# Tensor Perturbations and Thick Branes in Higher Dimensional Gauss-Bonnet
Gravity

Yongqun Xu  Department of Physics, South China University of Technology,
Guangzhou 510641, China  Xiangdong Zhang111Corresponding author.
<EMAIL_ADDRESS>Department of Physics, South China University of
Technology, Guangzhou 510641, China

###### Abstract

A thick brane model with Gauss-Bonnet action in a homogeneous anisotropic
$D=(4+1+d)$ spacetime is studied. By choosing a concrete metric ansatz, we
show that this spacetime is stable against linear tensor perturbations under
certain conditions. The graviton zero modes are also given. Besides, we
examine a particular example in six-dimensional spacetime with a given warp
factor. The coupled background scalar field and its potential are solved
analytically. Furthermore, the effective potential of the Kaluza-Klein modes
of the graviton is also discussed. We found that the effective potential can
have singularities under certain conditions, which are related to the non-
differentiability of the graviton zero modes.

##  I Introduction

General Relativity (GR) is the most successful gravitational theory which has
been proposed for over one century. Though GR has been tested by many
experiments, there are still a few unknown issues such as dark energy and dark
matter that remain a mystery which indicates GR is not perfect yet. These
issues strongly motivate us to consider the gravity theory beyond GR. During
the past several decades, many schemes of modified gravity have been put
forwardclifton_modified_2012 . Among all the potential directions, the extra
dimension scenario is attracted increasing attention. Historically, the very
first extra-dimensional gravity model is the so-called Kaluza-Klein (KK)
theory which aims to unify Einstein’s gravity and Maxwell’s electromagnetism
by introducing an extra compact circle spacial dimension. The radius scale is
small enough that it can not be detected. Moreover, extra dimensions are also
preferred by the string theory.

On the other hand, the extra dimension could also be very large and spread
infinitely. This possibility has been realized in the brane world model
langlois_brane_2002 ; liu_introduction_2017 . An outstanding advantage of the
brane world scenario is that it could nicely solve the hierarchy problem
arkani-hamed_hierarchy_1998 ; randall_large_1999 ; yang_gravity_2012 ;
barbosa-cendejas_mass_2014 ; guo_localization_2013 ; hundi_fermion_2011 .
Nowadays, most of the related works on brane world model are carried out in
five-dimensional spacetime. However, in five dimensions, the pure
gravitational trapping mechanism of vector fields is still problematic and
there is no remarkable proposal to solve the fermion mass hierarchy in the
Standard Model yet. To overcome these mentioned issues, one may resort to
higher dimensional brane world models. Particularly, brane solutions of a
higher dimensional $f(R)$ gravity with a real scalar field has been proposed
recently cui_tensor_2020 ; zhong_tensor_2011 ; gu_full_2017 ; zhong_pure_2016
; dzhunushaliev_thick_2020 ; PhysRevD.92.024011 ; cui_linear_2018 ;
gu_stable_2018 . Note that in the $f(R)$ brane world model, one usually needs
to solve higher derivative equations. One way to avoid this issue is to
consider a specific higher-order polynomial combination of the Riemann tensor
such that the field equation remains two derivatives. The resulted simplest
option is the famous Gauss-Bonnet gravity. The Gauss-Bonnet term is
particularly suitable for higher dimensional extension of GR because it is a
purely topological term in four dimensions and does not contribute to the
classical equation of motion. However, this term becomes dynamic in higher
dimensions. Besides, such quadratic Gauss-Bonnet terms of curvatures also
appears as a 1-loop correction of heterotic string theory GW86 . Hence
inspired by these works, in this paper, we pursue to investigate the Gauss-
Bonnet brane world model. Related works of Gauss-Bonnet gravity in brane world
model can be also seen in the literaturegiovannini_thick_2001 .

Moreover, it is well known that the stability of thick branes is a very
significant issue since brane systems should be stable at least under linear
perturbations. This issue has been extensively investigated cui_tensor_2020 ;
zhong_tensor_2011 ; giovannini_thick_2001 ; giovannini_gauge-invariant_2001 ;
giovannini_curvature_2005 ; zhong_pure_2016 ; PhysRevD.92.024011 ;
cui_linear_2018 ; gu_stable_2018 . In addition, to match the gravitational
experiments, the four-dimensional Newtonian potential should be recovered,
which means that the zero modes of the graviton should be localized on the
brane. Hence, we also devoted to studying the stability of tensor
perturbations and the zero modes of the graviton of Gauss-Bonnet brane world
model in this paper.

This paper is organized as follows: After introduction, we write down the
D-dimensional Gauss-Bonnet action in the Section II, and then we give a
particular anisotropic metric with two warp factors. The field equations under
this metric ansatz are also given. In Section III, we mainly focus on the
linear stability by computing the tensor perturbation equations. With these
equations in hand, we solve the field equations analytically with a given
scalar field and discuss the main property of the solution in Section IV. Some
necessary technical details can be found in the Appendix A.

Through out the paper we adopt the following symbol conventions: We use the
lowercase Latin letters $a,b,c,...$ running over $0,1,2,3,4,5...$ to stand for
the indices of the $D(=4+1+d)$-dimensional spacetime. The lowercase Greek
letters $\mu,\nu,\lambda,\sigma:0,1,2,3$ represent our four-dimensional
Minkowski spacetime $\mathcal{M}_{4}$ coordinates. Specially we use $x^{4}=y$
as the fifth extra spacial coordinate $\mathcal{R}_{1}$. The capital Latin
$A,B,C...:5,6,7...$ live in the higher extra dimensional Euclidean space
$\mathcal{E}_{d}$ with $d=1,2,...$. Moreover, through out the paper, we use
the definition
$R_{abc}{}^{d}=\partial_{b}\Gamma^{d}{}_{ac}-\partial_{a}\Gamma^{d}{}_{bc}+\Gamma^{e}{}_{ca}\Gamma^{d}{}_{be}-\Gamma^{e}{}_{cb}\Gamma^{d}{}_{ae}$
and the spacetime signature $(-,+,+...)$.

##  II Action, Field equation And Metric

Our discussion in this article is based on the following D-dimensional
Einstein-Hilbert action with a Gauss-Bonnet term
$\mathcal{L}_{GB}=R^{2}-4R_{ab}R^{ab}+R_{abcd}R^{abcd}$ as

$\displaystyle S=\frac{1}{2\kappa_{D}^{2}}\int
d^{D}x\sqrt{-g}\left(R+\alpha\mathcal{L}_{GB}\right)+S_{m},$ (1)

where $g=\det g_{ab}$, $\kappa^{2}_{D}=8\pi G^{(D)}$ with the $G^{(D)}$ being
the gravitional constant in D-dimensional spacetime, and $\alpha$ is the
Guass-Bonnet coupling constant which has dimension of $[lengh]^{2}$. By using
this action, we can directly derive the field equations by varing the action
with respect to the metric tensor $g_{ab}$:

$\displaystyle G_{ab}+\alpha H_{ab}=\kappa_{D}^{2}T_{ab},$ (2)

where
$T_{ab}=-\dfrac{2}{\sqrt{-g}}\dfrac{\partial\left(\sqrt{-g}\mathcal{L}_{m}\right)}{\partial
g^{ab}}$ is the energy-momentum tensor, and the lovelock tensor $H_{ab}$ is
coming from the contribution of the Gauss-Bonnet term clifton_modified_2012 :

$\displaystyle
H_{ab}=-4R_{a}{}^{c}R_{bc}+2R_{ab}R-4R^{cd}R_{acbd}+2R_{a}{}^{cde}R_{bcde}-\frac{1}{2}g_{ab}\mathcal{L}_{GB}.$
(3)

Parallel to the previous study such as in $f(R)$ brane world model
cui_tensor_2020 . In this paper, we also consider a special anisotropic
$(D=4+1+d)$-dimensional bulk manifold
$\mathcal{M}=\mathcal{M}_{4}\times\mathcal{R}_{1}\times\mathcal{E}_{d}$, which
is constructed by three subspaces:

  * •

A 4-dimentional Minkowski spacetime $\mathcal{M}_{4}$,

  * •

An extra 1-dimensional noncompact space $\mathcal{R}_{1}$,

  * •

An extra d-dimensional Euclidean space $\mathcal{E}_{d}$.

