Stable Palatini $f(\mathcal{R})$ braneworld

We consider the static domain wall braneworld scenario constructed from the Palatini formalism $f(\mathcal{R})$ theory. We check the self-consistency under scalar perturbations. By using the scalar-tensor formalism we avoid dealing with the higher-order equations. We develop the techniques to deal with the coupled system. We show that under some conditions, the scalar perturbation simply oscillates with time, which guarantees the stability. We also discuss the localization condition of the scalar mode by analyzing the effective potential and the fifth dimensional profile of the scalar mode. We apply these results to an explicit example, and show that only some of the solutions allow for stable scalar perturbations. These stable solutions also give nonlocalizable massless mode. This is important for reproducing a viable four-dimensional gravity.

In the smooth version of warped braneworld models [12][13][14][15][16][17][18], the warping along the fifth dimension is caused by background scalar field, and the extra dimension is infinitely large. So there is no need for stabilization. But the infinite range of the extra dimension also implies the existence of massless radion-like mode, in spite of the existence of bulk scalar field. The behavior of this massless mode is background-dependent. If the massless mode is localized, then it couples to the trace of the energy-momentum tensor of standard model particles. This is unviable since it leads to a long-range interaction. It is really a problem for domain wall braneworld models. This is one of our motivation of this work. We expect to study the scalar mode in the domain wall braneworld model in Palatini f (R) theory, and get the constraints under which the theory is phenomenologically consistent.
In addition to the localization problem, there is another problem. As a kind of toy model that allows for infinitely large extra dimension, the domain wall braneworld models are usually assumed to be static and have four-dimensional Poincaré symmetry, so that the background fields and the spacetime metric depend only on the fifth dimension. However, such kind of static model may not be consistent with time evolution. The model would be unstable if the perturbation grows with time. This is the stability problem of static braneworld models. It is usually referred to the tachyon instability problem since the instability can be described by the four-dimensional mass of the Kaluza-Klein modes. Clearly, physically viable models should be free of tachyon. Thus it is necessary to investigate the stability of the perturbations.
The Palatini f (R) domain wall braneworld model has been considered in previous literature [19,20]. The exact domain wall solutions were obtained, and it was shown that the tensor perturbations are stable [20]. The scalar perturbation remains unclear mostly because of the special structure of Palatini f (R) theory. The theory is assumed to have two independent variables, the spacetime metric and the independent connection. The connection can be eliminated thus one gets a metric theory, but with a modified source part. The complexity of the scalar perturbation mainly comes from this modified source. Such a theory has some special features on cosmology [21][22][23][24][25][26][27][28][29][30][31]. In braneworld scenario, it also has some interesting properties. In the previous braneworld models considered in other gravity theories [32][33][34][35][36], the warp factor decays exponentially at the boundaries of the fifth dimension. This is because the warp factor is related to the localization condition of the massless graviton. Usually, growing warp factor solutions are not allowed since this would give nonlocalizable massless graviton and localizable massless scalar mode. However, the warp factor in Palatini f (R) theory allows for both of decaying and growing solutions, and they all give localizable massless graviton [20]. This may provide some new mechanisms to localize standard model particle fields. In this work, we will deal with the scalar perturbations by using scalar-tensor theory since it is widely accepted that f (R) theory (both of metric formalism and Palatini formalism) has a mathematical equivalence with scalar-tensor theory.
The paper is organized as follows. In section II we give a model setup of the Palatini f (R) theory, and show how to remove the connection dependence. In section III we give the scalar-tensor formalism of the Palatini f (R) theory, and develop the techniques to deal with the perturbations of nonminimally coupled theory. The equations of perturbations are obtained for both of the single field theory and two-field theory. In section IV we analyze the localization problem and the stability against time evolution. At last, we give the conclusions in section V.

