Accelerated FRW Solutions in Chern-Simons Gravity

We consider a five-dimensional Einstein-Chern-Simons action which is composed of a gravitational sector and a sector of matter, where the gravitational sector is given by a Chern-Simons gravity action instead of the Einstein-Hilbert action and where the matter sector is given by the so called perfect fluid. It is shown that (i) the Einstein-Chern-Simons (EChS) field equations subject to suitable conditions can be written in a similar way to the Einstein-Maxwell field equations; (ii) these equations have solutions that describe accelerated expansion for the three possible cosmological models of the universe, namely, spherical expansion, flat expansion and hyperbolic expansion when $\alpha $, a parameter of theory, is greater than zero. This result allow us to conjeture that this solutions are compatible with the era of Dark Energy and that the energy-momentum tensor for the field $h^{a}$, a bosonic gauge field from the Chern-Simons gravity action, corresponds to a form of positive cosmological constant. It is also shown that the EChS field equations have solutions compatible with the era of matter: (i) In the case of an open universe, the solutions correspond to an accelerated expansion ($\alpha>0$) with a minimum scale factor at initial time that, when the time goes to infinity, the scale factor behaves as a hyperbolic sine function. (ii) In the case of a flat universe, the solutions describing an accelerated expansion whose scale factor behaves as a exponencial function when time grows. \item In the case of a closed universe it is found only one solution for a universe in expansion, which behaves as a hyperbolic cosine function when time grows.


I. INTRODUCTION
Some time ago was shown that the standard, five-dimensional General Relativity can be obtained from Chern-Simons gravity theory for a certain Lie algebra B [1], which was obtained from the anti de Sitter (AdS) algebra and a particular semigroup S by means of the S-expansion procedure introduced in Refs. [2], [3].
The five dimensional Chern-Simons Lagrangian for the B algebra is given by [1] L (5) ChS = α 1 l 2 ε abcde R ab R cd e e + α 3 ε abcde 2 3 R ab e c e d e e + 2l 2 k ab R cd T e + l 2 R ab R cd h e , where α 1 , α 3 are parameters of the theory [1], l is a coupling constant, R ab = dω ab + ω a b corresponds to the curvature 2−form in the first-order formalism related to the 1−form spin connection [4], [5], [6]. and e a , h a and k ab are others gauge fields presents in the theory [1].
We can see that i If one identifies the field e a with the vielbein, then, the covariant derivative of the field e a is the Torsion 2−form (T a = De a = de a + ω a b e b ). Therefore, the system consist of the Einstein-Hilbert action plus non-minimally coupled 1−form matter fields1 given by h a and k ab .
ii It is possible to recover the odd-dimensional Einstein gravity theory from Chern-Simons gravity theory in the limit where the coupling constant l equals to zero while keeping the effective Newton's constant fixed [1].
Recently was found [7] that the standard five-dimensional FRW equations and some of their solutions can be obtained, in a certain limit, from the so-called Chern-Simons-FRW field equations, which are the cosmological field equations corresponding to a Chern-Simons gravity theory.
It is the purpose of this paper to show that the Einstein-Chern-Simons (EChS) field equations, subject to (i) the torsion-free condition (T a = 0) and (ii) the variation of the matter Lagrangian with respect to (w.r.t.) the spin connection is zero (δL M /δω ab = 0) can be written in a similar way to the Einstein-Maxwell field equations. The interpretation of 3 the h a field as a perfect fluid allow us to show that the Einstein-Chern-Simons field equations have an universe in accelerated expansion as a of their solutions. This paper is organized as follows: In Section II we briefly review the Einstein-Chern-Simons field equations. In Section III we study the Einstein-Chern-Simons field equations in the range of validity of general relativity. In Section IV we consider accelerated solutions for Einstein-Chern-Simons field equations. We try to find solutions that describes accelerated expansion for cases of open universes, flat universes and closed universes. In Section V we consider the consistency of the solutions with the "Era of Matter". A summary and an appendix conclude this work.