Naturally, there should exist curved brane (like $\mathrm{d}S$ and
$A\mathrm{d}S$ brane) as well as flat brane. In this paper, we are
particularly interested in a flat and static brane, of which the line element
can be written as:

$\displaystyle\mathrm{d}s^{2}=a^{2}(y)\eta_{\mu\nu}\mathrm{d}x^{\mu}\mathrm{d}x^{\nu}+\mathrm{d}y^{2}+b^{2}(y)\delta_{AB}\mathrm{d}x^{A}\mathrm{d}x^{B}$
(4)

where $a(y)=\mathrm{e}^{A(y)}$ and $b(y)=\mathrm{e}^{B(y)}$ are two warped
factors. The Gauss-Bonnet term in this metric can be computed directly as

$\displaystyle\mathcal{L}_{GB}=$ $\displaystyle\frac{24a^{\prime
4}}{a^{4}}+\frac{96a^{\prime
2}a^{\prime\prime}}{a^{3}}+d\left(\frac{48a^{\prime
2}b^{\prime\prime}}{a^{2}b}+\frac{96a^{\prime
3}b^{\prime}}{a^{3}b}+\frac{96a^{\prime}a^{\prime\prime}b^{\prime}}{a^{2}b}+(d-1)\left(\frac{16a^{\prime\prime}b^{\prime
2}}{ab^{2}}\right.\right.$ (5) $\displaystyle\left.\left.+\frac{72a^{\prime
2}b^{\prime
2}}{a^{2}b^{2}}+\frac{32a^{\prime}b^{\prime}b^{\prime\prime}}{ab^{2}}+(d-2)\left(\frac{16a^{\prime}b^{\prime
3}}{ab^{3}}+\frac{4b^{\prime 2}b^{\prime\prime}}{b^{3}}+(d-3)\frac{b^{\prime
4}}{b^{4}}\right)\right)\right)$

Some higher order terms will arise as $d$ grows larger till $d=4$. Moreover,
the field equations under this metric ansatz are:

$\displaystyle\frac{3a^{\prime\prime}}{a}+\frac{3a^{\prime
2}}{a^{2}}+d\left(\frac{3a^{\prime}b^{\prime}}{ab}+\frac{b^{\prime\prime}}{b}-\frac{b^{\prime
2}}{2b^{2}}\right)+\frac{d^{2}b^{\prime
2}}{2b^{2}}-\alpha\left[\frac{12a^{\prime
2}a^{\prime\prime}}{a^{3}}-d\left(\frac{6a^{\prime\prime}b^{\prime
2}}{ab^{2}}-\frac{12a^{\prime
2}b^{\prime\prime}}{a^{2}b}-\frac{12a^{\prime}b^{\prime
3}}{ab^{3}}\right.\right.$ $\displaystyle\left.+\frac{18a^{\prime 2}b^{\prime
2}}{a^{2}b^{2}}-\frac{12a^{\prime
3}b^{\prime}}{a^{3}b}+\frac{12a^{\prime}b^{\prime}b^{\prime\prime}}{ab^{2}}-\frac{24a^{\prime}a^{\prime\prime}b^{\prime}}{a^{2}b}+\frac{3b^{\prime
4}}{b^{4}}-\frac{4b^{\prime
2}b^{\prime\prime}}{b^{3}}\right)-d^{2}\left(\frac{6b^{\prime
2}b^{\prime\prime}}{b^{3}}-\frac{18a^{\prime 2}b^{\prime
2}}{a^{2}b^{2}}-\frac{11b^{\prime 4}}{2b^{4}}\right.$
$\displaystyle\left.\left.+\frac{18a^{\prime}b^{\prime
3}}{ab^{3}}-\frac{6a^{\prime\prime}b^{\prime
2}}{ab^{2}}-\frac{12a^{\prime}b^{\prime}b^{\prime\prime}}{ab^{2}}\right)+d^{3}\left(\frac{6a^{\prime}b^{\prime
3}}{ab^{3}}-\frac{3b^{\prime 4}}{b^{4}}+\frac{2b^{\prime
2}b^{\prime\prime}}{b^{3}}\right)+\frac{d^{4}b^{\prime
4}}{2b^{4}}\right]=\frac{\kappa_{D}^{2}\eta^{\mu\nu}T_{\mu\nu}}{4a^{2}}$ (6)
$\displaystyle\frac{6a^{\prime
2}}{a^{2}}+d\left(\frac{4a^{\prime}b^{\prime}}{ab}-\frac{b^{\prime
2}}{2b^{2}}\right)+d^{2}\frac{b^{\prime
2}}{2b^{2}}+\alpha\left[-\frac{12a^{\prime 4}}{a^{4}}-\frac{d^{4}b^{\prime
4}}{2b^{4}}+d\left(-\frac{16a^{\prime}b^{\prime 3}}{ab^{3}}+\frac{36a^{\prime
2}b^{\prime 2}}{a^{2}b^{2}}-\frac{48a^{\prime
3}b^{\prime}}{a^{3}b}\right.\right.$ $\displaystyle\left.+\frac{3b^{\prime
4}}{b^{4}}\right)+d^{2}\left(\frac{24a^{\prime}b^{\prime
3}}{ab^{3}}-\frac{36a^{\prime 2}b^{\prime 2}}{a^{2}b^{2}}-\frac{11b^{\prime
4}}{2b^{4}}\right)\left.+d^{3}\left(\frac{3b^{\prime
4}}{b^{4}}-\frac{8a^{\prime}b^{\prime
3}}{ab^{3}}\right)\right]={\kappa_{D}^{2}}{T_{yy}}$ (7)
$\displaystyle\frac{b^{\prime
2}}{b^{2}}-\frac{b^{\prime\prime}}{b}+\frac{6a^{\prime
2}}{a^{2}}+\frac{4a^{\prime\prime}}{a}-\frac{4a^{\prime}b^{\prime}}{ab}+d\left(\frac{4a^{\prime}b^{\prime}}{ab}+\frac{b^{\prime\prime}}{b}-\frac{3b^{\prime
2}}{2b^{2}}\right)+d^{2}\frac{b^{\prime
2}}{2b^{2}}+\alpha\left[\frac{24a^{\prime
2}b^{\prime\prime}}{a^{2}b}-\frac{16a^{\prime\prime}b^{\prime
2}}{ab^{2}}\right.$ $\displaystyle+\frac{48a^{\prime
3}b^{\prime}}{a^{3}b}-\frac{72a^{\prime 2}b^{\prime
2}}{a^{2}b^{2}}+\frac{48a^{\prime}b^{\prime
3}}{ab^{3}}-\frac{32a^{\prime}b^{\prime}b^{\prime\prime}}{ab^{2}}-\frac{12a^{\prime
4}}{a^{4}}+\frac{48a^{\prime}a^{\prime\prime}b^{\prime}}{a^{2}b}+d\left(\frac{24a^{\prime\prime}b^{\prime
2}}{ab^{2}}-\frac{24a^{\prime
2}b^{\prime\prime}}{a^{2}b}-\frac{88a^{\prime}b^{\prime 3}}{ab^{3}}\right.$
$\displaystyle\left.+\frac{108a^{\prime 2}b^{\prime
2}}{a^{2}b^{2}}-\frac{48a^{\prime
3}b^{\prime}}{a^{3}b}+\frac{48a^{\prime}b^{\prime}b^{\prime\prime}}{ab^{2}}-\frac{48a^{\prime}a^{\prime\prime}b^{\prime}}{a^{2}b}+\frac{23b^{\prime
4}}{b^{4}}-\frac{22b^{\prime 2}b^{\prime\prime}}{b^{3}}\right)$
$\displaystyle+d^{2}\left(-\frac{8a^{\prime\prime}b^{\prime
2}}{ab^{2}}+\frac{48a^{\prime}b^{\prime 3}}{ab^{3}}-\frac{36a^{\prime
2}b^{\prime
2}}{a^{2}b^{2}}-\frac{16a^{\prime}b^{\prime}b^{\prime\prime}}{ab^{2}}-\frac{35b^{\prime
4}}{2b^{4}}+\frac{12b^{\prime 2}b^{\prime\prime}}{b^{3}}\right)$
$\displaystyle\left.-\frac{10b^{\prime 4}}{b^{4}}+\frac{12b^{\prime
2}b^{\prime\prime}}{b^{3}}-\frac{48a^{\prime
2}a^{\prime\prime}}{a^{3}}-\frac{d^{4}b^{\prime
4}}{2b^{4}}+d^{3}\left(-\frac{8a^{\prime}b^{\prime
3}}{ab^{3}}+\frac{5b^{\prime 4}}{b^{4}}-\frac{2b^{\prime
2}b^{\prime\prime}}{b^{3}}\right)\right]=\frac{\kappa_{D}^{2}\eta^{AB}}{db^{2}}{T_{AB}},$
(8)

