The structure of $f(R)$-brane model

Recently, a family of interesting analytical brane solutions were found in $f(R)$ gravity with $f(R)=R+\alpha R^2$ in Ref. [Phys. Lett. B 729, 127 (2014)]. In these solutions, inner brane structure can be turned on by tuning the value of the parameter $\alpha$. In this paper, we investigate how the parameter $\alpha$ affects the localization and the quasilocalization of the tensorial gravitons around these solutions. It is found that, in a range of $\alpha$, despite the brane has an inner structure, there is no graviton resonance. However, in some other regions of the parameter space, although the brane has no internal structure, the effective potential for the graviton KK modes has a singular structure, and there exists a series of graviton resonant modes. The contribution of the massive graviton KK modes to the Newton's law of gravity is discussed shortly.


I. INTRODUCTION
The braneworld scenarios [1][2][3][4][5], have attracted more and more attention. Because they present new insights and solutions for many issues such as the hierarchy problem, the cosmological problem, the nature of dark matter and dark energy, and so on. In some braneworld scenarios, all the fields in the standard model are assumed to be trapped on a submanifold (called brane) in a higher-dimensional spacetime (called bulk), and only gravity transmits in the whole bulk. So a natural and interesting question is how to reproduce the effective fourdimensional Newtonian gravity from a bulk gravity.
In models with compacted extra dimension, such as the ADD braneworld model [2,3], the effective gravity on the brane is transmitted by the zero mode of the discrete Kaluza-Klein (KK) modes of the bulk gravity. Since the zero mode is separated from the first KK excitation by a mass gap, gravity is effectively four-dimensional at low energy. While in models with infinitely large extra dimensions, like the one proposed by Randall and Sundrum (the RS2 model) [5], although the graviton spectrum is gapless now, the effective gravity on the brane is shown to be the Newtonian gravity plus an subdominant correction, so the low energy effective gravity is also four-dimensional. This is because the wave functions of all the massive modes are suppressed on the brane, so, massive modes only contribute a small correction to the * xuzg12@lzu.edu.cn † zhongy2009@lzu.edu.cn ‡ yuh13@lzu.edu.cn § liuyx@lzu.edu.cn, corresponding author Newtonian graivity at low energy scale.
Another class of interesting models with infinite extra dimensions is the so called thick branes. In these modes, the original singular thin brane in the RS2 model is replaced by some smooth thick domain walls generated by one or a few background scalars [6][7][8]. One of the interesting features of thick domain wall brane is that the massive gravitational modes feel an effective volcanolike potential, which might support some resonances. This feature was first noticed in Ref. [6].
Such resonant modes of graviton can be interesting both phenomenlogically and theoretically. Phenomenlogically, the wave function of a graviton resonance peaks at the location of brane and behaviors as a plain wave in the infinity of the extra dimension. As compared to the RS2 model, the massive modes of a thick brane might contribute a different correction to the Newtonian gravity. On the theoretical aspect, metastable massive graviton in thick brane models provides an alternative for massive gravity theory [9,10]. Unfortunately, the early proposal for thick brane [6][7][8] failed in finding massive graviton resonance.
To construct a model that support graviton resonance, one has to tune the shape of the effective potential for the graviton. For typical thick brane models, where the gravity is taken as general relativity, the only possible way is to tune the shape of the warp factor. Some successful models can be found in Refs. [11][12][13]37]. There is another way, however, to tune the effective potential for the graviton if the gravity is described by a more general theory. For example, in f (R) gravity (see [14][15][16] for comprehensive reviews on f (R) gravity and its applications in cosmology), the effective potential is determined by both the warp factor and the form of f (R) (see Ref. [17] for details). So, in principle, graviton reso-nances can be turned on by tuning gravity.
However, the construction of a thick f (R)-brane model is not easy in practice (see Refs. [17][18][19][20][21][22][23][24][25][26][27][28] for works on f (R)-branes). First of all, the dynamical equation in f (R)-gravity is of fourth order. So, traditional methods that help us to find analytical brane solutions in second order systems, such as the superpotential method (also known as the first order formalism [29,30]) do not work in a general f (R)-brane model. Usually one can only solve the system either by using numerical method [22], or by imposing strict constraints on the model, for example, by assuming the scalar curvature to be a constant [19].
