Configurational entropy as a bounding of Gauss-Bonnet braneworld models

R. A. C. Correa1∗, P. H. R. S. Moraes2†, A. de Souza Dutra1‡, W. de Paula2§, and T. Frederico2¶

1UNESP, Universidade Estadual Paulista,

12516-410, Guaratinguetá, SP, Brazil and

2ITA, Instituto Tecnológico de Aeronáutica,

12228-900, São José dos Campos, SP, Brazil

Abstract

Configurational entropy has been revealed as a reliable method for constraining some parameters of a given

model [Phys. Rev. D 92 (2015) 126005, Eur. Phys. J. C 76 (2016) 100]. In this letter we calculate the

configurational entropy in Gauss-Bonnet braneworld models. Our results restrict the range of acceptability of

the Gauss-Bonnet scalar values. In this way, the information theoretical measure in Gauss-Bonnet scenarios

opens a new window to probe situations where the additional parameters, responsible for the Gauss-Bonnet

sector, are arbitrary. We also show that such an approach is very important in applications that include p

and Dp-branes and various superstring-motivated theories.

Keywords: Entropy, Gauss-Bonnet, Gravity

∗ fis04132@gmail.com
† moraes.phrs@gmail.com
‡ dutra@feg.unesp.br
§ wayne@ita.br
¶ tobias@ita.br

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I. INTRODUCTION

In the late 90’s of the last century, observational evidences of an accelerated expansion of

the universe were found [1, 2]. It was proposed, as an explanation for such a counterintuitive

phenomenon, the existence of the vacuum quantum energy, which would manifest in large scales

as an “anti-gravitational force”. The vacuum quantum energy appears in the dynamical equations

of the universe in the ΛCDM (or standard) cosmological model as the cosmological constant (CC)

density parameter ΩΛ. However, when one compares the value of ΩΛ which can account for the

present dynamical scenario of the universe [3] with the value theoretically predicted in Particle

Physics [4], one realizes a huge discrepancy between them, which is known as the CC problem.

An alternative to evade the CC problem - among others surrounding the standard cosmology

model, which we shall quote below - is to consider modified theories of gravity. The f(R) [5]-

[7] and f(R, T ) [8, 9] theories of gravity have generated possibilities of describing the cosmic

acceleration with no need of a CC, evading, in this way, the CC problem. Those theories consider

the gravitational part of their action to be dependent on a function of the Ricci scalar R or of both

R and T , with T being the trace of the energy-momentum tensor.

Another possibility of describing cosmic acceleration without a CC comes from cosmology de-

rived from Gauss-Bonnet (GB) gravity [10]-[13]. Those are obtained from the consideration of the

GB invariant or a function of it in the gravitational part of the action. It is known that higher

orders in the GB term naturally arise in the low energy limit of string theory [14].

Some alternative gravity models are a powerful tool to evade the hierarchy problem as well,

which is related to the large discrepancy among aspects of the gravitational force and the other

fundamental forces. Those are the braneworld models [15]-[18], which consider our observable

universe as a 3 + 1 hypersurface (the brane) embedded in a five-dimensional space named bulk.

Gravity, departing from the other forces, would be able to propagate through the bulk, justifying

the hierarchy in the four-dimensional universe.

It is possible to unify some of the above formalisms in order to evade more than one standard

cosmology shortcoming simultaneously by invoking the brane set up in f(R) [19]-[24], f(R, T )

[25, 26] and GB [27]-[29] gravity models.

The alternative gravity models mentioned above carry some “free” parameters with them whose

values can be constrained, for instance, by cosmological observations. Observational constraints in

f(R) gravity parameters can be checked in [31]-[36]. For observational restrictions in f(R, T ) and

GB theories, check [37] and [38]-[41], respectively.

2



On the other hand, in a recent work [42], the concept of entropy has been reintroduced in the

literature, by taking into account the dynamical and informational contents of models with localized

energy configurations. Based on the Shannon’s information entropy, the so-called Configurational

Entropy (CE) was constructed. It can be applied to several nonlinear scalar field models featuring

solutions with spatially-localized energy. As pointed out in [42], the CE can resolve situations where

the energies of the configurations are degenerate. In this case, the CE can be used to select the

best configuration. Furthermore, the authors pointed out that this information-entropic measure

is an essential tool in the study of complex spatially-localized configurations.

