# General perturbations in D+1 standard and brane cosmology revisited

Zoé Delfín1 Rubén Cordero1 Tonatiuh Matos2 Miguel A. García-Aspeitia3
1Departamento de Física, Escuela Superior de Fíısica y Matemáticas del
Instituto Politécnico Nacional, Unidad Adolfo López Mateos, Edificio 9, 07738,
CDMX, México  2Departamento de Física, Centro de Investigación y de Estudios
Avanzados del IPN, A.P. 14-740, 07000 CDMX, México.  3Depto. de Física y
Matemáticas, Universidad Iberoamericana Ciudad de México, Prolongación Paseo
de la Reforma 880, 01219, CDMX, México.

###### Abstract

In the present work, we generalize the theory of perturbations in a
dimensional space-time $D+1$, where $D$ is the number of spatial dimensions,
with cosmological constant, studying scalar, vector and tensor perturbations,
as well as its structure in Newtonian and synchronous gauge. We also show the
theory of perturbations in the context of brane cosmology, where branes are
embedded in a set of D-spatial dimensions, a temporal dimension, and an
additional spatial dimension. In both standard and brane cosmology,
unperturbed spacetime is provided with a Friedmann-Lemaitre-Robertson-Walker
metric and arbitrary sectional curvature, the matter content has the shape of
a perfect fluid. In addition, we consider the arbitrary sectional curvature,
obtaining the respective equations in the Newtonian and synchronous norm. We
highlight that the results presented in this article can be used to treat
brane cosmology with a more efficient approach.

General Relativity, Brane Worlds, Cosmological perturbations, Newtonian gauge.

###### pacs:

##  I Introduction

Braneworlds have been extensively studied in the literature to solve problems
in scenarios associated with particle physics and cosmology. In cosmology, the
main objectives are to address the problem of the observed acceleration of the
Universe Riess  _et al._ (1998); Perlmutter  _et al._ (1999) and the nature of
dark matter, which have not yet been resolved. For now, the best candidate to
understand the puzzle of accelerating expansion is the well-known cosmological
constant (CC). For dark matter, the best candidates for this missing mass are
supposed to be dust or an ultralight scalar field Matos and Guzman (2000);
Matos and Urena-Lopez (2001). However, we do not have any detection of any of
them, and the CC is plagued of unsolved problems like the fine tuning and
coincidence problems (see Carroll (2001) for details), seeing ourselves in
need to change the paradigm and explore other alternatives for the Universe
acceleration Motta  _et al._ (2021) and the dark matter.

The idea of incorporate extra dimension was born due to the hierarchy problem
associated with the weakness of gravity in comparison with the other forces.
From an heuristic point of view, the addition of extra dimensions generates
corrections in the gravitational potential that scales as $\sim r^{-(d+1)}$
where $d$ is the number of dimensions. In this case, several experiments
contribute with stringent values for the existence of extra dimensions Long
_et al._ (2003). The Randall-Sundrum (RS) models incorporates one or two
cosmological branes into a five dimensional bulk with anti-de Sitter geometry,
with the aim of alleviate the hierarchy problem, in this case, the RS model
with only one brane is more successful due to its simplicity Randall and
Sundrum (1999a, b). Thus, under this background, it is possible to construct
the covariant approach in where a modified Einstein equation rise with extra
terms that provides with extra richness to the behavior of gravity in our
brane. We mention also the Arkani-Hamed-Dimopoulos-Dvali (ADD) model Arkani-
Hamed  _et al._ (1998); Antoniadis  _et al._ (1998); Akama (1982) in where the
idea of a large volume for the compact extra dimensions would generates
changes in the fundamental Planck scale. The Dvali-Gabadadze-Porrati (DGP)
Dvali  _et al._ (2000) is another brane model in where the transition to a
five dimensional space time mimics the observed Universe acceleration.
However, when the equations are transferred to a cosmological context and put
into the test using cosmological data samples, the DGP model present conflict
with observables Fang  _et al._ (2008).

The covariant form of the Einstein equation modifies the standard model of
cosmology, existing new fields like the well known dark radiation that comes
from the projection of the Weyl term in five dimensions. In the same fashion
as happens in standard cosmology, in braneworlds linear perturbations of the
modified field equations are used in order to study the acoustic peaks of the
cosmic microwave background radiation (CMB) noticing changes mainly in the
first acoustic peak Koyama (2003). From the astrophysics and cosmological
point of view, the gravitational waves generated in the braneworld context
also have been studied and in particular in the inflationary scenario Frolov
and Kofman (2002). Recent studies shown that braneworld cosmologies based in
RS context present difficulties due to the tension observed in the constraints
of the free parameters when it is exposed to cosmological data samples García-
Aspeitia  _et al._ (2017). However the braneworlds based in RS model but with
a variable brane tension are successful in alleviate the tension with
observables and additionally, the brane tension acts like the causative of the
current Universe acceleration Garcia-Aspeitia  _et al._ (2018). We encourage
to the reader the revision of Maartens and Koyama (2010) in where it is
presented several brane world models.

In the present work, we are interested in developing a formalism where the
branes when interacting generate perturbations that later give rise to the
formation of structures or gravitational waves. The branes studied in this
paper can be constituted of arbitrary matter (for example, scalar fields for
dark matter or spin 1 fields for the interactions of the standard model of
particles). Thus, we present a revision, beginning with perturbations with a
generalization of $D+1$ space-time, showing the scalar, vector and tensor
modes together with its expressions written in different gauges. In addition,
we show the brane perturbations embedded in a $D+1$ dimensional bulk assuming
a Friedmann-Lemaitre-Robertson-Walker (FLRW) metric geometry for the brane,
where the Newtonian and Synchronous gauges for perturbations are also shown.
Our expression are more explicit and can help with a more efficient
computational analysis when it is studied the cosmic background radiation,
structure formation, among others,

The outline of the paper is in the following form: in Sec. II we revisit the
perturbations in $D+1$ dimensions, adding new information lacked in other
papers, showing the scalar, vector and tensor modes. In Sec. III we present
the different gauges centering our attention in the Newtonian and Synchronous
gauges. In Sec. IV we develop a study of perturbations on a D+1-dimensional
brane embedded in a D+1+1-dimensional bulk, in similarity to that realized in
Standard Cosmology. Sec. V is devoted to present the Newtonian and Synchronous
gauges for a brane. In all sections, we consider that the spatial part of the
metrics is of $D$ dimension and with arbitrary sectional curvature. Finally in
Sec. VI we present the discussion and conclusions. We henceforth use units in
which $\hbar=c=k_{B}=1$ (natural units).

##  II Perturbations in $D+1$ dimensions

We initiate our analysis with the line element for this space-time which reads
as

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle\gamma_{\tau\tau}d\tau^{2}+\gamma_{ij}dx^{i}dx^{j}=-N^{2}d\tau^{2}+a^{2}\tilde{\gamma}_{ij}dx^{i}dx^{j}.$
(1)

Considering small perturbations around the metric $\gamma_{\mu\nu}$ in $D+1$
dimensions Weinberg (2008); Dodelson (2003), we have
$\bar{\gamma}_{\mu\nu}=\gamma_{\mu\nu}+\eta_{\mu\nu}$, where $\gamma_{\mu\nu}$
is the unperturbed metric tensor and $\eta_{\mu\nu}\ll 1$ is the perturbation
which is symmetric under index commutations as $\eta_{\mu\nu}=\eta_{\nu\mu}$.
All perturbed quantities will be denoted with a bar, in the form
$\bar{\gamma}^{\mu\nu}=\gamma^{\mu\nu}+\eta^{\mu\nu}$, $\bar{\gamma}^{\mu}_{\
\nu}=\gamma^{\mu}_{\ \nu}$. Considering the first order perturbation under the
Friedmann-Lemaitre-Robertson-Walker (FLRW) $D+1$ metric, with arbitrary
sectional curvature denoted by $K$, we have that the metric tensor can be
written as follows $\gamma_{\tau\tau}=-\textsl{N}^{2}$ ,
$\gamma_{i\tau}=\gamma_{\tau i}=0$, $\gamma_{ij}=a^{2}\tilde{\gamma}_{ij}$,
where $a\equiv a(\tau)$ is the scale factor and $N=a$ (for the conformal time)
and $N=1$ (for the cosmic time). We use the tilde notation to indicate that
the object in question is purely spatial. Perturbations fulfill the following
relations

$\displaystyle\eta^{\mu\nu}=-\gamma^{\mu\rho}\gamma^{\nu\sigma}\eta_{\rho\sigma},\quad\eta_{\mu\nu}=-\gamma_{\mu\rho}\gamma_{\nu\sigma}\eta^{\rho\sigma},\quad\gamma^{\mu\sigma}\eta_{\sigma\nu}=-\eta^{\mu\sigma}\gamma_{\sigma\nu}$
. (2)

In order to construct the perturbed $D+1$ Einstein field equation, we proceed
to find the perturbed Christoffel symbols through the formula

$\displaystyle\delta\Gamma^{\mu}_{\nu\lambda}=\frac{1}{2}\gamma^{\mu\rho}\left[-2\eta_{\rho\sigma}\Gamma^{\sigma}_{\nu\lambda}+\partial_{\lambda}\eta_{\rho\nu}+\partial_{\nu}\eta_{\rho\lambda}-\partial_{\rho}\eta_{\nu\lambda}\right],$
(3)

here $\delta$ indicates perturbation. Therefore the non-null perturbed symbols
are

$\displaystyle\delta\Gamma^{\tau}_{\tau\tau}$ $\displaystyle=$
$\displaystyle\frac{1}{2\textsl{N}^{2}}\left(2\frac{\dot{\textsl{N}}}{\textsl{N}}\eta_{\tau\tau}-\dot{\eta}_{\tau\tau}\right),$
$\displaystyle\delta\Gamma^{\tau}_{\tau i}$ $\displaystyle=$
$\displaystyle\frac{1}{2\textsl{N}^{2}}\left(2\frac{\dot{a}}{a}\eta_{i\tau}-\partial_{i}\eta_{\tau\tau}\right),$
$\displaystyle\delta\Gamma^{\tau}_{ij}$ $\displaystyle=$
$\displaystyle\frac{1}{2\textsl{N}^{2}}\left(2\frac{a\dot{a}}{\textsl{N}^{2}}\tilde{\gamma}_{ij}\eta_{\tau\tau}-\nabla_{j}\eta_{i\tau}-\nabla_{i}\eta_{j\tau}+\dot{\eta}_{ij}\right),\
\ \ $ $\displaystyle\delta\Gamma^{i}_{\tau\tau}$ $\displaystyle=$
$\displaystyle\frac{1}{2a^{2}}\tilde{\gamma}^{im}\left(-2\frac{\dot{\textsl{N}}}{\textsl{N}}\eta_{m\tau}+2\dot{\eta}_{m\tau}-\partial_{m}\eta_{\tau\tau}\right),$
$\displaystyle\delta\Gamma^{i}_{j\tau}$ $\displaystyle=$
$\displaystyle\frac{1}{2a^{2}}\tilde{\gamma}^{im}\left(-2\frac{\dot{a}}{a}\eta_{mj}+\dot{\eta}_{mj}+\nabla_{j}\eta_{m\tau}-\nabla_{m}\eta_{j\tau}\right),$
$\displaystyle\delta\Gamma^{i}_{jk}$ $\displaystyle=$
$\displaystyle\frac{1}{2a^{2}}\tilde{\gamma}^{im}\Bigl{(}-2\frac{a\dot{a}}{\textsl{N}^{2}}\tilde{\gamma}_{jk}\eta_{m\tau}+\nabla_{k}\eta_{mj}+\nabla_{j}\eta_{mk}-\nabla_{m}\eta_{jk}\Bigr{)},$
(4)

where $\dot{\eta}_{\mu\nu}$ and $\nabla_{\lambda}{\eta}_{\mu\nu}$ are the
temporal and covariant derivatives of $\eta_{\mu\nu}$, respectively. Moreover,
the perturbed Ricci curvature tensor can be calculated through the formula

$\displaystyle\delta R_{\mu\kappa}$ $\displaystyle=$
$\displaystyle\partial_{\kappa}\delta\Gamma^{\lambda}_{\lambda\mu}-\partial_{\lambda}\delta\Gamma^{\lambda}_{\mu\kappa}+\Gamma^{\lambda}_{\mu\sigma}\delta\Gamma^{\sigma}_{\kappa\lambda}+\Gamma^{\sigma}_{\kappa\lambda}\delta\Gamma^{\lambda}_{\mu\sigma}-\Gamma^{\lambda}_{\mu\kappa}\delta\Gamma^{\sigma}_{\sigma\lambda}-\delta\Gamma^{\lambda}_{\mu\kappa}\Gamma^{\sigma}_{\sigma\lambda},$
(5)

thus

$\displaystyle\delta R_{\tau\tau}$ $\displaystyle=$
$\displaystyle\frac{1}{2a^{2}}\tilde{\gamma}^{ml}\nabla_{m}\nabla_{l}\eta_{\tau\tau}+\frac{D\dot{a}}{2aN^{2}}\left(\dot{\eta}_{\tau\tau}-2\frac{\dot{N}}{N}\eta_{\tau\tau}\right)$
(6)
$\displaystyle+\frac{1}{2a^{2}}\tilde{\gamma}^{lm}\left[\ddot{\eta}_{ml}-\left(\frac{2\dot{a}}{a}+\frac{\dot{N}}{N}\right)\dot{\eta}_{ml}+2\left(\frac{\dot{a}^{2}}{a^{2}}+\frac{\dot{a}\dot{N}}{aN}-\frac{\ddot{a}}{a}\right)\eta_{ml}\right]$
$\displaystyle-\frac{1}{a^{2}}\tilde{\gamma}^{ml}\nabla_{m}\left(\dot{\eta}_{l\tau}-\frac{\dot{N}}{N}\eta_{l\tau}\right),$
$\displaystyle\delta R_{\tau j}$ $\displaystyle=$
$\displaystyle\frac{\left(D-1\right)\dot{a}}{2aN^{2}}\nabla_{j}\eta_{\tau\tau}+\frac{1}{2a^{2}}\tilde{\gamma}^{ml}\left(\nabla_{m}\nabla_{l}\eta_{j\tau}-\nabla_{j}\nabla_{l}\eta_{m\tau}\right)$
$\displaystyle-\left[\frac{\ddot{a}}{aN^{2}}+\left(D-1\right)\frac{\dot{a}^{2}}{a^{2}N^{2}}-\frac{\dot{a}\dot{N}}{aN^{3}}\right]\eta_{j\tau}+\frac{1}{2}\frac{\partial}{\partial
t}\left[\frac{1}{a^{2}}\gamma^{ml}\left(\nabla_{j}\eta_{ml}-\nabla_{l}\eta_{mj}\right)\right],$

$\displaystyle\delta R_{jk}$ $\displaystyle=$
$\displaystyle-\frac{1}{2N^{2}}\nabla_{j}\nabla_{k}\eta_{\tau\tau}-\left[\left(D-1\right)\frac{\dot{a}^{2}}{N^{4}}+\frac{a\ddot{a}}{N^{4}}-2\frac{a\dot{a}\dot{N}}{N^{5}}\right]\tilde{\gamma}_{jk}\eta_{\tau\tau}-\frac{1}{2}\frac{a\dot{a}}{N^{4}}\tilde{\gamma}_{jk}\dot{\eta}_{\tau\tau}$
$\displaystyle+\frac{1}{2a^{2}}\left[\tilde{\gamma}^{nm}\nabla_{n}\nabla_{m}\eta_{jk}-\nabla_{n}\nabla_{j}\left(\tilde{\gamma}^{nm}\eta_{mk}\right)\right.$
$\displaystyle\left.-\nabla_{n}\nabla_{k}\left(\tilde{\gamma}^{nm}\eta_{mj}\right)+\nabla_{j}\nabla_{k}\left(\tilde{\gamma}^{nm}\eta_{nm}\right)\right]$
$\displaystyle-\frac{1}{2N^{2}}\ddot{\eta}_{jk}+\left[\left(4-D\right)\frac{\dot{a}}{2aN^{2}}+\frac{\dot{N}}{2N^{3}}\right]\dot{\eta}_{jk}-\frac{\dot{a}}{2aN^{2}}\tilde{\gamma}_{jk}\tilde{\gamma}^{nm}\dot{\eta}_{nm}$
$\displaystyle+\frac{\dot{a}^{2}}{a^{2}N^{2}}\left(-2\eta_{jk}+\tilde{\gamma}_{jk}\tilde{\gamma}^{nm}\eta_{nm}\right)+\frac{\dot{a}}{aN^{2}}\tilde{\gamma}_{jk}\tilde{\gamma}^{nm}\nabla_{n}\eta_{m\tau}$
$\displaystyle+\frac{1}{2N^{2}}\left(\nabla_{j}\dot{\eta}_{k\tau}+\nabla_{k}\dot{\eta}_{j\tau}\right)+\frac{1}{2N^{2}}\left[\left(D-2\right)\frac{\dot{a}}{a}-\frac{\dot{N}}{N}\right]\left(\nabla_{j}\eta_{k\tau}+\nabla_{k}\eta_{j\tau}\right).$

Until now, we have developed the geometric side of the perturbed part of
Einstein’s equation. To continue with the matter side, we will first work with
the unperturbed equations.

