPH<0> 0 26 | 0 : 48 + 0 : 28 GLYPH<0> 0 16 |\n| E GLYPH<11> ective Aligned Spin: GLYPH<31> e GLYPH<11> | 0 : 34 + 0 : 19 GLYPH<0> 0 : 26 | 0 : 28 + 0 : 22 GLYPH<0> 0 : 28 | 0 : 36 + 0 : 19 GLYPH<0> 0 : 28 | 0 : 34 + 0 : 19 GLYPH<0> 0 : 27 | 0 : 27 + 0 : 21 GLYPH<0> 0 : 28 |\n| E GLYPH<11> ective Precession Spin: GLYPH<31> p | N / A | N / A | 0 : 44 + 0 : 35 GLYPH<0> 0 : 29 | 0 : 44 + 0 : 36 GLYPH<0> 0 : 30 | 0 : 42 + 0 : 39 GLYPH<0> 0 : 29 |\n| Luminosity Distance: DL / Mpc | 2749 + 1353 GLYPH<0> 1359 | 2241 + 1391 GLYPH<0> 1065 | 2831 + 1371 GLYPH<0> 1340 | 2797 + 1386 GLYPH<0> 1318 | 2270 + 1307 GLYPH<0> 974 |\n| redshift: z | 0 : 48 + 0 : 19 GLYPH<0> 0 : 21 | 0 : 40 + 0 : 20 GLYPH<0> 0 : 17 | 0 : 49 + 0 : 19 GLYPH<0> 0 : 21 | 0 : 48 + 0 : 19 GLYPH<0> 0 : 20 | 0 : 41 + 0 : 19 GLYPH<0> 0 : 16 | \nFIG. 8. one-dimensional marginal posterior distribution for the primary source-frame mass. The posteriors for three waveform models are shown: PhenomPv2 (blue), PhenomHM (orange) and PhenomPv3HM (purple). The 90% credible interval of each result is shown as the shaded area under their respective curves. We also plot as vertical black dashed lines the maximum BH mass from four different PPISN models, which were investigated in Ref. [90]. Ref. [1] predict a maximum mass of 58 : 4 M GLYPH<12> , Ref. [91] predicts 40 : 5 M GLYPH<12> and Refs. [92, 93] both predict 39 : 5 M GLYPH<12> . \n<!-- image --> \nintroduced in [105, 106] as well as implement a model for the GW suitable for neutron star-black hole binaries where the e GLYPH<11> ects of spin-precession, subdominant multipoles and tidal e GLYPH<11> ects could all become important. The model presented here could be used as a baseline for such a model. Fi- \n- [1] S. E. Woosley, The Astrophysical Journal 836 , 244 (2017).\n- [2] J. Aasi et al. (LIGO Scientific), Class. Quant. 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2017PhRvD..96b4002S

Distinguishing boson stars from black holes and neutron stars from tidal interactions in inspiraling binary systems

2017-01-01

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Binary systems containing boson stars—self-gravitating configurations of a complex scalar field—can potentially mimic black holes or neutron stars as gravitational-wave sources. We investigate the extent to which tidal effects in the gravitational-wave signal can be used to discriminate between these standard sources and boson stars. We consider spherically symmetric boson stars within two classes of scalar self-interactions: an effective-field-theoretically motivated quartic potential and a solitonic potential constructed to produce very compact stars. We compute the tidal deformability parameter characterizing the dominant tidal imprint in the gravitational-wave signals for a large span of the parameter space of each boson star model, covering the entire space in the quartic case, and an extensive portion of interest in the solitonic case. We find that the tidal deformability for boson stars with a quartic self-interaction is bounded below by Λ<SUB>min</SUB>≈280 and for those with a solitonic interaction by Λ<SUB>min</SUB>≈1.3 . We summarize our results as ready-to-use fits for practical applications. Employing a Fisher matrix analysis, we estimate the precision with which Advanced LIGO and third-generation detectors can measure these tidal parameters using the inspiral portion of the signal. We discuss a novel strategy to improve the distinguishability between black holes/neutrons stars and boson stars by combining tidal deformability measurements of each compact object in a binary system, thereby eliminating the scaling ambiguities in each boson star model. Our analysis shows that current-generation detectors can potentially distinguish boson stars with quartic potentials from black holes, as well as from neutron-star binaries if they have either a large total mass or a large (asymmetric) mass ratio. Discriminating solitonic boson stars from black holes using only tidal effects during the inspiral will be difficult with Advanced LIGO, but third-generation detectors should be able to distinguish between binary black holes and these binary boson stars.

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https://arxiv.org/pdf/1704.08651.pdf

{'Distinguishing Boson Stars from Black Holes and Neutron Stars from Tidal Interactions in Inspiraling Binary Systems': 'Noah Sennett, 1, 2 Tanja Hinderer, 1 Jan Steinhoff, 1 Alessandra Buonanno, 1, 2 and Serguei Ossokine 1 \n1 Max Planck Institute for Gravitational Physics (Albert Einstein Institute), \nAm Muhlenberg 1, Potsdam-Golm, 14476, Germany 2 Department of Physics, University of Maryland, College Park, Maryland 20742, USA \n(Dated: April 28, 2017) \nBinary systems containing boson stars-self-gravitating configurations of a complex scalar fieldcan potentially mimic black holes or neutron stars as gravitational-wave sources. We investigate the extent to which tidal effects in the gravitational-wave signal can be used to discriminate between these standard sources and boson stars. We consider spherically symmetric boson stars within two classes of scalar self-interactions: an effective-field-theoretically motivated quartic potential and a solitonic potential constructed to produce very compact stars. We compute the tidal deformability parameter characterizing the dominant tidal imprint in the gravitational-wave signals for a large span of the parameter space of each boson star model, covering the entire space in the quartic case, and an extensive portion of interest in the solitonic case. We find that the tidal deformability for boson stars with a quartic self-interaction is bounded below by Λ min ≈ 280 and for those with a solitonic interaction by Λ min ≈ 1 . 3. We summarize our results as ready-to-use fits for practical applications. Employing a Fisher matrix analysis, we estimate the precision with which Advanced LIGO and thirdgeneration detectors can measure these tidal parameters using the inspiral portion of the signal. We discuss a novel strategy to improve the distinguishability between black holes/neutrons stars and boson stars by combining tidal deformability measurements of each compact object in a binary system, thereby eliminating the scaling ambiguities in each boson star model. Our analysis shows that current-generation detectors can potentially distinguish boson stars with quartic potentials from black holes, as well as from neutron-star binaries if they have either a large total mass or a large (asymmetric) mass ratio. Discriminating solitonic boson stars from black holes using only tidal effects during the inspiral will be difficult with Advanced LIGO, but third-generation detectors should be able to distinguish between binary black holes and these binary boson stars.', 'I. INTRODUCTION': "Observations of gravitational waves (GWs) by Advanced LIGO [1], soon to be joined by Advanced Virgo [2], KAGRA [3], and LIGO-India [4], open a new window to the strong-field regime of general relativity (GR). A major target for these detectors are the GW signals produced by the coalescences of binary systems of compact bodies. Within the standard astrophysical catalog, only black holes (BHs) and neutron stars (NSs) are sufficiently compact to generate GWs detectable by currentgeneration ground-based instruments. To test the dynamical, non-linear regime of gravity with GWs, one compares the relative likelihood that an observed signal was produced by the coalescence of BHs or NSs as predicted by GR against the possibility that it was produced by the merger of either: (a) BHs or NSs in alternative theories of gravity or (b) exotic compact objects in GR. In this paper, we pursue tests within the second class. Several possible exotic objects have been proposed that could mimic BHs or NSs, including boson stars (BSs) [5, 6], gravastars [7, 8], quark stars [9], and axion stars [10, 11]. \nThe coalescence of a binary system can be classified into three phases- the inspiral, merger, and ringdowneach of which can be modeled with different tools. The inspiral describes the early evolution of the binary and can be studied within the post-Newtonian (PN) approximation, a series expansion in powers of the relative ve- \nlocity v/c (see Ref. [12] and references within). As the binary shrinks and eventually merges, strong, highlydynamical gravitational fields are generated; the merger is only directly computable using numerical relativity (NR). Finally, during ringdown, the resultant object relaxes to an equilibrium state through the emission of GWs whose (complex) frequencies are given by the object's quasinormal modes (QNMs), calculable through perturbation theory (see Ref. [13] and references within). Complete GW signals are built by synthesizing results from these three regimes from first principles with the effective-one-body (EOB) formalism [14, 15] or phenomenologically, through frequency-domain fits [16, 17] of inspiral, merger and ringdown waveforms. \nAn understanding of how exotic objects behave during each of these phases is necessary to determine whether GW detectors can distinguish them from conventional sources (i.e., BHs or NSs). Significant work in this direction has already been completed. The structure of spherically-symmetric compact objects is imprinted in the PN inspiral through tidal interactions that arise at 5PN order (i.e., as a ( v/c ) 10 order correction to the Newtonian dynamics). Tidal interactions are characterized by the object's tidal deformability, which has recently been computed for gravastars [18, 19] and 'mini' BSs [20]. During the completion of this work, an independent investigation on the tidal deformability of several classes of exotic compact objects, including exam- \nples of the BS models considered here, was performed in Ref. [21]; details of the similarities and differences to this work are discussed in Sec. VII below. Additional signatures of exotic objects include the magnitude of the spin-induced quadrupole moment and the absence of tidal heating. The possibility of discriminating BHs from exotic objects with these two effects was discussed in Refs. [22] and [23], respectively-we will not consider these effects in this paper. The merger of BSs has been studied using NR in head-on collisions [24, 25] and following circular orbits [26]. The QNMs have been computed for BSs [27-29] and gravastars [30-32]. \nIn this paper, we compute the tidal deformability of two models of BSs: 'massive' BSs [33] characterized by a quartic self-interaction and non-topological solitonic BSs [34]. The self-interactions investigated here allow for the formation of compact BSs, in contrast to the 'mini' BSs considered in Ref. [20]. We perform an extensive analysis of the BS parameter space within these models, thereby going beyond previous work in Ref. [21], which was limited to a specific choice of parameter characterizing the self-interaction for each model. Special consideration must be given to the choice of the numerical method because BSs are constructed by solving stiff differential equations-we employ relaxation methods to overcome this problem [35]. Our new findings show that for massive BSs, the tidal deformability Λ (defined below) is bounded below by Λ min ≈ 280 for stable configurations, while for solitonic BSs the deformability can reach Λ min ≈ 1 . 3. For comparison, the deformability of NSs is Λ NS glyph[greaterorsimilar] O (10) and for BHs Λ BH = 0. We compactly summarize our results as fits for convenient use in future gravitational wave data analysis studies. In addition, we employ the Fisher matrix formalism to study the prospects for distinguishing BSs from NSs or BHs with current and future gravitational-wave detectors based on tidal effects during the inspiral. Prospective constraints on the combined tidal deformability parameters of both objects in a binary were also shown for two fiducial cases in Ref. [21]. Our findings are consistent with the conclusions drawn in Ref. [21]; we discuss a new type of analysis that can strengthen the claims made therein on the distinguishability of BSs from BHs and NSs by combining information on each body in a binary system. \nThe paper is organized as follows. Section II introduces the BS models investigated herein. We provide the necessary formalism for computing the tidal deformability in Sec. III, and describe the numerical methods we employ in Sec. IV. In Sec. V, we compute the tidal deformability, providing results that range from the weak-coupling limit to the strong-coupling limit as well as numerical fits for the tidal deformability. Finally, in Sec. VI we discuss the prospects of testing the existence of stellar-mass BSs using GW detectors and provide some concluding remarks in Sec. VII. \nWe use the signature ( -, + , + , +) for the metric and natural units glyph[planckover2pi1] = G = c = 1, but explicitly restore factors of the Planck mass m Planck = √ glyph[planckover2pi1] c/G in places to \nimprove clarity. The convention for the curvature tensor is such that ∇ β ∇ α a µ -∇ β ∇ β a µ = R ν µαβ a ν , where ∇ α is the covariant derivative and a µ is a generic covector.", 'II. BOSON STAR BASICS': "Boson stars-self-gravitating configurations of a (classical) complex scalar field-have been studied extensively in the literature, both as potential dark matter candidates and as tractable toy models for testing generic properties of compact objects in GR. Boson stars are described by the Einstein-Klein-Gordon action \nS = ∫ d 4 x √ -g [ R 16 π -∇ α Φ ∇ α Φ ∗ -V ( | Φ | 2 ) ] , (1) \nwhere ∗ denotes complex conjugation. The only experimentally confirmed elementary scalar field is the Higgs boson [36, 37], which is an unlikely candidate to form a BS because it readily decays to W and Z bosons. However, other massive scalar fields have been postulated in many theories beyond the Standard Model, e.g., bosonic superpartners predicted by supersymmetric extensions [38]. \nThe Einstein equations derived from the action (1) are given by \nR αβ -1 2 g αβ R = 8 πT Φ αβ , (2) \nwith \nT Φ αβ = ∇ α Φ ∗ ∇ β Φ+ ∇ β Φ ∗ ∇ α Φ -g αβ ( ∇ γ Φ ∗ ∇ γ Φ+ V ( | Φ | 2 )) . (3) \nThe accompanying Klein-Gordon equation is \n∇ α ∇ α Φ = dV d | Φ | 2 Φ , (4) \nalong with its complex conjugate. \nThe earliest proposals for a BS contained a single noninteracting scalar field [39-41], that is \nV ( | Φ | 2 ) = µ 2 | Φ | 2 , (5) \nwhere µ is the mass of the boson. The free EinsteinKlein-Gordon action also describes the second-quantized theory of a real scalar field; thus, this class of BS can also be interpreted as a gravitationally bound Bose-Einstein condensate [41]. The maximum mass for BSs with the potential given in Eq. (5) is M max ≈ 0 . 633 m 2 Planck /µ , or in units of solar mass, M max /M glyph[circledot] ≈ 85peV /µ . The corresponding compactness for this BS is C max ≈ 0 . 08 [39]. 1 \nBecause this maximum mass scales more slowly with µ than the Chandrasekhar limit for a degenerate fermionic star M CH ∼ m 3 Planck /m 2 Fermion , this class of BSs is referred to as mini BSs . The tidal deformability was computed in this model in Ref. [20] \nSince the seminal work of the 1960s [39-41], BSs with various scalar self-interactions have been studied. We consider two such models in this paper, both which reduce to mini BSs in the weak-coupling limit. The first BS model we consider is massive BSs [33], with a potential given by \nV massive ( | Φ | 2 ) = µ 2 | Φ | 2 + λ 2 | Φ | 4 , (6) \nwhich is repulsive for λ ≥ 0. In the strong-coupling limit λ glyph[greatermuch] µ 2 /m 2 Planck , spherically symmetric BSs obtain a maximum mass of M max ≈ 0 . 044 √ λm 3 Planck /µ 2 [33]. In units of the solar mass M glyph[circledot] this reads M max /M glyph[circledot] ≈ √ λ (0 . 3GeV /µ ) 2 . Such configurations are roughly as compact as NSs, with an effective compactness of C max ≈ 0 . 158 [42, 43]. This BS model is a natural candidate from an effective-field-theoretical perspective because the potential in Eq. (6) contains all renormalizable self-interactions for a scalar field, i.e., other interactions that scale as higher powers of | Φ | are expected to be suppressed far from the Planck scale. The 'natural' values of λ ∼ 1 and µ glyph[lessmuch] m Planck yield the strong-coupling limit of the potential (6). Because it is the most theoretically plausible BS model, we investigate the strong-coupling regime of this interaction in detail in Section V A. \nThe second class of BS that we consider is the solitonic BS model [34], characterized by the potential \nV solitonic ( | Φ | 2 ) = µ 2 | Φ | 2 ( 1 -2 | Φ | 2 σ 2 0 ) 2 . (7) \nThis potential admits a false vacuum solution at | Φ | = σ 0 / √ 2. One can construct spherically symmetric BSs whose interior closely resembles this false vacuum state and whose exterior is nearly vacuum | Φ | ≈ 0; the transition between the false vacuum and true vacuum occurs over a surface of width ∆ r ∼ µ -1 . In the strong-coupling limit σ 0 glyph[lessmuch] m Planck , the maximum mass of non-rotating BSs is M max ≈ 0 . 0198 m 4 Planck / ( µσ 2 0 ), or M max /M glyph[circledot] ≈ ( µ/σ 0 ) 2 (0 . 7PeV /µ ) 3 [34]. The corresponding compactness C max ≈ 0 . 349 approaches that of a BH C BH = 1 / 2 [34]. 2 The main motivation for considering the potential (7) is as a model of very compact objects that could even possess a light-ring when C > 1 / 3. In this paper, we will only consider solitonic BSs as potential BH mimickers, as NSs could be mimicked by the more natural massive BS model. \nIn this paper, we restrict our attention to only nonrotating BSs. Axisymmetric (rotating) BSs have been constructed for the models we consider [45-48], but these solutions are significantly more complex than those that are spherically symmetric (non-rotating). The energy density of a rotating BS forms a toroidal topology, vanishing at the star's center. Because its angular momentum is quantized, a rotating BS cannot be constructed in the slow-rotation limit, i.e. by adding infinitesimal rotation to a spherically symmetric solution [49].", 'III. TIDAL PERTURBATIONS OF SPHERICALLY-SYMMETRIC BOSON STARS': 'We consider linear tidal perturbations of a nonrotating BSs. We work within the adiabatic limit, that is we assume that the external tidal field varies on timescales much longer than any oscillation period of the star or relaxation timescale to reach a microphysical equilibrium. These conditions are typically satisfied during the inspiral of compact binaries. Close to merger, the assumptions concerning the separation of timescales can break down and the tides can become dynamical [50-53]; we ignore these complications here. The computation of the tidal deformability of NSs in general relativity was first addressed in Refs. [54, 55] and was extended in Refs. [56, 57].', 'A. Background configuration': "Here we review the equations describing a spherically symmetric BS [5, 33, 39], which is the background configuration that we use to compute the tidal perturbations in the following subsection. We follow the presentation in Ref. [29]. The metric written in polar-areal coordinates reads \nds 2 0 = -e v ( r ) dt 2 + e u ( r ) dr 2 + r 2 ( dθ 2 +sin 2 θdϕ 2 ) . (8) \nAs an ansatz for the background scalar field, we use the decomposition \nΦ 0 ( t, r ) = φ 0 ( r ) e -iωt . (9) \nInserting Eqs. (8) and (9) into Eqs. (2)-(4) gives \ne -u ( -u ' r + 1 r 2 ) -1 r 2 = -8 πρ, (10a) \ne -u ( v ' r + 1 r 2 ) -1 r 2 = 8 πp rad , (10b) \nφ '' 0 + ( 2 r + v ' -u ' 2 ) φ ' 0 = e u ( U 0 -ω 2 e -v ) φ 0 , (10c) \nwhere a prime denotes differentiation with respect to r , U 0 = U ( φ 0 ), U ( φ ) = dV/d | Φ | 2 . Because the coefficients in Eq. (10c) are real numbers, we can restrict φ 0 ( r ) to be \na real function without loss of generality. We have also defined the effective density and pressures \nρ ≡-T Φ t t = ω 2 e -v φ 2 0 + e -u ( φ ' 0 ) 2 + V 0 , (11) \np rad ≡ T Φ r r = ω 2 e -v φ 2 0 + e -u ( φ ' 0 ) 2 -V 0 , (12) \np tan ≡ T Φ θ θ = ω 2 e -v φ 2 0 -e -u ( φ ' 0 ) 2 -V 0 , (13) \nwhere V 0 = V ( φ 0 ). Note that BSs behave as anisotropic fluid stars with pressure anisotropy given by \nΣ = p rad -p tan = 2 e -u ( φ ' 0 ) 2 . (14) \nAn additional relation derived from Eqs. (2)-(9) that will be used to simplify the perturbation equations discussed in the next subsection is \np ' rad = -( p rad + ρ ) 2 r [ e u ( 1 + 8 πr 2 p rad ) -1 ] -2Σ r . (15) \nWe restrict our attention to ground-state configurations of the BS, in which φ 0 ( r ) has no nodes. The background fields exhibit the following asymptotic behavior \nlim r → 0 m ( r ) ∼ r 3 , lim r →∞ m ( r ) ∼ M, (16a) \nlim r → 0 v ( r ) ∼ v ( c ) , lim r →∞ v ( r ) ∼ 0 , (16b) \nlim r → 0 φ 0 ( r ) ∼ φ ( c ) 0 , lim r →∞ φ 0 ( r ) ∼ 1 r e -r √ µ 2 -ω 2 , (16c) \nwhere M is the BS mass, v ( c ) and φ ( c ) 0 are constants, and m ( r ) is defined such that \ne -u ( r ) = ( 1 -2 m ( r ) r ) . (17)", 'B. Tidal perturbations': "We now consider small perturbations to the metric and scalar field defined such that \ng αβ = g (0) αβ + h αβ , (18) \nΦ = Φ 0 + δ Φ . (19) \nWe restrict our attention to static perturbations in the polar sector, which describe the effect of an external electric-type tidal field. Working in the Regge-Wheeler gauge [58], the perturbations take the form \nh αβ dx α dx β = ∑ l ≥| m | Y lm ( θ, ϕ ) [ e v h 0 ( r ) dt 2 + e u h 2 ( r ) dr 2 + r 2 k ( r )( dθ 2 + r sin 2 θdϕ 2 ) ] , (20a) \nand \nδ Φ = ∑ l ≥| m | φ 1 ( r ) r Y lm ( θ, ϕ ) e -iωt , (20b) \nwhere Y lm are scalar spherical harmonics. \nWe insert the perturbed metric and scalar field from Eqs. (18)-(20) into the Einstein and Klein-Gordon equations, Eqs. (2) and (4), and expand to first order in the perturbations. For the metric functions, the ( θ, φ )-component of the Einstein equations gives h 2 = h 0 , and the ( r, r )- and ( r, θ )-components can be used to algebraically eliminate k and k ' in favor of h 0 and its derivatives. Finally, the ( t, t )-component leads to the following second-order differential equation: \nh '' 0 + e u h ' 0 r ( 1 + e -u -8 πr 2 V 0 ) -32 πe u φ 1 r 2 [ φ ' 0 ( -1 + e -u -8 πr 2 p rad ) + rφ 0 ( U 0 -2 ω 2 e -v )] + h 0 e u r 2 [ -16 πr 2 V 0 -l ( l +1) -e u (1 -e -u +8 πr 2 p rad ) 2 +64 πr 2 ω 2 φ 2 0 e -v ] = 0 , (21) \nwhere we have also used the background equations (10a), (10b), and (15). From the linear perturbations to the Klein-Gordon equation, together with the results for the metric perturbations and the background equations, we obtain \nφ '' 1 + e u φ ' 1 r ( 1 -e -u -8 πr 2 V 0 ) -e u h 0 [ φ ' 0 ( -1 + e -u -8 πr 2 p rad ) + rφ 0 ( U 0 -2 ω 2 e -v )] + e u φ 1 r 2 [ 8 πr 2 V 0 -1 + e -u -l ( l +1) -r 2 ( U 0 +2 W 0 φ 2 0 ) + r 2 e -v ω 2 -32 πe -u r 2 ( φ ' 0 ) 2 ] = 0 , (22) \nwhere W 0 = W ( φ 0 ) with W ( φ ) = dU/d | Φ | 2 . These perturbation equations were also independently derived in Ref. [21] and are a special case of generic linear perturbations considered in the context of QNMs (see, e.g., \nRefs. [27-29]). As a check, we combined the three firstorder and one algebraic constraint for the spacetime perturbations from Ref. [29] into one second-order equation for h 0 , which agrees with Eq. (21) in the limit of static \nperturbations. For the special case of mini BSs, the tidal perturbation equations were also obtained in Ref. [20]. \nThe perturbations exhibit the following asymptotic behavior [29] \nlim r → 0 h 0 ( r ) ∼ r l , (23a) \nlim r →∞ h 0 ( r ) ∼ c 1 ( r M ) -( l +1) + c 2 ( r M ) l , (23b) \nlim r → 0 φ 1 ( r ) ∼ r l +1 , (23c) √ √ \nlim r →∞ φ 1 ( r ) ∼ r Mµ 2 / µ 2 -ω 2 e -r µ 2 -ω 2 . (23d)", 'C. Extracting the tidal deformability': "The BS tidal deformability can be obtained in a similar manner as with NSs [54, 56, 57]. Working in the (nearly) vacuum region far from the center of the BS, the formalism developed for NSs remains (approximately) valid. For simplicity, we consider only l = 2 perturbations for the remainder of this section. The generalization of these results to arbitrary l is detailed in Ref. [56]. \nAs shown in Eqs. (16) and (23), very far from the center of the BS, the system approaches vacuum exponentially. Neglecting the vanishingly small contributions from the scalar field, the metric perturbation reduces to the general form \nh vac 0 = c 1 ˆ Q 22 ( x ) + c 2 ˆ P 22 ( x ) + O [ ( φ 0 ) 1 , ( φ 1 ) 1 ] , (24) \nwhere we have defined x ≡ r/M -1, ˆ P 22 and ˆ Q 22 are the associated Legendre functions of the first and second kind, respectively, normalized as in Ref. [56] such that ˆ P 22 ∼ x 2 and ˆ Q 22 ∼ 1 /x 3 when x →∞ . The coefficients c 1 and c 2 are the same as in Eq. (23b). \nIn the BS's local asymptotic rest frame, the metric far from the star's center takes the form [59] \n¯ g 00 = -1 + 2 M r + 3 Q ij r 3 ( n i n j -1 3 δ ij ) + O ( 1 r 4 ) -E ij x i x j + O ( r 3 ) + O [ ( φ 0 ) 1 , ( φ 1 ) 1 ] , (25) \nwhere n i = x i /r , E ij is the external tidal field, and Q ij is the induced quadrupole moment. Working to linear order in E ij , the tidal deformability λ Tidal is defined such that \nQ ij = -λ Tidal E ij . (26) \nFor our purposes, it will be convenient to instead work with the dimensionless quantity \nΛ ≡ λ Tidal M 5 . (27) \nComparing Eqs. (24) and (25), one finds that the tidal deformability can be extracted from the asymptotic behavior of h 0 using \nΛ = c 1 3 c 2 . (28) \nFrom Eq. (24), the logarithmic derivative \ny ≡ d log h 0 d log r = rh ' 0 h 0 , (29) \ntakes the form \ny ( x ) = (1 + x ) 3Λ ˆ Q ' 22 ( x ) + ˆ P ' 22 ( x ) 3Λ ˆ Q 22 ( x ) + P 22 ( x ) , (30) \nor equivalently \nΛ = -1 3 ( (1 + x ) ˆ P ' 22 ( x ) -y ( x ) ˆ P 22 ( x ) (1 + x ) ˆ Q ' 22 ( x ) -y ( x ) ˆ Q 22 ( x ) ) . (31) \nStarting from a numerical solution to the perturbation equations (21) and (22), one obtains the deformability Λ by first computing y from Eq. (29) and then evaluating Eq. (31) at a particular extraction radius x Extract far from the center of the BS. Details concerning the numerical extraction are described in Sec. IV below.", 'IV. SOLVING THE BACKGROUND AND PERTURBATION EQUATIONS': "The background equations (10a)-(10c) and perturbation equations (21)-(22) form systems of coupled ordinary differential equations. These equations can be simplified by rescaling the coordinates and fields by µ (the mass of the boson field). To ease the comparison with previous work, we extend the definitions given in Ref. [29]: for massive BSs, we use \nr → m 2 Planck ˜ r µ , m ( r ) → m 2 Planck ˜ m (˜ r ) µ , λ → 8 πµ 2 ˜ λ m 2 Planck , ω → µ ˜ ω m 2 Planck , φ 0 ( r ) → m Planck ˜ φ 0 (˜ r ) (8 π ) 1 / 2 , φ 1 ( r ) → m 2 Planck ˜ φ 1 (˜ r ) µ (8 π ) 1 / 2 , (32) \nwhile for solitonic BSs, we use \nr → m 2 Planck ˜ r ˜ σ 0 µ , m ( r ) → m 2 Planck ˜ m (˜ r ) ˜ σ 0 µ , σ 0 → m Planck ˜ σ 0 (8 π ) 1 / 2 , ω → ˜ σ 0 µ ˜ ω m 2 Planck , φ 0 ( r ) → σ 0 ˜ φ 0 (˜ r ) (2) 1 / 2 , φ 1 ( r ) → m 2 Planck ˜ φ 1 (˜ r ) (16 π ) 1 / 2 µ , (33) \nwhere factors of the Planck mass have been restored for clarity. \nFinding solutions with the proper asymptotic behavior [Eqs. (16) and (23)] requires one to specify boundary conditions at both ˜ r = 0 and ˜ r = ∞ . To impose these boundary conditions precisely, we integrate over a compactified radial coordinate \nζ = ˜ r N + ˜ r , (34) \nas is done in Ref. [60], where N is a parameter tuned so that exponential tails in the variables ˜ φ 0 and ˜ φ 1 [see Eqs. (16) and (23)] begin near the center of the domain ζ ∈ [0 , 1]. For massive BSs, we use N ranging from 20 to 60 depending on the body's compactness; for solitonic BSs we use N between 1 and 10. \nGround-state solutions to the background equations (10a)-(10c) can be completely parameterized by the central scalar field ˜ φ ( c ) 0 and frequency ˜ ω . To determine the ground state frequency, we formally promote ˜ ω to an unknown constant function of ˜ r and simultaneously solve both the background equations and \n˜ ω ' (˜ r ) = 0 . (35) \nWe impose the following boundary conditions on this combined system: \nu (0) =0 , ˜ φ 0 (0) = ˜ φ ( c ) 0 , ˜ φ ' 0 (0) = 0 , v ( ∞ ) =0 , ˜ φ 0 ( ∞ ) = 0 . (36) \nHere, the inner boundary conditions ensure regularity at the origin, and the outer conditions guarantee asymptotic flatness. \nThe background and pertrubation equations are stiff, and therefore the shooting techniques usually used to solve two-point boundary value problems require signficant fine-tuning to converge to a solution [29]. To avoid these difficulties, we use a standard relaxation algorithm that more easily finds a solution given a reasonable initial guess [35]. Once a solution is found for a particular choice of the central scalar field ˜ φ ( c ) 0 and scalar coupling (i.e., λ for massive BSs or σ 0 for solitonic BSs), this solution can be used as an initial guess to obtain nearby solutions. By iterating this process, one can efficiently generate many BS configurations. \nAfter finding a background solution, we solve the perturbation equations (21) and (22). To improve numerical behavior of the perturbation equations near the boundaries, we factor out the dominant ˜ r dependence and instead solve for \n¯ h 0 (˜ r ) ≡ h 0 ˜ r -2 , (37) \n¯ φ 1 (˜ r ) ≡ ˜ φ 1 ˜ r -3 . (38) \nWe employ the boundary conditions \n¯ h 0 (0) = ¯ h ( c ) 0 , ¯ h ' 0 (0) = 0 , ¯ φ ' 1 (0) = 0 , ¯ φ 1 ( ∞ ) = 0 , (39) \nFIG. 1. Perturbations of a massive BS as a function of rescaled coordinate ˜ r and compactified coordinate ζ for a star of mass M = 3 . 78 m 2 Planck /µ with coupling ˜ λ = 300. Top panel: The background density ρ 0 (dashed) and its first-order perturbation δρ (solid), rescaled to fit on the same plot. Middle panel: Logarithmic derivative y of the metric perturbation. The tidal deformability Λ is calculated using the numerically computed solution (black) at the peak of y (dot-dashed vertical line). Using this value for Λ, we plot corresponding expected behavior in vacuum (red) as given by Eq. (30). Bottom panel: Tidal deformability computed from Eq. (31) as a function of extraction radius x Extract . \n<!-- image --> \nwhere the normalization ¯ h ( c ) 0 is an arbitrary non-zero constant. \nFinally, we compute the tidal deformability using Eq. (31) in the nearly vacuum region x glyph[greatermuch] 1. At very large distances, the exponential falloff of φ 0 and φ 1 is difficult to resolve numerically. This numerical error propagates through the computation of the tidal deformability in Eq. (31) for very large values of x . We find that extracting Λ at smaller radii provides more numerically stable results, with a typical variation of ∼ 0 . 1% for different choices of extraction radius x Extract . For consistency, we extract Λ at the radius at which y attains its maximum. \nFigure 1 demonstrates our procedure for computing the tidal deformability. The background and perturbation equations are solved for a massive BS with a coupling of ˜ λ = 300 using a compactified coordinate with N = 20. The profile of the effective density ρ , decomposed into its background value ρ 0 and first order correction δρ , is shown in the top panel for a star of mass 3 . 78 m 2 Planck /µ . Note that the magnitude of the perturbation is proportional to the strength of the external tidal field; to improve readability, we have scaled δρ to match the size of ρ 0 . \nThe middle panel of Fig. 1 shows the computed loga- \nrithmic derivative y across the entire spacetime (black). We calculate the deformability with Eq. (31) using the peak value of y , located at the dot-dashed line. Comparing with the top panel, one sees that the scalar field is negligible in this region, justifying our use of formulae valid in vacuum. The bottom panel depicts the typical variation of Λ computed at different locations x Extract -our procedure yields consistent results provided one works reasonably close to the edge of the BS. As a check, we insert the computed value of Λ back into the vacuum solution for y given in Eq. (30), plotted in red in the middle panel. As expected, this curve closely matches the numerically computed solution at large radii, but deviates upon entering a region with non-negligible scalar field.", 'A. Massive Boson Stars': "The dimensionless tidal deformability of massive BSs is given as a function of the rescaled total mass ˜ M [defined as in Eq. (32)] in the left panel of Fig. 2. The deformability in the weak-coupling limit ˜ λ = 0 is given by the dotted black curve; this limit corresponds to the mini BS model considered in Ref. [20]. 3 One finds that the tidal deformability of the most massive stable star (colored dots) decreases from Λ ∼ 900 in the weak-coupling limit towards Λ ∼ 280 as ˜ λ is increased. For large values of ˜ λ , the deformability exhibits a universal relation when written in terms of the rescaled mass ˜ M/ ˜ λ 1 / 2 in the sense that the results for large ˜ λ rapidly approach a fixed curve as the coupling strength increases. This convergence towards the ˜ λ = ∞ relation is illustrated in the right panel of Fig. 2, in which the x-axis is rescaled by an additional factor of ˜ λ 1 / 2 relative to the left panel; in both panels, we have added black arrows to indicate the direction of increasing ˜ λ . Employing this rescaling of the mass, we compute the relation Λ( ˜ M, ˜ λ ) in the strongcoupling limit ˜ λ →∞ below. The tidal deformability in this limit is plotted in Fig. 2 with a dashed black curve. \nThe gap in tidal deformability between BSs, for which the lowest values are Λ glyph[greaterorsimilar] 280, and NSs, where for soft equations of state and large masses Λ glyph[greaterorsimilar] 10, can be understood by comparing the relative size or compactness C = M/R of each object. From the definitions (26) \nand (27), one expects the tidal deformability to scale as Λ ∝ 1 /C 5 . In the strong-coupling limit, stable massive BSs can attain a compactness of C max ≈ 0 . 158; note that in the exact strong-coupling limit ˜ λ = ∞ , BSs develop a surface, and thus their compactness can be defined unambiguously. A NS of comparable compactness has a tidal deformability that is only ∼ 0-25% larger than that of BSs. However, NS models predict stable stars with approximately twice the compactness that can be attained by massive BSs, and thus, their minimum tidal deformability is correspondingly much lower. \nAs argued in Sec. II, the strong-coupling limit of massive BSs is the most plausible model investigated in this paper from an effective field theory perspective. We analyze the tidal deformability in this limit in greater detail. To study the strong-coupling limit of ˜ λ →∞ , we employ a different set of rescalings introduced, first in Ref. [33]: \nr → m 2 Planck ˜ λ 1 / 2 ˆ r µ , m ( r ) → m 2 Planck ˜ λ 1 / 2 ˆ m (ˆ r ) µ , λ → 8 πµ 2 ˜ λ m 2 Planck , ω → µ ˆ ω m 2 Planck , φ 0 ( r ) → m Planck ˆ φ 0 (ˆ r ) (8 π ˜ λ ) 1 / 2 , φ 1 ( r ) → m 2 Planck ˆ φ 1 (ˆ r ) µ (8 π ) 1 / 2 , (40) \nwhere we have kept the previous notation for ˜ λ to emphasize that it is the same quantity as defined in Eq. (32). \nKeeping terms only at leading order in ˜ λ -1 glyph[lessmuch] 1, Eqs. (10a)-(10c) become \ne -u ( -u ' ˆ r + 1 ˆ r 2 ) -1 ˆ r 2 = -2 ˆ φ 2 0 -3 ˆ φ 4 0 2 , (41) \ne -u ( v ' ˆ r + 1 ˆ r 2 ) -1 ˆ r 2 = ˆ φ 4 0 2 , (42) \nˆ φ 0 = ( ˆ ω 2 e -v -1 ) 1 / 2 , (43) \nwhere a prime denotes differentiation with respect to ˆ r . Note that in particular, Eq. (10c) becomes an algebraic equation, reducing the system to a pair of first order differential equations. \nTurning now to the perturbation equations, we use these rescalings and find that to leading order in ˜ λ -1 , Eqs. (21) and (22) become \n<!-- image --> \nFIG. 2. Dimensionless tidal deformability of a massive BS as a function of mass in units of (left) m 2 Planck /µ and (right) m 2 Planck ˜ λ 1 / 2 /µ . For each value of ˜ λ , the most compact stable configuration is highlighted with a colored dot. The arrows indicate the direction towards the strong-coupling regime, i.e. of increasing ˜ λ . \n<!-- image --> \nh '' 0 + e u h ' 0 ˆ r [ ˆ r 2 2 ( 1 -e -2 v ˆ ω 4 ) + e -u +1 ] -e u h 0 ˆ ˆ r 2 [ ˆ r 4 e u 4 ( 1 -e -v ˆ ω 2 ) 4 + ˆ r 2 ( e u (1 -e -v ˆ ω 2 ) 2 +10 e -v ˆ ω 2 (1 -e -v ˆ ω 2 ) -2 ) + e u (1 -e -u ) 2 + l ( l +1) ] = 0 , (44) ˆ φ 1 = h 0 ˆ r ( 1 + ˆ φ 2 0 ) 2 ˆ φ 0 . (45) \nAs with the background fields, the equation for the scalar field ˆ φ 1 becomes algebraic in this limit. Note that the scalar perturbation diverges as one approaches the surface of the BS, defined as the shell on which ˆ φ 0 vanishes. Nevertheless, the metric perturbation h 0 remains smooth over this surface. \nWe integrate the simplified background equations (41) and (42) and then the perturbation equation (44) using Runge-Kutta methods. We compute the tidal deformability using Eq. (31) evaluated at the surface of the BS, and plot the results in the right panel of Fig. 2 (dashed black).", 'B. Solitonic boson stars': 'The dimensionless tidal deformability of solitonic BSs is given as a function of the mass in Fig. 3. As in Fig. 2, the colored dots highlight the most massive stable configuration for different choices of the scalar coupling ˜ σ 0 . To aid comparison with the massive BS model, in the left panel we rescale the mass by an additional factor of ˜ σ 0 relative to the definition of ˜ M in Eq. (33). \nWhen the coupling ˜ σ 0 is strong, solitonic BSs can manifest two stable phases that can be smoothly connected through a sequence of unstable configurations [61]. The large plot in the left panel only shows stable configura- \nons on the more compact branch of configurations. In the weak-coupling limit ˜ σ 0 → ∞ , solitonic BSs reduce to the free field model considered in Ref. [20]. To illustrate this limit, we show in the smaller inset the tidal deformability for both phases of BSs as well as the unstable configurations that bridge the two branches of solutions. The weak-coupling limit is depicted with a dotted black curve. We find that the tidal deformability of the less compact phase of BSs smoothly transitions from Λ →∞ in the strong-coupling limit (˜ σ 0 → 0) 4 to Λ ∼ 900 in the weak-coupling limit (˜ σ 0 → ∞ ). Because their tidal deformability is so large, diffuse solitonic BSs of this kind would not serve as effective BH mimickers, and we will not discuss them for the remainder of this paper. However, it should be noted that only this phase of stable configurations exists when σ 0 glyph[greaterorsimilar] 0 . 23 m Planck . \nFocusing now on the more compact phase of solitonic BSs, one finds that the tidal deformability of the most massive stable star (colored dots) decreases towards Λ ∼ 1 . 3 as ˜ σ 0 is decreased. As before, the relation between a rescaled mass and Λ approaches a finite limit \nFIG. 3. Dimensionless tidal deformability as a function BS mass in units of (left) m 2 Planck /µ and (right) m 2 Planck / ( µ ˜ σ 2 0 ). For each value of ˜ σ 0 , the most compact stable BS is highlighted with a colored dot. The inset plot in the left panel shows both stable and unstable configurations over a larger range of Λ to illustrate the weak-coupling limit ˜ σ 0 →∞ (dotted black). The arrows indicate the direction towards the strong-coupling regime, i.e. of decreasing ˜ σ 0 ; while not plotted explicitly, the strong-coupling limit ˜ σ 0 → 0 corresponds the accumulation of curves in the right panel in the direction of the arrow. \n<!-- image --> \nin the strong coupling limit. We illustrate this in the right panel of Fig. 3 by rescaling the mass by an additional factor of ˜ σ -1 0 relative to the definition in Eq. (33). While we do not examine the exact strong-coupling limit ˜ σ 0 → 0 here, we find that the minimum deformability has converged to within a few percent of Λ = 1 . 3 for 0 . 03 m Planck ≤ σ 0 ≤ 0 . 05 m Planck .', 'C. Fits for the relation between M and Λ': 'In this section we provide fits to our results for practical use in data analysis studies, focusing on the regime that is the most relevant region of the parameter space for BH and NS mimickers. \nFor massive BS, it is convenient to express the fit in terms of the variable \nw = 1 1 + ˜ λ/ 8 , (46) \nwhich provides an estimate of the maximum mass in the weak-coupling limit ˜ M max ≈ 2 / ( π √ w ) [62] and has a compact range 0 ≤ w ≤ 1. A fit for massive BSs that is accurate 5 to ∼ 1% for Λ ≤ 10 5 and up to the maximum \nmass is given by \n√ w ˜ M = [ -0 . 529 + 22 . 39 log Λ -143 . 5 (log Λ) 2 + 305 . 6 (log Λ) 3 ] w + [ -0 . 828 + 20 . 99 log Λ -99 . 1 (log Λ) 2 + 149 . 7 (log Λ) 3 ] (1 -w ) . (47) \nThe maximum mass where the BSs become unstable can be obtained from the extremum of this fit, which also determines the lower bound for Λ. \nIn the solitonic case, a global fit for the tidal deformability for all possible values of σ 0 is difficult to obtain due to qualitative differences between the weak- and strong-coupling regimes. However, small values of σ 0 are most interesting, since they allow for the widest range for the tidal deformability and compactness. A fit for σ 0 = 0 . 05 m Planck accurate to better than 1% and valid for Λ ≤ 10 4 (and again up to the maximum mass) reads \nlog( σ 0 ˜ M ) = -30 . 834+ 1079 . 8 log Λ + 19 -10240 (log Λ + 19) 2 . (48) \nThis fit is expected to be accurate for 0 ≤ σ 0 glyph[lessorsimilar] 0 . 05 m Planck , i.e., including the strong coupling limit σ 0 = 0, within a few percent. Notice that this fit remains valid through tidal deformabilities of the same magnitude as that of NSs.', 'A. Estimating the precision of tidal deformability measurements': "Gravitational-wave detectors will be able to probe the structure of compact objects through their tidal interactions in binary systems, in addition to effects seen in the \nmerger and ringdown phases. In this section, we discuss the possibility of distinguishing BSs from NSs and BHs using only tidal effects. We emphasize that our results in this section are based on several approximations and should be viewed only as estimates that provide lower bounds on the errors and can be used to identify promising scenarios for future studies with Bayesian data analysis and improved waveform models. \nThe parameter estimation method based on the Fisher information matrix is discussed in detail in Ref. [63]. This approximation yields only a lower bound on the errors that would be obtained from a Bayesian analysis. We assume that a detection criterion for a GW signal h ( t ; θ ) has been met, where θ are the parameters characterizing the signal: the distance D to the source, time of merger t c , five positional angles on the sky, plane of the orbit, orbital phase at some given time φ c , as well as a set of intrinsic parameters such as orbital eccentricity, masses, spins, and tidal parameters of the bodies. Given the detector output s = h ( t ) + n , where n is the noise, the probability p ( θ | s ) that the signal is characterized by the parameters θ is \np ( θ | s ) ∝ p (0) e -1 2 ( h ( θ ) -s | h ( θ ) -s ) , (49) \nwhere p (0) represents a priori knowledge. Here, the inner product ( ·|· ) is determined by the statistical properties of the noise and is given by \n( h 1 | h 2 ) = 2 ∫ ∞ 0 ˜ h ∗ 1 ( f ) ˜ h 2 ( f ) + ˜ h ∗ 2 ( f ) ˜ h 1 ( f ) S n ( f ) df, (50) \nwhere S n ( f ) is the spectral density describing the Gaussian part of the detector noise. For a measurement, one determines the set of best-fit parameters ˆ θ that maximize the probability distribution function (49). In the regime of large signal-to-noise ratio SNR = √ ( h | h ), for a given incident GW in different realizations of the noise, the probability distribution p ( θ | s ) is approximately given by \np ( θ | s ) ∝ p (0) e -1 2 Γ ij ∆ θ i ∆ θ j , (51) \nwhere \nΓ ij = ( ∂h ∂θ i ∣ ∣ ∣ ∣ ∂h ∂θ j ) , (52) \nis the so-called Fisher information matrix. For a uniform prior p (0) , the distribution (51) is a multivariate Gaussian with covariance matrix Σ ij = ( Γ -1 ) ij and the root-meansquare measurement errors in θ i are given by \n√ 〈 (∆ θ i ) 2 〉 = √ ( Γ -1 ) ii , (53) \nwhere angular brackets denote an average over the probability distribution function (51). \nWe next discuss the model ˜ h ( f, θ ) for the signal. For a binary inspiral, the Fourier transform of the dominant mode of the signal has the form \n˜ h ( f, θ ) = A ( f, θ ) e iψ ( f, θ ) . (54) \nUsing a PN expansion and the stationary-phase approximation (SPA), the phase ψ is computed from the energy balance argument by solving \nd 2 ψ d Ω 2 = 2 d Ω /dt = 2 ( dE/d Ω) ˙ E GW , (55) \nwhere E is the energy of the binary system, ˙ E GW is the energy flux in GWs, and Ω = πf is the orbital frequency. The result is of the form \nψ = 3 128( π M f ) 5 / 2 [ 1 + α 1PN ( ν ) x + . . . + ( α Newt tidal + α 5PN ( ν ) ) x 5 + O ( x 6 ) ] , (56) \nwith x = ( πMf ) 2 / 3 , M = m 1 + m 2 , ν = m 1 m 2 /M 2 , M = ν 3 / 5 M , and the dominant tidal contribution is \nα Newt tidal = -39 2 ˜ Λ . (57) \nHere, ˜ Λ is the weighted average of the individual tidal deformabilities, given by \n˜ Λ( m 1 , m 2 , Λ 1 , Λ 2 ) = 16 13 [( 1 + 12 m 2 m 1 ) m 5 1 M 5 Λ 1 +(1 ↔ 2) ] . (58) \nThe phasing in Eq. (56) is known as the 'TaylorF2 approximant.' Specifically, we use here the 3 . 5PN pointparticle terms [12] and the 1PN tidal terms [64]. At 1PN order, a second combination of tidal deformability parameters enters into the phasing in addition to ˜ Λ. This additional parameter vanishes for equal-mass binaries and will be difficult to measure with Advanced LIGO [65, 66]. For simplicity, we omit this term from our analysis. \nThe tidal correction terms in Eq. (56) enter with a high power of the frequency, indicating that most of the information on these effects comes from the late inspiral. This is also the regime where the PN approximation for the point-mass dynamics becomes inaccurate. To estimate the size of the systematic errors introduced by using the TaylorF2 waveform model in our analysis, we compare the model against predictions from a tidal EOB (TEOB) model. The accuracy of the TEOB waveform model has been verified for comparable-mass binaries through comparison with NR simulations; see, for example, Ref. [50]. For our comparison, we use the same TEOB model as in Ref. [50]. The point-mass part of this model-known as 'SEOBNRv2'-has been calibrated with binary black hole (BBH) results from NR simulations. The added tidal effects are adiabatic quadrupolar tides including tidal terms at relative 2PN order in the EOB Hamiltonian and 1PN order in the fluxes and waveform amplitudes. The SPA phase for the TEOB model is computed by solving the EOB evolution equations to obtain Ω( t ), numerically inverting this result for t (Ω), and solving Eq. (55) to arrive at ψ (Ω). \nFIG. 4. Dephasing between the TEOB and tidal TaylorF2 models for non-spinning BNS systems including adiabatic quadrupolar tidal effects. The curves end at the prediction for the merger from NR simulations described in Ref. [69]. The labels denote the masses (in units of M glyph[circledot] ) and EoS of the NSs. \n<!-- image --> \nFigure 4 shows the difference in predicted phase from the TEOB model and the TaylorF2 model (56) for two nearly equal mass binary NS (BNS) systems. For our analysis, we consider two representative equations of state (EoS) for NSs: the relatively soft SLy model [67] and the stiff MS1b EoS [68]. Figure 4 illustrates that the dephasing between the TaylorF2 and TEOB waveforms remains small compared to the size of tidal effects, which is on the order of glyph[greaterorsimilar] 20 rad for MS1b (1 . 4+1 . 4) M glyph[circledot] . Thus, we conclude that the TaylorF2 approximant is sufficiently accurate for our purposes and leave an investigation of the measurability of tidal parameters with more sophisticated waveform models for future work. \nBesides the waveform model, the computation of the Fisher matrix also requires a model of the detector noise. We consider here the Advanced LIGO Zero-Detuned High Power configuration [70]. To assess the prospects for measurements with third-generation detectors we also use the ET-D [71] and Cosmic Explorer [72] noise curves. \nTo compute the measurement errors we specialize to the restricted set of signal parameters θ = { φ c , t c , M , ν, ˜ Λ } . The extrinsic parameters of the signal such as orientation on the sky enter only into the waveform's amplitude and can be treated separately; they are irrelevant for our purposes. Spin parameters are omitted because the TaylorF2 approximant inadequately captures these effects and one would instead need to use a more sophisticated model such as SEOBNR. We restrict our analysis to systems with low masses M glyph[lessorsimilar] 12 M glyph[circledot] [73] for which the merger occurs at frequencies f merger > 900Hz so that the information is dominated by the inspiral signal. The termination conditions for the inspiral signal employed in our analysis are the predicted merger frequencies from NR simulations: for BNSs the \nformula from Ref. [69], and for BBH that from Ref. [74]. \nFrom the tidal parameter ˜ Λ, we can obtain bounds on the individual tidal deformabilities. We adopt the convention that m 1 ≥ m 2 . For any realistic, stable selfgravitating body, we expect an increase in mass to also increase the body's compactness. Because the tidal deformability scales as Λ ∝ 1 /C 5 , we assume that Λ 1 ≤ Λ 2 . At fixed values of m 1 , m 2 , and ˜ Λ, the deformability of the more massive object Λ 1 takes its maximal value when it is exactly equal to Λ 2 , i.e. when ˜ Λ = ˜ Λ( m 1 , m 2 , Λ 1 , Λ 1 ). Conversely, Λ 2 takes its maximal value when Λ 1 vanishes exactly so that ˜ Λ = ˜ Λ( m 1 , m 2 , 0 , Λ 2 ). Substituting the expression for ˜ Λ from Eq. (58) and using that m 1 , 2 = M (1 ± √ 1 -4 ν ) / 2 leads to the following bounds on the individual deformabilities \nΛ 1 ≤ g 1 ( ν ) ˜ Λ , Λ 2 ≤ g 2 ( ν ) ˜ Λ , (59) \nwhere the functions g i are given by \ng 1 ( ν ) ≡ 13 16(1 + 7 ν -31 ν 2 ) , (60a) \ng 2 ( ν ) ≡ 13 8 [ 1 + 7 ν -31 ν 2 -√ 1 -4 ν (1 + 9 ν -11 ν 2 ) ] . (60b) \nThus, the expected measurement precision of ν and ˜ Λ provide an estimate of the precision with which Λ 1 and Λ 2 can be measured through \n∆Λ 1 ≤ [ ( g 1 ( ν )∆ ˜ Λ ) 2 + ( g ' 1 ( ν ) ˜ Λ∆ ν ) 2 ] 1 / 2 , (61a) \n∆Λ 2 ≤ [ ( g 2 ( ν )∆ ˜ Λ ) 2 + ( g ' 2 ( ν ) ˜ Λ∆ ν ) 2 ] 1 / 2 , (61b) \nFor simplicity, we have assumed in Eq. (61) that the statistical uncertainty in ν and ˜ Λ is uncorrelated. Note that for BBH signals, this assumption is unnecessary because ˜ Λ = 0, and thus the second terms in Eqs. (61a) and (61b) vanish. \nIn the following subsections, we outline two tests to distinguish conventional GW sources from BSs and discuss the prospects of successfully differentiating the two with current- and third-generation detectors. First, we investigate whether one could accurately identify each body in a binary as a BH/NS rather than a BS. This test is only applicable to objects whose tidal deformability is significantly smaller than that of a BS, e.g., BHs and very massive NSs. For bodies whose tidal deformabilities are comparable to that of BSs, we introduce a novel analysis designed to test the slightly weaker hypothesis: can the binary system of BHs or NSs be distinguished from a binary BS (BBS) system? For both tests, we will assume that the true waveforms we observe are produced by BBH or BNS systems and then assess whether the resulting measurements are also consistent with the objects being BSs. In our analyses we consider only a single detector \nand assume that the sources are optimally oriented; to translate our results to a sky- and inclination-averaged ensemble of signals, one should divide the expected SNR by a factor of √ 2 and thus multiply the errors on ∆ ˜ Λ by the same factor. \nWe consider two fiducial sets of binary systems in our analysis. First, we consider BBHs at a distance of 400 Mpc (similar to the distances at which GW150914 and GW151226 were observed [75, 76]) with total masses in the range 8 M glyph[circledot] ≤ M ≤ 12 M glyph[circledot] . This range is determined by the assumption that the lowest BH mass is 4 M glyph[circledot] and the requirement that the merger occurs at frequencies above ∼ 900Hz so the information in the signal is dominated by the inspiral. The SNRs for these systems range from approximately 20 to 49 given the sensitivity of Advanced LIGO. The second set of systems that we consider are BNSs at a distance of 200Mpc and with total masses 2 M glyph[circledot] ≤ M ≤ M max , where M max is twice the maximum NS mass for each equation of state. The lower limit on this mass range comes from astrophysical considerations on NS formation [77]. The BNS distance was chosen to describe approximately one out of every ten events within the expected BNS range of ∼ 300 Mpc for Advanced LIGO and translates to SNR ∼ 12 -22 for the SLy equation of state.", 'B. Distinguishability with a single deformability measurement': "A key finding from Sec. V is that the tidal deformability is bounded below by Λ glyph[greaterorsimilar] 280 for massive BSs and Λ glyph[greaterorsimilar] 1 . 3 for solitonic BSs. By comparison, the deformability of BHs vanishes exactly, i.e. Λ = 0, whereas for nearly-maximal mass NSs, the deformability can be of order Λ ≈ O (10). Thus, a BH or high-mass NS could be distinguished from a massive BS provided that a measurement error of ∆Λ ≈ 200 can be reached with GW detectors. Similarly, to distinguish a BH from a solitonic BS requires a measurement precision of ∆Λ ≈ 1. \nThe results for the measurement errors with Advanced LIGO for BBH systems at 400Mpc are shown in Fig. 5, for a starting frequency of 10Hz. The left panel shows the error in the combination ˜ Λ that is directly computed from the Fisher matrix as a function of total mass M and mass ratio q = m 1 /m 2 . As discussed above, the ranges of M and q we consider stem from our assumptions on the minimum BH mass and a high merger frequency. The right panel of Fig. 5 shows the inferred bound on the less well-measured individual deformability in the regime of unequal masses. We omit the region where the objects have nearly equal masses q → 1 because in this regime, the 68% confidence interval ν +2∆ ν exceeds the physical bound ν ≤ 1 / 4. Inferring the errors on the parameters of the individual objects requires a more sophisticated analysis [63] than that considered here. The coloring ranges from small errors in the blue shaded regions to large errors in the orange shaded regions; the labeled \nblack lines are representative contours of constant ∆Λ. Note that the errors on the individual deformability Λ 2 are always larger than those on the combination ˜ Λ. \nWe find that the tidal deformability of our fiducial BBH systems can be measured to within ∆Λ glyph[lessorsimilar] 100 by Advanced LIGO, which indicates that BHs can be readily distinguished from massive BSs. However, even for ideal BBHs-high mass, low mass-ratio binaries-the tidal deformability of each BH can only be measured within ∆Λ glyph[greaterorsimilar] 15 by Advanced LIGO. Therefore one cannot distinguish BHs from solitonic BSs using estimates of each bodies' deformability alone. Given these findings, we also estimate the precision with which the tidal deformability could be measured with third-generation instruments. Compared to Advanced LIGO, the measurement errors in the tidal deformability decrease by factors of ∼ 13 . 5 and ∼ 23 . 5 with Einstein telescope and Cosmic Explorer, respectively. Thus, the more massive BH in the binary would be marginally distinguishable from a solitonic BS with future GW detectors, as ∆Λ 1 ≤ ∆ ˜ Λ glyph[lessorsimilar] 1. These findings are consistent with the conclusions of Cardoso et al [21], although these authors considered only equalmass binaries at distances D = 100Mpc with total masses up to 50 M glyph[circledot] . However, we find that in an unequal-mass BBH case, the less massive body could not be differentiated from a solitonic BS even with third-generation detectors. \nNext, we consider the measurements of a BNS system, shown in Fig. 6 assuming the SLy EoS. We restrict our analysis to systems with individual masses 1 M glyph[circledot] ≤ m NS ≤ m max , where m max ≈ 2 . 05 M glyph[circledot] is the maximum mass for this EoS. Similar to Fig. 5, the left panel in Fig. 6 shows the results for the measurement error in the combination ˜ Λ directly computed from the Fisher matrix, and the right panel shows the error for the larger of the individual deformabilities. The slight warpage of the contours of constant ∆ ˜ Λ compared to those in Fig. 5, best visible for the ∆ ˜ Λ = 50 contour, is due to an additional dependence of the merger frequency on ˜ Λ for BNSs that is absent for BBHs, and a small difference in the Fisher matrix elements when evaluated for ˜ Λ glyph[negationslash] = 0. We see that the deformability of NSs of nearly maximal mass in BNS systems can be measured to within ∆Λ glyph[lessorsimilar] 200, and thus can be distinguished from massive BSs. However, the measurement precision worsens as one decreases the NS mass, rendering lighter NSs indistinguishable from massive BSs using only each bodies deformability alone. In the next subsection, we discuss how combining the measurements of Λ for each object in a binary system can improve distinguishability from BSs even when the criteria discussed above are not met. \nFor completeness, we also computed how well thirdgeneration detectors could measure the tidal deformabilities in BNS systems. As in the BBH case, we find that measurement errors in Λ decrease by factors of ∼ 13 . 5 and ∼ 23 . 5 with the Einstein Telescope and Cosmic Explorer, respectively. However, the conclusions reached above concerning the distinguishability of BHs or NSs \nFIG. 5. Estimated measurement error with Advanced LIGO of (left) the weighted average tidal parameter ˜ Λ and (right) the less well-constrained individual tidal parameter Λ 2 for BBH systems at 400 Mpc. The black lines are contours of constant ∆ ˜ Λ and ∆Λ 2 in the left and right plots, respectively. \n<!-- image --> \nand BSs remain unchanged.", 'C. Distinguishability with a pair of deformability measurements': "In the previous subsection we determined that compact objects whose tidal deformability is much smaller than that of BSs could be distinguished as such with Advanced LIGO, e.g., BHs versus massive BSs. In this subsection, we present a more refined analysis to distinguish compact objects from BSs when the deformabilities of each are of approximately the same size. In particular, we focus on the prospects of distinguishing NSs between one and two solar masses from massive BSs and distinguishing BHs from solitonic BSs. Throughout this section, we only consider the possibility that a single species of BS exists in nature; differentiation between multiple, distinct complex scalar fields goes beyond the scope of this paper. We show that combining the tidal deformability measurements of each body in a binary system can break the degeneracy in the BS model associated with choosing the boson mass µ . Utilizing the mass and deformability measurements of both bodies allows one to distinguish the binary system from a BBS system. \nIn Figs. 2 and 3, the tidal deformability of BSs was given as a function of mass rescaled by the boson mass and self-interaction strength. By simultaneously adjusting these two parameters of the BS model, one can produce stars with the same (unrescaled) mass and deformability. This degeneracy presents a significant obstacle in distinguishing BSs from other compact objects with com- \nparable deformabilities. For example, the boson mass can be tuned for any value of the coupling λ ( σ 0 ) so that the massive (solitonic) BS model admits stars with the exact same mass and tidal deformability as a solar mass NS. However, combining two tidal deformability measurements can break this degeneracy and improve the distinguishability between BSs and BHs or NSs. As an initial investigation into this type of analysis, we pose the following question: given a measurement ( m 1 , Λ 1 ) of a compact object in a binary, can the observation ( m 2 , Λ 2 ) of the companion exclude the possibility that both are BSs? We stress that our analysis is preliminary and that only qualitative conclusions should be drawn from it; a more thorough study goes beyond the scope of this paper. \nFrom the Fisher matrix estimates for the errors in ( M , ν, ˜ Λ) we obtain bounds on the uncertainty in the measurement ( m i , Λ i ) for each body in a binary, which we approximate as being characterized by a bivariate normal distribution with covariance matrix Σ = diag(∆ m i , ∆Λ i ). Figure 7 depicts such potential measurements by Advanced LIGO of ( m 1 , Λ 1 ) and ( m 2 , Λ 2 ), shown in black, for a (1 . 55 + 1 . 35) M glyph[circledot] BNS system at a distance of 200 Mpc with two representative equations of state for the NSs: the SLy and MS1b models discussed above. The dashed black curves in Figure 7 show the Λ( m ) relation for these fiducial NSs. Figure 8 shows the corresponding measurements in a 6 . 5-4 . 5 M glyph[circledot] BBH measured at 400 Mpc made by Advanced LIGO, Einstein Telescope, and Cosmic Explorer in blue, red, and black, respectively. \nThe strategy to determine if the objects could be BSs is the following. Consider first the measurement ( m 2 , Λ 2 ) \nFIG. 6. Estimated measurement error with Advanced LIGO of (left) the weighted average tidal parameter ˜ Λ and (right) the less well-constrained individual tidal parameter Λ 2 for BNS systems at 200 Mpc with the SLy equation of state. The black lines are contours of constant ∆ ˜ Λ and ∆Λ 2 in the left and right plots, respectively. \n<!-- image --> \nof the less massive body. For each point x = ( m, Λ) within the 1 σ ellipse, we determine the combinations of theory parameters ( µ, λ )[ x ] or ( µ, σ 0 )[ x ] that could give rise to such a BS, assuming the massive or solitonic BS model, respectively. As discussed above, in general, λ or σ 0 can take any value by appropriately rescaling µ . Finally, we combine all mass-deformability curves from Figs. 2 or 3 that pass through the 1 σ ellipise, that is we consider the model parameters ( µ, λ ) ∈ ⋃ x ( µ, λ )[ x ] or ( µ, σ 0 ) ∈ ⋃ x ( µ, σ 0 )[ x ] for massive and solitonic BS, respectively. These portions of BS parameter space are shown as the shaded regions in Figs. 7 and 8. If the tidal deformability measurements ( m 1 , Λ 1 ) of the more massive body-indicated by the other set of crosses-lie outside of these shaded regions, one can conclude that the measurements are inconsistent with both objects being BSs. \nFigure 7 demonstrates that an asymmetric BNS with masses 1 . 55-1 . 35 M glyph[circledot] can be distinguished from a BBS with Advanced LIGO by using this type of analysis. When considered individually, either NS measurement shown here would be consistent with a possible massive BS; by combining these measurements we improve our ability to differentiate the binary systems. This type of test can better distinguish BBSs from conventional GW sources than the analysis performed in the previous section because it utilizes measurements of both the mass and tidal deformability rather than just using the deformability alone. However the power of this type of test hinges on the asymmetric mass ratio in the system; with an equal-mass system, this procedure provides no more information than that described in Section VI B. \nFIG. 7. Dimensionless tidal deformability as a function of mass. Black points indicate hypothetical measurements of a (1 . 55 + 1 . 35) M glyph[circledot] binary NS system with the (left) MS1b and (right) SLy EoS; the error bars are estimated for a system observed at 200 Mpc. The shaded regions depict all possible massive BSs (i.e., all possible values of the boson mass µ and coupling λ ) consistent with the measurement of the smaller compact object. For the MS1b EoS, the tidal deformabilities of the binary are Λ 1 . 55 = 714 and Λ 1 . 35 = 1516. For the SLy EoS, the tidal deformabilites are Λ 1 . 55 = 150 and Λ 1 . 35 = 390. \n<!-- image --> \nA similar comparison between a BBH with masses 6 . 5-4 . 5 M glyph[circledot] and a binary solitonic BS system is illustrated in Fig. 8. For simplicity, the yellow shaded region depicts all possible solitonic BSs for a particular choice of coupling σ 0 = 0 . 05 m Planck that are consistent with the mea- \nurement of the smaller mass by Advanced LIGO (rather than all possible values of the coupling σ 0 ). We see that in contrast to the massive BS case, after fixing the boson mass µ with the measurement of one body, the measurement of the companion remains within that shaded region. As with the more simplistic analysis performed in Section VI B, we again find that Advanced LIGO will be unable to distinguish solitonic BSs from BHs. \nIn the previous section, we showed that thirdgeneration GW detectors will be able to distinguish marginally at least one object in a BBH system from a solitonic BS and thus determine whether a GW signal was generated by a BBS system. Using the analysis introduced in this section, we can now strengthen this conclusion. We repeat the procedure described above for a 6 . 5-4 . 5 M glyph[circledot] BBH at 400 Mpc but instead use the 3 σ error estimates in the measurements of the bodies' mass and tidal deformability. In Fig. 8, all possible solitonic BSs consistent with the measurement of the smaller mass are shown in green and pink for Einstein Telescope and Cosmic Explorer, respectively. We see that while the deformability measurements of each BH considered individually are consistent with either being solitonic BSs, they cannot both be BSs. Thus, we can conclude with much greater confidence that third-generation detectors will be able to distinguish BBH systems from binary systems of solitonic BSs. \nTo summarize, the precision expected from Advanced LIGO is potentially sufficient to differentiate between massive BSs and NSs or BHs, particularly in systems with larger mass asymmetry. Advanced LIGO is not sensitive enough to discriminate between solitonic BSs and BHs, but next-generation detectors like the Einstein Telescope or Cosmic Explorer should be able to distinguish between BBS and BBH systems. However, we emphasize again that our conclusions are based on several approximations and further studies are needed to make these precise. We also note that we have deliberately restricted our analysis to the parameter space where waveforms are inspiral-dominated in Advanced LIGO. Tighter constraints on BS parameters are expected for binaries where information can also be extracted from the merger and ringdown portion, provided that waveform models that include this regime are available.", 'VII. CONCLUSIONS': "Gravitational waves can be used to test whether the nature of BHs and NSs is consistent with GR and to search for exotic compact objects outside of the standard astrophysical catalog. A compact object's structure is imprinted in the GW signal produced by its coalescence with a companion in a binary system. A key target for such tests is the characteristic ringdown signal of the final remnant. However, the small SNR of that part of the GW signal complicates such efforts. Complementary information can be obtained by measuring a small but \nFIG. 8. Dimensionless tidal deformability as a function of mass. Hypothetical measurements of a (6 . 5 + 4 . 5) M glyph[circledot] binary BH system with error bars estimated for a system observed at 400 Mpc by Advanced LIGO, the Einstein Telescope, and Cosmic Explorer are given in blue, red, and black, respectively. The shaded regions depict all possible solitonic BSs with coupling σ 0 = 0 . 05 m Planck that are consistent with the measurements of the smaller compact object by each detector. \n<!-- image --> \ncumulative signature due to tidal effects in the inspiral that depend on the compact object's structure through its tidal deformability. This quantity may be measurable from the late inspiral and could be used to distinguish BHs or NSs from exotic compact objects. \nIn this paper, we computed the tidal deformability Λ for two models of BSs: massive BSs , characterized by a quartic self-interaction, and solitonic BSs , whose scalar self-interaction is designed to produce very compact objects. For the quartic interaction, our results span the entire two-dimensional parameter space of such a model in terms of the mass of the boson and the coupling constant in the potential. For the solitonic case, our results span the portion of interest for BH mimickers. We presented fits to our results for both cases that can be used in future data analysis studies. We find that the deformability of massive BSs is markedly larger than that of BHs and very massive NSs; in particular, we showed that the tidal deformability Λ glyph[greaterorsimilar] 280 irrespective of the boson mass and the strength of the quartic self-interaction. The tidal deformability of solitonic BSs is bounded below by Λ glyph[greaterorsimilar] 1 . 3. \nTo determine whether ground-based GW detectors can distinguish NSs and BHs from BSs, we first computed a lower bound on the expected measurement errors in Λ using the Fisher matrix formalism. We considered BBH systems located at 400 Mpc and BNS systems at 200 Mpc with generic mass ratios that merge above 900 Hz. We found that, with Advanced LIGO, BBHs could be distinguished from binary systems composed of massive BSs and that BNSs could be distinguished provided that the NSs were of nearly-maximal mass or of sufficiently different masses (i.e. a high mass ratio binary). We \nalso demonstrated that the prospects for distinguishing solitonic BSs from BHs based only on tidal effects are bleak using current-generation detectors; however, thirdgeneration detectors will be able to discriminate between BBH and BBS systems. We presented two different analyses to determine whether an observed GW was produced by BSs: the first relied on the minimum tidal deformability being larger than that of a NS or BH, while the second combined mass and deformability measurements of each body in a binary system to break degeneracies arising from the (unknown) mass of the fundamental boson field. \nRecent work by Cardoso et al. [21] also investigated the tidal deformabilities of BSs and the prospects of distinguishing them from BHs and NSs. Despite the topic being similar, the work in this paper is complementary: Cardoso et al. [21] performed a broad survey of tidal effects for different classes of exotic objects and BHs in modified theories of gravity, while our work focuses on an in-depth analysis of BSs. Additionally, these authors computed the deformability of BSs to both axial and polar tidal perturbations with l = 2 , 3, whereas our results are restricted to the l = 2 polar case. The l = 2 effects are expected to leave the dominant tidal imprint in the GW signal, with the l = 3 corrections being suppressed by a relative factor of 125Λ 3 / (351Λ 2 )( M Ω) 4 / 3 ∼ 4( M Ω) 4 / 3 [78] using the values from Table I of Ref. [21], where Ω is the orbital frequency of the binary. For reference, M Ω ∼ 5 × 10 -3 for a binary with M = 12 M glyph[circledot] at 900Hz. \nWe also cover several aspects that were not considered in Ref. [21], where the study of BSs was limited to a single example for a particular choice of theory parameters for each potential (quartic and solitonic). Here, we analyzed the entire parameter space of self-interaction strengths for the quartic potential and the regime of interest for BH mimickers in the solitonic case. Furthermore, we developed fitting formulae for immediate use in future data analysis studies aimed at constraining the BS parameters with GW measurements. Cardoso et al. [21] also discussed prospective constraints obtained from the Fisher matrix formalism for a range of future detectors, including the space-based detector LISA that we did not consider here. However, their analysis was limited to equal-mass systems, to bounds on ˜ Λ, and to the specific examples within each BS models. We went beyond this \n- [1] J. Aasi et al. (LIGO Scientific), Class. Quant. Grav. 32 , 074001 (2015), arXiv:1411.4547 [gr-qc].\n- [2] F. Acernese et al. (VIRGO), Class. Quant. Grav. 32 , 024001 (2015), arXiv:1408.3978 [gr-qc].\n- [3] Y. Aso, Y. Michimura, K. Somiya, M. Ando, O. Miyakawa, T. Sekiguchi, D. Tatsumi, and H. Yamamoto (KAGRA), Phys. Rev. D88 , 043007 (2013), arXiv:1306.6747 [gr-qc]. \nstudy by delineating a strategy for obtaining constraints on the BS parameter space from a pair of measurements and considering binaries with generic mass ratio. We also restricted our results to the regime where the signals are dominated by the inspiral. Although this choice significantly reduces the parameter space of masses surveyed compared to Cardoso et al., we imposed this restriction because full waveforms that include the late inspiral, merger and ringdown are not currently available. Another difference is that we took BBH or BNS signals to be the 'true' signals around which the errors were computed and used results from NR for the merger frequency to terminate the inspiral signals, whereas the authors of Ref. [21] chose BS signals for this purpose and terminated them at the Schwarzschild ISCO. \nThe purpose of this paper was to compute the tidal properties of BSs that could mimic BHs and NSs for GW detectors and to estimate the prospects of discriminating between such objects with these properties. Our analysis hinged on a number of simplifying assumptions. For example, the Fisher matrix approximation that we employed only yields lower bounds on estimates of statistical uncertainty. Additionally, we considered only a restricted set of waveform parameters, whereas including spins could also worsen the expected measurement accuracy. On the other hand, improved measurement precision is expected if one uses full inspiral-mergerringdown waveforms or if one combines results from multiple GW events. Our conclusions should be revisited using Bayesian data analysis tools and more sophisticated waveform models, such as the EOB model. Tidal effects are a robust feature for any object, meaning that the only change needed in existing tidal waveform models is to insert the appropriate value of the tidal deformability parameter for the object under consideration. However, the merger and ringdown signals are more difficult to predict, and further developments and NR simulations are needed to model them for BSs or other exotic objects.", 'ACKNOWLEDGMENTS': "N.S. acknowledges support from NSF Grant No. PHY1208881. We thank Ben Lackey for useful discussions. We are grateful to Vitor Cardoso and Paolo Pani for helpful comments on this manuscript. \n- [4] B. Iyer et al. 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2019JHEP...08..001H

Dark radiation and superheavy dark matter from black hole domination

2019-01-01

['-', '-', '-', '-']

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If even a relatively small number of black holes were created in the early universe, they will constitute an increasingly large fraction of the total energy density as space expands. It is thus well-motivated to consider scenarios in which the early universe included an era in which primordial black holes dominated the total energy density. Within this context, we consider Hawking radiation as a mechanism to produce both dark radiation and dark matter. If the early universe included a black hole dominated era, we find that Hawking radiation will produce dark radiation at a level Δ N <SUB>eff</SUB> ∼ 0 .03 - 0 .2 for each light and decoupled species of spin 0, 1/2, or 1. This range is well suited to relax the tension between late and early-time Hubble determinations, and is within the reach of upcoming CMB experiments. The dark matter could also originate as Hawking radiation in a black hole dominated early universe, although such dark matter candidates must be very heavy ( m <SUB>DM</SUB> ≳ 10<SUP>11</SUP> GeV) if they are to avoid exceeding the measured abundance.

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https://arxiv.org/pdf/1905.01301.pdf

{'Dark Radiation and Superheavy Dark Matter from Black Hole Domination': 'Dan Hooper a,b,c , ∗ Gordan Krnjaic a , † and Samuel D. McDermott a ‡ a Fermi National Accelerator Laboratory, Theoretical Astrophysics Group, Batavia, IL 60510 b University of Chicago, Kavli Institute for Cosmological Physics, Chicago, IL 60637 and c University of Chicago, Department of Astronomy and Astrophysics, Chicago, IL 60637 (Dated: May 6, 2019) \nIf even a relatively small number of black holes were created in the early universe, they will constitute an increasingly large fraction of the total energy density as space expands. It is thus well-motivated to consider scenarios in which the early universe included an era in which primordial black holes dominated the total energy density. Within this context, we consider Hawking radiation as a mechanism to produce both dark radiation and dark matter. If the early universe included a black hole dominated era, we find that Hawking radiation will produce dark radiation at a level ∆ N eff ∼ 0 . 03 -0 . 2 for each light and decoupled species of spin 0, 1/2, or 1. This range is well suited to relax the tension between late and early-time Hubble determinations, and is within the reach of upcoming CMB experiments. The dark matter could also originate as Hawking radiation in a black hole dominated early universe, although such dark matter candidates must be very heavy ( m DM > ∼ 10 11 GeV) if they are to avoid exceeding the measured abundance.', 'I. INTRODUCTION': "It has long been appreciated that inhomogeneties in the early universe could lead to the formation of primordial black holes [1]. In particular, inflationary models that predict a significant degree of non-Gaussianity could result in a cosmologically relevant abundance of black holes, typically with masses near the value enclosed by the horizon at or near the end of inflation, \nM hor ∼ M 2 Pl 2 H I ∼ 10 4 g ( 10 10 GeV H I ) , (1) \nwhere M Pl = 1 . 22 × 10 19 GeV is the Planck mass and H I is the Hubble rate at the time of black hole formation [220]. Alternatively, phase transitions in the early universe may have provided the conditions under which substantial quantities of black holes could have formed [21-24]. \nBlack holes with initial masses below M i < ∼ 5 × 10 8 grams will disappear through the process of Hawking evaporation prior to Big Bang Nucleosynthesis (BBN), and are thus almost entirely unconstrained by existing observations. Furthermore, black holes evolve like matter in the early universe, constituting an increasingly large fraction of the total energy density as the universe expands (up to the time of their evaporation). From this perspective, it would not be surprising if the early universe included an era in which primordial black holes made up a substantial fraction or even dominated the total energy density. \nUnlike most other mechanisms for particle production, the process of Hawking evaporation generates particles democratically, producing all particle species regardless of their assigned charges or couplings. If there exist any stable particles without significant couplings to the Standard Model (SM), they would be produced through Hawking evaporation without subsequently thermalizing, but still contributing to the universe's total energy density. This makes the evaporation of primordial black holes a particularly attractive mechanism for the production of both dark radiation and dark matter [77, 80, 84]. \nA 4 . 4 σ discrepancy has been reported between the value of the Hubble constant as determined from local measurements [25-27] and as inferred from the temperature anisotropies of the cosmic microwave background (CMB) [28]. This tension can be substantially relaxed if there exist one or more exotic particle species that contribute to the relativistic energy density of the universe in the period leading up to matter-radiation equality, as often parameterized in terms of the quantity, ∆ N eff [29-36] (for alternative explanations, see Refs [37-42]). If there exist any light and long-lived decoupled particles (such as axions, for example), then any black holes that are present in the early universe will produce a background of such states, contributing to the value of ∆ N eff . If the early universe included an era in which the energy density was dominated by black holes, we find that Hawking radiation will contribute to the \n∼ \ndark radiation at a level that can naturally address the tension between late and early-time Hubble determinations, ∆ N eff 0 . 03 -0 . 2 for each light and decoupled species of spin 0, 1/2, or 1. \n-The null results of direct detection [43-45] and collider searches for dark matter provide us with motivation to consider dark matter candidates that were never in thermal equilibrium in the early universe, but that were instead produced through other mechanisms, such as misalignment production [46-48], out-of-equilibrium decays [49-53], or gravitational production during inflation [54-56]. Hawking evaporation is a theoretically well motivated way to generate dark matter particles that would not lead to observable signals in existing experiments. As we will show, very heavy dark matter candidates ( m DM > ∼ 10 9 GeV) can naturally be generated with the measured abundance in scenarios in which the early universe included a black hole dominated era. \nIn this paper, we revisit primordial black holes in the early universe, focusing on scenarios which include a black hole dominated era prior to BBN. In such scenarios, we find that the products of Hawking evaporation can naturally contribute substantially to the abundance of dark radiation, at a level of ∆ N eff ∼ 0 . 03 -0 . 2 for each light and decoupled species. An abundance of dark radiation in this range would help to relax the reported Hubble tension, and is projected to be within the reach of upcoming stage IV CMB experiments. We also consider the production of dark matter through Hawking radiation in a black hole dominated early universe, finding that the measured dark matter abundance can be easily accommodated if the mass of the dark matter candidate lies in the range between m DM ∼ 10 9 GeV and the Planck scale.", 'A. Evaporation Preliminaries': 'A black hole with mass M BH loses mass through the process of Hawking evaporation [57], at a rate given by: \ndM BH dt = -G g glyph[star],H ( T BH ) M 4 Pl 30720 π M 2 BH glyph[similarequal] -7 . 6 × 10 24 g s -1 g glyph[star],H ( T BH ) ( g M BH ) 2 , (2) \nwhere G ≈ 3 . 8 is the appropriate graybody factor, the temperature of a black hole is \nT BH = M 2 Pl 8 πM BH glyph[similarequal] 1 . 05 × 10 13 GeV ( g M BH ) , (3) \nand g glyph[star],H ( T BH ) counts all existing particle species with masses below T BH [58, 59] according to the prescription \ng glyph[star],H ( T BH ) ≡ ∑ i w i g i,H , g i,H = 1 . 82 s = 0 1 . 0 s = 1 / 2 0 . 41 s = 1 0 . 05 s = 2 , (4) \nwhere w i = 2 s i +1 for massive particles of spin s i , w i = 2 for massless particles with s i > 0, and w i = 1 for s i = 0 species. At BH temperatures well above the electroweak scale BH evaporation emits the full SM particle spectrum according to their g glyph[star],H weights; at temperatures below the MeV scale, only photons and neutrinos are emitted, so in these limits we have \ng glyph[star],H ( T BH ) glyph[similarequal] { 108 , T BH glyph[greatermuch] 100 GeV , M BH glyph[lessmuch] 10 11 g 7 , T BH glyph[lessmuch] MeV , M BH glyph[greatermuch] 10 16 g . (5) \nAssuming g glyph[star],H ( T BH ) is approximately constant (which is always true for BH that evaporate entirely to SM radiation before BBN), integrating Eq. (2) yields the time dependence of a BH with initial mass M i \nM BH ( t ) = M i ( 1 -t τ ) 1 / 3 , (6) \nand its evaporation time τ can be written as \nτ ≈ 1 . 3 × 10 -25 s g -3 ∫ M i 0 dM BH M 2 BH g glyph[star],H ( T BH ) ≈ 4 . 0 × 10 -4 s ( M i 10 8 g ) 3 ( 108 g glyph[star],H ( T BH ) ) . \nAlthough black holes can undergo mergers to form larger black holes and gain mass through accretion in the early universe, we expect these processes to play an important role only at very early times, corresponding to T > ∼ 10 8 GeV × (10 8 g /M BH ) 3 / 4 (see Appendices A and B). With this in mind, one can think of the quantity, M i , as the mass of a given black hole after these processes have ceased to be efficient.', 'B. Radiation Dominated Early Universe': "Assuming for the moment that the universe was radiation dominated throughout its early history, the expansion rate is given by: \nH 2 ≡ ( ˙ a a ) 2 = 8 πGρ R 3 = 1 4 t 2 , ρ R ( T ) = π 2 g glyph[star] ( T ) 30 T 4 , (7) \nwhere G = M -2 Pl , ρ R is the energy density in radiation, and g glyph[star] ( T ) is the effective number of relativistic SM degreesof-freedom with equilibrium distributions \ng glyph[star] ( T ) = ∑ B g B ( T B T ) 4 + 7 8 ∑ F g F ( T F T ) 4 , (8) \nevaluated at the photon temperature T , where the sum is over all bosons/fermions B/F with temperatures T B/F and spin states g B/F . At the time of evaporation t = τ , the Hubble rate is H = 1 / 2 τ , which corresponds to a SM temperature of \nT τ glyph[similarequal] 40 MeV ( 10 8 g M i ) 3 / 2 ( g glyph[star],H ( T BH ) 108 ) 1 / 2 ( 14 g glyph[star] ( T τ ) ) 1 / 4 . (9) \nAs the universe expands, any population of primordial black holes that is present will evolve to constitute an increasingly large fraction of the universe's total energy density, with the ratio ρ BH /ρ R growing proportionally with the scale factor. For very massive and long-lived black holes, this will continue until the epoch of matter-radiation equality at which point the black holes will constitute some or all of the dark matter (see, for example, Refs [60, 61]). If the black holes are relatively light, however, they will evaporate long before this point in cosmic history. \nA universe that contains both radiation and a population of black holes will evolve as \nH 2 ≡ ( ˙ a a ) 2 = 8 πG 3 ( ρ R,i a 4 + ρ BH ,i a 3 ) , (10) \nwhere ρ R,i and ρ BH ,i denote the energy densities in radiation and in black holes at some initial time, respectively. If the universe starts out dominated by radiation with a temperature, T i , the fraction of the energy density that consists of black holes will grow by a factor of ∼ ( T i / 40 MeV)( M i / 10 8 g) 3 / 2 by the time that they evaporate. Thus, if the initial BH density faction f i satisfies \n(Eventual BH Domination) f i ≡ ρ BH ,i ρ R,i glyph[greaterorsimilar] 4 × 10 -12 ( 10 10 GeV T i )( 10 8 g M i ) 3 / 2 , (11) \nthe black hole population will ultimately come to dominate the energy density of the early universe. In this sense, black hole domination is at attractor solution to the evolution of the early universe. From this perspective, it would not be surprising if the early universe included a black hole dominated era.", 'C. Black Hole Dominated Early Universe': "In light of Sec.II B, it is generic to expect a period of BH domination even if the initial BH energy density is a small fraction of the initial radiation density, as shown in Eq. (11). If this condition is satisfied in the early universe, Hawking radiation plays a key role in reheating the universe and setting the initial conditions for all subsequent cosmological epochs, including Big Bang Nucleosynthesis (BBN). Thus, in this section and for the remainder of this work, we will assume that the early universe is black hole dominated at early times and the subsequent era of radiation domination, which contains BBN, arises entirely from Hawking radiation. \nOnce the black holes have entirely evaporated (at t = τ ), the universe will be filled with the products of Hawking radiation. In the case of SM evaporation products, these particles will thermalize to form a bath with the following temperature: \nT RH glyph[similarequal] 50 MeV ( 10 8 g M i ) 3 / 2 ( g glyph[star],H ( T BH ) 108 ) 1 / 2 ( 14 g glyph[star] ( T RH ) ) 1 / 4 . (12) \nTo preserve the successful predictions of BBN, we will limit ourselves to the case in which T RH > ∼ 3 MeV, corresponding to black holes lighter than M i < ∼ 6 × 10 8 g. For this range of masses, T BH glyph[greatermuch] 10 TeV through evaporation, so the SM contribution to g glyph[star],H is always glyph[similarequal] 108. One might na¨ıvely be concerned that scenarios with low reheating temperatures would pose a challenge for baryogenesis, as in standard cosmology this requires the existence of light particles that violate baryon number. The Hawking evaporation of light black holes, however, populates the universe with particles that are much heavier than the temperature of the SM bath, and thus provides a natural means by which to generate the universe's baryon asymmetry [19, 62-80]. \nThe Hawking evaporation of primordial black holes will produce all particle species, including any that exist beyond the limits of the SM. Unlike most other mechanisms for particle production in the early universe, Hawking radiation is universal across all species, depending only on the mass and spin of the radiated degrees-of-freedom. Any products of Hawking evaporation that possess significant couplings to the SM will rapidly thermalize with the surrounding bath. On the other hand, Hawking evaporation could also produce particle species with extremely feeble (perhaps only gravitational) couplings to the SM, which would not thermalize. If relatively light, such particles would constitute dark radiation, and would be observable through their contribution to the effective number of neutrino species, N eff . Heavier evaporation products, on the other hand, would quickly become non-relativistic and could plausibly contribute to or constitute the dark matter of the universe.", 'III. DARK HAWKING RADIATION AND THE CONTRIBUTION TO ∆ N eff': "As the universe expands, the energy density in black holes evolves as follows: \ndρ BH dt = -3 ρ BH H + ρ BH dM BH dt 1 M BH , (13) \nwhere the first term corresponds to dilution from Hubble expansion, while the second term is from Hawking evaporation as described in Eq. (2). In concert, the energy density in radiation evolves as follows: \ndρ R dt = -4 ρ R H -ρ BH dM BH dt ∣ ∣ ∣ ∣ SM 1 M BH , (14) \nwhere the second term is the result of energy being injected into the universe from the evaporating black holes. \nAfter the black holes have evaporated, the universe will be filled with SM radiation, along with any other Hawking radiation products that may have been produced. If there exist additional light states without significant couplings to the SM, so-called 'dark radiation' (DR), these particles will be produced through Hawking radiation and contribute to the radiation density of the universe during BBN and recombination. If the universe is black hole dominated during the era of evaporation, the fraction of the universe's energy density in such particles will be given by the proportion of their degrees-of-freedom, g DR ,H /g glyph[star],H . As the universe expands and cools, the energy density in dark radiation is diluted by four powers of the scale factor, \nρ DR ( T EQ ) ρ DR ( T RH ) = ( a RH a EQ ) 4 . (15) \nHowever, the energy density in the SM radiation bath is additionally diluted by a series of entropy dumps that occur when SM radiation temperature falls below the mass of a particle species, which transfers its entropy to the remaining radiation bath and increases the latter's temperature. To evaluate the impact of these transfers, we apply entropy conservation: \n( a 3 s ) RH = ( a 3 s ) EQ = ⇒ a 3 RH g glyph[star],S ( T RH ) T 3 RH = a 3 EQ g glyph[star],S ( T EQ ) T 3 EQ , (16) \nwhere g glyph[star],S is the effective number of relativistic degrees of freedom in entropy \ng glyph[star]S ( T ) = ∑ B g B ( T B T ) 3 + 7 8 ∑ F g F ( T F T ) 3 , (17) \nwhere the sum is over all bosons/fermions B/F with temperatures T B/F and spin states g B/F . Although g glyph[star],S = g glyph[star] at high temperatures, this is not the case at matter-radiation equality when T EQ glyph[similarequal] 0 . 75 eV and we have \ng glyph[star]S ( T EQ ) = 2 + 2 N ν ( 7 8 )( 4 11 ) ≈ 3 . 94 , g glyph[star] ( T EQ ) = 2 + 2 N ν ( 7 8 )( 4 11 ) 4 / 3 ≈ 3 . 38 , (18) \n<!-- image --> \nFIG. 1. In the left frame, we show the contribution to the effective number of neutrino species from Hawking evaporation in a scenario in which the early universe included a black hole dominated era. Results are shown for a single, light decoupled state that is either a Dirac fermion, Weyl Fermion, real scalar, massive vector, or massless spin-2 graviton, as a function of the temperature of the universe after black hole evaporation. Also shown are the current constraints [28] as well as the projected sensitivity of stage IV CMB measurements [81-83]. For ∆ N eff ∼ 0 . 1 -0 . 3 the tension between the value of the Hubble constant as determined from local measurements and as inferred from the temperature anisotropies in the cosmic microwave background can be substantially relaxed [25-27]. In the right frame, we show the relationship between the initial mass of the black holes and the temperature of the universe following their evaporation, assuming black hole domination. \n<!-- image --> \nwhere N ν glyph[similarequal] 3 . 046 is the effective number of SM neutrinos and we have used Eqs. (8) and (16). Thus, entropy conservation yields \nT EQ T RH = ( a RH a EQ )( g glyph[star],S ( T RH ) g glyph[star],S ( T EQ ) ) 1 / 3 , (19) \nand the energy density in SM radiation is diluted by the following factor: \nρ R ( T EQ ) ρ R ( T RH ) = ( a RH a EQ ) 4 ( g glyph[star] ( T EQ ) g glyph[star] ( T RH ) )( g glyph[star],S ( T RH ) g glyph[star],S ( T EQ ) ) 4 / 3 = ( a RH a EQ ) 4 ( g glyph[star] ( T EQ ) g glyph[star],S ( T RH ) 1 / 3 g glyph[star],S ( T EQ ) 4 / 3 ) . (20) \nUsing Eqs. (15) and (20), the dark and SM radiation density ratio at matter-radiation equality becomes \nρ DR ( T EQ ) ρ R ( T EQ ) = ( g DR ,H g glyph[star],H )( g glyph[star],S ( T EQ ) 4 / 3 g glyph[star] ( T EQ ) g glyph[star],S ( T RH ) 1 / 3 ) , (21) \nwhich is related to the effective number of neutrino species via \n∆ N eff = ρ DR ( T EQ ) ρ R ( T EQ ) [ N ν + 8 7 ( 11 4 ) 4 / 3 ] = ( g DR ,H g glyph[star],H )( g glyph[star]S ( T EQ ) g glyph[star]S ( T RH ) ) 1 / 3 ( g glyph[star]S ( T EQ ) g glyph[star] ( T EQ ) )[ N ν + 8 7 ( 11 4 ) 4 / 3 ] . (22) \nFor high reheat temperatures, we thus have \n∆ N eff ≈ 0 . 10 ( g DR ,H 4 )( 106 g glyph[star] ( T RH ) ) 1 / 3 , (23) \nwhich is one of our main results. Using Eq. (23) we see that, unlike relativistic thermal relics in equilibrium with the SM radiation bath, ∆ N eff glyph[lessorsimilar] 0 . 2 for any individual decoupled species produced via Hawking radiation, including Dirac fermions. This conclusion holds even for low values of T RH glyph[lessmuch] 100 GeV because the BH 'branching fraction' into dark radiation scales as g DR ,H ( T BH ) /g glyph[star],H ( T BH ) and T BH glyph[greatermuch] 100 GeV is always satisfied for BH masses that fully evaporate before BBN ( τ glyph[lessmuch] sec), so the relative DR contribution is always diluted by a factor of g glyph[star],H ( T BH ) glyph[similarequal] 108. \n<!-- image --> \nFIG. 2. Two examples of the evolution of the energy density in SM radiation, primordial black holes, and dark matter, which is produced through Hawking evaporation. In each frame, we have adopted an initial black hole mass of M i = 10 8 g. In the left frame, we show results for the case of an initial black hole density of f i = 8 × 10 -14 at T i = 10 10 GeV, and a dark matter particle mass of m DM = 10 9 GeV. In the right frame, the black holes come to dominate the energy density of the universe, and we have chosen m DM = 6 × 10 10 GeV. \n<!-- image --> \nWe summarize these results in Fig. 1, where we plot ∆ N eff as a function of the temperature of the universe after the black holes have evaporated, T RH . For a light and decoupled Dirac fermion, massive vector, or real scalar, we predict a contribution of ∆ N eff glyph[similarequal] 0 . 03 -0 . 2, below current constraints from measurements of the CMB and baryon acoustic oscillations (BAO) [28], but within the projected reach of stage IV CMB experiments [81-83]. \nIf black holes did not dominate the energy density of the early universe, the contribution to ∆ N eff will be reduced accordingly. In the limit in which ρ R glyph[greatermuch] ρ BH is maintained throughout the early universe, we arrive at: \n∆ N eff ≈ 3 . 5 × 10 -5 × ( g DR ,H 4 )( 106 g glyph[star] ( T τ ) ) 1 / 3 ( f i (10 10 GeV) 10 -15 )( M i 10 8 g ) 3 / 2 , (24) \nwhere f i (10 10 GeV) is the BH density fraction ρ BH /ρ R at a temperature of 10 10 GeV and T τ is the SM temperature at the time of evaporation in Eq. (7). In general, we expect Hawking evaporation to generate a sizable contribution to N eff only if the early universe included a black hole dominated era or if there exist a large number of light, decoupled species ( g DR ,H 1), as discussed in Sec. V. \nUp to this point, we have treated the particles that make up the dark radiation as if they were massless. In order for these Hawking radiation products to contribute towards dark radiation, their average kinetic energy evaluated at the time of matter-radiation equality must approximately exceed their mass, 〈 E DR 〉 > ∼ m DR . Under the simplifying assumption that all of the Hawking radiation is emitted at the initial temperature of the black hole, the average energy of these particles is given by: \nglyph[greatermuch] \n〈 E DR 〉 ∣ ∣ ∣ ∣ EQ ∼ αT BH ,i × T EQ T RH ( g glyph[star] ( T EQ ) g glyph[star] ( T RH ) ) 1 / 3 ∼ 3 . 9 MeV ( α 3 . 15 )( M i 10 8 g ) 1 / 2 ( 108 g glyph[star],H ( T BH ) ) 1 / 2 ( 14 g glyph[star] ( T RH ) ) 1 / 12 , \nwhere α ≈ 2 . 7 (3.15) for bosonic (fermionic) dark radiation. Numerically integrating the deposition of Hawking radiation over the lifetime of the black hole, we arrive at a slightly higher value, a factor of 1.4 larger than that given in the above expression. We thus conclude that in order to contribute towards dark radiation (as opposed to dark matter) at matter-radiation-equality, Hawking evaporation products must be lighter than ∼ 5 . 5 MeV × ( M i / 10 8 g) 1 / 2 .", 'IV. SUPERHEAVY DARK MATTER AS THE PRODUCTS OF HAWKING EVAPORATION': 'If there exist heavy, stable, decoupled particles, they will also be produced through Hawking radiation and will contribute to the abundance of dark matter (see, for example, Refs. [77, 80, 84, 85]). If the early universe included \nFIG. 3. The values of the black hole energy fraction ρ BH /ρ R (evaluated at T = 10 10 GeV) and initial black hole mass M i that lead to Ω DM h 2 glyph[similarequal] 0 . 1 for the case of dark matter in the form of a Dirac fermion with negligible couplings to the SM. Throughout the upper right portion of this plane, the early universe included a period in which black holes dominated the total energy density, and thus the results do not depend on the initial value of ρ BH /ρ R . \n<!-- image --> \nan era of black hole domination, the evaporation of the black holes leads to the following abundance of dark matter: \nΩ DM h 2 ≈ 0 . 1 ( g DM ,H 4 )( 6 × 10 10 GeV m DM )( 10 8 g M i ) 5 / 2 . (25) \nFrom Eq. (11), if the initial density ratio satisfies ρ BH ,i /ρ R,i glyph[lessorsimilar] 4 × 10 -12 × (10 10 GeV /T i )(10 8 g /M i ) 3 / 2 , the BH population never dominates the total energy density of the universe. In this case, the evaporation of the black holes leads to the following abundance of dark matter: \nΩ DM h 2 ≈ 0 . 1 ( f i (10 10 GeV) 8 × 10 -14 )( g DM ,H 4 )( 10 9 GeV m DM )( 10 8 g M i ) . (26) \nIn Fig. 2 we show two examples of the evolution of energy densities in radiation, black holes, and dark matter (as produced via Hawking radiation). In the left frame, we start our Boltzmann code (at T = 10 10 GeV) with only a small abundance of black holes, f i ≡ ρ R /ρ BH = 10 -14 . In this case, the universe never becomes black hole dominated, and the Hawking evaporation of Dirac fermions with a mass of 10 9 GeV make up the measured abundance of dark matter. In the right frame, we instead consider a case in which the early universe becomes black hole dominated, for which a heavier dark matter candidate is required. In each frame, we consider black holes with an initial mass of M i = 10 8 g. \nIn Fig. 3, we show the values of ρ BH /ρ R (evaluated at T = 10 10 GeV) and the initial black hole mass that lead to Ω DM h 2 glyph[similarequal] 0 . 1, for the case of dark matter in the form of a Dirac fermion with negligible couplings to the SM. In the upper right (lower left) portion of this plane, the early universe included (did not include) a period in which black holes dominated the total energy density. \nNext, we consider the case in which each black hole ends the process of evaporation leaving behind a remnant of mass M remnant = ηM Pl [86-88]. In a scenario in which black holes never dominate the energy density of the universe, the black holes lead to the following abundance of such remnants: \nΩ remnant h 2 ≈ 0 . 1 × ( η 1 )( f i (10 10 GeV) 1 . 5 × 10 -6 )( 10 8 g M i ) . (27) \nIn the opposite limit, in which there was a black hole dominated era, the evaporation of the black holes leads to the following abundance of Planck mass remnants: \nΩ remnant h 2 ≈ 0 . 1 × ( η 1 )( 6 × 10 5 g M i ) 5 / 2 . (28) \nDark matter in the form of a superheavy, gravitationally interacting state would be very challenging to observe or otherwise test. It has been proposed, however, that an array of quantum-limited impulse sensors could be used to detect gravitationally particles with a Planck-scale mass [89].', 'V. PRIMORDIAL BLACK HOLES IN THE PRESENCE OF LARGE HIDDEN SECTORS': "So far in this study, we have assumed that the black holes evaporate mostly into SM particles, possibly along with a small number of states that act as either dark radiation or dark matter. It seems highly plausible, however, that the SM describes only a small fraction of the degrees-of-freedom that constitute the universe's total particle content. If a large number of other particle degrees-of-freedom exist, black holes will evaporate more rapidly, producing the full array of particles that are kinematically accessible (all of those with masses below ∼ T BH ), independently of their couplings or other characteristics. \nAs a first case, we will consider a scenario in which there exist a large number of degrees-of-freedom associated with light particles with negligible couplings to the SM. Such a situation is motivated, for example, within the context of the string axiverse, in which a large number of light and feebly-coupled scalars are predicted [90-93]. If the early universe experienced a black hole dominated era, it follows from Eq. (23) that this would lead to ∆ N eff ∼ (0 . 04 -0 . 08) × N axion , where N axion is the number of axions that exist (see also Fig. 1). Given the current constraint of ∆ N eff < ∼ 0 . 28 [28], this indicates that N axion < ∼ 7, regardless of T RH . Thus the existence of a black hole dominated era appears to be inconsistent with the existence of a large axiverse (see also Ref. [84]). \nAs a second example, consider the Minimal Supersymmetric Standard Model (MSSM). In this case, approximately half of the Hawking evaporation products will be superpartners for all black holes with a temperature greater than the characteristic scale of superpartner masses, T BH > ∼ M SUSY , corresponding to M BH < ∼ 10 10 g × (TeV /M SUSY ). If R -parity is conserved, all such superpartners will decay to the lightest supersymmetric particle, producing a potentially large relic abundance. If the lightest superpartner is weakly interacting (such as a neutralino), it may or may not reach equilibrium with the SM bath, depending on the temperature at the time of evaporation. For M i > ∼ 10 -2 g × (TeV /M SUSY ) 2 / 3 , Hawking evaporation will finalize at a temperature below that of neutralino freeze-out, leading to a large relic abundance. If the early universe included a black hole dominated era, such a scenario is strongly excluded (see also Refs. [94-97]). To obtain an abundance of superpartners that is equal to the measured density of dark matter, we would require an initial black hole abundance (at T = 10 10 GeV) of only ρ BH /ρ R ∼ 10 -20 (TeV /M SUSY )(10 8 g /M i ).", 'VI. SUMMARY AND CONCLUSIONS': 'If even a small abundance of black holes were present in the early universe, this population would evolve to constitute a larger fraction of the total energy density, up to the point at which they evaporate or until matterradiation equality. From this perspective, it is natural to consider scenarios in which the early universe included an era of black hole domination. To avoid altering the light element abundances, such black holes must evaporate prior to BBN, corresponding to initial masses less than M i < 6 × 10 8 g. \n∼ \n× Unlike most other mechanisms for particle production in the early universe, Hawking evaporation generates particles democratically, producing all particle species including those with with small or negligible couplings to the SM. From this perspective, black holes provide a well-motivated mechanism to produce both dark radiation and dark matter. If \nFIG. 4. In contrast to thermal relics, Hawking radiation in a black hole dominated scenario can produce dark radiation in the form of particles with masses up to m DR ∼ 5 . 5 MeV × ( M i / 10 8 g) 1 / 2 , as well as dark matter for particles with masses between m DM ∼ 10 11 GeV × (10 8 g /M i ) 5 / 2 and the Planck scale. For thermal relics, masses below glyph[lessorsimilar] MeV spoil early universe cosmology [98] and masses above glyph[greaterorsimilar] 10 5 GeV overclose the universe in perturbative, unitary theories [99]. \n<!-- image --> \n∼ \nthe early universe included a black hole dominated era, then Hawking radiation will contribute to dark radiation at a level ∆ N eff ∼ 0 . 03 -0 . 2 for each light and decoupled species, depending on their spin and on the initial black hole mass. This range is well suited to relax the tension between late and early-time Hubble determinations, and is within the reach of upcoming stage IV CMB experiments. The dark matter could also originate as Hawking radiation in a black hole dominated early universe, although such dark matter candidates must be very heavy to avoid exceeding the measured abundance, m DM 10 11 GeV × (10 8 g /M i ) 5 / 2 . \n× In Fig. 4 we summarize some of key results of this analysis and contrast them with those found in the case of dark radiation or dark matter that originates as a thermal relic of the early universe. Whereas thermal relics can constitute dark radiation only if lighter than ∼ eV, Hawking radiation in a black hole dominated scenario can produce dark radiation in the form of particles with masses up to m DR < ∼ 5 . 5 MeV × ( M i / 10 8 g) 1 / 2 . 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For two black of holes of the same mass, this cross section is given by [60]: \nσ bc = π ( 85 π 3 ) 2 / 7 r 2 Schw v -18 / 7 glyph[similarequal] 45 ( M BH M 2 Pl ) 2 v -18 / 7 , (A1) \nwhere r Schw is the Schwarzschild radius of the black holes. The ratio of the binary capture rate to that of Hubble expansion is thus given by: \nΓ bc H ≈ 45 √ 3 ρ BH M BH v -11 / 7 √ 8 πρ 1 / 2 tot M 3 Pl (A2) \n≈ 0 . 02 × ( M BH 10 8 g )( T eff 10 7 GeV ) 2 ( v 10 -5 ) -11 / 7 ( ρ BH ρ tot ) , (A3) \nwhere T eff is defined such that ρ tot ≡ π 2 g glyph[star] ( T eff ) T 4 eff / 30. We conclude that even if the energy density of the early universe were dominated by black holes, the rate for binary capture exceeds the rate of Hubble expansion only at very early times, T eff > 10 8 GeV × (10 8 g /M BH ) 1 / 2 ( v/ 10 -5 ) 11 / 14 . \n∼ \n× Furthermore, even if primordial black holes form binaries efficiently in the early universe, it is not clear that they will merge before evaporating. Assuming that gravitational wave emission dominates the process by which black hole binaries loses energy, the inspiral time is given by [100]: \nt insp = 5 a 4 0 512 G 3 M 3 BH , (A4) \nwhere a 0 is the initial separation of the inspiring black holes. The ratio of the inspiral time to the evaporation time [see Eq. (7)] is then given by: \nt insp τ glyph[similarequal] 10 × ζ 4 ( 10 8 GeV T eff ) 8 ( 10 8 g M BH ) 6 , (A5) \nwhere ζ ≡ a 0 H bc and T eff is again defined such that ρ tot ≡ π 2 g glyph[star] ( T eff ) T 4 eff / 30, but in this case evaluated at the time of binary capture. Since ζ ∼ O (0 . 1), we expect the black holes to merge before evaporating ( t insp < ∼ t τ ) only at very early times, T bc > ∼ 4 × 10 7 GeV × ( ζ/ 0 . 1) 1 / 2 (10 8 g /M BH ) 3 / 4 . We do note, however, that in the early universe, when the ambient density is large, it is possible that more efficient mechanisms of angular momentum transfer were active, potentially leading to shorter inspiral times. \nTo summarize this section, if the universe were black hole dominated at very early times, corresponding to effective temperatures greater than ∼ 10 8 GeV × (10 8 g /M BH ) 3 / 4 , a substantial fraction of the black holes may have undergone mergers. With this in mind, one should interpret the initial black hole mass, M i , as used throughout this paper to denote the mass of the black holes after the processes of binary capture and inspiral have become inefficient.', 'Appendix B: Bondi-Hoyle Accretion': 'A black hole in a bath of radiation will undergo Bondi-Hoyle accretion, gaining mass at the following rate [101]: \ndM BH dt ∣ ∣ ∣ ∣ Accretion = 4 πλM 2 BH ρ R M 4 Pl (1 + c 2 s ) 3 / 2 , (B1) \nwhere λ is an O (1) constant and c s glyph[similarequal] 1 / √ 3 is the sound speed in the radiation bath. Combining this with the rate of mass loss from Hawking evaporation [see Eq. (2)], we can write the total rate of change as follows: \ndM BH dt = π G g ∗ ,H ( T BH ) T 2 BH 480 [ λg ∗ ( T R ) G g ∗ ,H ( T BH )(1 + c 2 s ) 3 / 2 ( T R T BH ) 4 -1 ] . (B2) \nSince λg ∗ / G g ∗ ,H (1 + c 2 s ) 3 / 2 is an order one quantity, we conclude that a black hole will generally gain mass when T R > ∼ T BH , and lose mass otherwise. When we compare the fractional accretion rate, (1 /M ) dM/dt , to that of Hubble expansion, we find that accretion plays an important role only when T R > ∼ 10 12 GeV × (10 8 g /M BH ) 1 / 2 .'}

2001NuPhB.608..375H

Randall-Sundrum II cosmology, AdS/CFT, and the bulk black hole

2001-01-01

['-', '-', '-']

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We analyse the cosmology of a brane world model where a single brane carrying the standard model fields forms the boundary of a 5-dimensional AdS bulk (the Randall-Sundrum II scenario). We focus on the thermal radiation of bulk gravitons, the formation of the bulk black hole, and the holographic AdS/CFT definition of the RSII theory. Our detailed calculation of bulk radiation reduces previous estimates to a phenomenologically acceptable, although potentially visible level. In late cosmology, in which the Friedmann equation depends linearly on the energy density /ρ, only about 1% of energy density is lost to the black hole or, equivalently, to the `dark radiation' (Ω<SUB>d,N</SUB>~=0.01 at nucleosynthesis). The preceding, unconventional ρ<SUP>2</SUP> period can produce up to 10% dark radiation (Ω<SUB>d,N</SUB>&lt;~0.1). The AdS/CFT correspondence provides an equivalent description of late RSII cosmology. We show how the AdS/CFT formulation can reproduce the ρ<SUP>2</SUP> correction to the standard treatment at low matter density. However, the 4-dimensional effective theory of CFT /+ gravity breaks down due to higher curvature terms for energy densities where ρ<SUP>2</SUP> behaviour in the Friedmann equation is usually predicted. We emphasize that, in going beyond this energy density, the microscopic formulation of the theory becomes essential. For example, the pure AdS<SUB>5</SUB> and string-motivated AdS<SUB>5</SUB>×S<SUP>5</SUP> definitions differ in their cosmological implications.

