Abstract
We study gravitational perturbations in the Randall-Sundrum two-brane background with scalar-curvature terms present in the action for the branes, allowing for positive as well as negative bulk gravitational constant. The case of negative bulk gravitational constant is motivated by recent braneworld models of dark energy. In the zero-mode approximation, we derive the linearized gravitational equations, which have the same form as in the original Randall-Sundrum model but with different expressions for the effective physical constants. In the case of negative bulk gravitational constant, if both brane Planck masses are nonzero, the theory contains one or two tachyonic mass eigenvalues in the gravitational sector. If one of the brane Planck masses is set to zero, then either a single tachyonic mass eigenvalue is present or tachyonic modes are totally absent depending on the relation between the nonzero brane Planck mass and brane separation.
I. INTRODUCTION
Linear gravitational perturbations of the flat braneworld models were studied beginning from the seminal papers by Randall & Sundrum [1, 2] , where their spectrum was shown to contain, besides the zero mode, also an infinite tower of Kaluza-Klein massive modes. Since then, perturbations in various types of braneworld scenarios and in various approximations were considered in the flat case as well as on the cosmological background (which is a much more complicated issue still far from being well understood; see [3] and references therein). In this paper, we study sertain aspects of linear perturbations on a particular simple background in a model which, to our knowledge, has not been completely investigated. It represents the Randall-Sundrum two-brane model (the so-called RS1 model [1] ) supplemented by scalarcurvature terms in the action for both branes [4] . We consider the version of this model in which the branes are timelike boundaries of the five-dimensional bulk. The effects of induced gravity on the branes in the setup under consideration were recently studied in [5] .
I. INTRODUCTION
Perturbations in the one-brane (RS2 [2] ) counterpart of this model were investigated in [6, 7] .
I. INTRODUCTION
The study of the braneworld models is usually confined to the case of positive bulk gravitational constant (or positive bulk Planck mass). This can be explained by the fact that only positive values of the bulk gravitational constant are allowed in the RandallSundrum model, since its sign coincides with the sign of the effective Newton's constant in that model [1, 2] . When one adds curvature terms in the action for the branes, this assumption can be relaxed, and, in this paper, we allow for positive as well as negative bulk gravitational constant. The possibility of negative bulk gravitational constant was motivated in [8] , especially in connection with the braneworld model of disappearing dark energy (DDE) [8, 9] . The DDE model is a braneworld model of expanding universe which, after the current period of acceleration, re-enters the matter-dominated regime continuing indefinitely in the future. The merit of this model of dark energy is the absence of the cosmological event horizon owing to the fact that the universe becomes flat, rather than De Sitter, in the asymptotic future. In the DDE model, the tension of the visible brane is negative, and, in order for the zero-mode graviton to be 'localized' around the brane, it is necessary that the value of the bulk Planck mass also be negative. Negative brane tension is required also for the existence of unusual 'quiescent' singularities [10] in the AdS-embedded braneworld models, which occur during the universe expansion and are characterized by finiteness of the scale factor, Hubble parameter, and matter density. Thus, it seems important to keep open the possibility of negative bulk Planck mass when generalizing the Randall-Sundrum model by including the curvature terms in the action for the branes. This paper is organized in the following manner. After describing the model, we review a suitable theory of linear gravitational perturbations on the Randall-Sundrum two-brane background. Then, in the zero-mode approximation, we derive the linearized system of gravitational equations with matter confined to the visible and/or hidden brane. These equations have the same form as in the original RS1 model but with different physical constants. After that, we investigate the case of negative bulk Planck mass and show that the linearized theory can contain tachyonic gravitational modes. In this case, one or two tachyonic mass eigenvalues are observed if both brane Planck masses are nonzero. If one of the brane Planck masses is zero, then either a single tachyonic mass eigenvalue is present or tachyonic modes are totally absent; which of these two possibility is realized depends upon the values of the nonzero brane Planck mass and brane separation. We determine the range of parameters for which tachyonic modes are present or absent. The theory without tachyonic modes appears to be physically reasonable; thus, we have before us an example of physically viable theory with negative bulk gravitational constant, in which the zero-mode graviton is 'localized' on the brane with negative tension. We also calculate the effective gravitational potentials of static matter sources on the visible and hidden brane.
