# Effects of quantum fluctuations of the metric on a braneworld

C. A. S. Almeida<EMAIL_ADDRESS>Institute of Cosmology, Department of
Physics and Astronomy, Tufts University, Medford, Massachusetts, USA.
Departamento de Física, Universidade Federal do Ceará, Fortaleza, CE, Brazil.
F. C. E. Lima<EMAIL_ADDRESS>Departamento de Física,
Universidade Federal do Maranhão, São Luís, MA, Brazil.

###### Abstract

Abstract: Adopting the premise that the expected value of the quantum
fluctuating metric is linear, i.e., $\langle g^{\mu\nu}\rangle=\alpha
g^{\mu\nu}$, we analyze the modified gravity theory induced by the Einstein-
Hilbert action coupled to a matter field. This approach engenders the $f(R,T)$
gravity used to investigate the braneworld. In this scenario, considering a
thick brane, the influence of metric fluctuations on brane dynamics is
investigated. Consequently, one shows how the metric fluctuations influence
the vacuum states. This influence has repercussions for modifying the brane
energy and the asymptotic profile of the matter field. After noticing these
modifications, we analyzed the most likely and stable structures from the
matter field. One performs this analysis considering the theoretical measure
of differential configurational entropy.

Keywords: Metric fluctuation; Einstein-Hilbert action; Braneworld.

##  I Introduction

A challenge to theoretical gravity models is their agreement with recent
observational data. For example, some data suggest the existence of a late
acceleration from the universe Harko  and the possibility of existing matter
and dark energy on it Riess , Perlmutter , Bernadis , Hanany . In this case,
the proposals that allow good agreement between theoretical models and
phenomenological data are the modified gravity theories Harko , Sotiriou1 ,
Sotiriou2 , Nojiri , Felice , Sharif . Physically, one assumes in the modified
gravity models that Einstein’s gravity of general relativity is modified to
formulate a more general action NojiriSDO . The simplest possibilities for
constructing a modified gravity theory are the models $f(R)$ Felice ,
Starobinsky , Olmo1 , Olmo2 , Olmo3 , $f(T)$ YFCai , SHChen , Dent , RJYang ,
and $f(R,T)$ Harko , Barrientos , JBarrientos , Alvarenga , Moraes , which
briefly are theories whose Einstein-Hilbert standard action is replaced by an
arbitrary function of the Ricci scalar $f(R)$ or by the trace of the stress-
energy tensor $f(T)$ or simultaneously by the Ricci scalar $R$ and the trace
of the stress-energy tensor $T$, i. e., a $f(R,T)$ gravity. Thus, motivated by
this, several studies have considered the modified gravity theory, see Refs.
ADas , Myrzakulov , RZaregonbadi , ZYousaf .

Naturally, a question arises when studying the modified gravity theories. That
issue is: what proposal is the most adequate to describe the modified gravity
theory? Some works propose specific models of modified gravity considering
weak field constraints obtained in the classical tests of general relativity
for models similar to the solar system TChiba1 , ALErickcek , TChiba2 ,
NojiriSDO00 , CapozzielloST1 , CapozzielloST2 . However, in this work, we will
consider a different approach, i.e., we will apply a quantum fluctuations
approach of the metric in the Einstein-Hilbert theory to obtain a modified
gravity theory. The quantum fluctuation approach to the metric entails
considering a system within quantum gravity where the metric comprises a
classical contribution ($g^{\mu\nu}$) and a quantum contribution
($\delta\hat{g}^{\mu\nu}$). In contemplating this approach, it is interesting
to note that the contribution from the expected value of the quantum part must
be non-null. This consideration arises from the role played by the quantum
component in the metric’s decomposition, which engenders modifications to
Einstein’s gravity structure, thereby enabling the emergence of a non-minimal
coupling between gravity and matter. For this approach, the expected value of
the metric adheres to certain physical assumptions, including the non-nullity
of the expected value of the quantum contribution and the feasibility of
expressing the metric $g^{\mu\nu}$ in terms of some symmetric second-order
tensor constructed from the metric. Naturally, a more fundamental scenario
that respects these impositions is when the expected value of the metric
exhibits a linear behavior, i.e., $\langle\hat{g}^{\mu\nu}\rangle=\alpha
g^{\mu\nu}$. As a consequence of this approach, one notes that the small
fluctuations induce linear $f(R,T)$ theories. This result will allow us to
study the impacts of the quantum fluctuations of metric on the braneworld
scenario in five dimensions.