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https://arxiv.org/pdf/hep-ph/0103214.pdf

{'Arthur Hebecker and John March-Russell': 'Theory Division, CERN, CH-1211 Geneva 23, Switzerland (March 20, 2001)', 'Abstract': "We analyse the cosmology of a brane world model where a single brane carrying the standard model fields forms the boundary of a 5-dimensional AdS bulk (the Randall-Sundrum II scenario). We focus on the thermal radiation of bulk gravitons, the formation of the bulk black hole, and the holographic AdS/CFT definition of the RSII theory. Our detailed calculation of bulk radiation reduces previous estimates to a phenomenologically acceptable, although potentially visible level. In late cosmology, in which the Friedmann equation depends linearly on the energy density ρ , only about 0.5% of energy density is lost to the black hole or, equivalently, to the 'dark radiation' (Ω d,N /similarequal 0 . 005 at nucleosynthesis). The preceding, unconventional ρ 2 period can produce up to 5% dark radiation (Ω d,N < ∼ 0 . 05). The AdS/CFT correspondence provides an equivalent description of late RSII cosmology. We show how the AdS/CFT formulation can reproduce the ρ 2 correction to the standard treatment at low matter density. However, the 4-dimensional effective theory of CFT + gravity breaks down due to higher curvature terms for energy densities where ρ 2 behaviour in the Friedmann equation is usually predicted. We emphasize that, in going beyond this energy density, the microscopic formulation of the theory becomes essential. For example, the pure AdS 5 and string-motivated AdS 5 × S 5 definitions differ in their cosmological implications.", '1 Introduction': "The brane world scenarios of Randall and Sundrum [1,2] have opened new perspectives in physics beyond the standard model. The one-brane scenario of [2] (RSII) appears to be particularly interesting from the point of view of gravitation and cosmology. \nIn RSII, the standard model fields are confined to the boundary of an infinitely extended AdS 5 bulk. Four-dimensional gravity is recovered at length scales above the AdS length /lscript . While related models with warped and non-compact extra dimensions have been considered before [3], an extensive discussion has been triggered by the specific model of [2]. An important reason for this is the proximity of this construction to recent more formal developments, such as D-branes and the AdS/CFT correspondence. On the phenomenological side, the RSII model demonstrates the extent to which gravity can be changed without violating experimental constraints. This is manifest in the existence of infinitely many, weakly coupled light modes and in the possibility of an unconventional cosmological evolution. \nAlthough there have been many previous treatments of 'brane cosmology', the majority of these focus on either the situation for flat extra dimensions [4], which is highly constrained because of either the problem of bulk gravitons or the radion problem [5], or on the situation in RSI with the additional 'TeV-brane' and radion problems [6]. \nIn the present paper, we develop the physical picture of the early universe in RSII, including in particular the phenomenologically relevant dynamics of bulk gravitons, the physics of the bulk black hole, the use of the AdS/CFT correspondence to make manifest some properties of the theory in the IR, and the varying early cosmology resulting from the possibility of quite different definitions of the theory in the UV. \nThe investigation of RSII cosmology and phenomenology was pioneered in [7,8,9,10] and [11,12,14,13] respectively. The cosmological evolution is most conveniently discussed in terms of a moving brane with static AdS-Schwarzschild metric in the bulk. This is equivalent to the description with an expanding bulk [15]. The two main unconventional cosmological features are the dark radiation (characterized by the size of the black hole in the bulk) and the ρ 2 term in the Friedmann equation. Both effects are governed by the 'AdS-cosmology scale' M c = √ M 4 //lscript , where M 4 is the 4-dimensional reduced Planck mass. In the most radical scenario with /lscript ∼ 1 mm, one has a 5-dimensional reduced Planck mass M 5 = 3 M 2 4 //lscript /similarequal 1 . 1 × 10 5 TeV and M c /similarequal 0 . 7 TeV. \nSo far, much attention has been devoted to the geometrical brane dynamics and its possible extensions [16] and to the evolution of inhomogeneities in the early universe [17] (see [18] for recent reviews). However, the discussion of radiation of gravitons by the brane has been less prominent. An estimate of this effect has been given in [10]. Other, model dependent, sources for dark radiation have been discussed in [19]. \n√ \nWe argue that graviton radiation by the hot brane will unavoidably produce a black hole in the bulk. If one assumes that the black hole was initially very small (e.g., because of inflation on the brane), then the size of the black hole is a calculable function of the reheating temperature. We present a detailed calculation of the rate at which energy is lost \nto the bulk due to graviton radiation. Our result allows for an extended ρ 2 period in the early universe. More specifically, we find that the ρ period produces dark radiation corresponding to Ω d,N /similarequal 0 . 005 at nucleosynthesis, while a maximally extended ρ 2 period can result in Ω d,N ∼ 0 . 05. These numbers are small since, parametrically, they are inversely proportional to the number of brane degrees of freedom (this number, which is O (100) in the standard model, enhances the expansion rate, thus leaving less time for graviton production). The contribution of the ρ 2 period is enhanced by a factor ∼ ln( T max /M c ) with the respect to the ρ period, where T max is the maximal brane temperature. It is interesting to note that gravitons emitted sufficiently early remain bound to the brane and fall into the horizon only at the end of the ρ 2 period. Our numerical results for Ω d,N can be lowered or enhanced by having additional degrees of freedom on the brane or in the bulk respectively. From a phenomenological point of view, it is amusing to note that the size of Ω d is potentially observable. \nGubser [10] has initiated an AdS/CFT description of RSII cosmology (see also [20]) in which the CFT corresponds to the bulk degrees of freedom. The CFT temperature, which is the bulk temperature near the brane, is responsible for the dark radiation. It is known that the leading corrections to 4d gravity arising from the bulk KaluzaKlein modes are reproduced by the CFT-gravity coupling. We point out that the direct coupling of brane fields to the CFT can be responsible for the known ρ 2 corrections in the Friedmann equation. In addition, at the same densities when these corrections become relevant, higher curvature terms induced by the CFT start to affect the 4d gravity theory. These effects, naively suppressed by M 4 , come in so early because of the large number of degrees of freedom of the CFT. Furthermore, we emphasize that the 'UV region' r < /lscript of RSII is very different in the string motivated case of AdS 5 × S 5 . \nOur paper is organized as follows. After discussing the basic cosmological setting and the origin of the bulk black hole in Sect. 2, we derive the energy loss through graviton radiation in Sect. 3. Section 4 deals with the AdS/CFT perspective and, in particular, with the UV definition of RSII. We summarize and outline interesting open questions in Sect. 5 and present some formulae related to the graviton production rate in the Appendix.", '2 Basics of late cosmological evolution': "We start from the 5-dimensional action [2] \nS = ∫ d 5 x √ -g 5 ( 1 2 M 3 5 RΛ 5 ) + ∫ brane d 4 x √ -g 4 ( L SM -Λ 4 ) , (1) \nwhich includes the contribution from a 4-dimensional brane with induced metric g 4 . We suppose that the standard model fields characterized by the Lagrangian L SM are localized on this brane. The situation is further simplified by identifying the regions on both sides of the brane, i.e., imposing Z 2 orbifold boundary conditions on the 5-dimensional spacetime. Our treatment of the dynamics of this system will follow closely the particularly clear and compact discussion of Kraus [8]. \nThe motion of the brane follows from the 5-dimensional Einstein equations. It can be characterized most conveniently in terms of the extrinsic curvature K µν of the brane (with brane indices µ, ν = 0 ... 3). K µν , also known as the second fundamental form, can be defined as the projection of \nK MN = ∇ M n N , (2) \n(with bulk indices M,N = 0 ... 4) onto the brane, where n N is the unit normal. Then the brane motion is determined by the Israel junction condition [21] (see also [22]) \nK + µν -K -µν = 2 K µν = -1 M 3 5 ( T µν -1 3 T ρ ρ g 4 ,µν ) , (3) \nwhere the index '+' refers to the side of the brane to which n N points and the index ' -' refers to the opposite side of the brane. The energy-momentum tensor T µν follows from the brane action in Eq. (1). \nFor negative Λ 5 , the AdS metric \nds 2 = -r 2 /lscript 2 dt 2 + r 2 ( d/vectorx ) 2 + /lscript 2 r 2 dr 2 , (4) \nwith /lscript 2 = 6 M 3 5 / | Λ 5 | , represents a vacuum solution of the 5-dimensional Einstein equations. If, in addition, the 4-dimensional cosmological constant, i.e., the brane tension, satisfies the relation Λ 4 = 6 M 3 5 //lscript , then the Israel junction conditions allow for an empty brane to be at rest in the above coordinates, e.g., at the position r = R . \nIn this situation, the brane observer sees 4-dimensional gravity at distance scales greater than /lscript . The corresponding reduced Planck mass, M 4 = 1 / √ 8 πG 4 , is fixed by the relation \nM 2 4 = M 3 5 /lscript , (5) \nwhere M 5 is the analogously defined 5-dimensional reduced Planck mass. Thus, in our following analysis, it will be convenient to consider M 4 and /lscript as our basic physical parameters and M 5 , Λ 4 and Λ 5 as derived quantities. \nStarting from Gauss's Theorema Egregium (see, e.g., [23]), which expresses the bulk curvature near the brane in terms of the intrinsic and extrinsic brane curvatures, and replacing the extrinsic brane curvature with the brane energy-momentum tensor according to Eq. (3), one arrives at [9] (see also [24]) \nM 2 4 R 4 µν = τ µν -1 2 τg 4 ,µν -1 4 M 4 c τ ρ µ ( τ ρν -1 3 τg 4 ,ρν ) + M 2 4 δ R 5 µρνσ g ρσ 4 . (6) \nHere τ µν = T µν + Λ 4 g 4 ,µν is the matter contribution to the brane energy-momentum tensor, R 4 µν is the brane Ricci tensor and δ R 5 µρνσ is the brane projection of the deviation of the bulk curvature tensor from its pure-AdS value. As will be discussed in detail below, the 'AdS-cosmology scale' \nplays a prominent role in the cosmology of the Randall-Sundrum model. It is intermediate with respect to the AdS-scale 1 //lscript and the 'strong gravity scale' M 5 = 3 √ M 2 4 //lscript since, in \nM c = √ M 4 //lscript (7) \nthe physically interesting limit 1 //lscript /lessmuch M 4 , one has 1 //lscript /lessmuch M c /lessmuch M 5 . Equation (6) reduces to the Einstein equation for the induced brane metric in the limit where the bulk is pure AdS and energy densities on the brane are much smaller than M 4 c . \nTo discuss cosmology, we have to introduce hot matter on the brane. In this situation, the bulk solution, Eq. (4), is not general enough. The reason for this is the unavoidable graviton radiation off the brane. No matter how small the amount of energy lost to the bulk might be, it will fall deep into the AdS bulk and produce, at least if back-reaction on the metric is neglected, an arbitrarily high local energy density. The only plausible physical resolution of this problem that we are aware of is offered by the creation of a black hole in the bulk. \nTo see this more explicitly, consider the trajectory of a graviton radiated at time t = 0 by a static brane at position R . At large t , this trajectory is given by \nr = R 1 + ( Rt//lscript 2 ) sin ϕ /similarequal /lscript 2 t sin ϕ , (8) \nwhere ϕ is the angle between the initial graviton momentum and the brane. (The dependence on this angle is obtained most easily by considering brane-perpendicular radiation first and then boosting the configuration along the brane.) Assume next that a 3-dimensional energy density ∆ ρ is lost to gravitons with angles between ϕ and ϕ +∆ ϕ . At large t , these gravitons will be found in a slice of the AdS bulk of invariant thickness \n/lscript r ∆ r /similarequal /lscript r · /lscript 2 cos ϕ t sin 2 ϕ · ∆ ϕ /similarequal cos ϕ sin ϕ · /lscript ∆ ϕ. (9) \nNote that the separation of gravitons in r direction due to different times of their emission from the brane is small compared to the separation due to different angles of emission. In the rest frame of the matter on the brane, the brane-parallel components of the graviton momenta are isotropically distributed. Therefore the radiation in the above slice ∆ r has its own 'rest frame'. In this frame, the momenta in r direction vanish and the momenta in brane direction are blue-shifted by a factor R/r with respect to their original values. Furthermore, the density in brane direction is increased by a factor ( R/r ) 3 due to AdS geometry. Thus, the 4-dimensional energy density of the radiation at position r , measured in its own rest frame, is given by \nρ r /similarequal ∆ ρ ( /lscript/r ) ∆ r ( R r ) 4 cos ϕ /similarequal ∆ ρ /lscript ∆ ϕ ( R r ) 4 sin ϕ. (10) \nThis local energy density diverges as r → 0 and we are forced to conclude that eventually a horizon will form, hiding any possible super-Planck-scale effects from our view. Thus, the relevant bulk metric is given by the AdS-Schwarzschild solution \nds 2 = -f ( r ) dt 2 + r 2 ( d/vectorx ) 2 + f ( r ) -1 dr 2 (11) \nwith \nf ( r ) = r 2 /lscript 2 ( 1 -r 4 h r 4 ) , (12) \nwhere r h is the position of the black hole horizon. \nNow the Israel junction conditions or, equivalently, Eq. (6) lead to the following equation of motion for the brane: \n3 M 2 4 ( ˙ R R ) 2 = ρ ( 1 + ρ 12 M 4 c ) +3 M 4 c ( r h R ) 4 , (13) \nwhere ˙ R = dR/dτ and τ is the proper time of the brane observer. Since the brane metric is given by \nds 2 b = -dτ 2 + R 2 ( d/vectorx ) 2 , (14) \nthe smallρ and smallr h limit of Eq. (13) reproduces the familiar 4-dimensional Friedmann equation. Deviations are characterized by M c . On the one hand, M c determines the scale at which the ρ 2 term in Eq. (13) becomes important. On the other hand, it sets the scale of the last term on the rhs of Eq. (13). This term, which contributes to the expansion rate like an energy density ρ d of 'dark radiation', is ∼ M 4 c if the brane is near the black hole horizon, R ∼ r h . Furthermore, as will be discussed in detail in the next section, the radiation of bulk gravitons competes with the Hubble expansion rate H = ˙ R/R at brane temperatures T > ∼ M c . \nLet us add a comment concerning the position of the horizon (or, equivalently, the size of the black hole) characterized by r h . As can be seen from Eqs. (11)-(14), the absolute value of this quantity is not physical. In fact, nothing changes if one rescales the bulk coordinates and the positions of brane and horizon according to \nx µ → αx µ , r → r/α and R → R/α , r h → r h /α, (15) \nwith some real number α > 0. The important physical parameter is the ratio r h /R (see Eq. (13)). Note furthermore that this degeneracy is lifted if one considers closed or open geometries, where f ( r ) = k + r 2 //lscript 2 (1 -r 2 h /r 2 ) with k = ± 1. However, since we are only interested in the early universe, we will primarily consider the flat case with k = 0.", '3 Dark radiation and graviton emission into the bulk': "Let us start with the cross section for the production of bulk gravitons by matter on the brane. If the cms-energy of the collision is large compared to particle masses, √ s /greatermuch m , and to the typical scale of the AdS-space curvature, √ s /greatermuch 1 //lscript , the only relevant scales in the process are √ s and the 5-dimensional Planck mass. Since the cross section has to be proportional to the 5-dimensional gravitational coupling, we expect σ ( s ) ∼ √ s/M 3 5 on dimensional grounds. 1 \nThe exact results, which are derived in the Appendix utilizing previous work of [14, 25], can be summarized by writing \nσ i ( s ) = c i √ s M 3 5 . (16) \nThe appropriate numerical constants for the cases of scalars, vector particles and fermions (with initial-state spin averaging included) are given by \nc s = 1 12 , c v = 1 4 , c f = 1 16 . (17) \nIn a hot plasma, the reaction rate per 3-volume is obtained by thermally averaging the cross section σ for the process under consideration and multiplying it with the squared number density n of the initial state particles (see, e.g., Ref. [26]). Analogously, the total rate of energy loss due to bulk graviton radiation is obtained by thermally averaging the product of cross section and lost energy (to avoid double counting the number of interactions between identical particles, a factor 1 / 2 has to be included): \n∆˙ ρ = -1 2 〈 σv rel · E 〉 n 2 = -1 2 ∫ d 3 p 1 d 3 p 2 f ( E 1 ) f ( E 2 ) σv rel · ( E 1 + E 2 ) . (18) \nHere f ( E ) = κ (2 π ) -3 (exp( E/T ) ± 1) -1 is the distribution function for bosons (minus sign) or fermions (plus sign) with κ spin degrees of freedom and v rel is the relative velocity of the colliding particles. Using the cross section of Eq. (16) and setting particle masses to zero, one finds \n∆˙ ρ = -2 T 8 c i 5 π 4 M 3 5 κ 2 i Γ( 7 2 ) ζ ( 7 2 )Γ( 9 2 ) ζ ( 9 2 ) , (19) \nin the bosonic case. An additional factor a f = (1 -2 -5 / 2 )(1 -2 -7 / 2 ) /similarequal 0 . 750 arises in the fermionic case. \nSumming the different particle species and normalizing to the total energy density of a relativistic gas one obtains \n∆˙ ρ ρ = -C T 4 M 3 5 , (20) \nC = 0 . 574 · g s c s +2 g v c v +2 g f a f c f g s + g v +(7 / 8) g f , (21) \nwhere \nfor g s scalar, g v vector, and g f fermionic degrees of freedom. The factors 2 in front of g v and g f are due to the 2 spin degrees of freedom of massless vector particles and fermions. In the standard model, g s = 4, g v = 24 and g f = 90, leading to C = 0 . 112. \nAt sufficiently late times, the expansion of the universe is governed by 4-dimensional gravity. To be more specific, this period is characterized by 1 /H /greatermuch /lscript or ρ /lessmuch M c , which means that the term linear in ρ dominates the rhs of Eq. (13) (' ρ period'). In this situation, effective 4-dimensional energy-momentum conservation ensures that the loss of energy-density on the brane is equal to the gain of dark energy-density ρ d , \n∆˙ ρ +∆˙ ρ d = 0 , (22) \nwhere both ρ and ρ d scale like radiation: ˙ ρ = -4 Hρ + ∆˙ ρ and ˙ ρ d = -4 Hρ d + ∆˙ ρ d . Thus, even if ρ d = 0 at some initial time τ 1 , it will develop a non-zero late-time value characterized by Ω d = ρ d / ( ρ + ρ d ). If ρ d /lessmuch ρ , we have \nΩ d = ∫ ∞ τ 1 dτ ( -∆˙ ρ ρ ) , (23) \nwhere the integrand is given by Eq. (20). Employing the relation ρ = g ∗ ( π 2 / 30) T 4 (where g ∗ is the effective number of degrees of freedom, g ∗ = g s + g v +(7 / 8) g f ), the temperature factor T 4 can be expressed through ρ . Now the integral in Eq. (23) is easily performed, giving the result \nΩ d = 15 C g ∗ π 2 √ 3 ρ 1 M 4 c , (24) \nwhere ρ 1 is the radiation density at τ = τ 1 . The largest value of ρ 1 compatible with linear ρ behaviour is ρ 1 /similarequal 12 M 4 c (cf. Eq. (13)), leading to \nΩ d = 90 C g ∗ π 2 . (25) \nAssuming that the evolution of the universe after the decoupling of bulk gravitons (which is complete soon after the beginning of the ρ period) respects entropy conservation, one finds that at the time of nucleosynthesis \nΩ d,N = ( g ∗ , N g ∗ ) 1 / 3 Ω d . (26) \nHere g ∗ , N /similarequal 10 . 75 is the number of light degrees of freedom at nucleosynthesis. Thus, one arrives at the result that bulk graviton production during the complete ρ period only gives rise to Ω d = 0 . 0044, i.e., somewhat less than 0.5% of energy density in dark radiation. The smallness of this contribution is a direct result of the large number of light fields on the brane, which leads to a fast expansion and a correspondingly fast cooling of the universe. \nThe above result is encouraging for two reasons. First it implies that late cosmological evolution in the RSII model is quite safe (although not without a potential signature in future accurate CMBR measurements) even assuming the most extreme values for /lscript and M 5 . Second it allows for a cosmological ρ 2 period with hot matter on the brane. However, also during the ρ 2 period matter is lost to dark radiation and we have to calculate the resulting contribution to Ω d . Since we do not have effective 4-dimensional gravity during the ρ 2 period, the energy lost by the brane is not necessarily the same as the energy found at late times in dark radiation, Ω lost /negationslash = Ω d . Equation (23) is still valid, but with the lhs replaced by Ω lost and with τ 1 as the upper limit of integration, \nΩ lost = ∫ τ 1 τ 0 dτ ( -∆˙ ρ ρ ) . (27) \nHere τ 0 is the earliest time at which our analysis is valid, e.g., the time of reheating. \nThe Friedmann equation in the ρ 2 period, \n6 M 3 5 H = ρ , (28) \nwhich follows from Eq. (13) in the limit ρ /greatermuch M 4 c , implies that ρ ∼ 1 /τ and H = (1 / 4 τ ). In this situation, Eqs. (20) and (27) give \nΩ lost = 45 C g ∗ π 2 ln( ρ 0 /ρ 1 ) /similarequal 90 C g ∗ π 2 · 1 3 ln ( M 4 /lscript/ 12 3 / 2 ) . (29) \nThe last approximate equality follows by assuming ρ 0 /similarequal M 4 5 , the highest density compatible with weakly interacting gravity. This result shows an enhancement by a potentially large factor (1 / 3) ln( M 4 /lscript ) compared to Eq. (25). \nWe still have to adress the question of how Ω lost at the ρ 2 period translates into Ω d at late times. This is most easily done from the brane perspective. During the ρ 2 period, the brane moves with almost speed-of-light through the AdS bulk (a natural bulk rest frame can be defined by the black-hole horizon or, equivalently, by the late-time limit, where the brane is almost at rest). The (absolute) acceleration of the brane can be calculated using the standard formula \na M = u N ∇ N u M , (30) \nwhere u M = dx M /dτ is the velocity of a particle at rest on the brane. Multiplying this acceleration with the brane normal vector n M (which is, in fact, the direction of the acceleration) and making use of the orthogonality relation n · u = n M u M = 0, one derives \na = n · a = -u M u N ∇ M n N . (31) \nThis is precisely the ττ component of the extrinsic curvature K µν . Using the Israel junction condition, Eq. (3), and defining n M to point into the direction of increasing r , the acceleration is found to be \na = -1 2 M 3 5 (Λ 4 + ρ ) . (32) \nNotice first that this provides a nice physical interpretation of the Israel junction condition: the brane motion is determined by the brane acceleration, which, in turn, is determined by the energy density on the brane. Furthermore, deep in the ρ 2 period, the brane can be thought of as accelerating into the direction of decreasing r with a ∼ ρ/M 3 5 . This implies that gravitons radiated by the brane are accelerated towards the brane with the same acceleration a . Since, in a thermalized situation, gravitons are emitted with smooth angular distribution, most of them will fall back onto the brane after having reached a maximal distance \nd ∼ 1 a ∼ M 3 5 ρ . (33) \nClearly, this discussion becomes invalid if d > ∼ /lscript , since then the AdS curvature takes over and determines the future path of the graviton. In this situation, the graviton will not return to the brane but fall into the black hole horizon. \nThe critical value of ρ is determined Eq. (33) with a ∼ 1 //lscript , \nρ ∼ M 3 5 /lscript ∼ M 4 c . (34) \nThis means that virtually no gravitons leave the vicinity of the brane during the ρ 2 period. Instead, the gravitons radiated by the brane remain gravitationally bound to the brane and fall into the black hole only at the end of the ρ 2 period, when the motion of the brane becomes non-relativistic. \nBetween its initial emission and its final fall into the horizon, each graviton can bounce off the brane many times. During this process, the graviton momentum parallel to the brane remains unchanged except for the trivial redshift factor associated with the AdS geometry. However, the momentum transverse to the brane decreases dramatically. This is seen most easily by observing that, in the black hole frame, the graviton is reflected many times by the retreating brane, losing part of its momentum in the r direction with each reflection. \nThus, we expect that the ρ 2 period contributes an amount Ω d = α Ω lost (with α < 1) to the late-time dark energy fraction. A lower bound on the constant α can be derived by assuming that only the energy going into the brane-parallel motion of the radiated gravitons will survive the multiple reflection phase and make it into the black hole horizon. This corresponds to replacing the factor ( E 1 + E 2 ) in Eq. (18) by | /vector p 1 + /vector p 2 | . Recalling that σ ∼ √ s and v rel = 1 -cos θ , one finds \nTo get an estimate, we use the relation \nα = ∫ d 3 p 1 d 3 p 2 f ( E 1 ) f ( E 2 ) √ s (1 -cos θ ) | /vector p 1 + /vector p 2 | ∫ d 3 p 1 d 3 p 2 f ( E 1 ) f ( E 2 ) √ s (1 -cos θ ) ( E 1 + E 2 ) . (35) \n| /vector p 1 + /vector p 2 | = √ E 2 1 + E 2 2 +2 E 1 E 2 cos θ ≥ ( E 1 + E 2 ) √ (1 + cos θ ) / 2 , (36) \nwhere θ is the angle between /vector p 1 and /vector p 2 and the equality is realized for E 1 = E 2 . Since √ s ∼ √ 1 -cos θ and since the angular integration is performed with the measure d (cos θ ), the following bound can be derived: \nα > ∫ 1 -1 d (cos θ ) √ (1 + cos θ ) / 2 (1 -cos θ ) 3 / 2 ∫ 1 -1 d cos θ (1 -cos θ ) 3 / 2 = 5 π 32 . (37) \nΩ d = 90 C g ∗ π 2 · α 3 ln ( M 4 /lscript/ 12 3 / 2 ) , (38) \nThus, we find 0 . 5 < ∼ α < 1, and the maximally extended ρ 2 period produces a dark energy contribution \nwhere C is defined in Eq. (21). Even for the conservative number α /similarequal 0 . 5, this clearly dominates over the contribution from the ρ period and corresponds, in the most optimistic scenario with /lscript /similarequal 1 mm = 1 / (0 . 2 meV), to Ω d,N /similarequal 0 . 05. Such a value is marginally consistent with present nucleosynthesis bounds and would probably be visible in future CMBR analyses. \nAs we discuss in the next section, the role of the black hole in late cosmology has a nice interpretation in the AdS/CFT correspondence. However, the ρ 2 -period depends crucially on which definition of RSII cosmology is chosen - that inherited from the AdS/CFT correspondence, or the above naive 'brane in AdS 5 ' definition.", '4 The AdS/CFT definition': "Many of the confusions concerning the early cosmology of RSII stem from the fact that there are various definitions of the theory, which accord with each other in the IR, but which can differ in their high-energy and high-temperature behaviour. The first definition is that employed by RSII and is the one used in the previous sections. Specifically, this involves a brane whose motion continues to be well-described by five-dimensional gravity at length scales below the AdS length /lscript , the 5d gravity description only breaking down at the fundamental 5d Planck length 1 /M 5 . \nThe second definition, which we now discuss, arises from the AdS/CFT correspondence [27,28,29]. Although more complicated in detail, this definition has the significant advantage that it is embedded in a richer context, allowing in principle a self-consistent discussion of high-energy (and high-temperature) effects via the connection to string theory. \nLet us recall some basic aspects of the AdS/CFT correspondence in the most studied case. IIB string theory on AdS 5 × S 5 is conjectured to be equivalent to an N = 4 supersymmetric Yang-Mills conformal field theory on the boundary of AdS 5 . Defining the 't Hooft coupling of the boundary SU ( N ) SYM theory λ = g 2 Y M N , the relation between the parameters of the theories is \ng 2 Y M = 4 πg s , ( /lscript /lscript s ) 4 = ( R S 5 /lscript s ) 4 = λ , (39) \nwhere g s and /lscript s are the string coupling and length, and R S 5 is the radius of the S 5 . That the boundary of the full 10-dimensional space is just 4-dimensional is easily seen from inspecting the AdS 5 × S 5 metric \nds 2 = /lscript 2 z 2 { dz 2 + dx 2 + z 2 d Ω 2 5 } (40) \nin the limit z → 0. Note that the radius of the S 5 part of the background, as measured by an observer located at z = /lscript , is /lscript . This feature is quite generic, applying to the broad class of backgrounds of the form AdS 5 × M 5 , since it derives from the ballancing of curvature contributions from the AdS 5 and M 5 in the bulk equations of motion. \nThe precise fashion in which these two theories correspond is most easily described in terms of the generating functional of correlation functions for the boundary theory \nZ [ ψ ( x )] = 〈 exp ∫ dxψ ( x ) O ( x ) 〉 (41) \nwhere O ( x ) are a complete set of operators of the boundary theory, and ψ are, for the moment, just the associated sources. The proposal of Refs. [29,28] is to identify Z [ ψ ( x )] with the functional integral of the AdS 5 × S 5 theory, with specified boundary behaviour for the fields: \nZ [ ψ ( x )] = ∫ [ d Ψ( z, x )] exp ( -S IIB [Ψ( z, x )]) ∣ ∣ ∣ Ψ( z → 0 ,x ) → ψ ( x ) . (42) \n∣ \nIn the limit of large N and large 't Hooft coupling λ , the bulk IIB string theory goes over to its (super)gravity limit in the tree approximation. In fact, a similar correspondence with a 4d boundary CFT is believed to apply much more generally for any gravity theory formulated on AdS 5 . In particular we will, for simplicity, consider the nonsupersymmetric case in the following. \nIt is important that, because of the short-distance singularities of the correlators, the above expressions must be regulated so that they are well-defined. A natural way of doing this is to impose the boundary conditions at a finite cutoff z = z c rather than at z = 0. By the Weyl rescaling symmetry of the coordinates ( z, x ), changes in the z coordinate of AdS 5 can be reinterpreted as changes in the energy scale of the CFT process under consideration, with z → 0 corresponding to the UV limit of the boundary conformal theory. Thus the cutoff at z = z c implements a UV regulation of the boundary theory. In order to get a well-defined limit as the cutoff is removed, z c → 0, the boundary theory must be supplemented by counterterms depending on the boundary values of the bulk fields. Moreover, the cutoff at z = z c also allows the normalizability of the bulk AdS graviton zero mode, the existence of such a mode implying that the regulated boundary theory now also couples to dynamical gravity. In the limit of z c → 0, the boundary theory becomes a pure CFT decoupled from gravity, the effective Planck mass of the boundary theory M 4 going to infinity as the graviton zero mode becomes non-normalizable. \nIn the specific case we are interested in, this cutoff is implemented by a physical 'Planck brane' located at z c . This Planck brane provides a physical cutoff for the bulk theory and the theory on the Planck brane is a 4-dimensional CFT coupled to dynamical 4-dimensional gravity arising from the graviton zero mode. The CFT degrees of freedom are just the new expression of the bulk Kaluza-Klein modes of the graviton in this 4d 'holographic' approach. \nFollowing Refs. [30,10,12], it is useful to be more explicit about the result of this procedure in the simple case where the only bulk field is the 5-dimensional metric g 5 ( x, z ), with boundary value g 4 ( x ) at the Planck (regulator) brane at z c . 2 The boundary operator for which g 4 is the source is just the energy-momentum tensor T µν CFT of the CFT, so the integration over all bulk metrics will lead to the generating functional Z [ g 4 ] = 〈 exp( -∫ g 4 ,µν T µν CFT ) 〉 CFT for correlation functions of T CFT . In addition, counterterms in the boundary metric g 4 ,µν must be added. Together this leads to a boundary \ntheory described by the effective action \nS 4d [ g 4 , ϕ ] = ∫ d 4 x √ g 4 { L ( ϕ ) -1 2 M 2 4 Rb 4 R 2 + . . . } . (43) \nHere R 2 = -R µν R µν / 8 + R 2 / 24 is the leading higher-derivative pure metric piece of the counterterm action, and b 4 = cf ( z c ) where f ( z c ) is a function that has a log( z c ) singularity as z c → 0, and c = 2 π 2 ( M 5 /lscript ) 3 is the central charge of the boundary CFT. \nAt long distances this effective action correctly reproduces the gravitational potential between two masses located on the Planck brane including leading corrections: \nV ( r ) = m 1 m 2 M 2 4 ( 1 r + a /lscript 2 r 3 + . . . ) , (44) \nfor r /greatermuch /lscript , and a a numerical coefficient. In the original bulk AdS 5 picture, the 1 /r 3 correction is due to the exchange of the continuous spectrum of Kaluza-Klein modes, while in the boundary CFT + 4-dimensional gravity description, the correction is due to the two-point function of the CFT energy-momentum tensor with two external 4d graviton propagators (1 /p 2 ) 〈 T ( p ) T ( -p ) 〉 (1 /p 2 ) ∼ log p 2 . Recall that the two-point function of T CFT satisfies \n〈 T ( p ) T ( -p ) 〉 ∼ c p 4 log p 2 (45) \nwith c the central charge. \nNote that an inspection of the calculation of Ref. [29] shows that for fixed 4d Planck mass, M 4 , the central charge c/ 2 π 2 = ( M 5 /lscript ) 3 = ( M 4 /lscript ) 2 of the boundary CFT depends only on the geometry of the AdS 5 (through the parameter /lscript ). In particular, for equal values of /lscript , the two theories defined by taking the geometry to be respectively either AdS 5 or AdS 5 × S 5 have the same 4d CFT + 4d gravity description in the IR. 3 Since the theories most certainly differ for length scales r < /lscript (the static gravitational potential behaving as V ( r ) ∼ 1 /r 2 or ∼ 1 /r 7 in the two cases), the 4d CFT + gravity description must break down at r /similarequal /lscript due to the existence of new strong couplings possibly involving new degrees of freedom. \nAs pointed out by Gubser [10], the CFT perspective is particularly useful for a physical understanding of the dark radiation term in the Friedmann equation. The AdSSchwarzschild geometry implies, by continuation to Euclidean space, a black hole temperature T BH = r h /π/lscript 2 . This corresponds to a local temperature T ( r ) = T BH / √ f ( r ) in the bulk [31]. For R /greatermuch r h , the temperature near the brane is T ( R ) /similarequal r h / ( π/lscriptR ). Thus, the brane observer interprets the CFT, represented by the bulk degrees of freedom, as being heated to this temperature. At weak coupling, the energy density of the heated CFT is given by ρ = 2 π 2 cT 4 . Taking into account the famous factor 3 / 4 (see, e.g., [32]), which is due to strong coupling effects in the CFT, one finds \nρ = 3 π 2 2 c [ T ( R )] 4 , (46) \nwhich is precisely the dark radiation term of Eq. (13). Note that, phenomenologically, T ( R ) has to be much smaller than the standard model temperature on the brane. Otherwise, the dark radiation completely dominates the total energy density due to the large number of degrees of freedom of the CFT. \nNear the horizon, f ( r ) goes to zero and the local bulk temperature diverges. Nevertheless, the brane observer in a cosmological setting sees an effective bulk temperature T = r h / ( π/lscriptR ), since the red-shift factor from the brane motion in the blackhole rest frame compensates the singular behaviour of 1 / √ f ( r ). We have, at present, no deeper understanding of this intriguing fact. Note, however, that the naive formula T ( r ) = T BH / √ f ( r ), which underlies the above discussion, should in itself be questioned near the horizon, since it is in principle affected by the ambiguities of the definition of the black-hole vacuum state (cf. [33]). \nIn the cosmological context of the earlier sections, an immediate question is if the ρ 2 correction in the Friedmann equation (13) is similarly reproduced by the AdS/CFT description Eq.(43). At first glance the answer appears to be no, as the ϕ -matter energymomentum tensor derived from L ( ϕ ) couples linearly with g 4 ,µν . However, the existence of the higher-derivative terms R 2 = -R µν R µν / 8 + R 2 / 24 in the 4d action Eq.(43) leads to an effective ρ 2 correction. 4 (Recall that these higher-derivative terms are predicted by the conformal anomaly of the boundary CFT and occur with coefficient b 4 enhanced by c .) To see this, consider the Friedmann equations in the limit that ρ /lessmuch M 4 c ; in this case the Hubble length H -1 /greatermuch /lscript . Then in the leading approximation H 2 ∼ ρ/M 2 4 , and substituting this back in to the higher-derivative R 2 terms gives \nb 4 ( -R µν R µν / 8 + R 2 / 24 ) ∼ cH 4 ∼ /lscript 2 ρ 2 M 2 4 = ρ 2 M 4 c , (47) \nthe required ρ 2 term. The coefficient of this effective ρ 2 correction can vary, though, depending on the definition of the UV theory. For example, the boundary action Eq.(43) can contain an additional higher-dimensional counterterm that is not forbidden by any symmetry: \n1 M 4 4 T CFT µν T µν ϕ . (48) \nSuch an operator directly linking the CFT with the brane-localized matter has recently been invoked [35] to explain the 1 /r 7 corrections to the gravitational potential found in the Lykken-Randall 'probe-brane' scenario [36] (see also [13]). Via the use of the T CFT two-point function, this operator in turn implies the additional interaction \n∆ S 4d ∼ 1 M 8 4 ∫ d 4 xd 4 yT ϕ ( x ) 〈 T CFT ( x ) T CFT ( y ) 〉 T ϕ ( y ) ∼ c M 8 4 ∫ d 4 xd 4 y T ϕ ( x ) T ϕ ( y ) | x -y | 8 (49) \nwhich, for homogeneous and isotropic radiation T ϕ = diag( ρ, -ρ/ 3 , -ρ/ 3 , -ρ/ 3), leads to a term (taking the cutoff for the Planck brane theory to be M 4 ) \nc M 4 4 ρ 2 ∼ ρ 2 M 4 c (50) \non the rhs of the Friedmann equation. \nThus it is possible to account for the ρ 2 term in the modified Friedmann equation, Eq.(13), from a CFT description, although the exact coefficient of this term depends on the UV definition of the theory. In the case that the UV ( r < /lscript ) theory is just the brane in AdS 5 , then, by definition, the CFT version must exactly reproduce the term in Eq.(13). Alternatively if the theory is the string-motivated AdS 5 × S 5 , then the ρ 2 term can receive further corrections. \nDespite the existence of the ρ 2 correction to the Friedmann equation, we wish to emphasize that the unusual cosmological ' ρ 2 -behavior' where this term dominates (occurring for ρ > M 4 c ) is not physically accessible in the string motivated AdS 5 × S 5 definition of the theory. The reason for this is simply that for the string coupling g s < 1 (which can always be arranged by suitable strong to weak coupling duality transforms) the string scale satisfies L -1 s ∼ g 1 / 4 s ( M 4 //lscript 3 ) 1 / 4 ∼ g 1 / 4 s ( M 3 c /M 4 ) 1 / 2 /lessmuch M c . Thus one encounters the full string theory well before one reaches the scale at which ρ 2 behaviour begins.", '5 Summary and Comments': "In this paper we have considered some of the cosmological issues raised by taking the Randall-Sundrum II framework to describe our world. The Randall-Sundrum II proposal assumes that the SM degrees of freedom are localized on the 'Planck brane' and are therefore described by a conventional (3+1)-d quantum field theory up to high scales; thus the successes of, for example, supersymmetric gauge coupling unification are not necessarily spoiled. However, RSII does not itself provide a solution to the hierarchy problem - one must still assume supersymmetry or technicolour, or some other as yet undiscovered mechanism, to solve this problem for the Planck-brane QFT; on the other hand, gravity is dramatically modified leading to new effects, especially in early universe cosmology, and associated constraints or signals. RSII provides an interesting laboratory for exploring modifications to gravity and early universe cosmology. \nThere are two immediate new effects in RSII cosmology. The first is that the motion of our Planck brane in the AdS bulk leads to a correction to the usual Friedmann equations for the scale factor - the well known ρ 2 -term. The second is the possibility of radiation into the AdS bulk, and the formation of a black hole in the AdS spacetime. Both of these issues have been previously studied. However, we argue that the amount of 'dark radiation' that could be produced during the linear-inρ and ρ 2 epochs in the early universe was overestimated in previous treatments. This leads to the false conclusion that the Planck brane temperature could never be high enough to reach the ρ 2 region. We give a detailed account of this radiation and further show that the formation of the bulk AdS-Schwarzschild background is a necessary consequence. The numbers that we find for the dark radiation are in a region that is accessible by future more precise cosmological measurements. In particular, we predict dark radiation at nucleosynthesis with Ω d,N /similarequal 0 . 005 from the ρ regime and Ω d,N < ∼ 0 . 05 from the ρ 2 regime. The precise amount of the contribution from the ρ 2 regime depends on the maximal brane temperature in the early universe. \nAnother important ingredient of our paper is the extended discussion of RSII cosmology in the framework of the AdS/CFT correspondence. We show how, for small brane energy densities, the manifestly 4-dimensional CFT description can reproduce the predictions of the standard definition in which the Planck brane moves in AdS 5 according to the Israel junction conditions, including the ρ 2 corrections to the Friedmann equations. However, we emphasize that there exist different definitions of the theory in the 'UV region' r < /lscript , and the precise size of the ρ 2 corrections depends on this UV theory. In the general case, the corrections to the cosmological evolution arise both from higher-derivative terms in the curvature and higher-dimension operators connecting the on-brane matter directly to the CFT. Because of the large central charge of the CFT, the coefficient of the higher curvature terms is greatly enhanced over its naive value, indicating a precocious breakdown of the 4d effective theory. In the case of the string-motivated definition on AdS 5 × M 5 , the ' ρ 2 -region' where the ρ 2 term dominates the evolution is not physically accessible. \nThere are many additional physical questions that either deserve further attention or have not been discussed at all. These include a further analysis of the near horizon, ( R -r h ) /r h /lessmuch 1, behavior of the 'Israel' brane theory, and the inclusion of the many more bulk fields (in addition to the graviton) that are likely to accompany the construction of even a semi-realistic model. At low energies, the manifestly 4-dimensional CFT+gravity description is likely to be useful to analyse such questions as the production and evolution of the cosmic microwave background fluctuations. Concerning the UV definition of the RSII theory, an important general point about the string-motivated AdS 5 × M 5 definition is that it encodes a consistent description of the theory in the high-energy regime, with different physical behaviour from the standard 'Israel' definition.", 'Acknowledgements': 'We would like to thank Tony Gherghetta, Yaron Oz and Riccardo Rattazzi for helpful discussions. We are also grateful to the authors of the later paper Ref. [38] (see also [39]) for pointing out two numerical errors in the first version of this paper. Special thanks go to Lorenzo Sorbo, who helped to compare the respective calculations.', 'Appendix': 'In this appendix, we briefly describe the calculation of the cross sections for the production of bulk gravitons by scalars, fermions and vector particles on the brane. The effective 4-dimensional Lagrangian for the graviton Kaluza-Klein modes and their coupling to matter has been derived, e.g., in Ref. [14]. To properly normalize the bulk excitations, one introduces a regulator brane at r = /lscript exp( -L//lscript ) and takes the limit L →∞ at the end. In the transverse traceless gauge, where the graviton fields H µν ( n ) satisfy H µν ( n ) , µ = 0 \nand g µν H µν ( n ) = 0, the Lagrangian reads 5 \nL H = -1 2 ∑ n ( H µν ( n ) , ρ H ( n ) , ρ µν + m 2 n H µν ( n ) H ( n ) µν ) + T µν ∑ n λ n H ( n ) µν , (A.1) \nwhere, in the limit m n /greatermuch 1 //lscript , \nm n = 1 /lscript nπe -L//lscript and λ n = 1 M 4 e -L/ 2 /lscript . (A.2) \nFor each n , the operator H ( n ) µν represents simply a massive spin-2 field with 5 physical degrees of freedom (see, e.g., [37]). Thus, introducing the 5 polarization tensors ε ( i ) µν ( q ) (with ε ( i ) µν ( q ) ε µν ( j ) ( q ) = δ i j ), the final state spin sum in the production of a graviton with momentum q can be taken multiplying the squared amplitude with \n∑ i ε ( i ) µν ( q ) ε ( i ) αβ ( q ) = g µα g νβ -1 3 g µν g αβ + (terms ∼ q µ , q ν , q α or q β ) (A.3) \nand summing over µν and αβ . Terms ∼ q are irrelevant since the energy-momentum tensor T µν in Eq. (A.1) is conserved. \nSince we will mainly be interested in cms-energies in the TeV region, we disregard the masses of brane degrees of freedom in the following calculation. In the case of one real scalar field, the energy-momentum tensor in Eq. (A.1) is given by \nT µν = ∂ µ ϕ∂ ν ϕ -1 2 g µν ( ∂ϕ ) 2 . (A.4) \nNow it is straightforward to derive the Feynman rule for the ϕϕH vertex and to calculate the total ϕϕ annihilation cross section after the sum over n is replaced by an integral over the final state mass m = √ s according to \n∑ n -→ /lscript π e L//lscript ∫ d √ s . (A.5) \nSimilarly, the annihilation cross sections for two vector particles (e.g., two photons) and for a fermion-antifermion pair can be obtained by substituting the energy-momentum tensors \nT µν = -F µρ F ρ ν + 1 4 g µν F 2 , (A.6) \nwith F µν = ∂ µ A ν -∂ ν A µ , and \nT µν = i 4 ψ ( γ µ ↔ ∂ ν + γ ν ↔ ∂ µ ) ψ (A.7) \ninto Eq. (A.1). The resulting cross sections are given in Eqs. (16) and (17) of the main part of the paper. \nNote that, since we are only interested in the limit √ s /greatermuch 1 //lscript , the AdS nature of the bulk has no physical significance. Therefore, our result can be compared with the graviton production cross section on a brane in a flat, 5-dimensional bulk (the ADD scenario with one extra dimension [4]) if the 5-dimensional gravitational constants are identified. Indeed, one can check explicitly that the one-graviton exchange amplitude calculated in the framework of this appendix (where i/ ( -q 2 -m 2 n ) ∑ j ε ( j ) µν ( q ) ε ( j ) αβ ( q ) ) is the graviton propagator) is identical to the result of Ref. [25] (see Eqs. (69)-(71) with δ = 1).', 'References': "- [1] L. Randall and R. Sundrum, Phys. Rev. Lett. 83 (1999) 3370 [hep-ph/9905221].\n- [2] L. Randall and R. Sundrum, Phys. Rev. Lett. 83 (1999) 4690 [hep-th/9906064].\n- [3] V. A. Rubakov and M. E. Shaposhnikov, Phys. Lett. B 125 (1983) 139;\n- P. van Nieuwenhuizen and N. P. 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1993PhRvD..48.3600A

Relating black holes in two and three dimensions

1993-01-01

['-', '-', '-', 'black hole physics', '-', '-']

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The three-dimensional black hole solutions of Bañados, Teitelboim, and Zanelli (BTZ) are dimensionally reduced in various different ways. Solutions are obtained to the Jackiw-Teitelboim theory of two-dimensional gravity for spinless BTZ black holes, and to a simple extension with a nonzero dilaton potential for black holes of fixed spin. Similar reductions are given for charged black holes. The resulting two-dimensional solutions are themselves black holes, and are appropriate for investigating exact ``S-wave'' scattering in the BTZ metrics. Using a different dimensional reduction to the string-inspired model of two-dimensional gravity, the BTZ solutions are related to the familiar two-dimensional black hole and the linear dilaton vacuum.

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https://arxiv.org/pdf/hep-th/9304068.pdf

{"Ana Ach'ucarro †": 'Dept. of Mathematics, Tufts University, Medford, MA 02174, USA. \nand \nDept. of Theoretical Physics, University of the Basque Country, Bilbao, Spain.', 'and': "Miguel E. Ortiz ‡ \nCenter for Theoretical Physics, \nLaboratory for Nuclear Science and Department of Physics, \nMassachusetts Institute of Technology, Cambridge, MA 02139, USA. \nThe three dimensional black hole solutions of Ba˜nados, Teitelboim and Zanelli (BTZ) are dimensionally reduced in various different ways. Solutions are obtained to the JackiwTeitelboim theory of two dimensional gravity for spinless BTZ black holes, and to a simple extension with a non-zero dilaton potential for black holes of fixed spin. Similar reductions are given for charged black holes. The resulting two dimensional solutions are themselves black holes, and are appropriate for investigating exact 'S-wave' scattering in the BTZ metrics. Using a different dimensional reduction to the string inspired model of two dimensional gravity, the BTZ solutions are related to the familiar two dimensional black hole and the linear dilaton vacuum.", 'January 1993': "Although examples of black hole solutions abound in two and four dimensions, it was until recently believed that no such solutions exist in three spacetime dimensions[1]. However, in a recent paper, Ba˜nados, Teitelboim and Zanelli (BTZ) [2], [3] found a vacuum solution to Einstein gravity with a negative cosmological constant which may be interpreted as a black hole. The solution has everywhere constant curvature, but the global topology is different to that of three dimensional Anti de Sitter space. As a result, the causal structure of the solution is closer to that of the Schwarzschild solution. However, the singularity hidden behind the horizon is of a weaker form than that of Schwarzschild [3], [4]. \nBelow we discuss how the BTZ solutions may be dimensionally reduced to solutions of various two dimensional theories of gravity. Our motivation is provided by the recent evidence that progress may be made in understanding black hole radiation and evaporation in the context of two dimensions [5]. The solutions we derive may all be interpreted as two dimensional black holes, and some of the corresponding two dimensional theories of gravity may in principle be used to exactly describe the scattering of rotationally symmetric matter ('S-waves') off the three dimensional black holes. Since our present understanding of three dimensional quantum gravity coupled to matter indicates that it is non-renormaliseable, these models may provide the only route for understanding the quantum behaviour of the BTZ solutions. \nFirst let us review the BTZ solutions [2], [3], [4]. They arise in a three dimensional theory of gravity \nwith a negative cosmological constant, i.e. Λ > 0. It is straightforward to check that the Einstein field equations \nS = ∫ d 3 x √ -g ( R +2Λ) (1) \nR µν -1 2 Rg µν +Λ g µν = 0 (2) \nare solved by the metric \nwhere \nds 2 = -N 2 ( r ) dt 2 + dr 2 N 2 ( r ) + r 2 ( N θ ( r ) dt + dθ ) 2 (3) \nN 2 ( r ) = Λ r 2 -M + J 2 4 r 2 , N θ ( r ) = -J 2 r 2 . (4) \nFor J = 0, this metric is similar to Schwarzschild. It has a single horizon at r = √ M/ Λ, and a singularity at r = 0. However there are two important differences. Firstly, \n(3) is not asymptotically flat - it is a constant curvature metric. Secondly, the singularity at r = 0 is much weaker than that of Schwarzschild spacetime. Whereas the singularity of Schwarzschild is manifested by the power law divergence of curvature scalars at small r , the BTZ solution has at most a delta function singularity at r = 0, since everywhere else the spacetime is of constant curvature [6]. An interesting special case occurs when M = 0. In this case the metric takes the form \nds 2 = -Λ r 2 dt 2 + dr 2 Λ r 2 + r 2 dθ 2 . (5) \nThis should be regarded as the extremal or vacuum solution of the J = 0 family. The spatial geometry of this vacuum solution is an infinite wormhole, whose radius shrinks to zero at r = 0 at an infinite spacelike proper distance from the asymptotic region. Geodesics reach the end of the wormhole at r = 0 within a finite proper time, and it is unclear how they should be continued beyond r = 0. However, since this solution is extremal, the expectation is that a horizon develops before even the lightest test particle reaches this point. \nIf J /negationslash = 0, (3) has two horizons, and its causal structure is similar to the ReissnerNordstrom spacetime. When M = √ Λ | J | , the two horizons coincide, and this should be regarded as the vacuum solution for fixed J . As before, the spacetimes are of constant curvature and have a singularity at r = 0, with at most distributional torsion and curvature at that point [3], [4]. Solutions with M < √ Λ | J | have no horizon. It has been conjectured in [2] that these should be discarded since they contain a naked singularity, except when M = -1, J = 0, when the spacetime is just AdS 3 . \nThe thermodynamics of the BTZ black holes suggest that evaporation of the black holes should take place. Since the temperature decreases with decreasing mass, and is zero for the extremal solutions, these seem to be the natural endpoints of evaporation [2]. \nLet us now discuss the possible dimensional reductions of the BTZ black holes. We begin with the straightforward dimensional reduction from Einstein gravity with a cosmological constant in three dimensions to the Jackiw-Teitelboim (JT) theory [7] in two. Suppose that the gravitational field in three dimensions is independent of a single coordinate, which we shall call θ , and that the metric may be written in the form, \nds 2 = g µν dx µ dx ν = h ij ( x i ) dx i dx j +Φ 2 ( x i ) dθ 2 (6) \nwhere µ, ν = 0 , 1 , 2 and i, j = 0 , 1. Then it is simple to see that the action (1) reduces to \nS = ∫ d 2 x √ -h Φ( R +2Λ) (7) \nwhich is precisely the JT action. It follows that any solution to the equations following from (1), of the form (6), yields a solution \nh ij ( x i ) , Φ( x i ) (8) \nof the equations following from (7). We shall henceforth refer to the field Φ as the dilaton. \nThe BTZ solutions with J = 0 are of the form (6), and they therefore yield a solution to the JT model [8], \nds 2 = -(Λ r 2 -M ) dt 2 + dr 2 Λ r 2 -M , Φ = r. (9) \nIt is important to note that the dimensional reduction, although trivial in appearance, has radically changed the properties of the metric. In three dimensions, the point r = 0 is singular. The two dimensional metric is perfectly well-behaved at r = 0. This is a result of the weak nature of the singularity in the three dimensional solution. The metric (9) may be analytically extended beyond r = 0, using the co-ordinate transformation \nr = √ M Λ cosh ρ sin ( √ Λ τ ) , tanh ( √ Λ Mt ) = tanh ρ cos ( √ Λ τ ) (10) \nand the maximally extended spacetime is the whole of two dimensional Anti-De Sitter spacetime, \nds 2 = -cosh 2 ρdτ 2 +Λ -1 dρ 2 , -∞ < τ, ρ < ∞ , (11) \nwhich of course has no horizons. The dilaton in these co-ordinates takes the form \nΦ = √ M Λ cosh ρ sin ( √ Λ τ ) , (12) \nwhich vanishes when √ Λ τ = nπ . The embeddings of (9) into (11) are shown in Fig. 1. \nIn order to interpret the two dimensional solution as a black hole, we must look at the behaviour of the dilaton. Recall that the dilaton is the θθ component of the three dimensional metric, and that the three dimensional solution is singular where it vanishes. It is therefore natural to cut the two dimensional spacetime off at this point, which we \ncall the 'strong coupling' region (although this name should not be taken too literally), if we wish to use the JT theory to model three dimensional physics. In this case, (9) does indeed represent a black hole, whose Penrose diagram, shown in Fig. 2, is identical to that of its three dimensional counterpart. \nThe two dimensional version of the extremal solution is also not geodesically complete unless it is extended (see also Ref. [9] for a discussion of this spacetime). The extended version of the spacetime is also identical to Anti-De Sitter spacetime, but for our purposes, we should again restrict our attention to the region where the dilaton is greater than zero. In this extremal case the spacetime \nds 2 = -Λ r 2 dt 2 + dr 2 Λ r 2 (13) \nhas a 'naked singularity' at r = 0 in the sense that the region where Φ = 0 is not hidden behind a horizon. However, this is of no consequence: In both two and three dimensions (as discussed above) we expect that no matter can probe this region of spacetime without a horizon developing. A Penrose diagram of the restricted extremal solution is shown in Fig. 3, which again is identical to the three dimensional Penrose diagram. Incidentally, this region is the Steady State Universe solution [10], but with timelike and spacelike directions interchanged. \nThe two dimensional reduction outlined above yields a two dimensional theory which can be used to model S-wave scattering off a spinless BTZ black hole. That solutions in two dimensions must be restricted by hand to end where the dilaton vanishes is both a blessing and a curse. On the one hand, there is no singular region to worry about, but on the other we must specify boundary conditions at r = 0 in some fashion. In principle, however, by coupling matter to (7) in the natural way for a dimensionally reduced theory, \nS = ∫ d 2 x √ -h Φ ( R +2Λ+ h ij ∂ i f∂ j f ) , (14) \nand imposing an appropriate boundary condition at Φ = 0, it may be possible to model the evaporation of a spinless BTZ black hole. \nIt is also possible to construct an effective two dimensional theory which arises from the dimensional reduction of the J /negationslash = 0 solutions of BTZ, and which can be used to study S-wave scattering for the spinning black holes. We begin by considering the reduction of a metric of the form \nds 2 = h ij ( x i ) dx i dx j +Φ 2 ( x i ) ( dθ + A i ( x i ) dx i ) 2 . (15) \nThe corresponding two dimensional theory involves the three fields h ij , A i and Φ. If we wish to consider spacetimes of fixed spin, however, we can use the following identity, \nΦ 3 /epsilon1 ij ∂ i A j √ -h = constant (16) \nwhich follows from the field equation for A i . The constant is precisely the spin J of the metric (15), since it is equal to the charge corresponding to asymptotic rotational invariance [11]. Using this identity, the action (6) for spacetimes of spin J may be dimensionally reduced to \nS = ∫ d 2 x √ -h Φ ( R +2Λ -J 2Φ 4 ) . (17) \nThis is an appropriate effective action for looking at S-wave scattering off a spinning BTZ black hole (interaction with rotationally symmetric matter will of course keep the black hole in the spin J sector). Again, any solution h ij , Φ to (17) corresponds to a solution to (1) of spin J , of the form (15). In particular, the t, r section of the BTZ black hole of spin J \n( \nds 2 = -( Λ r 2 -M + J 2 4 r 2 ) dt 2 + dr 2 Λ r 2 -M + J 2 4 r 2 ) (18) \nR +2Λ+ 3 J 2Φ 4 = 0 (19) \n) is a solution to (17), with Φ = r . However, in this case, the equation for the scalar curvature is \nso that R need not be constant. Indeed the two dimensional spinning black hole (18) and extremal solution have power law singularities in R at r = 0. The Penrose diagram for each of these spacetimes is identical to that of the three dimensional metric, and is shown in Fig. 4. \nIn addition to the BTZ solutions described above, it was also shown in [2] that charged black hole solutions similar to (3) exist. These are solutions following from the action \n∫ d 3 x √ -g ( R +2Λ+4 πGF µν F µν ) (20) \nand take the form (3), but with \nN 2 ( r ) = Λ r 2 -M -8 πGQ 2 ln( r/r 0 ) + J 2 4 r 2 and F rt = -Q r . (21) \nThese solutions have a power law curvature singularity at r = 0, where R ∼ 8 πGQ 2 /r 2 . They can have two, one or no horizons, depending on the relative values of Λ, J , GQ 2 , \nand M ' = M -8 πGQ 2 ln( √ Λ r 0 ). In the simplest case J = 0, these possibilities depend on whether \nM ' -4 πGQ 2 1 -ln [ 4 πGQ 2 ]) (22) \n( \n[ is greater than, equal to, or less than zero respectively. \nThe action (20) may be dimensionally reduced in a similar way to that described above, in both the J = 0 and the J /negationslash = 0 sectors, provided that we assume that F rt is independent of θ and that the other two components F µθ vanish. The resulting two dimensional action is \n∫ d 2 x √ -h Φ ( R +2Λ -J 2Φ 4 +4 πGF µν F µν ) (23) \nThe solution in two dimensions corresponding to the BTZ solution is the obvious analogue of (9) and (18) for Q /negationslash = 0. The electromagnetic field has F rt = -Q/r as before. This two dimensional spacetime has a curvature singularity at r = 0, even for J = 0, since R = -2Λ -8 πGQ 2 /r 2 -3 J 2 / 2 r 4 . As in three dimensions, this may be a naked singularity or may be shielded by one or two horizons. \nFinally, let us describe a third dimensional reduction of the uncharged BTZ black holes. This involves the reduction introduced in [12] from the three dimensional action (1) to the string inspired action for two dimensional gravity [5], [13], \n∫ d 2 x √ -ˆ he -2 φ ( ˆ R +4( ∇ φ ) 2 +2 λ ) (24) \nor rather, to the action [14] \nwhich is obtained from the string-inspired action by means of the identification ˆ h ij = h ij e 2 φ , Φ = e -2 φ . To get the action (25) from (1), consider the usual KaluzaKlein dimensional reduction used above to obtain the JT action (7) (or the equivalent action obtained from spinning metrics (15)) followed by a shift of Φ by a constant: \n∫ d 2 x √ -h (Φ R +2 λ ) , (25) \nΦ → Φ+ λ Λ , (26) \n(this procedure is in fact equivalent to implementing the shift (26) on the action (1) and then reducing [12]). It yields a two-parameter family of actions which include the stringinspired action in the limit Λ → 0. Notice that the potentially divergent term \n∫ d 2 x λ Λ R (27) \nis proportional to the Euler characteristic of the two dimensional manifold and, being a topological invariant, it does not affect the equations of motion. Also, note that the effect of the shift as seen from the three-dimensional point of view is to push all points in the 'extra' θ dimension an infinite distance away, making the proper length of a θ orbit diverge. \nUnlike the JT reductions, this procedure gives the same result irrespective of the value of J . This is best understood in the first order formalism, where the dreibein e a and the spin connection ω a are the dynamical variables (the action (1) is then replaced by the Chern-Simons action for the Anti de Sitter group SO(2,2) [15]). Inserting the condition that all fields be independent of θ into the equations of motion, we obtain two first integrals, \n2 e a θ ω aθ and Λ e a θ e aθ + ω a θ ω aθ , (28) \n( a = 0 , 1 , 2). The first of these is precisely (16), and is equal to the angular momentum of the spacetime. The second quantity is the first integral of the tt component of Einstein's equations, and may be identified with the mass of the spacetime [16]. A metric like (15) has e 2 θ = Φ, and in the gauge e 0 θ = e 1 θ = 0, \n2Φ ω 2 θ = J, ΛΦ 2 + ω a θ ω aθ = M. (29) \nThe BTZ black hole (3) has \ne 0 = Ndt, e 1 = dr N , e 2 = Φ( N θ dt + dθ ) , (30 a ) ω 0 = [ -1 2 Φ NN θ ,r -Φ ' NN θ ] dt -Φ ' Ndθ, ω 1 = -Φ N θ ,r dr 2 N , ω 2 = [ NN ,r + 1 2 Φ 2 N θ N θ ,r ] dt + 1 2 Φ 2 N θ ,r dθ. (30 b ) \nand in that case, given that Φ = r , the form of the functions N and N θ follow from equation (29): \nN θ ,r = J Φ 3 = J r 3 (31 a ) \n-N 2 = M -ΛΦ 2 -J 2 / 4Φ 2 (Φ ,r ) 2 = M -Λ r 2 -J 2 / 4 r 2 , (31 b ) \nup to an irrelevant integration constant. \nThe effect of the shift (26) on N and N θ may be computed immediately from (29) in a similar way, by replacing Φ by Φ + λ/ Λ, taking the limit Λ → 0 and then setting Φ = r [17]. It follows that \nN θ ,r = lim Λ → 0 [ J ( r + λ Λ ) 3 ] = 0 , (32 a ) -N 2 = lim Λ → 0 [ M -Λ ( r + λ Λ ) 2 -J 2 4 ( r + λ Λ ) 4 ] = ( M -λ 2 Λ ) -2 λr = m -2 λr. (32 b ) \nThe dependence on J disappears when the limit is taken, and the solutions depend only on the parameter m . \nFollowing Ref. [12], the two dimensional metric, determined in this case only by N , is ds 2 = -N 2 dt 2 + dr 2 /N 2 . This is flat, since dr/N = dN/λ . The familiar black hole solution, with a mass m = M -λ 2 / Λ, appears when we consider the 'string' metric, Φ -1 h ij : \nds 2 = 2 λ m + N 2 [ -N 2 dt 2 + dN 2 λ 2 ] (33) \nand can be brought to a more familiar form by the co-ordinate change λT = N sinh( λt ), λX = N cosh( λt ). Note that the shift relates two dimensional black holes with finite mass m to three dimensional ones with infinite mass M , for which the horizon has been pushed to infinity and all that remains is the black hole interior, and vice versa. (33) has been extensively studied in [5], [13], [14] and we shall not discuss it further, except to say that the linear dilaton vacuum solution occurs when m = 0 or M = λ 2 / Λ.", 'Acknowledgements': 'We are grateful to M. Crescimanno, D. Cangemi, G. Gibbons, C. Teitelboim and J. Zanelli for helpful comments.', 'References': "- [1] Solutions of pure Einstein gravity have been classified in S. Deser, R. Jackiw and G. 't Hooft, Ann. Phys. 152 (1984) 220, and no black hole solutions were found. It is straightforward, if tedious, to see that no analogues exist in three dimensions of the dilatonic black hole solutions of G. Gibbons and K. Maeda, Nucl. Phys. B298 (1988) 741, and D. Garfinkle, G. Horowitz, and A. Strominger, Phys. Rev. D43 (1991) 3140; Erratum: Phys. Rev. D45 (1992) 3888. Charged solutions with a horizon in 2+1 dimensions are known, but they do not conform to the usual notion of a black hole; see the discussion in B. Reznik, Phys. Rev. D45 2151 (1992).\n- [2] M. Ba˜nados, C. Teitelboim and J. Zanelli, Phys. Rev. Lett. 69 (1992) 1849\n- [3] M. Ba˜nados, M. Henneaux, C. Teitelboim and J. Zanelli, 'Geometry of the 2+1 black hole', Santiago preprint (1993).\n- [4] D. Cangemi, M. Leblanc and R. Mann, 'Gauge formulation of the spinning black hole in 2+1 dimensional Anti-De Sitter space', MIT preprint CTP #2162, gr-qc/9211013 (1992).\n- [5] See for example J. A. Harvey and A. Strominger, 'Quantum aspects of black holes', U. of Chicago preprint EFI-92-41, hep-th/9209055 (1992), and references therein.\n- [6] The singularity at r = 0 arises because (3) is the covering space of AdS 3 , identified by a discrete subgroup of the isometry group. It is due to a singularity in the causal structure, but it is unclear whether it should be regarded as a conical singularity in the usual sense [3], [4].\n- [7] R. Jackiw, in Quantum Theory of Gravity , ed. S. Christensen, Hilger, Bristol, 1984; C. Teitelboim, ibid .\n- [8] These may be related to the general solutions given in R. Jackiw, 'Gauge theories for gravity on a line', MIT preprint CTP# 2105, hep-th/9206093 (1992). The parameter M corresponds to the parameter α 0 of his equation (20), and the other two parameters α a correspond to a shift of origin not explicit in our solution. Although these parameters can be absorbed into the metric by a change in co-ordinates, they cannot be made to simultaneously disappear from both the metric and the dilaton.\n- [9] M. Crescimanno, 'Equilibrium two dimensional dilatonic spacetimes', MIT preprint CTP #2174 (1993).\n- [10] S. W. Hawking and G. F. R. Ellis, The large scale structure of space-time , CUP, Cambridge (1973).\n- [11] The charge is given in J. D. Brown and M. Henneaux, Comm. Math. Phys. 104 (1986) 207, as 2 πJ [ g αβ , π αβ ] = lim r →∞ ∮ dS α 2 ξ β π α β , where ξ = ∂/∂θ , π αβ is the usual momentum conjugate to g αβ in canonical gravity, and the integral is taken over the boundary of a spacelike hypersurface. For a metric of the form (15), the integrand over a circle at infinity is precisely the constant expression in (16). Note \nalso that (16) says that the integrand of the surface term is constant over the entire spacelike hypersurface of constant x 0 ; by Stokes's theorem it appears at first sight that J should therefore be zero. However, the contribution giving rise to J comes from a distributional source for angular momentum at the singularity of the three dimensional spacetime as could be expected. \n- [12] A. Ach'ucarro, 'Lineal gravity from planar gravity', Tufts preprint, hep-th/9207108 (1992), to appear in Phys. Rev. Lett.\n- [13] E. Witten, Phys. Rev. D44 (1991) 314; G. Mandal, A. Sengupta and S. Wadia, Mod. Phys. Lett. A6 (1991) 1685.\n- [14] H. Verlinde, proceedings of the Sixth Marcel Grossmann Meeting, to be published, (1992).\n- [15] A. Ach'ucarro and P. K. Townsend, Phys. Lett. 180B (1986) 89; Phys. Lett. 229B (1989) 383; E. Witten, Nucl. Phys. B311 (1988) 46; Nucl. Phys. B323 (1989) 113.\n- [16] The Anti de Sitter group SO(2,2) is a direct product of two SO(2,1) factors, each of which gives rise to a conserved charge M ± √ Λ J . It is interesting to note that the extremal black holes are solutions for which one of these charges is zero. \nFig. 1: Regions of the two dimensional spacetime with J = 0 covered by the co-ordinates r, t , embedded in Anti-De Sitter space. The diagonal and dotted lines have no special significance. Here r H = M/ Λ. \nFig. 2: The Penrose diagram for the region of the J = 0 spacetime for which Φ ≥ 0. The diagonal lines now represent event horizons, and the dotted line the 'strong coupling' region. \n√ \nFig. 3: The Penrose diagram for the region of the extremal two dimensional spacetime with M = J = 0, for which Φ ≥ 0. The dotted line represents the 'strong coupling' region. \nFig. 4: The Penrose diagram for the two dimensional spacetimes with J /negationslash = 0. (a) has M ≥ √ Λ | J | and r ± = ( M ± √ M 2 -Λ J 2 ) / 2Λ, and (b) is the extremal solution with M = √ Λ | J | and r H = √ M/ 2Λ."}

1999PhRvD..60j4039G

Topological censorship and higher genus black holes

1999-01-01

['-', '-', '-', '-', '-', '-']

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Motivated by recent interest in black holes whose asymptotic geometry approaches that of anti-de Sitter spacetime, we give a proof of topological censorship applicable to spacetimes with such asymptotic behavior. Employing a useful rephrasing of topological censorship as a property of homotopies of arbitrary loops, we then explore the consequences of topological censorship for the horizon topology of black holes. We find that the genera of horizons are controled by the genus of the space at infinity. Our results make it clear that there is no conflict between topological censorship and the nonspherical horizon topologies of locally anti-de Sitter black holes. More specifically, let D be the domain of outer communications of a boundary at infinity ``scri.'' We show that the principle of topological censorship (PTC), which is that every causal curve in D having end points on scri can be deformed to scri, holds under reasonable conditions for timelike scri, as it is known to do for a simply connected null scri. We then show that the PTC implies that the fundamental group of scri maps, via inclusion, onto the fundamental group of D: i.e., every loop in D is homotopic to a loop in scri. We use this to determine the integral homology of preferred spacelike hypersurfaces (Cauchy surfaces or analogues thereof) in the domain of outer communications of any four-dimensional spacetime obeying the PTC. From this, we establish that the sum of the genera of the cross sections in which such a hypersurface meets black hole horizons is bounded above by the genus of the cut of infinity defined by the hypersurface. Our results generalize familiar theorems valid for asymptotically flat spacetimes requiring simple connectivity of the domain of outer communications and spherical topology for stationary and evolving black holes.

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https://arxiv.org/pdf/gr-qc/9902061.pdf

{'Topological Censorship and Higher Genus Black Holes': 'G.J. Galloway , 1 K. Schleich , 2 D.M. Witt , 3 and E. Woolgar 4', 'Abstract': "Motivated by recent interest in black holes whose asymptotic geometry approaches that of anti-de Sitter spacetime, we give a proof of topological censorship applicable to spacetimes with such asymptotic behavior. Employing a useful rephrasing of topological censorship as a property of homotopies of arbitrary loops, we then explore the consequences of topological censorship for horizon topology of black holes. We find that the genera of horizons are controlled by the genus of the space at infinity. Our results make it clear that there is no conflict between topological censorship and the non-spherical horizon topologies of locally anti-de Sitter black holes. \nMore specifically, let D be the domain of outer communications of a boundary at infinity 'scri.' We show that the Principle of Topological Censorship (PTC), that every causal curve in D having endpoints on scri can be deformed to scri, holds under reasonable conditions for timelike scri, as it is known to do for a simply connected null scri. We then show that the PTC implies that the fundamental group of scri maps, via inclusion, onto the fundamental group of D , i.e. , every loop in D is homotopic to a loop in scri. We use this to determine the integral homology of preferred spacelike hypersurfaces (Cauchy surfaces or analogues thereof) in the domain of outer communications of any 4-dimensional spacetime obeying the PTC. From this, we establish that the sum of the \n2 \nDept. of Physics and Astronomy, University of British Columbia, 6224 Agriculture \nRoad, Vancouver, BC, Canada V6T 1Z1. e-mail: [email protected] 4 Dept. of Mathematical Sciences and Theoretical Physics Institute, University of \nAlberta, Edmonton, AB, Canada T6G 2G1. e-mail: [email protected] \ngenera of the cross-sections in which such a hypersurface meets black hole horizons is bounded above by the genus of the cut of infinity defined by the hypersurface. Our results generalize familiar theorems valid for asymptotically flat spacetimes requiring simple connectivity of the domain of outer communications and spherical topology for stationary and evolving black holes.", 'I. Introduction': "It is generally a matter of course that the gross features of shape, i.e. , the topology, of composite objects, from molecules to stars, are determined by their internal structure. Yet black holes are an exception. A black hole has little internal structure, but the topology of its horizon is nonetheless strongly constrained, seemingly by the external structure of spacetime. This was made apparent in early work of Hawking [1], who established via a beautiful variational argument the spherical topology of stationary horizons. Hawking's proof was predicated on the global causal theoretic result that no outer trapped surfaces can exist outside the black hole region, unless energy conditions or cosmic censorship are violated. As argued by Hawking in [1], the possibility of toroidal topology, which arises as a borderline case in his argument, can be eliminated by consideration of a certain characteristic initial value problem and the assumption of analyticity; see [2] for further discussion of this issue. \nIn recent years an entirely different approach to the study of black hole topology has developed, based on the notion of topological censorship . In 1994, Chru'sciel and Wald [3], improving in the stationary setting the result on black hole topology considered in [2] and [4], were able to remove the analyticity assumption in Hawking's theorem by making use of the active topological censorship theorem of Friedman, Schleich, and Witt [5] (hereinafter, FSW). This latter result states that in a globally hyperbolic, asymptotically flat (hereinafter, AF) spacetime obeying an Averaged Null Energy Condition (ultimately, in the modified form used below-see [6]), every causal curve beginning and ending on the boundary-at-infinity could be homotopically deformed to that boundary. When topological censorship holds, the domain of outer communications (or DOC-the region exterior to black and white holes) of an AF spacetime must be simply connected [6]. \nJacobson and Venkataramani [7], also using the topological censorship theorem of FSW, were able to extend the result of Chru'sciel and Wald on black hole topology \nbeyond the stationary case. The principle behind their arguments was that any horizon topology other than spherical would allow certain causal curves outside the horizon to link with it, and so such curves would not be deformable to infinity, which would contradict FSW. \nIn the early 1990s, new solutions with non-spherical black hole horizons were discovered in locally anti-de Sitter (adS) spacetimes [8-13]; for a recent review, see [14]. The original topological censorship theorem did not apply to these spacetimes since they were not AF and not globally hyperbolic. However, two improvements to the proof of topological censorship indicated that these differences ought not matter. Galloway [15] was able to produce a 'finite infinity' version of topological censorship that replaced the usual asymptotic conditions on the geometry with a mild geometrical condition on a finitely-distant boundary, and Galloway and Woolgar [16] were able to replace the assumption of global hyperbolicity by weak cosmic censorship. Moreover, it was soon observed [17] that topological censorship in the sense of FSW held true for each of the newly discovered black hole constructions in locally adS spacetime, although no general proof was known in this setting and obviously the aforementioned corollary implying spherical horizon topology could not hold. This is not paradoxical-the topology of the locally adS black hole spacetimes is such that no causal curve links with a non-spherical horizon in such a way as to preclude a homotopy deforming that curve to infinity. 5 In fact, we will see below that the corollary implying spherical topology of black holes in AF spacetime is merely a special case of a more general corollary of topological censorship which gives a relationship between the topology of the black hole horizons and the 'topology of infinity.' In this sense, the topology of the black hole horizons is governed by a structure that is as 'external' as possible, being entirely at infinity. \nIn this paper, we will consider spacetimes that obey the Principle of Topological Censorship (PTC). Let M be a spacetime with metric g ab . Suppose this spacetime can \nbe conformally included into a spacetime-with-boundary M ' = M∪I , with metric g ' ab whose restriction to M obeys g ' ab = Ω 2 g ab , and where I is the Ω = 0 surface, which defines I as the boundary-at-infinity. For a connected boundary component I , 6 the domain of outer communications D is defined by \nD := I + ( I ) ∩ I -( I ) . (1) \nThe PTC is the following condition on D : \nPrinciple of Topological Censorship (PTC). Every causal curve whose initial and final endpoints belong to I is fixed endpoint homotopic to a curve on I . 7 \nThe PTC has already been established for general, physically reasonable AF spacetimes, cf. [5], [16]. In the next section we present a proof of the PTC in a setting that includes many asymptotically locally anti-de Sitter black hole spacetimes. This generalization exploits the fact that PTC proofs generally follow from a condition on double-null components of the Ricci tensor. These components can be related to the double-null components of the stress energy tensor through the double-null components of the Einstein equations. This relation involves no trace terms and so clearly is insensitive to the cosmological constant. This corrects the impression that the PTC is invalid in the presence of a negative cosmological constant [13,18], an impression that, if it were true, would imply that the PTC in this case would impose no constraints at all on the topology of black hole horizons. Hawking's g = 0 restriction [1,19,20] on horizon topology, by way of contrast, does assume non-negative energy density on the horizon and so is not in force in the presence of a negative cosmological constant in vacuum but, as we will see, the PTC remains valid, and imposes constraints on the topology. \nIn Section III, we will prove that the mapping from the fundamental group of I to that of the domain of outer communications is a surjection for spacetimes satisfying the PTC. This theorem generalizes previous results on the DOC being simply connected in asymptotically flat spacetimes. This characterization allows us to shift attention from causal curves to arbitrary loops in further study of the consequences of the PTC. \nIn Section IV, by considering loops restricted to certain spacelike surfaces and using arguments from algebraic topology, we give a very direct derivation of the topology of black hole horizons and the integral homology of hypersurfaces in exterior regions. More precisely, we consider the topology of the closure of the Cauchy surfaces or analogues thereof for the DOC whose intersections with the horizons are closed 2-manifolds ('good cuts'). We find that \nk ∑ i =1 g i ≤ g 0 \nwhere the g i are the genera of the cuts of the black hole horizons and g 0 is the genus of a cut of I by the surface. Thus the topology of the black hole horizons is constrained by the topology at infinity. This result also pertains to any sub-domain of the DOC that lies to the future of a cut of I and whose Cauchy surfaces-or analogues thereofmeet the horizons at closed 2-manifolds. Therefore, it applies to the topology of the black hole horizons in the presence of black hole formation or evolution at times for which the appropriate sub-domain and surfaces can be found. Finally we demonstrate that the integral homology of these surfaces is torsion free and consequently completely determined by the Betti numbers. Furthermore, these are completely fixed in terms of the genera of the boundary. \nSection V contains a discussion concerning these results and their applicability to the case of non-stationary black holes.", 'II. Validity of the PTC': "The aim of this section is to present a version of the PTC applicable to spacetimes which are asymptotically locally anti-de Sitter. Hence, we consider a spacetime M , with metric g ab , which can be conformally included into a spacetime-with-boundary M ' = M∪I , with metric g ' ab , such that ∂ M ' = I is timelike ( i.e. , is a Lorentzian hypersurface in the induced metric) and M = M ' \\ I . We permit I to have multiple components. With regard to the conformal factor Ω ∈ C 1 ( M ' ), we make the standard assumptions that \n- (a) Ω > 0 and g ' ab = Ω 2 g ab on M , and\n- (b) Ω = 0 and d Ω /negationslash = 0 pointwise on I . \nJust as in the case of spacetimes without boundary, we say that a spacetimewith-boundary M ' is globally hyperbolic provided M ' is strongly causal and the sets J + ( p, M ' ) ∩ J -( q, M ' ) are compact for all p, q ∈ M ' . Note that when I is timelike, as in the situation considered here, M can never be globally hyperbolic. However in many examples of interest, such as anti-de Sitter space and the domain of outer communications of various locally adS spaces, M ' is. \nFor later convenience, we define a Cauchy surface for M ' to be a subset V ' ⊂ M ' which is met once and only once by each inextendible causal curve in M ' . Then V ' will be a spacelike hypersurface which, as a manifold-with-boundary, has boundary on I . It can be shown, as in the standard case, that a spacetime-with-timelike-boundary D ' which is globally hyperbolic admits a Cauchy surface V ' and is homeomorphic to R × V ' . (This can be shown by directly modifying the proof of Prop. 6.6.8 in [19]. Alternatively, arguments can be given which involve invoking Prop. 6.6.8. For example, one can consider the 'doubled spacetime' M '' of M ' through I , with metric on this double defined by the natural extension of that on M ' . Then one can apply Prop. 6.6.8 to M '' , which will be globally hyperbolic if M ' is.) Many of the locally anti-de Sitter and related models [8-11,13,14] which have been constructed have DOCs which admit Cauchy surfaces of this sort. \nThe proof of the PTC is a consequence of the following basic result. \nTheorem 2.1. Let M⊂M ' be as described above, and assume the following conditions hold. \n- ( i ) M ' = M∪I is globally hyperbolic.\n- ( ii ) There is a component I 0 of I which admits a compact spacelike cut.\n- ( iii ) For each point p in M near I 0 and any future complete null geodesic s → η ( s ) in M starting at p , ∫ ∞ 0 Ric( η ' , η ' ) ds ≥ 0 . \nThen I 0 cannot communicate with any other component of I , i.e. , J + ( I 0 ) ∩ ( I\\I 0 ) = ∅ . \nCondition ( iii ) is a modified form of the Average Null Energy Condition (ANEC). This term usually refers to a condition of the same form as ( iii ) except that the integral is taken over geodesics complete to both past and future. Note that if one assumes that the Einstein equations with cosmological constant R ab -1 2 Rg ab +Λ g ab = 8 πT ab hold, then for any null vector X, Ric( X,X ) = R ab X a X b = 8 πT ab X a X b . Then the integrand Ric( η ' , η ' ) in ( iii ) could be replaced by T ( η ' , η ' ). Clearly, the presence and sign of the cosmological constant is irrelevant to whether or not a spacetime satisfying the Einstein equations will satisfy condition ( iii ). \nFor the next theorem, if ∂ M ' is not connected, let I denote a single component of ∂ M ' . Let D = I + ( I ) ∩ I -( I ) be the domain of outer communications of M with respect to I . Assume that D does not meet any other components of ∂ M ' . Note that if the conditions of Theorem 2.1 hold, this latter assumption is automatically satisfied. Using in addition the fact that I is timelike, it follows that D is connected and the closure of D in M ' contains I . Then D ' := D ∪ I is a connected spacetime-with-boundary, with ∂ D ' = I and D = D ' \\ I . \nWe now state the following topological censorship theorem, applicable to asymptotically locally adS spacetimes. \nTheorem 2.2. Let D be the domain of outer communications with respect to I as described above, and assume the following conditions hold. \n- ( i ) D ' = D∪I is globally hyperbolic. \n- ( ii ) I admits a compact spacelike cut.\n- ( iii ) For each point p in M near I and any future complete null geodesic s → η ( s ) in D starting at p , ∫ ∞ 0 Ric( η ' , η ' ) ds ≥ 0 . \nThen the PTC holds on D . \nRemark. Let K be a cut of I , and let I K be the portion of I to the future of K , I K = I∩ I + ( K ). Let D K be the domain of outer of communications with respect to I K , D K = I + ( I K ) ∩ I -( I K ) = I + ( K ) ∩ I -( I ). Theorems 2.1 and 2.2 apply equally as well to D K . This procedure, first discussed by Jacobson and Venkataramani [7], allows one, by the methods of this paper, to study the topology of cuts on the future event horizon H = ∂I -( I ) of the form ∂I + ( I K ) ∩ H . By taking K sufficiently far to the future, this procedure enables one to consider cuts on H well to the future of the initial formation of the black hole, where one has a greater expectation that the intersection of H with ∂I + ( I K ) will be reasonable ( i.e. , a surface). See [7], and Sections 4 and 5 below for further discussion of this point. In what follows, it is worth keeping in mind that I may refer to the portion of scri to the future of a cut. \nProof of Theorem 2.1: The global hyperbolicity of M ' implies that I 0 is strongly causal as a spacetime in its own right, and that the sets J + ( x, I 0 ) ∩ J -( y, I 0 ) ( ⊂ J + ( x, M ' ) ∩ J -( y, M ' )) for x, y ∈ I 0 have compact closure in I 0 . This is sufficient to imply that I 0 is globally hyperbolic as a spacetime in its own right. Assumption ( ii ) then implies that I 0 is foliated by compact Cauchy surfaces. \nNow suppose that J + ( I 0 ) meets some other component I 1 of I , i.e. , suppose there exists a future directed causal curve from a point p ∈ I 0 to a point q ∈ I 1 . Let Σ 0 be a Cauchy surface for I 0 passing through p . Push Σ 0 slightly in the normal direction to I 0 to obtain a compact spacelike surface Σ contained in M . Let V be a compact spacelike hypersurface-with-boundary spanning Σ 0 and Σ. By properties of the conformal factor we are assured, for suitable pushes, that Σ is null mean convex . By this we mean that the future directed null geodesics issuing orthogonally from Σ which are 'inward pointing' with respect to I 0 ( i.e. , which point away from V ) have negative divergence. \nUnder the present supposition, J + ( V ) meets I 1 . At the same time, J + ( V ) cannot contain all of I 1 . If it did, there would exist a past inextendible causal curve in I 1 starting at q ∈ I 1 contained in the compact set J + ( V ) ∩ J -( q ), contradicting the strong causality of M ' . Hence, ∂J + ( V ) meets I 1 ; let q 0 be a point in ∂J + ( V ) ∩I 1 . Since M ' is globally hyperbolic, it is causally simple; cf. Proposition 6.6.1 in [19], which remains valid in the present setting. Hence, ∂J + ( V ) = J + ( V ) \\ I + ( V ), which implies that there exists a future directed null curve η ⊂ ∂J + ( V ) that extends from a point on V to q 0 . (Alternatively, one can prove the existence of this curve from results of [21], which are also valid in the present setting.) It is possible that η meets I several times before reaching q 0 . Consider only the portion η 0 of η which extends from the initial point of η on V up to, but not including, the first point at which η meets I . \nBy properties of achronal boundaries and the conformal factor, η 0 is a future complete null geodesic in M emanating from a point on V . Since η 0 cannot enter I + ( V ), it follows that: (1) η 0 actually meets V at a point p 0 of Σ, (2) η 0 meets Σ orthogonally at p 0 , and (3) η 0 is inward pointing with respect to I 0 ( i.e. , η 0 points away from V ). Since η 0 is future complete, the energy condition ( iii ) and null mean convexity of Σ imply that there is a null focal point to Σ along η 0 . But beyond the focal point, η 0 must enter I + (Σ) ( ⊂ I + ( V )), contradicting η 0 ⊂ ∂J + ( V ). \nProof of Theorem 2.2: The proof is an application of Theorem 2.1, together with a covering space argument. Fix p ∈ I . The inclusion map i : I → D ' induces a homomorphism of fundamental groups i ∗ : Π 1 ( I , p ) → Π 1 ( D ' , p ). The image G = i ∗ (Π 1 ( I , p )) is a subgroup of Π 1 ( D ' , p ). Basic covering space theory guarantees that there exists an essentially unique covering space ˜ D ' of D ' such that π ∗ (Π 1 ( ˜ D ' , ˜ p )) = G = i ∗ (Π 1 ( I , p )), where π : ˜ D ' →D ' is the covering map, and π ( ˜ p ) = p . Equip ˜ D ' with the pullback metric π ∗ ( g ' ) so that π : ˜ D ' →D ' is a local isometry. We note that when I is simply connected, ˜ D ' is the universal cover of D ' , but in general it will not be. In a somewhat different context ( i.e. , when D ' is a spacetime without boundary and I is a spacelike hypersurface), ˜ D ' is known as the Hawking covering spacetime, cf. [19], [22]. \nThe covering spacetime D ' has two basic properties: \n- (a) The component ˜ I of π -1 ( I ) which passes through ˜ p is a copy of I , i.e. , π ∗ | ˜ I : ˜ I → I is an isometry. \n˜ \n- (b) i ∗ (Π 1 ( ˜ I , ˜ p )) = Π 1 ( ˜ D ' , p ), where i : ˜ I ↪ → ˜ D ' . \n˜ \nNow let γ be a future directed causal curve in D ' with endpoints on I . Assume γ extends from x ∈ I to y ∈ I . Let ˜ γ be the lift of γ into ˜ D ' starting at ˜ x ∈ ˜ I . Note that the hypotheses of Theorem 2.1, with M = ˜ D := π -1 ( D ), M ' = ˜ D ' , and I 0 = ˜ I , are satisfied. Hence the future endpoint ˜ y of ˜ γ must also lie on ˜ I . Then we know that ˜ γ is fixed endpoint homotopic to a curve in ˜ I . Projecting this homotopy down to D ' , it follows that γ is fixed endpoint homotopic to a curve in I , as desired. \n˜ Property (b) says that any loop in ˜ D ' based at ˜ p can be deformed through loops based at ˜ p to a loop in ˜ I based at ˜ p . This property is an easy consequence of the defining property π ∗ (Π 1 ( ˜ D ' , ˜ p )) = i ∗ (Π 1 ( I , p )) and the homotopy lifting property. In turn, property (b) easily implies that any curve in ˜ D ' with endpoints on ˜ I is fixed endpoint homotopic to a curve in ˜ I . \nRemarks. Since many examples of locally adS spacetimes, for example those constructed by identifications of adS [9-11,13,14], do not obey the generic condition, our aim was to present a version of the PTC which does not require it. However if one replaces the energy condition ( iii ) by the generic condition and the ANEC: ∫ ∞ -∞ Ric( η ' , η ' ) ds ≥ 0 along any complete null geodesic s → η ( s ) in D , then Theorems 2.1 and 2.2 still hold, and moreover they do not require the compactness condition ( ii ). The proof of Theorem 2.1 under these new assumptions involves the construction of a complete null line (globally achronal null geodesic) which is incompatible with the energy conditions. The results in this setting are rather general, and in particular the proofs do not use in any essential way that I is timelike. We also mention that the global hyperbolicity assumption in Theorem 2.2 can be weakened in a fashion similar to what was done in the AF case in [16]. \nWe now turn our attention to a characterization of topological censorship which \nwill allow us to more easily explore the consequences of topological censorship for black hole topology.", 'III. Algebraic Characterization of Topological Censorship': "The main result of this section is a restatement of the properties of a spacetime satisfying the PTC in a language amenable to algebraic topological considerations. \nLet D and I be as in Section I. Then D ' = D ∪ I is a spacetime-with-boundary, with ∂ D ' = I . The inclusion map i : I → D ' induces a homomorphism of fundamental groups i ∗ : Π 1 ( I ) → Π 1 ( D ' ). Then \nProposition 3.1. If the PTC holds for D ' , then the group homomorphism i ∗ : Π 1 ( I ) → Π 1 ( D ' ) induced by inclusion is surjective. \nRemark. Note that the fundamental groups of D and D ' are trivially isomorphic, Π 1 ( D ) ∼ =Π 1 ( D ' ). Hence, Proposition 3.1 says roughly that every loop in D is deformable to a loop in I . Moreover, it implies that Π 1 ( D ) is isomorphic to the factor group Π 1 ( I ) / ker i ∗ . In particular, if I is simply connected then so is D . \nOur proof relies on the following straightforward lemma in topology: \nLemma 3.2. Let N be a manifold and S an embedded submanifold with inclusion mapping i : S → N . If in the universal covering space of N the inverse image of S by the covering map is connected, then i ∗ : Π 1 ( S ) → Π 1 ( N ) is surjective. \nProof: Strictly speaking, we are dealing with pointed spaces ( S, p ) and ( N,p ), for some fixed point p ∈ S , and we want to show that i ∗ : Π 1 ( S, p ) → Π 1 ( N,p ) is onto. Let [ c 0 ] be an element of Π 1 ( N,p ), i.e. , let c 0 be a loop in N based at p . Let ˜ N be the universal covering space of N , with covering map π : ˜ N → N . Choose ˜ p ∈ ˜ N such that π (˜ p ) = p . Let ˜ c 0 be the lift of c 0 starting at ˜ p ; then ˜ c 0 is a curve in ˜ N extending from ˜ p to a point ˜ q with π (˜ q ) = p . Since ˜ p, ˜ q ∈ π -1 ( S ) and π -1 ( S ) is path connected, there exists a curve ˜ c 1 in π -1 ( S ) from ˜ p to ˜ q . The projected curve c 1 = π (˜ c 1 ) is a loop in S based at p , i.e. , [ c 1 ] ∈ Π 1 ( S, p ). Since ˜ N is simply connected, ˜ c 0 is fixed endpoint homotopic to ˜ c 1 . But \nthis implies that c 0 is homotopic to c 1 through loops based at p , i.e. , i ∗ ([ c 1 ]) = [ c 0 ], as desired. \nProof of Proposition 3.1: We let ˜ N be the universal covering spacetime of N := D ' with projection π : ˜ N → N . Then ˜ N = ˜ D ∪ ˜ I , where ˜ D = π -1 ( D ) is the universal covering spacetime of D and ˜ I = π -1 ( I ) is the boundary of ˜ N . \nEvery point in ˜ D belongs to the inverse image by π of some point in D , and every point in D lies on some causal curve beginning and ending on I , so every point in ˜ D lies on some causal curve beginning and ending on ˜ I . By the PTC and basic lifting properties, no such curve can end on a different connected component of ˜ I than it began on. Therefore, if we label these connected components by α , the open sets I + ( ˜ I α ) ∩ I -( ˜ I α ) are a disjoint open cover of ˜ D . But ˜ D is connected, so α can take only one value, whence ˜ I is connected. It follows that D ' satisfies the conditions of Lemma 3.2 with S = I and therefore i ∗ : Π 1 ( I ) → Π 1 ( D ' ) is surjective. Thus i ∗ : Π 1 ( I ) → Π 1 ( D ) is surjective. \nAll known locally anti-de Sitter black holes and related spacetimes are in accord with Proposition 3.1. We conclude this section with the following simple corollary to Proposition 3.1. \nCorollary 3.3. If the PTC holds for D ' then D is orientable if I is. \nProof: In fact D ' is orientable, for if D ' were not orientable and I were, then D ' would possess an orientable double cover containing two copies I 1 and I 2 of I . Then a curve from p 1 ∈ I 1 to p 2 ∈ I 2 , where π ( p 1 ) = π ( p 2 ) = p ∈ I , would project to a loop in D ' not deformable to I , contrary to the surjectivity of i ∗ .", 'IV. Application to Black Hole Topology': "The boundary of the region of spacetime visible to observers at I by future directed causal curves is referred to as the event horizon. This horizon is a set of one or more null surfaces, also called black hole horizons, generated by null geodesics that have no future endpoints but possibly have past endpoints. The topology of these black \nhole horizons is constrained in spacetimes obeying the PTC because, as was seen in Section III, the topology of the domain of outer communications is constrained by the PTC-intuitively, causal curves that can communicate with observers at I cannot link with these horizons in a non-trivial way, rather they only carry information about the non-triviality of curves on I . \nUseful though this description is, it does not characterize the topology of these horizons themselves, but rather the hole that their excision leaves in the spacetime. However, one can obtain certain information about these horizons if one considers the topology of the intersection of certain spacelike hypersurfaces with the horizons, those whose intersection with the horizons are closed spacelike 2-manifolds (good cuts of the horizons). For example, if one has a single horizon of product form, as is the case for a stationary black hole, then the horizon topology is determined by that of the 2-manifold. Of course, black hole horizons are generally not of product form; however, the topology of any good cut is still closely related to that of the horizons. For example, if each horizon has a region with product topology, this topology is determined by that of a good cut passing through this region. As demonstrated by Jacobson and Venkataramani [7] for asymptotically flat spacetimes, the PTC constrains the topology of good cuts of the horizons by the closure of a Cauchy surface for the domain of outer communications to be 2-spheres. \nNow it is possible that a black hole horizon admits two good cuts by spacelike hypersurfaces, one entirely to the future of the other, such that the cuts are not homeomorphic. This can only happen if null generators enter the horizon between the two cuts. Physically, such a situation corresponds to a black hole with transient behavior, such as that induced by formation from collapse, collision of black holes or absorption of matter. Jacobson and Venkataramani observed that their theorem could also be applied to such situations for certain spacelike hypersurfaces that cut the horizons sufficiently far away from regions of black hole formation, collision or matter absorption. Precisely, when the domain of outer communications is globally hyperbolic to the future of a cut of I and \nif the PTC holds on this sub-domain, then the PTC will constrain the topology of this sub-domain, of its Cauchy surface and ultimately of good cuts of the horizons. Thus, though the PTC does not determine the topology of arbitrary embedded hypersurfaces or the cuts they make on the horizons, it does do so for hypersurfaces homeomorphic to Cauchy surfaces for these sub-domains that make good cuts of the horizons. Such sub-domains can be found for black hole horizons that settle down at late times. \nBelow we provide a generalization of these results applicable to a more general set of spacetimes satisfying the PTC than asymptotically flat spacetimes. This generalization is based on the observation that if one can continuously push any loop in the DOC down into an appropriate spacelike surface that cuts the horizons in spacelike 2-manifolds, it follows from Prop. 3.1 that the fundamental group of the spacelike surface is related to that of I . In the interests of a clear presentation that carefully treats all technical details, we first prove the following results for globally hyperbolic spacetimes with timelike I 8 and, we emphasize, for any globally hyperbolic sub-domain corresponding to the future of a cut of I . We then provide the results for the case of globally hyperbolic, asymptotically flat spacetimes. From the method of proof of these two cases, it is manifestly apparent that these theorems can be easily generalized to a wider class of spacetimes that satisfy the PTC. We conclude with a remark about how to provide the correct technical statement and proofs for these cases. We will also comment further on the transient behavior associated with black hole formation in the Discussion section. \nLet M be a spacetime with timelike infinity I and domain of outer communications D . Assume I is connected and orientable. Let K be a spacelike cut of I , and let I K be the portion of I to the future of K , I K = I ∩ I + ( K ). Let D K be the DOC with respect to I K , D K = I + ( I K ) ∩ I -( I K ) = I + ( K ) ∩ I -( I ). Note D ' K := D K ∪ I K is a connected spacetime-with-timelike-boundary, with ∂ D ' K = I K and D K = D ' K \\ I K . In the following we will assume that D ' K is globally hyperbolic and has a Cauchy surface \nV ' as in section II. The following theorem is the main result pertaining to the topology of black holes. \nTheorem 4.1. Let D K be the domain of outer communications to the future of the cut K on I as described above. Assume D ' K is globally hyperbolic and satisfies the PTC. Suppose V ' is a Cauchy surface for D ' K such that its closure V = V ' in M ' is a compact topological 3 -manifold-with-boundary whose boundary ∂V (corresponding to the edge of V ' in M ' ) consists of a disjoint union of compact 2-surfaces, \n∂V = k ⊔ i =0 Σ i , (2) \nwhere Σ 0 is on I and the Σ i , i = 1 , . . . , k , are on the event horizon. Then \nk ∑ i =1 g i ≤ g 0 , (3) \nwhere g j = the genus of Σ j , j = 0 , 1 , . . . , k . In particular, if Σ 0 is a 2 -sphere then so is each Σ i , i = 1 , . . . , k . \nRemark 1. Many known examples of locally adS black hole spacetimes have black hole horizons with genus equal of that of scri [9-11,13]. In fact, for these examples, V is a product space. \nRemark 2. Theorem 4.1 has been stated for spacetimes with timelike I . An analogous version for asymptotically flat spacetimes holds as well, but differs slightly in technical details. The AF case will be considered later in the section. \nTheorem 4.1 is established in a series of lemmas. Lemma 4.2 connects the fundamental group of the Cauchy surface to that of I . It is the only lemma that uses the conditions that the spacetime is globally hyperbolic and satisfies the PTC. The remaining lemmas are based purely on algebraic topology and are in fact applicable to any 3-manifold with compact boundary. \nLemma 4.2. Let the setting be as in Theorem 4.1. Then the group homomorphism i ∗ : Π 1 (Σ 0 ) → Π 1 ( V ) induced by inclusion i : Σ 0 → V is onto. \nProof: Let Z a be a timelike vector field on D ' K tangent to I K . Let r : D ' K → V ' be the continuous projection map sending each point p ∈ D ' K to the unique point in V ' determined by the integral curve of Z a through p . Note that r ( I K ) = Σ 0 . \nFix p ∈ Σ 0 . All loops considered are based at p . Let c 0 be a loop in V . By deforming c 0 slightly we can assume c 0 is in V ' . Since D ' K satisifies the hypotheses of Theorem 2.2, the PTC holds for D ' K . Then, by Proposition 3.1, c 0 can be continuously deformed through loops in D ' K to a loop c 1 in I K . It follows by composition with r that c 0 = r · c 0 can be continuously deformed through loops in V to the loop c 2 = r · c 1 in Σ 0 , and hence i ∗ ([ c 2 ]) = [ c 0 ], as desired. \nBy Corollary 3.3 and the assumptions of Theorem 4.1, V is a connected, orientable, compact 3-manifold-with-boundary whose boundary consists of k +1 compact surfaces Σ i , i = 0 , ..., k . For the following results, which are purely topological, we assume V is any such manifold. \nAll homology and cohomology below is taken over the integers. The j th homology group of a manifold P will be denoted by H j ( P ), b j ( P ) := the rank of the free part of H j ( P ) will denote the j th Betti number of P , and χ ( P ) will denote the Euler characteristic (the alternating sum of the Betti numbers). In the case of b j ( V ) (= b j ( V 0 )), we will simply write b j . \nIn general, for V as assumed above, b 0 = 1, b 3 = 0, and b 1 and b 2 satisfy the following inequalities: \nLemma 4.3. \nFor V \nas above, then \n(a) b 1 ≥ k ∑ i =0 g i , and (b) b 2 ≥ k . \nProof: The boundary surfaces Σ 1 , Σ 2 , . . . , Σ k clearly determine k linearly independent 2-cycles in V , and hence (b) holds. To prove (a) we use the formula, χ ( V ) = 1 2 χ ( ∂V ), valid for any compact, orientable, odd-dimensional manifold. This formula, together with the expressions, χ ( V ) = 1 -b 1 + b 2 and χ ( ∂V ) = k ∑ i =0 χ (Σ i ) = 2( k +1) -2 k ∑ i =0 g i , \nimplies the equation \nb 1 = b 2 + k ∑ i =0 g i -k . (4) \nThe inequality (a) now follows immediately from (b). \nLemma 4.4. If the group homomorphism i ∗ : Π 1 (Σ 0 ) → Π 1 ( V ) induced by inclusion is onto then the inequality (3), ∑ k i =1 g i ≤ g 0 , holds. In particular, if Σ 0 is a 2 -sphere then so is each Σ i , i = 1 , ..., k . \nProof: We use the fact that the first integral homology group of a space is isomorphic to the fundamental group modded out by its commutator subgroup. Hence, modding out by the commutator subgroups of Π 1 (Σ 0 ) and Π 1 ( V ), respectively, induces from i ∗ a surjective homomorphism from H 1 (Σ 0 ) to H 1 ( V ). It follows that the rank of the free part of H 1 ( V ) cannot be greater than that of H 1 (Σ 0 ), i.e. , \nb 1 ≤ b 1 (Σ 0 ) = 2 g 0 . (5) \nCombining this inequality with the inequality (a) in Lemma 4.3 yields the inequality (3). Since V is orientable, so are its boundary components. If Σ 0 is a 2-sphere, then each Σ i , i = 1 , . . . , k , is forced by (3) to have genus zero, and hence is a 2-sphere.", 'Proof of Theorem 4.1: Follows immediately from Lemma 4.2 and Lemma 4.4.': 'Next we, show that the condition on i ∗ in Lemma 4.4 completely determines the homology of V . \nProposition 4.5. If i ∗ : Π 1 (Σ 0 ) → Π 1 ( V ) is onto, then the integral homology H ∗ ( V, Z ) is torsion free, and hence is completely determined by the Betti numbers. Furthermore, the inequalities in Lemma 4.3 become equalities, \n(a) b 1 = k ∑ i =0 g i , and (b) b 2 = k . \nProof: We first prove that H ∗ ( V, Z ) is torsion free. Since V has boundary, H 3 ( V ) = 0. Also H 0 ( V ) is one copy of Z as V is connected. Thus, we need to show H 1 ( V ) and H 2 ( V ) are free.', 'Claim 1. H 2 ( V ) is free.': "To prove Claim 1 we recall the classic result that H n -1 ( N n ) is free for an orientable closed n-manifold N n . To make use of this, let V ' be a compact orientable 3-manifold without boundary containing V ( e.g. , take V ' to be the double of V ), and let B = V ' \\ V . \nAssume that W is a non-trivial torsion element in H 2 ( V ). Now view W as an element in H 2 ( V ' ). Suppose W = 0 in H 2 ( V ' ). Then W = 0 in H 2 ( V ' , B ). By excision, H 2 ( V ' , B ) = H 2 ( V, ∂V ), where ∂V is the manifold boundary of V . Hence W = 0 in H 2 ( V, ∂V ). This means that W = a sum of boundary components in H 2 ( V ). But a sum of boundary components cannot be a torsion element. Thus, W /negationslash = 0 in H 2 ( V ' ). Moreover, if nW = 0 in H 2 ( V ) then nW = 0 in H 2 ( V ' ). It follows that W is a non-trivial torsion element in H 2 ( V ' ), a contradiction. Hence, H 2 ( V ) is free.", 'Claim 2. H 1 ( V ) is free.': "To prove Claim 2 we first consider the relative homology sequence for the pair V ⊃ Σ 0 , \n· · · → H 1 (Σ 0 ) α → H 1 ( V ) β → H 1 ( V, Σ 0 ) ∂ → ˜ H 0 (Σ 0 ) = 0 . (6) \n(Here ˜ H 0 (Σ 0 ) is the reduced zeroeth-dimensional homology group.) Since, as discussed in Lemma 4.4, α is onto, we have ker β = im α = H 1 ( V ) which implies β ≡ 0. Hence ker ∂ = im β = 0, and thus ∂ is injective. This implies that H 1 ( V, Σ 0 ) = 0. \nNow consider the relative homology sequence for the triple V ⊃ ∂V ⊃ Σ 0 , \n· · · → H 1 ( ∂V, Σ 0 ) → H 1 ( V, Σ 0 ) = 0 → H 1 ( V, ∂V ) ∂ → H 0 ( ∂V, Σ 0 ) →··· . (7) \nSince H 0 ( ∂V, Σ 0 ) is torsion free and ∂ is injective, H 1 ( V, ∂V ) is torsion free. Next, Poincar'e-Lefschetz duality gives H 2 ( V ) ∼ = H 1 ( V, ∂V ). Hence H 2 ( V ) is torsion free. The universal coefficient theorem implies that \nH 2 ( V ) ∼ = Hom( H 2 ( V ) , Z ) ⊕ Ext( H 1 ( V ) , Z ) . (8) \nThe functor Ext( -, -) is bilinear in the first argument with respect to direct sums and Ext(Z k , Z) = Z k . Hence H 2 ( V ) cannot be torsion free unless H 1 ( V ) is. This completes the proof of Claim 2 and the proof that H ∗ ( V ) is torsion free. \nIt remains to show that the inequalities in Lemma 4.3 become equalities. We prove b 2 = k ; the equation b 1 = ∑ k i =0 g i then follows from equation (4). In view of Lemma 4.3, it is sufficient to show that b 2 ≤ k . Since H 2 ( V ) is finitely generated and torsion free, we have H 2 ( V ) ∼ = H 2 ( V ) ∼ = H 1 ( V, ∂V ), where we have again made use of Poincar'e-Lefschetz duality. Hence, b 2 = rank H 1 ( V, ∂V ). To show that rank H 1 ( V, ∂V ) ≤ k , we refer again to the long exact sequence (7). By excision, H 0 ( ∂V, Σ 0 ) ∼ = H 0 ( ∂V \\ Σ 0 , ∅ ) = H 0 ( ∂V \\ Σ 0 ). Hence, by the injectivity of ∂ , rank H 1 ( V, ∂V ) ≤ rank H 0 ( ∂V, Σ 0 ) = the number of components of ∂V \\ Σ 0 = k . This completes the proof of Proposition 4.5. \nThe conclusion of Proposition 4.5 applies to the spacelike 3-surface-with-boundary V of Theorem 4.1. Thus, we have completely determined the homology of the Cauchy surfaces of D ' K . \nWe now consider the asymptotically flat case with null infinity I = I + ∪ I -. For this case, let K be a spacelike cut of I -, and let I K be the portion of I to the future of K , I K = I ∩ J + ( K ). Let D K be the domain of outer of communications with respect to I K , D K = I + ( I K ) ∩ I -( I K ) = I + ( K ) ∩ I -( I + ). D ' K := D K ∪ I + is a connected spacetime-with-boundary, with ∂ D ' K = I + and D K = D ' K \\ I + . \nWe then have the following analogue of Theorem 4.1. \nTheorem 4.1 ' . Let D K be the domain of outer communications to the future of the cut K on I -of an asymptotically flat spacetime M as described above. Assume D ' K is globally hyperbolic and satisfies the PTC. Suppose V 0 is a Cauchy surface for D K such that its closure V = V 0 in M is a topological 3 -manifold-with-boundary, compact outside a small neighborhood of i 0 , with boundary components consisting of a disjoint union of compact 2 -surfaces, \n∂V = k ⊔ i =1 Σ i , \nwhere the Σ i , i = 1 , . . . , k , are on the event horizon. Then all Σ i , i = 1 , . . . , k are 2 -spheres. Moreover, V 0 has the topology of a homotopy 3 -sphere minus k +1 closed 3 -balls. \nRemark 1. In the AF case the asymptotic topology is spherical, which corresponds to g 0 = 0 in inequality (3). But since g i = 0, i = 1 , ..., k , inequality (3) is satisfied in the AF case, as well. Again, the topology of the event horizon is constrained by the topology at infinity. \nRemark 2. This theorem is a slightly strengthened version of the main theorem in [7]; it does not assume orientability of V 0 and we conclude a stronger topology for this Cauchy surface. \nProof of Theorem 4.1 ' : The arguments used to prove Theorem 4.1 can be easily adapted, with only minor technical changes involved, to prove in the AF case that each Σ i is a 2-sphere. Alternatively, one may argue as follows. By known results on topological censorship in the AF case ([5], [6]), D K is simply connected, and hence so is V 0 . It follows that ˜ V = V ∪ { i 0 } is a compact simply connected 3-manifold-withboundary, with boundary components Σ i , i = 1 , ..., k . Then, according to Lemma 4.9, p. 47 in Hempel [23], each Σ i is a 2-sphere. By attaching 3-cells to each Σ i we obtain a closed simply connected 3-manifold, which by well-known results (see [23]) is a homotopy 3-sphere. Removing the attached 3-cells and i 0 we obtain that V 0 is a homotopy 3-sphere minus k +1 closed balls. \nRemark. Although the above results were proved assuming global hyperbolicity, it is clear that the same results will hold for a more general set of spacetimes that satisfy the PTC and for which a version of Lemma 4.2 can be proved. Spacetimes that are not globally hyperbolic but satisfy a weaker condition such as weak cosmic censorship can still admit a projection onto a preferred spacelike surface. In particular, one can generalize the projection given by the integral flow of a timelike vector field on the domain of outer communications used to push loops into the Cauchy surface to be a retract. Recall a retract of X onto a subspace A is a continuous map r : X → A such that r | A = id. Thus, if V 0 is a regular retract of D , that is, if there exists a retract r : D∪I → V 0 ∪ Σ 0 such that r ( I ) ⊆ Σ 0 , then one can again establish Lemma 4.2.", 'V. Discussion': "We wish to emphasize that the results concerning black hole topology obtained in Section IV in no way contradict the numerical findings of [24] concerning the existence in principle of temporarily toroidal black holes in asymptotically flat spacetimes. The consistency of topological censorship with asymptotically flat models containing temporarily toroidal black hole horizons has been clearly elucidated in [25]. The acausal nature of cross-over sets, expected to be present in the early formation of the event horizon, permits slicings of the event horizon in asymptotically flat black hole spacetimes with exotic ( i.e. , non-spherical) topologies. See the recent papers [26,27] for further discussion. As described in Section IV, the method of topological censorship for exploring the topology of black hole horizons makes use of specific time slices, namely, Cauchy surfaces for the DOC or for the sub-region of the DOC to the future of a cut on I . Surfaces exhibiting temporarily toroidal black hole horizons are not such surfaces. Moreover, the method requires such a slice to have non-empty edge which meets the horizons in C 0 compact surfaces. We elaborate further on these points below. \nIt is important to keep in mind that not all Cauchy surfaces V 0 for the DOC are interiors of orientable manifolds with boundary V corresponding to the intersection of a spacelike slice with the black hole horizons. Consider the t = 0 slice of the RP 3 geon. As discussed in [5], this spacetime is constructed from the t = 0 slice of Schwarzschild spacetime by identifying antipodal points at the throat r = 2 M . The maximal evolution of this slice is a spacetime with spatial topology RP 3 -pt . Its universal covering space is the maximally extended Schwarzschild spacetime. \nThe t = 0 slice of the RP 3 geon contains a non-orientable RP 2 with zero expansion. This RP 2 is not a trapped surface as it does not separate the slice into two regions. It is not part of the DOC as any radially outward directed null geodesic from this surface does not intersect I ; thus it is clearly part of the horizon. The intersection of the DOC with the t = 0 slice produces a simply connected V 0 . The intersection of the horizon with this slice is RP 2 . However, we cannot attach this surface to V 0 to produce a manifold \nwith boundary V by the inclusion map; instead, this map reproduces the original t = 0 slice which has no interior boundary. Note, however, that any spacelike slice that does not pass through this RP 2 will intersect the horizon at an S 2 . In fact, this will be the generic situation. Moreover, the intersection of such a slice with the DOC will produce a simply connected V 0 which is the interior of a closed connected orientable V with an S 2 interior boundary. \nClearly this example does not contradict any results of Section IV, which assumes an orientable V with two or more boundaries. However, it does yield the important lesson that one must construct V to apply the theorem, not V 0 . It also gives an example of a badly behaved cut of the horizon, again illustrating the usefulness of taking slices to the future of a cut of I . \nFor our second example, we construct a toy model of a black hole spacetime that mimics a special case of topology change, namely that of black hole formation from a single collapse. This model illustrates several features; how a choice of a hypersurface can affect the description of horizon topology and how some cuts of I give rise to a Cauchy surface whose edge on the horizon is not a 2-manifold. \nWe begin with a 3-dimensional model; later a 4-dimensional example will be constructed by treating the 3-dimensional model as a hyperplane through an axis of symmetry in the larger spacetime. Our spacetime can have either anti-de Sitter or flat geometry; as both are conformal to regions of the Einstein static cylinder, we use as coordinates in the construction below those of the conformally related flat metric ( cf. [19], sections 5.1 and 5.2). We depict I as timelike in the accompanying figures, but it can equally well be null. \nWe begin in 3 dimensions with a line segment L defined by t = y = 0, | x | ≤ l . The future I + ( L ) of this line segment is a sort of elongated cone, whose traces in hyperplanes of constant t have the shape of rectangles with semi-circular caps attached to the two short sides. We foliate the spacetime by hyperboloids, \nt = a + √ r 2 + b , (9) \nwhere r 2 = x 2 + y 2 and a, b are conveniently chosen parameters. These hyperboloids cut I in a circle. We now remove from spacetime all points of J + ( L ) above a hyperplane t = d intersecting it to the future. What remains is a black hole spacetime and has a globally hyperbolic domain of outer communications D (again, cf. Section II). The black hole is the set of remaining points of I + ( L ). The horizon ∂I -( I + ) is generated by null geodesics that all begin on L . \nThe Cauchy surface for D will have topology R 2 and does not cross the horizon. Thus, to probe the topology of the horizon, one needs to consider spacetimes corresponding to the future of a cut of I . However, not every cut of I will produce a spacetime with a Cauchy slicing with the correct properties. Such a bad cut of I is illustrated in Figure 1. The boundary of the causal future of this cut intersects the horizon at a segment I of L . The topology of a Cauchy slice for its DOC is R 2 \\ I . Its closure intersects the horizon at I ; thus, as in the RP 3 geon above, the closure of this slice has no inner boundary, being in this case R 2 . \nThis 3-d spacetime corresponds to a 4-d axisymmetric spacetime. The correspondence between axisymmetric spatial hypersurfaces and the xy planes of the spacetime is generated by rotating each xy plane about the y axis. After this rotation, one sees that the line segment L becomes a disk in the 4-d axisymmetric spacetime. The Cauchy surface in question meets the horizon in a disk (a closed 2-ball) and has topology R 3 minus that disk. The closure of the Cauchy surface is R 3 and again has no internal boundary. Thus the results of Section IV do not apply to this Cauchy slice. \nA good cut of I is illustrated in Figure 2. This cut intersects the horizon at a sphere. The topology of a Cauchy surface for its DOC is R 2 \\ B 2 and the intersection of its closure is S 1 . The Cauchy surface in the corresponding 4-dimensional model has topology R 3 \\ B 3 with internal boundary S 2 . The results of Section IV clearly apply in the latter case. \nOf course, not all spacelike surfaces need be Cauchy surfaces of the spacetime to the future of a cut of I . The family of hyperboloids (9), two of which are illustrated \nin Figure 3, provides an example of such surfaces. As recognized in [25] in a similar model, this family exhibits formation of a temporarily toroidal black hole horizon as the parameter a in (9) increases; these surfaces intersect the horizon in a pair of topological circles, which by axial symmetry correspond to a toroidal horizon in the 4-dimensional spacetime. The circles increase in size and eventually meet, whence the horizon topology changes. After this point, these surfaces meet the horizon at a circle, corresponding to a sphere by axial symmetry. \nIn contrast, with respect to constantt surfaces, the horizon forms completely at the t = 0 instant. For every t > 0 hypersurface, the black hole has spherical topology, and inequality (3) holds. Any t = t 0 > 0 hypersurface is a Cauchy surface for the region of D that lies in the future of an appropriate cut of I . The apparent change of horizon topology from toroidal to spherical was an effect entirely dependent on the choice of hypersurface. The only unambiguous description of this black hole is that no causal curve was able to link with the horizon; i.e. , that the PTC was not violated. \nIn the introduction, we offered the view that the topology of the boundary at infinity constrained that of the horizons, but one could equally well reverse this picture. Let us contemplate a black hole considered as a stationary, causally well-behaved, isolated system cut off from the Universe by a sufficiently distant boundaryI . Then we have shown that topological censorship requires the genus of the horizon to be a lower bound for that of the boundary. As remarked above, this seems an intuitive result; as illustrated in Figure 4, when one visualizes placing a genus g 1 surface within a genus g 2 'box' with no possibility of entangling curves, it seems clear that g 2 ≥ g 1 . Yet, as is often the case with such things, the powerful machinery of algebraic topology was required to prove it. An advantage of using this powerful tool is that we were able to completely specify the homology of well-behaved exterior regions of black holes and, in virtue of Proposition 4.5, to say that all interesting homology of these exteriors, save that which is reflected in the topology of scri, is directly attributable to the presence of horizons.", 'Acknowledgements': "EW wishes to acknowledge conversations with R.B. Mann and W. Smith concerning locally anti-de Sitter spacetimes. GJG and EW thank Piotr Chru'sciel for conversations concerning the validity of topological censorship for these spacetimes. This work was partially supported by the Natural Sciences and Engineering Research Council of Canada and by the National Science Foundation (USA), Grant No. DMS-9803566.", 'References': "- [1] S.W. Hawking, Commun. Math. Phys. 25 (1972), 152.\n- [2] G.J. Galloway, in Differential Geometry and Mathematical Physics , Contemporary Math. 170 , ed. J. Beem and K. Duggal (AMS, Providence, 1994), 113.\n- [3] P.T. Chru'sciel and R.M. Wald, Class. Quantum Gravit. 11 (1994), L147.\n- [4] S.F. Browdy and G.J. Galloway, J. Math. Phys. 36 (1995), 4952.\n- [5] J.L. Friedman, K. Schleich, and D.M. Witt, Phys. Rev. Lett. 71 (1993), 1486.\n- [6] G.J. Galloway, Class. Quantum Gravit. 12 (1995), L99.\n- [7] T. Jacobson and S. Venkataramani, Class. Quantum Gravit. 12 (1995), 1055.\n- [8] J.P.S. Lemos, Phys. Lett. B 352 (1995), 46.\n- [9] M. Ba˜nados, C. Teitelboim, and J. Zanelli, Phys. Rev. Lett. 69 (1992), 1849; M. Ba˜nados, M. Henneaux, C. Teitelboim, and J. Zanelli, Phys. Rev. D 48 (1993), 1506.\n- [10] S. ˚ Aminneborg, I. Bengtsson, S. Holst, and P. Peld'an, Class. Quantum Gravit. 13 (1996), 2707.\n- [11] R.B. Mann, Class. Quantum Gravit. 14 (1997), 2927\n- 12 W.L. Smith and R.B. Mann, Phys. Rev. D 56 (1997), 4942.\n- [13] D.R. Brill, J. Louko, and P. Peld'an, Phys. Rev. D 56 (1997), 3600.\n- [14] R.B. Mann, in Internal Structure of Black Holes and Spacetime Singularities , eds. L. Burko and A. Ori, Ann. Israeli Phys. Soc. 13 (1998), 311.\n- [15] G.J. Galloway, Class. Quantum Gravit. 13 (1996), 1471. \n- [16] G.J. Galloway and E. Woolgar, Class. Quantum Gravit. 14 (1997), L1.\n- [17] S. ˚ Aminneborg, I. Bengtsson, D. Brill, S. Holst, and P. Peld'an, Class. Quantum Gravit. 15 (1998), 627.\n- [18] L. Vanzo, Phys. Rev. D 56 (1997), 6475.\n- [19] S.W. Hawking and G.F.R. Ellis, The Large Scale Structure of Space-Time , (Cambridge University Press, Cambridge, 1973).\n- [20] E. Woolgar, in progress.\n- [21] R.D. Sorkin and E. Woolgar, Class. Quantum Gravit. 13 (1996), 1971.\n- [22] B.C. Haggman, G.W. Horndeski, and G. Mess, J. Math. Phys. 21 (1980), 2412.\n- [23] J. Hempel, 3-Manifolds , (Princeton University Press, Princeton, 1976).\n- [24] S.A. Hughes et al., Phys. Rev. D 49 (1994), 4004.\n- [25] S.L. Shapiro, S.A. Teukolsky, and J. Winicour, Phys. Rev. D 52 (1995), 6982.\n- [26] S. Husa and J. Winicour, preprint gr-qc/9905039.\n- [27] M. Siino, Phys. Rev. D 58 (1998), 104016; D 59 (1999), 064006.", 'Figures': 'Figure 1. A bad cut of the horizon. The surface V K is the Cauchy surface for the domain of outer communications of the spacetime to the future of the cut K of I . The inner boundary of the causal future of K intersects the horizon at a spacelike line segment. Consequently the closure of V K has topology R 2 . \n<!-- image --> \nFigure 2. A good cut of the horizon. The surface V K is the Cauchy surface for the domain of outer communications of the spacetime to the future of the cut K of I . The intersection of the inner boundary of the causal future of K with the horizon is now S 1 . Consequently the closure of V K intersects the horizon at S 1 . \n<!-- image --> \n<!-- image --> \nFigure 3. Two slicings of the horizon by hyperboloids. Both illustrations concentrate on the region near the horizon. The top illustration is of a hyperboloid that intersects the horizon at two topological circles. The bottom illustration is of a hyperboloid that lies to the future of the first. It intersects the horizon at one topological circle. \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFigure 4. Entanglement and non-entanglement of curves on black hole horizons. Each illustration displays a cross-section of a cut of I . The cut-away reveals a cut of a black hole horizon inside. In the top illustration, the cut of I has genus zero, that of the horizon has genus 1, and, as illustrated, there are curves not deformable to I . In the middle illustration, the genus of the cut of I and that of the horizon are both 2, and they are linked in such a manner that every curve is deformable to I . In the illustration at bottom, the genus of the cut of I exceeds that of the horizon, and again every curve is deformable to I . \n<!-- image -->'}

2009ApJ...692..917M

Mergers of Stellar-Mass Black Holes in Nuclear Star Clusters

2009-01-01

['black hole physics', 'galaxies nuclei', 'gravitational waves', 'relativity', 'astrophysics']

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Mergers between stellar-mass black holes (BHs) will be key sources of gravitational radiation for ground-based detectors. However, the rates of these events are highly uncertain, given that such systems are invisible. One formation scenario involves mergers in field binaries, where our lack of complete understanding of common envelopes and the distribution of supernova kicks has led to rate estimates that range over a factor of several hundreds. A different, and highly promising, channel involves multiple encounters of binaries in globular clusters or young star clusters. However, we currently lack solid evidence of BHs in almost all such clusters, and their low escape speeds raise the possibility that most are ejected because of supernova recoil. Here, we propose that a robust environment for mergers could be the nuclear star clusters found in the centers of small galaxies. These clusters have millions of stars, BH relaxation times well under a Hubble time, and escape speeds that are several times those of globulars; hence, they retain most of their BHs. We present simulations of the three-body dynamics of BHs in this environment and estimate that, if most nuclear star clusters do not have supermassive BHs that interfere with the mergers, tens of events per year will be detectable with the advanced Laser Interferometer Gravitational-Wave Observatory.