II. THE MODEL
The action of the model in the neighbourhood of one brane has the form
II. THE MODEL
where the first part, proportional to the cube of the bulk Planck mass M 3 , describes the bulk bounded by the brane, and the remaining integrals are taken over the brane. Here, R is the curvature scalar in the bulk, R is the curvature scalar of the induced metric h ab on the brane, K is the trace of the tensor of the intrinsic curvature K ab of the brane, L (h ab , φ)
II. THE MODEL
is the Lagrangian of the matter fields φ on the brane, and integration in (1) implies natural volume elements in the bulk and on the brane. The action is similar in the neighbourhood of the other brane. In principle, two branes in our model may have different Planck masses m, and, to allow for solutions with flat vacuum branes, their tensions σ must have opposite signs and satisfy the well-known constraint [1, 2] 
II. THE MODEL
Note that, in this paper, we allow for positive as well as negative signs of the bulk Planck mass parameter M, the consequences of which will become clear later.
II. THE MODEL
Action (1) leads to the bulk described by the usual Einstein equation with cosmological constant:
II. THE MODEL
while the field equation on the brane is
II. THE MODEL
where τ ab is the stress-energy tensor on the brane stemming from the last term in action (1) .
II. THE MODEL
By contracting the Gauss identity
II. THE MODEL
on the brane and using Eq. (3), one obtains the 'constraint' equation
II. THE MODEL
which, together with (4), implies the following closed scalar equation on the brane:
II. THE MODEL
where τ = h ab τ ab .
II. THE MODEL
In the case of a vacuum brane (τ ab = 0), Eq. (7) takes the form
II. THE MODEL
where Λ RS is given by Eq. (2). It should be noted that the second term in Eq. (8) has precisely the form of one of the terms in the expression for the conformal anomaly, which describes the vacuum polarization at the one-loop level in curved space-time (see, e.g., [11] ).
1
Another useful identity is the Codazzi identity
1
which is valid at any timelike hypersurface in the bulk, in particular, on the branes, due to Eq. (3). Here, D a denotes the unique covariant derivative on the timelike hypersurface associated with the induced metric h ab .
1
The gravitational equations in the bulk can be integrated by using Gaussian normal coordinates, as described, e.g., in [12] . Specifically, in the Gaussian normal coordinates (x, y), where x = {x α } are the coordinates on the brane and y is the fifth coordinate in the bulk, the metric is written as
1
Introducing also the tensor of extrinsic curvature K ab of every hypersurface y = const, one can obtain the following system of differential equations for the components h αβ and K α β :
1
1 It is interesting that, while the conformal anomaly term R ab R ab − 1 3 R 2 cannot be obtained by the variation of a local four-dimensional Lagrangian, the very same term is obtained via the variation of a local Lagrangian in the five-dimensional braneworld theory under investigation [8] . Also note that this term is absent in the original Randall-Sundrum model which has m = 0.
1
where R α β are the components of the Ricci tensor of the metric h αβ induced on the hypersurface y = const, R = R α α is its scalar curvature, and K = K α α is the trace of the tensor of extrinsic curvature. The second equality in (11) is true by virtue of the 'constraint' equation (6) . Equations (11) and (12) together with the 'constraint' equation (6) represent the 4+1 splitting of the Einstein equations in Gaussian normal coordinates. The initial conditions for these equations are defined on the brane through Eq. (4).
