Configurational entropy in $f(R,T)$ brane models

In this work we investigate generalized theories of gravity in the so-called configurational entropy (CE) context. We show, by means of this information-theoretical measure, that a stricter bound on the parameter of $f(R,T)$ brane models arises from the CE. We find that these bounds are characterized by a valley region in the CE profile, where the entropy is minimal. We argue that the CE measure can open a new role and an important additional approach to select parameters in modified theories of gravitation.


I. INTRODUCTION
Although ΛCDM cosmological model provides a great match between theory and observation [1], what makes it to be usually referred to as the "concordance model", a number of shortcomings, as the cosmological constant and hierarchy problems, still await for convincing explanations.While by assuming general relativity as the gravitational theory, those problems cannot be solve straightforwardly, higher order derivative and extradimensional theories might contribute efficiently to solve these issues.
Note that f (R) and f (R, T ) cosmological models [2]- [10], with R and T being respectively the Ricci scalar and the trace of the energy-momentum tensor, are able to describe the cosmic acceleration our universe is undergoing (check [11]- [12]) with no need of invoking a cosmological constant.On the other hand, the hierarchy problem can be solved by assuming our observable universe is a sub-manifold which is embedded in an anti-de Sitter five-dimensional space (AdS 5 ), called bulk, as in Randall-Sundrum braneworld model [13].
In this work we will deal with braneworld models in the presence of scalar fields [14]- [17].Departing from the original Randall and Sundrum proposal, which leads to a thin braneworld scenario, the coupling with scalar fields leads to thick braneworlds.The presence of a scalar field makes the warp function to behave smoothly, yielding such a thickness.This possibility has opened a new area of study, and here we quote Refs.[18]- [25] for some works on the subject.
Specifically, we will consider the f (R, T ) gravity in such a thick braneworld scenario.Note that although f (R) brane models have already been presented in the literature [26]- [34], due to its recent elaboration, f (R, T ) gravity still lacks of a significant number of applications in the braneworld.Anyhow, it is remarkable the fact that in [35], the authors have pioneered such an approach.
Here, rather than a cosmological approach, we will investigate f (R, T ) brane models from the configurational entropy (CE) perspective.Gleiser and Stamatopoulos (GS) have proposed in [36] such a new physical quantity, which brings additional informations about some parameters of a given model for which the energy density is localized.They have shown that the higher the energy that approximates the actual solution, the higher its relative CE, which is defined as the absolute difference between the actual function CE and the trial function CE.The CE is able to solve situations where the energies of the configurations are degenerate.In this case, it can be used to select the best system configuration.
Although it has been recently proposed, the CE has already been used to study the non-equilibrium dynamics of spontaneous symmetry breaking [37], to obtain the stability bound for compact objects [38], to investigate the emergence of localized objects during inflationary preheating [39] and to distinguish configurations with energy-degenerate spatial profiles [40].Solitons, Lorentz symmetry breaking, supersymmetry, and entropy, were employed using the CE concept [41][42][43][44].The CE for travelling solitons reveals that the best value of the parameter responsible for breaking the Lorentz symmetry is that where the energy density is distributed equally around the origin.It was argued that the information-theoretical measure of travelling solitons in Lorentz symmetry violation scenarios can be very important to probe situations where the parameters responsible for breaking the symmetries are arbitrary, with the CE selecting the best value of the parameter in the model.Another interesting work about CE was presented in [45], where it is responsible for identifying the critical point in the context of continuous phase transitions.Finally, in braneworld scenarios it was shown that CE can be employed to demonstrate a high organizational degree in the structure of the system configuration for large values of a parameter of the sine-Gordon model [46].The entropic information-measure in the context of f (R) braneworlds with non-constant curvature has also been studied [47].
In this work, we calculate the CE in f (R, T ) brane arXiv:1509.00732v1[hep-th] 2 Sep 2015 models.We obtain its profile and reveal the information content of such models.The paper is organized as follows.In the next section we will review the concept of CE measure.In Section III, we outline the basic theoretical structure for generalized f (R, T ) theories of gravity.
In Section IV, we show two distinct models in f (R, T ) theory and its solutions.In Section V, we describe the calculations of the CE for f (R, T ) theories.We also show the corresponding CE profile.Finally, in Section VI, we present our conclusions and directions for future work.