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https://arxiv.org/pdf/0804.2783.pdf

{'Mergers of Stellar-Mass Black Holes in Nuclear Star Clusters': 'M. Coleman Miller and Vanessa M. Lauburg \nUniversity of Maryland, Department of Astronomy, College Park, Maryland 20742-2421', 'ABSTRACT': 'Mergers between stellar-mass black holes will be key sources of gravitational radiation for ground-based detectors. However, the rates of these events are highly uncertain, given that such systems are invisible. One formation scenario involves mergers in field binaries, where our lack of complete understanding of common envelopes and the distribution of supernova kicks has led to rate estimates that range over a factor of several hundred. A different, and highly promising, channel involves multiple encounters of binaries in globular clusters or young star clusters. However, we currently lack solid evidence for black holes in almost all such clusters, and their low escape speeds raise the possibility that most are ejected because of supernova recoil. Here we propose that a robust environment for mergers could be the nuclear star clusters found in the centers of small galaxies. These clusters have millions of stars, black hole relaxation times well under a Hubble time, and escape speeds that are several times those of globulars, hence they retain most of their black holes. We present simulations of the three-body dynamics of black holes in this environment and estimate that, if most nuclear star clusters do not have supermassive black holes that interfere with the mergers, at least several tens of events per year will be detectable with Advanced LIGO. \nSubject headings: black hole physics - galaxies: nuclei - gravitational waves relativity', '1. Introduction': "Ground-based gravitational wave detectors have now achieved their initial sensitivity goals (e.g., Abbott et al. 2007). In the next few years, these sensitivities are expected to improve by a factor of ∼ 10, which will increase the searchable volume by a factor of ∼ 10 3 and will lead to many detections per year. \nOne of the most intriguing possible sources for such detectors is the coalescence of a double stellar-mass black hole binary. Such binaries are inherently invisible, meaning that \nwe have no direct observational handle on how common they are or their masses, spin magnitudes, or orientations. Comparison of the observed waveforms (or of waveforms from merging supermassive black holes) with predictions based on post-Newtonian analysis and numerical relativity will be the most direct possible test of the predictions of strong-gravity general relativity. \nThe electromagnetic non-detection of these sources makes rate estimates highly challenging, because our only observational handles on BH-BH binaries come from possible progenitors. For example, a common scenario involves the effectively isolated evolution of a field binary containing two massive stars into a binary with two black holes that will eventually merge (e.g., Lipunov et al. 1997; Belczynski & Bulik 1999). There are profound uncertainties involved in calculations of these rates due to e.g., the lack of knowledge of the details of the common envelope phase in these systems and the absence of guides to the distribution of supernova kicks delivered to black holes. As a recent indication of the range of estimated rates, note that the Advanced LIGO detection rate of BH-BH coalescences is estimated to be anywhere between ∼ 1 -500 yr -1 by Belczynski et al. (2007), depending on how common envelopes are modeled. \nAnother promising location for BH-BH mergers is globular clusters or super star clusters, where stellar number densities are high enough to cause multiple encounters and hardening of binaries. Even though binaries are kicked out before they merge (Kulkarni, Hut, & McMillan 1993; Sigurdsson & Hernquist 1993; Sigurdsson & Phinney 1993, 1995; Portegies Zwart & McMillan 2000; O'Leary et al. 2006), these clusters can still serve as breeding grounds for gravitational wave sources. Indeed, O'Leary, O'Shaughnessy, & Rasio (2007) estimate a rate of 0.5 yr -1 for initial LIGO and 500 yr -1 for Advanced LIGO via this channel. There is, however, little direct evidence for black holes in most globulars (albeit they could be difficult to see). In addition, at least one black hole in a low-mass X-ray binary apparently received a > ∼ 100 km s -1 kick from its supernova (GRO J1655-40; see Mirabel et al. 2002). This is double the escape speed from the centers of even fairly rich globulars (Webbink 1985), leading to uncertainties about their initial black hole population and current merger rates. \nHere we propose that mergers occur frequently in the nuclear star clusters that may be in the centers of many low-mass galaxies (Boker et al. 2002; Ferrarese et al. 2006; Wehner & Harris 2006; note that some of these are based on small deviations from smooth surface brightness profiles and are thus still under discussion). It has recently been recognized that in these galaxies, which may not have supermassive black holes (for a status report on ongoing searches for low-mass central black holes, see Greene & Ho 2007), the nuclear clusters have masses that are correlated with the surrounding velocity dispersion σ as M ≈ 10 7 M /circledot ( σ/ 54 km s -1 ) 4 . 3 (Ferrarese et al. 2006). When the velocity dispersion is in \nthe range of ∼ 30 -60 km s -1 , the half-mass relaxation time is small enough that black holes (which have ∼ 20 × the average stellar mass) can sink to the center in much less than a Hubble time. In addition, although systems with equal-mass objects require roughly 15 half-mass relaxation times to undergo core collapse (Binney & Tremaine 1987), studies show that systems with a wide range of stellar masses experience core collapse within ∼ 0 . 2 × the half-mass relaxation time (Portegies Zwart & McMillan 2002; Gurkan et al. 2004). Therefore, clusters with masses less than ∼ few × 10 7 M /circledot will have collapsed by now and hence increased the escape speed from the center, allowing retention of most of their black holes. \nAs we show in this paper, nuclear star clusters are therefore excellent candidates for stellar-mass black hole binary mergers because they keep their black holes while also evolving rapidly enough that the holes can sink to a region of high density. If tens of percent of the black holes in eligible galaxies undergo such mergers, the resulting rate for Advanced LIGO is at least several tens per year. In § 2 we quantify these statements and results more precisely and discuss our numerical three-body method. We give our conclusions in § 3.", '2.1. Characteristic times and initial setup': "Our approach is similar to that of O'Leary et al. (2006), who focus on globular clusters with velocity dispersions σ ≤ 20 km s -1 . Here, however, we concentrate on the more massive and tightly bound nuclear star clusters. Our departure point is the relation found by Ferrarese et al. (2006) between the masses and velocity dispersions of such clusters: \nM nuc = 10 6 . 91 ± 0 . 11 ( σ/ 54 km s -1 ) 4 . 27 ± 0 . 61 M /circledot . (1) \nAssuming that there is no massive central black hole for these low velocity dispersions, the half-mass relaxation time for the system is (see Binney & Tremaine 1987) t rlx ≈ N/ 2 8 ln N t cross where N ≈ M nuc / 0 . 5 M /circledot is the number of stars in the system (assuming an average mass of 0 . 5 M /circledot ) and t cross ≈ R/σ is the crossing time. Here R = GM nuc /σ 2 is the radius of the cluster. Putting this together gives \nt rlx ≈ 1 . 3 × 10 10 yr ( σ/ 54 km s -1 ) 5 . 54 ± 0 . 61 . (2) \nThe relaxation time scales inversely with the mass of an individual star (Binney & Tremaine 1987), so a 10 M /circledot black hole will settle in roughly 1/20 of this time. Also note that large N-body simulations with broad mass functions evolve to core collapse within roughly 0.2 half-mass relaxation times (Portegies Zwart & McMillan 2002; Gurkan et al. 2004), hence in \nthe current universe clusters with velocity dispersions σ < 60 km s -1 will have had their central potentials deepened significantly. \nThe amount of deepening of the potential, and thus the escape speed from the center of the cluster, depends on uncertain details such as the initial radial dependence of the density and the binary fraction. Given that the timescale for segregation of the black holes in the center is much less than a Hubble time, we will assume that the escape speed is roughly 5 × the velocity dispersion, as is the case for relatively rich globular clusters (Webbink 1985). This may well be somewhat conservative, because the higher velocity dispersion here than in globulars suggests that a larger fraction of binaries will be destroyed in nuclear star clusters. This could lead to less efficient central energy production and hence deeper core collapse than is typical in globulars. \nWith this setup, our task is to follow the interactions of black holes in the central regions of nuclear star clusters, where we will scale by stellar number densities of n ∼ 10 6 pc -3 because of density enhancements caused by relaxation and mass segregation. Some black holes will begin their lives in binaries, but to be conservative we will assume that they start as single objects and have to exchange into binaries that contain main sequence stars or other objects. All binaries in the cluster will be hard, i.e., will have internal energies greater than the average kinetic energy of a field star, because otherwise they will be softened and ionized quickly (e.g., Binney & Tremaine 1987). If, for example, we consider binaries of two 1 M /circledot stars in a system with σ = 50 km s -1 , this means that the semimajor axis has to be less than a max ∼ 1 AU. Studies of main sequence binaries in globular clusters, which have σ ∼ 10 km s -1 , suggest that after billions of years roughly 5-20% of them survive, with the rest falling victim to ionization or collisions (Ivanova et al. 2005). The binary fraction will be lower in nuclear star clusters due to their enhanced velocity dispersion, but since when binaries are born they appear to have a constant distribution across the log of the semimajor axis from ∼ 10 -2 -10 3 AU (e.g., Abt 1983; Duquennoy & Mayor 1991) the reduction is not necessarily by a large factor. We will conservatively scale by a binary fraction f bin = 0 . 01. \nIf a black hole with mass M BH gets within a couple of semimajor axes of a main sequence binary, the binary will tidally separate and the BH will acquire a companion. The timescale on which this happens is t bin = ( n Σ σ ) -1 , where Σ = πr 2 p (1 + 2 GM tot / ( σ 2 r p )) is the interaction cross section for pericenter distances ≤ r p when gravitational focusing is included. Here M tot is the mass of the black hole plus the mass of the binary. If we assume that M BH = 10 M /circledot and it interacts with a binary with two 1 M /circledot members and an a = 1 AU semimajor axis, then the typical timescale on which a three-body interaction and capture of one of the stars occurs is \nt 3 -bod = ( n Σ σ ) -1 ≈ 1 . 2 × 10 9 yr( n/ 10 6 pc -3 ) -1 ( f bin / 0 . 01) -1 ( σ/ 50 km s)( a/ 1 AU) -1 . (3) \nThis is small enough compared to a Hubble time that we start our simulations by assuming that each black hole has exchanged into a hard binary, and follow its evolution from there. \nAnother important question is whether, after a three-body interaction, a black hole binary will shed the kinetic energy of its center of mass via dynamical friction and sink to the center of the cluster before another three-body encounter. If not, the kick speeds will add in a random walk, thus increasing the ejection fraction. \nTo compute this we note that the local relaxation time of a binary is \nt rlx = 0 . 339 ln Λ σ 3 G 2 〈 m 〉 M bin n (4) \n(Spitzer 1987) where σ is the local velocity dispersion, ln Λ ∼ 10 is the Coulomb logarithm, 〈 m 〉 is the average mass of interloping stars, n is their number density, and M bin is the mass of the binary. The timescale for a three-body interaction is t 3 -bod = ( n Σ σ ) -1 as above. For a gravitationally focused binary, which is of greatest interest because only these could in principle produce three-body recoil sufficient to eject binaries or singles, r p < GM bin /σ 2 . Therefore, Σ ≈ 2 πr p GM bin /σ 2 and \nt 3 -bod ≈ σ 2 πnr p GM bin . (5) \nIf we let r p = qGM bin /σ 2 , with q < 1, then \nt 3 -bod ≈ σ 3 2 πqG 2 M 2 bin n (6) \nso that \nt rlx /t 3 -bod ≈ 2 q ln Λ M bin 〈 m 〉 . (7) \nIn the center of a cluster, where mass segregation is likely to have flattened the mass distribution, we find that this quantity is typically less than unity (and it decreases as the binary hardens), meaning that after a three-body encounter a binary has an opportunity to share its excess kinetic energy via two-body encounters and thus settle back to the center of the cluster. We therefore treat the encounters separately rather than adding the kick speeds in a random walk. \nIn a given encounter, suppose that a binary of total mass M bin = M 1 + M 2 , a reduced mass µ = M 1 M 2 /M bin , and a semimajor axis a init interacts with an interloper of mass m int , and that the kinetic energy of the interloper at infinity is much less than the binding energy of the binary (i.e., this is a very hard interaction). If after the interaction the semimajor axis is a fin < a init , then energy and momentum conservation mean that the recoil speed of the \nbinary is given by v 2 bin = Gµ m int M bin + m int (1 /a fin -1 /a init ), and the recoil speed of the interloper is v int = ( M bin /m int ) v bin . For example, suppose that M 1 = M 2 = 10 M /circledot , M int = 1 M /circledot , a init = 0 . 1 AU, and a fin = 0 . 09 AU. The binary then recoils at v bin = 15 km s -1 and stays in the cluster, whereas the interloper recoils at v int = 300 km s -1 and is ejected.", '2.2. Results': "The central regions of the clusters undergo significant mass segregation, and thus the mass function will be at least flattened, and possibly inverted. This has, for example, been observed for globulars (Sosin 1997). To include this effect, when we consider the mass of a black hole, its companion, or the interloping third object in a binary-single encounter, we go through two steps. First we select a zero age main sequence (ZAMS) mass between 0 . 2 M /circledot and 100 M /circledot using a simple power law distribution dN/dM ∝ M -α . We allow α to range anywhere from 2.35 (the unmodified Salpeter distribution) to -1 . 0, where smaller values indicate the effects of mass segregation. Second, we evolve the ZAMS mass to a current mass. Our mapping is that for M ZAMS < 1 M /circledot , the star is still on the main sequence and retains its original mass; for 1 M /circledot < M ZAMS < 8 M /circledot the star has evolved to a white dwarf, with mass M WD = 0 . 6 M /circledot + 0 . 4 M /circledot ( M ZAMS /M /circledot -0 . 6) 1 / 3 ; for 8 M /circledot < M ZAMS < 25 M /circledot the star has evolved to a neutron star, with mass M NS = 1 . 5 M /circledot + 0 . 5 M /circledot ( M ZAMS -8 M /circledot ) / 17 M /circledot ; and for M ZAMS > 25 M /circledot the star has evolved to a black hole with mass M BH = 3 M /circledot + 17 M /circledot ( M ZAMS -25 M /circledot ) / 75 M /circledot . Therefore, we assume that black hole masses range from 3 M /circledot to 20 M /circledot . \nThese prescriptions are overly simplified in many ways. We therefore explore different mass function slopes, main sequence cutoffs, and so on, and find that our general picture is robust against specific assumptions. Note that, consistent with O'Leary et al. (2006), we find that there is a strong tendency for the merged black holes to be biased towards high masses. Therefore, if black holes with masses > 20 M /circledot are common, these will dominate the merger rates. This is important for data analysis strategies, because the low-frequency cutoff of ground-based gravitational wave detectors implies that higher-mass black holes will have proportionally more of their signal in the late inspiral, merger, and ringdown. \nThe three-body interactions themselves are assumed to be Newtonian interactions between point masses and are computed using the hierarchical N-body code HNBody (K. Rauch and D. Hamilton, in preparation), using the driver IABL developed by Kayhan Gultekin (see Gultekin, Miller, & Hamilton 2004, 2006 for a detailed description). These codes use a number of high-accuracy techniques to follow the evolution of gravitating point masses. Between interactions, we use the Peters equations (Peters 1964) to follow the gradual inspiral and \ncircularization of the binary via emission of gravitational radiation. This is negligible except near the end of any given evolution. \nWe begin by selecting the mass of the black hole and of its companion (which does not need to be a black hole) from the evolved mass function. We also begin with a semimajor axis that is 1/4 of the value needed to ensure that the binary is hard. We do this because soft binaries are likely to be ionized and thus become single stars rather than merge. We also select an eccentricity from a thermal distribution P ( e ) de = 2 ede . We then allow the binary to interact with single field stars drawn from the evolved mass function, one at a time, until either (1) the binary merges due to gravitational radiation, (2) the binary is split apart and thus ionized (this is exceedingly rare given our initial conditions), or (3) the binary is ejected from the cluster. The entire set of interactions until merger typically takes millions to tens of millions of years, and only rarely over a hundred million years, so it finishes in much less than a Hubble time. In the course of these interactions there are typically a number of exchanges, which usually swap in more massive for less massive members of the binary. This is the cause of the bias towards high-mass mergers that was also found by O'Leary et al. (2006). As shown in Table 1, for α < 1 most black holes acquire a black hole companion in the process of exchanges, and for α ≤ 0 . 5 virtually all do. \nThe results in Table 1 are focused on different mass function slopes and escape speeds. As expected, we find that for V esc > 150 km s -1 the overwhelming majority of black hole binaries merge in the nuclear star cluster rather than being ejected (see Figure 1). This is the difference from lowerσ globular clusters, where the mergers happen outside the cluster. Note also that in addition to few binaries being ejected, there are typically only 1-2 single black holes ejected per merger, suggesting that > 50% of holes will merge. In contrast, at the 50 km s -1 escape speed typical of globulars, > 20 single black holes are ejected per merger, suggesting an efficiency of < 10%. For well-segregated clusters (with α ≤ 0), the average mass of black holes that merge, binary ejection fraction and number of singles ejected, and number of black holes that merge with each other instead of other objects are all insensitive to the particular mass function slope. For less segregated clusters with α > 0, the retention fraction of black holes rises rapidly to unity because most of the objects that interact with the holes are less massive stars. In such cases there might be a channel by which the mass of the holes increases via accretion of stars, but we expect α > 0 to be rare for nuclear star clusters because of the shortness of the segregation times of black holes. Overall, there appears to be a wide range of realistic parameters in which fewer than 10% of binary black holes are ejected before merging.", '3. Discussion and Conclusions': "We have shown that nuclear star clusters with velocity dispersions around σ ∼ 30 -60 km s -1 are promising breeding grounds for stellar-mass black hole mergers. At significantly lower velocity dispersions, as found in globulars, the escape speed is low enough that the binaries are ejected before they merge. Significantly higher velocity dispersions appear correlated with the appearance of supermassive black holes (Gebhardt et al. 2000; Ferrarese & Merritt 2000). In such an environment there might also be interesting rates of black hole mergers, but the increasing velocity dispersion closer to the central object means that binary fractions are lower and softening, ionization, or tidal separation by the supermassive black hole itself are strong possibilities for stellar-mass binaries (Miller et al. 2005; Lauburg & Miller, in preparation). \nTo estimate the rate of detections with Advanced LIGO, we note that velocity dispersions in the σ ∼ 30 -60 km s -1 range correspond to roughly a factor of ∼ 10 in galaxy luminosity (Ferrarese et al. 2006). Galaxy surveys suggest (e.g., Blanton et al. 2003) that for dim galaxies the luminosity function scales as roughly dN/dL = φ ∗ ( L/L ∗ ) -α , where φ ∗ = 1 . 5 × 10 -2 h 3 Mpc -3 ≈ 5 × 10 -3 Mpc -3 for h = 0 . 71, and α ≈ -1. This implies that there are nearly equal numbers of galaxies in equal logarithmic bins of luminosity. A factor of 10 in luminosity is roughly e 2 , so the number density of relevant galaxies is approximately 10 -2 Mpc -3 . To get the rate per galaxy, we note that typical initial mass functions and estimates of the mass needed to evolve into a black hole combine to suggest that for a cluster of mass M nuc , approximately 3 × 10 -3 ( M nuc /M /circledot ) stars evolve into black holes (O'Leary, O'Shaughnessy, & Rasio 2007). That implies a few × 10 4 black holes per nuclear star cluster. If a few tens of percent of these merge in a Hubble time, and if the rate is slightly lower now because many of the original black holes have already merged (see O'Leary et al. 2006), that suggests a merger rate of > 0 . 1 × few × 10 4 / (10 10 yr) per galaxy, or few × 10 -9 Mpc -3 yr -1 . At the ∼ 2 Gpc distance at which Advanced LIGO is expected to be able to see mergers of two 10 M /circledot black holes (see, e.g., Mandel 2007), the available volume is 3 × 10 10 Mpc 3 , for a rate of > ∼ 100 per year. Roughly 50-80% of galaxies in the eligible luminosity range appear to have nuclear star clusters (see Ferrarese et al. 2006 for a summary). If the majority of the clusters do not have a supermassive black hole, this suggests a final rate of at least several tens per year for Advanced LIGO. This could be augmented somewhat by small galaxies that originally had supermassive black holes, but had them ejected after a merger and then reformed a central cluster (Volonteri 2007; Volonteri, Haardt, & Gultekin 2008). \nFor nearby ( z < 0 . 1) events of this type it might be possible to identify the host galaxy. However, for more typical z ∼ 0 . 5 ⇒ d ≈ 2 Gpc events the number of can- \ns is too large: even assuming angular localization of ∆Ω = (1 · ) 2 and a distance accuracy of ∆ d/d = 1%, the number of galaxies in the right luminosity range is N ∼ 4 π/ 3(2000 Mpc) 3 (∆Ω / 4 π )(∆ d/d )(0 . 01 Mpc -3 ) ≈ 80. Therefore, barring some unforseen electromagnetic counterpart, the host will usually not be obvious. \nWe anticipate that tens per year is a somewhat conservative number, because our simulations suggest that more like 50% of black holes will be retained, even as single objects, and because (unlike in a globular cluster) the central regions of galaxies are not devoid of gas, hence more black holes could form in the vicinity of the cluster and fall in. In addition, if stellar-mass black holes with masses beyond 20 M /circledot are common, this also increases the detection radius and hence the rate. Even for total masses ∼ 30 M /circledot and at redshifts z ∼ 0 . 5, the observer frame gravitational wave frequency at the innermost stable circular orbit is f ISCO ∼ 4400 Hz / [30(1 + z )] ∼ 100 Hz. This is close enough to the range where frequency sensitivity declines that detection of many of these events will rely strongly on the signal obtained from the last few orbits plus merger and ringdown. In much of this range, numerical relativity is essential. \nAs a final point, we note that for the same reason that nuclear star clusters are favorable environments for retention and mergers of stellar-mass black holes, they could also be good birthplaces for more massive black holes. This could be prevented, even for the relatively high escape speeds discussed here, if recoil from gravitational radiation during the coalescence exceeds ∼ 200 km s -1 . The key uncertainty here is the spin magnitudes of the holes at birth. Numerous simulations demonstrate that high spins with significant projections in the binary orbital plane can produce kicks of up to several thousand kilometers per second (Gonzalez et al. 2007). If there is significant processing of gas through accretion disks the spins are aligned in a way that reduces the kick to below 200 km s -1 (Bogdanovi'c, Reynolds, & Miller 2007), but stellar-mass black holes cannot pick up enough mass from the interstellar medium for this to be effective. For example, the BondiHoyle accretion rate is ˙ M Bondi ≈ 10 -13 M /circledot yr -1 ( σ/ 50 km s -1 ) -3 ( n/ 100 cm -3 )( M/ 10 M /circledot ) 2 , meaning that to accrete the ∼ 1% of the black hole mass needed to realign the spin (Bogdanovi'c, Reynolds, & Miller 2007) would require at least a trillion years. Current estimates of stellar-mass black hole spins suggest a/M > 0 . 5 in many cases (Shafee et al. 2006; McClintock et al. 2006; Miller 2007; Liu et al. 2008). If the spins are isotropically oriented and uniformly distributed in the range 0 < a/M < 1, and the mass ratios are in the m small /m big ∼ 0 . 6 -0 . 8 range typical in our simulations, then use of the Campanelli et al. (2007) or Baker et al. (2008) kick formulae imply that roughly 84% of the recoils exceed 200 km s -1 and 78% exceed 250 km s -1 . This suggests that multiple mergers are rare unless there is initially an extra-massive black hole as a seed (e.g., Holley-Bockelmann et al. 2008 for a discussion of the effects of gravitational wave recoil), but further study is important. \nIn conclusion, we show that the compact nuclear star clusters found in the centers of many small galaxies are ideal places to foster mergers between stellar-mass black holes. 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Simulations of nuclear star clusters a \n| V esc (km s - 1 ) b | M ms , max c | α d | 〈 M BH 〉 ( M /circledot ) e | f merge f | f notBH g | 〈 M bin , merge 〉 ( M /circledot ) h | 〈 N single , eject 〉 i |\n|-----------------------|----------------|-------|---------------------------------|-------------|-------------|------------------------------------------|----------------------------|\n| 50 | 1 M /circledot | 0 | 11.7 | 0.25 | 0 | 31.2 | 24.8 |\n| 62.5 | 1 M /circledot | 0 | 11.7 | 0.33 | 0 | 31.6 | 15.3 |\n| 75 | 1 M /circledot | 0 | 11.7 | 0.42 | 0 | 30.9 | 11.5 |\n| 87.5 | 1 M /circledot | 0 | 11.7 | 0.52 | 0 | 31.9 | 7.9 |\n| 20 | 1 M /circledot | 0 | 11.7 | 0.63 | 0.02 | 30.0 | 6.2 |\n| 112.5 | 1 M /circledot | 0 | 11.7 | 0.68 | 0 | 31.4 | 4.7 |\n| 125 | 1 M /circledot | 0 | 11.7 | 0.72 | 0.02 | 31.8 | 4.3 |\n| 137.5 | 1 M /circledot | 0 | 11.7 | 0.76 | 0.01 | 32.0 | 3 |\n| 150 | 1 M /circledot | 0 | 11.7 | 0.8 | 0.03 | 32.3 | 2.8 |\n| 162.5 | 1 M /circledot | 0 | 11.7 | 0.93 | 0.03 | 31.3 | 2 |\n| 175 | 1 M /circledot | 0 | 11.7 | 0.89 | 0.02 | 31.9 | 2 |\n| 187.5 | 1 M /circledot | 0 | 11.7 | 0.9 | 0.01 | 31.3 | 2.1 |\n| 200 | 1 M /circledot | 0 | 11.7 | 0.94 | 0.08 | 31.1 | 1.3 |\n| 212.5 | 1 M /circledot | 0 | 11.7 | 0.89 | 0.05 | 30.5 | 1 |\n| 225 | 1 M /circledot | 0 | 11.7 | 0.98 | 0.06 | 31.0 | 1.2 |\n| 237.5 | 1 M /circledot | 0 | 11.7 | 0.94 | 0.06 | 30.1 | 1 |\n| 250 | 1 M /circledot | 0 | 11.7 | 0.96 | 0.06 | 30.0 | 0.71 |\n| 200 | 1 M /circledot | -1 | 13.4 | 0.94 | 0 | 32.4 | 1.3 |\n| 200 | 1 M /circledot | -0.5 | 12.6 | 0.95 | 0.01 | 32.2 | 1.5 |\n| 200 | 1 M /circledot | 0.5 | 10.7 | 0.94 | 0.1 | 28.3 | 0.91 |\n| 200 | 1 M /circledot | 1 | 9.7 | 0.98 | 0.41 | 27.3 | 0.43 |\n| 200 | 1 M /circledot | 1.5 | 8.8 | 0.99 | 0.79 | 23.0 | 0.04 |\n| 200 | 1 M /circledot | 2 | 7.5 | 1 | 0.99 | - | 0 |\n| 200 | 1 M /circledot | 2.35 | 7.4 | 1 | 1 | - | 0 |\n| 200 | 3 M /circledot | -1 | 13.4 | 0.85 | 0.03 | 33.3 | 1.5 |\n| 200 | 3 M /circledot | -0.5 | 12.6 | 0.94 | 0.01 | 31.9 | 1.3 |\n| 200 | 3 M /circledot | 0 | 11.7 | 0.95 | 0.05 | 30.4 | 1.5 |\n| 200 | 3 M /circledot | 0.5 | 10.7 | 0.94 | 0.11 | 29.2 | 1 |\n| 200 | 3 M /circledot | 1 | 9.7 | 0.99 | 0.48 | 25.3 | 0.38 |\n| 200 | 3 M /circledot | 1.5 | 8.8 | 0.99 | 0.85 | 24.7 | 0.04 |\n| 200 | 3 M /circledot | 2 | 7.5 | 1 | 1 | - | 0 | \nTable 1-Continued \n| V esc (km s | M ms , max c | α d | 〈 M BH 〉 ( M /circledot ) e | f merge f | f notBH g | 〈 M bin , merge 〉 ( M /circledot ) h | 〈 N single , eject 〉 i |\n|---------------|----------------|-------|---------------------------------|-------------|-------------|------------------------------------------|----------------------------|\n| 200 | 3 M /circledot | 2.35 | 7.4 | 1 | 1 | - | 0 | \nFig. 1.- Fraction of binaries retained in the nuclear star cluster (solid line) and average number of black holes ejected per black hole merger (dotted line) as a function of the cluster escape speed. Here the zero age main sequence distribution of masses is dN/dM ∝ M 0 , to account for mass segregation in the cluster center, where most interactions occur. We also assume a maximum black hole mass of 20 M /circledot and a maximum main sequence mass of 1 M /circledot , but most results are robust against variations of these quantities. All runs are done with 100 realizations, which explains the lack of perfect smoothness. We see, as expected, that the retention fraction increases rapidly with escape speed, so that for nuclear star clusters most binaries stay in the cluster until merger. We also see that at V esc ∼ 200 km s -1 and above, most black hole singles also stay in the cluster. This suggests a high merger efficiency. \n<!-- image -->"}

2009ApJ...707..428B

Transport of Large-Scale Poloidal Flux in Black Hole Accretion

2009-01-01

['accretion', 'accretion disks', 'mhd', 'relativity', '-']

[]

We report on a global, three-dimensional GRMHD simulation of an accretion torus embedded in a large-scale vertical magnetic field orbiting a Schwarzschild black hole. This simulation investigates how a large-scale vertical field evolves within a turbulent accretion disk and whether global magnetic field configurations suitable for launching jets and winds can develop. We find that a "coronal mechanism" of magnetic flux motion, which operates largely outside the disk body, dominates global flux evolution. In this mechanism, magnetic stresses driven by orbital shear create large-scale half-loops of magnetic field that stretch radially inward and then reconnect, leading to discontinuous jumps in the location of magnetic flux. In contrast, little or no flux is brought in directly by accretion within the disk itself. The coronal mechanism establishes a dipole magnetic field in the evacuated funnel around the orbital axis with a field intensity regulated by a combination of the magnetic and gas pressures in the inner disk. These results prompt a re-evaluation of previous descriptions of magnetic flux motion associated with accretion. Local pictures are undercut by the intrinsically global character of magnetic flux. Formulations in terms of an "effective viscosity" competing with an "effective resistivity" are undermined by the nonlinearity of the magnetic dynamics and the fact that the same turbulence driving mass motion (traditionally identified as "viscosity") can alter magnetic topology.

[]

https://arxiv.org/pdf/0906.2784.pdf

{'Transport of Large Scale Poloidal Flux in Black Hole Accretion': 'Kris Beckwith \nInstitute of Astronomy University of Cambridge Madingley Road Cambridge CB3 0HA United Kingdom \[email protected] \nJohn F. Hawley \nAstronomy Department University of Virginia P.O. Box 400325 Charlottesville, VA 22904-4325 \[email protected] \nand \nJulian H. Krolik \nDepartment of Physics and Astronomy Johns Hopkins University Baltimore, MD 21218 \[email protected]', 'ABSTRACT': "We report on a global, three-dimensional GRMHD simulation of an accretion torus embedded in a large scale vertical magnetic field orbiting a Schwarzschild black hole. This simulation investigates how a large scale vertical field evolves within a turbulent accretion disk and whether global magnetic field configurations suitable for launching jets and winds can develop. We find that a 'coronal mechanism' of magnetic flux motion, which operates largely outside the disk body, dominates global flux evolution. In this mechanism, magnetic stresses driven by orbital shear create large-scale half-loops of magnetic field that stretch \nradially inward and then reconnect, leading to discontinuous jumps in the location of magnetic flux. In contrast, little or no flux is brought in directly by accretion within the disk itself. The coronal mechanism establishes a dipole magnetic field in the evacuated funnel around the orbital axis with a field intensity regulated by a combination of the magnetic and gas pressures in the inner disk. These results prompt a reevaluation of previous descriptions of magnetic flux motion associated with accretion. Local pictures are undercut by the intrinsically global character of magnetic flux. Formulations in terms of an 'effective viscosity' competing with an 'effective resistivity' are undermined by the nonlinearity of of the magnetic dynamics and the fact that the same turbulence driving mass motion (traditionally identified as 'viscosity') can alter magnetic topology. \nSubject headings: accretion, accretion discs - relativity - (magnetohydrodynamics) MHD", '1. Introduction': "Astrophysical jets are seen in such a wide range of astrophysical systems that their creation must not require particularly unusual conditions. A tentative consensus has emerged that jets are a natural consequence of accretion, rotation and magnetic fields (for a summary of this position, see Livio 2000). Although the presence of accretion and rotation are a given in an accreting system, the presence of magnetic fields has long been considered to be less certain. However, since magnetohydrodynamic (MHD) turbulence driven by the magnetorotational instability (MRI; see Balbus & Hawley 1991, 1998) accounts for the internal stresses that drive accretion, one is at least assured that the presence of accretion itself implies the existence of a tangled magnetic field of some reasonable strength. \nBut is that enough? Most jet launching mechanisms that have been developed depend on the existence of a large-scale, organized poloidal field, following in general terms the scenarios outlined analytically in the influential papers of Blandford & Znajek (1977) and Blandford & Payne (1982). In the Blandford-Payne model, a large-scale poloidal magnetic field is anchored in and rotates with the disk. If the fieldlines are angled outward sufficiently with respect to the disk, there can be a net outward force on the matter. As matter is accelerated along the rotating fieldlines, its angular momentum increases still further, accelerating and driving an outflow. In this model the power for the resulting jet comes from the rotation of the accretion disk. In the Blandford-Znajek model, the jet is powered by the rotating space-time of a black hole. (In the case of jets from systems with stars rather than black holes, stellar rotation can play a similar role.) Radial magnetic field lines lie along the hole's rotation \naxis and are anchored in the event horizon. Fieldline rotation is created by frame-dragging and this drives an out-going Poynting flux. Most simulations that have demonstrated jetlaunching have assumed the existence of a large-scale poloidal (mainly vertical) field in the initial conditions (see, e.g., the review by Pudritz et al. 2007). Although such simulations have provided considerable evidence that magnetic fields can be effective in powering jets, they cannot account for the origin of those fields. \nIf, as it seems, some sort of large-scale poloidal field is required for jet production, how can such a field be established in the near hole region? One possibility is an accretion disk dynamo, presumably working with the turbulent fields generated naturally by the MRI. A dynamo is an attractive possibility because it has the potential to be ubiquitous. A number of models have been put forward suggesting how some form of inverse cascade might occur within the MHD turbulence to generate a large scale field (e.g., Tout & Pringle 1996; Uzdensky & Goodman 2008), but these scenarios remain speculative and nothing definitive has been seen in global disk simulations done to date. In any case, the strict conservation of flux within a volume means that a disk dynamo could produce net flux only through interactions with boundaries, in this case either the black hole, or by expelling flux to large radius. \nThe global advection of net flux is an important process, therefore, whether the disk functions as a dynamo or net poloidal field is simply carried in to the near hole region from large radius. Even without any dynamo action, a weak net field might become locally strong it were concentrated by the accretion process. In the absence of a single field polarity at large distance, a near-hole net field could nevertheless be built up due to a random walk process as field is accreted (e.g., Thorne et al. 1986). Whether or not net field can be so accreted, however, is a matter of some uncertainty. The main concern is that the field would not be accreted if, as seems intuitively likely, the field intensity declines outward and the rate of diffusion of the field through the matter exceeds the rate at which matter accretes (van Ballegooijen 1989; Lubow et al. 1994; Heyvaerts et al. 1996; Ogilvie & Livio 2001). This competition between inward advection and outward diffusion is typically described in terms of an effective turbulent accretion viscosity ν t determining the accretion rate, and an effective magnetic diffusivity, η t that sets the field diffusion rate. The fate of the field is then determined by the effective magnetic Prandtl number P m = ν t /η t . In this picture, it is argued that the relevant scale for effective resistivity is ∼ H whereas the relevant scale for effective viscosity is ∼ R ; consequently, field can be brought inward only if P m > R/H . \nIt is far from clear, however, that field transport within MHD turbulence can be described in such terms. The picture just described assumes that the fluid motion is independent of the field, yet in Nature the dynamical coupling between fluid and field assures \nthat even weak fields are amplified, and that the effective viscosity ν t is, in fact, the result of magnetic stresses. Thus, there is no kinematic regime in which the Lorentz forces can be neglected. The resulting MRI-driven turbulence is inherently three-dimensional; indeed, three-dimensional studies are essential to generating long-term sustained MHD turbulence due to the fundamental restriction of the anti-dynamo theorem. By contrast, the framework of effective turbulent viscosity and diffusivity is cast in axisymmetry, in the hope that suitable time- and space-averaging of the field transport process in a fully turbulent, threedimensional disk can lead to some value of η t in much the same way that suitable averaging of correlations in the magnetic field, normalized to some suitable pressure term (either gas, magnetic or total pressure) leads to a value of ν t . But the primary mechanism by which magnetic fields change the fluid elements to which they are attached is through magnetic reconnection driven by the fluid turbulence, not gradual slippage via resistivity. The effective viscosity and magnetic diffusivity are derived from making the ansatz that stress and reconnection can be modeled as a viscosity or resistivity, i.e., a constant coefficient multiplying the gradient in orbital rotation rate or magnetic field strength. Whether or not such a model is appropriate to accretion flows remains to be determined. Finally, net flux and field topology are global concepts; a purely local description of field motion in terms of transport coefficients may not be sufficient. \nNumerical simulations offer a promising approach to investigate these and other issues related to global field evolution within accretion flows. Of course, simulations entail their own set of difficulties: global simulations require adequately fine resolution over a wide range of radii in the disk body and throughout a good deal of the surrounding coronal region as well. The numerical grid boundaries should be sufficiently remote so as to minimize artificial effects on the disk. Although axisymmetric simulations can be of interest, the problem ultimately must be done in three dimensions because any axisymmetric calculation fails to describe properly essential properties of both the fluid turbulence and the magnetic field evolution. Finally, because we need to follow the disk long enough to observe net accretion in an approximate inflow equilibrium, the duration of the simulation must be long compared to typical local dynamical times in the disk. \nWork along these lines is ongoing. Global, three dimensional black hole accretion simulations without initial fields threading the disk were carried out and described in a series of papers beginning with De Villiers et al. (2003). In these simulations initial weak dipole loops are contained within an isolated gas torus. The subsequent MRI-driven accretion flow carries the inner half of the initial field loop (with a single sign for the net vertical field component) down to the black hole, creating radial field as the loop is stretched. Differential rotation along that radial field leads to rapid growth of the toroidal field, especially within the plunging region of the flow. This growing toroidal field causes the ejection of a magnetic \ntower into the low density funnel region and the establishment of a global dipole field anchored in the black hole. This field can then produce a Poynting flux jet if the black hole is rotating. \nBeckwith et al. (2008a) examined several alternative initial field configurations in comparison to the dipolar loop. Quadrupolar initial field loops produce a much weaker funnel field. A similar result was found by McKinney & Gammie (2004) for axisymmetric simulations, and McKinney & Blandford (2009) in a three-dimensional model. A variation proposed by Hawley & Krolik (2006) and further elaborated upon by Beckwith et al. (2008a) is a series of dipolar loops contained within the accretion flow. Such loops can lead to the creation of a temporary net sense of vertical field that can support a significant field in a relativistic outflow along the rotation axis for a sizable period of time. Beckwith et al. (2008a) also computed a model that began with a purely toroidal field. In this case the MRI creates turbulence, field and angular momentum transport, but without creating any overall organization to the poloidal field. As a consequence, no funnel field is generated. \nIn this paper we continue our study of the influence of field topology on accretion and potential jet formation through a three-dimensional general relativistic MHD simulation with an initial field configuration consisting of a net vertical field passing through an orbiting ring of gas at modestly large radius. We will investigate the evolution of the disk in the presence of a net field while determining if such an initial condition can lead to the establishment of field configurations conducive to jet formation. From this study we hope to gain more insight into whether or not large scale fields can be advected along with the accreting gas and to lay the groundwork for further work. \nThe rest of this paper is structured as follows: In § 2, we describe the methodology and the initial conditions for our simulations. In § 3 we discuss their results as they pertain to magnetic flux motion. Finally, in § 4, we summarize our results, compare them with current models from the literature, and explain their implications for observations of relativistic jets in Nature.", '2. Numerical Details': 'For this work we employ the general relativistic MHD code GRMHD developed in De Villiers & Hawley (2003); De Villiers (2006). GRMHD has already been used in several studies to simulate black hole accretion in three spatial dimensions (De Villiers et al. 2003; Hirose et al. 2004; De Villiers et al. 2005; Krolik et al. 2005; Hawley & Krolik 2006; Beckwith et al. 2008a,b). GRMHD solves the equations of ideal non-radiative MHD in the static Kerr \nmetric of a rotating black hole using Boyer-Lindquist coordinates. Values are expressed in gravitational units ( G = M = c = 1) with line element ds 2 = g tt dt 2 + 2 g tφ dtdφ + g rr dr 2 + g θθ dθ 2 + g φφ dφ 2 and signature ( -, + , + , +). The relativistic fluid at each grid zone is described by its density ρ , specific internal energy glyph[epsilon1] , 4-velocity U µ , and isotropic pressure P . The relativistic enthalpy is h = 1 + glyph[epsilon1] + P/ρ . The pressure is related to ρ and glyph[epsilon1] via the equation of state for an ideal gas, P = ρglyph[epsilon1] (Γ -1). The magnetic field is described by two sets of variables. The first is the constrained transport (CT; Evans & Hawley 1988) magnetic field B i = [ ijk ] F jk , where [ ijk ] is the completely anti-symmetric symbol, and F jk are the spatial components of the electromagnetic field strength tensor. From the CT magnetic field components, we derive the magnetic field four-vector, (4 π ) 1 / 2 b µ = ∗ F µν U ν , and the magnetic field scalar, || b 2 || = b µ b µ . The electromagnetic component of the stress-energy tensor is T µν (EM) = 1 2 g µν || b || 2 + U µ U ν || b || 2 -b µ b ν .', '2.1. Initial and Boundary Conditions': "In this study we carry out both axisymmetric and fully three dimensional simulations around a Schwarzschild black hole. The initial condition, as in previous works (De Villiers et al. 2003; Hirose et al. 2004; De Villiers et al. 2005; Krolik et al. 2005; Hawley & Krolik 2006; Beckwith et al. 2008a,b), consists of an isolated gas torus upon which a weak magnetic field is imposed. We use the torus model described by De Villiers et al. (2003) with an adiabatic index of Γ = 5 / 3. The initial angular momentum distribution is slightly subKeplerian, with a specific angular momentum at the inner edge of the torus (located at r = 30M) glyph[lscript] in ≡ U φ /U t = 6 . 10. For this choice of glyph[lscript] in , the pressure maximum is at r ≈ 40 M and the outer edge of the torus is located at r ≈ 60 M . The angular period at the pressure maximum is Ω -1 ≈ 250 M . With this inner edge radius, the resulting accretion flow should have sufficient radial range to permit a detailed study of the advection of vertical flux. The choice of angular momentum parameter at the inner edge of the torus, glyph[lscript] in , was made so that the initial torus (and hence the steady state accretion flow) would have a scale height similar to that of previous simulations (see e.g. Beckwith et al. 2008b). The torus is initially surrounded by zones filled with zero-velocity gas set to a constant value of density and pressure. The ratios of these 'vacuum' values to the torus maximum are 2 × 10 -4 for the pressure, and 5 × 10 -6 for the density. Because the code is relativistic, the timestep is always limited by the time for light to cross the smallest cell; high Alfv'en speeds in the low-density coronal region do not create any unique difficulty. \nThe initial magnetic field is homogeneous and vertical, filling the annular cylinder 35 M ≤ R ≤ 55 M (where R is the cylindrical radius). This configuration differs from \nthe vertical field used in some previous studies (see e.g. Koide et al. 1999; McKinney & Gammie 2004; De Villiers 2006) which fills all space, not just through the initial torus. By isolating the field in this manner, we know that any vertical flux that ends up crossing the black hole event horizon was brought in from field initially passing through the torus at large radius, rather than already being present at the horizon in the initial conditions. Also, because equilibrium field strengths scale strongly with radius, a global vertical field that is strong enough to be interesting at one radius is likely to be overwhelming at some radii while negligible at others. We prefer to see the global field strength emerge as an outcome of the evolution. In our configuration the initial vertical field is subthermal within the torus and is unstable to the MRI. It should be noted, though, that this initial field configuration is not in dynamic equilibrium outside of the torus. Also, this is but one very specific realization of a torus embedded in large scale field. Future experiments will need to explore a wider variety of initial configurations. \nThis initial field is computed from the curl of the four-vector potential, i.e., \nF αβ = ∂ α A β -∂ β A α = A β,α -A α,β . (1) \nwith only A φ glyph[negationslash] = 0. The vector potential corresponding to a uniform (Newtonian) vertical field geometry is proportional to the cylindrical radius, i.e., \nA φ ∝ r sin θ. (2) \nThe vector potential for the initial field is \nA φ = A 0 R in R < R in r sin θ R in ≤ R ≤ R out R out R > R out (3) \nwhere R in and R out are 35 M and 55 M . Outside of those cylindrical radii the field is zero. A 0 was specified so that the ratio of the average thermal to average magnetic pressure inside the torus is β = 100. At the center of the torus β = 275 and β declines smoothly toward 1 near the edge of the torus. In the surrounding corona, β = 0 . 038. The initial mass, field and β distribution is shown in Figure 1. \nThe simulation we will focus on is three-dimensional with 256 × 256 × 64 zones in ( r, θ, φ ). The simulation is labeled 'VD0m3d.' Given the challenge of adequately resolving such models and of doing anything like a convergence study, we also carried out two-dimensional simulations at two different resolutions. In two dimensions we used 256 × 256 and 512 × 512 zones in ( r, θ ). The two-dimensional simulations are labeled 'VD0m2d' and 'VD0h2d' where the 'm' designates medium resolution and the 'h' designates the higher resolution case. \nFor all models the radial grid extends from an inner boundary at r in = 2 . 104 to an outer boundary located at r out = 500 M . The radial grid was graded using a logarithmic distribution that concentrates zones near the inner boundary. An outflow condition is applied at both the inner and outer radial boundaries. In contrast to the zero net flux simulations that we have performed previously, magnetic fields pierce the outer boundary in the initial state. This net flux is kept fixed throughout the simulation, but otherwise the field at the boundary can evolve. When the fluid velocity normal to the boundary is directed outward we perform a simple copy of fluid variables into the ghost zones that lie outside of the boundary. In the case where the fluid velocity is inward, the ghost zones are filled with a cold, low density, zero momentum fluid at the vacuum value so that only cold, very low density gas can enter the computational grid. To minimize the influence of the outer boundary it is located as far away from the region of interest as is feasible. Any torques that arise from the interaction of the outer boundary with the large scale vertical field should have minimal influence on the turbulent disk itself. \nThe θ -grid spans the range 0 . 01 π ≤ θ ≤ 0 . 99 π . This creates a conical cutout that surrounds the coordinate axis. Such a cutout will prevent flow through the coordinate axis, but it is computationally advantageous to avoid the coordinate singularity. The size of this θ -cutout is determined by two considerations. First, light travel times across narrow φ -zones near the axis sets the timestep, so the cutout should not be too small. In this simulation, ∆ t = 2 . 53 × 10 -3 M . The θ cutout must not be too large; initial vertical field should not cross the θ -cutout before reaching the outer radial boundary at r = 500 M . A reflecting boundary condition is used along the θ -cutout. The θ zones were distributed logarithmically so as to concentrate zones around the equator. \nThe φ -grid spans the quarter plane, 0 ≤ φ ≤ π/ 2. The φ grid is uniform and periodic boundary conditions are applied. The use of this restricted angular domain significantly reduces the computational requirements of the simulation. Schnittman et al. (2006) examined the variance in surface density as a function of φ and found the characteristic size of perturbations to be around 25 · . Thus, while some global features may be lost by using a restricted domain, the character of the local MRI turbulence is captured.", '2.2. Magnetic Flux': "In analyzing the evolution of the large scale poloidal flux we will make use of the poloidal flux function, which corresponds to the φ -component of the magnetic vector potential (van Ballegooijen 1989; Heyvaerts et al. 1996; Reynolds et al. 2006), which we must calculate from the simulation data. In GRMHD, we directly evolve the components of the Faraday \ntensor, F µν using the alternative form of the induction equation (for further details see De Villiers & Hawley 2003): \n∂ δ F αβ + ∂ α F βδ + ∂ β F δα = 0 (4) \nThe space-space components of F µν are identified with the constrained-transport (CT) magnetic fields via B i = [ ijk ] F jk : \nB r = F φθ ; B θ = F rφ ; B φ = F θr . (5) \nThis identification allows us to write the induction equation in the familiar form \n∂ t B i -∂ j ( V i B j -B i V j ) = 0 , (6) \nwhere V i = U i /U t . The tensor F µ,ν is related to the magnetic induction in the fluid frame, B α by the relation \nF µν = glyph[epsilon1] αβµν B α U β , (7) \nwhere glyph[epsilon1] µν δ γ = √ -g [ µν δ γ ]. \nThe Faraday tensor can also be written in terms of a vector potential: \nF αβ = ∂ α A β -∂ β A α = A β,α -A α,β . (8) \nIt is convenient to define an azimuthally-averaged poloidal flux function, A φ . This can be derived from the azimuthally-averaged CT field data by noting that the total (spatial) derivative of A φ is \ndA φ = ∂A φ ∂r dr + ∂A φ ∂θ dθ. (9) \n¿From the definition of the Faraday tensor, we have that \nF φθ = A θ,φ -A φ,θ ; F rφ = A φ,r -A r,φ . (10) \nAfter azimuthal averaging, this reduces to \nF φθ = -A φ,θ ; F rφ = A φ,r . (11) \nThe space-space components of F µν are identified with the CT magnetic fields via B i = [ ijk ] F jk so that \nB r = F φθ = -A φ,θ ; B θ = F rφ = A φ,r . (12) \nWe therefore have that \ndA φ = F rφ dr -F φθ dθ = B θ dr -B r dθ. (13) \nTo find the flux from this differential, we could in principle integrate over any poloidal surface. For the radial motion of vertical flux through the disk, the surface of greatest significance is the equatorial plane. Because this surface is interrupted by the event horizon, we must stretch it over half of the event horizon. We will therefore define a flux function that is the sum of two parts: the radial flux through the inner horizon boundary, \nΨ( θ ) | r = r in = ∫ θ 0 dθ ' B r ( r in , θ ' ) , (14) \nand the vertical flux through the equator, \nΦ( r ) | θ = π/ 2 = ∫ r r in dr ' B θ ( r ' , θ = π/ 2) . (15) \nUsing these functions, we can determine how the accumulated flux through the top half of the event horizon depends on polar angle, Ψ( θ ), and how much total vertical flux has been brought to within radius r . The full flux function along the total path is \nA ( r, θ = π/ 2) = Ψ( π/ 2) -Φ( r ) . (16) \nThis corresponds to the net flux piercing the surface covering the top hemisphere of the black hole horizon and the equatorial plane out to radius r . The full vector potential component A φ ( r, θ ) is obtained by integrating the CT variables throughout the computational domain starting from r = r in and θ = 0. A ( r, θ = π/ 2) is therefore identical to A φ ( r, θ = π/ 2). For the three-dimensional simulation, VD0m3d, B r and B θ are integrated over over the φ -dimension prior to computing A φ ( r, θ ). Given the periodic nature of the φ direction this is sufficient to compute the total flux piercing the volume bounded by the equatorial plane and the black hole horizon. We note that this procedure is consistent with the approach adopted in the calculations of van Ballegooijen (1989), Heyvaerts et al. (1996), and Reynolds et al. (2006).", '3.1. Overall evolution': "The simulation begins with an isolated torus threaded with a column of vertical field. The first part of the simulation is marked by relatively rapid evolution of the coronal field and the surface layers of the torus. Figure 2 shows the evolution of the poloidal magnetic flux and the density distribution over the time period 1000 M ≤ t ≤ 2500 M . This process, along with the subsequent evolution of the flow is also shown in an animation included in the online \nedition of this work. We encourage the reader to refer to this material as it clarifies many of the issues that we shall discuss at length in the remainder of the text. As noted above, the initial state is not in equilibrium within the corona. There is differential rotation between the coronal gas and the torus, and the magnetic pressure within the vertical field column is not balanced by pressure within the initial atmosphere. The field expands radially, and stresses at the boundary of the disk begin to drive low density gas from the torus outward along field lines. In the inner half of the torus, low density surface layers move inward in two thin accretion streams above and below the equatorial plane, dragging poloidal field lines down to the black hole. This behavior was dubbed an 'avalanche flow' when it was observed in the axisymmetric MHD accretion torus simulations of Matsumoto et al. (1996), and it is a common feature to accretion models that begin with vertical fields threading the disk (e.g., Koide et al. 1999). These surface features develop well before field deeper within the torus is significantly amplified by the MRI. The surface layers in the outer half of the torus also evolve, but here mass flows radially outward along the field lines, carrying the field along as it does so. \nBy t = 2000 M net flux has begun to build up on the black hole horizon, well before any significant mass accretion has begun. This flux arrives there by a process we call the 'coronal mechanism,' which we discuss briefly here and will discuss again in greater detail in § 3.2. A typical field line participating in the coronal mechanism may be traced from the outer boundary at high latitude more or less vertically down to the upper surface of the torus, where it bends sharply toward the hole near the boundary of the denser region. It goes in some distance before doubling back (forming a 'hairpin' shape) and returning to the torus. There it passes through the equator, and turns inward to mirror its behavior on the other side of the equator. A short time later, the apex of each hairpin moves inward and closer to the equator, where, in the early stages of this simulation when there is little mass at small radii, it can encounter an oppositely-oriented hairpin from the other hemisphere. Reconnection ensues, causing the magnetic flux distribution to change essentially instantaneously. \nThe four panels of Fig. 2 show these structures in differing ways. In the first panel, an isolated hairpin field line can be clearly seen approaching very close to the horizon. By the time of the second panel, the bends in this field line have reconnected, attaching it to the event horizon and liberating a pair of closed loops. Also in the second panel, new hairpins can be seen forming along the top and bottom surfaces of the disk in the region 20 M < r < 30 M . These hairpins evolve further and become embroiled in the beginnings of disk turbulence (the channel modes) in the third and fourth panels. \nMagnetic field enters the extremely low-density funnel around the rotation axis by an offshoot of the coronal mechanism just described. There is initially no field within the \nfunnel. Once horizontal field lines have been brought close to the black hole, their radial, near-horizontal components become unstable to the formation of ballooning half-loops that rise upward into the funnel. These loops are the initial source of field for the funnel. Only the inner half of the rising loop enters the funnel and, as a result, the field direction changes sign abruptly at the funnel's outer edge along the centrifugal barrier. \nThe coronal process also occurs at radii exterior to the initial pressure maximum of the torus. There it acts to move net flux progressively outward. Above the disk surface, the field is stretched out radially, but subsequently bends back down toward the equator, where reconnection can occur. This transfers the net flux from field lines going through the torus (those lines now close in large loops) to vertical field lines at large radius. \nThus, in the early phase of the evolution the global topology of the field is rapidly rearranged through coronal motions without any significant evolution within the main disk. Although these features result from the initial conditions, the process illustrates how (in principle) coherent large scale poloidal flux can be rapidly moved over large radial distances. \nAlthough low density surface layers participate in the initial avalanche flow, the dense part of the torus is largely unaffected by these coronal motions. By t = 2500 M , however, the MRI reaches a nonlinear amplitude throughout the torus and the characteristic structures associated with the 'channel mode' have become visible within the torus. The next stage of evolution is illustrated in Figure 3 which shows snapshots every 500 M in time from 3000 to 4500 M . Gas in the torus moves radially both inward and outward in the characteristic channels as angular momentum is transported along field lines, which are themselves increasingly stretched into long radial filaments. The channel modes are unstable to nonaxisymmetric perturbations and rapidly break up, resulting in a turbulent disk without prominent radial features. In the axisymmetric simulations, these relatively coherent large-scale radial flows continue to dominate throughout the evolution. This artificial persistence of coherent motion is a significant limitation of two-dimensional simulations with vertical field. \nAccretion into the black hole does not begin until 1650 M , and the mass accretion rate at the inner edge of the disk reaches its long-term mean for the first time shortly after 4500 M . The mass accretion rate then continues to vary by factors of ∼ 2 around this mean through the remainder of the simulation. The disk achieves an approximate inflow equilibrium within r glyph[similarequal] 25 M from time 10 4 M onward. Prior to 10 4 M , the mass in the inner disk ( r ≤ 25 M ) rises steadily, but after that time, it changes by about 10%, and without any prevailing trend. The mass accretion rate averaged from 10 4 M until the end of the simulation at 2 × 10 4 M is flat to within 10% for r ≤ 25 M . The total mass on the grid at the end of the simulation is 78% of its initial value; of this, 18 . 8% has gone into the black hole. \nThe divergence-free nature of the magnetic field means that the total flux is rigorously conserved except for losses at the boundaries of the computational domain. To measure the evolution of the total flux we integrate along the black hole horizon from the axis to the midplane and then out along the equator to the outer boundary. Figure 4 shows this integral as a function of time. The figure shows that there is no flux lost through the outer radial boundary during the simulation, but considerable net flux is deposited on the horizon at the expense of flux through the equator. The majority of the flux brought to the horizon arrives by t glyph[similarequal] 5000 M , which implies that the net horizon field is created mostly by coronal motions, not through disk accretion. During the later stages of more or less steady-state accretion, the horizon flux grows much more slowly. Relative to its magnitude at t = 5000 M , the horizon flux increases by only 28% over the next 5000 M and adds only another 11% during the last 10 4 M . By contrast, the total mass accreted onto the black hole grows by a factor of 9 from 5000 M to 2 × 10 4 M , with a rate of increase that is approximately constant from glyph[similarequal] 10 4 M onward. \nThe appearance of the overall flow at the end of the simulation is given in Figure 5 which shows the azimuthally-averaged density distribution and the β parameter, and in Figure 6, which shows the vector potential component A φ at time t = 2 × 10 4 M . There are several distinct regions. First, there is a low density, strongly magnetized funnel in which the field lines are primarily radial. Second, there is a corona characterized by large-scale field lines that loop about creating islands and hairpins, with numerous current sheets separating closely-spaced regions of oppositely-directed field. Here there are regions of both relatively strong and relatively weak magnetization; β ranges from ∼ 0 . 1 to ∼ 10. A thin region of high β fluid running along the funnel wall marks a current sheet between field lines directed in opposite radial directions as well as the boundary of the region with strong toroidal field. Finally, the main disk body remains weakly magnetized, with β ranging from ∼ 10-100. The disk thickness H/R ranges from 0.1 to 0.15, using the vertical density moment (Noble et al. 2008) for the definition of H . As can be seen from Figures 3 and 6, substantial parts of the disk body (most of the disk inside r glyph[similarequal] 40 M ) are actually disconnected from the large-scale field, their magnetic connections having been broken by the reconnection events that transferred field both to the horizon and to the outside of the torus. In the outer half of this region (20 M glyph[lessorsimilar] r glyph[lessorsimilar] 40 M ), however, some field lines do pass through the equator and out through the disk and into the corona. \nThis initial magnetic field and torus configuration was also simulated in axisymmetry. While the initial evolutions of the two- and three-dimensional models are similar, once the MRI sets in within the torus the subsequent evolutions differ dramatically. In axisymmetry, the channel modes dominate throughout the simulation, creating coherent radial field lines that extend over large distances. These field structures strongly influence the accretion \nof both mass and magnetic flux. Figure 7 compares the flux through the horizon, the mass accretion rate and the specific angular momentum accreted in simulations VD0m3d and VD0m2d. The axisymmetric simulation is characterized by very large fluctuations in mass accretion rate. Similar large fluctuations are seen in the value of the magnetic flux through the horizon. Note the contrast between the net specific angular momentum j in carried into the hole by the accretion flow in the two- and three-dimensional simulations. For both plots the dashed line indicates the specific angular momentum of the innermost stable circular orbit at 6 M . The deep minima in the axisymmetric run correspond to strong torques created by radial field extending through the plunging region. These extended radial fields are associated with the coherent channel flows seen in axisymmetry, and they have an obvious strong effect on mass, angular momentum and magnetic fluxes carried into the black hole. The three-dimensional simulation is qualitatively different. In three dimensions, the channel modes quickly break up, yielding to genuine turbulence. The accretion rate into the hole fluctuates, but not nearly as strongly as in axisymmetry. Similarly, only in 3-d can one speak sensibly about a well-defined net accreted angular momentum per unit rest-mass, j in . On the basis of these contrasts, we conclude that axisymmetric simulations, while useful for preliminary investigations of various initial configurations, are of only limited utility in investigating the long term behavior of MHD turbulent disks.", '3.2. The coronal mechanism': 'The dominant mechanism for bringing flux to the event horizon in simulation VD0m3d is one that operates primarily in the corona, and not in the disk accretion flow proper. Although most of the flux is brought in during the first 5000 M in time, there is a slow increase in flux throughout the remainder of the simulation. The majority of this flux appears to be transported through the continuing operation of the coronal mechanism. The key aspect is that magnetic field is carried inward by infalling low density fluid. Although this coronal mechanism operates outside the disk where most of the mass accretion happens, it, too, depends on angular momentum transport to proceed, and the basic physical mechanism for angular momentum transport is likewise the same: magnetic torques. Wherever the vertical field is perturbed so as to create a radial component, orbital shear acting on the radial field in turn creates toroidal field. Moreover, whenever the orbital frequency decreases outward, the relative signs of the radial and toroidal fields are always such as to make the r -φ element of the magnetic stress tensor, b r b φ / 4 π , negative. Such a stress transports angular momentum outward, tending to accentuate inflow, and therefore growth in the radial field component; this is, of course, the mechanism of the magneto-rotational instability. \nAn example of the coronal mechanism in action is given in Figure 8. In this figure, color contours show the strength of the azimuthally averaged Maxwell stress per unit mass, overlaid with field lines derived from A φ ( r, θ ). For clarity we focus on those field lines that still pass through the equator in the accretion disk. The first frame, corresponding to t = 6000 M , features a field that runs from the equator vertically upward through the disk and then down toward the hole before reversing and moving radially outward. The strong stress per unit mass in the vicinity of the hairpin carries angular momentum away from the associated matter, enabling it to move inward. In the next frame at t = 6200, the hairpin moves in radially toward the black hole. The next two frames show a close up view of the innermost part of the accretion flow at times t = 6440 and 6460 M . By the time of the first one, the inner portion of the hairpin has reached and crossed the horizon. Only 20 M later, the inside hairpin has reconnected, forming a closed field loop. Thus, the net result of the coronal mechanism at this stage is to attach those fieldlines forming the exterior of the hairpin to the black hole. These fieldlines then extend out into the corona, ultimately attaching to the disk at much larger radius. \nAlthough net local inflow of magnetic field is a necessary condition for flux accretion, it is not a sufficient one; the net flux changes only when reconnection occurs. Local motion alone is insufficient because magnetic flux is an essentially global concept, so local motion is not enough to determine net flux transport. This distinction is emphasized by contrasting the local magnetic inflow rate with global measures. We define the local rate by \nV Φ ( r, θ ) ≡ 1 2 π ∫ dφV r B θ (17) \naveraged over the portion of the simulation during which the trapped flux grows slowly (5000-2 × 10 4 M ). This is shown in Figure 9, for which the radius was chosen to be 15 M . As this figure shows, the local flux inflow speed is considerably greater just outside the funnel ( | π -θ | glyph[similarequal] 0 . 2 π ), where it is glyph[similarequal] 0 . 5-1 × 10 -4 , than in the midplane, where it is nearly two orders of magnitude smaller. \nThis local rate of flux inflow is, however, much larger than the global rate of flux accumulation. The global rate is ∂A φ /∂t , which can be inferred from Figure 4. For the purpose of this figure, the magnetic flux was separated into the flux piercing the midplane, Φ, and the flux through the upper hemisphere of the black hole, Ψ. During the early stages of the simulation (before 5000 M ), ∂ Ψ /∂t glyph[similarequal] 1 × 10 -5 , but this rate of increase diminishes at later times by at least an order of magnitude. Thus, the local inflow rate, V Φ ( θ glyph[similarequal] 0 . 2 π, 0 . 8 π ) glyph[similarequal] 1 × 10 -4 at r = 15 M during the latter 3/4 of the simulation, is as much as two orders of magnitude greater than the net global rate. In other words, local field inflow alone does not suffice to create global flux inflow. \nLikewise, reconnection on its own is also a necessary condition for flux accretion, but not a sufficient one. In order for flux accretion to occur, reconnection must result in a global rearrangement of the field topology. This fact is illustrated by the field structures displayed in Figure 8. In that figure, a narrow current sheet lies between the two sides of the hairpin, permitting reconnection to happen there. In order to transfer magnetic flux inward through the accretion flow and increase the net flux in the funnel, reconnection must preferentially destroy the side of the magnetic hairpin that lies closest to the midplane. This is possible because reconnection can also occur across the equator, something that the funnel field cannot do. Figure 10 illustrates the idea. Coronal hairpin field loops bring field down to the inner disk, returning along the disk back out to large radius where they eventually pass through the equator and connect to another hairpin in the opposite hemisphere. The innermost bend of these hairpins can attach to the horizon. When this happens, the field appears to be almost a closed loop, with one end of the loop inside the horizon (e.g., the two inner-disk field lines in the gray region of the diagram). If the two oppositely-directed portions of one of these loops can meet at intermediate radius, the equatorial flux jumps inward. This inward jump drastically shortens the time until the entire field loop can accrete onto the black hole. When that happens, flux of one sign is removed from the horizon above the equator and flux of the other sign is removed from below the equator, leaving a net increase in flux piercing each horizon hemisphere. \nAn example of a reconnection event within VD0m3d that leads to this sort of global flux rearrangement is shown in Figure 11. In this figure, color contours denote the radial component of the CT-field, B r , indicating the direction of the fieldline, which are again overlaid with white contours derived from the azimuthally averaged A φ ( r, θ ). Initially ( t = 14600 M ), fieldlines connect large radii to the black hole event horizon through the funnel region (deep blue region below midplane) and then proceed back out into the disk close to the midplane (yellow region below midplane). These fieldlines cross the midplane at large radius and proceed back to the black hole north of the midplane (light blue region north of midplane), before finally proceeding out to large radius through the funnel region (red region north of midplane). At time t = 14620 M , the oppositely directly fieldlines that lie just above and below the midplane have reconnected at r ∼ 6 M , and the resulting closed fieldloop (which is still attached to the black hole) is rapidly accreted. This process transfers flux from the outer disk to r ∼ 6 M (where the reconnection event occurs) and then down to the black hole as the field loop is accreted, arriving at the black hole with the outer edge of the loop. \nThe coronal mechanism resembles the flux advection mechanism proposed by Rothstein & Lovelace (2008) in the sense that most flux transport takes place outside of the disk body, and that flux in the corona moves inward because magnetic stresses send angular momentum \noff to infinity. They emphasize the contrast between the turbulent disk and non-turbulent flow outside of the disk. We likewise observe that large-scale, relatively laminar flows within the low-density corona easily move field about; flux-freezing is an excellent approximation there. On the other hand, there are also several points of contrast between our results and the model of Rothstein & Lovelace (2008). They argue that in the disk corona, defined as the region where β ∼ 1, the T z φ component of the magnetic stress tensor could transport angular momentum outward as effectively as the T R φ component to which attention is more often directed, allowing the disk surface to move inward. Although this is similar to what we see in our simulation during the initial avalanche phase, it is different from the dynamics prevailing at later times. During the accretion phase of the simulation, the coronal field lines develop MRI-like bends at high altitudes above the disk ( >> H ), so that the inflow occurs in the corona rather than along the disk surface. In addition, as shown by Figure 12, which contrasts the azimuthal averages of -b r b φ /ρ and -b θ b φ /ρ , there is very little net angular momentum flux per unit mass in the polar angle direction except in the funnel (at latitudes not too far from the midplane, b θ b φ glyph[similarequal] b z b φ ). Although the magnitude of T θ φ is often comparable to T r φ , it has very nearly the same probability of having either sign, so that its mean value is small. On the funnel edge, T θ φ is more likely to take angular momentum away from the disk but has magnitude on average somewhat smaller than T r φ , while in the outer corona, on average it brings angular momentum toward the midplane.', '3.3. Inflow in the disk body': "The majority of the net flux on the black hole is brought there by the coronal mechanism during the initial 5000 M of time. However, the flux on the horizon does increase over the last 1 . 5 × 10 4 M and, while the coronal mechanism continues to operate during this interval, it is possible that part of the flux might be brought in through the accretion disk itself. As the right panel of Figure 9 makes plain, the local magnetic inflow rate V Φ is roughly two orders of magnitude smaller in the midplane than it is at the funnel edge. Nonetheless, its magnitude is still consistent with the total flux inflow rate during the latter part of the simulation. It may therefore be possible for flux advection directly associated with the accretion flow in the disk body to account for some part of the flux accumulation during the quasi-steady accretion phase. In this section we attempt to estimate the size of that contribution. \nWe begin by comparing the rate of magnetic flux accretion to the rate of mass accretion. If magnetic flux and mass moved inward in lock-step, the functions \nM ( r, t ) ≡ ∫ t 0 dt ' ˙ M ( r, t ' ) /M 0 (18) \nand \nA ( r, t ) ≡ A φ ( r, t, π/ 2) /A φ ( r max , 0 , π/ 2) (19) \nwould be identical. These quantities are normalized to the initial mass, M 0 , and the initial flux; r max is the outer radius of the initial torus. In Figure 13, we show the time-dependence of both these quantities at two fiducial radii, just outside the event horizon (left panel) and at r = 20 M (right panel). The former, of course, tells about the flux attached directly to the black hole and the mass deposited there. The difference between the two plots is the mass and flux within the accretion disk between the horizon and r = 20 M ; here we will refer to this region as the 'inner disk.' \nIn evaluating Figure 13, one must allow for an important distinction between the histories of mass and magnetic flux: the mass of the black hole must increase monotonically (and the mass of the inner disk, although not required to do so, generally does), but the net magnetic flux can increase or decrease, both by magnetic reconnection and by the radial motion of closed loops. Thus, the inflow of magnetic flux is highly intermittent, and the timederivative of flux within some radius can easily be negative. On short timescales, the ratio of the rate of flux accretion to mass accretion can vary by at least an order of magnitude, as well as fluctuate in sign. For this reason, a sensible comparison of the rate of magnetic flux inflow relative to that of mass can be made only in regard to long-term time averages. In addition, in order to evaluate the efficiency of this process in real disks, we must restrict these averages to times when the inner disk was in approximate inflow equilibrium: for this simulation, that means the latter half of the data, from t = 10 4 M to t = 2 × 10 4 M . Within this span, we smooth A ( t ) by taking a running average 1000 M wide and then compose the ratio of the change in A to M from 10 4 M to 2 × 10 4 M using the smoothed data. We find that ∆ A / ∆ Mglyph[similarequal] 0 . 47 at the horizon, and glyph[similarequal] 0 . 50 at r = 20 M . In other words, relative to the initial amount embedded in the accretion flow, magnetic flux moves inward roughly half as fast as mass, and these rates are the same at both the horizon and r = 20 M . If the mass is in a state of inflow equilibrium, the equality of these two ratios shows that the magnetic flux in the inner disk is also in equilibrium. \nThis ratio of inflow rates may be contrasted with the local ratio of magnetic flux to mass. Looking beyond the short-timescale fluctuations in the plots of A ( t ) in Figure 13, one sees that the average values of the flux at the horizon and the flux contained within r = 20 M are very similar. In other words, the vertical magnetic flux through the inner disk, Φ( r = 20 M ), contributes very little to A . Time-averaging over the last 10 4 M of data, we find that it is only glyph[similarequal] 0 . 86% of the initial flux, and it appears that what flux there is is confined to the outer part of the inner disk. By contrast, averaged over the same period, 5% of the initial mass can be found in the inner disk. Thus, in the inner disk, the time-averaged flux/mass ratio is only 17% of the initial value. \nTo reconcile this flux/mass ratio with the accreted flux/mass ratio, there are only two possibilities: either the magnetic flux moves in at the same rate (or slower) than the mass and (at least) 2 / 3 of the flux reaching the horizon between 10 4 M and 2 × 10 4 M arrived there via a different route (most likely the coronal mechanism), or the magnetic flux on average moved inward three times faster than the mass. We view the former alternative as much more plausible. Note, however, any flux delivery from outside the inner disk must be balanced by flux losses from the inner disk to the horizon because the magnetic inflow rate at r = 20 M is nearly the same as at the horizon. \nNext we further explore the nature of the magnetic flux equilibrium in the inner disk, using two different views of this quantity. Figure 14 shows the distribution of the vertical flux function Φ( r ) time-averaged over the last 5000 M of the simulation and normalized to the initial flux value. For comparison, we also plot in that figure both the initial flux distribution and the final mass distribution in the disk, both likewise normalized to the initial total. The radial derivative d Φ /dr shows the magnitude and sign of the vertical magnetic field piercing the equator. We see that, like the mass, vertical flux has spread away from its initial location, both in and out, but, as already mentioned, a much smaller part of the net magnetic flux than the mass resides in the inner disk. \nThe time-dependence of the magnetic flux in the inner disk over the last 10 4 M in time is illustrated in Figure 15. As this figure shows, not only is the magnetic flux contained within the inner disk a rather small part of the total, it fluctuates in time, frequently going negative. Little sign of any long-term trend can be seen, consistent with our contention that the inner disk magnetic flux has reached a statistical equilibrium. The frequent sign changes of the local net flux suggest that the poloidal magnetic field lines in the inner disk predominantly close within the inner disk. \nMoreover, the magnetic field corresponding to the net flux is a very small fraction of the total field within the disk. Figure 16 shows the azimuthally averaged value of B θ through the equator at the end time as a function of radius, along with the initial value. The MRI has generated considerable field of both signs throughout the disk. Local regions of the disk have a net vertical flux, but their contrasting signs lead to little contribution to the total disk flux. The global net flux is present only as a slight positive excess moving both inward and outward with time. \nTo reach the state described by these figures, most of the large-scale magnetic connections to the disk matter must have been reconnected away. But this should come as no surprise because we have already seen that a significant part of the original flux has moved by reconnection from the disk to the horizon. The matter in the inner disk has little net flux because it has been largely bypassed through the coronal mechanism. Thus, while the \nMRI has created a large effective turbulent viscosity in the sense that considerable mass has accreted from the location of the initial torus, very little net flux has moved with it. In other words, within the disk, turbulence created by the MRI transports angular momentum rapidly and drives accretion, but very little flux moves with the accreting mass.", '3.4. Strength of the Funnel Field': 'Lastly, we raise the question of whether in the very long-term the magnetic flux on the horizon would continue to grow. If it is regulated by a combination of the magnetic and gas pressure near r = 6 M , and these are in an approximately time-steady state, one might expect that further flux accumulation on the horizon would have to stop. Figure 17 supports this idea. This figure shows that the magnetic pressure inside the funnel, the magnetic pressure in the inner accretion flow, and the gas pressure in the inner accretion flow are well correlated with one another in a temporal sense, even while all three change by more than two orders of magnitude over the course of the simulation. The time-dependent data show fluctuations that are so large, however, that the much slower accumulation of flux seen in Figure 13 is well below the noise, so our data on magnetic and gas pressures cannot answer any questions about long-term saturation of magnetic flux attached to the horizon. \nThat the magnetic and gas pressures should be closely related is an old idea (Rees et al. 1982; MacDonald & Thorne 1982), but there is also a long history of controversy about whether the magnetic field pressure in the funnel should be closer in magnitude to the (poloidal) magnetic pressure or the gas pressure in the inner disk. For example, Ghosh & Abramowicz (1997) and Livio et al. (1999) argued that as the poloidal magnetic field strength in the disk is subthermal, the funnel field strength should be regulated by it rather than the gas pressure. In the simulation presented here, the total , i.e. poloidal plus toroidal magnetic pressure in the disk is smaller than the gas pressure for the duration of the simulation (i.e. the field remains subthermal), and the magnetic pressure from the field in the funnel lies between these two candidate regulators, sometimes closer to one and sometimes closer to the other. Thus, the funnel field is consistently stronger than the poloidal field in the disk. \nThe relevance of the inner disk poloidal field to the strength of the funnel field seems to be rather limited based upon what we observe about the process carrying flux to the horizon. The field in the funnel is nearly radial, and its total intensity is determined by the poloidal flux that is delivered to the black hole over the course of the simulation. This flux delivery system is the coronal mechanism, and so it should not be surprising that the poloidal flux within the accretion disk itself plays at most a secondary role (e.g., by determining the rate of reconnection and accretion of flux loops). Instead, simple pressure-matching in the \ninner regions of the accretion flow-be it gas, (total, mostly toroidal) magnetic or radiation pressure-appears to be important in regulating how much flux may be attached to the black hole. We speculate that when the flux on the horizon has a total field intensity larger than would be consistent with the gas and magnetic pressure in the nearby accretion flow, the rate of reconnection along the funnel field boundary rises, so that the field approaching the black hole is entirely in closed loops and does not add to the net flux on the horizon.', '4. Summary, Discussion & Conclusions': "In this paper we investigate the evolution of an accretion torus embedded in a large-scale vertical magnetic field orbiting a Schwarzschild black hole, with a view toward studying how magnetic flux moves relative to the accretion flow. The simulation stretched over 2 . 0 × 10 4 M in time, corresponding to 80Ω -1 at the initial torus pressure maximum, long enough to establish inflow equilibrium in the inner disk for the second half of the simulation. Of particular interest is how the net vertical field evolves, and whether or not a field distribution consistent with the formation of jets or winds can develop. We trace the evolution of the net poloidal flux distribution with a particular focus on net flux that becomes attached to the black hole. Our primary result is that a significant fraction of the initial fluxglyph[similarequal] 27%-is brought to the black hole horizon, even though only a rather smaller fraction of the initial massglyph[similarequal] 19% is accreted. The flux attached to the horizon supports a coherent poloidal field within the evacuated axial funnel, a requirement for the creation of a Blandford-Znajek type jet (if the black hole rotates, which it did not in this simulation). The mass and flux are carried inward through distinct mechanisms: the mass by turbulent stresses within the accretion disk, and the flux by large-scale motions in the low density corona. \nMost of the global magnetic flux motion is mediated by a novel mechanism that we have dubbed the coronal mechanism. Rather than the gradual 'diffusive' process that had been the focus of most previous discussions of magnetic flux motion in accretion flows, in this mechanism the flux is brought directly down to the horizon as the net flux jumps discontinuously when large-scale magnetic loops reconnect across the equator. The same orbital shear that creates angular momentum transport in the disk body by correlating radial and azimuthal magnetic field also acts in the corona; the difference is that in the corona the resulting stress leads to the formation of large-scale loops stretched rapidly inward rather than to MRI-driven turbulence. These stretched loops can reconnect to oppositely-directed field at much smaller radius than their footpoint, leading to sudden macroscopic radial changes in the location of magnetic flux. Although the coronal mechanism acts particularly rapidly during the initial transient phase of the simulation, it continues to dominate flux \nmotion throughout the simulation, including the long period during which the inner disk is in approximate inflow equilibrium. We might therefore reasonably expect it to be a property of actual accretion disks. \nThe same reconnection that creates global relocation of net flux simultaneously creates closed field loops within the disk body. For this reason, the flux/mass ratio within the accretion flow is suppressed by a factor of order unity relative to what it is in the initial state. Earlier global disk simulations have, in most cases, assumed zero net flux for reasons of simplicity and computational convenience; the coronal mechanism might, in fact, make this assumption a reasonable approximation to the magnetic field in typical disks. The suppression of flux/mass in the disk body also makes conventional advection of flux in direction association with mass accretion relatively inefficient. \nAs seen in previous simulations (particularly Beckwith et al. 2008a), if the overall field topology at the black hole is dipolar, the funnel field can be relatively immune to reconnection and hence long-lived. We conjecture that the funnel field amplitude is determined by pressure balance with the gas and (total, mostly toroidal) magnetic pressures in the inner disk and coronal regions. Future experiments could test this hypothesis by running much longer simulations with more available net field to see if the field strength levels off or continues to build. \nAs with any simulation, our results are subject to some uncertainty due to the limitations inherent in a numerical solution. One concern is the importance of reconnection to the overall flux evolution. In our numerical simulation, reconnection occurs at the grid scale when oppositely signed field components are brought together. Although leaving the microscopic rate of such an important process to numerical effects is a concern, the overall motions that lead to the formation of current sheets and subsequent reconnection area driven by events on much larger scales. In this sense, we might argue that the rate of reconnection seen in the simulation is relatively independent of our gridscale. We have not, however, demonstrated that our results are numerically converged, although the global field motions in the axisymmetric simulations carried out at two different resolutions were very similar. Furthermore, the thermodynamics of these simulations are incomplete because energy lost to numerical magnetic reconnection and numerical cancellation of fluid velocities is not captured; conversely, we do not account for radiation in any way, either as a cooling agent or as a contributor to the pressure. One thing that is clear, regardless of any other numerical limitation, is that axisymmetric simulations are of limited utility. In addition to the constraints associated with the anti-dynamo theorem, the vertical field channel modes remain dominant throughout the simulation, giving a qualitatively distinct evolution at late time. Three dimensional simulations are essential for studying long term, steady-state behavior. \nTo place our somewhat surprising results in context, it is worthwhile contrasting them with previous suggestions about how magnetic flux moves through accretion flows. These are quite disparate, for they have in general been based on approximate analytic arguments. We examine three principal approaches, of which one has received much more attention than the others. \nThe greatest amount of attention has been given to a picture in which the matter moves inward by an unspecified angular momentum transport process (called 'viscosity', but not thought to be literally that) while the magnetic field diffuses relative to the matter through another mechanism called 'turbulent resistivity', but similarly unattached to specific microphysics (e.g., van Ballegooijen 1989; Lubow et al. 1994; Heyvaerts et al. 1996; Ogilvie & Livio 2001). The ratio between this effective viscosity and resistivity (the nominal Prandtl number), would then determine whether the flux mostly moves inward with the mass, or instead diffuses outward relative to the mass fast enough to avoid much net inward motion. \nThis approach rests upon two little-examined assumptions: First, that the overall flow can be regarded as if it were laminar and time-steady, and second, that the behavior of the underlying turbulence can be reduced to parameters (an advection rate and a diffusion, i.e., resistivity, coefficient) whose values are independent of the magnetic field. With these assumptions, field motion in the disk body governs field motion far away from the disk, and one may describe the field evolution in the language of linear diffusion. Unfortunately, neither assumption applies to real accretion flows. \nIt is now well-established that angular momentum transport in accretion disks is due to turbulent magnetic stresses driven by the MRI. Consequently, the magnetic field structure is neither laminar nor time-steady. Even in the corona, where the MRI per se does not occur, the same basic physics leads (as we have earlier discussed) to irregular field motions. There is therefore no direct connection between local field motion deep inside the disk (e.g., resistive diffusion) and the position of the corresponding field lines far away. Moreover, as we have emphasized, flux is a global concept, not a local one; the coronal mechanism, for example, could never be described by a local theory of this sort. \nThe second assumption, that the evolution of the field can be described in terms of simple resistive and viscous diffusion operating on a background field gradient is similarly problematic. Accretion is driven by nonlinear magnetic stresses; because the effective inflow velocity arises only from an average of a strongly fluctuating velocity field, it is non-local in both time and space. Similarly, the breakdown in ideal MHD happens primarily by driven reconnection, the rate of which depends on the structure and magnitude of both the magnetic and velocity fields. As a result, neither the mean rate at which magnetic field is carried from place to place nor any tendency for its structure to smooth is either independent of or linear \nin the field strength. \nIn fact, the formulation in terms of a competition between 'viscous' and 'resistive' diffusion breaks down in an even more fundamental way. Where there is net flux passing through the disk body, the very same MHD turbulence that transports angular momentum simultaneously decouples the flux from the matter through turbulence-driven reconnection. Field lines associated with flux passing through the equator develop bends that readily break off as closed loops. The matter moves inward with the closed loops while the flux stays in place. Thus, in dramatic contrast to the conventional prediction, rapid 'viscous' accretion can, and apparently does, co-exist with largely stationary flux, while the 'resistivity' that permits this decoupling has no particular impact on smoothing the field distribution. \nMoreover, there is relatively little net flux passing through the disk because reconnection associated with the coronal mechanism detaches it efficiently. In other words, most of the flux motion occurs in an 'end-run' that bypasses the disk altogether, so that it has rather little to do with any activity in the disk body, 'viscous', 'resistive', or other. \nLocal smoothing of field structures can occur, but it is questionable whether one can define an overall diffusion coefficient to describe its rate. For example, Guan & Gammie (2009) have recently attempted to quantify magnetic diffusion in the context of MRI-driven MHD turbulence using shearing box simulations. They imposed a sinusoidally varying vertical field with an amplitude above the background due to the MRI-turbulent flow and observed the subsequent decrease in this mode's Fourier power. The peak vertical field amplitudes studied vary from 10-80% of the initial toroidal field strength. They found that the decay time for the imposed feature is several tens of Ω -1 , with the decay rate an increasing function of the perturbation amplitude. These results indicate that local field gradients can be smoothed within the turbulent flow, but it is unclear to what degree the inferred average diffusion coefficient depends on the specific situation studied. In particular, in accretion disks, the net field, the part determining the flux, is, especially in the disk body, small in magnitude compared to the fluctuating turbulent field. When the dynamics are strongly nonlinear, spreading of structures in this small net field may proceed in a way very different from spreading of larger amplitude structures; in fact, the motion of the net field may have much more to do with the dynamics of the fluctuating field than to its own disposition. \nFor all these reasons, therefore, we see no useful way to describe the results of the present simulation in the traditional language of arbitrarily specified transport coefficients applied to gradients of the time-averaged magnetic field. \nTwo other concepts have recently been studied. One of these (Rothstein & Lovelace 2008; Lovelace et al. 2009) in some respects is a blend of the 'advection/diffusion' picture \nand the coronal mechanism. On the one hand, it uses the conventional formalism of fixed transport coefficients; on the other hand, it relies heavily on field line motions in the corona. These two approaches are united by supposing that the MRI and its resulting turbulence are suppressed in the corona, so that it suffers neither 'turbulent viscosity' nor 'turbulent resistivity'. Consequently, coronal motion is dominated by vertical transport of angular momentum, in a magneto-centrifugal wind when the effective Prandtl number is large, and in a Poynting flux-dominated outflow otherwise. The result is that (for most parameters), the upper surface of the disk moves inward, carrying the footpoints of the large-scale flux with it, while the flow in the midplane is outward . \nAs we have previously discussed, the fundamental physical element of the MRI, the creation of substantial magnetic stresses by orbital shear, does operate within the corona, and strongly-this is the foundation of the coronal mechanism. Despite these coronal motions, this is not a significant source of angular momentum transport for the main accretion disk. Although the instantaneous amplitude of the magnetic T θ φ stress is typically sizable, outside the funnel its sign fluctuates so that we see no significant net vertical transport of angular momentum from the disk. In addition, the net fluid radial velocity in the midplane, although small compared to its rms value, is inward. Nonetheless, our results are consistent with their suggestion that magnetic stresses carrying angular momentum outward through the corona can be effective in moving field lines inward. Because the mass density in the corona is so low, removing a comparatively small amount of angular momentum can lead to large-scale field motions. Because the subsequent motions are not dominated by turbulence, flux freezing remains effective within the coronal plasma. \nIn the second of the two recent suggestions for how flux moves in accretion flows, Spruit & Uzdensky (2005) propose that the flux motion is not controlled by a simple gradient, but rather by the dynamics of intermittent field bundles. These authors suggest that magnetized patches could accrete relatively rapidly from angular momentum losses through a wind. Further, if such fields were strong enough to suppress the MRI, then turbulence, and subsequently magnetic reconnection, would be greatly reduced for those patches. While such magnetized patches would not themselves lead to a significant net magnetic flux passing through the disk per se , they could accumulate at the central black hole until the field strength itself suppressed further accretion. Such a scenario is similar to that of the magnetically arrested accretion model of Narayan et al. (2003). Strong nonaxisymmetric field structures appear in a simulation of Igumenshchev (2008) where a strong field is continually injected at the outer boundary, providing some support for the Spruit & Uzdensky (2005) concept. In our simulation, we do see nonaxisymmetric variations in the vertical flux throughout the disk, but not in the form of local pockets of intense vertical flux, nor do we see either any significant disk wind or any interruption of the accretion flow due to \n'magnetic arrest.' Nonetheless, we agree with Spruit & Uzdensky (2005) in emphasizing the intermittency of the flow and the potential importance of nonaxisymmetries. \nThe study carried out here is, of course, only a first step, and the results presented here point to several avenues of investigation with future numerical experiments. For example, simulations that continue for longer times and begin with net field at larger radii can better explore the efficacy of the coronal mechanism. In particular longer simulations with more available net field are required to see if the field strength in the funnel levels off or continues to build over time. Simulations in the Kerr metric with non-zero spin parameters could also study directly how magnetic flux evolution and the coronal mechanism relate to jet launching and collimation. The requirements of three spatial dimensions and increased spatial resolution will continue to make these investigations challenging. \nThis work was supported by NSF grant PHY-0205155 and NASA grant NNX09AD14G (JFH), and by NSF grant AST-0507455 (JHK). We thank Sean Matt, Jim Pringle, Chris Reynolds, Charles Gammie and Scott Noble for useful discussions. We acknowledge JeanPierre De Villiers for improvements to the algorithms used in the GRMHD code. The simulations described were carried out on the Teragrid Ranger system at TACC, supported by the National Science Foundation.", 'REFERENCES': "Balbus, S. A., & Hawley, J. 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A., & Goodman, J. 2008, ApJ, 682, 608, 0803.0337 \nvan Ballegooijen, A. A. 1989, in Astrophysics and Space Science Library, Vol. 156, Accretion Disks and Magnetic Fields in Astrophysics, ed. G. Belvedere, 99-106 \n<!-- image --> \nFig. 1.- Initial torus and field configuration. White contours denote magnetic field lines, color contours density distribution (left panel) and the gas β parameter (right panel). \n<!-- image --> \nFig. 2.- Initial evolution of the flow in simulation VD0m3d showing the advection of the vertical field from the initial location of the torus to the black hole. White contours are magnetic field lines, color contours show the density distribution. From left to right, the panels correspond to times t = 1000 , 1500 , 2000 , 2500 M . The color scale is the same as in Figure 1. By t = 2500 M , the funnel region has become strongly magnetic pressure dominated and two large scale channel modes have formed within the initial torus. \n<!-- image --> \nFig. 3.- Growth and subsequent break up of the vertical field MRI channel mode in simulation VD0m3d. White contours are magnetic field lines, color contours show the density distribution. From left to right, the panels correspond to times t = 3000 , 3500 , 4000 , 4500 M . The color scale is the same as in Figure 1. By t = 4500 M , the disk has become turbulent; large scale MRI modes persist within the corona. \n<!-- image --> \nFig. 4.- Total flux A φ integrated from the axis along the horizon to the equator and then out along the equator to the outer boundary. The equatorial component, Φ, and the horizon component, Ψ are shown separately. The vertical scale is in units of the total initial flux. The total flux is unchanged through the evolution; losses in the equatorial flux are accounted for in net flux through the horizon. \n<!-- image --> \n<!-- image --> \nFig. 5.- Azimuthally-averaged gas density (left panel) and plasma β (right panel) at time 2 × 10 4 M in simulation VD0m3d. \n<!-- image --> \nFig. 6.- Contours of A φ ( r, θ ), displayed both in color and as poloidal field lines (isosurfaces of A φ ( r, θ )) at time 2 × 10 4 M in simulation VD0m3d. \n<!-- image --> \nFig. 7.- Comparison between results from the two-dimensional (VD0m2d) and threedimensional (VD0m3d) simulation. Top panel shows the flux through the horizon, Ψ for VD0m2d (solid line) and VD0m3d (dashed line). The middle panel shows the net mass accretion rate through the horizon in units of the initial torus mass. The bottom panel shows the specific angular momentum carried into the hole as a function of time for simulations VD0m2d and VD0m3d. The strong coherent radial fields in the two-dimensional system create large fluctations in these quantities. \n<!-- image --> \nFig. 8.- Evolution of a magnetic 'hairpin' in the corona during the turbulent steady state. Color contours show the azimuthal average of stress per unit mass, -b r b φ /ρ , and are overlaid with the poloidal field structure (white lines), which are kept fixed between the panels. From left to right, the top panels show times t = 6000, and 6200 M (top row). The bottom row shows the near-hole region at times t = 6440 and 6460 M . During this time a reconnection event takes place within the field hairpin. \n<!-- image --> \nFig. 9.- Azimuthal mean of -V r B θ (15 M ), averaged over the time span 5000-1 . 5 × 10 4 M at r = 15 M . Two θ ranges and scales are used: a scale appropriate for the corona and excluding only the funnel region (top), and only the disk proper plotted on a scale appropriate to the value there. \n<!-- image --> \nFig. 10.- Schematic of the field line structure in the coronal mechanism. Field lines within the coronal are carried in toward the black hole, forming a hairpin-like structure. When the hairpin connects to the horizon flux is added to the funnel field and opposite-signed flux is added to the disk (shaded region). Reconnection across the equator (dashed line) allows this field to form loops that accrete. Accretion of those field loops results in an increase of the net horizon flux. \n<!-- image --> \nFig. 11.- Reconnection event that occurs in the disk and leads to the rapid transport of poloidal magnetic flux from the outer disk to the black hole. Color contours show B r and are overlaid with the poloidal field structure (white lines), which are kept fixed between the panels. From left to right the panels show t = 14600, 14620, and 14640 M . \n<!-- image --> \nFig. 12.- Azimuthal average of -b r b φ /ρ (left panel) and -b θ b φ /ρ (right panel) at the same time as shown in Fig. 5. Regions with -b i b φ /ρ > 1 are colored dark red, regions with -b i b φ /ρ < 10 -5 (including -b i b φ /ρ < 0) are deep blue. The sign of -b θ b φ /ρ is adjusted so that positive values correspond to electromagnetic angular momentum flux directed away from the disk. The azimuthal average of -b θ b φ is much smaller than that of -b r b φ everywhere outside the funnel. \n<!-- image --> \nFig. 13.- Fractional accretion of magnetic flux A (solid curve) and mass M (dashed curve) as functions of time. Because we stored shell-integrated mass accretion rates every 1 M in time, but the full 3-d data required to compute the vector potential only every 10 M , the latter curve has greater time resolution than the former. (Top) At r = 2 . 1 M . (Bottom) At r = 20 M . \n<!-- image --> \nFig. 14.- The net vertical flux through the equator Φ( r ), integrated from the black hole horizon outward, averaged over the last 5000 M of the simulation and normalized to the inital total value ('Vertical Flux, Φ'). The initial Φ distribution is also shown (dashed line). The net flux has spread through the disk over the course of the simulation. For comparison, the mass distribution M ( r ) is shown (labeled 'Mass') for the end time, t = 2 × 10 4 M , likewise normalized to its initial total value. \n<!-- image --> \nFig. 15.- Space-time contours of the net vertical flux through the equator Φ( r ), integrated from the black hole horizon outward for the inner disk during the second half of the simulation, normalized to the total initial flux. Never very large in magnitude, the integrated magnetic flux in this region fluctuates in sign. \n<!-- image --> \nFig. 16.- Azimuthal average of the absolute value of |B θ | (solid line) and the average of B θ (dot-dashed line) along the equatorial plane at t = 2 × 10 4 M (solid line) along with the initial B θ distribution (dashed line). The vertical field remains relatively strong but experiences fluctuations in sign throughout the disk body. The net flux is a small positive bias in the vertical field value. \n<!-- image --> \nFig. 17.- Azimuthally-averaged magnetic pressure as a function of time at two locations: inside the funnel near its base ( r = 6 M , θ = 0 . 044 π ; solid curve); and in the inner accretion flow ( r = 6 M , θ = π/ 2; dotted curve), and azimuthally-averaged gas pressure in the inner accretion flow (also r = 6 M , θ = π/ 2; dashed curve). \n<!-- image -->"}

1996NuPhB.478..181C

Non-extreme black holes from non-extreme intersecting M-branes

1996-01-01

['-']

[]

We present non-extreme generalisations of intersecting p-brane solutions of eleven-dimensional supergravity which upon toroidal compactification reduce to non-extreme static black holes in dimensions D = 4, D = 5 and 6 ⩽ D ⩽ 9, parameterised by four, three and two charges, respectively. The D = 4 black holes are obtained either from a non-extreme configuration of three intersecting five-branes with a boost along the common string or from a non-extreme intersecting system of two two-branes and two five-branes. The D = 5 black holes arise from three intersecting two-branes or from a system of an intersecting two-brane and five-brane with a boost along the common string. The five-brane and two-brane with a boost along one direction reduce to black holes in D = 6 and D = 9, respectively, while a D = 7 black hole can be interpreted in terms of a non-extreme configuration of two intersecting two-branes. We discuss the expressions for the corresponding masses and entropies.

[]

https://arxiv.org/pdf/hep-th/9606033.pdf

{'Non-Extreme Black Holes from Non-Extreme Intersecting M-branes': 'Mirjam Cvetiˇc 1 ∗ and A.A. Tseytlin 2 † \n1 \nDepartment of Applied Mathematics and Theoretical Physics, \nUniversity of Cambridge, Cambridge CB3 9EW, U.K. \nand \n2 \nTheoretical Physics Group, Blackett Laboratory, \nImperial College, London SW7 2BZ, U.K. \n(June 1996)', 'Abstract': 'We present non-extreme generalisations of intersecting p-brane solutions of eleven-dimensional supergravity which upon toroidal compactification reduce to non-extreme static black holes in dimensions D = 4, D = 5 and 6 ≤ D ≤ 9, parameterized by four, three and two charges, respectively. The D = 4 black holes are obtained either from a non-extreme configuration of three intersecting five-branes with a boost along the common string or from non-extreme intersecting system of two two-branes and two five-branes. The D = 5 black holes arise from three intersecting two-branes or from a system of intersecting two-brane and five-brane with a boost along the common string. Five-brane and two-brane with a boost along one direction reduce to black holes in D = 6 and D = 9, respectively, while D = 7 black hole can be interpreted in terms of non-extreme configuration of two intersecting two-branes. We discuss the expressions for the corresponding masses and entropies. \n04.50.+h,04.20.Jb,04.70.Bw,11.25.Mj \nTypeset using REVT E X', 'I. INTRODUCTION': "Recently, black holes in string theory have become a subject of intensive research, due, in part, to the fact that microscopic properties, e.g., the statistical origin of the entropy of certain black holes can be addressed either by using a conformal field theory description of the NS-NS backgrounds [1-7] or a D -brane representation of U-dual R-R backgrounds [8-15]. \nBy now, the classical solutions for the BPS-saturated as well as non-extreme static and rotating black holes of N = 4 , 8 supersymmetric superstring vacua are well understood. In particular, the explicit form of the generating solution for general rotating black holes has been obtained. The generating solution is specified by the canonical choice of the asymptotic values of the scalar fields, the ADM mass, [ D -1 2 ] components of angular momentum and by five, three and two charges, in dimensions D = 4, D = 5 and 6 D ≤ 9, respectively. 1 \n≤ \n≤ The explicit from of the generating solutions was first determined in the case of toroidally compactified heterotic string ( N = 4 superstring vacua) or in the NS-NS sector of the toroidally compactified type IIA superstring. 2 By applying U -duality transformations such solutions are mapped onto backgrounds with R-R charges which have an interpretation in terms of D -brane configurations. These BPS-saturated black holes have regular horizons and finite semiclassical Bekenstein-Hawking (BH) entropy in dimensions D = 4 [23,21,24,3] and D = 5 [8,4]. In D ≥ 6 the BPS-saturated axi-symmetric solutions have singular horizons and zero BH entropy [25-27,6]. \nEntropy of certain non-extreme or near-extreme black holes was discussed (both in the static and rotating cases) in D = 5 [10,28,29] and D = 4 [31] and for rotating black holes with NS-NS (electric) charges in 6 D ≤ 9 [6]. \n≤ \n≤ A unifying treatment of string-theory black hole properties may arise by identifying such black holes as (toroidally) compactified configurations of intersecting two-branes and fivebranes of eleven-dimensional M-theory [32,33]. The D = 10 backgrounds with NS-NS and R-R charges appear on an equal footing when viewed from eleven dimensions. A discussion \n2 The most general black hole configuration is obtained by applying to the generating solution a subset of T - and S -duality transformations (for N = 4 superstring vacua) [1] or U -duality transformations (for N = 8 superstring vacua) [22], which do not affect the canonical choice of the asymptotic values of the scalar fields. These transformations do not change the D -dimensional Einstein-frame metric, and thus the metric of a general black hole in this class is thus fully specified by the parameters of the generating solution. A solution with arbitrary asymptotic values of the scalar fields is found by appropriate rescalings of the physical parameters, i.e. of the ADM mass, the angular momenta and the charges, by the asymptotic values of the scalar fields. \nof intersections of certain BPS-saturated M-branes along with a proposal for intersection rules was first given in [34]. A generalization to a number of different harmonic functions specifying intersecting BPS-saturated M-branes which led to a better understanding of these solutions and a construction of new intersecting p-brane solutions in D ≤ 11 was presented in [35] (see also related work [33,36-40]). Specific configurations of that type reduce to the BPS-saturated black holes with regular horizons in D = 5 [35] and D = 4 [33] whose properties are determined by three and four charges (or harmonic functions), respectively. \nThe purpose of the present paper is to relate the non-extreme static black holes to nonextreme versions of intersecting M-brane solutions of [35,33]. This approach may shed light on the structure of non-extreme black holes from the point of view of M-theory, and, in particular, clarify the origin of their BH entropy. Our interpretation of non-extreme black holes as non-extreme intersecting M-branes (or p-branes in D = 10) does not seem to be related to the 'brane-antibrane' picture suggested in [9,28,31,41]. \nAs we shall discuss in Section II, there exists a procedure allowing one to construct a non-extreme version of a given BPS-saturated intersecting M-brane solution which generalises the approach of [35]. The resulting eleven-dimensional metric and the four-form field strength depend on the 'non-extremality' parameter, representing a deviation from the BPS-saturated limit, and the 'boosts', specifying charges of the configuration. Upon dimensional reduction these parameters determine the ADM mass and the charges of nonextreme static black holes in 4 D ≤ 9. 3 \n≤ \n≤ In Sections III, IV, and V we shall consider examples of non-extreme configurations of intersecting M-branes and relate them, via dimensional reduction along internal M-brane directions, to non-extreme black holes in dimensions D = 4, D = 5 and 6 ≤ D ≤ 9, respectively. In Section VI we shall present the general expressions for the mass and BH entropy of these solutions and comment on some of their consequences.", 'II. NON-EXTREME INTERSECTING M-BRANE SOLUTIONS': "The aim is to generalise the extreme supersymmetric (BPS-saturated) intersecting Mbrane solutions of [35,33] to the non-extreme case. Even though non-extreme solutions are no longer supersymmetric, it turns out that they can be constructed as a 'deformation' of extreme solutions, parameterised by several one-center harmonic functions H i , one for each constituent M-brane, and the Schwarzschild solution, parameterised by the function f ( r ) = 1 -µ/r D -3 . 4 Here D is the dimension of the space-time transverse to the configuration, and \n4 Similar 'product structure' was found previously for non-extreme versions of isotropic extreme black p-brane solutions in [42,27]. \nthe 'non-extremality' parameter µ specifies a deviation from the BPS-saturated limit. The same type of construction applies also to intersecting p-brane solutions in ten dimensions. \nIt should be stressed that the solutions presented in this paper should not be interpreted as intersections of non-extreme M-branes, even though they reduce to single non-extreme Mbrane solutions when all other charge parameters are set equal to zero. Each of non-extreme M-branes is parameterised, in general, by independent masses and charges while the nonextreme version of intersecting M-brane solutions has only one common mass parameter - the non-extremality parameter µ . Such configurations should be viewed as non-extreme 'bound-state' configurations. Note also that the non-extreme solutions below are oneparameter 'deformations' of a special type of supersymmetric solutions [35,33], for which all of the harmonic functions are chosen to have a simple spherically symmetric one-center form. The non-extreme version of multi-center solutions (corresponding to individual pbranes 'placed' at different points in transverse space) are expected to be unstable, i.e. described by time-dependent background fields. On the contrary, the non-extreme versions of single-center solutions discussed below are static. \nOne way of understanding why the structure of the non-extreme solutions is similar to that of extreme ones is based on first doing a dimensional reduction to D = 10, applying T -duality transformation and then lifting the solution back to D = 11. Since a non-extreme solution has the same number of isometries as the extreme one, it can be 'generated' by starting from the Schwarzschild background instead of the flat space one by T -duality considerations similar to the ones used in the extreme case in [37,38]. Alternatively, one may start with extreme solution and consider its deformation caused by turning on the non-extremality parameter µ . \nAsimple algorithm which leads to non-extreme version of a given extreme solution (which indeed can be checked to satisfy the eleven-dimensional supergravity equations of motion and also corresponds upon dimensional reduction to known non-extreme black hole solutions) consists of the following steps: \n- (1) Make the following replacements in the D -dimensional transverse space-time part of the metric: \ndt 2 → f ( r ) dt 2 , dx n dx n → f -1 ( r ) dr 2 + r 2 d Ω 2 D -2 , f ( r ) = 1 -µ r D -3 , (1) \nand also use the special one-center form of the harmonic functions, \nH i = 1 + Q i r D -3 , Q i = µ sinh 2 δ i , (2) \nfor the constituent two-branes, and \nH i = 1 + P i r D -3 , P i = µ sinh 2 γ i , (3) \nfor the constituent five-branes. \n(2) In the expression for the field strength F 4 of the three-form field make the following replacements: \nH ' i → H ' i = 1 + Q i r D -3 + Q i -Q i = [1 -Q i r D -3 H -1 i ] -1 , Q i = µ sinh δ i cosh δ i , (4) \nin the 'electric' (two-brane) part, and \nH i → H ' i = 1 + P i r D -3 , P i = µ sinh γ i cosh γ i , (5) \nin the 'magnetic' (five-brane) part. Here Q i and P i are the respective 'electric' and 'magnetic' charges of the configuration. In the extreme limit µ → 0 , δ i →∞ , and γ i →∞ , while the charges Q i and P i are kept fixed. In this case Q i = Q i and P i = P i , so that H ' i = H i . The form of F 4 and the actual value of its 'magnetic' part does not change compared to the extreme limit. \n(3) In the case when the extreme solution has a null isometry, i.e. intersecting branes have a common string along some direction y , one can add momentum along y by applying the coordinate transformation \nt ' = cosh β t -sinh β y , y ' = -sinh β t + cosh β y , (6) \nto the non-extreme background obtained according to the above two steps. Then \n-f ( r ) dt 2 + dy 2 → -f ( r ) dt ' 2 + dy ' 2 = -dt 2 + dy 2 + µ r D -3 ( cosh β dt -sinh β dy ) 2 \n= -K -1 ( r ) f ( r ) dt 2 + K ( r ) ̂ dy 2 , ̂ dy ≡ dy +[ K ' -1 ( r ) -1] dt , (7) \nK = 1 + ˜ Q r D -3 , K ' -1 = 1 -˜ Q r D -3 K -1 , ˜ Q = µ sinh 2 β , ˜ Q = µ sinh β cosh β , (8) \nwhere the boost β is related to the new electric charge parameter ˜ Q , i.e. momentum along direction y . In the extreme limit µ → 0 , β →∞ , the charge Q is held fixed, K = K ' and thus this part of the metric (7) becomes dudv +( K 1) du 2 , where v, u = y ± t . \n- \n± Below we shall illustrate this algorithm on several examples. Let us start with basic non-extreme M-brane solutions found in [43]. The two-brane background has the form 5 \nds 2 11 = T -1 / 3 ( r ) ( T ( r )[ -f ( r ) dt 2 + dy 2 1 + dy 2 2 ] + f -1 ( r ) dr 2 + r 2 d Ω 2 7 ) , (9) \nF 4 = -3 dt ∧ dT ' ∧ dy 1 ∧ dy 2 , (10) \nf = 1 -µ r 6 , T -1 = H = 1 + Q r 6 , T ' = H ' -1 = 1 -Q r 6 T , (11) \nQ = µ sinh 2 δ , Q = µ sinh δ cosh δ . \nwhere \nAgain, the extreme solution is obtained by setting f = 1 , Q = Q, T = T ' . 6 \nThe five-brane solution is \nds 2 11 = F -2 / 3 ( r ) ( F ( r )[ -f ( r ) dt 2 + dy 2 1 + ... + dy 2 5 ] + f -1 ( r ) dr 2 + r 2 d Ω 2 4 ) , (12) \nF 4 = 3 ∗ dF '-1 , (13) \nwhere the dual form is defined with respect to the flat transverse space. The parameters of the five-brane solution are 'magnetic' analogues of the 'electric' parameters δ, Q , Q of the two-brane solution (11) and are denoted by γ, P , P , i.e. \nf = 1 -µ r 3 , F -1 = H = 1 + P r 3 , F ' -1 = H ' = 1 + P r 3 , (14) \nP = µ sinh 2 γ , P = µ sinh γ cosh γ . \nThe two other non-extreme solutions found in [43] correspond to two and three intersecting two-branes with equal values of parameters δ i = δ . \nA generalisation to the case of different parameters δ i can be easily found using the above algorithm. For example, the non-extreme version of 2 ⊥ 2 configuration, i.e. two two-branes intersecting at a point, is thus given by 7 \nds 2 11 = ( T 1 T 2 ) -1 / 3 [ -T 1 T 2 fdt 2 + T 1 ( dy 2 1 + dy 2 2 ) + T 2 ( dy 2 3 + dy 2 4 ) + f -1 dr 2 + r 2 d Ω 2 5 ] , (15) \nF 4 = -3 dt ∧ ( dT ' 1 ∧ dy 1 ∧ dy 2 + dT ' 2 ∧ dy 3 ∧ dy 4 ) , (16) \nwhere ( i = 1 , 2) \nf = 1 -µ r 4 , T -1 i = 1 + Q i r 4 , T ' i = 1 -Q i r 4 T i , (17) \nQ i = µ sinh 2 δ i \n, \nQ i = µ sinh δ i \ncosh δ i \n. \nFor T 1 = T 2 , T ' 1 = T ' 2 this reduces to the anisotropic four-brane solution of [43]. The nonextreme version of extreme 2 ⊥ 2 ⊥ 2 configuration (three two-branes intersecting at a point) [34,35] has a similar form and will be discussed below in Section IV. \n7 For the sake of simplicity, in what follows we often do not indicate explicitly the argument r (the radial transverse coordinate) of the functions T i , F i and f . \nIn the following Sections we shall construct the non-extreme configurations of intersecting M-branes which in D ≤ 9 reproduce the generating solutions for non-extreme static black hole backgrounds. They will be built in terms of basic M-branes according to the above algorithm. As in [34,35] we shall consider only intersections which in the extreme limit preserve supersymmetry: two two-branes can intersect at a point, two five-branes can intersect at a three-brane (with three-branes allowed to intersect over a string) and five-brane and two-brane can intersect at a string.", 'III. D = 4 NON-EXTREME BLACK HOLES': "Four-dimensional black holes with four independent charges 8 can be obtained upon toroidal compactification from two different intersecting M-brane configurations [33], 2 ⊥ 2 ⊥ 5 ⊥ 5 and 'boosted' 5 ⊥ 5 ⊥ 5. While the two resulting black hole backgrounds are related by four-dimensional U -duality, the underlying intersecting M-brane solutions should be related by a symmetry transformation of the M-theory. Such a symmetry transformation should be obtained as a combination of T -duality and SL (2 , Z ) symmetry of the D = 10 type IIB theory 'lifted' to D = 11.", 'A. Intersection of two Two-Branes and two Five-Branes': "The first of the above eleven-dimensional configurations corresponds to the two twobranes intersecting at a point and two five-branes intersecting at a three-brane, with each of the two-branes intersecting with each of the five-branes at a string. The non-extreme version of the BPS-saturated solution found in [33] is given by \nds 2 11 = ( T 1 T 2 ) -1 / 3 ( F 1 F 2 ) -2 / 3 [ -T 1 T 2 F 1 F 2 fdt 2 + F 1 ( T 1 dy 2 1 + T 2 dy 2 3 ) + F 2 ( T 1 dy 2 2 + T 2 dy 2 4 ) + F 1 F 2 ( dy 2 5 + dy 2 6 + dy 2 7 ) + f -1 dr 2 + r 2 d Ω 2 2 ] , (18) F 4 = -3 dt ∧ ( dT ' 1 ∧ dy 1 ∧ dy 2 + dT ' 2 ∧ dy 3 ∧ dy 4 ) + 3( dF '-1 1 ∧ dy 2 dy 4 + dF '-1 2 ∧ dy 1 dy 3 ) . (19) \n∗ \n∧ \n∗ \n∧ \nThe coordinates y 1 , ..., y 7 describe the toroidally compactified directions. The function f , parameterising a deviation from the extremality, and functions T i , T ' i and F i , F ' i , specifying the (non-extreme) two-brane and five-brane configurations depend on the radial coordinate r of (1 + 3)-dimensional (transverse) space-time, \nf = 1 -µ r , T -1 i = 1 + Q i r , T ' i = 1 -Q i r T i , (20) \nQ i = µ sinh 2 δ i , Q i = µ sinh δ i cosh δ i , i = 1 , 2 , \nF -1 i = 1 + P i r , F ' -1 i = 1 + P i r , (21) \nP i = µ sinh 2 γ i , P i = µ sinh γ i cosh γ i , i = 1 , 2 . \nIn the extreme limit µ → 0, δ i → ∞ and γ i → ∞ , while the charges Q i and P i are held fixed. Again, in this limit f = 1, T i = T ' i , F i = F ' i . \nThe nine-area of the regular outer horizon r = µ of the anisotropic seven-brane metric (18) is \nA 9 = 4 πL 7 [ r 2 ( T 1 T 2 F 1 F 2 ) -1 / 2 ] r = µ = 4 πL 7 µ 2 cosh δ 1 cosh δ 2 cosh γ 1 cosh γ 2 , (22) \nwhere the internal directions y 1 , ..., y 7 are assumed to have periods L . In the BPS-saturated limit the area reduces to \n( A 9 ) BPS = 4 πL 7 √ Q 1 Q 2 P 1 P 2 . (23) \nUpon toroidal compactification to four dimensions one finds the following Einstein-frame metric \nds 2 4 = -λ ( r ) f ( r ) dt 2 + λ -1 ( r )[ f -1 ( r ) dr 2 + r 2 d Ω 2 2 ] , (24) \nwhere \nλ ( r ) = ( T 1 T 2 F 1 F 2 ) 1 / 2 = r 2 [( r + Q 1 )( r + Q 2 )( r + P 1 )( r + P 2 )] 1 / 2 . (25) \nIn the BPS limit f = 1 and i → Q i , i → P i . \nQ \n→ → The four-dimensional metric (24) is precisely the one of the non-extreme four-dimensional black hole with two electric and two magnetic charges found in [44]. \nP", 'B. Intersection of three Five-Branes with a Boost': "The second relevant configuration [33] is that of three five-branes, each pair intersecting at a three-brane, with an extra boost along a string common to three three-branes. The corresponding non-extreme background has the form \nds 2 11 = ( F 1 F 2 F 3 ) -2 / 3 [ F 1 F 2 F 3 ( -K -1 fdt 2 + K ̂ dy 2 1 ) + F 2 F 3 ( dy 2 2 + dy 2 3 ) + F 1 F 3 ( dy 2 4 + dy 2 5 ) + F 1 F 2 ( dy 2 6 + dy 2 7 ) + f -1 dr 2 + r 2 d Ω 2 2 ] , (26) \nF 4 = 3( ∗ dF '-1 1 ∧ dy 2 ∧ dy 3 + ∗ dF '-1 2 ∧ dy 4 ∧ dy 5 + ∗ dF '-1 3 ∧ dy 6 ∧ dy 7 ) , (27) \nwhere (cf. (7)) \n̂ dy 1 = dy 1 +( K ' -1 -1) dt , (28) \nand K and K ' depend on the boost parameter β along the string ( y 1 ) direction. The background is thus parameterised by f ( r ) and the following functions of r ( i = 1 , 2 , 3) \nK = 1 + ˜ Q r , K ' -1 = 1 -˜ Q r K , F -1 i = 1 + P i r , F ' -1 i = 1 + P i r , (29) \n˜ Q = µ sinh 2 β , ˜ Q = µ sinh β cosh β , P i = µ sinh 2 γ i , P i = µ sinh γ i cosh γ i . (30) \nThe four charges ˜ Q and P 1 , P 2 , P 3 are held fixed in the limit µ → 0 , β →∞ , γ i →∞ . The area of nine-surface at r = µ is \nA 9 = 4 πL 7 [ r 2 ( K -1 F 1 F 2 F 3 ) -1 / 2 ] r = µ = 4 πL 7 µ 2 cosh β cosh γ 1 cosh γ 2 cosh γ 3 , (31) \n( A 9 ) BPS = 4 πL 7 √ ˜ QP 1 P 2 P 3 . (32) \nThe four-dimensional Einstein-frame metric resulting upon toroidal compactification is of the form (24) with \nλ ( r ) = ( K -1 F 1 F 2 F 3 ) 1 / 2 = r 2 [( r + ˜ Q )( r + P 1 )( r + P 2 )( r + P 3 )] 1 / 2 . (33) \nAgain, this four-dimensional metric is the same as in [44], but now it depends on one electric and three magnetic charges. \nNote that the dimensional reduction to ten dimensions (along y 1 direction) gives a nonextreme generalisation of a configuration of intersecting R-R p-branes of type IIA theory, namely, a zero-brane and three four-branes [36,33,39]. Applying T -duality and SL (2 , Z ) symmetry of type IIB theory one is able to construct various other non-extreme D = 10 pbrane configurations which in the extreme limit have a representation in terms of intersecting D -branes. Their form is always consistent with the algorithm of Section II. In particular, it is straightforward to write down the non-extreme version of the maximally symmetric 3 ⊥ 3 ⊥ 3 ⊥ 3 solution (four three-branes intersecting at a point, with each pair of three-branes intersecting at a string), found in [33,39].", 'IV. D = 5 NON-EXTREME BLACK HOLES': "The extreme D = 5 black holes with three independent charges [8,4] (generating solution for general extreme D = 5 black holes with regular horizons) can be obtained from the two different intersecting M-brane configurations [35]: 2 ⊥ 2 ⊥ 2, i.e. three two-branes intersecting at a point, and 'boosted' 2 ⊥ 5, i.e. intersecting two-brane and five-brane with a momentum along the common string. Below we shall present the non-extreme versions of these D = 11 solutions, which serve as generating solutions for non-extreme static D = 5 black hole solutions. \nwhere", 'A. Intersection of three Two-Branes': "This O (4) symmetric background is a straightforward generalisation of the non-extreme 2 ⊥ 2 solution (15),(16) \nds 2 11 = ( T 1 T 2 T 3 ) -1 / 3 [ -T 1 T 2 T 3 fdt 2 + T 1 ( dy 2 1 + dy 2 2 ) + T 2 ( dy 2 3 + dy 2 4 ) + T 3 ( dy 2 5 + dy 2 6 ) (34) \n+ f -1 dr 2 + r 2 d Ω 2 3 ] , \nF 4 = -3 dt ∧ ( dT ' 1 ∧ dy 1 ∧ dy 2 + dT ' 2 ∧ dy 3 ∧ dy 4 + dT ' 3 ∧ dy 5 ∧ dy 6 ) , (35) \nf = 1 -µ r 2 , T -1 i = 1 + Q i r 2 , T ' i = 1 -Q i r 2 T i , Q i = µ sinh 2 δ i , Q i = µ sinh δ i cosh δ i , i = 1 , 2 , 3 . (36) \nFor δ 1 = δ 2 = δ 3 , i.e. equal T i and equal T ' i , this solution coincides with the anisotropic six-brane solution of [43]. \nThe nine-area of the regular horizon at r = µ 1 / 2 is \nA 9 = 2 π 2 L 6 [ r 3 ( T 1 T 2 T 3 ) -1 / 2 ] r = µ 1 / 2 = 2 π 2 L 6 µ 3 / 2 cosh δ 1 cosh δ 2 cosh δ 3 . (37) \nIn the extreme limit it becomes \n( A 9 ) BPS = 2 π 2 L 6 √ Q 1 Q 2 Q 3 . (38) \nThe five-dimensional Einstein-frame metric obtained by reduction along y 1 , ..., y 6 is \nds 2 5 = -λ 2 ( r ) f ( r ) dt 2 + λ -1 ( r )[ f -1 ( r ) dr 2 + r 2 d Ω 2 3 ] , (39) \nwhere \nλ ( r ) = ( T 1 T 2 T 3 ) 1 / 3 = r 2 [( r 2 + Q 1 )( r 2 + Q 2 )( r 2 + Q 3 )] 1 / 3 . (40) \nThis is the metric of non-extreme five-dimensional black holes found in [16,28] (where one of the electric charges was replaced by a magnetic one). In the BPS limit Q i → Q i , f → 1 and we get a solution which is U -dual to the solution of [4].", 'B. Intersection of Two-Brane and Five-Brane with a Boost': "The non-extreme generalisation of the supersymmetric configuration of a two-brane intersecting five-brane with a 'boost' along the common string [35] has the form \nds 2 11 = T -1 / 3 F -2 / 3 [ TF ( -K -1 fdt 2 + K ̂ dy 2 1 ) + Tdy 2 2 + F ( dy 2 3 + dy 2 4 + dy 2 5 + dy 2 6 ) (41) \n+ f -1 dr 2 + r 2 d Ω 2 3 ] , F 4 = -3 dt ∧ dT ' ∧ dy 1 ∧ dy 2 +3 ∗ dF '-1 1 ∧ dy 2 , (42) \nwhere 9 ̂ dy 1 = dy 1 + ( K ' -1 -1) dt . The relevant functions of the radial coordinate r of the (1 + 4)-dimensional space-time are \nK = 1 + ˜ Q r 2 , K ' -1 = 1 -˜ Q r 2 K -1 , ˜ Q = µ sinh 2 β , ˜ Q = µ sinh β cosh β , \nT -1 = 1 + Q r 2 , T ' = 1 -Q r 2 T , Q = µ sinh 2 δ , Q = µ sinh δ cosh δ , \nF -1 = 1 + P r 2 , F ' -1 = 1 + P r 2 , P = µ sinh 2 γ , P = µ sinh γ cosh γ , (43) \nand f is the same as in (36). The three charges ˜ Q,Q,P are held fixed in the extreme limit µ → 0 , β , δ →∞ , γ →∞ . \n→∞ \nWe find again \nA 9 = 2 π 2 L 6 [ r 3 ( K -1 TF ) -1 / 2 ] r = µ 1 / 2 = 2 π 2 L 6 µ 3 / 2 cosh β cosh δ cosh γ , (44) \n( A 9 ) BPS = 2 π 2 L 6 √ ˜ QQP . (45) \nThe corresponding five-dimensional Einstein-frame metric is (39) with \nλ ( r ) = ( TFK -1 ) 1 / 3 = r 2 [ ( r 2 + ˜ Q )( r 2 + Q )( r 2 + P ) ] 1 / 3 , (46) \ni.e. is precisely the space-time metric found in [16,28]. In the extreme limit ˜ Q → ˜ Q, Q → Q, P → P , f → 1 we get back to the solution of [4].", 'V. 6 ≤ D ≤ 9 NON-EXTREME BLACK HOLES': 'The generating solution for black holes in dimensions D ≥ 6 can be parameterised by two charges. The boosted non-extreme two-brane naturally reduces to D = 9 black hole. The D = 7 non-extreme black hole can be described as a dimensional reduction of a configuration of two two-branes intersecting at a point. The boosted non-extreme five-brane represents the two-charge black hole in D = 6. Black holes in D = 10 do not have a natural M -brane description.', 'A. Two-Brane with a Boost': "It is possible to describe all two-charge 6 ≤ D ≤ 9 non-extreme black holes as dimensional reductions of a non-extreme generalisation of boosted two-brane solution which has nonmaximal rotational isometry, i.e. O ( D -1) × [ O (2)] 9 -D symmetry, instead of O (8). Adding a boost along one of the two directions of the two-brane we find from (9),(10) 10 \nds 2 11 = T -1 / 3 [ T ( -K -1 fdt 2 + K ̂ dy 2 1 + dy 2 2 ) + dy 2 3 + ... + dy 2 11 -D (47) \nF \n∧ \n∧ \n+ f -1 dr 2 + r 2 d Ω 2 D -2 ] , 4 = -3 dt dT ' ∧ dy 1 dy 2 , (48) \nf \nwhere ̂ dy 1 = dy 1 +( K ' -1 -1) dt , and \n= 1 -µ r D -3 \n, \nK = 1 + ˜ Q r D -3 , K ' -1 = 1 -˜ Q r D -3 K -1 , ˜ Q = µ sinh 2 β , ˜ Q = µ sinh β cosh β , \nT -1 = 1 + Q r D -3 , T ' = 1 -Q r D -3 T , Q = µ sinh 2 δ , Q = µ sinh δ cosh δ . (49) \nQ and ˜ Q are the two electric charges which are held fixed in the extreme limit. \nThe area of the horizon at r = µ 1 / ( D -3) is \nA 9 = ω D -2 L 11 -D [ r D -2 ( K -1 T ) -1 / 2 ] r = µ 1 / ( D -3) = ω D -2 L 11 -D µ D -2 D -3 cosh β cosh δ , (50) \nwhere all internal coordinates y 1 , ..., y 11 -D are assumed to have period L and ω D -2 = 2 π D -1 2 / Γ( D -1 2 ). The area (50) vanishes in the BPS-saturated limit, in agreement with the fact that there are no BPS-saturated black holes with regular horizons of finite area in D ≥ 6 [27,6]. \n≥ The corresponding D -dimensional Einstein-frame metric is \nds 2 D = -λ D -3 ( r ) f ( r ) dt 2 + λ -1 ( r )[ f -1 ( r ) dr 2 + r 2 d Ω 2 D -2 ] , (51) \nλ ( r ) = ( K -1 T ) 1 D -2 = r 2( D -3) D -2 [( r D -3 + ˜ Q )( r D -3 + Q )] 1 D -2 . (52) \n≥ \nIt coincides with the metric of non-extreme black hole solutions in D 6 [25,26,6]. \nNote also that for D = 10 and Q = 0 the metric (51) describes also the electric R-R black hole (0-brane) in ten dimensions [45] which can be obtained by dimensional reduction of 'boosted' Schwarzschild solution in D = 11.", 'B. Five-Brane with a Boost': "The D = 6 black hole with one electric and one magnetic charge has a natural interpretation as a dimensional reduction of a boosted five-brane. Boosting the metric (12) along y 1 we find \nds 2 11 = F -2 / 3 [ F ( -K -1 fdt 2 + K ̂ dy 2 1 + dy 2 2 + ... + dy 2 5 ) + f -1 dr 2 + r 2 d Ω 2 4 ] , (53) \nwhere f and F are the same as in (14), ̂ dy 1 = dy 1 +( K ' -1 -1) dt and \nK = 1 + ˜ Q r 3 , K ' -1 = 1 -˜ Q r 3 K -1 , ˜ Q = µ sinh 2 β , ˜ Q = µ sinh β cosh β . (54) \nThe black hole background resulting upon dimensional reduction along internal five-brane directions y 1 , ..., y 5 is parameterised by one electric and one magnetic charge.", 'C. Intersection of two Two-Branes': "The D = 7 non-extreme black hole admits also a description in terms of non-extreme version of 2 ⊥ 2 configuration. 11 Dimensional reduction of the background (15),(16) along y 1 , ..., y 4 leads to the D = 7 black hole background, with the role of Q , ˜ Q , Q, ˜ Q played by 1 , Q 2 , Q 1 , Q 2 . \nQ It is also possible to give an alternative description of D = 6 black hole (now with two electric charges) by using O (5)-symmetric version of (15). i.e. the non-extreme version of 2 ⊥ 2 solution with one of the transverse space coordinates ( y 1 ) treated as an isometric internal space one: \nds 2 11 = ( T 1 T 2 ) -1 / 3 [ -T 1 T 2 fdt 2 + dy 2 1 + T 1 ( dy 2 2 + dy 2 3 ) + T 2 ( dy 2 4 + dy 2 5 ) (55) \n+ f -1 dr 2 + r 2 d Ω 2 4 ] . \nThe corresponding nine-area and the D = 6 , 7 Einstein-frame metrics reduce to the expressions in (50) and (51), with the role of T , K -1 now played by T 1 , T 2 . \nIt is also of interest to compare the metrics of the eleven-dimensional solutions which reduce to black holes in D = 4 , 5 , 6 with respective n = 4 , 3 , 2 charges in the case when all charges (boost parameters) are equal, i.e. T i = F i = K -1 = H -1 . For n = 4 and n = 3, i.e. the respective cases of D = 4 and D = 5 black holes, whose extreme limits have regular horizons, we get \nQ \nds 2 11 = H n -2 [ -H -n fdt 2 + f -1 dr 2 + r 2 d Ω 2 6 -n ] + ̂ dy 2 1 + ... + dy 2 n +3 , (56) \nwhere ̂ dy 1 = dy 1 for the unboosted configurations (18), (34), and ̂ dy 1 = dy 1 +( H ' -1 -1) dt for the boosted ones (26), (41). At the same time, in the case of n = 2, i.e. black holes whose extreme limits have singular horizons, e.g., for D = 6 black hole, we get the metric \nds 2 11 = H 2 / 3 [ H -1 ( -fdt 2 + dy 2 2 + ... + dy 2 5 ) + ̂ dy 2 1 + f -1 dr 2 + r 2 d Ω 2 4 ] , (57) \nwhere ̂ dy 1 = dy 1 for the intersecting two-brane representation (55) and ̂ dy 1 = dy 1 +( H ' -1 -1) dt for the boosted five-brane one (53). While the radii of the internal coordinates are constant in the case (56), i.e. D = 4 , 5 black holes with equal n = 3 , 4 charges, this is no longer so for black holes in D > 5 with equal n = 2 charges. 12", 'VI. UNIVERSAL EXPRESSIONS FOR MASS AND ENTROPY': "It is possible to write down the formulas for the mass and the entropy which apply to all eleven-dimensional 'anisotropic p-brane' solutions discussed in previous Sections. Similar expressions for the corresponding non-extreme black holes were given in [44,6]. Note also that analogous relations for isotropic black brane solutions appeared in [42,27]. \nLet p = 11 -D be the common internal dimension of intersecting M-branes, i.e. the dimension of an anisotropic p-brane. Then D is the dimension of the black hole obtained by dimensional reduction. For D = 4, D = 5 and 6 ≤ D ≤ 9 the black hole has the metric of the form (51) with λ ( r ) = ( H 1 ...H n ) -1 / ( D -2) , H i = 1 + Q i /r D -3 , i = 1 , ..., n , with n ≤ 4, n ≤ 3 and n ≤ 2, respectively (cf. (25),(33), (40),(46),(52)). Here we shall use the same notation Q i for all n charges, some of which may be electric, and some magnetic. Let G 11 = 8 πκ 2 be the Newton's constant in eleven dimensions (the Newton's constant in D dimensions is then G D = G 11 /L p ). Then for a O ( D -1) - symmetric solution the normalised charges per unit volume are 13 \nq i = aQ i , a ≡ ω D -2 √ 2 κ ( D -3) , Q i = µ sinh β i cosh β i , Q i = µ sinh 2 β i . (58) \nAs follows from (52), the ADM mass is \nM ADM = ω D -2 2 κ 2 L p [( D -2) µ +( D -3) n ∑ i =1 Q i ] . (59) \n13 The relation of our notation to that of [27] is the following: D -3 → d , Q → r d 0 , Q → r d -, µ → µ d . The charges q i correspond to constituent objects of an N -charge bound state (hence there is no √ n factor in the expression for the charges). \nIt can be expressed in terms of the non-extremality parameter µ and charges Q i (which are fixed in the extreme limit µ → 0) as follows (note that Q i = -µ 2 + √ Q 2 i +( µ 2 ) 2 ) \nM ADM = b [ n ∑ i =1 √ Q 2 i +( µ 2 ) 2 + λµ ] , b ≡ ω D -2 2 κ 2 ( D -3) L p , (60) \nwhere the parameter λ (not to be confused with the function λ ( r ) in D -dimensional metric (51)) is the same as in [27] \nλ ≡ D -2 D -3 -n 2 . (61) \nExplicitly, λ n =1 = D -1 2( D -3) , λ n =2 = 1 D -3 , λ n =3 = 5 -D 2( D -3) , λ n =4 = 4 -D 2( D -3) , i.e. λ ≥ 0 and vanishes only for D = 4 , n = 4 and D = 5 , n = 3, i.e. the cases with regular horizons in the extreme (BPS-saturated) limit. 14 \nThe Bekenstein-Hawking entropy S BH , which follows from the obvious generalisation of the expressions for the area (50), (22),(31),(37) and (44) (with K -1 T → ( H 1 ...H n ) -1 ) is \nS BH = 2 πA 9 κ 2 = cµ D -2 D -3 n ∏ i =1 cosh δ i , c ≡ 2 πω D -2 κ 2 L p , (62) \nor, in terms of the Hawking temperature T H , \nS BH = b µ T H , T H = 1 4 π ( D -3) µ -1 D -3 n ∏ i =1 ( cosh δ i ) -1 . (63) \nThese general formulas apply also to the case of D = 10 black hole [45] where D = 10 and n = 1. \nExpressed in terms of µ and Q i , the entropy becomes \nS BH = cµ λ n ∏ i =1 [√ Q 2 i +( µ 2 ) 2 + µ 2 ] 1 / 2 . (64) \nIt has non-zero extreme limit ( µ → 0) only when λ = 0. In this case M ADM and S BH take simple forms \nM ADM = b n ∑ i =1 √ Q 2 i +( µ 2 ) 2 , S BH = c n ∏ i =1 [√ Q 2 i +( µ 2 ) 2 + µ 2 ] 1 / 2 . (65) \nM ADM resembles the energy of a system of relativistic particles with masses Q i (masses of individual constituents in extreme limit), all having the same momentum proportional to µ . This suggests a 'bound-state' interpretation of this non-extreme system (cf. [47]). \nFor all fixed values of λ and Q i the mass (60) and the entropy (64) satisfy the following relation \n∂ ln S BH ∂ ln µ = b -1 ∂M ADM ∂µ , i.e. T H ∂S BH ∂µ = ∂M ADM ∂µ . (66) \nThis thermodynamic relation is valid both in the Schwarzschild ( Q i = 0) case as well as in the extreme limit ( µ = 0). This explains why the proportionality between the entropy and the area of the horizon can be assumed to be true also in the extreme limit (cf. [15]): this proportionality certainly holds in non-extreme (or near-extreme) case (see, e.g., [48]) and thus should be meaningful also in the limit µ 0. \nOther intersecting M-brane configurations (with λ /negationslash = 0) have zero entropy in the extreme limit ( µ → 0). Representative examples of such configurations are unboosted p = 7 configurations 5 ⊥ 5 ⊥ 5, 5 ⊥ 5 ⊥ 2, 5 ⊥ 2 ⊥ 2, corresponding to D = 4 black holes with n = 3 charges, unboosted p = 6 configuration 2 ⊥ 5, corresponding to D = 5 black hole with n = 2 charges, and p = 4 configuration 2 ⊥ 2, corresponding to D = 7 black hole with n = 2 charges. In this case it is of interest to study the near-extreme limit, where \n→ \nM ADM = M 0 +∆ M + O ( µ 2 ) , M 0 = b n ∑ i =1 Q i , ∆ M = bλµ , (67) \nS BH = c 1 n ∏ i =1 Q 1 / 2 i E λ , E ≡ ∆ M , c 1 = c ( bλ ) -λ . (68) \nUsing the thermodynamic relation dE = TdS it follows from (68) that \nS BH = c 2 n ∏ i =1 ( Q i ) 1 2(1 -λ ) T λ 1 -λ , c 2 = ( c 1 λ λ ) 1 1 -λ = ( cb -λ ) 1 1 -λ , (69) \nwhere \nT = ( c 1 λ ) -1 n ∏ i =1 Q i -1 / 2 E 1 -λ (70) \nis the near-extreme limit of the Hawking temperature T H in (63). \nGeneralising the discussion in [27], we may enquire when this entropy has a massless ideal gas entropy form. The power ν = λ/ (1 -λ ) of the temperature in (69) is equal to 2 and 5 for unboosted two-brane and five-brane, respectively [27]. 15 The only other cases when ν is integer are configurations with D = 5 , n = 2, i.e. the anisotropic six-brane corresponding to unboosted 2 ⊥ 5 intersection, and D = 4 , n = 3, i.e. the anisotropic sevenbranes corresponding to unboosted 5 ⊥ 5 ⊥ 5, 2 ⊥ 5 ⊥ 5 and 2 ⊥ 2 ⊥ 5 intersections. In these cases \nλ = 1 / 2 and ν = 1, so that, not unexpectedly [33], here S BH has a 'string'-like form, i.e. the form of the entropy of a gas of massless particles in (1+1)-dimensions. For other anisotropic eleven-dimensional p-branes, e.g., those reducing to isotropic dilatonic p-branes in lower dimensions [42,27], S BH does not have an ideal gas scaling. \nTo conclude, we have constructed non-extreme versions of intersecting M-brane solutions which correspond to one parameter 'deformations' of the supersymmetric intersecting M-brane solutions, and maintain the simple 'product' structure. This product structure implies that the non-extreme static black holes obtained upon dimensional reduction have a form which provides a straightforward interpolation between the Schwarzschild and BPSsaturated backgrounds. \nThe M-brane interpretation of non-extreme black hole solutions may provide an insight into the problem of statistical understanding of their properties. In particular, it would be of interest to study in more detail the statistical origin of the BH entropy of near-extreme anisotropic p-branes and interpret them in terms of massless modes living on near-extreme intersections, along the lines of [27,33].", 'ACKNOWLEDGMENTS': 'We would like to thank I. Klebanov, G. Papadopoulos, J. Russo and P. Townsend for useful discussions and remarks. M.C. acknowledges the hospitality of the Department of Applied Mathematics and Theoretical Physics of Cambridge University. The work is supported by U.S. DOE Grant No. DOE-EY-76-02-3071 (M.C.), the National Science Foundation Career Advancement Award No. PHY95-12732 (M.C.), the PPARC (A.T.), ECC grant SC1 ∗ -CT92-078 (A.T.) and the NATO collaborative research grant CGR No. 940870.', 'REFERENCES': '- [1] A. Sen, Nucl. Phys. B440 (1995) 421, hep-th/9411187.\n- [2] F. Larsen and F. Wilczek, PUPT-1576, hep-th/9511064.\n- [3] M. Cvetiˇc and A.A. Tseytlin, Phys. Rev. D53 (1996) 5619, hep-th/ 9512031.\n- [4] A.A. Tseytlin, Mod. Phys. Lett. A11 (1996) 689, hep-th/9601177.\n- [5] F. Larsen and F. Wilczek, PUPT-1614, hep-th/9604134.\n- [6] M. Cvetiˇc and D. Youm, IASSNS-HEP-96/43, hep-th/9605051.\n- [7] A.A. Tseytlin, Imperial/TP/95-96/48, hep-th/9605091.\n- [8] A. Strominger and C. Vafa, HUTP-96-A002, hep-th/9601029.\n- [9] C.G. Callan and J.M. Maldacena, PUPT-1591, hep-th/9602043.\n- [10] G.T. Horowitz and A. Strominger, hep-th/9602051.\n- [11] J.C. Breckenridge, R.C. Myers, A.W. Peet and C. Vafa, HUTP-96-A005, hepth/9602065.\n- [12] J.M. Maldacena and A. Strominger, hep-th/9603060.\n- [13] C.V. Johnson, R.R. Khuri and R.C. Myers, CERN-TH-96-62, hep-th/9603061.\n- [14] J. Maldacena and L. Susskind, hep-th/9604042.\n- [15] G. Horowitz, UCSBTH-96-07, gr-qc/9604051.\n- [16] M. Cvetiˇc and D. Youm, IASSNS-HEP-96-24, hep-th/9603100.\n- [17] P.M. Llatas, UCSBTH-96-05, hep-th/9605058.\n- [18] M. Cvetiˇc and D. Youm, IASSNS-HEP-95-107, hep-th/9512127.\n- [19] D.P. Jatkar, S. Mukherji and S. Panda, MRI-PHY-13-95, hep-th/9512157.\n- [20] M. Cvetiˇc and D. Youm, IASSNS-HEP-96-27, hep-th/9603147.\n- [21] M. Cvetiˇc and D. Youm, Phys. Rev. D53 (1996) 584, hep-th/9507090.\n- [22] M. Cvetiˇc and C.M. Hull, DAMTP-R-96-31, hep-th /9606193.\n- [23] R. Kallosh, A. Linde, T. Ortin, A. Peet, and A. Van Proeyen, Phys. Rev. D46 (1992) 5278, hep-th/9205027.\n- [24] M. Cvetiˇc and A.A. Tseytlin, Phys. Lett. B366 (1996) 95, hep-th/9510097.\n- [25] A.W. Peet, PUPT-1548, hep-th/9506200.\n- [26] G.T. Horowitz and A. Sen, Phys. Rev. D53 (1996) 808, hep-th/9509108.\n- [27] I.R. Klebanov and A.A. Tseytlin, PUPT-1613, hep-th/9604089.\n- [28] G. Horowitz, J. Maldacena and A. Strominger, hep-th/9603109.\n- [29] J.C. Breckenridge, D.A. Lowe, R.C. Myers, A.W. Peet, A. Strominger and C. Vafa, HUTP-96/A008, hep-th/9603078.\n- [30] A. Sen, Int. J. Mod. Phys. A9 (1994) 3707, hep-th/9402002.\n- [31] G.T. Horowitz, D.A. Lowe and J.M. Maldacena, PUPT-1608, hep-th/9603195.\n- [32] R. Dijkgraaf, E. Verlinde and H. Verlinde, hep-th/9603126.\n- [33] I.R. Klebanov and A.A. Tseytlin, PUPT-1616, hep-th/9604166.\n- [34] G. Papadopoulos and P. Townsend, DAMTP-R-96-12, hep-th/9603087.\n- [35] A.A. Tseytlin, Imperial/TP/95-96/38, hep-th/9604035.\n- [36] M. Cvetiˇc and A. Sen, unpublished.\n- [37] K. Behrndt, E. Bergshoeff and B. Janssen, UG-4-96, hep-th/9604168.\n- [38] J.P. Gauntlett, D.A. Kastor and J. Traschen, CALT-68-2055, hep-th/9604179.\n- [39] V. Balasubramanian and F. Larsen, PUPT-1617, hep-th/9604189.\n- [40] N. Khviengia, Z. Khviengia, H. Lu and C.N. Pope, CTP-TAMU-19-96, hep-th/9605077. \n- [41] R. Kallosh and A. Rajaraman, SU-ITP-96-17, hep-th/9604193.\n- [42] M.J. Duff, H. Lu and C.N. Pope, CTP-TAMU-14-96, hep-th/9604052.\n- [43] R. Guven, Phys. Lett. B276 (1992) 49.\n- [44] M. Cvetiˇc and D. Youm, contribution to the Proceedings of STRINGS 95: Future Perspectives in String Theory, hep-th/9508058.\n- [45] G.T. Horowitz and A. Strominger, Nucl. Phys. B360 (1991) 197.\n- [46] S.S. Gubser, I.R. Klebanov and A.W. Peet, hep-th/9602135.\n- [47] M.J. Duff and J. Rahmfeld, CTP-TAMU-17/96, hep-th/9605085.\n- [48] S.W. Hawking and G.T. Horowitz, Class. Quantum. Grav. 13 (1996) 1487.'}

2007PhRvD..76b4005V

Observation of incipient black holes and the information loss problem

2007-01-01

['-', '-', '-', 'astrophysics', '-', '-']

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We study the formation of black holes by spherical domain wall collapse as seen by an asymptotic observer, using the functional Schrödinger formalism. To explore what signals such observers will see, we study radiation of a scalar quantum field in the collapsing domain wall background. The total energy flux radiated diverges when backreaction of the radiation on the collapsing wall is ignored, and the domain wall is seen by the asymptotic observer to evaporate by nonthermal “pre-Hawking radiation” during the collapse process. Evaporation by pre-Hawking radiation implies that an asymptotic observer can never lose objects down a black hole. Together with the nonthermal nature of the radiation, this may resolve the black hole information loss problem.