II. EINSTEIN-CHERN-SIMONS FIELD EQUATIONS
In Ref. [7] was found that in the presence of matter the lagrangian is given by where L For simplicity, we will assume that the torsion vanishes (T a = 0) and k ab = 0. In this case the Eqs.(3-6) takes the form This field equations system can be written in the form where we introduce κ 5 = κ/8α 3 and α = −α 1 /α 3 . Imposing the condition δL M /δω ab = 0 (for consistency with the condition T a = 0) we find that the equations (7-9) can be written in the form This means that the Einstein-Chern-Simons field equations, subject to the conditions T a = 0, k ab = 0 and δL M /δω ab = 0, can be re-written in a way similar to the Einstein- Introducing (13) into (10) we obtain the Einstein's field equation 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 (10-12) 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 (10-12).
IV. ACELERATED SOLUTION FOR EINSTEIN-CHERN-SIMONS FIELD EQUA-TIONS From Ref. [7] we know that the vielbein for five dimensional FRW metric is given by where a(t) is the scale factor of the universe and k is the sign of the curvature of spacetime: Following Ref. [7], we postulate that the bosonic field h a is given by where h(0) is a constant and h(t) is a function of time t that must be determined.
In accordance with the equation (10), we will consider a fluid composed of two perfect fluids, the first one related to ordinary energy-momentum tensor and the second one related to field h a . The energy-momentum tensors in the comoving frame, where the fluids are at rest, are given by where ρ is the matter density and p is the pressure of fluid. Then, the energy-momentum tensor for the composed fluid is = diag(ρ,p,p,p,p).
Introducing (17-23) into eqs. (10-12) we find the following field equations (see Ref. [7] and A) We should note that equation (24) was studied in Ref. [9] in the context of inflationary cosmology . In this work we consider the study of the present acceleration for the complete Einstein-Chern-Simons field equations.
The Equations (24) and (25) are very similar to the Friedmann equations in five dimensions. However now ρ and p are subject to restrictions imposed by the remaining equations.
We can consider the case where T µν = 0, i.e., when the contribution from the ordinary matter is negligible compared to the contribution from the field h a . In this case, the energymomentum tensorT µν fluid is given bỹ and the equations (24 -28) take the form Introducing (32) into (30) we obtain which can be rewritten 1. Solutionä = 0 Consider the solutionä = 0, i.e., a solution without accelerated expansion. For the first term in left side of (36) we haveȧ 2 + k a 2 = 0, The solution is In this case a(t) is increase linearly, i.e., there is no accelerated expansion.
Replacing this solutions into equations (30 -33) we find and equation (34) is satisfied for h(t) arbitrary.

Solutionä = 0
From (36) we obtain we obtainȧ From (41) we can see two options (i) α > 0 and (ii) a. Case α > 0 : Consider the case where the constant α is positive. Using the following ansatz 1 1 This ansatz can be obtained fromȧ using an hyperbolic substitution where t ′ is a constant of integration, we obtain and therefore the initial condition a 0 = a(t = t 0 ) leads This results shows that if α > 0, then there is an accelerated expansion (see Fig. 2).
On the another hand, from (45) and (46) we can see thaẗ replacing (45), (46) and (47) into (30 -33) we obtain i.e., we have an accelerated expansion when the energy density is positive and pressure is negative (like a cosmological constant positive).
Integrating, we find where C is a constant of integration. The initial condition h 0 = h(t 0 ) leads Consider now the case when the constant α is negative. The ansatz with t ′ a contant of integration, leads therefore The initial condition a 0 = a(t = t 0 ), leads Therefore if a(t) > 0 thenä(t) < 0, which shows that if α < 0, then there is a decelerated expansion (see Fig. 3).
On the another hand, replacing (54) and (55) into (30 -33) we obtain Since the energy momentum tensor is given bỹ we have that the corresponding energy density and pressure are (α < 0) i.e., the energy density is negative and the pressure is positive (like a cosmological constant negative).

From equation (34) we find
Integrating, we find The solution for an static universe is given by which leads and the equation (34) is satisfied for all h(t). a. Case α > 0 : In this case we have an expanding universe The initial condition a 0 = a(t 0 ) leads and Replacing (69) into equation (34), solving for h(t) and using the initial condition h 0 = h(t 0 ), we find b. Case α < 0 : In this case it is not possible to find a solution.
16 C. Case T µν = 0 and k = 1 Introducing (32) into (30) we obtain which can be rewritten as In this case it is not possible to find a solution.

Caseä = 0
From equation (72) we obtain 2 From (73) we can see two cases: a. Case α > 0 : If α > 0 we can postulate a solution given by where t ′ is a constant of integration, which leads The initial condition a 0 = a(t = t 0 ) leads which shows an accelerated expansion (see Replacing (76) and (77) into (30 -33) we obtain i.e., we have an accelerated expansion when the energy density is positive and pressure is negative (like a cosmological constant positive) so that where C is a constant of integration. The initial condition h 0 = h(t 0 ) leads If α < 0 the equation (73) have no solution.

D. Era of Dark Energy from Einstein-Chern-Simons gravity
The results in the previous section are summarized in Tables I, II and III.
So that we have found solutions that describe accelerated expansion for the three possible cosmological models of the universe. Namely, spherical expansion (k = 1), flat expansion (k = 0) and hyperbolic expansion (k = −1) when the constant α is greater than zero. This means that the Einstein-Chern-Simons field equations have as a of their solutions a universe We have also shown that the EChS field equations have solutions that allows us to identify the energy-momentum tensor for the field h a with a negative cosmological constant.