II. AN OVERVIEW ABOUT CONFIGURATIONAL ENTROPY MEASURE
Here, we will review the very recent work by GS [36], where it was shown that, in analogy with Shannon's information entropy, a CE measure in a functional space can be used to select the best fit solution with spatiallylocalized energy.Another interesting consequence highlighted by GS is the fact that the CE relates the dynamical and informational content of physical models.In this case, this information-theoretical measure can be able to extract information about the different solutions which is related to their spatial profiles.Thus, following [36], let us begin by writing the CE as where d is the number of spatial dimensions, f (ω) = f (ω)/f max (ω), where f max (ω) is the maximal modal fraction, that is, the mode giving the highest contribution.
In this case, the function f (ω) which was defined as the modal fraction is represented as The function F(ω) represents the Fourier Transform of the energy density of the configuration.It is important to remark that the energy density must be square-integrable since in such cases the entropy can be well defined.
We will extend the procedure presented in [36], which is absolutely general when applied to systems with spatially-localized energy for a scalar field theory that describes generalized gravity coupled to a scalar field in five-dimensional space-time.Following that work, it is possible to obtain the entropy of the configurations assuming that F(ω) obeys the Fourier transform: We are working with spatially-localized energy densities, which are part of a set of square-integrable bounded functions ρ(y) ∈ L 2 (R) and their Fourier transforms F(ω).It is also important to remark that the Plancherel's theorem states that As argued by GS, the information-entropic measure approach can be capable of taking into account the dynamical and the informational contents of models with localized energy configurations.In this case, the CE provides a complementary perspective to situations where strictly energy-based arguments are inconclusive.In fact, as pointed out in [36], higher CE correlates with higher energy, and consequently the configuration is more disordered.Therefore, the information-theoretical measure is responsible to point out which solution is the most ordered one among a family of infinite solutions.
Thus, we will apply this new approach to investigate generalized gravity theories coupled to a real scalar field.As we will see, important consequences will arise from the CE concept.Furthermore, we will show that the CE provides a stricter bound on the parameters of the f (R, T ) models.

III. f (R, T ) THEORIES OF GRAVITY
In this section, we present a brief review about f (R, T ) theories of gravity.In this context, it was presented some years ago by Harko and collaborators the seminal work entitled f (R, T ) gravity [5].In that context the authors showed that it is possible to construct a modified theory of gravity, where the gravitational Lagrangian is given by an arbitrary function of the Ricci scalar R and of the trace of the energy-momentum tensor T .On a physical viewpoint, the dependence on T may be originated by the presence of exotic imperfect fluids or quantum effects.Therefore, as shown in [5], in (3+1) dimensions the action S for the f (R, T ) modified theories of gravity can be written in the form where f (R, T ) is the arbitrary function of R and T , g is the determinant of the metric g µν with µ, ν assuming the values 0, 1, 2, 3, and L m is the matter Lagrangian density.Note that we are using the natural system of units with G = c = 1, so that the Einstein gravitational constant is defined as κ 2 = 8π.We emphasize that the f (R, T ) modified theory of gravity has been analysed in the literature in several scenarios.For instance, we can find studies in cosmological models [6]- [10], [48], in quantum chromo dynamics [49], in a Bianchi Type-II string with magnetic field [50] and in thermodynamics [51,52].
On the other hand, recently, Bazeia and collaborators, motivated by the works [5,53], have found thick brane models in f (R, T ) generalized theories of gravity in an AdS 5 warped geometry with an extra dimension of infinite extent [35].It has been shown that two distinct choices of the functional form of f (R, T ) induce quantitative modifications in the thick brane profile, without changing its qualitative behavior.Another interesting result obtained in [35] is the fact that the gravity sector remains linearly stable in each choice of f (R, T ) analized.
The theory used by Bazeia and collaborators for describing generalized gravity coupled to a scalar field in five-dimensional space-time, with an extra dimension y of infinite extent, is given by the following action where L s is the Lagrange density, g = det(g ab ), with the indices a and b running from 0 to 4, the signature of the metric is (+ − − − −) and we are using units such that 4πG (5) = 1.In this case, the Lagrangian density is put into the form From the equation above, the energy-momentum tensor of the scalar field in this theory is given by In addition, the corresponding equation of motion for the scalar field and the modified Einstein equation can be written as where V φ ≡ dV /dφ, f R ≡ df /dR and f T ≡ df /dT .Following the approach presented in [35], we will work with the metric where y is the extra dimension, η µν is the usual Minkowski metric in the four-dimensional spacetime with signature (+, −, −, −), and e 2A is the so-called warp factor.In particular, when the warp function A and the field φ are static and depend only on the extra dimension y, the equation of motion and the modified Einstein equations become where the prime denote derivatives with respect to the extra dimension.
Here, it is important to highlight that this particular approach, where we are considering the static case with A = A(y) and φ = φ(y), makes possible the finding of innovative analytical solutions for the theory.A good example are the interesting solutions found in [35].In that work, two distinct models for f (R, T ) theories of gravity have been studied, and it was shown that the gravity sector of the thick braneworld configurations remains linearly stable.Moreover, each model induces new quantitative modifications in the thick brane profile.Thus, in order to make the present work more comprehensive and didactical, in the next section we will present those two examples and their solutions.