Einstein equation can take the following form

$\displaystyle R_{\mu\nu}=\kappa S_{\mu\nu},$ (9)

where $\kappa=-8\pi G$, $G$ is the Newton gravitational constant and
$S_{\mu\nu}$ is the source tensor and can be written in terms of the momentum-
energy tensor and $\Lambda$ as the cosmological constant,

$\displaystyle S_{\mu\nu}\equiv
T_{\mu\nu}-\frac{1}{D-1}\left(\gamma_{\mu\nu}T^{\lambda}_{\
\lambda}-\frac{2}{\kappa}\Lambda\gamma_{\mu\nu}\right).$ (10)

We consider that the unperturbed energy-momentum tensor must be in the form of
a perfect fluid as

$\displaystyle T_{\mu\nu}=P\gamma_{\mu\nu}+\left(P+\rho\right)u_{\mu}u_{\nu},$
(11)

where $\rho$ is the energy density, $P$ is the pressure and the D+1 velocity
components are $u_{\tau}=-N$ and $u_{i}=0$ in a comoving system, then, the
components of the energy-momentum tensor are

$\displaystyle T_{\tau\tau}=N^{2}\rho,\quad T_{\tau i}=T_{i\tau}=0,\quad
T_{ij}=T_{ji}=a^{2}P\tilde{\gamma}_{ij},$ (12) $\displaystyle T^{\tau}_{\
\tau}=-\rho,\quad T^{i}_{\ j}=P\delta^{i}_{j},\quad T^{k}_{\ k}=DP,\quad
T^{\lambda}_{\ \lambda}=-\rho+DP$ ,

therefore the components of the source tensor are

$\displaystyle
S_{ij}=\frac{1}{D-1}\left(\rho-P+\frac{2}{\kappa}\Lambda\right)a^{2}\tilde{\gamma}_{ij},\quad
S_{i\tau}=0,\quad
S_{\tau\tau}=\frac{1}{D-1}\left[\left(D-2\right)\rho+DP-\frac{2}{\kappa}\Lambda\right]N^{2},$
(13)

and the components of the Ricci tensor are

$\displaystyle
R_{ij}=\left[\left(1-D\right)\left(K+\frac{\dot{a}^{2}}{N^{2}}\right)-\frac{a\ddot{a}}{N^{2}}\right]\tilde{\gamma}_{ij},\quad
R_{i\tau}=0,\quad
R_{\tau\tau}=D\left(-\frac{\dot{a}\dot{N}}{aN}+\frac{\ddot{a}}{a}\right).$
(14)

By substituting (13) and (14) into Eq. (9) we obtain the components of the
Einstein equation

$\displaystyle\left(1-D\right)\left(K+\frac{\dot{a}^{2}}{N^{2}}\right)-\frac{a\ddot{a}}{N^{2}}=\frac{\kappa}{D-1}\left(\rho-P+\frac{2}{\kappa}\Lambda\right)a^{2},$
$\displaystyle
D\left(-\frac{\dot{a}\dot{N}}{aN}+\frac{\ddot{a}}{a}\right)=\frac{\kappa}{D-1}\left[\left(D-2\right)\rho+DP-\frac{2}{\kappa}\Lambda\right]N^{2},$
(15)

thus we obtain the Friedmann equations in $D+1$ dimensions as

$\displaystyle\rho=\frac{D}{2\kappa}\left[\left(\frac{1-D}{a^{2}}\right)\left(K+\frac{\dot{a}^{2}}{N^{2}}\right)-\frac{\dot{a}\dot{N}}{aN^{3}}-\frac{2\Lambda}{D}\right],$
(16)

$\displaystyle
P=\frac{1}{2\kappa}\left[\frac{\left(D-2\right)\left(D-1\right)}{a^{2}}\left(K+\frac{\dot{a}^{2}}{N^{2}}\right)-D\frac{\dot{a}\dot{N}}{aN^{3}}+2\left(D-1\right)\frac{\ddot{a}}{aN^{2}}+2\Lambda\right].$
(17)

We continue developing the perturbation of the source tensor similarly as the
previous objects in the form

$\displaystyle\delta S_{\mu\nu}=\delta
T_{\mu\nu}-\frac{1}{D-1}\left[\gamma_{\mu\nu}\delta T^{\lambda}_{\
\lambda}+\left(T^{\lambda}_{\
\lambda}-\frac{2}{\kappa}\Lambda\right)\eta_{\mu\nu}\right],$ (18)

and substituting (16) and (17) in (18) we have

$\displaystyle\delta S_{\tau\tau}$ $\displaystyle=$ $\displaystyle\delta
T_{\tau\tau}+\frac{1}{D-1}N^{2}\delta T^{\lambda}_{\
\lambda}-\frac{D}{2\kappa}\left[\frac{\left(D-1\right)}{a^{2}}\left(K+\frac{\dot{a}^{2}}{N^{2}}\right)-\frac{\dot{a}\dot{N}}{aN^{3}}+2\frac{\ddot{a}}{aN^{2}}-\frac{4}{D\left(D-1\right)}\Lambda\right]\eta_{\tau\tau},$

$\displaystyle\delta S_{\tau j}=\delta S_{j\tau}$ $\displaystyle=$
$\displaystyle\delta
T_{j\tau}-\frac{D}{2\kappa}\left[\frac{\left(D-1\right)}{a^{2}}\left(K+\frac{\dot{a}^{2}}{N^{2}}\right)-\frac{\dot{a}\dot{N}}{aN^{3}}+2\frac{\ddot{a}}{aN^{2}}-\frac{4}{D\left(D-1\right)}\Lambda\right]\eta_{j\tau},$
(20)

$\displaystyle\delta S_{ij}$ $\displaystyle=$ $\displaystyle\delta
T_{ij}-\frac{1}{D-1}a^{2}\tilde{\gamma}_{ij}\delta T^{\lambda}_{\
\lambda}-\frac{D}{2\kappa}\left[\frac{\left(D-1\right)}{a^{2}}\left(K+\frac{\dot{a}^{2}}{N^{2}}\right)-\frac{\dot{a}\dot{N}}{aN^{3}}+2\frac{\ddot{a}}{aN^{2}}-\frac{4}{D\left(D-1\right)}\Lambda\right]\eta_{ij}.$

Putting together all the elements to present the perturbed version of Einstein
equations give us

$\displaystyle\kappa\left(\delta T_{\tau\tau}+\frac{1}{D-1}N^{2}\delta
T^{\lambda}_{\ \lambda}\right)=$
$\displaystyle\frac{D}{2}\left[\frac{\left(D-1\right)}{a^{2}}\left(K+\frac{\dot{a}^{2}}{N^{2}}\right)-3\frac{\dot{a}\dot{N}}{aN^{3}}+2\frac{\ddot{a}}{aN^{2}}-\frac{4}{D\left(D-1\right)}\Lambda\right]\eta_{\tau\tau}$
(22)
$\displaystyle+\frac{1}{2a^{2}}\tilde{\gamma}^{ml}\nabla_{m}\nabla_{l}\eta_{\tau\tau}+\frac{D\dot{a}}{2aN^{2}}\dot{\eta}_{\tau\tau}$
$\displaystyle+\frac{1}{2a^{2}}\tilde{\gamma}^{lm}\left[\ddot{\eta}_{ml}-\left(\frac{2\dot{a}}{a}+\frac{\dot{N}}{N}\right)\dot{\eta}_{ml}+2\left(\frac{\dot{a}^{2}}{a^{2}}+\frac{\dot{a}\dot{N}}{aN}-\frac{\ddot{a}}{a}\right)\eta_{ml}\right]$
$\displaystyle-\frac{1}{a^{2}}\tilde{\gamma}^{ml}\nabla_{m}\left(\dot{\eta}_{l\tau}-\frac{\dot{N}}{N}\eta_{l\tau}\right),$
$\displaystyle\kappa\delta T_{\tau j}=$
$\displaystyle\frac{\left(D-1\right)\dot{a}}{2aN^{2}}\nabla_{j}\eta_{\tau\tau}+\frac{1}{2a^{2}}\tilde{\gamma}^{ml}\left(\nabla_{m}\nabla_{l}\eta_{j\tau}-\nabla_{j}\nabla_{l}\eta_{m\tau}\right)$
(23)
$\displaystyle+\frac{1}{2}\left[\frac{D\left(D-1\right)}{a^{2}}K+\left(D-2\right)\left(D-1\right)\frac{\dot{a}^{2}}{a^{2}N^{2}}-\left(D-2\right)\frac{\dot{a}\dot{N}}{aN^{3}}+2\left(D-1\right)\frac{\ddot{a}}{aN^{2}}-\frac{4}{\left(D-1\right)}\Lambda\right]\eta_{j\tau}$
$\displaystyle+\frac{1}{2}\frac{\partial}{\partial
t}\left[\frac{1}{a^{2}}\gamma^{ml}\left(\nabla_{j}\eta_{ml}-\nabla_{l}\eta_{mj}\right)\right],$

$\displaystyle\kappa\left(\delta
T_{jk}-\frac{1}{D-1}a^{2}\tilde{\gamma}_{jk}\delta T^{\lambda}_{\
\lambda}\right)$ $\displaystyle=$
$\displaystyle-\frac{1}{2N^{2}}\nabla_{j}\nabla_{k}\eta_{\tau\tau}-\left[\left(D-1\right)\frac{\dot{a}^{2}}{N^{4}}+\frac{a\ddot{a}}{N^{4}}-2\frac{a\dot{a}\dot{N}}{N^{5}}\right]\tilde{\gamma}_{jk}\eta_{\tau\tau}-\frac{1}{2}\frac{a\dot{a}}{N^{4}}\tilde{\gamma}_{jk}\dot{\eta}_{\tau\tau}$
$\displaystyle+\frac{1}{2a^{2}}\left[\tilde{\gamma}^{nm}\nabla_{n}\nabla_{m}\eta_{jk}-\nabla_{n}\nabla_{j}\left(\tilde{\gamma}^{nm}\eta_{mk}\right)\right.$
$\displaystyle\left.-\nabla_{n}\nabla_{k}\left(\tilde{\gamma}^{nm}\eta_{mj}\right)+\nabla_{j}\nabla_{k}\left(\tilde{\gamma}^{nm}\eta_{nm}\right)\right]$
$\displaystyle-\frac{1}{2N^{2}}\ddot{\eta}_{jk}+\left[\left(4-D\right)\frac{\dot{a}}{2aN^{2}}+\frac{\dot{N}}{2N^{3}}\right]\dot{\eta}_{jk}-\frac{\dot{a}}{2aN^{2}}\tilde{\gamma}_{jk}\tilde{\gamma}^{nm}\dot{\eta}_{nm}$
$\displaystyle+\frac{\dot{a}^{2}}{a^{2}N^{2}}\tilde{\gamma}_{jk}\tilde{\gamma}^{nm}\eta_{nm}+\frac{\dot{a}}{aN^{2}}\tilde{\gamma}_{jk}\tilde{\gamma}^{nm}\nabla_{n}\eta_{m\tau}$
$\displaystyle+\frac{1}{2N^{2}}\left(\nabla_{j}\dot{\eta}_{k\tau}+\nabla_{k}\dot{\eta}_{j\tau}\right)+\frac{1}{2N^{2}}\left[\left(D-2\right)\frac{\dot{a}}{a}-\frac{\dot{N}}{N}\right]\left(\nabla_{j}\eta_{k\tau}+\nabla_{k}\eta_{j\tau}\right)$
$\displaystyle+\frac{D}{2}\left[\left(D-1\right)\frac{K}{a^{2}}+\left(D-1-\frac{4}{D}\right)\frac{\dot{a}^{2}}{a^{2}N^{2}}-\frac{\dot{a}\dot{N}}{aN^{3}}+2\frac{\ddot{a}}{aN^{2}}-\frac{4}{D\left(D-1\right)}\Lambda\right]\eta_{jk},$

The energy-momentum tensor is restricted to the conservation condition
$T^{\sigma}_{\ \nu;\sigma}=0$. The respective perturbed version is given by

$\displaystyle\partial_{\sigma}\delta T^{\sigma}_{\
\nu}+\Gamma^{\sigma}_{\sigma\rho}\delta T^{\rho}_{\
\nu}+\delta\Gamma^{\sigma}_{\sigma\rho}T^{\rho}_{\
\nu}-\Gamma^{\rho}_{\sigma\nu}\delta T^{\sigma}_{\
\rho}-\delta\Gamma^{\rho}_{\sigma\nu}T^{\sigma}_{\ \rho}=0,$ (25)

setting $\nu=j$ gives the equation of momentum conservation

$\displaystyle\partial_{\tau}\delta T^{\tau}_{\ j}+\nabla_{l}\delta T^{l}_{\
j}+\left[\frac{\dot{N}}{N}+\left(D-1\right)\frac{\dot{a}}{a}\right]\delta
T^{\tau}_{\ j}-\frac{a\dot{a}}{N^{2}}\tilde{\gamma}_{lj}\delta T^{l}_{\ \tau}$
$\displaystyle+\frac{1}{2N^{2}}\left(2\frac{\dot{a}}{a}\eta_{j\tau}-\nabla_{j}\eta_{\tau\tau}\right)\left(\rho+P\right)=0,$
(26)

and setting $\nu=0$ gives the equation of energy conservation

$\displaystyle\partial_{\tau}\delta T^{\tau}_{\ \tau}+\nabla_{l}\delta
T^{l}_{\ \tau}+D\frac{\dot{a}}{a}\delta T^{\tau}_{\
\tau}-\frac{\dot{a}}{a}\delta T^{l}_{\
l}+\frac{1}{2a^{2}}\tilde{\gamma}^{lm}\left(2\frac{\dot{a}}{a}\eta_{ml}-\dot{\eta}_{ml}\right)\left(\rho+P\right)=0.$
(27)

