# Generalized Approach for the Perturbative Dynamical Braneworld in $D$
dimensions

Zi-Chao<EMAIL_ADDRESS>Hao<EMAIL_ADDRESS>Yungui
<EMAIL_ADDRESS>(corresponding author) aSchool of Physics,
Huazhong University of Science and Technology, Wuhan, Hubei 430074, China
bCollege of Physics, Chongqing University, Chongqing 401331, China
cDepartment of Physics, School of Physical Science and Technology, Ningbo
University, Ningbo, Zhejiang 315211, China

###### Abstract

In this paper, we propose an approach to derive the brane cosmology in the
$D$-dimensional braneworld model. We generalize the “bulk-based” approach by
treating the 4-brane as a small perturbation to the $D$-dimensional
spherically symmetric spacetime. The linear corrections from a static 4-brane
to the metric are derived from the linearized perturbation equations, while
the nonlinear corrections are found by a parameterization of the perturbed
metric solution. We use a time-dependent generalization to give the
nonlinearly perturbed metric solution for the dynamical braneworld model, and
analyse the stability of the model under the motion of the 4-brane. Through
the fine tuning, we can recover the Friedmann equations for the universe with
and without an effective cosmological constant. More importantly, the de
Sitter expansion of the universe can be reproduced.

##  I Introduction

The idea that there exist extra (spatial) dimensions is compelling. It has
sparked numerous discussions regarding the nature and the structure of extra
dimensions. In the early higher-dimensional theories, it was commonly believed
that these extra dimensions are compact and of small size Kaluza1 ; Klein1 ;
Klein2 ; ADD1 ; ADD2 ; Randall1 . An opposing viewpoint, initially proposed by
Rubakov and Shaposhnikov Rubakov1 ; Rubakov2 , emerged in the model developed
by Randall and Sundrum (i.e., the RS-2 model) Randall2 , in which the extra
dimensions could be large and infinite. As a braneworld model, the universe in
the RS-2 (and many other) model is supposed to be a 4-brane embedded in a
static anti-de Sitter (AdS) bulk. All the particles in the standard model are
confined on the 4-brane and the linearized gravity on the 4-brane is proved to
be Einstein gravity Randall2 ; Lykken:1999nb ; Garriga1 ; Giddings1 . However,
it is challenging to directly apply the RS-like model to brane cosmology. In
fact, if the embedded 4-brane has a de Sitter (dS) expansion, the bulk
spacetime is generally time-dependent. While in the model the fine tuning
between the bulk cosmological constant and the energy density and pressure of
the 4-brane could keep the bulk spacetime static, it will result in a
vanishing effective brane cosmological constant. Thus, the induced metric on
the 4-brane inevitably becomes flat Minkowski, which fails to describe the dS
expansion.

To find the cosmological brane solutions for the RS spacetime, a “brane-based”
approach was proposed in Refs. Kaloper:1999sm ; Chung:1999zs ; Csaki:1999jh ;
Cline:1999ts ; Shiromizu:1999wj ; Maartens:2000fg . Notably, it was the
Binétruy-Deffayet-Langlois (BDL) model that exactly solved the cosmological
equations on a single 4-brane embedded in a 5-dimensional bulk governed by a
negative bulk cosmological constant Binetruy1 ; Binetruy2 . In the BDL model,
the 4-brane is fixed in the bulk in order to get Gaussian normal coordinates,
with which the extra diagonal metric component becomes normal to the 4-brane.
Unlike the RS-like single-wall model, the bulk spacetime in the BDL model is
supposed to be time-dependent to induce the cosmological expansion of the
4-brane. The evolution of the bulk spacetime is driven by the matter on the
4-brane through the singularity part of the field equations. So the background
spacetime built in the “brane-based” approach could be understood as a time-
dependent generalization of the static bulk spacetime with time-dependent
corrections sourced by the energy density and pressure of the 4-brane (see
also Cline:1999ts ).

An equivalent description of the “brane-based” approach can be achieved by
applying an explicit coordinate transformation to the Gaussian normal
coordinates Mukohyama:1999wi ; Bowcock:2000cq . Instead of keeping static in
the bulk, the 4-brane in the alternative “bulk-based” approach could be
dynamical due to the energy density and pressure of the 4-brane Kraus:1999it ;
Ida:1999ui ; Chamblin:1999ya ; Chamblin:1999ea ; Gubser:1999vj ;
Hawking:2000kj ; Anchordoqui:2000du . The time-dependent cosmological
equations on such a dynamical 4-brane are usually derived from the Israel
joint condition Israel1 . In this approach, the static bulk spacetime
structure is more transparent because it exhibits maximal symmetry under the
motion of the 4-brane. Taking advantage of this, several researches, including
brane cosmology Collins:2000yb ; Banerjee:2018qey ; Csaki:2000dm , AdS/CFT
Garriga:1999bq ; Savonije:2001nd ; Cai:2001ja , shortcuts Caldwell1 ; Wang2002
; Visinelli1 ; Yu1 ; Lin1 ; Lin2 , etc., have been conducted from the “bulk-
based” perspective.

Note that neither of the two approaches considers both the back-reaction and
the dynamics of the 4-brane. In this paper, we aim to provide a generalized
version of the “bulk-based” approach by considering the back-reaction from a
dynamical 4-brane to the bulk spacetime. The contributions from the 4-brane
are treated as small perturbations to the bulk, enabling us to analytically
calculate the linear corrections to the metric. The nonlinear corrections are
introduced through the parameterization of the perturbed metric to stabilize
the dynamical braneworld model. We also study the cosmological equations on
the 4-brane and prove that there exists a dS expansion on the 4-brane.

The paper is arranged as follows. In Sec. II, we derive the Einstein equations
for a $D$-dimensional spherically symmetric spacetime with a bulk cosmological
constant and a 4-brane fixed in the bulk. Then, treating the static 4-brane as
a small perturbation, we obtain a nonlinearly perturbed metric solution by a
specific parameterization. By introducing a time-dependent generalization of
the parameterization, we establish a perturbative dynamical braneworld model
in Sec. III. We also study the cosmological equations on the 4-brane, and show
the normal expansions of the universe. Finally, our conclusion and discussion
are given in Sec. IV.

##  II Static 4-brane

###  II.1 Field equations for the $D$-dimensional bulk

We start by considering a $D(=4+d)$-dimensional spacetime with a bulk
cosmological constant $\Lambda$ and an embedded 4-brane. The gravity is
described by the $D$-dimensional general relativity with its action written as

$S_{g}^{~{}}=M^{D-2}_{~{}}\int\text{d}^{D}_{~{}}x\sqrt{-g}\,(R^{(D)}_{~{}}-2\Lambda).$
(1)

Here, $M$ is the $D$-dimensional fundamental Planck scale. With the existence
of the 4-brane, the $D$-dimensional spacetime is not static in general. The
metric could be given by

$\text{d}s^{2}_{D}=A(R,T)\text{d}R^{2}_{~{}}+B(R,T)\text{d}\Omega^{2}_{d-1}+C(R,T)\text{d}\Sigma^{2}_{4}\,,$
(2)

where

$\text{d}\Omega^{2}_{d-1}=\tilde{g}_{mn}^{~{}}\text{d}y^{m}_{~{}}\text{d}y^{n}_{~{}}$
(3)

is the line element of a $(d-1)$-dimensional subspace, $T$ is the bulk time,
and $(R,y^{m}_{~{}})$ are the coordinates of the extra space. We use the line
element

$\text{d}\Sigma^{2}_{4}=\hat{g}_{\mu\nu}^{~{}}\text{d}x^{\mu}_{~{}}\text{d}x^{\nu}_{~{}}$
(4)

to describe a static and maximally symmetric 4-dimensional submanifold, where
$x^{\mu}_{~{}}=(T,r,\theta,\phi)$ represents the 4-dimensional coordinates
spanning on it. It is convenient to set $\hat{g}_{TT}^{~{}}=-1$ by the
coordinate transformation for the following calculations.

Without the loss of generality, we assume that the 4-brane can move in the
bulk. However, as we shall see in Sec. III, the evolution of the bulk metric
is governed by both the brane’s motion and the matter confined on it. The
contributions from the two sources are coupled, making it challenging to
obtain the metric solutions directly. As a result, we would like to consider a
static 4-brane at first. Under the metric assumption (2), the 4-brane is
embeded through the condition

$R=R_{0}^{~{}}\,,~{}~{}~{}~{}y^{m}_{~{}}=y_{0}^{m}\,,$ (5)

where $R_{0}^{~{}}$ and $y_{0}^{m}$ denote a fixed position in the bulk. With
this condition, the 4-dimensional fields confined on the brane will couple to
the following induced metric:

$g^{(4)}_{\mu\nu}\equiv
g_{\mu\nu}^{~{}}(x^{\mu}_{~{}},R=R_{0}^{~{}},y^{m}_{~{}}=y_{0}^{m}).$ (6)

The energy-momentum tensor of the matter confined on the 4-brane is

$T_{MN}^{~{}}=\big{[}(\rho+p)u_{M}^{~{}}u_{N}^{~{}}+p\,h_{MN}^{~{}}\big{]}\delta(R-R_{0}^{~{}})\delta^{(d-1)}_{~{}}(y^{m}_{~{}}-y_{0}^{m}),$
(7)

where $\rho$ and $p$ are respectively the energy density and pressure of the
matter, $u^{M}_{~{}}=(1/\sqrt{C},0,\ldots,0)$ is the unit velocity vector of
the comoving observer on the 4-brane, and
$h_{MN}^{~{}}=g^{(4)}_{\mu\nu}\delta^{\mu}_{M}\delta^{\nu}_{N}$ is the
projection tensor for the 4-brane. The singular behavior of the energy-
momentum tensor means that it will contribute a boundary condition to the bulk
metric on the location of the 4-brane. When taking it into account, the
Einstein equations
$\sqrt{-g}\,G_{MN}^{~{}}=\frac{1}{M_{~{}}^{D-2}}\sqrt{-g^{(4)}_{~{}}}\,T_{MN}^{~{}}-\Lambda\sqrt{-g}\,g_{MN}^{~{}}$
become