II. MODEL SETUP
We start from the general Palatini formalism f (R) theory with a background scalar field χ, where R = g M N R M N (Γ), and the Ricci tensor R M N (Γ) is defined by The connection Γ is not the Christoffel symbol constructed from the spacetime metric, but an independent variable. Note that the source field only couples to the spacetime metric. One can immediately get the field equations for the metric, and for the connection, The covariant derivative ∇ is compatible with the connection Γ. This formula has an analogy to that of general relativity, in which ∇ P g M N = 0 (and ∇ P √ −gg M N = 0). Indeed, the condition (4) allows one to define an auxiliary metric q M N such that The comparison with (4) gives the solution q M N = f 2 D−2 R g M N , which is just a conformal transformation of the spacetime metric. Now it can be easily checked that this auxiliary metric is the one that defines the connection Γ and the covariant derivative ∇. Once again, as we mentioned in previous context, the source field couples to g M N other than q M N . This is somewhat an assumption, however, which implies that the independent connection Γ(q M N ) does not define the spacetime parallel transport since the covariant derivatives in the source part are defined by the Christoffel symbol. In this sense, Palatini f (R) theory is also a metric theory [37]. To be specific, expressing the Ricci scalar R in terms of q M N one gets the transformation where R is the usual curvature scalar defined by the spacetime metric. Thus the field equations (3) and (4) can be combined to get It seems that the field equation still depends on the connection. However, there is a subtlety here. The trace of the equation (3) gives an algebraic relation between R and the trace of the energy-momentum tensor. This implies that R is fully determined once we know the energy-momentum tensor (the source). Thus we see that the right-hand side of (7) is nothing but a modified source. This is actually a general feature of Palatini theories. The well-known Eddington-inspired Born-Infeld theory [35,[38][39][40] has a similar structure, but the formalism is much more complicated after the connection dependence is removed. Since the energy-momentum tensor contains first-order derivatives, the modified source then contains third-order derivatives through R, f R , and f (R). For example, for scalar field we have T ∼ (∂φ) 2 , thus ∇ 2 f R gives ∂ 3 φ∂φ. Note that the term like (∂φ) 2 ∂ 2 g also appears. From this observation we see that the theory may give some special physics.

III. SCALAR PERTURBATIONS IN THICK BRANEWORLD
A. Scalar-tensor formalism Now let us turn to the braneworld perturbations of this theory. The tensor perturbations have been considered in Ref. [20]. The tensor modes satisfy a second-order equation, with a tiny modification from that of general relativity. This can be expected from Eq. (7), in which all of the covariant derivatives on the right-hand side act on scalars. For scalar modes, as we have mentioned in previous section, there are third-order derivatives.
We start from Eq. (7), which can be derived from the action Note that we dropped the total derivative terms. Recall that there is an equivalence between f (R) theory (both of the metric formalism and the Palatini formalism) and the scalar-tensor theory, which cast the higher-order theory into an ordinary second-order theory by introducing an extra scalar field. This scalar field actually describes the extra degree of freedom in theory with higher-order derivatives. For our model (8), it is still a theory with higher-order derivatives if we regard the R (thus f (R) and f R ) terms as functions of the metric and the source T . Now we define φ ≡ f R , then we have where ). This is a theory without higher-order derivatives, but with one more degree of freedom. The field equation is then the equation (7) with f R replaced by φ. The exact background solutions and the tensor perturbations were discussed in [20]. It was shown that the tensor modes are stable under time evolution. The profile along the fifth dimension depends on the background, and there exists a localizable massless graviton. However, the scalar modes remain unclear. We mainly deal with this problem in this work.

B. Scalar perturbations
Since we are considering the thick braneworld model, we require the five-dimensional background fields to have only y dependence so that the four-dimensional Poincaré symmetry is conserved, i.e. φ 0 ≡ φ 0 (y), χ 0 ≡ χ 0 (y). For the scalar perturbations, we will work in the longitudinal gauge, The perturbations of the scalar fields are defined as where the lower index 0 implies the background quantities. The 5µ part and the off-diagonal part of the µν components of the perturbed equation (7) simply give two constraints on the scalar modes, Note that the constraint (12) has the anisotropic contribution coming from the effective energy momentum tensor, which is absent in general relativity. The scalar modes Φ and δφ couple to each other, hence one can only chose to eliminate the scalar modes Ψ and δχ. However, this would lead to coupled perturbation equations. There is a novel technique that can largely simplify these constraints. Let us consider the new variables such that the constraint (12) becomes Here a is a dimensionless parameter. In terms of the new variables, the constraint (13) can be expressed as Now we set a = 1/3, then the constraint becomes Clearly, the constraints (15) and (17) have the same formalism as those obtained in general relativity.

Single field
We first consider the single field case, namely, L(g, χ) = 0. The constraint (17) is simply It can be used to solve the mode Φ and the solution is An intuitive idea is to solve the other one scalar mode δφ with the perturbation equations. However, it can be easily checked that the perturbation equations reduce to be 0 × δφ = m 2 φ 0 (y). This means that it is impossible to solve δφ. Instead, we get a constraint, m 2 φ 0 (y) = 0 with m 2 = (x σ )/ (x σ ). The solution is m 2 = 0 or φ 0 (y) = 0. Using the constraint (14) we show that the solution φ 0 (y) = 0 leads to δφ = 0 and the divergence of Φ, which is unviable, hence we have m 2 = 0 and φ 0 (y) = 0. This implies that there exists only a massless Kaluza-Klein mode. By considering the background equations we show that the only solution is This is the model of general relativity with a cosmological constant, which is exactly the Randall-Sundrum model [41] if we insert a thin brane at the origin. The scalar perturbation Φ is sort of a radion-like mode. There is no need to stabilize this mode since the extra dimension is infinitely large. One can easily show that this massless mode cannot be localized. So there is no extra long-range force contributing to the four-dimensional gravity. Recall that the single field case corresponds to the Palatini f (R) theory without source. Hence, we conclude that it is impossible to get a thick braneworld model in pure geometric Palatini f (R) theory, and there are only thin brane solutions in this case.