Perturbation of the metric can be decomposed into scalars, divergenceless
vectors and divergenceless traceless symmetric tensors, which are not coupled
to each other

$\displaystyle\eta_{\tau\tau}$ $\displaystyle=$ $\displaystyle-N^{2}E,$
$\displaystyle\eta_{\tau i}$ $\displaystyle=$
$\displaystyle\eta_{i\tau}=aN\left[\nabla_{i}F+G_{i}\right],$
$\displaystyle\eta_{ij}$ $\displaystyle=$ $\displaystyle
a^{2}\left[A\tilde{\gamma}_{ij}+\nabla_{i}\nabla_{j}B+\nabla_{j}C_{i}+\nabla_{i}C_{j}+D_{ij}\right],$
(28)

where $A$, $B$, $C_{i}$, $D_{ij}$, $E$, $F$ y $G_{i}$ are functions of
$\vec{x}$ and $t$, satisfying the conditions

$\displaystyle\varepsilon_{ijk}\tilde{\gamma}^{jl}\nabla_{l}\tilde{\gamma}^{km}\nabla_{m}F=0,\quad\varepsilon_{ijk}\tilde{\gamma}^{jl}\nabla_{l}\tilde{\gamma}^{km}\nabla_{m}B=0,$
$\displaystyle\nabla_{i}G^{i}=0,\quad\nabla_{i}C^{i}=0,\quad\nabla_{i}D^{i}_{\
j}=0,$ $\displaystyle D_{ij}=D_{ji},\quad D^{i}_{\ i}=0.$ (29)

Let us consider that the perturbed moment-energy tensor is also a perfect
fluid, then

$\displaystyle\delta T_{\mu\nu}$ $\displaystyle=$ $\displaystyle
P\eta_{\mu\nu}+\left(\rho+P\right)u_{\mu}\delta
u_{\nu}+\left(\rho+P\right)\delta u_{\mu}u_{\nu}$ (30)
$\displaystyle+\left(\delta\rho+\delta
P\right)u_{\mu}u_{\nu}+\gamma_{\mu\nu}\delta P,$

for this, we have the $D+1$ perturbed velocity $\bar{u}_{\mu}=u_{\mu}+\delta
u_{\mu}$, which must comply with the normalization conditions
$\gamma^{\mu\nu}u_{\mu}u_{\nu}=-1$, so we have the following relation

$\displaystyle\gamma^{\mu\nu}\delta
u_{\mu}=-\frac{1}{2}\eta^{\mu\nu}u_{\mu}\quad{\rm or}\quad\delta
u^{\mu}=\gamma^{\mu\sigma}\delta
u_{\sigma}-\gamma^{\mu\lambda}\gamma^{\eta\varsigma}\eta^{\lambda\varsigma}u_{\eta},$
(31)

then we have $\delta u_{\tau}=\eta_{\tau\tau}/2N$.

In general, the perturbation of velocity $\delta u_{i}$ can be decomposed into
the gradient of a scalar potential $\delta u$ and a divergenceless vector
$\delta u^{V}_{i}$. For the perturbation of the momentum-energy tensor we can
add the terms $\nabla_{i}\nabla_{j}\pi^{S}$,
$\nabla_{i}\pi^{V}_{j}\,+\,\nabla_{j}\pi^{V}_{i}$, and $\pi^{T}_{ij}$, which
represent dissipative corrections to the inertia tensor and the following
conditions are satisfied

$\nabla_{i}\pi^{iV}=\nabla_{i}\delta u^{iV}=0,\quad\nabla_{i}\pi^{iT}_{\
j}=0,\quad\pi^{iT}_{\ i}=0,$ (32)

with these corrections, the perturbation of the moment-energy tensor for a
perfect fluid takes the following form

$\displaystyle\delta
T_{\tau\tau}=-\rho\eta_{\tau\tau}+N^{2}\delta\rho,\quad\delta T_{i\tau}=\delta
T_{\tau i}=P\eta_{i\tau}-N\left(\rho+P\right)\left(\nabla_{i}\delta u+\delta
u^{V}_{i}\right),$ $\displaystyle\delta
T_{ij}=P\eta_{ij}+a^{2}\tilde{\gamma}_{ij}\delta
P+a^{2}\left(\nabla_{i}\nabla_{j}\pi^{S}+\nabla_{i}\pi^{V}_{j}+\nabla_{j}\pi^{V}_{i}+\pi^{T}_{ij}\right),$
(33)

From the mixed perturbed energy-momentum tensor $T^{\mu}_{\
\nu}=\gamma^{\mu\lambda}T_{\lambda\nu}$, we can obtain the perturbed part

$\displaystyle\delta T^{\mu}_{\ \nu}=\gamma^{\mu\lambda}\left(\delta
T_{\lambda\nu}-\eta_{\lambda\sigma}T^{\sigma}_{\ \nu}\right),$ (34)

in which we have the components

$\displaystyle\delta T^{\tau}_{\ \tau}=-\delta\rho,\quad\delta T^{i}_{\
j}=\delta^{i}_{j}\delta P,\quad\delta T^{\tau}_{\
i}=\frac{1}{N}\left(\rho+P\right)\delta u_{i},$ $\displaystyle\delta T^{i}_{\
\tau}=\frac{1}{a^{2}}\tilde{\gamma}^{im}\left[\left(\rho+P\right)\left(\eta_{m\tau}-N\delta
u_{m}\right)\right],$ $\displaystyle\delta T^{\lambda}_{\
\lambda}=-\delta\rho+D\delta P.$ (35)

###  II.1 Scalar Modes

In these modes the eight scalars $E$, $F$, $A$, $B$, $\delta\rho$, $\delta P$,
$\pi^{S}$ and $\delta u$ are involved. The part of Eq.
(LABEL:eq:EcEinsteinTensor) proportional to $\tilde{\gamma}_{jk}$ gives

$\displaystyle\frac{\kappa a^{2}}{D-1}\left[\delta\rho-\delta
P-\tilde{\gamma}^{nm}\nabla_{n}\nabla_{m}\pi^{S}\right]$ $\displaystyle=$
$\displaystyle\left[\left(D-1\right)\frac{\dot{a}^{2}}{N^{2}}+\frac{a\ddot{a}}{N^{2}}-\frac{a\dot{a}\dot{N}}{N^{3}}\right]E+\frac{1}{2}\frac{a\dot{a}}{N^{2}}\dot{E}$
$\displaystyle+\frac{1}{2}\tilde{\gamma}^{nm}\nabla_{n}\nabla_{m}A+\left(\frac{a^{2}\dot{N}}{2N^{3}}-D\frac{a\dot{a}}{N^{2}}\right)\dot{A}-\frac{a^{2}}{2N^{2}}\ddot{A}$
$\displaystyle-\frac{a\dot{a}}{2N^{2}}\tilde{\gamma}^{nm}\nabla_{n}\nabla_{m}\dot{B}+\frac{\dot{a}}{N}\tilde{\gamma}^{nm}\nabla_{n}\nabla_{m}F$
$\displaystyle+\left[\left(D-1\right)K+\frac{a\dot{a}\dot{N}}{N^{3}}-\frac{2}{D-1}\Lambda
a^{2}\right]A,$

additionally, the part of Eq. (LABEL:eq:EcEinsteinTensor) of the form
$\nabla_{j}\nabla_{k}S$ (where $S$ is any scalar) is

$\displaystyle\nabla_{j}\nabla_{k}\left\\{-2\kappa
a^{2}\pi^{S}+E+\left(D-2\right)A-\frac{a^{2}}{N^{2}}\ddot{B}+\left(\frac{a^{2}\dot{N}}{N^{3}}-D\frac{a\dot{a}}{N^{2}}\right)\dot{B}+2\frac{a}{N}\dot{F}\right.$
$\displaystyle+\left.2\left(D-1\right)\frac{\dot{a}}{N}F+\left[2\left(D-1\right)K+2\frac{a\dot{a}\dot{N}}{N^{3}}-\frac{4}{D-1}\Lambda
a^{2}\right]B\right\\}=0,$ (37)

for the part of Eq. (23) of the form $\nabla_{j}S$ (where $S$ is any scalar)
gives

$\displaystyle-\frac{2\kappa N}{D-1}\left(\rho+P\right)\nabla_{j}\delta
u=-\frac{\dot{a}}{a}\nabla_{j}E+\nabla_{j}\dot{A}+\left(2\frac{N}{a}K+\frac{2}{D-1}\frac{\dot{a}\dot{N}}{N^{2}}-\frac{4}{\left(D-1\right)^{2}}\Lambda
aN\right)\nabla_{j}F,$ (38)

for Eq. (22) we have

$\displaystyle\frac{\kappa N^{2}}{D-1}\left[\left(D-2\right)\delta\rho+D\delta
P+\tilde{\gamma}^{ml}\nabla_{m}\nabla_{l}\pi^{S}\right]$ $\displaystyle=$
$\displaystyle-\frac{N^{2}}{2a^{2}}\tilde{\gamma}^{ml}\nabla_{m}\nabla_{l}E-\frac{D\dot{a}}{2a}\dot{E}-\frac{N}{a}\tilde{\gamma}^{ml}\nabla_{m}\nabla_{l}\dot{F}$
(39)
$\displaystyle-\frac{\dot{a}}{a^{2}}N\tilde{\gamma}^{ml}\nabla_{m}\nabla_{l}F+\frac{D}{2}\ddot{A}-D\left(\frac{\ddot{a}}{a}-\frac{\dot{a}\dot{N}}{aN}+\frac{2}{D\left(D-1\right)}\Lambda
N^{2}\right)E$
$\displaystyle+\frac{1}{2}\tilde{\gamma}^{ml}\nabla_{m}\nabla_{l}\ddot{B}+\left(\frac{\dot{a}}{a}-\frac{\dot{N}}{2N}\right)\left(D\dot{A}+\tilde{\gamma}^{ml}\nabla_{m}\nabla_{l}\dot{B}\right).$

Finally, the momentum conservation condition (26) which is a derivative
$\nabla_{j}$ is

$\displaystyle\nabla_{j}\left[\delta
P+\tilde{\gamma}^{nm}\nabla_{n}\nabla_{m}\pi^{S}+\frac{1}{N}\partial_{\tau}\left[\left(\rho+P\right)\delta
u\right]+D\frac{\dot{a}}{aN}\left(\rho+P\right)\delta
u+\frac{1}{2}\left(\rho+P\right)E\right]=0,$ (40)

and the energy conservation condition (27) is

$\displaystyle\delta\dot{\rho}+\frac{D\dot{a}}{a}\left(\delta\rho+\delta
P\right)+\tilde{\gamma}^{ml}\nabla_{m}\nabla_{l}\left[\frac{N}{a}\left(\rho+P\right)\left(-F+a^{-1}\delta
u\right)+\frac{\dot{a}}{a}\pi^{S}\right]$
$\displaystyle+\frac{1}{2}\left(\rho+P\right)\partial_{\tau}\left[DA+\tilde{\gamma}^{ml}\nabla_{m}\nabla_{l}B\right]=0.$
(41)

Notice that Eqs. (40), (41), $\delta\rho$, $\delta P$, and $\pi^{S}$ are
elements of the perturbation to the total energy-momentum tensor, but the same
equations apply to each constituent of the universe that does not exchange
energy and momentum with other constituents.

###  II.2 Vector Modes

These modes that involve the divergenceless vectors are $G_{i}$, $C_{i}$,
$\delta u^{V}_{i}$, and $\pi^{V}_{i}$. The part of Eq.
(LABEL:eq:EcEinsteinTensor) for the form $\nabla_{k}V_{j}$ (where $V_{j}$ is a
divergenceless vector) give us

$\displaystyle\nabla_{k}\left\\{2\kappa
a^{2}\pi^{V}_{j}+\left(D\frac{a\dot{a}}{N^{2}}-\frac{a^{2}\dot{N}}{N^{3}}\right)\dot{C}_{j}+\frac{a^{2}}{N^{2}}\ddot{C}_{j}-\left(D-1\right)\frac{\dot{a}}{N}G_{j}\right.$
$\displaystyle-\left.\frac{a}{N}\dot{\gamma}_{j}+\left[-2\left(D-1\right)K+\frac{4}{D-1}\Lambda
a^{2}\right]C_{j}\right\\}=0,$ (42)

while the part of Eq. (23) of the form $V_{j}$ (where $V_{j}$ is any
divergenceless vector) reads

$\displaystyle-\kappa\left(\rho+P\right)N\delta
u^{V}_{j}=\frac{1}{2}\frac{N}{a}\tilde{\gamma}^{ml}\nabla_{m}\nabla_{l}G_{j}-\frac{1}{2}\tilde{\gamma}^{ml}\nabla_{m}\nabla_{l}\dot{C}_{j}+\left[\left(D-1\right)K\frac{N}{a}+\frac{\dot{a}\dot{N}}{N^{2}}-\frac{2}{D-1}\Lambda
aN\right]G_{j}.$ (43)

The equation associated to the conservation of momentum (26) that takes the
form of a divergenceless vector as

$\displaystyle\tilde{\gamma}^{lm}\nabla_{m}\nabla_{l}\pi^{V}_{j}+\frac{1}{N}\partial_{\tau}\left[\left(\rho+P\right)\delta
u^{V}_{j}\right]+D\frac{\dot{a}}{aN}\left(\rho+P\right)\delta u^{V}_{j}=0.$
(44)

In particular, for a perfect fluid $\pi^{V}_{i}=0$, and Eq. (44) tells us that
$\left(\rho+P\right)\delta u^{V}_{j}$ decays as $a^{-D}$. In this case with
$K=0$ and $\Lambda=0$, the Eq. (42) imply that the quantity
$G_{i}-a\dot{C}_{i}/N$ decays as $a^{-D+1}$. Notice that they decay of vector
modes do not play an important role in cosmology and therefore do not emerge
from inflationary models.

###  II.3 Tensor Modes

These modes involve the 2-symmetric traceless and divergenceless tensors
$D_{ij}$ and $\pi^{T}_{ij}$. The part of Eq. (LABEL:eq:EcEinsteinTensor) of
the form of a traceless and divergenceless tensor is the wave equation for
gravitational radiation, given by the expression

$\displaystyle 2\kappa
a^{2}\pi^{T}_{ij}=\tilde{\gamma}^{ml}\nabla_{m}\nabla_{l}D_{ij}-\frac{a^{2}}{N^{2}}\ddot{D}_{ij}-\left(D\frac{a\dot{a}}{N^{2}}-\frac{a^{2}\dot{N}}{N^{3}}\right)\dot{D}_{ij}+2\left[\left(D-1\right)K+\frac{a\dot{a}\dot{N}}{N^{3}}-\frac{2}{D-1}\Lambda
a^{2}\right]D_{ij}.$ (45)

Finally we have the condition of conservation of the current
$\nabla_{\mu}(nu^{\mu})=0$. This condition tells us that the unperturbed
density number complies $n\propto a^{-D}$, therefore from the perturbed part
we have

$\displaystyle\frac{n}{2a^{2}N}\tilde{\gamma}^{lm}\dot{h}_{ml}+\frac{n}{N}\partial_{\tau}\left(\frac{\delta
n}{n}\right)+\nabla_{l}\left(n\delta u^{l}\right)=0.$ (46)

From the previous equation when considering only the scalar part of the
perturbation of the metric, we have

$\displaystyle\frac{1}{N}\partial_{\tau}\left(\frac{\delta
n}{n}\right)+\frac{1}{2N}\left(D\dot{A}+\tilde{\gamma}^{lm}\nabla_{m}\nabla_{l}\dot{B}\right)+\frac{1}{a^{2}}\tilde{\gamma}^{lm}\nabla_{m}\nabla_{l}\left(\delta
u-aF\right)=0.$ (47)

Thus, equations (40), (41) and (47) form a complete set for scalar modes.