$\displaystyle\hat{G}_{ij}^{(4)}-\frac{C}{2B}\hat{g}^{~{}}_{ij}\tilde{R}^{(d-1)}_{~{}}\\!\\!$
$\displaystyle\\!=\\!$
$\displaystyle\\!\\!\frac{1}{M^{D-2}_{~{}}}\frac{\sqrt{-g^{(4)}_{~{}}}}{\sqrt{-g}}T_{ij}^{~{}}-C\Lambda\hat{g}_{ij}^{~{}}+\bigg{[}\frac{\ddot{C}}{C}-\frac{3}{2}\frac{C^{\prime\prime}}{A}+\frac{3}{4}\frac{A^{\prime}C^{\prime}}{A^{2}_{~{}}}-\frac{3}{4}\frac{\dot{C}^{2}_{~{}}}{C^{2}_{~{}}}+\frac{1}{4}\frac{\dot{A}\dot{C}}{AC}+\frac{1}{2}\frac{\ddot{A}}{A}-\frac{1}{4}\frac{\dot{A}^{2}_{~{}}}{A^{2}_{~{}}}$
(8a)
$\displaystyle\\!\\!+\frac{d-1}{4}\frac{\dot{A}\dot{B}}{AB}+\frac{d-1}{4}\frac{A^{\prime}B^{\prime}C}{A^{2}_{~{}}B}+\frac{d-1}{2}\frac{\ddot{B}}{B}-\frac{d-1}{2}\frac{B^{\prime\prime}C}{AB}-\frac{(d-4)(d-1)}{8}\frac{B^{\prime
2}_{~{}}C}{AB^{2}_{~{}}}+\frac{d-1}{4}\frac{\dot{B}\dot{C}}{BC}$
$\displaystyle\\!\\!-\frac{3(d-1)}{4}\frac{B^{\prime}C^{\prime}}{AB}+\frac{(d-4)(d-1)}{8}\frac{\dot{B}^{2}_{~{}}}{B^{2}_{~{}}}\bigg{]}\hat{g}_{ij}^{~{}},$
$\displaystyle\hat{G}_{TT}^{(4)}+\frac{C}{2B}\tilde{R}^{(d-1)}_{~{}}\\!\\!$
$\displaystyle\\!=\\!$
$\displaystyle\\!\\!\frac{1}{M^{D-2}_{~{}}}\frac{\sqrt{-g^{(4)}_{~{}}}}{\sqrt{-g}}T_{TT}^{~{}}+C\Lambda+\frac{3}{2}\frac{C^{\prime\prime}}{A}-\frac{3}{4}\frac{\dot{C}^{2}_{~{}}}{C^{2}_{~{}}}-\frac{3}{4}\frac{A^{\prime}C^{\prime}}{A^{2}_{~{}}}-\frac{3}{4}\frac{\dot{A}\dot{C}}{AC}-\frac{d-1}{4}\frac{\dot{A}\dot{B}}{AB}-\frac{(d-2)(d-1)}{8}\frac{\dot{B}^{2}_{~{}}}{B^{2}_{~{}}}$
(8b)
$\displaystyle\\!\\!-\frac{d-1}{4}\frac{A^{\prime}B^{\prime}C}{A^{2}_{~{}}B}+\frac{d-1}{2}\frac{B^{\prime\prime}C}{AB}+\frac{(d-4)(d-1)}{8}\frac{B^{\prime
2}_{~{}}C}{AB^{2}_{~{}}}-\frac{3(d-1)}{4}\frac{\dot{B}\dot{C}}{BC}+\frac{3(d-1)}{4}\frac{B^{\prime}C^{\prime}}{AB},$
$\displaystyle-\frac{A}{2B}\tilde{R}^{(d-1)}_{~{}}-\frac{A}{2C}\hat{R}^{(4)}_{~{}}\\!\\!$
$\displaystyle\\!=\\!$
$\displaystyle\\!\\!-A\Lambda+\frac{3}{2}\frac{A\ddot{C}}{C^{2}_{~{}}}-\frac{3}{2}\frac{C^{\prime
2}_{~{}}}{C^{2}_{~{}}}-\frac{3}{4}\frac{A\dot{C}^{2}_{~{}}}{C^{3}_{~{}}}+\frac{d-1}{2}\frac{A\ddot{B}}{BC}-\frac{(d-2)(d-1)}{8}\frac{B^{\prime
2}_{~{}}}{B^{2}_{~{}}}+\frac{(d-4)(d-1)}{8}\frac{A\dot{B}^{2}_{~{}}}{B^{2}_{~{}}C}$
(8c)
$\displaystyle\\!\\!+\frac{d-1}{2}\frac{A\dot{B}\dot{C}}{BC^{2}_{~{}}}-(d-1)\frac{B^{\prime}C^{\prime}}{BC},$
$\displaystyle\tilde{G}^{(d-1)}_{mn}-\frac{B}{2C}\tilde{g}^{~{}}_{mn}\hat{R}^{(4)}_{~{}}\\!\\!$
$\displaystyle\\!=\\!$
$\displaystyle\\!\\!-B\Lambda\tilde{g}_{mn}^{~{}}+\bigg{[}\frac{3}{2}\frac{B\ddot{C}}{C^{2}_{~{}}}-2\frac{BC^{\prime\prime}}{AC}-\frac{1}{2}\frac{BC^{\prime
2}_{~{}}}{AC^{2}_{~{}}}-\frac{3}{4}\frac{B\dot{C}^{2}_{~{}}}{C^{3}_{~{}}}+\frac{A^{\prime}BC^{\prime}}{A^{2}_{~{}}C}+\frac{1}{2}\frac{\dot{A}B\dot{C}}{AC^{2}_{~{}}}+\frac{1}{2}\frac{\ddot{A}B}{AC}+\frac{d-2}{4}\frac{\dot{A}\dot{B}}{AC}$
(8d)
$\displaystyle\\!\\!-\frac{1}{4}\frac{\dot{A}^{2}_{~{}}B}{A^{2}_{~{}}C}+\frac{d-2}{4}\frac{A^{\prime}B^{\prime}}{A^{2}_{~{}}}+\frac{d-2}{2}\frac{\ddot{B}}{C}-\frac{d-2}{2}\frac{B^{\prime\prime}}{A}-\frac{(d-5)(d-2)}{8}\frac{B^{\prime
2}_{~{}}}{AB}+\frac{d-2}{2}\frac{\dot{B}\dot{C}}{C^{2}_{~{}}}$
$\displaystyle\\!\\!-(d-2)\frac{B^{\prime}C^{\prime}}{AC}\bigg{]}\tilde{g}_{mn}^{~{}},$
$\displaystyle\frac{3}{2}\frac{\dot{C}C^{\prime}}{C^{2}_{~{}}}\\!\\!$
$\displaystyle\\!=\\!$
$\displaystyle\\!\\!\frac{3}{2}\frac{\dot{C}^{\prime}}{C}+\frac{d-1}{2}\frac{\dot{B}^{\prime}}{B}-\frac{d-1}{4}\frac{\dot{B}B^{\prime}}{B^{2}_{~{}}}-\frac{3}{4}\frac{\dot{A}C^{\prime}}{AC}-\frac{d-1}{4}\frac{\dot{B}C^{\prime}}{BC}-\frac{d-1}{4}\frac{\dot{A}B^{\prime}}{AB},$
(8e)

where primes denote derivatives with respect to $R$, $\tilde{R}^{(d-1)}_{~{}}$
is the curvature scalar of the $(d-1)$-dimensional subspace constructed by
$\tilde{g}_{mn}^{~{}}$, $\hat{R}^{(4)}_{~{}}$ is the curvature scalar of the
4-dimensional submanifold associated with $\hat{g}_{\mu\nu}^{~{}}$, and
$\tilde{G}_{mn}^{(d-1)}$ and $\hat{G}^{(4)}_{\mu\nu}$ are induced Einstein
tensors related to $\tilde{R}^{(d-1)}_{~{}}$ and $\hat{R}^{(4)}_{~{}}$,
respectively. However, it is hard to solve these field equations directly due
to the numerous parameters involved. To decouple the effect of the 4-brane
from the background spacetime, we will regard the matter on the 4-brane as a
small perturbation to the spacetime for simplicity. Under this assumption, the
background solutions are static, satisfying the Einstein’s equations without
the energy-momentum tensor of the matter. In this case, the bulk metric will
become time-dependent only if we introduce (non)linear perturbations sourced
from either the brane’s motion or the evolution of the matter on the 4-brane.