We are going to discourse about the CE mechanism below. For now, it is interesting to highlight

some of its applications which reveal CE as a powerful physical tool nowadays. For instance, it

was shown in [43] that the CE quantifies the emergence of spatially-localized, time-dependent,

long-lived structures known as oscillons [44–46]. In that case, the CE is responsible for providing

the informational content of nonequilibrium field structures, in particular of coherent states that

emerge during spontaneous symmetry breaking. By considering a Starobinsky functional form for

f(R), i.e., f(R) = R+ αR2, in brane models with nonconstant curvature, the α parameter values

were constrained by CE consideration in [47]. The free parameters of the f(R, T ) = R − αT and

f(R, T ) = R+βR2−αT brane models were constrained in [48]. It has been shown that, indeed, CE

can be used in order to extract a rich information about the structure of the model configurations.

Moreover, CE was applied to pure brane models in [49] and restrictions to the anti-de-Sitter bulk

curvature and domain wall thickness were obtained.

Studies regarding CE can also be found in solitonic Q-balls [50], in the context of (2 + 1)-

dimensional Ginzburg-Landau models [51], in astrophysical objects [52], in two interacting scalar

fields theories [53], in traveling solitons in Lorentz and CPT breaking systems [54], and in topo-

logical Abelian string-vortex and string-cigar context [55].

Our intention in this letter is to obtain some restrictions to the GB braneworld parameters via

CE approach. The letter is organized as follows. In Section II we discourse about the information

content which can be obtained from the CE approach. Since we are interested in restricting GB

braneworld models, we present a brief review of those in Section III. In Section IV we calculate

the CE in GB braneworld and we discuss our results in Section V.

3



II. THE CONFIGURATIONAL ENTROPY

Gleiser and Stamatopoulos (GS) [42] have recently proposed a detailed picture of the so-called

CE for the structure of localized solutions in classical field theories. In this section, analogously to

that work, we formulate a CE measure in the functional space, from the field configurations where

the GB braneworld scenarios can be studied.

There is an intimate link between information and dynamics, where the entropic measure plays

a prominent role. The entropic measure is well known to quantify the informational content

of physical solutions to the equations of motion and their approximations, namely, the CE in

functional space [42]. GS proposed that nature optimizes not solely by optimizing energy through

the plethora of a priori available paths, but also from an informational perspective [42].

The starting point is to consider structures with spatially-localized energy and a modal fraction

f(ω) which measures the relative weight of each mode ω such that

f(ω) =
|F|ω||2∫
dω|F|ω||2

, (1)

with F(ω) being the Fourier transform.

The CE is defined as

SC(f) = −
∑

fn ln fn (2)

and provides the informational content of configurations compatible with the particular constraints

of a given physical system. We can say that when all N modes carry the same weight, fn = 1/N

and the discrete CE presents a maximum at SC = lnN . Alternatively, if only one mode is present,

SC = 0.

For general, non-periodic functions in an open interval, the continuous CE reads

SC(f) = −
∫
dωf̂(ω) ln |f̂(ω)|, (3)

with f̂(ω) ≡ f(ω)/fmax(ω) defined as the normalized modal fraction, whereas fmax(ω) is the

maximum fraction. In this case, this condition ensures the positivity of SC .

4



III. THE GAUSS-BONNET BRANEWORLD GRAVITY MODEL

Since we are interesting in the calculation of the CE in GB brane models, it is worth to briefly

review such an alternative gravity theory. This can be appreciated in the following.

An alternative to extend standard gravity is through the addition of the GB term

G = R2 − 4RµνRµν +RµνλρR
µνλρ, (4)

or a function of it, in the usual Einstein-Hilbert gravity lagrangian. In (4), Rµν is the Ricci tensor

and Rµνλρ is the Riemann tensor. Since we are considering a five-dimensional braneworld model,

the Greek indices above assume the values 0, 1, 2, 3, 4.

The GB brane gravity action for a general function of the GB term reads:

S =
1

2

∫
d4xdy

√
−g[R+ h(G)], (5)

with g being the determinant of the metric and h(G) being a function of the GB scalar.

We consider as the matter source of the universe a scalar field φ specified by the lagrangian

density

L =
1

2
gµν∂

µφ∂νφ− V (φ) (6)

to be added to (5), with V (φ) being the scalar field potential.

Moreover we will work with the following metric

ds2 = e2A(y)ηabdx
adxb − dy2, (7)

with A(y) representing the warp function, ηab the Minkowski metric and a, b running from 0 to 3.

In [28], the authors showed that for the metric (7),

G = 24(4A′′A′2 + 5A′4) (8)

and the energy-momentum tensor

5



Tab = ηab

[
1

2
φ′2 + V (φ)

]
e2A, (9)

with primes denoting derivatives with respect to y.