##  III Gauges in Perturbations

The solutions of the equations derived so far are non-physical scalar and
vector modes, they correspond to a coordinate transformation of the FLRW
metric and the momentum-energy tensor, to avoid this problem it is possible to
adopt a suitable conditions for the perturbed metric and moment-energy tensor.

Let start considering a space-time coordinate transformation as Weinberg
(2008)

$\displaystyle x^{\mu}\rightarrow
x^{\prime\mu}=x^{\mu}+\epsilon^{\mu}\left(x\right),$ (48)

with $\epsilon^{\mu}\left(x\right)\ll 1$. We work with the gauge
transformations, which affect only the field perturbations. For this purpose,
after making the coordinate transformation (115), we relabel coordinates by
dropping the prime on the coordinate argument, and we attribute the whole
change in $\bar{\gamma}_{\mu\nu}(x)$ to a change in the perturbation
$\eta_{\mu\nu}(x)$. The field equations should thus be invariant under the
gauge transformation
$\eta_{\mu\nu}\left(x\right)\rightarrow\eta_{\mu\nu}\left(x\right)+\Delta\eta_{\mu\nu}\left(x\right)$,
where

$\displaystyle\Delta\eta_{\mu\nu}\left(x\right)\equiv\bar{\gamma}^{\prime}_{\mu\nu}(x)-\bar{\gamma}_{\mu\nu}(x).$
(49)

To first order in $\epsilon(x)$ and $\eta_{\mu\nu}(x)$, Eq. (49) in terms of
unperturbed FLRW metric is

$\displaystyle\Delta\eta_{\mu\nu}\left(x\right)$ $\displaystyle=$
$\displaystyle\bar{\gamma}^{\prime}_{\mu\nu}(x^{\prime})-\frac{\partial\bar{\gamma}_{\mu\nu}(x)}{\partial
x^{\lambda}}\epsilon^{\lambda}\left(x\right)-\bar{\gamma}_{\mu\nu}(x)$
$\displaystyle=$
$\displaystyle-\gamma_{\lambda\mu}(x)\frac{\partial\epsilon^{\lambda}\left(x\right)}{\partial
x^{\nu}}-\gamma_{\lambda\nu}\left(x\right)\frac{\partial\epsilon^{\lambda}\left(x\right)}{\partial
x^{\mu}}-\frac{\partial\gamma_{\mu\nu}(x)}{\partial
x^{\lambda}}\epsilon^{\lambda}\left(x\right),$

or in more detail

$\displaystyle\Delta\eta_{\tau\tau}=2\frac{\dot{N}}{N}\epsilon_{\tau}-2\frac{\partial\epsilon_{\tau}}{\partial
t},$ $\displaystyle\Delta\eta_{\tau i}=-\frac{\partial\epsilon_{i}}{\partial
t}-\nabla_{i}\epsilon_{\tau}+2\frac{\dot{a}}{a}\epsilon_{i},$
$\displaystyle\Delta\eta_{ij}=-\nabla_{j}\epsilon_{i}-\nabla_{i}\epsilon_{j}+2\frac{a\dot{a}}{N^{2}}\tilde{\gamma}_{ij}\epsilon_{\tau},$
(51)

with all quantities evaluated at the same space-time coordinate point. The
field equations will be invariant only if the same gauge transformation is
applied to all tensors, and in particular to the energy-momentum tensor. A
more generalized formula analogous to Eq. (LABEL:eq:Deltah1) is

$\displaystyle\Delta\delta Z_{\mu\nu}\left(x\right)$ $\displaystyle=$
$\displaystyle-
Z_{\lambda\mu}(x)\frac{\partial\epsilon^{\lambda}\left(x\right)}{\partial
x^{\nu}}-Z_{\lambda\nu}\left(x\right)\frac{\partial\epsilon^{\lambda}\left(x\right)}{\partial
x^{\mu}}-\frac{\partial Z_{\mu\nu}(x)}{\partial
x^{\lambda}}\epsilon^{\lambda}\left(x\right),$ (52)

or explicitly

$\displaystyle\Delta\delta
Z_{\tau\tau}=-2\frac{\dot{N}}{N}\zeta\epsilon_{\tau}+2\zeta\frac{\partial\epsilon_{\tau}}{\partial
t}+\dot{\zeta}\epsilon_{\tau},$ $\displaystyle\Delta\delta Z_{\tau
i}=-\xi\frac{\partial\epsilon_{i}}{\partial
t}+\zeta\nabla_{i}\epsilon_{\tau}+2\xi\frac{\dot{a}}{a}\epsilon_{i},$
$\displaystyle\Delta\delta
Z_{ij}=-\xi\left(\nabla_{j}\epsilon_{i}+\nabla_{i}\epsilon_{j}\right)+\frac{1}{N^{2}}\frac{\partial}{\partial
t}\left(a^{2}\xi\right)\tilde{\gamma}_{ij}\epsilon_{\tau}.$ (53)

Notice that this applies to a tensor of second order, with components

$\displaystyle Z_{\tau\tau}=\zeta(t)N^{2};\quad\quad Z_{\tau
i}=Z_{i\tau}=0;\quad\quad Z_{ij}=\xi(t)a^{2}\tilde{\gamma}_{ij}.$ (54)

Note also, that we use $\delta$ for a perturbation, while $\Delta$ denotes the
change in a perturbation associated with a gauge transformation. In
particular, for the energy-momentum tensor $\zeta=\rho$ and $\xi=P$.

In order to write these gauge transformations in terms of the scalar, vector,
and tensor components, it is necessary to decompose the spatial part of
$\epsilon^{\mu}$ into the gradient of a spatial scalar, plus a divergenceless
vector, in the form

$\displaystyle\epsilon_{i}=\nabla_{i}\epsilon^{S}+\epsilon^{V}_{i},\quad\quad\nabla_{i}\epsilon_{i}^{V}=0.$
(55)

Thus the transformations (51) and (53) give us the gauge transformations of
the metric perturbation components defined by Eqs. (28)

$\displaystyle\Delta
E=\frac{2}{N^{2}}\left(\dot{\epsilon}_{\tau}-\frac{\dot{N}}{N}\epsilon_{\tau}\right),\quad\Delta
F=\frac{1}{aN}\left(-\epsilon_{\tau}-\dot{\epsilon}^{S}+2\frac{\dot{a}}{a}\epsilon^{S}\right),\quad\Delta
G_{i}=\frac{1}{aN}\left(-\dot{\epsilon}_{i}^{V}+2\frac{\dot{a}}{a}\epsilon_{i}^{V}\right),$
$\displaystyle\Delta A=2\frac{\dot{a}}{aN^{2}}\epsilon_{\tau},\quad\Delta
B=-\frac{2}{a^{2}}\epsilon^{S},\quad\Delta
C_{i}=-\frac{1}{a^{2}}\epsilon_{i}^{V},\quad\Delta D_{ij}=0,$ (56)

and from (33) for the pressure, energy density, and velocity potential, reads

$\displaystyle\Delta\delta\rho=\frac{\dot{\rho}\epsilon_{\tau}}{N^{2}},\quad\Delta\delta
P=\frac{\dot{P}\epsilon_{\tau}}{N^{2}},\quad\Delta\delta
u=-\frac{\epsilon_{\tau}}{N}.$ (57)

The other ingredients of the energy-momentum tensor are gauge invariant

$\displaystyle\Delta\pi^{S}=\Delta\pi^{i}_{V}=\Delta\pi^{ij}_{T}=\Delta\delta
u^{V}_{i}=0.$ (58)

Note in particular that the conditions $\pi^{S}=\pi^{i}_{V}=\pi^{ij}_{T}=0$
for a perfect fluid and the condition $\delta u^{V}_{i}=0$ for the potential
flow are gauge invariant. In general, any space-time-scalar $\vartheta(x)$ for
which $\vartheta^{\prime}(x^{\prime})=\vartheta(x)$ under arbitrary coordinate
transformations would undergo the change
$\Delta\delta\vartheta(x)\equiv\vartheta^{\prime}(x)-\vartheta(x)=\vartheta^{\prime}(x)-\vartheta^{\prime}(x^{\prime})$,
which to first order in perturbations is

$\displaystyle\Delta\delta\vartheta(x)=\vartheta(x)-\vartheta(x^{\prime})=-\frac{\partial\vartheta(t)}{\partial
x^{\mu}}\epsilon^{\mu}(x)=\frac{\dot{\vartheta}(t)}{N^{2}}\epsilon_{\tau}.$
(59)

This applies for instance to the number density $n$ or a scalar field
$\varphi$. For a perfect fluid both $\rho$ and $P$ are defined as scalars, and
the gauge transformations in Eq. (57) of $\delta\rho$ and $\delta P$ are other
special cases of Eq. (59). Likewise, for a perfect fluid the gauge
transformation in Eq. (57) of $\delta u$ can be derived from the vector
transformation law of $u_{\mu}$. Because the gauge transformation properties
of $\delta\rho$, $\delta P$, $\delta u$, etc. do not depend on the
conservation laws, Eqs. (57) and (58) apply to each individual constituent of
the universe in any case in which the energy-momentum tensor is a sum of terms
for different constituents of the universe, even if these individual terms are
not separately conserved.

It is possible to neglect the gauge degrees of freedom either by working only
with gauge-invariant quantities, or by choosing a gauge. The tensor quantities
$\pi_{ij}$ and $D_{ij}$ appearing in Eq. (45) are already gauge invariant, and
non gauge-fixing is necessary or possible. For the vector quantities
$\pi^{V}_{i}$, $\delta u^{V}_{i}$, $C_{i}$ and $G_{i}$, we can write Eqs.
(42)-(44) in terms of the gauge-invariant quantities $\pi^{V}_{i}$, $\delta
u^{V}_{i}$ and $G_{i}-(a/N)\dot{C}_{i}$, or we can fix a gauge for these
quantities by choosing $\epsilon^{V}_{i}$, so that either $C_{i}$ or $G_{i}$
vanishes. For the scalar perturbations it is somewhat more convenient to fix a
gauge.

###  III.1 Newtonian Conformal Gauge

In this gauge only scalar perturbations are considered Weinberg (2008);
Dodelson (2003); Ma and Bertschinger (1995). The line element of such gauge is

$\displaystyle
ds^{2}=-N^{2}\left(1+2\Phi\right)dt^{2}+a^{2}(1-2\Psi)\tilde{\gamma}_{ij}dx^{i}dx^{j},$
(60)

here we choose $\epsilon^{S}$ so that $B=0$, and then choose $\epsilon_{\tau}$
so that $F=0$. Both choices are unique, so that after choosing Newtonian
gauge, there is no remaining freedom to make gauge transformations. Thus in
this gauge, we have $A=-2\Psi$, $E=2\Phi$, $B=0$ and $F=0$.

The gravitational field equations (LABEL:eq:MScalar1)-(39), takes the form

$\displaystyle\frac{\kappa a^{2}}{D-1}\left[\delta\rho-\delta
P-\tilde{\gamma}^{nm}\nabla_{n}\nabla_{m}\pi^{S}\right]$ $\displaystyle=$
$\displaystyle\left[\left(D-1\right)\frac{\dot{a}^{2}}{N^{2}}+\frac{a\ddot{a}}{N^{2}}-\frac{a\dot{a}\dot{N}}{N^{3}}\right]2\Phi+\frac{a\dot{a}}{N^{2}}\dot{\Phi}$
(61)
$\displaystyle-\tilde{\gamma}^{nm}\nabla_{n}\nabla_{m}\Psi-\left(\frac{a^{2}\dot{N}}{2N^{3}}-D\frac{a\dot{a}}{N^{2}}\right)2\dot{\Psi}+\frac{a^{2}}{N^{2}}\ddot{\Psi}$
$\displaystyle-\left[\left(D-1\right)K+\frac{a\dot{a}\dot{N}}{N^{3}}-\frac{2}{D-1}\Lambda
a^{2}\right]2\Psi,$

$\displaystyle\nabla_{j}\nabla_{k}\left\\{-\kappa
a^{2}\pi^{S}+\Phi-\left(D-2\right)\Psi\right\\}=0,$ (62)

$\displaystyle-\frac{\kappa N}{D-1}\left(\rho+P\right)\nabla_{j}\delta
u=-\frac{\dot{a}}{a}\nabla_{j}\Phi-\nabla_{j}\dot{\Psi},$ (63)

$\displaystyle\frac{\kappa N^{2}}{D-1}\left[\left(D-2\right)\delta\rho+D\delta
P+\tilde{\gamma}^{ml}\nabla_{m}\nabla_{l}\pi^{S}\right]$ $\displaystyle=$
$\displaystyle-\frac{N^{2}}{a^{2}}\tilde{\gamma}^{ml}\nabla_{m}\nabla_{l}\Phi-\frac{D\dot{a}}{a}\dot{\Phi}-D\left(\frac{\ddot{a}}{a}-\frac{\dot{a}\dot{N}}{aN}+\frac{2}{D\left(D-1\right)}\Lambda
N^{2}\right)2\Phi$ (64)
$\displaystyle-D\ddot{\Psi}-D\left(2\frac{\dot{a}}{a}-\frac{\dot{N}}{N}\right)\dot{\Psi}.$

The conservation conditions (40), (41) and (47) becomes

$\displaystyle\delta
P+\tilde{\gamma}^{nm}\nabla_{n}\nabla_{m}\pi^{S}+\frac{1}{N}\partial_{\tau}\left[\left(\rho+P\right)\delta
u\right]+D\frac{\dot{a}}{aN}\left(\rho+P\right)\delta
u+\left(\rho+P\right)\Phi=0,$ (65)

$\displaystyle\delta\dot{\rho}+\frac{D\dot{a}}{a}\left(\delta\rho+\delta
P\right)+\tilde{\gamma}^{ml}\nabla_{m}\nabla_{l}\left[\frac{N}{a^{2}}\left(\rho+P\right)\delta
u+\frac{\dot{a}}{a}\pi^{S}\right]-D\left(\rho+P\right)\dot{\Psi}=0,$ (66)

$\displaystyle\frac{1}{N}\partial_{\tau}\left(\frac{\delta
n}{n}\right)+\frac{1}{a^{2}}\tilde{\gamma}^{lm}\nabla_{m}\nabla_{l}\delta
u-\frac{D}{N}\dot{\Psi}=0.$ (67)

By performing algebra in the previous equations we arrive to the following
constraint