where the prime ′ denotes the derivative with respect to the extra-dimensional
coordinate $y$. With a given matter distribution we can solve the equations
above. But before we proceed, we should wonder whether this set of equations
gives a reasonable solution, or under what condition they do so. To this aim,
we need to first analyze the behavior under linear perturbation carefully.

##  III Linear Stability Under Tensor Perturbation

The perturbed equation can be very complicated for such a Gauss-Bonnet
action(1) and bulk metric (4). The perturbed tensor equations of motion can be
formally written as:

$\displaystyle\delta G_{ab}+\alpha\delta H_{ab}=\kappa_{D}^{2}\delta T_{ab}.$
(9)

Usually, a linear perturbation can be decomposed into scalar, vector and
tensor parts. Since these three perturbations are decoupled to each other.
Hence here for simplicity, we just let alone the others and just focus on the
tensor perturbation. Note that the tensor perturbation can impose various
kinds of gauge conditions. Here we consider the most popular one which is the
so-called the transverse-traceless(TT) gauge condition:

$\displaystyle\mathrm{d}s^{2}=a^{2}(y)(\eta_{\mu\nu}+h_{\mu\nu})\mathrm{d}x^{\mu}\mathrm{d}x^{\nu}+\mathrm{d}y^{2}+b^{2}(y)\delta_{ij}\mathrm{d}x^{i}\mathrm{d}x^{j},$
(10)

where $h_{\mu\nu}$ obeys the TT gauge condition:

$\displaystyle\partial_{\mu}h_{\nu}^{\mu},\quad h=\eta^{\mu\nu}h_{\mu\nu}=0.$
(11)

The tensor perturbation of $f(R)$ brane-world model with TT gauge is carried
out in cui_tensor_2020 ; zhong_tensor_2011 . Similarly we compute the
perturbed quantities step by step, from the perturbed Christoffel connection
to the Riemann Curvature. The non-vanishing perturbed Christoffel connection
are listed below:

$\displaystyle\delta\Gamma^{\lambda}{}_{\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(\partial_{\mu}h_{\nu}^{\lambda}+\partial_{\nu}h_{\mu}^{\lambda}-\partial^{\lambda}h_{\mu\nu})$
$\displaystyle\delta\Gamma^{y}{}_{\mu\nu}$ $\displaystyle=$ $\displaystyle-
aa^{\prime}h_{\mu\nu}-\frac{a^{2}}{2}\partial_{y}h_{\mu\nu}$
$\displaystyle\delta\Gamma^{A}{}_{\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\frac{a^{2}}{b^{2}}\partial^{A}h_{\mu\nu}$
$\displaystyle\delta\Gamma^{\lambda}{}_{\mu y}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\partial_{y}h_{\mu}^{\lambda}$
$\displaystyle\delta\Gamma^{\lambda}{}_{\mu A}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\partial_{A}h_{\mu}^{\lambda}$ (12)

By directly using the formula
$\delta{R_{abc}}^{d}=\nabla_{b}\delta{\Gamma^{d}}_{ac}-\nabla_{a}\delta{\Gamma^{d}}_{bc}$
, one can computed all of the needed perturbation quantities which can be
found in Appendix A. For example, the pertured Ricci tensor can be written as
:

$\displaystyle\delta
R_{\mu\nu}=-\frac{a^{2}}{2}\left[\left(\frac{4a^{\prime}}{a}+\frac{db^{\prime}}{b}\right)\partial_{y}+\left(\partial_{y}^{2}+\frac{\square^{(4)}}{a^{2}}+\frac{\hat{\Delta}^{(d)}}{b^{2}}\right)+2\left(\frac{3a^{\prime
2}}{a^{2}}+\frac{a^{\prime\prime}}{a}+\frac{da^{\prime}b^{\prime}}{ab}\right)\right]h_{\mu\nu},$
(13)

where
$\square^{(4)}=\eta^{\mu\nu}\partial_{\mu}\partial_{\nu},\quad\hat{\Delta}^{(d)}=\delta^{ij}\partial_{i}\partial_{j}$
are the d’Alembert and Laplace operator of Minkowski space $\mathcal{M}_{4}$
and extra d-dimensional spacial Euclidean space $\mathcal{E}_{d}$
respectively. Finally, after tedious but straightforward computation, we end
up with $(\mu,\nu)$ components of the perturbed field equations:

$\displaystyle\left[-\frac{1}{2a^{2}}+\alpha\left(\frac{2a^{\prime\prime}}{a^{3}}+d\left(\frac{2a^{\prime}b^{\prime}}{a^{3}b}+\frac{2b^{\prime\prime}}{a^{2}b}-\frac{b^{\prime
2}}{a^{2}b^{2}}\right)+\frac{d^{2}b^{\prime
2}}{a^{2}b^{2}}\right)\right]\square^{(4)}h_{\mu\nu}$
$\displaystyle-\left[\frac{1}{2b^{2}}-\alpha\left(\frac{4a^{\prime\prime}}{ab^{2}}-\frac{4a^{\prime}b^{\prime}}{ab^{3}}+\frac{2a^{\prime
2}}{a^{2}b^{2}}-\frac{2b^{\prime\prime}}{b^{3}}+\frac{2b^{\prime
2}}{b^{4}}+d\left(\frac{4a^{\prime}b^{\prime}}{ab^{3}}+\frac{2b^{\prime\prime}}{b^{3}}-\frac{3b^{\prime
2}}{b^{4}}\right)+d^{2}\frac{b^{\prime
2}}{b^{4}}\right)\right]\hat{\Delta}^{(d)}h_{\mu\nu}$
$\displaystyle+\left[-\frac{2a^{\prime}}{a}-\frac{db^{\prime}}{2b}+\alpha\left(\frac{4a^{\prime}a^{\prime\prime}}{a^{2}}+\frac{4a^{\prime
3}}{a^{3}}+d\left(\frac{4a^{\prime\prime}b^{\prime}}{ab}+\frac{4a^{\prime}b^{\prime\prime}}{ab}-\frac{8a^{\prime}b^{\prime
2}}{ab^{2}}+\frac{14a^{\prime 2}b^{\prime}}{a^{2}b}+\frac{2b^{\prime
3}}{b^{3}}-\frac{2b^{\prime}b^{\prime\prime}}{b^{2}}\right)\right.\right.$
$\displaystyle\left.\left.+d^{2}\left(\frac{8a^{\prime}b^{\prime
2}}{ab^{2}}-\frac{3b^{\prime
3}}{b^{3}}+\frac{2b^{\prime}b^{\prime\prime}}{b^{2}}\right)+\frac{d^{3}b^{\prime
3}}{b^{3}}\right)\right]h^{\prime}_{\mu\nu}$
$\displaystyle+\left[-\frac{1}{2}+\alpha\left(\frac{2a^{\prime
2}}{a^{2}}+d\left(\frac{4a^{\prime}b^{\prime}}{ab}-\frac{b^{\prime
2}}{b^{2}}\right)+\frac{d^{2}b^{\prime
2}}{b^{2}}\right)\right]h^{\prime\prime}_{\mu\nu}=0.$ (14)