Here, we derive one of the static vacuum solutions to the field equations (8)
with $\Lambda<0$:

$\text{d}s^{2}_{D}=A(R)\text{d}R^{2}_{~{}}+B(R)\text{d}\Omega^{2}_{d-1}-C(R)\text{d}T^{2}_{~{}}+C(R)\text{d}\Sigma^{2}_{3}\,,$
(9)

where

$\displaystyle A(R)\\!\\!$ $\displaystyle\\!=\\!$
$\displaystyle\\!\\!\frac{(d+3)(d+2)}{-2\Lambda}\frac{1}{R^{2}_{~{}}},$ (10a)
$\displaystyle B(R)\\!\\!$ $\displaystyle\\!=\\!$
$\displaystyle\\!\\!\frac{(d+3)(d+2)}{-2\Lambda}R^{2}_{~{}},$ (10b)
$\displaystyle C(R)\\!\\!$ $\displaystyle\\!=\\!$
$\displaystyle\\!\\!R^{2}_{~{}}.$ (10c)

This metric describes an $\text{AdS}_{D}^{~{}}$ bulk spacetime with a
$(d-1)$-dimensional sphere,

$\text{d}\Omega^{2}_{d-1}=\text{d}\theta^{2}_{1}+\text{sin}^{2}_{~{}}\theta_{1}^{~{}}\text{d}\theta_{2}^{~{}}+\ldots+\text{sin}^{2}_{~{}}\theta_{1}^{~{}}\ldots\text{sin}^{2}_{~{}}\theta_{d-2}^{~{}}\text{d}\theta_{d-1}^{~{}},$
(11)

and a 3-dimensional flat subspace,

$\text{d}\Sigma^{2}_{3}=\text{d}r^{2}_{~{}}+r^{2}_{~{}}(\text{d}\psi^{2}_{~{}}+\text{sin}^{2}_{~{}}\psi\,\text{d}\phi^{2}_{~{}}).$
(12)

The static vacuum solution (9) is smooth in the whole bulk. It does not
contribute any singular terms to the field equations. So, for an
$\text{AdS}_{D}^{~{}}$ bulk with a 4-brane, it could be applicable everywhere
except on the brane’s location $R=R_{0}^{~{}}$ and $y^{m}_{~{}}=y^{m}_{0}$. To
ensure the self-consistency of the field equations (8), we can additionally
introduce a nonsmooth part, $\Delta g_{MN}^{~{}}$, into the metric (9). Since
the 4-brane is static, these corrections are solely sourced from the confined
matter on the 4-brane. They can be regarded as perturbations to the background
spacetime and do not govern the spacetime structure at the leading order. For
the sake of simplification, we will reserve the spherical symmetry of the
extra space under the perturbations. Thus, the corrections are independent of
$y^{m}_{~{}}$ and could be simplified to

$\displaystyle\Delta g_{mn}^{~{}}\\!\\!$ $\displaystyle\\!=\\!$
$\displaystyle\\!\\!\mathcal{B}(R,T)\tilde{g}_{mn}^{~{}},$ (13a)
$\displaystyle\Delta g_{TT}^{~{}}\\!\\!$ $\displaystyle\\!=\\!$
$\displaystyle\\!\\!\mathcal{C}_{T}^{~{}}(R,T)\hat{g}_{TT}^{~{}},$ (13b)
$\displaystyle\Delta g_{ij}^{~{}}\\!\\!$ $\displaystyle\\!=\\!$
$\displaystyle\\!\\!\mathcal{C}_{s}^{(ij)}(R,T)\hat{g}_{ij}^{~{}}.$ (13c)

Here, we assume that the contributions of the matter to the $TT$ and $ij$
components of the background metric (9) are different, i.e.,
$\mathcal{C}_{T}^{~{}}\neq\mathcal{C}_{s}^{(ij)}$. We also set the same
correction ($\mathcal{C}_{s}^{(ij)}=\mathcal{C}_{s}^{~{}}$) to each of the
$ij$ components on account of the embedding of a homogeneous and isotropic
4-brane. Note that, the functions $\mathcal{B}$, $\mathcal{C}_{T}^{~{}}$, and
$\mathcal{C}_{s}^{~{}}$ need not be time-independent because they are sourced
by $\rho$ and $p$. In addition, there is no correction to $A$ because
$A^{\prime\prime}$ does not appear in the field equations (8). With these
metric corrections (13), the perturbed field equations reduce to

$\displaystyle\frac{1}{VM^{D-2}_{~{}}}\frac{1}{\sqrt{A}}\tilde{T}_{ij}^{(5)}\\!\\!$
$\displaystyle\\!=\\!$
$\displaystyle\\!\\!\bigg{[}-\frac{\ddot{C}_{s}^{~{}}}{C_{T}^{~{}}}+\frac{C_{s}^{\prime\prime}}{A}+\frac{1}{2}\frac{C_{s}^{~{}}C_{T}^{\prime\prime}}{AC_{T}^{~{}}}-\frac{1}{2}\frac{A^{\prime}C_{s}^{\prime}}{A^{2}_{~{}}}-\frac{1}{4}\frac{C_{s}^{~{}}A^{\prime}C_{T}^{\prime}}{A^{2}_{~{}}C_{T}^{~{}}}+\frac{1}{4}\frac{\dot{C}_{s}^{2}}{C_{s}^{~{}}C_{T}^{~{}}}+\frac{1}{2}\frac{\dot{C}_{s}^{~{}}\dot{C}_{T}^{~{}}}{C_{T}^{2}}-\frac{d-1}{4}\frac{A^{\prime}\mathfrak{B}^{\prime}C_{s}^{~{}}}{A^{2}_{~{}}\mathfrak{B}}-\frac{1}{4}\frac{C_{s}^{\prime
2}}{AC_{s}^{~{}}}$ (14a)
$\displaystyle\\!\\!-\frac{d-1}{2}\frac{C_{s}^{~{}}\ddot{\mathfrak{B}}}{\mathfrak{B}C_{T}^{~{}}}+\frac{d-1}{2}\frac{C_{s}^{~{}}\mathfrak{B}^{\prime\prime}}{A\mathfrak{B}}+\frac{(d-4)(d-1)}{8}\frac{C_{s}^{~{}}\mathfrak{B}^{\prime
2}_{~{}}}{A\mathfrak{B}^{2}_{~{}}}-\frac{d-1}{2}\frac{\dot{\mathfrak{B}}\dot{C}_{s}^{~{}}}{\mathfrak{B}C_{T}^{~{}}}+\frac{d-1}{4}\frac{\dot{\mathfrak{B}}C_{s}^{~{}}\dot{C}_{T}^{~{}}}{\mathfrak{B}C_{T}^{2}}+\frac{1}{2}\frac{C_{s}^{\prime}C_{T}^{\prime}}{AC_{T}^{~{}}}$
$\displaystyle\\!\\!-\frac{d-1}{2}\frac{\mathfrak{B}^{\prime}C_{s}^{\prime}}{A\mathfrak{B}}+\frac{d-1}{4}\frac{C_{s}^{~{}}\mathfrak{B}^{\prime}C_{T}^{\prime}}{A\mathfrak{B}C_{T}^{~{}}}-\frac{(d-4)(d-1)}{8}\frac{C_{s}^{~{}}\dot{\mathfrak{B}}^{2}_{~{}}}{\mathfrak{B}^{2}_{~{}}C_{T}^{~{}}}-\frac{1}{4}\frac{C_{s}^{~{}}C_{T}^{\prime
2}}{AC_{T}^{2}}+C_{s}^{~{}}\Lambda\bigg{]}\hat{g}_{ij}^{~{}}\,,$
$\displaystyle\frac{1}{VM^{D-2}_{~{}}}\frac{1}{\sqrt{A}}\tilde{T}_{TT}^{(5)}\\!\\!$
$\displaystyle\\!=\\!$
$\displaystyle\\!\\!-\frac{3}{2}\frac{C_{T}^{~{}}C_{s}^{\prime\prime}}{AC_{s}^{~{}}}+\frac{3}{4}\frac{\dot{C}_{s}^{2}}{C_{s}^{2}}+\frac{3}{4}\frac{A^{\prime}C_{T}^{~{}}C_{s}^{\prime}}{A^{2}_{~{}}C_{s}^{~{}}}+\frac{d-1}{4}\frac{A^{\prime}\mathfrak{B}^{\prime}C_{T}^{~{}}}{A^{2}_{~{}}\mathfrak{B}}-\frac{d-1}{2}\frac{\mathfrak{B}^{\prime\prime}C_{T}^{~{}}}{A\mathfrak{B}}-\frac{(d-4)(d-1)}{8}\frac{\mathfrak{B}^{\prime
2}_{~{}}C_{T}^{~{}}}{A\mathfrak{B}^{2}_{~{}}}$ (14b)
$\displaystyle\\!\\!+\frac{(d-2)(d-1)}{8}\frac{\dot{\mathfrak{B}}^{2}_{~{}}}{\mathfrak{B}^{2}_{~{}}}+\frac{3(d-1)}{4}\frac{\dot{\mathfrak{B}}\dot{C}_{s}^{~{}}}{\mathfrak{B}C_{s}^{~{}}}-\frac{3(d-1)}{4}\frac{\mathfrak{B}^{\prime}C_{s}^{\prime}C_{T}^{~{}}}{A\mathfrak{B}C_{s}^{~{}}}-C_{T}^{~{}}\Lambda,$
$\displaystyle-A\Lambda\\!\\!$ $\displaystyle\\!=\\!$
$\displaystyle\\!\\!-\frac{3}{2}\frac{A\ddot{C}_{s}^{~{}}}{C_{s}^{~{}}C_{T}^{~{}}}+\frac{3}{4}\frac{C_{s}^{\prime
2}}{C_{s}^{2}}+\frac{3}{4}\frac{C_{s}^{\prime}C_{T}^{\prime}}{C_{s}^{~{}}C_{T}^{~{}}}+\frac{3}{4}\frac{A\dot{C}_{s}^{~{}}\dot{C}_{T}^{~{}}}{C_{s}^{~{}}C_{T}^{2}}-\frac{d-1}{2}\frac{A\ddot{\mathfrak{B}}}{\mathfrak{B}C_{T}^{~{}}}+\frac{(d-2)(d-1)}{8}\frac{A\dot{\mathfrak{B}}^{2}_{~{}}}{\mathfrak{B}^{2}_{~{}}C_{T}^{~{}}}+\frac{d-1}{4}\frac{A\dot{\mathfrak{B}}\dot{C}_{T}^{~{}}}{\mathfrak{B}C_{T}^{2}}$
(14c)
$\displaystyle\\!\\!-\frac{3(d-1)}{4}\frac{A\dot{\mathfrak{B}}\dot{C}_{s}^{~{}}}{\mathfrak{B}C_{s}^{~{}}C_{T}^{~{}}}+\frac{3(d-1)}{4}\frac{\mathfrak{B}^{\prime}C_{s}^{\prime}}{\mathfrak{B}C_{s}^{~{}}}+\frac{d-1}{4}\frac{\mathfrak{B}^{\prime}C_{T}^{\prime}}{\mathfrak{B}C_{T}^{~{}}},$
$\displaystyle-\mathfrak{B}\Lambda\\!\\!$ $\displaystyle\\!=\\!$
$\displaystyle\\!\\!-\frac{3}{2}\frac{\mathfrak{B}\ddot{C}_{s}^{~{}}}{C_{s}^{~{}}C_{T}^{~{}}}+\frac{3}{2}\frac{\mathfrak{B}C_{s}^{\prime\prime}}{AC_{s}^{~{}}}+\frac{1}{2}\frac{\mathfrak{B}C_{T}^{\prime\prime}}{AC_{T}^{~{}}}+\frac{3}{4}\frac{\mathfrak{B}C_{s}^{\prime}C_{T}^{\prime}}{AC_{s}^{~{}}C_{T}^{~{}}}-\frac{1}{4}\frac{\mathfrak{B}C_{T}^{\prime
2}}{AC_{T}^{2}}+\frac{3}{4}\frac{\mathfrak{B}\dot{C}_{s}^{~{}}\dot{C}_{T}^{~{}}}{C_{s}^{~{}}C_{T}^{2}}-\frac{3}{4}\frac{A^{\prime}\mathfrak{B}C_{s}^{\prime}}{A^{2}_{~{}}C_{s}^{~{}}}-\frac{1}{4}\frac{A^{\prime}\mathfrak{B}C_{T}^{\prime}}{A^{2}_{~{}}C_{T}^{~{}}}$
(14d)
$\displaystyle\\!\\!-\frac{d-2}{4}\frac{A^{\prime}\mathfrak{B}^{\prime}}{A^{2}_{~{}}}-\frac{d-2}{2}\frac{\ddot{\mathfrak{B}}}{C_{T}^{~{}}}+\frac{d-2}{2}\frac{\mathfrak{B}^{\prime\prime}}{A}+\frac{(d-5)(d-2)}{8}\frac{\mathfrak{B}_{~{}}^{\prime
2}}{A\mathfrak{B}}-\frac{3(d-2)}{4}\frac{\dot{\mathfrak{B}}\dot{C}_{s}^{~{}}}{C_{s}^{~{}}C_{T}^{~{}}}+\frac{d-2}{4}\frac{\dot{\mathfrak{B}}\dot{C}_{T}^{~{}}}{C_{T}^{2}}$
$\displaystyle\\!\\!+\frac{3(d-2)}{4}\frac{\mathfrak{B}^{\prime}C_{s}^{\prime}}{AC_{s}^{~{}}}+\frac{d-2}{4}\frac{\mathfrak{B}^{\prime}C_{T}^{\prime}}{AC_{T}^{~{}}},$
$\displaystyle\frac{3}{4}\frac{\dot{C}_{s}^{~{}}C_{s}^{\prime}}{C_{s}^{2}}\\!\\!$
$\displaystyle\\!=\\!$
$\displaystyle\\!\\!\frac{3}{2}\frac{\dot{C}_{s}^{\prime}}{C_{s}^{~{}}}+\frac{d-1}{2}\frac{\dot{\mathfrak{B}}^{\prime}}{\mathfrak{B}}-\frac{d-1}{4}\frac{\dot{\mathfrak{B}}\mathfrak{B}^{\prime}}{\mathfrak{B}^{2}_{~{}}}-\frac{d-1}{4}\frac{\dot{\mathfrak{B}}C_{T}^{\prime}}{\mathfrak{B}C_{T}^{~{}}}-\frac{3}{4}\frac{\dot{C}_{s}^{~{}}C_{T}^{\prime}}{C_{s}^{~{}}C_{T}^{~{}}},$
(14e)