III. LINEAR PERTURBATIONS
Linear perturbations of the RS1 model are well studied (see, e.g., [13, 14] and references therein). Here, we would like to see the modifications arising from the presence of the scalarcurvature terms in the action for the branes (nonzero values of the masses m and m * ). Our treatment in this and in the subsequent section is similar to that of [14] .
III. LINEAR PERTURBATIONS
The perturbed metric of our solution in Gaussian normal cordinates has the form (10) with
III. LINEAR PERTURBATIONS
where
III. LINEAR PERTURBATIONS
and we emphasize that k can be positive as well as negative depending on the signs of M and σ. The perturbations of the tensor of extrinsic curvature and of the Einstein tensor have the form
III. LINEAR PERTURBATIONS
Here and below, the indices of ∂ α and γ αβ are raised and lowered with respect to the flat metric η αβ .
III. LINEAR PERTURBATIONS
Using the freedom of choice of the coordinates x α on the brane, one can choose the harmonic gauge in which ∂ αγ αβ = 0 on one of the branes. In this gauge, we have
III. LINEAR PERTURBATIONS
on that brane.
III. LINEAR PERTURBATIONS
In the unperturbed solution, the first (visible) brane is assumed to be at y = 0, and the second (hidden) brane is at y = ρ. First, we consider the situation where the hidden brane does not have matter on it (stress-energy tensor equal to zero). Then, when studying perturbations, it is convenient to choose Gaussian normal coordinates with respect to the hidden brane. Thus, the hidden brane remains at y = ρ, while the position of the visible brane is linearly perturbed to become y = φ(x), which is the so-called radion degree of freedom. Let m and σ denote the Planck mass and tension of the visible brane, and let those of the hidden brane be m * and σ * = −σ, respectively. The linearly perturbed boundary equation (4) on the second (hidden) brane becomes
III. LINEAR PERTURBATIONS
where
III. LINEAR PERTURBATIONS
β K, and we have taken into account that the extrinsic curvature is calculated with respect to the normal in the positive direction of y. Choosing harmonic coordinates on the hidden brane, for which (16) is satisfied, we have
III. LINEAR PERTURBATIONS
where a * = a(ρ) = e −kρ andγ =γ α α .
III. LINEAR PERTURBATIONS
Linearization of the vacuum constraint equation (8) implies the condition 2γ = 0 on the hidden brane if
III. LINEAR PERTURBATIONS
which we assume to be the case. This condition and Eq. (18) implies the condition ∂γ/∂y = 0 at the hidden brane. Then the Codazzi relation (9) implies the condition ∂ (∂ αγ αβ ) /∂y = 0 at the same brane.
III. LINEAR PERTURBATIONS
Now we turn to Eqs. (11) and (12) . Using (15), we can write the second-order differential equations for perturbationsγ αβ in the bulk. First, we verify that the Gaussian normal coordinates x α remain harmonic in the bulk. We introduce the quantity
III. LINEAR PERTURBATIONS
which is an indicator of the harmonicity of the coordinates x α on the hypersurface y = const.
III. LINEAR PERTURBATIONS
Then we can write the system of differential equations for v α andγ that stems from system (11), (12):
III. LINEAR PERTURBATIONS
with the following boundary conditions at the hidden brane (y = ρ):
III. LINEAR PERTURBATIONS
The unique solution of system (21) in the bulk with the boundary conditions (22) is
III. LINEAR PERTURBATIONS
In particular, this means that the Gaussian normal coordinates which we are using remain harmonic with respect to x all over the bulk.