We need to add that, the $h_{\mu\nu}$ term of Eq.(III) has been eliminated
with the consideration of the field equation(II). We will apply some
mathematical methods to get some physical results from the equation above.
First of all, we introduce a coordinate transformation
$\mathrm{d}y=a\mathrm{d}z,\ \mathrm{d}x^{A}=\dfrac{a}{b}\mathrm{d}w^{A}$ to
obtain a conformal-flat metric:

$\displaystyle\mathrm{d}s^{2}=a^{2}(z)(\eta_{\mu\nu}\mathrm{d}x^{\mu}\mathrm{d}x^{\nu}+\mathrm{d}z^{2}+\delta_{AB}\mathrm{d}x^{A}\mathrm{d}x^{B}).$
(15)

Then the perturbation equation (III) in such conformal flat coordinate reads:

$\displaystyle\left[\frac{1}{2a^{2}}-\alpha\left(-\frac{2a^{\prime
2}}{a^{6}}+\frac{2a^{\prime\prime}}{a^{5}}+d\left(-\frac{b^{\prime
2}}{a^{4}b^{2}}+\frac{2b^{\prime\prime}}{a^{4}b}\right)+d^{2}\frac{b^{\prime
2}}{a^{4}b^{2}}\right)\right]\square^{(4)}h_{\mu\nu}+\left[\frac{1}{2a^{2}}-\alpha\left(\frac{4a^{\prime\prime}}{a^{5}}\right.\right.$
$\displaystyle\left.\left.-\frac{2a^{\prime
2}}{a^{6}}-\frac{2a^{\prime}b^{\prime}}{a^{5}b}+\frac{2b^{\prime
2}}{a^{4}b^{2}}-\frac{2b^{\prime\prime}}{a^{4}b}+d\left(\frac{2a^{\prime}b^{\prime}}{a^{5}b}-\frac{3b^{\prime
2}}{a^{4}b^{2}}+\frac{2b^{\prime\prime}}{a^{4}b}\right)+\frac{d^{2}b^{\prime
2}}{a^{4}b^{2}}\right)\right]\hat{\Delta}^{(d)}h_{\mu\nu}+\left[\frac{3a^{\prime}}{2a^{3}}\right.$
$\displaystyle\left.+\frac{db^{\prime}}{2a^{2}b}-\alpha\left(-\frac{2a^{\prime
3}}{a^{7}}+\frac{4a^{\prime}a^{\prime\prime}}{a^{6}}+d\left(\frac{2a^{\prime
2}b^{\prime}}{a^{6}b}-\frac{5a^{\prime}b^{\prime
2}}{a^{5}b^{2}}+\frac{2b^{\prime
3}}{a^{4}b^{3}}+\frac{4b^{\prime}a^{\prime\prime}}{a^{5}b}+\frac{4a^{\prime}b^{\prime\prime}}{a^{5}b}-\frac{2b^{\prime}b^{\prime\prime}}{a^{4}b^{2}}\right)\right.\right.$
$\displaystyle\left.\left.+d^{2}\left(\frac{5a^{\prime}b^{\prime
2}}{a^{5}b^{2}}-\frac{3b^{\prime
3}}{a^{4}b^{3}}+\frac{2b^{\prime}b^{\prime\prime}}{a^{4}b^{2}}\right)+\frac{d^{3}b^{\prime
3}}{a^{4}b^{3}}\right)\right]h^{\prime}_{\mu\nu}$
$\displaystyle+\left[\frac{1}{2a^{2}}-\alpha\left(\frac{2a^{\prime
2}}{a^{6}}+d\left(\frac{4a^{\prime}b^{\prime}}{a^{5}b}-\frac{b^{\prime
2}}{a^{4}b^{2}}\right)+\frac{d^{2}b^{\prime
2}}{a^{4}b^{2}}\right)\right]h^{\prime\prime}_{\mu\nu}=0.$ (16)

This is a second-order linearized tensor equation that can be abstractly
written as follows:

$\displaystyle
J(z)\square^{(4)}h_{\mu\nu}+K(z)\hat{\Delta}^{(d)}h_{\mu\nu}+2P(z)\partial_{z}h_{\mu\nu}+Q(z)\partial_{z}^{2}h_{\mu\nu}=0$
(17)

where $J,K,P,Q$ are functions of coordinate $z$ defined by Eq.(III).
Naturally, are we want to solve the equation, we are willing to decompose it
into three different parts respectively:

$\displaystyle h_{\mu\nu}=\epsilon_{\mu\nu}(x^{\mu})\Phi(z)\xi(x_{A}).$ (18)

We can firstly obtain the $(\mu,\nu)$ part, which is a Klein-Gordon equation:

$\displaystyle(\square^{(4)}-m^{2})\epsilon_{\mu\nu}(x^{\mu})=0$ (19)

and the Helmholtz equation for the extra d-dimentions:

$\displaystyle\left(\hat{\Delta}^{(d)}+l^{2}\right)\xi(x_{A})=0.$ (20)

What we left behind is a second-order ordinary differential equations for the
extra fifth dimension:

$\displaystyle\left(2\frac{P}{Q}\partial_{z}+\partial_{z}^{2}\right)\Phi(z)=\left(\frac{K}{Q}l^{2}-\frac{J}{Q}m^{2}\right)\Phi(z).$
(21)

We can always eliminate the one-order derivative $\partial_{z}$ term with one
more factor $\Phi(z)=\psi(z)\exp(-\int P/Q\mathrm{d}z)$, one can check that
the Eq.(21) will be turned into a Shrodinger-like equationgu_full_2017 :

$\displaystyle\left(-\partial_{z}^{2}+U(z)\right)\psi(z)=\left(\frac{J}{Q}m^{2}-\frac{K}{Q}l^{2}\right)\psi(z)$
(22)

where $U(z)$ is the effective potential defined as

$\displaystyle
U(z)=\left(\frac{P}{Q}\right)^{2}+\partial_{z}\left(\frac{P}{Q}\right).$ (23)

After that, we we can introduce a superpential $W(x)=-P/Q$ and re-written the
Hamiltonian of Eq.(22) into two operators $A^{\dagger}=\partial_{z}+W$ and
$A=-\partial_{z}+W$ as what we do in supersymmetric quantum mechanics
cooper_supersymmetry_1995 :

$\displaystyle
A^{\dagger}A\psi(z)=\left(\frac{J}{Q}m^{2}-\frac{K}{Q}l^{2}\right)\psi(z).$
(24)

With all this process, we can finally give the condition that makes sure the
stability under linear perturbation. It immediately follows that we must have
$m^{2}\geq 0$ and $l^{2}\geq 0$ in Eq.(19) and Eq.(20) for stability. And more
importantly, we must have the graviton zero mode normalizable to avoid tachyon
states.