Theoretical models that consider the existence of extra dimensions start from
the premise that the universe is in a higher-dimensional spacetime Rubakov ,
Koyama , Rubakov2 . Considering this, these theories have attracted the
attention of several researchers. An interesting theory in this scenario is
the braneworld theory Ida , Tan , Gauy , Germani , Sahni . The idea of
braneworld begins to gain supporters with the proposal of the Horava-Witten
theory, which relates the heterotic string theory
$\mathrm{E}_{8}\times\mathrm{E}_{8}$ coupled to $M$ theory with eleven-
dimensions compacts Koyama . In this scenario, supergravity lives at the
5-dimensional Anti-de-Sitter (AdS) spacetime. Meantime, Standard Model
particles are confined to the 3-brane Koyama . Phenomenologically, this
hypothesis opens a way to resolve the mass hierarchy problem between the
fundamental scales of particle physics and gravity. Based on this, Randall and
Sundrum RS1 , RS2  proposed the five-dimensional braneworld model that became
known by their name. In his theory, one assumes the existence of four-
dimensional domain walls contained in a five-dimensional AdS spacetime RS1 ,
RS2 .

The braneworld models on modified gravity scenarios have been a topic of
increasing interest, see Refs. Wang , Saavedra , Hoff , Guo2 . Generally
speaking, one believes that these braneworld models in modified gravity
scenarios can provide sophisticated solutions to hierarchy problems, further
answering some issues about the description of dark matter Aspeitia  and dark
energy Koyama2 , as well as other questions Brax , Alam . Due to this,
braneworld theories have gained space in some investigations Biggs ,
Pourhassan , Sokoliuk . That is because, in this scenario, one can more easily
notice the effects of modified gravity on the brane, see Ref. MSLA . Thus,
motivated by this, we will investigate a thick brane scenario in a $f(R,T)$
gravity, seeking to understand how the metric fluctuations influence a five-
dimensional braneworld.

Not far from these theories, we will consider the Configurational Entropy (CE)
initially proposed by Gleiser et al. Gleiser1 , Gleiser2 , Gleiser3 , Gleiser4
, Gleiser5 , Gleiser6  to find the values of the metric fluctuations that
describe the most likely and stable braneworld. Thus, we hope to obtain the
most likely behavior of the brane in an $f(R,T)$ gravity theory induced by
quantum fluctuations from the metric. Indeed, to reach these results, let us
adopt a variant of CE, i.e., the Differential Configurational Entropy (DCE).
We will do this because DCE has proven appropriate in studying the localized
structures that arise in braneworld theories, e. g., see Ref. Roldao4 .
Moreover, the DCE is a good approach in investigations from models that admit
topological defects type domain walls Lima3 . That is because DCE can provide
adequate information about the parameters that describe a stable field
configuration Gleiser1 , Gleiser2 , Gleiser3 , Gleiser4 , Gleiser5 , Gleiser6
. Examples of the application of this approach appear in Ref. Gleiser2 , where
the authors show that the energy variations of the structures are proportional
to theoretical measures of CE and its variants. Furthermore, this approach has
reported significant results on the dynamics of spontaneous symmetry breaking
Gleiser1 , of compact objects Gleiser3 , Gleiser7 , and the stability of
modified gravity models on braneworlds FMoreira1 , FMoreira2 .

Based on all the applications and concepts presented throughout this
introduction, the question naturally arises: what is the influence of metric
fluctuations in a modified gravity scenario? Also, how do these fluctuations
are felt at the braneworld? In this paper, let us answer these questions.

We organized our work as follows: in section II, one considers the quantum
fluctuations approach from the metric to induce an f(R) gravity. In section
III, one builds a braneworld theory using the modified gravity theory. In
section IV, we adopt the approach of configurational entropy to select the
most likely and stable regimes associated with the braneworld. To finalize, in
section V our discoveries are announced.

##  II Quantum fluctuations inducing a modified gravity

A major open problem in theoretical physics is the quantization of gravity. In
this scenario, to quantize gravity, several steps are required. The first step
towards a quantized theory of gravity is to assume that metric is an ordinary
field. Thus, in this section, let us adopt this premise for the metric
profile. Indeed, this will allow, in principle, us to build a gravity-
effective theory. To apply this approach, we employ a metric fluctuation on
the Einstein-Hilbert Lagrangian density, i. e.,

$\displaystyle\mathcal{L}=\sqrt{-g}\bigg{[}-\frac{1}{4}R+\mathcal{L}_{\text{matter}}\bigg{]}$
(1)

where the Lagrangian density of the matter field is

$\displaystyle\mathcal{L}_{\text{matter}}=\frac{1}{2}\nabla_{\mu}\phi\,\nabla^{\mu}\phi-V(\phi).$
(2)

To apply a non-perturbative quantization approach, let us promote the
classical fields to field operators. Thus, Einstein’s equation takes the form:

$\displaystyle\hat{G}_{\mu\nu}=\hat{R}_{\mu\nu}-\frac{1}{2}g_{\mu\nu}\hat{R}=\kappa^{2}\hat{T}_{\mu\nu}.$
(3)

Indeed, one can find a similar approach in Refs. Dzhunushaliev ,
Dzhunushaliev1 . For example, in Ref. Dzhunushaliev , one shows that the
metric in a gravity quantum extension has classical and quantum contributions.
Meanwhile, Dzhunushaliev Dzhunushaliev1  et al. apply this approach to explain
the acceleration of the Universe.