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https://arxiv.org/pdf/gr-qc/0609024.pdf

{'Observation of Incipient Black Holes and the Information Loss Problem': "Tanmay Vachaspati and Dejan Stojkovic \nCERCA, Department of Physics, Case Western Reserve University, Cleveland, OH 44106-7079 \nLawrence M. Krauss \nCERCA, Department of Physics, Case Western Reserve University, \nCleveland, OH \n44106-7079 (permanent address) and \nalso, Department of Physics and Astronomy, Vanderbilt University, Nashville, TN \nWe study the formation of black holes by spherical domain wall collapse as seen by an asymptotic observer, using the functional Schrodinger formalism. To explore what signals such observers will see, we study radiation of a scalar quantum field in the collapsing domain wall background. The total energy flux radiated diverges when backreaction of the radiation on the collapsing wall is ignored, and the domain wall is seen by the asymptotic observer to evaporate by non-thermal 'preHawking radiation' during the collapse process. Evaporation by pre-Hawking radiation implies that an asymptotic observer can never lose objects down a black hole. Together with the non-thermal nature of the radiation, this may resolve the black hole information loss problem.", 'I. INTRODUCTION': "Black holes embody the long-standing theoretical challenge of combining general relativity and quantum mechanics, with various proposals being advocated over the years to resolve paradoxes associated with black hole formation, evaporation and information loss. Resolution of these issues has become even more timely with the possible formation and evaporation of black holes in particle accelerators in the framework of higher dimensional models that have recently garnered much attention. The process of black hole formation is generally discussed from the viewpoint of an infalling observer. However, in all physical settings it is the viewpoint of the asymptotic observer that is relevant. More concretely, if a black hole is formed in the Large Hadron Collider, it has to be observed by physicists sitting on the CERN campus. The physicists have clocks in their offices and they watch the process of formation and evaporation in this coordinate frame. They must address questions such as: At what time did a black hole form? Is any information lost into the black hole? How long did it take for the black hole to evaporate? What is the spectrum of the decay products? \nThe process of gravitational collapse has been studied extensively over the last few decades, from many different viewpoints, including 1+1 dimensional models and modifications of general relativity ( e.g. see [1]). Unlike a large subset of this work, our analysis is in 3+1 dimensions and within conventional general relativity. We model the general problem by choosing to study a collapsing spherical shell of matter, more specifically a vacuum domain wall. The physical setup of the problem and the functional Schrodinger formalism are described in Sec. II. \nA crucial aspect of our analysis is that we address the question of black hole formation and evaporation as seen by an asymptotic observer . Initially, when the domain wall is large, the spacetime is described by the Schwarzschild metric, just as for a static star. From here on, the wall and the metric are evolved forward in time, \nalways using the Schwarzschild time coordinate. We emphasize that all our discussion, unless explicitly stated, refers to the Schwarzschild time, t , and this defines the time slicing of the spacetime. As is well known, the Schwarzschild coordinate system breaks down at a black hole horizon, and there is danger that our analysis will also break down at some point during the gravitational collapse. However, we do not encounter any such difficulties, suggesting that our calculation is self-consistent. A second danger is that the coordinate system may provide an incomplete description of the gravitational collapse spacetime. This remains a possibility. However, we find that Schwarzschild coordinates are sufficient to answer the very specific set of questions we ask from the asymptotic observer's viewpoint. Namely, does the asymptotic observer see objects disappear into a black hole in the time that he sees the collapsing body evaporate? And, is the spectrum of the radiation received ever truly thermal (even in the semiclassical approximation)? \nIn Sec. III we verify the standard result that the formation of an event horizon takes an infinite (Schwarzschild) time if we consider classical collapse. This is not surprising and is often viewed as a limitation of the Schwarzschild coordinate system. To see if this result changes when quantum effects are taken into account, we address the problem of quantum collapse using a minisuperspace version of the functional Schrodinger equation [2] in Sec. IV. We find that even in this case the black hole takes an infinite time to form, contrary to some speculations in the literature [3]. \nIn Sec. V we consider the possible radiation associated with the collapsing shell by considering the interaction of a quantum scalar field and the classical background of a collapsing domain wall. We treat the problem using the functional Schrodinger picture, which we relate to the standard Bogolubov treatment carried out in Sec. VI. Here we find that the shell, even as it collapses, radiates away its energy in a finite amount of time. With some assumptions about the metric close to the incipient horizon, we conclude that the evaporation time is shorter \nthan what would be taken by objects to fall through a black hole horizon. This leads us to the conclusion that the asymptotic observer will see the evaporation of the collapsing shell before he can see any objects disappear. \nWe discuss our results from the point of view of an infalling observer in Sec. VII, where we attempt to reconcile the fact that such an observer will not see substantial radiation with the observations made by an asymptotic observer. Our conclusions are summarized in Sec. VIII, where we elucidate a possible resolution of the information loss problem suggested by our results, together with a discussion of possible loopholes and future directions.", 'II. SETUP AND FORMALISM': "To study a concrete realization of black hole formation we consider a spherical Nambu-Goto domain wall that is collapsing. To include the possibility of (spherically symmetric) radiation we consider a massless scalar field, Φ, that is coupled to the gravitational field but not directly to the domain wall. The action for the system is \nS = ∫ d 4 x √ -g [ -1 16 πG R + 1 2 ( ∂ µ Φ) 2 ] -σ ∫ d 3 ξ √ -γ + S obs (1) \nwhere the first term is the Einstein-Hilbert action for the gravitational field, the second is the scalar field action, the third is the domain wall action in terms of the wall world volume coordinates, ξ a ( a = 0 , 1 , 2), the wall tension, σ , and the induced world volume metric \nγ ab = g µν ∂ a X µ ∂ b X ν (2) \nThe coordinates X µ ( ξ a ) describe the location of the wall and Roman indices go over internal domain wall worldvolume coordinates ζ a , while Greek indices go over spacetime coordinates. The term S obs in Eq. (1) denotes the action for the observer. \nWe will begin first with the Wheeler-de Witt equation in order to explore and contrast quantum vs classical collapse of the domain wall, but we will eventually utilize the functional Schrodinger formalism to study both collapse and radiation in this system. \nThe Wheeler-de Witt equation for a closed universe is \nH Ψ = 0 (3) \nwhere H is the Hamiltonian and Ψ[ X α , g µν , Φ , O ] is the wave-functional for all the ingredients of the system, including the observer's degrees of freedom denoted by O . Note that the wave-functional, Ψ, is a functional of the fields but not of the spacetime coordinates. We will separate the Hamiltonian into two parts, one for the system and the other for the observer \nH = H sys + H obs (4) \nAny (weak) interaction terms between the observer and the wall-metric-scalar system are included in H sys . The observer is assumed not to significantly affect the evolution of the system and similarly for the system vis a vis the observer. The total wave-functional can be written as a sum over eigenstates \nΨ = ∑ k c k Ψ k sys (sys , t )Ψ k obs ( O , t ) (5) \nwhere k labels the eigenstates, c k are complex coefficients, and we have introduced the observer time, t , via \ni ∂ Ψ k obs ∂t ≡ H obs Ψ k obs (6) \nWith the help of an integration by parts, and the fact that the total wave-functional is independent of t , the Wheeler-de Witt equation implies the Schrodinger equation \nH sys Ψ k sys = i ∂ Ψ k sys ∂t (7) \nFor convenience, from now on we will denote the system wave-function simply by Ψ and drop the superscript k and the subscript 'sys'. Similarly H will now denote H sys , and the Schrodinger equation reads \nH Ψ = i ∂ Ψ ∂t (8) \nA general treatment of the full Wheeler-de Witt equation is very difficult and we shall utilize the frequently employed strategy of truncating the field degrees of freedom to a finite subset. In other words, we will consider a minisuperspace version of the Wheeler-de Witt equation. As long as we keep all the degrees of freedom that are of interest to us, this is a useful truncation. With this in mind, we only consider spherical domain walls and assume spherical symmetry for all the fields. So the wall is described by only the radial degree of freedom, R ( t ). The metric is taken to be the solution of Einstein equations for a spherical domain wall. The metric is Schwarzschild outside the wall, as follows from spherical symmetry [4] \nds 2 = -(1 -R S r ) dt 2 +(1 -R S r ) -1 dr 2 + r 2 d Ω 2 , r > R ( t ) \n(9) \nwhere, R S = 2 GM is the Schwarzschild radius in terms of the mass, M , of the wall, and \nd Ω 2 = dθ 2 + r 2 sin 2 θdφ 2 (10) \nIn the interior of the spherical domain wall, the line element is flat, as expected by Birkhoff's theorem, \nds 2 = -dT 2 + dr 2 + r 2 dθ 2 + r 2 sin 2 θdφ 2 , r < R ( t ) (11) \nThe interior time coordinate, T , is related to the observer time coordinate, t , via the proper time, τ , of the domain wall. \ndT dτ = [ 1 + ( dR dτ ) 2 ] 1 / 2 (12) \nand \nwhere \nB ≡ 1 -R S R (14) \nThe ratio of these equations gives \ndT dt = (1 + R 2 τ ) 1 / 2 B ( B + R 2 τ ) 1 / 2 = [ B -(1 -B ) B ˙ R 2 ] 1 / 2 (15) \nwhere R τ = dR/dτ and ˙ R = dR/dt . Integrating Eq. (15) still requires knowing R ( τ ) (or R ( t )) as a function of τ (or t ). \nSince we are restricting our minisuperspace to fields with spherically symmetry, we need not include any other metric degrees of freedom. The scalar field can also be truncated to the spherically symmetric modes (Φ = Φ( t, r )). \nBy integrating the equations of motion for the spherical domain wall, Ipser and Sikivie [4] found that the mass is a constant of motion and is given by \nM = 1 2 [ √ 1 + R 2 τ + √ B + R 2 τ ]4 πσR 2 (16) \nwhere it is assumed that max( R ) > 1 / 4 πGσ [4]. This expression for M is implicit since R S = 2 GM occurs in B . Solving for M explicitly in terms of R τ gives \nM = 4 πσR 2 [ 1 + R 2 τ -2 πGσR ] (17) \nand with the relations between T and τ given above we get \n√ \nM = 4 πσR 2 [ 1 √ 1 -R 2 T -2 πGσR ] (18) \nwhere R T denotes dR/dT . \nWe now discuss the classical collapse of the domain wall.", 'III. CLASSICAL TREATMENT OF DOMAIN WALL COLLAPSE': 'A naive approach to obtaining the dynamics for the spherical domain wall is to insert the spherical ansatz for both the wall and the metric in the original action. This would lead to an effective action for the radial coordinate R ( t ). However, it is known that this approach does not give the correct dynamics for gravitating systems. We find that this approach does not straightforwardly lead to mass conservation as given in Eq. (16). So we take the alternative approach of finding an action that leads \ndt dτ = 1 B [ B + ( dR dτ ) 2 ] 1 / 2 (13) \nto the correct mass conservation law. The form of the action can be deduced from Eq. (18) quite easily \nS eff = -4 πσ ∫ dTR 2 [ √ 1 -R 2 T -2 πGσR ] (19) \nwhich can now be written in terms of the external time t \nS eff = -4 πσ ∫ dtR 2 [ √ B -˙ R 2 B -2 πGσR √ B -1 -B B ˙ R 2 ] (20) \nand the effective Lagrangian is \nL eff = -4 πσR 2 √ B -˙ R 2 B -2 πGσR √ B -1 -B B ˙ R 2 (21) \nThe generalized momentum, Π, can be derived from Eq. (21) \nΠ = 4 πσR 2 ˙ R √ B 1 √ B 2 -˙ R 2 -2 πGσR (1 -B ) √ B 2 -(1 -B ) ˙ R 2 (22) \nThe Hamiltonian (in terms of ˙ R ) is \nH = 4 πσB 3 / 2 R 2 1 √ B 2 -˙ R 2 -2 πGσR √ B 2 -(1 -B ) ˙ R 2 (23) \nTo obtain H as a function of ( R, Π), we need to eliminate ˙ R in favor of Π using Eq. (22). This can be done but is messy, requiring solutions of a quartic polynomial. Instead we consider the limit when R is close to R S and hence B → 0. In this limit the denominators of the two terms in Eqs. (22) (also in (23)) are equal and \nwhere \nΠ ≈ 4 πµR 2 ˙ R √ B √ B 2 -˙ R 2 (24) \nµ ≡ σ (1 -2 πGσR S ) (25) \nand \n- \nH ≈ 4 πµB 3 / 2 R 2 √ B 2 ˙ R 2 (26) \n√ = [ ( B Π) 2 + B (4 πµR 2 ) 2 1 / 2 (27) \n] \nThe Hamiltonian is a conserved quantity and so, from Eq. (26), \n[ The Hamiltonian has the form of the energy of a relativistic particle, √ p 2 + m 2 , with a position dependent mass. \nB 3 / 2 R 2 √ B 2 -˙ R 2 = h (28) \nwhere h = H/ 4 πµ is a constant. (Up to the approximation used to obtain the simpler form of the Hamiltonian in Eq. (26), the Hamiltonian is the mass.) \nSolving Eq. (28) for ˙ R we get \n˙ R = ± B ( 1 -BR 4 h 2 ) 1 / 2 , (29) \nwhich, near the horizon, takes the form \n˙ R ≈ ± B ( 1 -1 2 BR 4 h 2 ) (30) \nsince B 0 as R → R S . \n→ \n→ The dynamics for R ∼ R S can be obtained by solving the equation ˙ R = ± B . To leading order in R -R S , the solution is \nR ( t ) ≈ R S +( R 0 -R S ) e ± t/R S . (31) \nwhere R 0 is the radius of the shell at t = 0. As we are interested in the collapsing solution, we choose the negative sign in the exponent. This solution implies that, from the classical point of view, the asymptotic observer never sees the formation of the horizon of the black hole, since R ( t ) = R S only as t →∞ . This result is similar to the well-known result (for example, see [5]) that it takes an infinite time for objects to fall into a pre-existing black hole as viewed by an asymptotic observer [6]. In our case there is no pre-existing horizon, which is itself taking an infinite amount of time to form during collapse. To see if this conclusion will change when quantum effects are taken into account ( e.g. Sec. 10.1.5 of [3]) we now explore the quantum dynamics of the collapsing domain wall.', 'IV. QUANTUM TREATMENT OF DOMAIN WALL COLLAPSE': 'The classical Hamiltonian in Eq. (27) has a square root and so we first consider the squared Hamiltonian \nH 2 = B Π B Π+ B (4 πµR 2 ) 2 (32) \nwhere we have made a choice for ordering B and Π in the first term. In general, we should add terms that depend on the commutator [ B, Π]. However, in the limit R → R S , we find \n[ B, Π] ∼ 1 R S \nEstimating H by the mass, M , of the domain wall, the terms due to the operating order ambiguity will be negligible provided \nM /greatermuch 1 R S ∼ m 2 P M \nwhere m P is the Planck mass. Hence the operator ordering ambiguity can be ignored for domain walls that are much more massive than the Planck mass. \nNow we apply the standard quantization procedure. We substitute \nΠ = -i ∂ ∂R (33) \nin the squared Schrodinger equation, \nH 2 Ψ = -∂ 2 Ψ ∂t 2 (34) \nThen \n-B ∂ ∂R ( B ∂ Ψ ∂R ) + B (4 πµR 2 ) 2 Ψ = -∂ 2 Ψ ∂t 2 (35) \nTo solve this equation, define \nwhich gives \nu = R + R S ln ∣ ∣ ∣ ∣ R R S -1 ∣ ∣ ∣ ∣ (36) \nB Π = -i ∂ ∂u (37) \nEq. (34) then gives \n∂ 2 Ψ ∂t 2 -∂ 2 Ψ ∂u 2 + B (4 πµR 2 ) 2 Ψ = 0 (38) \nThis is just the massive wave equation in a Minkowski background with a mass that depends on the position. Note that R needs to be written in terms of the coordinate u and this can be done (in principle) by inverting Eq. (36). However, care needs to be taken to choose the correct branch since the region R ∈ ( R S , ∞ ) maps onto u ( -∞ , + ) and R ∈ (0 , R S ) onto u (0 , -∞ ). \n∈ \n-∞ ∈ -∞ We are interested in the situation of a collapsing wall. In the region R ∼ R S , the logarithm in Eq. (36) dominates and \n∞ \n∈ \nR ∼ R S + R S e u/R S \nWe look for wave-packet solutions propagating toward R S , that is, toward u →-∞ . In this limit \nB ∼ e u/R S → 0 \nand the last term in Eq. (38) can be ignored. Wave packet dynamics in this region is simply given by the free wave equation and any function of light-cone coordinates ( u ± t ) is a solution. In particular, we can write a Gaussian wave packet solution that is propagating toward the Schwarzschild radius \nΨ = 1 √ 2 πs e -( u + t ) 2 / 2 s 2 (39) \nwhere s is some chosen width of the wave packet in the u coordinate. The width of the Gaussian wave packet remains fixed in the u coordinate while it shrinks in the R coordinate via the relation dR = Bdu which follows from \nEq. (36). This fact is of great importance, since if the wave packet remained of constant size in R coordinates, it might cross the event horizon in finite time. \nThe wave packet travels at the speed of light in the u coordinate - as can be seen directly from the wave equation Eq. (38) or from the solution, Eq. (39). However, to get to the horizon, it must travel out to u = -∞ , and this takes an infinite time. So we conclude that the quantum domain wall does not collapse to R S in a finite time, as far as the asymptotic observer is concerned, so that quantum effects do not alter the classical result that an asymptotic observer does not observe the formation of an event horizon. \nThe above analysis shows that the collapsing wall takes an infinite time to reach R = R S . The analysis leaves room for processes by which the wave packet can jump from the ( R S , ∞ ) region to the (0 , R S ) region, without ever going through R S . Note that this is different from tunneling through a barrier. In that case, the wave function is non-zero within the barrier, and a small part of it leaks over to the other side of the barrier. In the present case, R S occurs at u = -∞ and so, if there is any barrier, it is infinitely far away. If there is to be a jump from outside to inside R S , it does not show up in the present description using the Wheeler-de Witt equation. \nWe have obtained the massive wave equation, Eq. (38), by first squaring the classical Hamiltonian, Eq. (27). This procedure eliminated the square root occurring in the Hamiltonian. It is possible that some other quantization procedure will yield different conclusions. In this context, we note, in fact, that we need not square the Hamiltonian to get rid of the square root provided we work in the near horizon limit. In that case \nH = ( B Π) 2 + B (4 πµR 2 ) 2 ] 1 / 2 ≈ ± B Π (40) \n[ \n] where the sign is chosen to make H non-negative. Then the Schrodinger equation again yields wave packets propagating at the speed of light in the ( t, u ) coordinate system and with the horizon located at u = -∞ .', 'V. RADIATION - SEMICLASSICAL TREATMENT': "If an external observer never sees the formation of an event horizon, we need to explore what radiation might be observed that characterizes gravitational collapse. To do so we consider a quantum scalar field in the background of the collapsing domain wall. We do not consider gravitational radiation since this is excluded by our restriction to spherically symmetric modes in minisuperspace. In this section, we approach the problem using the functional Schrodinger equation since (i) we have already set up this approach and used it in the previous section, (ii) we believe the approach is more suited to treating the backreaction problem, and (iii) it allows us to calculate the total radiation of which Hawking radiation may only be a subset. To connect with earlier work, we discuss \nthe problem of Hawking radiation using the conventional Bogolubov transformations in Sec. VI. \nThe action for the scalar field is \nS = ∫ d 4 x √ -g 1 2 g µν ∂ µ Φ ∂ ν Φ (41) \nWe decompose the (spherically symmetric) scalar field into a complete set of real basis functions denoted by { f k ( r ) } \nΦ = ∑ k a k ( t ) f k ( r ) (42) \nThe exact form of the functions f k ( r ) will not be important for us. We will be interested in the wavefunction for the mode coefficients a k } . \n{ \n} In the functional Schrodinger picture, we wish to find the wavefunctional Ψ[Φ; t ] by solving a functional Schrodinger equation. This is equivalent to finding the wavefunction of an infinite set of variables, ψ ( { a k } , t ), by solving an ordinary Schrodinger equation in an infinite dimensional space. The procedure (detailed below) is to find independent eigenmodes, { b k } , for the system for which the Hamiltonian is a sum of terms, one for each independent eigenmode. Then the wavefunction factorizes and can be found by solving a time-dependent Schrodinger equation of just one variable. \nSince the metric inside and outside the shell have different forms, we split the action into two parts \nS = S in + S out (43) \nwhere \nS in = 2 π ∫ dt ∫ R ( t ) 0 dr r 2 [ -( ∂ t Φ) 2 ˙ T + ˙ T ( ∂ r Φ) 2 ] (44) \nS out = 2 π ∫ dt ∫ ∞ R ( t ) dr r 2 [ -( ∂ t Φ) 2 1 -R S /r + ( 1 -R S r ) ( ∂ r Φ) 2 ] (45) \n˙ T is given by Eq. (15), which with Eq. (29), gives \n˙ T = dT dt = B [ 1 + (1 -B ) R 4 h 2 ] 1 / 2 (46) \nAs R → R S , ˙ T ∼ B → 0. Therefore the kinetic term in S in diverges as ( R -R S ) -1 in this limit and dominates over the softer logarithmically divergent contribution to the kinetic term from S out . Similarly the gradient term in S in vanishes in this limit and is sub-dominant compared to the contribution coming from S out . Hence, \nS ∼ 2 π ∫ dt [ -1 B ∫ R S 0 dr r 2 ( ∂ t Φ) 2 + ∫ ∞ R S dr r 2 ( 1 -R S r ) ( ∂ r Φ) 2 ] (47) \nwhere we have changed the limits of the integrations to R S since we are working in the regime R ( t ) ∼ R S . This approximation is valid provided the contribution from r ∈ ( R S , R ( t )) to the integrals remains subdominant, and also the time variation introduced by the true integration limit ( R ( t )) can be ignored. These requirements are not arduous. \nNow, we use the expansion in modes in Eq. (42) to write \nS = ∫ dt [ -1 2 B ˙ a k M kk ' ˙ a k ' + 1 2 a k N kk ' a k ' ] (48) \nwhere M and N are matrices that are independent of R ( t ) and are given by \nM kk ' = 4 π ∫ R S 0 dr r 2 f k ( r ) f k ' ( r ) (49) \nN kk ' = 4 π ∫ ∞ R S dr r 2 ( 1 -R S r ) f ' k ( r ) f ' k ' ( r ) (50) \nUsing the standard quantization procedure, the wave function ψ ( a k , t ) satisfies \n[( 1 -R S R ) 1 2 Π k ( M -1 ) kk ' Π k ' + 1 2 a k N kk ' a k ' ] ψ = i ∂ψ ∂t (51) \nwhere \nΠ k = -i ∂ ∂a k (52) \nis the momentum operator conjugate to a k . \nSo the problem of radiation from the collapsing domain wall is equivalent to the problem of an infinite set of coupled harmonic oscillators whose masses go to infinity with time. Since the matrices M and N are symmetric and real ( i.e. Hermitian), it is possible to do a principal axis transformation to simultaneously diagonalize M and N (see Sec. 6-2 of Ref. [7] for example). Then for a single eigenmode, the Schrodinger equation takes the form \n[ -( 1 -R S R ) 1 2 m ∂ 2 ∂b 2 + 1 2 Kb 2 ] ψ ( b, t ) = i ∂ψ ( b, t ) ∂t (53) \nwhere m and K denote eigenvalues of M and N , and b is the eigenmode. \nWe re-write Eq. (53) in the standard form \n[ -1 2 m ∂ 2 ∂b 2 + m 2 ω 2 ( η ) b 2 ] ψ ( b, η ) = i ∂ ∂η ψ ( b, η ) (54) \nwhere \nη = ∫ t 0 dt ( 1 -R S R ) (55) \nω 2 ( η ) = K m 1 1 -R S /R ≡ ω 2 0 1 -R S /R (56) \nand \nFIG. 1: Model for R ( t ). \n<!-- image --> \nWe have chosen to set η ( t = 0) = 0. \nTo proceed further, we need to choose the background spacetime i.e. the behavior of R ( t ). The classical solution in Eq. (31), tells us that 1 -R S /R ∼ exp( -t/R S ) at late times. We are mostly interested in the particle production during this period. At early times, the behavior depends on how the spherical domain wall was created and we are free to choose a behavior for R ( t ) that is convenient for calculations and interpretation. To be able to interpret particle production at very late times it is easiest to have a static situation. This can be obtained if we artificially take the collapse to stop at some time, t f . Eventually we can take t f →∞ to go over to the eternal collapse case. So our choice for R will be \n1 -R S R = 1 t ∈ ( -∞ , 0) e -t/R S t ∈ (0 , t f ) e -t f /R S t ∈ ( t f , ∞ ) (57) \nThis choice does have the issue that the derivative of R has discontinuities at t = 0 and t = t f . However, we shall show below that these discontinuities do not affect particle production. \nWith the chosen behavior of R , the spacetime is static at early times and the initial vacuum state for the modes is the simple harmonic oscillator ground state, \nψ ( b, η = 0) = ( mω 0 π ) 1 / 4 e -mω 0 b 2 / 2 (58) \nThen the exact solution to Eq. (54) at later times is [8] \nψ ( b, η ) = e iα ( η ) ( m πρ 2 ) 1 / 4 exp [ i m 2 ( ρ η ρ + i ρ 2 ) b 2 ] (59) \nwhere ρ η denotes derivative of ρ ( η ) with respect to η , and ρ is given by the real solution of the ordinary (though non-linear) differential equation \nρ ηη + ω 2 ( η ) ρ = 1 ρ 3 (60) \nwith initial conditions \nρ (0) = 1 √ ω 0 , ρ η (0) = 0 (61) \nThe phase α is given by \nα ( η ) = -1 2 ∫ η 0 dη ' ρ 2 ( η ' ) (62) \nIn Appendix A we discuss the behavior of ρ as η → R S ( t → ∞ ). Also note that the solution for ρ and ρ η is continuous. \nConsider an observer with detectors that are designed to register particles of different frequencies for the free field φ at early times. Such an observer will interpret the wavefunction of a given mode b at late times in terms of simple harmonic oscillator states, { ϕ n } , at the final frequency, \n¯ ω = ω 0 e t f / 2 R S (63) \nThe number of quanta in eigenmode b can be evaluated by decomposing the wavefunction (Eq. (59)) in terms of the states, { ϕ n } , and by evaluating the occupation number of that mode. To implement this evaluation, we start by writing the wavefunction for a given mode at time t > t f in terms of the simple harmonic oscillator basis at t = 0. \nψ ( b, t ) = ∑ n c n ( t ) ϕ n ( b ) (64) \nwhere \nc n = ∫ db ϕ ∗ n ( b ) ψ ( b, t ) (65) \nwhich is an overlap of a Gaussian with the simple harmonic oscillator basis functions. The occupation number at eigenfrequency ¯ ω ( i.e. in the eigenmode b ) by the time t > t f , is given by the expectation value \nN ( t, ¯ ω ) = ∑ n n | c n | 2 (66) \nIn Appendix B we evaluate the occupation number in the eigenmode b and the result is given in Eq. (B12) \nN ( t, ¯ ω ) = ¯ ωρ 2 √ 2 [ ( 1 -1 ¯ ωρ 2 ) 2 + ( ρ η ¯ ωρ ) 2 ] (67) \nfor t > t f . \nBy calculating ˙ N , it can be checked that N remains constant for t < 0 and also t > t f . Hence all the particle production occurs for 0 ≤ t ≤ t f . There is a possibility that the particle production is due to discontinuities in the derivative of R at t = 0 , t f . However, as we shall see below, the particle number grows with increasing t f , while the discontinuity at t = 0 is fixed, and that at t = t f gets weaker. This indicates that particle production occurs only during 0 < t < t f and is a consequence of the gravitational collapse. \nNow we can take the t f →∞ limit. In Appendix A we have shown that ρ remains finite but ρ η → -∞ as t > \nt f → ∞ , provided ω 0 /negationslash = 0. However, we are interested in the behavior of N for fixed frequency, ¯ ω . Since ¯ ω = ω 0 e + t f / 2 R S , t f → ∞ also implies ω 0 → 0. From the discussion in Appendix A, we also know that ρ →∞ as ω 0 → 0. Hence we find \nN ( t, ¯ ω ) ∼ ¯ ωρ 2 √ 2 ∼ e t/ (2 R S ) √ 2 , t > t f →∞ (68) \nThis is confirmed by our numerical evaluation of N as a function of time t > t f for several different values of ω (see Fig. 2). \nTherefore the occupation number at any frequency diverges in the infinite time limit when backreaction is not taken into account. This implies that backreaction due to particle creation will have important consequences for gravitational collapse. \nWe have also numerically evaluated the spectrum of mode occupation numbers at any finite time and show the results in Fig. 3 for several different values of t . The similar shapes of the different curves suggest that there might be a simple relationship between them. By rescaling both axes we find that the curves roughly (though not completely) collapse into a single curve as shown in Fig. 4. Hence, knowing the spectrum at time t i approximately gives us the spectrum at all times via \nλ -1 ( t ) N ( t, ¯ ω/λ ' ( t )) = λ -1 ( t i ) N ( t i , ¯ ω/λ ' ( t i )) (69) \nwhere, we can determine the function λ ( t ) by considering the time variation of N ( t, 0), and λ ' by Eq. (63). The result is \nλ ( t ) = 1 √ 2 [ e t/ 2 R S + e -t/ 2 R S -2 ] (70) \nλ ' ( t ) = e t/ 2 R S (71) \nWe can compare the curve in Fig. 4 with the occupation numbers for the Planck distribution \nN P ( ω ) = 1 e βω -1 (72) \nwhere β is the inverse temperature. It is clear that the spectrum of occupation numbers is non-thermal. In particular, there is no singularity in N at ω = 0 at finite time, there are oscillations in N , and the rescaling law of Eq. (69) is not applicable to a thermal distribution. As t → ∞ , the peak at ω = 0 does diverge and the distribution becomes more and more thermal. Even at finite times, at small frequencies \nN P ( ω /lessmuch β -1 ) ≈ 1 βω (73) \nand the rescaling law amounts to rescaling the temperature by a factor λλ ' . \nNow, from Eq. (54), since the time derivative of the wavefunction on the right-hand side is with respect to η , ω is the mode frequency with respect to η and not with \nFIG. 2: N versus t/R S for various fixed values of ¯ ωR S . The curves are lower for higher ¯ ωR S . At late times the behavior is given by Eq. (68). \n<!-- image --> \nFIG. 3: N versus ¯ ωR S for various fixed values of t/R S . The occupation number at any frequency grows as t/R S increases. \n<!-- image --> \nrespect to time t . Eq. (55) tells us that the frequency in t is (1 -R S /R ) times the frequency in η , and at time t f , this implies \nω ( t ) = e -t f /R s ¯ ω (74) \nwhere the superscript ( t ) on ω refers to the fact that this frequency is with respect to time t . This rescaling of the frequency implies that the temperature for the asymptotic observer (with time coordinate t ) can be obtained by find the 'best fit temperature' β -1 and then rescaling by (1 -R S /R ). So the temperature seen by the asymptotic observer is \nT = e -t f /R S β -1 ( t f ) (75) \n(The temperature T is not to be confused with the time coordinate within the spherical domain wall, also denoted by T in Sec. II.) By using the scaling in Eq. (71), it is easy to see that β -1 grows as e + t f / 2 R S at late times and so T is constant. We can fit a thermal spectrum to the collapsed spectrum of Fig. 4, as shown in Fig. 5 to obtain \nT ≈ 0 . 19 R S = 2 . 4 4 πR S = 2 . 4 T H (76) \nwhere T H = 1 / 4 πR S ∼ . 08 /R S is the Hawking temperature. Since there is ambiguity in fitting the non-thermal spectrum by a thermal distribution, we can only say that the constant temperature, T , and the Hawking temperature are of comparable magnitude. \nFIG. 4: The same as Fig. 3 but with the axes rescaled as in Eq. (69). This graph shows that the spectrum at different times is approximately self-similar. \n<!-- image --> \nFIG. 5: Ln(1 + 1 /N ) versus ¯ ωR S for t = 8 R S . The dashed line shows Ln(1 + 1 /N P ) versus ¯ ωR S where N P is a Planck distribution. The slope gives β -1 and the temperature in Eq. (76). \n<!-- image --> \nThe occupation number N ( t, ω ) can be related to the asymptotic flux of radiation following standard procedures (e.g. Chapter 8 of Ref. [1]) and will result in the usual greybody factors. \nWe thus see that in the context of the Schrodinger formalism there is evidence of Hawking-like, but nonthermal radiation emitted during gravitational collapse before any event horizon is formed. There are several possible sources that one can imagine for this radiation, including radiation due to a time-dependent metric, and also Hawking emission [9]. Since the Schrodinger method in principle accounts for all such sources of radiation, it is worthwhile reexamining the original Hawking calculation, done using the Heisenberg picture and Bogolubov machinery, in the context of our above results.", "VI. HAWKING'S CALCULATION": "In Hawking's pioneering paper [9], he considered a collapsing body. By matching asymptotic field operators, he could find the Bogolubov coefficients, and then the particle emission rate. The result is the famous Hawking thermal radiation at temperature \nT H = κ 2 π (77) \nwhere κ = 1 / 2 R S is the surface gravity. \nSince Hawking radiation is calculated in the t → ∞ limit (asymptotic field operators), the result does not provide an answer to our original question: what will an experimentalist observe at a finite time? So we must re-calculate the radiation from a collapsing domain wall which is close to, but still larger than, the Schwarzschild radius. Stated in a slightly different way - does the experimentalist see Hawking radiation before the event horizon is formed? \nAs Hawking showed, the mode functions of a massless scalar field in the black hole spacetime have a 'phase pile-up' near the event horizon [9]. In other words, if we retrace the mode functions from I + back in time up to I -, the phase of the mode function diverges on I -at the point v 0 in Fig. 6, where the coordinate v is defined by \n∣ \nv = t + r + R S ln ∣ ∣ ∣ r R S -1 ∣ ∣ ∣ (78) \n∣ ∣ The radial part of the ingoing mode functions on I -are (Eq. (2.11) of [9]) \n∣ \nf ω ' = F ω ' ( r ) √ 2 πω ' r e iω ' v (79) \nThe relevant part of the outgoing mode function at frequency ω when extended back to I -is given in Eq. (2.18) of [9] \np (2) ω ∼ P -ω √ 2 πωr exp ( -i ω κ ln ( v 0 -v CD )) , v < v 0 (80) \nand zero for v > v 0 , where P -ω , C , D are constants, and κ = 1 / 2 R S . The expression in Eq. (80) is only valid for small v 0 v , and for large ω ' (geometrical optics limit). \nThe overlaps of p (2) ω with f ω ' and ¯ f ω ' determine the Bogolubov coefficients. This is equivalent to taking the Fourier transform of p (2) ω . Following Hawking's calculation, the Bogolubov coefficients for large ω ' are (see Eq. (2.19), (2.20) of [9]; also see [5]) \n- \nα (2) ωω ' ≈ P -ω 2 π ( CD ) iω/κ e i ( ω -ω ' ) v 0 √ ω ' ω × Γ ( 1 -iω κ ) ( -iω ' ) -1+ iω/κ (81) \nβ (2) ωω ' ≈ -iα (2) ω ( -ω ' ) (82) \nEven though the expression for α (2) ωω ' is only valid for large ω ' , Hawking argues on analyticity grounds that the singularity at ω ' = 0 should be present. So to obtain β (2) ωω ' it becomes necessary to go around the pole at ω ' = 0 to negative values of ω ' . The choice of deformation of the contour around the pole is determined on the grounds of analyticity, and the result is \n| β (2) ωω ' | = | α (2) ω ( -ω ' ) | = exp ( -πω κ ) | α (2) ωω ' | (83) \nFIG. 6: Spacetime of a blackhole with null rays originating at I -and going to I + . The last ray that makes it to I + is emitted at v 0 . An observer, O, far from the collapsing wall will attempt to detect a flux of radiation over a finite but large time interval. The last ray to get to the observer originates at I -at v 1 < v 0 and arrives at I + at u = u 1 . We are interested in finding the particle flux in the section of I + between the points marked i 0 and u 1 . \n<!-- image --> \nFIG. 7: The integration contour in the calculation of the Bogolubov coefficients runs from v = -∞ to v = 0 in the complex v plane. In Hawking's calculation v 0 -v 1 = 0, and the branch cut starts at the origin. For ω ' < 0, the integration contour is rotated to the upper imaginary axis, and for ω ' > 0 to the negative imaginary axis. In Hawking's case, this relates the Bogolubov coefficients α ωω ' and β ωω ' as pedagogically described in Ref. [5]. (As pointed out to us by F. Dowker, care is required in comparing the calculations in [9] and [5] since Hawking considers modes e + iω ' v while Townsend considers e -iω ' v . We are following Hawking's calculation.) In our case, these very rotations can also be done. However, the branch cut starts at v 0 -v 1 > 0 and the simple relation between the Bogolubov coefficients needed for thermal emission is not obtained. \n<!-- image --> \nFrom here, the calculation of the thermal flux of Hawking radiation follows. \nNow consider an observer who only sees the collapsing object for a finite time (see Fig. 6). The last ray detected by such an observer emerges from I -at v = v 1 < v 0 . For this observer, the phase of the mode functions have a tendency to pile-up but there is no divergence as in Eq. (80) because v ≤ v 1 < v 0 . As far as this observer is concerned, the behavior in Eq. (80) holds for v ≤ v 1 , while for v > v 1 the back-tracked mode functions vanish. The Fourier transform of p (2) ω now gives the Bogolubov coefficients the following ω ' dependence \nα (2) ωω ' ≈ ∫ v 1 dv exp [ -iω ' v -i ω κ ln( v 0 -v ) ] (84) \nFollowing Ref. [5], for ω ' > 0 we rotate the integration contour to the negative imaginary axis ( v → -iy ) and for ω ' < 0 to the positive imaginary axis (see Fig. 7). Simple manipulations for ω ' > 0 then give \n| α (2) ωω ' | = e πω/ 2 κ ∣ ∣ ∣ ∫ ∞ 0 dy e -ω ' y -ωθ/κ ( y 2 + δ 2 ) iω/ 2 κ ∣ ∣ ∣ \n∣ \n∣ where δ = ( v 0 -v 1 ) and θ = tan -1 ( δ/y ). Similarly, for ω ' < 0 we get \n| α (2) ωω ' | ≈ e -πω/ 2 κ ∣ ∣ ∣ ∣ ∫ ∞ 0 dy e -| ω ' | y + ωθ/κ ( y 2 + δ 2 ) iω/ 2 κ ∣ ∣ ∣ \n∣ ∣ Since β (2) ωω ' = α (2) ω ( -ω ' ) the above expressions yield both Bogolubov coefficients. \nThe crucial difference between Hawking's asymptotic result and the finite time result is the factor exp( ± ωθ/κ ) within the integral. Because of this factor, the relation in Eq. (83) does not hold and thermality is lost. However, if this extra factor is nearly unity, we can expect the spectrum to be nearly thermal. The integral is cut-off exponentially for y > 1 /ω ' and hence we estimate that the spectrum will be nearly thermal provided ωω ' δ/κ /lessmuch 1. Hence the spectrum is thermal at low frequencies and gets closer to being thermal as time goes on ( δ → 0), both of which seem plausible on physical grounds. \nIt is difficult however, to go beyond these qualitative statements in attempting to compare our results with what one might derive in the Hawking approach, in particular to determine possibly how much of the effect we obtain might be due to Hawking radiation, as opposed to particle creation by a changing metric. This is because the spectrum depends on a sum over all ω ' , while the Hawking analysis is done in the geometrical optics limit, at large frequencies. Hence to find the spectrum in this approach, we need a more complete solution to the equations of motion for all the modes of the scalar field in the domain wall background. Such solutions are more difficult to obtain (as described in [1] for example).", 'VII. INFALLING OBSERVER': "So far we have considered the wall collapse from the point of view of an asymptotic observer. From the point of view of an infalling observer, the time coordinate is τ of Sec. II and the collapse appears to proceed differently. For example, if we ignore radiation, the classical equation of motion can be written from the conservation of M in Eq. (17). Then, as the wall approaches the Schwarzschild radius, \nR 2 τ ≈ [ M 4 πσR 2 S +2 πGσR S ] 2 -1 (85) \nThe right-hand side is a non-zero constant, implying that the wall is collapsing with constant velocity in the τ coordinate. This shows that the collapse into a black hole occurs in a finite time interval for the infalling observer. Further, Hawking has argued [9] that the infalling observer does not detect significant Hawking radiation since the emission is dominantly at low frequencies compared to 1 /R S , while the infalling observer can only have local detectors of size less than R S . Thus the infalling observer would appear to see event horizon formation in a finite time, with no significant radiation emanating from the black hole. \nThese paradoxical views of the asymptotic and infalling observers need to be reconciled, and the conventional way to reconcile them is summarized in the spacetime diagram of an evaporating black hole shown in Fig. 8. The diagram is drawn so that the asymptotic observer sees evaporation in a finite time and the infalling observer falls into the black hole in a finite time also. \nWe have to note that the diagram in Fig. 8 does not follow from a rigorous solution to the problem of radiation from a collapsing object with backreaction included. There are some analyses of this problem in (1 + 1)dimensional models [10, 11] whose connection with the (3 + 1)-dimensional problem is unclear (e.g. [12]). Thus, the diagram in Fig. 8 is a conjectured diagram that is widely used in the literature. While this diagram may well be the correct one once the full problem of gravitational collapse with backreaction is solved, we have to emphasize that it also has some puzzling features that indicate that it may not be the best conjecture to make in the absence of a backreaction analysis. \nThe conventionally drawn spacetime of an evaporating black hole has features that are not consistent with our findings. Since the asymptotic observer sees Hawkinglike radiation from the collapsing wall prior to event horizon formation, the mass of the collapsing wall must be decreasing, and at the point denoted by F in Fig. 8 the entire energy of the wall has been radiated to I + . However, in the spacetime of Fig. 8, it is at precisely this instant that the asymptotic observer sees infalling objects disappear into the event horizon, even though there is nothing left of the collapsing wall to form the singularity. A spacetime region such as the triangular region \nFIG. 8: The conventional spacetime diagram for an evaporating black hole. The observer A will register a flux of quantum radiation even during collapse, and will be able to account for the entire energy of the shell by the time he/she gets to the line EF. From this point on, A will conclude that there is no energy left in the region of the collapsing shell. Yet A will see objects and other observers (such as I) disappear into what can at most be a Planck scale object and this is a puzzling feature of this picture. \n<!-- image --> \nbehind the event horizon only seems reasonable if not all of the collapsing shell energy has been lost to I + up to the point F, and there is some energy-momentum source left behind to crunch up in the singularity. Also, if the spacetime near the event horizon is described by the Schwarzschild metric, there is infinite gravitational redshfit of signals escaping to infinity, while the diagram shows that signals escape to infinity in a finite time. Finally, as is well known, the diagram in Fig. 8 also gives rise to the information loss paradox. While these features of the diagram in Fig. 8 are not inconceivable, they are sufficiently strange as to cast doubt on the validity of the picture. \nInstead it may happen that the true event horizon never forms in a gravitational collapse. We saw that an outside observer never sees formation of a horizon in finite time, not even in the full quantum treatment. What about an infalling observer? As in Hawking's case, the infalling observer does not see radiation, but this is due to size limitations of his detectors. The mode occupation numbers we have calculated will also be the mode occupation numbers that the infalling observer will calculate, even if they be associated with frequency modes that he cannot personally detect. The infalling observer never crosses an event horizon, not because it takes an infinite time, but because there is no event horizon to cross. As the infalling observer gets closer to the collapsing wall, the wall shrinks due to radiation back-reaction, evaporating before an event horizon can form. The evaporation appears mysterious to the infalling observer since his detectors don't register any emission from the collapsing \nwall. Yet he reconciles the absence of radiation with the evaporation as being due to a limitation of the frequency range of his detectors. Both he and the asymptotic observer would then agree that the spacetime diagram for an evaporating black hole is as shown in Fig. 9. In this picture a global event horizon and singularity never form. A trapped surface (from within which light cannot escape) may exist temporarily, but after all of the mass is radiated, the trapped surface disappears and light gets released to infinity. \nThe spacetime picture that we are advocating is similar to that described in Refs. [13, 14] and, more recently, Refs. [15, 16, 17].", 'VIII. DISCUSSION': "In this paper we have studied the collapse of a gravitating spherical domain wall using the functional Schrodinger equation. We would like to clearly delineate our analysis of the collapse and the emitted non-thermal quantum radiation from the interpretational issues about the formation of an event horizon. \nFirst, we studied the collapse of a gravitating spherical domain in both classical and quantum theory, ignoring any evaporative processes. It has been suggested in the literature that quantum fluctuations can cause the collapse and formation of a black hole in a finite (Schwarzschild) time [3]. However, our results show that this is not the case and the horizon does not form in a finite time even in the full quantum treatment. \nThen we studied radiation from the collapsing shell as seen by the asymptotic observer. In the process of gravitational collapse, there are two, perhaps related, sources of radiation: first is the radiation from particle creation in the changing gravitational field of the collapsing ball, and second may be Hawking-like radiation due to a mismatch of vacua at early and late times. The functional Schrodinger analysis takes all such sources into account and therefore gives the total particle production. We have found a non-thermal distribution of particle occupation numbers, with departures from thermality as illustrated in Fig. 3 and discussed toward the end of Sec. V. In a limited range of frequencies, the spectrum is approximately thermal and the temperature fitted in a restricted range of frequencies is constant and roughly equal to the Hawking temperature 1 / 4 πR S . The radiation becomes thermal in the entire range of frequencies only in the limit t → ∞ , i.e. when the horizon is formed. Further, the mode occupation number diverges in the infinite time limit, if the backreaction is neglected ( i.e. the background is held fixed). Since an outside observer never sees formation of a horizon in a finite time, radiation observed by him is never quite thermal. (Non-thermal features also get greatly amplified once the background is also treated quantum mechanically [18].) This nonthermal radiation has strong implications for the information loss paradox since it can carry information about \nthe collapsing matter. \nWithout a rigorous calculation that includes backreaction, one can not give a definite answer to the final fate of a collapsing object. It may happen that the diagram in Fig. 8 is correct and some radical and elaborate solutions to the problems we mentioned in Sec. VII are needed. However, one can imagine an alternative picture, different from the one in Fig. 8, which seems to have fewer problems, and that is that an event horizon never forms. Since the mass of the shell is decreasing during the collapse, the shell will be chasing its own Schwarzschild radius, and the question is whether the shell will catch up to its own Schwarzschild radius or completely evaporate before that happens [14]. \nWith backreaction included, the radiation should lead to a continual reduction of the Schwarzschild radius, R S , occurring in the Ipser-Sikivie metric (see Sec. II). Then, as seen by the asymptotic observer, one of two possibilities occurs: either the collapsing domain wall evaporates and R S → 0 in a finite time, or else backreaction causes the radiation rate to slow down and vanish in a finite time. This latter possibility is unlikely, as our estimates suggest that the rate of emission increases as R S decreases [24]. We therefore conjecture that the backreaction due to particle production will cause the collapsing domain wall spacetime to completely evaporate in a finite time. In this case, the spacetime can either be as given in Fig. 8, or have the same global spacetime structure as Minkowski space, as shown in Fig. 9. If the latter picture is correct, it also means that the infalling observer will not encounter an event horizon, because this feature is simply absent from the spacetime. Another way to see this is to note that the causal relation between two events is the same for all observers. Hence if the asymptotic observer sees a signal from an infalling object after he sees the last radiation ray emitted by the evaporating wall, this will also be the sequence of signals seen by the infalling observer. As discussed in Sec. VII, the infalling observer would expect to see an intense burst of radiation as the wall approaches the Schwarzchild radius, but can fail to do so because his detectors are too small to detect the emitted range of frequencies. \nIn the absence of an exact backreaction calculation, we also have to allow for the possibility that a value of the critical mass exists above which Fig. 8 applies and below which Fig. 9 holds. Also, as discussed by Hawking [19], the question of 'whether a black hole forms' is not sharp enough and may not make sense in the full quantum theory since all of the measurements are made by an asymptotic observer at infinity, while a collapsing object exists for a finite time and disappears by emitting radiation in the strong field region in the middle. An asymptotic observer can never be sure if a black hole formed because of underlying quantum uncertainty [19]. \nThe broad picture we have obtained is consistent with that proposed in Refs. [13, 14], though there are differences in the analysis and the conclusions. In particular, we find a non-thermal spectrum whereas Gerlach argues \nFIG. 9: The spacetime of a collapsing domain wall. During collapse the wall emits non-thermal (quasi-Hawking) radiation as depicted by the arrows. Our calculations indicate that the total energy flux between the point i 0 to some point indicated by F is equal to the energy of the initial domain wall. Hence we conjecture that the domain wall evaporates completely at point D. Between F and i + , there is no radiation flux arriving at I + . The event horizon and singularity present in the customary treatment are not formed and the spacetime structure is the same as that of Minkowski spacetime. \n<!-- image --> \nfor thermality. Our picture also supports the interpretation of Hawking radiation given in Ref. [5] whereby particles are created during the process of gravitational collapse and are then radiated slowly to form what we call Hawking radiation. We have indeed found particle production during the collapse but the radiation is not quite thermal. It is only in the frequency range where the occupation number spectrum can be approximated by T/ω (Eq. (73)) that thermality holds at finite time. Also note that the non-thermality we find is in the mode occupation numbers. Propagation of the radiation in the background metric will cause further non-thermality due to greybody factors. \nIf we live in a world of low scale gravity, the collision of particles in high energy accelerators will lead to a situation where the particles are in a continual state of gravitational collapse from which non-thermal radiation is being emitted. The life-time of such a state can be estimated once we know the details of the radiation more precisely from an analysis which includes backreaction. However, on dimensional grounds, Hawking's estimate for the lifetime of a black hole ( ∼ R 3 S /G ) may well apply to the colliding particles as well. \nIn reality the collapse is further complicated by the fact that the collapsing object is not kept in isolation and there are external forces that can disrupt the collapse at any point in time. From the perspective of potential information loss, note that any infalling encyclopedias can be returned to the asymptotic observer if the collapse \nis disrupted at any time, as it could be, for example by a bomb set to go off at some late, but finite time. Most importantly, since we calculate that the radiation emitted during the gravitational collapse is never truly thermal, the classic information loss issue in black holes should, in this case, be a non-problem for the asymptotic observer [25]. \nOur primary result, that no event horizon forms in gravitational collapse as seen by an asymptotic observer is suggestive of the possibility of using the number of local event horizons to classify and divide Hilbert space into superselection sectors, labeled by the number of local event horizons. Our result suggests that no operator could increase the number of event horizons, but the possibility of reducing the number of pre-existing primordial event horizons is not so clear and would require that Hawking radiation not cause any primordial black hole event horizons to evaporate completely. \nOur conclusions have been derived on the basis of a number of assumptions which we now discuss. The first is the truncation of superspace to minisuperspace. We have only included spherically symmetric field configurations. Even then, the metric is restricted to be the classical solution sourced by a spherical domain wall. A more general analysis would include more metric degrees of freedom, though it is hard to see how this would make a difference to our conclusion. Similarly, we have restricted ourselves to a zero thickness domain wall. A more general analysis would allow for a thick wall. Finally, the Wheeler-de Witt formalism as we have used it, does not allow for the creation and annihilation of domain walls ('third quantization'). Perhaps third quantization could allow for the spontaneous creation of a black hole and the annihilation of the wall, effectively leading to black hole formation. A fourth possibility is that our Lagrangian breaks down near the Schwarzschild horizon and 'quantum gravity' effects become important. This is usually thought not to be the case since the spacetime curvature near the horizon is small for large black holes. \nPerhaps the most serious drawback of our analysis is that it does not include backreaction on the gravitational collapse due to radiation. While we do not expect such inclusion to alter our conclusions regarding the nonexistence of event horizons for asymptotic observers, we are currently exploring ways to extend our treatments to include backreaction. \nNo theoretical idea is complete without the possibility of experimental verification and so it is important to ask if the picture we have developed in this paper can also be tested experimentally. We have already mentioned the relevance of our conclusions to black hole production in particle accelerators provided low scale gravity is correct. However, there is an even more accessible experimental system where these theoretical ideas can be put to the test. These are condensed matter systems in which sonic black holes (dumbholes) may exist [20]. It is very hard to realize a dumbhole in the laboratory for various experimental reasons; the closest known realiza- \ntion seems to be the propagating He-3 AB interface in the experiment of Ref. [21] as discussed in [22]. Yet the crucial aspect of our work in this paper is that there is no need to produce a dumbhole in order to see acoustic 'pre-Hawking' radiation . The process of collapse toward a dumbhole will give off radiation. This is also the conclusion of Ref. [23] though the details of the analysis and conclusions are different - for example, we find non-thermal emission whereas these authors claim thermal emission with a modified temperature that is lower than the Hawking temperature. In any case, it should be much easier to do experiments in the laboratory that do not go all the way to forming a dumbhole, and this could be an ideal arena to test pre-Hawking radiation. \nOur conclusions are important not only for the general issue of the breakdown of unitarity via information loss, but also for more general studies of black hole formation, whether they be in the context of astrophysics ( e.g. galactic black holes) or in future accelerator experiments. In all these situations, we are asymptotic observers watching the gravitational collapse of matter, and we may never see effects associated with a black hole event horizon. Only effects occurring during the gravitational collapse itself appear to be visible.", 'Acknowledgments': 'We are grateful to Fay Dowker, Larry Ford, Alan Guth, Jonathan Halliwell, Irit Maor, Harsh Mathur, Don Page, Paul Townsend, Alex Vilenkin, Bob Wald, Frank Wilczek, and Serge Winitzki for their interest, advice, and feedback. TV is also grateful to the participants of the COSLAB meeting at the Lorentz Center (Leiden), including Brandon Carter, Tom Kibble, Bill Unruh and Matt Visser, for discussion. TV acknowledges hospitality by Imperial College and Leiden University where some of this work was done. DS is grateful to participants of COSMO 06 meeting for very useful comments and discussions. LMK acknowledges hospitality of Vanderbilt University during the completion of this work. This work was supported by the U.S. Department of Energy, NASA at Case Western Reserve University, and NWO (Netherlands) at Leiden University.', 'APPENDIX A: ρ EQUATION': "In the range t < 0, ω is a constant and the solution to Eq. (60) is \nρ ( η ) = 1 √ ω 0 (A1) \nIn the range 0 < t < t f , we do not have an analytic solution but we can derive certain useful properties. First note that in terms of η \nω 2 = ω 2 0 1 -η/R S (A2) \nThen the equation for ρ after rescalings can be written as: \nd 2 f dη ' 2 = -( ω 0 R S ) 2 [ f 1 -η ' -1 f 3 ] (A3) \nwhere η ' = η/R S , f = √ ω 0 ρ . The boundary conditions are \nf (0) = 1 , df (0) dη ' = 0 (A4) \nThe last term with the 1 /f 3 becomes singular as f → 0. Let us consider another equation with the 1 /f 3 replaced by something better behaved. For example, \nd 2 g dη ' 2 = -( ω 0 R S ) 2 [ g 1 -η ' -g ] (A5) \nwith boundary conditions \ng (0) = 1 , dg (0) dη ' = 0 (A6) \nEq. (A5) implies that g ( η ' ) is monotonically decreasing as long as g ( η ' ) > 0. Furthermore, it is decreasing faster than the solution for f as long as f < 1, since the 1 /f 3 term in Eq. (A3) is a larger 'repulsive' force than the g term in the Eq. (A5). So \ng ( η ' ) ≤ f ( η ' ) (A7) \nfor all η ' such that g ( η ' ) > 0. \nEqs. (A5) with initial conditions (A6) can be solved in terms of degenerate hypergeometric functions. For us, the important point is that the solution for g is positive for all η ' and, in particular, g (1) > 0 for all the values of ω 0 R S that we have checked. Therefore f ( η ' ) is positive, at least for a wide range of ω 0 R S . \nLet f 1 = f (1) /negationslash = 0. Then the equation for f can be expanded near η ' = 1. \nd 2 f dη ' 2 ∼ -( ω 0 R S ) 2 [ f 1 1 -η ' -1 f 3 1 ] (A8) \nThis shows that \ndf dη ' ∼ ( ω 0 R S ) 2 f 1 ln(1 -η ' ) →-∞ (A9) \nas η ' → 1. Hence ρ ( η = R S ) is strictly positive and finite while ρ η ( η = R S ) = -∞ for finite and non-zero ω 0 . Since f = √ ω 0 ρ , and f → 1 for ω 0 → 0, we also see that ρ and ρ η → 0 as ω 0 0. \n→∞ \n→ \n→ In the range t f < t , ω is a constant. However, the solution for ρ is not a constant, unlike in the range t < 0, since the constant solution 1 / √ ω ( t f ) does not necessarily match up with ρ ( t f -) to ensure a continuous solution. Yet it is easy to check that in this region ˙ N = 0 and so there is no change in the occupation numbers. So we need only find N ( t f -, ¯ ω ) to determine N ( t →∞ , ¯ ω ).", 'APPENDIX B: NUMBER OF PARTICLES RADIATED AS A FUNCTION OF TIME': "We use the simple harmonic oscillator basis states but at a frequency ¯ ω to keep track of the different ω 's in the calculation. To evaluate the occupation numbers at time t > t f , we need only set ¯ ω = ω ( t f ). So \nφ n ( b ) = ( m ¯ ω π ) 1 / 4 e -m ¯ ωb 2 / 2 √ 2 n n ! H n ( √ m ¯ ωb ) (B1) \nwhere H n are Hermite polynomials. Then Eq. (65) together with Eq. (59) gives \nc n = ( 1 π 2 ¯ ωρ 2 ) 1 / 4 e iα √ 2 n n ! ∫ dξe -Pξ 2 / 2 H n ( ξ ) ≡ ( 1 π 2 ¯ ωρ 2 ) 1 / 4 e iα √ 2 n n ! I n (B2) \nwhere \nP = 1 -i ¯ ω ( ρ η ρ + i ρ 2 ) (B3) \nTo find I n consider the corresponding integral over the generating function for the Hermite polynomials \nJ ( z ) = ∫ dξe -Pξ 2 / 2 e -z 2 +2 zξ = √ 2 π P e -z 2 (1 -2 /P ) (B4) \nSince \nTherefore \ne -z 2 +2 zξ = ∞ ∑ n =0 z n n ! H n ( ξ ) (B5) \n∫ dξe -Pξ 2 / 2 H n ( ξ ) = d n dz n J ( z ) ∣ ∣ ∣ ∣ z =0 (B6) \nI n = √ 2 π P ( 1 -2 P ) n/ 2 H n (0) (B7) \nSince \nH n (0) = ( -1) n/ 2 √ 2 n n ! ( n -1)!! √ n ! , n = even (B8) \nand H n (0) = 0 for odd n , we find the coefficients c n for even values of n , \nc n = ( -1) n/ 2 e iα (¯ ωρ 2 ) 1 / 4 √ 2 P ( 1 -2 P ) n/ 2 ( n -1)!! √ n ! (B9) \nFor odd n , c n = 0. \nNext we find the number of particles produced. Let \nThen \nχ = ∣ ∣ ∣ ∣ 1 -2 P ∣ ∣ ∣ ∣ (B10) \nN ( t, ¯ ω ) = ∑ n =even n | c n | 2 = 2 √ ¯ ωρ 2 | P | χ d dχ ∑ n =even ( n -1)!! n !! χ n = 2 √ ¯ ωρ 2 | P | χ d dχ 1 √ 1 -χ 2 = 2 √ ¯ ωρ 2 | P | χ 2 (1 -χ 2 ) 3 / 2 (B11) \n- [1] 'Quantum Fields in Curved Space', N.D. Birrell and P.C.W. Davies, Cambridge University Press (1982).\n- [2] B.S. DeWitt, Phys. Rev. 160 , 1113 (1967).\n- [3] 'Black Hole Physics', V.P. Frolov and I.D. Novikov (Kluwer Academic Publishers, Dordrecht, 1998).\n- [4] J. Ipser and P. Sikivie, Phys. Rev. D 30 , 712 (1984).\n- [5] P. Townsend, gr-qc/9707012.\n- [6] 'Gravitation', C.W. Misner, K.S. Thorne and J.A. Wheeler, (W.H. Freeman, New York, 1973).\n- [7] 'Classical Mechanics', H. Goldstein, Addison-Wesley 1980.\n- [8] C. M. A. Dantas, I. A. Pedrosa and B. Baseia, Phys. Rev. A 45 , 1320 (1992).\n- [9] S. W. Hawking, Commun. Math. Phys. 43 , 199 (1975) [Erratum-ibid. 46 , 206 (1976)].\n- [10] C. G. . Callan, S. B. Giddings, J. A. Harvey and A. Strominger, Phys. Rev. D 45 , 1005 (1992) [arXiv:hep-th/9111056].\n- [11] P. C. W. Davies, S. A. Fulling and W. G. Unruh, Phys. Rev. D 13 , 2720 (1976).\n- [12] V. P. Frolov, P. Sutton and A. Zelnikov, Phys. Rev. D 61 , 024021 (2000) [arXiv:hep-th/9909086]. \nInserting the expressions for χ and P , leads to \nN ( t, ¯ ω ) = ¯ ωρ 2 √ 2 [ ( 1 -1 ¯ ωρ 2 ) 2 + ( ρ η ¯ ωρ ) 2 ] (B12) \nIn summary, we have found the occupation number of modes as a function of ρ which is a function of time as given by the non-linear differential equation Eq. (60). The equation connecting ρ and time t has only been solved numerically but we have discussed the behavior of ρ and ρ η as η → R S ( t →∞ ) in Appendix A. \n- [13] D. G. Boulware, Phys. Rev. D 13 , 2169 (1976).\n- [14] U. H. Gerlach, Phys. Rev. D 14 , 1479 (1976).\n- [15] P. Hajicek, Phys. Rev. D 36 , 1065 (1987).\n- [16] A. Ashtekar and M. Bojowald, Class. Quant. Grav. 22 , 3349 (2005) [arXiv:gr-qc/0504029].\n- [17] G. L. Alberghi, R. Casadio, G. P. Vacca and G. Venturi, Phys. Rev. D 64 , 104012 (2001) [arXiv:gr-qc/0102014].\n- [18] T. Vachaspati and D. Stojkovic, arXiv:gr-qc/0701096.\n- [19] S. W. Hawking, Phys. Rev. D 72 , 084013 (2005) [arXiv:hep-th/0507171].\n- [20] W. G. Unruh, Phys. Rev. Lett. 46 , 1351 (1981).\n- [21] M. Bartkowiak et al, Physica B 284-288 , 240 (2000).\n- [22] T. Vachaspati, Journal of Low Temperature Physics 136 , Nos. 5/6 (2004).\n- [23] C. Barcelo, S. Liberati, S. Sonego, and M. Visser, gr-qc/0607008.\n- [24] Though our analysis only holds for objects of mass greater than Planck mass (see Sec. IV) and there could be qualitative changes in the collapse and radiation as the Planck mass is approached.\n- [25] To be sure, one must explicitly evaluate the information contained in the non-thermal radiation."}

2009arXiv0907.1190B

Better Late than Never: Information Retrieval from Black Holes

2009-01-01

['-', '-', '-']

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We show that, in order to preserve the equivalence principle until late times in unitarily evaporating black holes, the thermodynamic entropy of a black hole must be primarily entropy of entanglement across the event horizon. For such black holes, we show that the information entering a black hole becomes encoded in correlations within a tripartite quantum state, the quantum analogue of a one-time pad, and is only decoded into the outgoing radiation very late in the evaporation. This behavior generically describes the unitary evaporation of highly entangled black holes and requires no specially designed evolution. Our work suggests the existence of a matter-field sum rule for any fundamental theory.

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https://arxiv.org/pdf/0907.1190.pdf

{'Better Late than Never: Information Retrieval from Black Holes': 'Samuel L. Braunstein and Stefano Pirandola \nDepartment of Computer Science, University of York, York YO10 5GH, United Kingdom', 'Karol ˙ Zyczkowski': "Institute of Physics, Jagiellonian University, 30-059 Krakow, Poland and Center for Theoretical Physics, Polish Academy of Science, 02-668 Warszawa, Poland (Dated: Received 17 April 2012) \nWe show that, in order to preserve the equivalence principle until late times in unitarily evaporating black holes, the thermodynamic entropy of a black hole must be primarily entropy of entanglement across the event horizon. For such black holes, we show that the information entering a black hole becomes encoded in correlations within a tripartite quantum state, the quantum analogue of a one-time pad, and is only decoded into the outgoing radiation very late in the evaporation. This behavior generically describes the unitary evaporation of highly entangled black holes and requires no specially designed evolution. Our work suggests the existence of a matter-field sum rule for any fundamental theory. \nPACS numbers: 04.70.Dy, 03.65.Xp, 03.67.-a, 03.70.+k \nBlack hole evaporation as tunneling .-Although pair creation provides the conventional heuristic picture of the microscopic process by which a black hole evaporates [1], it has come under increasing suspicion due to intrinsic difficulties. In particular, pair creation necessarily requires the dimensionality of the interior Hilbert space of a black hole to be increasing while simultaneously its physical size is decreasing [2, 3]. \nBy contrast, quantum tunneling, which operates by moving quantum subsystems across the classically forbidden barrier of the event horizon, naturally avoids this difficulty [3]. Furthermore, quantum tunneling invites an elegant Hilbert space description of the evaporation process across event horizons [3]: We start with the standard decomposition of a black hole Hilbert space into a tensor product between the interior (int) and exterior (ext) by H int ⊗H ext [4] and note that an event horizon's tensor product structure in no way implies that its spatial location cannot be fuzzy [3]. \nTunneling now operates [3] by selecting some subsystem from the black hole interior and moving it to the exterior H int →H B ⊗H R by \n| i 〉 int → ( U | i 〉 ) BR , (1) \nwhere U denotes the unitary process that might be thought of as 'selecting' the subsystem to eject, | i 〉 is the initial state of the black hole interior, B denotes the reduced size subsystem corresponding to the remaining interior after evaporation, and R denotes the subsystem that escapes as radiation [3, 5, 6]. \nEquation (1) has been used before to study black hole evaporation [3, 5, 6]; however, with the exception of Ref. [3], it has not been used as a process associated with any underlying physical mechanism. Indeed, Ref. [3] showed that the symmetries implicit in this equation, in conjunction with global conservation laws for the no-hair \nquantities (energy, charge, and angular momentum), suffice to completely determine black hole tunneling probabilities for any black hole and particle type, reproducing and even extending the predictions of field theory on curved spacetime. This work therefore strongly supports Eq. (1) as a pertinent microscopic formulation of unitary black hole evaporation. Its implications for the retrieval of information about in-fallen matter will be further studied here. \nDynamical evaporation with entanglement .-It is now well accepted that entanglement across boundaries is generic [7]. Therefore, our key point of departure from previous work [3, 5, 6, 8, 9] will be to allow for entanglement across the event horizon. Incorporated into the evaporative dynamics of Eq. (1), but making no assumption of how much or how little transevent horizon entanglement there may be, this entanglement gives \nN ∑ i =1 √ p i | i 〉 int ⊗| i 〉 ext → N ∑ i =1 √ p i ( U | i 〉 ) BR ⊗| i 〉 ext . (2) \nNow, the nature of the black hole as a compact object of a given mass constrains any interior evolution to only access an effectively finite dimensional Hilbert space [3]. Quantities defined within (the support of) this finite Hilbert space will similarly be finite, including, for example, any von Neumann entropies, measures of entanglement, etc. Indeed, it has been argued [10] that the dimensionality for the initial black hole Hilbert space should be well approximated by the thermodynamic entropy S BH = A / (4 ln 2) for a black hole of area A , giving a dimensionality N ≡ dim( H int ) = BR = 2 S BH , where we reuse subsystem labels for Hilbert space dimensionalities and for later convenience we evaluate entropies using base-two logarithms. We might say that the black hole interior comprises log 2 N = S BH qubits. (Throughout, the term 'qubits' is used merely as a unit of information \ncontent and does not literally imply a set of two-level systems.) That the number of qubits initially within the black hole is well approximated by S BH is supported by the holographic principle [11] and independently by the amount of Hawking radiation that would be generated, consistent with energy conservation. \nNaively, to make quantitative predictions based on this description, we would need to know the detailed dynamics U within the black hole. In fact, the behavior of information flow in a high-dimensional system under a specific unitary will be in excellent agreement with the Haar average over all unitaries acting on dimension N . This follows from Levy's lemma [12], which states that the logarithm of the probability of any such difference /epsilon1 scales as -N/epsilon1 2 . For a stellar mass black hole, such dimensionalities N must be at least 10 10 77 , so even the smallest deviations from the average behavior should occur with vanishingly small probability. Numerical simulations in even very low dimensions show this to be well supported, and similar results are well known beyond black hole physics [13]. Thus, here we replace the behavior of the specific unitary in Eq. (2) by the Haar average. \nVanishing of transevent horizon entanglement .-Moving to the average behavior allows one to rigorously interpret the evaporative dynamics of a black hole in terms of the properties of random quantum error correcting codes [6]. In this interpretation, one-half of an entangled state is encoded into a larger Hilbert space via a random unitary encoding. Decoupling theorems [14] tell us how much (how many qubits) of the encoded state one must have access to, in principle, in order to reconstruct the original unencoded state, including its entanglement. We derive a generalized decoupling theorem and use it to address a broader set of questions. (See the Supplemental Material in Ref. [15] for proofs and a discussion of both quantum and classical decoupling theorems.) \nFor example, for an entangled black hole evolving via Eq. (2), this generalized decoupling theorem shows that, for any positive number c , once \nlog 2 R = 1 2 S BH + 1 2 H (1 / 2) ( ρ ext ) + c (3) \nqubits have radiated away, the transevent horizon entanglement will have vanished, appearing instead, with virtually unit fidelity (at least 1 -2 -c ), as entanglement between the external neighborhood and radiation. Here, the entropy of entanglement is quantified by a R'enyi entropy H ( q ) ( ρ ) ≡ log 2 (tr ρ q ) / (1 -q ) with q of order unity, for the reduced density matrix of the (ext) state ρ ext = N i =1 p i | i ext ext 〈 i neighboring the event horizon. \n∑ \n〉 \n| \n| 〈 Entanglement and the equivalence principle .-We will now explicitly link the presence of transevent horizon entanglement with the equivalence principle. Specifically, the equivalence principle is expected to be preserved for black holes larger than the Planck scale. We will argue below that the presence of this entanglement must \nbe similarly preserved until such scales. We then use the tunneling dynamics to calculate the initial amount of transevent horizon entanglement. \nWe start by recalling the equivalence principle, which tells us that a freely falling observer sees no local effects due to gravity. Applied to black holes, it has been argued [10] that the equivalence principle implies that an observer freely falling past the event horizon would see no Hawking radiation, only a zero temperature vacuum state-just as an unaccelerated observer in flat spacetime. Now, the well-understood quantum physics of condensed matter systems tells us that entanglement across boundaries is generic in or near the ground state [7]. Furthermore, in axiomatic quantum field theory, entanglement across boundaries for fields in their vacuum state is implicit in the Reeh-Schlieder theorem [16]. In the Supplemental Material [15], we derive a lower bound for the energy of a free scalar field when the quantum state is restricted to have no entanglement across an arbitrary hypothetical boundary. This disentanglement energy diverges as a power of the UV regulator [15], and hence is far above the vacuum state. Applied to black holes, this means that the loss of entanglement across the event horizon would force the quantum fields across it to be arbitrarily far from the vacuum state-an energetic curtain would have descended around the black hole [17]signaling a manifest failure of the equivalence principle. \nNext, we use the epoch for the loss of transevent horizon entanglement, given by Eq. (3), to quantify how much transevent horizon entanglement was in the initial black hole. Here, we rely on the observation that a black hole's size may be directly quantified by its area or, equivalently, its entropy. For black holes in the latter stages of evaporation via Eq. (2), their entropy is well approximated by S BH -log 2 R [15]. Therefore, an evaporating black hole can be said to approach the Planck scale (see Ref. [15] for a detailed discussion) when, to high precision, log 2 R ≈ S BH . From Eq. (3), preserving entanglement until such late times implies that \nH (1 / 2) ( ρ ext ) ≈ S BH . (4) \nIn other words, preserving transevent horizon entanglement up until an evaporating black hole approaches the Planck scale requires that its initial entanglement entropy be almost exactly its initial thermodynamic entropy [18]. This result is insensitive to where we place the entry point to the Planck scale [15]. Furthermore, this equality does not change when the quantum state of the matter that originally collapsed to form the black hole is taken into account [15]. Finally, we note that, for the special case where the transevent horizon entangled state in Eq. (2) reduces to uniform entanglement (where all nonzero probabilities are equal), Eq. (3) may be replaced by established results [6], allowing a straightforward check of our analysis (see Ref. [15]). \nIncorporating in-fallen matter .-Naively, one might expect the entropy of ordinary matter S matter that collapses to form a black hole to be a large fraction of a black hole's thermodynamic entropy. However, this is not the case: 't Hooft [11] has shown that S matter /lessorsimilar S 3 / 4 BH . Thus, for anything but Planck scale black holes, the entropic contribution from in-fallen matter is negligible, S matter ≪ S BH . This then raises the question of when and in what fashion the information about the in-fallen matter can be retrieved. The remainder of this Letter addresses this question. We proceed from our result, Eq. (4), that a black hole's thermodynamic entropy is almost entirely entropy of transevent horizon entanglement. In so doing, we need not further appeal to the equivalence principle or the specific state of quantum fields across the event horizon. \nWe tag the matter by entanglement with some distant reference (ref) subsystem [6, 9] and use the decoupling theorem to track its flow. It is conventional to assume that there is no 'bleaching' mechanism [19] that can strip away any of the information about the in-fallen matter as it collapses to form a black hole. In that case, the exterior Hilbert space can contain no information about it. Now, the no-hiding theorem [9] gives a unique description for a quantum state where information is not available within some specific subsystem. No-hiding implies that the quantum state of a newly formed black hole interior (int) and its surroundings must have the form \n1 √ K K ∑ i =1 | i 〉 ref ⊗ ∑ j √ p j ( | i 〉 ⊗ | j 〉 ⊕ 0) int ⊗| j 〉 ext , (5a) \nup to overall int-local and ext-local unitaries. Here, ⊕ 0 means we pad unused dimensions of the interior space by zero vectors [9], and log 2 K ≡ S matter is the number of qubits describing the quantum state of the matter collapsing to form the black hole. \nApplying the dynamics of Eq. (1) to our entangled black hole, in the presence of in-fallen matter, gives \n→ 1 √ K K ∑ i =1 | i 〉 ref ⊗ ∑ j √ p j [ U ( | i 〉 ⊗ | j 〉 ⊕ 0)] BR ⊗| j 〉 ext . (5b) \nInformation retrieval from entangled black holes .-We now apply our generalized decoupling theorem to the evaporative dynamics of Eq. (5). In order to state our results, it will be convenient to roughly quantify the number of unentangled (pure) qubits within the initial black hole state in Eq. (5a); we define this 'excess' as \nχ ( q ) ≡ S BH -S matter -H ( q ) ( ρ ext ) ≥ 0 . (6) \nNote that Eq. (4) implies χ (1 / 2) ≪ S BH . \nWe now summarize the results about information encoding and retrieval. Since, in each application of the theorem, an independent dummy variable appears [ c in \nEq. (3)] that is dwarfed by other entropies, here we omit reference to them (the complete statements can be found in Ref. [15]). \nThermalization: Initially, one might suppose that any in-fallen matter would be well within the interior of the black hole, far inside the event horizon, and so would not be selected by U to participate in tunneling across this boundary. Only after the black hole had sufficiently 'scrambled' the internal states (after what might be called the global thermalization time [6] for the black hole) would the subsystem encoding the state of the infallen matter be accessible for selection and ejection by tunneling [20]. Note that estimates of scrambling times vary. Some recent analyses suggest that black holes are fast scramblers [6, 21] (with the scrambling time being little more than the time for a single Hawking photon to evaporate), whereas other estimates are slow [22]. \nEncoding: During the global thermalization time and for the next 1 2 χ (1 / 2) qubits radiated, all the information about the state of the in-fallen matter is encoded with virtually unit fidelity within the black hole interior. For the next S matter + 1 2 ( χ (2) -χ (1 / 2) ) qubits radiated, this information becomes encoded into the tripartite correlations of a quantum one-time pad [9] among the black hole interior, the external neighborhood, and the radiation. In other words, it is the evaporation via tunneling (across the event horizon) that encodes the information as tripartite entanglement. After encoding and until the last S matter + 1 2 χ (2) qubits radiated the information remains within this quantum one-time pad; it is inaccessible from any subsystem individually, but it is accessible from any two of them. The quantum one-time pad is a random quantum error correction code. The properties of such codes dictate the size of subsystems one must have access to in order to be able to reconstruct the original state of the in-fallen matter. \nDecoding: At this point in the evaporation process, entanglement within the black hole becomes so depleted that it can no longer contain the correlations of all the in-fallen matter. The final S matter + 1 2 χ (2) qubits to be radiated marks the start of information release into the radiation. From here until the final 1 2 χ (1 / 2) qubits radiated from the black hole, the full information about the in-fallen matter is decoded and becomes available in the outgoing radiation for the first time. This decoding takes the same amount of time as the encoding. Since typically χ (2) -χ (1 / 2) /lessorsimilar O (1) and this quantity cannot be negative, the encoding or decoding occurs at roughly the radiation emission rate; recall, Hawking quanta typically carry around one (qu)bit of thermal entropy. (See Ref. [15] for a heuristic picture of the flow of information.) \nThis completes our analysis of information retrieval for unitarily evaporating highly entangled black holes. As is evident from this summary, decoding of the information about in-fallen matter is very brief, occurring within the final and vanishingly small fraction S matter /S BH /lessorsimilar \nO ( S -1 / 4 BH ) of a large black hole's lifetime (as measured in Hawking quanta radiated). This is so late that its timing is unaffected by even very long scrambling times [22]. That said, within this very brief epoch, decoding is also very slow, occurring at the radiation emission rate; thus, information about the in-fallen matter is decoded over the time scale required for S matter /lessorsimilar O ( S 3 / 4 BH ) Hawking quanta to evaporate. Because the number of qubits radiated during decoding is so vast, essentially all the information has been retrieved long before the black hole shrinks to the Planck scale. Note that the time scales above follow from 't Hooft's entropic bound [11]; however, none of the mechanisms or mathematical results in this Letter rely on this bound [15]. \nDiscussion and wider implications .-The application of information theoretic approaches to the physics of black holes is relatively new [2, 3, 5, 6, 8-10]. Here, we have shown that this approach offers a description of black holes as highly entangled, with direct consequences for the time course of information retrieval therefrom. This approach necessarily requires an explicit formulation of the microscopic evaporation process, which, here, we take to be quantum tunneling [3]. The analysis and the results are grounded in black hole physics, and hence cannot be taken to apply to arbitrary horizons, but the tunneling mechanism invoked should apply more universally. \nTo ground our approach in the physics of black holes, we have relied on a number of key principles and results from classical general relativity and field theory, including the implications of the equivalence principle [10] for the field-theoretic state at the event horizon, the nonexistence of a 'bleaching' mechanism [19], and the requirement for some thermalization or scrambling mechanism [6, 21, 22] that allows information from deep inside a black hole to reach the surface before radiating away [20] (although our results are largely insensitive to the time scale and hence the underlying scrambling mechanism). \nPreviously, the no-hiding theorem [9, 23] was used to prove that Hawking's prediction of featureless radiation implied that the information about the in-fallen matter could not be in the radiation field but must reside in the remainder of Hilbert space-then presumed to be the black hole interior. That work presented a strong form of the black hole information paradox pitting the predictions of general relativity against those of quantum mechanics [9]. Here, we have shown that transevent horizon entanglement provides a way out, since now the 'remainder of Hilbert space' comprises both the black hole interior and external neighborhood. Because the evaporating black hole actually involves three subsystems, the information may be encoded within them as pure correlations via a quantum one-time pad [9], so the information remains inaccessible from any one subsystem. \nImportantly, the detailed physics of black holes (inside the event horizon) remains beyond the scope of this \nLetter. Thus, this Letter leaves mysterious those longstanding questions about the internal dynamics of black holes that would require knowledge of the geometry well within the black hole and extensive field-theoretic calculations or even a theory of quantum gravity to be addressed. The very assumption of unitarity is one such question. Another is our positing of a finite entanglement entropy across the event horizon, without a detailed field-theoretic description of how this should be calculated [24-28]. Similarly, the dynamics of the entangled degrees of freedom exterior to the black hole remains unclear. Finally, we assume the existence of some global thermalization process that leads to complete scrambling of the information encoded within the black hole. \nThe simultaneous encoding of information externally (in the combined radiation and external neighborhood) and 'internally' (if one slightly stretches the horizon to envelop the bulk of the external neighborhood entanglement in addition to the black hole interior) is reminiscent of the principle of black hole complementarity [10]. This principle was introduced to account for the apparent cloning suggested by the possibility of choosing a 'nice time' slice through the black hole spacetime that crosses most of the outgoing radiation as well as the collapsing body well inside the event horizon but still far from the singularity [29]. Interpreted in the context of our work here, if such slices are drawn after the encoding of the information into the tripartite quantum one-time pad, the 'cloning' would be a manifestation of the multiple ways of reading out the information from the tripartite structure. If such slices are drawn before the encoding occurred, then too little of the outgoing radiation would be crossed for a potential violation of the no-cloning theorem (note that the number of qubits radiated may be used as a surrogate for a time coordinate). \nOur results indicate that, except for the very final, vanishingly small fraction of a (large) black hole's lifetime, the Hawking radiation is completely uncorrelated with the state of the in-fallen matter. Thus, the behavior Hawking found so indicative of a loss of unitarity is in fact completely generic for unitarily evolving, entangled black holes, requiring no specially designed evolution. Of course, by assuming unitarity from the outset, we cannot directly address the black hole information paradox. Rather, our result dissociates completely information-free radiation from a loss of unitarity and hence undermines the very logic used to formulate the paradox. \nFinally, in light of their curious equality, it has previously been conjectured that a black hole's thermodynamic entropy is actually entropy of entanglement [2427]. Indeed, it unavoidably holds for some types of extremal black holes [25, 26] and even allows their entropy to be computed at the microscopic level [27]. The conventional riposte to this conjecture is made by noting that the entropy of entanglement of quantum fields pierc- \ning a black hole's event horizon would be proportional to the number of matter fields that exist, but, since a black hole's thermodynamic entropy is purely geometric, there should be no a priori relationship between these quantities (see, e.g., Ref. [30]; for a counterargument see Ref. [31]). By studying dynamically evolving black holes, not merely static ones [24-27], we now counter this conventional riposte. Equating a black hole's entropy with entropy of entanglement suggests the existence of a sum rule to constrain the number and types of matter fields in any fundamental theory. \nS.L.B. acknowledges the kind hospitality of the A. Watssman Institute for Innovative Thinking, where this work was initiated. K. ˙ Z. acknowledges support by the Deutsche Forschungsgemeinschaft under Project No. SFB/TR12 and under the Project No. DEC2011/02/A/ST1/00119, financed by the Polish National Science Center. The authors thank N. Cohen, M. Patra, and H.-J. Sommers for fruitful discussions. \nNote added .-Our 'energetic curtain' (first coined in Ref. [17]) appears to be the same phenomenon recently called a 'firewall' [32].", 'I. STRUCTURE AND INTERPRETATION OF DECOUPLING THEOREMS': "Decoupling theorems effectively describe the performance of random quantum error correction codes (QECC), whereby the quantum state to be protected | ψ 〉 input is embedded into a larger 'code' Hilbert space by | ψ 〉 → | ψ 〉 ⊗ | φ 0 〉 followed by its encoding by a Haarrandom unitary U acting on the code space. Thus \nQECC : | ψ 〉 input →| Ψ 〉 code = ( U | ψ 〉 ) code , (7) \nwhere | Ψ 〉 code is the larger dimension code state and in the last expression we have suppressed the ancillary subsystem in standard state φ 0 〉 . \n| \n〉 A subtlety we should mention is that the proof of the quantum decoupling theorem relies on the input state, which we wish to later locate, being entangled with a reference subsystem (ref). Thus, for example, Eq. (7) becomes \n1 1 ⊗ QECC : ∑ i | i 〉 ref ⊗| i 〉 input → ∑ i | i 〉 ref ⊗ ( U | i 〉 ) code (8) \nwhere we have suppressed normalization for convenience. This is a powerful step because it effectively allows us to utilize entanglement monogamy to precisely pin down where our encoded subsystem may be located. \nWith access to a sufficiently large piece of the code subspace, decoupling theorems tell us how well we can, \nin principle, reconstruct the original state (with its full entanglement to the reference in tact). In particular, for a k -qubit input state encoded into an n -qubit code state, there exists a threshold of 1 2 ( n + k ) qubits above which access to more than this number of qubits of the code state allows near ideal reconstruction of the original state. More precisely, access to any 1 2 ( n + k ) + c qubits from the code state allows reconstruction [1 SM ] of the original state with a mean fidelity of reconstruction (averaged over random encodings) bounded below by 1 -2 -c . Since the 'excess' unaccessed qubits of the code are not needed for the reconstruction, they have effectively decoupled from the original state and so the reconstruction protocol is unaffected by any errors that occur on these excess qubits. Thus the decoupling theorem quantifies the performance of random quantum error correcting codes. \nImportantly, the proofs of (quantum) decoupling theorems are non-constructive. They only demonstrate the existence of a reconstructing unitary with the claimed performance, but do not say how it may be made.", 'II. DECOUPLING IN A CLASSICAL SETTING': 'Here we paraphrase the discussion in Ref. [1 SM ] for a simple version of a decoupling result in a classical setting. Consider a k -bit plaintext message randomly encoded into an n -bit ciphertext string. The codebook for this code will consist of 2 k n -bit random codewords and their associated k -bit messages. Obviously, anyone with access to the codebook and any specific encoded message will be able to exactly decode it. However, knowing the codebook allows one to do almost as well with just a few more than k bits of the encoded message (indeed any k plus a few bits) of the encoded message. \nAs noted in Ref. [1 SM ], given access to only k + c bits of (and their location in) the encoded message, one can eliminate many of the potential entries in the codebook, thus narrowing down the possible message. To estimate the probability for this procedure to identify any particular message from k + c bits we may treat matches as uniformly random (for our randomly generated codebook). The probability of a random match between k + c bits from a specific encoded message and the identically located k + c bits from any specific codeword in the codebook will then be 2 -( k + c ) . Given therefore that there are 2 k possible messages to distinguish between, the probability of failure to identify the correct message will be 2 k 2 -( k + c ) = 2 -c . Finally, the probability with which access to any k + c bits of the encoded message (and the codebook) allows one to have successfully reconstructed the original plaintext message (what one might call the fidelity of reconstruction) is just 1 2 -c . \nWe might note some significant differences between this classical decoupling and quantum decoupling results. \n- \nFirst, for the case where k /lessmuch n , classical decoupling allows many reconstructions of the original message from completely distinct subsets of bits from the encoded message - the classical information can be cloned in this manner. By contrast, for the analogous quantum encoding [as in Eq. (8)] access to 1 2 ( n + k ) + c qubits of the encoded state are needed to achieve a reconstruction fidelity of 1 -2 -c . So quantum cloning is strictly prohibited. Nonetheless, any 1 2 ( n + k ) + c qubits are adequate for this purpose. Second, in the classical setting the reconstruction protocol is trivial given access to the codebook, whereas the analogous reconstruction protocol in the quantum case is only shown to exist (the proof is non-constructive) but may require full knowledge of the encoding random unitary U , which would be an extreme burden in any even moderately high-dimensional scenario.', 'III. PENALTY FOR DISENTANGLEMENT ACROSS A HYPOTHETICAL BOUNDARY': "Here we investigate the energy penalty one must pay for creating a disentangled (i.e., separable) state across a hypothetical boundary. We are not here going to consider the effects of real boundary conditions on the state of a quantum system, merely the effect of a constraint on the state space so as to exclude entangled states across a non-physical (fictive) boundary. Indeed, the equivalence principle has been argued [2 SM ] to imply that freely falling observers see nothing physical as they pass the event horizon. \nConsider M coupled Harmonic oscillators with Hamiltonian \nH = 1 2 M ∑ i =1 p 2 i + 1 2 M ∑ i,j =1 K ij x i x j , (9) \nwhere [ x i , p j ] = i δ ij and K is a real symmetric (nonnegative definite) matrix. The ground state wavefunction as a function of /vectorx ≡ ( x 1 , . . . , x M ) T is \nΨ( /vectorx ) = (det √ K ) 1 / 4 π M/ 4 exp( -/vectorx · √ K · /vectorx ) , (10) \nwith ground state energy 1 2 tr ( √ K ). \nLet us introduce a hypothetical boundary at index b < M . We assign all oscillators with indices i ≤ b as 'inside' this fictive boundary and all other oscillators as 'outside'. It is natural to partition the coupling matrix K into blocks as \nK = ( K in Q Q T K out ) . (11) \nwhere K in is a b × b symmetric matrix and K out is an ( M -b ) × ( M -b ) symmetric matrix. The Hamiltonian \nof Eq. (9) may be rewritten as \nH = 1 2 ( /vector p 2 in + /vectorx in · K in · /vectorx in + /vector p 2 out + /vectorx out · K out · /vectorx out ) + /vectorx in · Q · /vectorx out , (12) \nwhere /vectorx = /vectorx in ⊕ /vectorx out decomposes /vectorx into a b -dimensional vector /vectorx in and an ( M -b )-dimensional vector /vectorx out . This effectively decomposes the full M -oscillator Hilbert space H total into a tensor product H total = H in ⊗H out . \nTheorem: The general separable state across H in ⊗H out with lowest energy for Hamiltonian (12) has energy above the ground state of \nE penalty ≡ 1 2 [ tr ( √ K in ) + tr ( √ K out ) -tr ( √ K ) ] . (13) \nWe call this the minimal 'energy penalty' for ensuring the separability of a state across a hypothetical boundary. \nProof: A general separable state is just the convex sum over states of the form ρ in ⊗ ρ out , where without loss of generality we may treat ρ in and ρ out as pure states. A lower bound to the energy expectation of a general separable state is therefore given by the lower bound for the energy expectation over a single such tensor product of pure states. \nConsider now a general product of pure states. We may always write its wavefunction as a displaced product \nΨ prod ( /vectorx in , /vectorx out ) ≡ (14) D in ( /vectorx 0 in + i/vectorp 0 in )Ψ in ( /vectorx in ) D out ( /vectorx 0 out + i/vectorp 0 out )Ψ out ( /vectorx out ) , \nwhere Ψ zero ≡ Ψ in Ψ out is taken to have zero mean positions and momenta. The expectation 〈 H 〉 prod of Hamiltonian (12) with respect to the general state of Eq. (14) may now be rewritten as an expectation over this 'zero mean' state Ψ zero as \n1 2 [ 〈 /vector p 2 in + /vectorx in · K in · /vectorx in + /vector p 2 out + /vectorx out · K out · /vectorx out 〉 zero + /vector p 0 2 in + /vectorx 0 in · K in · /vectorx 0 in + /vector p 0 2 out + /vectorx 0 out · K out · /vectorx 0 out +2 /vectorx 0 in · Q · /vectorx 0 out ] . (15) \nSince K is non-negative definite, for any vector /vectorx 0 = /vectorx 0 in ⊕ /vectorx 0 out we have /vectorx 0 · K · /vectorx 0 ≥ 0. Thus \n〈 H 〉 prod ≥ (16) 1 2 〈 /vector p 2 in + /vectorx in · K in · /vectorx in + /vector p 2 out + /vectorx out · K out · /vectorx out 〉 zero . \nNote that the right-hand-side is just the expectation of the sum of a pair of independent oscillators with individual ground state energies 1 2 tr ( √ K in ) and 1 2 tr ( √ K out ) respectively. Thus, \n〈 H 〉 prod ≥ 1 2 [ tr ( √ K in ) + tr ( √ K out ) ] . (17) \nFurther, since these independent product ground states have zero means, this lower bound is achieved. \nIn order to see how this separability penalty appears in a field theoretic setting consider a free scalar field with Hamiltonian \nH = 1 2 ∫ d 3 x [ π 2 +( /vector ∇ ϕ ) 2 ] , (18) \nwhere π = ∂ t ϕ is the conjugate momentum for the quantum field ϕ and these satisfy the equal-time canonical commutation relations \n[ ϕ ( t, /vectorx ) , π ( t, /vectorx ' ) ] = i δ ( /vectorx -/vectorx ' ) . (19) \nFollowing Srednicki [3 SM ], we introduce a lattice of discrete points with equal spacing a in the radial direction. Furthermore, the field is placed in a spherical box of radius A = ( N + 1) a and the field is taken to vanish at the (real) boundary at A . The field and its conjugate momentum can be decomposed into partial waves ϕ j,lm and π j,lm satisfying the equal time commutation relation \n[ ϕ j,lm , π j ' ,l ' m ' ] = i δ jj ' δ ll ' δ mm ' , (20) \nwhere ja gives the discrete radial coordinate and { l, m } label the partial waves' angular momentum. The discretized Hamiltonian then becomes H = ∑ l,m H lm , with [3 SM ] \nH lm = 1 2 a N ∑ j =1 [ π 2 j,lm +( j + 1 2 ) 2 ( ϕ j,lm j -ϕ j +1 ,lm j +1 ) 2 + l ( l +1) j 2 ϕ 2 j,lm ] . (21) \nNumerical calculations of E penalty from Eq. (13) for this discretized Hamiltonian yield \nE penalty /similarequal 0 . 05 ( N +1) 2 a = 0 . 05 A 2 a 3 , (22) \nwhere the hypothetical boundary index is chosen as b = N/ 2 (across a range of even N from 50 to 100). This penalty diverges as the cube of the ultraviolet regulator 1 /a . Thus we expect pure quantum states where entanglement has essentially vanished across a hypothetical boundary to have very large energies.", 'IV. UNIFORM ENTANGLEMENT': "Because one of the key claims in the paper is about loss of trans-event horizon entanglement, we shall repeat the key calculation here for a black hole with trans-event horizon entanglement, but where, for simplicity, that entanglement is taken to be uniform. This allows us to \nrepeat the analysis solely using results already available in the literature. \nConsider black hole evaporation with uniform transevent horizon entanglement as \n1 √ E E ∑ j =1 | j 〉 int ⊗| j 〉 ext → 1 √ E E ∑ j =1 ( U | j 〉 ) BR ⊗| j 〉 ext . (23) \nHere log 2 E is the entropy of entanglement between the external (ext) neighborhood and the interior of the black hole. Except for the interpretation of the source of entanglement, this model has been recently analyzed by Ref. [1 SM ]. We may therefore quote their key result in our terms: For any positive c , once 1 2 S BH + 1 2 log 2 E + c qubits have radiated away [this is just the 1 2 ( n + k ) + c qubits required as discussed in the first section of this Supplementary Material], the trans-event horizon entanglement between the external neighborhood and the interior subsystems will have virtually vanished, with it appearing instead (with a fidelity of at least 1 -2 -c ) as entanglement between the external neighborhood and the outgoing radiation. Here (as in our manuscript) c is a free parameter, but will be dwarfed by any of the entropies involved. \nRepeating the argument from our manuscript, this loss must be delayed until the black hole has evaporated to roughly the Planck scale. (Indeed, section III of this Supplementary Material provides energy estimates for the departure from vacuum across the event horizon when trans-event horizon entanglement is lost.) Such a delay implies that roughly S BH qubits must have already been radiated before such loss occurs, in which case \nlog 2 E ≈ S BH . (24) \nIn section V below, we shall see that when the uniform entanglement of the above analysis is replaced with general trans-event horizon entanglement, the measure of entanglement log 2 E is replaced by the R'enyi entropy H (1 / 2) ( ρ ext ). This replacement is unchanged in the presence of in-fallen matter (also section VI).", 'V. FORMALISM FOR GENERAL ENTANGLEMENT': "Note that all R'enyi entropies are bounded above by the logarithm of the Hilbert space dimension, so 0 ≤ H ( q ) ( ρ ext ) ≤ n ≡ S BH for the state we study. Of particular interest to us here will be two R'enyi entropies for q = 1 2 , 2, so \nH (1 / 2) ( ρ ext ) = log 2 [ (tr √ ρ ext ) 2 ] H (2) ( ρ ext ) = -log 2 (tr ρ 2 ext ) . (25) \n(In the limit of q → 1 the R'enyi entropy reduces to the more familiar von Neumann entropy.) \nOur key result is based on our generalization (theorem below) of the decoupling theorem of Ref. [4 SM ]. Consider now the tripartite state \nρ XYZ = ρ XY 1 Y 2 Z , (26) \nwhere the joint subsystems Y = Y 1 Y 2 will be decomposed as either the radiation subsystem and interior black hole subsystem RB or vice-versa BR . This allows us to define \nσ U XY 2 Z ≡ tr Y 1 ( U Y ρ XYZ U † Y ) . (27) \nIn keeping with the naming convention of Ref. [4 SM ], we call the result below the Mother-in-law decoupling theorem.", 'Generalized decoupling theorem:': '(∫ U ∈ U ( Y ) dU ∥ ∥ σ U XY 2 Z -σ U X ⊗ σ U Y 2 Z ∥ ∥ 1 ) 2 ≤ tr ρ 2 ν X tr ρ 2 µ Z { [ tr ρ 2 XZ ( ρ -2 ν X ⊗ ρ -2 µ Z ) -2 tr ρ XZ ( ρ 1 -2 ν X ⊗ ρ 1 -2 µ Z ) +tr ρ 2 -2 ν X tr ρ 2 -2 µ Z ] + Y 2 Y 1 [ tr ρ 2 XYZ ( ρ -2 ν X ⊗ ρ -2 µ Z ) +tr ρ 2 -2 ν X tr ρ 2 Y Z ρ -2 µ Z ] } (28) ≤ Y 2 Y 1 tr ρ 2 ν X tr ρ 2 µ Z [ tr ρ 2 XYZ ( ρ -2 ν X ⊗ ρ -2 µ Z ) +tr ρ 2 -2 ν X tr ρ 2 Y Z ρ -2 µ Z ] (29) ≤ 2 Y 2 Y 1 2 H X + H Z , (30) \nwhere H A ≡ H (1 / 2) ( ρ A ), 0 ≤ 2 ν, 2 µ ≤ 1, and the trace norm is defined by ‖ X ‖ 1 ≡ tr | X | . Recall from our manuscript, that we reuse subsystem labels for Hilbert space dimensionalities, thus here Y 2 /Y 1 denotes the ratio of their Hilbert space dimensions. Note that here, to go from Eq. (28) to Eq. (29), we would require ρ XZ = ρ X ⊗ ρ Z ; and to go from Eq. (29) to Eq. (30), we would require ρ XYZ is pure and we take 2 ν = 2 µ = 1 2 . \nProof: Using the Cauchy-Schwarz inequality we may write \n∥ ∥ σ U XY 2 Z -σ U X ⊗ σ U Y 2 Z ∥ ∥ 1 (31) ≤ ∥ ∥ ρ ν X ⊗ 1 1 Y 2 ⊗ ρ µ Z ∥ ∥ 2 × ∥ ρ -ν X ⊗ ρ -µ Z ( σ U XY 2 Z -σ U X ⊗ σ U Y 2 Z ) ∥ ∥ 2 , \n∥ \n∥ where without loss of generality we may assume that ρ ν X and ρ µ Z are invertible; then using the methods already outlined in Ref. [4 SM ] the results are easily obtained. \nWe note that the statement of the result reduces to the conventional decoupling theorem for the choice ν = 0 and subsystem Z is one-dimensional. \nOf particular interest here is the case where 2 ν = 1 2 and ρ ext ,Y is pure, which gives \n∫ U ∈ U ( Y ) dU ∥ ∥ σ U ext ,Y 2 -σ U ext ⊗ σ U Y 2 ∥ ∥ 1 ≤ ( 2 Y 2 Y 1 2 H ext ) 1 2 , (32) \nwith H ext ≡ H (1 / 2) ( ρ ext ). \n≡ Now 1 -F ( ρ, σ ) ≤ 1 2 ‖ ρ -σ ‖ 1 , where the fidelity is defined by F ( ρ, σ ) ≡ ‖ √ ρ √ σ ‖ 1 . As a consequence, the fidelity with which the initial trans-event horizon entanglement is encoded within the combined ext , Y 1 subsystem is bounded below by 1 -√ 2 H ext Y 2 /Y 1 . Now allowing this in turn to be bounded from below by 1 -2 -c and choosing Y 1 = R and Y 2 = B and recalling that BR = 2 S BH gives the result quoted in our manuscript [Eq. (3) there]. \nInterestingly, the opposite choice Y 1 = B and Y 2 = R tells us that, for any positive c , for fewer than 1 2 [ S BH -H (1 / 2) ( ρ ext )] -c qubits radiated away, the initial transevent horizon entanglement remains encoded between the external neighborhood and the interior subsystems with fidelity of at least 1 -2 -c . This effectively gives the number of qubits that must be radiated before transevent horizon entanglement begins to be reduced from its initial value. Of particular interest is the case when H (1 / 2) ( ρ ext ) ≈ S BH for which we would conclude that the trans-event horizon entanglement begins to be depleted by radiation almost immediately.', 'VI. ENTANGLEMENT LOSS IN THE PRESENCE OF IN-FALLEN MATTER': "The unitary evaporation of an entangled black hole in the presence of in-fallen matter was argued in our manuscript to be described by \n1 √ K K ∑ i =1 | i 〉 ref ⊗ ∑ j √ p j ( | i 〉 ⊗ | j 〉 ⊕ 0) int ⊗| j 〉 ext (33) → 1 √ K K ∑ i =1 | i 〉 ref ⊗ ∑ j √ p j [ U ( | i 〉 ⊗ | j 〉 ⊕ 0)] BR ⊗| j 〉 ext , \nwhere log 2 K ≡ S matter is the number of qubits of quantum information in the in-fallen matter. \nIt is now straightforward to apply the generalized decoupling theorem above to show that, for an arbitrary positive number c , when the number of qubits radiated reaches \nlog 2 R = S BH -1 2 χ (1 / 2) + c, (34) \nthen the trans-event horizon entanglement has effectively vanished and instead has been transferred to entangle- \nment between the external neighborhood and the outgoing radiation, with a fidelity of at least 1 -2 -c . Recall from our manuscript that the number of unentangled qubits initially within the black hole is roughly quantified by \nχ ( q ) ≡ S BH -S matter -H ( q ) ( ρ ext ) ≥ 0 , (35) \nwith q of order unity. \nRepeating the argument from our manuscript, unless this occurs when log 2 R ≈ S BH then a noticeable violation of the equivalence principle will occur. This implies that \nχ (1 / 2) ≪ S BH , (36) \nor equivalently, that \nS BH ≈ H (1 / 2) ( ρ ext ) + S matter ≈ H (1 / 2) ( ρ ext ) , (37) \nsince from 't Hooft's bound, the entropic content of matter is only a vanishingly small fraction of the thermodynamic entropy of the black hole, i.e., S matter ≪ S BH .", "VII. WHERE'S THE PLANCK SCALE?": "In the first part of the manuscript, we show that a black hole's thermodynamic entropy must be very well approximated by its entropy of entanglement across the event horizon. The proof relied on preservation of the equivalence principle prior to the black hole having evaporated to the Planck scale. However, what defines the beginning of the Planck scale for black holes? \nA universal feature of black holes is their thermodynamic entropy or (essentially up to a constant prefactor) their surface area. We shall therefore use the entropy (in bits) as a measure of size of a black hole. As we wish to avoid making claims about the physics of Planck scale black holes, we shall suppose there is some size, above which Planck scale effects are negligible (in particular, above which the predictions of the equivalence principle are left in tact). Stated conversely, we shall suppose that entry into the Planck scale regime, where effects on the equivalence principle begin to become non-negligible, occurs at some generic size (or equivalently entropy). In particular, we take this entry into the Planck scale for black holes of entropy smaller than \nS Planckian BH /lessorsimilar 2 p . (38) \nIt will turn out that virtually any choice for p makes no difference to our analysis since the entry-point entropy so defined will be dwarfed by those of that of typical large black holes (e.g., a stellar mass black hole has thermodynamic entropy of 10 10 77 ). \nTo see how the argument runs, we must determine the thermodynamic entropy of a black hole evaporating according to Eq. (33). We shall suppose the von Neumann \nentropy computed from Eq. (33) is a good estimate for the thermodynamic entropy. Evolution corresponds to the radiation subsystem R becoming an ever larger portion of the initial black hole Hilbert space and the remaining black hole interior subsystem B becoming an ever shrinking portion, subject to the constraint that \n2 S BH = BR, (39) \nwhere one should recall that we reuse subsystem labels as their corresponding Hilbert space dimensionalities. During evaporation, a simple upper bound to the von Neumann entropy of the black hole interior S ( B ) is given by the logarithm of its dimensionality. Hence \nS ( B ) ≤ S BH -log 2 R. (40) \nFor a lower bound we can use the negative logarithm of the so-called purity \nS ( B ) ≥ -log 2 〈〈 tr ( ρ U B ) 2 〉〉 U , (41) \nwhere ρ U B is the reduced substate on the black hole interior of Eq. (33), and 〈〈· · · 〉〉 U denotes averaging over the random unitary U with the Haar measure. Using standard methods [4 SM ], the purity can be easily estimated in the latter stages of evaporation to be \n〈〈 tr ( ρ U B ) 2 〉〉 U /similarequal B ( R 2 -1) ( BR ) 2 -1 /similarequal R 2 S BH , (42) \nand hence S ( B ) /greaterorsimilar S BH -log 2 R . These bounds imply that during the latter stages of evaporation via Eq. (33), a black hole's von Neumann entropy will be \nS ( B ) /similarequal S BH -log 2 R. (43) \nCombining Eqs. (38) and (43) we see that for a large black hole to have evaporated to just above the Planck scale (prior to any need to invoke Planck-scale physics), it must have emitted virtually all its initial entropy as Hawking radiation. In other words, as one approaches the Planck scale, one has to very high precision that \nlog 2 R ≈ S BH . (44)", 'VIII. INFORMATION RETRIEVAL FROM PURE-STATE BLACK HOLES': "As noted in our manuscript, the description of an evaporating black hole via \n| i 〉 int → ( U | i 〉 ) RB . (45) \nis not new. This was originally formulated [5 SM ] assuming that all the in-falling matter (and the black-hole itself) was in a pure state | i 〉 . In other words, it is assumed \nthat initially there is no trans-event horizon entanglement. It should be noted, that prior to Ref. [6 SM ] this evolution was not connected to or claimed to be supported by any microscopic mechanism. \nThe original analysis suggested that a 'discernible information' (corresponding to the deficit of the entropy of a subsystem from its maximal value) would yield a suitable metric for information content in the radiation [5 SM ]. In order to find the 'typical' behavior of an evaporating black hole it calculated the mean discernible information averaged over random unitaries [5 SM ]. \nStarting with a pure-state interior, the mean discernible information of the radiation remains almost zero until half the qubits of the initial black hole had been radiated, after which it rises at the rate of roughly two bits for every qubit radiated [5 SM ]. This behavior suggests that first entanglement is created, followed by dense coding [7 SM ] of classical information about the initial state. \nIn order to get a much clearer picture of quantum information flow in Eq. (45) we can rely on the decoupling theorem. In particular, entangling the state of the in-fallen matter with some distant reference (ref) subsystem, allows one to track the flow of quantum information [8 SM , 1 SM ]. In this way Eq. (45) becomes \n1 √ K K ∑ i =1 | i 〉 ref ⊗| i 〉 int → 1 √ K K ∑ i =1 | i 〉 ref ⊗ ( U | i 〉 ) BR , (46) \nrecall that log 2 K ≡ S matter is the number of qubits describing the quantum state of the matter used to form the otherwise pure-state black hole. Using the decoupling theorem [4 SM ] we may show that, for any positive number c , prior to 1 2 ( S BH -S matter ) -c qubits having been radiated, the quantum information about the in-fallen matter is encoded within the black hole interior with fidelity at least 1 -2 -c ; whereas after a further S matter +2 c ' qubits have been radiated, for arbitrary positive c ' , the information about the in-fallen matter is encoded within the radiation with fidelity at least 1 -2 -c ' (see also Ref. [1 SM ] for this latter result). The quantum information about the in-fallen matter naively appears to leave in a narrow 'pulse' at the radiation emission rate; this pulse occurs just as half of the black hole's qubits have radiated away. \nNow consider what happens if additional matter is dumped into the black hole after its creation. Following Ref. [1 SM ], we model this process via cascaded random unitaries on the black hole interior - one unitary before each radiated qubit. (Naturally, any such analysis relies on a very short global thermalization time for the black hole. An assumption which was not needed for any of the results quoted in our manuscript itself.) Within the pure-state black hole of Eq. (46), it was argued [1 SM ] that after half of the initial qubits had radiated away, any information about matter subsequently falling into the black hole would be 'reflected' immediately at roughly the radiation emission rate [1 SM ]. By \ncontrast, in the early stages of evaporation information about matter subsequently thrown in would only begin to emerge after half of the initial qubits of the black hole had radiated away [1 SM ]. These very different behaviors in the first and second halves of its life suggest that such a black hole acts almost as two different species: as storage during the first half of its radiated qubits and as a reflector during the second half. \nA subtle flaw to this argument of Ref. [1 SM ] is due to the omission of the fact that a black hole's entropy is nonextensive, e.g., scaling as the square of the black hole's mass M 2 for the Schwarzshild family of black holes: for every k qubits dumped into such a black hole, the entropy typically increases by O ( kM ) /greatermuch k . Likewise, the number of unentangled qubits within the (initially pure-state) black hole will increase by O ( kM ). Therefore, within the cascaded unitary pure-state black hole, the reflection described in Ref. [1 SM ] would not begin immediately, but only after a large delay in time of O ( kM 2 ). Notwithstanding the delay, the pure-state black hole behaves effectively as two distinct species as described above. \nBecause this behavior seems so bizarre it is worth going back over the key assumptions that went into it: i) that the behavior of a specific unitary in Eq. (45) is well described by Haar averages over all random unities; ii) that the number of qubits comprising the initial black hole Hilbert space is n /similarequal S BH . (These two assumptions are discussed in some detail in our manuscript.) Finally, iii) that the black hole is initially in a pure state up to a negligible amount of entanglement that may come from the matter content. In fact, it is this last assumption which is weakest and at odds with the well known quantum physics of condensed matter systems and rigorous results from axiomatic field theory as discussed in our manuscript.", 'IX. INFORMATION RETRIEVAL FROM AN ENTANGLED BLACK HOLE': "Here we give explicit statements of results from decoupling summarized in our manuscript. \nApplying the decoupling theorem [4 SM ] to the entangled-state black hole of Eq. (33) allows us to show that, for any positive number c , for all but the final S matter + 1 2 χ (2) + c qubits radiated, the information about the in-fallen matter is encoded in the combined space of the external neighborhood and black hole interior with fidelity at least 1 -2 -c . Similarly, for any positive c ' , for all but the initial S matter + 1 2 χ (2) + c ' qubits radiated, this information is encoded in the combined radiation and external neighborhood subsystems with fidelity at least 1 -2 -c ' . In addition, at all times this information is encoded with unit fidelity within the joint radiation and interior subsystems. \nIn other words, between the initial and final roughly \nS matter + 1 2 χ (2) qubits radiated, the information about the in-fallen matter is effectively deleted from each individual subsystem [8 SM , 9 SM ], instead being encoded in any two of the three of subsystems (consisting of the out-going radiation, the external neighborhood, and the black hole interior). During this time, the information about the in-fallen matter is to an excellent approximation encoded within the perfect correlations of a quantum one-time pad [10 SM , 8 SM ] of these three subsystems. \nFurthermore, using our generalized decoupling theorem we may show that, for any positive c '' , that prior to the first 1 2 χ (1 / 2) -c '' qubits radiated, the information about the in-fallen matter is still encoded solely within the black hole interior, with a fidelity of at least 1 -2 -c '' . Similarly, for any positive c ''' , within the final 1 2 χ (1 / 2) -c ''' qubits radiated, the information about the in-fallen matter is encoded within the out-going radiation, with a fidelity of at least 1 -2 -c ''' . Combining these with the above results we see that both the encoding and decoding of the tripartite quantum one-time pad occur during the radiation of roughly S matter + 1 2 ( χ (2) -χ (1 / 2) ) /similarequal S matter qubits, i.e., the black hole's quantum one-time pad encoding (and decoding) occurs at roughly the radiation emission rate. \nHow does this entangled-state description of black hole evaporation respond to matter subsequently swallowed after its formation? Instead of the two distinct behaviors of storage and reflection found in the pure-state black hole, here, any additional qubits thrown in will immediately begin to be encoded into the tripartite one-time pad. The decoding into the radiation subsystem of the information about all the in-fallen matter will only occur at the very end of the evaporation. (The non-extensive increase in black hole entropy is taken up as entanglement with the external neighborhood so no further delays occur.) Thus, instead of behaving almost as two distinct species, a highly entangled-state black hole has one principle behavior - forming a tripartite quantum one-time pad between the black hole interior, the external neighborhood and the radiation from the black hole, with release of that information only at the end of the evaporation. \nCan we reconcile the information retrieval behavior of the pure-state black hole with its entangled counterpart? Naively, if the pure-state black hole analysis were run on twice as many qubits, but stopped just after the information about the in-fallen matter had escaped as a narrow pulse then there would be broad agreement between the two types of black hole. This doubling of the number of qubits would make some crude sense if we supposed that the pure-state black hole was not making a split between interior and exterior at the event horizon, but somewhat further out at some arbitrary boundary where trans-boundary entanglement would not be participating in the evaporation. The dimensionality of the Hilbert space within this extended boundary would then be dom- \ninated by the product of the dimensionality of the original black hole interior, and the nearby external neighborhood entangled with them. This would be roughly twice the number of qubits within the black hole interior itself. Once the original number of qubits had evaporated away (now half the total for our extended boundary pure-state black hole) the black hole interior would be exhausted of Hilbert space and evaporation would cease. This suggests that despite the general incompatibility between the two types of black hole, a pure-state analysis, if thoughtfully set up, could capture important features of information retrieval from an entangled-state black hole.", 'X. HEURISTIC FLOW VIA CORRELATIONS': "The rigorous results from our manuscript may be heuristically visualized by following how the correlations with the distant reference system behave. For a pure tripartite state XYZ , these correlations satisfy \nC ( X : Y ) + C ( X : Z ) = S ( X ) , (47) \nHere S ( X ) is the von Neumann entropy for subsystem X and C ( X : Y ) ≡ 1 2 [ S ( X ) + S ( Y ) -S ( X,Y )], onehalf the quantum mutual information, is a measure of correlations between subsystems X and Y . Relation (47) is additive for a pure tripartite state, so the correlations with subsystem X smoothly move from subsystems Y to Z and vice-versa. \nFor simplicity, here we restrict ourselves to the case where \nρ ext = 1 M M ∑ j =1 | j 〉 ext ext 〈 j | , (48) \nand where we assume no excess unentangled qubits, i.e., χ ( q ) = 0. Thus, the initial number of qubits within the black hole interior is given by log 2 N = log 2 ( BR ) = S matter + log 2 M , for S matter qubits of in-fallen matter. We computed the above measure of correlations, Eq. (47), from von Neumann entropies approximated using the average purity (see next section); numerical calculations showed this as a good approximation for systems of even a few qubits. Fig. 1 shows a typical scenario: A black hole is assumed to be created from in-fallen matter comprising S matter qubits of information and negligible excess unentangled qubits. Within the first S matter qubits radiated, information about the in-fallen matter (a) vanishes from the black hole interior at roughly the radiation emission rate and (b) appears in the joint radiation and external neighborhood subsystem. From then until just before the final S matter qubits are radiated, the in-fallen matter's information is encoded in a tripartite state, involving the radiation, external neighborhood and interior subsystems, subplots (b) and (c). In the final S matter \nFIG. 1: Correlations to the reference subsystem as a function of the number of qubits radiated (log 2 R ). Correlations between the reference (ref) subsystem and: (a) black hole interior, B ; (b) radiation, R , and external (ext) neighborhood; (c) black hole interior and external neighborhood; and (d) radiation alone. Note that, as expected from Eq. (47), the sum of C 's in subplots (a) and (b) is a constant, as is that of subplots (c) and (d). In each subplot, the in-fallen matter consists of S matter = 10 qubits and the black hole initially consists of log 2 BR = 100 qubits with χ ( q ) = 0. (Entropies are evaluated using base-two logarithms.) \n<!-- image --> \nqubits radiated the information about the in-fallen matter is released from its correlations and appears in the radiation subsystem alone, subplot (d). This qualitative picture is in excellent agreement with the results from the decoupling theorem and its generalization.", 'Evaluation of purities': "In order to approximate the computation of the correlation measure described above, we use a lower bound for a subsystem with density matrix ρ \n〈〈 S ( ρ ) 〉〉 ≥ -〈〈 log 2 p ( ρ ) 〉〉 ≥ -log 2 〈〈 p ( ρ ) 〉〉 . (49) \nHere S ( ρ ) = -tr ρ log 2 ρ is the von Neumann entropy of ρ , p ( ρ ) = tr ρ 2 is its purity, and here 〈〈· · · 〉〉 denotes averaging over random unitaries with the Haar measure. The former inequality above is a consequence of the fact that the R'enyi entropy is a non-increasing function of its argument [11 SM ], and the latter follows from the concavity of the logarithm and Jensen's inequality. We may estimate the von Neumann entropies required then by the rather crude approximation 〈〈 S ( ρ ) 〉〉 ≈ -log 2 〈〈 p ( ρ ) 〉〉 , which turns out to be quite reasonable for spaces with even a few qubits. \nAlthough traditional methods [12 SM ] may be used to compute these purities, a much simpler approach is to use the approach from Ref. [4 SM ]. In particular, for a typical \npurity of interest we use the following decomposition \ntr σ U 2 R, ext = tr ( σ U R, ext ⊗ σ U R ' , ext ' S R, ext; R ' , ext ' ) (50) = tr ( ρ ref ,BR, ext ⊗ ρ ref ' ,B ' R ' , ext ' × U † BR ⊗ U † B ' R ' S R ; R ' U BR ⊗ U B ' R ' S ext;ext ' ) \nwhere S A ; A ' is the swap operator between subsystems A and A ' , similarly, S AB ; A ' B ' = S A ; A ' S B ; B ' . Then the average over the Haar measure is accomplished by an application of Schur's lemma [4 SM ] \n〈〈 U † A ⊗ U † A ' S A 2 ; A ' 2 U A ⊗ U A ' 〉〉 = A 2 ( A 2 1 -1) A 2 -1 1 1 A ; A ' + A 1 ( A 2 2 -1) A 2 -1 S A ; A ' . (51) \nThis approach allows us to straight-forwardly compute the required purities as \np (ref) = 1 K , p (ext) = 1 N , p (ref,ext) = 1 KN , p ( R ) = 1 ( BR ) 2 -1 ( R ( B 2 -1) + B ( R 2 -1) KN ) , (52) p ( R, ext) = 1 ( BR ) 2 -1 ( R ( B 2 -1) N + B ( R 2 -1) K ) , \nwith p ( B, ext) and p ( B, ext) given by the above expressions under the exchange R ↔ B , similarly the exchange K ↔ N gives us expressions for p (ref , R ), etc.", 'XI. BLACK HOLES VERSUS LUMPS OF COAL': "Bekenstein [13 SM ] tells of a thought experiment he attributes to Sidney Coleman: A cold piece of coal (initially in its ground state) is illuminated by a laser beam. The system is thus prepared in a pure state and radiates thermally after the laser is switched off. Eventually the lump of coal returns to its initial state, so presumably the radiation subsystem has merely encoded any information in the subtle correlations between the individual thermal photons. Can this differ from the overall behavior of a unitarily evaporating black hole? \nSuch a hot coal model of a black hole will correspond very closely to the pure-state model of a black hole. As such, information about the state of the laser beam that has heated up the coal will become accessible from the radiation field shortly after half of the total number of thermal photons (each carrying roughly one bit's worth of information) have radiated away. This behavior, however, will be very different from the entangled black hole analyzed in our manuscript. For such highly entangled black holes there is another component of the system to include in the dynamics: The entanglement across the boundary corresponding to the event horizon. This entanglement is not merely static as it would be across a fixed boundary, but must itself escape from the black \nhole in order for the boundary itself to shrink. As was uncovered in our manuscript, entangled black holes encode the information about the in-fallen matter into a quantum one-time pad. The information is in principle accessible from any two of three subsystems (the interior of the black hole, the modes just external to the black hole but entangled with it across the event horizon and the Hawking radiation itself) within a very short time after the black hole begins to radiate. Once that encoding into the quantum one-time pad has occurred, this information becomes inaccessible from any one of these subsystems alone (and in particular from the Hawking radiation). \nOnly when the quantum one-time pad becomes decoded will the full information become accessible within the Hawking radiation. For a highly entangled black hole, as shown in our manuscript, this occurs within the final and vanishingly small fraction of the black hole's lifetime. Before this time, the Hawking radiation is completely uncorrelated from the information about the in-fallen matter. This behavior is therefore very different from that of information return from a hot coal. \nThe authors gratefully acknowledge H.-J. Sommers's original calculation of Eq. (52) and several fruitful discussions with him, Netta Cohen and Manas Patra. \n- [1] S. W. Hawking, Commun. Math. Phys. 43 , 199 (1975).\n- [2] H. Nikoli'c, Int. J. Mod. Phys. D 14 , 2257 (2005); S. D. Mathur, Classical Quantum Gravity 26 , 224001 (2009).\n- [3] S. L. Braunstein and M. K. Patra, Phys. Rev. Lett. 107 , 071302 (2011).\n- [4] S. W. Hawking, Phys. Rev. D 14 , 2460 (1976).\n- [5] D. N. Page, Phys. Rev. Lett. 71 , 3743 (1993).\n- [6] P. Hayden and J. Preskill, J. High Energy Phys. 09 (2007) 120.\n- [7] J. Eisert, M. Cramer, and M. B. Plenio, Rev. Mod. Phys. 82 , 277 (2010).\n- [8] J. A. Smolin and J. Oppenheim, Phys. Rev. Lett. 96 , 081302 (2006).\n- [9] S. L. Braunstein and A. K. Pati, Phys. Rev. Lett. 98 , 080502 (2007).\n- [10] L. Susskind, L. Thorlacius, and J. Uglum, Phys. Rev. D 48 , 3743 (1993).\n- [11] G. 't Hooft, in Salamfestschrift: A Collection of Talks , edited by A. Ali, J. Ellis, and S. Randjbar-Daemi (World Scientific, Singapore, 1993), Vol. 4.\n- [12] V. Milman and G. Schechtman, Asymptotic Theory of Finite Dimensional Normed Spaces (Springer, New York, 2001).\n- [13] See, e.g., M. L. Mehta, Random Matrices (Elsevier, Amsterdam, 2004).\n- [14] A. Abeyesinghe, I. Devetak, P. Hayden, and A. Winter, Proc. R. Soc. A 465 , 2537 (2009).\n- [15] See Supplemental Material at http://link.aps.org/ \n- supplemental/10.1103/PhysRevLett.110.101301 for proofs and a discussion of decoupling theorems. (This material is included in full as the appendix.)\n- [16] S. Schlieder, Commun. Math. Phys. 1 , 256 (1965).\n- [17] S. L. Braunstein, arXiv:0907.1190v1.\n- [18] Our analysis is silent as to whether or not this relationship between thermodynamic and entanglement entropy also holds for acceleration or cosmological horizons.\n- [19] J. Preskill, in Black Holes, Membranes, Wormholes and Superstrings , edited by S. Kalara and D. V. Nanopoulos (Singapore, World Scientific, 1993) p. 22.\n- [20] All incoming degrees of freedom are required to be scrambled for our information retrieval mechanism to apply in full. For black holes with nontrivial interiors, WKB tunneling analysis attributes a temperature to inner horizons that always exceeds that of the outer (event) horizon. Thus, the existence of an inner horizon is no absolute barrier to global scrambling.\n- [21] Y. Sekino and L. Susskind, J. High Energy Phys. 10 (2008) 065.\n- [22] S. B. Giddings, Phys. Rev. D 76 , 064027 (2007).\n- [23] D. Kretschmann, D. Schlingemann, and R. F. Werner, J. Funct. Anal. 255 , 1889 (2008); J. R. Samal, A. K. Pati, and A. Kumar, Phys. Rev. Lett. 106 , 080401 (2011).\n- [24] G. 't Hooft, Nucl. Phys. B256 , 727 (1985); L. Bombelli, R. K. Koul, J. Lee, and R. D. Sorkin, Phys. Rev. D 34 , 373 (1986); M. Srednicki, Phys. Rev. Lett. 71 , 666 (1993).\n- [25] S. Hawking, J. Maldacena, and A. Strominger, J. High Energy Phys. 05 (2001) 001.\n- [26] R. Brustein, M. B. Einhorn, and A. Yarom, J. High Energy Phys. 01 (2006) 098.\n- [27] R. Emparan, J. High Energy Phys. 06 (2006) 012.\n- [28] T. Padmanabhan, Phys. Rev. D 82 , 124025 (2010).\n- [29] D. A. Lowe, J. Polchinski, L. Susskind, L. Thorlacius, and J. Uglum, Phys. Rev. D 52 , 6997 (1995).\n- [30] T. Nishioka, S. Ryu, and T. Takayanagi, J. Phys. A 42 , 504008 (2009).\n- [31] R. Brout, Int. J. Mod. Phys. D 17 , 2549 (2008).\n- [32] A. Almheiri, D. Marolf, J. Polchinski, and J. Sully, J. High Energy Phys. 02 (2013) 062.", 'Supplemental Material bibliography': '[1 SM ] See Ref. [6]. \n- [2 SM ] See Ref. [10].\n- [3 SM ] See Srednicki, Ref. [24].\n- [4 SM ] See Ref. [14].\n- [5 SM ] See Ref. [5].\n- [6 SM ] See Ref. [3].\n- [7 SM ] C. H. Bennett and S. J. Wiesner, Phys. Rev. Lett. 69 , 2881 (1992).\n- [8 SM ] See Ref. [9].\n- [9 SM ] See Ref. [23].\n- [10 SM ] D. W. Leung, Quantum Inf. Comput. 2 , 14 (2001).\n- [11 SM ] I. Bengtsson and K. ˙ Zyczkowski, Geometry of Quantum States: An Introduction to Quantum Entanglement. (Cambridge University Press, Cambridge, 2006).\n- [12 SM ] P. A. Mello, J. Phys. A 23 , 4061 (1990).\n- [13 SM ] J. D. Bekenstein, Contemp. Phys. 45 , 31 (2004).'}

2011PhRvD..83j4027V

Bumpy black holes in alternative theories of gravity

2011-01-01

['-', '-', '-', '-', '-', '-', '-']

[]

We generalize the bumpy black hole framework to allow for alternative theory deformations. We construct two model-independent parametric deviations from the Kerr metric: one built from a generalization of the quasi-Kerr and bumpy metrics and one built directly from perturbations of the Kerr spacetime in Lewis-Papapetrou form. We find the conditions that these “bumps” must satisfy for there to exist an approximate second-order Killing tensor so that the perturbed spacetime still possesses three constants of the motion (a deformed energy, angular momentum and Carter constant) and the geodesic equations can be written in first-order form. We map these parametrized metrics to each other via a diffeomorphism and to known analytical black hole solutions in alternative theories of gravity. The parametrized metrics presented here serve as frameworks for the systematic calculation of extreme mass-ratio inspiral waveforms in parametrized non-general relativity theories and the investigation of the accuracy to which space-borne gravitational wave detectors can constrain such deviations.