where we have defined $\mathfrak{B}\equiv B+\mathcal{B}$, $C_{s}^{~{}}\equiv
C+\mathcal{C}_{s}^{~{}}$, and $C_{T}^{~{}}\equiv C+\mathcal{C}_{T}^{~{}}$.
Note that in the above field equations, we have integrated over the extra
$(d-1)$-dimensional space to eliminate
$\delta^{(d-1)}_{~{}}(y^{m}_{~{}}-y^{m}_{0})$ in the energy-momentum tensor.
$\tilde{T}_{MN}^{(5)}$ is thus an effective energy-momentum tensor defined on
the rest 5-dimensional submanifold. The parameter $V$ indicates the volume of
the extra $(d-1)$-dimensional space, which is related to the quantities for
the extra $(d-1)$-dimensional space, such as the spatial curvature and the
matter distribution. Our method of integrating the field equations is
equivalent to using the dimensionality reduction directly to the
$D$-dimensional action to arrive at an effective 5-dimensional theory. The
difference is that in our method, we have an extra field equation to constrain
the extra freedom arising from the dimensionality reduction.

###  II.2 Parameterization of perturbed solutions

In the perturbed equations (14), the bulk metric (9) is coupled with its
perturbations. Although we could eliminate the background field equations by
assuming $\mathcal{B}\ll B$, $\mathcal{C}_{s}^{~{}}\ll C$, and
$\mathcal{C}_{T}^{~{}}\ll C$, the remaining equations governing the
perturbations are still nonlinear. It is not practical to solve them directly.
Thus, we would like to first derive the linear perturbations. To linearize the
perturbed field equations (14), the following decompositions of $\mathcal{B}$,
$\mathcal{C}_{s}^{~{}}$, and $\mathcal{C}_{T}^{~{}}$ are useful:

$\displaystyle\mathcal{B}\\!\\!$ $\displaystyle\\!=\\!$
$\displaystyle\\!\\!\mathcal{B}^{(1)}_{~{}}+\mathcal{B}^{(2)}_{~{}}+\mathcal{B}^{(3)}_{~{}}+\ldots\,,$
(15a) $\displaystyle\mathcal{C}_{s}^{~{}}\\!\\!$ $\displaystyle\\!=\\!$
$\displaystyle\\!\\!\mathcal{C}^{(1)}_{s}+\mathcal{C}^{(2)}_{s}+\mathcal{C}^{(3)}_{s}+\ldots\,,$
(15b) $\displaystyle\mathcal{C}_{T}^{~{}}\\!\\!$ $\displaystyle\\!=\\!$
$\displaystyle\\!\\!\mathcal{C}^{(1)}_{T}+\mathcal{C}^{(2)}_{T}+\mathcal{C}^{(3)}_{T}+\ldots\,,$
(15c)

where $\mathcal{B}^{(1)}_{~{}}$, $\mathcal{C}_{s}^{(1)}$, and
$\mathcal{C}_{T}^{(1)}$ correspond to the linear perturbations, and the other
terms denote the higher-order perturbations. The constraints on
$\mathcal{B}^{(1)}_{~{}}$, $\mathcal{C}_{s}^{(1)}$, and
$\mathcal{C}_{T}^{(1)}$ are given by the singular part of the linearized
perturbation equations from Eqs. (14a), (14b), and (14d),

$\displaystyle\frac{d-1}{2}\frac{(\mathcal{B}^{(1)}_{~{}})^{\prime\prime}}{B}\\!\\!$
$\displaystyle\\!=\\!$
$\displaystyle\\!\\!-\frac{3}{2}\frac{(\mathcal{C}_{s}^{(1)})^{\prime\prime}}{C}-\frac{\sqrt{A}\,\rho\,\delta(R-R_{0}^{~{}})}{VM^{D-2}_{~{}}},$
(16a)
$\displaystyle\frac{d-1}{2}\frac{(\mathcal{B}^{(1)}_{~{}})^{\prime\prime}}{B}\\!\\!$
$\displaystyle\\!=\\!$
$\displaystyle\\!\\!-\frac{(\mathcal{C}_{s}^{(1)})^{\prime\prime}}{C}-\frac{1}{2}\frac{(\mathcal{C}_{T}^{(1)})^{\prime\prime}}{C}-\frac{\sqrt{A}\,p\,\delta(R-R_{0}^{~{}})}{VM^{D-2}_{~{}}},$
$\displaystyle\frac{d-2}{2}\frac{(\mathcal{B}^{(1)}_{~{}})^{\prime\prime}}{B}\\!\\!$
$\displaystyle\\!=\\!$
$\displaystyle\\!\\!-\frac{3}{2}\frac{(\mathcal{C}_{s}^{(1)})^{\prime\prime}}{C}-\frac{1}{2}\frac{(\mathcal{C}_{T}^{(1)})^{\prime\prime}}{C}.$
(16c)

So the local solutions of $\mathcal{B}^{(1)}_{~{}}$, $\mathcal{C}_{s}^{(1)}$,
and $\mathcal{C}_{T}^{(1)}$ near the 4-brane can be approximated as

$\displaystyle\mathcal{B}^{(1)}_{~{}}\\!\\!$ $\displaystyle\\!=\\!$
$\displaystyle\\!\\!-\frac{\rho-3p}{VM^{D-2}_{~{}}}\frac{2}{d+2}\bigg{[}\frac{(d+3)(d+2)}{-2\Lambda}\bigg{]}^{\frac{3}{2}}_{~{}}R\,|R-R_{0}^{~{}}|,$
$\displaystyle\mathcal{C}_{T}^{(1)}\\!\\!$ $\displaystyle\\!=\\!$
$\displaystyle\\!\\!\frac{(d+1)\rho+3p}{VM^{D-2}_{~{}}}\frac{2}{d+2}\sqrt{\frac{(d+3)(d+2)}{-2\Lambda}}R\,|R-R_{0}^{~{}}|,$
$\displaystyle\mathcal{C}_{s}^{(1)}\\!\\!$ $\displaystyle\\!=\\!$
$\displaystyle\\!\\!-\frac{\rho+(d-1)p}{VM^{D-2}_{~{}}}\frac{2}{d+2}\sqrt{\frac{(d+3)(d+2)}{-2\Lambda}}R\,|R-R_{0}^{~{}}|.$

Obviously, they are the linear-in-$|R-R_{0}^{~{}}|$ corrections to the metric.
One could check that it is hard to describe the normal expansion of the
4-brane solely using the linear perturbation equations in this model. So it is
necessary to introduce corrections beyond the order of
$\mathcal{O}(|R-R_{0}^{~{}}|^{2}_{~{}})$ to the field equations. These
corrections include the higher-order metric perturbations (e.g.
$\mathcal{B}^{(2)}_{~{}}$ and $\mathcal{B}^{(3)}_{~{}}$) sourced by lower-
order perturbations and the higher-order products (e.g.
$\mathcal{B}^{(1)}_{~{}}\mathcal{B}^{(1)}_{~{}}$ and
$\mathcal{B}^{(1)}_{~{}}\mathcal{B}^{(2)}_{~{}}$) directly constructed by the
lower ones. They do not contribute singular terms near the brane, but smooth
nonlinear terms to the field equations. In the next section, we will show how
these nonlinear perturbations stabilize the model and help us to recover the
standard cosmology on the 4-brane.