A feature of this approach is that the quantities $\Gamma^{\rho}\,_{\mu\nu}$,
$R^{\rho}\,_{\lambda\mu\nu}$ and $R_{\mu\nu}$ are initially promoted to
operators so that their classic definitions are not changed. Thus, assuming
these quantities as operators $\hat{\Gamma}^{\rho}\,_{\mu\nu}$,
$\hat{R}^{\rho}\,_{\lambda\mu\nu}$ and $\hat{R}_{\mu\nu}$ one can considering
Heisenberg’s formalism, solve the equation of operator (3) by averaging over
all possible products of the metric operator $\hat{g}(x_{i})$. That gives us
an infinite set of equations, namely,

$\displaystyle\begin{matrix}\langle\mathcal{Q}|\hat{g}(x_{1})\cdot\hat{G}_{\mu\nu}|\mathcal{Q}\rangle=\kappa^{2}\langle\mathcal{Q}|\hat{g}(x_{1})\cdot\hat{T}_{\mu\nu}|\mathcal{Q}\rangle\\\
\langle\mathcal{Q}|\hat{g}(x_{1})\cdot\hat{g}(x_{2})\cdot\hat{G}_{\mu\nu}|\mathcal{Q}\rangle=\kappa^{2}\langle\mathcal{Q}|\hat{g}(x_{1})\cdot\hat{g}(x_{2})\cdot\hat{T}_{\mu\nu}|\mathcal{Q}\rangle\\\
\vdots\\\
\langle\mathcal{Q}|\hat{g}(x_{1})\cdot...\cdot\hat{g}(x_{i})\hat{G}_{\mu\nu}|\mathcal{Q}\rangle=\kappa^{2}\langle\mathcal{Q}|\hat{g}(x_{1})\cdot...\cdot\hat{g}(x_{i})\cdot\hat{T}_{\mu\nu}|\mathcal{Q}\rangle.\end{matrix}$
(4)

In this scenario, we interpret $|\mathcal{Q}\rangle$ as a quantum state. An
approach to solving the system of Eq. (4) is to decompose the metric operator
as follows:

$\displaystyle\hat{g}_{\mu\nu}=g_{\mu\nu}+\delta\hat{g}_{\mu\nu},$ (5)

so that $g_{\mu\nu}$ is the average metric and $\delta\hat{g}_{\mu\nu}$ is a
fluctuation of the non-trivial metric. For more details, see Refs.
Dzhunushaliev , Dzhunushaliev1 , Dzhunushaliev2 . Using the perturbation (5),
let us expand the Lagrangian (1). In this case, disregarding higher-order
perturbations, one obtains

$\displaystyle\mathcal{L}\approx-\frac{1}{4}\sqrt{-g}(R+G_{\mu\nu}\langle\hat{g}^{\mu\nu}\rangle)+\sqrt{-g}\bigg{(}\mathcal{L}_{\text{matter}}+\frac{1}{2}T_{\mu\nu}\langle\delta
g^{\mu\nu}\rangle\bigg{)}.$ (6)

As suggested by Yang in Ref. RYang , it is convenient to assume linear metric
fluctuations due to symmetries of the Lagrangian density (1). Thereby,
adopting a linear metric fluctuation, i.e.,
$\langle\hat{g}^{\mu\nu}\rangle\approx\alpha g^{\mu\nu}$ 111We adopt the
expected value of the linear metric $\langle g^{\mu\nu}\rangle\approx\alpha
g^{\mu\nu}$ once our interest consists in developing a $f(R,T)$ gravity.
However, the constraint on $\langle g^{\mu\nu}\rangle$ are $\langle
g^{\mu\nu}\rangle$ is non-zero and a symmetric two-rank tensor. Thus, one has
several possibilities a priori. For instance, adopting $\langle
g^{\mu\nu}\rangle\simeq\alpha R^{\mu\nu}$, one obtains a quadratic gravity
theory, see Refs. Dzhunushaliev , Dzhunushaliev1 ., we obtain the modified
Lagrangian density, namely,

$\displaystyle\mathcal{L}=\sqrt{-g}\bigg{[}-\frac{1}{4}(1-\alpha)R-\frac{1}{2}\alpha
T+\mathcal{L}_{\text{matter}}\bigg{]}.$ (7)

In this formulation, $\alpha$ denotes a dimensionless parameter responsible
for adjusting the quantum fluctuations of the metric. Seeking a theory that
approaches the usual theory, i.e., the Einstein-Hilbert theory, let us assume
that the parameter $\alpha$ describes the small fluctuations of the metric,
consequently, $|\alpha|\ll 1$. Furthermore, the $R$ displayed on the action is
the Ricci scalar, and $T$ is the trace of the stress-energy tensor
$T=g^{\mu\nu}T_{\mu\nu}$, so that the stress-energy tensor is