The linearization of a f (R)-brane model is also a challenging task. Naively, the linear perturbations around an arbitrary f (R)-brane solution should also satisfy some fourth order differential equations. However, in Ref. [17], the authors found that the tensor perturbation perturbation equation can be finally rewritten as a second order Schrödinger-like equation. The results of Ref. [17] enable us to analyse the graviton mass spectrum of any thick f (R)-branes solutions. For example, in Ref. [23], the authors constructed an analytical thick brane solution in a model with f (R) = R + αR 2 . The solution is stable against the tensor perturbation. The localization of zero modes of both graviton and fermion is also shown to be possible. Unfortunately, no graviton resonance was found in the model of Ref. [23]. Recently, a more general analytical thick f (R)-brane solution was reported in Ref. [28]. The solution of Ref. [28] is a generalization of the one in [23]. The model in Ref. [28] has an interesting feature: inner brane structure appears for a particular range of the parameter α. Usually, the appearance of inner brane structure is accompanied by graviton resonances [11][12][13]37]. So it is interesting to see if it is possible to find graviton resonances in the model of [28], and what is the relation between the inner brane structure and the graviton resonance. These two questions constitute the motivation for the present work.
In the next section II, we briefly review the model and corresponding solutions in Refs. [23] and [28]. Then, in section III, we study the localization of the graviton zero mode and the condition for graviton resonances in the model of [28]. The conclusion and discussions will be given in section IV.

II. REVIEW OF THE f (R)-BRANE MODEL AND SOLUTIONS
We start with the five-dimensional action of the f (R) gravity minimally coupled with a canonical scalar field where f (R) is a function of the scalar curvature R, and κ 2 5 = 8πG 5 with G 5 the five-dimensional Newton constant. In the following, we set κ 2 5 = 2. The signature of the metric is taken as (−, +, +, +, +), and the bulk coordinates are denoted by Capital Latin indices, M, N, · · · = 0, 1, 2, 3, 4, and the brane coordinates are denoted Greek indices, µ, ν, · · · = 0, 1, 2, 3.
We are interested in the static Minkowski brane embedded in a five-dimensional spacetime, so the line element is assumed as where e 2A(y) is the warp factor, y = x 4 stands for the extra dimension, and η µν is the induced metric on the brane. For static brane solutions with the setup (2), the background scalar field φ is only the function of the extra dimension, i.e., φ = φ(y). Therefore, the equations of motion for the f (R)-brane system are where the prime denotes the derivative with respect to y, f R ≡df (R)/dR, and V φ ≡dV (φ)/dφ. In Ref. [23], a toy model with and the φ 4 potential was considered, where λ > 0 is the self coupling constant of the scalar field, and v is the vacuum expectation value of the scalar field. An analytical solution was found in Ref. [23] e A(y) = sech(ky), where the parameters are given by The scalar field satisfies φ(0) = 0 and φ(±∞) = ±v, and the potential reaches the minimum (the vacuum) at φ = ±v. The energy density ρ = T 00 = e 2A ( 1 2 φ ′2 + V (φ)) peaks at the location of the brane, y = 0, and trends to vanish at the boundary of the extra dimension. The brane is embedded in an anti-de Sitter spacetime.
The tensor perturbation of the above brane solution has been analyzed in Ref. [23]. It was shown that the solution is stable against the tensor perturbation and the gravity zero mode is localized on the brane. Furthermore, it was found that although there are fermion resonant KK modes on the brane, there are no graviton resonant modes [23].
Recently, a general solution was constructed in Ref. [28] with the same f (R) given in Eq. (6). The warp factor is assumed as the general form of (8) with two positive parameters B and k: The corresponding Ricci tensor at the boundary of the extra dimension is given by Then from Eq. (4) we can give the derivative of the scalar field: Note that the α here corresponds to −α in Ref. [28].
The energy-momentum tensor density is Solving d 2 ρ , where α 1 < α s < α 2 for finite B. So y = 0 is an inflection point of ρ when α = α s , and the brane will have an internal structure when α ≤ α s . The energy density ρ(y) of the brane system is shown in Fig. 1.
For an arbitral B and a fixed α, the analytical solutions for the scalar field and scalar potential were found in Ref. [28]: when α = α 1 , the solution is when α = α 2 , the result is just the one found in Ref. [23]: Here c i , v i , and λ i are positive parameters determined by B and k. Here, we only list the expressions of v i : Now, it is clear that, when B = 1 and α = α 2 = 3 232k 2 , the exact solution described by Eqs. (12), (20), (21), and (23) is just the one given in Eqs. (6)- (11), for which the brane has no internal structure.