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https://arxiv.org/pdf/1102.3706.pdf

{'Bumpy Black Holes in Alternative Theories of Gravity': "Sarah Vigeland, 1 Nicol'as Yunes, 1 and Leo C. Stein 1 \n1 Department of Physics and MIT Kavli Institute, Cambridge, MA 02139, USA. (Dated: November 5, 2018) \nWe generalize the bumpy black hole framework to allow for alternative theory deformations. We construct two model-independent parametric deviations from the Kerr metric: one built from a generalization of the quasi-Kerr and bumpy metrics and one built directly from perturbations of the Kerr spacetime in Lewis-Papapetrou form. We find the conditions that these 'bumps' must satisfy for there to exist an approximate second-order Killing tensor so that the perturbed spacetime still possesses three constants of the motion (a deformed energy, angular momentum and Carter constant) and the geodesic equations can be written in first-order form. We map these parameterized metrics to each other via a diffeomorphism and to known analytical black hole solutions in alternative theories of gravity. The parameterized metrics presented here serve as frameworks for the systematic calculation of extreme mass-ratio inspiral waveforms in parameterized non-GR theories and the investigation of the accuracy to which space-borne gravitational wave detectors can constrain such deviations. \nPACS numbers: 04.30.-w,04.50.Kd,04.70.-s", 'I. INTRODUCTION': "Gravitational waves (GWs) will be powerful tools for learning about source astrophysics and testing strong field gravity [1, 2]. The Laser Interferometer Space Antenna (LISA) [3-6], for example, is expected to be sensitive to roughly a few radians or better of GW phase in a one-year observation. Such levels of precision are achieved through matched filtering, where the data is cross-correlated with a family of waveform models (see eg. [7]). If a signal is present in the data, this crosscorrelation acts as a filter that selects the member of the waveform family that most closely resembles the signal. \nExtreme mass-ratio inspirals (EMRIs) are ideal astrophysical sources to perform precision GW tests of strong field gravity with LISA [1, 8]. These sources consist of a small compact object (SCO) that spirals into a supermassive black hole (SMBH) in a generic orbit, producing millions of radians in GW phase inside LISA's sensitivity band. These waves carry detailed information about the spacetime geometry in which the SCO moves, thus serving as a probe to test general relativity (GR). \nTests of strong field gravity cannot rely on waveform families that assume GR is correct a priori . Instead, such tests must employ more generic waveforms that allow for GR deviations. Recently, Yunes and Pretorius [9] proposed the parameterized post-Einsteinian (ppE) framework, in which analytic waveforms that represent comparable mass-ratio coalescences in GR are parametrically deformed. For specific values of the ppE parameters, one recovers GR waveforms; otherwise they describe waveforms in non-GR theories. The ppE scheme has been shown to be sufficiently flexible to map all known alternative theory predictions for comparable mass-ratio coalescences [9]. \nPpE deformations of EMRI waveforms have not yet been constructed because of the complexities associated with the computation of such waveforms. EMRIs are not amenable to post-Newtonian (PN) approximation \nschemes [10-14], from which many GR templates are today analytically constructed. Instead, to build EMRI waveforms one must solve geodesic-like equations, enhanced with a radiation-reaction force that induces an inspiral. The solution of such differential equations can only be found numerically (see eg. [15] for a recent review of the self-force problem). \nOne can parametrically deform EMRI waveforms by introducing non-GR deviations in the numerical scheme used to build such waveforms. Two main ingredients can then be modified: the conservative sector, which controls the shape of non-radiative orbits, and the dissipative sector, which controls the rate of inspiral and the GW generation mechanism. The conservative sector, to leading order in the mass-ratio, depends only on the background spacetime metric, the Kerr metric, on which the SCO moves like a test-particle. \nThe goal of this work is to find a parametric deformation of the Kerr metric that allows for non-GR deviations while retaining a smooth Kerr limit , i.e. as the deformation parameters go to zero, the deformed Kerr metric reduces smoothly and exactly to Kerr. This limit then guarantees that the main properties of the Kerr background survive to the deformation, such as the existence of an event horizon, an ergosphere, and, in particular, three constants of the motion, such that the geodesic equations can be separated into first-order form. \nAlthough motion in alternative theories of gravity need not be geodesic, we will here assume that the geodesic equations hold to leading order in the mass ratio. This then allows us to focus only on metric deformations that correct the conservative sector of EMRI waveforms. This assumption is justified for theories that derive from certain diffeomorphism-covariant Lagrangians [16], as is the case, for example, in dynamical Chern-Simons (CS) modified gravity [17], in Einstein-Dilaton-Gauss-Bonnet theory [18-22], and in dynamical quadratic gravity [23-25]. We are not aware of any alternative theory where this is not the case to leading order in the mass ratio (and \nneglecting spins), although it is not hard to imagine that one may exist. \nA framework already exists to parametrically deform the metric tensor through the construction of so-called bumpy spacetimes [26-28]. Such metrics are deformations of the Kerr metric, where the bumps are required to satisfy the Einstein equations. Because of this last condition, bumpy black holes (BHs) are not sufficiently generic to represent BHs in certain alternative theories of gravity (e.g. solutions that are not Ricci flat). A better interpretation is to think of these bumps as representing exterior matter distributions. The bumpy BH formalism then allows tests of whether compact objects are truly described by the vacuum Kerr metric or by some more general tensor with external matter sources, assuming GR still holds. \nWe here propose two generalizations of the bumpy BH formalism to allow for metric deformations that can represent vacuum BH solutions in alternative theories. The first approach, the generalized bumpy Kerr (BK) scheme, takes the standard bumpy metric and relaxes the requirement that the bumps satisfy the Einstein equations. The second approach, the generalized deformed Kerr (DK) metric, perturbs the most general stationary, axisymmetric metric in Lewis-Papapetrou form and transforms it to Boyer-Lindquist coordinates without assuming the deformations satisfy the Einstein equations. In both cases, the metric is exactly Kerr to zeroth order in the deformation parameter, but it deviates from Kerr at first order through a set of arbitrary functions. \nThe metric deformation is then restricted by requiring that the full metric still possesses three constants of the motion: a conserved energy (associated with time invariance), a conserved angular momentum component (associated with azimuthal rotational invariance), and an approximate second-order Killing tensor that leads to a conserved Carter constant. These conditions restrict the set of arbitrary functions that parameterize the metric deformations to have a certain functional form. It is these conditions that select the appropriate 'bumps' instead of the imposition of the Einstein equations. \nThe restrictions imposed above are not strictly necessary, as there is no guarantee that BHs in alternative theories of gravity will continue to have three conserved quantities. However, all BH solutions in alternative theories of gravity known to date and that are not pathological (i.e. they are stationary, axisymmetric, asymptotically flat, and contain no spacetime regions with closed time-like curves outside the horizon) possess three constants of the motion [29]. Examples include the slowlyrotating solution found in dynamical CS gravity [30] and the spherically symmetric solution found in dynamical quadratic gravity [25]. \nMuch effort has gone into finding spacetimes with an exact second-order (or higher-order) generalized 'Carter constant,' thus allowing for the separation of the equations of motion [31-35]. That work demonstrates that, for a broad class of spacetimes, such separation can \nbe done provided that the Carter constant is quartic in the orbit's 4-momentum (i.e., the constant is C = ξ αβγδ p α p β p γ p δ , where ξ αβγδ is a 4th-rank Killing tensor and p α is the 4-momentum). In this analysis, we show that one can in fact find an approximate Carter constant that is quadratic in the 4-momentum, C = ξ αβ p α p β , for many relevant spacetimes, provided they differ from Kerr only perturbatively . \nFinally, we gain insight on the different proposed parameterizations by studying certain key limits. First, we show that the BK and DK deformations are related to each other by a gauge transformation. Second, we show that both the BK and DK metrics can be exactly mapped to specific non-GR BH metrics, the dynamical CS gravity one and the dynamical quadratic one. Third, we study the structure that the deformations must take when only deforming frame-dragging or the location of the event horizon. Fourth, we separate the geodesic equations into first-order form in both the BK and DK metrics, and calculate the modified Kepler law. This modification corrects the dissipative dynamics through the conversion of radial quantities to frequency space. \nThe parameterized metrics (BK and DK) proposed in this paper lay the foundations for a systematic construction and study of ppE EMRI waveforms. With these metrics and their associated separated equations of motion, one can now study modified SCO trajectories and see how these impact the GW observable. Numerical implementation and a detailed data analysis study will be presented in a forthcoming publication. \nThis paper is organized as follows. Sec II reviews BHs in GR and derives the first-order form of the geodesic equations. Sec. III introduces the BK formalism, which generalizes the bumpy BH formalism. Sec. IV presents the DK parameterized metric. Sec. V compares the parameterizations to each other and to alternative theory predictions. Sec. VI discusses some general properties of the parameterizations and points to future work. \nIn this paper we primarily follow the notation of Misner, Thorne and Wheeler [36]. Greek letters stand for spacetime indices, while Latin letters in the middle of the alphabet ( i, j, k, . . . ) stand for spatial indices only. Parenthesis and brackets in index lists stand for symmetrization and antisymmetrization respectively, i.e. A ( ab ) = ( A ab + A ba ) / 2 and A [ ab ] = ( A ab -A ba ) / 2. Background quantities are denoted with an overhead bar, such that the full metric g µν = ¯ g µν + /epsilon1h µν can be decomposed into a background ¯ g µν plus a small deformation h µν , where /epsilon1 /lessmuch 1 is a book-keeping parameter. The background spacetime is always taken to be the Kerr metric. We use geometric units with G = c = 1.", 'II. BLACK HOLES AND TEST-PARTICLE MOTION IN GENERAL RELATIVITY': 'In this section, we review some basic material on the Kerr BH solution in GR and the motion of test-particles \n(see e.g. [36] for more details). The most general stationary, axisymmetric BH solution in GR is the Kerr metric, which in Boyer-Lindquist coordinates ( t, r, θ, φ ) takes the form \n¯ g tt = -( 1 -2 Mr ρ 2 ) , ¯ g tφ = -2 Mar ρ 2 sin 2 θ , ¯ g rr = ρ 2 ∆ , ¯ g θθ = ρ 2 , ¯ g φφ = Σ ρ 2 sin 2 θ , (1) \nwhere we have introduced the quantities \nρ 2 ≡ r 2 + a 2 cos 2 θ , (2) \n∆ ≡ r 2 f + a 2 , f ≡ 1 -2 M r , (3) \nΣ ≡ ( r 2 + a 2 ) 2 -a 2 ∆sin 2 θ . (4) \nThe overhead bars are to remind us that the Kerr metric will be taken as our background spacetime. This background describes a BH with mass M and spin angular momentum S i = (0 , 0 , Ma ), where a is the Kerr spin parameter with units of length.', 'A. Equations of Motion': "With this background, we can now compute the equations of motion for a test particle of mass µ moving on a background worldline ¯ x µ = ¯ z µ ( λ ), where λ is an affine parameter. One way to derive these equations is via the action for a non-spinning test-particle (see e.g. [37]) \nS Mat = -µ ∫ γ dλ √ -¯ g αβ (¯ z ) ˙ ¯ z α ˙ ¯ z β , (5) \nwhere ˙ ¯ z µ = d ¯ z µ /dλ is the tangent to ¯ z µ and ¯ g is the determinant of the background metric. \nThe contribution of this action to the field equations can be obtained by varying it with respect to the metric tensor. Doing so, we obtain the stress-energy for the test-particle: \nT αβ Mat (¯ x µ ) = µ ∫ dτ √ -¯ g ¯ u α ¯ u β δ (4) [¯ x -¯ z ( τ )] , (6) \nwhere τ is proper time, related to λ via dτ = dλ √ -¯ g αβ (¯ z ) ˙ ¯ z α ˙ ¯ z β , ¯ u µ = d ¯ z µ /dτ is the particle's 4-velocity, normalized via ¯ g µν ¯ u µ ¯ u ν = -1 and δ (4) is the four-dimensional Dirac density, defined via d 4 x √ -¯ g δ (4) ( x ) = 1. \n∫ \n-The divergence of this stress-energy tensor must vanish in GR, which implies that test-particles follow geodesics: \nD dτ d ¯ z α dτ = 0 , (7) \nwhere D/dτ is a covariant derivative. The divergence of the stress-energy tensor vanishes due to local energymomentum conservation and the equivalence principle [36]. We have assumed here that the particle is nonspinning, so that geodesic motion is simply described by \n] \nEq. (7); otherwise other terms would arise in the action that would lead to spin-dependent modifications.", 'B. Constants of the Motion and Separability': "The geodesic equations as written in Eq. (7) are of second-order form, but they can be simplified to firstorder form if there exist at least three constants of the motion plus a normalization condition on the fourmomentum. If the metric is sufficiently symmetric, then Killing vectors ξ α and Killing tensors ξ αβ exist that satisfy the vectorial and tensorial Killing equations \n( α ξ β ) = 0 , (8) \n∇ ( α ξ βδ ) = 0 . (9) \n∇ \nThe Kerr metric possesses two Killing vectors and one Killing tensor. The vectors, ¯ t α = [1 , 0 , 0 , 0] and ¯ φ α = [0 , 0 , 0 , 1], are associated with the stationarity and axisymmetry of the metric respectively. In addition, the Kerr metric also possesses the Killing tensor: \n¯ ξ αβ = ∆ ¯ k ( α ¯ l β ) + r 2 ¯ g αβ , (10) \nwhere ¯ k α and ¯ l α are principal null directions: \n¯ k α = [ r 2 + a 2 ∆ , 1 , 0 , a ∆ ] , ¯ l α = [ r 2 + a 2 ∆ , -1 , 0 , a ∆ ] . (11) \nThe contraction of these Killing vectors and tensors allows us to define constants of the motion, namely the energy E , z-component of the angular momentum L , and the Carter constant C : \nE ≡ -t α u α , (12) \n≡ -L φ α u α (13) \nC ≡ ξ αβ u α u β . (14) \n≡ \nIt is conventional to define another version of the Carter constant, namely \nQ ≡ C -( L -aE ) 2 . (15) \nWith these three constants of the motion [either ( E,L,C ) or ( E,L,Q )] and the momentum condition p α p α = -µ 2 , or simply u α u α = -1, we can separate the geodesic equations and write them in first-order form: \nρ 2 ˙ ¯ t = [ -a ( aE sin 2 θ -L + ( r 2 + a 2 P ∆ ] , (16) \n( ( ρ 2 ˙ ¯ φ = [ -( aE -L sin 2 θ ) + aP ∆ ] , (17) \nρ 4 ˙ ¯ r 2 \n= \nr 2 + a 2 \nE \naL \n) \n) \n2 \n[ ρ 4 ˙ ¯ θ 2 = Q -cot 2 θL 2 -a 2 cos 2 θ ( 1 -E 2 ) , (19) \n[( ) -] -∆ [ Q +( aE -L ) 2 + r 2 , (18) \nwhere overhead dots stand for differentiation with respect to proper time and P ≡ E ( r 2 + a 2 ) -aL . Notice that not only have the equations been written in first-order form, but they have also been separated. \nOnce the geodesic equations have been derived, one can compute Kepler's third law, relating radial separations to orbital frequencies for an object moving in an equatorial, circular orbit. For such an orbit, the equatorial, circular orbit conditions ˙ ¯ r = 0 and d ˙ ¯ r/dr = 0 lead to [38] \n| ¯ Ω | ≡ | ˙ ¯ φ/ ˙ ¯ t | = M 1 / 2 r 3 / 2 + aM 1 / 2 . (20) \nIn the far field limit, M/r /lessmuch 1, this equation reduces to the familiar Kepler law: Ω ∼ ( M/r 3 ) 1 / 2 .", 'III. BLACK HOLES AND TEST-PARTICLE MOTION IN ALTERNATIVE THEORIES: GENERALIZED BUMPY FORMALISM': 'We here introduce the standard bumpy BH formalism [26-28] and generalize it to the BK framework. The \nbumps are constrained by requiring that an approximate, second-order Killing tensor exists. We then rewrite the geodesic equations in first-order form.', 'A. From Standard to Generalized Bumpy Black Holes': "Let us first review the basic concepts associated with the bumpy BH formalism [26-28]. This framework was initially introduced by Collins and Hughes [26] to model deformations of the Schwarzschild metric and expanded by Vigeland and Hughes [27] to describe deformations on a Kerr background. In its standard formulation, generalized to spinning BHs [27], the metric is perturbatively expanded via \ng µν = ¯ g µν + /epsilon1 h SBK µν , (21) \nwhere /epsilon1 is a book-keeping parameter that reminds us that | h SBK µν | / | g µν | /lessmuch 1. The background metric ¯ g µν is the Kerr solution of Eq. (1), while in the standard bumpy formalism, the metric functions h SBK µν are parameterized via \nh SBK tt = -2 ψ SBK 1 ( 1 + 2 Mr ρ 2 ) -4 aMr ρ 4 σ SBK 1 , h SBK tr = -γ SBK 1 2 a 2 Mr sin 2 θ ∆ ρ 2 , h SBK tφ = -(2 ψ SBK 1 -γ SBK 1 ) 2 aMr sin 2 θ ρ 2 -2 σ SBK 1 [ ( r 2 + a 2 ) 2 Mr ρ 4 + ∆ ρ 2 -∆ 2 ρ 2 -4 Mr ] , h SBK rr = 2( γ SBK 1 -ψ SBK 1 ) ρ 2 ∆ , h SBK rφ = γ SBK 1 [ 1 + 2 Mr ( r 2 + a 2 ) ∆ ρ 2 ] a sin 2 θ , h SBK θθ = 2 ( γ SBK 1 -ψ SBK 1 ) ρ 2 , h SBK φφ = [ ( γ SBK 1 -ψ SBK 1 ) 8 a 2 M 2 r 2 sin 2 θ ρ 2 -2 Mr -2 ψ SBK 1 ρ 4 ∆ ρ 2 -2 Mr + 4 aMr ρ 2 -2 Mr σ SBK 1 ( ∆+ 2 Mr ( r 2 + a 2 ) ρ 2 )] sin 2 θ ρ 2 , (22) \nwhere ( ψ SBK 1 , γ SBK 1 , σ SBK 1 ) are functions of ( r, θ ) only that represent the 'standard bumps.' \nA few comments are due at this point. First, notice that this metric contains only three arbitrary functions ( ψ SBK 1 , γ SBK 1 , σ SBK 1 ), instead of four, as one would expect from the most general stationary and axisymmetric metric. This is because of the specific way Eq. (22) is derived (see below), which assumes a Ricci-flat metric in the a = 0 limit. Second, note that many metric components are non-vanishing: apart from the usual ( t, t ), ( t, φ ), ( r, r ), ( θ, θ ) and ( φ, φ ) metric components that are non-zero in the Kerr metric, Eq. (22) also has nonvanishing ( t, r ) and ( r, φ ) components. \nThe derivation of this metric is as follows. One begins with the most general stationary and spherically symmetric metric in Lewis-Pappapetrou form that is Ricci-flat. One then perturbs the two arbitrary functions in this metric and maps it to Schwarschild coordinates. Through \nthe Newman-Janis procedure, one then performs a complex rotation of the tetrad associated with this metric to transform it into a deformed Kerr spacetime. At no stage in this procedure is one guaranteed that the resulting spacetime will still possess a Carter constant or that it will remain vacuum. In fact, the metric constructed from Eq. (22) does not have a Carter constant, nor does it satisfy the vacuum Einstein equations, leading to a non-zero effective stress-energy tensor. \nSuch features make the standard bumpy formalism not ideal for null-tests of GR. One would prefer to have a framework that is generic enough to allow for non-GR tests while still possessing a smooth GR limit, such that as the deformations are taken to zero, one recovers exactly the Kerr metric. This limit implies that the deformed metric will retain many of the nice properties of the Kerr background, such as the existence of an event horizon, an ergosphere, and a Carter constant. \nThis can be achieved by generalizing the bumpy BH formalism through the relaxation of the initial assumption that the metric be Ricci-flat prior to the NewmanJanis procedure. Inspired by [26-28], we therefore promote the non-vanishing components of the metric perturbation, i.e., ( h BK tt , h BK tr , h BK tφ , h BK rr , h BK rφ , h BK θθ , h BK φφ ), to arbitrary functions of ( r, θ ). These functions are restricted only by requiring that an approximate second-order Killing tensor still exists. \nThis restriction is not strictly necessary, but it is appealing on several fronts. First, the few analytic non-GR BH solutions that are known happen to have a Carter constant, at least perturbatively as an expansion in the non-Kerr deviation. Second, a Carter constant allows for the separation of the equations of motion into first-order form, which then renders the system easily integrable with already developed numerical techniques.", 'B. Existence Conditions for the Carter Constant': "Let us now investigate what conditions must be enforced on the metric perturbation h BK αβ so that a Killing tensor ξ αβ and its associated Carter constant exist, at least perturbatively to O ( /epsilon1 ). Killing's equations can be used to infer some fairly general properties that such a tensor must have. First, this tensor must be nonvanishing in the same components as the metric perturbation. For the generalized bumpy metric, this means that ( ξ tθ , ξ rθ , ξ θφ ) must all vanish. Furthermore, if we expand the Killing tensor as \nξ αβ = ¯ ξ αβ + /epsilon1 δξ αβ , (23) \nwe then find that δξ αβ must have the same parity as ¯ ξ αβ for the full Killing tensor to have a definite parity. For the generalized bumpy metric, this means that ( ξ tt , ξ rr , ξ θθ , ξ φφ , ξ tφ ) must be even under reflection: θ → θ π . \nThese conditions imply that of the 10 independent degrees of freedom in δξ αβ , only 7 are truly necessary. With this in mind, we parameterize the Killing tensor as in Eq. (10), namely \n- \nξ αβ = ∆ k ( α l β ) + r 2 g αβ . (24) \nNotice that ξ αβ depends here on the full metric g αβ and on null vectors k α and l α that are not required to be the Kerr ones or the principal congruences of the full \nspacetime: k α /negationslash = ¯ k α and l α /negationslash = ¯ l α . We decompose these vectors via \nk α = ¯ k α + /epsilon1 δk α , l α = ¯ l α + /epsilon1 δl α , (25) \nwhere ¯ k α and ¯ l α are given in Eq. (11), and the tensor ξ αβ into \nξ αβ = ¯ ξ αβ + /epsilon1 δξ αβ , (26) where ¯ ξ αβ is given in Eq. (10), while \nδξ αβ ≡ ∆ [ δk ( α l β ) + δl ( α k β ) +2 h BK δ ( α ¯ k β ) ¯ l δ ] +3 r 2 h BK αβ . (27) \nAll lowering and raising of indices is carried out with the background metric. Although this particular ansatz only allows 6 independent degrees of freedom (as the vectors are assumed to be null), we will see it suffices to find a Carter constant. \nWith this ansatz, the tensor Killing equation [Eq. (9)] becomes \n∂ ( µ δξ αβ ) -2 ¯ Γ δ ( µα δξ β ) δ = 2 δ Γ δ ( µα ¯ ξ β ) δ . (28) \nThe system of equations one must solve is truly formidable and in fact overconstrained. Equation (28) is a set of 20 partial differential equations, while the normalization conditions l α l α = 0 = k α k α add two additional algebraic equations. This means there are a total of 13 degrees of freedom (7 in h BK αβ and 6 in δl α and δk α after imposing the normalization condition) but 20 partial differential equations to solve. \nIn spite of these difficulties, we have solved this system of equations with Maple and the GrTensorII package [39] and found that the perturbation to the null vectors must satisfy \nδk α BK = [ r 2 + a 2 ∆ δk r BK + δ 1 , δk r BK , 0 , a ∆ δk r BK + δ 2 ] , δl α BK = [ r 2 + a 2 ∆ δk r BK + δ 3 , δk r BK + δ 4 , 0 , a ∆ δk r BK + δ 5 ] , (29) \nwhere δ i ≡ δ i ( r ) are arbitrary functions of r , generated upon solving the differential system. These functions are fully determined by the metric perturbation, which is given by \nh BK tt = -a 2 M ρ 4 ∆ P BK 1 ∂h BK tφ ∂r -a M P BK 2 P BK 1 h BK tφ -a 2 sin 2 θ 4 M ( ρ 2 -4 Mr )∆ 2 P BK 1 ( ∂δ 1 ∂r + ∂δ 3 ∂r ) -a 4 M ∆ 2 sin 2 θ P BK 3 P BK 1 ( ∂δ 2 ∂r + ∂δ 5 ∂r ) + ∆ ρ 2 P BK 4 P BK 1 ( δ 1 + δ 3 ) -a 2 M ∆ ρ 2 P BK 5 P BK 1 ( δ 2 + δ 5 ) + ∆ ρ 2 ( r 2 + a 2 )¯ ρ 2 P BK 1 Θ 1 , \nh BK tr = 1 2 ( 1 -2 Mr ρ 2 ) ( δ 1 -δ 3 ) + aMr sin 2 θ ρ 2 ( δ 2 -δ 5 ) -1 2 δ 4 , h BK rr = -Θ 1 ∆ + ρ 2 ∆ δ 4 , h BK rφ = aMr sin 2 θ ρ 2 ( δ 1 -δ 3 ) -sin 2 θ 2 ρ 2 ( P BK 3 -2 a 2 Mr sin 2 θ )( δ 2 -δ 5 ) + a 2 sin 2 θδ 4 , h BK θθ = -Θ 1 + ρ 2 Θ 2 , h BK φφ = -( r 2 + a 2 ) 2 a 2 h BK tt -2( r 2 + a 2 ) a h BK tφ + ∆ a 2 Θ 1 + ∆ 2 a 2 ( δ 1 + δ 3 ) -∆ 2 sin 2 θ a ( δ 2 + δ 5 ) , ∂ 2 h BK tφ ∂r 2 = ( ∂ 2 δ 2 ∂r 2 + ∂ 2 δ 5 ∂r 2 ) P BK 6 + ( ∂ 2 δ 1 ∂r 2 + ∂ 2 δ 3 ∂r 2 ) P BK 7 + ( ∂δ 2 ∂r + ∂δ 5 ∂r ) P BK 8 ,n P BK 8 ,d + ( ∂δ 1 ∂r + ∂δ 3 ∂r ) P BK 9 ,n P BK 9 ,d +( δ 2 + δ 5 ) P BK 10 ,n P BK 10 ,d +( δ 1 + δ 3 ) P BK 11 ,n P BK 11 ,d + ∂h BK tφ ∂r P BK 12 ,n P BK 12 ,d + h BK tφ P BK 13 ,n P BK 13 ,d +Θ 1 P BK 14 , (30) \nwhere ¯ ρ 2 ≡ r 2 -a 2 cos 2 θ , Θ 1 , 2 ≡ Θ 1 , 2 ( θ ) are functions of θ and P BK i are polynomials in r and cos θ , which are given explicitly in Appendix A. Notice that δk r BK does not appear in the metric perturbation at all, so we are free to set it to zero, i.e. δk r BK = 0. \nWith this at hand, given some metric perturbation h BK αβ that satisfies Eq. (30), one can construct the arbitrary functions δ i and Θ i and thus build the null directions of the perturbed spacetime, such that a Killing tensor and a Carter constant exist perturbatively to O ( /epsilon1 ). We have verified that the conditions described above satisfy the generic Killing tensor properties, described at the beginning of this subsection.", '1. Non-Rotating Limit': 'Let us now take the non-rotating limit, i.e. a → 0, of the DK Carter conditions. As before, we expand all arbitrary functions as in Eqs. (31) and (32). Imposing the constraints \nΘ 3 , 0 = 0 , γ 1 , 0 = -h DK rr, 0 f γ 3 , 0 = 0 , γ 3 , 1 = h DK rr, 0 2 r 2 -r -4 M 2 r 3 f 2 h DK tt, 0 γ 4 , 0 = h DK tt, 0 2 f 2 + h DK rr, 0 2 , (58) \nforces the metric components to take the form \nh DK tt = h DK tt, 0 , h DK rr = h DK rr, 0 , h DK φφ = 0 . (59) \nAll other pieces of O ( a 0 ) in the remaining metric components can be set to zero by setting ( h DK rθ , h DK θθ , h DK tφ ) to zero. With these choices, the differential conditions are automatically satisfied, where we have set all integration constants to zero. \nThe Killing tensor is then given by Eq. (24), with the perturbed null vector components \nδk α DK = [ -h DK rr, 0 2 + h DK tt, 0 2 f 2 , -h DK rr, 0 f, 0 , O ( a ) ] , (60) δl α DK = [ h DK rr, 0 2 + h DK tt, 0 2 f 2 , 0 , 0 , O ( a ) ] , \nwhere here we have set δl r DK = 0. As before, the Killing tensor reduces exactly to that of the Schwarzschild spacetime, with the only non-zero components being ( ¯ ξ θθ , ¯ ξ φφ ) = r 4 (1 , sin 2 θ ).', 'C. Equations of Motion and Geodesics in First-Order Form': "The equations of motion for test-particles in alternative theories of gravity need not be geodesic. But as we review in Appendix B, they can be approximated as geodesics in the test-particle limit for a wide class of alternative theories, provided we neglect the spin of the small body. Here, we restrict attention to theories where the modified equations of motion remain geodesic, but of the generalized bumpy Kerr background constructed above. \nThe geodesic equations can be rewritten in firstorder form provided that three constants of the motion exist. In Sec. III B, we showed that provided the generalized bumpy functions satisfy certain conditions, then an approximate, second-order Killing tensor and a conserved Carter constant exist. Furthermore, it is easy to see that the metric in Eqs. (21) \nand (22) remains stationary and axisymmetric, since ( h BK tt , h BK tr , h BK tφ , h BK rr , h BK rφ , h BK θθ , h BK φφ ) are functions of only ( r, θ ). \nThe constants of the motion then become \nE = ¯ E + /epsilon1 ( h µν t µ ¯ u ν +¯ g µν t µ δu ν ) , (36) \nL = ¯ L + /epsilon1 ( h µν φ µ ¯ u ν + ¯ g µν φ µ δu ν ) , (37) \nC = ¯ C + /epsilon1 ( δξ µν ¯ u µ ¯ u ν +2 ¯ ξ µν ¯ u ( µ δu ν ) ) , (38) \nwhile the normalization condition becomes \n0 = h µν ¯ u µ ¯ u ν +2¯ g µν ¯ u µ δu ν , (39) \nsince by definition -1 = ¯ g µν ¯ u µ ¯ u ν . In these equations, ¯ u µ = [ ˙ ¯ t, ˙ ¯ r, ˙ ¯ θ, ˙ ¯ φ ] is the unperturbed, Kerr four-velocity, while δu µ is a perturbation. \nEquations (36)-(39) can be decoupled once we make a choice for ( E,L,Q ). This choice affects whether the turning points in the orbit are kept the same as in GR, or whether the constants of the motion are kept the same. We here choose the latter for simplicity by setting E = ¯ E , L = ¯ L and C = ¯ C , which then implies that δu µ must be such that all terms in parenthesis in Eqs. (36)-(38) vanish. Using this condition and Eq. (39), we then find that \nρ 2 ˙ t = T K ( r, θ ) + δT ( r, θ ) , ρ 4 ˙ r 2 = R K ( r ) + δR ( r, θ ) , ρ 2 ˙ φ = Φ K ( r, θ ) + δ Φ( r, θ ) , ρ 4 ˙ θ 2 = Θ K ( θ ) + δ Θ( r, θ ) , (40) \nwhere ( T K , R K , Θ K , Φ K ) are given by the right-hand sides of Eqs. (16), (18), (19), and (17) respectively. The perturbations to the potential functions ( δT, δR, δ Θ , δ Φ) are given by \nδT ( r, θ ) = [ ( r 2 + a 2 ) 2 ∆ -a 2 sin 2 θ ] h tα ¯ u α + 2 aMr ∆ h φα ¯ u α , (41) \nδ Φ( r, θ ) = 2 aMr ∆ h tα ¯ u α -ρ 2 -2 Mr ∆sin 2 θ h φα ¯ u α , (42) \nδR ( r, θ ) = ∆ A ( r, θ ) r 2 + B ( r, θ ) ] , (43) \n[ \n] δ Θ( r, θ ) = A ( r, θ ) a 2 cos 2 θ -B ( r, θ ) , (44) \nwhere the functions A ( r, θ ) and B ( r, θ ) are proportional to the perturbation: \nA ( r, θ ) = 2 h αt ¯ ˙ t + h αφ ¯ ˙ φ ] ¯ u α -h αβ ¯ u α ¯ u β , (45) \n[ \n] B ( r, θ ) = 2 [( ¯ ξ tt ¯ ˙ t + ¯ ξ tφ ¯ ˙ φ ) δ ˙ t + ( ¯ ξ tφ ¯ ˙ t + ¯ ξ φφ ¯ ˙ φ ) δ ˙ φ ] + δξ αβ ¯ u α ¯ u β . (46) \nAs before, we have dropped the superscript BK here. Interestingly, notice that the perturbation automatically \ncouples the ( r, θ ) sector, so that the first-order equations are not necessarily separable. One can choose, however, the γ i functions in such a way so that the equations remain separable, as will be shown elsewhere. \nThe perturbations to the potential functions result in a modification of Kepler's law [Eq. (20)]: \n| Ω | = | ¯ Ω | -M 1 / 2 ( r 1 / 2 ( r -3 M ) + 2 aM 1 / 2 ) r 5 / 4 ( r 3 / 2 + aM 1 / 2 ) 2 δT ( r ) + ( r 1 / 2 ( r -3 M ) + 2 aM 1 / 2 ) 1 / 2 r 5 / 4 r 3 / 2 + aM 1 / 2 ) δ Φ( r ) , (47) \n( \n) where we recall that | ¯ Ω | was already given in Eq. (20), δT and δ Φ are given by Eq. (41) and (42), and we have used that E = ¯ E and L = ¯ L . In the far field limit ( M/r /lessmuch 1), this becomes \n| Ω | ∼ | ¯ Ω | + 1 r 2 δ Φ . (48) \nwhere we have assumed that δ Φ and δT are both of order unity for simplicity. Given any particular metric perturbation, one can easily recalculate such a correction to Kepler's law from Eq. (47). Clearly, a modification of this type in the Kepler relation automatically modifies the dissipative dynamics when converting quantities that depend on the orbital frequency to radius and vice-versa.", 'IV. BLACK HOLES AND TEST-PARTICLE MOTION IN ALTERNATIVE THEORIES: DEFORMED KERR FORMALISM': 'In the previous section, we generalized the bumpy BH framework at the cost of introducing a large number of arbitrary functions, later constrained by the requirement of the existence of a Carter constant. In this section, we investigate a different parameterization (DK) that isolates the physically independent degrees of freedom from the start so as to minimize the introduction of arbitrary functions.', 'A. Deformed Kerr Geometry': 'Let us first consider the most general spacetime metric that one can construct for a stationary and axisymmetric spacetime. In such a geometry, there will exist two Killing vectors, t a and φ a , that represent invariance under a time and azimuthal coordinate transformation. Because these Killing vectors are independent, they will commute satisfying t [ a φ b ∇ c t d ] = 0 = t [ a φ b ∇ c φ d ] . Let us further assume the integrability condition \nt a R a [ b t c φ d ] = 0 = φ a R a [ b t c φ d ] . (49) \nThe condition in Eq. (49) guarantees that the 2-planes orthogonal to the Killing vectors t a and φ a are integrable. \nGeneric stationary and axisymmetric solutions to modified field equations do not need to satisfy Eq. (49). However, all known analytic solutions in GR and in alternative theories do happen to satisfy this condition. We will thus assume it holds here as well. \nGiven these conditions, the most general stationary and axisymmetric line element can be written in LewisPapapetrou canonical form, using cylindrical coordinates ( t, ρ, φ, z ): \nds 2 = -V ( dt -wdφ ) 2 + V -1 ρ 2 dφ 2 +Ω 2 dρ 2 +Λ dz 2 ) , (50) \n( \n) where ( V, w, Ω , Λ) are arbitrary functions of ( ρ, z ). In GR, by imposing Ricci-flatness, one can eliminate one of these functions via a coordinate transformation, and we can thus set Λ = 1. The Einstein equations then further restrict the form of these arbitrary functions [40] \n¯ V = ( r + + r -) 2 -4 M 2 + a 2 M 2 -a 2 ( r + -r -) 2 ( r + + r -+2 M ) 2 + a 2 M 2 -a 2 ( r + -r -) 2 , ¯ w = 2 aM ( M + r + + r -2 )( 1 -( r + -r -) 2 4( M 2 -a 2 ) ) 1 4 ( r + + r -) 2 -M 2 + a 2 ( r + -r -) 2 4( M 2 -a 2 ) , \n¯ Λ = 1 , \n¯ Ω = V ( r + + r -) 2 -4 M 2 + a 2 M 2 -a 2 ( r + -r -) 2 4 r + r -, (51) \nwhere r ± = √ ρ 2 + ( z ± √ M 2 -a 2 ) 2 . One can map from this coordinate system to Boyer-Lindquist coordinates ( t, r, θ, φ ) via the transformation ρ = √ ∆sin θ and z = ( r M ) sin θ to find the metric in Eq. (1). \nWe will not impose Ricci-flatness here, as we wish to model alternative theory deviations from Kerr. We thus work directly with Eq. (50), keeping ( V, w, Ω , Λ) as arbitrary functions, but transforming this to BoyerLindiquist coordinates. Let us then re-parametrize Ω 2 ≡ γ , λ ≡ γ Λ, and q ≡ V w , and perturb the metric functions via \n- \nV = ¯ V + δV , q = ¯ q + δq , λ = ¯ λ + δλ, γ = ¯ γ + δγ . (52) \nThe metric then becomes \ng µν = ¯ g µν + h DK µν , (53) \nwhere the perturbation is given by \nh DK tt = -δV , h DK tφ = δq , h DK rr = δλ cos 2 θ + δγ ( r -M ) 2 sin 2 θ ∆ , h DK rθ = ( r -M ) cos θ sin θ ( δγ -δλ ) , h DK θθ = δλ sin 2 θ ( r -M ) 2 + δγ cos 2 θ ∆ , h DK φφ = sin 2 θ ρ 2 -2 Mr { 4 aMrδq -[ ( r 2 + a 2 ) 2 -a 2 sin 2 θ ∆ ] δV } , (54) \nand all other components vanish. \nAs is clear from the above expressions, the most general perturbation to a stationary, axisymmetric metric yields five non-vanishing metric components that depend still only on four arbitrary functions ( δV, δq, δγ, δλ ) of two coordinates ( r, θ ). This is unlike the standard bumpy formalism that introduces six non-vanishing metric components that depend on four arbitrary functions (courtesy of the Newman-Janis algorithm applied to a nonKerr metric). Furthermore, this analysis shows that two of the six arbitrary functions introduced in the generalized bumpy scheme of the previous section must not be independent, as argued earlier. \nFollowing the insight from the previous section, we will henceforth allow these 5 metric components ( h DK tt , h DK rr , h DK rθ , h DK θθ , h DK φφ ) to be arbitrary functions of ( r, θ ), although technically there are only four independent degrees of freedom. We do this because it eases the analytic calculations to come when one investigates which conditions h DK αβ must satisfy for there to exist an approximate second-order Killing tensor. Moreover, it allows us to relax the undesirable requirement, implicit in Eq. (54), that when δV /negationslash = 0, then both h tt and h φφ must be non-zero in the a → 0 limit.', 'B. Existence Conditions for the Carter constant': 'Let us now follow the same methodology of Sec. III B to determine what conditions the arbitrary functions must satisfy in order for there to be a second-order Killing tensor. We begin by parameterizing the Killing tensor just as in Eq. (24), with the expansion of the null vectors as in Eq. (25) and the replacement of BK → DK everywhere. With this at hand, the Killing equation acquires the same structure as Eq. (28). \nWe have solved the Killing equations with Maple and the GrTensorII package [39] to find that the null vectors must satisfy: \nδk α DK = [ r 2 + a 2 ∆ ( δl r DK + γ 1 ) + γ 4 , δl r DK + γ 1 , -h DK rθ ρ 2 , a ∆ ( δl r DK + γ 1 ) + γ 3 ] , δl α DK = [ -r 2 + a 2 ∆ δl r DK + γ 4 , δl r DK , h DK rθ ρ 2 , -a ∆ δl r DK + γ 3 ] , (55) \nwhere γ i ≡ γ i ( r ) are arbitrary functions of radius, while δl r DK and h DK rθ are arbitrary functions of both ( r, θ ). As before with δk r BK , we find below that the function δl r DK does not enter the metric perturbation, so we can set it to zero. \nThe functions γ i are completely determined by the metric perturbation: \nh DK tt = -a M P DK 2 P DK 1 h DK tφ -a 2 M ρ 4 ∆ P DK 1 ∂h DK tφ ∂r -2 a 2 r ( r 2 + a 2 )∆sin θ cos θ ρ 2 P DK 1 h DK rθ + ( r 2 + a 2 )¯ ρ 2 ∆ ρ 2 P DK 1 I + 2 a 2 r 2 ∆sin 2 θ P DK 1 γ 1 + ¯ ρ 2 ( r 2 + a 2 )∆ ρ 2 P DK 1 Θ 3 -a M ∆sin 2 θ ρ 2 P DK 3 P DK 1 γ 3 + 2∆ ρ 2 P DK 4 P DK 1 γ 4 -a 2 2 M ρ 2 ∆ 2 sin 2 θ P DK 1 dγ 1 dr -a 2 M ∆ 2 (Σ + 2 a 2 Mr sin 2 θ ) sin 2 θ P DK 1 dγ 3 dr -a 2 2 M ∆ 2 ( ρ 2 -4 Mr ) sin 2 θ P DK 1 dγ 4 dr , h DK rr = -1 ∆ I 1 ∆ Θ 3 , h DK φφ = -( r 2 + a 2 ) 2 a 2 h DK tt + ∆ a 2 I 2( r 2 + a 2 ) a h DK tφ + ∆ a 2 Θ 3 -2∆ 2 sin 2 θ a γ 3 + 2∆ 2 a 2 γ 4 , ∂h DK θθ ∂r = 2 r ρ 2 h DK θθ + 2 a 2 sin θ cos θ ρ 2 h DK rθ +2 ∂h DK rθ ∂θ + 2 r ρ 2 I 2 r γ 1 + 2 r ρ 2 Θ 3 , ∂ 2 h DK tφ ∂r 2 = 8 aM sin θ cos θ ρ 8 P DK 5 P DK 1 h DK rθ -4 aMr ( r 2 + a 2 ) sin θ cos θ ρ 6 ∂h DK rθ ∂r + 2 a 2 sin 2 θ ρ 4 P DK 6 P DK 1 h DK tφ -2 r ρ 2 P DK 7 P DK 1 h DK tφ + 4 aMr sin 2 θ ρ 4 P DK 15 P DK 16 I 4 aMr sin 2 θ ρ 4 P DK 8 P DK 1 γ 1 + 4 aMr ρ 4 P DK 9 P DK 1 Θ 3 + 2 sin 2 θ ρ 4 P DK 10 P DK 1 γ 3 -16 aM sin 2 θ ρ 4 P DK 11 P DK 1 γ 4 -2 a ρ 4 P DK 12 P DK 1 dγ 1 dr -2 sin 2 θ ρ 4 P DK 13 P DK 1 dγ 3 dr -2 a sin 2 θ ρ 4 P DK 14 P DK 1 dγ 4 dr -a ∆sin 2 θ ρ 2 d 2 γ 1 dr 2 -∆sin 2 θ ρ 4 (Σ + 2 a 2 Mr sin 2 θ ) d 2 γ 3 dr 2 -a ∆( ρ 2 -4 Mr ) sin 2 θ ρ 4 d 2 γ 4 dr 2 , (56) \nwhere recall that ¯ ρ 2 ≡ r 2 -a 2 cos 2 θ , Θ r = Θ r ( θ ) is an arbitrary function of polar angle, while P DK i are polynomials in r and cos θ , given explicitly in Appendix A. Notice that many of these relations are integro-differential, as I is defined as \nI = ∫ dr [ 2 a 2 sin θ cos θ ρ 2 h DK rθ +2 r γ 1 + ρ 2 dγ 1 dr ] . (57) \nNotice also that the component h DK rθ is free and thus there are here truly only four independent metric components. \nWe see then that any metric perturbation h DK αβ that satisfies Eq. (56) will possess a Carter constant. If given a specific non-Kerr metric, one can then use these equations to reconstruct the γ i and Θ 2 functions to automatically obtain the perturbative second-order Killing tensor associated with this spacetime. \nWith this at hand, one can now construct the constants of the motion. The metric in Eqs. (53) and (54) remains stationary and axisymmetric since all metric components h DK αβ depend only on ( r, θ ). Therefore, the constants of the motion can be expanded in exactly the same way as in Eqs. (36)-(38). With this at hand, the geodesic equations can be rewritten in first-order form in exactly the same way as in the BK case, except that here different metric components are non-vanishing. Of course, all perturbed quantities must here have a DK superscript.', 'A. To Each Other': "In the previous sections, we proposed two different parameterizations of deformed spacetimes suitable for modeling alternative theory predictions. These parameterizations are related via a gauge transformation. Under a general diffeomorphism, the metric transforms according to h DK µν → h DK ' µν ≡ h DK µν + ∇ ( µ ξ ν ) , where we parameterize the generating vectors as \nξ = [ ξ 0 ( r ) , ξ 1 ( r, θ ) , ξ 2 ( r, θ ) , ξ 3 ( r )] . (61) \nThe question is whether a generating vector exists that could take a generic h DK µν metric perturbation to one of h BK µν form. The DK parameterization has an ( r, θ ) component that is absent in the BK one, while the BK one has ( t, r ) and ( r, φ ) components that are absent in the DK parameterization. \nLet us assume that we have a certain metric in DK form. The first task is to remove the ( r, θ ) component, i.e. to find a diffeomorphism such that h DK ' rθ = 0. This can be achieved by requiring that \nh DK rθ + 1 2 Σ [ 1 ∆ ∂ξ 1 ∂θ + ∂ξ 2 ∂r ] = 0 , (62) \nwhose solution is \nξ 1 = F 1 ( r ) -∆ ∫ ( ∂ξ 2 ∂r + 2 h DK rθ ρ 2 ) dθ , (63) \nwhere F 1 ( r ) is a free integration function and ξ 2 ( r, θ ) is free. \nThe generating vector of Eq. (61) with the condition in Eq. (63) not only guarantees that h DK ' rθ = 0, but also ensures that the only non-vanishing components of the gauge-transformed metrics are the ( t, t ), ( t, r ), ( t, φ ), ( r, r ), ( r, φ ), ( θ, θ ) and ( φ, φ ), exactly the same non-zero components as in the BK metric. The new components \nh DK ' tt = h DK tt -M ¯ ρ 2 Σ 2 ξ 1 + 2 a 2 Mr sin θ cos θ Σ 2 ξ 2 , h DK ' tr = -1 2 ( 1 -2 Mr Σ ) dξ 0 dr -aMr sin 2 θ Σ dξ 3 dr , h DK ' tφ = h DK tφ + aM ¯ ρ 2 sin 2 θ Σ 2 ξ 1 -2 aMr ( r 2 + a 2 ) cos θ sin θ Σ 2 ξ 2 , h DK ' rr = h DK rr + a 2 r ∆ 2 ξ 1 -a 2 sin θ cos θ ∆ ξ 2 + Σ ∆ ∂ξ 1 ∂r , h DK ' rφ = -sin 2 θ 2Σ { 2 aMr dξ 0 dr + [ ( r 2 + a 2 ) 2 + a 2 ∆sin 2 θ ] dξ 3 dr } , \nh DK ' θθ = h DK θθ + r ξ 1 -a 2 sin θ cos θ ξ 2 +Σ ∂ξ 2 ∂θ , h DK ' φφ = h DK φφ + sin 2 θ Σ 2 { r 5 -a 2 Mr 2 + a 2 cos 2 θ [ 2 r ( r 2 + M 2 ) +∆ M ] + a 4 cos 4 θ ( r -M ) } ξ 1 +sin θ cos θ × [ ∆+ 2 Mr ( r 2 + a 2 ) 2 Σ 2 ] ξ 2 , (64) \nwhere recall that ¯ ρ 2 ≡ r 2 -a 2 cos 2 θ and ξ 1 is given by Eq. (63). \nThe above result can be simplified somewhat by setting ξ 2 = 0 and F 1 ( r ) = 0 in Eq. (63). Notice that the ( t, r ) and ( r, φ ) components are not modified by these vector components. The modified components then become \nh DK ' tt = h DK tt -M ¯ ρ 2 Σ 2 ξ 1 , h DK ' tφ = h DK tφ + aM ¯ ρ 2 sin 2 θ Σ 2 ξ 1 h DK ' rr = h DK rr + a 2 r ∆ 2 ξ 1 + Σ ∆ ∂ξ 1 ∂r , h DK ' θθ = h DK θθ + r ξ 1 , h DK ' φφ = h DK φφ + sin 2 θ Σ 2 { r 5 -a 2 Mr 2 + a 2 cos 2 θ [ 2 r ( r 2 + M 2 ) +∆ M ] + a 4 cos 4 θ ( r -M ) ξ 1 . (65) \nWe have thus found a generic diffeomorphism that maps a DK metric to one that has the same non-zero components as a BK metric. Notice that ξ 0 and ξ 3 only enter to generate ( t, r ) and ( r, φ ) components. If we know the form of the components that we are trying to map to, then we could solve for these two vector components. For example, let us assume that h BK tr and h BK rφ are given and we wish to find ξ 0 , 3 such that h DK ' tr = h BK tr and h DK ' rφ = h BK rφ . \n} \nThis implies \n0 = h BK tr + 1 2 ( 1 -2 Mr Σ ) dξ 0 dr -aMr sin 2 θ Σ dξ 3 dr , (66) 0 = h BK rφ + sin 2 θ 2Σ { 2 aMr dξ 0 dr + [ ( r 2 + a 2 ) 2 + a 2 ∆sin 2 θ ] dξ 3 dr } . (67) \nLet us, for one moment, allow ξ 0 , 3 to be arbitrary functions of ( r, θ ). The solution to the above system is then \nξ 0 \n= \n- \nF 2 ( θ ) + 2 ∆ ∫ dr ρ 2 [ h BK tr cos 2 θ ( 2 a 2 Mr -a 2 r 2 -a 4 ) h BK tr 2 a 2 Mr + a 2 r 2 + a 4 -2 Mrah BK rφ , (68) \n] \n( ) ] ξ 3 = F 3 ( θ ) + 2 ∆ ∫ dr ρ 2 [ h BK rφ ( ρ 2 -2 Mr ) -2 aMr sin 2 θh BK tr . (69) \nBut of course, to avoid introducing other spurious components to the DK ' metric, such as h DK tθ , we must require that ξ 0 , 3 be functions of r only. This implies that the integrands must be themselves also functions of r only. Setting the integrands equal to F 2 ( r ) and F 3 ( r ), we find the conditions \nh BK tr = F 2 ( r ) ( ρ 2 -2 Mr ) +2 Mra sin 2 θF 3 ( r ) ρ 2 , h BK rφ = 2 Mra sin 2 θF 2 ( r ) ρ 2 -F 3 ( r ) ρ 2 [ ρ 4 + a 4 ρ 2 +2 Mr )] sin 2 θ . (70) \n[ )] Provided h BK tr and h BK rφ components can be written in the above form, the generating function becomes \n( \nξ 0 = -2 ∫ F 2 ( r ) dr , ξ 3 = -2 ∫ F 3 ( r ) dr . (71)", '1. Dynamical Chern-Simons Modified Gravity': "The BK and DK parameterizations can also be mapped to known alternative theory BH metrics. In dynamical CS gravity [17], one can employ the slow-rotation approximation, where one assumes the BH's spin angular momentum is small, | S | /M 2 /lessmuch 1, and the small-coupling approximation, where one assumes the theory's corrections are small deformations away from GR to find an analytical BH solution. Yunes and Pretorius [30] found that this solution is simply g µν = ¯ g µν + h CS µν , where ¯ g µν is the Kerr metric (expanded in the slow-rotation limit), while the only non-vanishing component of the deformation tensor in Boyer-Lindquist coordinates is \nh CS tφ = 5 8 ζ CS a M M 5 r 4 sin 2 θ ( 1 + 12 M 7 r + 27 M 2 10 r 2 ) , (72) \nwhere a is the Kerr spin parameter, M is the BH mass and ζ CS is a dimensionless constant that depends on the CS couplings and is assumed to be small. \nSuch a non-GR BH solution can be mapped to the generalized bumpy parameterization of Eqs. (21) and (30) via δ 2 = δg CS φ = δ 5 , and all other δ i and Θ i vanish. The quantity δg CS φ is given by [41] \nδg CS φ = -ζ CS aM 4 70 r 2 +120 Mr +189 M 2 112 r 8 (1 -2 M/r ) . (73) \nWith these choices, the full null vectors become \nk α = [ r 2 + a 2 ∆ , 1 , 0 , a ∆ + δg CS φ ] , l α = [ r 2 + a 2 ∆ , -1 , 0 , a ∆ + δg CS φ ] , (74) \nwhich agrees with Eq. (35) of [41]. Moreover, with these choices all the metric components vanish except for the ( t, φ ) one, which must be equal to Eq. (72) for the differential constraint in Eq. (30) to be satisfied. \nSimilarly, we can also map it to the DK parameterization of Eqs. (53) and (54) via γ 3 = δg CS φ , and all other arbitrary functions vanish. With this, the perturbation to the principal null directions are equal to Eq. (74), which agrees with [41]. Moreover, we have checked that Eq. (72) satisfies the differential constraint found in Sec. IV.", '2. Dynamical, Quadratic Gravity': 'We can similarly map the BK and DK parameterizations to the BH solution found in dynamical quadratic gravity [25]. Treating GR corrections as small deformations (i.e. working in the small-coupling limit), Yunes and Stein [25] found that the unique non-spinning BH solution is g µν = ¯ g µν + h QG µν , where ¯ g µν is the Schwarzschild metric, while h QG µν in Schwarzschild coordinates is given by [25] \nh QG tt = -ζ QG 3 M 3 r 3 ( 1 + 26 M r + 66 5 M 2 r 2 + 96 5 M 3 r 3 -80 M 4 r 4 ) , h QG rr = -ζ QG f 2 M 2 r 2 ( 1 + M r + 52 3 M 2 r 2 + 2 M 3 r 3 + 16 5 M 4 r 4 -368 3 M 5 r 5 ) , (75) \nwhere M is the mass of the Schwarzschild BH and ζ QG is a dimensionless constant that depends on the couplings of the theory and is assumed to be small \nFor a spherically symmetry background, one can easily show that the Schwarzschild Killing tensor (i.e. one whose only non-vanishing components are ξ θθ = r 4 and ξ φφ = r 4 sin 2 θ ) satisfies the Killing equation, regardless of the \nfunctional form of the ( t, t ) and ( r, r ) components of the metric. This means that the perturbed vectors δk α and δl α must adjust so that the above is true. We find that the following vector components do just that: \nδk t = ζ QG M 2 r 4 2 fr 6 ( 1 + 8 3 M r + 14 M 2 r 2 + 128 5 M 3 r 3 + 48 M 4 r 4 ) , δk r = ζ QG M 2 r 4 fr 6 ( 1 + M r + 52 3 M 2 r 2 + 2 M 3 r 3 + 16 5 M 4 r 4 -368 3 M 5 r 5 ) , δl t = -ζ QG M 2 r 4 2 f 2 r 6 ( 1 + 4 3 M r + 26 M 2 r 2 + 32 5 M 3 r 3 + 48 5 M 4 r 4 -448 3 M 5 r 5 ) , (76) \nand all other components vanish. \nThat perturbed null vectors exist to reduce the Killing tensor to its Schwarzschild counterpart must surely be the case, as we have already shown in Sec. III B 1 and IVB1 that in the non-rotating limit both the BK and DK parameterizations allow for generic ( t, t ) and ( r, r ) deformations.', '3. Arbitrary perturbations: perturbation to γ 3 , perturbations to γ 1 and γ 4': 'We can understand the functions that parametrize the metric perturbation by considering perturbations where we allow only a few of the parametrizing functions to be nonzero. Consider a perturbation in the deformed Kerr parametrization of Eqs. (53) and (54) described by γ 1 = γ 4 = Θ 3 = 0 and h rθ = h θθ = 0. In the small a limit, the only nonzero component of the metric is the ( t, φ ) component: \nh tφ = -f r 2 sin 2 θ γ 3 ( r ) + C ) , (77) \n( \n) where C is a constant. Now consider a perturbation in the DK parametrization whose only nonzero parameters are γ 1 and γ 4 . In the small a limit, this produces perturbations to the metric given by \nh tt = fγ 1 ( r ) + 2 f 2 γ 4 ( r ) , h rr = -f -1 γ 1 ( r ) , h φφ = f 2 sin 2 θ [ r 4 2 M dγ 1 dr +2 r 2 γ 4 ( r ) + r 3 ( r -4 M ) 2 M dγ 4 dr ] . (78) \nWe then see that in the DK parameterization γ 3 controls modifications to frame-dragging (and thus the ergosphere), while γ 1 and γ 4 control modifications to the location of the event-horizon and the innermost-stable circular orbit.', 'VI. DISCUSSION': 'The detection of GWs from EMRIs should allow for the detailed mapping of the spacetime metric on which the SCOs move. Such a mapping in turn allows for null tests of GR, as one would in principle be able to constrain deviations of the background spacetime from the Kerr metric. To carry out such tests, however, one requires a parameterization of the metric tensor that can handle model-independent, non-GR deviations. This was the purpose of this paper. \nWe have constructed two such parameterizations: one as a generalization of the bumpy BH formalism (the BK scheme) and one as a generalized deformation of the Kerr metric (the DK scheme). The former promotes the nonvanishing components of the metric perturbation that contain bumps to arbitrary functions of radius and polar angle. The latter takes the most general stationary, axisymmetric line element without assuming Ricci-flatness and constructs a generic metric deformation from it. In both cases, the arbitrary functions introduced are constrained by requiring that there exists an approximate second-order Killing tensor. \nThese schemes differ from each other in the metric components that are assumed not to vanish. We have found a gauge transformation, however, that relates them. That is, we have shown that a generating vector exists such that the BK metric can be mapped to the DK one. We have also mapped both parameterizations to known analytical non-GR solutions, thus automatically finding their respective Killing tensors and Carter constants. \nThe perturbative nature of our approach puts some limitation on the generality of the deformation. All throughout we restricted the metric deformation to be a small deviation away from the GR solution. But due to the structure of the perturbation, there can be regions in spacetime where this deformation dominates over the GR background. For example, the full metric tensor might lose Lorentzian signature if the metric perturbation dominates over the Kerr background and it possesses a certain definite sign, i.e., the metric will no longer have det( g ) < 0. Clearly, such regions are unphysical, and the coupling constants that control the magnitude of the perturbation should be adjusted appropriately to ensure that if they exist, they are hidden inside the horizon. \nWe should emphasize that we were guided by two principles in the construction of the parameterized schemes proposed here. First, we wanted to ensure that the parameterizations would easily map to known analytic solutions. Second, we wished to retain a smooth Kerr limit, such that when the deformation parameters are taken to zero, the deformed metric goes smoothly to Kerr. This in turn guaranteed that certain properties of the Kerr background were retained, such as the existence of a horizon. We additionally required that the metric tensors allowed a second-order perturbative Killing tensor, so that there exists a perturbative Carter constant. This latter re- \nquirement is not strictly necessary to perform null tests, as one could build GWs from the evolution of the full second-order equations of motion without rewriting them in first-order form. \nAn interesting avenue of future research would be to investigate the Petrov type [42] of the BK and DK metrics. Brink has shown that GR spacetimes that admit a Carter constant must also be of Type D [33]. In alternative theories of gravity, however, a formal mathematical proof of the previous statement is lacking. We have here found Ricci-curved metrics that admit a Carter constant. One could now attempt to prove that the perturbative Carter existence conditions found in this paper also automatically imply that the spacetime is approximately Petrov Type D. This is a purely mathematical result that is beyond the scope of this paper. \nAn interesting physical question that could now be answered with either the DK or BK frameworks is whether LISA could place interesting constraints on non-GR Kerr deformations. To answer this question, one would have to evolve geodesics in either framework and construct \n- [1] B. F. Schutz, J. Centrella, C. Cutler, and S. A. Hughes (2009), 0903.0100.\n- [2] C. F. Sopuerta, GW Notes, Vol. 4, p. 3-47 4 , 3 (2010).\n- [3] P. Bender et al., LISA Pre-Phase A Report, Max-PlanckInstitut fur Quantenoptik, Garching (1998), mPQ 233.\n- [4] K. Danzmann and A. Rudiger, Class. Quant. Grav. 20 , S1 (2003).\n- [5] K. 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D78 , 102002 (2008), 0807.1179.\n- [33] J. Brink, Phys. Rev. D81 , 022001 (2010), 0911.1589.\n- [34] J. Brink, Phys. Rev. D81 , 022002 (2010), 0911.1595.\n- [35] J. Brink (2009), 0911.4161.\n- [36] C. W. Misner, K. Thorne, and J. A. Wheeler, Gravitation (W. H. Freeman & Co., San Francisco, 1973).\n- [37] E. Poisson, Living Rev. Relativity 7 , 6 (2004), gr-qc/0306052, URL http://www.livingreviews.org/lrr-2004-6 .\n- [38] J. M. Bardeen, W. H. Press, and S. A. Teukolsky, Astrophys. J. 178 , 347 (1972).\n- [39] Grtensorii , this is a package which runs within Maple but \ndistinct from packages distributed with Maple. It is distributed freely on the World-Wide-Web from the address: http://grtensor.org . \n- [40] G. C. Jones and J. E. Wang, Phys. Rev. D71 , 124019 (2005), hep-th/0506023.\n- [41] C. F. Sopuerta and N. Yunes, Phys. Rev. D 80 , 064006 (2009), 0904.4501.\n- [42] A. Z. Petrov, Einstein spaces (1969).", 'Appendix A: Polynomial Functions': 'We here provide explicit expressions for the polynomials appearing in the conditions that guarantee the existence of a Carter constant. In Eq. (30), the polynomials P DK i are given by \n- P BK 1 = r 6 +5 a 2 r 4 +2 a 4 r 2 2 a 2 cos 2 θ (2 r 4 + a 2 r 2 + a 4 ) -a 4 cos 4 θ ( r 2 a 2 ) , \n- \n- \n- -P BK 4 = r 3 ( r 2 ( r 2 M ) + a 2 (3 r -4 M )) 2 a 2 r 2 cos 2 θ ( r 2 -2 Mr a 2 ) -a 4 cos 4 θ (3 r 2 2 Mr + a 2 ) ,\n- -P BK 3 = r ( r 3 + a 2 ( r +4 M )) + a 2 cos 2 θ ( r 2 -4 Mr + a 2 ) , \n- \n- \n- \n- -P BK 2 = r 4 M + a 2 r 2 ( r +2 M ) -a 2 cos 2 θ ( r 2 ( r +2 M ) a 2 ( r -2 M )) -a 4 cos 4 θ ( r -M ) , \n- \n- \n- \n- \n----P BK 6 = -1 2 ∆ ( ρ 2 ∆ -2 Mrρ 2 +4 Mra 2 +4 Mr 3 ) sin 2 θ ρ 4 , P BK 7 = 1 2 a ∆ (sin ( θ )) 2 ( -ρ 2 +4 Mr ) ρ 4 , (A1) \n- \n- ---P BK 5 = r 3 ( r -2 M )( r 2 ( r +2 M ) + a 2 ( r +4 M )) -r 3 cos 2 θ ( r 4 -4 M 2 r 2 -a 2 ( r 2 -6 Mr +16 M 2 ) -2 a 4 ) -a 2 cos 4 θ (2 r 3 ( r 2 2 Mr +4 M 2 ) + a 2 r ( r 2 +4 Mr -4 M 2 ) a 4 ( r -2 M )) a 4 cos 6 θ ( r -2 M )∆ , \nP BK 8 ,n \n= \n- \n8 r 11 -4 r 7 ρ 4 +16 r 9 M 2 +2 r 9 ρ 2 -44 a 6 Mρ 2 ∆ -8 a 10 M -68 a 4 Mr 4 ρ 2 +128 a 4 M 2 r 3 ρ 2 -56 a 6 rM 2 ρ 2 \n-16 r 8 Ma 2 -16 r 7 a 2 M 2 -48 r 3 a 6 M 2 +16 r 4 Ma 6 -56 a 2 M 2 r 3 ρ 4 +64 a 2 M 2 r 5 ρ 2 +72 a 4 rM 2 ρ 4 +48 r 7 a 4 + 32 r 5 a 6 +8 r 3 a 8 +60 a 4 Mρ 4 ∆ -18 a 2 Mρ 6 ∆ -80 a 4 M 2 r 5 +8 a 8 M ∆+16 a 8 M 2 r +36 a 8 Mρ 2 -32 r 7 M 2 ρ 2 -4 r 8 Mρ 2 -12 r 3 a 4 ρ 4 -12 a 2 r 5 ρ 4 -4 Mr 6 ρ 4 -4 a 6 rρ 4 -44 a 6 Mρ 4 +12 Mr 4 ρ 6 -8 M 2 r 3 ρ 6 +24 M 2 r 5 ρ 4 + 2 r 3 a 2 ρ 6 +5 a 4 rρ 6 +12 a 4 Mρ 6 sin 2 θ , \n- ( -12 a 4 r 5 ρ 2 -16 r 3 a 6 ρ 2 -6 a 8 rρ 2 -36 a 2 Mr 6 ρ 2 +40 a 2 Mr 4 ρ 4 +32 r 9 a 2 -3 r 5 ρ 6 -8 r 10 M -20 a 2 rM 2 ρ 6\n- ) P BK 8 ,d = 4 r 6 +2 ρ 4 a 2 -4 a 2 ρ 2 ∆ -8 a 2 ρ 2 Mr +2 a 4 ρ 2 +8 r 4 a 2 -ρ 4 ∆ -2 ρ 4 Mr -2 r 4 ρ 2 +4 a 4 ∆+8 Mra 4 -4 a 6 ) ρ 4 ,\n- ( \n+ 56 a 2 Mr 6 +40 a 4 Mr 4 -24 r 7 a 2 -8 r 3 a 6 +8 a 6 M ∆+16 a 6 M 2 r -12 a 4 ρ 4 M +28 a 6 Mρ 2 -36 a 4 Mρ 2 ∆ + 12 a 2 ρ 4 M ∆ -48 a 4 M 2 r 3 +8 ra 2 M 2 ρ 4 -3 a 2 rρ 6 +16 r 7 M 2 +2 a 2 Mρ 6 +18 a 2 r 5 ρ 2 +6 r 7 ρ 2 +18 r 3 a 4 ρ 2 + 6 a 6 rρ 2 + r 3 ρ 6 +48 a 2 M 2 r 3 ρ 2 -40 a 4 M 2 rρ 2 -48 a 2 Mr 4 ρ 2 , \n- P BK 9 ,n = a sin 2 θ ( -16 M 2 r 5 ρ 2 +8 M 2 r 3 ρ 4 -4 Mr 4 ρ 4 -20 Mr 6 ρ 2 -32 a 2 M 2 r 5 +24 r 8 M -24 a 4 r 5 -8 a 8 M -8 r 9 \nP BK 10 ,n \n= sin 2 θ \n- ) P BK 9 ,d = ρ 4 ( 4 r 6 +2 ρ 4 a 2 -4 a 2 ρ 2 ∆ -8 a 2 ρ 2 Mr +2 a 4 ρ 2 +8 r 4 a 2 -ρ 4 ∆ -2 ρ 4 Mr -2 r 4 ρ 2 +4 a 4 ∆+8 Mra 4 -4 a 6 ) , \n-12 rMa 4 ρ 4 -48 a 2 r 8 +6 r 4 a 2 ρ 4 -48 a 4 r 6 +16 a 6 M 2 ∆+32 a 6 M 3 r +8 a 6 ρ 2 ∆ -8 a 2 M 2 ρ 6 -16 r 10 \n- \n- ( -16 M 2 r 8 +6 a 2 ρ 6 Mr +24 r 4 a 4 ρ 2 +32 r 9 M 32 a 2 ρ 4 Mr 3 -8 a 8 ρ 2 +8 a 6 M 2 ρ 2 +40 M 2 r 6 ρ 2 +16 rMa 6 ρ 2 \n+ 24 a 2 r 6 ρ 2 -6 Mr 3 ρ 6 +3 r 4 ρ 6 +2 r 6 ρ 4 +5 a 4 ρ 6 +64 r 7 Ma 2 +24 Mr 5 ρ 4 -24 M 2 r 4 ρ 4 -48 Mr 7 ρ 2 24 a 6 M 2 ρ 2 sin 2 θ 6 a 2 ρ 6 ∆+6 a 4 ρ 4 ∆ 4 a 6 ρ 4 , \n- ---32 r 5 Ma 2 ρ 2 +8 r 8 ρ 2 -16 a 8 M 2 +96 a 2 M 2 r 4 ρ 2 -16 r 4 a 6 +32 r 5 Ma 4 -80 r 6 a 2 M 2 -48 r 4 a 4 M 2 +16 r 3 Ma 4 ρ 2\n- ---) P BK 10 ,d = ρ 4 4 r 6 +2 ρ 4 a 2 -4 a 2 ρ 2 ∆ -8 a 2 ρ 2 Mr +2 a 4 ρ 2 +8 r 4 a 2 -ρ 4 ∆ -2 ρ 4 Mr -2 r 4 ρ 2 +4 a 4 ∆+8 Mra 4 -4 a 6 ) , \n) \n- ) P BK 11 ,n = -8 aM sin 2 θ ( r 3 ρ 4 -5 ρ 2 r 5 +4 r 7 -3 ra 2 ρ 4 +6 a 2 r 3 ρ 2 + a 2 ρ 4 M + a 2 ρ 2 M ∆+2 a 2 ρ 2 M 2 r -3 a 4 Mρ 2 -8 a 2 Mr 4 + 3 a 4 rρ 2 -2 Mr 6 -4 a 4 r 3 +3 Mr 4 ρ 2 +2 a 4 M ∆+4 a 4 M 2 r -2 a 6 M , \n( \n- P BK 11 ,d = ρ 4 ( 4 r 6 +2 ρ 4 a 2 -4 a 2 ρ 2 ∆ -8 a 2 ρ 2 Mr +2 a 4 ρ 2 +8 r 4 a 2 -ρ 4 ∆ -2 ρ 4 Mr -2 r 4 ρ 2 +4 a 4 ∆+8 Mra 4 -4 a 6 ) , \n( \n- ) P BK 12 ,d = ρ 2 4 r 6 +2 ρ 4 a 2 -4 a 2 ρ 2 ∆ -8 a 2 ρ 2 Mr +2 a 4 ρ 2 +8 r 4 a 2 -ρ 4 ∆ -2 ρ 4 Mr -2 r 4 ρ 2 +4 a 4 ∆+8 Mra 4 -4 a 6 ) ,\n- ( P BK 12 ,n = -2 r ( -28 a 2 ρ 2 ∆ -56 a 2 ρ 2 Mr +18 a 4 ρ 2 + ρ 6 +2 ρ 4 ∆+4 ρ 4 Mr +4 ρ 4 a 2 -18 r 4 ρ 2 +16 r 6 + 32 r 4 a 2 +16 a 4 ∆+32 Mra 4 16 a 6 ) , \n- \n- P BK 13 ,n = 2 a 2 (sin ( θ )) 2 ( 8 a 2 ρ 2 ∆+16 a 2 ρ 2 Mr -8 a 4 ρ 2 -ρ 6 -6 ρ 4 ∆ -12 ρ 4 Mr +8 ρ 4 a 2 +24 r 4 ρ 2 -16 r 6 -16 r 4 a 2 , \nP BK 14 \n= \n- ( P BK 13 ,d = ρ 4 ( 4 r 6 +2 ρ 4 a 2 -4 a 2 ρ 2 ∆ -8 a 2 ρ 2 Mr +2 a 4 ρ 2 +8 r 4 a 2 -ρ 4 ∆ -2 ρ 4 Mr -2 r 4 ρ 2 +4 a 4 ∆+8 Mra 4 -4 a 6 ) , \n) \n-4 aMr (sin ( θ )) 2 ( -4 r 4 + ρ 2 ∆+2 ρ 2 Mr -4 a 2 ρ 2 +4 r 2 a 2 ) ρ 4 (4 r 6 +2 ρ 4 a 2 -4 a 2 ρ 2 ∆ -8 a 2 ρ 2 Mr +2 a 4 ρ 2 +8 r 4 a 2 -ρ 4 ∆ -2 ρ 4 Mr -2 r 4 ρ 2 +4 a 4 ∆+8 Mra 4 -4 a 6 ) . \n- P DK 1 = r 2 ( r 4 +5 a 2 + r 2 +2 a 4 ) 2 a 2 cos 2 θ (2 r 4 + a 2 r 2 + a 4 ) -a 4 cos 4 θ ( r 2 a 2 ) ,\n- -P DK 2 = r 2 ( r 2 M + a 2 r +2 a 2 M ) -a 2 cos 2 θ ( r 2 ( r +2 M ) a 2 ( r -2 M )) a 4 cos 4 θ ( r -M ) , \n- \n- \n- \n- ---P DK 3 = r 3 ( r -2 M )( r 2 ( r +2 M ) + a 2 ( r +4 M )) + 2 r 3 a 2 cos 2 θ ( r 2 -2 Mr +4 M 2 + a 2 ) + a 4 cos 4 θ ( r 2 M )∆ , \n- \n- \n- P DK 4 = r 3 ( r 2 ( r -2 M ) + a 2 (3 r 4 M )) -2 r 2 a 2 cos 2 θ ( r 2 2 Mr -a 2 ) a 4 cos 4 θ (3 r 2 -2 Mr + a 2 ) , \n- \n- ----P DK 5 = r 4 ( r 6 +3 a 2 r 4 +8 a 4 r 2 +2 a 6 ) -3 a 4 r 4 cos 2 θ (3 r 2 -a 2 ) + a 4 cos 4 θ (5 r 6 -3 a 2 r 4 +6 a 4 r 2 +2 a 6 ) + a 6 cos 6 θ (2 r 4 3 a 2 r 2 -a 4 ) , \n- \n- \n- \n- \n- -P DK 6 = r 4 ( r 2 6 a 2 ) + 3 a 2 r 2 cos 2 θ (3 r 2 +4 a 2 ) -a 4 cos 2 θ (9 r 2 2 a 2 ) -a 6 cos 6 θ ,\n- --P DK 7 = r 6 +10 a 2 r 4 +6 a 4 r 2 a 2 cos 2 θ (11 r 4 +16 a 2 r 2 +10 a 4 ) + a 4 cos 4 θ (5 r 2 +6 a 2 ) + a 6 cos 6 θ ,\n- P DK 8 = r 2 (3 r 2 -a 2 ) a 2 cos 2 θ ( r 2 -3 a 2 ) , \n- \n- -P DK 10 = r 2 (3 r 2 -a 2 ) 3 cos 2 θ ( r 4 -a 4 ) + a 2 cos 4 θ ( r 2 3 a 2 ) , \n- \n- --P DK 9 = r 2 (3 r 4 +5 a 2 r 2 2 a 4 ) -2 a 2 cos 2 θ ( r 2 + a 2 )(2 r 2 3 a 2 ) + a 4 cos 4 θ ( r 2 -3 a 2 ) , \n- \n- \n- --P DK 11 = -r 4 (3 r 6 -2 Mr 5 +24 a 2 r 4 -18 a 2 Mr 3 -8 a 2 M 2 r 2 +19 a 4 r 2 -24 a 4 Mr +48 a 4 M 2 +6 a 6 ) + a 2 cos 2 θ (21 r 8 -18 Mr 7 -8 M 2 r 6 +18 a 2 r 6 -42 a 2 Mr 5 +72 a 2 M 2 r 4 +33 a 4 r 4 -32 a 4 Mr 3 +24 a 4 M 2 r 2 +12 a 6 r 2 -16 a 6 M 2 ) + a 4 cos 4 θ (11 r 6 +6 Mr 5 -24 M 2 r 4 -12 a 2 r 4 +22 a 2 Mr 3 -24 a 2 M 2 r 2 +3 a 4 r 2 -24 a 4 Mr +24 a 4 M 2 +2 a 6 ) + a 6 cos 6 θ (3 r 4 6 Mr 3 -6 a 2 r 2 +18 a 2 Mr 8 a 2 M 2 -a 4 ) , \n--- \n-- \n- ---P DK 12 = r 3 ( r 3 M +3 a 2 r 2 -6 a 2 Mr -a 4 ) -a 2 cos 2 θ (3 r 5 -3 Mr 4 -3 a 2 Mr 2 -3 a 4 r +2 a 4 M ) + a 4 cos 4 θ ( r 3 3 a 2 r + a 2 M ) ,\n- P DK 13 = r 3 ( r 6 +9 a 2 r 4 -10 a 2 Mr 3 +6 a 4 r 2 -8 a 4 Mr +2 a 6 ) \nP DK 14 \n= \nP DK 15 \n= \n- \n-cos 2 θr ( r 8 +18 a 2 r 6 -22 a 2 Mr 5 +15 a 4 r 4 -26 a 4 Mr 3 +20 a 6 r 2 -12 a 6 Mr +6 a 8 ) + a 2 cos 4 θ (9 r 7 -12 Mr 6 +6 a 2 r 5 -22 a 2 Mr 4 +27 a 4 r 3 -18 a 4 Mr 2 +6 a 6 r +4 a 6 M ) + a 4 cos 6 θ (3 r 5 +4 Mr 4 -10 a 2 r 3 +6 a 2 Mr 2 +3 a 4 r -6 a 4 M ) + a 6 cos 8 θ ( r 3 -3 a 2 r +2 a 2 M ) , r 3 (3 r 8 -4 Mr 7 +22 a 2 r 6 -30 a 2 Mr 5 +8 a 2 M 2 r 4 +29 a 4 r 4 -14 a 4 Mr 3 +12 a 6 r 2 -12 a 6 Mr -16 a 6 M 2 +2 a 8 ) -a 2 cos 2 θ (15 r 9 -24 Mr 8 +8 M 2 r 7 +10 a 2 Mr 6 +21 a 4 r 5 -34 a 4 Mr 4 -32 a 4 M 2 r 3 +24 a 6 r 3 +12 a 6 Mr 2 -32 a 6 M 2 r +6 a 8 r +8 a 8 M ) -a 4 cos 4 θ (13 r 7 -32 Mr 6 +6 a 2 r 5 +14 a 2 Mr 4 +8 a 2 M 2 r 3 -3 a 4 r 3 -42 a 4 Mr 2 +48 a 4 M 2 r +4 a 6 r -16 a 6 M ) -a 6 cos 6 θ (3 r 5 -12 Mr 4 +8 M 2 r 3 -2 a 2 r 3 +18 a 2 Mr 2 -16 a 2 M 2 r -5 a 4 r +6 a 4 M ) , r 3 ( r 6 -8 M 2 r 4 +9 a 2 r 4 -22 a 2 Mr 3 +6 a 4 r 2 -4 a 4 Mr +16 a 4 M 2 +2 a 6 ) -a 2 cos 2 θ (9 r 7 -28 Mr 6 +9 a 2 r 5 -18 a 2 Mr 4 +16 a 2 M 2 r 3 +16 a 2 M 2 r 3 +18 a 4 r 3 -36 a 4 Mr 2 +6 a 6 r -8 a 6 M ) -a 4 cos 4 θr (3 r 4 -4 Mr 3 + r 2 (8 M 2 -9 a 2 ) + 18 a 2 Mr -16 a 2 M 2 ) -a 6 cos 6 θ ( r 3 -3 a 2 r +2 a 2 M ) . (A2)', 'Appendix B: Equations of Motion in Alternative Theories': 'The derivation of the equations of motion in Sec. II A is fairly general. In particular, this derivation is independent of the metric used; in no place did we use that the spacetime was Kerr. We required the divergence of the test particle stress-energy tensor vanishes; this condition is always true in GR, because of a combination of local stress-energy conservation and the equivalence principle. \nOne might wonder whether the divergence-free condition of the stress-energy tensor holds in more general theories. In any metric GR deformation, the field equations will take the form \nG αβ + H αβ = T Mat αβ + T H αβ , (B1) \nwhere G αβ is the Einstein tensor, H αβ is a tensorial de- \nation of the Einstein equations, and T H αβ is a possible stress-energy modification, associated with additional fields. The divergence of this equation then leads to \n∇ α H αβ = ∇ α T Mat αβ + ∇ α T H αβ , (B2) \nsince the Bianchi identities force the divergence of the Einstein tensor to vanish. The Bianchi identities hold in alternative theories, as this is a geometric constraint and not one that derives from the action. We see then that the divergence of the matter stress-energy tensor vanishes independently provided \n∇ α H αβ = ∇ α T H αβ . (B3) \nWhether this condition [Eq. (B3)] is satisfied depends somewhat on the theory of interest. Theories that include additional degrees of freedom that couple both to \nthe geometry and have their own dynamics usually satisfy Eq. (B3). This is because additional equations of motion arise upon variation of the action with respect to these additional degrees of freedom, and these additional equations reduce to Eq. (B3). Such is the case, for example, in dynamical CS modified gravity [41]. If no additional degrees of freedom are introduced, the satisfaction of Eq. (B3) depends on whether the divergence of the new tensor H αβ vanishes, which need not in general be the case. \nRecently, it was shown that the equations of motion are geodesic to leading order in the mass-ratio for any classical field theory that satisfies the following constraints [16]: \n- · It derives from a diffeomorphism-covariant Lagrangian, ensuring a Bianchi identity;\n- · It leads to second-order field equations. \nThe second condition seems somewhat too stringent, as we know of examples where third-order field equations still lead to geodesic motion, i.e. dynamical CS gravity [41]. Therefore, it seems reasonable to assume that this condition could be relaxed in the future. Based on this, we take the viewpoint that the equations of motion are geodesic even in the class of alternative theories we consider here.'}

2010ApJ...714..713S

The NGC 404 Nucleus: Star Cluster and Possible Intermediate-mass Black Hole

2010-01-01

['galaxies elliptical lenticular;cd', 'cd', 'galaxies formation', 'galaxies', 'galaxies kinematics and dynamics', 'galaxies nuclei', 'galaxies structure', '-', '-']

[]

We examine the nuclear morphology, kinematics, and stellar populations in nearby S0 galaxy NGC 404 using a combination of adaptive optics assisted near-IR integral-field spectroscopy, optical spectroscopy, and Hubble Space Telescope imaging. These observations enable study of the NGC 404 nucleus at a level of detail possible only in the nearest galaxies. The surface brightness profile suggests the presence of three components: a bulge, a nuclear star cluster (NSC), and a central light excess within the cluster at radii &lt; 3 pc. These components have distinct kinematics with modest rotation seen in the NSC and counter-rotation seen in the central excess. Molecular hydrogen emission traces a disk with rotation nearly orthogonal to that of the stars. The stellar populations of the three components are also distinct, with half of the mass of the NSC having ages of ~1 Gyr (perhaps resulting from a galaxy merger), while the bulge is dominated by much older stars. Dynamical modeling of the stellar kinematics gives a total NSC mass of 1.1 × 10<SUP>7</SUP> M <SUB>sun</SUB>. Dynamical detection of a possible intermediate-mass black hole (BH) is hindered by uncertainties in the central stellar mass profile. Assuming a constant mass-to-light ratio, the stellar dynamical modeling suggests a BH mass of &lt;1 × 10<SUP>5</SUP> M <SUB>sun</SUB>, while the molecular hydrogen gas kinematics are best fitted by a BH with a mass of 4.5<SUP>+3.5</SUP> <SUB>-2.0</SUB> × 10<SUP>5</SUP> M <SUB>sun</SUB>. Unresolved and possibly variable dust emission in the near-infrared and active galactic nucleus-like molecular hydrogen emission-line ratios do suggest the presence of an accreting BH in this nearby LINER galaxy.

[]