Two fields
For general theory (8), the scalar χ plays the role of source. As can be seen from the constraints (15) and (17), we are not able to solve δφ in terms of Φ and δχ in this case. So the only choice of variables to be eliminated is ( Ψ, δχ). Varying the quadratic order of the action (9) with respect to Ψ and δφ, and replacing the scalar modes Φ and Ψ with Φ and Ψ, we get the perturbation equations The coefficients P i , Q i , and S i are listed in the Appendix section. We find that there is only one independent scalar mode in this theory. Clearly, δφ has the same mass spectrum with Φ, so we only need to discuss Φ. It satisfies It has a Shrödinger-like formalism of equation in coordinate r, The coordinate r is defined by dy = ζdr. The variable η is defined by the decomposition Φ(x σ , r) = (x σ )γ(r)η(r) in such a way that the first order derivative term can be eliminated. Note that we used the relation (x σ ) = m 2 (x σ ). There are two reasons why we write this formalism equation. First, we want to know how does the background affect the mass spectrum, and the Shrödinger-like equation helps. Second, this formalism implies that the quantity υ = (x σ )η(r) is the canonically normalized scalar mode. In other words, the quadratic order perturbation of action (9) is simply The expressions of γ(r) and ζ(r) are Here we have a constraint for the background solutions, (S 3 − S 2 )(S 3 − 4S 2 ) > 0. If it is negative then the coordinate transformation from y to r does not hold. In fact, there would be a gradient instability if ζ(r) is imaginary, so this is not viable. Now we can write the effective potential in Eq. (24) as