Setting $m^{2}=l^{2}=0$ in Eq.(22), or $A\psi(z)=0$, at least, we can obtain
the ground state, or the graviton zero modes of this system:

$\displaystyle\psi_{0}(z)\propto\exp\int\frac{P}{Q}\mathrm{d}z.$ (25)

With the help of these equations, we can set several restrictions for the
perturbation. In the following, we will show the whole procedure through a
specific example in the next section.

##  IV Brane Solution In 6d Spacetime

After perturbation analysis, we now turn to search for a particular brane
solution. We choose $d=1$ as a non-trivial example. In 6-dimensions or the
case $d=1$, the line element reads

$\displaystyle\mathrm{d}s^{2}=a^{2}(y)\eta_{\mu\nu}\mathrm{d}x^{\mu}\mathrm{d}x^{\nu}+\mathrm{d}y^{2}+L^{2}b^{2}(y)\mathrm{d}\theta^{2}.$
(26)

where $\theta\in[0,2\pi)$ is the coordinate of the sixth dimension which is
compact.

Consider a background scalar field $\phi(y)$ which is only the function of $y$
for a static flat brane, with the following action of the matter:

$\displaystyle
S_{m}=\int\mathrm{d}^{6}x\sqrt{-g}\left(-\frac{1}{2}\partial^{a}\phi\partial_{a}\phi-V(\phi)\right),$
(27)

where $V(\phi)$ is the self-interacting potential. Set $d=1$ in Eq.
(II),(II),(II), the components of the field equations can be written as
follows :

$\displaystyle\frac{3a^{\prime}b^{\prime}}{ab}+\frac{3a^{\prime
2}}{a^{2}}+\frac{3a^{\prime\prime}}{a}+\frac{b^{\prime\prime}}{b}-12\alpha\left(\frac{a^{\prime
2}b^{\prime\prime}}{a^{2}b}+\frac{2a^{\prime}a^{\prime\prime}b^{\prime}}{a^{2}b}+\frac{a^{\prime
3}b^{\prime}}{a^{3}b}+\frac{a^{\prime
2}a^{\prime\prime}}{a^{3}}\right)=-\kappa_{6}^{2}\left[V(\phi)+\frac{1}{2}\phi^{\prime
2}\right]$ $\displaystyle\frac{4a^{\prime}b^{\prime}}{ab}+\frac{6a^{\prime
2}}{a^{2}}-12\alpha\left(\frac{4a^{\prime
3}b^{\prime}}{a^{3}b}+\frac{a^{\prime
4}}{a^{4}}\right)=\kappa_{6}^{2}\left[\frac{1}{2}\phi^{\prime
2}-V(\phi)\right]$ $\displaystyle\frac{4a^{\prime\prime}}{a}+\frac{6a^{\prime
2}}{a^{2}}-12\alpha\left(\frac{4a^{\prime
2}a^{\prime\prime}}{a^{3}}+\frac{a^{\prime
4}}{a^{4}}\right)=-\kappa_{6}^{2}\left[\frac{1}{2}\phi^{\prime
2}+V(\phi)\right]$ (28)

Moreover, the equation of motion for the background scalar field is:

$\displaystyle\phi^{\prime\prime}+\left(4\frac{a^{\prime}}{a}+\frac{b^{\prime}}{b}\right)\phi^{\prime}-\frac{\partial
V}{\partial\phi}=0.$ (29)

In order to solve these equations, we give linearly combinations from Eq. (IV)

$\displaystyle\alpha\left(\frac{48a^{\prime
2}a^{\prime\prime}}{a^{3}}-\frac{48a^{\prime
3}b^{\prime}}{a^{3}b}\right)+\frac{4a^{\prime}b^{\prime}}{ab}-\frac{4a^{\prime\prime}}{a}=\kappa_{6}^{2}\phi^{\prime
2}$ $\displaystyle\alpha\left(\frac{12a^{\prime
2}b^{\prime\prime}}{a^{2}b}+\frac{24a^{\prime}a^{\prime\prime}b^{\prime}}{a^{2}b}-\frac{36a^{\prime
3}b^{\prime}}{a^{3}b}+\frac{12a^{\prime
2}a^{\prime\prime}}{a^{3}}-\frac{12a^{\prime
4}}{a^{4}}\right)+\frac{a^{\prime}b^{\prime}}{ab}+\frac{3a^{\prime
2}}{a^{2}}-\frac{3a^{\prime\prime}}{a}-\frac{b^{\prime\prime}}{b}=\kappa_{6}^{2}\phi^{\prime
2}$ $\displaystyle\alpha\left(\frac{12a^{\prime
2}b^{\prime\prime}}{a^{2}b}+\frac{24a^{\prime}a^{\prime\prime}b^{\prime}}{a^{2}b}+\frac{12a^{\prime
3}b^{\prime}}{a^{3}b}-\frac{36a^{\prime
2}a^{\prime\prime}}{a^{3}}-\frac{12a^{\prime
4}}{a^{4}}\right)-\frac{3a^{\prime}b^{\prime}}{ab}+\frac{3a^{\prime
2}}{a^{2}}+\frac{a^{\prime\prime}}{a}+\frac{b^{\prime\prime}}{b}=0.$ (30)

Then the main perturbation equation for this case reads:

$\displaystyle\hat{\Delta}^{(d)}h_{\mu\nu}\left(\frac{1}{2L^{2}b^{2}}-\alpha\left(\frac{2a^{\prime
2}}{a^{2}b^{2}}+\frac{4a^{\prime\prime}}{ab^{2}}\right)\right)+\square^{(d)}h_{\mu\nu}\left(\frac{1}{2a^{2}}-\alpha\left(\frac{2a^{\prime}b^{\prime}}{a^{3}b}+\frac{2a^{\prime\prime}}{a^{3}}+\frac{2b^{\prime\prime}}{a^{2}b}\right)\right)$
$\displaystyle+h_{\mu\nu}^{\prime}\left(\frac{2a^{\prime}}{a}+\frac{b^{\prime}}{2b}-\alpha\left(\frac{4a^{\prime
3}}{a^{3}}+\frac{14a^{\prime
2}b^{\prime}}{a^{2}b}+\frac{4a^{\prime}a^{\prime\prime}}{a^{2}}+\frac{4b^{\prime}a^{\prime\prime}}{ab}+\frac{4a^{\prime}b^{\prime\prime}}{ab}\right)\right)$
$\displaystyle+h_{\mu\nu}^{\prime\prime}\left(\frac{1}{2}-\alpha\left(\frac{2a^{\prime
2}}{a^{2}}+\frac{4a^{\prime}b^{\prime}}{ab}\right)\right)=0.$ (31)

We choose a particular smooth function for $a(y)$ and $b(y)$ which is also
widely adopted in the literature liu_introduction_2017 ; gu_full_2017 ;
yu_gravitational_2016 ; cui_tensor_2020 ; zhong_tensor_2011 ; guo_tensor_2016
; yang_thick_2012 ; csaki_universal_2000 ; liu_domain_2011 ; zhong_pure_2016 ;
PhysRevD.92.024011 ; cui_linear_2018  to solve and analyze the brane. Here we
assume that

$\displaystyle a(y)=b(y)=\mathrm{sech}^{n}(ky),$ (32)

with $n>0$, so that the metric(4) can be reduced to an $A\mathrm{d}S$ at
infinity. While the parameter $k>0$ was set to adjust the thickness of the
brane. Substitute Eq.(32) into Eq.(IV), we will get the only one admissable
equation as:

$\displaystyle\kappa_{6}^{2}\phi^{\prime 2}=48\alpha
k^{4}n^{3}\text{sech}^{4}(ky)+4k^{2}n\left(1-12\alpha
k^{2}n^{2}\right)\text{sech}^{2}(ky)$ (33)

Note that it is easy to check that the ordinary differential equation :

