Tensor Perturbations and Thick Branes in Higher-dimensional $f(R)$ Gravity

We study brane worlds in an anisotropic higher-dimensional spacetime within the context of $f(R)$ gravity. Firstly, with a concrete metric ansatz, we demonstrate that this spacetime is stable against linear tensor perturbations under certain conditions. We provide a conclusion for higher-dimensional $f(R)$ gravity in this particular metric. Moreover, the Kaluza-Klein modes of the graviton are analyzed. Secondly, we investigate thick brane solutions in six dimensions and their properties. We further exhibit two sets of solutions for thick branes. At last, the effective potential of the Kaluza-Klein modes of the graviton is discussed for the two solvable $f(R)$ models in higher dimensions.


Introduction
General relativity (GR) is successful in many fields but leaves a few issues up in the air. Both phenomenological and theoretical investigations reveal modifications to GR under certain circumstances. Higher curvature terms may serve as these modifications and many investigations start from them but out of various considerations. However, they may lead to higher derivatives in equations of motion rather than second ones, which are generally difficult to be solved. f (R) theory of gravitation [1,2], one of the simplest higher derivative generalizations of GR, is possible to be analytically explored in some situations. Although f (R) theory is an effective theory, its success in many fields has attracted extensive attention.
On the other hand, the brane world scenario provides an alternative approach to address outstanding issues in four dimensions. Particularly, the brane world model with a warped extra dimension pioneered by Randall and Sundrum [3] has drawn wide attention since it exhibits the possibility of an infinite fifth dimension without violating known experiments of gravitation.
The general brane world sum rules indicate the existence of some particular classes of fivedimensional brane world models in f (R) gravity [4]. Moreover, due to the inevitable appearance of higher derivatives in equations of motion, investigation on junction conditions of f (R) gravity in the brane world scenarios opens the possibility of a new class of thin brane solutions [5][6][7][8][9][10][11].
Thick branes, domain walls [12] with a warp spacetime background, can preferably circumvent the requirements of the junction conditions and naturally remove the divergence of curvature. Recent reviews including f (R) thick branes refer to Refs. [13][14][15]. To avoid solving higher derivative equations, f (R) gravity within the context of brane worlds has been considered via the conformal equivalence between f (R) theory and GR with a scalar field [5,16].
Most of the related works are investigated in five-dimensional spacetime. It is worth noting that the known thick f (R) branes in higher spacetime dimensions [18,19,22] are constructed in pure gravity without background scalar fields. However, to localize a bulk fermion field on the brane, the Yukawa coupling between the fermion field and a background scalar field is usually needed. Moreover, in five dimensions, the pure gravitational trapping mechanism of vector fields remains problematic and there is no remarkable proposal for the fermion mass hierarchy in the Standard Model. These issues may employ higher dimensions to address. Our goal is to investigate brane solutions with nonconstant curvature in the context of six-dimensional f (R) gravity with a real scalar field.
Before exploring solutions in a background spacetime, we should first consider its perturbation stability. Considering four-dimensional Poincaré symmetry, the decomposition of perturbations will give rise to massless and massive graviton Kaluza-Klein (KK) modes. A localized massless graviton KK mode contributes to the four-dimensional Newtonian potential and the massive ones lead to corrections to the Newtonian potential. Gravitational resonant modes, a class of massive KK modes, have been studied for various solvable f (R) models in brane world scenarios [25,34]. More complete analyses on perturbations with extra spatial dimensions in the context of GR has been achieved in Refs. [35][36][37][38][39][40][41]. The linear stability of the tensor perturbation of f (R) brane models was firstly investigated in Ref. [42]. Other related investigations can also be seen in Refs. [16,27,[29][30][31][32]43]. Scalar perturbations within the context of pure f (R) gravity have been elaborated by the transformation of the f (R) theory to a scalar-tensor theory [16,32] and directly studied in the higher-order frame [43]. However, scalar perturbations in f (R) gravity with background scalar fields are difficult to be analyzed because of the coupling between the scalar modes of the metric perturbations and background scalar fields, and the resolution was given in Ref. [44]. For our research, we investigate the linear stability of higher-dimensional f (R) gravity in brane world scenarios against tensor perturbations. This paper is organized as follows. In section 2, we consider f (R) gravity in a D-dimensional spacetime and give the equations of motion under a concrete metric describing flat branes. In section 3, we investigate the linear stability of this background spacetime against tensor perturbations. In section 4, the graviton KK modes are discussed. In section 5, we seek for analytic solutions of thick branes in a six-dimensional bulk spacetime. Exact solutions of the scalar field with a domain wall configuration are studied. In section 6, we discuss the effective potential in solvable f (R) models in higher dimensions. Finally, discussions and conclusions are given in section 7.

