# Brane worlds in gravity with auxiliary fields

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## Abstract

Recently, Pani et al. explored a new theory of gravity by adding nondynamical fields, i.e., gravity with auxiliary fields (Phys Rev D 88:121502, 2013). In this gravity theory, higher-order derivatives of matter fields generically appear in the field equations. In this paper we extend this theory to any dimensions and discuss the thick braneworld model in five dimensions. Domain wall solutions are obtained numerically. The stability of the brane system under tensor perturbations is analyzed. We find that the system is stable under tensor perturbations and the gravity zero mode is localized on the brane. Therefore, the four-dimensional Newtonian potential can be realized on the brane.

## Keywords

Gravity Theory Warp Factor Auxiliary Field Domain Wall Solution Brane Model## 1 Introduction

In order to solve the problems such as singularity, nonrenormalizability, dark energy, and dark matter in general relativity, modified gravity theories have been presented and investigated. For recent reviews of modified gravities and related topics, see Refs. [1, 2, 3]. One of such modifications studied extensively is the Palatini extension of the modified gravity. For the Einstein–Hilbert action, the Palatini theory is the same as the original metric theory. But if the action differs from the Einstein–Hilbert action, one usually gets a different gravity theory. Palatini \(f(R)\) [4], Eddington-inspired Born–Infeld (EiBI) [5], and Born–Infeld-\(f(R)\) [6] gravities are well known theories of such a kind. Interestingly, it was found that EiBI gravity is identical to a bimetric version of gravity with an auxiliary field [7]. These theories can be considered as adding auxiliary fields to the action, and they have attracted much attention in recent years. Besides, adding auxiliary fields is always helpful to construct a Lagrangian formalism of some theories.

Recently, Pani et al. [8] suggested a new gravity theory that modifies general relativity by adding nondynamical auxiliary fields. This theory satisfies the weak equivalence principle and the corresponding modified Einstein equations contain higher-order derivatives of the matter fields. Although the details of the nature of the auxiliary fields and the way they enter to the action are unclear, this work provides a generic framework to study the phenomena and observational constrains of the gravity theory with auxiliary fields. This theory is determined by only two parameters up to the next to leading order in the derivative expansion [8]. In some approximations, EiBI and Palatini \(f(R)\) gravities correspond to the special cases of the theory. For gravity with tensor auxiliary fields, see a subsequent note by Ba\(\tilde{\text {n}}\)ados and Cohen [9].

On the other hand, the extra dimension theory gives a new view of our universe, and opens a new way to solve the gauge hierarchy and cosmology problems. One of the famous models of this theory is the Randall–Sundrum (RS) braneworld model [10, 11], which is considered in general relativity. It provides the theoretical predictions of extra dimension effects, which may be detected in future experiments and observations. Along with the progress in the modified gravity and the braneworld models, many efforts have been made considering the braneworld model in modified gravities [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36]. In this paper, we will apply the gravity theory with auxiliary fields to braneworld model and investigate the deformation and stability problems of the brane system. We hope that this work will shed light on future studies of applying certain gravity theory with auxiliary fields to braneworld models.

For a thin brane model, the energy-momentum tensor of the brane is a delta function of the extra dimension. In the original RS model, the field equations are second order and hence the thickness of the brane can be neglected. However, if a gravity theory contains higher-order derivatives of matter fields, such as the gravity theory with auxiliary fields, it is very hard to solve the field equations for a thin brane model. On the other hand, the thick brane model, which can be used to study the inner structure of the brane, is a nature extension of the RS model [12, 13, 14]. The brane configuration is usually generated by a smooth scalar field which connects two nontrivial vacua [15]. So the energy-momentum tensor of the matter field is a smooth function of the extra dimension and it is convenient to apply to the gravity theory with auxiliary fields.

Theoretically, the important problems in a braneworld model include the stability problem and the localization of the massless graviton on the brane, which are essential to recover the effective four-dimensional Newtonian potential on the brane. Experimentally, the interest mainly focuses on the phenomenology of braneworld models, such as the deviation from the Newton potential caused by the massive Kaluza–Klein (KK) gravitons, and the high-energy particle scattering process involving KK particles. However, the spectrum of the gravity KK modes is determined by the brane configuration, which depends on the braneworld model. In this paper, we study the braneworld model in the new gravity theory with auxiliary fields. We find that domain wall solutions are supported in this theory. The brane system is stable under tensor perturbations, and the massless graviton is localized on the brane. Furthermore, we also find some new phenomena that do not appear in general relativity.

## 2 Brane model with auxiliary fields and numerical solutions

Region I: \(c_1^{2}-4c_2c_0<0\), there is no solution.

Region II: \(A''(0)<0\).

Region III: \(A''(0)>0\).

In next section, we mainly focus on the stability of the gravitational perturbation and localization of gravity zero mode, which are two important issues for a brane model that should be investigated before considering their application.

## 3 Tensor perturbation and localization of gravity zero mode

## 4 Conclusion

In this paper, we investigated the thick braneworld model in gravity with auxiliary fields. By numerically studying the model with a kink configuration of the scalar field, we found that the solutions of the brane system with nondeformed and deformed warp factors even though the energy density of the brane does not split. This new phenomenon is connected to the presence of auxiliary fields. We also presented the parameter spaces corresponding to these two types of warp factors.

The tensor perturbation of the background metric of the flat brane system was analyzed. Although the gravity with auxiliary fields we considered in this paper is a fourth-order derivative theory, the linear equation of the tensor perturbation is of second order. The equation of motion of the KK mode of the tensor perturbation was turned into a Schrödinger-like equation. The effective potential \(U(z)\) has the shape of a volcano with one or more potential wells, which do not depend on whether or not the warp factor is deformed. It was shown that the tensor perturbation of the flat brane model is stable. The massless mode of the tensor perturbation is localized on the brane, while the massive modes are continuous and nonlocalized. Therefore, four-dimensional Newton gravity can be realized on the brane.

## Notes

### Acknowledgments

We thank Prof. Y.Q. Wang and Drs. B.M. Gu, X.L. Du, and F.W. Chen for helpful discussions. This work was supported by the National Natural Science Foundation of China (Grant No. 11375075). K. Yang was supported by the Scholarship Award for Excellent Doctoral Student granted by Ministry of Education.

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