IV. TWO SPECIFIC MODELS AND THEIR SOLUTIONS
In the present section, we consider two particular classes of f (R, T ) modified gravity models, which were investigated in [35].In that reference, it has been found that in such particular cases, both the field φ and warp function A can be obtained analytically.
In the first case, the f (R, T ) function is given by where α and n are real parameters.In this case, when n = 1 the exact solutions presented in [35] for the field φ and the warp function A are written as where γ, a and b are real parameters [35], and B ≡ 4γab 2 /(4 − 3α).Furthermore, the energy density can be put in the form Now, let us investigate another model, where f (R, T ) is represented by with α and β being real parameters.In this case the solutions for the field and warp function can be written in the following form where k is a positive parameter and E(Φ, Ξ) is the elliptic integral of the second kind with Φ ≡ φ s = arcsin tanh(ky) , and Ξ ≡ 392βk 2 160βk 2 −3 .
Here, the energy density is given by where S = sech(ky).

V. CE IN f (R, T ) THEORIES
In this section, we will study the entropic profile of the generalized gravity theories here presented.
For this first case, we can rewrite the energy density (17) in the following form where δ 1 ≡ δ and δ 2 ≡ δ + 2, with δ ≡ 4γa/B.Furthermore, we have Therefore substituting the energy density given by Eq.( 23) into Eq.(3), we can obtain, after some arduous calculation, the following Fourier transform where 2 H 1 [δ j , µ j, ; µ j, + 1; −1] is the hypergeometric function, and we are using the definitions Moreover, we have with Now, we can write the modal fraction in the form where (m,n) 1 Figure 1 below depicts f (ω) for different values of α.As it can be seen, the modal fraction profile is influenced by the α parameters.In this case, we can check that the α parameter is responsible for the appearance of a split in the modal fraction, where there are two peaks which are symmetric with respect to the origin.
where δ1 = 2, δ2 = 4 and δ3 = 6, with From the energy density (34) we can find the Fourier transform where we are using the following definitions Thus, we can find that with Therefore, we can write where   In this work we investigated f (R, T ) braneworld models in the CE context.We showed, by means of this information-theoretical measure, that a stricter bound on the parameter of f (R, T ) theories of gravity arises from the CE.We found that these bounds are characterized by a valley region in the CE profile, where the entropy is minimal.
From inspection of such results, we can see that in the case where f (R, T ) = R − α T n , the CE shows a rich structure for varying α.In this case, there is a sharp minimum at the value α 1.07, where the corresponding CE is given by σ c 1.4.This result leads us to conclude that the CE can be used in order to extract a rich information about the structure of the configurations, which is clearly related to their profiles.Therefore, we found that the best ordering for the solutions are those given by α 1.07.
Moreover, when f (R, T ) = R+βR 2 −αT , the minimal CE corresponds to a valley region where the best values for α and β are localized in the bottom of the valley.In this way, the information-theoretical measure of generalized theories of gravity, such as f (R, T ) theories, opens a new window to probe situations where the parameters responsible for the control of the theory are, at first sight, arbitrary.In this case, the CE selects the best values.
Thus, in this work, we can conclude that the CE provides a complementary perspective to investigate generalized theories of gravity such as Gauss-Bonnet, Weyl and Brans-Dicke theories, as well as several other modified theories of gravitation.Another interesting line of investigation in which CE can play an important rule is the evolution of the domain walls in Euclidean space [54].In this case, the information-theoretical measure can be used as a discriminant to understand the phase transition processes.

FIG. 1 :
FIG. 1: Modal fractions for different values of α.The CE for f (R, T ) = R − α T n is plotted in Fig.2below.

Figures 3 -
Figures 3-4 below show the modal fraction for α = 0.1 and α = 0.98, respectively.From that figures, we can note that the α parameters induce the formation of two peaks which are symmetric in relation to the origin.