f (R) Gravity in D-dimensional Spacetime
We start with the following D-dimensional action within the context of f (R) gravity (for reviews see Refs. [1,2]), where g (D) ≡ det g MN , and κ 2 D = 8πG (D) N = 1/M D−2 (D) with G (D) N the D-dimensional gravitational constant and M (D) the D-dimensional fundamental scale. The variation of the action (1) with respect to the metric g MN yields the following field equation where Alembert operator, and δS m δg MN is the energy-momentum tensor. Specifically, a particular example of an anisotropic (4 manifold with a noncompact extra spatial dimension, and E d is an Euclidean manifold with d extra spatial dimensions. In this paper, we are interested in a four-dimensional static flat spacetime embedded in the (4 + 1 + d)-dimensional bulk, which takes the form ds 2 = e 2A(y) η µν dx µ dx ν + dy 2 + e 2B(y) δ i j dx i dx j .
Here e A(y) and e B(y) are warp factors which give rise to the warped geometry, η µν and δ i j are metrics in the M 4 and the E d , respectively, and y = x 5 is the spacial extra-dimensional coordinate.
With the coordinate transformations dz = e −A(y) dy and dw i = e B(y)−A(y) dx i , the above metric can be rewritten as Denoting a(y) = e A(y) and b(y) = e B(y) and using the metric ansatz (3), Eq. (2) is reduced to (µ, ν) : (y, y) : where the prime denotes the derivative with respect to the extra-dimensional coordinate y. To explore the stability of this configuration and effective gravitation in the M 4 , we would like to examine perturbations concerning this background spacetime.

Linear Stability
In D-dimensional spacetime, pure f (R) = R + αR n gravity (without matter) has been considered for thick brane solutions [18,19]. Nevertheless, the perturbative stability of spacetime is still unknown. We mainly focus on the linear stability of spacetime under perturbations and begin with gravitational perturbations for general f (R) gravity in D-dimensional spacetime (for general perturbations see appendix A). For a deformation of Eq. (2), the linearized field equation where the δ denotes a linear perturbation. This equation can also be derived by variation with respect to the action including quadratic terms of perturbations.
With the help of the expansions of ∇ M ∇ N f R and g MN ∇ A ∇ A f R : one can write the two terms δ (∇ M ∇ N f R ) and δ g MN ∇ A ∇ A f R on the right hand side of Eq. (6) as We investigate perturbations under the background metric (3) where h µν represents the TT tensor mode obeying Some fundamental quantities of spacetime under such a perturbation can be calculated accordingly. They are collected in appendix B.
The (µ, ν) components of δ (∇ M ∇ N f R ) and δ g MN (D) f R under the perturbations (9) are calculated as Plugging the expression of δG µν (70) (see appendix B) into Eq. (6), and considering δ ∇ µ ∇ ν f R and δ g µν (D) f R in Eq. (11), the (µ, ν) component of Eq. (6) is worked out to be Here, (4) = η µν ∂ µ ∂ ν and∆ (d) = δ i j ∂ i ∂ j are the d'Alembert operator in the M 4 and the Laplace operator in the E d , respectively. Taking the background equation (5a) into account, the main perturbed equation (12) for the TT tensor mode is reduced to for the curved background (3), the above equation (13) can be written as In the coordinates x µ , z, w i , Eq. (13) turns into where Next, we perform the KK decomposition Then we obtain the Klein-Gordon equation for the four-dimensional part ǫ µν (x ρ ): the Schrödinger-like equation for the fifth-dimensional spatial part ψ(z): and the Helmholtz equation for the extra d-dimensional spatial part ξ w i : Here the effective potential W(z) is of the following form with Both m 2 < 0 and l 2 < 0 are not physically reasonable, they will lead to the solution (17) either evolving exponentially in time or increasing exponentially in space. Essentially, the Schrödinger-like equation (19) can be factorized as a supersymmetric quantum mechanics form with which ensures m 2 − l 2 0. We conclude that this system is stable under tensor perturbations.
However, the condition m 2 − l 2 0 does not ensure that there are no lower energy states which are apparent tachyons. If one demands that apparent tachyon states are absent in the M 4 , m 2 0 should be satisfied. The two conditions m 2 − l 2 0 and m 2 0 are significant for the stability. We give a brief discussion for several cases: • there are no any apparent tachyon states in the M 4 if m 2 − l 2 0 and m 2 0 are satisfied; • the case m 2 −l 2 0 and m 2 < 0 will result in some apparent tachyon states of the graviton in the M 4 ; • it is worth stressing that m 2 = 0 and l 2 = 0 will lead to a zero energy state which stands for the graviton zero mode; however, a zero energy state does not indicate m 2 = 0 and l 2 = 0 for the case of m 2 = l 2 .
Until now, the above analyses of tensor perturbations are independent of the explicit background spacetime. It is worth noting that the above results are applicable for d 0. For d = 0, namely the five-dimensional case, it was reported in Ref. [42]. In the case of d > 0, the effective potential compared with the one with d = 0 will be corrected due to the presence of the warp factor b. Moreover, the above derivation can also be done with the line element (4). Some related works in the context of GR can be seen in Refs. [35,37,38,40,41].
Stability conditions, the ghost-free condition d f (R) dR > 0 and the tachyon-free condition d 2 f (R) dR 2 > 0, should be fulfill physically. If these stability conditions are broken down, the effective potential (21) will occur some divergences which are not physical.