$\displaystyle
T_{\mu\nu}=-\frac{2}{\sqrt{-g}}\frac{\delta\sqrt{-g}\mathcal{L}_{\text{matter}}}{\delta
g^{\mu\nu}}.$ (8)

##  III The braneworld in $f(R,T)$ gravity

Assuming the corrections arising from the metric fluctuations, let us build a
braneworld considering a modified gravity scenario type
$f(R,T)=\lambda(R)+\xi(T)$, i. e.,

$\displaystyle
S=\int\,d^{5}x\,\bigg{[}-\frac{1}{4}\lambda(R)-\xi(T)+\mathcal{L}_{\text{matter}}\bigg{]}$
(9)

with

$\displaystyle\lambda(R)=(1-\alpha)R\hskip 28.45274pt\text{and}\hskip
28.45274pt\xi(T)=\frac{1}{2}\alpha T.$ (10)

The $\lambda(R)$ and $\xi(T)$ functions chosen in Eq. (10) is the result
presented in the Lagrangian (7). Furthermore, we will adopt $4\pi G_{5}=1$ and
$g=$det$(g_{ab})$.

To study the braneworld in $f(R,T)$ gravity, allow us to assume the line
element

$\displaystyle ds^{2}=\text{e}^{2A(y)}\eta_{ab}dx^{a}dx^{b}-dy,$ (11)

where e2A is called the warp factor. Here, the indices $a$ and $b$ are varying
from 0 to 3 with the extra-dimension being $y=x^{4}$ and metric signature
$(+,-,-,-,-)$.

Being the matter field described by Lagrangian density (2), the stress-energy
tensor will be

$\displaystyle T_{\mu\nu}=\nabla_{\mu}\phi\,\nabla_{\nu}\phi-
g_{\mu\nu}\mathcal{L}_{\text{matter}}\hskip 28.45274pt\text{with}\hskip
28.45274pt\mu,\nu=0,1,\dots,4;$ (12)

so that the trace of the stress-energy tensor is

$\displaystyle
T=g^{\mu\nu}T_{\mu\nu}=-\frac{3}{2}\nabla_{\mu}\phi\,\nabla^{\mu}\phi+5V(\phi).$
(13)

Let us now investigate the equations of motion of our five-dimensional
braneworld in $f(R,T)$ gravity. For this, we start by varying the action
concerning the scalar field $\phi$. That leads us to

$\displaystyle\nabla_{\mu}\nabla^{\mu}\phi+3\nabla_{\mu}[\xi_{T}\nabla^{\mu}\phi]+(5\xi_{T}+1)V_{\phi}=0,$
(14)

where $\xi_{T}=d\xi/dT$ and $V_{\phi}=dV/d\phi$.

Subsequently, varying the action (9) concerning the metric, one arrives at
Einstein’s equation, namely,

$\displaystyle\lambda_{R}R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}\lambda+(g_{\mu\nu}\square-\nabla_{\mu}\nabla_{\nu})\lambda_{R}=2T_{\mu\nu}+2g_{\mu\nu}\lambda+6\lambda_{T}\nabla_{\mu}\phi\,\nabla_{\nu}\phi$
(15)

where $\lambda_{R}=d\lambda/dR$ with $R=20A^{\prime}(y)+8A^{\prime\prime}(y)$.
Here the prime notation refers to the derivative concerning the extra-
dimension coordinate.

Exposing the Eqs. (14) and (15) in terms of the matter field $\phi$ and the
warp function $A(y)$, one obtains:

$\displaystyle\bigg{(}1+\frac{3}{2}\alpha\bigg{)}\phi^{\prime\prime}+4\bigg{(}1+\frac{3}{2}\alpha\bigg{)}A^{\prime}\phi^{\prime}=\bigg{(}1+\frac{5}{2}\alpha\bigg{)}\frac{\partial
V}{\partial\phi};$ (16)
$\displaystyle(1-\alpha)A^{\prime\prime}=-\frac{2}{3}\bigg{(}1+\frac{3}{2}\alpha\bigg{)}\phi^{\prime
2};$ (17) $\displaystyle 3(1-\alpha)A^{\prime
2}=\frac{1}{2}\bigg{(}1+\frac{3}{2}\alpha\bigg{)}\phi^{\prime
2}-\bigg{(}1+\frac{5}{2}\alpha\bigg{)}V.$ (18)

Algebraically, one can reduce this set of equations to the expressions:

$\displaystyle\phi^{\prime\prime}+4A^{\prime}\phi^{\prime}=\bigg{(}\frac{1+5\alpha/2}{1+3\alpha/2}\bigg{)}\frac{\partial
V}{\partial\phi},$ (19)

and

$\displaystyle A^{\prime\prime}+4A^{\prime
2}=-\frac{4}{3}\bigg{(}\frac{1+5\alpha/2}{1-\alpha}\bigg{)}V.$ (20)