$\displaystyle\kappa
f^{\prime}(y)=\pm\sqrt{a_{1}(1-a_{2})\text{sech}^{4}(ky)+a_{1}a_{2}\text{sech}^{2}(ky)}$
(34)

has the solution:

$\displaystyle f(y)=\pm\frac{\sqrt{a_{1}}}{\kappa
k}\left(i\left[E(iky|a_{2})-F(iky|a_{2})\right]+\tanh(ky)\sqrt{a_{2}\sinh^{2}(ky)+1}\right),$
(35)

where $F$ and $E$ are the first and second elliptic integrals respectively.
Compare the above equation with the Eq. (33) we have

$\displaystyle a_{1}$ $\displaystyle=4k^{2}n,\quad a_{2}=1-12k^{2}n^{2}\alpha$
(36) $\displaystyle\phi(y)$ $\displaystyle=\pm
2\frac{\sqrt{n}}{\kappa_{6}}\left(\mathrm{i}[E(iky|p)-F(iky|p)]+\tanh(ky)\sqrt{a_{2}\sinh^{2}(ky)+1}\right).$
(37)

Moreover, with the help of Eq.(29) to obtain the potential of the scalar field
as:

$\displaystyle
V(\phi(y))=2\frac{k^{2}n}{\kappa_{6}^{2}}[(5n+1)a_{2}\mathrm{sech}^{2}(ky)+6\alpha
k^{2}n^{2}(5n+2)\mathrm{sech}^{4}(ky)].$ (38)

Since the background scalar field $\phi(y)$ should be real-valued, it proposes
a restriction for the parameters in the ellipse integral, which means
$a_{1}>0,\ a_{2}\geq 0$ or :

$\displaystyle\frac{1}{12n^{2}}\geq k^{2}\alpha$ (39)

The feasible region of the Gauss-Bonnet coupling constant $\alpha$ is shown in
Fig.1

Figure 1: The real-value condition for the background scalar field sets a
restriction for the Gauss-Bonnet coupling constant $\alpha$. Unlike $f(R)$
brane, the parameter $\bar{\alpha}$ in this context can be sufficiently large
for small $n$.

Figure 2: The background scalar field $\bar{\phi}(\bar{y})$.

Figure 3: The potential $\bar{V}(\bar{\phi})$.

To make things clear, we need to introduce the following dimensionless
quantities:

$\displaystyle\bar{y}=ky,\qquad\bar{\alpha}=k^{2}\alpha,\qquad\bar{\phi}=\kappa_{6}\phi,\qquad\bar{V}=\kappa_{6}^{2}V/k^{2}.$
(40)

Furthermore, the scalar field $\phi(y)$ and its potential $V(\phi)$ are shown
in Fig.3 and Fig.3 respectively. The parameter $n$ is set to 1 in these two
figures.

From Fig. 3, we should notice that the scalar field $\bar{\phi}(\bar{y})$ has
two kinks for small $\bar{\alpha}$ and only one kink for large $\bar{\alpha}$.
So there must be a critical value of $\bar{\alpha}$ which
$\bar{\phi}(\bar{y})$ will have one more kink once $\alpha$ exceed.

This value can be found by solving the following equations:

$\displaystyle\left.\frac{\mathrm{d}^{3}\bar{\phi}}{\mathrm{d}\bar{y}^{3}}\right|_{\bar{y}=0}=0\quad\Rightarrow\quad\bar{\alpha}_{c}=-\frac{1}{12n^{2}}.$
(41)

Besides, it is remarkable that $\bar{\alpha}$ has the upper bound
$\bar{\alpha}=1/12n^{2}$, which leads to a simpler and concise solution, and
are displayed in Fig. 5 and Fig. 5:

$\displaystyle\bar{\phi}(\bar{y})=\pm 2\sqrt{n}\tanh(\bar{y})$ (42)
$\displaystyle\bar{V}(\bar{\phi})=\frac{(5n+2)}{16n}\left(\bar{\phi}^{2}-4n\right)^{2}.$
(43)

Figure 4: The scalar field $\bar{\phi}(\bar{y})$ when
$\bar{\alpha}=1/12n^{2}$.

Figure 5: The potential $\bar{V}(\bar{\phi})$when $\bar{\alpha}=1/12n^{2}$.

Finally, we also want to explore the distribution of the energy density
$\rho=T_{ab}U^{a}U^{b}$ along the extra dimension $y$, where
$U^{M}=(1/a(y),0,0,0,0,0)$. We introduce the dimensionless energy density:

$\displaystyle\bar{\rho}=\frac{\kappa_{6}^{2}}{k_{6}^{2}}\rho=\left[(2n+5n^{2}-24n^{3}\alpha-60n^{4}\alpha)\cosh
2\bar{y}+(2n+5n^{2}+24n^{3}\alpha)\right]\text{sech}^{4}\bar{y}.$ (44)

Figure 6: $\bar{\rho}(\bar{y},\bar{\alpha})$ with $n=1/10$.

Figure 7: The energy density $\bar{\rho}(\bar{y})$ will be split into two
different peaks once $\alpha$ is less than $\alpha_{s}$.

A lot of information can be obtained from the Eq.(44). First, we had better to
stress that, the energy density is always positive definite under the
condition (39). Second, the energy density behaves differently in various
regions of the parameter space of $n$ and $\bar{\alpha}$. Third, the peak of
energy density will increase fastly when $\alpha<0$. Importantly, we notice
that the energy density of the brane will be split into two different peaks as
the value of $\alpha$ decreases. This behavior is displayed in Fig. 7 and Fig.
7. The critical value of $\alpha$ is determined by:

$\displaystyle\left.\frac{\mathrm{d}^{2}\bar{\rho}}{\mathrm{d}\bar{y}^{2}}\right|_{\bar{y}=0}=0\quad\Rightarrow\quad\alpha_{s}=-\frac{2+5n}{24n^{2}}.$
(45)

The split of the brane strongly implies that the brane may have some fine
inner structure, which may have some physical results. It is worth stressing
that, various kinds of reasons will result in a splitting brane, for example,
a split brane caused by spacetime torsion is investigated in yang_thick_2012
and caused by a couple of multiple background scalar fields is study in
bazeia_bloch_2004 .

##  V Effective Potential

Now we going to discuss the effective potential. With the given warped factor
(32), we are able to explore the effective potential (23). The results are
depicted in Fig. 9 and Fig. 9. As a comparison, the effective potential of GR
is drawn with a thin line in Fig. 9, we found that the well goes deeper as
$\bar{\alpha}$ increases. By examining the second derivative with respect to
$\bar{z}$, we found that the potential may split into two wells due to the
Gauss-Bonnet correction term as shown in Fig. 9. In Fig. 9, a strange
potential with two singularities is shown, compared to a normal smooth well.
Note that although in the above section, we only discuss the brane with $d=1$,
the procedure can actually be carried out for general dimensions. Hence, the
singularities in D-dimensions (D=4+1+d) are determined by the following
equation:

$\displaystyle\xi^{3}=2(1+d)(2+d)n^{2}\alpha(\xi-1)\xi^{1+n},\qquad\xi=1+z^{2}\geq
1.$ (46)

So we can always find two symmetric singularities when $n>1,\bar{\alpha}>0$.

Figure 8: The picture of dimensionless effective potential
$\bar{U{\bar{z}}}=U(z)/k^{2},\bar{z}=kz$. A double well caused by Guass-Bonnet
correction term when $\bar{\alpha}^{2}$ is larger.

Figure 9: Singularities of the potential may appear for positive
$\bar{\alpha}$ value when $n>1$.

The singularities of the potential should be accused of the non-
differentiability of the graviton zero mode $\psi_{0}(\bar{z})$, from which we
can see that they can still be normalized. The split effective potential is
related to a graviton mode with two different peaks. These behaviors are shown
in Fig. 10.