MODES OF THE TENSOR FLUCTUATION
In this section, we investigate the stability of solution against tensor fluctuation of the metric as well as localization of gravity on the brane. Especially, we will find that the effective potential for the KK modes of the tensor fluctuation has a rich structure and it will support some resonant KK modes when B is large enough and α > 0.
We start with the transverse-traceless (TT) tensor per-turbations of the background spacetime: where h µν = h µν (x ρ , y) depends on all the spacetime coordinates and satisfies the TT condition It can be shown that the TT tensor perturbations h µν is decoupled from the scalar and vector perturbations of the metric as well as the perturbation of the scalar field δφ =φ(x ρ , y). The perturbed Einstein equations for the TT tensor perturbations are given by [17] or, equivalently, where a(y) = e A(y) . By making the coordinate transformation dz = a −1 dy, Eq. (27) becomes Then, by doing the decomposition h µν (x ρ , z) where ǫ µν (x ρ ) satisfies the TT condition η µν ǫ µν = 0 = ∂ µ ǫ µ ν , we can get from (28) the equation for the KK modes ψ(z) of the tensor perturbations, which is a Schrödinger-like equation [17]: where the effective potential W (z) is Equation (29) can be factorized as with which indicates that there is no gravitational mode with m 2 < 0. For Eq. (29), the solution of the zero mode with m = 0 is It is easy to show that ψ (0) (z) is normalizable, i.e., which implies that the zero mode is localized on the brane. Here we note that if f R (z) = 1 + 2αR(z) = 0 has solution z = ±z 0 , then the effective potentials W would has singularities at z = ±z 0 and the corresponding zero mode vanishes at that point. There is no analytical expression for the potential W with respect to the coordinate z. However, we can express it in the coordinate y as a(y)a ′2 (y) + a(y) 2 a ′′ (y) a(y) .
From now on, we define dimensionless variablesᾱ ≡ αk 2 , y ≡ ky,z ≡ kz,W ≡ W/k 2 andm ≡ m/k. Thenz is a function of justȳ and Eq. (29) becomes where the effective potential is . (38) Now the parameter k does not appear, which implies that we only need to consider different values of the dimensionless varieties B andᾱ to search for resonant modes. For convenience, we remove the bars on all variables, which is equivalent to let k = 1. So we have .
Note that f R = 1+2αR(y) appears in the denominator of the expression of W (z) (30) or W (z(y)) (36). For general relativity, we have f R = 1 and so the effective potential W (z(y)) is always regular for a smooth solution of the thick brane. However, for the f (R) gravity with f (R) = R + αR 2 , f R may be vanishing and the effective potential will be divergent, i.e., 1+2αR(y) = 1−8αB[−2+(2+5B) tanh 2 (y)] = 0 (42) will lead to the singularities of the effective potential W . This will result in two δ-like potential wells which are related to the appearance of the resonant KK modes of the tensor perturbations. Now we analyze the parameter space of (B, α) where Eq. (42) has solutions. Since 1 + 2αR(y) is an even function of y, we only consider the region y ∈ (0, +∞). We first discuss the case of α > 0. It is clear that 1 + 2αR(y) decreases with y and 1 + 2αR(0) = 1 + 16αB > 0. So the sufficient and necessary condition that Eq. (42) has a solution is 1 + 2αR(y → +∞) = 1 − 40αB 2 < 0. Therefore, the relation between α and B is On the other hand, according to the constrain on α (14), we should have 1 40B 2 < α 2 , which results in B > 2. Next, we discuss the case of α < 0, for which 1 + 2αR(y) increases with y and 1 + 2αR(y → +∞) = 1 − 40αB 2 > 0. So the sufficient and necessary condition that Eq. (42) has a solution is 1 + 2αR(0) = 1 + 16αB ≤ 0. Therefore, the relation between α and B is Furthermore, the constrain on α (14) requires − 1 16B ≥ α 1 , namely B ≤ − 2 5 , which contradicts with B > 0. So Eq. (42) has no solution for negative α. In conclusion, only when the constrain conditions are satisfied, does Eq. (42) have a solution. The relations among α 1 , α s , α k , α 2 are shown in Fig. 2. Because of the monotony of it, f R (the derivative of f (R) with respect of R) is negative in some range of y as long as Eq. (42) has a solution. This will lead to the existence of ghosts. In order to judge whether there are resonant modes, we consider the partner equation of the Schrödinger-like equation (31): K † Kψ(z) = m 2 ψ(z), for which the corresponding potential is given by Similar to W (z), there is no analytical expression for W s (z). In the y coordinate, we have . (47) The structure of W and W s is shown in Fig. 3. Because dz dy = a(y) is positive, the shape of W s (z) is the same with that of W s (z(y)). Thus, in order to estimate whether there are resonant modes we just need to consider the shape of W s (z(y)). The appearance of "quasiwell" of the dual potential may lead to the resonances of the gravitational KK modes. Noticing it is convenient to judge whether there are resonant modes for Eq. (29) from W s . Solving ∂ 2 y W | y=0 = 0 and ∂ 2 y W s | y=0 = 0 respectively results in This implies that both the effective potential and the corresponding dual one have an interval structure in the range of α 1 < α < α ′ and the range of α 1 < α < α ′′ , respectively (see Fig. 3). Furthermore, it can be seen from Fig. 3 that there is also another interesting quasiwell with singularity for large B and α satisfying the constrain conditions (45).