Taking all these higher-order corrections into account, we finally arrive at
the following parameterization of the perturbed metric:

$\displaystyle\text{d}s_{D}^{2}\\!\\!$ $\displaystyle\\!=\\!$
$\displaystyle\\!\\!A\,\text{d}R^{2}_{~{}}+B\,e^{\mathcal{B}_{0}^{~{}}|R-R_{0}^{~{}}|}_{~{}}\,\text{d}\Omega_{d-1}^{2}-C\,e^{\mathcal{C}_{T0}^{~{}}|R-R_{0}^{~{}}|}_{~{}}\,\text{d}T^{2}_{~{}}$
(18)
$\displaystyle\\!\\!+\,C\,e^{\mathcal{C}_{s0}^{~{}}|R-R_{0}^{~{}}|}_{~{}}\,\text{d}\Sigma^{2}_{3}\,,$

where

$\displaystyle\mathcal{B}_{0}^{~{}}\\!\\!$ $\displaystyle\\!=\\!$
$\displaystyle\\!\\!-\frac{\rho-3p}{VM^{D-2}_{~{}}}\frac{2}{d+2}\sqrt{\frac{(d+3)(d+2)}{-2\Lambda}}\frac{1}{R}\,,$
(19a) $\displaystyle\mathcal{C}_{T0}^{~{}}\\!\\!$ $\displaystyle\\!=\\!$
$\displaystyle\\!\\!\frac{(d+1)\rho+3p}{VM^{D-2}_{~{}}}\frac{2}{d+2}\sqrt{\frac{(d+3)(d+2)}{-2\Lambda}}\frac{1}{R}\,,$
(19b) $\displaystyle\mathcal{C}_{s0}^{~{}}\\!\\!$ $\displaystyle\\!=\\!$
$\displaystyle\\!\\!-\frac{\rho+(d-1)p}{VM^{D-2}_{~{}}}\frac{2}{d+2}\sqrt{\frac{(d+3)(d+2)}{-2\Lambda}}\frac{1}{R}\,.~{}$
(19c)

Note that since each exponent in the metric components yields a constant
factor at infinity, which would prevent (18) from recovering the background
metric (9), one can introduce the coordinate transformations to solve the
problem. Such coordinate transformations only need to eliminate those constant
factors, so they are trivial and easily to be found. Under these
transformations, the nonsmooth part of the parameterization still satisfies
the boundary condition of the field equations at $R=R_{0}^{~{}}$, and gives a
non-trivial correction to the bulk metric near the brane. Far away from the
brane, the non-trivial correction is carried by the higher-order terms of the
parameterization, while the background bulk metric remains the leading order.
In fact, the parameterization (18) extents the validity of the perturbed
metric solution into the whole bulk, since it carries all the nonlinear-
in-$|R-R_{0}^{~{}}|$ terms. Although a more accurate matching prefers a more
general parameterization, it will make the calculation much more complicated.
For the sake of simplification, our following discussions will be based on the
parameterization (18).

##  III Dynamical 4-brane

In the last section, we assume that the background spacetime keeps static
under the embedding of the 4-brane. It is a very strong restriction, and could
make it hard to describe a normal expansion of the 4-brane. Indeed, the matter
on the 4-brane could result in the motion of the 4-brane in the bulk, and
could cause the spacetime time-dependent at the
$\sim|R-R_{0}^{~{}}|^{0}_{~{}}$ order as well. In this section, we will show
how the motion of a 4-brane is driven by the energy density and pressure of
the matter confined on it, and how the balancing between the bulk and bare
cosmological constants recovers the standard cosmology.

###  III.1 Stable time-dependent generalization

We let the 4-brane move in $R$ direction while still fixing it in
$y^{m}_{~{}}$ direction for convenience. The location of the 4-brane is then
described by

$R=\mathcal{R}(T),~{}~{}~{}y^{m}_{~{}}=y^{m}_{0}\,.$ (20)

With this condition, we can construct a series of spacelike normal vectors
$n_{M}^{(\mathfrak{m})}$’s to each of the $\mathfrak{m}$-dimensional
hypersurface by

$n_{M}^{(\mathfrak{m})}\equiv\nabla_{M}^{~{}}f^{\mathfrak{m}}_{~{}},$ (21)

where $\mathfrak{m}$ ranges from $4$ to $D-1$. According to the condition
(20), the parameter function for the 4-brane can be given by

$f^{\mathfrak{m}}_{~{}}=y^{\mathfrak{n}}_{0}\,\,\,\text{for}\,\,\,\mathfrak{m}>4$
(22)

and

$f^{\mathfrak{m}}_{~{}}=R-\mathcal{R}(T)\,\,\,\text{for}\,\,\,\mathfrak{m}=4.$
(23)

Here, we have set $\mathfrak{n}=\mathfrak{m}-3$ for convenience. Then, the
normalization
$\tilde{n}_{M}^{(\mathfrak{m})}\tilde{n}^{(\mathfrak{m}),M}_{\color[rgb]{1,1,1}{none}}=1$
yields

$\tilde{n}_{M}^{(\mathfrak{m})}=\delta_{~{}M}^{\mathfrak{n}}\sqrt{g_{\mathfrak{n}\mathfrak{n}}^{~{}}}\,\,\,\text{for}\,\,\,\mathfrak{m}>4$
(24)

and

$\tilde{n}_{M}^{(4)}=\bigg{(}-\sqrt{\frac{AC}{C-A\dot{\mathcal{R}}^{2}_{~{}}}},0,0,0,\sqrt{\frac{AC}{C-A\dot{\mathcal{R}}^{2}_{~{}}}},0,\ldots,0\bigg{)},$
(25)

where dots denote derivatives with respect to the bulk time $T$. With these
unit normal vectors $\tilde{n}_{M}^{(\mathfrak{m})}$, the projection tensor
for the 4-brane turns into

$\displaystyle\tilde{h}_{MN}^{~{}}\\!\\!$ $\displaystyle\\!=\\!$
$\displaystyle\\!\\!g_{MN}^{~{}}-\sum_{\mathfrak{m}=4}^{D-1}\tilde{n}_{M}^{(\mathfrak{m})}\tilde{n}_{N}^{(\mathfrak{m})}$
(26) $\displaystyle\\!=\\!$
$\displaystyle\\!\\!g_{MN}^{~{}}-\tilde{n}_{M}^{(4)}\tilde{n}_{N}^{(4)}-\delta_{~{}M}^{m}\delta_{~{}M}^{n}\,g_{mn}^{~{}}\,.$

For the observer on the 4-brane, the unit velocity vector can be denoted as

$\tilde{u}^{M}_{~{}}=\bigg{(}\frac{1}{\sqrt{C-A\dot{\mathcal{R}}^{2}_{~{}}}},0,0,0,\frac{\dot{\mathcal{R}}}{\sqrt{C-A\dot{\mathcal{R}}^{2}_{~{}}}},0,\ldots,0\bigg{)},$
(27)

which involves the motion of the 4-brane in $R$ direction. One can check that
the velocity vector is already projected on the 4-brane and orthogonal to the
brane’s normal vector by $\tilde{h}_{MN}^{~{}}\tilde{u}^{M}_{~{}}=0$ and
$\tilde{n}_{M}^{(\mathfrak{m})}\tilde{u}^{M}_{~{}}=0$. Recalling the condition
(20), we can construct the energy-momentum tensor of the matter on the moving
4-brane as follows:

$T_{MN}^{~{}}=\big{[}(\rho+p)\tilde{u}_{M}^{~{}}\tilde{u}_{N}^{~{}}+p\,\tilde{h}_{MN}^{~{}}\big{]}\delta(R-\mathcal{R})\delta^{(d-1)}_{~{}}(y^{m}_{~{}}-y^{m}_{0}).$
(28)

Plugging the condition (20) into the line element (9), the induced metric
$\tilde{g}_{\mu\nu}^{(4)}$ that the 4-dimensional fields couple to is also
modified to

$\text{d}s_{4}^{2}=-\mathcal{R}^{2}_{~{}}\mathcal{H}^{2}_{~{}}\text{d}T^{2}_{~{}}+\mathcal{R}^{2}_{~{}}\text{d}\Sigma_{3}^{2}$
(29)

with

$\mathcal{H}^{2}_{~{}}=1-\frac{(d+3)(d+2)}{-2\Lambda}\frac{\dot{\mathcal{R}}^{2}_{~{}}}{\mathcal{R}^{4}_{~{}}}.$
(30)

In this case, a time-dependent generalization of the perturbed metric (18) is
written as