III. LINEAR PERTURBATIONS
Taking into account relations (23), from (11), (12) one obtains the system of equations for perturbations in the bulk:
III. LINEAR PERTURBATIONS
with the boundary condition at y = ρ which stems from (18):
III. LINEAR PERTURBATIONS
To obtain the boundary equations on the visible brane, one must take into account its 'bending' in the bulk: y = φ(x). The induced metric on the visible brane in the linear approximation becomes
III. LINEAR PERTURBATIONS
so that its perturbation is
III. LINEAR PERTURBATIONS
Substituting it to the boundary condition at the visible brane
III. LINEAR PERTURBATIONS
we obtain the boundary condition at y = 0:
III. LINEAR PERTURBATIONS
Taking trace of this equation, we obtain the equation for the radion field φ:
III. LINEAR PERTURBATIONS
where τ ≡ η αβ τ αβ is the trace of the stress-energy tensor, and A = M 3 + 2km 2 . Thus, the radion field is coupled to the trace of the stress-energy tensor, as is the case in the Randall-Sundrum model [13] , but with different coupling constant. Using Eq. (30), from
III. LINEAR PERTURBATIONS
Now we have to solve the bulk equations (24) with the boundary conditions (30), (25) and (31). Proceeding to the Fourier transform with momenta p α in the coordinates x α and omitting the tensor indices, we have for the Fourier image ψ(q, y) ofγ αβ (x, y):
III. LINEAR PERTURBATIONS
where the prime denotes the derivative with respect to y, and q = √ −p 2 (here we assume p 2 ≡ p α p α ≤ 0; the tachyonic case will be studied in Sec. V). After the standard change of variable and function
III. LINEAR PERTURBATIONS
we get the equation
III. LINEAR PERTURBATIONS
in which the prime denotes the derivative with respect to z. Note that z(y) is a monotonic function of y for both signs of k, but the sign of z coincides with the sign of the constant k.
III. LINEAR PERTURBATIONS
The boundary conditions follow from (31) and (25):
III. LINEAR PERTURBATIONS
where T stands for the Fourier transform of the expression
III. LINEAR PERTURBATIONS
with tensor indices omitted.
III. LINEAR PERTURBATIONS
The general solution of the Bessel equation (34) is given by
III. LINEAR PERTURBATIONS
where J 2 and Y 2 are the Bessel functions, P and Q are constants, and the modulus of z reflects the fact that the domain of z is positive or negative depending on the sign of k.
III. LINEAR PERTURBATIONS
Using the recurrence relations
III. LINEAR PERTURBATIONS
we obtain from (35) and (36):
III. LINEAR PERTURBATIONS
solving which, one finds the constants P and Q and obtains the solution for ψ(z):
III. LINEAR PERTURBATIONS
where the constants are given by
III. LINEAR PERTURBATIONS
These results differ from the similar results [13, 14] The spectrum of the model is determined by the equality of the denominator of (42) to zero. Introducing the dimensionless variable s = q/|k| and parameters µ = km 2 /M 3 , µ * = km 2 * /M 3 , and α = e kρ , we obtain the following equation for the spectrum:
III. LINEAR PERTURBATIONS
where
III. LINEAR PERTURBATIONS
If both masses m and m * are nonzero, then the ultraviolet asymptotics of the spectrum for the Kaluza-Klein modes is determined by the zeros of the last term in (44), so that
III. LINEAR PERTURBATIONS
which coincides with the asymptotics of the spectrum in the Randall-Sundrum model, determined by the zeros of the first term in (44).
III. LINEAR PERTURBATIONS
If m = 0, m * = 0, then the asymptotics of the spectrum is determined by the second term in (44):
III. LINEAR PERTURBATIONS
If m = 0, m * = 0, then it is determined by the third term in (44):
III. LINEAR PERTURBATIONS
In these last two cases, the spectrum is somewhat shifted.
IV. LINEARIZED EQUATIONS IN THE ZERO-MODE APPROXIMATION
In the zero-mode approximation [13, 14] , one considers the limit as q → 0. In this limit, using Eqs. 
IV. LINEARIZED EQUATIONS IN THE ZERO-MODE APPROXIMATION
where it should be stressed that G αβ is the Einstein tensor of the induced metric (26) on the brane, and the constants A and B are given by
IV. LINEARIZED EQUATIONS IN THE ZERO-MODE APPROXIMATION
The effective Newton's constant G N is given by the relation
IV. LINEARIZED EQUATIONS IN THE ZERO-MODE APPROXIMATION
and one should note the extra contribution from the radion in (49), which involves the trace 2 The constant A is the same as in Eq. (30).