Now, after some straightforward calculation it is possible to obtain the following equations of

motion [28]:

− 3A′′ = 2φ′2 + 12hG(G)A′′A′2, (10)

6A′2 = φ′2 − 2V (φ)− 1

2
h(G) + 48hG(G)(A′4 +A′′A′2), (11)

with hG(G) ≡ dh(G)/dG. Note that within this methodology, the potential V (φ) is a variable to

be determined instead of a quantity introduced as a basic characteristic of the theory.

For the case h(G) = αGn, with α and n being constants, it can be shown that the explicit form

assumed by the potential is

V (φ) = −3

4
A′ + 3A′2 − 1

4
αGn + 3αnA′2(8A′2 + 7A′′)Gn−1

−αn(n− 1)[A′(14A′2 + 2A′′)G′ + (2A′2 −A′′)G′′]Gn−2

−αn(n− 1)(n− 2)G′2(2A′2 −A′′)Gn−3. (12)

Furthermore, we have

φ′2 = −3

2
A′′ − 6αnA′2A′′Gn−1

−2αn(n− 1)
[
(2A′2 −A′′)G′′ − 2A′(A′2 − 5A′′)G′

]
Gn−2

−2αn(n− 1)(n− 2)G′2(2A′2 −A′′)Gn−3. (13)

On the other hand, from (9) the energy density T00 is

ρ = e2A

[
1

2
φ′2 + V (φ)

]
. (14)

Following references [28, 56], we can choose the ansatz

A(y) = B ln [sech(y)] , (15)

6



where B > 0.

Now, using the Eqs.(8), (12), (13) and (15), the energy density (14) is written in the form

ρ (y) =

5∑
`=1

s` (n,B, α)Q` (n,B, α; y) , (16)

where we are using the following definitions

s1 (n,B, α) ≡ 3B

2
, Q1 (n,B, α; y) ≡ sech2B(y), (17)

s2 (n,B, α) ≡ −3B2, Q2 (n,B, α; y) ≡ sech2B(y) tanh2(y), (18)

s3 (n,B, α) ≡ αn23n3nB3n+1, Q3 (n,B, α; y) ≡ sech2B(y) tanh2n+2(y) [ΨB (y)]n−1 , (19)

s4 (n,B, α) ≡ −αn23n−23n+1B3n−1, Q4 (n,B, α; y) ≡ sech2B+2(y) tanh2n(y) [ΨB (y)]n−1 , (20)

s5 (n,B, α) ≡ −α23n−23nB3n, Q5 (n,B, α; y) ≡ sech2B(y) tanh2n(y) [ΨB (y)]n , (21)

with

ΨB (y) ≡ 5B tanh2(y)− 4 sech2(y). (22)

The profiles for the energy density and warp factor are depicted in Fig.1, which shows the

influence of the B parameter on the configurations. It is important to explain the reason for

working with small values of B. In Fig.1, the reader can note that when B increases the brane is

narrowed down. Moreover, simultaneously, the energy density develops lateral peaks and a central

valley which goes rapidly to zero as B increases. Such a critical phenomenon of thick brane models

is called “brane splitting” and it was first presented in [57]. It has already appeared in GB [20]

and f(R) [21] models. Since in the brane splitting case, the field will not be confined to the brane,

we avoid large values of B.

IV. INFORMATION CONTENT IN GAUSS-BONNET BRANEWORLD

Now we will apply the new concept of CE in the functional space from the field configurations

where the GB braneworld scenarios can be studied. To begin, we write the Fourier transform

F [ω] =
1√
2π

∫
dy eiωyρ(y), (23)

7



B = 0.2
B = 0.3
B = 0.4

-15 -10 -5 0 5 10 15

0.0

0.2

0.4

0.6

0.8

1.0

y

Ρ
HyL

;
e2

A

FIG. 1: Energy density (thin continuous line) and warp factor (dashed line) with α = 1 and n = 1.

where ρ(y) is the standard energy density. Here, it is important to remark that the energy density

given by Eq.(16) is localized. In fact, as shown by GS, the condition necessary so that the CE is

well-defined in physical applications is that the energy density is represented by a spatially-confined

structure.