The KK Modes of the Graviton
In what follows, we will discuss the KK modes of the tensor perturbations which will help us analyze the four-dimensional effective theory. Here, we assume that the effective potential (21) has a volcano-like shape and vanishes at infinity (see Sec. 6 for the case considered in this paper). Then, there is a bound state with m 2 − l 2 = 0 and a series of continuous free states with m 2 − l 2 > 0 in terms of the Schrödinger-like equation (19). The continuum spectrum of these free states starts from m 2 − l 2 = 0. For a four-dimensional observer in the M 4 , the KK modes of the tensor perturbations reflect the configuration of extra dimensions. We use h (ml) µν (x M ) to denote the KK modes while ψ ml (z) their z-coordinate parts. Regardless of the value of l 2 , these modes are called massless if m 2 = 0 and massive if m 2 0. One can find a special set of modes with m 2 = l 2 : where N is a normalization constant. If we impose periodic boundary conditions on the E d , the condition l 2 0 will be satisfied. There is a special bound state (the graviton zero mode) ψ 00 (z) when m 2 = l 2 = 0, which reads The graviton zero mode (26) can be normalized if In order to get a four-dimensional effective gravitational theory, we need a reasonable background solution such that the normalization condition (27) is satisfied. Here, we clarify the normalization condition (27) from the perspective of action reduction. Noting that g µν (x ρ ) ≡ η µν + h (00) µν (x ρ ) depends on the four-dimensional coordinate x ρ (see Eqs. (17) and (26)), we can connect the D-dimensional curvature scalar R and the four-dimensional oneR through where the quantity with bar is computed from the metricḡ µν . By substituting the four-dimensional metricḡ µν and the relation (28) into the gravitational part of the action (1), we can obtain the four-dimensional effective action We can explicitly perform the y integral to obtain a four-dimensional action. If we require the four-dimensional effective theory is GR, the four-dimensional effective Planck mass M Pl is expressed as follows whereV = d dx is the volume of the E d . If the volumeV is finite, the condition of localizing the graviton zero mode on the M 4 is which is just the normalization condition (27). This is also the no-go theorem as shown in Ref. [45]. Therefore, a localized graviton zero mode (26) will lead to GR and hence the fourdimensional Newtonian potential. In addition, other KK modes will make a correction. We do not discuss this issue here.

Brane Solutions in Six-dimensional Spacetime
In this section, we will seek brane solutions satisfying the above restrictive conditions. As an example, we consider a background real scalar field within the context of six-dimensional f (R) gravity. Different from the line element with two compact extra dimensions [45,46], we consider the following one where θ ∈ [0, 2π) is a compact dimension. The background scalar field φ is merely a function of y for a static flat brane. From Eq. (5), we obtain the following equations in six dimensions (µ, ν) : (y, y) : (θ, θ) : The equation of motion for the background scalar field is The combination of Eq. (33) yields three transformed equations Here, only three of the four equations in (34) and (35) are independent because the energy momentum tensor is conserved. If the two warp factors satisfy a(y) = b(y), we have only two independent equations.
For the following tensor perturbations ds 2 = a 2 (y)(η µν + h µν )dx µ dx ν + dy 2 + L 2 b 2 (y)dθ 2 , the main perturbed equation is The relationship between the four-dimensional effective Planck scale M Pl and the six-dimensional fundamental scale M (6) is So far, for six-dimensional f (R) gravity within brane world scenarios, we have obtained background equations of motion and have analyzed linear stability for tensor perturbations. We will solve analytical solutions in the following subsections with Eqs. (34) and (35).