V. CONSISTENCY OF THE SOLUTIONS WITH THE "ERA OF MATTER"
In the previous section, we find that the solutions of EChS field equations, with T µν = 0, can be useful as models of the era of dark energy. In this section we review the consistency of this equations with the era of matter.
In the era of matter, ordinary matter is modeled as dust, i.e., pressure corresponding to the era of matter is zero. In this case the field equations (24 -28) takes the form and the conservation equations (divergence-free energy-momentum tensor ) for each fluids are given byρ The equation (85) have as solution where the initial conditions a 0 = a(t 0 ) and ρ 0 = ρ(t 0 ) has been set.
Replacing (87) and (82) into equation (80) we have where we defined In this case, the equation (88) can be rewritten where we findȧ 1. Case α > 0 In this case From (91) we can see thatȧ is well defined if where a min = 4 κ 5 αl 2 ρ 0 3 a 0 .

21
On the other hand a 0 must satisfy and therefore These results allow us to analyze the radicand in (91) i.e., which is satisfied for all a.
a. Plus or minus sign?
The choice of the sign into the radicand has information about the allowed values ofȧ.
Let us considerȧ > 0 (the analysis of the caseȧ < 0 is very similar) The functionȧ(a) is monotonically increasing (decreasing) if we consider the plus (minus) sign in front of the square root.
From (99) we can see that there existȧ cri a cri :=ȧ(a 4 min ) = If we consider the plus (minus) sign in front of the square root,ȧ cri is the minimum (maximum) value ofȧ.
If there is a limit to a ≫ a min , then where k = −1.
b. Case where the sign is "+" In this caseȧ whose approximate solution is where we use A = 1 αl 2 and k = −1. c. Case where the sign is "−" 24 whose approximate solution is

Case α < 0
In this case From (91) we can see thatȧ is well defined if but this condition is satisfied for all a.
a. Case where the sign is '+' In this case The left side of the last equation must be positive, i.e., From (108) we obtain and again, the left side of the last equation must be positive, i.e., and from (110) we find Since i. e., a max is a local maximum. It is direct to prove thatȧ = 0 for a = a max . If a has a maximum value a max then (see (87)) 26 This means that ρ has a minimum value ρ min given by Consider the case whereȧ > 0. We just considerȧ > 0 because the analysis of the casė a < 0 looks very similar. In this casė 27 is a decreasing function. We can see that the minimum value ofȧ is given bẏ a min =ȧ(a max ) = 0 (119) and the maximum value ofȧ is given bẏ b. Case where the sign is "−" In this case we obtain the following condition where A = − 1 αl 2 and k = −1. This condition is trivially satisfied for all a.
This result implies thatȧ = 0. This means that a has no local maximums/minimums, so a is monotonically increasing or monotonically decreasing.
If there is a limit to a ≫ 4 − B A a 0 , theṅ whose approximate solution is where we use k = −1.
In this caseȧ is a decreasing function. The maximum value ofȧ is given bẏ and we can see thatȧ tends to a minimum value given bẏ

Case α > 0
In this case From (128) we can see thatȧ is well defined if and therefore a minimum value for a is given by On the other hand a 0 ≥ a min , so that These results leads i.e., a has no local maximums/minimums 2 , so that a is monotonically increasing or monotonically decreasing.
The choice of the sign into the radicand has information about the allowed values ofȧ.
Let us considerȧ > 0, the analysis of the caseȧ < 0 is very similaṙ The functionȧ(a) is monotonically increasing(decreasing) if we consider the plus(minus) sign in front of the square root.
If we consider the plus (minus) sign in front of the square root,ȧ cri is the minimum(maximum) value ofȧ. If there is a limit to a ≫ a min theṅ a. Case where the sign is "+"

33
In this caseȧ whose approximate solution is b. Case where the sign is "−" whose approximate solution is where we use A = 1 αl 2 > 0 and a min = 4 κ 5 αl 2 ρ 0 In this case From (128) we can see thatȧ is well defined if This condition is only satisfied if we use the minus sign "−" for all a, i.e., and therefore a has no local maximums/minimums, so a is monotonically increasing or monotonically decreasing. So thatȧ has a maximum value in a = 0, i.e., andȧ tends to a minimum value given bẏ If exist a limit for a ≫ 4 − B A a 0 theṅ whose approximate solution is where we use B = κ 5 ρ 0 3 .