Furthermore, it is interesting to highlight that the braneworld described by
equations (19) and (20) has energy due to the propagation of the matter field
along the extra-dimension. In this case, this energy is

$\displaystyle E=\int\,\text{e}^{2A}\,\bigg{[}\frac{1}{2}\phi^{\prime
2}+V(\phi)\bigg{]}\,dy,$ (21)

where brane’s energy density ($\rho_{E}$) is

$\displaystyle\rho_{E}(\phi;y)=\text{e}^{2A}\,\bigg{[}\frac{1}{2}\phi^{\prime
2}+V(\phi)\bigg{]}.$ (22)

###  III.1 The thick-brane model

Allow us to continue our study assuming a particular geometry for spacetime.
To choose a specific profile of spacetime, let us adopt an appropriate form
for the warp function $A(y)$. Indeed, one bases this choice on some
requirements. For example, in general, it is interesting to assume an $A(y)$
that reproduces a Randall-Sundrum type warp factor far from the brane, i.e.,
$\lim_{y\to\infty}\text{e}^{2A(y)}=0$. Meanwhile, in the neighborhood of the
brane, it should have a smooth profile (no singularity). That allows us to
bypass the thin-brane energy scale problem and leads us to a thick-brane
model. We found two distinct braneworld behaviors, i.e., the thin and thick
brane. In both cases, one requests that the warp factor be symmetrical.
Mathematically, $\text{e}^{2A(y)}$=$\text{e}^{2A(-y)}$ is required for the
matter field to have $Z_{2}$ symmetry preserved, while the symmetry break
occurs in the matter sector. Furthermore, another condition that restricts the
behavior of the warp factor is the zero-mode normalization of the graviton
$\int_{-\infty}^{\infty}\,dy\,\text{e}^{8A(y)}$ which must be finite and non-
null. In addition to these requirements, let us assume a warp function profile
that falls into the approximate thin-brane theory and bypasses the brane
energy scaling problem. A warp function used that fulfills these requirements
is the model in which the warp function takes the form:

$\displaystyle A(y)=-\text{ln}[\cosh(\sigma y)].$ (23)

In this context, the independent variable $y$ has a length dimension.
Consequently, the parameter $\sigma$ is the inverse of length. So, the
parameter $\sigma$ modifies the brane’s width. Furthermore, one highlights
that the warp function (23) has been used extensively in several models of
braneworlds, see Refs. Moreira1 , CCLi , TTSui , Moreira2 . For example, in
Ref. Moreira1 , one assumes the function (23) to study braneworld theories in
gravity $f(T,B)$. Meanwhile, adopting the warp function (23), the Ref. CCLi
presents an investigation of the effective theories with self-interacting.
These applications are interesting because they show that the warp function of
the type chosen in Eq. (23) accurately describes the behavior of the brane and
can give us predictions about a thin-brane theory. That is possible because
the brane thickness is adjustable by changing the $\sigma$ parameter.

We expose the behavior of the warp function $A(y)$ and the warp factor
$\text{e}^{2A(y)}$ respectively in Figs. 1(a) and 1(b). Note that the warp
function locates the brane at $y=0$ and reduces to zero over large distances,
as required initially, but reproducing a thick brane behavior. However, for
large values of $\sigma$, one obtains an effective theory of type thin brane.
For convenience, in this study, we will consider the thick-brane case, i.e.,
when $|\sigma|\ll 1$.

(a) (b)

Figure 1: (a) Plots of the warp function $A(r)$. (b) The behavior of the warp
factor e2A(y).

Considering the profile of the function $A(y)$ (and consequently, the warp
factor e2A(y)) and Eq. (17), it is obtained that the solution of the matter
field in terms of the extra dimension, the fluctuation parameters and the
brane thickness. In this case, the matter field solution is

$\displaystyle\phi(\sigma,\alpha;\,y)=2\sqrt{\frac{3(1-\alpha)}{2+3\alpha}}\arctan\bigg{[}\frac{\tanh(\sigma
y)}{2}\bigg{]}.$ (24)

Meantime, considering the matter field solution and Eqs. (19) and (20), one
concludes that the interaction is

$\displaystyle V(\sigma,\alpha;\,\phi)=$
$\displaystyle-\frac{3(1-\alpha)\sigma^{2}}{4+10\alpha}\bigg{[}4-5\,\text{sech}\bigg{(}2\,\text{arctanh}\bigg{(}\text{tan}\bigg{(}\frac{\phi}{2\sqrt{\frac{5}{2+3\alpha}-1}}\bigg{)}\bigg{)}\bigg{)}\bigg{]}$
(25)

with $|\phi|<\pi\sqrt{\frac{3(1-\alpha)}{2+3\alpha}}$.