Starobinsky Gravity
First, we consider the Starobinsky gravity which is the first model of inflation in four dimensions and also gives rise to cosmic acceleration which ends when the term αR 2 is smaller than R [47]. Here, we seek thick brane solutions in the six-dimensional Starobinsky gravity.
In general, the superpotential method is not available for higher-order differential equations.
In order to avoid solving higher-order equations directly, we would like to adopt the reconstruction technique [31,48,49], which seeks a reasonable action of background fields to satisfy fixed configurations reasonable for brane models. As mentioned above, for simplicity we set the warp factors a(y) and b(y) as where k is a parameter with dimension mass. For ky → ±∞, a(y) = b(y) → e −nk|y| , which implies that the metric (32) reduces an AdS one at the boundary. The AdS curvature relates to the parameters k and n. The length scale 1/k corresponds to the thickness of the thick brane.
The parameter space allowing brane solutions is shown in Fig. 1. Moreover, the scalar field φ(y)  solution is characterized by a functionφ(ȳ) that approaches to the two different vacua of the potential asȳ → ±∞. This solution also indicates a length scale corresponding to the thickness of the brane.
From Fig. 2(a), the scalar field (41a) is a double kink for small values ofᾱ but is a single kink for large values ofᾱ. There is a critical value at which the double kink becomes a single one. We can seek the critical value by requiring the third-order derivative of the dimensionless scalar field (41a) with respect toȳ to satisfy the following condition The critical value ofᾱ is given byᾱ at whichφ This means that whenᾱ exceeds the critical valueᾱ c , the scalar field (41a) becomes a single kink. We exhibit the critical solution of the scalar field and the critical scalar potential in Fig. 3.  For the special value ofᾱ:ᾱ we obtain a concise solution where v = 1 κ 6 4n(n + 3)(3n + 1) (n + 1)(3n + 2) .
We depict the kink solution and the scalar potential in Fig. 4.   Next we explore the distribution of the energy density ρ along the extra dimension y: where U M = (1/a(y), 0, 0, 0, 0, 0). It is worth mentioning that the vacuum energy density has to been deducted from the total energy density. We introduce the dimensionless energy densitȳ ρ = ρκ 2 6 /k 2 . It is shown that there is a critical value ofᾱ blow which the energy density will be split. Such critical value ofᾱ isᾱ which is solved from the following condition For the above solution (41), we plot the energy density of the thick brane in Fig. 5. It is shown that, for various values ofᾱ, the maximum of the energy density increases with n. On the other hand, the brane splits only whenᾱ is less than the critical valueᾱ 0 (see Figs. 5(a) and 5(b)). For a fixed n, as shown in Fig. 6, the thickness of the brane decreases with increasingᾱ, which can also be seen from Figs. 5(a) and 5(b), and the brane gradually splits as the decrease ofᾱ.  ofᾱ. The energy density in Fig. 5(a) exhibits split behavior. A non-split energy density is shown in Fig. 5(b). The energy density at the critical valueᾱ 0 is shown in Fig. 5(c).
This solution requires n 1/5. We plot the scalar field φ and the potential V(φ) in Fig. 8. For a small n, φ is a double kink solution.   It is interesting that a simplified solution is obtained if n = 1/5: V(φ(y)) = 32k 2 3κ 2 6 9 cosh(2ky) − 7 sech 6 (ky) , where a newly defined coupling constantκ 6 = 3 √ 5κ 6 is adopted. In this special case, we can obtain an analytical expression for the scalar potential V(φ): According to Eq. (49), we plot the energy density in Fig. 9. It is shown that the brane is  split for a small n. There is a critical value n c = ( √ 65 + 3)/10 ≃ 1.106 , which satisfies the condition (51). The brane is not split for n > n c . Besides, the maximum of the energy density increases with n.