$\text{d}s_{D}^{2}=\tilde{\mathfrak{A}}\text{d}R^{2}_{~{}}+\tilde{\mathfrak{B}}\text{d}\Omega_{d-1}^{2}-\tilde{C}_{T}^{~{}}\text{d}T^{2}_{~{}}+\tilde{C}_{s}^{~{}}\text{d}\Sigma_{3}^{2},$
(31)

where

$\displaystyle\tilde{\mathfrak{B}}\\!\\!$ $\displaystyle\\!=\\!$
$\displaystyle\\!\\!B\,\mathcal{B}_{1}^{~{}}(R,T)\,e^{\mathcal{B}_{0}^{~{}}|R-\mathcal{R}|}_{~{}},$
(32a) $\displaystyle\tilde{C}_{T}^{~{}}\\!\\!$ $\displaystyle\\!=\\!$
$\displaystyle\\!\\!C\,\mathcal{C}_{T1}^{~{}}(R,T)\,e^{\mathcal{C}_{T0}^{~{}}|R-\mathcal{R}|}_{~{}},$
(32b) $\displaystyle\tilde{C}_{s}^{~{}}\\!\\!$ $\displaystyle\\!=\\!$
$\displaystyle\\!\\!C\,\mathcal{C}_{s1}^{~{}}(R,T)\,e^{\mathcal{C}_{s0}^{~{}}|R-\mathcal{R}|}_{~{}}.$
(32c)

Note that on account of the appearance $\ddot{A}$ in the $ij$ and $mn$
components of the field equations, the $RR$ component of the metric has been
generalized to

$\tilde{\mathfrak{A}}=A\,\mathcal{A}_{1}^{~{}}(R,T)\,e^{\mathcal{A}_{0}^{~{}}|R-\mathcal{R}|}_{~{}},$
(33)

where the smooth function $\mathcal{A}_{0}^{~{}}$ is sourced from the brane
matter. Unlike $\mathcal{B}_{0}^{~{}}$, $\mathcal{C}_{T0}^{~{}}$, and
$\mathcal{C}_{s0}^{~{}}$, $\mathcal{A}_{0}^{~{}}$ cannot be constrained by the
joint condition for the case of a static 4-brane due to the absence of
$A^{\prime\prime}$. In other words, $\mathcal{A}_{0}^{~{}}$ should vanish when
the 4-brane is fixed in the bulk. A tricky way is to parameterize it as
$\mathcal{A}_{0}^{~{}}=\mathcal{A}_{0}^{(1)}(\rho,p)\dot{\mathcal{R}}+\mathcal{A}_{0}^{(2)}(\rho,p)\dot{\mathcal{R}}^{2}_{~{}}+\ldots\,$,
where the coefficients should satisfy the field equations. Compared with the
metric (18) in the last section, the bulk spacetime now becomes time-dependent
at the $|R-\mathcal{R}|^{0}_{~{}}$ order with the new functions
$\mathcal{A}_{1}^{~{}}$, $\mathcal{B}_{1}^{~{}}$, $\mathcal{C}_{T1}^{~{}}$,
and $\mathcal{C}_{s1}^{~{}}$. These functions are smooth and dominated by
$\rho$, $p$, and the motion of the 4-brane. As before, we can still use the
singular terms of the field equations to derive expressions for them. What is
different is that field equations now include extra singular terms arising
from, for example, second time derivatives acting on $\tilde{\mathfrak{A}}$,
$\tilde{\mathfrak{B}}$, and other variables.

Taking the generalized metric (31) into account, the singular parts of
linearized perturbation equations at the $|R-\mathcal{R}|^{1}_{~{}}$ order
become

$\displaystyle\frac{p}{VM^{D-2}_{~{}}}\frac{\mathcal{R}^{4}_{~{}}}{C^{2}_{~{}}}\sqrt{A}\,\mathcal{H}\delta(R-\mathcal{R})\\!\\!$
$\displaystyle\\!=\\!$
$\displaystyle\\!\\!\Big{(}\frac{d-1}{2}\mathcal{B}_{0}^{~{}}\mathcal{B}_{1}^{~{}}+\mathcal{C}_{s0}^{~{}}\mathcal{C}_{s1}^{~{}}\Big{)}\Big{(}1-\frac{A}{C}\dot{\mathcal{R}}^{2}_{~{}}\Big{)}\delta(R-\mathcal{R})+\frac{1}{2}\mathcal{C}_{T0}^{~{}}\mathcal{C}_{T1}^{~{}}\delta(R-\mathcal{R})$
(34a)
$\displaystyle\\!\\!-\frac{1}{2}\frac{A}{C}\mathcal{A}_{0}^{~{}}\mathcal{A}_{1}^{~{}}\dot{\mathcal{R}}^{2}_{~{}}\delta(R-\mathcal{R}),$
$\displaystyle-\frac{\rho}{VM^{D-2}_{~{}}}\frac{\mathcal{R}^{4}_{~{}}}{C^{2}_{~{}}}\sqrt{A}\,\mathcal{H}\delta(R-\mathcal{R})\\!\\!$
$\displaystyle\\!=\\!$
$\displaystyle\\!\\!\Big{(}\frac{d-1}{2}\mathcal{B}_{0}^{~{}}\mathcal{B}_{1}^{~{}}+\frac{3}{2}\mathcal{C}_{s0}^{~{}}\mathcal{C}_{s1}^{~{}}\Big{)}\Big{(}1-\frac{A}{C}\dot{\mathcal{R}}^{2}_{~{}}\Big{)}\delta(R-\mathcal{R}),$
(34b)
$\displaystyle\frac{1}{2}\frac{A}{C}\mathcal{A}_{0}^{~{}}\mathcal{A}_{1}^{~{}}\dot{\mathcal{R}}^{2}_{~{}}\delta(R-\mathcal{R})\\!\\!$
$\displaystyle\\!=\\!$
$\displaystyle\\!\\!\Big{(}\frac{d-2}{2}\mathcal{B}_{0}^{~{}}\mathcal{B}_{1}^{~{}}+\frac{3}{2}\mathcal{C}_{s0}^{~{}}\mathcal{C}_{s1}^{~{}}\Big{)}\Big{(}1-\frac{A}{C}\dot{\mathcal{R}}^{2}_{~{}}\Big{)}\delta(R-\mathcal{R})+\frac{1}{2}\mathcal{C}_{T0}^{~{}}\mathcal{C}_{T1}^{~{}}\delta(R-\mathcal{R}).$
(34c)

Here, the number of the independent equations are less than that of the
functions we should constrain on the boundary. Here, we can set
$\mathcal{A}_{1}^{~{}}=1$ for convenience, and let
$\mathcal{C}_{T1}^{~{}}=(1-A\dot{\mathcal{R}}^{2}_{~{}}/C)^{1/2}_{~{}}+(A\dot{\mathcal{R}}^{2}_{~{}}/C)^{2}_{~{}}$
by choosing the gauge for the bulk time. Since we assume that the 4-brane has
a little back-reaction to the bulk spacetime, the functions
$\mathcal{B}_{1}^{~{}}$, $\mathcal{C}_{s1}^{~{}}$, and
$\mathcal{C}_{T1}^{~{}}$ should follow
$\mathcal{B}_{1}^{~{}}=\mathcal{C}_{s1}^{~{}}=\mathcal{C}_{T1}^{~{}}=1$ far
away from the 4-brane. The local solutions near the 4-brane, consistent with
our assumption and the boundary conditions (34), are given by

$\displaystyle\mathcal{B}_{1}^{~{}}\\!\\!$ $\displaystyle\\!=\\!$
$\displaystyle\\!\\!\mathcal{C}_{s1}^{~{}}=\Big{(}1-\frac{A}{C}\dot{\mathcal{R}}^{2}_{~{}}\Big{)}^{-1/2}_{~{}},$
(35a) $\displaystyle\mathcal{A}_{0}^{~{}}\\!\\!$ $\displaystyle\\!=\\!$
$\displaystyle\\!\\!\mathcal{A}_{0}^{(2)}\dot{\mathcal{R}}^{2}_{~{}}=\frac{A}{C}\mathcal{C}_{T0}^{~{}}\dot{\mathcal{R}}^{2}_{~{}}.$
(35b)

So, recalling Eq. (30), we have
$\mathcal{B}_{1}^{~{}}=\mathcal{C}_{s1}^{~{}}=\mathcal{C}_{T1}^{-1}=1/\mathcal{H}$,
$\mathcal{A}_{0}^{~{}}=\mathcal{C}_{T0}^{~{}}(1-\mathcal{H}^{2}_{~{}})$, and
$\mathcal{C}_{T1}^{~{}}=\mathcal{H}+(1-\mathcal{H}^{2}_{~{}})^{2}_{~{}}$ on
the 4-brane. One can check that
$\mathcal{B}_{1}^{~{}}=\mathcal{C}_{s1}^{~{}}=\mathcal{C}_{T1}^{~{}}=1$ when
the 4-brane is fixed in the bulk. That is to say, when the contributions from
$\rho$ and $p$ to the brane’s motion is negligible, the background spacetime
can be considered static in this model. Once again, the contributions from the
brane’s back-reaction are treated as perturbations in our model. We prefer the
scenario in which the model can maintain stability under these perturbations.
It can be proved that the stability of the model requires
$A\dot{\mathcal{R}}^{2}_{~{}}/C\ll 1$ near the 4-brane. Reminding that our
induced metric (29) on the 4-brane is consistent with the Friedmann-Lemaître-
Robertson-Walker (FLRW) metric with vanishing curvature:

$\text{d}s_{4}^{2}=-\text{d}t^{2}_{~{}}+a(t)^{2}_{~{}}\text{d}\Sigma_{3}^{2},$
(36)

where $t$ is the cosmic time and $a(t)$ is the scale factor. So the brane’s
location (20) in $R$ direction is related to the expansion of the 4-brane
itself through

$a^{2}_{~{}}=\mathcal{R}^{2}_{~{}}.$ (37)

In this case, comparing the two metrics, one can obtain a mapping between the
bulk time and the cosmic time,

$\text{d}t^{2}_{~{}}=\mathcal{R}^{2}_{~{}}\mathcal{H}^{2}_{~{}}\text{d}T^{2}_{~{}}.$
(38)

With the relations (37) and (38), it is found that, near the brane,
$A\dot{\mathcal{R}}^{2}_{~{}}/C$ satisfies

$\frac{A(\mathcal{R})}{C(\mathcal{R})}\dot{\mathcal{R}}^{2}=\frac{(d+2)(d+3)}{-2\Lambda}H^{2}_{~{}}\Big{[}1+\frac{(d+2)(d+3)}{-2\Lambda}H^{2}_{~{}}\Big{]}^{-1}_{~{}},$
(39)

where $H$ is the Hubble parameter. Thus, in the low-energy case
$\left(\frac{(d+3)(d+2)}{-2\Lambda}H^{2}_{~{}}\ll 1\right)$, the corrections
to the metric consistently remain small. They could be treated as small
perturbations to the background spacetime. These perturbations do not exhibit
divergence over time, so the model is always stable, which is consistent with
our assumptions. The result is expected, since we have introduced nonlinear
perturbations through the parameterization (18) to cure the instability of the
linear perturbation (17) that possibly occurs under our dynamic generalization
to the perturbed metric.

