5D EChS Cosmology with Perfect Fluid

We consider a Chern-Simons gravity theory in the cosmological context for a flat Friedmann-Robertson-Walker metric in five dimensions, chosen barotropic perfect fluids filled the Universe, that is, an equation of state with constant state parameter for the fluid representing the h field of Einstein-Chern-Simons cosmology and an equation of state with variable state parameter for the fluid representing the matter field.


Introduction
The Einstein-Chern-Simons (EChS) cosmology was introduced by [1] in order to have an dynamical description to a five dimensional Friedmann-Robertson-Walker (FRW) cosmology from Chern-Simons gravity theory for a certain Lie algebra B [2], which was obtained from the AdS algebra and a particular semigroup S by means of the S-expansion procedure development by [3,4].
Using the extended Cartan's homotopy formula as in Ref. [5], and integrating by parts, the five dimensional EChS Lagrangian for the B algebra can be written as where α 1 , α 3 are parameters of the theory, l is a coupling constant, T a = de a + ω a b e b is the torsion, R ab = dω ab + ω a c ω cb corresponds to the curvature 2-form in the first-order formalism related to the 1-form spin connection [6], [7], [8], and e a , h a and k ab are others gauge fields presents in the theory [2].
We consider now the fields equations for the lagrangian given by where L

(5) ChS
is the five-dimensional Chern-Simons Lagrangian given by (1), L M = L M (e a , h a , ω ab ) is the matter Lagrangian and κ is a coupling constant related to the effective Newton's constant. If T a = 0 and k ab = 0. The variation of the lagrangian (2), leads to the following field equations consider here δL M δω ab = 0, imposed for consistency with T a = 0. Notice that from this fields equations we recover the odd-dimensional Einstein gravity theory when the curvature R ab takes values not excessively large and the parameter l takes small values (l −→ 0) [2] and the constant α takes values not excessively large, namely where If R ab is not large then δL M /δe a is also not large. This means that General Relativity can be seen as a low energy limit of Einstein-Chern-Simons gravity. So that, in the range of validity of the General Relativity, the equations (3-6) are given by On the another hand, if R ab is large enough, so that when it is multiplied by l 2 (which is very small) will have a non-negligible results, then we will find that δL M /δh a is not negligible. This means that, in this case, we must consider the entire system of equations (3-6).
Next we consider a flat five dimentional FRW metric and given a equation of state for the h field of the EChS theory like a barotropic perfect fluid with constant state parameter, this generates a variable state parameter in the equation of state chosen for the matter field. We think that is important because this no was explored in [1] and this open a new family of exact solutions to the EChS cosmology.
where ω (h) is a constant. In order to obtain solutions for the scale factor, energy density of the field h, field matter matter, we analyze the case when ω (h) = −1 and when ω (h) = −1 separately.
Writing the balance equation for the field h (16) in terms of the Hubble parameter H and using the equation of state (18), we obtain from this result we can find the scalar factor Now introducing (20) in (15), we obtain and introducing (20) in (13), we obtain If we now propose an equation of state with variable state parameter P = ω(t)ρ for the field matter, we can find from (14) that is from this result we can find the scalar factor Now introducing (25) in (15), we obtain and introducing (25) in (13), we obtain If we now propose an equation of state with variable state parameter P = ω(t)ρ for the field matter, we can find from (14) that This result agrees with that found in Ref. [1].

Conclusions
In this paper we have considered a Chern-Simons gravity theory in the cosmological context for a flat FRW metric in five dimensions, considering an equation of state with ω (h) constant for the fluid that represents the h field of EchS theory and an equation of state with ω(t) variable for the fluid representing the matter field. With this we have obtained an exact solution for our cosmological model, finding the Hubble parameter, the scalar factor, pressures and energy densities for the matter field and h field, and the variable parameter of state for the matter field, considering the cases ω (h) = −1 and ω (h) = −1.