IV. LINEARIZED EQUATIONS IN THE ZERO-MODE APPROXIMATION
of the stress-energy tensor. 3 If k > 0, this contribution is exponentially suppressed for large separations between the branes, kρ ≫ 1.
IV. LINEARIZED EQUATIONS IN THE ZERO-MODE APPROXIMATION
If matter is present only on the hidden brane, then it still induces curvature on the visible brane [13, 14] . In our theory, we obtain the result
IV. LINEARIZED EQUATIONS IN THE ZERO-MODE APPROXIMATION
where τ * αβ is the stress-energy tensor on the hidden brane, and τ * is its trace. If both branes contain matter, then the results on the right-hand sides of (49) and (52) to A [the same property is observed in the general equation (7)]. In particular, the nonzero value of the constant B was already assumed in the linearization scheme [see Eq. (19)]. The special cases where either A or B is equal to zero must be studied separately.
IV. LINEARIZED EQUATIONS IN THE ZERO-MODE APPROXIMATION
Our second remark is that, unlike in the original Randall-Sundrum model (m = m * = 0), in our theory the sign of the constant M can be negative as well as positive: apart from the scalar contribution described by the trace of the stress-energy tensor, matching with the general-relativity limit fixes only the sign of the overall constant in (49) and (52). In particular, for a sufficiently small absolute value of M, namely, |M 3 /2k| ≪ m 2 , m 2 * , the sign of M does not matter. In the formal limit M → 0 with k > 0 (hence, k → +∞), the equation for the visible brane (49) turns to the usual linearized Einstein equation. In the formal simultaneous limit M → 0 and σ → 0 so that k = σ/M 3 is fixed, expressions (49) and (52) become
IV. LINEARIZED EQUATIONS IN THE ZERO-MODE APPROXIMATION
and
IV. LINEARIZED EQUATIONS IN THE ZERO-MODE APPROXIMATION
respectively.
IV. LINEARIZED EQUATIONS IN THE ZERO-MODE APPROXIMATION
Finally, we note that our result does not explicitly contain the constant σ but contains it only in the combination k = σ/3M 3 . Therefore, for one and the same effective law of gravity (49) [8, 9] , as discussed in the introduction. However, unlike in the pure Randall-Sundrum case, the presence of the curvature terms in the action for the branes leads to a possibility of unwanted tachyonic modes in the gravitational sector of the theory if M < 0. In this section, we demonstrate that there can be only one or two tachyonic mass eigenvalues in the theory under consideration and determine the range of parameters for which tachyonic modes do not exist so that the theory is physically viable.
IV. LINEARIZED EQUATIONS IN THE ZERO-MODE APPROXIMATION
In looking for tachyonic modes, one needs to solve Eq. (32) for q 2 = −p 2 < 0, i.e.,
IV. LINEARIZED EQUATIONS IN THE ZERO-MODE APPROXIMATION
where the prime denotes the derivative with respect to y. After the standard change of variable and function
IV. LINEARIZED EQUATIONS IN THE ZERO-MODE APPROXIMATION
we get the equation for the new function χ(z):
IV. LINEARIZED EQUATIONS IN THE ZERO-MODE APPROXIMATION
in which the prime denotes the derivative with respect to z. With matter present on the visible brane only, the boundary conditions are similar to (35) and (36):
IV. LINEARIZED EQUATIONS IN THE ZERO-MODE APPROXIMATION
where T is the Fourier transform of expression (37), with tensor indices omitted.