In Fig. 2(a), one displays the matter field for several values of the brane
thickness. Moreover, Fig. 2(b) shows the behavior of the matter field for
several values of the fluctuation parameter. Posteriorly, in Fig. 3, we expose
the behavior of the interaction that satisfies Eqs. (16), (17) and (18) in
terms of the field $\phi$.

(a) (b)

Figure 2: (a) Matter field as a function of the extra dimension for several
values of the brane thickness and keeping $\alpha=0.1$. (b) Matter field as a
function of the extra dimension for several values of the metric fluctuation
and keeping $\sigma=0.2$.

(a) (b)

Figure 3: (a) Potential as a function of the matter field for several values
of brane thickness and keeping $\alpha=10^{-1}$. (b) Potential as a function
of the matter field for several values of the metric fluctuation and keeping
$\sigma=0.2$.

The results found in Eqs. (24 and 25) and shown in Figs. 2 and 3 show the
field of matter described by solitonic solutions so that far from the brane,
the theory reaches the vacuum of matter’s topological sector. In fact, this
vacuum value is
$\phi_{0}=\pm\frac{\pi}{2}\sqrt{\frac{3(1-\alpha)}{2+3\alpha}}$. Therefore,
the vacuum that arises due to spontaneous symmetry breaking in the matter
sector modifies due to the metric fluctuations. Note that if the fluctuation
of the metric reaches the value of $\alpha=1$, we will have $\phi_{0}=0$, and
thus there will no longer be a vacuum. In other words, when $\alpha\to 1$,
there will be no spontaneous symmetry breaking, so that matter field with
symmetry $Z_{2}$ interpolating between the minimum energy configurations will
not exist. Meantime, if the metric fluctuations are small, i.e., $|\alpha|<1$,
spontaneous symmetry breaking is preserved. Thus, metric fluctuations will be
perceived far from the brane regardless of the thickness $\sigma$. Besides,
the results suggest that the variation in thickness will contract the matter
field so that the thicker (greater the $\sigma$) the brane is, the smoother
the topological transition of the matter field between the vacuums $\phi_{0}$
of the theory. In contrast, if the brane tends to a thin brane-like behavior,
i.e., $|\sigma|\ll 1$. Then, we will have contraction (or compactification) of
the field of matter in a way that the matter field tends to evolve quickly
into a vacuum.

To finish this discussion of the topological brane theory, allow us to
investigate the behavior of brane energy density. In this case, considering
Eq. (22) and the solutions found [Eqs. (24) and (25)], one obtains the brane
energy density. We display the brane energy density in Figs. 4(a) and 4(b).

(a) (b)

Figure 4: (a) Energy density of the brane for several values of its thickness
and keeping $\alpha=10^{-1}$. (b) Brane energy density for several values of
the metric fluctuation and keeping $\sigma=0.2$.

The brane energy density suggests a kink-like profile of the matter field.
Also, as the quantum fluctuations of metric must be small, i.e., $|\alpha|\ll
1$, one notes the absence of internal structures in the thick brane when
$0<\sigma<1$. As it was possible to predict, the brane has higher energy when
the brane thickness decreases. This behavior occurs because there is a
divergence when $\sigma\to 0$. So, to bypass this energy scaling problem,
$\sigma$ must be contained in the range $0<\sigma\leq 1$. Moreover, we note
that the modified gravity changes the vacuum value influencing the brane
energy, i.e., when $\alpha$ decreases, the brane energy increases.

##  IV Configurational information theoretical-measurement in braneworld in
modified gravity

The theoretical measure of information first appears with Claude E. Shannon in
his seminal work on the mathematical theory of communication Shannon . In his
work, Shannon seeks to describe the best way to encode the information that an
emitter transmits to a receiver. After Shannon’s work, the definition of
information entropy was reformulated and applied in several scenarios to
obtain the information entropy in several theories, e.g., see to Refs.
Gleiser1 , Gleiser2 , Gleiser3 , Gleiser4 , Gleiser5 , Gleiser6 , Roldao4 ,
Roldao1 , Roldao2 , Roldao3 , Roldao5 . Although Shannon’s approach is
suitable for investigating the information from systems of particles, in field
theories, one needs to reformulate this due to the infinite degrees of
freedom. Thus, based on Shannon’s theory, Gleiser et al. Gleiser1 , Gleiser2
propose an information entropy approach (or information theoretical measure)
for the continuous limit Gleiser1 . This approach is also called
Configurational Entropy (CE). In this scenario, the CE describes a
representation of information-theoretic measures that detail the
configurational complexity of the fields of a system Gleiser1 , Gleiser2 ,
Gleiser3 , Gleiser4 , Gleiser5 . In this work, we will use one variant of
Configurational Entropy (CE), i.e., Differential Configurational Entropy
(DCE). The use of this approach justifies by its applications. Indeed, the DCE
has shown to be a good approach that gives us information about the
informational content of the fields so that one can identify the most likely
stable structures of the theory, see Refs. Roldao4 , Correa1 , Correa2 ,
Correa3 . Motivated by the vast applications of DCE in the study of field
theory Lima3 , Lima1 , Lima2 , in high energy physics Roldao6 , Roldao7 , we
are encouraged to adopt DCE to identify the most likely structures of our
brane in $f(R,T)$ gravity.