Effective Potentials
The above six-dimensional thick brane solutions can be generalized to a higher-dimensional bulk. To explore the effective gravity on the brane in such background, we discuss the effective potential (21) for the above solvable f (R) models in higher dimensions with the warp factors (40). Here we only consider the case n = 1. For these two solvable f (R) models, the graviton zero mode (26) satisfies the normalization condition (27) when n > 0 for the warp factors (40). As an example, we show a normalized graviton zero mode in Fig. 10(d). The difference of the effective potential (21) between GR and f (R) gravity is demonstrated in Fig. 10.
From Fig. 10, one can find that the effective potential in Fig. 10(b) has a double well due to the curvature scalar correction concerning GR. Therefore, a correction term for GR alters the effective potential for the graviton KK modes. This implies that the graviton mass spectrum will be different for various gravity theories. Besides, the depth of the effective potential is increased with the increase of the dimensions d of the extra space. Therefore, the KK modes of the graviton may partially reflect the configuration of extra dimensions.

Conclusions
In summary, we have investigated f (R) gravity with extra spatial dimensions in brane world scenarios. It was shown that the spacetime configuration is perturbatively stable. Further, we found two sets of thick brane solutions in a six-dimensional spacetime. We showed that the distribution of the energy density of the background scalar field concentrates on the vicinity of the thick brane, which implies that there is no divergence of curvature. Moreover, it was shown that the graviton zero mode is localized on the brane embedded in any dimension. This implies that the four-dimensional Newtonian potential can be recovered on the brane.
In this paper, we considered an extra d-dimensional Euclidean space E d besides the extra dimension y. A general extra d-dimensional Riemannian space is still interesting. It is well known that f (R) theory is equivalent to a second-order scalar-tensor theory of gravitation. In our exploration, we investigated branes only in f (R) gravity rather than transform it into a scalar- (d) f (R) = R + αR 2 Figure 10: Plots of the dimensionless effective potentialW(z) in the Schrödinger-like equation (19) for various f (R) models and the zero mode for f (R) = R + αR 2 , wherez = kz andW(z) = W(z)/k 2 .
All the effective potentials approach to zero atz → ±∞.
tensor theory. So far, we considered a special six-dimensional example. Models in different dimensions can lead to novel features of the bulk spacetime. The background metric ansatz (3) or (32) can also be replaced by other forms which are from some interesting configurations of the bulk. We leave this in future works. In this paper, we considered the brane with four-

A General Perturbations
Suppose that a D-dimensional spacetime undergoes a small perturbation δg MN on a fixed background g MNg where ǫ is a real parameter to indicate the degree of perturbation. In light of perturbation theory, the inverse of the perturbed metric is where δg MN = g MP g NQ δg PQ and ellipses correspond to the more higher-order perturbations.
Quantities determined by metric can be obtained accordingly. For the sake of simplicity, one can omit the real parameter ǫ without causing confusion. We list some quantities under the above metric perturbations as follows: where and square brackets "[M| |N]" appearing in indices signify antisymmetrization in terms of M and N. In the region of weak gravitational field, one can decompose the metric into a small perturbation around a flat spacetime.

B Explicit Forms of Perturbations of the Fundamental Quantities
To linear order, the perturbed forms of the fundamental quantities will be collected in this appendix. Firstly, we keep both the zero-order and linear order of perturbed quantities without considering the condition (10). Secondly, we give necessary simplified quantities by introducing the condition (10), which will be used in this paper.
We list perturbed quantities up to the linear order without considering condition (10) in the following. The nonvanishing components of the perturbation of the connection arẽ The nonvanishing components of the perturbation of the Ricci tensor read as ∂ y a a η µν ∂ y h where (4) = η µν ∂ µ ∂ ν and∆ (d) = δ i j ∂ i ∂ j are the d'Alembert operator in the M 4 and the Laplace operator in the E d , respectively. The perturbed curvature scalar is The nonvanishing components of the perturbation of the Einstein tensor arẽ G µν =a 2 3 (∂ y a) 2 a 2 + ∂ y ∂ y a a + d (d − 1) 2 (∂ y b) 2 b 2 + ∂ y b b + 3d ∂ y a∂ y b ab (η µν + h µν ) ∂ y a∂ y b ab ∂ y a∂ y b ab Taking into account the condition (10), we list all components of the curvature scalar, the Ricci tensor, and the Einstein tensor only in the linear order perturbation in the following. All components of the perturbed Ricci tensor are simplified as δR µy =0 , δR µi = 0 , δR yy = 0 , δR yi = 0 , δR i j = 0 , and the perturbed curvature scalar vanishes δR =0 .
It is worth noting that the δ used here merely stands for linear order quantities under perturbations.