Figure 10: Different possible graviton zero modes. This plot shows the
graviton zero modes with one and two peaks, and a special case with non-
differentiability at $\bar{z}=\pm 1$.

##  VI Conclusions

In summary, we investigate the Gauss-Bonnet thick-brane model in this paper.
The behavior of this system under linear tensor perturbation with TT gauge is
investigated in detail in this paper. The tensor perturbation equation can be
reduced into the Klein-Gordon equation, Helmholz equation, and a second-order
linear ordinary differential equation (Schr$\ddot{\mathrm{o}}$dinger-like
equation). We conclude that the system is stable in tensor perturbation under
certain conditions. Besides, the graviton zero mode is given.

Furthermore, with a coupling background scalar field and given warp factors,
an analytical brane solution in six-dimensional spacetime was constructed.
With which we can set a restriction for the Gauss-Bonnet coupling constant in
the bulk action. The solution can be simplified a step further. In this case,
we claim that the energy density will be split into two peaks with some
particular parameter value. The critical value of the coupling constant is
given to determine whether it will be split. At the end of the paper, we
investigate the effective potential of the Schr$\ddot{\mathrm{o}}$dinger-like
equation with respect to various parameters. The potential has a double-well
at the bottom which is generated by the Gauss-Bonnet term, compared to one
simple well of GR. Moreover, we found that the singularities of the potential
should be accused of the non-differentiability of the graviton zero mode
$\psi_{0}(\bar{z})$, and the split effective potential is related to a
graviton mode with two different peaks.

###### Acknowledgements.

This work is supported by NSFC with Grants No.11775082.

##  Appendix A Perturbational Quantities

What we want to obtain is the $(\mu,\nu)$ part of the perturbed field
equation(9). We choose a method with brute force to evaluate. Theoretically,
we can first find out the non-vanishing perturbed Christoffel connection, and
then use it to construct the perturbed Riemann curvature tensor. But we do it
inversely, we start from the needed quantities like the perturbed Ricci
tensor, we might firstly decompose the contraction $\delta R_{\mu\nu}=\delta
R_{\mu a\nu}{}^{a}$ into:

$\displaystyle\delta R_{\mu\nu}=\delta R_{\mu\sigma\nu}{}^{\sigma}+\delta
R_{\mu y\nu}{}^{y}+\delta R_{\mu A\nu}{}^{A}.$ (47)

It is the same for $H_{\mu\nu}$, but we need to expand it into more basic
quantities firstly:

$\displaystyle\delta H_{\mu\nu}$ $\displaystyle=-\delta(4R_{\mu}{}^{c}R_{\nu
c})+\delta(2R_{\mu\nu}R)-\delta(4R^{cd}R_{\mu c\nu
d})+\delta(2R_{\mu}{}^{cde}R_{\nu cde})$ (48)
$\displaystyle-\frac{a^{2}}{2}\mathcal{L}_{GB}h_{\mu\nu}-\frac{a^{2}}{2}\eta_{\mu\nu}\delta\mathcal{L}_{GB}.$

And naturally, we may continue to expand the first four terms:

$\displaystyle\delta(-4R_{\mu}{}^{c}R_{\nu c})$ $\displaystyle=-4(\delta
g^{cd}R_{\mu d}R_{\nu c}+g^{cd}\delta R_{\mu d}R_{\nu c}+g^{cd}R_{\mu d}\delta
R_{\nu c})$ (49)
$\displaystyle=-4(-\frac{h^{\sigma\rho}}{a^{2}}R_{\mu\rho}R_{\nu\sigma}+g^{\sigma\rho}\delta
R_{\mu\rho}R_{\nu\sigma}+g^{\sigma\rho}R_{\mu\rho}\delta R_{\nu\sigma})$
$\displaystyle=4\frac{f^{2}}{a^{2}}h_{\mu\nu}-8\frac{f}{a^{2}}\delta
R_{\mu\nu}$ $\displaystyle\delta(2R_{\mu\nu}R)=2(R\delta
R_{\mu\nu}+R_{\mu\nu}\delta R)$ (50) $\displaystyle\delta(-4R^{cd}R_{\mu c\nu
d})$ $\displaystyle=-4(\delta g^{cf}R_{fd}R_{\mu c\nu}{}^{d}+g^{cf}\delta
R_{fd}R_{\mu c\nu}{}^{d}+g^{cf}R_{fd}\delta R_{\mu c\nu}{}^{d})$
$\displaystyle=$ $\displaystyle-4\left(-\frac{a^{\prime
2}}{a^{2}}fh_{\mu\nu}+\frac{a^{\prime 2}}{a^{2}}\delta
R_{\mu\nu}+\frac{f}{a^{2}}\delta R_{\mu\sigma\nu}{}^{\sigma}+R_{yy}\delta
R_{\mu y\nu}{}^{y}+\frac{s}{b^{2}}\delta R_{\mu A\nu}{}^{A}\right)$ (51)
$\displaystyle\delta(2R_{\mu}{}^{cde}R_{\nu cde})$ $\displaystyle=2(\delta
g^{df}R_{\mu cf}{}^{e}R_{de\nu}{}^{c}+g^{df}\delta R_{\mu
cf}{}^{e}R_{de\nu}{{}^{c}}+g^{df}R_{\mu cf}{}^{e}\delta R_{de\nu}{}^{c})$ (52)
$\displaystyle=2(\delta g^{\sigma\rho}R_{\mu c\rho}{{}^{e}}R_{\sigma
e\nu}{{}^{c}}+R^{f}{}_{e\nu}{}^{c}\delta R_{\mu cf}{{}^{e}}+R_{\mu
c}{{}^{de}\delta R_{de\nu}{}^{c}}).$

The three terms in (52) can be expanded as follow:

$\displaystyle\delta g^{\sigma\rho}R_{\mu c\rho}{{}^{e}}R_{\sigma
e\nu}{{}^{c}}=$ $\displaystyle-\left(\frac{a^{\prime
4}}{a^{4}}+d\left(\frac{a^{\prime}b^{\prime}}{ab}\right)^{2}+\frac{a^{\prime\prime
2}}{a^{2}}\right)a^{2}h_{\mu\nu}$ (53) $\displaystyle R^{f}{}_{eb}{}^{c}\delta
R_{acf}{{}^{e}}=$ $\displaystyle-\frac{a^{\prime 2}}{a^{2}}\left(\delta
R_{\mu\sigma\nu}{}^{\sigma}-\eta_{\varepsilon\nu}\eta^{\lambda\sigma}\delta
R_{\mu\sigma\lambda}{}^{\varepsilon}\right)-\frac{a^{\prime\prime}}{a}\delta
R_{\mu y\nu}{}^{y}+aa^{\prime\prime}\eta_{\varepsilon\nu}\eta^{yy}\delta
R_{\mu 44}{}^{\varepsilon}$ (55)
$\displaystyle+\frac{a^{2}}{b^{2}}\frac{a^{\prime}b^{\prime}}{ab}\eta_{\varepsilon\nu}\eta^{AB}\delta
R_{\mu AB}{}^{\varepsilon}-\frac{a^{\prime}b^{\prime}}{ab}\delta R_{\mu
E\nu}{}^{E}$ $\displaystyle R_{\mu c}{{}^{de}\delta R_{deb}{}^{c}}$
$\displaystyle=-2\left(\frac{a^{\prime 2}}{a^{2}}\delta
R_{\mu\sigma\nu}{}^{\sigma}+\frac{a^{\prime\prime}}{a}\delta R_{\mu
4\nu}{}^{4}+\frac{a^{\prime}b^{\prime}}{ab}\delta R_{\mu A\nu}{}^{A}\right).$
(56)

Some basic elements, like $\delta R_{\mu
a\nu}{}^{b},\eta_{\varepsilon\nu}\eta^{ab}\delta R_{\mu ab}{}^{\varepsilon}$,
appear repeatedly all the way, they can be decomposed into forms like
$\nabla_{a}\delta\Gamma^{b}{}_{cd}$. After expanding them into coordinate
basis, all we need to do is comparing with the quantities given in (III).