$\displaystyle a^{3}\delta\rho-DH\frac{a^{3}}{N}\left(\rho+P\right)\delta
u+\left(D-1\right)\frac{a}{\kappa}\tilde{\gamma}^{lm}\nabla_{m}\nabla_{l}\Psi+\frac{D}{\kappa}\left[\left(D-1\right)Ka+\frac{a^{2}\dot{a}\dot{N}}{N^{3}}\right]\Psi=0.$
(68)

###  III.2 Synchronous gauge

In the synchronous gauge we have $A\neq 0$, $B\neq 0$, $E=0$ and $F=0$. The
line element of such a gauge is

$\displaystyle
ds^{2}=-N^{2}dt^{2}+a^{2}\left[(1+A)\tilde{\gamma}_{ij}+\nabla_{i}\nabla_{j}B\right]dx^{i}dx^{j}.$
(69)

Hence, the gravitational field equations (LABEL:eq:MScalar1)-(39), for this
gauge takes the form

$\displaystyle\frac{\kappa a^{2}}{D-1}\left[\delta\rho-\delta
P-\tilde{\gamma}^{nm}\nabla_{n}\nabla_{m}\pi^{S}\right]=\frac{1}{2}\tilde{\gamma}^{nm}\nabla_{n}\nabla_{m}A+\left(\frac{a^{2}\dot{N}}{2N^{3}}-D\frac{a\dot{a}}{N^{2}}\right)\dot{A}-\frac{a^{2}}{2N^{2}}\ddot{A}$
$\displaystyle-\frac{a\dot{a}}{2N^{2}}\tilde{\gamma}^{nm}\nabla_{n}\nabla_{m}\dot{B}+\left[\left(D-1\right)K+\frac{a\dot{a}\dot{N}}{N^{3}}-\frac{2}{D-1}\Lambda
a^{2}\right]A,$ (70)

$\displaystyle-2\kappa
a^{2}\pi^{S}+\left(D-2\right)A-\frac{a^{2}}{N^{2}}\ddot{B}+\left(\frac{a^{2}\dot{N}}{N^{3}}-D\frac{a\dot{a}}{N^{2}}\right)\dot{B}+\left[2\left(D-1\right)K+2\frac{a\dot{a}\dot{N}}{N^{3}}-\frac{4}{D-1}\Lambda
a^{2}\right]B$ $\displaystyle=0,$ (71)

$\displaystyle-\frac{2\kappa N}{D-1}\left(\rho+P\right)\delta u=\dot{A},$ (72)

$\displaystyle\frac{\kappa N^{2}}{D-1}\left[\left(D-2\right)\delta\rho+D\delta
P+\tilde{\gamma}^{ml}\nabla_{m}\nabla_{l}\pi^{S}\right]$ $\displaystyle=$
$\displaystyle\frac{D}{2}\ddot{A}+\frac{1}{2}\tilde{\gamma}^{ml}\nabla_{m}\nabla_{l}\ddot{B}+\left(\frac{\dot{a}}{a}-\frac{\dot{N}}{2N}\right)\left(D\dot{A}+\tilde{\gamma}^{ml}\nabla_{m}\nabla_{l}\dot{B}\right).$

The conservation conditions given by Eqs. (40), (41) and (47) becomes

$\displaystyle\delta
P+\tilde{\gamma}^{nm}\nabla_{n}\nabla_{m}\pi^{S}+\frac{1}{N}\partial_{\tau}\left[\left(\rho+P\right)\delta
u\right]+D\frac{\dot{a}}{aN}\left(\rho+P\right)\delta u=0,$ (74)

$\displaystyle\delta\dot{\rho}+\frac{D\dot{a}}{a}\left(\delta\rho+\delta
P\right)+\tilde{\gamma}^{ml}\nabla_{m}\nabla_{l}\left[\frac{N}{a^{2}}\left(\rho+P\right)\delta
u+\frac{\dot{a}}{a}\pi^{S}\right]$
$\displaystyle+\frac{1}{2}\left(\rho+P\right)\partial_{\tau}\left[DA+\tilde{\gamma}^{ml}\nabla_{m}\nabla_{l}B\right]=0,$
(75)

$\displaystyle\frac{1}{N}\partial_{\tau}\left(\frac{\delta
n}{n}\right)+\frac{1}{2N}\left(D\dot{A}+\tilde{\gamma}^{lm}\nabla_{m}\nabla_{l}\dot{B}\right)+\frac{1}{a^{2}}\tilde{\gamma}^{lm}\nabla_{m}\nabla_{l}\delta
u=0.$ (76)

##  IV Branes perturbations: A brane embedded in a $D+1+1$ bulk

Our configuration is a bulk of dimension $M$, in this space-time there is a
embedded purely spatial hypersurface of dimension $D$, whose coordinate system
is $\left(...x^{i},x^{j},x^{k}...\right)$, there is one extra spatial
dimension, whose coordinate system is $a$, and one temporal dimension. So
$M=D+1+1$ and the coordinate system for the bulk is
$\left(t,a,x^{i},x^{j},x^{k}...\right)$ Cordero and Vilenkin (2002); García-
Aspeitia and Matos (2010). Let us consider that in our bulk has a cosmological
constant. The gravitational interaction inside the brane is not taken into
account, so we have the following action

$\displaystyle S=S_{gravity}+S_{brane},$ (77)

where

$\displaystyle S_{gravity}=\int
dt\;da\;d^{D}x\frac{\sqrt{-g}}{\kappa_{B}}\left(-\Lambda+\frac{1}{2}R\right),$
(78)

$\displaystyle S_{brane}=\int
dt\;d^{D}x\sqrt{-\gamma}\left(\varepsilon\frac{1}{\kappa}\left\langle
trK\right\rangle+\mathcal{L}\right).$ (79)

In the bulk we have that, $g$ is the determinant of the metric, $R$ is the
scalar of curvature of Ricci and $\kappa_{B}\propto G_{B}$, $G_{B}$ being a
gravitational constant for bulk. On the brane we have to, $\gamma$ is the
determinant of the metric, $trK$ is the trace of the extrinsic curvature
$K_{\mu\nu}$, $\mathcal{L}$ es the Lagrangian of the content of the fields on
the brane, $\varepsilon$ is $+1$ if the brane is temporalloid and $-1$ if it
is spatialloid, for our case $\varepsilon=-1$, the bracket $\left\langle
trK\right\rangle=trK^{+}-trK^{-}$, it would be the same for all objects, the
subscript $+$ or $-$ indicates whether it is considering the metric above or
below the brane respectively.

The line element in the bulk is,

$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle
g_{00}dt^{2}+g_{aa}da^{2}+g_{ij}dx^{i}dx^{j}$ (80) $\displaystyle=$
$\displaystyle-\mathcal{A}_{\pm}dt^{2}_{\pm}+\mathcal{B}_{\pm}da^{2}+\mathcal{C}\tilde{g}_{ij}dx^{i}dx^{j},$

where $\mathcal{A}_{\pm},\,\mathcal{B}_{\pm},$ and $\mathcal{C}$ are functions
of the scale factor $a$, furthermore, $g_{ta}=g_{ti}=g_{ai}=0$. The metric for
space-time complies with $g_{ML}=g_{LM}$, where
$M,\,L=t,\,a,\,,\,i,\,j,\,k,\,...$.

The line element in the brane, is

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle\gamma_{\tau\tau}d\tau^{2}+\gamma_{ij}dx^{i}dx^{j}=-N^{2}d\tau^{2}+a^{2}\tilde{\gamma}_{ij}dx^{i}dx^{j},$
(81)

where $N$ and $a$ are functions of the time $\tau$. The spatial hypersurface
is endowed with an arbitrary sectional curvature $K$ and of dimension $D$,
which is the same metric as in section II. $\mathcal{A}_{\pm}$ and
$\mathcal{B}_{\pm}$ are also functions of $K$. Notice that the temporal
coordinate for the bulk is $t$ and for the brane is $\tau$. Therefore the
equation of motion of the branes is

$\displaystyle\left\langle K\right\rangle\gamma_{\mu\nu}-\left\langle
K_{\mu\nu}\right\rangle=\kappa T_{\mu\nu},$ (82)

equation (82) can be rewritten in terms of the source tensor $\left\langle
K_{\mu\nu}\right\rangle=\kappa S_{\mu\nu}$, Weinberg (2008) where

$\displaystyle S_{\mu\nu}=-T_{\mu\nu}+D^{-1}T^{\lambda}_{\
\lambda}\gamma_{\mu\nu}.$ (83)

In addition, the extrinsic curvature is defined as

$\displaystyle
K_{\mu\nu}=-\gamma^{A}_{\mu}\gamma^{B}_{\nu}\nabla_{B}n_{A}=-\gamma^{A}_{\mu}\partial_{\nu}n_{A}+\gamma^{A}_{\mu}\gamma^{B}_{\nu}n_{C}\Gamma^{C}_{AB},$
(84)

The subscript $\pm$ will be omitted, and it will be shown explicitly when
necessary, however it must be understand that every quantity that depends on
$\mathcal{A},\,\mathcal{B}$, depends at the same time on the subscript $\pm$.

Taking into account that $\gamma^{0}_{\tau}=\dot{t}$ and
$\gamma^{a}_{\tau}=\dot{a}$, we find that the extrinsic curvatures are

$\displaystyle
K^{\pm}_{\tau\tau}=\mp\frac{N}{2\sqrt{\mathcal{AB}}\dot{t}}\left(\dot{t}^{2}\partial_{a}\mathcal{A}+\dot{a}^{2}\partial_{a}\mathcal{B}+2\mathcal{B}\left(\ddot{a}-\frac{\dot{a}\dot{N}}{N^{2}}\right)+\frac{\dot{a}^{2}\dot{t}^{2}}{N^{2}}\left(1-N\right)\left(\mathcal{B}\partial_{a}\mathcal{A}+\mathcal{A}\partial_{a}\mathcal{B}\right)\right),$
(85)

$\displaystyle
K^{\pm}_{ij}=\pm\frac{\dot{t}}{2N}\sqrt{\frac{\mathcal{A}}{\mathcal{B}}}\tilde{\gamma}_{ij}\partial_{a}\mathcal{C},$
(86)

where

$\displaystyle\dot{t}=\sqrt{\frac{N^{2}+\mathcal{B}\dot{a}^{2}}{\mathcal{A}}},$
(87)

we found this by taking into consideration

$\displaystyle U_{M}U^{M}=-1,\quad n_{M}U^{M}=0,\quad n_{M}n^{M}=1,$ (88)

where $U^{A}=\left(\frac{\dot{t}}{N},\frac{\dot{a}}{N},0,...,0\right)$ and
$n_{A}$ is the normal vector to the brane, where

$\displaystyle n^{\pm}_{0}=\pm\sqrt{\mathcal{AB}}\frac{\dot{a}}{N},\quad
n^{\pm}_{a}=\mp\sqrt{\mathcal{AB}}\frac{\dot{t}}{N}.$ (89)

We consider that the unperturbed energy-moment tensor must have the form of a
perfect fluid, therefore its components are those found in (13). Then the
source tensor is

$\displaystyle S_{\tau\tau}=-\left(\frac{D-1}{D}\rho+P\right)N^{2},\quad
S_{i\tau}=0,\quad S_{ij}=-\frac{1}{D}\rho a^{2}\tilde{\gamma}_{ij},$

and we find that the density and pressure are

$\displaystyle\rho=-\frac{D}{2a^{2}\kappa}\left(\sqrt{\mathcal{B}^{-1}_{+}+\frac{\dot{a}^{2}}{N^{2}}}+\sqrt{\mathcal{B}^{-1}_{-}+\frac{\dot{a}^{2}}{N^{2}}}\right)\partial_{a}\mathcal{C},$
(91)

$\displaystyle P=\frac{1}{2N^{2}\kappa}$
$\displaystyle\left[\frac{1}{\dot{t}_{+}\sqrt{\mathcal{A}_{+}\mathcal{B}_{+}}}\left(\dot{t}^{2}_{+}\partial_{a}\mathcal{A}_{+}+\dot{a}^{2}\partial_{a}\mathcal{B}_{+}+2\mathcal{B}_{+}\left(\ddot{a}-\frac{\dot{a}\dot{N}}{N^{2}}\right)+\frac{\dot{a}^{2}\dot{t}^{2}}{N^{2}}\left(1-N\right)\left(\mathcal{B}_{+}\partial_{a}\mathcal{A}_{+}+\mathcal{A}_{+}\partial_{a}\mathcal{B}_{+}\right)\right)\right.$
(92)
$\displaystyle+\left.\frac{1}{\dot{t}_{-}\sqrt{\mathcal{A}_{-}\mathcal{B}_{-}}}\left(\dot{t}^{2}_{-}\partial_{a}\mathcal{A}_{-}+\dot{a}^{2}\partial_{a}\mathcal{B}_{-}+2\mathcal{B}_{-}\left(\ddot{a}-\frac{\dot{a}\dot{N}}{N^{2}}\right)+\frac{\dot{a}^{2}\dot{t}^{2}}{N^{2}}\left(1-N\right)\left(\mathcal{B}_{-}\partial_{a}\mathcal{A}_{-}+\mathcal{A}_{-}\partial_{a}\mathcal{B}_{-}\right)\right)\right]$
$\displaystyle+\frac{D-1}{2a^{2}\kappa}\left(\sqrt{\mathcal{B}^{-1}_{+}+\frac{\dot{a}^{2}}{N^{2}}}+\sqrt{\mathcal{B}^{-1}_{-}+\frac{\dot{a}^{2}}{N^{2}}}\right)\partial_{a}\mathcal{C}.$

The bulk metric perturbation is $h_{AB}$ and the brane metric perturbation is
$\eta_{\mu\nu}$. Thus, the source perturbation tensor is

$\displaystyle\delta S_{\mu\nu}=-\delta
T_{\mu\nu}+\frac{1}{D}\left(\gamma_{\mu\nu}\delta T^{\lambda}_{\
\lambda}+T^{\lambda}_{\ \lambda}\eta_{\mu\nu}\right),$ (93)

and the extrinsic curvature perturbation is

$\displaystyle\delta K_{\mu\nu}=-\gamma^{A}_{\mu}\partial_{\nu}\delta
n_{A}+\gamma^{A}_{\mu}\gamma^{B}_{\nu}\left(\Gamma^{C}_{AB}\delta
n_{C}+n_{C}\delta\Gamma^{C}_{AB}\right).$ (94)

Hence, the perturbed part of the equation of motion is

$\displaystyle\left\langle\delta K_{\mu\nu}\right\rangle=\kappa\delta
S_{\mu\nu},$ (95)

or explicitly

$\displaystyle\left\langle-\gamma^{A}_{\mu}\partial_{\nu}\delta
n_{A}+\gamma^{A}_{\mu}\gamma^{B}_{\nu}\left(\Gamma^{C}_{AB}\delta
n_{C}+n_{C}\delta\Gamma^{C}_{AB}\right)\right\rangle=\kappa\left[-\delta
T_{\mu\nu}+\frac{1}{D}\left(\gamma_{\mu\nu}\delta T^{\lambda}_{\
\lambda}+T^{\lambda}_{\ \lambda}\eta_{\mu\nu}\right)\right].$