C. Case k = 1
In this case, the equation (88) can be rewritten as from whereȧ with k = 1.

Case α > 0
In this case From (148) we can see thatȧ is well defined if so that With these considerations we can analyze if the radicand is positive in (148)

a. Plus or minus sign?
Let us considerȧ > 0, the analysis of the caseȧ < 0 is very similaṙ The functionȧ(a) is monotonically increasing (decreasing) if we consider the plus (minus) sign in front of the square root.
If we consider the plus(minus) sign in front of the square root,ȧ cri is the minimum(maximum) value ofȧ. b. Case where the sign is "+" In this case so that but (see equation (87)) It is direct to prove thatȧ = 0 for a > a min , then a has no local maximums/minimums, and therefore a is monotonically increasing or monotonically decreasing.
If there is a limit to a ≫ a min , theṅ whose approximate solution is where we use A = 1 αl 2 and k = 1. c. Case where the sign is "−"

In this case
therefore This condition must be also satisfied by a min so that, but (see equation (87) and therefore From (160) we obtain i.e., From (165)  − k with A > 0, k = 1 andȧ 0 <ȧ cri .

From (148) we can see thatȧ is well defined if
this constrain exclude the case with plus sign "+" in front of square root. This condition leads where k = 1 and A = 1 αl 2 < 0. There is a maximum value for a a max = 3αl 2 k 2 + κ 5 ρ 0 a 4 0 6k , this maximum leads but (see equation (87)) so that, If there is a maximum a max then, must exist a minimum for ρ ρ min = 6ka

D. Solutions for era of matter
We have found a family of solutions for era of matter.
If we consider an open space (k = −1), the solutions found include (i) an accelerated expansion (α > 0) with a minimum scale factor at initial time that, when the time goes to infinity, the scale factor behaves as a hyperbolic sine function (Fig. 7) (ii) a decelerated expansion (α < 0), with a Big Crunch in a finite time t max (Fig. 9) (iii) and a couple of solutions without accelerated expansion, whose scale factor tends to a constant value: α > 0 ( Fig. 8) and α < 0 (Fig. 11) . See Table IV and Table V.

Decelerated
No accelerated From models found in Section V A we can see that there are solutions with α > 0 for accelerated contracting universe and no accelerated contracting universe (see Figure 6, a < 0). These solutions were not studied.
Solutions found for a flat universe (k = 0) in expansion are (i) an accelerated expansion whose scale factor behaves as a exponencial function when time grows and starts from a minimum value (Fig. 15) (ii) and a couple of solutions with decelerated expansion whose scale factor tends to square root function: α > 0 (Fig. 16) and α < 0 (Fig. 18) . See Table   46 VI and Table VII.
In this case there are also solutions of contraction universe (ȧ < 0) (i) one ends with a minimum value a min when α is positive (Fig. 14) (ii) and other ends with a Big Crunch when α is negative (Fig. 17).
Finally, we only found one solution for a closed universe (k = 1) in expansion. This solution is found when α is greater than zero. It behaves as a hyperbolic cosine function when time grows and starts from a minimum value (Fig 20). See Table VIII.
Furthermore, there are two contracting universe solutions, both ends in a finite time (i) one ends with a minimun value a min , when α is positive (Fig. 22) (ii) and other ends with a Big Crunch, when α is negative (See Table IX and Fig. 24).  ρ 0 , ρ max = 3 κ 5 αl 2 andȧ cri = κ 5 ρ 0 3αl 2 a 2 0 − k. A decelerated solution describes an expanding universe, which then stops the expansion and then contracts until scale factor reaches a minimum a min > 0, in a finite time t max .

Accelerated
Decelerated  In fact, in Section IV we have found solutions that describes accelerated expansion for the three possible cosmological models of the universe. Namely, spherical expansion (k = 1), flat expansion (k = 0) and hyperbolic expansion (k = −1) when the constant α is greater than zero. This mean that the Einstein-Chern-Simons field equations have as a of their solutions an universe in accelerated expansion. This result allow us to conjeture that this solutions are compatible with the era of Dark Energy and that the energy-momentum tensor for the field h a corresponds to a form of positive cosmological constant. We have also shown that the EChS field equations have solutions that allows us to identify the energy-momentum tensor for the field h a with a negative cosmological constant.
On the other hand, in Section V we have found a family of solutions for era of matter. In the case k = −1 (open universe), the solutions correspond to (i) an accelerated expansion (α > 0) with a minimum scale factor at initial time that, when the time goes to infinity, the scale factor behaves as a hyperbolic sine function (ii) a decelerated expansion (α < 0), with a Big Crunch in a finite time t max (iii) and a couple of solutions without accelerated expansion, whose scale factor tends to a constant value. In the case k = 0 (flat universe), the solutions describing (i) an accelerated expansion whose scale factor behaves as a exponencial function when time grows and starts from a minimum value (ii) and a couple of solutions with decelerated expansion whose scale factor tends to square root function. In the case k = 1 it is found only one solution for a closed universe in expansion, which behaves as a hyperbolic cosine function when time grows and starts from a minimum value. However there are two contracting universe solutions, both ends in a finite time. One ends with a minimun value a min , when α is positive and other ends with a Big Crunch, when α is negative.