$\displaystyle\nabla_{\sigma}\delta\Gamma^{\sigma}{}_{\mu\nu}=-\frac{1}{2}\partial_{\sigma}\partial^{\sigma}h_{\mu\nu}+a^{\prime}a\partial_{y}h_{\mu\nu}+4\frac{a^{\prime}}{a}\delta\Gamma^{y}{}_{\mu\nu}$
(57)
$\displaystyle\nabla_{y}\delta\Gamma^{y}{}_{\mu\nu}=\partial_{y}\delta\Gamma^{y}{}_{\mu\nu}-2\frac{a^{\prime}}{a}\delta\Gamma^{y}{}_{\mu\nu}$
(58)
$\displaystyle\nabla_{A}\delta\Gamma^{A}{}_{\mu\nu}=\frac{1}{2}\frac{a^{2}}{b^{2}}\partial_{A}\partial^{A}h_{\mu\nu}+d\frac{a^{\prime}}{a}\delta\Gamma^{y}{}_{\mu\nu}$
(59)
$\displaystyle\nabla_{\mu}\delta\Gamma^{\sigma}{}_{\sigma\nu}=-\Gamma^{a}{}_{\mu\sigma}\delta\Gamma^{\sigma}{}_{a\nu}+\Gamma^{\sigma}{}_{\mu
a}\delta\Gamma^{a}{}_{\sigma\nu}=-a^{\prime 2}h_{\mu\nu}$ (60)
$\displaystyle\nabla_{\mu}\delta\Gamma^{y}{}_{y\nu}=a^{\prime 2}h_{\mu\nu}$
(61) $\displaystyle\nabla_{\mu}\delta\Gamma^{A}{}_{A\nu}=0.$ (62)

Combining the quantities listed above to construct the perturbed Riemann
curvature:

$\displaystyle\delta
R_{\mu\sigma\nu}{}^{\sigma}=-\frac{1}{2}\partial_{\sigma}\partial^{\sigma}h_{\mu\nu}+a^{\prime}a\partial_{y}h_{\mu\nu}+4\frac{a^{\prime}}{a}\delta\Gamma^{y}{}_{\mu\nu}+a^{\prime
2}h_{\mu\nu}$ (63) $\displaystyle\delta R_{\mu
y\nu}{}^{y}=\partial_{y}\delta\Gamma^{y}{}_{\mu\nu}-2\frac{a^{\prime}}{a}\delta\Gamma^{y}{}_{\mu\nu}-a^{\prime
2}h_{\mu\nu}$ (64) $\displaystyle\delta R_{\mu
A\nu}{}^{A}=\frac{1}{2}\frac{a^{2}}{b^{2}}\partial_{A}\partial^{A}h_{\mu\nu}+d\frac{a^{\prime}}{a}\delta\Gamma^{y}{}_{\mu\nu}$
(65)

And with the same process:

$\displaystyle\nabla_{A}\delta\Gamma^{\varepsilon}{}_{\mu
B}=\frac{1}{2}\partial_{A}\partial_{B}h_{\mu}^{\varepsilon}+\frac{1}{2}\eta_{AB}b^{\prime}b\partial_{y}h_{\mu}^{\varepsilon}$
(66) $\displaystyle\nabla_{y}\delta\Gamma^{\varepsilon}{}_{\mu
y}=\frac{1}{2}\partial_{y}\partial_{y}h_{\mu}^{\varepsilon}$ (67)
$\displaystyle\nabla_{\sigma}\delta\Gamma^{\varepsilon}{}_{\mu\lambda}=\partial_{\sigma}\delta\Gamma^{\varepsilon}{}_{\mu\lambda}+\frac{1}{2}a^{\prime}a\eta_{\sigma\mu}\partial_{y}h_{\lambda}^{\varepsilon}+\frac{1}{2}a^{\prime}a\eta_{\sigma\lambda}\partial_{y}h_{\mu}^{\varepsilon}+\frac{a^{\prime}}{a}\delta^{\varepsilon}_{\sigma}\delta\Gamma^{y}{}_{\mu\lambda}$
(68) $\displaystyle\nabla_{\mu}\delta\Gamma^{\varepsilon}{}_{AB}=0$ (69)
$\displaystyle\nabla_{\mu}\delta\Gamma^{\varepsilon}{}_{yy}=-\frac{a^{\prime}}{a}\partial_{y}h_{\mu}^{\varepsilon}$
(70)
$\displaystyle\nabla_{\mu}\delta\Gamma^{\varepsilon}{}_{\sigma\lambda}=\partial_{\mu}\delta\Gamma^{\varepsilon}{}_{\sigma\lambda}+\frac{1}{2}a^{\prime}a\eta_{\sigma\mu}\partial_{y}h_{\lambda}^{\varepsilon}+\frac{1}{2}a^{\prime}a\eta_{\mu\lambda}\partial_{y}h_{\sigma}^{\varepsilon}+\frac{a^{\prime}}{a}\delta^{\varepsilon}_{\mu}\delta\Gamma^{y}{}_{\sigma\lambda}.$
(71)

We can get another set of perturbed Riemann Curvatue:

$\displaystyle\eta_{\varepsilon\nu}\eta^{AB}\delta
R_{\mu\nu}{}^{A}=\frac{1}{2}\partial_{A}\partial^{A}h_{\mu\nu}+\frac{1}{2}db^{\prime}b\partial_{y}h_{\mu\nu}$
(72) $\displaystyle\eta_{\varepsilon\nu}\eta^{44}\delta
R_{\mu\nu}{}^{y}=\frac{1}{2}\partial_{y}\partial^{y}h_{\mu\nu}+\frac{a^{\prime}}{a}\partial_{y}h_{\mu\nu}$
(73) $\displaystyle\eta_{\varepsilon\nu}\eta^{\lambda\sigma}\delta
R_{\mu\nu}{}^{\sigma}=\frac{1}{2}\partial_{\sigma}\partial^{\sigma}h_{\mu\nu}+\frac{3}{2}a^{\prime}a\partial_{y}h_{\mu\nu}+\frac{a^{\prime}}{a}\delta\Gamma^{y}{}_{\mu\nu}$
(74)

With all these elements given above, we can obtain all the necessary perturbed
curvature.

In the end, we need to add that:

$\displaystyle\delta R_{\mu y}=\delta R_{\mu A}=\delta R_{yy}=\delta
R_{yA}=\delta R_{AB}=0.$ (75)

They can be evaluated in a similar way (see Appendix of Ref cui_tensor_2020 ).
So the perturbed Ricci scalar vanished as $\delta R=0$. And for the perturbed
Gauss-Bonnet term:

$\displaystyle\delta\mathcal{L}_{GB}=(-4R_{a}{}^{c}R_{bc}+2R_{ab}R-4R^{cd}R_{acbd}+2R_{a}{}^{cde}R_{bcde})\delta
g^{ab}=0.$ (76)

Since the first factor is proportional to $\delta g_{ab}$ and its derivative,
the whole term will finally vanish because of the TT gauge condition (11).

For the perturbation of energy-momentum tensor $\delta T_{\mu\nu}$, we only
consider the perturbation caused by the fluctuation of the background metric
csaki_universal_2000 . That is to say:

$\displaystyle\delta T_{\mu\nu}=T_{\mu}^{\alpha}h_{\alpha\nu}$ (77)

or practically $\delta
T_{\mu\nu}=1/4\eta^{\alpha\beta}T_{\alpha\beta}h_{\mu\nu}$.

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