Furthermore, it is pertinent to emphasize that the DCE formalism proves
advantageous as it can provide appropriate values for the parameters governing
domain walls LimaCasana . In this section, by applying the principles of the
DCE formalism, we will examine the set of values of the parameters $\alpha$
for generating the topological kinks that configure the matter sector of the
braneworld in $f(R,T)$ gravity theory. It is worth noting that, in principle,
the parameters of metric quantum fluctuations $\alpha$ can take any value,
provided they are subject to the restriction $|\alpha|<1$. This analysis
remains consistent once it allows for finding the most likely values for these
parameters. Such inference is justified by the reduction in the complexity of
the matter field as the DCE decreases, increasing the probability of obtaining
specific field configurations Gleiser1 , Gleiser2 , Gleiser6 . Additionally,
one employs this formalism in several studies of systems with localized energy
configurations NBraga1 , NBraga2 , NBraga3 . Furthermore, it is pertinent to
highlight that configurational entropy proves to be a good tool for studying
the mass spectrum of kaon vector resonances RochaSilva .

###  IV.1 Conceptual review of DCE applied to braneworld

Before studying this theoretical measure of information from the braneworld in
$f(R,T)$ gravity induced by metric fluctuations, let us start by presenting
some concepts that underlie our study. To carry out the theoretical
measurement of information, allow us to use the DCE concept. In this case, one
defines the DCE in terms of Fourier’s transform of the brane energy density,
i.e.,

$\displaystyle
G[\omega]=\frac{1}{\sqrt{2\pi}}\int\,\rho_{E}(y)\,\text{e}^{i\omega y}\,dy.$
(26)

Substituting Eq. (22) into Fourier’s transform (26), one obtains

$\displaystyle
G[\omega]=\frac{1}{\sqrt{2\pi}}\int\,\bigg{[}\frac{1}{2}\phi^{\prime}(y)^{2}+V(\phi(y))\bigg{]}\,\text{e}^{i\omega
y+2A(y)}\,dy.$ (27)

Considering Fourier’s transform (27), the modal fraction of the theory is
constructed. This quantity is

$\displaystyle
g(\omega)=\frac{|G(\omega)|^{2}}{\int\,|G(\omega)|^{2}\,d\omega}.$ (28)

By definition, the modal fraction is the weight relative to each wave mode at
the reciprocal space. Thus, this quantity is always $\leq 1$. For more
details, see Refs. Gleiser1 , Gleiser2 , Gleiser3 , Gleiser4 , Gleiser5 ,
Gleiser6 .

Assuming the modal fraction defined in Eq. (28), we define the DCE as

$\displaystyle
S_{C}[g(\omega)]=-\int\,\bar{g}(\omega)\,\text{ln}[\bar{g}(\omega)]\,d\omega,$
(29)

where the integrand of Eq. (29) is called entropic density and
$\bar{g}(\omega)$ is the modal fraction normalized.

Once defined the DCE, we are now ready to study the differential
configurational entropy of the thick brane presented in the previous section.

###  IV.2 Thick brane DCE in $f(R,T)$ gravity

The first step in calculating the DCE is to obtain the modal fraction of the
system (22). To find the modal fraction, we substitute the warp function (23),
the matter field solution (24), and the interaction (25) in terms of the extra
dimension in Fourier’s transform (27). Posteriorly, considering the solution
of $G(\omega)$, one obtains, after an extensive calculation, the modal
fraction of the brane. In this case, the modal fraction is

$\displaystyle g(\alpha,\,\sigma;\,\omega)=$
$\displaystyle\frac{105\pi[2\alpha\sigma^{2}\omega+(3+5\alpha)\omega^{3}]^{2}}{16\sigma^{5}[35\alpha^{2}+28\alpha(3+5\alpha)\sigma+20(3+5\alpha)^{2}\sigma^{2}]}\bigg{[}-1+\cosh\bigg{(}\frac{\pi\omega}{\sigma}\bigg{)}\bigg{]}^{-1}$
(30)

Therefore, one can note that the DCE will be changed when the metric
fluctuation and brane thickness varies.

Using the modal fraction (30), we calculate the DCE (29) from the brane. To
perform this calculation, a numerical investigation of the solution of the
integral (29) is considered. The numerical result of DCE is shown in Fig.
5(b). Meanwhile, the entropic density associated with numerical solutions is
found in Fig. 5(a).

(a) (b)

Figure 5: (a) Entropic density of the DCE for several values of metric
fluctuation. (b) DCE of the braneworld in terms of the fluctuation.