The perturbations of the Christoffel symbols are

$\displaystyle\delta\Gamma^{0}_{00}$ $\displaystyle=$
$\displaystyle\frac{1}{2\mathcal{A}}\left(\frac{1}{\mathcal{B}}h_{0a}\partial_{a}\mathcal{A}-\partial_{0}h_{00}\right),$
$\displaystyle\delta\Gamma^{0}_{0a}$ $\displaystyle=$
$\displaystyle\frac{1}{2\mathcal{A}}\left(\frac{1}{\mathcal{A}}h_{00}\partial_{a}\mathcal{A}-\partial_{a}h_{00}\right),$
$\displaystyle\delta\Gamma^{0}_{0i}$ $\displaystyle=$
$\displaystyle-\frac{1}{2\mathcal{A}}\partial_{i}h_{00},$
$\displaystyle\delta\Gamma^{0}_{aa}$ $\displaystyle=$
$\displaystyle\frac{1}{2\mathcal{A}}\left(\frac{1}{\mathcal{B}}h_{0a}\partial_{a}\mathcal{B}-2\partial_{a}h_{0a}+\partial_{0}h_{aa}\right),$
$\displaystyle\delta\Gamma^{0}_{ij}$ $\displaystyle=$
$\displaystyle\frac{1}{2\mathcal{A}}\left(-\frac{1}{\mathcal{B}}h_{0a}\tilde{g}_{ij}\partial_{a}\,\mathcal{C}-\nabla_{j}h_{0i}-\nabla_{i}h_{0j}+\partial_{0}h_{ij}\right),$
$\displaystyle\delta\Gamma^{0}_{ai}$ $\displaystyle=$
$\displaystyle\frac{1}{2\mathcal{A}}\left(\frac{1}{\mathcal{C}}h_{0i}\partial_{a}\mathcal{C}-\partial_{i}h_{0a}-\partial_{a}h_{0i}+\partial_{0}h_{ai}\right),$
(97)

$\displaystyle\delta\Gamma^{a}_{aa}$ $\displaystyle=$
$\displaystyle\frac{1}{2\mathcal{B}}\left(-\frac{1}{\mathcal{B}}h_{aa}\partial_{a}\mathcal{B}+\partial_{a}h_{aa}\right),$
$\displaystyle\delta\Gamma^{a}_{0a}$ $\displaystyle=$
$\displaystyle\frac{1}{2\mathcal{B}}\left(-\frac{1}{\mathcal{A}}h_{0a}\partial_{a}\mathcal{A}+\partial_{0}h_{aa}\right),$
$\displaystyle\delta\Gamma^{a}_{ai}$ $\displaystyle=$
$\displaystyle\frac{1}{2\mathcal{B}}\left(-\frac{1}{\mathcal{C}}h_{ai}\partial_{a}\mathcal{C}+\partial_{i}h_{aa}\right),$
$\displaystyle\delta\Gamma^{a}_{00}$ $\displaystyle=$
$\displaystyle\frac{1}{2\mathcal{B}}\left(-\frac{1}{\mathcal{B}}h_{aa}\partial_{a}\,\mathcal{A}+2\partial_{0}h_{0a}-\partial_{a}h_{00}\right),$
$\displaystyle\delta\Gamma^{a}_{ij}$ $\displaystyle=$
$\displaystyle\frac{1}{2\mathcal{B}}\left(\frac{1}{\mathcal{B}}h_{aa}\tilde{g}_{ij}\partial_{a}\,\mathcal{C}+\nabla_{j}h_{ai}+\nabla_{i}h_{aj}-\partial_{a}h_{ij}\right),$
$\displaystyle\delta\Gamma^{a}_{0i}$ $\displaystyle=$
$\displaystyle\frac{1}{2\mathcal{B}}\left(\partial_{i}h_{0a}+\partial_{0}h_{ai}-\partial_{a}h_{0i}\right),$
(98)

$\displaystyle\delta\Gamma^{i}_{jk}$ $\displaystyle=$
$\displaystyle\frac{1}{2\mathcal{C}}\tilde{g}^{il}\left(\frac{1}{\mathcal{B}}h_{al}\tilde{g}_{jk}\partial_{a}\mathcal{C}+\nabla_{k}h_{lj}+\nabla_{j}h_{lk}-\nabla_{l}h_{jk}\right),$
$\displaystyle\delta\Gamma^{i}_{0j}$ $\displaystyle=$
$\displaystyle\frac{1}{2\mathcal{C}}\tilde{g}^{il}\left(\nabla_{j}h_{0l}+\partial_{0}h_{lj}-\nabla_{l}h_{0j}\right),$
$\displaystyle\delta\Gamma^{i}_{aj}$ $\displaystyle=$
$\displaystyle\frac{1}{2\mathcal{C}}\tilde{g}^{il}\left(-\frac{1}{\mathcal{C}}h_{lj}\partial_{a}\mathcal{C}+\nabla_{j}h_{al}+\partial_{a}h_{lj}-\nabla_{l}h_{aj}\right),$
$\displaystyle\delta\Gamma^{i}_{00}$ $\displaystyle=$
$\displaystyle\frac{1}{2\mathcal{C}}\tilde{g}^{il}\left(-\frac{1}{\mathcal{B}}h_{al}\partial_{a}\,\mathcal{A}+2\partial_{0}h_{0l}-\partial_{l}h_{00}\right),$
$\displaystyle\delta\Gamma^{i}_{aa}$ $\displaystyle=$
$\displaystyle\frac{1}{2\mathcal{C}}\tilde{g}^{il}\left(-\frac{1}{\mathcal{B}}h_{al}\partial_{a}\mathcal{B}+2\partial_{a}h_{la}-\partial_{l}h_{aa}\right),$
$\displaystyle\delta\Gamma^{i}_{0a}$ $\displaystyle=$
$\displaystyle\frac{1}{2\mathcal{C}}\tilde{g}^{il}\left(-\frac{1}{\mathcal{A}}h_{0l}\partial_{a}\mathcal{A}+\partial_{a}h_{0l}+\partial_{0}h_{al}-\partial_{l}h_{0a}\right),$
(99)

It is possible to calculate the perturbations of extrinsic curvature

$\displaystyle\delta K^{\pm}_{\tau\tau}$ $\displaystyle=$
$\displaystyle\pm\left[-\dot{t}\delta\dot{n}_{0}-\dot{a}\delta\dot{n}_{a}+\frac{1}{2\mathcal{AB}}\left(2\mathcal{B}\dot{a}\dot{t}\partial_{a}\mathcal{A}\delta
n_{0}+\mathcal{A}\left(\dot{t}^{2}\partial_{a}\mathcal{A}+\dot{a}^{2}\partial_{a}\mathcal{B}\right)\delta
n_{a}\right)\right.$ (100)
$\displaystyle+\frac{\sqrt{\mathcal{AB}}}{2N\mathcal{A}^{2}\mathcal{B}^{2}}\left(2\mathcal{B}^{2}\dot{a}^{2}\dot{t}\partial_{a}\mathcal{A}h_{00}+\mathcal{A}^{2}\dot{t}\left(\dot{t}^{2}\partial_{a}\mathcal{A}+\dot{a}^{2}\partial_{a}\mathcal{B}\right)h_{aa}+\mathcal{AB}\dot{a}\left(3\dot{t}^{2}\partial_{a}\mathcal{A}+\dot{a}^{2}\partial_{a}\mathcal{B}\right)h_{0a}\right)$
$\displaystyle-\frac{1}{2N\sqrt{\mathcal{AB}}}\left(\mathcal{B}\dot{a}\dot{t}^{2}\partial_{0}h_{00}+2\mathcal{A}\dot{t}^{3}\partial_{0}h_{0a}+\left(2\mathcal{A}\dot{t}^{2}\dot{a}-\mathcal{B}\dot{a}^{3}\right)\partial_{0}h_{aa}\right)$
$\displaystyle\left.-\frac{1}{2N\sqrt{\mathcal{AB}}}\left(2\mathcal{B}\dot{a}^{3}\partial_{a}h_{0a}+\mathcal{A}\dot{t}\dot{a}^{2}\partial_{a}h_{aa}+\left(2\mathcal{B}\dot{a}^{2}\dot{t}-\mathcal{A}\dot{t}^{3}\right)\partial_{a}h_{00}\right)\right],$

$\displaystyle\delta K^{\pm}_{\tau i}$ $\displaystyle=$
$\displaystyle\pm\left[-\delta\dot{n}_{i}+\frac{\dot{a}\partial_{a}\mathcal{C}}{2\mathcal{C}}\delta
n_{i}+\frac{1}{2N\sqrt{\mathcal{AB}}}\left(\frac{\dot{a}\partial_{a}\mathcal{C}}{\mathcal{C}}\left(\mathcal{B}\dot{a}h_{0i}+\mathcal{A}\dot{t}h_{ai}\right)+N^{2}\left(\partial_{a}h_{0i}-\partial_{0}h_{ai}\right)\right.\right.$
(101)
$\displaystyle\left.-\left.\frac{}{}\left(\mathcal{B}\dot{a}\dot{t}\partial_{i}h_{00}+\left(\mathcal{A}\dot{t}^{2}+\mathcal{B}\dot{a}^{2}\right)\partial_{i}h_{0a}+\mathcal{A}\dot{a}\dot{t}\partial_{i}h_{aa}\right)\right)\right]$

$\displaystyle\delta K^{\pm}_{ij}$ $\displaystyle=$
$\displaystyle\pm\left[\frac{1}{2N\sqrt{\mathcal{AB}}}\left(-\frac{\partial_{a}\mathcal{C}}{\mathcal{B}}\tilde{g}_{ij}\left(\mathcal{B}\dot{a}h_{0a}+\mathcal{A}\dot{t}h_{aa}\right)-\mathcal{B}\dot{a}\left(\nabla_{j}h_{0i}+\nabla_{i}h_{0j}-\partial_{0}h_{ij}\right)-\mathcal{A}\dot{t}\left(\nabla_{j}h_{ai}+\nabla_{i}h_{aj}-\partial_{a}h_{ij}\right)\right)\right.$
(102)
$\displaystyle\left.-\frac{\partial_{a}\mathcal{C}}{2\mathcal{B}}\tilde{g}_{ij}\delta
n_{a}-\nabla_{i}\delta n_{j}\right].$

The relationship between brane perturbations and bulk perturbations are
$\eta_{\mu\nu}=\gamma^{M}_{\mu}\gamma^{N}_{\nu}h_{MN}$ and it is the same
relationship for all tensors involved, thus

$\displaystyle\eta_{\tau\tau}$ $\displaystyle=$
$\displaystyle\dot{t}^{2}h_{00}+2\dot{a}\dot{t}h_{0a}+\dot{a}^{2}h_{aa}$
$\displaystyle\eta_{\tau i}$ $\displaystyle=$
$\displaystyle\dot{t}h_{0i}+\dot{a}h_{ai}$ $\displaystyle\eta_{ij}$
$\displaystyle=$ $\displaystyle h_{ij},$ (103)

The perturbations of the equations of Einstein for a perfect fluid takes the
form

$\displaystyle\left\langle\delta K_{\tau\tau}\right\rangle$ $\displaystyle=$
$\displaystyle\kappa\left(-\delta T_{\tau\tau}+\frac{1}{D}\left(-a^{2}\delta
T^{\lambda}_{\lambda}+T^{\lambda}_{\lambda}\eta_{\tau\tau}\right)\right)$
$\displaystyle=$
$\displaystyle\kappa\left(\frac{1-D}{D}a^{2}\delta\rho-a^{2}\delta
P-\frac{a^{2}}{D}\tilde{\gamma}^{ml}\nabla_{m}\nabla_{l}\pi^{S}+\left(\frac{D-1}{D}\rho+P\right)\left(\dot{t}^{2}h_{00}+2\dot{a}\dot{t}h_{0a}+\dot{a}^{2}h_{aa}\right)\right),$

$\displaystyle\left\langle\delta K_{\tau i}\right\rangle$ $\displaystyle=$
$\displaystyle\kappa\left(-\delta T_{\tau
i}+\frac{1}{D}T^{\lambda}_{\lambda}\eta_{\tau
i}\right)=\kappa\left(a\left(\rho+P\right)\delta
u_{i}-\frac{1}{D}\rho\left(\dot{t}h_{0i}+\dot{a}h_{ai}\right)\right),$ (105)

$\displaystyle\left\langle\delta K_{ij}\right\rangle$ $\displaystyle=$
$\displaystyle\kappa\left(-\delta
T_{ij}+\frac{1}{D}\left(a^{2}\tilde{\gamma}_{ij}\delta
T^{\lambda}_{\lambda}+T^{\lambda}_{\lambda}\eta_{ij}\right)\right)$
$\displaystyle=$
$\displaystyle\kappa\left(-\frac{a^{2}}{D}\tilde{\gamma}_{ij}\delta\rho+\frac{a^{2}}{D}\tilde{\gamma}_{ij}\tilde{\gamma}^{ml}\nabla_{m}\nabla_{l}\pi^{S}-a^{2}\left(\nabla_{i}\nabla_{j}\pi^{S}+\nabla_{i}\pi^{V}_{j}+\nabla_{j}\pi^{V}_{i}+\pi^{T}_{ij}\right)-\frac{1}{D}\rho
h_{ij}\right).$

Perturbation of the bulk metric can be decomposed into scalars, divergenceless
vectors and divergenceless traceless symmetric tensors,

$\displaystyle h_{00}$ $\displaystyle=$ $\displaystyle-\mathcal{A}W^{B},\quad
h_{0a}=\sqrt{\mathcal{AB}}V^{B},\quad h_{aa}=\mathcal{B}U^{B},$ $\displaystyle
h_{0i}$ $\displaystyle=$
$\displaystyle\sqrt{\mathcal{AC}}\left(\nabla_{i}P^{B}+Q^{B}_{i}\right),\quad
h_{ai}=\sqrt{\mathcal{BC}}\left(\nabla_{i}R^{B}+S^{B}_{i}\right),$
$\displaystyle h_{ij}$ $\displaystyle=$
$\displaystyle\mathcal{C}\left(A^{B}\tilde{g}_{ij}+\nabla_{i}\nabla_{j}B^{B}+\nabla_{i}C^{B}_{j}+\nabla_{j}C^{B}_{i}+D^{B}_{ij}\right),$
(107)

where same conditions to those seen in (29) are complied, also similarly as
$\delta u_{i}$ we have that $\delta n^{\pm}_{i}=\nabla_{i}\delta
n^{\pm}+\delta n^{V\pm}_{i}$. Perturbations of the energy-momentum tensor on
the brane, as well as their respective conservation conditions, have the form
that was developed in Sec. II.

Hereafter we show the scalar, vector and tensor modes of Eqs.
(LABEL:eq:EcEinsteinScalarKExt)-(LABEL:eq:EcEinsteinTensorKExt).