By using the Plancherel theorem, it follows that∫
dω |F [ω]|2 =

∫
dr |ρ(y)|2 . (24)

Now, substituting the energy density given by Eq.(16) into Eq.(23), we can obtain, after some

arduous calculation, the following Fourier transform

F(ω) =
5∑
`=1

2∑
j=1

s̃` (n,B, α) I`,j (n,B, α;ω) , (25)

where s̃` (n,B, α) ≡ s` (n,B, α) /
√

2π and I`,j [n,B, α;ω] are the following corresponding functions

I1,1 (n,B, α;ω) =
4

c+ iω
G
[
c,
c+ iω

2
;
c+ iω

2
+ 1;−1

]
, (26)

I1,2 (n,B, α;ω) =
4

c− iω
G
[
c,
c− iω

2
;
c− iω

2
+ 1;−1

]
, (27)

8



I2,1 (n,B, α;ω) =
1

2 + b̃0 + iω
G

[
b̃0,

2 + b̃0 + iω

2
;
4 + b̃0 + iω

2
;−1

]

+
1

b0 − iω
G
[
b̃0,

b0 − iω
2

;
b0 − iω

2
+ 1;−1

]
, (28)

I2,2 (n,B, α;ω) =
1

b0 + iω
G
[
b̃0,

b0 + iω

2
;
b0 + iω

2
+ 1;−1

]
+

1

iω − 2− b̃0
G

[
b̃0,

iω − 2− b̃0
2

;
iω − b̃0

2
;−1

]
, (29)

I3,1 (n,B, α;ω) =
(α1α2)n−1 Γ[γ1 + 1]Γ[γ2 + 1]

2Γ[γ1 + γ2 + 2]
H(3)
D [γ1 + 1; γ3,−γ4,−γ4; γ1 + γ2 + 2;−1, v, u] ,

(30)

I3,2 (n,B, α;ω) =
(α1α2)n−1 Γ[1− γ̄0]Γ[γ̄1 + 1]

2Γ[γ̄1 − γ̄0 + 2]
H(3)
D [1− γ̄0; γ̄2,−γ̄3,−γ̄4; γ̄1 − γ̄0 + 2;−1, v, u] ,

(31)

I4,1 (n,B, α;ω) =
2δ−1ξθ1 (ε1ε2)θ Γ[R]Γ[2ζ +R+ 1]

Γ[2ζ +R+ 1]
H(3)
D [R; Φ,−θ,−θ; 2ζ +R+ 1;−1, ṽ, ũ] , (32)

I4,2 (n,B, α;ω) =
2δ̃−1ξ̃θ̃1 (ε1ε2)θ̃ Γ[R̃]Γ[λ̃+ 1]

Γ[λ+ R̃+ 1]
H(3)
D

[
R̃; Φ,−θ,−θ;λ+R+ 1;−1, v̂, û

]
, (33)

I5,1 (n,B, α;ω) =
2δ̂−1ξ̂θ1 (ε̃1ε̃2)θ Γ[R̂]Γ[2ζ̂ + R̂+ 1]

Γ[2ζ +R+ 1]
H(3)
D

[
R̂; Φ̂,−θ̂,−θ̂; 2ζ̂ + R̂+ 1;−1, U, Y

]
,

(34)

I5,2 (n,B, α;ω) =
2δ̊−1ξ̊θ̊1 (ε̃1ε̃2)θ̃ Γ[R̊]Γ[̊λ+ 1]

Γ[λ+ R̊+ 1]
H(3)
D

[
R̊; Φ̊,−θ̊,−θ̊;λ+ R̊+ 1;−1, U, Y

]
, (35)

9



with the definitions

c ≡ 2B + 2, b0 ≡ 2B, b̃0 ≡ 2B + 2, α1 ≡
d0

2
+

√
d2

0

4
− 1, α2 ≡

d0

2
−
√
d2

0

4
− 1,

d0 ≡
5B

2
+ 4, ε1 ≡

D0

2
+

√
D2

0

4
− 1, ε2 ≡

D0

2
−
√
D2

0

4
− 1, D0 ≡ 2 +

4

5B
,

γ1 ≡
2B − 4n+ 4 + iω

2
− 1, γ2 ≡ 2n+ 2, γ3 ≡ 2B + 4, γ4 ≡ n− 1, v ≡ 1

α1
, u ≡ 1

α2
,

γ̄0 ≡
2B − 4n+ 4− iω

2
− 1, γ̄1 ≡ 2n+ 2, γ̄2 ≡ −2B − 4, γ̄3 = γ̄4 ≡ n− 1, ξ1 ≡ 5B,

ζ ≡ n, θ ≡ n− 1, Φ ≡ 2θ + 2B + 2 + 2n, R ≡ iω − 2ζ − 2θ + Φ, U ≡ 1

ε̃2
, Y ≡ 1

ε̃1
,

δ ≡ 2B + 2, ṽ = v̂ ≡ 1

ε1
, ũ = û ≡ 1

ε2
, R̃ = −R̂ ≡ iω − 2ζ + 2θ + Φ, ,

Φ̂ = −Φ̊ ≡ 2θ − 2B + 2 + 2n, θ̂ = −θ̊ ≡ 2n− 1, R̊ ≡ −iω − 2ζ + 2θ + Φ.