IV. STABILITY PROBLEM AND LOCALIZATION PROBLEM
We have mentioned the stability problem in the introduction section. Now let us explain this problem in detail. Recall that we used the relation (x σ ) = m 2 (x σ ) in Eq. (24). It was obtained by considering the fourier expansion in momentum space and using p µ p µ = −m 2 . Clearly, m plays the role of the four-dimensional mass of the scalar perturbation mode (x σ ), and the spectrum is determined by the background spacetime through (27). This is actually an assumption of plane wave nature for the scalar mode (x σ ). However, it is really a problem whether it is consistent to assume that the scalar mode oscillates with time. The background system would be unstable if the scalar mode has nontrivial time evolution like growing or damping solutions. This instability can be described by the sign of m 2 . If there exist perturbation modes with m 2 < 0, then we would have an imaginary frequency ω = m 2 + p 2 , which gives rise to instability and destroys the static background. Therefore, to get a consistent model it is necessary to have nonnegative m 2 . We see that this is actually a tachyon instability problem.
For tensor modes, it is straightforward to know that the effective potential supports a nonnegative m 2 , since the perturbation equation can be factorized as Q † Qh µν = m 2 h µν [20] for the theory with f R > 0. In some other models considered in the previous literature [14,15,42], the effective potential for the scalar mode can also be factorized. However, for our case (27), it is not clear whether it can be factorized for general Palatini f (R) theory. If it can be factorized, then we have Comparing this with Eq. (27), we get If this equation has regular solution for λ(r), then we can factorize the Shrödinger-like equation to be which obviously has the formalism Q † Qη = m 2 η and implies m 2 ≥ 0. It also gives the solution for the massless scalar mode by The localization condition of the massless scalar mode is |η 0 | 2 dr = |η 0 / ζ| 2 dy < ∞.
We see that it is the mode η 0 / √ ζ that describes the localization condition in y coordinate. There is another possibility which also gives nonnegative m 2 . Note that Eq. (27) can be written as then the operator on the left-hand side gives a positive definite eigenvalue, i.e. m 2 > 0. This is also a possibility that avoids the tachyon instability problem. Note that there is no massless scalar mode in this case. Now let us apply the above discussion to the exact solution for Palatini f (R) braneworld model given by reference [20]. The model is f (R) = R + αR 2 , which is a simple modification to general relativity, and the modification is described by the parameter α. The solutions for the background quantities are χ(y) = 2n(6n − 1) 3(3n + 2)(n − 1)κ 5 dy sech 4 (ky) + 2sech 2 (ky).
To make the background solutions consistent, the parameter n is restricted to be n < −2/3 or 0 < n < 1/6 or n > 1.
This transformation gives the coordinate r = dyζ(y). We give the plot of r(y) in Fig. 1. Note that if ζ is imaginary then we would have negative coefficient for Φ and υ, which leads to gradient instability. To make the transformation regular, and to avoid gradient instability, we require ζ to be real. This requirement rules out the solution with n < −2/3. Further more, we have 3(n−1) (ky) (n − 1)(cosh(2ky) + 2)((9n + 11) cosh(2ky) + 30n + 10) .
For 0 < n < 1/6, the quantity (39) is negative and it may give negative m 2 . Let us check this by considering the effective potential V eff directly. The lengthy expression of V eff is given in the Appendix section. Note that we write it in y coordinate, which does not affect our analyses. Clearly, for 0 < n < 1/6, the potential blows up. We have V eff < 0 at the origin y = 0 (r = 0). In fact, since the transformation function defined in (26) diverges, the new coordinate r has a finite range. It implies that V eff (r) is actually an infinitely high potential well. That means all of the scalar modes, including the massless mode, are localized. This would contribute a long-range force to the gravity, and cause a violation to the observed four-dimensional Newtonian gravity, which is unacceptable. Hence the solution with 0 < n < 1/6 should also be ruled out. We give a typical plot of V eff (r) and the numerical solution of η 0 / √ ζ corresponding to the massless mode in Fig. 2. Obviously, such a profile is localizable. The nodes reveal the existence of lower states, the tachyon modes. So the solutions with 0 < n < 1/6 should also be ruled out. This result also implies that the effective potential may not be able to be factorized into the form (30), since if it can be factorized then there are no tachyon states.
For n > 1, the effective potential is positive definite. There is no scalar mode with m 2 < 0 in this case. Therefore, the model is stable under scalar perturbations. About the massless mode, we failed to get the analytic solution. But it is straightforward to conclude that it cannot be localized since the potential is positive definite. Hence there is no violation to the four-dimensional gravity. The plot of the potential and the numerical solution of η 0 / √ ζ corresponding to the massless mode are given in Fig. 2. As can be seen, the massless mode diverges at infinity so it cannot be localized.
The above analysis shows that the model should be constrained if scalar perturbations are considered. This is one of our goal of this work. The previous work is not enough to constrain the model. For the solutions given in [20], the constraints are n > 1 and ζ 2 > 0. respectively. For n = 1/7, V eff (r) is an infinitely high potential well, and it has boundary since r has finite range. The plot of η0/ √ ζ reveals that the massless mode η0 has nodes, so it is not the lowest state, which implies the existence of tachyon. For n = 5/3, the potential is positive definite. The mode η0/ √ ζ cannot be normalized.

V. CONCLUSIONS
To summarize, we studied the scalar perturbations of thick braneworld model in Palatini f (R) theory. By taking the advantages of scalar-tensor theory, we avoided dealing with the perturbations with third-order derivatives. For a pure geometric theory, we showed that it is impossible to get a smooth version of braneworld model. This is contrary to the metric f (R) theory [43][44][45][46][47]. For the theory with a source field, we used some techniques for the constraints, which can largely simplify the perturbation equations. In this case, there is only one independent scalar perturbation mode, although we have a nonminimally coupled scalar field φ and a source scalar field χ. Essentially, this is because the extra scalar degree of freedom is just an auxiliary field which is not dynamically independent. This can also be understood from the fact that the Palatini f (R) theory corresponds to the Brans-Dicke theory with the Brans-Dicke parameter ω BD = 4/3 in five dimensions, in which the nonminimally coupled scalar does not have its own dynamics [37]. So there is only one independent scalar degree of freedom in this system. We also analyzed the stability problem and the localization problem. For a general theory, we failed to get a factorized formalism of perturbation equation. If it can be factorized into a formalism like Q † Qη = m 2 η, then we can conclude that there are no tachyon states, thus the scalar perturbation would be stable under time evolution, which makes the static system consistent. As an explicit example, we analyzed some exact background solutions given in the previous literature. We showed that only the solutions with n > 1 are stable and the other solutions should be ruled out if scalar perturbation is considered. The corresponding massless mode cannot be localized, which guarantees the recovering of four-dimensional gravity. This gives some further constraints on the original Palatini f (R) theory.
It is interesting to consider general scalar-tensor theory. There are more degrees of freedom, so there are more independent perturbation equations. Obviously, the scalar modes couple to each other. The braneworld models with multiple scalar fields has been considered in reference [48][49][50][51]. It is interesting to consider the theory with nonminimally coupled scalar field, and find out the conditions under which the theory is viable, and that will be our future work.