###  IV.1 Scalar Modes

The part of Eq. (LABEL:eq:EcEinsteinTensorKExt) proportional to
$\tilde{g}_{jk}$ gives

$\displaystyle\left\langle\frac{1}{2N\sqrt{\mathcal{AB}}}\left(-\frac{\partial_{a}\mathcal{C}}{\mathcal{B}}\left(\mathcal{B}\sqrt{\mathcal{AB}}\dot{a}V^{B}+\mathcal{A}\mathcal{B}\dot{t}U^{B}\right)+\mathcal{BC}\dot{a}\partial_{0}A^{B}+\mathcal{A}\dot{t}\left(A^{B}\partial_{a}\mathcal{C}+\mathcal{C}\partial_{a}A^{B}\right)\right)-\frac{\partial_{a}\mathcal{C}}{2\mathcal{B}}\delta
n_{a}\right\rangle$ $\displaystyle=-\kappa\frac{a^{2}}{D}\left(\delta\rho+\rho
A^{B}-\tilde{\gamma}^{lm}\nabla_{m}\nabla_{l}\pi^{S}\right).$ (108)

The part of Eq. (LABEL:eq:EcEinsteinTensorKExt) of the form
$\nabla_{j}\nabla_{k}S$ (where $S$ is any scalar) gives

$\displaystyle\nabla_{j}\nabla_{k}\left\\{\left\langle\frac{1}{2N\sqrt{\mathcal{AB}}}\left(-\mathcal{B}\dot{a}\left(2\sqrt{\mathcal{AC}}P^{B}-\mathcal{C}\partial_{0}B^{B}\right)-\mathcal{A}\dot{t}\left(2\sqrt{\mathcal{BC}}R^{B}-\mathcal{C}\partial_{a}B^{B}-B^{B}\partial_{a}\mathcal{C}\right)\right)-\delta
n\right\rangle\right.$ $\displaystyle\left.+\kappa
a^{2}\left(\pi^{S}+\frac{\rho}{D}B^{B}\right)\right\\}=0.$ (109)

The part of Eq. (105) of the form $\nabla_{j}S$ (where $S$ is any scalar)
gives

$\displaystyle\left\langle-\nabla_{i}\delta\dot{n}+\frac{\dot{a}\partial_{a}\mathcal{C}}{2\mathcal{C}}\nabla_{i}\delta
n+\frac{1}{2N\sqrt{\mathcal{AB}}}\left(\frac{\dot{a}\partial_{a}\mathcal{C}}{\mathcal{C}}\left(\mathcal{B}\sqrt{\mathcal{AC}}\dot{a}\nabla_{i}P^{B}+\mathcal{A}\sqrt{\mathcal{BC}}\dot{t}\nabla_{i}R^{B}\right)\right.\right.$
$\displaystyle+N^{2}\left(\nabla_{i}P^{B}\partial_{a}\sqrt{\mathcal{AC}}+\sqrt{\mathcal{AC}}\nabla_{i}\partial_{a}P^{B}-\sqrt{\mathcal{BC}}\nabla_{i}\partial_{0}R^{B}\right)$
$\displaystyle\left.\left.-\left(-\mathcal{AB}\dot{a}\dot{t}\nabla_{i}W^{B}+\sqrt{\mathcal{AB}}\left(\mathcal{A}\dot{t}^{2}+\mathcal{B}\dot{a}^{2}\right)\nabla_{i}V^{B}+\mathcal{AB}\dot{a}\dot{t}\nabla_{i}U^{B}\right)\right)\right\rangle$
$\displaystyle=-\kappa\left(\frac{\sqrt{\mathcal{C}}\rho}{D}\nabla_{i}\left(\dot{t}\sqrt{\mathcal{A}}P^{B}+\dot{a}\sqrt{\mathcal{B}}R^{B}\right)-a\left(P+\rho\right)\nabla_{i}\delta
u\right).$ (110)

Eq. (LABEL:eq:EcEinsteinScalarKExt) gives

$\displaystyle\left\langle-\dot{t}\delta\dot{n}_{0}-\dot{a}\delta\dot{n}_{a}+\frac{1}{2\mathcal{AB}}\left(2\mathcal{B}\dot{a}\dot{t}\partial_{a}\mathcal{A}\delta
n_{0}+\mathcal{A}\left(\dot{t}^{2}\partial_{a}\mathcal{A}+\dot{a}^{2}\partial_{a}\mathcal{B}\right)\delta
n_{a}\right)\right.$
$\displaystyle+\frac{\sqrt{\mathcal{AB}}}{2N\mathcal{A}^{2}\mathcal{B}^{2}}\left(-2\mathcal{B}^{2}\dot{a}^{2}\dot{t}\mathcal{A}W^{B}\partial_{a}\mathcal{A}+\mathcal{A}^{2}\mathcal{B}\dot{t}\left(\dot{t}^{2}\partial_{a}\mathcal{A}+\dot{a}^{2}\partial_{a}\mathcal{B}\right)U^{B}+\mathcal{AB}\sqrt{\mathcal{AB}}\dot{a}\left(3\dot{t}^{2}\partial_{a}\mathcal{A}+\dot{a}^{2}\partial_{a}\mathcal{B}\right)V^{B}\right)$
$\displaystyle-\frac{1}{2N\sqrt{\mathcal{AB}}}\left(-\mathcal{A}\mathcal{B}\dot{a}\dot{t}^{2}\partial_{0}W^{B}+2\mathcal{A}\sqrt{\mathcal{AB}}\dot{t}^{3}\partial_{0}V^{B}+\mathcal{B}\left(2\mathcal{A}\dot{t}^{2}\dot{a}-\mathcal{B}\dot{a}^{3}\right)\partial_{0}U^{B}\right)$
$\displaystyle\left.-\frac{1}{2N\sqrt{\mathcal{AB}}}\left(2\mathcal{B}\dot{a}^{3}\left(\sqrt{\mathcal{AB}}\partial_{a}V^{B}+V^{B}\partial_{a}\sqrt{\mathcal{AB}}\right)+\mathcal{A}\dot{t}\dot{a}^{2}\left(\mathcal{B}\partial_{a}U^{B}+U^{B}\partial_{a}\mathcal{B}\right)-\left(2\mathcal{B}\dot{a}^{2}\dot{t}-\mathcal{A}\dot{t}^{3}\right)\left(\mathcal{A}\partial_{a}W^{B}+W^{B}\partial_{a}\mathcal{A}\right)\right)\right\rangle$
$\displaystyle=\kappa\left(\frac{1-D}{D}a^{2}\delta\rho-a^{2}\delta
P-\frac{a^{2}}{D}\tilde{\gamma}^{ml}\nabla_{m}\nabla_{l}\pi^{S}+\left(\frac{D-1}{D}\rho+P\right)\left(-\dot{t}^{2}\mathcal{A}W^{B}+2\dot{a}\dot{t}\sqrt{\mathcal{AB}}V^{B}+\dot{a}^{2}\mathcal{B}U^{B}\right)\right).$
(111)

###  IV.2 Vector Modes

The part of Eq. (LABEL:eq:EcEinsteinTensorKExt) of the form $\nabla_{k}V_{j}$
(where $V_{j}$ is a divergenceless vector) gives

$\displaystyle\left\langle\frac{1}{2N\sqrt{\mathcal{AB}}}\left(-\mathcal{B}\dot{a}\left(\sqrt{\mathcal{AC}}\nabla_{i}Q^{B}_{j}-\mathcal{C}\partial_{0}\nabla_{i}C^{B}_{j}\right)-\mathcal{A}\dot{t}\left(\sqrt{\mathcal{BC}}\nabla_{i}S^{B}_{j}-\left(\mathcal{C}\nabla_{i}\partial_{a}C^{B}_{j}+\nabla_{i}C^{B}_{j}\partial_{a}\mathcal{C}\right)\right)\right)\right.$
$\displaystyle\left.-\nabla_{i}\delta
n^{V}_{j}\frac{}{}\right\rangle=-\kappa\left(a^{2}\nabla_{i}\pi^{V}_{j}+\frac{a^{2}\rho}{D}\nabla_{i}C_{j}\right),$
(112)

while the part of Eq. (105) of the form $V_{j}$ (where $V_{j}$ is any
divergenceless vector) gives

$\displaystyle\left\langle-\delta\dot{n}^{V}_{i}+\frac{\dot{a}\partial_{a}\mathcal{C}}{2\mathcal{C}}\delta
n^{V}_{i}+\frac{1}{2N\sqrt{\mathcal{AB}}}\left(\frac{\dot{a}\partial_{a}\mathcal{C}}{\sqrt{\mathcal{C}}}\left(\mathcal{B}\sqrt{\mathcal{A}}\dot{a}Q^{B}_{i}+\mathcal{A}\sqrt{\mathcal{B}}\dot{t}S^{B}_{i}\right)\right.\right.$
$\displaystyle\left.\left.+N^{2}\left(\left(\sqrt{\mathcal{AC}}\partial_{a}Q^{B}_{i}+Q^{B}_{i}\partial_{a}\sqrt{\mathcal{AC}}\right)-\sqrt{\mathcal{BC}}\partial_{0}S^{B}_{i}\right)\right)\right\rangle=\kappa\left(a\left(\rho+P\right)\delta
u^{V}_{i}-\frac{1}{D}\rho aNG_{i}\right).$

###  IV.3 Tensor Modes

The part of Eq. (LABEL:eq:EcEinsteinTensorKExt) of the form of a traceless and
divergenceless tensors is

$\displaystyle\left\langle\frac{1}{2N\sqrt{\mathcal{AB}}}\left[\mathcal{B}\dot{a}a^{2}\partial_{0}D_{ij}+\mathcal{A}\dot{t}\left(a^{2}\partial_{a}D_{ij}+2aD_{ij}\right)\right]\right\rangle=-\kappa
a^{2}\left(\pi^{T}_{ij}+\frac{\rho}{D}D_{ij}\right),$ (114)

the conditions of conservation of the current are internal properties of the
brane, therefore they are the same as in previous equations.

##  V Gauges in Perturbations of branes

Here, we consider a space-time coordinate transformation as Weinberg (2008)

$\displaystyle x^{A}\rightarrow x^{\prime A}=x^{A}+e^{A}\left(x\right).$ (115)

From Eqs. (LABEL:eq:Deltah1) and (52) we have the brane fashion

$\displaystyle\Delta
h_{00}=-2\partial_{0}e_{0}+\frac{1}{\mathcal{B}}e_{a}\partial_{a}\mathcal{A},\quad\Delta
h_{0a}=-\partial_{0}e_{a}-\partial_{a}e_{0}+\frac{1}{\mathcal{A}}e_{0}\partial_{a}\mathcal{A},\quad\Delta
h_{aa}=-2\partial_{a}e_{a}+\frac{1}{\mathcal{B}}e_{a}\partial_{a}\mathcal{B},$
$\displaystyle\Delta h_{0i}=-\partial_{0}e_{i}-\nabla_{i}e_{0},\quad\Delta
h_{ai}=-\partial_{a}e_{i}-\nabla_{i}e_{a}+\frac{1}{\mathcal{C}}e_{i}\partial_{a}\mathcal{C},$
$\displaystyle\Delta
h_{ij}=-\nabla_{j}e_{i}-\nabla_{i}e_{j}-\frac{1}{\mathcal{B}}\tilde{g}_{ij}e_{a}\partial_{a}\mathcal{C}.$
(116)

and the gauge transformation for a second order tensor in general is

$\displaystyle\Delta\delta Z_{00}=-\zeta\Delta
h_{00}+e_{0}\partial_{0}\zeta-\frac{\mathcal{A}}{\mathcal{B}}e_{a}\partial_{a}\zeta,\Delta\delta
Z_{0a}=\zeta\partial_{a}e_{0}-\frac{1}{\mathcal{A}}e_{0}\zeta\partial_{a}\mathcal{A}-\chi\partial_{0}e_{a},\Delta\delta
Z_{aa}=\chi\Delta
h_{aa}-e_{a}\partial_{a}\chi+\frac{\mathcal{B}}{\mathcal{A}}e_{0}\partial_{0}\chi,$
(117) $\displaystyle\Delta\delta
Z_{0i}=\zeta\partial_{i}e_{0}-\xi\partial_{0}e_{i},\quad\Delta\delta
Z_{ai}=-\chi\partial_{i}e_{a}+\frac{1}{\mathcal{C}}e_{i}\chi\partial_{a}\mathcal{C}-\xi\partial_{a}e_{i},$
$\displaystyle\Delta\delta Z_{ij}=\xi\Delta
h_{ij}+\mathcal{C}\tilde{g}_{ij}\left(\frac{1}{\mathcal{A}}e_{0}\partial_{0}\xi-\frac{1}{\mathcal{B}}e_{a}\partial_{a}\xi\right),$
.

where $Z_{00}=\zeta(t,a)\mathcal{A}$, $Z_{aa}=\chi(t,a)\mathcal{B}$,
$Z_{ij}=\xi(t,a)\mathcal{C}\tilde{g}_{ij}$.

The covariant transformation
$T_{\mu\nu}=\gamma^{A}_{\mu}\gamma^{B}_{\nu}T_{AB}$ Dodelson (2003) between
energy-momentum tensors indicates that in particular $\zeta=-\chi=\rho$ and
$\xi=P$. Then, Eq. (116) gives the gauge transformations of the metric
perturbation components defined by Eqs. (107):

$\displaystyle\Delta
W^{B}=\frac{1}{\mathcal{A}}\left(2\partial_{0}e_{0}-\frac{1}{\mathcal{B}}e_{a}\partial_{a}\mathcal{A}\right),\quad\Delta
V^{B}=\frac{1}{\sqrt{\mathcal{AB}}}\left(-\partial_{0}e_{a}-\partial_{a}e_{0}+\frac{1}{\mathcal{A}}e_{0}\partial_{a}\mathcal{A}\right),\quad\Delta
U^{B}=\frac{1}{\mathcal{B}}\left(-2\partial_{a}e_{a}+\frac{1}{\mathcal{B}}e_{a}\partial_{a}\mathcal{B}\right),$
$\displaystyle\Delta
P^{B}=-\frac{1}{\sqrt{\mathcal{AC}}}\left(\partial_{0}e^{S}+e_{0}\right),\quad\Delta
Q^{B}_{i}=-\frac{1}{\sqrt{\mathcal{AC}}}\partial_{0}e^{V}_{i},$
$\displaystyle\Delta
R^{B}=\frac{1}{\sqrt{\mathcal{BC}}}\left(-\partial_{a}e^{S}-e_{a}+\frac{1}{\mathcal{C}}e^{S}\partial_{a}\mathcal{C}\right),\quad\Delta
S^{B}_{i}=\left(\frac{1}{\sqrt{\mathcal{BC}}}\partial_{a}e^{V}_{i}+\frac{1}{\mathcal{C}}e^{V}_{i}\partial_{a}\mathcal{C}\right),$
$\displaystyle\Delta
A^{B}=-\frac{1}{\mathcal{BC}}e_{a}\partial_{a}\mathcal{C},\quad\Delta
B^{B}=-\frac{2}{\mathcal{C}}e^{S},\quad\Delta
C^{B}_{i}=-\frac{1}{\mathcal{C}}e_{i}^{V},\quad\Delta D^{B}_{ij}=0,$ (118)

The transformation $\Delta\delta
T_{\mu\nu}=\gamma^{A}_{\mu}\gamma^{B}_{\nu}\Delta\delta T_{AB}$ and (117),
indicates that the gauge transformations for the perturbations to the
pressure, energy density, velocity potential and dissipative terms, are the
same as those shown in (57) and (58).