Furthermore, in the above expressions, G [�,�;�;�] stand for the well-known hypergeometric

functions and H(3)
D [⊗;⊗,⊗;⊗;⊗,⊗,⊗] is the so-called Lauricella functions of three variables [58].

Thus, the modal fraction (1) becomes

f(ω) =

∑5
`,`′=1

∑2
j,j′=1 s̃` (n,B, α) s̃∗`′ (n,B, α) I`,j (n,B, α;ω) I∗`′,j′ (n,B, α;ω)∑5

`,`′=1

∑2
j,j=1 s̃` (n,B, α) s̃∗`′ (n,B, α)

∫
dωI`,j (n,B, α;ω) I∗`′,j′ (n,B, α;ω)

. (36)

In Fig.2 the modal fraction is depicted for different values of the parameter B.

B = 0.10

B = 0.12

B = 0.14

-4 -2 0 2 4

0

5

10

15

Ω

fHΩ
L

(a) Modal fraction with n = 1 and α = 1.

B = 0.10

B = 0.12

-4 -2 0 2 4

0

10 000

20 000

30 000

40 000

50 000

60 000

Ω

fHΩ
L

(b) Modal fraction with n = 2 and α = 1.

FIG. 2: Modal fractions

In Fig.3 it can be seen that given a value of B there is a value of α for which CE is a minimum.

Furthermore when B decreases, that minimum value also decreases. This results are in agreement

with the CE concept found in [42]. Here, we can observe that lower CE correlates with lower

energy. Thus, the most prominent solutions to the GB theory under analysis are given for some

specific values of α. This will be analysed further in the next section.

10



B = 0.4
B = 0.6
B = 0.8

-10 -5 0 5 10

1.0

1.2

1.4

1.6

1.8

Α

S C
HΑ

L

(a) CE with n = 1.

B = 0.6

B = 0.4

-0.4 -0.2 0.0 0.2 0.4
1.4

1.5

1.6

1.7

1.8

Α

S C
HΑ

L

(b) CE with n = 2.

FIG. 3: Configurational entropy

V. DISCUSSION AND CONCLUSIONS

The CE has been revealed as a reliable method for constraining some given model parameters,

as one can check, for instance, in [47, 48]. The entropic information has been here studied in

braneworld models, with emphasis on the GB scenario, which has been chosen by its very physical

content and usefulness. We have shown that the information theoretical measure of GB braneworld

models opens new possibilities to physically constrain, for example, parameters that are related to

the GB term. The CE provides the most appropriate value of these parameters that are consistent

with the best organizational structure.

The information measure of the system organization is related to modes in the braneworld

model. Hence the constraints of the parameters that we obtained for the GB model provide the

range of the parameters associated to the most organized braneworld models with respect to the

information content of these models.

By analysing Fig.3 of the previous section, we can see that for high values ofB, the CE minimum,

which is related to the reliable solutions of the system, is found for α = 0 or α ' 0. In other words,

there is a specific value for B which makes the GB term to vanish. Therefore, the CE serves as a

new tool to specify the dynamic constraints of the GB model.

As we can see, the CE provides a complementary perspective to investigate alternative theories

of gravity. Further interests that concerns to CE and which we are interested in, can be found in

dynamical bound in alternatives theory of gravity, such as Brans-Dicke [59], Kaluza-Klein [60] and

f(R,Lm) [61] gravity models, among many others.

11



Acknowledgments

RACC thanks CAPES for financial support and to the Physics Graduate Program at Unesp-

Campus Guaratinguetá for hospitality. PHRSM would like to thank São Paulo Research Foun-

dation (FAPESP), grant 2015/08476-0, for financial support. ASD and WP are thankful to the

CNPq for financial support. TF is thankful to the CNPq, FAPESP, and CAPES.

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13


	I Introduction
	II The configurational entropy
	III The Gauss-Bonnet braneworld gravity model
	IV Information content in Gauss-Bonnet braneworld
	V Discussion and conclusions
	 Acknowledgments
	 References