https://arxiv.org/pdf/1003.0680.pdf

{'THE NGC 404 NUCLEUS: STAR CLUSTER AND POSSIBLE INTERMEDIATE MASS BLACK HOLE': 'Anil C. Seth 1,2 , Michele Cappellari 3 , Nadine Neumayer 4 , Nelson Caldwell 1 , Nate Bastian 5 , Knut Olsen 6 , Robert D. Blum 6 , Victor P. Debattista 7 , Richard McDermid 8 , Thomas Puzia 9 , Andrew Stephens 8 Accepted to ApJ, Mar. 1, 2010', 'ABSTRACT': 'We examine the nuclear morphology, kinematics, and stellar populations in nearby S0 galaxy NGC 404 using a combination of adaptive optics assisted near-IR integral-field spectroscopy, optical spectroscopy, and HST imaging. These observations enable study of the NGC 404 nucleus at a level of detail possible only in the nearest galaxies. The surface brightness profile suggests the presence of three components, a bulge, a nuclear star cluster, and a central light excess within the cluster at radii < 3 pc. These components have distinct kinematics with modest rotation seen in the nuclear star cluster and counter-rotation seen in the central excess. Molecular hydrogen emission traces a disk with rotation nearly orthogonal to that of the stars. The stellar populations of the three components are also distinct, with half of the mass of the nuclear star cluster having ages of ∼ 1 Gyr (perhaps resulting from a galaxy merger), while the bulge is dominated by much older stars. Dynamical modeling of the stellar kinematics gives a total nuclear star cluster mass of 1 . 1 × 10 7 M /circledot . Dynamical detection of a possible intermediate mass black hole is hindered by uncertainties in the central stellar mass profile. Assuming a constant mass-to-light ratio, the stellar dynamical modeling suggests a black hole mass of < 1 × 10 5 M /circledot , while the molecular hydrogen gas kinematics are best fit by a black hole with mass of 4 . 5 +3 . 5 -2 . 0 × 10 5 M /circledot . Unresolved and possibly variable dust emission in the near-infrared and AGN-like molecular hydrogen emission line ratios do suggest the presence of an accreting black hole in this nearby LINER galaxy. \nSubject headings: galaxies:nuclei - galaxies: elliptical and lenticular, cD - galaxies: kinematics and dynamics - galaxies: formation - galaxies: structure - galaxies: individual (NGC 404)', '1. INTRODUCTION': "The centers of galaxies contain both massive black holes (MBHs) and nuclear star clusters (NSCs). The presence of MBHs has been dynamically measured in about 50 galaxies and it appears that most massive galaxies probably have a MBH (e.g. Richstone et al. 1998; Graham 2008). Nuclear star clusters are compact ( r eff ∼ 5 pc), massive ( ∼ 10 7 M /circledot ) star clusters found at the center of a majority of spirals and lower mass ellipticals (Boker et al. 2002; Carollo et al. 2002; Cˆot'e et al. 2006). Unlike normal star clusters, they have multiple stellar populations with a wide range of ages (Long et al. 2002; Walcher et al. 2006; Rossa et al. 2006; Seth et al. 2006; Siegel et al. 2007). Nuclear star clusters coexist with MBHs in some galaxies (Filippenko & Ho 2003; Seth et al. 2008a; Shields et al. 2008; Graham & Spitler 2009). However, the nearby galaxies M33 and NGC 205 have NSCs but no apparent central black hole (Gebhardt et al. 2001; Valluri et al. 2005), while some high mass core elliptical galaxies have MBHs but lack NSCs (Cˆot'e et al. 2006). \n- 1 Harvard-Smithsonian Center for Astrophysics, 60 Garden Street Cambridge, MA 02138 2\n- 2 OIR Fellow, [email protected]\n- 3 Sub-department of Astrophysics, University of Oxford, \nDenys Wilkinson Building, Keble Road, Oxford OX1 3RH 4 \n- European Southern Observatory, Garching\n- European Southern Observatory, Garching\n- 5 Institute of Astronomy, Cambridge University\n- 6 National Optical Astronomy Observatory, Tucson\n- 7 Jeremiah Horrocks Institute, University of Central Lancashire\n- 8 Gemini Observatory, Hilo\n- 9 Herzberg Institute of Astrophysics, Victoria \nOccupation fractions and masses for MBHs in lower mass galaxies retain the imprint of the seed black holes (BHs) from which they form, information which has been lost due to subsequent accretion in higher mass galaxies (e.g. Volonteri et al. 2008). However, the presence and mass of MBHs in lower mass galaxies is very poorly constrained. Most galaxies are too far away to measure the dynamical effect of a BH with mass /lessorsimilar 10 6 M /circledot (often referred to as intermediate mass black holes; IMBHs) with current instrumentation. Thus the presence of IMBHs in galaxy centers has only been inferred when AGN activity is observed (e.g. Filippenko & Sargent 1989; Greene & Ho 2004; Satyapal et al. 2007). These AGN provide only a lower limit on the number of MBHs in lower mass galaxies, and the BH mass estimates from the AGN are quite uncertain. \nDynamical measurements of MBHs in nearby massive galaxies have revealed that the mass of a galaxy's central MBH is correlated with its bulge mass (Kormendy & Richstone 1995; Magorrian et al. 1998; Haring & Rix 2004). The scaling of BH mass with the large-scale properties of galaxies extends to many measurable quantities including the bulge velocity dispersion (Ferrarese et al. 2000; Gebhardt et al. 2000; Graham 2008; Gultekin et al. 2009). More recently, Ferrarese et al. (2006), Wehner & Harris (2006), and Rossa et al. (2006) have presented evidence that NSCs scale with bulge mass and dispersion in elliptical and early-type spiral galaxies in almost exactly the same way as MBHs. \nThe similarity between the NSC and MBH scaling relationships led Ferrarese et al. (2006) and Wehner & Harris (2006) to suggest that MBHs and NSCs are two differ- \nent types of central massive objects (CMO) both of which contain a small fraction ( ∼ 0.2%) of the total galaxy mass. This correlation of the CMO mass with the large scale properties of galaxies suggests a link between the formation of the two. However, the nature of this connection is unknown, as is the relation between NSCs and MBHs. Studies of NSCs can help address these issues. Their morphology, kinematics, and stellar populations contain important clues about their formation and the accretion of material into the center of galaxies (e.g. Hopkins & Quataert 2010). \nThe scaling relationships of large-scale galaxy properties with the mass of the CMO might indicate that galaxy centers should be simple systems. The Milky Way center clearly shows this not to be the case. It is an incredibly complicated environment with a black hole ( M BH = 4 × 10 6 M /circledot ; Ghez et al. 2008) and a nuclear star cluster ( M NSC ∼ 3 × 10 7 M /circledot Genzel et al. 1996; Schodel et al. 2007; Trippe et al. 2008) that contains stars of many ages and in different substructures including disks of young stars in the immediate vicinity of the black hole (e.g. Lu et al. 2009; Bartko et al. 2009). Only through understanding these complex structures in the Milky Way and finding other examples in nearby galaxies can we hope to fully understand MBH and NSC formation, and the links between these objects and their host galaxies. \nIn our current survey, we are looking at a sample of the nearest galaxies ( D < 5 Mpc) that host NSCs. We are resolving their properties using a wide range of observational data to understand how NSCs form and their relation to MBHs. In our first paper on nearby edge-on spiral NGC 4244 (Seth et al. 2008b), we showed evidence that the NSC kinematics are dominated by rotation, suggesting that it was formed by episodic accretion of material from the galaxy disk. \nThis paper focuses on the NSC and possible MBH in NGC 404, the nearest S0 galaxy. Table 1 summarizes its properties. The stellar populations of the galaxy are predominantly old and can be traced out to 600 '' (9 kpc) (Tikhonov et al. 2003; Williams et al. 2010). However, HI observations show a prominent nearly face-on HI disk at radii between 100-400 '' , with detectable HI out to 800 '' (del R'ıo et al. 2004). CO observations and optical color maps show the presence of molecular gas and dust within the central ∼ 20 '' (300 pc) of the galaxy (Wiklind & Henkel 1990; Tikhonov et al. 2003), lying primarily to the NE of the nucleus. A nuclear star cluster in the central arcsecond of NGC 404 was noted by Ravindranath et al. (2001) from an analysis of NICMOS data. \nThe presence of an AGN in NGC 404 is a controversial topic. The optical spectrum of the nucleus has line ratios with a LINER classification (Ho et al. 1997). Assuming a distance of ∼ 3 Mpc, NGC 404 is the nearest LINER galaxy; other nearby examples include M81 and NGC 4736 (M94). Recent studies have shown that most LINERS do in fact appear to be AGN, with a majority of them having detected X-ray cores (Dudik et al. 2005; Gonz'alez-Mart'ın et al. 2006; Zhang et al. 2009), radio cores (Nagar et al. 2005), and many of them possesing mid-IR coronal lines (Satyapal et al. 2004) and UV variable cores (Maoz et al. 2005). However, NGC 404 is quite unusual in its properties. No radio core is ob- \nTABLE 1 NGC 404 Properties \n| Distance a | 3.06 Mpc |\n|--------------------------|-------------------------|\n| m - M | 27.43 |\n| pc/ '' | 14.8 |\n| Galaxy M V, 0 , M I, 0 b | -17.35, -18.36 |\n| Bulge/Total Luminosity c | 0.76 |\n| Bulge M V, 0 , M I, 0 d | -17.05, -18.06 |\n| Bulge Mass e | 9.2 × 10 8 M /circledot |\n| HI Gas Mass f | 1.5 × 10 8 M /circledot |\n| Molecular Gas Mass g | 6 × 10 6 M /circledot |\n| Central Velocity h | -58.9 km s - 1 | \nNote . - ( a ) TRGB measurement from Karachentsev et al. (2004), ( b ) at r < 200 '' from Tikhonov et al. (2003) corrected for foreground reddening, ( c ) for r < 200 '' from Baggett et al. (1998), ( d ) combining above values; we derive a slightly brighter bulge M I, 0 = -18 . 20 in § 3, ( e ) using M/L I = 1 . 28 from § 5.4, ( f ) del R'ıo et al. (2004), ( g ) Wiklind & Henkel (1990), corrected to D = 3 . 06 Mpc, ( h ) heliocentric velocity, see § 4.1. \nserved down to a limiting flux of 1.3 mJy at 15 GHz with the VLA in A array (Nagar et al. 2005), however, del R'ıo et al. (2004) do detect an unresolved 3 mJy continuum source at 1.4 GHz using the C array. A compact X-ray source is detected (Lira et al. 2000; Eracleous et al. 2002), but its low luminosity and soft thermal spectrum indicates that it could be the result of a starburst. Signatures of O stars are seen in the UV spectrum of the nucleus, however dilution of these lines suggests that ∼ 60% of the UV flux could result from a non-thermal source (Maoz et al. 1998). This suggestion is supported by more recent UV observations which show that the UV emission is variable, declining by a factor of ∼ 3 between 1993 and 2002 (Maoz et al. 2005). HST observations of H α show that the emission occurs primarily in a compact source 0.16 '' north of the nucleus and in wispy structures suggestive of supernova remnants (Pogge et al. 2000). The [O III ] emission has a double lobed structure along the galaxy major-axis (Plana et al. 1998), and has a higher velocity dispersion than H α near the galaxy center (Bouchard et al. 2010). The mid-IR spectrum of NGC 404 shows evidence for high excitation consistent with other AGN (Satyapal et al. 2004). In particular the ratio of the [O IV ] flux relative to other emission lines ([Ne II ], [Si II ]) is higher than any other LINERs in the Satyapal et al. (2004) sample and is similar to other known AGN. However, [NeV] lines, which are a more reliable indicator of AGN activity (Abel & Satyapal 2008), are not detected. In summary, the case for an accreting MBH in NGC 404 remains ambiguous, with the variable UV emission providing the strongest evidence in favor of its existence. \nIn this paper we take a detailed look at the central arcseconds of NGC 404 and find evidence that it contains both a massive nuclear star cluster and a black hole. In § 2 we describe the data used in this paper including adaptive optics Gemini/NIFS observations. We then use this data to determine the morphology in § 3, the stellar and gas kinematics in § 4, and the stellar populations in § 5. In § 6 we present dynamical modeling from which we derive a NSC mass of 1 . 1 × 10 7 M /circledot and find mixed results on detecting a possible MBH with mass < 10 6 M /circledot . In § 7 we discuss these results, concluding in § 8.", '2. OBSERVATIONS & REDUCTION': 'This paper presents a combination of a wide variety of data including: (1) high spatial resolution near-infrared IFU spectroscopy with Gemini NIFS, (2) optical longslit spectra from the MMT, and (3) HST imaging data from the UV through the NIR. Using these data we have determined the morphology, gas and stellar kinematics, and stellar populations of the central regions of NGC404. In this section we describe these data and their reduction.', '2.1. Gemini NIFS data': "Observations of NGC 404 with Gemini's Near-Infrared Integral Field Spectrometer (NIFS) are part of an ongoing survey of nearby nuclear star clusters. The data were taken using the Altair laser-guide star (LGS) system on the Gemini North telescope on the nights of 21st and 22nd September, 2008. The nucleus itself was used as a tip-tilt star. A total of twelve 760 second exposures were obtained; half were dithered on-source exposures while the others were sky exposures offset by 131' to the northwest taken after each on-source exposure. A telluric calibrator, HIP 1123, an A1 star, was observed both nights at an airmass similar to the observations. \nThe data were reduced using the Gemini v. 1.9 IRAF package, utilizing pipelines based on the NIFSEXAMPLES scripts. Ronchi mask and arc lamp images were used to determine the spatial and spectral geometry of the images. The sky images were subtracted from their neighboring on-source images, and then these images were flat-fielded, had bad pixels removed, and were split apart into long-slit slices. The spectra were then divided by the telluric calibrator (using NFTELLURIC) after correcting for the Br γ absorption in the calibrator. We made minor alterations to the Gemini pipeline to enable propogation of the variance spectrum. This also required creation of our own IDL version of the NIFCUBE to make the final data cubes. In this process the initial pixel size of the data, 0.103 '' in the vertical direction and 0.045 '' in the horizontal direction, was rebinned to 0.05 '' × 0.05 '' pixels in the final data cube for each on-source exposure. Data cubes for the six dithered exposures were then shifted spatially and combined with cosmic-ray rejection using our own IDL program. Data from the 2nd night was spectrally shifted to the wavelength solution of the first night before combining. The heliocentric correction was nearly identical on both nights (+13.25 km s -1 ). The NIFS spectrum within a 1 '' aperture is shown in Fig. 1 and shows strong absorption and emission lines. \nWe also processed the sky frames in a manner identical to the on-source exposures. We used the resulting data to calibrate the spectral resolution of our observations. Using thirteen isolated, strong, doublet sky lines, we measured the spectral resolution of each pixel in our final data cube. We found significant variations in the spectral resolution across the detector, with a median resolution of 4.36 ˚ A FWHM ( λ/ ∆ λ = 5275), and values ranging from 3.9 to 5.1 ˚ A FWHM. \nA K -band image was made by collapsing all channels after multiplying by (1) a 9500K black-body to correct for the shape of the telluric spectrum, and (2) the 2MASS response curve. Flux calibration was obtained from 2MASS by deriving a zeropoint from im- \nages of both sources and calibrators in our sample. We flux calibrated our spectra using synthetic 2MASS photometry of our telluric calibrator HIP 1123, assuming its spectral shape resembles that of a 9500K black body. From this flux calibration we obtained a flux calibrated spectrum of our NGC404 NIFS data cube with units of ergs s -1 cm -2 ˚ A -1 . The flux calibration of the image and spectrum is good to ∼ 10%.", '2.1.1. PSF Determination': "Understanding the point spread function (PSF) is important for interpreting the kinematic observations presented in § 4, and is even more critical for the dynamical modeling in § 6. We used two methods for measuring the PSF of the calibrator: (1) we created images from the telluric calibrator for both nights to estimate the PSF, and (2) we compared the final NIFS K -band images of the NGC404 nucleus to HST F814W data. \nWe initially tried modeling the PSF as a double Gaussian (as in Krajnovi'c et al. 2009). However, double Gaussians were a very poor fit to the wings of the PSF in the telluric calibrator images. Using an inner Gaussian + outer Moffat profile (Σ( r ) = Σ 0 / [(1 + ( r/r d ) 2 ] 4 . 765 ) adequately described the telluric calibrator PSF. The best fit on both nights was a PSF with an inner Gaussian FWHM of 0 . '' 12, and an outer Moffat profile r d of 0 . '' 95, each containing about half the light. The two nights PSFs were very similar ( < 10% differences) in both shape and fraction of light in each component. Unfortunately, our telluric images were not well-dithered and thus the central PSF is not well sampled. \nWe further refined the PSF by fitting a convolved version of the F814W band HST image to the NIFS K -band image. Although HST/NIC2 and NIC3 F160W data were also available, the larger pixel scale and the limited resolution of HST in the NIR (FWHM ∼ 0.14 '' ) make the F814W data a better image to use as a PSF reference. Due to the dust around the nucleus, we fit only the western, apparently dust-free, half of the nucleus (Fig. 2). Also, we find in § 3.1 that the NIFS image contains compact non-stellar emission from hot dust at the center; we therefore correct the K -band image for this dust correction before determination of the PSF. Finally, because the profile of the galaxy is much shallower than the PSF, we fix the functional form of the outer profile to that derived for the telluric star, but allow the inner PSF width and fraction of light in each component to vary. The best-fitting PSF has a core with FWHM ∼ 0 . '' 09 (assuming an HST resolution of 0 . '' 065 based on TinyTim models; Krist (1995)) containing 51% of the light, with the outer Moffat (with r d = 0 . '' 95) function containing 49% of the light. \nThis PSF suffers from a number of uncertainties. Most notably, the presence of a bluer population at the center of the cluster implied by Fig. 2 may make the F814W image more centrally concentrated relative to the dustcorrected K -band image. Due to our fitting the NIFS PSF core to a convolution of the F814W image, this effect could lead to an over-estimate in the core FWHM.", '2.1.2. Stellar Kinematics Determination': "Derivation of the stellar kinematics was done as described in Seth et al. (2008b) with a couple of additions. \nFig. 1.Left Optical spectrum of the NGC 404 nuclear star cluster (1 '' × 2.3 '' aperture) taken with the blue channel spectrograph on the MMT. Right K -band spectrum of the NGC 404 nuclear star cluster using a 1 '' radius aperture from our Gemini/NIFS data. Emission lines of H 2 are labeled above the spectrum (see § 3.2), while strong absorption features are labeled below the spectrum (including Br γ which is seen weakly in absorption and emission, see Fig. 6). \n<!-- image --> \nIn brief, we first spatially binned the data to a target signal-to-noise (S/N) of 25 using the Voronoi tesselation method described in Cappellari & Copin (2003). The S/N of the data was as high as 175 in the center, and data within ∼ 0 . '' 7 was left unbinned. Next, we determined the stellar line-of-sight velocity distribution (LOSVD) using the penalized pixel-fitting of Cappellari & Emsellem (2004) to derive the velocity, dispersion, h 3, and h 4 components in the wavelength region from 22850 ˚ A to 23900 ˚ A. We used high-resolution templates from Wallace & Hinkle (1996) which has supergiant, giant and main sequence stars between spectral types G and M; we used eight of these stars with the most complete wavelength coverage. These templates were convolved to the instrumental resolution measured in each pixel from the sky lines before fitting the LOSVD. Nearly identical kinematic results and fit qualities were obtained using the larger GNIRS library (Winge et al. 2008) as templates. We performed LOSVD fits over different ranges of wavelengths and with different subsets of templates and achieved consistent results, suggesting template mismatch is not a significant issue. The higher spectral resolution and more accurate velocity zeropoints made the Wallace & Hinkle (1996) better than the Winge et al. (2008) templates for deriving the kinematics. Errors on the LOSVD were calculated by taking the standard deviation of values derived from 100 Monte Carlo simulations in which appropriate levels of Gaussian random noise was added to each spectral pixel before remeasuring the LOSVD. Errors on the radial velocities ranged from 0.5 to 6 km s -1 depending primarily on the S/N.", '2.2. Optical Spectroscopy': "Optical spectroscopy of the NGC 404 cluster was obtained (along with spectra of other nearby NSCs) with the blue channel spectrograph (BCS) on the MMT 6.5m telescope on the night of 19 Jan 2009. We used the 500 \nline grating and a 1 arcsecond slit to obtain a resolution of ∼ 3.6 ˚ A FWHM between 3680 ˚ Aand 6815 ˚ A with a pixel size of 0 . '' 3 spatially and 1.17 ˚ A spectrally. Two 1200s exposures were taken, with one being affected by clouds. Due to variations in the slit width, flat-fielding of the data was accomplished by carefully constructing a composite flat field using the pattern from quartz flats taken just after the exposure with pixel-to-pixel information drawn from flats throughout the night. Wavelength calibration was done using arcs taken just after the science exposures. Spectral extraction was accomplished using the standard IRAF DOSLIT routine using optimal extraction over the central 2 . '' 3 of the slit while sky regions were selected to avoid regions of emission near the nucleus. Guide camera seeing was around 0 . '' 8, however within the instrument it appeared to be significantly larger ( ∼ 1 . '' 7). Three flux calibrators, Gliese 191B2B, HZ44, and VMa2 (Bessell 1999) were observed on the same night with the same observing set up. Flux calibration was achieved by combining information from all three flux calibrators using SENSFUNC . Final residuals between the fitted sensitivity function and the 3 calibrators were < 0.03 mags between 3680 ˚ Aand 6200 ˚ A. We note that 2nd order contamination was present beyond 6200 ˚ A. One of the NGC 404 exposures and 2/3 calibrator exposures were affected by clouds. However, by scaling to the clear exposures, the absolute calibration of the spectra appears to be quite good; comparison of emission line fluxes to those in Ho et al. (1997) shows agreement to within 5% through a similar size aperture. The S/N per pixel of the final combined spectrum ranged from 100 at 3700 ˚ A to 250 around 5000 ˚ A. The extracted nuclear spectrum is shown in Fig. 1.", '2.3. HST data': 'A wide variety of HST data are available for the nucleus of NGC 404 taken with the FOS, WFPC2, NIC- \nFig. 2.The HST WFPC2 F 547 M -F 814 W (approximately V -I ) color map of the nuclear region. Contours show the NIFS continuum emission at µ K of 11.2, 12.2, 13.2, 14.2 and 15.2 mag/arcsec 2 . The image is centered on the dust-corrected peak of the continuum light with North up and East to the left. Redder regions result from dust extinction. The western half of the nucleus appears to have little internal extinction (see § 5.2). \n<!-- image --> \nMOS and ACS HRC and WFC. In this paper we will use WFPC2 data in the F547M, F656N (PID: 6871), F555W, and F814W (PID: 5999) filters, ACS WFC data in F814W filter (PID: 9293) and NICMOS NIC2 and NIC3 data in the F160W filter (PID: 7330 & 7919). Data were downloaded from the HST archive and the Hubble Legacy Archive (HLA). The WFPC2 data all had the nucleus in the PC chip and were unsaturated, while the ACS F814W data is saturated at the center but covers the galaxy to larger radii than the WFPC2 data. Astrometric alignment of this data and the NIFS data was tied to the F814W WFPC2 data. The centroided position of the nucleus was used to align each image to the F814W data. Although dust clearly affects the area around the nucleus, the F547M-F814W color map (Fig. 2) and the alignment of sources in the NIFS Br γ map with the HST H α image (see § 3.2) suggests that the very center of the galaxy does not suffer from significant internal dust extinction.', '3. MORPHOLOGY': "To determine the extent and luminosity of the nuclear star cluster in NGC 404, it is necessary to fit the relatively steep surface brightness profile of the inner part of the galaxy. Using S'ersic fits to I and H/K -band data, we find evidence that a nuclear star cluster dominates the central 1 '' (15 pc) of the galaxy, and that an additional emission component appears to be present within the central few parsecs (see Fig. 3). \nBefore describing our results on the central parts of the galaxy, we summarize previous studies of the large scale light-profile of NGC 404. Using ground-based V -band data out to nearly 300 '' , Baggett et al. (1998) decompose the SB profile of NGC404 into a r 1 / 4 bulge and exponential disk. They find a bulge effective radius of 64 '' and a disk with scale length of 130 '' . The bulge dominates the light out to ∼ 150 '' and the bulge-to-total ratio is ∼ 0.7. Tikhonov et al. (2003) found that the number counts of red giant branch stars between 100-500 '' (1.5-7.5 kpc) are also well explained by an exponential disk with a similar, \n106 '' scale length. \nWe used archival HST data, our NIFS K -band image, and the 2MASS Large Galaxy Atlas (Jarrett et al. 2003) to construct the two surface brightness profiles shown in Fig. 3. Surface brightness measurements were obtained on all images using the ellipse program in IRAF. The F814W-band ( I ) profile combines HST WFPC2 data from the PC chip at radii below 5 '' with ACS/WFC data at larger radii (the ACS image was saturated at smaller radii). Diffraction spikes from β And (7' to the SE) and bright foreground stars and background galaxies were masked. Dust is obviously present near the center and this was masked out as well as possible using a F 555 W -F 814 W color map at radii between 0 . '' 3 and 5 '' from the center. Comparing this profile with the NIR light profile suggests that some dust extinction remains within the inner 1 '' . The NIR profile combines the NIFS K -band image (at r ≤ 1 '' ), with the F160W-band ( H ) NICMOS/NIC3 image (1 < r ≤ 5 '' ) and H band data from the 2MASS Large Galaxy Atlas (at r > 5 '' ). The NIFS data was scaled to match the H -band zeropoint; no scaling was needed to match the 2MASS and NICMOS data. The final F814W and NIR surface brightness profiles are shown in Fig. 3. \n∼ \nWe first fit the outer part of both profiles to a S'ersic profile (see Graham & Driver 2005, for a complete discussion of S'ersic profiles). During our fitting, the S'ersic profiles were convolved with the PSF. The 2MASS data was only usable to 70 '' due to low S/N, and even within this radius the sky subtraction is suspect due to nearby β And. The F814W data is more reliable at these larger radii, however we fit only the measurements within 80 '' due to uncertainty in the sky background and possible contribution from a disk at larger radii. At radii beyond 5 '' , the profiles are well fit by similar S'ersic profiles with n ∼ 2 . 5 and effective (half-light) radius of ∼ 43 '' (640 pc). Exact values for all fits are given in Table 2. We identify this outer component as the bulge given the results of Baggett et al. (1998), who show the bulge component dominates at r /lessorsimilar 150 '' . It was not possible to fit either profile down to smaller radii with a single S'ersic. \nFollowing Graham & Guzm'an (2003); Cˆot'e et al. (2006), we identify the excess light over the bulge at small radii to be a NSC. We discuss the nature of this component more in § 7.1. In both profiles, the NSC component is brighter than the bulge at r < 1 '' . We fit this component with an additional S'ersic component (following Graham & Spitler 2009) and find it has an effective radius of ∼ 0 . '' 7 (10 pc). The NSC S'ersic n value is quite sensitive to the functional form of the PSF, therefore we prefer the F814W value of n =1.99 due to the betterknown PSF. These parameters are in the ranges typical of NSCs (Cˆot'e et al. 2006; Graham & Spitler 2009) in other galaxies. The absolute magnitude of this component (based on the model) is M I , M H = -13 . 64 , -15 . 34 within the central 3 '' , and M I , M H = -13 . 17 , -14 . 93 within 1 '' . \nIn the inner 0 . '' 2, there is a clear excess luminosity above the double-S'ersic fit in both profiles. Adding up the light in excess of the profile we find an M I , M H = -11 . 81 , -12 . 61. Note that these magnitudes are quite sensitive to the PSF and S'ersic n indices of the NSC fit. We will show below that in the NIR, a fraction of this light appears to be emission by hot 950 K dust. A \n<!-- image --> \nFig. 3.Surface brightness profiles of NGC 404 created from HST, Gemini/NIFS, and 2MASS data. In both panels, the observed surface brightness profiles are plotted with symbols, the solid line shows the best-fit two-S'ersic model, the dashed line the outer (bulge) component, and the dot-dashed line the inner (NSC) component. The PSF is shown as a dotted red line in both plots. Residuals disappear off the plot at small radii to the unmodeled extra-light component. Left the F814W profile constructed from both WFPC2/PC (diamonds) and ACS/WFC (squares) data. This data is the most reliable data available at large radii. Right the NIR profile constructed from three images: NIFS K -band (diamonds), HST NICMOS/NIC3 (triangles) and 2MASS large galaxy atlas data (squares Jarrett et al. 2003). Blue diamonds show a correction for hot dust emission discussed in § 3.1. \n<!-- image --> \ncorrection for this dust emission is shown with blue diamonds in Fig. 3. However, dust alone cannot explain the luminosity of this excess light, accounting for only ∼ 35% of the excess in the NIR profile. Furthermore, at I -band, no significant dust emission would be expected at all. Two additional clues paint a complicated picture. First, the color in the central 0 . '' 2 is somewhat bluer in the HST F 547 M -F 814 W images (Fig. 2) and has compact UV emission (Maoz et al. 2005), both suggestive of the presence of young stars. Second, the stars in this area appear to be counter-rotating relative to the NSC and galaxy (see § 4.1). Thus the central excess appears to result from a combination of hot dust emission, younger stars, and perhaps AGN continuum emission in the optical and UV. \nIn summary, we find evidence for 3 components at the center of NGC 404. \n- 1. A bulge with S'ersic n ∼ 2 . 5 and effective radius of 640 pc that dominates the light between 1 '' and 80 '' (and beyond).\n- 2. A nuclear star cluster which dominates the light in the central arcsecond (15 pc).\n- 3. A central excess at r < 0 . '' 2. This excess appears to result from the combination of hot dust, young stars, and perhaps AGN continuum emission. \nWithin the central 1 '' in the I -band (roughly the area included in the MMT spectrum analyzed in § 5), ∼ 55% of the light comes from the NSC, 29% from the underlying bulge, and 16% from the central excess. \nThe isophotes in the ACS F814W data at 10-80 '' have a position angle of ∼ 80 degrees and an ellipticity of ∼ 0.08. This matches well with the morphology of the HI disk on a large scale; del R'ıo et al. (2004) finds an HI inclination of ∼ 10 degrees and a position angle of about ∼ 80 degrees \nTABLE 2 Morphological Fits \n| Component | S'ersic n | F814W Profile r eff [ '' ] | µ eff,F 814 W [mag/ '' | M I |\n|----------------|----------------|------------------------------|--------------------------|--------|\n| Bulge | 2.43 | 41.5 | 20.18 | -18.09 |\n| NSC | 1.99 | 0.74 | 16.02 | -13.64 |\n| Central Excess | Central Excess | | NIR Profile | -11.81 | \n| Component | S'ersic n | r eff [ '' | µ eff,H [mag/ '' | M H |\n|---------------------------------|---------------------------------|---------------------------------|--------------------|--------|\n| Bulge | 2.71 | 44.7 | 18.82 | -19.62 |\n| NSC | 2.61 | 0.68 | 14.15 | -15.34 |\n| Central Excess | Central Excess | | | -12.61 |\n| Central Excess (Dust Corrected) | Central Excess (Dust Corrected) | Central Excess (Dust Corrected) | | -12.14 | \nNote . - Bulge magnitudes given for r ≤ 80 '' , and NSC magnitudes for r ≤ 3 '' . Central excess magnitudes are for the total flux above the fit in the central arcsecond. Distance modulus is assumed to be 27.43 giving 1 '' =14.8 pc. Note that the magnitudes are not corrected for foreground extinction ( A I = 0 . 11, A H = 0 . 03; Schlegel et al. 1998). \nfor radii between 100-300 '' . Due to dust absorption, the clearest picture of the inner arcsecond of NGC 404 comes from the NIFS K -band image. As can be seen from the contours in Fig. 2, the position angle at r < 0 . '' 25 has a PA of 60-70 degrees. The PA then abruptly changes to 25 degrees at r = 0 . '' 3 and gradually increases to the large scale PA of 80 degrees by ∼ 1 '' . The ellipticity decreases from ∼ 0.2 near the center to 0.07 at 1 '' . These changes fit well with our picture of three distinct components, each having their own distinct morphology.", '3.1. A Hot Dust Component': "Within the central 0.2', the NIFS spectra show a significant spectral flattening and a reduction in the depths of the CO lines by about 10%. A comparison of the central spectrum to single stellar populations models from Maraston (2005) shows that its spectral slope is flatter/redder than any stellar population models. Further- \nFig. 5.The surface brightness profile of the NICMOS F160W observations (NIC2 in red, NIC3 in orange), 2MASS H -band (dashed), and NIFS K -band observations scaled to match the F160W NICMOS observations at r > 1 '' . The dashed line shows the NIFS observations after correction for dust emission. \n<!-- image --> \nFig. 4.Fit of the central NIFS spectrum of NGC 404 to a model including emission from hot dust. The black line shows the central spectrum, with the overplotted line showing the best fit. This fit consists of a template stellar spectrum taken from an annulus of radius 0 . '' 5-1 . '' 0, plus a black-body of temperature 950 K (shown in gray). Note that emission lines were excluded from the fit and plot. \n<!-- image --> \nmore, the younger stellar populations expected in the central excess should have a somewhat bluer, not redder, continuum. This redder continuum suggests two possibilities (1) strong dust absorption or (2) emission from hot dust. \nWe rule out dust absorption as a possibility because the required dust absorption to flatten the spectrum is A V ∼ 6 magnitudes assuming the reddening law from Cardelli et al. (1989). The F 547 M -F 814 W shows that the nucleus is actually bluer than the surrounding areas ( F 547 M -F 814 W ∼ 0.8). Furthermore, the area to the west of the nucleus appears unreddened with F 547 W -F 814 W ∼ 0 . 95, consistent with the expected color of the stellar population based on spectroscopic fits (see § 5.2), while the eastern half of the nucleus is at most 0.8 magnitudes redder in F 547 M -F 814 W (approximately V -I ). This suggests a maximum extinction in the region around the nucleus of A V 2. \nTherefore, hot dust appears to be the cause of the flattened continuum in the central pixels. Hot dust is a common feature in Seyfert nuclei at NIR wavelengths, and is typically found to have temperatures of 800-1300 K (Alonso-Herrero et al. 1996; Winge et al. 2000; Riffel et al. 2009). Studies of Seyfert galaxies with NICMOS by Quillen et al. (2000, 2001) show that the hot-dust component is frequently unresolved and variable on months to year timescales. The presence of hot dust components in LINER galaxies appears to be unstudied. \n∼ \nWe fit the contribution and temperature of this hot dust emission by assuming that the central spectrum is made up of a nuclear star cluster spectrum drawn from an annulus of 0 . '' 5-1 . '' 0 plus a pure blackbody spectrum. As shown in Fig. 4, the central spectrum ( r ≤ 0 . '' 05) is best fit by a dust temperature of 950 K, contributing about 20% of the light. The residuals between the central spectra and model have a standard deviation of 1.4% across the band and the fit completely eliminates the difference in spectral slopes between the annular and central spectrum. \nFitting spectrum in radial annuli, we find no evidence for dust emission beyond 0 . '' 3 from the nucleus. Assuming a uniform dust temperature of 950 K, we calculated the contribution of dust in each individual spaxel. Using \nthis, we correct the K -band image for the hot dust contribution. We use the dust-corrected K -band image and central position for all our subsequent analysis. The dust emission accounts for only ∼ 35% of the central excess in the NIR, as is shown with the blue points in right panel of Fig. 3. \nThe dust emission is slightly offset to the NE of the photocenter. This offset is in the direction of the dust absorption clearly seen on larger scales to the E of the cluster in the HST color image (Fig. 2). The corrected image has a photocenter offset by about a half-pixel from the uncorrected image, while the dust emission is offset by ∼ 1.5 pixels from the corrected photocenter. The small shift in the photocenter reduces the asymmetry in the velocity field, moving the photocenter to the center of the central counter-rotation (see § 4.1). As found in other AGN, the hot dust component is unresolved, with a measured FWHM of 0 . '' 14 (single-Gaussian PSF fits give a FWHM of ∼ 0 . '' 18 for our NIFS data). \n∼ The total dust emission in K band has a magnitude of K ∼ 15 . 4 which translates to a flux of 1 . 3 × 10 -13 ergs s -1 cm -2 and luminosity of 1 . 4 × 10 38 ergs s -1 . Assuming the hot dust emission is the result of accretion onto a BH of ∼ 10 5 M /circledot ( § 6), this suggests L bol /L Edd /greaterorsimilar 10 -5 ; typical LINERS have L bol /L Edd ∼ 3 × 10 -5 (Ho 2004). The hot dust luminosity is somewhat larger than the [O III ] luminosity (assuming D=3.06 Mpc) of 2 × 10 37 ergs s -1 (Ho et al. 1997) and the possible hard X-ray component with 2-10 keV luminosity of ∼ 3 × 10 37 ergs s -1 noted by Terashima et al. (2002). The 1.6 µ m dust emission, [O III ], and hard X-ray luminosities of Seyfert galaxies are discussed in detail by Quillen et al. (2001). Although the luminosities presented here are much lower than those in their sample (as would be expected for a system with a lower accretion rate), the relative luminosities in these three bands fall within the scatter of their data suggesting a comparable SED. \nWe also find tentative evidence for variability in the dust emission between our NIFS images taken in 2008 and HST NICMOS images taken in 1998. This variability may be expected given the variability in the nuclear UV flux reported by Maoz et al. (2005). They find that the flux at 2500 ˚ A decreasing by a factor of ∼ 3 between 1994 and 2002. However, this comparison is based on \nFigure 5 shows a comparison of the surface brightness profiles of all the NIR data, with the NIFS K -band image scaled to fit the SB profile of the NICMOS F160W images at r > 1 '' . Within the central 0 . '' 2, the scaled NIFS luminosity is clearly fainter than the NIC2 image. This change in brightness could be explained by a bluer stellar population within the central excess, but the effect is much larger than expected - the F 547 M -F 814 W image shows a change of ∆( F 547 M -F 814 W ) ∼ 0 . 1 mags between the center and the NSC at larger radii, implying a ∆( H -K ) < 0 . 1. However, the observed ∆( H -K ) = 0 . 4 after attempting to correct both the NIC2 image and NIFS dust-subtracted image for the effects of the PSF. Using the original NIFS image (without dust subtraction), we still get ∆( H -K ) = 0 . 3. We suggest that this large ∆( H -K ) may result from variable dust emission, with the NIC2 observations being taken at an earlier epoch when the dust-emission was more prominent. Multi-epoch HST observations of the nucleus in the UV and NIR using the same instrument will be extremely useful for verifying a possible variable AGN contribution and constraining its SED. \n<!-- image --> \nFig. 6.Left a WFPC2 F656N-F814W map of the nuclear region. Negative values are areas with strong H α emission. Right Br γ emission line map. Negative values mean strong Br γ emission, while positive values result from Br γ absorption. Contours and orientation are identical to those in Fig. 2. \n<!-- image --> \nobservations by three different cameras, the earliest of which was an 0 . '' 86 aperture spectrum, thus spatial resolution of the UV variability is not possible.", '3.2. NIR Emission Line Excitation': "In this section we discuss the emission lines observed in the Gemini/NIFS K -band spectrum. We find that they imply that the emission originates in dense molecular gas with line ratios typical of those observed in known AGN. \nTo obtain estimates for line emission in the NGC 404 nucleus, we summed together flux-calibrated integrated spectra from our NIFS data with radii ranging between 1 and 25 pixels. We then used both the Wallace & Hinkle (1996) and Winge et al. (2008) templates to fit the stellar continuum from 20200 ˚ A to 24100 ˚ A excluding the parts of the spectrum near known emission lines. Subtracting this continuum off left us with a pure emis- \non spectrum. We measured numerous H 2 emission lines as well as Br γ emission. The only observable coronal line commonly seen in AGN is [CaVIII], which falls directly on a CO bandhead; an 3 -σ upper limit of 3 . 6 10 -17 ergs s -1 cm -2 was placed on this line. \nDetected H 2 lines (and their vacuum wavelengths) included 1-0 S(2) (20337.6 ˚ A), 2-1 S(3) (20734.7 ˚ A), 1-0 S(1) (21218.3 ˚ A), 2-1 S(2) (21542.1 ˚ A), 1-0 S(0) (22232.9 ˚ A), 21 S(1) (22477.2 ˚ A) and 1-0 Q(1) (24065.9 ˚ A). Line ratios of these lines can be used to determine the excitation mechanisms of the gas. In particular, the ratio of the 2-1 S(1) and 1-0 S(1) lines is between 0.11 and 0.16 with smaller values at larger radii. These values are indicative of thermal excitation from shocks or X-ray photons in a dense cloud ( n /greaterorsimilar 10 4 cm -3 ), while fluorescent emission in a low density medium has a higher 2-1 S(1)/1-0 S(1) of ∼ 0.55 (e.g. Mouri 1994). The observed ratio is typical of those observed in other AGN (e.g. Reunanen et al. 2002; Riffel et al. 2008; Storchi-Bergmann et al. 2009). \n× \nThe rotational and vibrational temperatures can be estimated from ratios of H 2 lines; using the relation from Reunanen et al. (2002) T vib = 5600 /ln ( F (1 -0 S (1)) /F (2 -1 S (1)) ∗ 1 . 355), we find T vib = 2340 ± 160 K within an 1 '' radius aperture. Similarly, the rotational temperature can be determined using the relation T rot = -1113 /ln ( F (1 -0 S (2)) /F (1 -0 S (0)) ∗ 0 . 323) (Reunanen et al. 2002) which gives a somewhat lower temperature of T rot = 1640 ± 360 K within a 1 '' aperture. The higher vibrational temperature may be indicative of some contribution of fluorescence (e.g. Riffel et al. 2008). \nUnfortunately the Br γ emission line flux can not be accurately estimated due to uncorrected telluric absorption in the vicinity of the Br γ line. However, the distribution of Br γ flux is much patchier than the H 2 (see Fig. 6) suggesting different sites for the two types of emission. Note that in starbursting galaxies, the Br γ is stronger than the H 2 1-0 S(1) line (Rodr'ıguez-Ardila et al. 2004), a possibility which is clearly excluded by our data (see Fig. 1). This again is consistent with emission originat- \n<!-- image --> \nFig. 7.Velocity and dispersion of the stellar component derived from the CO bandhead. Contours show contours of the K -band image with contours separated by a magnitude in surface brightness (as in Fig. 2). Radial Velocities are shown relative to the central velocity of -58.9 km s -1 . \n<!-- image --> \nFig. 8.The LOSVD along the major axis of the innermost component at PA=80 · . Panels show the radial velocity, LOS dispersion, h 3 and h 4 components of the LOSVD (from top to bottom). Bin centers falling within 0 . '' 1 of the major axis are plotted. \n<!-- image --> \ne to thermal excitation in dense gas as suggested by the H 2 2-1 S(1)/1-0 S(1) ratio. \nThe distribution of Br γ and H α visible in Fig. 6 emission is primarily concentrated in two clumps, one just to the N of the nucleus and one ∼ 1 '' to the SE. These are \nFig. 9.The LOS dispersion values as a function of radius for all data points with dispersion errors less than 10 km s -1 . Error bars on individual points can be seen in light gray. The thick line shows a binned average and standard deviation as a function of radius. \n<!-- image --> \nlikely the sites of the most recent star formation in the nucleus. In § 5 we find a total young stellar mass at ages /lessorsimilar 10 Myr of ∼ 10 4 M /circledot . The central excess has bluer optical colors and UV emission but lacks strong H α or Br γ emission, suggesting the young stars there have an age of /greaterorsimilar 10 Myr (Gogarten et al. 2009).", '4. KINEMATICS': 'Our NIFS data enable us to derive kinematics for both the absorption lines, using the CO bandheads from 22900-24000 ˚ A and the strongest of the H 2 lines at 21218 ˚ A.', '4.1. Stellar Kinematics': "The velocity and dispersion across the NIFS FOV is shown in Fig. 7. At radii of ∼ 1 '' , we see a blue-shift to the W and red-shift to the E, rotation that is in the same direction as the HI rotation seen at larger radii \nFig. 10.Results of kinemetric analysis of the stellar velocity field as function of radius. Top the systemic velocity, middle the position angle, bottom the amplitude of rotation. Triangles in bottom two panels indicate counter-rotation . \n<!-- image --> \n(del R'ıo et al. 2004). At r /lessorsimilar 0 . '' 2, there appears to be modest counter-rotation. An alternative view of the velocity field is seen in Fig. 8 which shows the LOSVD profile along a PA=80 · . The error bars in the velocity range from < 1 kms -1 near the center to ∼ 5 kms -1 at large radii. The counter-rotation seen in the central excess (at r < 0 . '' 2) is clearly significant. However, given the nearly face-on orientation of the larger scale of the galaxy, this counter-rotation does not necessarily imply a large ( > 90 · ) difference in the angle of the rotation axes of the two stellar components. \nTo quantify the position angle (PA) and amplitude of the rotation as a function of radius, we used the kinemetry program of Krajnovi'c et al. (2006). This kinemetry program uses a generalized version of ellipse fitting to analyze and quantify the velocity map or other higher order moments of the LOSVD. We present the results of the kinemetry analysis of the velocity maps in Fig. 10, showing the systemic velocity, PA, and rotation speed as a function of radius. We limited the axial ratio to between 0.8 and 0.95 (consistent with our ellipse fits to the K band image), and fixed the center to the photocenter of the dust-corrected K band image. We find an amplitude for the central counter-rotation of 3.3 km s -1 (shown in triangles in Fig. 10), while the rotation at larger radii has a maximum rotation amplitude of 8.4 km s -1 at 1 . '' 2. The PA of the counter-rotation and rotation at larger radii are quite similar, with values of 80-90 · (E of N). Our observed rotation is at a similar PA to the HI gas, which has a rotation axis of 70-90 · from 100-300 '' (del \nR'ıo et al. 2004). The maximum observed rotation velocity of the HI gas is ∼ 45 kms -1 . Recent observations of the stellar kinematics along the major axis by Bouchard et al. (2010) suggest that increasing counter-rotation relative to the HI is seen between radii of 1-20 '' , with rotation in the sense of the HI again taking over at larger radii. Clearly, large-scale IFU kinematics of this galaxy would be very interesting. \nA radial profile of the stellar dispersion is shown in Fig. 9 for bins with uncertainties in the dispersion of < 10 kms -1 . While the scatter of values is large at most radii, the velocity dispersion clearly declines from ∼ 40 kms -1 at the center to a mean value of ∼ 33 kms -1 at 0 . '' 6. Our dispersion values are consistent with the 40 ± 3 kms -1 determined by Barth et al. (2002) for the NGC 404 center using the calcium triplet. Dynamical modeling based on the stellar kinematics is detailed in § 6.1.", '4.2. H 2 Gas Kinematics and Morphology': "The H 2 gas kinematics and morphology were derived by fitting the 21218 ˚ A1-0 S(1) emission line, the strongest of the H 2 emission lines in the K -band. In each spaxel, we fit this line with a Gaussian to derive a flux, radial velocity, and velocity dispersion (after correction for the instrumental resolution). The results of these fits are shown in Fig. 11. The physical condition of this gas is discussed more in § 3.2. We note that the mass of luminous excited H 2 gas is very small ( < 1 M /circledot ) and although it likely tracers a larger mass of colder molecular gas, this gas is not expected to be dynamically significant in the nucleus (Wiklind & Henkel 1990). \nThe H 2 emission peaks slightly to the SW of the continuum emission, ∼ 0 . '' 07 from the centroid of the dust emission corrected continuum (see white contours in Fig. 11). The FWHM of the distribution is ∼ 0 . '' 4 (6 pc). At r /lessorsimilar 0 . '' 3, it is elongated ( q 0 . 6) with PA ∼ 25 · . \n§ \n∼ \n∼ The H 2 gas velocity field provides a significant contrast with the stellar velocity field. It shows a clear rotation signature at a position angle nearly orthogonal to the stellar rotation. Using the kinemetric analysis program (Krajnovi'c et al. 2006), we find a kinematic PA of ∼ 15 · out to 0 . '' 2, which then twists to PA ∼ -10 at larger radii. Aradial profile of the rotation velocity and dispersion are shown in Fig. 12. The rotation peaks at r ∼ 0 . '' 15 (2.2 pc) with an amplitude of 36 km s -1 . Beyond 0 . '' 3, the rotation flattens out to an amplitude of 20-25 km s -1 . The central dispersion ( r = 0 . '' 05) is 39 km s -1 , which likely includes unresolved rotation. At r = 0 . '' 1 it drops to 33 kms -1 , then slowly declines to ∼ 25 kms -1 at larger radii. Dynamical modeling of the gas kinematics are presented in 6.2. \nThe axial ratio of the intensity distribution and our gas dynamical model suggest an inclination of ∼ 30 · at r < 0 . '' 3, with smaller inclinations at larger radii. Assuming this geometry, the H 2 disk appears to be rotation dominated at all radii with V rot /σ ∼ 2 near the center. H 2 emission line disks are commonly observed in nearby AGN (Neumayer et al. 2007; Riffel et al. 2008; Hicks et al. 2009), although the size of these disks are typically much larger than in NGC 404. For instance, in the Hicks et al. (2009) sample of 11 nearby AGN, they find H 2 disks traced by the 21218 ˚ A line with half-width \n<!-- image --> \nFig. 11.Velocity and dispersion of the molecular hydrogen data determined from the 1-0 S(1) line at 21218 ˚ A. Black contours indicate the surface brightness of the K -band image with contours separated by a magnitude in surface brightness. White contours show the H 2 total intensity 5, 10, 20, 30, 40, 50, 70,and 90% of the peak. Radial velocities are shown relative to the central stellar radial velocity (-58.9 kms -1 ). \n<!-- image --> \nFig. 12.The rotation velocity (black line) and dispersion (blue dashed line) derived from H 2 21218 ˚ A line using the kinemetry program of Krajnovi'c et al. (2006). \n<!-- image --> \nhalf-maximum values ranging from 7-77 pc (vs. 3 pc in NGC 404). Most of these disks have V rot /σ values of ∼ 1, although a couple have V rot /σ > 2 at larger radii.", '5. STELLAR POPULATIONS': 'In this section we use the optical spectrum of the NGC 404 nuclear star cluster to constrain its formation history. We first use the optical emission lines to constrain the reddening and derive a maximum current SFR of 1 . 0 × 10 -3 M /circledot /yr. Fits to the absorption line spectrum show that the nucleus has a range of stellar ages, with stars ∼ 1 Gyr in age contributing more than half the light of the spectrum. Next, we consider the range in massto-light ratios ( M/L s) allowed by the stellar populations fits. Finally, we show that the bulge has a distinctly older stellar population than the nucleus.', '5.1. Optical Emission Line Information': "Before discussing our fits to the optical absorption line spectrum, we first use the optical emission lines to estimate the reddening, current metallicity and star formation rate (SFR). We note again that our line fluxes and ratios agree very well with those in Ho et al. (1997). \nAfter correcting for the underlying stellar population using our population fits described below, we find a Balmer decrement ( F ( Hα ) /F ( Hβ )) of 3.59. For an AGN the expected Balmer decrement is 3.1 while for an HII region it 2.85 (Osterbrock 1989). Given the ambiguous nature of the nuclear emission, this suggests extinctions of A V = 0 . 43 -0 . 73. As noted in § 3.1, the F 547 W -F 814 W color-map indicates extinctions of A V /lessorsimilar 2 covering about half the nucleus, thus these Balmer decrement extinction values seem reasonable. We note that the patchy distribution of the H α emission (see Fig. 6) is significantly different from the stellar light distribution, thus the Balmer decrement extinction may not exactly match the extinction of the stellar light. \nIf we assume that the emission is primarily powered by star-formation, then we can estimate the metallicity based on strong-line indicators. Using the metallicity calibration of the [OII]/[NII] ratio by Kewley & Dopita (2002), we find a metallicity of 12+log(O/H)=8.84, roughly solar. We can also estimate the gas-phase metallicity from the Tremonti et al. (2004) metallicityluminosity relationship; assuming M B ∼ -16 . 75 (using values from Tikhonov et al. 2003), we get a metallicity 12+log(O/H)=8.33. These results suggest a solar or slightly sub-solar current gas-phase metallicity. \nFinally, we can place an upper limit on the current star-formation rate in the nucleus using the H α luminosity. Correcting for extinction, we get an H α flux of 10.8 × 10 -14 ergs s -1 cm -2 corresponding to L Hα = 1 . 2 × 10 38 ergs s -1 . This gives an upper limit to the current star formation rate of 1 . 0 × 10 -3 M /circledot yr -1 using the Kennicutt (1998) relation. The value is an upper limit \nas some of the H α flux may originate from AGN activity. The radio continuum detection of 3 mJy at 1.4 cm by (del R'ıo et al. 2004) implies a SFR of ∼ 2 × 10 -4 M /circledot yr -1 if it is the result of recent star formation. \nThe UV spectra of Maoz et al. (1998) in an 0 . '' 86 aperture also suggests the presence of O stars in the nucleus. We correct their estimate of the number of O stars to a distance of 3 Mpc and for extinction of A V = 0 . 5 -1 . 0. This gives a range of 17-170 O stars, suggesting a total stellar mass of 4000-40000 M /circledot assuming a Kroupa et al. (1993) IMF. The low end of these estimates are consistent with the 10 3 -4 M /circledot of recent stars implied by the H α and radio SFRs assuming formation over the last ∼ 10 Myr.", '5.2. Nuclear Cluster Stellar Populations': "In this section we fit the integrated nuclear optical spectrum to determine the stellar populations of the NSC. We find that stars with a range of ages are present, but that stars 1-3 Gyr make up /greaterorsimilar 50% (6 × 10 5 M /circledot ) of the NSC mass. This dominant population has a solar or slightly subsolar metallicity ([Z]=-0.4). About 10 4 M /circledot of young stars are also present. These results are consistent with other studies of the NGC 404 nucleus (Cid Fernandes et al. 2004; Bouchard et al. 2010). \nWe fit the flux-calibrated spectrum between 37406200 ˚ A, avoiding the second order UV contamination at wavelengths above 6200 ˚ A. For stellar population models we use both (1) the Bruzual & Charlot (2003) models (BC03) which are based on Padova stellar models (Bertelli et al. 1994; Girardi et al. 2000) and the STELIB library of stellar spectra (Le Borgne et al. 2003) and (2) the updated Charlot & Bruzual 2007 (CB07) version of these models incorporating the AGB models of Marigo & Girardi (2007) and Marigo et al. (2008) obtained from St'ephane Charlot. These models provide spectra for single age simple stellar populations (SSPs) over a finely spaced age grid and at metallicities of Z=0.0001, 0.0004, 0.004, 0.008, 0.02, and 0.05 at a spectral resolution similar to our observations. We find that typically the CB07 models more accurately describe the data; they differ from the BC03 models by up to 10% across the wavelength range we are considering here. \nTo remove known problems in the wavelength calibration of the STELIB library (Koleva et al. 2008) and any wavelength issues in our own spectrum, we determined the relative velocity between our spectrum and the SSP models every 100 ˚ A. Variations of up to ± 35 kms -1 around the systemic velocity were found and corrected. We also made a flux calibration correction of ≤ 3% across the spectrum that was determined from residuals of our fits to the nuclear clusters in NGC 404, NGC 205, M33, and NGC 2403 - this is described in more detail below. We fit the models to our spectrum using Christy Tremonti's simplefit program (Tremonti et al. 2004), which uses a Levenberg-Marquardt algorithm to perform a χ 2 minimization to find the best-fit extinction (using the prescription of Charlot & Fall 2000) and scalings of the input model spectra. The code excludes the regions around expected emission lines from the fit. \nWe started our fitting by considering just single age stellar populations. Over the full range of 37406200 ˚ A, the best-fitting SSP is the CB07 model with \nAge=1.14 Gyr, Z=0.004, and an A V = 0 . 85 mags. The reduced χ 2 is 21.8 corresponding to typical residuals of 3%. This fit is shown in Fig. 13 as the blue line. The residuals are dominated by a mismatch between the large-scale shape of the nuclear cluster continuum with the SSP model. To mitigate these effects and get a better sense of the metallicity, we also tried fitting just the metal line rich region from 5050-5700 ˚ A. The best fit is again a CB07 model with an age of 1.14 Gyr, but with Z=0.008 and A V = 0 . 91. This metallicity and reddening are quite similar to the values expected based on the emission lines and the mass-metallicity relationship. \nPrevious studies have found significant evidence for multiple stellar populations in nuclear star clusters (Long et al. 2002; Rossa et al. 2006; Walcher et al. 2006; Seth et al. 2006; Siegel et al. 2007). While most of these studies have focused on spiral galaxies, there is also direct evidence that the NGC 404 nuclear star cluster contains multiple stellar populations; Maoz et al. (1998) finds evidence for massive O stars, suggesting a population with age /lessorsimilar 10 Myr, while based on the color of the nucleus and its spectrum, older stars are clearly present as well. Also, as discussed in § 3.1, we expect contamination of 30% from the bulge in our MMT spectrum. \nWe tried a number of approaches to our multi-age fits. All of these involved fitting a small subsample ( ∼ 10) of the SSPs with ages ranging from 1 Myr to 20 Gyr (see Walcher et al. 2006; Koleva et al. 2008, for a more complete discussion of stellar population synthesis). We tried three types of models with both the CB07 and BC03 spectra: \n∼ \n- 1. Single-metallicity models using the 10 default ages in the simplefit code (5, 25, 101, 286, 640, 904, 1434, 2500, 5000, 10000 Myr) or the 14 age bins used by Walcher et al. (2006) (1, 3, 6, 10, 30, 57, 101, 286, 570, 1015, 3000, 6000, 10000, 20000 Myrs).\n- 2. Inspired by observations of the Sgr dSph nucleus (a.k.a. M54; Monaco et al. 2005; Siegel et al. 2007) we also tried fitting models with chemical evolution from low metallicity (Z=0.0004 / [M/H]=-1.7) at old ages (13 Gyr) to solar metallicity for young ages (following Fig. 2 from Siegel et al. 2007). Specifically we used a model with Z=0.02 at ages of 1, 10, 50, 101, 286, 570, 1015, and 2500 Myr plus Z=0.008 models at 5000 and 10000 Myr and Z=0.0004 models at 13000 Myr.\n- 3. Assuming a continuous or exponential star formation history with chemical evolution similar to the previous item. \nOf all the single metallicity models (model 1), the best fit is obtained with the CB07 spectra using the default ages with Z=0.008 yielding a reduced χ 2 of 4.3 and typical residuals of 1%. This is a vast improvement over the best fit SSP model. The fit is dominated by intermediate age stars, with 72% of the light in the 1.434 Gyr SSP. Many other models also produce nearly as good fits as this one including models with solar and super-solar metallicity. \nAll the best-fitting ( χ 2 red < 5) models have fairly similar age distributions with 50-85% of the light coming \nFig. 13.(Top panel -) Normalized spectrum of the NGC 404 nucleus (black) with the best-fitting single age (blue) and multi-age (red) fits to the spectrum. Vertical lines show emission line regions that were excluded from the fit. Gray spectra indicate the different aged SSPs contributing to the multi-age fit. (Bottom panel) residuals of the best fit single age (blue) and multi-age (red) fits. Also shown is the flux calibration correction described in the text. \n<!-- image --> \nTABLE 3 Best-Fit NSC Stellar Population Model \n| Age [Myr] | 1 | 10 | 50 | 101 | 286 | 570 | 1015 | 2500 | 5000 | 10000 | 13000 |\n|----------------|-------|-------|------|-------|-------|-------|--------|--------|--------|---------|---------|\n| Z | 0.02 | 0.02 | 0.02 | 0.02 | 0.02 | 0.02 | 0.02 | 0.02 | 0.008 | 0.008 | 0.0004 |\n| Light Fraction | 0.013 | 0.09 | 0 | 0 | 0.157 | 0 | 0.277 | 0.239 | 0.166 | 0.058 | 0 |\n| Mass Fraction | 0 | 0.003 | 0 | 0 | 0.035 | 0 | 0.152 | 0.317 | 0.316 | 0.177 | 0 | \nfrom ages between 600 and 3000 Myrs, 5-6% of the light coming from ages < 10 Myr, and 0-25% of the light coming from ages > 5 Gyr. The A V values range from 0.84 to 1.16 and the implied V I colors range from 0.84-1.04. \nThe spectrum is not well fit by a constant star formation history (model 3), giving a reduced χ 2 of 22. The constant SFH clearly has too many young stars, with the continuum mismatched in a way that cannot be compensated for by the reddening. We note that a constant SFH does provide good fits to the spectra of nuclei in many late-type galaxies (Seth et al. 2006; Walcher et al. 2006). An exponentially declining SFH with a timescale of ∼ 8 Gyr does a better job of matching the overall continuum of the observations, but still yields a reduced χ 2 double that of our best fitting models. \n- \nThe best-fit model is obtained from model 2, which contains metallicity evolution. The reduced χ 2 of the \nbest fit is 4.23, slightly better than any of the single metallicity models. This model is shown in red in Fig. 13 with the individual SSP components shown in gray in the top panel. The best-fit model has significant contributions from many of the individual ages, these are shown in Table 3. It is once again dominated by 1-3 Gyr old stars, which make up half the mass of the NSC. This appears to be a robust result, with all our best-fit models containing a similar result. The luminosity weighted mean age is 2.3 Gyr, the mass weighted mean age is 4.3 Gyr. The best fit A V is 0.88 and the M/L s are 0.89, 0.73, and 0.34 in the V -, I -, and H -bands. NGC 404 does not seem to have a dominant old metal-poor population like that seen in M54 (Monaco et al. 2005). From the M/L analysis presented in § 5.3, we find that the maximum contribution for an old (13 Gyr) metal-poor (Z=0.0004) population that keeps the reduced χ 2 below \n5 is ∼ 17% of the luminosity in the V -band and ∼ 35% of the total mass. \nWe can compare our best-fit model to other observed properties of the nucleus. The integrated colors of the best-fit model are I -H =1.59 and V -I =0.96. An estimate of the I -H ∼ 1 . 7 (Table 2) comes from the SB profiles, while the F 547 W -F 814 W ∼ V -I ∼ 0 . 95 comes from the unreddened W portion of the nucleus in Fig. 2. Both of these quantities need to be corrected for foreground extinction of E ( I -H ) and E ( V -I ) of 0.06 and 0.08 magnitudes; after this correction the measured integrated colors agree with the best-fit spectral synthesis model to within 10%. \nThe best-fit model also suggests that the young stars ( ≤ 10 Myr) make up 6%/0.2% of the light/mass in H -band and 8%/0.3% of the light/mass I -band. This gives a mass of of about 1 . 1 × 10 4 M /circledot in young stars, in very good agreement with H α SFR integrated over 10 Myr and the estimates from the UV spectra (see discussion in 5.1; Maoz et al. 1998). \nFinally, we compared the spectrum of NGC 404 to other nuclear star clusters for which we took spectra on the same night, including M33, NGC 205, NGC 2403, and NGC 2976. Both M33 and NGC 205 do better than any single stellar population model at matching the NGC 404 spectrum, with reduced χ 2 values of 9.3 and 15.0. The stellar populations in the M33 nucleus have been previously discussed by Long et al. (2002). They find the spectrum requires a large contribution of stars with an age of ∼ 1 Gyr and a lesser contribution of stars at 40 Myr. This dominant 1 Gyr population is consistent with our findings above. In our initial fits of these nuclear star clusters we noted that the residual pattern was quite similar regardless of the details of the fit, with the most dramatic feature being a change of a few percent between 4700-4800 ˚ A. Our flux calibration errors are expected to be at this level based on the residuals from the flux calibrators. Given the similarities of the residuals in all of the nuclear spectra we medianed the best fit model 1 residual spectra from NGC 205, M33, NGC 404, NGC 2403, then smoothed it and used it as a correction to our flux calibration. The correction is shown as the green line in the residual plot in Fig. 13. This change improved the χ 2 of the single-stellar population and multiage fits substantially at all metallicities but had little effect on the best-fit parameters. \n§ \nThe stellar population of NGC 404 has been previously studied by Cid Fernandes et al. (2004) who analyzed the blue spectrum (3400-5500 ˚ A) of the NGC 404 nucleus as part of a study of a large number of LINER nuclei. Using empirical templates of five nuclei ranging from young to old they find a best fit for NGC 404 of 78.5% of light in an intermediate age template (for which they used a spectrum of NGC 205) and 21.5% from old templates, with an A V of 0.90 and residuals of 6.6%; this residual is much larger than for our fits (with ∼ 1% residuals) due primarily to the lower S/N of their spectrum. Their fit agrees quite well with our finding of a ∼ 1 Gyr old population in the nucleus and our best-fit reddening. Unlike us, they find no evidence for young stars, probably because the NGC 205 template spectrum used for intermediate ages also includes a young stellar component (e.g. Monaco et al. 2009). The nuclear stellar popula- \nhave also been recently studied by Bouchard et al. (2010) using a different fitting code and stellar library. Their best-fit four-component model matches ours very well, with 4% of the light in stars younger than 150 Myr, 20% at 430 Myr, 62% at 1.7 Gyr, and 14% at 12 Gyr.", '5.3. Nuclear Cluster Mass to Light Ratio': "The multi-age population fits to the NGC 404 nuclear spectrum are degenerate, with many possible combinations of ages and extinctions giving similar quality fits. This translates into an uncertainty in the stellar population M/L , with the presence of more young stars decreasing the M/L and the presence of more old stars increasing it. For comparison with the dynamical models ( § 6.1), we'd like to quantify the range of possible M/L s allowed by spectral synthesis fits. \nTo try to quantify the possible variation in the M/L , we took the best-fit model described in the previous section (model 2, with CB07 and current solar metallicity) and fixed the age of each individual SSP (e.g. just the 10 Gyr population) to have between 0-100% of the flux of the unconstrained best-fit model, while the extinction and contribution of the rest of the SSPs were allowed to vary. We then took all 'acceptable' fits (defined below) and examined the range in M/L . Because we have SB profiles in both I - and H -bands (Fig. 3), we focus on the M/L s in those bands. \nWe choose to consider all fits with reduced χ 2 values of < 5 as acceptable, a value 18% higher than our best fitting models. This choice is informed by the χ 2 probability distribution; our fits have ∼ 660 degrees of freedom and ignoring any model errors, the 99% confidence limit for this distribtion is a ∆ χ 2 of 13%. Unfortunately, since our best fit has a reduced χ 2 is 4.2, there must be significant deficiencies in the models or flux calibration at the 1% level. Nonetheless, the range of M/L s with reduced χ 2 < 5 should reflect the a conservative estimate of the acceptable values, roughly equivalent to 3 -σ error bars. \n-We find that the range of acceptable M/L s is 0.681.22 in I band and 0.29-0.61 in H band. We note that the best-fit M/L of 0.73 in I band and 0.34 in H band is near to the lower range of these values, suggesting a larger degeneracy in the mass of old stars than for the younger populations. Our best-fit values are consistent with with the dynamical M/L I = 0 . 70 ± 0 . 04 derived in § 6.1.", '5.4. Bulge Stellar Populations': "In Cid Fernandes et al. (2005), they analyze the radial variations in the NGC 404 spectrum and find increasing age with radius, with the old and intermediate age templates having equal weight at radii between 5 and 10 arcseconds. We use our MMT spectra of NGC 404 to extract a spectra at radii between 5 '' and 10 '' from the nucleus. From our profile fits (Fig. 3), we expect that the bulge component should dominate this spectrum. This spectrum has a peak S/N of 70 around 5000 ˚ A dropping to 25 at 3700 ˚ A. \nThe models that best fit the NGC 404 bulge spectrum are the same sets of models as the nuclear spectrum (i.e. model 1 with Z=0.008 and model 2). However, like Cid Fernandes et al. (2005) we find that the bulge has a significantly older population with all the \n§ \nbest fits having > 50% of the light in an old ( ≥ 5 Gyr) component. The best fits have A V between 0.16 and 0.24 which corresponds well to the A V of 0.19 expected for the foreground extinction (Schlegel et al. 1998) and is consistent with the dust absorption being primarily restricted to the inner 5 '' of the galaxy as suggested by HST color images. The best fits all have residuals of ∼ 2.6%, corresponding to reduced χ 2 of 2.0. All the fits have ∼ 2% of the light in a young ( < 100 Myr) component and a significant contribtution ( ∼ 30%) from a ∼ 1 Gyr component. This analysis clearly shows a significant population difference between the bulge and nuclear spectrum which supports the separation of the SB profiles into separate components in 3. \nAs all the models have nearly identical fits, we infer M/L ratio using model 2, which best fit the nuclear star cluster spectrum. We find a M/L of 1.69, 1.28, and 0.63 in the V -, I -, and H -bands. \nTwo contemperaneous papers suggest that a dominant old population extends out to larger radii as well. At r ∼ 25 '' , Bouchard et al. (2010) find two-thirds of the mass is an old component using spectral synthesis. And using a deep color-magnitude diagram at a radius of ∼ 160 '' (near the disk-bulge transition), (Williams et al. 2010) have found ∼ 80% of the stellar mass is > 10 Gyr old. The nucleus, with its dominant intermediate age population, is therefore quite distinct from the rest of the galaxy.", '6. DYNAMICAL MODELING': 'In this section we present dynamical modeling of the central mass distribution of NGC 404 using both the stellar (absorption line) and gas (H 2 emission) kinematics. From modeling of the stellar kinematics, we find a M/L I = 0 . 70 ± 0 . 04. We can use this to derive a total mass for the nuclear star cluster by combining this estimate with the total luminosity of the NSC and central excess (Table 2) to get a mass of 1 . 1 ± 0 . 2 × 10 7 M /circledot . The error on the NSC mass includes additional uncertainty due to the possible variation in M/L of the central excess. We note that the dynamical M/L I matches the best-fit M/L I derived from the stellar population fits to within 10%. \nWe can also constrain the presence of a MBH using the dynamical models. The two methods give inconsistent results for the BH mass, with the stellar kinematics suggesting an upper limit of < 1 × 10 5 M /circledot , while the gas models are best fit with a MBH of mass 4 . 5 +3 . 5 -2 × 10 5 M /circledot (3 σ errors). These measurements assume a constant M/L within the NSC; taking into account possible variations of M/L at very small radii, the BH mass estimates/upper limits can increase by up to 3 × 10 5 M /circledot . If there is a MBH in NGC 404, its mass is constrained to be < 10 6 M /circledot , smaller than any previous dynamical BH mass measurement in a present-day galaxy.', '6.1. Stellar Dynamical Model': "We constructed a dynamical model to estimate the mass and M/L of the nuclear star cluster and of a possible central supermassive black hole inside it. The first step in this process is developing a model for the light distribution. To parametrize the surface brightness distribution of NGC 404 and deproject the surface brightness into three dimensions, we adopted a Multi-Gaussian \nFig. 14.χ 2 contours describing the agreement between the JAM dynamical models and the NIFS integral-field observations of V rms . The models optimize the three parameters M BH , β z and M/L but the contours are marginalized over the M/L , which is overplotted as dashed contours with labels. Dots show the grid of models. The three lowest χ 2 contour levels represent the ∆ χ 2 = 1, 4 and 9 (thick red line) corresponding to 1, 2, and 3 σ confidence levels for one parameter. \n<!-- image --> \nFig. 15.The observed NIFS profiles of V rms , biweight averaged over circular annuli (data points), are compared to JAM model predictions of the same quantity, for different M BH = 0 (lowest solid line), 1, 3, 5, and 10 × 10 5 M /circledot (top dashed line). All models have M/L I = 0 . 70 M /circledot /L /circledot and β z = 0 . 5 as the best fitting solution. \n<!-- image --> \nExpansion (MGE; Emsellem et al. 1994). The MGE fit was performed with the method and software 10 of Cappellari (2002). We fit a 2-D form of the I -band data as presented in § 3: at r < 5 '' , we use the WFPC2 I -band image with reddened regions masked out, while outside of 5 '' , we use the ACS data (which is saturated in the nucleus). Although the use of NIR data would be preferable given the presence of dust in the nucleus, the uncertainty in the NIFS PSF and the poor resolution and sampling of the available HST NICMOS data makes the F814W data the best available image. The MGE fit takes into account the PSF: we use a Tiny Tim PSF (Krist 1995). In the fit we forced the observed axial ratios q ' of the MGE Gaussians to be as big as possible, not to artificially constrain the allowed inclination of the model. The PSF-convolved MGErepresents a reasonable match to the F 814 W -band images with some mismatch seen in the inner parts due to dust. We note that modeling results based on NICMOS F160W mass models are consistent with those presented here. \nThe primary uncertainty in the determination of M BH results from uncertainties in the central stellar mass profile. The F 814 W luminosity at r < 0 . '' 05 (0.7 pc) is ∼ 1 . 3 × 10 6 L /circledot , while the MGE model has a luminosity of ∼ 4 × 10 5 L /circledot within a sphere of the same radius. Translating this luminosity into a stellar mass density is complicated given the possible presence of dust, stellar population gradients, non-thermal emission and possible nuclear variability. Furthermore, the available kinematics sample only the very nuclear R /lessorsimilar 1 . '' 5 region of the galaxy. This spatial coverage is not sufficient to uniquely constrain the orbital distribution and M BH in NGC 404 (Shapiro et al. 2006; Krajnovi'c et al. 2009). This complex situation does not justify the use of general brute-force orbital superposition dynamical modeling methods (e.g. Richstone & Tremaine 1988; van der Marel et al. 1998). This implies we will not be able to uniquely prove the existence of a BH in this galaxy, but we can still explore the ranges of allowed M BH by making some simplifying observationally-motivated assumptions. A similar, but less general approach was used in other studies of BHs in NSCs from stellar kinematics (Filippenko & Ho 2003; Barth et al. 2009). We note that because we kinematically resolve the NSC and can determine its M/L , we can provide stricter BH upper limits than those provided by these papers, in which the NSC kinematics were unresolved. \nThe adopted dynamical model uses the Jeans Anisotropic MGE (JAM) software 10 which implements an axisymmetric solution of the Jeans equations which allow for orbital anisotropy (Cappellari 2008). Once a cylindrical orientation for the velocity ellipsoid has been assumed, parameterized by the anisotropy parameter β z = 1 -σ 2 z /σ 2 R , the model gives a unique prediction for the observed second velocity moments V rms = √ V 2 + σ 2 , where V is the mean stellar velocity and σ is the velocity dispersion. The JAM models appear to provide good descriptions of integral-field observations of large samples of real lenticular galaxies (Cappellari 2008; Scott et al. 2009) and give similar estimates for BH masses as Schwarzschild models (Cappellari et al. 2010). \nThe luminous matter likely dominates in the extreme high-density nucleus of NGC 404. For this dynamical model we assume light traces mass with a constant M/L . For the model we assumed a nearly face-on inclination of i = 20 · ( i = 90 · being edge on) as indicated by the regular HI disk kinematics and geometry (del R'ıo et al. 2004), and further supported by the nearly circular isophotes at small radii. However, unlike the gas dynamical model, our results do not depend strongly on this inclination assumption. The model has three free parameters: (i) The anisotropy β z , (ii) the mass of a central supermassive black hole M BH and (iii) the I -band total dynamical M/L . To find the best fitting model parameters we constructed a grid for the two nonlinear parameters ( β z , M BH ) values and for each pair we linearly scaled the M/L to match the data in a χ 2 sense. The resulting contours of χ 2 , marginalized over M/L are presented in Fig. 14. They show that with the assumed stellar mass distribution, the models suggest a M BH < 1 × 10 5 M /circledot with β z ∼ 0 . 5 and M/L = 0 . 70 ± 0 . 04 M /circledot /L /circledot in I -band (3 σ levels or ∆ χ 2 = 9). A corresponding data-model comparison is shown in Fig. 15. The dynamical M/L I \n(which includes a correction for foreground extinction) is quite close to our best-fit stellar population estimate of M/L I = 0 . 73. \nAs noted above, the largest uncertainty in our MBH measurements comes from uncertainty in the central stellar mass density hidden by the assumption of a constant M/L . The bluer color in the F 547 M -F 814 W color image (Fig. 2) and the presence of UV emission at r /lessorsimilar 0 . '' 05 suggests a possible decrease in M/L which could mask the presence of a possible BH. Because our optical spectroscopy does not resolve the central stuctures, we can only use the HST imaging data to try constraining possible variations in the M/L . We now determine a limit on how much the central M/L may vary by quantitatively examining the color difference between the NSC and the central pixels. We measured the NSC colors just to the Wof the nucleus and find a F 330 W -F 547 M ( U -V ) of 0.48 and an F 547 -F 814 W ( V -I ) of 0.87 after correction for foreground reddening. The colors at r < 0 . '' 05 are both about 0.1 magnitudes bluer than the NSC colors. In V -I the presence of a ∼ 10 Myr population will give the largest reduction in M/L with the least affect on the color, since these populations are dominated by red supergiants at long wavelengths (e.g. see Fig. 4 in Seth et al. 2008b). For a solar metallicity population, the V -I color at 10 Myr is about 0.5 (Girardi et al. 2000), meaning the observed nuclear color difference is consistent with about 30% of the light in young stars. This would result in a ∼ 30% reduction in the M/L , as the M/L for a 10 Myr population is very small. However, this large fraction of a young population would imply a larger change in U -V color than observed, so 30% represents an upper limit on the reduction in M/L . If we take the I -band luminosity at r < 0 . '' 05 of 1 . 3 × 10 6 L /circledot with our default M/L of 0.70, we get a mass at r < 0 . '' 05 of 9 × 10 5 M /circledot . Reducing the M/L by 30% at the center relative to the NSC would therefore reduce the central stellar mass and increase the BH mass by ∼ 3 × 10 5 M /circledot . Therefore, even including the effects of a possible decline in the central M/L , the BH mass is constrained to be /lessorsimilar 4 × 10 5 M /circledot by the stellar dynamical model. Future HST spectroscopy that can resolve the stellar populations gradient in the nucleus should produce a more accurate mass model that will enable us to robustly test the presence of a MBH in NGC 404.", '6.2. Gas Dynamical Model': "In addition to the stellar dynamical model we constructed a dynamical model to reproduce the kinematics of molecular hydrogen emission ( § 4.2) to obtain an additional constraint on the presence of a possible central BH. We follow the modeling approach of Neumayer et al. (2007) that assumes the gas to be settled in a thin disk in the joint potential of the stars and a putative BH. We assume the gas is moving on circular orbits and we do not include a pressure term in the gas model, as the velocity dispersion of the gas is small. We have tested the effect of an additional pressure term and find it to be very small, well below our derived errors. The stellar potential is fixed using the MGE parameters and the stellar M/L I derived § 5.2, and thus suffers from the same uncertainty in the central stellar mass profile. \nThe gas velocity map appears quite complex with sud- \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFig. 16.Velocity field of the best-fitting H 2 dynamical model (middle, top), with a black hole mass of M BH = 4 . 5 × 10 5 M /circledot and a disk inclination of 37 · , in comparison to the symmetrized data (top left). The velocity residual (data-model) is shown in the right panel. The velocity curves in the bottom panels are extracted along the line of nodes (overplotted to the velocity maps), and represent the peak velocity curves. The diamonds correspond to the model velocity curve, while the crosses correspond to the data. \n<!-- image --> \nFig. 17.Constraining the mass of the central BH: the figure indicates the grid of models (in black hole mass, M BH , and disk inclination) that was calculated, and the contours show χ 2 in the vicinity of the best-fit dynamical models for matching the H 2 kinematics. The minimum χ 2 model is at M BH ∼ 4 . 5 × 10 5 M /circledot and a disk inclination of 37 · . The contours indicate the 1, 2, and 3 σ confidence levels, respectively (see text for details). \n<!-- image --> \nden changes in position angle of the major kinematic axis and some of the kinematic features might be due to non-gravitational motions driven by the LINER nucleus. However, as described in § 4.2, there also appears to be clear rotation in the central 0 . '' 4 × 0 . '' 4 with a major rotation axis of ∼ 15 · , and a twist to slightly negative PAs at larger radii. We interpret this twist in the line-ofnodes as a change in inclination angle of the molecular gas disk and model the kinematics using a tilted ring model. The inclination angle of the gas disk for radii > 0 . '' 2 is fixed to the value of 20 · that is also used for \nthe stellar model and consistent with inclinations derived by del R'ıo et al. (2004) from HI. The dynamical model has two free parameters: (i) The inclination angle of the inner (r ≤ 0 . '' 2) H 2 disk i and (ii) the mass of a central supermassive black hole M BH . To find the best fitting model parameters we constructed a grid for the two parameters ( i , M BH ) and matched the data (velocity and velocity dispersion) minimising χ 2 . Figure 17 shows the contours of χ 2 and indicates the best fit model parameters i = 37 · ± 10 · and M BH = 4 . 5 +3 . 5 -2 . 0 × 10 5 M /circledot (3σ errors). The best-fitting model for the H 2 velocity field is compared to the data in Fig. 16. We note that the best-fit inclination is consistent with the axial ratios of the emission ( 4.2). \nAs Figure 17 shows, the inclination angle i and the central mass M BH are strongly coupled, since the amplitude of the rotation curve is proportional to √ M BH × sin( i ). This degeneracy in the gas model leads to the relatively large uncertainties in inclination angle and black hole mass. However, the gas kinematics cannot be reproduced without a central black hole, even for an almost edge-on central inclination angle. The best-fitting black hole mass is significantly different from the non-detection of < 1 × 10 5 M /circledot derived from the stellar kinematics. One possible source of this discrepancy would be nongravitational motions that are not accounted for by the gas-kinematical model. Another possible problem is the assumption of a thin gas disk, although thicker gas distributions provided worse fits to the gas kinematic data. We note that both determinations use the same MGE mass model, and thus the uncertainty due to the central M/L applies to these results as well. This could increase the BH mass estimates by as much as 3 × 10 5 M /circledot . \n§ \n<!-- image --> \nFig. 18.Scaling relations for NSCs and MBHs in early-type galaxies. Data for MBHs is from Haring & Rix (2004), while for the NSCs it is from Cˆot'e et al. (2006) with mass estimates from Seth et al. (2008a). The solid line shows the joint MBH/NSC fit from Ferrarese et al. (2006), while the dashed line shows the fit to the MBH data from Haring & Rix (2004). Just the bulge mass is plotted for NGC 404, the total galaxy mass would be ∼ 30% larger. \n<!-- image -->", '7.1. Nuclear Star Cluster: Scaling Relationships and Formation': "We compare the NGC 404 NSC to the scaling relationships between galaxy/bulge mass and BH/NSC mass in Fig. 18. We have derived the mass of the NSC using the dynamical M/L determined from the NIFS stellar kinematics and the luminosity of the NSC (and central excess) above the underlying bulge. The NGC 404 NSC mass is a large fraction (1-2%) of the bulge/galaxy mass, making it fall significantly above the relationship for MBHs and early-type Virgo NSCs given by Ferrarese et al. (2006), however, there are several other similar outliers in the Virgo NSC sample. Using the Ferrarese et al. (2006) NSC mass vs. σ relation also gives a mass nearly an order of magnitude smaller than the measured NSC mass. \nThe NSCs high mass relative to its galaxy raises the question of whether our interpretation of this component as an NSC is correct. As an alternative, we could be seeing some form of 'extra-light component' (Hopkins et al. 2009; Kormendy et al. 2009) or an 'extended nuclear source' (Balcells et al. 2007) that are typically seen on somewhat larger scales than NSCs in coreless elliptical, lenticular, and early-type spirals. The mass fraction of ∼ 1% places the NGC 404 NSC in the regime of overlap between NSCs and the extra-light components (Fig. 46 of Hopkins et al. 2009). \nUsing data collected in Seth et al. (2008a), we plot the size and mass of clusters in Fig. 19, and find that the NSC in NGC 404 is typical of those in other early-type galaxies. The NGC 404 NSC is also typical in the mass-density plane, and the NSC S'ersic index of ∼ 2 is similar to that found for the M32 and Milky Way NSC in (Graham & Spitler 2009). The morphology of the NGC 404 NSC is therefore completely consistent with other NSCs. Its mass and density is also similar to the lowest mass extralight components (Hopkins et al. 2009, Fig. 45), and follows the size-surface brightness scaling of the extended nuclear components (Fig. 3 of Balcells et al. 2007). In Hopkins et al. (2009), they argue that while the extralight and NSCs can overlap in morphological properties, \nFig. 19.Mass vs. size for NSCs collected in Seth et al. (2008a). The large solid star shows the position of the NGC 404 NSC, showing that it has a size and mass typical of other early type galaxies. Data is from Carollo et al. (1997, 1998, 2002), Boker et al. (2002), Cˆot'e et al. (2006) and Seth et al. (2006). Points with underlying gray dots have dynamical or spectroscopic M/L estimates from (Walcher et al. 2005) and Rossa et al. (2006). \n<!-- image --> \nthe two can still be distinguished based on their kinematics and stellar populations. We disagree with their argument. Kinematically, both NSCs and extra-light can be disky and rotating (Krajnovi'c et al. 2008; Seth et al. 2008b; Trippe et al. 2008) and in early-type galaxies, the stellar populations of many NSCs are not significantly different from their underlying galaxy (Cˆot'e et al. 2006; Rossa et al. 2006). We therefore suggest that the distinction between extra-light components and NSCs may be ambiguous in some cases such as NGC 404. \nThe extra-light components in elliptical galaxies are thought to result from gas funneled to the center of the galaxy during gas-rich mergers (e.g. Mihos & Hernquist 1994; Hopkins et al. 2009). Given the complicated kinematics and stellar population of the NGC 404 NSC, it must be formed from multiple episodes of nuclear accretion. However, it formed about half its stars 1-3 Gyr ago. In del R'ıo et al. (2004), they suggest NGC 404 underwent a merger with a gas-rich dwarf ∼ 1 Gyr ago to explain the presence of the HI gas disk observed at large radii. The age estimate of the merger was based on the radius beyond which the gas had not yet settled onto the galaxy plane. It is therefore plausible that the ∼ 6 × 10 6 M /circledot of stars formed at about this same age in the NSC resulted from accumulation of merger gas at the center of the galaxy. A merger origin is also consistent with the presence of counter-rotation in the central excess. This scenario is quite different from the formation mechanism suggested by observations of late-type galaxies (Seth et al. 2006; Walcher et al. 2006; Seth et al. 2008b), which appear to form primarily by episodic accretion of material (gas or stars) from the galaxy disk. Resolved observations of a larger sample of NSCs in early-type galaxies is required to determine if NGC 404 is an unusual case, or if NSCs in early-type galaxies typically show some evidence gas accreted from mergers.", '7.2. The Possible NGC 404 MBH': 'Our dynamical modeling in § 6 provides mixed evidence for the presence of an MBH in NGC 404 with mass less than 10 6 M /circledot (making it an IMBH candidate). If verified through follow-up observations, this BH would be the lowest mass dynamically detected BH at the center of a \ngalaxy, with previous determinations of low mass BHs in M32 (2 . 5 ± 0 . 5 × 10 6 M /circledot ; Verolme et al. 2002) and Circinus (1 . 7 ± 0 . 3 × 10 6 M /circledot ; Greenhill et al. 2003) being significantly more massive. A reverberation mapping mass of the BH in nearby Seyfert 1 galaxy NGC 4395 (Peterson et al. 2005), and indirect mass measurements in other low mass Seyfert 1 galaxies (e.g. Greene & Ho 2007) provide additional evidence for the presence of IMBHs at the centers of lower mass galaxies. Possible IMBHs at the center of star clusters (and possible former nuclei) G1 (Gebhardt et al. 2005), ω Cen (Noyola et al. 2008; van der Marel & Anderson 2009), and M54 (Ibata et al. 2009) suggest that IMBHs with masses of 10 4 -5 M /circledot may be widespread, but these results remain controversial. Measuring the occupation fraction and mass of BHs in galaxies like NGC 404 is key to understanding the formation of MBHs in general (Volonteri et al. 2008). \nFrom the latest version of the M BH -σ relations for different galaxy samples from Gultekin et al. (2009), the predicted M BH ranges from 0 . 8 -3 × 10 5 M /circledot for a bulge σ of 35-40 km s -1 . This range of masses is consistent with all our dynamical BH mass estimates. A somewhat higher mass (8 × 10 5 M /circledot ) is implied by the Haring & Rix (2004) relationship (see Fig. 18) and is consistent only with the gas-dynamical estimate, while the Ferrarese et al. (2006) relationships have masses > 10 6 M /circledot and are thus inconsistent with our data. If a black hole with a few × 10 5 M /circledot resides in the center of NGC 404, better knowledge of the mass model from future HST observations should enable us to detect it. \nThe relative mass of NSCs and BHs within a single galaxy can probe the relationship of these two objects. Seth et al. (2008a) find that the relative mass of BHs and NSCs is of order unity (with a scatter of at least one order of magnitude), while Graham & Spitler (2009) suggest a possible evolution in the relative mass with the least massive galaxies being NSC dominated, and the most massive galaxies being BH dominated. The BH in NGC 404 is < 10% of the NSC mass and thus is not an obvious outlier from other galaxies where the mass of both objects is known. \nFinally, we note that the evidence of variability in the UV spectrum (Maoz et al. 2005) and possibly the NIR dust emission ( § 3.1), suggests that the NGC 404 black hole accretion is variable. In both cases, the observed flux decreased from observations taken in the 1990s to those taken in the last decade. This might help explain the ambiguous nature of the AGN indicators in the NGC 404 nucleus; emission that originates near the BH (e.g. hard X-ray emission, broad H α ) may have recently disappeared, while the narrow-line region located at tens of parsecs, may continue to show signs of the earlier accretion. However, the observed hot dust emission suggests that the BH may still be accreting at a very low level.', '8. CONCLUSIONS AND FUTURE PROSPECTS': "This paper is the second resulting from our survey of nearby nuclear star clusters, and demonstrates the rich detail we can obtain for these objects. The NGC 404 nucleus is a complicated environment, with both a nuclear star cluster and a possible black hole. We have found that the surface brightness profile of the inner part of NGC 404 suggests the galaxy can be broken into three \ncomponents: (1) a bulge that dominates the light beyond 1 '' , with M bulge ∼ 9 × 10 8 M /circledot , r eff = 640 pc, and a S'ersic index of ∼ 2.5, (2) a NSC that dominates the light in the central arcsecond with r eff = 10 pc and a dynamical mass of 1 . 1 ± 0 . 2 × 10 7 M /circledot , and (3) a central excess at r < 0 . '' 2, composed of younger stars, dust emission, and perhaps AGN continuum. NIFS IFU spectroscopy shows that the NSC has modest rotation along roughly the same axis as the HI gas at larger radii, while the central excess counter-rotates relative to the NSC. Furthermore, molecular gas traced by H 2 emission shows rotation perpendicular to the stellar rotation. A stellar population analysis of optical spectra indicate that half of the stars in the NSC formed ∼ 1 Gyr ago. Some ancient and very young ( < 10 Myr) stars are also present. This star formation history is dramatically different from the rest of NGC 404, which is dominated by stellar populations > 5 Gyr in age. We suggest a possible scenario where the burst of star formation in the NSC ∼ 1 Gyr ago resulted from the accretion of gas into the galaxy center during a merger. This formation scenario is quite different from the episodic disk accretion suggested by observations NSCs in late-type galaxies. \nOur dynamical modeling of the stellar and gas kinematics provide mixed evidence for the presence of a black hole in NGC 404. Assuming a constant M/L within the nucleus, the stellar dynamical model suggests an upper limit of 1 × 10 5 M /circledot , as well as measuring a M/L I = 0 . 70 ± 0 . 04 for the NSC. The gas kinematics are best fit by models including the presence of a black hole with M BH = 4 . 5 +3 . 5 -2 . 0 × 10 5 M /circledot . Both dynamical BH mass estimates rely on a model for the stellar mass that we construct from HST F 814 W -band imaging. Uncertainties in the light profile (due to variability) and M/L (due to stellar population changes) within the central excess are of the same order as the difference between the two black hole mass estimates. We have proposed to measure the mass model by using HST/STIS spectroscopy to resolve the stellar populations within the nucleus and additional multi-band imaging to extend this model to two dimensions. If successful we will combine this mass model with the kinematic observations presented here as well as larger-scale kinematics obtained from the MMT to more robustly determine the BH mass. \nIn addition to the direct evidence of the BH from the dynamical models, we find two other properties of the nucleus which suggest the presence of an AGN. First, we find unresolved hot dust emission at the center of the NSC with a luminosity of ∼ 1 . 4 × 10 38 ergs s -1 . Comparison of the NIFS light profile to previous NICMOS observations suggests that this dust emission may be variable as is seen in other AGN. Second, the H 2 line ratios within the central arcsecond indicate thermal excitation in dense gas similar to what is seen in other AGN. Our proposed HST observations request multi-epoch UV through NIR imaging to search for definitive evidence of BH accretion in NGC 404. \nOur nearby NSC survey includes 13 galaxies within 5 Mpc with M B between -15.9 and -18.8 for which we will have comparable data to that presented here obtained using MMT, VLT, and Gemini. These galaxies span a wide-range of Hubble types in which we can examine the process of NSC formation; there are four early type E/S0 \ngalaxies and 9 spirals of type Sc and later. NGC 404 represents one of the stronger cases for finding a black hole, given its LINER emission and proximity. However, we expect to be able detect or place upper limits of /lessorsimilar 10 5 M /circledot on black holes in each of the sample galaxies. \nAcknowledgments: We thank the referee, Jenny Greene, for helpful comments, St'ephane Charlot for sharing his models, Christy Tremonti for sharing her code, the staff at Gemini and MMT, NED, and ADS. AS acknowledges support from the Harvard-Smithsonian CfA as a CfA \nand OIR fellow, and helpful conversations with Margaret Geller, Pat Cˆot'e, and Davor Krajnovi'c. MC and NB acknowledge support from STFC Advanced Fellowships. NN acknowledges support from the Cluster of Excellence 'Origin and Evolution of the Universe'. Partially based on observations obtained at the Gemini Observatory, which is operated by the Association of Universities for Research in Astronomy, Inc.. Gemini data was taken as part of program GN-2008B-Q-74. \nFacilities: Gemini:Gillett (NIFS/ALTAIR), HST (ACS/WFC), MMT", 'REFERENCES': "Abel, N. P., & Satyapal, S. 2008, ApJ, 678, 686 Alonso-Herrero, A., Ward, M. J., & Kotilainen, J. K. 1996, MNRAS, 278, 902 \n- Baggett, W. E., Baggett, S. M., & Anderson, K. S. J. 1998, AJ, 116, 1626\n- Balcells, M., Graham, A. W., & Peletier, R. F. 2007, ApJ, 665, 1084 \nBarth, A. J., Ho, L. C., & Sargent, W. L. W. 2002, AJ, 124, 2607 Barth, A. J., Strigari, L. E., Bentz, M. 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2011PhRvD..83d4045P

High accuracy binary black hole simulations with an extended wave zone

2011-01-01

['-', '-', '-', '-', 'methods numerical', 'waves', 'waves', '-', '-']

[]

We present results from a new code for binary black hole evolutions using the moving-puncture approach, implementing finite differences in generalized coordinates, and allowing the spacetime to be covered with multiple communicating nonsingular coordinate patches. Here we consider a regular Cartesian near-zone, with adapted spherical grids covering the wave zone. The efficiencies resulting from the use of adapted coordinates allow us to maintain sufficient grid resolution to an artificial outer boundary location which is causally disconnected from the measurement. For the well-studied test case of the inspiral of an equal-mass nonspinning binary (evolved for more than 8 orbits before merger), we determine the phase and amplitude to numerical accuracies better than 0.010% and 0.090% during inspiral, respectively, and 0.003% and 0.153% during merger. The waveforms, including the resolved higher harmonics, are convergent and can be consistently extrapolated to r→∞ throughout the simulation, including the merger and ringdown. Ringdown frequencies for these modes (to (ℓ,m)=(6,6)) match perturbative calculations to within 0.01%, providing a strong confirmation that the remnant settles to a Kerr black hole with irreducible mass M<SUB>irr</SUB>=0.884355±20×10<SUP>-6</SUP> and spin S<SUB>f</SUB>/M<SUB>f</SUB><SUP>2</SUP>=0.686923±10×10<SUP>-6</SUP>.

[]

https://arxiv.org/pdf/0910.3803.pdf

{'High accuracy binary black hole simulations with an extended wave zone': 'Denis Pollney, 1 Christian Reisswig, 1 Erik Schnetter, 2, 3 Nils Dorband, 1 and Peter Diener 3, 2 \n- 1 Max-Planck-Institut fur Gravitationsphysik, Albert-Einstein-Institut, Potsdam-Golm, Germany \n2 Center for Computation & Technology, Louisiana State University, Baton Rouge, LA, USA 3 Department of Physics & Astronomy, Louisiana State University, Baton Rouge, LA, USA \n(Dated: 2009-10-13) \nWe present results from a new code for binary black hole evolutions using the moving-puncture approach, implementing finite differences in generalised coordinates, and allowing the spacetime to be covered with multiple communicating non-singular coordinate patches. Here we consider a regular Cartesian near zone, with adapted spherical grids covering the wave zone. The efficiencies resulting from the use of adapted coordinates allow us to maintain sufficient grid resolution to an artificial outer boundary location which is causally disconnected from the measurement. For the well-studied test-case of the inspiral of an equal-mass non-spinning binary (evolved for more than 8 orbits before merger), we determine the phase and amplitude to numerical accuracies better than 0 . 010% and 0 . 090% during inspiral, respectively, and 0 . 003% and 0 . 153% during merger. The waveforms, including the resolved higher harmonics, are convergent and can be consistently extrapolated to r → ∞ throughout the simulation, including the merger and ringdown. Ringdown frequencies for these modes (to ( glyph[lscript], m ) = (6 , 6) ) match perturbative calculations to within 0 . 01% , providing a strong confirmation that the remnant settles to a Kerr black hole with irreducible mass M irr = 0 . 884355 ± 20 × 10 -6 and spin S f /M 2 f = 0 . 686923 ± 10 × 10 -6 . \nPACS numbers: 04.25.dg, 04.30.Db, 04.30.Tv, 04.30.Nk', 'I. INTRODUCTION': "The numerical solution of Einstein's equations has made great progress in recent years. Orbits and mergers of binary systems of black holes and neutron stars are now routinely published by a number of research groups, using independent codes and methodologies [1, 2, 3, 4]. A number of important astrophysical phenomena associated with binary black hole mergers have been studied in some detail. In particular, the recoil of the merger remnant has been studied for a number of different initial data models [5, 6, 7, 8, 9, 10, 11, 12], and its final mass and spin has been mapped out for fairly generic merger models involving spinning and unequal mass black holes [13, 14, 15, 16, 17, 18]. Since these quantities are determined by the last few quasi-circular orbits before merger, they can be calculated to good approximation with fairly short evolutions, and with minimal influence of an artificial outer boundary. \nOf particular topical relevance, however, is the construction of long waveforms which can be used for gravitationalwave analysis of the binary [19], and also to construct a family of templates [20, 21, 22, 23], so to inform and improve gravitational wave detection algorithms. Here the requirements are particularly challenging for numerical simulations, requiring waveforms which are accurate in phase and amplitude over multiple cycles to allow for an unambiguous matching to post-Newtonian waveforms at large separation. Some recent studies have shown very promising results in this direction for particular binary black hole models [24, 25, 26, 27, 28, 29, 30, 31, 32]. However, they have also highlighted the problems associated with producing long waveforms of sufficient accuracy. \nFirst of all, for binaries with a larger separation, systematic errors associated with gravitational waveform extraction at a finite radius become more pronounced. Typically a num- \nber of extraction radii are used, and the results extrapolated to infinite radius (assuming such a consistent extrapolation exists given potential issues of gauge). In order to have some confidence in the results, the outermost 'extraction sphere' needs to be at a large radius, say on the order of 150 -200 M (where M is the mass of the system and sets the fiducial length scale). Even at this radius, the amplitude of the extrapolated waveform differs significantly from the measured waveform. Unfortunately, extracting at larger radii comes at a computational expense. One of the standard methods in use today is finite differencing in conjunction with 'mesh refinement', in which the numerical resolution is chosen based on the length scale of the problem. A minimum number of discrete data points are required to resolve a feature of a given size accurately, which sets a limit on the minimum resolution which should be applied in a region. Thus, even with mesh refinement there is a limit on the coarseness of the grid which can be allowed in the wave-zone. For a Cartesian grid, the number of computational points scales as r 3 , so that requiring a sufficient resolution to 200 M already comes at significant expense, and increasing this distance further becomes impractical. \nAn additional difficulty arises from the requirement that the outer boundary have minimal influence on the interior evolution, since it is (in all current binary black hole codes) an artificial boundary. This places an additional requirement on the size of the computational grids, so that even outside the wave-zone region where the physics is accurately resolved, it is conventional to place several even coarser grids. This is done in the knowledge that the physical variables can not be resolved in these regions, but the grids are helpful as a numerical buffer between the outer boundary and interior domain. Again, adding these outer zones comes at a computational expense. The boundaries with under-resolved regions also lead to unphysical reflections which can contaminate the solution. The problem of increasing the grid size can be significantly re- \nduced if, rather than a Cartesian coordinate system, one uses a discretisation which has a radial coordinate. Then, for a fixed angular resolution, the number of points on the discrete grid increases simply as a linear function of r , rather than the r 3 of the Cartesian case. This has two advantages. The gravitational wave-zone has spherical topology and therefore, a numerical approximation would be most efficiently represented by employing a spherical grid. A further computational motivation comes from the fact that non-synchronous mesh-refinement (such as the Berger-Oliger algorithm) can greatly complicate the parallelisation of an evolution scheme, and thus having many levels of refinement clearly has an impact on the efficiency of large scale simulations. This will become particularly relevant for the coming generations of peta-scale machines which strongly emphasise parallel execution (possibly over several thousand cores) over single processor performance. \nThe use of spherical-polar coordinates has largely been avoided in 3-dimensional general relativity due to potential problems associated with the coordinate singularity at the poles. Additionally, even if regularisation were simple, the inhomogeneous areal distribution of latitude-longitude grid points over the sphere make spherical-polar coordinates suboptimal. A number of alternative coordinate systems have been proposed and implemented for studies of black holes in 3D. The Pittsburgh null code avoids the problem of regularisation at the poles by implementing a 2D stereographic patch system [33]. Cornell/Caltech have developed a multipatch system which has been used for long binary black hole evolutions [4, 34] 1 . This code, using spectral spatial differentiation, uses an intricate patch layout in order to reduce the overall discretisation error. The boundary treatment between patches is based on the transfer of characteristic variables. A similar approach was implemented by the LSU group, for the case of finite differences with penalty boundary conditions [38], and used to successfully evolve single perturbed black holes with a fixed background [39] and have recently been attempted for binary black hole systems [40]. \nIn this paper we describe a binary black hole evolution code based on adapted radial coordinates in the wave zone, for generic evolution systems. In particular, we demonstrate stable and accurate binary black hole evolutions using BSSNOKin conjunction with this coordinate system. The grids in the wave zone follow a prescription which was first used by Thornburg [41], in which six regular patches cover the sphere, and data at the boundaries of the patches are filled by interpolation. The six patch wave zone is coupled to an interior Cartesian code, which covers the domain in which the bodies move, and optionally allows for mesh refinement around each of the individual bodies. The resulting code has the advantages of making use of established techniques for moving puncture evolutions on Cartesian grids, while having excellent efficiency (and consequently accuracy) in the wave zone due \nto the use of adapted radially-oriented grids. \nIn the following sections we detail the coordinate structures which we use. We then describe our Einstein evolution code, which is largely based on conventional techniques common to Cartesian puncture evolutions. Finally we perform evolutions of a binary black hole system in order to validate the code against known results, as well as demonstrate the ability to extract accurate waves at a large radius with comparatively low computational cost.", 'II. SPACETIME DISCRETISATION': "This section describes the implementation of a generic code infrastructure for evolving spacetimes which are covered by multiple overlapping grid patches. A key feature of our approach is its flexibility. It is not restricted to any particular formulation of the Einstein equations; the mechanism for passing data between patches (interpolation) is also formulation independent (though characteristic [42] or penalty-patch boundaries [40, 43, 44] are also an option); the size, placement and local coordinates of individual patches are completely adaptable, provided that there is sufficient overlap between neighbours to transfer boundary data. Further, each patch is a locally Cartesian grid with the ability to perform mesh-refinement to better resolve localised steep gradients, if necessary. The particular application demonstrated in this paper is to provide a more efficient covering of the wave-zone of an isolated binary black hole inspiral. \nAt the same time, we would like to take advantage of the fact that black hole evolutions via the 'moving puncture' approach are well established as a simple and effective method for stably evolving black hole spacetimes [2, 3]. By this method, gauge conditions are applied to prevent the spacetime from reaching the curvature singularity, so that an artificial boundary is not required within the horizons [45]. The usual approach is to discretise using Cartesian grids which cover the black holes with an appropriate resolution, without special treatment or boundary conditions for the black hole interiors, relying rather on the causal structure of the evolution system to prevent error modes from emerging [46]. The Cartesian grids are extended to cover the wave zone (at reduced resolution for the sake of efficiency), extending to a cubical grid outer boundary where an artificial condition is applied. \nA principal difficulty faced by this method is that the discretisation is not well suited to model radial waves at large radii. In order to resolve the wave profile, a certain minimum radial resolution is required and must be maintained as the wave propagates to large radii. The angular resolution, however, can remain fixed - if a wave is resolved at a certain angular resolution as small radii, then it is unlikely to develop significant angular features as it propagates to large distances from the isolated source. Cartesian grids with fixed spacing, however, resolve spheres with an angular resolution which scales according to r 2 . Thus, to maintain a given required radial resolution, the angular directions become extremely overresolved at large radii, and this comes at a large computational cost. Namely, for a Cartesian grid to extend in size or increase \nit's resolution by a factor n , the cost in memory and number of computations per timestep increase by n 3 , while for a radial grid with fixed angular resolution, the increase is linear, n 2 . \nFor the near-zone, in the neighbourhood of the orbits of the individual bodies, the geometrical situation is not as straightforward, since there is no clearly defined radial propagation direction between the bodies. If the local geometry is reasonably well known (for instance, the location of horizon surfaces), adapted coordinates can also be considered in this regime. The technical requirements of such coordinate systems can, however, be high. Since the bodies are moving, the grids must move with them, or dynamical gauges chosen such that the bodies remain in place relative to the numerical coordinates. Potential problems arise from the coordinate singularity if the grids are extended to r = 0 , as is the case with the standard puncture approach. Thus, in the near-zone, Cartesian coordinates can provide significant simplification to the overall infrastructure requirements, while the previously mentioned drawbacks of Cartesian coordinates are less prevalent, as it is useful to have homogeneous resolution in each direction in situations where there is no obvious symmetry. \nThe evolution code which we have constructed for the purpose of modelling waveforms from an isolated system is based on a hybrid approach, involving a Cartesian mesh-refined region covering the near zone in which the bodies orbit, and a set of adapted radial grids which efficiently cover the wave zone. The overall structure is illustrated in Fig. 1 (top), which shows an equatorial slice of the numerical grid. Computations on each local patch are carried out in a globally Cartesian coordinate system. In the particular implementation considered here, the grids overlap by some distance so that data at the boundaries between each local coordinate patch can be communicated by interpolation from neighbouring patches. The resulting code is both efficient, but also simple in structure and able to take advantage of well established methods for evolving moving puncture black holes. If suitable interpolation methods are used, then such a system can also be used for solutions with discontinuities and shocks as are present in hydrodynamics. \nThe code has been implemented within the Cactus framework [47, 48] via extensions to the Carpet driver [49, 50, 51], which handles parallelisation via domain decomposition of grids over processors, as well as providing the required interpolation operators for boundary communication and analysis tools.", 'A. Coordinate systems': "The configuration displayed in Fig. 1 consists of seven local coordinate patches: an interior Cartesian grid, and six outer patches corresponding to the faces of the interior cube. Each patch consists of a uniformly spaced (in local coordinates) \ngrid which can be refined (though in practise we will only use this feature for the interior grid). The outer patches have a local coordinate system ( ρ, σ, R ) corresponding to the 'inflated cube' coordinates which have previously been used in relativity for single black hole evolutions [41], and are displayed in the lower plot of Fig. 1. The local angular coordinates ( ρ, σ ) range over ( -π/ 4 , + π/ 4) × ( -π/ 4 , + π/ 4) and can be related to global angular coordinates ( µ, ν, φ ) which are given by \nµ ≡ rotation angle about the x-axis = arctan( y/z ) , (1a) \nφ ≡ rotation angle about the z-axis = arctan( y/x ) . (1c) \nν ≡ rotation angle about the y-axis = arctan( x/z ) , (1b) \nFor example, on the + z patch, the mapping between the local ( ρ, σ, R ) and Cartesian ( x, y, z ) coordinates is given by: \nρ ≡ ν = arctan( x/z ) , (2a) \nR = f ( r ) , (2c) \nσ ≡ µ = arctan( y/z ) , (2b) \nwith appropriate rotations for each of the other cube faces, and where r = √ x 2 + y 2 + z 2 . As emphasised by Thornburg [41], in addition to avoiding pathologies associated with the axis of standard spherical polar coordinates, this choice of local coordinates has a number of advantages. In particular, the angular coordinates on neighbouring patches align so that interpolation is only 1-dimensional, in a line parallel to the face of the patch. This potentially improves the efficiency of the interpolation operation as well as the accuracy. The coordinates also cover the sphere more uniformly than, say, a stereographic 2-patch system. \nThe local radial coordinate, R , is determined as a function of the global coordinate radius, r . We can use this degree of coordinate freedom to increase the physical (global) extent of the wave-zone grids, at the cost of some spatial resolution. In practise, we use a function of the form \nf ( r ) = A ( r -r 0 ) + B √ 1 + ( r -r 0 ) 2 /glyph[epsilon1], (3a) \nwith \nR = f ( r ) -f (0) . (3b) \nin order to transition between two almost constant resolutions (determined by the parameters A and B ) over a region whose width is determined by glyph[epsilon1] , centred at r 0 . \nThe effect of the radial transformation is illustrated in Fig. 2. The coordinate R is a nearly constant rescaling of r at small and large radii. The change in the scale factor is largely confined to a transition region. Note that since we apply the same global derivative operators (described below) to analysis tools as are used for the the evolution, it is possible to do analysis (e.g., measure waveforms, horizon finding) within regions where the radial coordinate is non-uniform. The regions of near-constant resolution are, however, useful in order to make comparisons of measurements at different radii without the additional complication of varying numerical error due to the underlying grid spacing. \n<!-- image --> \nFIG. 1: A schematic view of the z = 0 slice of the grid setup that is used. The upper plot shows the central Cartesian grid surrounded by six 'inflated-cube' patches (the four equatorial patches are shown, shaded). The boundaries of the nominal grids owned by each patch are indicated by thick lines. The lower plot shows an r = constant surface of the exterior patches, indicating their local coordinate lines. \n<!-- image --> \nData on each patch are evaluated at grid-points which are placed at uniformly spaced points of a Cartesian grid. Thus, local derivatives can be calculated via standard finite difference techniques. These are then transformed to a common underlying Cartesian coordinate system by applying an appropriate Jacobian which relates the local to global coordinates. That is, the global (Cartesian) coordinates, x i , are related to the local coordinates, a i , by \nx i = x i ( a j ) , i, j = 0 , 1 , 2 . (4) \nand derivatives, ∂/∂a i , of fields are determined using finite differences in the regularly spaced a i coordinates, which are \nFIG. 2: The local radial coordinate, R (solid line), can be stretched as a function of the global coordinate, r , in order to increase the effective size of the grid. The function specified by Eqs. (3) transitions between two almost constant radial resolutions over a distance glyph[epsilon1] centred at r 0 . \n<!-- image --> \nthen transformed using \n∂ ∂x i = ( ∂a j ∂x j ) ∂ ∂a j , (5a) \n∂ 2 ∂x i ∂x j = ( ∂ 2 a k ∂x i ∂x j ) ∂ 2 ∂a 2 k + ( ∂a k ∂x i ∂a l ∂x j ) ∂ 2 ∂a k ∂a l , (5b) \nin order to determine their values in the global frame. We store and evaluate tensor components and their evolution equations in the common global frame, so that there is no need to apply transformations when relating data across patch boundaries. In addition to the obvious simplification of the inter-patch boundary treatment, this has a number of other advantages, particularly when it comes to analysis tools (surface finding, gravitational wave measurements, visualisation) which may reference data on multiple patches. Since the data is represented in the common global basis, these tools do not need to know anything about the local grid structures or coordinates.", 'B. Inter-patch interpolation': "Data is communicated between patches by interpolating onto overlapping points. Each patch, p , is responsible for determining the numerical solution for a region of the spacetime. The boundaries of these patches can overlap neighbouring patches, q , (and in fact must do so for the case of the interpolating boundaries considered here), creating regions of the spacetime which are covered by multiple patches. We define three sets of points on a patch. The nominal regions, N p , contain the points where the numerical solution is to be determined. The nominal regions of the patches do not overlap, ⋂ p N p = ∅ , so that if data is required at any point in the spacetime, it can be obtained without ambiguity by referencing the single patch in whose nominal region it resides. A patch, p , is bounded by a layer of ghost points, G p , which overlap the nominal zones of neighbouring patches, q , G p ∩ ⋃ q N q = G p , and are filled by interpolation. (These points are conceptually similar to the inter-processor ghost-zones used by domain decomposition parallelisation algorithms in order to divide grids over processors.) The size of these regions is determined by the width of the finite difference stencil to be used in finite \nFIG. 3: Schematic of interpolating patch boundaries in 1-dimension, assuming 4-point finite difference and interpolation stencils. Points in the nominal zones, N p,q , are indicated by filled circles, points in ghost zones, G p,q , by open squares, and points in overlap zones, O p,q , by closed squares. The vertical dotted line demarcates the boundary between nominal zones on each patch. Ghost points on patch p are evaluated by centred interpolation operations from points in S q on the overlapping patch (arrows) and vice versa . \n<!-- image --> \ndifferencing the evolution equations on the nominal grid. Finally, an additional layer of overlap points, O q , is required: i) to ensure that the set of stencil points, S q ⊂ O q ∪ N q , used to interpolated to the ghost zone does not itself originate from the ghost zone of the interpolating patch, S q ∩ G q = ∅ , O q ∩ ⋃ p N p = O q ; and ii) to compensate for any difference in the grid spacing between the local coordinates on the two patches. An illustration of the scheme in 1-dimension the scheme is provided in Fig. 3. \nNote that points in ⋃ q O q ⊂ ⋃ p N p are not interpolated, but rather are evolved in the same way as nominal grid points within ⋃ p N p . That is, in these regions points on each grid are evolved independently, and is in principle multi-valued. However, since the union of set of nominal points on each patch ⋃ p N p uniquely and unambiguously covers the entire simulation domain, i.e. ⋂ p N p = ∅ , and since the overlap regions are a subset of the nominal grid, if data is required at a point within these overlap zones, there is exactly one patch owing this point on its nominal grid, and it will be returned uniquely from this patch. The differences between evolved field values evaluated in these overlap points converge away with the finite difference order of the evolution scheme. \nThe use of additional overlap points makes this inter-patch interpolation algorithm somewhat simpler than the one implemented by Thornburg in [41]. That algorithm required interpatch boundary conditions to be applied in a specific order to ensure that all interpolation stencils are evaluated without using undefined grid points, and requires off-centring interpolation stencils under certain circumstances, which is not necessary when overlap points are added. It also relies on the particular property of the inflated-cube coordinates which ensured that the ghost-zones could be filled using 1-dimensional interpolation in a direction orthogonal to the boundary. This property would be non-trivial (and often impossible) to generalise to match arbitrary patch boundaries, such as that between the Cartesian and radially oriented grids of Fig. 1. \nAnother significant difference between Thornburg's approach and the approach presented here is that former stores tensor components in the patch-local frame, while we store them in the global coordinate frame. Evaluating components in the patch-local frame requires a basis transformation while interpolating. This is further complicated in the case of nontensorial quantities (such as the ˜ Γ i of the BSSNOK formula- \non) which have quite complicated basis transformation rules involving spatial derivatives. Instead, we store tensor components in the global coordinate frame, which requires no basis transformation during inter-patch interpolations. \nThe number of ghost points in G p can be reduced using finite difference stencils which become lop-sided towards the boundaries of the patch, and may provide an important optimisation since interpolation between grids can be expensive, particularly if processor communication is involved. However, this tends to be at the cost of increased numerical error in the finite difference operations towards the grid boundaries. We have generally found it preferable to use centred stencils throughout the nominal, N p , and overlap, O p , zones and have applied certain optimisations to the interpolation operators as described below. Another optimisation can be achieved by using lower order interpolation so that it is possible to reduce the number of overlapping points in p . \nThe interpolation scheme for evaluating data in ghost zones is based on Lagrange polynomials using data from the overlapping patch. In 1-dimension, the Lagrange interpolation polynomial can be written as \nO \nL x [ φ ]( x ) = N ∑ i b i ( x ) φ i , (6a) \nwhere the coefficients are \nb i ( x ) = ∏ k glyph[negationslash] = i ( x -x k ) ( x i -x k ) . (6b) \nIn these expressions, x ∈ G p is the coordinate of the interpolation point and φ i ∈ S q ⊂ N q ∪O q are the values at grid-points in the interpolation molecule surrounding x . The number of grid-points in the interpolation molecule, N , determines the interpolation order, and interpolation of n -th order accuracy is given by N = n +1 stencil points in the molecule. \nFor interpolation in d -dimensions, the interpolation polynomial can be constructed as a tensor product of 1-dimensional Lagrange interpolation polynomials along coordinate directions, x = ( x 1 , ..., x d ) : \nL [ φ ]( x ) = L x 1 [ φ ]( x 1 ) ⊗ . . . ⊗L x d [ φ ]( x d ) = ( N ∑ i b i ( x 1 ) φ i , ) · · · N ∑ j c j ( x d ) φ j . (7) \nTherefore, for d -dimensional interpolation of order n , one has to determine N d neighbouring stencil points and associated interpolation coefficients, Eq. (6b), for each point in the ghostzone of a given patch. Most generally, full 3-dimensional interpolation is required, though in particular cases coordinates between two patches can be constructed such that they align locally so that only 1-dimensional interpolation is needed. This is, for instance, the case for the overlap region between the inflated-cube spherical patches used here (see Fig. 1). We have optimised the current code to automatically take advantage of this. \nIn order to interpolate to a point for which the coordinates a p i given in the basis of patch p are given, we need to know \nthe patch owning the nominal region containing this point. For this we first convert a p i to the global coordinate basis x i , then determine which patch q owns the corresponding nominal region N q , and then convert x i to the local coordinate bases this patch a q i . By construction, patch q has sufficient overlap points to evaluate the interpolation stencil there: \nx i := local-to-global p ( a p i ) , (8a) \nq := owning-patch ( x i ) , (8b) \na q i := global-to-local q ( x i ) . (8c) \nThe three operations 'local-to-global', 'owning-patch', and 'global-to-local' depend on the patch system and their local coordinate systems. \nWe can then find the points of patch q that are closest to the interpolation point a q i in the local coordinates this patch. In order to find these points, we exploit the uniformity of the grid in local coordinates. The grid indices of the stencil points in a given direction are determined via \ni ∈ { floor ( j + k ) ∣ ∣ ∣ ∣ j = x -x 0 ∆ x , k = -n 2 , · · · , n 2 } , (9) \nwhere x 0 is the origin of the local grid, n is the interpolation order, and 'floor' denotes rounding downwards to the nearest integer. \nThere remains to be determined the refinement level which contains the region surrounding the interpolation point, as well as the processor that owns this part of the grid. For this purpose, an efficient tree-search algorithm has been implemented. In this algorithm, the individual patches and refinement levels are defined as 'super-regions', i.e., bounding boxes that delineate the global grid extent before processor decomposition. Each of these super-regions is recursively split into smaller regions. The leaves of the resulting tree structure represent the individual local components of the processor decomposition. Locating a grid point in this tree structure requires O (log n ) operations on n processors, whereas a linear search (that would be necessary without a tree structure) would require O ( n ) operations. \nWhile the corresponding tree structure is generic, the actual algorithm used in Carpet splits the domain into a kd tree of depth d in d = 3 dimensions. That is, the domain is first split into k sub-domains in the x direction, each of these sub-domains is then independently split into several in the y direction, and each of these is then split in the z direction. This leads to cuboid sub-domains for each processor, where the sub-domains do not overlap, and where each sub-domain can have a different shape. Carpet balances the load so that each processor receives approximately the same number of grid points, while keeping the sub-domains' shapes as close to a cube as possible. \nOur implementation pre-calculates and stores most of the above information when the grid structure is set up, saving a significant amount of time during interpolation. In particular, the following are stored: \n- · For each ghost-point, the source patch (where the interpolation is performed), and the local coordinates on this patch; \n- · For each ghost-point, the interpolation stencil coefficients (6b);\n- · For each processor, the communication schedule specifying which interpolation points need to be sent to what other processor. \nWhen the grid structure changes, for example, when a meshrefinement grid is moved or resized, these quantities have to be recalculated. \nAltogether, the inter-patch interpolation therefore consists of applying processor-local interpolation stencils, sending the results to other processors, receiving results from other processors, and storing these results in the local ghost-points. These are all operations requiring no look-up in complex data structures, and which consequently execute very efficiently on modern hardware.", 'C. Finite differencing': "Spatial derivatives are computed using standard finite difference stencils, which have currently been implemented up to 8th-order [44]. The stencils are centred, except for the terms corresponding to an advection by the shift vector, of the form β i ∂ i u (see Sec. III, below). These derivatives are calculated using an 'upwind' stencil which is shifted by one point in the direction of the shift, and of the same order. We find that these upwind stencils provide a significant increase in the numerical accuracy of the puncture motion at a given resolution (see Appendix A). The particular stencils which we use can be generated via a recursion algorithm, as outlined in [52]. \nOn each patch we allow the local grids to be refined in order to increase the accuracy of computations in localised regions. For the application of the evolution of an isolated binary considered here, we only refine the central Cartesian grid in the neighbourhood the bodies. This is done using standard 2 : 1 Berger-Oliger mesh refinement techniques via the Carpet infrastructure [49, 50, 51]. The time step for the outer patches is taken to be the same as the coarse grid step of the interior patch, so that no time-interpolation is required at inter-patch boundaries. \nTime integration is carried out using standard method-oflines integrators. We find that for the time resolution we are using, a 4th-order Runge-Kutta (RK4) method provides a good compromise between sufficient accuracy and a low memory footprint. We set the time resolution of the outer grids according to the timestep of the coarsest Cartesian grid, which is limited by the Courant condition at the specified spatial resolution. By placing the Cartesian-spherical boundary at a small radius (and thus extending to finer Cartesian grids) we attain a high time resolution in the wave zone, potentially important for determining higher harmonics.", 'D. Surface integration and harmonic decomposition': 'Anumber of quantities of physical interest are measured by projecting them onto closed surfaces surrounding the source. \nIn particular, gravitational wave measurements rely on computations on constant coordinate spheres S 2 , parameterized by local spherical-polar coordinates ( θ, φ ) with constant coordinate radius r . In principle, it would be possible to construct coordinates on these 2-dimensional spheres which correspond to the underlying grid coordinates of the evolution, for instance as portrayed in the lower figure of Fig. 1. In this case, data can be mapped directly onto the spheres. More generally, however, interpolation can be used to obtain data at points on the measurement spheres, according to the procedure outlined in Sec. II B, above. \nFor the purpose of analysis, it is often convenient to decompose the data on S 2 in terms of (spin-weighted) spherical harmonic modes, \nA glyph[lscript]m = ∫ d Ω √ -gA (Ω) s ¯ Y glyph[lscript]m (Ω) , (10) \nwhere g is the determinant of the surface metric and Ω angular coordinates. In standard spherical-polar coordinates ( θ, φ ) , \n√ -g = sin 2 θ . (11) \nThe integral, Eq. (10), is solved numerically as follows. In the spherical polar case, we can take advantage of an highly accurate Gauss quadrature scheme which is exact for polynomials of order up to 2 N -1 , where N is the number of gridpoints. More specifically, we use Gauss-Chebyshev quadrature. The scheme can be written out as \n∫ d Ω f (Ω) = N θ ∑ i N φ ∑ j f ij w j + O ( N θ ) , (12) \nwhere N θ and N φ are the number of angular gridpoints and w j are weight functions [53, 54], \nw j = 2 π N φ 1 N θ √ 2 π N θ / 2 -1 ∑ l =0 1 2 l +1 sin ( [2 l +1] πj N ) , j = 0 , ..., N θ -1 . (13) \nIn our implementation, the weight functions are pre-calculated for fast surface integration.', 'III. EVOLUTION SYSTEM': "We evolve the spacetime using a variant of the 'BSSNOK' evolution system [55, 56, 57, 58] and a specific set of gauges [59, 60], which have been shown to be effective at treating the coordinate singularities of Bowen-York initial data. \nThe 4-geometry of a spacelike slice Σ (with timelike normal, n α ) is determined by its intrinsic 3-metric, γ ab and extrinsic curvature, K ab , as well as a scalar lapse function, α , and shift vector, β a which determine the coordinate propagation. The standard BSSNOK system defines modified variables by performing a conformal transformation on the 3metric, \nφ := 1 12 ln det γ ab , ˜ γ ab := e -4 φ γ ab , (14) \nsubject to the constraint \ndet ˜ γ ab = 1 , (15) \nand by removing the trace of K ab , \nK := tr K ij = g ij K ij , (16) \n˜ A ij := e -4 φ ( K ij -1 3 γ ij K ) , (17) \nwith the constraint \n˜ A := ˜ γ ij ˜ A ij = 0 . (18) \nAdditionally, three new variables are introduced, defined in terms of the Christoffel symbols of ˜ γ ab by \n˜ Γ a := ˜ γ ij ˜ Γ a ij . (19) \nIn principle the ˜ Γ a can be determined from the ˜ γ ab , on a slice however their introduction is key to establishing a strongly hyperbolic (and thus stable) evolution system. In practise, only the constraint Eq. (18) is enforced during evolution [61], while Eq. (15) and Eq. (19) are simply monitored as indicators of numerical error. Thus, the traditional BSSNOK prescription evolves the variables \nφ, ˜ γ ab , K, ˜ A ab , ˜ Γ a , (20) \naccording to evolution equations which have been written down a number of times (see [62, 63] reviews). \nIn the context of puncture evolutions, it has been noted that alternative scalings of the conformal factor may exhibit better numerical behaviour in the neighbourhood of the puncture as compared with φ . In particular, a variable of the form \nˆ φ κ := (det γ ab ) -1 /κ , (21) \nhas been suggested [3, 64]. In [3], it is noted that certain singular terms in the evolution equations for Bowen-York initial data can be corrected using χ := ˆ φ 3 . Alternatively, [64] notes that W := ˆ φ 6 has the additional benefit of ensuring γ remains positive, a property which needs to be explicitly enforced with χ . \nIn terms of ˆ φ κ , the BSSNOK evolution equations become: \n∂ t ˆ φ κ = 2 κ ˆ φ κ αK + β i ∂ i ˆ φ κ -2 κ ˆ φ κ ∂ i β i , (22a) \n∂ t ˜ γ ab = -2 α ˜ A ab + β i ∂ i ˜ γ ab +2˜ γ i ( a ∂ b ) β i (22b) -2 3 ˜ γ ab ∂ i β i , \n∂ t K = -D i D i α + α ( A ij A ij + 1 3 K 2 ) + β i ∂ i K, (22c) \n∂ t ˜ A ab =( ˆ φ κ ) κ/ 3 ( -D a D b α + αR ab ) TF + β i ∂ i ˜ A ab (22d) +2 ˜ A i ( a ∂ b ) β i -2 3 A ab ∂ i β i , \n∂ t ˜ Γ a =˜ γ ij ∂ i β j β a + 1 3 ˜ γ ai ∂ i ∂ j β j -˜ Γ i ∂ i β a (22e) \n+ 2 3 ˜ Γ a ∂ i β i -2 ˜ A ai ∂ i α \n+2 α ( ˜ Γ a ij ˜ A ij -κ 2 ˜ A ai ∂ i ˆ φ κ ˆ φ κ -2 3 ˜ γ ai ∂ i K ) , \nwhere D a is the covariant derivative determined by ˜ γ ab , and 'TF' indicates that the trace-free part of the bracketed term is used. \nWe have implemented the traditional φ form of the BSSNOK evolution equations, as well as the χ and W variants implicit in the evolution system, Eqs. (22). All three evolution systems produce stable evolutions of binary black holes, though the χ variant requires some special treatment if, due to numerical error particularly in the neighbourhood of the punctures, ˆ φ 3 ≤ 0 [65]. Generally we find that the advection of the puncture (and thus the phase accuracy of the simulation) exhibits lower numerical error when using the χ and W variants (see Appendix C). Convergence properties of physical variables (such as measured gravitational waves, or constraints measured outside of the horizons), however, are not affected by the choice of conformal variable. \nThe Einstein equations are completed by a set of four constraints which must be satisfied on each spacelike slice: \nH ≡ R (3) + K 2 -K ij K ij = 0 , (23a) \nM a ≡ D i ( K ai -γ ai K ) = 0 . (23b) \nAgain, we do not actively enforce these equations, but rather monitor their magnitude in order to determine whether our numerical solution is deviating from a solution to the Einstein equations. \nThe gauge quantities, α and β a , are evolved using the prescriptions that have been commonly applied to BSSNOK black hole, and particularly puncture, evolutions. For the lapse, we evolve according to the ' 1 + log ' condition [66], \n∂ t α -β i ∂ i α = -2 αK, (24) \nwhile the shift is evolved using the hyperbolic ' ˜ Γ -driver' equation [59], \n∂ t β a -β i ∂ i β a = 3 4 αB a , (25a) \n∂ t B a -β j ∂ j B i = ∂ t ˜ Γ a -β i ∂ i ˜ Γ a -ηB a , (25b) \nwhere η is a parameter which acts as a (mass dependent) damping coefficient, and is typically set to values on the order of unity for the simulations carried out here. The advective terms in these equations were not present in the original definitions of [59], where co-moving coordinates were used, but have been added following the experience of more recent studies using moving punctures [2, 60].", 'A. Wave extraction': "The Newman-Penrose formalism [67] provides a convenient representation for a number of radiation related quantities as spin-weighted scalars. In particular, the curvature component \nψ 4 ≡ -C αβγδ n α ¯ m β n γ ¯ m δ , (26) \nis defined as a particular component of the Weyl curvature, C αβγδ , projected onto a given null frame, { l , n , m , ¯ m } . \nThe identification of the Weyl scalar ψ 4 with the gravitational radiation content of the spacetime is a result of the peeling theorem [67, 68, 69], which states that in an appropriate frame and for sufficiently smooth and asymptotically flat initial data near spatial infinity, the ψ 4 component of the curvature has the slowest fall-off with radius, (1 /r ) . \nThe most straight-forward way of evaluating ψ 4 in numerical (Cauchy) simulations is to define an orthonormal basis in the three space (ˆ r , ˆ θ , ˆ φ ) , centered on the Cartesian grid center and oriented with poles along ˆ z . The normal to the slice defines a time-like vector ˆ t , from which we construct the null frame \nO \nl = 1 √ 2 ( ˆ t -ˆ r ) , n = 1 √ 2 ( ˆ t +ˆ r ) , m = 1 √ 2 ( ˆ θ -i ˆ φ ) . (27) \nNote that in order to make the vectors { l , n , m , ¯ m } null, (ˆ r , ˆ θ , ˆ φ ) have to be orthonormal relative to the spacetime metric. In practice, we fix ˆ r and then apply a Gram-Schmidt orthonormalization procedure to determine ˆ θ and ˆ φ ) 3 . It is then possible to calculate ψ 4 via a reformulation of (26) in terms of the geometrical variables on the slice [71] via the electric and magnetic parts of the Weyl tensor, \nψ 4 = C ij ¯ m i ¯ m j , (28) \nwhere \nC ij ≡ E ij -iB ij = R ij -KK ij + K i k K kj -i glyph[epsilon1] i kl ∇ l K jk . (29) \nThe remaining Weyl scalars can be similarly obtained and read \nψ 3 = 1 √ 2 C ij ¯ m i e j r , (30a) \nψ 2 = 1 2 C ij e i r e j r , (30b) \nψ 1 = -1 √ 2 C ij m i e j r , (30c) \nψ 0 = C ij m i m j , (30d) \nwhere ( e j r ) ≡ ˆ r is the unit radial vector. \n≡ In relating ψ 4 to the gravitational radiation, one is limited by the fact that the measurements have been taken at a finite radius from the source. Local coordinate and frame effects can complicate the interpretation of ψ 4 . These problems can largely be alleviated by taking measurements at several radii and performing polynomial extrapolations to r → ∞ . Procedures for doing so have been studied in [72, 73]. In [73] we have shown that if a sufficiently large outermost extrapolation radius is used, the variation in this procedure is of the order ∆ A = 0 . 03% and ∆ φ = 0 . 003 rad in amplitude \nand phase respectively, and is consistent throught the evolution, including inspiral, merger and ringdown. The extrapolation error is larger than the numerical error determined in Sec. IV C 2, below, even if it is performed using data at r = 1000 M distant from the source, highlighting the need for measurements at large radii. For the 'extrapolated' data plotted in this paper, we have performed polynomial extrapolations as detailed in [73], using the six outermost measurements at r = 280 M, 300 M, 400 M, 500 M, 600 M, 1000 M } . \n} In a companion paper [74], we use the same dataset to calculate ψ 4 directly at J + using characteristic extraction [75, 76]. Here the traditional approach (which is gauge dependent and has a finite-radius cut-off error) presented here is replaced by a characteristic formulation of the Einstein equations in order to determine the fields out to future null infinity. In this paper, we restrict ourselves to a discussion of the numerical error inherent in the evolution procedure via the multi-patch code, and will report in more detail on systematic measurement errors due to finite radius effects and the characteristic extraction procedure elsewhere [74, 77].", 'A. Initial data': 'To demonstrate the efficacy of the infrastructure described in the previous sections, we have carried out an evolution of the by now well-studied case of the late-inspiral and merger of a pair of non-spinning equal-mass black holes (see, for example, [78] for an extensive discussion of numerical results involving this model). The particular numerical evolution which we have carried out starts from an initial separation d/M = 11 . 0 and goes through approximately 8 orbits (a physical time of around 1360 M ), merger and ring-down. The masses of the punctures are set to m = 0 . 4872 and are initial placed on the x -axis with momenta p = ( ± 0 . 0903 , ∓ 0 . 000728 , 0) , giving the initial slice an ADM mass M ADM = 0 . 99051968 ± 2 × 10 -8 . These initial data parameters were determined using a post-Newtonian evolution from large initial separation, following the procedure outlined in [79], with the conservative part of the Hamiltonian accurate to 3PN, and radiationreaction to 3.5PN, and determines orbits with a measured eccentricity of e = 0 . 004 ± 0 . 0005 .', 'B. Grid setup': 'The binary black hole evolution was carried out on a 7patch grid structure, as described in Sec. II, incorporating a Cartesian mesh-refined region which covers the near-zone, and six radially oriented patches covering the wave-zone. \nThe inner boundary of the radial grids was placed at r t = 35 . 2 M relative to the centre of the Cartesian grid. As a general rule, this boundary should be made as small as possible to improve efficiency in terms of memory usage. However other factors may make it preferable to move it further out. In particular, since we do not perform time interpolation at grid \n{ \nboundaries, the time step dt of the coarsest Cartesian grid determines the timestep of the radial grids, and thus the wave zone. Updates of the radial grids tend to be expensive, as they are large, so that if dt is too small, computation time may be spent over-resolving (in time) the wave zone. Particularly if the principle interest is in the lower order wave modes, it may be optimal to add an additional Cartesian mesh-refinement grid with a coarser time-step, and thus move r t outwards. \nThe outer boundary for the spherical grids was chosen based on the expected time duration of the measurement and radius of the furthest detector, in order to remove any influence of the artificial outer boundary condition. In particular, given that the evolution takes a time T m for the entire inspiral, merger and ringdown, and gravitational wave measurements taken at a finite radius r d , we would like to ensure that a disturbance travelling at the speed of light from the outer boundary does not reach the measurement radius (see Fig. 4). That is, noting that the physical modes travel at the speed of light, c = 1 [80, 81] 4 , we place the boundary at \nr b > T m +2 r d . (31) \nFor the particular evolution considered here, T m glyph[similarequal] 1350 M , and our outermost measurements are taken at r d = 1000 M . We have placed the outer boundary of the evolution domain at r b = 3600 M . \nThe near-zone grids incorporate 5 levels of 2:1 mesh refinement, covering regions centred around each of the black holes. For the highest resolution we have considered here, the finest grid (covering the black hole horizon) has a grid spacing of dx = 0 . 02 M . The wave-zone grids have an inner radial resolution which is commensurate with the coarse Cartesian grid resolution, dr = 0 . 64 M in this case. This resolution is maintained essentially constant to the outermost measurement radius ( r = 1000 M ), at which point we apply a gradual decrease in resolution (as described in Sec. II A) over a distance of r = 500 M . From r = 1500 M to the outer boundary, we maintain a resolution of dx = 2 . 56 M , sufficient to resolve the inspiral frequencies of the dominant ( glyph[lscript], m ) = (2 , 2) mode of the gravitational wave signal. The transition between the resolutions is performed over a distance of 500 M between r = 1000 M and r = 1500 M . The angular coordinates have 31 points (30 cells) in ν and φ on each of the 6 patches. The time-step of the wave-zone grids is dt = 0 . 144 , and we take wave measurements at each iteration. \nWe have carried out evolutions at three resolutions in order to estimate the convergence of our numerical methods. The grid described above is labelled h 0 . 64 . The lower resolutions, labelled h 0 . 80 and h 0 . 96 have each of the specified grid spacings scaled by 0 . 80 / 0 . 64 and 0 . 96 / 0 . 64 , respectively. \nFIG. 4: Schematic of the causal propagation of information during the evolution. The gravitational wave source is located in the vicinity of r = 0 , with waves propagating outward at the speed of light c = 1 , and are measured at radius r d for a time of interest which would include the inspiral, merger and ringdown of a binary system. The unphysical outer boundary of the grid is located at r b , which is chosen to be sufficiently far removed that the future Cauchy horizon of the domain of dependence of the initial slice does not reach r b until the measurement is complete. \n<!-- image -->', 'C. Results': 'The binary black hole initial data described in Sec. IV A evolves for about 8 orbits ( glyph[similarequal] 1350 M ) before merger. Various ( glyph[lscript], m ) modes of ψ 4 are plotted in Fig. 5. We find that for the grids we have used, the modes to ( glyph[lscript], m ) = (4 , 4) mode are quite well resolved throughout the evolution. The (6 , 6) mode is also measurable, and shows a clear signal, particularly during ringdown. The (8 , 8) mode is dominated by noise for most of the inspiral, though during the merger and ringdown phase, a clear signal is present and the amplitude and frequency can be estimated. Tests with an analytical solution confirm that the angular resolutions which we have used are at best marginal for resolving this mode. \nIn the following sections, we report results regarding the convergence and accuracy of these measurements, as well as determine the parameters of the merger remnant. By analysing the ring-down behaviour of the waves we conclude that the remnant is indeed a Kerr black hole (see Sec. IV C 4, below).', '1. Numerical convergence': 'We can establish the consistency of our discretisation by showing that it does indeed converge to a unique solution in the continuum limit. Ideally, an exact solution can be used to test this. However, since there are no exact solutions which adequately model the physical scenario which we wish to consider (inspiralling black hole binaries), an alternative is to evaluate numerical solutions at several (at least three) different resolutions and establish that the differences decrease as \nresolution is increased. For an implementation in which all of the discrete operations are carried out with the same order of accuracy, the convergence test should yield a clear exponent corresponding to that order. \nThe evolution code incorporates a number of discrete operations, which for various practical reasons, are carried out to different orders of accuracy. These are listed in Table I. The primary operation which is carried out over the bulk of the grid is the computation of finite difference derivative operations in order to evaluate the right-hand side of the evolution equations (22a)-(22e). For the tests carried out in this paper, 8th-order stencils are used for this operation, including the upwinded advection terms. It is common to apply a small amount of artificial dissipation in order to smooth highfrequency effects. This is done at one higher order (9th) than the interior finite differencing in order to maintain the correct continuum limit. (In our experiments, however, we have noted that dissipation at this high order has a negligible impact on the solution, and can effectively be omitted.) Various boundary operations (inter-patch boundary communication, meshrefinement boundaries) are carried out at lower order. This is done largely for efficiency reasons, as the communication involved in boundary interpolation can be time-consuming if the stencil widths are large. Intuitively, the numerical error associated with these operations may have reduced influence in any case, as they are applied only at 2D interfaces. In practise this does seem to be the case, for instance, as experiments with 4th and 5th order interpolation operators between patches show similar accuracy in the final solution. Similarly, operations involving different time-levels are at lower order, again for efficiency reasons. The time resolution of our evolutions tends to be high enough that one might expect a small error coefficient of the RK4 integrator. The lowest order operation which we use is the 2nd-order time interpolation at meshrefinement boundaries. Applying higher order here would require keeping more time levels in memory (currently we store three). Our results are consistent with previous studies using mesh-refinement for black hole evolution which suggest that the influence of the low order time-interpolation boundary conditions is negligible for the time resolutions which we apply (see, for example, [65]). \nFor test cases involving a single non-spinning black hole, in fact we find 8th-order convergence in the Hamiltonian constraints. This is likely due to the relatively constant values (except for some gauge evolution) maintained by the evolution variables during the evolution, which minimises error due to time-integration or propagation across boundaries. \nA more relevant situation is that of a binary black hole inspiral, which we have tested using the parameters described above in Sec. IV A. For this model, we have measured the gravitational waveform, ψ 4 , integrated over spheres at radii from r = 100 M to r = 1000 M , at the three resolutions h 0 . 96 , h 0 . 80 and h 0 . 64 . Results for the ( glyph[lscript], m ) = (2 , 2) mode are shown in Fig. 6. The evolution lasts for about 1350 M before merger, and the plots encompass the inspiral, merger (at t = 0 M on this time axis), and ringdown. The figure plots the error in phase ∆ φ and relative amplitude ∆ A for the (2 , 2) mode extracted at r = 100 M and r = 1000 M , respec- \nFIG. 5: The dominant spherical harmonic modes of ψ 4 for glyph[lscript] = 2 , 4 , 6 , 8 , measured at r = 200 M from the coordinate centre. The plots on the right show amplitude and frequency evolution during the late inspiral, merger and ringdown.. \n<!-- image --> \nTABLE I: Table of convergence order of various numerical aspects of the evolution code. Spatial restriction is carried out by a direct copy. The surface integration is exact for polynomials up to degree 2 N -1 , where N is the number of grid-points along one direction on the sphere. \n| Numerical method | Order |\n|-----------------------------------|---------|\n| Grid interior finite differencing | 8 |\n| Inter-patch interpolation | 5 |\n| Kreiss-Oliger dissipation | 9 |\n| Time integration (RK4) | 4 |\n| Mesh-refinement: | |\n| Spatial prolongation | 5 |\n| Spatial restriction | n/a |\n| Time interpolation | 2 |\n| Analysis tools: | |\n| Interpolation | 4 |\n| Finite differencing | 8 |\n| Surface integration | 2 N - 1 | \n, between medium h 0 . 80 and low h 0 . 96 resolutions and high h 0 . 64 and medium h 0 . 80 resolutions in the wave-zone. The latter error is scaled such that the curves will overlap in the case of a 4th-order convergent solution. At both radii, we \nfind that during the inspiral phase, the rescaled error of the higher resolutions lies below that of the lower resolution, suggesting better than 4th-order convergence (in fact, closer to 8th-order over significant portions of the plot). At later times, around the peak of the waveform, the curves are more closely aligned, indicating 4th-order convergence. The plot suggests that during the very dynamical late stages of the inspiral, the lower order boundary conditions and/or the time integration, play a more important role relative to the early inspiral phase of the evolution, where the convergence order is closer to that of the interior finite differencing. The results are, however, convergent over the entire evolution (including merger and ringdown). As we will see in the next section, the accuracy is excellent for these resolutions so that the rate of convergence is not a particular issue. \nWe have verified convergence for a number of different modes of the ψ 4 waveform at different radii. For instance, Fig. 7 shows similar results for the ( glyph[lscript], m ) = (6 , 6) mode, which is some two orders of magnitude smaller in peak amplitude than the ( glyph[lscript], m ) = (2 , 2) mode (see Fig. 5). During the early inspiral, it is difficult to evaluate a convergence order due to high frequency noise which is large relative to the waveform amplitude. However, a measurable signal is clear in the last orbit, merger and ringdown phase, and converges at a clear 3rd order. \n<!-- image --> \nFIG. 6: Convergence in amplitude (top) and phase (bottom) of the ( glyph[lscript], m ) = (2 , 2) mode of ψ 4 for detectors at r = 100 M and r = 1000 M . The higher resolution difference, h 0 . 80 -h 0 . 64 , is scaled for 4th-order convergence. \n<!-- image --> \nFIG. 7: Convergence in amplitude (top) and phase (bottom) of the ( glyph[lscript], m ) = (6 , 6) mode of ψ 4 for detector at r = 100 M during the late through merger. The higher resolution difference, h 0 . 80 -h 0 . 64 , is scaled for 3rd-order convergence. \n<!-- image -->', '2. Accuracy': 'We estimate the numerical phase and amplitude error by means of a Richardson expansion at a given resolution ∆ , \nu ∆ ( t, x ) = u ( t, x ) + ∆ e 1 ( t, x ) + ∆ 2 e 2 ( t, x ) + · · · , (32) \nwhere u ( t, x ) is the solution of the original differential equation, and the e i ( t, x ) are error terms at different orders in ∆ . Assuming convergence at a fixed order, n , we can expect some of these error functions to vanish. Using solutions, u , obtained at two resolutions, ∆ 1 and ∆ 2 , the Richardson expansion implies \nu ∆ 1 -u ∆ 2 = e n (∆ n 1 -∆ n 2 ) + O (∆ n +1 ) = e n ∆ n 2 ( C n -1) + O (∆ n +1 ) ∼ glyph[epsilon1] ∆ 2 ( C n -1) , (33) \nwhere glyph[epsilon1] ∆ 2 is the estimated solution error on the higher resolution grid, and where \nC n := ( ∆ 1 ∆ 2 ) n . (34) \nWe thus obtain an estimate for the solution error that is at least accurate to order n +1 , \nglyph[epsilon1] ∆ 2 ∼ 1 C n -1 ( u ∆ 1 -u ∆ 2 ) , (35) \nwhich we use as an estimate of the numerical error in our solutions. \nDuring the inspiral phase (which for this purpose we regard as being the period t ≤ -100 M ), we have found roughly 8thorder convergence in the amplitude and phase, as described above. The remaining relative error for the ( glyph[lscript], m ) = (2 , 2) mode can be estimated as \nmax T ∈ [ -1350 , -100] err ( A ) inspiral = 0 . 090% , (36a) \nmax T ∈ [ -1350 , -100] err ( φ ) inspiral = 0 . 010% . (36b) \nwhere err ( A ) := ∆ A/A and err ( φ ) := ∆ φ/φ , i.e., the rate of loss of phase with φ . During merger and ring-down ( t > -100 M ), we observe 4th-order convergence in the amplitude, while maintaining 8th-order convergence in the phase. This results in the estimate \nmax T ∈ ( -100 , 150] err ( A ) merger = 0 . 153% , (37a) \nmax T ∈ ( -100 , 150] err ( φ ) merger = 0 . 003% . (37b) \nThe time evolution of the numerical error in phase and amplitude is shown in Fig. 8. \nWe note that these errors are of comparable order to the errors inherent in the extrapolation [73]. Moreover, as is pointed out in [74], the error between extrapolated waveforms and those determined at future null infinity, J + , by characteristic extraction, is an order of magnitude larger than the numerical error determined here. This highlights the importance of reducing systematic errors inherent in finite radius measurements of ψ 4 . \nFIG. 8: Absolute numerical error in the amplitude (top) and phase (bottom) accumulated over the course of the evolution for the highest resolution run, determined according to Eq. (35) for the pointwise differences in amplitude and phase between medium and high resolution runs. For the phase we assume the measured 8th-order convergence over the entire evolution, while for the amplitude we use 8th-order before t ≤ -100 , and 4th-order thereafter (see text). \n<!-- image -->', '3. Properties of the merger remnant': 'The merger remnant can be measured with high accuracy, using either the isolated horizon formalism [82, 83], or geometrical measures of the apparent horizon [84, 85]. Some results are reported in Table II, along with estimated numerical errors. The results agree well with previous highaccuracy measurements, such as those obtained by spectral evolution [4, 78], with the spin and irreducible mass agreeing within three decimal and four decimal places, respectively. places. While this is larger than the reported errors, we note that we have evolved a different initial data set than [4]. As reported in Sec. IV A our evolution has somewhat more eccentricity, and the level of agreement can be used to judge the influence of small amounts of eccentricity on the result. \nBy comparing the properties of the merger remnant with the integrated radiated energy, E rad , and angular momentum, J rad , determined from the gravitational waveforms, we find the residuals \n| M f + M rad -M ADM | = 2 . 6 × 10 -4 , (38a) \n| S f + J rad -J ADM | = 3 . 1 × 10 -4 . (38b) \nHere we have used the extrapolations of the gravitational waveforms to r →∞ based on the 6 outermost measurement radii. A more detailed discussion of this procedure is given in [73]. The results can be further improved through taking measurements at J + , as outlined in [74, 77].', '4. Quasi-normal modes of the merger remnant': "In Fig. 5, we have shown the late-time behaviour of the amplitude and frequency for the dominant spherical harmonic modes of ψ 4 , to ( glyph[lscript], m ) = (8 , 8) . We note that during ringdown, the frequencies settle to a constant value. If the final black hole is a Kerr black hole, these frequencies are given by \nTABLE II: Properties of the merger remnant as measured on the apparent horizon ( M irr , S f /M 2 f ) and from the gravitational radiation ( E rad , J rad ). Ranges indicate the estimated numerical error. For the error in J ADM , we have simply quoted machine precision (it is an analytical expression of the input momenta on the conformally flat initial slice). \n| Total ADM mass, M ADM | 0 . 99051968 ± 20 × 10 - 9 |\n|-----------------------------------|------------------------------|\n| Total ADM angular momentum, J ADM | 0 . 99330000 ± 10 × 10 - 17 |\n| Irreducible mass, M irr | 0 . 884355 ± 20 × 10 - 6 |\n| Spin, S f /M 2 f | 0 . 686923 ± 10 × 10 - 6 |\n| Christodoulou mass, M f | 0 . 951764 ± 20 × 10 - 6 |\n| Angular momentum, S f | 0 . 622252 ± 10 × 10 - 6 |\n| Radiated energy, E rad | 0 . 038546 ± 51 × 10 - 6 |\n| Radiated angular momentum, J rad | 0 . 370391 ± 17 × 10 - 6 | \nthe quasi-normal modes of a Kerr black hole with given spin a . \nAs reported in the previous section, our evolution leads to a merger remnant with a = 0 . 686923 ± 1 × 10 -5 (see Table II), as measured on the horizon. The real part of the prograde quasi-normal mode (QNM) frequencies for modes up to ( glyph[lscript], m ) = (7 , 7) , can be found tabulated in [86]. For example, Mω 22 = 0 . 526891 for the ( glyph[lscript], m ) = (2 , 2) mode, given a final black hole of the measured mass M f and spin S f . \nAt this point it is worth noting that the QNM determined from perturbations of a Kerr black hole are most naturally expressed in terms of a basis of spin-weighted spheroidal harmonics. By contrast, our waveforms have been decomposed relative to a basis of spin-weighted spherical harmonics, which are easily calculated via Legendre functions. In order to make an appropriate comparison between these modes with the perturbative results we need to apply a transformation to the wave-modes. We have \nˆ ψ glyph[lscript] ' m ' 4 = ∑ glyph[lscript],m ψ glyph[lscript],m 4 〈 glyph[lscript], m | glyph[lscript] ' , m ' 〉 , (39) \nwhere a dash denotes labelling of the spheroidal harmonic modes, and 〈 glyph[lscript], m | glyph[lscript] ' , m ' 〉 is the overlap defined by \n〈 glyph[lscript], m | glyph[lscript] ' , m ' 〉 = ∫ Ω d Ω -2 ¯ S glyph[lscript] ' m ' ( c glyph[lscript] ' m ' ) -2 Y glyph[lscript]m . (40) \nThe spheroidal harmonics parameter c glyph[lscript] ' m ' = aω glyph[lscript] ' m ' depends on the spin a of the black hole and the corresponding prograde or retrograde QNM frequency ω glyph[lscript] ' m ' of the ( glyph[lscript] ' m ' ) spheroidal harmonic mode 5 . If c = 0 (as is the case for non-spinning black holes), the spheroidal harmonics reduce to the spherical harmonics. The spin-weighted spheroidal harmonics used here have been implemented following Leaver [87] and are reviewed in [86]. \nThe frequencies measured during the ringdown are plotted in Fig. 9 for the modes ( glyph[lscript], m ) = (2 , 2) , (4 , 4) and (6 , 6) . We \nTABLE III: Prograde N = 0 QNM frequencies for different modes and spin a = 0 . 6869 as determined by perturbative methods [86], ω lit . , and as measured during ringdown in the numerical relativity simulation, ω NR . \n| ( glyph[lscript], m ) | M f ω lit . | M f ω NR | ω NR - M f ω lit . | |\n|-------------------------|---------------|---------------------|------------------------|\n| (2 , 2) | 0 . 526891 | 0 . 5267 ± 0 . 0011 | 1 . 9 × 10 - 4 |\n| (4 , 4) | 1 . 131263 | 1 . 1312 ± 0 . 0028 | 6 . 3 × 10 - 5 |\n| (6 , 6) | 1 . 707630 | 1 . 7074 ± 0 . 0662 | 2 . 3 × 10 - 4 | \nhave plotted data for the r = 1000 M measurement, as well as the value obtained by extrapolating the waveforms extracted at the outermost 6 measurement spheres to r →∞ , and find that in fact the extrapolation has little effect on the frequency of the lower order modes at these distances from the source. We note that there is a modulation of the ringdown frequency, particularly apparent in the (2 , 2) mode. This is a result of mode mixing, which stems from the use of the spherical harmonic basis for the ψ 4 measurements. By transforming the r = 1000 M result to spheroidal harmonics, this modulation visible in the t < 40 M signal is largely removed (dashed line). \nAs the amplitude of the wave declines exponentially to the level of numerical error, the frequencies become difficult to measure accurately. We estimate the ringdown frequency for each mode by performing a least-squares fit of a horizontal line through the measured spheroidal harmonic frequency over the range t ∈ [40 , 80] M (dotted line) with the standard deviation of the fit as a gauge of the error (grey region). These constant lines represent the estimated frequency of the associated QNM modes, and are tabulated as ω NR in Table III. They agree to high precision with the prograde QNM frequencies, ω lit . , determined Kerr black holes by perturbative methods [86]. We conclude that the merger remnant is compatible with a Kerr black hole within the given error estimates.", 'V. DISCUSSION': 'The results of this paper provide a demonstration of the usefulness of adapted coordinates in numerical relativity simulations. The precision of the calculations have allowed us to obtain convergent modes to glyph[lscript] = 6 , through merger and ringdown, with accurate predictions of the quasi-normal ringdown frequencies of the remnant. \nOur implementation of non-singular radially adapted coordinates for the wave zone is based on the use of multiple grid patches with interpolating boundaries, coupled to a BSSNOK evolution code. Thornburg [41] first demonstrated that such a setup could lead to stable evolutions in the case of a spinning black hole in Kerr-Schild coordinates. We have demonstrated that the approach is also effective and robust for dynamical puncture evolutions, and in particular the problem of binary black holes. \nThe implementation described here has a number of advantages, principle among them being its flexibility. While we have presented results for a particular grid structure adapted to radially propagating waves, there are no principle prob- \n<!-- image --> \n<!-- image --> \nFIG. 9: The ringdown frequencies for the dominant ψ 4 modes to glyph[lscript] = 6 of the merger remnant. From top to bottom, the plots show the frequencies of the ( glyph[lscript], m ) = (2 , 2) , (4 , 4) and (6 , 6) modes respectively, over a timescale from the (2 , 2) waveform peak to 100 M later, at which point the waveform amplitude is too small to measure an accurate frequency. The ψ 4 data measured at r = 1000 M is plotted, in addition to the value extrapolated to r → ∞ , and the transformation to spheroidal harmonics. The expected quasi-normal mode frequency is plotted as a dotted line, as well as a fit to the spheroidal harmonic data over the range t ∈ [40 M, 80 M ] , with error-bars determined by the standard deviation of the fit. \n<!-- image --> \nlems with restructuring the grids to cover any required domain, for instance adapted to excision boundaries or toroidal fields. Since data is stored in the underlying Cartesian basis, and passed by interpolation across boundaries, the coordinates used on each patch are largely independent of the others, and there is no need for numerical grid generating schemes. While we have used the BSSNOK formalism to evolve the Einstein equations, in principle any stable strongly hyperbolic system can be substituted. The BSSNOK system has, however, proven particularly useful for evolving black holes via the puncture approach, which itself has proven to be a very flexible methodology. We have demonstrated results for the most well-studied test case, non-spinning equal-mass black holes, the same techniques can be applied to different mass ratios and spinning black holes, simply by changing the physical input parameters. (The appendices include some examples of spinning black hole evolutions.) \nFinally, we emphasise again the accuracies which can be attained by this approach. Our finite difference results show numerical error estimates which are on par with those achieved using spectral spatial discretisation [4]. The adapted radial coordinate allows us to take measurements at radii much larger than have been used before, as well as obtain accurate measurements of higher glyph[lscript] modes during merger, which have an amplitude more than two orders of magnitude smaller than the dominant ( glyph[lscript], m ) = (2 , 2) mode. One of the aspects which makes this possible is the fact that we are able to extend our grids to a distance such that the measurements are included in the future domain of dependence of the initial data (causally disconnected outer boundaries), and the waves are reasonably well resolved over this entire domain so that internal reflections are minimised. Furher, we note that our results are consistent with other puncture-method calculation in that the results are convergent and can be consistently extrapolated to r → ∞ throughout the entire evolution, including late inspiral and ringdown [73], where other approaches have had difficulties. \nThe absence of artificial boundaries, as well as dissipative regions in the wave zone, removes an important source of potential error in solving the Einstein equations as an initialboundary value problem. The remaining errors can be categorised in three forms. First, numerical error due to the discretisation. This can be reduced through the use of higher order methods for the operations performed in various parts of the code, and fortunately is also easy to quantify by performing tests at multiple resolutions. We note that for finite differences, the largest improvement in accuracy occurs in going from 2nd to 4th-order for the interior computations, and beyond that there are diminishing returns [88]. While it does not yet seem to be a limiting factor, except possibly during the merger, the RK4 time-stepping will at some level of resolution be a determining factor in the accuracy regardless of the spatial order (and this is also the case for current implementations of spectral methods). The second source of error is a physical error, inherent in the choice of initial data parameters for the binary evolution. At the separations which are practical for numerical relativity (say d < 20 M ), the physical model is expected to have shed all of its eccentricity. We have used post-Newtonian orbital parameters to attempt to place our black holes in low eccentricity trajectories, and this is quite effective. Alternative approaches, involving iteratively correcting the initial data parameters until a tolerable eccentricity has been reached, are able to reduce the eccentricity still further [89]. This technique can in principle also be adapted to the moving puncture approach. The final source of error arises in the measurement of ψ 4 , which is done at a finite radius, and then extrapolated to r → ∞ by some procedure. We have attempted to minimise this error by placing detectors at large radii, well into the region where the perturbations are linear, and have shown that the extrapolations are consistent with measurements at larger radii, as well as with each other in the r → ∞ limit [73]. However, there remain ambiguities particularly in gauge-dependent quantities such as the choice of surface on which measurements are taken, and the definition of time and radial distance to be used in the ex- \non. In a companion paper [74], we have demonstrated that these ambiguities can be removed entirely by the procedure of characteristic extraction , whereby evolution data on a world-tube is used as an inner boundary condition for a fully relativistic characteristic evolution, extending to null infinity, J + . The results suggest that systematic errors inherent in finite radius measurements of ψ 4 are more than an order of magnitude larger than the numerical errors reported here.', 'Acknowledgments': 'We dedicate this paper to the memory of Thomas Radke, who has made invaluable contributions to the development and optimisation of Cactus, Carpet and the code described here. The authors are pleased to thank: Ian Hinder, Sascha Husa, Badri Krishnan, Philipp Moesta, Christian D. Ott, Luciano Rezzolla, Jennifer Seiler, Jonathan Thornburg, and Burkhard Zink for their helpful input; the developers of Cactus [47, 48] and Carpet [49, 50, 51] for providing an open and optimised computational infrastructure on which we have based our code; Nico Budewitz for optimisation work with our local compute cluster, damiana ; support from the DFG SFB/Transregio 7, the VESF, and by the NSF awards no. 0701566 XiRel and no. 0721915 Alpaca . Computations were performed at the AEI, at LSU, on LONI (numrel03), on the TeraGrid (TG-MCA02N014), and the Leibniz Rechenzentrum Munchen (h0152).', 'APPENDIX A: THE INFLUENCE OF UPWINDED ADVECTION STENCILS': "It has long been recognised that for BSSNOK evolutions employing a shift vector, β a , the overall accuracy can be improved by 'upwinding' the finite difference stencils for advective terms of the form β i ∂ i u [59]. The upwind derivatives employ stencils which are off-centred by some number of grid points in the direction of β a . The drawback of the method is that in order to maintain the same order of accuracy in the derivatives, the stencil must have the same width as a centred stencil, but since it is offset in either a positive or negative direction, it effectively requires an additional number of points to be available to the derivative operator equal to the size of the offset. For parallel codes which physically decompose the grid over processors and communicate ghost-zone boundaries, this means that a larger number of points must be communicated and can impact the overall efficiency. Further, a larger number of points must be translated at inter-patch and refinement level boundaries. \nThe original observation that upwinding is helpful was made with a code that used 2nd-order spatial finite differences. In that case, the centred stencils are small (three points) and the upwind derivatives correspond to sideways derivatives in the direction of the shift, i.e., no 'downwind' information is used. For higher order schemes, the importance of upwinding may be less significant, since the stencils are large relative to the size of the shift vector. In practise, some implementations \nFIG. 10: Trajectories of the two inspiralling punctures for a spinning configuration a 1 = -a 2 = 0 . 8 , with upwinded advection terms (solid lines) and without (dashed lines). In the case where no upwinding has been used, the black holes do not inspiral, due to the accumulation of numerical error. \n<!-- image --> \nhave empirically determined that upwinding by 1 point at 6thorder is helpful [79]. However, this is not done universally, particularly in conjunction with 8th-order centred differencing [12, 90]. \nWe have found upwinding to be important in reducing numerical error in the black hole motion for every order of accuracy we have tried. The effect is demonstrated in Fig. 10, which plots the motion of the black hole punctures for a data set involving a pair of equal-mass binaries with spins a 1 = -a 2 = 0 . 8 evolved at a relatively low resolution with 8th-order spatial finite differencing. The results of two evolutions are plotted, one using fully centred stencils, and the other upwinding the advection terms with a one-point offset. Whereas the latter evolution displays the expected inspiral behaviour, at this resolution the binary evolved with centred advection actually flies apart. The is purely a result of accumulated numerical error, and at higher resolutions both tracks can be made to inspiral and merge. Our observation, however, is that for a given fixed resolution, the one-point offset advection has a significantly reduced numerical error in the phase as compared to the fully centred derivatives. \nBased on some limited experimentation with larger offsets, we have the general impression that the one point offset provides the optimal accuracy for each of the finite difference orders we have tried (4th, 6th, 8th). We do not exclude the possibility that there may be situations in which the fully centred stencils perform as well as upwinded advection, however we have not come across a situation where the latter method performs worse. \nAs an alternative, we have also tested lower order upwinded derivatives as a potential scheme which would allow us to \nFIG. 11: Phase evolution of the ( glyph[lscript], m ) = (2 , 2) mode ψ 4 for the aligned-spin model with a 1 = -a 2 = 0 . 8 h = 0 . 64 M . The 6thorder case at h 0 . 64 has a trajectory between the low resolution ( h 0 . 80 ) and high resolution ( h 0 . 64 ) 8th-order evolution. \n<!-- image --> \nmaintain a smaller stencil width. We generally find that the resultant numerical errors are of the same magnitude or larger than if we had not done the upwind at all. \nWe note parenthetically the fact that the off-centering is most important in the immediate neighbourhood of the black holes, where the shift has a non-trivial amplitude. It is possible that a scheme where the stencils are off-centred only on grids where the shift is larger than some threshold would also be effective, and not suffer the drawbacks mentioned above over the bulk of the grid. We have not experimented with such a scheme, however.", 'APPENDIX B: HIGH ORDER FINITE DIFFERENCING': 'A recent trend in the implementation of finite difference codes for relativity has been the push towards higher order spatial derivatives, It is now common to use 6th or 8th-order stencils. The benefit of higher order stencils is that the convergence rate can be dramatically increased, so that a small increase in resolution leads to a large gain in accuracy. And while not guaranteed, it is often the case that for a given fixed resolution, a higher order derivative will be more accurate, requiring fewer points to accurately represent a wavelength [88]. \nIn moving to high order stencils, there is a trade-off between the possible accuracy improvements, and the extra computational cost. High order stencils generally involve two extra floating point operations per order. Since they require a larger stencil width, they also incur a cost in communication of larger ghost zones, as well as requiring wider overlap zones at grid boundaries. In practice, we find that higher order stencils can also have a more strict Courant limit, requiring a smaller timestep (and thus more computation to reach a given physical time). While it is possible to demonstrate a large gain in accuracy in switching from 2nd to 4th-order operators, there are diminishing returns in the transition to 6th and higher order [88]. \nWe have experimented with 4th, 6th and 8th-order finite differencing for the evolution equations. Generally we find that the 8th-order operators can indeed provide a notable benefit, particularly in the phase accuracy, at low resolution. In \nFig. 11, we plot the phase evolution for an equal mass model with spins a 1 = -a 2 = 0 . 8 . The evolution covers the last three orbits and ringdown. We find that for this high-spin case, even over this short duration, a significant dephasing takes place. Assuming 8th-order convergence, the 6th-order evolution at the h 0 . 64 resolution would be comparable to the 8thorder at approximately h 0 . 77 resolution. We can get some idea of the relative amount of work required for each calculation by noting there would be N = (0 . 64 / 0 . 77) 3 fewer grid points in the h 0 . 77 evolution, but the 8th-order derivatives require 9 / 7 times as many floating point computations for a derivative in one coordinate direction, and requires a Courant factor which is 0 . 9 times that of the 6th-order run. Taken together, this suggests an 8th-order run at h 0 . 77 would require a factor 0 . 68 of the amount of work of the 6th-order case to achieve comparable accuracy. Note that this computation does not take into account potential additional communication overhead associated with the wider 8th-order stencils. But assuming this is not dominant, the conclusion seems to be that for this level of accuracy, the 6th-order evolution is somewhat less efficient than the 8th-order version would be. \nFor a given situation, it may be that these factors change significantly. Implementation, and even hardware, details can shift the balance of costs between various operations. Further, the test case considered here involves a fairly high spin. Lower spin models (such as that considered in the main body of the paper), are accurate at modest resolutions, and in such cases the 6th-order evolutions may in fact prove to be relatively more efficient if the accuracy is already sufficient for a given purpose. On the other hand, if grid sizes and memory consumption are limiting factors, the 8th-order operators do give a consistent accuracy benefit for a fixed grid size. Our expectation, however, is that implementing yet higher order stencils (for example, 10th-order) may not be justified on the basis of efficiency. \nAs a final point, we note that the required high-order accuracy appears to be largely a consequence of the field gradients in the near-zone, immediately surrounding the black holes. An alternative scheme, then, could be to apply highorder finite differencing in this region, while using a lower order (and thus more efficient) scheme in the wave zone. Results from such a test are displayed in Fig. 12, where we have used 8th-order only on the finest refinement level, i.e. , the mesh surrounding the black holes, but 4th-order on all coarser Cartesian and radial wave-zone grids. This, in turn, allows for a slightly less restrictive Courant limit, so that it becomes possible to run with a slightly larger time-stepping. The phase evolution of ψ 4 is almost identical to that of the fully 8th-order case, but the we found that the speed of the run was increased by more than 25% (similar to that of the fully 6th-order evolution). Further optimisations, such as decreasing ghost-zone sizes of the 4th-order grids and consequently the communication overhead, might improve this further. While the errors and convergence order of this scheme have not been tested in detail, we suggest it as a potentially quite effective scheme for the impatient. \nFIG. 12: Amplitude and phase evolution of the ( glyph[lscript], m ) = (2 , 2) mode of ψ 4 for the equal-mass aligned-spin model, comparing 8th-order spatial finite differencing with a scheme in which 8th-order is used only on the fine meshes surrounding the bodies, and 4th-order on the wave-zone grids. \n<!-- image --> \nFIG. 13: Differences in phase of a spinning configuration with resolution h = 0 . 80 M and conformal variables φ and W against a simulation with h = 0 . 64 M and conformal variable W . The dephasing is significant as we are on the coarse limit of resolution for this particular configuration. \n<!-- image -->', 'APPENDIX C: CHOICE OF CONFORMAL VARIABLE': "In Sec. III, we have described our implementation of the BSSNOK evolution system, and note that currently three variations are in use, based on the use of different variables to represent the conformal scalar. The original formulation is based on the use of φ := log γ/ 12 . An issue with this variable in the context of puncture evolutions is that it has an O (ln r ) singularity which can lead to large numerical error in finite \ndifferences calculated in the neighbourhood of the puncture. More recently, the use of alternative variables χ = γ -1 / 3 [3] and W = γ -1 / 6 [64] have been proposed as a means of improving this situation by replacing φ with variables that are regular everywhere on the initial data slice. In terms of the evolution system outlined in Eqs. (22), the χ and W options correspond to the choices κ = 3 and κ = 6 , respectively. \nThe influence of this change of variable can be seen in improved phase accuracy of binary evolutions carried out with either χ or W . 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2013MNRAS.432..530R