IV. LINEARIZED EQUATIONS IN THE ZERO-MODE APPROXIMATION
Solution of (57) is now given by the modified Bessel functions I 2 and K 2 :
IV. LINEARIZED EQUATIONS IN THE ZERO-MODE APPROXIMATION
where P and Q are constants, and the modulus of z again reflects the fact that the domain of z is positive or negative depending on the sign of k. The recurrence relations of type (39) are valid also for the modified Bessel functions:
IV. LINEARIZED EQUATIONS IN THE ZERO-MODE APPROXIMATION
and we can use them in deriving the solution similar to (42):
IV. LINEARIZED EQUATIONS IN THE ZERO-MODE APPROXIMATION
where
IV. LINEARIZED EQUATIONS IN THE ZERO-MODE APPROXIMATION
Tachyonic modes correspond to those values of p for which the denominator of (62) turns to zero:
IV. LINEARIZED EQUATIONS IN THE ZERO-MODE APPROXIMATION
Since the transformation p → e 
IV. LINEARIZED EQUATIONS IN THE ZERO-MODE APPROXIMATION
where
IV. LINEARIZED EQUATIONS IN THE ZERO-MODE APPROXIMATION
Since α > 1 for k > 0, all functions in (66) are strictly positive for positive s. This implies that tachyonic modes are absent in the case µ, µ * > 0, or, equivalently, M > 0.
IV. LINEARIZED EQUATIONS IN THE ZERO-MODE APPROXIMATION
However, tachyonic modes may be present in the opposite case M < 0. It is possible to indicate the corresponding range of parameters where tachyonic modes are present or absent.
IV. LINEARIZED EQUATIONS IN THE ZERO-MODE APPROXIMATION
To do this, it is convenient to introduce the following auxiliary function of two variables s ands:
IV. LINEARIZED EQUATIONS IN THE ZERO-MODE APPROXIMATION
[to be compared with (65)]. By construction,Ē(s, s) ≡ E(s).
IV. LINEARIZED EQUATIONS IN THE ZERO-MODE APPROXIMATION
First, we consider the case where both µ and µ * are nonzero (in the present case, they are then both negative). Then the equation
IV. LINEARIZED EQUATIONS IN THE ZERO-MODE APPROXIMATION
gives the two branches of solutions with respect tos:
IV. LINEARIZED EQUATIONS IN THE ZERO-MODE APPROXIMATION
and solving the original equation (65) is equivalent to solving one of the equations
IV. LINEARIZED EQUATIONS IN THE ZERO-MODE APPROXIMATION
It is easy to verify that the expression under the square root of (69) is strictly positive for positive s so that the two solutionss ± (s) exist for all s > 0 and are positive. The asymptotic behaviour of these solutions for small and large s can easily be found:
IV. LINEARIZED EQUATIONS IN THE ZERO-MODE APPROXIMATION
From these expressions it is clear that the graph ofs + (s) crosses the graph of f (s) = s at least once for any values of parameters in the range µ, µ * < 0 under consideration. Thus, Eq. (70) has a solution, and at least one tachyonic mass eigenvalue is present in this range of parameters. Numerical computation indicates that there is exactly one solution connected with the branchs + (s) in this case.
IV. LINEARIZED EQUATIONS IN THE ZERO-MODE APPROXIMATION
It is also clear that the graph ofs − (s) definitely crosses the graph of f (s) = s in the case
IV. LINEARIZED EQUATIONS IN THE ZERO-MODE APPROXIMATION
Again, numerical computation indicates that there is only one tachyonic solution connected with the branchs − (s) in this case. They also indicate that tachyonic modes connected with the branchs − (s) are absent in the case of the opposite inequality in (73).
IV. LINEARIZED EQUATIONS IN THE ZERO-MODE APPROXIMATION
Thus, tachyonic modes exist for all values of parameters in the range µ, µ * < 0. In the case under consideration, M < 0, one can expect tachyonic modes to be absent only if one of the brane Planck masses m or m * is zero. 4 We show that this is indeed the case and determine the range of masses and brane separations for which tachyonic modes are absent.