###  III.2 Cosmology on the 4-brane

The dynamics of the 4-brane is described by the non-singular part of the field
equations near the 4-brane. Substituting the metric (31) into the perturbation
equations (14), we have

$\displaystyle\mathcal{F}_{1}^{~{}}\frac{\ddot{\mathcal{R}}}{\mathcal{R}}\\!\\!$
$\displaystyle\\!=\\!$
$\displaystyle\\!\\!\big{[}\mathcal{F}_{2}^{~{}}(1-\mathcal{H}_{~{}}^{2})-\mathcal{F}_{3}^{~{}}\mathcal{H}^{2}_{~{}}-\mathcal{F}_{4}^{~{}}\mathcal{H}_{~{}}^{2}(1-\mathcal{H}_{~{}}^{2})\big{]}\mathcal{R}_{~{}}^{2}$
(40a)
$\displaystyle\\!\\!-\mathcal{F}_{5}^{~{}}(1-\mathcal{H}_{~{}}^{2})\mathcal{H}\frac{4\mathcal{H}-1-4\mathcal{H}_{~{}}^{3}}{\mathcal{H}+(1-\mathcal{H}_{~{}}^{2})^{2}_{~{}}}\mathcal{R}_{~{}}^{2},$
$\displaystyle-\frac{\ddot{\mathcal{R}}}{\mathcal{R}^{3}_{~{}}}\\!\\!$
$\displaystyle\\!=\\!$
$\displaystyle\\!\\!\frac{\mathcal{F}_{8}^{~{}}+\mathcal{F}_{9}^{~{}}\mathcal{H}^{2}_{~{}}+\mathcal{F}_{10}^{~{}}(1-\mathcal{H}_{~{}}^{2})}{\mathcal{F}_{6}^{~{}}-\mathcal{F}_{7}^{~{}}(1-\mathcal{H}_{~{}}^{2})}\big{[}\mathcal{H}+(1-\mathcal{H}_{~{}}^{2})^{2}_{~{}}\big{]}$
(40b)
$\displaystyle\\!\\!+\mathcal{F}_{11}^{~{}}\frac{\mathcal{H}^{2}_{~{}}}{1-\mathcal{H}_{~{}}^{2}}\frac{\mathcal{H}+(1-\mathcal{H}_{~{}}^{2})^{2}_{~{}}}{\mathcal{F}_{6}^{~{}}-\mathcal{F}_{7}^{~{}}(1-\mathcal{H}_{~{}}^{2})}$
$\displaystyle\\!\\!+\frac{\mathcal{F}_{12}^{~{}}+\mathcal{F}_{13}^{~{}}(1-\mathcal{H}_{~{}}^{2})}{\mathcal{F}_{6}^{~{}}-\mathcal{F}_{7}^{~{}}(1-\mathcal{H}_{~{}}^{2})}\mathcal{H}^{2}_{~{}},$

where $\mathcal{F}_{i}^{~{}}$’s are the parameter functions in terms of $d$,
$\Lambda$, $\rho$, and $p$. The dynamics of the 4-brane is governed by $\rho$
and $p$. Note that with (37) and (38), the variables of the 4-brane could be
converted into the cosmological quantities through

$\displaystyle\frac{1}{\mathcal{H}^{2}_{~{}}}\\!\\!$ $\displaystyle\\!=\\!$
$\displaystyle\\!\\!1+\frac{(d+3)(d+2)}{-2\Lambda}H^{2}_{~{}},$ (41a)
$\displaystyle\frac{\ddot{\mathcal{R}}}{\mathcal{R}}\\!\\!$
$\displaystyle\\!=\\!$
$\displaystyle\\!\\!\bigg{[}\frac{\partial^{2}_{~{}}\mathcal{R}}{\partial
t^{2}_{~{}}}\frac{1}{\mathcal{R}}+H^{2}_{~{}}+2\frac{(d+3)(d+2)}{-2\Lambda}H^{4}_{~{}}\bigg{]}\mathcal{R}^{2}_{~{}}\mathcal{H}^{4}_{~{}}.~{}~{}~{}~{}~{}~{}~{}$
(41b)

So the cosmology on the 4-brane could be derived from the brane’s dynamic
equations (40). However, it is not easy to recover the standard form of
cosmology directly because of the higher-order terms
$\mathcal{O}(\mathcal{H}^{3}_{~{}})$ in the equations (40). We therefore
employ the following parameterization for each of the cosmological quantities:

$\mathcal{Q}=\mathcal{Q}^{(0)}_{~{}}+\mathcal{Q}^{(1)}_{~{}}(\rho,p)+\mathcal{Q}^{(2)}_{~{}}(\rho^{2}_{~{}},p^{2}_{~{}})+\ldots\,,$
(42)

where we assume that the effects of extra dimensions are entirely incorporated
within the higher-order terms $\mathcal{O}(\rho^{2}_{~{}},p^{2}_{~{}})$. The
lower-order terms should satisfy the present cosmological observations. They
should be consistent with the prediction from the standard cosmology on the
4-brane. Thus, they would be the dominant terms in the expression (42) for the
late universe. On the contrary, the higher-order terms are expected to
contribute considerable corrections primarily during the early universe. They
are therefore treated as perturbations to $Q$ in the context of the late
universe.

Substituting Eqs. (41) and (42) into Eq. (40), we find the following modified
Friedmann equation:

$\displaystyle\frac{\partial^{2}_{~{}}\mathcal{R}}{\partial
t^{2}_{~{}}}\frac{1}{\mathcal{R}}+H^{2}_{~{}}\\!\\!$ $\displaystyle\\!=\\!$
$\displaystyle\\!\\!\frac{\rho-3p}{6M_{\text{pl}}^{2}}+\frac{\mathcal{P}_{1}^{~{}}\,p^{2}_{~{}}+\mathcal{P}_{2}^{~{}}\,p\,\rho+\mathcal{P}_{3}^{~{}}\,\rho^{2}_{~{}}}{M^{2D-4}_{~{}}V^{2}_{~{}}}$
(43) $\displaystyle\\!\\!+\mathcal{O}(\rho^{3}_{~{}},p^{3}_{~{}}),$

where $\mathcal{P}_{1}^{~{}}$, $\mathcal{P}_{2}^{~{}}$, and
$\mathcal{P}_{3}^{~{}}$ are parameter functions only related to the
dimensionality of the extra space. The bulk cosmological constant has been
finely tuned to

$\displaystyle\Lambda\\!\\!$ $\displaystyle\\!=\\!$
$\displaystyle\\!\\!-\frac{(7d+17)(d+3)}{36(d+2)}\mathcal{P}_{4}^{~{}}\lambda$
(44) $\displaystyle\\!=\\!$
$\displaystyle\\!\\!-\frac{(d+3)(d+2)}{2}\mathcal{P}_{4}^{2}\bigg{(}\frac{M^{D-2}_{~{}}V}{M_{\text{pl}}^{2}}\bigg{)}^{2}_{~{}}$

to reproduce the predictions of general relativity at the leading order (see
the first term on the r.h.s. of Eq. (43)). Here, $\mathcal{P}_{4}^{~{}}$ is
also the parameter function solely dependent on $d$.

Figure 1: Parameter functions $\mathcal{P}_{i}^{~{}}$’s with respect to the
dimension of the extra space. (a) Analytical results for the 4-brane with a
vanishing effective cosmological constant. (b) Fitting results for the
$\text{dS}_{4}^{~{}}$ brane. The gray area indicates the region where singular
behaviors appear in the numerical results.

We plot Fig. 1 to show the behaviors of $\mathcal{P}_{i}^{~{}}$’s. The
parameter $\lambda$ is supposed to be the bare cosmological constant (on the
4-brane) introduced by $\rho\rightarrow\rho+M_{\text{pl}}^{2}\lambda$. As is
shown in Eq. (43), the modified Friedmann equation adopts a vanishing
effective cosmological constant $\Lambda_{\text{eff}}^{~{}}$ on the 4-brane,
which, as we will see later, is related to both the bulk cosmological constant
and the bare cosmological constant. It can describe the normal expansion of
the late universe with $\Lambda_{\text{eff}}^{~{}}=0$ through the leading-
order term, and provides nontrivial corrections to the expansion of the
universe during the early universe.