###  V.1 Newtonian gauge

The bulk line element for this gauge is Weinberg (2008); Dodelson (2003); Ma
and Bertschinger (1995)

$\displaystyle
ds^{2}=-\mathcal{A}_{\pm}\left(1+2\Xi_{\pm}\right)dt^{2}_{\pm}+4\sqrt{\mathcal{A_{\pm}B_{\pm}}}\Upsilon_{\pm}dtda+\mathcal{B}_{\pm}\left(1+2\Theta_{\pm}\right)da^{2}+a^{2}(1-2\Psi)\tilde{g}_{ij}dx^{i}dx^{j},$
(119)

then $P^{B}$, $R^{B}$ and $B^{B}$ are null. The gravitational field equations
(108)-(111), takes the form

$\displaystyle\left\langle\frac{1}{N\sqrt{\mathcal{AB}}}\left(-\frac{\partial_{a}\mathcal{C}}{\mathcal{B}}\left(\mathcal{B}\sqrt{\mathcal{AB}}\dot{a}\Upsilon+\mathcal{A}\mathcal{B}\dot{t}\Theta\right)-\mathcal{BC}\dot{a}\partial_{0}\Psi-\mathcal{A}\dot{t}\left(\Psi\partial_{a}\mathcal{C}+\mathcal{C}\partial_{a}\Psi\right)\right)-\frac{\partial_{a}\mathcal{C}}{2\mathcal{B}}\delta
n_{a}\right\rangle$
$\displaystyle=-\kappa\frac{a^{2}}{D}\left(\delta\rho-2\rho\Psi-\tilde{\gamma}^{lm}\nabla_{m}\nabla_{l}\pi^{S}\right),$
(120)

$\displaystyle\nabla_{j}\nabla_{k}\left\\{-\delta n^{+}+\delta n^{-}+\kappa
a^{2}\pi^{S}\right\\}=0,$ (121)

$\displaystyle\left\langle-\nabla_{i}\delta\dot{n}+\frac{\dot{a}\partial_{a}\mathcal{C}}{2\mathcal{C}}\nabla_{i}\delta
n-\frac{1}{N\sqrt{\mathcal{AB}}}\left(-\mathcal{AB}\dot{a}\dot{t}\nabla_{i}\Xi+\sqrt{\mathcal{AB}}\left(\mathcal{A}\dot{t}^{2}+\mathcal{B}\dot{a}^{2}\right)\nabla_{i}\Upsilon+\mathcal{AB}\dot{a}\dot{t}\nabla_{i}\Theta\right)\right\rangle=\kappa
a\left(P+\rho\right)\nabla_{i}\delta u,$ (122)

$\displaystyle\left\langle-\dot{t}\delta\dot{n}_{0}-\dot{a}\delta\dot{n}_{a}+\frac{1}{2\mathcal{AB}}\left(2\mathcal{B}\dot{a}\dot{t}\partial_{a}\mathcal{A}\delta
n_{0}+\mathcal{A}\left(\dot{t}^{2}\partial_{a}\mathcal{A}+\dot{a}^{2}\partial_{a}\mathcal{B}\right)\delta
n_{a}\right)\right.$
$\displaystyle+\frac{\sqrt{\mathcal{AB}}}{N\mathcal{A}^{2}\mathcal{B}^{2}}\left(-2\mathcal{B}^{2}\dot{a}^{2}\dot{t}\mathcal{A}\Xi\partial_{a}\mathcal{A}+\mathcal{A}^{2}\mathcal{B}\dot{t}\left(\dot{t}^{2}\partial_{a}\mathcal{A}+\dot{a}^{2}\partial_{a}\mathcal{B}\right)\Theta+\mathcal{AB}\sqrt{\mathcal{AB}}\dot{a}\left(3\dot{t}^{2}\partial_{a}\mathcal{A}+\dot{a}^{2}\partial_{a}\mathcal{B}\right)\Upsilon\right)$
$\displaystyle-\frac{1}{N\sqrt{\mathcal{AB}}}\left(-\mathcal{A}\mathcal{B}\dot{a}\dot{t}^{2}\partial_{0}\Xi+2\mathcal{A}\sqrt{\mathcal{AB}}\dot{t}^{3}\partial_{0}\Upsilon+\mathcal{B}\left(2\mathcal{A}\dot{t}^{2}\dot{a}-\mathcal{B}\dot{a}^{3}\right)\partial_{0}\Theta\right)$
$\displaystyle\left.-\frac{1}{N\sqrt{\mathcal{AB}}}\left(2\mathcal{B}\dot{a}^{3}\left(\sqrt{\mathcal{AB}}\partial_{a}\Upsilon+\Upsilon\partial_{a}\sqrt{\mathcal{AB}}\right)+\mathcal{A}\dot{t}\dot{a}^{2}\left(\mathcal{B}\partial_{a}\Theta+\Theta\partial_{a}\mathcal{B}\right)-\left(2\mathcal{B}\dot{a}^{2}\dot{t}-\mathcal{A}\dot{t}^{3}\right)\left(\mathcal{A}\partial_{a}\Xi+\Xi\partial_{a}\mathcal{A}\right)\right)\right\rangle$
$\displaystyle=\kappa\left(\frac{1-D}{D}a^{2}\delta\rho-a^{2}\delta
P-\frac{a^{2}}{D}\tilde{\gamma}^{ml}\nabla_{m}\nabla_{l}\pi^{S}-\left(\frac{D-1}{D}\rho+P\right)2N^{2}\Phi\right).$
(123)

In this last equation, we have used:
$\eta_{\tau\tau}=\gamma^{A}_{\tau}\gamma^{B}_{\tau}h_{AB}$ or
$-N^{2}\Phi=-\dot{t}^{2}\mathcal{A}\Xi+2\dot{a}\dot{t}\sqrt{\mathcal{AB}}\Upsilon+\dot{a}^{2}\mathcal{B}\Theta$.

###  V.2 Synchronous Gauge

In the synchronous gauge we have $A^{B}\neq 0$, $B^{B}\neq 0$, all other
scalars are null. The line element of such gauge is

$\displaystyle
ds^{2}=-\mathcal{A}_{\pm}dt^{2}_{\pm}+\mathcal{B}_{\pm}da^{2}+a^{2}\left[(1+A)\tilde{g}_{ij}+\nabla_{i}\nabla_{j}B\right]dx^{i}dx^{j}.$
(124)

Hence, the gravitational field equations (108)-(111), takes finally the form

$\displaystyle\left\langle\frac{1}{2N\sqrt{\mathcal{AB}}}\left(\mathcal{BC}\dot{a}\partial_{0}A^{B}+\mathcal{A}\dot{t}\left(A^{B}\partial_{a}\mathcal{C}+\mathcal{C}\partial_{a}A^{B}\right)\right)-\frac{\partial_{a}\mathcal{C}}{2\mathcal{B}}\delta
n_{a}\right\rangle$ $\displaystyle=-\kappa\frac{a^{2}}{D}\left(\delta\rho+\rho
A^{B}-\tilde{\gamma}^{lm}\nabla_{m}\nabla_{l}\pi^{S}\right),$ (125)

$\displaystyle\nabla_{j}\nabla_{k}\left\\{\left\langle\frac{1}{2N\sqrt{\mathcal{AB}}}\left(-\mathcal{B}\dot{a}\left(-\mathcal{C}\partial_{0}B^{B}\right)-\mathcal{A}\dot{t}\left(-\mathcal{C}\partial_{a}B^{B}-B^{B}\partial_{a}\mathcal{C}\right)\right)-\delta
n\right\rangle\right.$ $\displaystyle\left.+\kappa
a^{2}\left(\pi^{S}+\frac{\rho}{D}B^{B}\right)\right\\}=0,$ (126)

$\displaystyle-\nabla_{i}\delta\dot{n}^{+}+\nabla_{i}\delta\dot{n}^{-}+\frac{\dot{a}\partial_{a}\mathcal{C}}{2\mathcal{C}}\left(\nabla_{i}\delta
n^{+}-\nabla_{i}\delta n^{-}\right)=\kappa
a\left(P+\rho\right)\nabla_{i}\delta u,$ (127)

$\displaystyle\left\langle-\dot{t}\delta\dot{n}_{0}-\dot{a}\delta\dot{n}_{a}+\frac{1}{2\mathcal{AB}}\left(2\mathcal{B}\dot{a}\dot{t}\partial_{a}\mathcal{A}\delta
n_{0}+\mathcal{A}\left(\dot{t}^{2}\partial_{a}\mathcal{A}+\dot{a}^{2}\partial_{a}\mathcal{B}\right)\delta
n_{a}\right)\right\rangle$
$\displaystyle=\kappa\left(\frac{1-D}{D}a^{2}\delta\rho-a^{2}\delta
P-\frac{a^{2}}{D}\tilde{\gamma}^{ml}\nabla_{m}\nabla_{l}\pi^{S}\right).$ (128)

##  VI Discussion and Conclusions

Here we present a formulation of perturbations in more than the standard 3+1
dimensions, where the background metric is homogeneous and isotropic and
consists of an arbitrary sectional curvature where the content of matter is
endowed with a cosmological constant, being the geometry of space-time
determined by the Ricci tensor as usual. Thus, we develop a set of equations
that indicate the evolution of perturbations, where we also propose that the
equations of general relativity are fulfilled satisfied with the number of
dimensions of space, such equations can be particularized and be very useful
for important models such as the case of structure formation of the universe
where initial conditions are considered and the content of matter such as
photons, baryons and neutrinos, or also the case of generation of
gravitational waves, either primordial or in the vicinity of a black hole,
both due to tensor perturbations.

In that same line of thought and looking for a formulation of perturbations in
extra dimensions, we present a development of perturbations for branes where
we start from evolutionary equations and in where the geometry of space-time
is determined by the extrinsic curvature tensor. We have that the bulk where
the branes are embedded has a particular distribution of matter, however the
homogeneity and isotropy is fulfilled in the internal space of the brane,
having in turn an arbitrary sectional curvature. For this case, we also find
equations for the evolution of perturbations in branes, and their respective
scalar, vector and tensor modes, choosing also the Newtonian conformal and
synchronous gauges, these in their version of extra dimensions. When we study
our equations with the Ricci tensor, we observe the universe as a whole, but
when we work with the extrinsic curvature tensor, we observe the universe as
part of a whole; in both cases we consider the energy-momentum tensor as a
perfect fluid, which already has the structure to also be treated as a scalar
fields.

Our work shown in this paper will be very useful for the development of
perturbations in the Bulk in a more efficient way, where both the Ricci tensor
and the extrinsic curvature will be considered in the same treatment to
determine the geometry of the space-time in where we have generalized
evolutionary equations for perturbations. Among the possible applications of
this generalized model are: i) the generation of gravitational waves in the
bulk or in extra dimensions, ii) the formation of structure into the brane due
to the interaction with another brane (these may have different matter
contents) or even with the same bulk.

###### Acknowledgements.

Z.E.M.D appreciates the support of CONACYT and the collaborations of the
Escuela Superior de Física y Matemáticas (ESFM) and Instituto Avanzado de
Cosmología (IAC). M.A.G-A acknowledges support from cátedra Marcos Moshinsky
(MM) and Universidad Iberoamericana for the support with the National Research
System (SNI) grant. The numerical analysis was carried out by the Numerical
Integration for Cosmological Theory and Experiments in High-energy
Astrophysics ”Nicte Ha” cluster at IBERO University, acquired through cátedra
MM support.

## References

  * Riess  _et al._ (1998) A. G. Riess, A. V. Filippenko, P. Challis, A. Clocchiatti, A. Diercks, _et al._ , The Astronomical Journal 116, 1009 (1998).
  * Perlmutter  _et al._ (1999) S. Perlmutter, G. Aldering, G. Goldhaber, R. A. Knop, P. Nugent, others, and T. S. C. Project, The Astrophysical Journal 517, 565 (1999).
  * Matos and Guzman (2000) T. Matos and F. S. Guzman, Class. Quant. Grav. 17, L9 (2000), arXiv:gr-qc/9810028 .
  * Matos and Urena-Lopez (2001) T. Matos and L. A. Urena-Lopez, Phys. Rev. D 63, 063506 (2001), arXiv:astro-ph/0006024 .
  * Carroll (2001) S. M. Carroll, Living Rev. Rel. 4, 1 (2001), arXiv:astro-ph/0004075 [astro-ph] .
  * Motta  _et al._ (2021) V. Motta, M. A. García-Aspeitia, A. Hernández-Almada, J. Magaña, and T. Verdugo, (2021), arXiv:2104.04642 [astro-ph.CO] .
  * Long  _et al._ (2003) J. C. Long, H. W. Chan, A. B. Churnside, E. A. Gulbis, M. C. M. Varney, and J. C. Price, Nature 421, 922 (2003), arXiv:hep-ph/0210004 .
  * Randall and Sundrum (1999a) L. Randall and R. Sundrum, Phys. Rev. Lett. 83, 3370 (1999a), arXiv:hep-ph/9905221 .
  * Randall and Sundrum (1999b) L. Randall and R. Sundrum, Phys. Rev. Lett. 83, 4690 (1999b), arXiv:hep-th/9906064 .
  * Arkani-Hamed  _et al._ (1998) N. Arkani-Hamed, S. Dimopoulos, and G. R. Dvali, Phys. Lett. B 429, 263 (1998), arXiv:hep-ph/9803315 .
  * Antoniadis  _et al._ (1998) I. Antoniadis, N. Arkani-Hamed, S. Dimopoulos, and G. R. Dvali, Phys. Lett. B 436, 257 (1998), arXiv:hep-ph/9804398 .
  * Akama (1982) K. Akama, Lect. Notes Phys. 176, 267 (1982), arXiv:hep-th/0001113 .
  * Dvali  _et al._ (2000) G. R. Dvali, G. Gabadadze, and M. Porrati, Phys. Lett. B 485, 208 (2000), arXiv:hep-th/0005016 .
  * Fang  _et al._ (2008) W. Fang, S. Wang, W. Hu, Z. Haiman, L. Hui, and M. May, Phys. Rev. D 78, 103509 (2008), arXiv:0808.2208 [astro-ph] .
  * Koyama (2003) K. Koyama, Phys. Rev. Lett. 91, 221301 (2003), arXiv:astro-ph/0303108 .
  * Frolov and Kofman (2002) A. V. Frolov and L. Kofman, (2002), arXiv:hep-th/0209133 .
  * García-Aspeitia  _et al._ (2017) M. A. García-Aspeitia, J. Magaña, A. Hernández-Almada, and V. Motta, Int. J. Mod. Phys. D 27, 1850006 (2017), arXiv:1609.08220 [astro-ph.CO] .
  * Garcia-Aspeitia  _et al._ (2018) M. A. Garcia-Aspeitia, A. Hernandez-Almada, J. Magaña, M. H. Amante, V. Motta, and C. Martínez-Robles, Phys. Rev. D 97, 101301 (2018), arXiv:1804.05085 [gr-qc] .
  * Maartens and Koyama (2010) R. Maartens and K. Koyama, Living Rev. Rel. 13, 5 (2010), arXiv:1004.3962 [hep-th] .
  * Weinberg (2008) S. L. Weinberg,  _Cosmology_ (OUP Oxford, 2008).
  * Dodelson (2003) S. Dodelson,  _Modern Cosmology_ (Academic Press, 2003).
  * Ma and Bertschinger (1995) C.-P. Ma and E. Bertschinger, The Astrophysical Journal 455, 7 (1995).
  * Cordero and Vilenkin (2002) R. Cordero and A. Vilenkin, Physical Review D 65 (2002), 10.1103/physrevd.65.083519.
  * García-Aspeitia and Matos (2010) M. A. García-Aspeitia and T. Matos, General Relativity and Gravitation 43, 315 (2010).