Interesting results arise when analyzing the DCE of the brane in $f(R,T)$
gravity induced by metric fluctuations, i.e., DCE reaches maximum values when
$\sigma$ increases [see Fig. 4(b)]. Furthermore, there is a critical point of
the DCE at $\alpha=0$ regardless of the thickness of the brane. That suggests
that the most likely structures appear when the metric fluctuations are zero,
i.e., in the usual theory without gravity modifications. However, other local
critical points occur in $\alpha=0.04$. Thereby, we note, numerically, that
the most likely and stable field configurations are kink-like and emerge when
$|\alpha|=0.4$ and $\sigma=0.8$, i. e., in the modified gravity scenario.

##  V Final remarks

In this work, one studies the influence of metric fluctuations in a braneworld
scenario. We noted that when applying the metric quantum perturbations in the
Einstein-Hilbert action coupled to the matter field, a modified gravity theory
is obtained and described by the function
$f(R,T)=-\frac{1}{4}(1-\alpha)R+\frac{1}{2}\alpha T$. This result is
interesting once it allows us to recover the usual case, i.e., the Einstein-
Hilbert theory, when the metric fluctuations are null, i.e., $\alpha\to 0$.

Considering the results obtained by the perturbative approach, we build a
braneworld in $f(R,T)$ gravity. In this scenario, one notes that the vacuum
states depend on the quantum fluctuations of the metric. Consequently, due to
these dependencies, the asymptotic value of the matter field is changed.
Furthermore, it is possible to notice that when the fluctuation reaches
maximum values, the domain wall will no longer exist. That is because when
$\alpha\to 1$, the vacuum expected value is null. Meantime, for small metric
fluctuations, i.e., $|\alpha|<1$, the domain walls arise. So that when
$|\alpha|\ll 1$, the fluctuation effects will feel far from the brane. These
results influence the brane energy, where critical points increase their
intensity around $y=0$ when $|\alpha|\ll 1$.

Finally, we use the differential configurational entropy formalism to study
the most likely, and stable matter field configurations. In this analysis,
some attractive results emerge, i.e., the DCE reaches absolute maximums (or
minimum) at $\alpha=0$ independent of the brane thickness. This critical point
is indicated in $\alpha=0$ [Figs. 4(a) and (b)], regardless of brane
thickness, suggests that the most likely structures appear when the metric
fluctuations are zero, i.e., when we recover the usual theory. However,
another local critical point occurs at $\alpha\simeq 0.04$, indicating
configurations more likely and stable are kink-like structures and appear in
the modified gravity scenario when $|\alpha|\simeq 0.4$ and $\sigma=0.8$.

Although we have developed a theoretical discussion, one can confront these
results experimentally. That is possible because the modified gravity theories
presented can be verified through cosmography SCapozziello1 , SCapozziello2 .
This analysis is feasible, given that the cosmological principle postulates a
single factor as the primary degree of freedom governing the universe
SCapozziello1 , SCapozziello2 , i.e.,

$\displaystyle
a(t)=1+\sum_{k=1}^{\infty}\frac{1}{k!}\frac{d^{k}a}{dt^{k}}\bigg{|}_{t=t_{0}}(t-t_{0})^{k}.$
(31)

Using $a(t)$, one can estimate the Hubble, deceleration, jerk, and snap
parameters, respectively, as

$\displaystyle H(t)=\frac{1}{a}\frac{da}{dt},\hskip
14.22636ptq(t)=-\frac{1}{aH^{2}}\frac{d^{2}a}{dt^{2}},\hskip
14.22636ptj(t)=\frac{1}{aH^{3}}\frac{d^{3}a}{dt^{3}},\hskip
14.22636pt\text{and}\hskip
14.22636pts(t)=\frac{1}{aH^{4}}\frac{d^{4}a}{dt^{4}}.$ (32)

In this form, it is possible to confront the data obtained in our study with
observational data. That is feasible because it allows for the correlation of
cosmological parameters with the parameter of modified gravity arising from
quantum fluctuation.

A future perspective of this study is to understand how these fluctuations
influence cosmological objects and their properties. We hope to perform this
study soon.

## ACKNOWLEDGMENT

The authors thank the Brazilian research agencies, i.e., CNPq, CAPES, and
FAPEMA, for financial support. C. A. S. Almeida thanks the Conselho Nacional
de Desenvolvimento Cientifico e Tecnologico (CNPq), grant no 309553/2021-0.
Furthermore, C. A. S. Almeida is grateful to the Print-UFC CAPES program,
project number 88887.837980/2023-00, and acknowledges the Department of
Physics and Astronomy at Tufts University for hospitality. F. C. E. Lima is
supported by FAPEMA BPD-05892/23 and grateful to the Department of Physics
from the Universidade Federal do Ceará (UFC) for hospitality and support.

## CONFLICTS OF INTEREST/COMPETING INTEREST

All the authors declared that there is no conflict of interest in this
manuscript.

## DATA AVAILABILITY

No data was used for the research described in this article

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