Radiative efficiency, variability and Bondi accretion on to massive black holes: the transition from radio AGN to quasars in brightest cluster galaxies

2013-01-01

['accretion', 'accretion disks', 'galaxies active', 'galaxies jets', 'astronomy x rays', '-']

[]

We examine unresolved nuclear X-ray sources in 57 brightest cluster galaxies to study the relationship between nuclear X-ray emission and accretion on to supermassive black holes. The majority of the clusters in our sample have prominent X-ray cavities embedded in the surrounding hot atmospheres, which we use to estimate mean jet power and average accretion rate on to the supermassive black holes over the past several hundred Myr. We find that roughly half of the sample have detectable nuclear X-ray emission. The nuclear X-ray luminosity is correlated with average accretion rate determined using X-ray cavities, which is consistent with the hypothesis that nuclear X-ray emission traces ongoing accretion. The results imply that jets in systems that have experienced recent active galactic nucleus (AGN) outbursts, in the last ∼10<SUP>7</SUP> yr, are `on' at least half of the time. Nuclear X-ray sources become more luminous with respect to the mechanical jet power as the mean accretion rate rises. We show that nuclear radiation exceeds the jet power when the mean accretion rate rises above a few per cent of the Eddington rate, or a power output of {∼ }10^{45} {erg s^{-1}}, where the AGN apparently transitions to a quasar. The nuclear X-ray emission from three objects (A2052, Hydra A, M84) varies by factors of 2-10 on time-scales of 6 months to 10 years. If variability at this level is a common phenomenon, it can account for much of the scatter in the relationship between mean accretion rate and nuclear X-ray luminosity. We find no significant change in the spectral energy distribution as a function of luminosity in the variable objects. The nuclear X-ray luminosity is consistent with emission from either a jet, an advection-dominated accretion flow, or a combination of the two, although other origins are possible. We also consider the longstanding problem of whether jets are powered by the accretion of cold circumnuclear gas or nearly spherical inflows of hot keV gas. For a subset of 13 nearby systems in our sample, we re-examine the relationship between the jet power and the Bondi accretion rate. The results indicate weaker evidence for a trend between Bondi accretion and jet power, due to uncertainties in the cavity volumes and gas densities at the Bondi radius. We suggest that cold gas fuelling could be a likely source of accretion power in these objects; however, we cannot rule out Bondi accretion, which could play a significant role in low-power jets.

[]

https://arxiv.org/pdf/1211.5604.pdf

{'H. R. Russell 1 ∗ , B. R. McNamara 1 , 2 , 3 , A. C. Edge 4 , M. T. Hogan 4 , R. A. Main 1 , A. N. Vantyghem 1': '- 1 Department of Physics and Astronomy, University of Waterloo, Waterloo, ON N2L 3G1, Canada\n- 2 Perimeter Institute for Theoretical Physics, Waterloo, Canada\n- 3 Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA\n- 4 Department of Physics, Durham University, Durham DH1 3LE \n29 October 2018', 'ABSTRACT': "We examine unresolved nuclear X-ray sources in 57 brightest cluster galaxies to study the relationship between nuclear X-ray emission and accretion onto supermassive black holes. The majority of the clusters in our sample have prominent X-ray cavities embedded in the surrounding hot atmospheres, which we use to estimate mean jet power and average accretion rate onto the supermassive black holes over the past several hundred Myr. We find that roughly half of the sample have detectable nuclear X-ray emission. The nuclear X-ray luminosity is correlated with average accretion rate determined using X-ray cavities, which is consistent with the hypothesis that nuclear X-ray emission traces ongoing accretion. The results imply that jets in systems that have experienced recent AGN outbursts, in the last ∼ 10 7 yr, are 'on' at least half of the time. Nuclear X-ray sources become more luminous with respect to the mechanical jet power as the mean accretion rate rises. We show that nuclear radiation exceeds the jet power when the mean accretion rate rises above a few percent of the Eddington rate, or a power output of ∼ 10 45 erg s -1 , where the AGN apparently transitions to a quasar. The nuclear X-ray emission from three objects (A2052, Hydra A, M84) varies by factors of 2 -10 on timescales of 6 months to 10 years. If variability at this level is a common phenomenon, it can account for much of the scatter in the relationship between mean accretion rate and nuclear X-ray luminosity. We find no significant change in the spectral energy distribution as a function of luminosity in the variable objects. The relationship between accretion and nuclear X-ray luminosity is consistent with emission from either a jet, an ADAF, or a combination of the two, although other origins are possible. We also consider the longstanding problem of whether jets are powered by the accretion of cold circumnuclear gas or nearly spherical inflows of hot keV gas. For a subset of 13 nearby systems in our sample, we re-examine the relationship between the jet power and the Bondi accretion rate. The results indicate weaker evidence for a trend between Bondi accretion and jet power, primarily due to the uncertainty in the cavity volumes. We suggest that cold gas fuelling could be a likely source of accretion power in these objects however we cannot rule out Bondi accretion, which could play a significant role in low power jets. \nKey words: X-rays: galaxies: clusters - galaxies:active - galaxies:jets - accretion, accretion discs", '1 INTRODUCTION': "Energetic feedback from supermassive black holes (SMBHs) plays an important role in the formation and evolution of galaxies. \n- ∗ E-mail: [email protected] \nWith this realisation, the exploration and understanding of active galactic nuclei has taken a new emphasis. Key lines of evidence include the relationship between nuclear black hole mass and the mass of the host galaxy (Mσ relation; Magorrian et al. 1998; Kormendy & Gebhardt 2001; Merritt & Ferrarese 2001), which was likely imprinted through the quasar era ('quasar mode feedback'; Silk & Rees 1998; Haehnelt, Natarajan & Rees 1998; \nDi Matteo, Springel & Hernquist 2005), and the prevalence of radio bubbles in the X-ray atmospheres of giant elliptical and brightest cluster galaxies ('radio mode feedback'; McNamara et al. 2000; Churazov et al. 2000; Fabian et al. 2000; McNamara & Nulsen 2007). These forms of feedback are ultimately powered by binding energy released by accretion onto massive black holes. However, energy is released primarily in the form of radiation from quasars, while radio jets release their energy in a mechanical form. \nThe reason why black holes release their binding energy in different forms is poorly understood. Clues have come from the so-called fundamental plane of black holes (Merloni, Heinz & di Matteo 2003; Falcke, Kording & Markoff 2004). This relationship between the radio and X-ray power emerging from the vicinity of the black hole and the black hole mass extends over nine orders of magnitude from stellar mass black holes to supermassive black holes. The continuity of this relationship indicates that the emergent properties of black holes are fundamentally the same regardless of their mass. Furthermore, accreting black hole binaries undergo changes in emission states that seem to correlate with the accretion rate normalized to the black hole mass (eg. Fender et al. 1999; Maccarone, Gallo & Fender 2003; Remillard & McClintock 2006). \nChanges in the specific accretion rate apparently lead to structural changes in the accretion disk which govern the release of binding energy in the form of a jetted outflow or radiation. During periods of high accretion rate a geometrically thin and optically thick disk forms that dissipates its energy primarily in the form of radiation (Shakura & Sunyaev 1973; Novikov & Thorne 1973; Frank, King & Raine 2002). During periods of more modest accretion, a hot, radiatively inefficient accretion flow (RIAF) or advection-dominated accretion flow (ADAF) forms that releases its energy in the form of a jetted outflow or wind (eg. Narayan & Yi 1994; Abramowicz et al. 1995; Narayan & McClintock 2008). ADAFs are geometrically thick, optically thin disks where the ion temperature substantially exceeds the electron temperature. In these disks, the inflow timescale is much shorter than the cooling timescale. Thus the accretion energy cannot be radiated and is either advected inward with the flow or is released in a wind or radio jet (ADIOS; Blandford & Begelman 1999, 2004). In the context of SMBHs, a radiatively efficient disk is formed when the accretion rate approaches the Eddington value giving rise to a Seyfert nucleus or quasar. When the accretion rate falls below a few percent of the Eddington rate, the nucleus becomes faint and a radio galaxy is formed (eg. Churazov et al. 2005). \nGalaxy formation models incorporating AGN feedback distinguish between radiatively-dominated quasar feedback at early times and a mechanically-dominated radio mode at late times (eg. Springel, Di Matteo & Hernquist 2005; Croton et al. 2006; Sijacki & Springel 2006; Hopkins et al. 2006; Bower et al. 2006). Quasar feedback operates through intense radiation that is expected to couple to the gas and strong winds which drive gas from the host galaxy, quenching star formation and regulating the growth of the SMBH. This will eventually starve the SMBH of fuel and, as the accretion rate drops, a transition to mechanically-dominated radio mode feedback is expected. The SMBH launches jets which regulate radiative cooling in the surrounding hot atmosphere and the growth of the most massive galaxies. The AGN activity is closely correlated with the properties of the host halo indicating that they form a feedback loop (eg. Bˆırzan et al. 2004; Dunn & Fabian 2004; Rafferty et al. 2006). \nDespite evidence for rapid accretion onto their SMBHs, giant elliptical and brightest cluster galaxies in the nearby Universe \nrarely harbour quasars. The exceptions include H1821+643, 3C 186 and IRAS09104+4109 (Crawford et al. 1999; Belsole et al. 2007; Russell et al. 2010; Siemiginowska et al. 2010, Cavagnolo et al. submitted). Instead most harbour low luminosity AGN ie. ADAFs (Fabian & Rees 1995; Sambruna et al. 2000; Di Matteo et al. 2000, Hlavacek-Larrondo & Fabian 2011). These low luminosities imply that their host galaxies harbour massive black holes exceeding ∼ 10 9 M /circledot . Nuclear emission likely signals ongoing accretion. However, any dependence of the amplitude and form of the emission emerging from the nucleus on the accretion rate, as found in X-ray binaries, requires an independent means of estimating the accretion rate itself. Therefore it is difficult to test nuclear emission models that predict a strong dependence on the form of nuclear power output with nuclear accretion rate (Falcke, Kording & Markoff 2004; Churazov et al. 2005). Here we examine the nuclear emission properties in a sample of over 50 BCGs using the energy demands of X-ray cavities as a measure of mean nuclear accretion rate. \nWeassume H 0 = 70 km s -1 Mpc -1 , Ω m = 0 . 3 and Ω Λ = 0 . 7. All errors are 1 σ unless otherwise noted.", '2.1 Sample selection': "We intend to explore the emergent properties of accreting black holes at the centres of clusters. Therefore we have selected objects with large X-ray cavities which we use to estimate the mean accretion rate of each object over the past 10 7 -10 8 yr. Sources with a range of cavity powers, and thus accretion rates, were selected. The sources were selected from cluster, group and elliptical galaxy samples which show evidence of AGN activity in the form of cavities in X-ray images (Bˆırzan et al. 2004; Rafferty et al. 2006; Allen et al. 2006; Cavagnolo et al. 2010; O'Sullivan et al. 2011a). These objects were supplemented with other recently discovered X-ray point sources in cavity systems (RXCJ0352.9+1941, RXCJ1459.4-1811, RXCJ1524.2-3154, RXCJ1558.3-1410, Zw 348) and three noncavity systems (Zw 2089, A2667, A611) each with a bright point source. \nWe have also included three quasars taken from the literature for comparison, H1821+643, IRAS09104+4109 and 3C186 (Russell et al. 2010; Cavagnolo et al. submitted; Siemiginowska et al. 2005, 2010). These sources have very different spectral energy distributions (SEDs) from the low luminosity AGN dominating this sample. All three quasars have broad optical emission lines and bolometric luminosities of ∼ 10 47 erg s -1 , far greater than the rest of the sample. Both H1821+643 and 3C186 have strong big blue bumps in the optical-UV band whereas IRAS09104+4109 is a heavily obscurred quasar with most of its bolometric luminosity emerging in the IR. The big blue bump emission is usually interpreted as thermal emission from an accretion disk around the SMBH, which has then been re-radiated in the infrared in IRAS09104+4109. This component is absent in radiatively inefficient low luminosity AGN (eg. Chiaberge, Capetti & Macchetto 2005). \nThis sample is neither complete nor unbiased and we are mindful of this in our interpretation and analysis. Our sample overlaps the 15 objects analysed by Merloni & Heinz (2007) and we have quadrupled the sample size, including a large number of upper limits. In total, 57 sources were selected covering a redshift range from the nearest Virgo ellipticals at a distance of only 17 Mpc to 3C186 at z = 1 . 06 and a mass range from single elliptical galaxies \nto rich clusters. We sample mean accretion rates from 2 × 10 -6 to 0 . 6 ˙ M Edd for the first time. The full list of sources, excluding the three quasars, is shown in Table 1.", '2.2 Chandra data reduction': 'For each object in this sample, we selected the deepest Chandra observation in the archive for analysis (Table 1). Each observation was reprocessed using CIAO 4.4 and CALDB 4.4.7 provided by the Chandra X-ray Center (CXC). The level 1 event files were reprocessed to apply the latest gain and charge transfer inefficiency correction and then filtered to remove photons detected with bad grades. The improved background screening provided by VFAINT mode was also applied where available. Background light curves were extracted from the level 2 event files of neighbouring chips for observations on ACIS-I and from ACIS-S1 for observations on ACIS-S3. The background light curves were filtered using the LC CLEAN script 1 provided by M. Markevitch to identify periods affected by flares. The final cleaned exposure times of each observation are detailed in Table 1. Standard blank-sky backgrounds were extracted for each observation, processed identically to the events file and reprojected to the corresponding sky position. The blank-sky background was then normalized to match the count rate in the 9 . 5 -12 keV energy band in the observed dataset. This correction was less than 10 per cent for the majority of the observations. Each normalized blank sky background was also checked against the observed background spectrum extracted from a sourcefree region of each dataset to ensure it was a good match.', '2.3 X-ray point source flux': "Fig. 1 shows an example of a clear detection of surface brightness depressions indicating cavities and an X-ray point source at the centre of A2052 (Blanton, Sarazin & McNamara 2003; Blanton et al. 2009, 2011). We required at least a 3 σ detection above the background, where the background error considers only Poisson statistics, in a hard 3 -7 keV energy band image to confirm the detection of an X-ray point source. The majority of the sources were not detected with sufficient counts above the cluster background to generate a reasonable spectrum. Therefore, we used two alternative methods, based on those described by Hlavacek-Larrondo & Fabian (2011), to calculate the X-ray flux of the confirmed sources and 1 σ upper limits on the non-detections. \nThe photometric method sums the point source flux in a 1 arcsec radius region using an exposure-corrected image, which was weighted by a spectral model for the point source. This region was centred on the peak in the X-ray cluster emission if no point source was detected. The only exception was the Centaurus cluster where we used the position of the radio source as the centre (Taylor et al. 2006). For the majority of the sample, we used an absorbed powerlaw model PHABS(ZPHABS(POWERLAW)) with no intrinsic absorption, a photon index Γ = 1 . 9 (eg. Gilli, Comastri & Hasinger 2007) and Galactic absorption from Kalberla et al. (2005). For sources detected with several hundred source counts or more, this model was fitted to an extracted spectrum using XSPEC version 12 (Arnaud 1996) with the intrinsic absorption and the photon index left free. These model parameters are detailed in Table 1. We used the modified version of \nthe C-statistic available in XSPEC to determine the best-fit parameters for spectra with a low number of counts (Cash 1979; Wachter, Leach & Kellogg 1979). Note that the intrinsic absorption was set to zero for sources with only upper limits and the photon index was set to 1.9 where this was consistent with the best-fit value within the error. The cluster background was subtracted using an annulus around the point source from 1 . 5 -2 . 5 arcsec and an exposure-corrected image that was uniformly weighted at the average peak in the cluster spectrum around 1 . 5 keV. The point source flux was also corrected for the fraction of the point spread function (PSF) falling in the 1 arcsec region ( 90 per cent 2 ). A larger region encompassing more of the PSF also included a greater fraction of cluster background, which increased the measurement uncertainty. \nHowever, for galaxy clusters and groups with steep central surface brightness peaks this method is likely to significantly undersubtract the background cluster emission. We therefore also employed a spectroscopic method where we fit a spectrum extracted from the 1 arcsec radius point source region with a model for both the point source and cluster emission. The parameters for the cluster model were determined by extrapolating profiles of the projected cluster properties in to the point source region. These profiles were generated by extracting spectra from a series of circular annuli centred on the point source and fitting them with a suitable spectral model to determine the temperature, metallicity and normalization. We required a minimum of ∼ 2000 source counts per region to ensure good constraints on the cluster parameters. However, fewer counts per region were allowed for the low temperature sources in the sample where the Fe L line emission improves temperature diagnostics. Point sources were identified using the CIAO algorithm WAVDETECT, visually confirmed and excluded from the analysis (Freeman et al. 2002). Regions of nonthermal jet emission were also excluded. The cluster spectra were grouped to contain a minimum of 20 counts per spectral channel, restricted to the energy range 0 . 5 -7 keV and fit in XSPEC with appropriate responses, ancillary responses and backgrounds. \nFor the majority of the clusters, an absorbed single temperature PHABS(MEKAL) model (Balucinska-Church & McCammon 1992; Mewe, Gronenschild & van den Oord 1985; 1986; Kaastra 1992; Liedahl, Osterheld & Goldstein 1995) provided a good fit to the cluster emission in each annulus. This model was insufficient for some of the bright, nearby clusters with multiphase gas signatures, such as Centaurus, M87 and A2052. In these cases, a second MEKAL component or a MKCFLOW component was added to the spectral model. The cluster redshift and Galactic column density were fixed to the values given in Table 1. Abundances were measured assuming the abundance ratios of Anders & Grevesse (1989). For the multi-temperature models, the metallicity was tied between the two components and, for the MKCFLOW model, the lower temperature was fixed to 0 . 1 keV and the higher temperature was tied to that of MEKAL component. We produced radial profiles for the best fit temperature, metallicity and normalization parameters and used powerlaw fits to the inner points to extrapolate these properties into the point source region. The cluster spectra were also used to determine the Galactic absorption for sources located on regions of the sky where the absorption is highly variable. The spectral fits were repeated for annuli at large radii, excluding cooler gas components in the cluster core, leaving the n H parameter free (Table 1). \nThe spectrum extracted from the point source region was then \nFig. 2 compares our spectroscopic point source fluxes with measurements for the same sources available in the literature. The majority of the fluxes are consistent within the errors. Small variations are expected due to differences in background subtraction and the position selected for the upper limits but there are three sources, Centaurus, NGC4782 and RBS797, with significantly different values which we have considered in detail. There is only a modest discrepancy for RBS797 given the large errors and this is likely due to the additional model components used for the Cavagnolo et al. (2011) result. The differences for Centaurus and NGC4782 are due to our use of a two temperature rather than a single temperature cluster model. The best-fit single temperature falls midway between the preferred higher and lower temperature values of the two component model and therefore significantly underestimates the cluster surface brightness in the 2 -10 keV band used to determine the point source flux. Our two component model therefore finds a higher cluster background and a significantly lower nuclear point source upper limit. \n<!-- image --> \nIE-07 \n2E-07 AE-07 \nFigure 1. Exposure-corrected Chandra images covering the same field of Abell 2052 (see Blanton, Sarazin & McNamara 2003; Blanton et al. 2009, 2011). The colour bar has units photons cm -2 s -1 pixel -1 . Left: 0 . 5 -7 keV energy band showing the X-ray cavities. Right: 3 -7 keV energy band showing the AGN point source detection. \n<!-- image --> \nfitted with a combined model for both the point source and cluster emission. The similarity between the powerlaw and thermal model components, particularly for higher temperature clusters, can make it difficult to distinguish them with Chandra 's spectral resolution. Most of the model parameters were therefore constrained for the fit. The parameters for the point source model were set to those detailed in Table 1 with only the normalization left free. The cluster model temperature was fixed to the value determined by extrapolating from neighbouring annuli. The metallicity was fixed to the value in the neighbouring cluster annulus as it was generally not found to vary significantly at these small radii. The normalization of the cluster component was problematic because it was strongly affected by cavity substructure in the core, which produced large variation. We therefore constrained the cluster normalization in the point source region to be no less than the normalization from the neighbouring annulus, scaled by the ratio of their respective areas. The XSPEC CFLUX model was used to determine the flux or an upper limit for the unabsorbed point source component (Table 1). \nThe main source of error in both the photometric and spectroscopic measurements of the point source flux is the subtraction of the cluster emission. The photometric method is likely to underestimate the background cluster emission and should therefore be treated as an upper limit on the point source flux. The spectroscopic method improves on this by allowing for the increase in cluster surface brightness towards the cluster centre but may significantly overestimate the cluster background because it is indistinguishable from the powerlaw component. For strongly obscured point sources and low temperature cluster emission, the spectroscopic method is likely to be a significant improvement over the photometric method. For higher temperature clusters, the photometric method may be more accurate. We have therefore listed both the photometric and spectroscopic fluxes in Table 1 but used the generally more accurate spectroscopic flux in our analysis. The possible bias in this measurement for higher temperature clusters with strong 3 -7 keV emission is discussed in section 3.2. \nSeveral of the brighter point sources in the sample were significantly piled up in the longest Chandra exposures initially selected for analysis. Pile up occurs whenever two or more photons, arriving in the same detector region and within a single ACIS frame inte- \ngration time, are detected as a single event (Davis 2001). For M87 and Cygnus A, there were alternative observations available in the archive with shorter 0 . 4 s frame times for which the point source was not piled up. These short frame time observations were used to calculate the point source flux and the cluster background was analysed using the deeper exposures. All the archival observations of the point source in Perseus, where the cluster centre is not positioned far off axis distorting the PSF, were found to be significantly piled up. Perseus was therefore excluded from this sample.", '2.4 Cavity power': 'The cavities observed in the X-ray images of this sample allow a direct measurement of the mechanical output from the AGN (Churazov et al. 2000; Dunn & Fabian 2004; Bˆırzan et al. 2004; McNamara & Nulsen 2007). For a bubble filled with relativistic plasma, the energy required to inflate it is given by E = 4 PV , where the bubble is assumed to be in pressure equilibrium with the surrounding ICM. The bubble energy is then divided by the sound speed timescale or the buoyant rise time to estimate the power input to the ICM (see eg. Bˆırzan et al. 2004). Cavity powers for the \nFigure 2. Comparison of the spectroscopic point source fluxes with values from the literature (see Table 1). Note that the point source fluxes shown here were evaluated in the same energy band as those given in the literature and may differ significantly from those in Table 1. \n<!-- image --> \nmajority of the targets in this sample were available in the literature and otherwise estimated following Bˆırzan et al. (2004) (see Table 1). These values should be treated generally as lower limits on the total mechanical energy input by the central AGN. The datasets available in the Chandra archive for this cluster sample vary from snapshot 10 ks exposures to almost complete orbit exposures over 100 ks. Even for the nearest and brightest galaxy clusters, deeper exposures continue to discover new cavities (eg. Fabian et al. 2011). Some cavities will therefore have been missed from these estimates of the total power. In addition, weak shocks and sound waves have also been found to contribute significantly to the power output of the central AGN (eg. Fabian et al. 2003, 2006; Forman et al. 2005). \nEstimates of the cavity power can be complicated by the presence of significant X-ray emission from jets. For example, the synchrotron self-Compton emission in the centre of 3C 295 makes it difficult to determine the extent of the inner cavities and therefore the total power is significantly underestimated for this system (Harris et al. 2000; Allen, Ettori & Fabian 2001). The cavity power estimate for 3C 295 is therefore shown as a lower limit. Three sources in our sample, Zw 2089, A611 and A2667, have point sources but no detected cavities in the X-ray images. Ideally, these sources would be included in the analysis with upper limits on the cavity power. However, it is difficult to place meaningful constraints on the possible size of non-detected cavities, particularly at large radius where the noise increases and the X-ray surface brightness drops off rapidly (see Birzan et al. 2012). These sources have therefore been included with only illustrative upper limits, not quantitative. \nBy comparing the radiative output from the X-ray point source with the cavity power, we can calculate the radiative efficiency of the AGN. The total power output from the AGN can also be used to infer a mean accretion rate, given an assumption of the accretion efficiency. We refer throughout to a mean rather than instanta- \nneous accretion rate as this method estimates the average accretion requirements over the ∼ 10 7 yr age of each cavity. We have also calculated the theoretical Eddington and Bondi accretion rates for comparison with the inferred accretion rates for the sample.', '2.5 Eddington luminosity': "The Eddington luminosity indicates the limiting luminosity of the SMBH when the outward pressure of radiation prevents the gravitational infall of accreting material. For a fully ionized plasma, the Eddington luminosity can be expressed as \nL Edd erg s -1 = 1 . 26 × 10 47 ( M BH 10 9 M /circledot ) , (1) \nwhere M BH is the SMBH mass. Dynamical estimates of the black hole mass were used where these were available (Cygnus A, Tadhunter et al. 2003; Rafferty et al. 2006; M84, Walsh, Barth & Sarzi 2010; M87, Gebhardt et al. 2011; NGC4261, Ferrarese, Ford & Jaffe 1996). For the majority of the sample without dynamical masses, we relied on well-studied relations between the black hole mass and the properties of the host galaxy. Following Graham (2007), the apparent K-band magnitude of the host galaxy from the 2MASS catalogue 3 (Skrutskie et al. 2006) was converted to an absolute magnitude, corrected for Galactic extinction (Schlegel, Finkbeiner & Davis 1998), redshift and evolution (Poggianti 1997), and used to estimate the black hole mass. 3C401, RBS797 and Zw348 have detected X-ray point sources but no available apparent K-band magnitudes for this analysis and are not included in plots with Eddington accretion rate or luminosity scaling. Dalla Bont'a et al. (2009) find an upper limit on the central black hole mass in A2052 from dynamical measurements, which is consistent with the value calculated from the apparent K-band magnitude. \nThere are however potential problems with the use of Kband magnitudes to estimate black hole masses. Lauer et al. (2007) suggest that the apparent magnitudes from 2MASS are not deep enough to capture the extent of the BCG envelope and therefore will underestimate the total luminosity and black hole mass. Batcheldor et al. (2007) instead suggest that the BCG extended envelope, formed of debris from tidal stripping, is unlikely to be closely associated with the central galaxy dynamics. We have therefore considered these black hole masses to be estimates and, where possible, present the results with and without scaling by the Eddington accretion rate.", '2.6 Bondi accretion rate': "The primary aim of this paper is to examine the relationship between accretion and power output in SMBHs. One of the outstanding problems is whether the jets are powered by cold accretion from circumnuclear accretion disks of atomic and molecular gas or whether they are fuelled by a nearly spherical inflow of hot keV gas. It is impossible to prove either case because we lack imaging on the scale of and below the Bondi radius for almost all objects in our sample. However this has been done in nearby ellipticals where Chandra 's resolution is close to the size of the Bondi sphere (eg. Di Matteo et al. 2001, 2003; Churazov et al. 2002; Pellegrini et al. 2003). This problem has been addressed by Allen et al. (2006) in a \nsample of nearby ellipticals and by Rafferty et al. (2006) in a more distant sample with properties similar to our own and these studies arrived at somewhat different conclusions. Allen et al. (2006) found a strong trend between jet power and an estimate of the Bondi accretion rate based on circumnuclear X-ray properties whereas the sample analysed by Rafferty et al. (2006) required greater extrapolation in to the Bondi sphere making their conclusions on Bondi accretion highly uncertain. Nevertheless, using energetic arguments, they found Bondi power accretion was unable to fuel the most powerful jets unless their black holes were much larger than implied by the Mσ relation. This question is crucial for understanding the relationship between jet power and nuclear emission therefore here we re-analyse data for a subsample of 13 systems where we have resolution close to the Bondi sphere. This subsample includes the 9 galaxies targeted by Allen et al. (2006) and 4 additional objects with new, deeper Chandra observations (Table 2). \nAssuming spherical symmetry and negligible angular momentum, the Bondi rate, ˙ M B , is the accretion rate for a black hole embedded in an atmosphere of temperature, T , and density, ne , (Bondi 1952) and can be expressed as \n˙ M B M /circledot yr -1 = 0 . 012 ( k B T keV ) -3 / 2 ( ne cm -3 )( M BH 10 9 M /circledot ) 2 (2) \nfor an adiabatic index γ = 5 / 3. This accretion occurs within the Bondi radius, r B , where the gravitational potential of the black hole dominates over the thermal energy of the surrounding gas, \nr B kpc = 0 . 031 ( k B T keV ) -1 ( M BH 10 9 M /circledot ) . (3) \nThe Bondi accretion rate is therefore an estimate of the rate of accretion from the hot ICM directly onto the black hole and depends on the temperature and density of the cluster atmosphere at the Bondi radius. \nThe projected cluster spectra, extracted from a series of annuli as described in section 2.3, were deprojected using the model-independent spectral deprojection routine DSDEPROJ (Sanders & Fabian 2007; Russell, Sanders & Fabian 2008). Assuming only spherical symmetry, DSDEPROJ starts from the background-subtracted spectra and uses a geometric method (Fabian et al. 1981; Kriss, Cioffi & Canizares 1983) to subtract the projected emission off the spectrum from each successive annulus. The resulting deprojected spectra were each fitted in XSPEC with an absorbed single temperature MEKAL model to determine the temperature and density of the gas, as described in section 2.3. Several of the selected clusters have clear evidence for multiple temperature components in the inner regions (eg. M87, Forman et al. 2005; Centaurus, Fabian et al. 2005) and it is likely that deeper exposures of other clusters in the subsample will also produce robust detections of multi-phase gas. However, given the available range in exposure depth, a uniform method with a single temperature model was used to determine the emission-weighted average temperature of the ICM in the cluster centre. We also generated deprojected electron density profiles with finer radial binning from the surface brightness profiles and incorporating the temperature and metallicity variations (eg. Cavagnolo et al. 2009). \nThese profiles trace the cluster parameters to radii within an order of magnitude of the Bondi radius and therefore some extrapolation is required. Note that although the Bondi radius in M87 is resolved by Chandra , a significant region is affected by pileup and the PSF from the jet knot HST-1 and must be excluded. The \ntemperature profiles were generally found to flatten in the central regions of these systems. We have therefore assumed that the temperature at the accretion radius has only decreased by a further factor of 2 from the innermost temperature bin. Using equation 3, the Bondi radius was calculated from this innermost temperature value and the black hole mass. Dynamical black hole mass estimates were used where available. For comparison with Allen et al. (2006), stellar velocity dispersions from the HyperLeda Database 4 were then used to calculate black hole masses for the rest of the subsample (Tremaine et al. 2002). Only HCG62 did not have a dynamical or velocity mass estimate and the K band magnitude was used (section 2.5). Tremaine et al. (2002) find an intrinsic dispersion of 0 . 25 -0 . 3 dex in log ( M BH ) in the M BH -σ relation, which dominates over the error in the measurement of the velocity dispersion. \nThe deprojected electron density continues to increase in the cluster centre and was therefore extrapolated to the Bondi radius using several different model profiles. Three different models were considered: a powerlaw model continuing a steep density gradient to r B , a β -model flattening to a constant and a shallowing S'ersic profile with n = 4. These models were fitted to the density profile and used to calculate the gas density at r B . The density at r B is shown as a range of likely values from the powerlaw model upper limit to the β -model or S'ersic model lower limit. Using equation 2, the Bondi accretion rate was calculated from the density and temperature at r B and the black hole mass.", '2.7 Radio point source flux': 'We also compared the nuclear X-ray flux with the radio core flux for a subset of the sample to try to determine the origin of the nuclear X-ray emission. Radio observations were available for 22 sources in our sample which allowed us to reliably distinguish ongoing core activity. If the sole source of the X-ray flux is in the base of a jet, a direct relationship between the radio and X-ray core flux would be expected. \nNine of the sources in this subsample were observed simultaneously at C and X bands with the ATCA (project C1958, PI Edge). All but one of the remaining sources were observed at C-band with the VLA-C array (various projects, PI Edge), with the last source (A478) having been observed simultaneously at L and X band with the VLA-A array (project AE117). For each BCG, the radio-SED was further populated with data from the major radio catalogs (including but not limited to: AT20G at 20 GHz, NVSS/FIRST at 1 . 4 GHz, SUMSS at 0 . 843 GHz, WENSS/WISH at 0 . 325 GHz, VLSS at 0 . 074 GHz). Additional fluxes were found by searches around the radio-peak coordinates in both the NED and HEASARC online databases. All literature fluxes were individually scrutinised to ensure reliable matches. Where the synthesised beam size was considered limiting, leading to source confusion, data were discarded. Four of the sources have VLBA observations at C-band. \nCore flux contributions were calculated by considering both the morphology and SEDs of each of the sources. The VLBA Cband observations provided direct measurements of the core flux. For the remainder, the SEDs were decomposed into two major components; a flatter spectrum, active component attributed to ongoing activity within the core of the AGN and a steeper spectrum component, most dominant at lower frequencies attributed to either \nFigure 3. Histogram showing the distribution of the ratio of X-ray point source luminosity to cavity power for the low luminosity AGN in the sample. Three point sources and one upper limit have no detected cavities and are therefore not included. \n<!-- image --> \npast AGN activity or alternate acceleration mechanisms (e.g. radio lobes, mini-haloes, etc.). \nWhere a clearly resolved core was present in the observations, two component SEDs were fitted directly. For sources which were resolution limited at at C-band, spectral breakdown of the SEDs was performed on a case-by-case basis. Consideration was given to extent seen at other wavelengths, spectral shape and variability, with the proviso that variable sources are more likely contain a strong currently active core. Simple mathematical models were fit to the SEDs using IDL routines where a strong case could be made for believing distinct components were present. For sources where past and current activity could not be reliably distinguished, limits were placed on the core contribution. Full details of the SED analysis will be presented in Hogan et al. (in prep).', '3 RESULTS': 'In total, 27 out of 54 BCGs in this sample were found to have Xray central point sources detected above 3 σ in the 3 -7 keV energy band. Although this is not a complete sample of objects, central X-ray point sources appear to be common in BCGs with detected X-ray cavities. Fig. 3 shows the distribution of the ratio of X-ray nuclear luminosity in the 2 -10 keV energy range to cavity power, L X / P cav, for the cluster sample. The detected point sources in this sample cover a broad range in radiative efficiency from L X / P cav > 0 . 1 (eg. 3C 295) to L X / P cav < 10 -3 (eg. A2199). This distribution presumably reflects a broad range in average accretion rate with the majority accreting in a radiatively inefficient mode ie. they are ADAFs (see section 1).', '3.1 Radiative and cavity power output': "Fig. 4 (left) shows a correlation between the nuclear point source 2 -10 keV luminosity and the cavity power injected \ninto the surrounding ICM. This correlation was first found by Merloni & Heinz (2007) for a sample of 15 AGN with measured cavity powers, 13 of which also had X-ray point source detections. The scatter in this correlation covers three orders of magnitude therefore the generalised Kendall's τ rank correlation coefficient for censored data (Brown, Hollander & Korwar 1974) from the survival analysis package ASURV Rev 1.3 (Isobe, Feigelson & Nelson 1986; Isobe & Feigelson 1990; Lavalley, Isobe & Feigelson 1992) was used to evaluate its significance. We find a probability of accepting the null hypothesis that there is no correlation between the point source luminosity and the cavity power of P null = 4 × 10 -4 . Given the substantial difference in the timescales, ie. six orders of magnitude, it is surprising to observe a trend at all between these properties and suggests that AGN feedback is persistant. \nIt is not clear if the systems with upper limits on the point source luminosity form part of this trend, and are only just too faint to detect, or are currently 'off'. We have therefore considered these two scenarios separately using the BCES estimators (Akritas & Bershady 1996) for a linear regression fit to the data points only and the ASURV non-parametric Buckley-James linear regression fit to both the points and upper limits (Buckley & James 1979). Note that this analysis excluded the point sources with no detected cavities in the surrounding ICM as no effective upper limits for the cavity power could be estimated (see section 2.4). The inclusion of the upper limits produces a significant shift in the correlation but only a small change in the observed slope, which is consistent within the error. The BCES orthogonal best fit was found to be log ( P cav ) = ( 0 . 69 ± 0 . 08 ) log ( L X ) + 15 ± 3 compared to the ASURV Buckley-James best fit of log ( P cav ) = ( 0 . 7 ± 0 . 1 ) log ( L X )+ 14. The majority of the sample analysed are in a radiatively inefficient accretion mode where the mechanical cavity power dominates over the radiative output from the AGN. The observed slope shows that the radiative efficiency of the X-ray nucleus increases with increasing cavity power. \nFor comparison, we have included three quasars from the literature that are located in luminous cool core clusters to illustrate this increase in radiative efficiency. H1821+643 (Russell et al. 2010), IRAS 09104+4109 (Cavagnolo et al. submitted) and 3C 186 (Siemiginowska et al. 2005, 2010) each have a total radiative power of L ∼ 10 47 erg s -1 that exceeds their cavity power by at least an order of magnitude. Fig. 4 (left) shows that these sources appear to form an extension of the trend to much higher cavity powers and presumably much higher accretion rates. This implies that as the accretion rate rises black holes become more radiatively efficient. \nFollowing Merloni & Heinz (2007), we also estimate the bolometric point source luminosity for all sources in the sample and scale the luminosity and cavity power by the Eddington luminosity (Fig. 4 right). The SED for low luminosity AGN lacks the 'big blue bump' of emission dominating higher accretion rate sources and is likely to be dominated by the emission at hard X-ray energies. Vasudevan & Fabian (2007) find a typical bolometric correction for low luminosity sources of order ∼ 10, however for ease of comparison we adopt the same value as Merloni & Heinz (2007). Fig. 4 (right) shows the correlation between the bolometric point source luminosity and the cavity power where both quantities are scaled by the Eddington luminosity (section 2.5). The non-parametric Buckley-James linear regression method from ASURV software was used to determine the best-fit relation for both the points and the upper limits (Buckley & James 1979). The best-fit slope of log ( L cav / L Edd ) ∝ 0 . 9log ( L nuc / L Edd ) is significantly steeper than that found by Merloni & Heinz (2007) (Fig. 4 right). Using the BCES estimators linear regression fit to only the data points we \n<!-- image --> \nFigure 4. Left: Nuclear X-ray luminosity in the energy range 2 -10 keV calculated using the spectroscopic method versus cavity power from the literature. Right: bolometric nuclear luminosity versus cavity power where both quantities are scaled by the Eddington luminosity. Clusters with confirmed point source detections are shown by the filled circles and upper limits are shown by the open circles. Three quasar sources from the literature are included for comparison (red crosses). The variable sources are shown as blue stars (see section 3.5). The best-fit to both points and upper limits using the ASURV Buckley & James (1979) estimator for censored data is shown by the solid line. The BCES orthogonal fit to the data points only is shown by the dashed line. The best-fit from Merloni & Heinz (2007) is shown as a dotted line and is consistent with the BCES fit to the data points within the errors. \n<!-- image --> \ndetermine that the best fit is consistent with the Merloni & Heinz (2007) result within the errors. The inclusion of a large number of upper limits results in a significant difference in the slope of the correlation. We show in section 3.5 that 3 systems vary by up to an order of magnitude over a ten year timespan. This is apparently a significantly contributing factor to the scatter to which the Merloni & Heinz (2007) sample was not sensitive. Another factor that may be contributing is beaming of the central X-ray source, which is discussed in detail in Merloni & Heinz (2007).", '3.2 Selection effects': 'It is also clear that the scatter in this correlation between the nuclear X-ray luminosity and the cavity power is underestimated by our sample selection. Sources were selected primarily on the detection of cavities in X-ray observations and therefore sources with point sources but no cavities are generally missed from the lower right of Fig. 4. Three such sources, Zw 2089, A611 and A2667, were added to the sample to illustrate this selection bias. Note that the upper limits on the cavity powers are only illustrative, not quantitative (section 2.4). Whilst the observation of A2667 is shallow at only 8 ks, both A611 and Zw 2089 have sufficiently deep X-ray observations that would detect signs of any cavities or feedback-related substructure in the cluster cores. In particular, Zw 2089 contains a bright X-ray point source and appears very relaxed with smooth extended X-ray emission and no likely cavity structures. However, if cavities in this system are emerging along our line of sight they will be particularly difficult to detect. There could be a significant number of similar systems with bright point sources but no cavity structures, which will tend to increase the scatter in the observed correlation further. \nThe brightness and temperature of the surrounding cluster \nemission could also introduce another selection effect to this analysis. Point sources are identified by a significant detection of emission in a hard X-ray band, 3 -7 keV, above the background cluster (section 2.3). Bright, high temperature clusters will have more emission in this energy band than fainter, cooler systems therefore potentially making it more difficult to detect a point source above this background. Fig. 5 shows that for systems with strong background cluster emission in the 3 -7 keV band there are several sources (labelled) with higher upper limits. This suggests that an AGN in these BCGs would have to be brighter to be detected than in BCGs with fainter emission in this energy band. However, only a handful of sources in our sample appear to be affected so this will only slightly reduce the scatter in Fig. 4. \nThe selection bias in our sample will have a significant impact on the best-fit linear relation determined for the correlation in Fig. 4 (right). This is illustrated by the difference in slope between the Buckley-James linear regression in this analysis and the result found by Merloni & Heinz (2007). We have therefore not drawn any further conclusions from the slope of this correlation but considered the possible sources of the scatter, which covers at least three orders of magnitude.', '3.3 Uncertainty in cavity power and black hole masses': 'The measured cavity powers available in the literature for a particular system are often found to differ by factors of a few up to an order of magnitude. This can be due to new observations of an object, which reveal more cavities or better constrain the shapes of previously known cavities. It also reflects the inherent systematic uncertainty and judgement of the extent of the cavity volume (see eg. McNamara & Nulsen 2012). Cavities with bright rims, such as those in the Hydra A (eg. McNamara et al. 2000) and A2052 (eg. \nFigure 5. Nuclear X-ray flux in the 2 -10 keV energy band versus the cluster flux from a surrounding annulus in the 3 -7 keV energy band. \n<!-- image --> \nBlanton et al. 2011), have a well-defined shape, although some uncertainty still exists over the line of sight extent and whether the inner, outer or middle of the rims should be used. Most cavities do not have complete rims and their extent is difficult to constrain given the rapid decline in X-ray surface brightness with radius (Birzan et al. 2012). The cavity power for A2390 has a particularly large error, P cav = 1 . 0 + 1 . 0 -0 . 9 × 10 46 erg s -1 , because the extent of cavities is difficult to determine from the X-ray image. As discussed in section 2.4, the cavity power for 3C295 is likely to have been significantly underestimated and this is therefore an outlier. Therefore, for the majority of the sources, uncertainty in the cavity power is unlikely to introduce scatter greater than an order of magnitude. Also, although the black hole masses are likely to be a significant source of additional error for Fig. 4 (right), the scatter in Fig. 4 (left) is comparably large and does not depend on black hole mass.', '3.4 Absorption': "Another significant source of uncertainty in the X-ray point source fluxes may be attributable to photoelectric absorption by intervening gas within the galaxy or in a circumnuclear torus. This would cause a systematic underestimate of the point source luminosity. The amount of intrinsic absorption can be determined from spectral fitting but only for the brighter sources in our sample. Fig. 6 shows the intrinsic absorbing column density as a function of nuclear Xray luminosity for 25 of the 27 detected low luminosity AGN. There were insufficient counts for the detections of 3C388 and A2667 to constrain the intrinsic absorption. The intrinsic n H values cover a range from relatively unobscured sources with n H < 10 21 cm -2 , such as M87, up to heavily obscured narrow line radio galaxies with n H > 10 23 cm -2 , such as 3C 295. It is therefore plausible that a number of the non-detected sources could be moderately or heavily absorbed. This would account for some of the scatter in Fig. 4 and potentially render these sources undetectable in the 0 . 5 -7 keV energy band accessible to Chandra . \nFig. 6 has interesting astrophysical implications as well. The \nFigure 6. Nuclear X-ray luminosity versus the intrinsic absorption of the point source for a subset of objects with sufficient counts for a reasonable spectral fit. \n<!-- image --> \nmost luminous sources in our sample often contain large columns of cold intervening gas along the line of sight, in some cases exceeding 10 23 atoms cm -2 . It is unknown where along the line of sight this gas lies but it is likely to be close to the nucleus. This implies the existence of a supply of cold gas to fuel the nucleus. The range of intrinsic absorption may be related in part to the geometry of cold gas relative to the central point source. For example the point sources in Hydra A and NGC4261 are peering through circumnuclear gas disks that are highly inclined to the plane of the sky (eg. Jaffe & McNamara 1994; Dwarakanath, Owen & van Gorkom 1995). Furthermore the path towards the core in Cygnus A is strongly reddened by intervening dust and presumably accompanying gas located within 800 parsecs of the nucleus (Vestergaard & Barthel 1993). \nUsing a subset of example upper limits at different redshifts, we determined that if these undetected sources had an intrinsic absorption of 10 22 -10 23 cm -2 their true luminosity could be factors of up to a few greater than the observed upper limit. Intrinsic absorption of ∼ 10 24 cm -2 and above is required for the true luminosity of the source to be an order of magnitude or more greater than the observed upper limit. For a significant amount of the scatter in Fig. 4 to be generated by intrinsic absorption, a large fraction of the upper limits must therefore be Compton thick. These sources would then represent a very different population to the detected point sources in this sample. \nAlthough a large fraction of the upper limits in this sample could be significantly absorbed (eg. Maiolino et al. 1998; Risaliti, Maiolino & Salvati 1999; Fabian & Iwasawa 1999; Brandt & Hasinger 2005; Guainazzi, Matt & Perola 2005), this still seems unlikely to completely account for the three to four orders of magnitude scatter in Fig. 4. Reliably identifying Compton-thick AGN and determining their intrinsic luminosity is difficult (eg. Comastri 2004; Nandra & Iwasawa 2007; Alexander et al. 2008). Gandhi et al. (2009) (see also Hardcastle, Evans & Croston 2009) found a linear relation between the 2 -10 keV X-ray luminosity of \nthe AGN and the 12 µ m mid-infrared luminosity. Somewhat surprisingly, Gandhi et al. (2009) also found that the 8 Compton-thick sources in their sample did not deviate significantly from this trend. We might therefore expect that a large fraction of the undetected sources in our sample have high infrared luminosities as the absorbed power from the nucleus is re-radiated at these energies. \nHowever, for BCGs at the centre of cool core clusters, a significant fraction of the observed infrared luminosity is likely due to star formation (eg. Egami et al. 2006; Quillen et al. 2008; O'Dea et al. 2008). Egami et al. (2006) found that the BCGs in A1835, A2390 and Zw3146, all with point source upper limits in our sample, are infrared bright and have SEDs typical of starforming galaxies. Quillen et al. (2008) identify several BCGs as having very strong AGN contributions in the infrared, such as Zw2089, but in more modest cases this is difficult to disentangle from star formation. Interestingly, Quillen et al. (2008) find that the BCG in S'ersic 159-03 is infrared faint with only an upper limit on the IR star formation rate. Therefore, despite being one of the strongest upper limits in our sample (Fig. 4), S'ersic 159-03 does not appear to host a heavily absorbed AGN. Without detailed SEDs to disentangle the AGN contribution from star formation in these objects it is difficult to systematically determine if a significant fraction of the point source upper limits are heavily absorbed sources. However, even if this is the case, it is unlikely to explain the three orders of magnitude scatter in Fig. 4 (see also Evans et al. 2006).", '3.5 X-ray variability': "Another source of scatter is variability in the nuclear X-ray luminosity. The cavity power is averaged over the cavity ages, which are estimated from the sound crossing time or the buoyant rise time and are typically 10 7 -10 8 yr (Bˆırzan et al. 2004). However, the nuclear power is expected to be variable on timescales much shorter than this. The effect of variability on the trends in Fig. 4 is very much like that of relativistic beaming (Merloni & Heinz 2007). Merloni & Heinz (2007) further pointed out that variability is likely only to be a problem in the most luminous AGN such as Seyferts and quasars, which can vary on timescales of weeks. A subset of the clusters in our sample have multiple observations spaced by several years in the Chandra archive, which we searched for possible variations in the X-ray point source flux. \nThe point source flux was calculated for each observation using both the photometric and spectroscopic methods described in section 2.3. For the spectroscopic method, the differing depth of the observations could produce variations in the cluster parameters determined from surrounding annuli, which would add scatter to the measured point source fluxes. However, by fixing the cluster parameters in each observation to those determined from the deepest exposure available, we verified that this did not significantly alter the results. The point source fluxes for this subsample are shown in Table 3. The best-fit intrinsic absorption and photon index are shown for each exposure but the fluxes were calculated using the values of these parameters from the deepest exposure of that source. No significant change in the intrinsic absorption or the photon index was found for the sources analysed. \nFig. 7 shows that the point source flux was found to significantly vary in A2052, Hydra A and M84. The flux in A2052 was observed to decline by an order of magnitude over the ten years traced by the Chandra archive. Hydra A shows a more modest decline by a factor of ∼ 2 over 5 years, whereas for M84 a decline by a factor of ∼ 3 . 5 drop in flux occurs in only 6 months. Note that for Hydra A the earliest ACIS-I observation in 2000 was not \nincluded (obs. ID 575) as this dataset was taken during the soft proton damage to the detector. This analysis is generally limited by the availability of suitably spaced observations of sufficient depth in the Chandra archive but suggests that a significant fraction of AGN in BCGs may be varying on timescales of months to a few years. The bright central point sources in the Perseus cluster and M87 have long been known to be variable at X-ray wavelengths (eg. Rothschild et al. 1981; Harris, Biretta & Junor 1997). Another source in our sample, NGC4261, has been found to be variable on short 3 -5 ks timescales in a study by Sambruna et al. (2003). This variability is likely to be a significant source of scatter in the correlation between nuclear X-ray luminosity and cavity power. The cavity power is an average of the AGN activity over 10 7 -10 8 yr whereas the point source luminosity is likely to fluctuate significantly on much shorter timescales potentially by orders of magnitude. \nThe shape of the nuclear spectrum from Hydra A is dramatically different from those of A2052 and M84. Hydra A's spectrum falls sharply below 2 keV while the flux below 2 keV in A2052 and M84 continues to rise. This strong decline in flux shortward of 2 keV in Hydra A is due to a large column of intervening gas that may be associated with the large circumnuclear disk (eg. Dwarakanath, Owen & van Gorkom 1995; Hamer et al. in prep). We also searched for a change in the shape of the nuclear spectrum as the sources varied. Fig. 8 shows the spectrum for each observation of each point source found to have significantly varying X-ray flux. The cluster background was subtracted from each spectrum using a surrounding annulus. The spectra are remarkably consistent between the observations and suggest that despite the large variations in flux, particularly in A2052, there has been no significant change in the shape of the spectrum.", '3.6 Nuclear radio luminosity': 'Fig. 9 (left) shows no apparent correlation between the nuclear Xray flux and the 5 GHz radio core flux. There does appear to be an approximately linear trend between the nuclear X-ray luminosity and the radio luminosity (Fig. 9, right), although the X-ray flux is on average an order of magnitude larger. However, it is highly likely that this trend is due to redshift selection effects given the lack of a correlation in the flux-flux plot. \nWhilst care was taken to provide reliable core contributions to the overall radio flux density at C band, there are of course limitations. A variety of facilities were used to obtain the flux measurements used in the SEDs. Whilst this variety was considered in the decompositions, there will undoubtedly be situations where the true core contribution is lower than found in this analysis. This is due to contamination from extended emission in the lower resolution observations which is not adequately accounted for in the models. Similarly, for the highly core-dominated sources, large observed variability may lead to the radio core flux being underestimated at the epoch of the X-ray observations. These shortcomings will be a contributing factor to the scatter seen in Fig. 9. It should be noted however that the radio core contributions used here are taken from a larger sample of radio-loud BCGs analysed by Hogan et al (in prep). Of this larger sample, 26 are observed with the VLBA and strong agreement is seen between the direct VLBA core measurements and the SED-breakdown derived core contributions. Finally, many of the radio cores are self-absorbed so the 5 GHz flux may significantly underestimate the total radio power of the core. There is also likely to be significant scatter due to variability in both the X-ray and the radio flux. With no clear trend between the X-ray \nFigure 7. AGN flux variability in A2052 (left), Hydra A (centre) and M84 (right). The point source fluxes are calculated using both the photometric (open crosses) and spectroscopic methods (filled circles). Note that a spectroscopic flux measurement could not be produced for M84 obs. ID 401 because the 1 ks exposure was too short. \n<!-- image --> \nFig. 11 (left) shows the Bondi power plotted against the cavity power generated by the inner two cavities in each system. This shows a significant weakening of the trend found by Allen et al. (2006) driven mainly by a difference in the estimates of cavity volume. For most of these sources, our cavity powers are consistent with Allen et al. (2006) within the large errors of a factor of 2 -3 on these values. However, for M84, M89, NGC4472 and NGC507 our cavity powers are lower than those of Allen et al. (2006) by factors of up to an order of magnitude (see also Merloni & Heinz 2007). This was partly due to the availability of new, deeper observations of M84, NGC507 and NGC4472, which more clearly \n<!-- image --> \n<!-- image --> \nFigure 8. Point source spectra normalized to the flux at 2 keV for A2052 (left), Hydra A (centre) and M84 (right). Note that a spectrum could not be produced for M84 obs. ID 401 because the 1 ks exposure was too short. \n<!-- image --> \nand radio nuclear flux it appears less likely that the X-ray emission originates solely from the base of a jet.', '3.7 Bondi accretion': "The deprojected temperature and electron density profiles for the Bondi subsample of 13 systems are shown in Fig. 10. For each cluster we have marked the location of the Bondi radius and shown that the radial profiles are within roughly an order of magnitude of this. The two methods of calculating the deprojected density profile are consistent as expected. For the clusters that overlap with the Allen et al. (2006) sample, we generally find good agreement between the density and temperature profiles. The Centaurus cluster profiles were found to differ significantly because Allen et al. (2006) used a 35 · wide sector to the NE of the nucleus compared to our full annuli, which included the complex structure W of the nucleus. The temperature profile for NGC5846 is also significantly different in shape but we note that the central values used for the Bondi analysis are consistent. We used a more recent, deep observation of this source and the results are consistent with the Machacek et al. (2011) analysis. \nInner cavity substructure produced some sharp decreases \nin the deprojected density profile in several clusters, including NGC4636 and NGC5044 (Fig. 10). The density models were therefore fitted to all points within the central few kpc to smooth over substructure that is difficult to correctly deproject. In general, the inner radii of the density profiles were well-described by the three models used to extrapolate to the Bondi radius. The Bondi radius, accretion rate and cavity powers calculated from the temperature and density profiles for each of the selected systems are shown in Table 2. Following Allen et al. (2006), we calculated the cavity power for only the inner two cavities of each object that are currently being inflated by the central AGN. \nFigure 9. X-ray 2 -10 keV vs. radio 5 GHz flux (left) or luminosity (right). Nuclear X-ray detections are shown by the solid points and X-ray upper limits are shown by the open points. \n<!-- image --> \nrevealed the cavity extent. Allen et al. (2006) also used 1 . 4 GHz radio images to determine the edges of the cavities and this may have caused significant differences from our primarily X-ray method. Our new estimates of X-ray cavity power agree with estimates from Cavagnolo et al. (2010), Rafferty et al. (2006) and O'Sullivan et al. (2011a). It is not clear whether cavity powers will be more accurate when calculated using the X-ray or the radio observations, therefore we have also included Fig. 11 (right) showing our analysis of the Bondi accretion power versus the cavity powers from Allen et al. (2006). This plot shows a larger scatter than Allen et al. (2006) found and this scatter is due solely to differences in how we and Allen et al. (2006) calculated the central density. The vertical 'error bars' should not be interpreted as such. Instead they represent the range of Bondi powers from the three best-fit models and the midpoint is marked as no model provides a significantly better fit. The exception is the Centaurus cluster where the β -model is significantly preferred over the powerlaw and S'ersic profiles. We therefore used the Bondi power from the best-fit β -model and its associated errors. \nThe Kendall's τ rank correlation was used to determine if these two measures of cavity power are significantly correlated with the Bondi accretion rate. For our estimates of cavity power, we find no significant correlation with τ = 0 . 2. For the estimates of cavity power from Allen et al. (2006), we calculate τ = 0 . 7 and reject the null hypothesis of no correlation at 95% confidence but not at 99% confidence. This analysis therefore suggests weaker evidence for a trend between the cavity power and Bondi accretion power, primarily due to the uncertainty in estimates of the cavity volumes.", '4 DISCUSSION': 'X-ray central point sources appear to be common in BCGs hosting X-ray cavities. We find a detection fraction of ∼ 50 per cent for the BCGs in our sample. The majority of these sources are radiatively inefficient with required average accretion rates of only 10 -5 -10 -2 ˙ M Edd . The nuclear X-ray luminosity for these sources was observed to correlate with the AGN cavity power, which is surprising given the vastly different timescales for these quantities. Cavity power is averaged over the bubble ages, typically 10 7 -10 8 yr, while the nuclear X-ray luminosity is an instantaneous measurement and we have shown that this can vary significantly on shorter timescales of months to years. The scatter in this correlation covers over three orders of magnitude. A significant fraction of this scatter is likely due to X-ray variability but absorption and uncertainty in the cavity power estimates will also contribute. \nThe interpretation of these results is complicated by the uncertainty in the origin of the nuclear X-ray emission. The X-ray emission may originate from the accretion disk corona, from the base of a parsec-scale jet or a combination of the two although another origin is also possible. However, the nuclear X-ray emission is generally considered a probe of accretion power, whether it is from the accretion flow or from the jet (eg. Falcke & Biermann 1995; Heinz & Sunyaev 2003). Therefore, the observed L X -P cav correlation suggests the accretion power roughly scales with the cavity power over long timescales with the large scatter reflecting variability on shorter timescales.', '4.1 Duty cycle of activity': "It is also not clear if systems with only upper limits on the point source luminosity are simply faint or currently 'off'. As shown in Fig. 4, whether the upper limits form the faint end of the detected \nFigure 10. Deprojected temperature and electron density profiles of a subset of the cluster sample for which the cluster properties can be resolved at radii within an order of magnitude of the Bondi radius (shown by the vertical dashed line). \n<!-- image --> \n<!-- image --> \nsource population or are instead a different population of 'off' systems can have a significant impact on the correlation's slope and scatter. Approximately ∼ 50 per cent of the sample do not have detected nuclear X-ray emission. It is likely that at least some sources are simply a little too faint to be detected, particularly if they are embedded in bright cluster emission (see section 3.2). These objects may therefore still be consistent with the observed P cav -L X correlation. However, objects such as MS0735 and Sersic 159 have upper limits on their radiative luminosities a factor of thousand below that expected from this trend and are effectively 'off'. \nHlavacek-Larrondo & Fabian (2011) considered a sample of highly radiatively inefficient nuclei in clusters with powerful AGN outbursts, including MS0735, and suggested several explanations including absorption and variability. We have found significant variability for several sources in a subset of our sample, which could indicate a cycle of activity, but don't find absorption to be as important. Although we caution that this sample is by no means complete, the fraction of detections to non-detections indicates a duty cycle of at least ∼ 50 per cent in systems with recent AGN outbursts. If we consider only the 31 sources that overlap with the Rafferty et al. (2006) sample, we find a similar detection fraction of \nat least 40 per cent. This suggests that roughly half of all systems undergoing an AGN outburst in the last ∼ 10 8 yr have evidence of ongoing accretion. Mendygral, Jones & Dolag (2012) found that simulations with a jet duty cycle of 50 per cent, cycling on and off with a 26 Myr period, produced multiple cavity pairs with a similar morphology to observations (see also O'Neill & Jones 2010; Mendygral, O'Neill & Jones 2011). For complete samples of clusters, the fraction with detected X-ray cavities implies a duty cycle of at least ∼ 60 -70 per cent (Dunn & Fabian 2006; Bˆırzan et al. 2009; Bˆırzan et al. 2012).", '4.2 Radiative efficiency and evidence for a transition luminosity?': "Although it is not clear if the upper limits form the faint end of the detected source population or are a separate population of 'off' systems, the best fit models for these two possibilities have a consistent slope in Fig. 4 (left). This slope shows an increase in radiative efficiency with the mean accretion rate. The quasars included for comparison form an extension of this trend from cavity power-dominated to radiation-dominated sources. Studies have \nFigure 11. Bondi accretion power versus cavity power. Left: cavity power from this analysis. Right: cavity power from Allen et al. (2006). The best-fit relation from Allen et al. (2006) is shown as a dashed line. \n<!-- image --> \nalso shown that the radio loudness of low luminosity AGN to luminous quasars is inversely correlated with the mass accretion rate (eg. Ho 2002; Terashima & Wilson 2003; Panessa et al. 2007). Supermassive black holes appear to become more efficient at releasing energy through jets as their accretion rate drops. HlavacekLarrondo et al. (submitted) also find strong evolution in the nuclear X-ray luminosities of SMBHS hosted by BCGs such that the fraction of BCGs with radiatively-efficient nuclei is decreasing over time. \nObservational evidence suggests that the accretion process is largely similar for both stellar mass and supermassive black holes and therefore we could potentially use studies of X-ray binaries to understand accretion in AGN (eg. Maccarone, Gallo & Fender 2003; Merloni, Heinz & di Matteo 2003; Falcke, Kording & Markoff 2004). X-ray binaries are broadly classified into low-hard and high-soft states, which relate to the accretion disk properties and variation in the accretion rate can trigger state transitions (eg. Remillard & McClintock 2006). In the low-hard state, the accretion rate is low, the accretion disk is optically thin and radiatively inefficient. The mechanical power of the radio jet dominates over the radiative power and the X-ray and radio fluxes are correlated (eg. Gallo, Fender & Pooley 2003; Fender & Belloni 2004). Observations of X-ray binaries have shown that as the accretion rate rises above ∼ 0 . 01 -0 . 1 ˙ M Edd the source makes a spectral transition from the low-hard to the highsoft state (eg. Nowak 1995; Done, Gierli'nski & Kubota 2007). In this state the X-ray emission is dominated by an optically thick, geometrically thin accretion disk and the radio emission drops dramatically suggesting the outflow is suppressed (eg. Fender et al. 1999). \nFig. 12 shows the radiative and cavity power output as a function of the required mean accretion rate for our AGN sample, where all quantities are scaled by the Eddington rate. The mean accretion rate was calculated from the cavity power plus the bolometric lu- \nminosity of the point source and scaled by the Eddington accretion rate, \n˙ M ˙ M Edd = ( P cav + L bol ) L Edd . (4) \nL bol was calculated as shown in section 3.1 for the low luminosity AGN and was taken from the literature for the quasars. Note that for most of the sources considered L bol is insignificant compared to P cav and the required mean accretion rate is dictated by the cavity power. The quasars are the obvious exceptions. There are two points for each source on the plot showing both the cavity power and the radiative power. For sources where the radiative power or the cavity power dominates the output, the corresponding points will, by definition, lie on a line of equality between Power / L Edd and ˙ M / ˙ M Edd . This produces a clear line of points along y = x in Fig. 12. \nThe illustrative model from Churazov et al. (2005) of a change from a radiatively inefficient, outflow dominated mode to a radiation dominated mode has been shown for comparison in Fig. 12. Fig. 12 shows a trend of increasing radiative efficiency with mean accretion rate (see also Fig. 4). The radiative and mechanical power outputs converge and become comparable at an Eddington rate of a few per cent. For accretion rates below ∼ 0 . 1 ˙ M Edd the cavity power dominates over the radiative output, which is a factor of 10 -1000 times lower. Above ∼ 0 . 1 ˙ M Edd , a transition apparently occurs where mechanical power drops suddenly and the radiative power strongly dominates. This strong transition is seen in three objects: H1821+643, IRAS09104+4109 and 3C 186. These are quasars in the centres of galaxy clusters, few are known but they show this intriguing and potentially very important effect where they transition from mechanically dominated to radiation dominated AGN. This is precisely the behaviour expected when the accretion rate increases and an object transitions from an ADAF to a geometrically thin and optically thick disk. The AGN in our sam- \nFigure 12. The required mean accretion rate scaled by the Eddington rate, ˙ M / ˙ M Edd, plotted against the cavity power (blue circles) and the radiative power (red triangles) scaled by the Eddington luminosity. Therefore, there are two points for each source. Detected nuclear X-ray point sources are shown by the solid symbols and upper limits are shown by the open symbols. Quasar sources are labelled. The radiation and outflow model lines are illustrative only and show a transition from outflow domination at low accretion rates to radiative domination at high accretion rates (from Churazov et al. 2005, Fig. 1). \n<!-- image --> \nerefore appear to show the same qualitative behaviour with variation in accretion rate found for stellar mass black holes. \nHowever, there is significant scatter in the trend for radiatively inefficient sources, some of which may be due to variability in the X-ray flux. There are also outliers on the plot, notably MS0735 and A2390. A2390 may have overestimated cavity power and thus could move to the left. MS0735 was noted as anomalous to Churazov's scenario in Churazov et al. (2005). In this case the issue could be related to powering by the spin of the black hole (McNamara et al. 2009) where the spin energy is tapped more efficiently than mc 2 (eg. McNamara, Rohanizadegan & Nulsen 2011; Tchekhovskoy, Narayan & McKinney 2011; Cao 2011; McKinney, Tchekhovskoy & Blandford 2012). This would imply greater jet power per accreted mass than objects powered directly by accretion, moving it to the left in Fig. 12. It is also possible that the unknown value of ε , the conversion efficiency between mass and energy P = ε ˙ Mc 2 , is a large source of scatter, particularly as it is applied to mechanical power. We have assumed in eq. 4 that ε is tied between the radiation and cavity power and divides out but this is of course not necessarily true. Nevertheless, the increasing nuclear brightness relative to mechanical power is solid. And the transition to quasars does depend on power output and by inference, ˙ M . This picture is also a simplification of stellar mass black hole state transitions. Observed transitions from the low-hard to the high-soft state in X-ray binaries are accompanied by an intense and rapid radio outburst, which has no obvious analogy in our AGN model (eg. Fender, Belloni & Gallo 2004).", '4.3 Accretion power': "The source of fuel for the observed AGN activity has been the subject of considerable debate. There is, in general, sufficient cold gas in BCGs to fuel the range of observed jet powers (Edge 2001; Salom'e & Combes 2003; Soker 2008; Donahue et al. 2011). However, Bondi accretion directly from the cluster's hot atmosphere is appealing because it can provide both a steady fuel supply and a simple feedback mechanism. Although whilst the gas density is high enough to supply sufficient fuel to low power jet systems through Bondi accretion (eg. Allen et al. 2006), this is difficult to achieve for high power jets ( > 10 45 erg s -1 ; Rafferty et al. 2006; Hardcastle, Evans & Croston 2007; McNamara, Rohanizadegan & Nulsen 2011). Using a sample of nearby systems, Allen et al. (2006) found a correlation between the cavity power and Bondi accretion rate suggesting that a few per cent of the rest mass energy of material crossing the Bondi radius emerges in the jets. \nIn our analysis of 13 systems, including the Allen et al. (2006) sample, we found a significantly larger scatter in the correlation between the cavity power and Bondi accretion rate. This was primarily due to differences in our calculation of the density at the Bondi radius and estimates of the cavity power. Calculation of the density at the Bondi radius required an extrapolation of this profile over an order of magnitude in radius for each object in our sample. We used three different model density profiles for the extrapolation and found each provided a similarly good fit, with the exception of the Centaurus cluster where a β -model was significantly better. This fit was extended to cover a few kpc rather than just the central points as cavity substructure close to the centre affected the deprojection of the density profile. The use of three equally plausible extrapolations of the density profile produced different estimates of the density at the Bondi radius and increased the scatter in the Bondi accretion rate compared to Allen et al. (2006). Deeper observations of several clusters, showing the cavity extent more clearly, significantly altered the cavity power measured for those objects. We have therefore found weaker evidence for a trend between the cavity power and Bondi accretion rate. \nBondi accretion is energetically a plausible mechanism for fuelling the lower-powered radio sources in our sample. However, it is insufficient to fuel the most powerful systems (Hardcastle, Evans & Croston 2007; McNamara, Rohanizadegan & Nulsen 2011). There are also theoretical issues with Bondi accretion that include the ability to shed angular momentum and the zero central pressure requirement (Proga & Begelman 2003; Pizzolato & Soker 2005, 2010; Soker 2008; Narayan & Fabian 2011). For a more complete discussion see McNamara & Nulsen (2012). \nIn view of the high column densities found for many objects in this sample, which are consistent with significant levels of cold circumnuclear gas, and the prevalence of cold molecular gas in cD galaxies (eg. Edge 2001; Salom'e & Combes 2003) we suggest that cold gas fuelling is a likely source of accretion power in these objects. Nevertheless we cannot rule out or exclude Bondi accretion, which could play a significant role, particularly in low power jets (Allen et al. 2006).", '4.4 Nuclear X-ray emission mechanism': "The origin of the observed nuclear X-ray emission is not currently understood. Observed correlations between the X-ray and radio core luminosities provide the strongest support for a \nnon-thermal jet-related origin (eg. Fabbiano, Gioia & Trinchieri 1989; Canosa et al. 1999; Hardcastle & Worrall 1999). Radio and optical luminosity correlations for FR I nuclei also support this conclusion (Chiaberge, Capetti & Celotti 1999) and multi-wavelength spectral energy distributions for these sources can be modelled by synchrotron and synchrotron self-Compton emission from a jet (eg. Capetti et al. 2002; Yuan et al. 2002; Chiaberge et al. 2003a). However, Donato, Sambruna & Gliozzi (2004) show that a significant fraction of sources with strong optical jet emission do not have an X-ray component potentially indicating different physical origins. The detection of broad Fe K α lines and rapid variability on ks timescales favours an accretion flow origin (eg. Gliozzi, Sambruna & Brandt 2003). Radio and X-ray correlations do not necessarily imply a common origin for the emission as accretion processes and jets are likely to be correlated phenomena (eg. Begelman, Blandford & Rees 1984). Merloni, Heinz & di Matteo (2003) and Falcke, Kording & Markoff (2004) argued that these correlations are part of a 'fundamental plane' linking radio and Xray emission to black hole mass but differ on whether this reveals trends in accretion or jet physics (eg. Kording, Falcke & Corbel 2006; Hardcastle, Evans & Croston 2009; Plotkin et al. 2012).", '4.4.1 Jet origin?': "Wu, Yuan & Cao (2007) analysed the spectral energy distributions of eight FR I sources including two of the variable sources in our sample, A2052 and M84 (3C 272.1). They found that the emission in M84 is dominated by a jet and the ADAF model predicts too hard a spectrum at X-ray energies. A2052 appears to have a comparable contribution from the jet and the ADAF. These two sources have very different accretion rates: for A2052 ˙ M / ˙ M Edd = 9 + 10 -2 × 10 -4 and for M84 ˙ M / ˙ M Edd = 5 + 8 -4 × 10 -6 . Yet they both experience significant variations in nuclear flux on timescales of months to years. \nStrong flux variability on timescales of 1 -2 months is seen from both the core and the jet knot HST-1 in M87 (eg. Harris, Biretta & Junor 1997; Harris et al. 2006). From 2000 to 2009, HST-1 is the site of a massive X-ray, UV and radio flare. During this period, the X-ray emission from HST-1 dominates over the nucleus and rises and falls by an order of magnitude (Harris et al. 2009). The nuclear variability is characterised by Harris et al. 2009 as 'flickering' with changes in flux of order a factor of a few over timescales of months to years. It is not clear if the nuclear X-ray emission is due to the inner unresolved jet or the accretion flow. The magnitude and timescales of the X-ray flux variability found in A2052, Hydra A and M84 are therefore consistent with that observed in M87. Interestingly, HST UV observations of the nucleus in A2052 find the luminosity increased by a factor of ten from 1994 to 1999 (Chiaberge et al. 2002). This period was then followed by a decrease in the X-ray luminosity by a similar factor from 2000 to 2010, which could indicate a flaring event similar to that experienced by HST-1. \nSambruna et al. (2003) found variability with XMM-Newton on 3 -5 ks timescales in the FR I radio galaxy NGC4261, which is also part of our sample. For an ADAF, the X-ray emission is radiated from a relatively large volume and variation is expected on timescales longer than around a day (Ptak et al. 1998; Terashima et al. 2002). The observed variability timescale in NGC4261 is around two orders of magnitude shorter than the ADAF light crossing time suggesting that the variable component is more likely to be associated with the inner jet. Unfortunately, the \ncount rate in the Chandra observations of our sample is generally not large enough to search the light curve of each individual observation for flux variation. We also do not find any significant variation in the spectral properties of the three variable sources identified in our subsample.", '4.4.2 ADAF origin?': 'ADAF models predict trends between the nuclear radio and Xray luminosities that scale as L R ∝ L 0 . 6 X and ˙ M ∝ L 0 . 5 X (Yi & Boughn 1998). Assuming ˙ M ∝ P cav, this scaling is consistent with the slope shown in Fig. 4. Although the slope between L R and L X shown in Fig. 9 appears to be steeper than L R ∝ L 0 . 6 X , the lack of a flux-flux correlation and the large uncertainties suggest this trend is unreliable (see section 3.6). It is therefore worthwhile to consider the consequences of emission from an ADAF. \nFig. 13 compares the X-ray point source luminosities with predictions from ADAF models at different accretion rates (Merloni, Heinz & di Matteo 2003). The nuclear X-ray luminosity scales very close to linearly with black hole mass allowing us to scale up these models to the required ∼ 10 9 M /circledot . For low accretion rates, the 2 -10 keV emission includes inverse Compton scattering of soft synchrotron or disk photons and a bremsstrahlung component at higher energies. At higher accretion rates, the Comptonscattered component dominates as the optical depth rises and cooling processes become more efficient. The exact scaling of L 2 -10 keV with ˙ M / ˙ M Edd will depend on the parameters chosen for the model, such as the viscosity, magnetic pressure and electron heating fraction. Therefore, Merloni, Heinz & di Matteo (2003) obtain a single powerlaw fit L 2 -10 keV ∝ ( ˙ M / ˙ M Edd ) 2 . 3 from their ADAF models for comparison with observational data. \nThe required mean accretion rate was calculated from the cavity power plus the radiative power as shown in eq. 4. For most of the sources considered L bol is insignificant compared to P cav and the required mean accretion rate is dictated by the cavity power. The three quasars are the obvious exceptions to this. The majority of the observed sources are consistent with the emission expected from an ADAF for black hole masses from 5 × 10 8 M /circledot to 5 × 10 9 M /circledot . Four sources in our sample have more accurate dynamical black hole masses and therefore provide a more reliable constraint when compared with ADAF model predictions. Cygnus A and NGC4261 have mass accretion rates a factor of 3 -5 smaller than the ADAF model predictions for their respective black hole masses. The difference is even greater for M84 and M87 with over an order of magnitude and close to two orders of magnitude discrepancy, respectively. There are several possible reasons for this. The cavity power in M84 is particularly difficult to estimate as the outburst appears to have blown out most of the X-ray atmosphere. The cavity volume and surrounding pressure may therefore have been underestimated. The total mechanical power in M87 has been significantly underestimated as the shock produces an additional 2 . 4 × 10 43 erg s -1 , which is four times greater than the cavity power. This would cause M84 and M87 to move to the right in Fig. 13 and closer to the ADAF models. So this translates to a similar increase in the mean accretion rate. The nuclear X-ray emission from M84 and M87 is therefore likely to be consistent with an ADAF given the large uncertainties in the ADAF models. As previously discussed in section 2.4, the cavity power is likely to have been underestimated for the majority of the systems in this sample and this will tend to move points to the right in Fig. 13. \nWe therefore conclude that it is plausible that the X-ray point source emission is due to an ADAF but we cannot distinguish be- \nFigure 13. Point source X-ray luminosity versus the inferred accretion rate scaled by the Eddington rate. The lines are ADAF model predictions from Merloni, Heinz & di Matteo (2003) for black hole masses of 5 × 10 8 M /circledot (dashed), 1 × 10 9 M /circledot (solid) and 5 × 10 9 M /circledot (dash-dotted). The variable point sources are shown by the blue stars. Sources with dynamical black hole masses are shown by the open symbols (Cygnus A, M84, M87 and NGC4261). \n<!-- image --> \ntween this and a jet origin with the available data. Given the lack of a clear trend between the nuclear radio and X-ray flux it is likely that further progress on this problem will require modelling of the AGN spectral energy distribution (eg. Wu, Yuan & Cao 2007).', '5 CONCLUSIONS': "Using archival Chandra observations of 57 BCGs, we have investigated the relationship between nuclear X-ray emission and AGN radio jet (cavity) power. Although this is not a complete sample of objects, we find that nuclear X-ray emission is common with roughly half of the sample hosting a detectable X-ray point source. Assuming nuclear X-ray emission indicates active accretion, our study implies that the AGN in systems with recent outbursts are 'on' at least 50 per cent of the time. Furthermore, we examine the correlation between the nuclear X-ray luminosity and the average accretion rate determined from the energy required to inflate the Xray cavities. This correlation is consistent with the hypothesis that the nuclear X-ray emission traces active accretion in these systems. The majority of the sources in this sample are radiatively inefficient with required mean accretion rates of only 10 -5 -10 -2 ˙ M Edd . The nuclear X-ray sources become more luminous compared to the cavity power as the average accretion rate increases. The nuclear X-ray emission exceeds the cavity power when the average accretion rate rises above a few percent of the Eddington rate, where the AGN power output appears to transition from cavity power-dominated to radiation-dominated in the three BCGs hosting quasars. \nA small subset of the clusters in our sample had multiple archival Chandra observations of sufficient depth to search for variability in the nuclear X-ray flux. We found that A2052, M84 and Hydra A were significantly varying by factors of 2 -10 on \ntimescales of 6 months to ten years. Despite the large variations in flux, we did not find significant change in the shape of the nuclear spectra. This analysis is generally limited by the availability of suitably spaced observations of sufficient depth in the Chandra archive but suggests that a significant fraction of AGN in BCGs may be varying on timescales of months to a few years. This variability is likely to be a significant source of the large scatter in the observed correlation between the nuclear luminosity and cavity power. Our results suggest that the accretion power roughly scales with the cavity power over long timescales with the large scatter reflecting the variability on shorter timescales. \nThe interpretation of these results is complicated by the uncertainty of the nuclear X-ray emission origin. This emission may originate from the accretion disk corona, from the base of a jet or a combination of both, although other mechanisms are also possible. We discuss the similarity in magnitude and timescale of the X-ray variability found in A2052, M84 and Hydra A to that observed from both the core and jet knot HST-1 in M87. We also show that the X-ray nuclear luminosity and required mean accretion rate of the systems analysed are consistent with the predictions from ADAF models. We conclude that an ADAF is a plausible origin of the Xray point source emission but we cannot distinguish between this and a jet origin with the available data. \nWe have also considered the longstanding problem of whether jets are powered by the accretion of cold circumnuclear gas or accretion from the hot keV atmosphere. For a subsample of 13 nearby systems, the Bondi accretion rate was calculated using three equally plausible model extrapolations of the cluster density profile to the Bondi radius. The results suggest weaker evidence for a trend between the cavity power and the Bondi accretion rate, primarily due to the uncertainty in the cavity volumes. Cold gas fuelling may therefore be a more likely source of accretion power given the high column densities found for many objects in our sample, which are consistent with significant quantities of cold circumnuclear gas, and the prevalance of cold molecular gas in BCGs. However, we cannot rule out Bondi accretion, which may play a significant role, particularly in low power jets.", 'ACKNOWLEDGEMENTS': 'HRR and BRM acknowledge generous financial support from the Canadian Space Agency Space Science Enhancement Program. RAM and ANV acknowledge support from the Natural Sciences and Engineering Research Council of Canada. We thank Paul Nulsen, Avery Broderick, Roderick Johnstone and James Taylor for helpful discussions. This publication makes use of data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation. 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Russell et al. \nc e s \n] \n] ] \n] \n] \n] \n] \n] \n] ] \n] \n7 ] \n5 ] \n5 ] \n| | R e f e r e ) [ 1 | [ 2 ] , [ [ 4 ] , [ [ 2 ] , [ | [ 5 ] , [ [ 1 [ 1 | [ 1 [ 6 ] , [ - | [ 1 [ 6 ] , [ [ 1 [ 7 ] , [ | [ 8 ] , [ - [ 1 - | [ 1 [ 6 ] , [ [ 6 ] , [ [ 1 | [ 1 [ 9 ] , [ [ 1 0 ] , [ 1 1 ] , [ 1 2 ] , [ [ 1 | [ 1 4 [ 1 5 [ 1 5 [ 1 5 [ 1 6 ] , [ [ 1 5 [ 1 | 5 [ 1 8 ] , [ [ 1 9 ] , [ |\n|-----------|---------------------|---------------------------------|-------------------------|-------------------------|---------------------------------|---------------------------------|---------------------------------|-----------------------------------------------------|-------------------------------------------------|-----------------------------|\n| c | 2 s - 1 | | | | | | | | | |\n| | 0 k e V c m - . 1 | ± 3 1 0 . 3 | 0 . 3 0 . 5 . 1 . 2 | . 0 4 . 4 0 . 2 0 . 8 | . 6 . 2 . 4 0 . 2 0 . 3 | 0 . 6 0 . 1 0 . 2 . 3 0 . 4 | 0 8 1 0 ± 6 0 0 3 | . 5 3 0 . 8 - 0 . 7 + 1 - 3 ± 4 2 . 2 | . 1 2 . 1 2 . 2 . 2 ± 2 . 1 | . 3 0 . 1 0 . 4 0 . 3 |\n| | , 2 - 1 e r g < 0 | 5 5 2 ± . 7 ± | 6 . 9 + - < 2 < 0 | < 0 < 1 3 . 1 + - | < 1 < 1 < 1 3 . 1 + - | 0 . 9 ± 1 . 0 + - < 1 2 . 8 ± | < 0 . < 0 . 5 6 0 < 0 . | < 0 1 4 . 5 + 1 5 6 6 3 ± < 0 | < 0 < 0 < < 0 7 4 < 0 | < 0 . 7 ± 1 . 4 + - |\n| | F S - 1 4 | 3 | | | | | 2 | | | 0 |\n| | ( 1 0 | | | | | | | | | |\n| | ) | | | | | | | | | |\n| | s - 1 | | | | | | | | | |\n| c | | | | 9 . 2 | 1 | | | | | |\n| | k e V m - 2 . 3 | 4 0 . 3 0 . 3 | 0 . 3 . 2 . 8 | 4 . 2 0 | . 8 . 1 . 5 0 . | 0 . 3 0 . 1 . 3 0 . 4 | 7 2 . 6 ± 7 0 6 7 | 9 4 1 0 . 4 1 0 . 2 . 0 | 9 5 . 1 . 0 . 6 ± 2 . 4 | . 9 0 . 0 9 0 . 2 |\n| | F P , 2 1 4 e r < | 7 4 . 0 3 . 7 | 7 . 3 < < | < < 3 . 0 | < < < 3 . 8 | 3 . 1 1 . 4 < 2 . 4 | < < 3 1 6 < | < 2 1 7 . 6 5 . 5 < | < < < < 1 2 < | < 0 . 8 4 2 . 6 |\n| | 1 0 - | | | | | | | | | |\n| e s . | ( | | | | | | | | | |\n| fl u x | | | . 0 9 . 0 8 | | | | | 0 . 0 9 0 . 0 7 . 1 | . 2 | |\n| r c e | Γ 1 . 9 | 1 . 9 1 . 9 1 . 9 | 5 + 0 - 0 1 . 9 1 . 9 | 1 . 9 1 . 9 1 . 9 | 1 . 9 1 . 9 1 . 9 1 . 9 | 1 . 9 1 . 9 1 . 9 1 . 9 | 1 . 9 1 . 9 1 . 9 1 . 9 | 1 . 9 1 . 9 3 ± 7 ± 3 ± 0 1 . 9 | 1 . 9 1 . 9 1 . 9 1 . 9 9 ± 0 1 . 9 | 1 . 9 1 . 9 1 . 9 |\n| s o u | | | 1 . 5 | | | | | 2 . 1 2 . 3 2 . | 0 . | |\n| p o i n t | 2 ) | | | | | | | 1 3 | | |\n| n d | , z m - | + 6 - 5 | | | | | ± 0 . 7 | 0 . 3 0 . 0 3 0 . 0 2 0 . 0 0 . 0 | 0 . 6 0 . 5 | 0 . 0 8 0 . 0 7 |\n| i e s a | n H 0 2 2 c 0 | 4 3 0 0 | 0 0 0 | 0 0 0 | 0 0 0 0 | 0 0 0 0 | 0 0 9 . 6 0 | 0 3 . 5 ± . 1 2 + - 0 6 ± 9 2 ± 0 | 0 0 0 0 3 . 0 + - 0 | 0 0 . 2 3 + - |\n| p e r t | ( 1 | | | | | | 1 | 0 0 . 0 . 0 | | 0 |\n| p r o | ) | | | | | | | | | |\n| p l e b | m - 2 1 8 ∗ | 3 4 5 5 8 2 | 2 9 7 8 5 9 | 6 7 8 ∗ 4 7 | 1 9 0 4 2 5 7 2 | 8 9 8 ∗ 4 8 7 3 | 2 1 8 ∗ 2 3 2 | 6 1 8 4 6 8 9 9 9 4 6 2 2 6 8 | 2 8 2 5 4 0 4 9 7 5 5 3 | 9 0 3 7 8 7 |\n| S a m n | H 2 2 c . 2 2 | 0 . 0 1 0 . 0 5 0 . 0 5 | 0 . 0 4 0 . 0 2 0 . 0 1 | 0 . 0 5 . 2 7 7 0 . 0 4 | 0 . 0 1 . 0 2 . 0 3 0 . 0 2 | 0 . 0 0 . 0 7 6 0 . 0 2 0 . 0 1 | 0 . 0 1 . 0 9 4 0 . 2 7 0 . 0 3 | 0 . 0 0 . 0 0 . 0 2 0 . 0 1 0 . 0 2 0 . 0 | 0 . 0 3 0 . 0 5 0 . 0 2 0 . 0 3 0 . 0 1 0 . 0 1 | 0 . 0 1 . 0 3 0 . 0 4 |\n| 1 . | ( 1 0 0 | | | 0 | 0 0 | 0 | 0 | | | 0 |\n| b l e a | | | | | | | | | | |\n| T a | s u r e s ) . 5 | 5 . 6 6 . 3 2 . 7 | 4 . 5 8 . 2 . 9 | 8 . 6 . 3 . 0 | . 8 9 . 2 . 7 3 . 8 | . 6 . 6 . 3 . 4 | . 9 . 2 . 2 . 1 | . 5 4 . 6 . 1 . 8 . 4 1 . 9 | 3 . 8 . 0 . 2 . 5 . 9 . 7 | . 8 . 3 . 5 |\n| | x p o ( k 4 9 | 7 2 2 | 6 3 6 7 | 1 0 3 9 3 2 | 1 5 1 0 7 4 1 2 | 4 0 7 0 5 6 8 | 8 5 8 7 9 6 5 | 4 8 9 4 5 1 2 5 3 5 | 1 3 4 0 2 0 2 0 9 9 3 9 | 6 7 4 9 8 2 |\n| | E | | | | | | | | | |\n| o i n t | | | | | | | | | | |\n| i m p | S 3 | I 3 I 3 S 3 | S 3 I 0 I 3 | S 3 S 3 S 3 | S 3 I 3 S 3 S 3 | I 3 S 3 S 3 S 3 | S 3 S 3 S 3 S 3 | S 3 S 3 S 3 S 3 S 3 I 3 | I 3 I 3 S 3 S 3 S 3 S 3 | I 3 S 3 S 3 |\n| A | | | | | | | | | | |\n| I D | 9 | 4 5 0 | 0 4 7 | 1 9 4 | 0 0 0 7 7 | 4 8 3 9 4 | 5 4 7 2 | 7 7 0 8 8 2 0 | 7 0 2 2 3 9 7 4 | 6 0 9 |\n| O b s . | 7 9 3 | 2 2 5 5 2 9 4 3 7 | 4 9 4 9 0 9 8 9 | 7 9 2 1 6 6 3 1 9 | 1 0 9 6 8 8 4 9 7 5 8 0 | 0 7 4 1 9 7 3 2 2 2 1 | 5 7 8 4 9 5 1 7 0 1 0 4 6 | 6 2 5 4 9 5 9 0 1 8 0 2 0 7 9 0 | 1 0 4 2 8 8 2 0 2 4 2 8 9 5 6 1 1 2 | 3 9 2 3 2 2 9 3 9 |\n| | | | | | | 1 | | | | |\n| | h i f t 4 9 | 4 1 1 7 1 1 | 1 2 5 1 6 6 | 6 6 8 1 8 0 | 2 5 2 3 7 3 5 1 | 3 0 2 8 0 5 2 3 0 0 | 7 5 1 4 6 1 3 7 | 4 0 5 4 9 3 5 4 4 1 1 5 0 | 6 0 1 6 5 5 9 5 6 7 5 3 3 | 3 1 5 4 9 3 |\n| | R e d s 0 . 0 3 | 0 . 4 6 0 . 0 9 0 . 2 0 | 0 . 2 4 0 . 0 5 0 . 0 5 | 0 . 0 1 0 . 0 8 0 . 2 8 | 0 . 0 6 0 . 2 5 0 . 0 7 0 . 0 3 | 0 . 0 0 . 2 2 0 . 0 8 0 . 2 | 0 . 0 4 0 . 0 1 0 . 0 5 0 . 0 1 | . 1 5 . 0 0 . 0 0 0 . 0 0 0 . 0 0 . 0 4 | 0 . 2 1 . 0 0 . 0 0 0 . 0 1 0 . 0 0 . 0 0 | 0 . 0 0 0 . 0 1 0 . 0 0 |\n| | | | | | | | | 0 0 0 | 0 0 | |\n| | 6 | | | | | | | | 7 4 2 1 | | \nT a r g e t \n2 A 0 3 3 5 + 0 9 \n3 C 2 9 5 \n3 C 3 8 8 3 C 4 0 1 \n4 C 5 5 . 1 6 \nA 2 0 5 2 \nA 2 1 9 9 \nA 2 3 9 0 \nA 2 5 9 7 A 2 6 6 7 A 4 0 5 9 C e n t a u r u s C y g n u s A H C G 6 2 H e r c u l e s A \nH y d r a A \n1 ] 3 ] \nM 8 7 M 8 9 M K W 3 S M S 0 7 3 5 . 6 + N G C 5 0 7 N G C 1 3 1 6 N G C 1 6 0 0 \nN G C 4 2 6 1 \nN G C 4 4 7 2 N G C 4 6 3 6 N G C 4 7 8 2 \nN G C 5 0 4 4 \nc © 0000 RAS, MNRAS 000 , 000-000 \nA 8 5 A 1 3 3 A 2 6 2 A 4 7 8 \nA 6 1 1 \nA 1 7 9 5 A 1 8 3 5 A 2 0 2 9 \n1 ] \nM 8 4 \nn c e s \n1 ] \n0 . 0 \nAGN in brightest cluster galaxies \n23 \n2 . 3 ) . 0 0 6 ) , 0 0 3 ) , 0 0 6 ) , [ 2 0 ] \n| R e f e r | ) [ 2 | [ 1 [ 1 [ 1 | [ 1 [ 2 1 ] [ 2 | | [ 1 [ 2 | [ 6 ] , [ 6 ] | ( s e e s e c t i o & C r o s t o n | c N a m a r a l l e n e t a l . e t a l . ( 2 0 0 9 |\n|---------------|-------------------|-----------------|-------------------------------|-------------------------------------|---------------------------------------|-----------------|---------------------------------------|-------------------------------------------------------|\n| c | 2 s - | | 7 | | | | 0 0 5 ) a n s & M | 1 3 ] A a v i d |\n| k e V | c m - . 3 . 0 5 | . 0 8 . 4 0 . 2 | 0 . 4 ± 0 . ± 2 ± | 2 ± 1 0 . 4 ± 2 | . 0 7 0 . 2 0 . 4 + 6 | - 5 . 3 . 3 | . ( 2 , E v a z i n | ) , [ 9 ] D |\n| , 2 - 1 0 | e r g < 0 < | 0 < 0 < 1 + | 2 . 6 - 9 . 8 4 1 | 3 2 1 3 . 2 ± 2 2 | < 0 . 8 ± . 9 ± | 1 2 7 < 0 < 3 | e t a l a s t l e S a r | 2 0 0 6 [ 1 |\n| F S | - 1 4 | | 1 | 3 | 2 2 | | e r l a a r d c t o n , | l . ( 0 7 ) , |\n| | ( 1 0 | | | | | | a l b ] H l a n | e t a ( 2 0 |\n| 1 | ) | | | | | | b y K ) , [ 2 [ 7 ] B | a c e k t a l . |\n| c | s - | | | | | | r e d 2 0 0 6 1 1 ) , | a c h e k e |\n| k e V - 2 | m 0 2 | 7 2 . 4 . 2 | 0 . 3 0 . 4 2 | 2 1 0 . 5 3 | . 0 0 . 1 0 . 4 | 7 5 3 . 0 | a s u a l . ( ( 2 0 | 2 ] M c h a c |\n| 2 - 1 0 | r g c < 1 . < 0 . | < 2 < 2 | 8 ± . 4 ± 6 1 ± ± | 3 6 2 1 ± 7 ± 4 1 ± | < 1 3 ± 4 ± 0 3 ± | < 0 . < 5 | e m e y e t i a n | , [ 1 M a |\n| F P , - 1 4 | e | | 4 . 2 1 | 5 . | 3 . 4 . 2 | | v a l u f e r t & F a b | 2 0 0 3 ) [ 1 8 ] |\n| | ( 1 0 | | | | | | t i c R a f o | l . ( 1 b ) , |\n| | | | 6 | | | | a l a c : [ 1 ] r o n d | e t a 0 1 |\n| | 9 | 9 9 9 | 9 0 . 0 0 . 2 | 9 9 9 9 | 9 0 . 2 9 | 9 9 9 | h e G n c e s - L a r | a t t e o a l . ( 2 p |\n| Γ | 1 . | 1 . 1 . 1 . | 1 . 0 1 ± . 4 ± | 1 . 1 . 1 . 1 . | 1 . . 2 ± 1 . | 1 . 1 . 1 . | a n t f e r e a c e k | i M n e t p r e |\n| | | | 2 . 1 | | 2 | | r t h . R e H l a v | 1 ] D l i v a l . i n |\n| ) | 2 ) | | 0 2 | 4 2 | 6 5 | | i g h e e 1 σ 6 ] | , [ 1 O ' S u l e t a |\n| n t ' d H , z | c m - 0 | 0 0 0 | 0 ± 0 . 8 + 0 . 9 - 0 . 8 ± . | 0 4 + 1 . 0 - 0 . 8 ± 0 . 5 + 5 - 3 | 0 3 + 0 . 0 - 0 . 0 8 + 0 . 7 - 0 . 4 | ± 2 0 0 | b e h e s a r 1 ) , [ | 0 0 6 ) 7 ] m a r a | \n( \na b l e 1 \n( 1 0 \n2 ) \nT E x p o s u r e a n H b ( k s ) ( 1 0 2 2 c m -9 8 . 8 0 . 0 9 1 8 ∗ 8 4 . 7 0 . 0 4 2 9 3 5 . 3 0 . 0 5 1 0 4 4 . 5 0 . 0 2 2 3 1 1 6 . 0 0 . 3 8 8 3 . 5 0 . 0 5 7 8 ∗ 3 8 . 3 0 . 0 2 2 8 2 7 . 2 0 . 1 4 4 0 ∗ 3 9 . 5 0 . 0 7 3 6 4 0 . 9 0 . 1 1 0 7 ∗ 3 6 . 0 0 . 1 0 6 0 9 3 . 3 0 . 0 1 1 4 9 3 . 8 0 . 0 2 5 6 4 8 . 9 0 . 0 2 5 0 3 8 . 6 0 . 0 2 8 6 9 5 . 8 0 . 0 0 7 5 3 4 . 5 0 . 0 2 4 6 i s m a r k e d w i t h ∗ w e r e s p e c t r o s c o p i c F S , 2 -1 0 k e , [ 5 ] H l a v a c e k -L a r r o n d o ] B a l m a v e r d e , C a p e t t i & , [ 1 6 ] C h i a b e r g e e t a l . ( a v a g n o l o e t a l . ( 2 0 1 1 ) , [ \nA i m p o i n t \nO b s . I D \nR e d s h i f t \nT a r g e t \nS 3 I 3 I 3 I 3 \n9 5 1 7 7 9 2 3 4 9 7 2 4 1 9 4 \n0 . 0 0 6 6 0 . 0 0 5 7 0 . 0 3 4 8 0 . 0 2 7 4 \nN G C 5 8 1 3 N G C 5 8 4 6 N G C 6 2 6 9 N G C 6 3 3 8 \nS 3 \n1 2 8 8 1 \n0 . 1 0 2 8 \nP K S 0 7 4 5 -1 9 1 \nc © 0000 RAS, MNRAS 000 , 000-000 \nS 3 \n1 2 8 8 4 \n0 . 0 2 3 0 \nP K S 1 4 0 4 -2 6 7 \nS 3 \n7 9 0 2 \n0 . 3 5 4 0 \nR B S 7 9 7 \nS 3 \n1 0 4 6 6 \n0 . 1 0 9 0 \nR X C J 0 3 5 2 . 9 + 1 9 4 1 \nS 3 \n9 4 2 8 \n0 . 2 3 5 7 \nR X C J 1 4 5 9 . 4 -1 8 1 1 \nS 3 \n9 4 0 1 \n0 . 1 0 2 8 \nR X C J 1 5 2 4 . 2 -3 1 5 4 \nS 3 \n9 4 0 2 \n0 . 0 9 7 0 \nR X C J 1 5 5 8 . 3 -1 4 1 0 \nI 3 \n1 1 7 5 8 \n0 . 0 5 8 0 \nS ' e r s i c 1 5 9 -0 3 \nS 3 \n1 1 3 8 9 \n0 . 0 1 4 7 \nU G C 4 0 8 \nS 3 \n1 0 4 6 5 \n0 . 2 5 3 5 \nZ w 3 4 8 \nS 3 \n1 0 4 6 3 \n0 . 2 3 5 0 \nZ w 2 0 8 9 \nS 3 I 3 \n1 2 9 0 3 9 3 7 1 \n0 . 2 1 5 0 0 . 2 9 0 6 \nZ w 2 7 0 1 Z w 3 1 4 6 \n3 \n0 \nf o u n d t V fl u x v a l e t a l . ( 2 G r a n d i 2 0 0 3 b ) , 2 3 ] M c N \nN o t e . -a F i n a l c l e a n e d e x p o s u r e t i m e s . b C o l u m n d e n s i t e c E r r o r s a n d u p p e r l i m i t s o n t h e p h o t o m e t r i c F P , 2 -1 0 k e V a n d [ 3 ] H l a v a c e k -L a r r o n d o e t a l . ( 2 0 1 2 ) , [ 4 ] E v a n s e t a l . ( 2 0 0 6 ) [ 8 ] D i M a t t e o e t a l . ( 2 0 0 1 ) , [ 9 ] M c N a m a r a e t a l . ( 2 0 0 0 ) , [ 1 0 [ 1 4 ] M c N a m a r a e t a l . i n p r e p , [ 1 5 ] C a v a g n o l o e t a l . ( 2 0 1 0 ) O ' S u l l i v a n e t a l . ( 2 0 1 1 a ) , [ 2 1 ] J o h n s t o n e e t a l . ( 2 0 0 5 ) , [ 2 2 ] C \n3 \n5 \n0 . \n0 \n3 ] 1 ] \nTable 2. Bondi parameters for the selected subsample of targets.Table 3. Point source fluxes and key parameters for each source tested for variability. The photometric point source flux, F P, and the spectroscopic point source flux, F S, are both given in the 2 -10 keV energy band. A spectroscopic flux measurement could not be produced for M84 obs. ID 401 because the 1 ks exposure was too short. \n| Target | D L (Mpc) | T ( r B ) (keV) | r B (kpc) | ne ( r B ) ( cm - 3 ) | ˙ M B (M /circledot yr - 1 ) | P B (10 43 erg s - 1 ) | P cav (10 42 erg s - 1 ) |\n|-----------|-------------|-------------------|---------------------|--------------------------|--------------------------------|--------------------------|----------------------------|\n| A2199 | 132.3 | 1 . 10 ± 0 . 07 | 0 . 020 ± 0 . 001 | 1 . 1 + 0 . 8 - 0 . 6 | 0 . 006 + 0 . 004 - 0 . 003 | 4 ± 2 | 9 + 4 - 2 |\n| Centaurus | 49.3 | 0 . 41 ± 0 . 04 | 0 . 028 ± 0 . 002 | 0 . 4 + 0 . 3 - 0 . 2 | 0 . 002 + 0 . 002 - 0 . 001 | 0 . 51 ± 0 . 06 | 14 + 7 - 4 |\n| HCG62 | 59.3 | 0 . 31 ± 0 . 03 | 0 . 058 ± 0 . 006 | 0 . 3 + 0 . 3 - 0 . 004 | 0 . 006 + 0 . 006 - 0 . 000 | 5 ± 2 | 6 + 4 - 2 |\n| M84 | 17 | 0 . 34 ± 0 . 01 | 0 . 084 ± 0 . 003 | 0 . 41 + 0 . 10 - 0 . 01 | 0 . 020 + 0 . 005 - 0 . 001 | 13 ± 1 | 1 . 1 + 0 . 9 - 0 . 4 |\n| M87 | 17 | 0 . 52 ± 0 . 04 | 0 . 42 ± 0 . 04 | 0 . 23 ± 0 . 05 | 0 . 35 ± 0 . 07 | 200 ± 40 | 8 + 7 - 3 |\n| M89 | 17 | 0 . 35 ± 0 . 01 | 0 . 041 ± 0 . 002 | 0 . 6 + 0 . 7 - 0 . 5 | 0 . 007 + 0 . 008 - 0 . 006 | 5 ± 4 | 0 . 3 + 0 . 2 - 0 . 1 |\n| NGC507 | 71.6 | 0 . 49 ± 0 . 02 | 0 . 048 ± 0 . 002 | 0 . 8 + 1 . 8 - 0 . 5 | 0 . 02 + 0 . 04 - 0 . 01 | 20 ± 10 | 19 + 14 - 7 |\n| NGC1316 | 25.4 | 0 . 343 ± 0 . 007 | 0 . 0267 ± 0 . 0005 | 1 . 3 + 1 . 6 - 0 . 9 | 0 . 006 + 0 . 008 - 0 . 004 | 5 ± 4 | 0 . 9 + 0 . 8 - 0 . |\n| NGC4472 | 17 | 0 . 372 ± 0 . 007 | 0 . 061 ± 0 . 001 | 0 . 5 ± 0 . 2 | 0 . 014 + 0 . 006 - 0 . 005 | 9 ± 3 | 4 0 . 7 + 0 . 5 - 0 . 3 |\n| NGC4636 | 17 | 0 . 23 ± 0 . 02 | 0 . 028 ± 0 . 002 | 0 . 20 + 0 . 03 - 0 . 05 | 0 . 0009 + 0 . 0001 - 0 . 0002 | 0 . 5 ± 0 . 1 | 0 . 27 + 0 . 15 - 0 . 09 |\n| NGC5044 | 40.1 | 0 . 31 ± 0 . 01 | 0 . 0290 ± 0 . 0009 | 0 . 3 + 0 . 5 - 0 . 1 | 0 . 002 + 0 . 003 - 0 . 001 | 2 ± 1 | 1 . 3 + 0 . 8 - 0 . 4 |\n| NGC5813 | 28.4 | 0 . 33 ± 0 . 01 | 0 . 0288 ± 0 . 0009 | 0 . 28 + 0 . 18 - 0 . 09 | 0 . 0016 + 0 . 0010 - 0 . 0005 | 1 . 0 ± 0 . 4 | 0 . 7 + 0 . 6 - 0 . |\n| NGC5846 | 24.5 | 0 . 378 ± 0 . 009 | 0 . 0256 ± 0 . 0006 | 0 . 4 + 0 . 4 - 0 . 3 | 0 . 002 + 0 . 002 - 0 . 001 | 1 . 3 ± 1 . 0 | 3 1 . 6 + 1 . 2 - 0 . 6 | \n| Target | Obs. ID | Date | Aimpoint | Exposure (ks) | n H , z (10 22 cm - 2 ) | Γ | F P (10 - 14 erg cm - 2 s - 1 ) | F S (10 - 14 erg cm - 2 s - 1 ) |\n|----------|-----------|------------|------------|-----------------|---------------------------|--------------------------|-----------------------------------|-----------------------------------|\n| A2052 | 890 | 03/09/2000 | S3 | 30.5 | < 0 . 014 | 2 . 06 + 0 . 08 - 0 . 06 | 12 . 6 ± 0 . 4 | 10 ± 1 |\n| A2052 | 5807 | 24/03/2006 | S3 | 123.8 | < 0 . 019 | 2 . 00 + 0 . 09 - 0 . 07 | 3 . 8 ± 0 . 1 | 3 . 1 + 0 . 2 - 0 . 3 |\n| A2052 | 10879 | 05/04/2009 | S3 | 80 | 0 . 16 + 0 . 08 - 0 . 07 | 3 . 2 ± 0 . 4 | 2 . 2 ± 0 . 1 | 1 . 1 ± 0 . 3 |\n| A2052 | 10478 | 25/05/2009 | S3 | 119 | 0 . 03 + 0 . 05 - 0 . 03 | 2 . 2 ± 0 . 2 | 2 . 1 ± 0 . 1 | 1 . 4 ± 0 . 3 |\n| A2052 | 10477 | 05/06/2009 | S3 | 59 | 0 . 11 + 0 . 09 - 0 . 08 | 2 . 8 + 0 . 5 - 0 . 4 | 2 . 4 ± 0 . 1 | 1 . 4 + 0 . 3 - 0 . 4 |\n| A2052 | 10479 | 09/06/2009 | S3 | 63.9 | 0 . 08 + 0 . 08 - 0 . 07 | 2 . 8 ± 0 . 4 | 2 . 2 ± 0 . 1 | 1 . 3 + 0 . 3 - 0 . 3 |\n| A2390 | 500 | 08/10/2000 | S3 | 8.8 | 0 | 1.9 | 0 . 9 ± 0 . 4 | < 1 . 1 |\n| A2390 | 4193 | 11/09/2003 | S3 | 70.6 | 0 . 1 + 0 . 2 - 0 . 1 | 1.9 | 1 . 4 ± 0 . 1 | 0 . 9 + 0 . 1 - 0 . 2 |\n| Hydra A | 576 | 02/11/1999 | S3 | 17.4 | 4 . 3 + 0 . 8 - 0 . 6 | 1.9 | 36 ± 3 | 26 ± 2 |\n| Hydra A | 4969 | 13/01/2004 | S3 | 62.1 | 3 . 0 + 0 . 8 - 0 . 7 | 1 . 7 + 0 . 4 - 0 . 3 | 27 ± 1 | 18 ± 1 |\n| Hydra A | 4970 | 22/10/2004 | S3 | 94.6 | 2 . 2 + 0 . 6 - 0 . 5 | 1 . 2 ± 0 . 3 | 24 ± 1 | 14 . 5 + 0 . 8 - 0 . 7 |\n| M84 | 401 | 20/04/2000 | S3 | 1.7 | 0 | 1.9 | 7 ± 1 | - |\n| M84 | 803 | 19/05/2000 | S3 | 25.5 | 0 . 17 ± 0 . 04 | 2 . 1 ± 0 . 1 | 8 . 6 ± 0 . 4 | 8 . 2 + 0 . 4 - 1 . 2 |\n| M84 | 5908 | 01/05/2005 | S3 | 45.1 | 0 . 12 + 0 . 03 - 0 . 02 | 2 . 13 ± 0 . 09 | 17 . 5 ± 0 . 4 | 15 + 1 - 3 |\n| M84 | 6131 | 07/11/2005 | S3 | 35.8 | 0 . 17 + 0 . 08 - 0 . 07 | 2 . 2 ± 0 . 3 | 4 . 9 ± 0 . 3 | 4 . 2 + 0 . 3 - 0 . 4 |\n| NGC5044 | 798 | 19/03/2000 | S3 | 14.3 | 0 | 1.9 | 2 . 4 ± 0 . 4 | < 0 . 2 |\n| NGC5044 | 9399 | 07/03/2008 | S3 | 82.5 | 0 . 23 + 0 . 08 - 0 . 07 | 1.9 | 2 . 6 ± 0 . 2 | 1 . 4 + 0 . 4 - 0 . 3 |\n| PKS0745 | 2427 | 16/06/2001 | S3 | 17.9 | 0 | 1.9 | 3 . 0 ± 0 . 6 | 2 . 0 + 0 . 6 - 1 . 8 |\n| PKS0745 | 12881 | 27/01/2011 | S3 | 116 | < 0 . 08 | 1 . 8 ± 0 . 2 | 4 . 8 ± 0 . 3 | 2 . 6 + 0 . 2 - 0 . 4 |\n| PKS1404 | 1650 | 07/06/2001 | S3 | 7.1 | 0 . 07 ± 0 . 05 | 2 . 4 ± 0 . 2 | 24 ± 1 | 22 ± 2 |\n| PKS1404 | 12884 | 03/01/2011 | S3 | 83.5 | 0 . 04 ± 0 . 02 | 2 . 01 ± 0 . 06 | 21 . 4 ± 0 . 4 | 19 . 8 ± 0 . 7 |"}

2015MNRAS.454.1848R

Electron thermodynamics in GRMHD simulations of low-luminosity black hole accretion

2015-01-01

['mhd', 'stars black holes', 'galaxy center', 'galaxies jets', 'galaxies nuclei', '-']

[]

Simple assumptions made regarding electron thermodynamics often limit the extent to which general relativistic magnetohydrodynamic (GRMHD) simulations can be applied to observations of low-luminosity accreting black holes. We present, implement, and test a model that self-consistently evolves an entropy equation for the electrons and takes into account the effects of spatially varying electron heating and relativistic anisotropic thermal conduction along magnetic field lines. We neglect the backreaction of electron pressure on the dynamics of the accretion flow. Our model is appropriate for systems accreting at ≪10<SUP>-5</SUP> of the Eddington accretion rate, so radiative cooling by electrons can be neglected. It can be extended to higher accretion rates in the future by including electron cooling and proton-electron Coulomb collisions. We present a suite of tests showing that our method recovers the correct solution for electron heating under a range of circumstances, including strong shocks and driven turbulence. Our initial applications to axisymmetric simulations of accreting black holes show that (1) physically motivated electron heating rates that depend on the local magnetic field strength yield electron temperature distributions significantly different from the constant electron-to-proton temperature ratios assumed in previous work, with higher electron temperatures concentrated in the coronal region between the disc and the jet; (2) electron thermal conduction significantly modifies the electron temperature in the inner regions of black hole accretion flows if the effective electron mean free path is larger than the local scaleheight of the disc (at least for the initial conditions and magnetic field configurations we study). The methods developed in this work are important for producing more realistic predictions for the emission from accreting black holes such as Sagittarius A* and M87; these applications will be explored in future work.