The function |ψ(z)| 2 can be interpreted as the probability of finding the massive KK modes at the position z along extra dimension [38]. In order to find the resonant modes, we definite the relative probability [38] where 2z b is approximately the width of the brane, and z max is taken as z max = 10z b . It is clear that large relative probabilities P (m 2 ) of finding massive KK modes within a narrow range −z b < z < z b around the brane location indicate the existence of resonant modes. From Fig. 3, we can see that there is a quasiwell when α → α 1 , which is consistent with what we analyse. So, it seems that we may find resonant modes for small α.
When α → α 2 and B ≥ 4 which satisfies the condition (45), the effective potential has two δ-like potential wells and there are the resonant modes found.
Firstly, we discuss the case of B ≤ 2. When α approaches to α 1 , the dual potential W s (z) has a quasiwell in the middle, so resonant modes of Eq. (29) may appear. We use the numerical method to search for resonant modes for B ≤ 2 but no resonant mode is found. The reason is that the quasiwell is not deep or wide enough. The potential W (z) for B = 2 is shown in Figs. 4(a) and 4(b). The relations between the relative probability P and the mass square m 2 for B = 2 are shown in Fig. 5.
Secondly, we discuss the case of B > 2. Similar to the case of B ≤ 2, there is a quasiwell in the middle of the dual potential W s (z), but no resonant modes are found when α 1 ≤ α ≤ α k (see Figs. 4(c), 4(d), and Figs. 6(a)-6(d)). When α k < α ≤ α 2 , the potential W (z), which has Z 2 symmetry and is shown in Fig. 4(d), will have two δ-like potential wells. A series of resonant modes come forth (see Figs. 6(e) and 6(f)) and their wave functions are shown in Fig. 7 for B = 4.

IV. CONCLUSION AND DISCUSSION
In this paper, we investigated localization and resonant KK modes of the tensor fluctuation for the f (R)-brane model with f (R) = R + αR 2 . This investigation was based on the interesting analytical brane solution found in Ref. [28], where the warp factor was given by e A(y) = sech B (ky) and the parameter α is constrained as α 1 ≤ α ≤ α 2 (see Eq. (14)). It was found that, when α 1 ≤ α ≤ α s (< 0), the brane has an internal structure, but the effective potential W (z) for the KK modes of the tensor fluctuation has only a simple structure (volcano-like potential with a single or double well) and no resonances were found. The reason is that the quasiwell of the effective potential is not deep enough. This is different from the brane models in general relativity, for which there are usually resonant modes when the brane has an internal structure [11,12,37].
However, the effective potential may have a rich structure with singularities and it will support a series of resonant KK modes when B is large enough (B > 2) and α > 0, although the brane has no inner structure any-more. The reason is that the effective potential for the f (R)-brane model is decided by both the warp factor and the function f (R). It was found that, when B > 2 and 1 40B 2 k 2 < α ≤ α 2 , the effective potential has two δ-like potential wells and there are a series of resonant modes on the brane.
The existence of resonant modes will contribute a correction to the Newtonian gravitational potential, which will be discussed in the future work. But f R is negative in some range so long as the effective potential has singularities because of the monotony of f R . However, this anomaly will result in the appearance of ghosts. So we hope to construct a f (R)-brane model with positive f R and graviton resonances in the future.