IV. LINEARIZED EQUATIONS IN THE ZERO-MODE APPROXIMATION
In the case µ < 0, µ * = 0, the functionĒ(s,s) given by Eq. (67) takes the simple form
IV. LINEARIZED EQUATIONS IN THE ZERO-MODE APPROXIMATION
and Eq. (68) has a single solution with respect tos:
IV. LINEARIZED EQUATIONS IN THE ZERO-MODE APPROXIMATION
It can be verified that the function D 1 (s)/D(s) is convex upwards, so that the equation s(s) = s has exactly one solution or no solutions in the range s > 0 depending on the value of the derivatives ′ (0). Specifically, a solution exists ifs ′ (0) > 1, and there are no solutions in the opposite cases ′ (0) ≤ 1. Calculating the derivatives ′ (0), we obtain that exactly one tachyonic mass eigenvalue is present in the theory if
IV. LINEARIZED EQUATIONS IN THE ZERO-MODE APPROXIMATION
and tachyonic modes are absent if the value of the Planck mass m is sufficiently large, namely, if
IV. LINEARIZED EQUATIONS IN THE ZERO-MODE APPROXIMATION
In the limit of infinite separation between branes, α = e kρ → ∞, the condition of absence of tachyonic modes becomes |µ| ≥ 1/2, which coincides with the condition A ≥ 0, where A is given by (50). Interestingly, this is also the condition of positivity of the effective Newton's constant in the zero-mode approximation (49), (52) in the same limit.
IV. LINEARIZED EQUATIONS IN THE ZERO-MODE APPROXIMATION
The case µ = 0, µ * < 0 is analysed in quite a similar way. Now the functionĒ(s,s)
IV. LINEARIZED EQUATIONS IN THE ZERO-MODE APPROXIMATION
given by Eq. (67) takes the form
IV. LINEARIZED EQUATIONS IN THE ZERO-MODE APPROXIMATION
and Eq. (68) has one solution with respect tos:
IV. LINEARIZED EQUATIONS IN THE ZERO-MODE APPROXIMATION
Again, it can be verified that the equations(s) = s has exactly one solution in the range s > 0 ifs ′ (0) > 1, and there are no solutions in the opposite cases ′ (0) ≤ 1. Calculating the derivatives ′ (0), we obtain that exactly one tachyonic mass eigenvalue is present in the theory if
IV. LINEARIZED EQUATIONS IN THE ZERO-MODE APPROXIMATION
and tachyonic modes are absent if the value of the Planck mass m * is sufficiently large, namely, if
VI. CORRECTIONS TO NEWTON'S LAW
In this section, we compute the gravitational potential V (r) on the visible brane induced by a static point source located on the visible or hidden brane and determine corrections to the Newton's law in the physically reasonable case k > 0, i.e., where the zero-mode graviton is localized around the visible brane. Our starting formula is h (vis) 00
VI. CORRECTIONS TO NEWTON'S LAW
where h (vis) αβ is the induced metric on the visible brane.
A. Mater source on the visible brane
If matter source is on the visible brane only, then the induced metric h We take into account (30) and (62) for the Fourier transform of the solution for the induced metric on the first brane with the source on the same brane. Collecting all expressions together, we obtain the Fourier image of the gravitational potential
A. Mater source on the visible brane
where the function f (p) denotes the second fraction in expression (62) taken at the position of the visible brane (z = z 0 ):
A. Mater source on the visible brane
The potential V (r) is obtained by taking the inverse Fourier transform of (83):
A. Mater source on the visible brane
where
A. Mater source on the visible brane
and E(s) is given by Eq. (65).
A. Mater source on the visible brane
The integral in (86) cannot be evaluated exactly, but it can be approximated in different regions of r, as it is done, e.g., in [6, 15] .
1.
On very large spatial scales kr ≫ α ≡ e kρ , we need the asymptotics of the function Ψ(s) in the region αs ≪ krs ∼ 1. It is given by the expression
1.
and the integral in (86) is approximated by using the regularization [6, 15] with the result
1.