[]

https://arxiv.org/pdf/1509.04717.pdf

{'S. M. Ressler 1 , A. Tchekhovskoy 1 /star , E. Quataert 1 , M. Chandra 2 , C. F. Gammie 2 , 3': '1 Departments of Astronomy & Physics, Theoretical Astrophysics Center, University of California, Berkeley, CA 94720 \n- 2 Department of Astronomy, University of Illinois, 1002 West Green Street, Urbana, IL 61801 \n3 Department of Physics, University of Illinois, 1002 West Green Street, Urbana, IL 61801 \n6 August 2018', 'ABSTRACT': 'Simple assumptions made regarding electron thermodynamics often limit the extent to which general relativistic magnetohydrodynamic (GRMHD) simulations can be applied to observations of low-luminosity accreting black holes. We present, implement, and test a model that self-consistently evolves an entropy equation for the electrons and takes into account the e ff ects of spatially varying electron heating and relativistic anisotropic thermal conduction along magnetic field lines. We neglect the back-reaction of electron pressure on the dynamics of the accretion flow. Our model is appropriate for systems accreting at /lessmuch 10 -5 of the Eddington accretion rate, so radiative cooling by electrons can be neglected. It can be extended to higher accretion rates in the future by including electron cooling and proton-electron Coulomb collisions. We present a suite of tests showing that our method recovers the correct solution for electron heating under a range of circumstances, including strong shocks and driven turbulence. Our initial applications to axisymmetric simulations of accreting black holes show that (1) physically-motivated electron heating rates that depend on the local magnetic field strength yield electron temperature distributions significantly di ff erent from the constant electron to proton temperature ratios assumed in previous work, with higher electron temperatures concentrated in the coronal region between the disc and the jet; (2) electron thermal conduction significantly modifies the electron temperature in the inner regions of black hole accretion flows if the e ff ective electron mean free path is larger than the local scale-height of the disc (at least for the initial conditions and magnetic field configurations we study). The methods developed in this work are important for producing more realistic predictions for the emission from accreting black holes such as Sagittarius A* and M87; these applications will be explored in future work. \nKey words: MHD-general relativity - black hole accretion', '1 INTRODUCTION': "A wide variety of low luminosity accreting black holes are currently interpreted in the context of a Radiatively Ine ffi cient Accretion Flow (RIAF) model that describes a geometrically thick, optically thin disc with a low accretion rate and luminosity. In particular, this is true of the black hole at the center of our galaxy, Sagittarius A* (Narayan et al. 1998), the black hole at the center of Messier 87 (Reynolds et al. 1996), and other low luminosity Active Galactic Nuclei (AGN), as well as a number of X-ray binary systems (see Remillard & McClintock 2006 for a review). The gas densities in these systems are low enough that the time scale for electron-ion collisions is much longer than the time scale for accretion to occur, so a one-temperature model of the gas is no longer valid (as originally recognised by Shapiro, Lightman & Eardley \n1976, Ichimaru 1977, and Rees, Begelman, Blandford, & Phinney 1982). Instead, a better approximation is to treat the electrons and ions as two di ff erent fluids, each with its own temperature. \nCalculating the emission from accreting plasma requires predicting the electron distribution function close to the black hole. To date, time dependent numerical models of RIAFs that attempt to directly connect to observations often assume a Maxwellian distribution with a constant electron to proton temperature ratio, Te / Tp , and take the results of GRMHD simulations as the solution for the total gas temperature, Tg = Tp + Te (Dibi et al. 2012; Drappeau et al. 2013; Mo'scibrodzka et al. 2009). This neglects, however, several physical processes that have di ff erent e ff ects on the electron and proton thermodynamics and that are currently only included in onedimensional semi-analytic models. Such e ff ects include electron thermal conduction (e.g., Johnson & Quataert 2007), electron cooling (e.g., Narayan & Yi 1995), and non-thermal particle acceleration and emission (e.g., Yuan, Quataert & Narayan 2003). To date, \nextensions of the simple Tp / Te = const. prescription have been limited to post-processing models that do not self-consistently evolve the electron thermodynamics over time. Examples include the prescription of Mo'scibrodzka et al. (2014) which takes Tp / Te = const. in the disc proper but sets Te = const. in the jet outflow region, as well as the model of Shcherbakov, Penna & McKinney (2012), who solve a 1-D radial equation for Tp -Te at a single time-slice in the midplane to obtain a functional relationship Tp / Te = f ( Tg ) that is then applied to the rest of the simulation. To enable a more robust connection between observations of accreting black holes and numerical models of black hole accretion, it is critical to extend the detailed thermodynamic treatment of electrons used in 1D calculations to multi-dimensional models. This is the goal of the current paper. In particular, we describe numerical methods for separately evolving an electron energy equation in GRMHD simulations. We focus on including heating and anisotropic thermal conduction in these models. Future work will include electron radiative cooling and Coulomb collisions between electrons and protons. \nIn a turbulent, magnetised plasma, electrons and ions are heated at di ff erent rates depending on the local plasma conditions (e.g., Quataert & Gruzinov 1999; Cranmer et al. 2009; Howes 2010; Sironi 2015). Furthermore, since the electron-to-proton mass ratio is small, electrons will both conduct and radiate their heat much more e ffi ciently than the ions. The combination of these effects leads to the expectation that, in general, Te < Tp . In the present paper, we thus neglect the e ff ect of the electron thermodynamics on the overall dynamics of the accretion flow. This allows us to treat the simulation results as a fixed background solution on top of which we independently evolve the electrons. Even if we find that Te ∼ Tp in some regions of the disc, this treatment may still be a reasonable first approximation given the uncertainties in the electron physics. \nThe neglect of electron cooling in the present paper is reasonable for systems accreting at /lessorsimilar 10 -5 of the Eddington rate, ˙ M Edd, so that the synchrotron cooling time is much longer than the accretion time (Mahadevan & Quataert 1997). In particular, this likely includes Sagittarius A* in the galactic center. The application of our methodology to Sgr A* is particularly important given the wealth of multi-wavelength data (e.g., Serabyn et al. 1997, Zhao et al. 2003, Genzel et al. 2003, Bagano ff et al. 2003, Barri'ere et al. 2014) and current and forthcoming spatially resolved observations with the Event Horizon Telescope (Doeleman et al. 2008) and Gravity (Gillessen et al. 2010). \nThe goal of this paper is to present our formalism and methodology for evolving the electron thermodynamics and to apply the results to 2D (axisymmetric) GRMHD simulations of an accreting black hole. We show the range of possible electron temperature distributions in the inner region of the disc, which directly impacts the predicted emission. Future work will explore the impact that these results have on the emission, spectra, and images of Sagittarius A*. \nThe remainder of this paper is organised as follows. § 2 describes our theoretical model of electron heating and anisotropic electron conduction while § 3 describes the numerical implementation of this model. § 4 contains tests of the numerical implementation, § 5 applies the model to a 2D simulation of an accretion disc around a rotating black hole, and § 6 discusses the implications of this application and concludes. Boltzmann's constant, kb , and proton mass, mp , are taken to be 1 throughout. We use cgs units, with Lorentz-Heaviside units for the magnetic field (e.g., magnetic pressure is b 2 / 2), and a metric signature of ( -+++ ). We also assume that the gas is mostly hydrogen and ideal. Since we also assume \nne ≈ np ≡ n , then ρ = mene + mpnp ≈ mpn = n (setting mp = 1), so we use ρ and n interchangeably.", '2 ELECTRON THERMODYNAMICS': "The accreting plasmas of interest are su ffi ciently low density that the electron-proton Coulomb collision time is much longer than the dynamical time and so a two-temperature structure can develop, with the protons and electrons having di ff erent temperatures (and, indeed, di ff erent distribution functions). Moreover, at the low accretion rates where radiative cooling can be neglected, the electronelectron and proton-proton Coulomb collision times are also much longer than the dynamical time (Mahadevan & Quataert 1997). However, the plasma densities are high enough that the plasma is nearly charge-neutral and so we assume that ne ≈ np . We further assume that the electron flow velocity is the same as that of the protons. 1 This need not strictly be true (e.g., in the solar wind the relative velocities of particle species can be of order the Alfven speed; e.g. Bourouaine et al. 2013 ), but is a reasonable first approximation. A similar approach is often used in modeling the global dynamics of the low-collisionality solar wind (e.g., Chandran et al. 2011). \nUnder these assumptions, the key di ff erence in the electron and proton physics lies in their di ff erent thermodynamics: the protons and electrons have very di ff erent heating and cooling processes that need to be separately accounted for. Formally, because of the low collisionality conditions we should separately solve the electron and proton Vlasov equations. This is computationally extremely challenging, however, particularly in the global geometry required to predict the emission from accreting plasmas (even local shearing box calculations using the particle-in-cell technique to solve the Vlasov equation require an unphysical electron-proton mass ratio, thus making it di ffi cult to reliably model the electron thermodynamics; e.g., Riquelme et al. 2012). As a result, we assume a fluid model in this paper. Our fluid approximation corresponds to taking moments of the Vlasov equation and applying closures on higher moments of the distribution function. As we shall describe, our closure corresponds to specific models for the conductive heat flux, the viscous momentum flux, and the turbulent heating rate of each particle species. \nOur basic model is thus to take a single-fluid GRMHD solution (e.g., Komissarov 1999; Gammie, McKinney & T'oth 2003; De Villiers & Hawley 2003) as an accurate description of the total fluid (composed of both the electron and proton gas) dynamics and thus the accretion flow density, magnetic field strength, and velocity field. We evolve the electrons as a second fluid on top of this background solution. The GRMHD solution may itself include viscosity and conduction as in Chandra et al. (2015). Our assumption that the electrons do not back react on the flow dynamics is formally valid in the limit that Te /lessmuch Tp , but should be a reasonable approximation so long as Te /lessorsimilar Tp in regions of large plasma β /greaterorsimilar 1, i.e., where gas pressure forces are dynamically important. One advantage of not coupling the electron pressure to the GRMHD solution is that we can run multiple electron models in one simulation, allowing us to explore systematic uncertainties with a minimum of computational time. \nIn this paper, we focus on implementing electron heating and \nanisotropic conduction. Coulomb collisions are straightforward to include but are negligible for the low accretion rates at which electron cooling can be neglected. In future work, electron cooling will be self-consistently incorporated building on the BHlight code developed by Ryan, Dolence & Gammie (2015).", '2.1 Basic Model': "The stress-energy tensors for the electron and proton fluids in our model take the form: \nT µν e = ( ρ e + ue + Pe ) u µ e u ν e + Peg µν + τ µν e + q µ e u ν e + u µ e q ν e T µν p = ( ρ p + up + Pp ) u µ p u ν p + Ppg µν + τ µν p , (1) \nwhere ρ k , uk , and Pk are the fluid frame density, internal energy, and pressure, respectively, u µ k is the fluid four-velocity in the coordinate frame, τ µν k is a general stress tensor that accounts for viscous e ff ects, and q µ e is the heat flux carried by the electrons. The subscript k denotes p or e (and will also denote the total gas quantities labeled by g below). We leave τ µν k as a general tensor that will be modelspecific. For each species, ignoring electron-electron, electron-ion, and ion-ion collisions, one can take the zeroth and first moment of the Vlasov equation to show that \n∇ µ ( ρ ku µ k ) = 0 (2) \nand \n∇ µ T µν e = -enu µ e F ν µ ∇ µ T µν p = enu µ p F ν µ , (3) \nwhere F µν is the electromagnetic field tensor. In ideal, single-fluid GRMHD in the absence of shocks, the conservation of entropy equation, ρ Tgu µ ∂µ sg = 0, where sg is the entropy per particle, follows directly from the conservation of particle number and the stress-energy (see page 563 in Misner, Thorne & Wheeler 1973). To derive entropy equations for the electron and proton fluid used in our model, we perform the same series of manipulations; namely, contracting both equations (3) with u ν (the total fluid velocity, which we take to be ≈ u µ p ≈ u µ e ) and invoking equation (2), which give us: \nρ Teu µ ∂µ sp = Qp , (4) \nand \nρ Teu µ ∂µ se = Qe - ∇ µ q µ e -a µ q µ e , (5) \nwhere we have defined the heating rate per unit volume for each species as a sum of viscous and Ohmic resistance terms, Qe ≡ u ν ∇ µτ µν e + enu µ e u ν F ν µ and Qp ≡ u ν ∇ µτ µν p -enu µ pu ν F ν µ , and where a µ ≡ u ν ∇ ν u µ is the four-acceleration, which accounts for gravitational redshifting of the temperature by the metric. We can write the heating rates in terms of the electric field four-vector, e µ ≡ u ν F νµ , \nand the four-currents, J µ e ≡ -neu µ e , J µ p ≡ neu µ p as 2 : \nQe = u ν ∇ µτ µν e + J µ e e µ Qp = u ν ∇ µτ µν p + J µ p e µ. (6) \nThe intuitive understanding of the Ohmic heating terms on the right-hand side of equation (6) is that they are ∼ /vector Jk · E evaluated in the rest frame of the total fluid. To derive the entropy equation for the total fluid, we first define several total fluid variables as a sum of electron and proton terms: ρ = ρ p + ρ e ≈ n , ug = up + ue , Pg = Pp + Pe , Tg = Tp + Te , J µ = J µ p + J µ e = en ( u µ p -u µ e ) and τ µν g = τ µν e + τ µν e , denoting total gas mass density, internal energy, pressure, temperature, current, and viscous stress. Then, using the thermodynamic identity, ρ Tku µ ∂µ sk = u µ ∂µ uk -( uk + Pk ) u µ ∂µ log( ρ ), we find that the entropy per particle of the total gas, sg , satisfies the relation ρ Tgu µ ∂µ sg = ρ Tpu µ ∂µ sp + ρ Teu µ ∂µ se , resulting in \nρ Tgu µ ∂µ sg = Q - ∇ µ q µ e -a µ q µ e , (7) \nwith the total heating rate per unit volume: \nQ = Qp + Qe = u ν ∇ µτ µν g + J µ e µ. (8) \nIn practice, we use the electron entropy equation (5) to evolve the electron thermodynamics. To determine the overall dynamics of the electron + proton gas, we use Maxwell's equations in addition to a standard, single-fluid GRMHD evolution representing the total gas. The equations for the latter are obtained by separately summing the electron and proton parts of equation (2) and equation (3), resulting in a mass conservation equation, \n∇ µ ( ρ u µ ) = 0 , (9) \nand an energy-momentum equation, \n∇ µ T µν g + T µν EM ) = -∇ µτ µν g , (10) \n) with the total gas stress-energy tensor, \nT µν g = ρ + ug + Pg ) u µ u ν + Pgg µν , (11) \n( ( \n) and the electromagnetic stress energy tensor 3 , T µν EM = F µα F ν α -g µν F αβ F αβ / 4. We have used the identity ∇ µ T µν EM = -J µ F ν µ in equation (10), the assumption that u µ e ≈ u µ p ≈ u µ in equation (11), and the charge-neutrality assumption ne = np = n throughout. Furthermore, we have dropped the electron thermal conduction terms in the evolution of the total gas properties, though we keep them in the evolution of the electron entropy (equation 5). This is consistent if Te /lessorsimilar Tp since electron conduction will then a ff ect the electron thermodynamics but not the overall stress-energy of the fluid. Finally, we take T µν EM to be given by the ideal MHD limit (i.e., e µ → 0): \nT µν EM = b 2 u µ u ν + b 2 2 g µν -b µ b ν , (12) \nwhere b µ ≡ /epsilon1 µνκλ u ν F λκ/ 2 is the magnetic field four-vector defined \n2 We have kept the subscripts e and p for the four-currents (and thus four-velocities) in equation (6) because the details of the Ohmic heating depend on the small but non-zero velocity di ff erence between the proton and electron fluids (or equivalently the velocity di ff erence between the electron / proton fluid and the total fluid). An explicit expression for these terms would require a detailed kinetic theory calculation beyond the scope of the present work (i.e., some form of 'generalized Ohm's Law,' as in, e.g., Koide 2010). In our model, as described in the text, numerical resistivity provides the Ohmic heating that is then distributed to electrons and protons according to a closure model obtained from previous work in kinetic theory. \n3 Here we have chosen to absorb a factor of (4 π ) -1 / 2 into the definition of F µν . \nin terms of the Levi-Civita tensor, /epsilon1 µνκλ , and b 2 ≡ b µ b µ is twice the magnetic pressure. With these assumptions, equations (9), (10), and Maxwell's equations are simply the standard single-fluid equations of ideal GRMHD except with an explicit viscosity tensor. In standard conservative GRMHD codes (including the one used in this work), this viscosity tensor is not included explicitly but implicitly generated numerically by the Riemann solver. Furthermore, the Riemann solver also introduces a finite numerical resistivity into Maxwell's equations, allowing for a nonzero e µ (and thus nonzero Ohmic heating). For further discussion of these points, see 3.1. \nTo summarise, we take a standard single fluid GRMHD evolution of u µ , ρ, ug , and Pg as a reasonable estimate of the total gas properties. This corresponds to assuming that electron conduction has a negligible contribution to the dynamics of the total gas and that the adiabatic index is independent of the electron thermodynamic quantities (e.g.; Pg / ug ≡ [ Pp + Pe ] / [ ue + up ] ≡ γ -1 ≈ some function of total gas quantities only). Formally, this assumption requires that the electron internal energy is small compared to the proton internal energy. From this, we can calculate the heating directly from equation (7) (dropping the conduction terms) without requiring an analytic expression for Q . Finally, we use knowledge of the nature of heating in a collisionless plasma obtained from kinetic theory (described in § 5.1) to relate the heating rate per unit volume of the electrons, Qe , to that of the total fluid, Q , and directly add it to the electron entropy equation (equation 5), as described in § 2.2. This completes our model. \n§ \n§ For simplicity, we assume that the adiabatic indices of the electron, γ e , proton, γ p , and total gas, γ , are constants, where Pk = ( γ k -1) uk for k = e , p , or g . This simplifies the numerical implementation of the model, as it allows us to write the entropy per particle in a simple form, sk = ( γ k -1) -1 log( Pk ρ -γ k ), and avoids the complication of having to evaluate Tp / Te when updating the total fluid variables. This can be seen by noting that \nPg ug ≡ Pe + Pp ue + up = ( γ e -1)( γ p -1) 1 + Tp / Te ( γ p -1) + ( γ e -1) Tp / Te , (13) \nwhich is only constant in the limits that Te /lessmuch Tp or Tp /lessmuch Te . From this, we see that this simplification of γ = const. is formally inconsistent if γ e /nequal γ p , which is generally the case in the accreting systems of interest, where the electrons are typically relativistically hot ( γ e ≈ 4 / 3) but the protons are nonrelativistic ( γ p ≈ 5 / 3). However, since equation (13) is bounded between 1 / 3 and 2 / 3 and we expect Te /lessorsimilar Tp ⇒ γ ≈ γ p , we do not anticipate that this approximation will a ff ect our results significantly.", '2.2 Electron Heating': 'We parameterise the heating term, Qe in equation (6) by writing Qe = feQ , where fe ( β, Te , Tp , .... ) ≡ Qe / Q is the fraction of the total dissipation, Q , received by the electrons. This function, in general, depends on the local plasma environment and our model is not limited to any particular choice of fe . As knowledge in the field develops we can readily incorporate di ff erent assumptions about electron heating. A more detailed discussion of one physically-motivated prescription for fe is given in § 5.1. Given a GRMHD solution, the total heating rate of a fluid element moving with four-velocity u µ in the coordinate frame can be computed (from equation 7, dropping the conduction terms): \nQ = ρ Tgu µ ∂µ sg , (14) \nwhere sg is the entropy per particle. We can rewrite equation (14) in terms of κ g ≡ Pg ρ -γ , where sg = ( γ -1) -1 log( κ g ), as Q = ρ γ ( γ - \n1) -1 u µ ∂µκ g . We use κ g to avoid the undesirable numerical properties of logarithms as the argument goes to 0. Likewise, we will often use κ e in place of se in equation (5).', '2.3 Anisotropic Electron Conduction': 'Some care must be taken when generalising the theory of anisotropic conduction along magnetic field lines to a relativistic and covariant formulation. In particular, the theory must be consistent with causality in that the heat flux should not respond instantly to temperature gradients. Our formulation of anisotropic electron conduction draws heavily on the treatment of Chandra et al. (2015), who consider a single fluid model in which the heat flux is coupled to the dynamics via the stress-energy tensor. We give a brief summary of our approach here, highlighting those aspects of our electron-only treatment that di ff er from the formulation in Chandra et al. (2015). \nOne can derive a perturbation solution for the heat flux, q µ e , by expanding the entropy current in powers of q µ e and imposing the second law of thermodynamics. The most straightforward relativistic generalisation of the classical, isotropic heat flux first written down by Eckart (1940) is first order in this expansion and was later shown by Hiscock & Lindblom (1985) to be unconditionally unstable, precisely because it violated causality (Chandra et al. 2015 showed the same for anisotropic conduction). Israel & Stewart (1979) derived a second order solution for q µ e which was later shown to be conditionally stable (Hiscock & Lindblom 1985; Chandra et al. 2015). Here we use a first order reduction of that second order model that has been shown to be both stable and self-consistent (Andersson & Lopez-Monsalvo 2011). We refer the reader to Chandra et al. (2015) for more details. \nWe parameterise the heat flux as \nq µ e = φ ˆ b µ , (15) \nwhere ˆ b µ is a unit vector ( ˆ b µ ˆ b µ = 1) along the magnetic field fourvector, b µ , and the scalar φ is given by the following evolution equation: \nwhere g is the determinant of the metric, and we used an identity, ∇ µ A µ = ∂µ ( √ -gA µ ) / √ -g , to convert covariant derivatives into partial ones (eq. 86.9 in Landau & Lifshitz 1975). Here φ eq is the equilibrium value of the heat flux given by \n∇ µ ( φρ u µ ) = 1 √ -g ∂µ ( √ -g φρ u µ ) = -ρ [ φ -φ eq τ ] , (16) \nφ eq = -ρχ e ˆ b µ ∂µ Te + ˆ b µ a µ Te ) . (17) \n( \n) where χ e is the thermal di ff usion coe ffi cient of the electrons and τ is the relaxation time scale for the heat flux over which it responds to temperature gradients. Note that equation (16) is a relaxation equation in which the heat flux relaxes on a timescale τ to the equilibrium value. \nThe equilibrium heat flux in equation (17) is the natural relativistic extension of anisotropic conduction along the magnetic field (analogous to the isotropic heat flux of Eckart 1940). The heat flux, q µ e , then contributes to the electron energy equation as in equation (5). Physically motivated prescriptions for the parameters χ e and τ are all that are required to complete the model. We discuss one choice of these in § 5.1.', '2.3.1 Stability of Anisotropic Electron Conduction Theory': 'In our formalism, we assume that the fluid velocity, u µ , and the electron number density, ne = ρ/ mp , are independent of the elec- \non thermodynamics. Thus, in order to do a perturbative analysis we need only perturb the electron temperature, Te , and the heat flux, φ , in equations (5) and (16). Doing this in the fluid rest frame in Minkowski space, where u µ = (1 , 0 , 0 , 0), and writing the perturbations in Fourier space as ∝ exp( λ t + i /vector k · /vector x ), we find the dispersion relation: \nλ 2 + λ τ + ( γ e -1) χ e τ ( ˆ b · /vector k ) 2 = 0 (18) \nwith the solutions: \nλ = 1 2 τ ( -1 ± √ 1 -4( γ e -1) χ e τ ( ˆ b · /vector k ) 2 ) . (19) \nThe theory is unstable if Re( λ ) > 0, which can only occur if the term under the square root is both real and greater than unity. However, this is impossible for any value of k when γ e /greaterorequalslant 1, so we conclude that equations (5) and (16) are unconditionally stable. This is in contrast to the case where equations (5) and (16) are coupled to the ideal MHD equations, which is unstable to small perturbations unless the relaxation time is larger than a critical value (Hiscock & Lindblom 1985; Chandra et al. 2015).', '3 NUMERICAL IMPLEMENTATION OF ELECTRON HEATING AND CONDUCTION': "The method outlined above can, in general, be applied to any GRMHD'background' simulation. For the rest of this work, however, we will consider only conservative codes, as the equations of ideal MHD can be naturally written in that form. Because of this, in what follows we will seek to put all of our evolution equations in a conservative form, namely: \n∂ U ∂ t + ∂ F i ∂ x i = S , (20) \nwhere U is a 'conserved' variable, F i is the corresponding flux in the i th direction, and S is the source, which in general includes the contribution from the connection coe ffi cients. Equation (20) can then be approximated in one spatial dimension by the following discretisation: \nU n + 1 = U n -∆ t F n + 1 / 2 j + 1 / 2 -F n + 1 / 2 j -1 / 2 ∆ x -S n + 1 / 2 , (21) \n \n where the fluxes are evaluated at face centres using the chosen Riemann solver. The generalisation to higher dimensions is straightforward. \nWith that in mind, we can rewrite equation (5): \n∂µ ( √ -g ρ u µ κ e ) = √ -g ( γ e -1) ρ γ e -1 [ feQ - ∇ µ q µ e -a µ q µ e ] , (22) \nwhere we have used the definition κ e ≡ exp[( γ e -1) se ]. Note that equation (22) is a quasi-conservative equation with U κ e = √ -g ρ u t κ e and F i κ e = √ -g ρ u i κ e ('quasi' conservative because the standard definition of conservative equations excludes source terms with derivatives). To solve equation (22), we use operator splitting in the following sequence of steps: \n- 1. Solve the conservative equation with S κ e = 0.\n- 2. Explicitly update κ e with the heating term (the first term in the brackets in eq. 22).\n- 3. Implicitly solve a matrix equation to include the conduction source terms (the rest of the terms in square brackets in eq. 22). \nSteps 2 and 3 are described in detail in § 3.3 and § 3.4, respectively, while step 1 will be specific to the choice of the background numerical scheme.", '3.1 Heating in Conservative Codes': "Formally, the equations of ideal MHD used by conservative GRMHD simulations imply that the heating rate per unit volume, Q , in equation (8) is identically zero. However, conservative codes implicitly add numerical viscosity and resistivity terms to the stress-energy tensor and Maxwell's equations, respectively. The former implies that the numerically evolved stress tensor is in fact T µν g , num = T µν g + τ µν g = T µν MHD + O (truncation error) for some numerical viscosity tensor τ µν g , while the latter implies J µ e µ = 0 + O (truncation error). The numerical resistivity can be thought of as implicitly introducing a form of Ohm's law that allows for a nonzero electric field four-vector, e µ . Thus, even though the energy implied by T µν MHD , num = T µν g , num + T µν EM , num is conserved to machine precision (see below for details), T µν MHD experiences truncationlevel heating. This manifests itself as entropy generation: truncation errors lead to dissipation of magnetic and kinetic energy close to the grid scale that is captured as internal energy. We use this change in entropy to directly calculate the heating rate per unit volume of the gas, Q . \nAlthough T µν MHD , num is conserved to machine precision, the second law of thermodynamics is satisfied only to truncation error. Thus there can be locally regions with Q < 0. In particular, the truncation error can be positive or negative, so in places with small actual change in entropy or large truncation error the change in entropy can be negative. This is the case even in test problems in which our methods of calculating the heating give the correct, converged, answer for the fluid variables (see § 4). Thus, while Q may be instantaneously or locally negative, when integrated over a sufficient length of time and / or space in the fluid frame it will satisfy the second law of thermodynamics. \nWe choose this method of calculating the heating rate as opposed to introducing an explicit functional form for Q because it seems reasonable to assume that for several applications, the gridscale dissipation in conservative codes is a well-defined quantity determined by the converged large-scale physics of the problem. Turbulence, for example, takes kinetic and / or magnetic energy at the largest scales and cascades it down to a small dissipative scale where it is converted into internal energy. For a numerical scheme with no explicit viscosity, the scale at which dissipation occurs depends entirely on the resolution of the simulation, but we expect the heating rate itself will be fixed (in an averaged sense; see, e.g., Davis, Stone & Pessah 2010). We expect the same for forced reconnection at high β values, as in the disc midplane, where the largescale dynamics sets the rate at which the field lines of opposite sign are brought together. \nThe above argument relies on the conservation of energy. In an arbitrary space-time, however, conservation of energy is only welldefined if the metric is stationary (time-independent) and therefore possesses a time-like Killing vector, K µ . If such a vector exists (as it does for the Kerr metric of interest in this work), we can construct a conserved current from the stress-energy tensor via J µ ≡ -K µ T ν µ , where this current satisfies ∇ µ J µ = 0. This allows us to define a conserved energy in a coordinate basis: \nE = ∫ J t √ -gdx 1 dx 2 dx 3 , (23) \nwhere the integral is over all space (i.e. the space orthogonal to the \ntime coordinate). Often, the Killing vector takes the form K = ∂ t , which simplifies equation (23) to \nE = ∫ -T t t √ -gdx 1 dx 2 dx 3 . (24) \nThus, -T t t can be thought of as the conserved energy per unit volume for a particular choice of coordinates. The total energy, E , is conserved to machine precision, modulo fluxes of energy through the boundaries, so entropy can only be generated by conversion of one form of energy to another.", '3.2 Calculating the Total Heating Rate': 'To calculate the heating generated at each time step, we introduce an entropy-conserving equation as a reference to compare with the energy conservation equation. The entropy-conserving equation is simply a conservation equation like equation (20) with U κ g = √ -g ρ u t κ g , F i κ g = √ -g ρ u i κ g , and S κ g = 0. If we call the solution to this equation ˆ κ g (the ˆ denotes the solution corresponding to entropy conservation), then we show in Appendix B1 that the total heating rate of the fluid (measured in the fluid rest frame) incurred over an interval ∆ t (measured in the coordinate frame), \nQ = ( ρ γ -1 γ -1 ) n + 1 / 2 [ ρ u t ( κ g -ˆ κ g ) ∆ t ] n + 1 , (25) \nwhere u t accounts for the transformation of ∆ t from the coordinate frame to the fluid rest frame, n and n + 1 denote the values at the beginning and the end of the time step, respectively, so that tn + 1 = tn + ∆ t , and n + 1 / 2 denotes the values when calculating the fluxes. To compute the dissipation rate via eq. (25), we set ˆ κ n g = κ n g at the beginning of each time step and use ˆ κ n + 1 / 2 g = κ n + 1 / 2 g when calculating the fluxes. Physically, equation (25) means that the Lagrangian heating rate is set by the di ff erence between the entropy implied by the total energy conserving solution ( κ g ) and the entropy implied by the entropy conserving solution (ˆ κ g ).', '3.3 Electron Heating Update': 'Let us call ˆ κ e the solution to equation (22) without any source terms. On top of this adiabatic evolution, electrons receive a fraction, fe , of the heating of the gas, Qe = feQ . In discrete form, this can be written as follows, \n( ρ γ e ) n + 1 / 2 γ e -1 ( κ e -ˆ κ e ) n + 1 = f n + 1 / 2 e ( ρ γ ) n + 1 / 2 γ -1 ( κ g -ˆ κ g ) n + 1 . (26) \nTherefore, the heating update to the electrons, ˆ κ n + 1 e → κ n + 1 e , takes the following form: \nκ n + 1 e = ˆ κ n + 1 e + γ e -1 γ -1 ( ρ γ -γ e fe ) n + 1 / 2 ( κ g -ˆ κ g ) n + 1 . (27)', '3.4 Electron Conduction Update': "Wenote that the evolution equation for the heat flux φ (equation 16) is already in a quasi-conservative form if we define U φ = √ -g ρ u µ φ and F i φ = √ -g ρ u i φ . We treat the evolution of φ in an operator split way similar to the evolution of κ e with the following series of steps: \n- 1 Solve the conservative equation with S φ = 0.\n- 2 Implicitly solve a matrix equation to include the source terms. \nThe source terms in the electron entropy and φ equation due to conduction are given by: \n \nS κ e , cond = √ -g ( γ e -1) ρ 1 -γ e ( -∇ µ q µ e -a µ q µ e ) S φ = -ρ √ -g φ + ρχ e ( ˆ b µ ∂µ Te + Te ˆ b µ a µ ) τ , (28) \n which are discretised in space by using slope-limited derivatives across three grid cells and discretised in time by centring the time derivatives at tn + 1 / 2. The latter discretisation gives us an implicit equation for the variables κ e and φ at time tn + 1. If we call ˆ φ n + 1 the heat flux after being updated by step 1, and κ e , H the electron entropy after being updated by heating, then this matrix equation takes the form: \n \n( a 11 a 12 a 21 a 22 ) ( κ n + 1 e φ n + 1 ) = ( b 1 b 2 ) , (29) \nwith components \na 11 = ( √ -g ρ u t ∆ t ) n + 1 a 12 = [ √ -g ( γ e -1) ρ 1 -γ e ] n + 1 / 2 [ ˆ b t ∆ t ] n + 1 a 21 = √ -g ρ 2 ˆ b t χ e τ n + 1 / 2 [ ρ γ e -1 ∆ t ] n + 1 a 22 = ( √ -g ρ u t ∆ t ) n + 1 (30) \nb 1 = ( √ -g κ e , H ρ u t ∆ t ) n + 1 + [ ( γ e -1) ρ 1 -γ e ] n + 1 / 2 × [ √ -g ( q t e ∆ t ) n -( ∂ i ( √ -gq i e ) -√ -gq µ e a µ ) n + 1 / 2 ] b 2 = ( √ -g ˆ φρ u t ∆ t ) n + 1 -√ -g ( ρφ τ ) n + 1 / 2 + √ -g ( ρ 2 χ e τ ) n + 1 / 2 × [ ( ˆ b t ) n + 1 / 2 ( Te ∆ t ) n -( ˆ b µ ∂µ Te + ˆ b µ a µ Te ) n + 1 / 2 ] . (31) \nand \nThe system of equations has a straightforward solution, \n( κ n + 1 e φ n + 1 ) = 1 a 11 a 22 -a 21 a 12 ( b 2 a 11 -b 1 a 21 b 1 a 22 -b 2 a 12 ) . (32) \nTo ensure that the heat flux, φ , does not reach unphysically large values, we apply a limiting scheme to keep | φ | /lessorsimilar ( ue + ρ ec 2 ) v t , e ≡ φ max, where ρ e = ρ me / mp and v t , e is the electron thermal speed. Since we are considering physical systems in which the electrons are always at least mildly relativistic, this limit e ff ectively reduces to | φ | /lessorsimilar uec / √ 3, which corresponds to a 'saturated' heat flux in which the heat is redistributed at the electron thermal speed. The numerical implementation of this limit is to replace the values of the thermal di ff usivity, χ e , and the relaxation time-scale, τ , with 'e ff ective' values (Chandra et al. 2015): \nχ e ff = χ e f ( | φ | φ max ) , (33) \nτ e ff = τ f ( | φ | φ max ) , (34) \nand \nwhere \nf ( x ) = 1 -1 1 + exp ( -x -1 0 . 1 ) + /epsilon1 , (35) \nwhich sharply transitions from 1 → /epsilon1 for some small /epsilon1 as | φ | → φ max. Thus, according to equation (16), when | φ | > φ max, | φ | decays exponentially on a timescale ∼ /epsilon1 τ until it drops below φ max. The parameter /epsilon1 is chosen such that the criterion for numerical stability is always satisfied (see § 3.4.1 and Appendix B3).", '3.4.1 Numerical Stability of Electron Conduction': 'Adetailed derivation of the criteria for numerical stability is in Appendix B3. The basic result is that for a Courant-Friedrichs-Lewy (CFL) number, C , reasonably chosen between 0 and 1, the relaxation time, τ , must satisfy \nτ > f ( C ) ( ∆ t ∆ x ) 2 χ e , (36) \nwhere f ( C ) is a function of the CFL number. This can be understood as a requirement that the relaxation time τ (which we are free to choose as arbitrarily large, though which should correspond to a physical time scale), must be larger than the time step ∆ t (which is limited by computational expense) by the ratio between ∆ t and the standard Courant limit for a di ff usive process ∆ t di ff = ∆ x 2 /χ e .', '3.5 Treatment of the Floors': 'Conservative codes deal poorly with vacua of internal energy and density. Because of this, many schemes employ floors on internal energy and density to ensure that the errors in solving for the primitive variables from the conservative variables do not produce unphysically small or negative values. The nature of the model outlined above requires special care to be taken when these floors are activated, as they introduce artificial changes in internal energy, which act as a source of heat, and density, which change the conversion between entropy and internal energy.', '3.5.1 Electron Energy Floors': 'Though the second law of thermodynamics states that the heating term u µ ∂µκ g should be positive definite, numerically we find that u µ ∂µκ g can be locally negative because of truncation errors. This introduces the possibility of the electron internal energy going to zero (or even becoming negative) due to truncation error fluctuations in our heating term. To correct for this, we implement a floor on the electron internal energy that is 1% of the floor on the total gas internal energy. That is, if ue drops below 0 . 01 ug , we reset ue to 0 . 01 ug .', '3.5.2 Total Gas Internal Energy Floors': 'When the floor on internal energy of the total gas is activated, there is an artificial increase in ug which then shows up in our heating term. We treat this addition of energy as if it were a physical, isochoric addition to the energy of the gas and add it to the electrons as described above. We emphasise that the internal energy floor does not a ff ect the system dynamics in any significant way because it is only activated in magnetically-dominated regions where the value of the internal energy is dynamically irrelevant.', '3.5.3 Density Floors': 'When the floor on density is activated, the total gas internal energy remains unchanged. However, the value of ˆ ug ≡ ˆ κ g ρ γ / ( γ -1) increases by a factor of ( ρ floor /ρ init) γ , where ρ init is the pre-floor density. To correct for this, we require conservation of the evolved gas entropy when the density floor is activated by decreasing ˆ κ g by a factor of ( ρ floor /ρ init) γ . Furthermore, we enforce that the evolved electron entropy remains unchanged by the density floor in the same manner by decreasing κ e by a factor of ( ρ floor /ρ init) γ e . Similar to the internal energy floors, the density floors do not a ff ect the dynamics of the system.', '4 TESTS OF NUMERICAL IMPLEMENTATION': "In this section we describe a series of tests that demonstrate the robustness and accuracy of our method of evolving the electron internal energy. We implemented the model described in § 2 and § 3 into the conservative GRMHD code, HARM2D (High-Accuracy Relativistic Magnetohydrodynamics; Gammie, McKinney & T'oth 2003; Noble et al. 2006). To speed up the computations, we parallelised the code using OpenMP and MPI via domain decomposition.", '4.1 Tests of Electron Heating': "In what follows we demonstrate the validity and convergence of our implementation of electron heating using a number of tests. The 2nd order convergence of HARM in smooth flows and 1st order convergence in discontinuous flows is well documented in Gammie, McKinney & T'oth (2003) and we will not reproduce it here.", '4.1.1 Explicit Heating in a Hubble-Type Flow': 'To test whether our discretizations of the heating is correctly time centred, i.e., converges at the expected 2nd order in time, we focus here on solving the electron equation when we introduce an explicit heating term to the total energy equation. We do this in an unmagnetised, 1D Hubble-type flow with v ∝ x (restricting ourselves to non-relativistic velocities). In the local rest frame of a fluid element, this velocity field gives an outflow in both directions that is homogenous and isotropic, causing the density to uniformly decrease with time as matter leaves the computational domain. The velocity profile also scales with time to satisfy the momentum equation ( ∂ v /∂ t + v ∂ v /∂ x = 0). In the absence of heating, the internal energy and pressure evolve according to entropy conservation ( P ∝ ρ γ ), so that the solution at later times is given by (Tchekhovskoy, McKinney & Narayan 2007): \nv = v 0 x 1 + v 0 t ug = ug , 0 (1 + v 0 t ) γ ρ = ρ 0 1 + v 0 t . (37) \nIf we now add a cooling term to the energy equation, of the form: \nQ = -ug , 0 v 0 ( γ -2) (1 + v 0 t ) 3 , (38) \nthe internal energy should evolve as \nug = ug , 0 (1 + v 0 t ) 2 . (39) \nFigure 1. L1 norm of the error in the electron entropy for heating in a 1D Hubble-type flow (see § 4.1.1). Above a resolution of ∼ 1000 the relativistic errors in the analytic result are comparable to the numerical truncation errors, so convergence is no longer seen. \n<!-- image --> \nPlugging these solutions in the electron entropy equation (22) for fe = 1 and ue ( t = 0) = u 0, we obtain: \nκ e = ( γ -2) ( γ e -1) γ e -2 u 0 ρ γ e 0 1 (1 + v 0 t ) 2 -γ e . (40) \nFor the numerical test, we set these analytic solutions as the boundary and initial conditions in a one-dimensional grid and check if we maintain this solution after a dynamical time of L / max[ v ( t = 0)]. We set γ = 5 / 3, γ e = 4 / 3, max( v 0 x ) = 10 -3 c , and max( ρ v 0 x / ug ) = 1, on a computational domain of 0 /lessorequalslant x /lessorequalslant 1. Formally, since Θ e ≡ kTe / mec 2 /lessmuch 1, the choice of γ e = 4 / 3 is unphysical. However the motivation for this choice stems from the fact that our primary application is to the inner regions of an accretion disc around a black hole, where we expect γ e ≈ 4 / 3 /nequal γ ≈ 5 / 3. We find that our calculation converges at second order (see Figure 1), up until the point at which the errors in the analytic solution due to relativistic e ff ects become important (which, for max( v 0 x ) = 10 -3 is δκ e /κ e ∼ v 2 / c 2 ∼ 10 -6 ).', '4.1.2 1D (Noh) Shock Test': "In Appendix B4, we show that for a high Mach number shock in which the electrons are assumed to receive a constant fraction fe of the 'viscous' heating in the shock, the post shock electron internal energy ue is given by: \nu f e u f g = fe 2 [( γ + 1 γ -1 ) γ e ( 1 -γ γ e ) + 1 + γ γ e ] γ 2 -1 γ 2 e -1 , (41) \nwhere u f g and γ are the post-shock internal energy and the adiabatic index of the fluid. Equation (41) assumes that the electrons do not back react on the shock structure, consistent with the model developed in this paper. When γ = γ e , equation (41) is equal to fe , while for γ = 5 / 3 and γ e = 4 / 3 it is ∼ 0 . 76 fe . Here we check whether our numerical implementation of electron heating is consistent with this result. \nThe initial conditions for this test are an unmagnetised, nonrelativistic ( γ = 5 / 3), uniform density and internal energy fluid. The velocity profile is discontinuous at the center of the grid, with a left and a right state given by v l = -v r = constant > 0. The resulting solution is two shocks propagating outwards with a static region in between. We focus on a non-relativistic ( | v | = 10 -3 c ) flow of initially cold gas so that the Mach number of the flow satisfies \nFigure 3. Convergence of the post-shock electron internal energy in the 1D shock test to the analytic solution (equation 41). The shock's Mach number is ∼ 49 (see Sec. 4.1.2). The γ e = γ = 5 / 3 electrons converge at 1st order, as expected, but the γ e = 4 / 3 electrons do not converge to the correct solution to better than ∼ 3% (see Figure 2). This is because our calculation of the heating requires a well-resolved shock structure, which is not the case for modern shock-capturing conservative codes (see § 4.1.2 for details). Introducing an explicit bulk viscosity to resolve the shock structure leads to convergence for γ e /nequal γ (see Appendix C). For γ e = γ , a convenient cancellation makes the evolution of the electron entropy independent of the shock structure. \n<!-- image --> \nM /greatermuch 1. For this test we fix fe = 0 . 5 and show the results for both γ e = 4 / 3 and γ e = 5 / 3. \nFigure 2 shows the density and electron internal energy as a function of position for a shock with M ∼ 49 at t = 0 . 6 L / | v l | , where L is the size of the computational domain. Figure 3 shows that our simulation converges at 1st order to the analytic result for the post-shock electron internal energy when γ = γ e but to a value di ff ering from the expected result by ∼ 3% when γ e /nequal γ (Figure 2). This di ff erence is smaller at lower Mach number, as shown explicitly in Figure 4. The modest discrepancy between the analytic post-shock electron temperature and the HARM solution is because an accurate, converged calculation of the heating term requires a well-resolved shock structure that gets better resolved at higher resolution. This is not the case for modern shock capturing techniques, for which the numerical width of the shock is always a few grid points and our heating calculation is never able to resolve the shock. This is not an issue when γ = γ e because the factors of density in equation (27) cancel, removing the dependence on the shock structure. We show in Appendix C that introducing an explicit bulk viscosity leads to convergence to the analytical result at 2nd order for γ /nequal γ e . However, the /lessorsimilar 3% error as seen in Figures 2 and 4 is su ffi cient for our purposes so we do not include a bulk viscosity in our calculations.", '4.1.3 2D Forced MHD Turbulence Test': "Another test problem with a known, converged heating rate is driven turbulence in a periodic box. If we inject the fluid with a constant energy input rate of ˙ E in at large-scales, we should find that ∫ QdV = ˙ E in after saturation of kinetic and magnetic energies has been reached. Thus, for the electron heating model outlined above, after this saturation point the electrons should receive a fraction fe of ˙ E in. Furthermore, if we have a periodic box in which the par- \nFigure 5 shows our results for the electron and total internal energies as a function of time in the box at 512 2 . We see that once approximate saturation of the turbulence is reached at t ∼ L / cs , 0 (or ˜ t ∼ 0 in the figure), the internal energies are in very good agreement with a linear fit, as expected given the constant rate of energy injection. For a parameterization of ∫ uedV = get + be and ∫ ugdV = ggt + bg , ge and gg represent the electron and total heating rates, respectively. These can be compared to the energy injection rate, ˙ Ein , which is a fixed constant of 0 . 5¯ ρ c 3 s , 0 . At a resolution of 512 2 , we find the total heating rate di ff ers from ˙ Ein by ∼ 4%, while \n<!-- image --> \n<!-- image --> \nFigure 2. High Mach number shock results for an electron heating fraction fe = 0 . 5 at a resolution of 2000 cells. Top: solid blue line shows the fluid density in a numerical simulation. Density undergoes a jump of ρ 2 /ρ 1 = ( γ + 1) / ( γ -1) = 4 at the two shocks, located at x ≈ 0 . 35 and x = 0 . 65. Left: electron internal energy relative to total fluid internal energy for γ e = 4 / 3. The analytic solution is shown with the solid red line and the numerical solution with the dotted black line. Right: the same for γ e = 5 / 3. The analytic solution uses the functional form for ue / ug ( ρ ) (see Appendices B4 and C for details) and applies it to the density returned by the simulation. At this resolution all the fluid variables are essentially converged. The γ e = 5 / 3 electrons show convergence to the expected result of ue = feug (the numerical and analytical lines are essentially on top of each other) while the γ e = 4 / 3 electrons converge to a value that is greater than the analytic result ( ue = 0 . 379 ug for fe = 0 . 5; equation 41) by ∼ 3%. This is because the internal shock structure is never well resolved without an explicit bulk viscosity (see § 4.1.2 and Appendix C for details). \n<!-- image --> \nnumber is fixed, then the total internal energy change from adiabatic expansion / compression will sum to zero. Thus the analytic result we expect for our model of electron heating is that ∫ ρ Te ˙ sedV = fe ∫ QdV = fe ˙ E in. This test checks whether our model satisfies this result numerically. \nWe start with a static, uniform density fluid with β = 6 and sound speed cs , 0 = 8 . 6 × 10 -4 c in a 2D periodic box. The initial magnetic field is uniform: the magnetic field lines are straight and lie in the plane of the simulation. Then, at each time step, we give Gaussian random kicks to the velocity such that the wave number satisfies /vector k · δ/vector v = 0 (i.e. the driving is incompressible), and σ 2 v ∝ k 6 exp( -8 k / k peak) (compare to Lemaster & Stone 2009). The normalisation is fixed such that the rate of energy injection is equal to ˙ E in = 0 . 5¯ ρ c 3 s , 0 . This leads to a rms turbulent velocity that is ∼ 0 . 8 cs , 0 ∼ 1 . 8 v A , so that the turbulence is subsonic and roughly \nAlfv'enic. The peak driving wave number is set to half the box size: k peak = 4 π/ L . Furthermore, we ensure that no net momentum is added to the box by subtracting o ff (from the kicks) any average velocity that would have been generated by the kicks. For this test we fix fe = 0 . 5, γ = 5 / 3 and γ e = 4 / 3. \nFigure 4. Percent error in the post-shock electron internal energy for γ e = 4 / 3 and γ = 5 / 3 as a function of Mach number in the 1D shock test as computed by HARM at a resolution of 2000 for a fractional heat given to the electrons of fe = 1 / 2 and an initial ue / ug = 0 . 1. The analytic solution is given by equation (B27). The final time was fixed such that the two shocks were located at x = 0 . 25 and x = 0 . 75 in a 0 /lessorequalslant x /lessorequalslant 1 domain. Note that the fractional errors are always /lessorsimilar 3 . 3%. The change in the percent error as Mach number goes to 1 is because the flow becomes increasingly smooth and the electron internal energy is no longer converged at a resolution of 2000. \n<!-- image --> \nTable 1. Turbulence Test Linear Fits ( § 4.1.3) \n| Resolution: | 128 | 256 | 512 |\n|--------------------|-------------------|----------------|------------------|\n| ge - 0 . 5 ˙ Ein a | 0 . 0027 ˙ Ein | 0 . 0042 ˙ Ein | 0 . 0024 ˙ Ein |\n| gg - ˙ Ein b | - 0 . 00017 ˙ Ein | 0 . 0012 ˙ Ein | - 0 . 0016 ˙ Ein | \nthe electron heating rate di ff ers from fe ˙ Ein by ∼ 2%. Unfortunately, a rigorous convergence study of these quantities is not possible because of the nature of turbulence in 2D. Due to the inverse energy cascade, the kinetic and magnetic energies never truly saturate and convergence of any of the fluid variables is never achieved. This can be seen from Table 1, where the values of ge and gg are quoted at various resolutions, neither of which display significant convergence to the expected 0 . 5 ˙ Ein and ˙ Ein , respectively. Nevertheless, we find the percent level error found at all resolutions to be su ffi -ciently small to satisfy our error tolerance in the full accretion disc simulation.", '4.1.4 Shadow Solution': "For our two-temperature model, we can seek a solution in which the electron fluid simply 'shadows' the total gas, in that ue ∝ ug . Such a solution is found by setting ue ( t = 0) = feug ( t = 0) at some initial time, since we can solve the first law of thermodynamics for the electrons with ue = feug for all time, assuming that γ e = γ and \nFigure 5. Electron and total gas internal energies summed over the grid as a function of time at a resolution of 512 2 for the forced subsonic MHD turbulence simulation with an assumed electron heating fraction of fe = 0 . 5 and initial β = 10. We define ˜ t = ( t -ti ) cs , 0 / L and ˜ ug , ( e ) = ug , ( e ) -ug , ( e )( t = ti ), where ti ∼ 4 L / cs , 0 is the time at which kinetic and magnetic energy roughly saturate. We normalised the integrated internal energy by the energy injection rate, ˙ E in = 0 . 5 ρ c 3 s , 0 , , so that the y -axis has dimensions of time which we measure in units of L / cs , 0. In these variables, the analytic solutions for the total gas and electron internal energies are lines with slopes of 1 and fe = 0 . 5, which are plotted as solid lines, to be compared to the simulation results which are represented by points. We find that the electron heating rate is 0 . 5 of the total heating rate, consistent with the analytic solution given the input value of fe = 0 . 5. For a numerical comparison at di ff erent resolutions, see Table 1, which shows the results of applying a linear regression fit to the internal energies. \n<!-- image --> \nfe is a constant. This can be seen from the first law: \nu µ ∂µ ue = fe ρ Tgu µ ∂µ sg -ue + Pe ρ u µ ∂µρ, (42) \nbecause the electrons will always get a fraction, fe of the entropygenerated heat (the first term on the RHS of equation 42), while the compression term is directly proportional to ue ∝ feug . This solution is valid regardless of the details of the overall fluid evolution, so we can apply it to an arbitrarily complicated system. \nFor this test, we evolve the electron internal energy in the full accretion disc simulation around a rotating black hole as outlined in § 5. We initially apply small ( δ ue / ue ∼ 0 . 04) perturbations to the electron internal energy about the average value of ug , e , 0 = 0 . 5 ug , 0, and set fe = 0 . 5 and γ e = γ = 5 / 3. For this test alone, we set the floor on electron internal energy to be a fraction fe of the floor on the total fluid (as opposed to our usual choice of 1%). If this latter step were neglected, the floors would cause the polar regions to di ff er significantly from the expected result, though leaving the disc and corona una ff ected (i.e. they still satisfy the analytic result). The test is whether or not our simulation can maintain this result over the run time of 2000 M . \nRunning this test at a resolution of 512 2 gives an average fractional error of \n∣ ∣ \n∣ \n1 N 2 N -1 ∑ j = 0 N -1 ∑ i = 0 ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ( [ ue / ug ] i j -fe ) fe ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∼ 0 . 8% , (43) \n∣ ∣ ∣ which is smaller than our initial perturbations and shows that our \nnumerical solution correctly evolves equation (42) even in a complex problem with MHD turbulence, weak shocks, and other heating processes in the presence of a curved metric.", '4.2 Tests of Electron Conduction': 'Our model and testing suite for conduction closely resembles that of Chandra, Foucart & Gammie (2015), so we leave the details to Appendix A. In summary, we have found second order convergence for linear modes, for a static, 1D atmosphere in the Schwarzschild metric, and for a relativistic, spherically symmetric Bondi accretion flow. We also show that the electrons properly conduct along field lines in a 2D test.', '5 APPLICATION TO AN ACCRETING BLACK HOLE IN 2D GRMHD SIMULATIONS': "We apply the new methods discussed in § 2 and the numerical implementation described in § 3 to the astrophysical environment of an accretion disc surrounding a spinning black hole as described by the Kerr Metric with a spin parameter of a = 0 . 9375. For this spin the last stable circular orbit is ≈ 2 . 04 rg and the thin disc radiative e ffi ciency is ≈ 0 . 18 (Novikov & Thorne 1973). We use the conservative code HARM (Gammie, McKinney & T'oth 2003) as our background GRMHD solution. Our initial conditions for the total fluid are the Fishbone & Moncrief (1976) equilibrium torus solution (see Appendix D) with inner radius r in = 6 rg and with the maximum density of the disc occurring at r max = 12 rg . Note that here and throughout r and θ refer to the Boyer-Lindquist coordinates. This equilibrium solution has a temperature maximum of ≈ 7 . 5 × 10 10 K and a thickness 4 of h / r ∼ 0 . 18 at r max. We normalise the torus density distribution such that the maximum value of density in the torus is ρ max c 2 = 1 and perturb the internal energy of the gas with random kicks on the order of δ ug / ug ∼ 0 . 04 to provide the perturbations for the magnetorotational instability (MRI, Balbus & Hawley 1991) to develop. 5 We overlay this equilibrium solution with an initial magnetic field with 2 P max / b 2 max = 100 (where max refers to the maximum value inside the torus), defined by the scalar vector potential: \nA ϕ ∝ ( ρ/ρ max -0 . 2) cos θ, (44) \nif ρ > 0 . 2 ρ max and 0 otherwise. This vector potential defines two meridional loops contained in the torus that are antisymmetric about the equator. This choice ensures that the field lines are not along constant density. Since constant density implies constant temperature when entropy is constant, field lines along constant density would be isothermal in the initial condition (as would happen if we dropped the factor of cos θ in eq. 44). 2D MHD torus simulations are unable to reach a statistical steady state in which the initial conditions are forgotten, so initially isothermal field lines could artificially suppress electron conduction even at later times. We choose the 2-loop initial condition to avoid this. \n4 \nHere we define h / r \n/iintegtext ρ u t θ \nπ/ 2 √ \ng d θ d φ /iintegtext ρ u t √ \ng d θ d φ . \nFor the electrons, we start with ue / ug = 0 . 1, and run two different models for fe , described below. For conduction runs, we set the initial heat flux to zero. \nOur conduction runs are all in 256 2 grids with a physical size of the domain in spherical polar coordinates of ( R in , R out) × ( θ in , θ out) = (0 . 8 r H , 1000 rg ) × (0 , π ), where r H = rg (1 + √ 1 -a 2 ) is the black hole event horizon radius. For a = 0 . 9375, r H ≈ 1 . 35 rg . In the regions with r < 50 rg , the code uses modified Kerr-Schild coordinates ( t , x 1 , x 2 , and ϕ ) of Gammie, McKinney & T'oth (2003), so that the regions with the highest resolution are near the midplane close to the horizon. For r > 50 rg , we use hyper-exponential coordinates to move out the outer radial boundary r = R out and limit unphysical reflection e ff ects by defining the internal code coordinate x 1 implicitly by the equation (Tchekhovskoy, Narayan & McKinney 2011): \nr / rg = { exp( x 1 ) : r /lessorequalslant 50 rg exp x 1 + [ x 1 -x 1 ( r = 50 rg )] 4 } : r > 50 rg . \n{ \n} The electron heating-only (i.e. without conduction) runs have the same parameters but a higher resolution of 512 2 . At the inner and outer radial boundaries we apply the standard outflow (copy) boundary conditions, at the polar boundaries we apply the standard antisymmetric boundary conditions (with all quantities symmetric across the polar axis except u θ and B θ , whose signs are reversed). \nFigure 6 shows the background HARM solution for the density, magnetic field, temperature, plasma β ≡ 2 Pg / b 2 , and the heating rate per unit volume in the coordinate frame, -Qut , averaged over the time interval 900 -1100 rg / c , as well as the initial field configuration. After ∼ 1200 rg / c the turbulence starts to decay, an artefact of 2D simulations in which MRI turbulence is not sustainable. \nAs noted in § 3.1, we find locally that Q < 0 (violating the second law of thermodynamics) in many regions due to truncation errors. This is because HARM satisfies the total energy equation to machine precision but only satisfies the second law of thermodynamics to truncation error. However, while Q may be instantaneously or locally negative, when integrated over a su ffi cient length of time and / or space in the fluid frame it will satisfy the second law of thermodynamics. In our torus simulation, for instance, Figure 7 shows that when averaged over θ and time (900 -1100 rg / c ), the heating rate is entirely positive definite within the region of interest. Furthermore, when integrated over the volume enclosed between the event horizon, rH , and r = 6 rg (roughly the radius at which the accretion time ∼ 1000 rg / c ), we find \n2 π ∫ 0 π ∫ 0 6 rg ∫ rH -utQ ( r 2 + a 2 cos 2 θ ) sin θ d r d θ d ϕ ≈ 0 . 17 ˙ Mc 2 , (45) \nwhere the factor of -ut converts Q to the coordinate frame. In equation (45), ˙ M is the accretion rate of the black hole in terms of coordinate time (corresponding to time measured by a distant observer) at the event horizon radius, r = r H, \n˙ M = ∫ r = r in ρ u r ( r 2 + a 2 cos 2 θ ) sin θ d θ d ϕ. (46) \nThe heating rate in equation (45) is in excellent agreement with that expected for a rapidly spinning black hole (e.g., the Novikov & Thorne 1973 model predicts a radiative e ffi ciency of ≈ 0 . 18 for a = 0 . 9375). \nIn this work, all mass-weighted averages are computed using the weighting function: ρ u t √ -g , which represents the conserved \nmass per unit coordinate volume. For example, a radial average of a function f ( x 1 , x 2 , x 3 , t ) is computed as: \nx 1 max ∫ x 1 min f ( x 1 , x 2 , x 3 , t ) ρ u t √ -gdx 1 x 1 max ∫ x 1 min ρ u t √ -gdx 1 . (47)", '5.1 Electron Parameter Choices': 'Here we describe physically motivated estimates of the electron heating fraction, fe , and the electron thermal di ff usivity, χ e , appropriate for low-collisionality accretion flows such as that of Sagittarius A*. A more comprehensive exploration of physical models will be explored in future work. \nWe consider two simple models for the electron heating fraction fe . The first sets fe = 1 / 8, a constant. Because the electron adiabatic index is not the same as the proton (total) adiabatic index, and the heating is not spatially uniform, a constant fe model does not necessarily lead to a constant Tp / Te . The second, more physical model, sets fe based on theoretical models of the dissipation of MHD turbulence in low-collisionality plasmas. These generically predict that electrons receive most of the turbulent heating at low β while protons receive most of the turbulent heating at high β . This is true both for reconnection (Numata & Loureiro 2015) and collisionless damping of turbulent fluctuations (Quataert & Gruzinov 1999). This dependence on β is the key qualitative feature of our chosen model of fe . For concreteness, we use the specific calculations of Howes (2010) who provided a simple fitting function for the electron to proton heating rate as a function of plasma parameters in calculations of the collisionless damping of turbulent fluctuations in weakly compressible MHD turbulence like that expected in accretion discs. These models do a reasonable job of explaining the measured proton and electron heating rates in the near-Earth solar wind (Howes 2011). The functional form of fe is derived from the relations: \nQp Qe = c 1 c 2 2 + β 2 -0 . 2log 10 ( Tp / Te ) p c 2 3 + β 2 -0 . 2log 10 ( Tp / Te ) p √ mpTp meTe e -1 /β p , (48) \nwith c 1 = 0 . 92, c 2 = 1 . 6 / ( Tp / Te ), and c 3 = 18 + 5 log 10 ( Tp / Te ) for Tp / Te > 1, while c 2 = 1 . 2 / ( Tp / Te ) and c 3 = 18 for Tp / Te < 1. The corresponding result for fe is simply \nfe ≡ Qe Qp + Qe = 1 1 + Qp / Qe . (49) \nThe critical assumption used in deriving equation (48) is that the turbulent fluctuations on the scale of the proton Larmor radius have frequencies much lower than the proton cyclotron frequency. This is believed to be well-satisfied for weakly compressible MHD turbulence in accretion disks (e.g., Quataert 1998). For concreteness, we note that for Tp / Te = 1 and β p = (0 . 1 , 0 . 3 , 1 , 10), we have Qp / Qe = (0 , 0 . 01 , 0 . 16 , 8 . 6), while for Tp / Te = 10 and β p = (0 . 1 , 0 . 3 , 1 , 10), Qp / Qe = (0 , 0 . 001 , 0 . 09 , 12), respectively. This demonstrates the strong transition from predominantly electron to predominantly proton heating with increasing β p , with the transition happening at a value of β p that depends weakly on the proton to electron temperature ratio. This implies that we expect strong electron heating in the corona and jet regions but suppressed electron heating in the bulk of the disc. \nWe reiterate that the key feature of equation (48) is not the \nprecise value of the predicted Qp / Qe , but rather the transition from Qp /greaterorsimilar Qe for β p /greatermuch 1 to Qp /lessmuch Qe for β e /lessmuch 1. This qualitative transition is much more robust than the specific functional form in equation (48) (e.g., Quataert & Gruzinov 1999; Numata & Loureiro 2015). \nFor the electron thermal di ff usion parameters, since χ e is a di ff usion coe ffi cient, we assume that it has the form \nχ e = α ecr , (50) \nwhere α e is a dimensionless thermal di ff usivity, and r is the radial distance from the center of the black hole, which is comparable to the density scale height of the disc, H . Since we are interested in fairly relativistic electrons, we choose the relevant velocity to be c in our di ff usivity estimate. In what follows, we consider a range of dimensionless di ff usivities, α e ∼ 0 . 1 -10. A typical value of α e ∼ 1 is motivated by the idea that particles scatter roughly after moving a distance comparable to the length-scale over which the magnetic field strength, density, etc. change. In fact, for high beta plasmas, the mean free path due to wave-particle scattering can be significantly lower, reducing the thermal di ff usivity significantly. In Appendix B2 we discuss the specific limits imposed by electron temperature anisotropy instabilities present in a turbulent plasma. In particular, the whistler and firehose instabilities lead to limits on ∆ Te / Te (eq. B9) and thus the electron viscosity and thermal di ff usivity, where the temperature anisotropy is defined with respect to the local magnetic field. In terms of the electron thermal di ff usivity, this becomes χ e = min( α erc , χ max), where χ max is set by velocity space instabilities and is estimated in Appendix B2. Finally, we choose the relaxation time scale, τ to be given by the thermal time scale: \nτ ∼ χ e v 2 th ∼ χ e c 2 . (51) \nComparing this to the stability condition given by equation (36), we see that stability is ensured if \nv th /lessorsimilar ∆ x ∆ t . (52) \nSince we have a non-uniform grid, the time step ∆ t is essentially set by the light crossing time of the smallest grid cell (i.e. that nearest the horizon), meaning that ∆ t /lessorsimilar c ∆ x near the horizon and ∆ t /lessmuch c ∆ x further from the horizon. For a reasonable choice of a CFL number of 0.5, we find that equation (52) is satisfied everywhere and is not a limiting factor in our simulation. Moreover, we find that the exact value of τ is not critical as long as it satisfies numerical stability and is not too long (e.g. is less than a local dynamical time).', '5.2 Electron Heating Only': 'In this section we focus solely on the e ff ects of separately evolving the electron internal energy equation without conduction in our black hole torus simulation and compare the results for di ff erent electron heating models.', '5.2.1 Constant Electron Heating Fraction': 'Figure 8 shows the temperature ratio, Te / Tg , averaged over the interval 900 -1100 rg / c for fe = 1 / 8. We reiterate that Tg here is the temperature inferred from the underlying single fluid GRMHD solution (approximately the proton temperature in our model) while \nFigure 6. Properties of our 2D black hole accretion simulations. The top panel shows the density over-plotted with magnetic field lines in the initial conditions (left) and averaged over 900 -1100 rg / c (right). The remaining panels are the total gas temperature in units of mpc 2 (middle left), the plasma parameter, β ≡ 2 Pg / b 2 (middle right), and the absolute value of the heating rate per unit volume in the coordinate frame, | Qut | , in units of ˙ Mc 2 / ( √ -g ) (bottom), all averaged over time in the interval 900 -1100 rg / c . Note that for calculating the average β , we use 2 〈 Pg 〉 / 〈 b 2 〉 , where 〈〉 denotes an average over time. These plots represent the background GRMHD solution on top of which we separately solve the electron entropy equation. \n<!-- image --> \nTe is the electron temperature determined from our separate electron entropy equation. We include this constant fe result primarily because it is conceptually similar (although quantitatively di ff erent) to the constant Tp / Te assumption often used in the literature. Notice that the resulting Te / Tg ratio, seen in Fig. 8, is non-uniform \ndespite the constant fe . Also note that due to the fact that MHD turbulence is unsustainable in 2D simulations, the heating dies o ff after ∼ 1200 rg / c of evolution and prevents the outer r /greaterorsimilar 10 rg region of the disc from ever being heated substantially. However, as we will see later, conduction can occur at a much faster (electron \nFigure 7. Mass-weighted average (see eq. 47) of the heating rate per unit volume in the coordinate frame, averaged over time in the interval 900 -1100 rg / c and over θ from 0 to π . Note that for our metric sign convention, ut /lessorequalslant 0. The total volume integrating heating out to ∼ 6 rg is ∼ 0 . 17 ˙ Mc 2 (equation 45), comparable to the Novikov & Thorne (1973) heating rate for this black hole spin. \n<!-- image --> \nFigure 8. Ratio of 〈 Te 〉 / 〈 Tg 〉 in our black hole accretion simulation, where 〈〉 denotes an average over time in the interval 900 -1100 rg / c (where Tg is the temperature of the single fluid GRMHD simulations and Te is the electron temperature). These results assume a constant fraction of dissipated heat is given to the electrons ( fe = 1 / 8). Compare with the more physical β -dependent heating results in Figure 9. \n<!-- image --> \nthermal) speed along the magnetic field lines and can a ff ect the solution at somewhat larger radii.', '5.2.2 β -Dependent Electron Heating': 'Figure 9 shows the temperature ratio, Te / Tg , the electron temperature itself, Θ e , and the electron heating fraction, fe , averaged over the interval 900 -1100 rg / c for the β -dependent heating model of equation (48), which we regard as a more physical electron heating model than fe = const. We note that this leads to hot electrons being strongly concentrated in the corona of the torus in between the disc and the jet, where β is the smallest (and fe ∼ 1 from equation 48). \nThis is also clear from the 1D profiles of electron temperature as a function of polar angle in Figure 10. \nFigure 10 shows the mass-weighted average over radius ( r = 5 -7 rg ) of the electron and gas temperatures, plotted versus the polar angle, θ . The fe = 1 / 8 electrons and total gas temperatures have mild variation in T with θ , while the fe = fe ( β ) electrons have significantly higher temperatures in the polar regions. This demonstrates that the non-uniformity of the electron temperature in the fe ( β ) model is primarily caused by the strong β -dependence of our model of fe as opposed to any non-uniformity of the heating rate itself (Figure 6).', '5.3 Conduction and Electron Heating': 'We now consider the e ff ects of electron conduction on the electron temperature structure of black hole accretion discs. We focus on the more physical model of β -dependent heating described in § 5.1. In all of our calculations, we include the velocity space instability limit on the electron thermal conductivity (Appendix B2), although runs without this limit produce similar results because β is modest ( /lessorsimilar 1 -10) in the inner regions of these simulations (Figure 6). Figure 11 shows the electron temperature as a function of radius at the mid-plane in the simulations with and without conduction. Figure 12 shows the e ff ects of conduction more quantitatively via the fractional change in temperature between the electron temperature solution with conduction and that without. \nTo summarise Figures 11 and 12, conduction has little e ff ect on the electron temperature for α e /lessorsimilar 1. However, for α e /greaterorsimilar 1, conduction leads to a significant radial redistribution of heat such that the electron temperature is factors of a few larger at large radii. Even for α e > 1, however, the angular redistribution of heat is much less e ffi cient, as seen in the radially and time-averaged electron temperatures in Figure 10 for α e = 10. This is primarily because of the structure of the magnetic field, as can be seen by noting that the regions where conduction modifies the temperature in Figure 12 largely follow magnetic field contours which do not e ffi ciently connect the polar and equatorial regions. To aid the interpretation of these results, Figure 13 shows the heat flux φ normalised to the maximum value φ max = ( ue + ρ ec 2 ) v t , e ; even for α e = 10 the heat flux is still well below the saturated value in significant parts of the domain. We now summarise and interpret these results in more detail. \nFor α e /lessorsimilar 1 we find conduction to have only a small e ff ect on the electron thermodynamics in the accretion disc, despite the relatively high conductivity. We can understand this result as being due to the suppression of the isotropic heat flux by being projected along field lines, quantified by the ratio, \n/epsilon1 2 ≡ ( q µ q µ ) aniso ( q µ q µ ) iso , (53) \nwhere q µ iso and q µ aniso are evaluated using the electron temperature as evolved without conduction and which we now define. For this diagnostic, we use \nq ν iso = -ρχ eh µν ∂µ Te + Tea µ ) , (54) \n( \n) where h µν = u µ u ν + g µν is the projection tensor that projects along a space-like direction perpendicular to the fluid velocity u µ . This projection ensures that the heat flux in the fluid frame has a zero timecomponent. Likewise, for q ν aniso , we use the first order anisotropic heat flux: q ν aniso = ( ˆ b µ q iso µ ) ˆ b ν . Note that in equation (53), /epsilon1 is always \n<!-- image --> \n<!-- image --> \nFigure 9. Ratio of 〈 Te 〉 / 〈 Tg 〉 (top), electron temperature, 〈 Te 〉 , in units of mec 2 (middle), and electron heating fraction, 〈 fe 〉 (bottom), where 〈〉 denotes an average over time in the interval 900 -1100 rg / c . These results are for β -dependent heating (see § 5.2.2). Compare to Figure 8 for a constant electron to proton heating ratio. The highly non-uniform distribution of β (see Figure 6) and the strong β dependence of the electron-to-total heating ratio (equation 48) lead to a strong angular dependence of Te / Tg . \n<!-- image --> \nFigure 10. Mass-weighted average of total gas and electron temperature (in units of mec 2 ) as a function of the polar angle, θ . We show the electron temperature with and without conduction for a β -dependent electron heating fraction, fe , as well as without conduction for a constant electron heating fraction fe = 1 / 8. The results are averaged over time from 900 -1100 rg / c and averaged over r from 5 -7 rg . Note that the total gas temperature has been multiplied by a constant fraction to more clearly compare to the electron temperatures. The electron temperature with β -dependent heating displays much stronger θ variation because the electron heating fraction itself varies with θ (see Figure 9). Conduction has only a modest e ff ect on redistributing heat in θ due to the geometry of the field. \n<!-- image --> \n/lessorequalslant 1 because both heat fluxes are mutually orthogonal to u µ . Figure 14 shows | q aniso | / | q iso | in our torus simulation, where we find the suppression of the isotropic heat flux to be around /epsilon1 ∼ 0 . 2. The simplest explanation for this small number is that the field is predominantly in the ϕ direction, where the temperature gradient is identically 0 in 2D simulations. For instance, in local shearing box calculations, Guan et al. (2009) found that the typical angle between /vector B and ˆ ϕ was ∼ 10 -15 · , corresponding to a suppression of the heat flux with /epsilon1 0 . 25. \nContrary to the α e < 1 cases, setting α e /greaterorequalslant 1 causes conduction to have a significant e ff ect by redistributing the electron heat from the coronal regions to the bulk of the torus at larger radii. This redistribution of heat causes the electron temperature to actually exceed the total gas temperature in certain regions, which formally violates our assumption that Te /lessmuch Tp . \n∼ \n/lessmuch While the calculation with α e = 10, or with χ e = 10 rc , might seem to use an unphysically large conductivity, roughly corresponding to a length scale for conduction of ∼ 10 H , where H is the disc density scale height, the heat flux in these calculations is limited to be smaller than the value set by the physically motivated whistler criterion in equation (B10) and to be less than the saturated heat flux ∼ uec . As Figure 13 shows, the heat flux is saturated in only part of the domain. Furthermore, the appropriate length scale for conduction should be the scale height along field lines , which could be significantly greater than the overall density scale height if the field has a large toroidal component. For these reasons, we believe that the larger α e solutions may in fact be physical because they correspond to a heat flux closer to the saturated value ∼ uec expected in low-collisionality plasmas. \nFigure 11. Electron temperature in the mid-plane ( θ = π/ 2) in units of mec 2 for black hole accretion simulations with β -dependent heating and for electron conduction with dimensionless conductivity α e = 0 , 0 . 1 , 1 , 10 (where the electron thermal di ff usivity is χ e = α erc ; see § 5.1). The results are time averaged over the interval 900 -1100 rg / c . For alphae /greaterorsimilar 1, conduction redistributes energy from small to large radii, increasing the electron temperature at larger radii. Compare to Figure 10, which shows that redistribution of heat in the polar direction is less e ffi cient. \n<!-- image -->', '6 CONCLUSIONS': "We have presented a method for evolving a separate electron entropy equation in parallel to the standard equations of ideal General Relativistic MHD. Our motivation is the study of two-temperature radiatively ine ffi cient accretion flows (RIAFs) onto black holes, in which the electron-proton Coulomb collision time is su ffi ciently long that the proton and electron thermodynamics decouple (e.g., Rees et al. 1982). Understanding the electron temperature distribution close to the black hole is necessary for robustly predicting the radiation from the numerical simulations of black hole accretion (and outflows) in the sub-Eddington regime. \nThe long-term goal of the present work is to incorporate the key processes that influence the electron thermodynamics in RIAFs into GRMHD simulations: heating, thermal conduction, radiative cooling, and electron-proton Coulomb collisions. In the present paper we have focused on the first two of these processes. Specifically, we have developed, implemented, and tested a model that quantifies the rate of heating in a conservative GRMHD simulation ( § 2). We then assign a fraction fe of this heating to the electrons based on a microphysical model of the key heating processes (e.g., turbulence, reconnection, shocks; see, e.g., § 5.1). In addition, we have implemented and tested a model of relativistic anisotropic conduction of heat (by electrons) along magnetic field lines, based on the Chandra et al. (2015) formulation of anisotropic relativistic conduction ( § 2.3). The electron thermal di ff usivity is a free parameter in this calculation. For the black hole accretion disc applications of interest, we advocate a 'saturated' heat flux in which the thermal di ff usivity is ∼ rc , subject to additional constraints imposed by velocity space instabilities and scattering by wave-particle interactions (Appendix B2). \nWe implemented our electron energy model in a conservative GRMHD code HARM2D (Gammie, McKinney & T'oth 2003), though the model we have developed can be applied to any underlying GRMHD scheme. For simplicity, the implementation in this paper neglects the back reaction of the electron pressure on \n<!-- image --> \n<!-- image --> \nFigure 12. Fractional di ff erence in electron temperature between solutions with and without electron conduction shown in colour (see colour bar for details) over-plotted with magnetic field lines shown as solid black lines. The fractional di ff erence is calculated as 〈 Te , c 〉 / 〈 Te , 0 〉 -1, where 〈〉 denotes an average over time from 900 -1100 rg / c . The results include β -dependent electron heating for α e = 0 . 1 (top), α e = 1 (middle), and α e = 10 (bottom panel), where the electron thermal di ff usivity is χ e = α erc ( § 5.1). Higher α e allows more heat to flow from the inner regions to larger radii. For α e = 0 . 1 conduction has a negligible e ff ect on the electron temperature, while for α e /greaterorsimilar 1 conduction leads to order unity changes in Te . \n<!-- image --> \nFigure 13. 〈| φ |〉 / 〈 φ max 〉 , the ratio of the electron heat flux to the maximum value φ max = ( ue + ρ ec 2 ) v t , e , where 〈〉 denotes an average over time from 900 -1100 rg / c . This is calculated based on the results of a black hole accretion simulation with β -dependent electron heating and a dimensionless electron thermal conductivity of α e = 10 (where the electron thermal di ff usivity is χ e = α erc ; see § 5.1). Comparison with Figure 12 shows that conduction has a significant e ff ect on redistributing heat only in the regions where the heat flux is saturated or nearly saturated. However, even for a high electron thermal conductivity of α e = 10, the heat flux is still well below the saturated value over much of the domain. \n<!-- image --> \nFigure 14. 〈| q aniso |〉 / 〈| q iso |〉 , the ratio of the anisotropic (field-aligned) heat flux to the isotropic heat flux , where 〈〉 denotes an average over time from 900 -1100 rg / c . This is calculated based on the results of a black hole accretion simulation with β -dependent electron heating but without conduction. The factor of ∼ 5 -10 suppression of the field aligned heat flux is roughly consistent with that expected from local shearing box calculations of MRI turbulence, where /vector B is aligned with the ˆ ϕ direction (e.g. Guan et al. 2009). \n<!-- image --> \nthe dynamics of the accretion flow. We believe that this is a reasonable first approximation given some of the uncertainties in the electron physics. Formally, this approximation is is valid only when Te /lessmuch Tp though we expect it to be a reasonable first approximation when Te /lessorsimilar Tp in regions with plasma β /greaterorsimilar 1, i.e., in the regions where gas thermal pressure forces are dynamically important. \nWe have demonstrated that our implementations of electron heating and conduction are accurate and second order convergent in several smooth test problems ( § 4 and Appendix A). For shocks, \nthe heating converges at first order but to a post shock temperature that di ff ers from the analytic solution by /lessorsimilar 3% when the electron adiabatic index di ff ers from the adiabatic index of the fluid in the GRMHD solution (e.g., Figure 2). This discrepancy arises because standard Riemann solvers 'resolve' the shock structure with only a few grid points. Including an explicit bulk viscosity to broaden and resolve the shock leads to a converged numerical solution for the post-shock electron energy that agrees with the analytic solution (Appendix C). In practice, the /lessorsimilar 3% discrepancy between the numerical and analytic solutions for standard Riemann solvers is sufficiently accurate given other uncertainties in the electron physics. For this reason, we do not use bulk viscosity in our calculations. Moreover, strong shocks are rare and account for a negligible fraction of the dissipation in accretion disc simulations with aligned black hole and accretion disc angular momentum. \nIn addition to formulating and testing our electron energy equation model, we have also presented a preliminary application of these new methods to simulations of black hole accretion. Specifically, we have studied the impact of realistic electron heating and electron thermal conduction on the spatial distribution of the electron temperature in 2D (axisymmetric) simulations of black hole accretion onto a rotating black hole. We find that the resulting electron temperatures di ff er significantly from the assumption of a constant electron to proton temperature ratio used in previous work to predict the emission from GRMHD simulations (Mo'scibrodzka et al. 2009; Dibi et al. 2012; Drappeau et al. 2013); see, e.g. Figures 9-11. This is due to the strong β -dependence of the electron heating fraction, fe , described in § 5.1: electrons are preferentially heated in regions of lower β , causing Te / Tp to be larger in the coronal regions compared to the midplane. In addition, we find that the e ff ect of thermal conduction on the electron temperatures is suppressed by the fact that the heat flux must travel along field lines, which are predominantly toroidal and thus not aligned with the temperature gradient. Specifically, we find that electron conduction modifies the temperature distribution only if the e ff ective electron mean free path along the magnetic field is /greaterorsimilar the local radius in the flow (see Figure 12). In this case, there is a net transfer of heat from the corona to the bulk of the disc. This increases the electron temperature at larger radii by a factor of 2. \nIt is important to stress that the unsustainability of MHD turbulence in 2D simulations (e.g., Guan & Gammie 2008) limits how thoroughly we can interpret the accretion disc results presented in this paper. Since a steady state is never truly reached, the bulk of the disc retains memory of the initial conditions and only the innermost regions ( r /lessorsimilar 10 rg ) develop significant turbulence. This could artificially limit the e ff ects of electron conduction because the thermal time for relativistic electrons is ∼ r / c and is thus substantially shorter than the local dynamical time only at large radii. Future work will use the methods developed here in 3D simulations. \n∼ \nIt is also important to stress that, as in previous work, our results for both the gas and electron temperature are not reliable when b 2 /greatermuch ρ c 2 . In these regions the ratios of b 2 /ρ c 2 and b 2 / ug are so large that the evolution of the density and internal energy are dominated by truncation errors in the magnetic field, to which they are nonlinearly coupled by the total energy equation. This requires the use of density and internal energy floors. Because our calculation of the electron heating rate relies on quantifying the entropy changes in the underlying GRMHD solution, our predicted electron temperatures also become unreliable when b 2 /greatermuch ρ c 2 . In the accretion disc simulations, this only a ff ects the regions close to the pole where there is very little matter, not the evolution of the electrons in the bulk of the accretion disc or corona. We have specifically tested \nseveral treatments of the internal energy and density floors which produce dramatically di ff erent results in the poles but are all consistent in the higher density regions for both the fluid variables and the electron temperature. \nFuture applications of the methods developed in this paper will center on using our electron temperature calculations to predict the emission from accreting black holes. In particular, we hope to produce more accurate images of the radio and IR emission of Sagittarius A* (and M87) that can be used to interpret the forthcoming spatially resolved observations by the Event Horizon Telescope (Doeleman et al. 2009) and Gravity (Gillessen et al. 2010).", 'ACKNOWLEDGEMENTS': 'We thank F. Foucart for useful discussions, as well as all the members of the horizon collaboration, horizon.astro.illinois.edu, for their advice and encouragement. We also thank Dmitri Uzdensky for a useful and thorough referee report. This work was supported by NSF grant AST 13-33612 and NASA grant NNX10AD03G, and a Romano Professorial Scholar appointment to CFG. EQ is supported in part by a Simons Investigator Award from the Simons Foundation and the David and Lucile Packard Foundation. MC is supported by the Illinois Distinguished Fellowship from the University of Illinois. Support for AT was provided by NASA through Einstein Postdoctoral Fellowship grant number PF3-140131 awarded by the Chandra X-ray Center, which is operated by the Smithsonian Astrophysical Observatory for NASA under contract NAS8-03060, and by NSF through an XSEDE computational time allocation TG-AST100040 on TACC Stampede. This work was made possible by computing time granted by UCB on the Savio cluster.', 'REFERENCES': "| Anderson J. L., Witting H. R., 1974, Physica, 74, 466 |\n|--------------------------------------------------------------------------------------------------------------------|\n| Andersson N., Lopez-Monsalvo C. 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M., 1976, ApJ, 204, 187 Sharma P., Quataert E., Hammett G. W., Stone J. M., 2007, ApJ, 667, 714 Shcherbakov R. V., Penna R. F., McKinney J. C., 2012, ApJ, 755, 133 \nSironi L., 2015, ApJ, 800, 89 \nTchekhovskoy A., McKinney J. C., Narayan R., 2007, MNRAS, 379, 469 \nTchekhovskoy A., Narayan R., McKinney J. C., 2011, MNRAS, 418, L79 \nYuan F., Quataert E., Narayan R., 2003, ApJ, 598, 301 \nZhao J.-H., Young K. H., Herrnstein R. M., Ho P. T. P., Tsutsumi T., Lo K. Y., Goss W. M., Bower G. C., 2003, ApjL, 586, L29", 'APPENDIX A: TESTS OF ELECTRON CONDUCTION': 'This Appendix outlines tests of our numerical implementation of electron conduction that demonstrate that our calculations are robust and second-order accurate. The tests are taken directly from Chandra, Foucart & Gammie (2015), to which we refer the reader for more details.', 'A1 Conduction Along Field Lines': 'This test is simply to check whether the electrons conduct heat along field lines properly. The initial conditions are a 2D, periodic box of physical size 1 × 1 with uniform pressure and a small, density variation (and hence temperature variation) of the form: \nρ = ρ 0 ( 1 -e -( x -0 . 5) 2 + ( y -0 . 5) 2 0 . 005 ) . (A1) \nThe field lines are sinusoidal and given by \nBx = B 0 By = B 0 sin(8 π x ) , (A2) \nderived from a scalar potential of \nAz = B 0 ( y + 1 8 π cos(8 π x ) ) . (A3) \nFor the conduction parameters, we choose χ e = 0 . 5 /ρ and τ = 1 and run the simulation for 10 τ .', 'A2 Linear Modes Test': 'This test checks whether our implementation of conduction gives the correct eigenmodes corresponding to Equation (19). Writing λ = -α ± i ω , we initialise perturbations in an otherwise uniform box about the equilibrium solution with wave number k = 2 √ 2 π and run the simulation for one period: t = 2 π/ω . The analytic solution is that the perturbations, δ , should obey δ ( t = 2 π/ω ) = δ ( t = 0) e -2 πα/ω . We choose ˆ b = 1 / √ 3ˆ x + √ 2 / √ 3ˆ y and /vector k = 2 π ˆ x + 2 π ˆ y . We find that both φ and ue converge at second order to the analytical solution as shown in Figure A2.', 'A3 1D Atmosphere in a Schwarzschild Metric': 'This test checks whether our implementation of the electron conduction gives the correct analytic result in a non-trivial space-time. In the Schwarzschild metric, the solution for a fluid in hydro-static \nFigure A1 shows that the final state of the fluid is that of isothermal field lines, exactly as expected, with heat flux equilibrating the temperature along the magnetic field lines. This shows that our implementation of conduction properly limits the heat flux to be parallel to the magnetic field. \n<!-- image --> \nFigure A1. Temperature profiles over-plotted with magnetic field lines in the 2D anisotropic conduction test from Chandra, Foucart & Gammie (2015), adapted for electron conduction (see § A1). The top panel is at the initial time while the bottom panel is at the end of the run ( t = 10 τ ). The field lines become isothermal, consistent with heat conduction only along the magnetic field. \n<!-- image --> \nequilibrium reduces to a system of two ordinary di ff erential equations, which can be solved for any given temperature profile (see Chandra, Foucart & Gammie 2015 for details). For this test, we initialise the temperature and heat flux of the electrons to be this equilibrium solution for a purely radial field and see if the code can maintain it over a time of 100 rg / c in a computational domain of 1 . 4 rg /lessorequalslant r /lessorequalslant 90 rg . To compute the error, we again use the L1 norm and find 2nd order convergence for both φ and ue , as shown in Figure A3.', 'A4 Relativistic Bondi Accretion': 'This test checks whether our implementation of the electron conduction gives the correct analytic result in a fluid with u i /nequal 0, which activates terms that were not present in the 1D atmosphere test. For the standard, spherically symmetric, steady-state Bondi solution for an accreting black hole (Hawley, Smarr & Wilson 1984), we can \nFigure A2. L1 Norm of errors in the 2D linear modes test after one period as computed from the eigenfrequencies given in equation (19). See § A2. \n<!-- image --> \nFigure A3. L1 norms of the error in both the heat flux and electron internal energy for the 1D atmosphere test in the Schwarzschild metric ( § A3). \n<!-- image --> \nsolve equation (16) by numerical integration if we assume that the heat flux does not back-react on the electron temperature. For this test, we set the initial condition of the fluid variables to be the Bondi solution and the initial conditions of φ to be given by the solution to equation (16) with Dirichlet boundary conditions. We choose the sonic point to occur at rc = 20 M and fix the outer boundary at a spherical radius of R out = 40 M to have φ ( r = 40 M ) = 0. The inner radius of the grid is inside the event horizon at r = 1 . 6 M . The test is whether or not the code can maintain this state over a period of t = 200 M . We find second order convergence of the heat flux to the analytical solution, as shown in Figure A4. \nFigure A4. L1 norms of the error in the magnitude of the heat flux for the relativistic Bondi accretion test ( § A4). \n<!-- image -->', 'B1 Total Heating Rate': "This section derives the result quoted in equation (25). \nFirst, we introduce the variable ˆ κ g , which is equivalent to κ g ≡ Pg ρ -γ at the beginning of the time step and at the n + 1 / 2 'predictor' step, but which is evolved over a time step according to: \n∂µ ( √ -g ρ ˆ κ gu µ ) = 0 . (B1) \nWe discretise equation (B1) in a standard way (i.e. equation 21): \n( √ -g ρ ˆ κ gu t ) n + 1 -( √ -g ρκ gu t ) n ∆ t + [ √ -g ρκ gu x ] n + 1 / 2 i + 1 -[ √ -g ρκ gu x ] n + 1 / 2 i ∆ x = 0 , (B2) \nwhere the square brackets indicate fluxes computed via the Riemann solver at cell interfaces and the generalisations to higher dimensions is straightforward. Note that we have dropped the ˆ in the n + 1 / 2 and n terms because ˆ κ g = κ g at the beginning of the time step and at the n + 1 / 2 step. We obtain the new value of entropy, ˆ κ n + 1 g , at tn + 1 ≡ tn +∆ t via solving equation (B2). We emphasise that ˆ κ g is not the true entropy at t n + 1 but the entropy evolved according to equation (B1) [or its discretised equivalent equation B2] and thus does not include any heating. \nAt the end of the time step (i.e. at t = tn + 1), we compute the 'true' value of the entropy due to the full GRMHD evolution, according to the definition of κ g : \nκ n + 1 g = ( Pg ρ γ ) n + 1 . (B3) \nUnlike ˆ κ n + 1 g , which does not include any heating, κ n + 1 g accounts for the heating as implied by the conservative evolution of the underlying GRMHD scheme. The di ff erence ( κ g -ˆ κ g ) n + 1 is related to the heating incurred during time step n , and we will use it below. \nTo compute the heating rate we evaluate the quantity: \n1 - \n( \nwhere the third equality holds because \nQ ≡ ρ Tu µ ∂µ sg = ρ γ γ -1 u µ ∂µκ g ≡ ρ γ -1 γ -1 ( ρκ gu µ ); µ ≡ γ 1 ρ γ -1 √ -g ∂µ √ -g ρκ gu µ ) , (B4) \n( ρκ gu µ ); µ = ∂µ ( √ -g ρκ gu µ ) / √ -g ≡ κ g ∂µ ( √ -g ρ u µ ) / √ -g + ρ u µ ∂µκ g \nand the first term vanishes due to conservation of mass. We evaluate eq. (B4) at the n + 1 / 2 time step in a discretised form by centring the time derivatives at n + 1 / 2 but evaluating the prefactor at the n + 1 time step: \nQ n + 1 / 2 = ( 1 γ -1 ρ γ -1 √ -g ) n + 1 / 2 × { ( √ -g ρκ gu t ) n + 1 -( √ -g ρκ gu t ) n ∆ t + [ √ -g ρκ gu x ] n + 1 / 2 i + 1 -[ √ -g ρκ gu x ] n + 1 / 2 i ∆ x . (B5) \n \nQ n + 1 / 2 = ( ρ γ -1 γ -1 ) n + 1 / 2 { ρ u t ( κ g -ˆ κ g ) } ∆ t n + 1 . (B6) \n Now, multiplying eq. (B2) by ( 1 γ -1 ρ γ -1 / √ -g ) n + 1 / 2 and adding the result to eq. (B5), we obtain equation (25) of the main text:", 'B2 Whistler Instability Limit on Conduction': 'We assume that the electrons are relativistic with Θ e = kTe / mec 2 /greaterorsimilar 1. If the electrons relax to thermal equilibrium with a scattering rate ν e , relativistic kinetic theory implies that the electron viscosity η e and thermal di ff usivity χ e satisfy (Anderson & Witting 1974) \nη e /similarequal Θ e c 2 ν e χ e /similarequal 1 . 6 c 2 ν e (B7) \nVelocity space instabilities set an upper limit on the electron thermal conductivity in a turbulent plasma. Physically, as the magnetic field in the accretion disc fluctuates in time, this generates pressure anisotropy, which is resisted by velocity space instabilities that isotropise the distribution function and thus limit the magnitude of the thermal di ff usivity. Chandra et al. (2015) show that the theory of relativistic anisotropic viscosity implies that the pressure anisotropy and scattering rate are related by \nν e ∆ Pe Pe /similarequal u µ ∂µ ln [ B 3 ρ 2 ] . (B8) \nwhere ∆ Pe = P ⊥ -P ‖ and we have neglected some general relativistic terms for simplicity. \nElectrons satisfy limits on pressure anisotropy of \n∆ Pe Pe /greaterorsimilar -1 . 3 β e ∆ Pe Pe /lessorsimilar 0 . 25 β 0 . 8 e (B9) \nThe second term on the right hand side of equation (B9) is a fit to the whistler instability threshold for relativistically hot electrons (based on numerical solutions of the dispersion relation derived in Gladd 1983). The coe ffi cient in the numerator technically depends \nweakly on Θ e , varying from /similarequal 0 . 125 for non-relativistic electrons to /similarequal 0 . 25 for Θ e /similarequal 10 (Lynn 2014). Note that the slope of the β e term for the whistler instability in equation (B9) is a fit for β e /similarequal 0 . 1 -30. Gary & Wang (1996) and Sharma et al. (2007) found a somewhat shallower slope ∝ β -1 / 2 e in non-relativistic calculations but this is not a good fit over the large dynamic range of β e considered here. The first limit in equation (B9) is the electron firehose instability which is an electron-scale resonant analogue of the fluid firehose instability (Gary & Nishimura 2003). This limit is based on nonrelativistic calculations and should to be extended to the relativistic limit in future work. However, based on our whistler calculations this is unlikely to be a significant e ff ect. \nSharma et al. (2007) found that the typical pressure anisotropy satisfied ∆ P / P /greaterorequalslant 0 in simulations that explicitly evolved a pressure tensor. Physically, this sign of the pressure anisotropy corresponds to outward angular momentum transport. Assuming that the RHS of equation (B8) is ∼ Ω , the whistler instability limit in equation (B9) thus implies χ e ∼ crg ( r / rg ) 3 / 2 (4 β e ) -0 . 8 . This is not a significant constraint on the conductivity relative to the saturated value ( χ e ∼ crg ), for β e /lessorsimilar 1, which can occur either in the corona / outflow or because Te /lessmuch Tp . However, this estimate does suggest that the electron conductivity may be modest in the bulk of the disc at ∼ 10 rg if β e /greatermuch 1. \n/greatermuch Equation (B9) can be implemented by calculating ∆ Pe / Pe using equation (B8) given an assumed χ e (and using equation (B7) to relate ν e and χ e ). If equation (B9) is violated, ν e should be increased and χ e decreased such that equation (B9) is satisfied. Alternatively, an even simpler first approximation would be to simply limit \nχ e /lessorsimilar crg ( r / rg ) 3 / 2 (4 β e ) -0 . 8 ≡ χ max (B10) \nmotivated by the estimate in the preceding paragraph for the whistler instability. This is the limit we have used in the accretion disc simulations in § 5 of the main text.', 'B3 Electron Conduction Numerical Stability': 'Non-relativistically, an explicit implementation of thermal conduction is stable only if the time step, ∆ t , satisfies the condition ∆ t /lessorsimilar ∆ x 2 /χ , where ∆ x is the grid spacing in 1-dimension and χ is the thermal di ff usivity. The relativistic theory outlined in § 2 . 3, however, where the heat flux φ is evolved according to equation (16), di ff ers from the non-relativistic case in that it is no longer di ff usive. This alters the criterion for stability to be a condition on the relaxation time, τ , given by equation (36), which we derive here. \nTo check the numerical stability of our conduction theory we assume that we are in Minkowski space in the rest frame of the fluid, and further simplify our analysis to one dimension in which ˆ b = ˆ i . \nUnder these assumptions, a Von Neumann stability analysis on equations (16) and (5) leads to a quadratic equation for the amplification factor, G , with the following solutions: \nG = 1 - C [1 -cos( k )] -1 2 ∆ t τ ( 1 ± √ 1 -4( γ e -1) χ e τ ∆ x 2 sin ( k ) 2 ) , (B11) \nwith the condition for stability being that | G | /lessorequalslant 1. Here, as before, C denotes the Courant factor. To analyse equation (B11), we consider two cases: 1) when the square root term is real, and 2) when the square root term is imaginary. \nWhen the square root term is real, the condition for stability \nbecomes: \nτ /greaterorequalslant ∆ t 2 - C [1 -cos( k )] -( γ e -1) χ e ∆ t ∆ x 2 sin ( k ) 2 (2 - C [1 -cos( k )]) 2 (B12) \n The right hand side is a maximum for k = π modes, which gives, simply: \n \nτ /greaterorequalslant ∆ t 2(1 - C ) . (B13) \nThe more interesting case is when the square root term in Equation (B11) is imaginary, where the criterion for stability becomes: \nτ /greaterorequalslant ∆ t ( γ e -1) χ e ∆ t ∆ x 2 sin ( k ) 2 + C [1 -cos( k )] -1 C (2 - C [1 -cos( k )]) [1 -cos( k )] , (B14) \ncos( k ) = 1 -C -√ 4 K (1 - C ) - C 2 a 2 -2 K (1 - C ) (B15) \n∆ t > ∆ x 2 ( γ e -1) χ e 1 -4 C (1 - C ) + √ 1 -4 C (2 C 1) 8(1 - C ) ≡ ∆ tcrit , (B16) \nand a maximum at k = π otherwise. So if ∆ t < ∆ tcrit , our criterion becomes: \nτ /greaterorequalslant ∆ t [ 2 C 1 4 C (1 - C ) ] ≡ τ max , 1 . (B17) \nFinally, if ∆ t > ∆ tcrit , then we have \nτ /greaterorequalslant \n∆ t × 2 K ( C 2 -4 C (1 - C ) ) 4 K C (1 - C ) ( √ 4 K (1 - C ) - C 2 -2 C ) + 2 C 4 + ( 4 K 2 (1 - C ) + 4 K C (1 - C ) - C 3 ) √ 4 K (1 - C ) - C 2 4 K C (1 - C ) ( √ 4 K (1 - C ) - C 2 -2 C ) + 2 C 4 ≡ τ max , 2 . (B18) \nThe general behaviour of Equation (B18) is complicated, but the result is roughly consistent with \nτ /greaterorsimilar ( γ e -1) ( ∆ t ∆ x ) 2 χ e . (B19) \nfor most reasonable choices of the Courant factor. This is the result quoted in equation (36) of the main text. \nTo summarise, our scheme is stable when: \n \nτ /greaterorequalslant max [ ∆ t 2(1 - C ) , τ max , 1 ] : ∆ t < ∆ tcrit max [ ∆ t 2(1 - C ) , τ max , 2 ] : ∆ t /greaterorequalslant ∆ tcrit , \n for ∆ tcrit , τ max , 1, and τ max , 2 as defined in equations (B16), (B17), and (B18), respectively.', 'B4 Electron Heating in a 1D Shock': "Formally, for an ideal shock in a zero-viscosity fluid there is no unique path in ( P , ρ ) space that connects the pre and post-shock values given by the Rankine-Hugoniot conditions, meaning that the dissipation per unit volume, ∫ ρ Tds , is not a well-defined quantity. \n \n which, defining K ≡ ( γ e -1) ∆ t χ e / ∆ x 2 , has a maximum at \nif \nHowever, by introducing any non-zero viscosity, the degeneracy is broken and there exists a unique path in ( P , ρ ) space and hence a well-defined dissipation. To see this, we take the 1D RankineHugoniot relations for a static shock, given some prescription for the viscous stress, τ ≡ 4 / 3 µ /vector ∇· v ( µ is the dynamic viscosity coe ffi -cient, and can be an arbitrary function of plasma parameters), \n˙ m = ρ v ˙ p = ρ v 2 + P + τ ˙ E = 1 2 ρ v 3 + γ γ -1 P v + τ v , (B20) \nwhere ˙ m , ˙ p , and ˙ E are constants representing the mass, momentum, and energy flux across the shock. Absent τ , we could combine these three equations in several di ff erent ways to get a relationship of the form P = P ( ρ ). With non-zero viscosity, however, there is only one unique way to do this, namely, by taking ˙ p v -˙ E and solving for ˙ m , which gives: \nP ( ρ ) = ( γ -1) ( 1 2 ˙ m 2 ρ -˙ p + ˙ E ˙ m ρ ) , (B21) \nor, in terms of κ ≡ P ρ -γ , \nκ g ( ρ ) = ( γ -1) ( 1 2 ˙ m 2 ρ γ + 1 -˙ p ρ γ + ˙ E ˙ m ρ γ -1 ) . (B22) \nWe assume that the electrons receive a constant fraction of the total heat: \nρ Teu µ ∂µ se = fe ρ Tgu µ ∂µ sg ⇒ ρ γ e γ e -1 u µ ∂µκ e = fe ρ γ γ -1 u µ ∂µκ g , (B23) \nor in quasi-conservative form (using the mass continuity equation and assuming a flat space metric): \n∂ ∂ x µ ( ρ u µ κ e ) = fe γ e -1 γ -1 ρ γ -γ e ∂ ∂ x µ ( ρ u µ κ g ) . (B24) \nThe final electron entropy is given by integrating this equation from the initial to the final density, which, for a 1D shock reduces to \n∞ ∫ -∞ ∂ ∂ x ( ˙ m κ e ) = fe γ e -1 γ -1 ρ f ∫ ρ i ρ γ -γ e ˙ m ∂κ ∂ρ d ρ, (B25) \ngiving: \nu f e = u i e ( ρ f ρ i ) γ e + fe γ -1 ( ˙ m 2 ρ f γ + 1 γ e + 1 -˙ p γ γ e + ˙ E ρ f ˙ m γ -1 γ e -1 ) -fe γ -1 ( ρ f ρ i ) γ e ( ˙ m 2 ρ i γ + 1 γ e + 1 -˙ p γ γ e + ˙ E ρ i ˙ m γ -1 γ e -1 ) , (B26) \nwhere ρ f is determined from the Rankine-Hugoniot conditions. For a strong shock with Mach number /greatermuch 1, this simplifies to \nu f e = ˙ m v i fe γ 2 e -1 [( γ + 1 γ -1 ) γ e ( 1 -γ γ e ) + 1 + γ γ e ] , (B27) \nwhere v i is the pre-shock fluid velocity in the shock's rest frame. Dividing by u f g = 2 ˙ m v i ( γ 2 -1 ) -1 yields equation 41. \n ", 'APPENDIX C: ELECTRON HEATING IN A VISCOUS SHOCK': 'In this appendix we show that by introducing an explicit bulk viscosity to the non-relativistic hydrodynamic equations, our electron heating calculation outlined in § 3.3 give an electron internal energy that converges to the analytic result derived in Appendix B4 for electron heating at a shock. \nWe treat viscosity by explicitly adding the 1D viscous energy and momentum fluxes to the ideal MHD fluxes for a constant kinematic viscosity, ν : \nFE , v isc = -4 ν 3 ρ v d v d x Fp , v isc = -4 ν 3 ρ d v d x . (C1) \nNote that these are non-relativistic fluxes which are formally inconsistent with the relativistic code in which they are used. However, our goal here is simply to show that with a resolved shock structure the electron heating calculation converges to the correct answer. The non-relativistic limit is fine for this purpose. The fluxes in equation (C1) smooth out discontinuities to a continuous profile of finite width, determined by ν and the velocity scale. The solution for the profile of a viscous shock, now defined as a smooth transition from an initial to final state as opposed to a discontinuity, can be computed analytically for a constant kinematic viscosity, ν . In the shock frame, taking x → -∞ as the initial state, this solution takes the form: \nv ( x ) = ( γ + 2 M -1 ) + exp [ -3( x -x 0) v i 4 ν ( 1 -1 M )] exp [ -3( x -x 0) v i 4 ν ( 1 -1 M )] + ( γ + 1) , (C2) \nwhere M is the pre-shock Mach number, v i is the pre-shock speed at x → -∞ , and x 0 is a constant determining the location of the shock. For a pre-shock density ρ i , the density profile is obtained from the mass conservation equation: ρ i v i / v ( x ), which determines the pressure profile from equation (B21). Similarly, the profile for the internal energy of the electrons in terms of ρ ( x ) is given by equation (B26) with the substitution ρ f ρ ( x ). \nFor our numerical test, we do not use the standard Noh test as outlined in § 4.1.2 due to the problems noted by the original paper (Noh 1987). For any numerical scheme that gives the shock a finite width, the formation of the shock from the converging flow undershoots the density at the center of the grid by a finite amount that does not disappear at higher resolution. Given this di ffi culty, our numerical test is instead to set the initial and boundary conditions of both the fluid and electron variables equal to the analytic solution for a stationary shock (e.g., equation C2) and evolve for a dynamical time of L / v i , where L is the grid size. We choose γ = 5 / 3, v i = 10 -2 c , M ∼ 49, and ν = 0 . 01 v i L . Figure C1 shows both the density profile and the ratio of the electron internal energy to the total internal energy for both γ e = 4 / 3 and γ e = 5 / 3 electrons at the end of the run as compared to the analytic solution (equation B26). We find good agreement with the analytic solution and second order convergence (Figure C2) up to the resolution at which relativistic errors in the analytic solution become important ( δ ug / ug ∼ ( v / c ) 2 ∼ 10 -4 ). \n→ \n<!-- image --> \nFigure C1. High Mach number ( ∼ 49), stationary, viscous shock results for an electron heating fraction fe = 0 . 5 at a resolution of 2000 cells. Top: fluid density. Bottom: electron internal energy relative to total fluid internal energy. Both the γ e = 4 / 3 and γ e = 5 / 3 electrons display good agreement with the analytic solution, converging at 2nd order (see Figure C2). This is in contrast to the formulation without explicit viscosity used in § 4.1.2, in which the shock structure is always just a few grid points. An accurate calculation of the shock heating requires a well-resolved shock structure (i.e., a shock with a finite width), which is provided by adding explicit bulk viscosity to the fluid equations. Given that the error incurred by our numerical scheme without explicit viscosity ( ∼ 3%) is acceptable for our purposes, we do not use explicit viscosity in our calculations. \n<!-- image --> \nFigure C2. Convergence results for the electron internal energy in a steadystate, 1D, high Mach number, viscous shock as compared to the analytic solution (see Appendix C). Both the γ e = 5 / 3 and γ e = 4 / 3 electrons converge at 2nd order, as opposed to the non-viscous shock of § 4.1.2 where only the γ e = 5 / 3 electrons converged to the analytic solution. Second order convergence is achieved in this test problem because the shock profile is wellresolved and continuous. This shows that our method correctly captures the dissipation in strong shocks when the shock profile can be resolved. At the highest resolution, relativistic corrections to the (non-relativistic) analytic solution become important so the error no longer converges at second order. \n<!-- image -->', 'APPENDIX D: TORUS INITIAL CONDITIONS': 'In this appendix we describe in more detail the initial configuration of the torus in our simulations of an accreting black hole. In all expressions that follow we measure radii in units of the gravitational radius rg GM / c 2 (or equivalently set G = M = c = 1). \nFishbone & Moncrief (1976) derived an equilibrium solution (their equation 3.6) of the general relativistic hydrodynamic equations in the Kerr metric in terms of the relativistic enthalpy, h ≡ ( ρ + Pg + ug ) /ρ , and the constant angular momentum per unit mass, l ≡ u ϕ u t . We use their equation 3.6 exactly as presented when r > rin and when the right-hand side is positive, otherwise we set ρ = P = 0. Additionally, we assume an adiabatic equation of state, P = κ 0 ρ γ , for some choice of κ 0, and fix l such that the density maximum occurs at r max: \n≡ \nl = [ a 2 -2 a √ r max + r 2 max ] [ -2 ar max ( a 2 -2 a √ r max + r 2 max )] √ 2 a √ r max + r 2 max -3 r max + ( a + √ r max( r max -2) ) ( r 3 max + a 2 ( r max + 2) ) ( a 2 + r 2 max -2 r max ) √ 1 + 2 ar -3 / 2 max -3 / r max × 1 r 3 max √ 2 a √ r max + r 2 max -3 r max , (D1) \nwhere a is the dimensionless spin parameter of the black hole. This expression for l is equivalent to the Keplerian value at r = r max.'}