For the (A)dS 4-brane, it is more difficult to simplify the modified Friedmann
equation even if we apply the parameterization (42). Here, we can introduce
the low-energy approximation to Eqs. (40). Then, with the following fine-
tuning on the bulk cosmological constant and the bare cosmological constant:

$\displaystyle\Lambda\\!\\!$ $\displaystyle\\!=\\!$
$\displaystyle\\!\\!-\frac{(d+3)(d+2)}{2}\bigg{(}\frac{M^{D-2}_{~{}}V}{M_{\text{pl}}^{2}}\bigg{)}^{2}_{~{}}\mathcal{Y}^{2}_{~{}},$
(45a) $\displaystyle\lambda\\!\\!$ $\displaystyle\\!=\\!$
$\displaystyle\\!\\!\frac{-2\Lambda}{(d+3)(d+2)}\mathcal{X},$ (45b)

the modified Friedmann equation becomes

$\displaystyle\frac{\partial^{2}\mathcal{R}}{\partial
t^{2}}\frac{1}{\mathcal{R}}+H^{2}\\!\\!$ $\displaystyle\\!=\\!$
$\displaystyle\\!\\!\frac{2}{3}\Lambda_{\text{eff}}^{~{}}+\frac{\rho-3p}{6M_{\text{pl}}^{2}}+\frac{\mathcal{P}_{6}^{~{}}\,p^{2}_{~{}}+\mathcal{P}_{7}^{~{}}\,p\,\rho+\mathcal{P}_{8}^{~{}}\,\rho^{2}}{\mathcal{P}_{5}^{~{}}M^{2D-4}_{~{}}V^{2}_{~{}}}$
(46) $\displaystyle\\!\\!+\mathcal{O}(\rho^{3}_{~{}},p^{3}_{~{}}),$

where $\mathcal{P}_{5}^{~{}}$, $\mathcal{P}_{6}^{~{}}$,
$\mathcal{P}_{7}^{~{}}$, and $\mathcal{P}_{8}^{~{}}$ are parameter functions
in terms of $\mathcal{X}$, $\mathcal{Y}$, and $d$. The parameter functions
$\mathcal{X}$ and $\mathcal{Y}$ could be numerically solved from Eq. (40).

Figure 2: Numerical results of the effective cosmological constant with
respect to the dimension of the extra space. (a) For an $\text{AdS}_{4}^{~{}}$
brane, the blue and black lines correspond to two different branches of
fitting results. (b) For a $\text{dS}_{4}^{~{}}$ brane, the black line is the
unique fitting result. In both two cases, red points are the numerical
results. The rescaled effective cosmological constant is dimensionless, and is
defined by
$\tilde{\Lambda}_{\text{eff}}^{~{}}=\big{(}\frac{M_{\text{pl}}^{2}}{M^{D-2}_{~{}}V}\big{)}^{2}_{~{}}\Lambda_{\text{eff}}^{~{}}$.

The effective cosmological constant is related to $\Lambda$ and $\lambda$ by

$\Lambda_{\text{eff}}^{~{}}=\frac{-6\Lambda}{(d+3)(d+2)}\frac{1+\mathcal{X}\mathcal{Y}}{(d-3)-\mathcal{X}\mathcal{Y}}\,.$
(47)

As is shown in Fig. 2, it has two branches of negative solutions, referring to
two kinds of $\text{AdS}_{4}^{~{}}$ brane in the model. Although our numerical
calculation fails to read out the result for $\Lambda_{\text{eff}}^{~{}}<0$
when $1<d<8$, we can introduce the fitting function matching the numerical
result to give a semi-analytical analysis. Our model also allows the embedding
of a $\text{dS}_{4}^{~{}}$ brane in the bulk. As is shown in Fig. 2, the
effective cosmological constant is positive for $\Lambda$ and $\lambda$ given
by (45). For the sake of simplification, here we only show our fitting results
of the parameter functions $\mathcal{P}_{5}^{~{}}$, $\mathcal{P}_{6}^{~{}}$,
$\mathcal{P}_{7}^{~{}}$, and $\mathcal{P}_{8}^{~{}}$ for the
$\text{dS}_{4}^{~{}}$ brane in Fig. 1.

##  IV Conclusion and discussion

In this paper, we employed a new approach to construct a dynamical braneworld
model with the 4-brane having nonlinear back-reaction to the
$\text{AdS}_{D}^{~{}}$ spacetime.

We considered a $D$-dimensional spacetime with a bulk cosmological constant.
The ordinary matter is confined on a 4-brane embedded in this spacetime. The
underlying gravity is described by the $D$-dimensional general relativity.
First, we assume that the 4-brane is fixed in the bulk through the condition
(5). The singularity of the energy-momentum tensor (7) then makes the metric
nonsmooth on the location of the 4-brane. Since the 4-brane is treated as a
small perturbation, it only contributes nonsmooth corrections to the metric at
the perturbation level. In this case, the background field equations are the
vacuum Einstein equations with a smooth background metric. The local solution
(17) of the linearized metric perturbation in the $\text{AdS}_{D}^{~{}}$ bulk
(9) was derived from the singularity part of the linearized perturbation
equations. Analyzing the behavior of the nonlinear perturbations, we found the
parameterization (18) of the nonlinearly perturbed metric. This
parameterization extends the validity of the metric perturbation solution to
the whole bulk by taking all the nonlinear-in-$|R-R_{0}^{~{}}|$ terms into
account. Under a coordinate transformation, it can recover the background
metric at infinity while introducing a correction to the metric near the
4-brane. Note that the parameterization (18) is only a subset of a more
general form:

$\displaystyle\text{d}s_{D}^{2}\\!\\!$ $\displaystyle\\!=\\!$
$\displaystyle\\!\\!A\,\text{d}R^{2}_{~{}}+B\,e^{f_{B}}_{~{}}\,\text{d}\Omega_{d-1}^{2}-C\,e^{f_{T}}_{~{}}\,\text{d}T^{2}_{~{}}+\,C\,e^{f_{s}}_{~{}}\,\text{d}\Sigma^{2}_{3}\,,$

where

$\displaystyle f_{B}^{~{}}\\!\\!$ $\displaystyle\\!=\\!$
$\displaystyle\\!\\!\mathcal{B}_{0}^{(1)}|R-R_{0}^{~{}}|+\mathcal{B}_{0}^{(2)}(R-R_{0}^{~{}})^{2}_{~{}}+\ldots\,,$
(49a) $\displaystyle f_{T}^{~{}}\\!\\!$ $\displaystyle\\!=\\!$
$\displaystyle\\!\\!\mathcal{C}_{T0}^{(1)}|R-R_{0}^{~{}}|+\mathcal{C}_{T0}^{(2)}(R-R_{0}^{~{}})^{2}_{~{}}+\ldots\,,$
(49b) $\displaystyle f_{s}^{~{}}\\!\\!$ $\displaystyle\\!=\\!$
$\displaystyle\\!\\!\mathcal{C}_{s0}^{(1)}|R-R_{0}^{~{}}|+\mathcal{C}_{s0}^{(2)}(R-R_{0}^{~{}})^{2}_{~{}}+\ldots\,.$
(49c)

Apparently, the more terms we reserve in the exponents, the more precise the
parameterization becomes. However, this improvement in precision comes at the
expense of increasing computational complexity.

The metric perturbation is generally time-dependent, since its source can
evolute with the bulk time. In addition, the evolution of the matter on the
4-brane could also trigger the motion of the 4-brane in the extra space.
Unlike the static 4-brane, a dynamical 4-brane can contribute extra
corrections to the background spacetime. We thus employed the parameterization
(31) of the perturbed metric by a time-dependent generalization of (18).
Similarly to the previous case, the parameters therein were derived from the
singular part of the linearized perturbation equations. It does not imply that
the parameterization (31) only refers to a linearized perturbed metric.
Indeed, the linearized perturbation only corresponds to the leading-order
approximation of (31) with respect to $|R-\mathcal{R}|$. Therefore, the
parameters in the full solutions also satisfy the linearized perturbation
equations. Furthermore, we proved that under the low-energy approximation, the
metric perturbation is stable and remains small during the motion of the
4-brane. In fact, this nature is reserved by the nonlinear part of the
perturbation. It can be verified that, with a time-dependent generalization of
(17), the perturbed metric can diverge to infinity along with the motion of
the 4-brane. So a perturbative dynamical braneworld model with only the
linearized perturbation would be unstable.

In addition to the stability of the dynamical braneworld model, the cosmology
on the 4-brane is also an important topic. In our model, the brane cosmology
is related to the equations that govern the dynamics of the 4-brane through
the mapping between two metrics defined on the 4-brane, i.e., the induced
metric (29) and the FLRW metric (36). The relation (38) does not alter the
nature of spacetime. Thus the accelerating expansion of the 4-brane clearly
indicates the presence of dynamics for the 4-brane in the bulk. Although a
delicate cancellation of the dynamics terms in the field equations might
assist in achieving the accelerated expansion of the universe on the 4-brane
fixed in the (static) bulk, it would be so tricky that we would not expect it
to happen. Assuming that extra dimensions give small corrections to the
cosmology for the late-time universe, we introduced the parameterization (42)
to each of the cosmological parameters in the field equations, and then
derived the modified Friedmann equations. Unlike 4-dimensional theories, the
curvature of spacetime in the context of cosmology is determined by the
effective cosmological constant rather than the bare one. It was found that by
balancing the bare cosmological constant and the bulk cosmological constant
[see Eq. (44)], the standard Firedmann equations with a vanishing effective
cosmological constant could be recovered at leading order. Note that in this
case, the parameters in the modified Friedmann equations could still be solved
analytically. However, once the effective cosmological constant is
nonvaninshing, the calculation becomes significantly more complicated.
Therefore we introduced the low-energy approximation to simplify the field
equations. In this case, the fine-tuning (45) is required, and the parameters
in the modified Friedmann equations can only be solved numerically. We found
that when the $\text{AdS}_{4}^{~{}}$ brane is embedded in the bulk, there are
two branches of solutions for the effective cosmological constant. Our
numerical results also support the existence of the universe with a positive
effective cosmological constant, which is consistent with the dS expansion of
our universe. In conclusion, our model allows for the presence of a 4-brane
with a vanishing effective cosmological constant, an $\text{AdS}_{4}^{~{}}$
brane, and more importantly, a $\text{dS}_{4}^{~{}}$ brane in the bulk.

## Acknowledgements

We thank Yu-Xiao Liu and Wu-Zhong Guo for useful discussions. This work was
supported by the National Natural Science Foundation of China (Grants No.
12247142, No. 12047564, and No. 11875136).

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