The potential in this region has Newtonian form
1.
where G N given by (51), which is in complete agreement with the zero-mode approximation (49). The theory has continuous Newtonian limit as α → ∞.
2.
On the scales 1 ≪ kr ≪ α, it is the region of integration 1/α ≪ s ≪ 1 that substantially contributes to the integral in (86). In this region, we have
2.
substituting which to (86), we obtain
2.
and
2.
3. In the case kr ≪ 1, we can use the asymptotics for the functon Ψ(s) at infinity:
2.
This case is further partitioned into two asymptotic regions, depending on the magnitude of µ.
2.
3a. kr ≪ |µ|. Here, if µ is negative, then it cannot be small by absolute value since, in this latter case, the theory contains tachyons. We obtain
2.
and
2.
The logarithmic corrections in these expressions assume that the expression 15/8 + 1/µ is not very small by absolute value.
2.
3b. µ ≪ kr ≪ 1. Here we must consider only positive µ. In this case,
2.
and the gravitational law is five-dimensional:
2.
B. Mater source on the hidden brane
2.
In a similar way one can consider the case where the stationary matter resides on the hidden brane with mass M * defined as τ * 00 = M * δ( r ). In this case, we obtain the following expression for the gravitational potential on the visible brane:
2.
where
2.
and E(s) is given by Eq. (65).
2.
The asymptotic expressions for Ψ(s) can be found in various regions:
2.
Ψ(s) ≈ √ 2παs e −αs
2.
(1 + 2µ) 1 + 3 8αs 
2.
Ψ(s) ≈ 2 √ α e −αs , s ≫ 1 , µ = µ * = 0 .
2.
Using these expression, it is not difficult to obtain the estimates for the gravitational potential V (r) caused by the presence of the static source on the hidden brane in varios regions. We have
2.
where G N is given by (51). Again, in this distance range, the result can be obtained by using the zero-mode approximation (52).
2.
In the case kr ≪ α, the result crucially depends on whether µ * is zero or not: 
2.
Here the constants c n take the following approximate values: 
2.
Expression (108) is not valid for sufficiently small µ * since we know that, in the limit of µ * → 0, the asymptotics changes to (107). In fact, comparison with the exact numerical integration of (100) shows that our approximate result (108) is only good for |µ * | > ∼ 0.5.
VII. DISCUSSION
It is known that the braneworld model becomes rather rich in its cosmological manifestations if curvature term is present in the action for the brane (see [4, [6] [7] [8] [9] ). In this paper, we studied the model with curvature terms for the branes on the original Randall-Sundrum two-brane background. The linearized gravitational equations (49) and (52) in this case have the same structure as in the original Randall-Sundrum model but with different physical constants. In the limit of vanishing brane Planck masses m and m * , they tend to the known results [6, 15, 14] , which are physically reasonable only for M > 0. In the opposite limit of M → 0 while k = σ/M 3 is fixed, they produce reasonable results (53) and (54) independently of the sign of M.
VII. DISCUSSION
For negative values of M, the linearized theory typically contains tachyonic modes in the gravitational sector. If both brane Planck masses are nonzero, then we have one or two tachyonic mass eigenvalues depending on the constants of the theory, the conditions of which were determined in Sec. V. However, in the case where one of the brane Planck masses is zero, tachyonic modes are absent if the other brane has sufficiently high Planck mass [given by
VII. DISCUSSION
Eq. (77) for the visible brane, and by Eq. (81) for the hidden brane]. It is this case that gives an example of an apparently physically viable theory with negative value of the bulk Planck mass M. Note that, in the case of negative M, the zero-mode graviton is 'localized' around the brane with negative tension. This may be important for braneworld cosmological models, in particular, for the model of disappearing dark energy (DDE) recently discussed in [8, 9] 

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