Accelerated FRW Solutions in Chern-Simons Gravity

# Accelerated FRW Solutions in Chern-Simons Gravity

Juan Crisóstomo    Fernando Gomez    Patricio Salgado Departamento de Física, Universidad de Concepción, Casilla 160-C, Concepción, Chile    Cristian Quinzacara Departamento de Física, Universidad de Concepción, Casilla 160-C, Concepción, Chile and
Facultad de Ingeniería y Tecnología, Universidad San Sebastián, Campus Las Tres Pascualas, Lientur 1457, Concepción, Chile
Mauricio Cataldo Departamento de Física, Universidad del Bío-Bío, Casilla 5-C, Concepción, Chile.    Sergio del Campo Instituto de Física, Pontificia Universidad Católica de Valparaíso, Av. Universidad 300, Campus Curauma, Valparaíso, Chile.
July 31, 2019
###### Abstract

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 , 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 , 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 () 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. (iii) 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.

FRW, Acelerated, Chern-Simons
###### pacs:
04.50.-h, 04.50.Kd, 98.80.-k, 98.80.Jk

## 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 Izaurieta2009213 (), whose generators satisfy the commutation relationships

This algebra was obtained from the anti de Sitter (AdS) algebra and a particular semigroup by means of the S-expansion procedure introduced in Refs. izaurieta:123512 (), izaurieta:073511 ().

In order to write down a Chern–Simons lagrangian for the algebra, we start from the one-form gauge connection

 A=12ωabJab+1leaPa+12kabZab+1lhaZa, (1)

and the two-form curvature

 F =12RabJab+1lTaPa+12(Dωkab+1l2eaeb)Zab +1l(Dωha+ka \ \ beb)Za. (2)

Consistency with the dual procedure of S-expansion in terms of the Maurer-Cartan forms izaurieta:073511 () demands that inherits units of length from the fünfbein; that is why it is necessary to introduce the parameter again, this time associated with .

It is interesting to observe that are still Lorentz generators, but are no longer AdS boosts; in fact, . However, still transforms as a vector under Lorentz transformations, as it must be in order to recover gravity in this scheme.

A Chern-Simons lagrangian in dimensions is defined to be the following local function of a one-form gauge connection :

 L(5)ChS(A)=k⟨AF2−12A3F+110A5⟩, (3)

where denotes a invariant tensor for the corresponding Lie algebra is the corresponding the two-form curvature and is a constant Zanelli:2005sa ().

Using theorem VII.2 of Ref. izaurieta:123512 (), it is possible to show that the only non-vanishing components of a invariant tensor for the   algebra are given by

 ⟨Ja1a2Ja3a4Pa5⟩ = α14l33ϵa1⋯a5, (4) ⟨Ja1a2Ja3a4Za5⟩ = α34l33ϵa1⋯a5, ⟨Ja1a2Za3a4Pa5⟩ = α34l33ϵa1⋯a5,

where and are arbitrary independient constants of dimensions .

Using the extended Cartan’s homotopy formula as in Ref. irs1 (), and integrating by parts, it is possible to write down the Chern-Simons Lagrangian in five dimensions for the algebra as

 L(5)EChS =α1l2ϵabcdeeaRbcRde +α3ϵabcde(23Rabecedee+2l2kabRcdTe+l2RabRcdhe) +dB(4)EChS (5)

where the suface term is given by

 B(4)EChS =α1l2ϵabcdeeaωbc(23dωde+12ωd2fωfe) +α3ϵabcde[l2(haωbc+kabec)(23dωde+12ωd2fωfe) (6)

and where , are parameters of the theory, is a coupling constant, corresponds to the curvature -form in the first-order formalism related to the -form spin connection Zanelli:2005sa (), Chamseddine1989291 (), Chamseddine1990213 (), and , and are others gauge fields presents in the theory Izaurieta2009213 ().

From (5) we can see that the third term is a surface term and can be removed from this Lagrangian. So that,

 L(5)EChS =α1l2εabcdeRabRcdee +α3ϵabcde(23Rabecedee+2l2kabRcdTe+l2RabRcdhe) (7)

is the Einstein-Chern-Simons Lagrangian studied in Ref Izaurieta2009213 ().

It should be noted the absence of kinetic terms for the fields and in equation (7). The term kinetic for the and fields are present in the surface term of the Lagrangian (5) given by (6).

The Lagrangian (7) show that standard, five-dimensional General Relativity emerges as the limit of a CS theory for the generalized Poincaré algebra . Here is a length scale, a coupling constant that characterizes different regimes within the theory. The  algebra, on the other hand, is constructed from the AdS algebra and a particular semigroup by means of the S-expansion procedure. The field content induced by the algebra includes the vielbein , the spin connection and two extra bosonic fields and , which can be interpreted as boson fields coupled to the field curvature and the parameter can be interpreted as a kind of coupling constant.

Recently was found PhysRevD.84.063506 () 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 () and (ii) the variation of the matter Lagrangian with respect to (w.r.t.) the spin connection is zero () can be written in a similar way to the Einstein-Maxwell field equations. The interpretation of the 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 V 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 VI 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. PhysRevD.84.063506 () was found that in the presence of matter the lagrangian is given by

 L=L(5)ChS+κLM (8)

where is the five-dimensional Chern-Simons lagrangian given by (7), is the matter Lagrangian and is a coupling constant related to the effective Newton’s constant. The variation of the lagrangian (8) w.r.t. the dynamical fields vielbein , spin connection , and , leads to the following field equations

 εabcde(2α3Rabeced+α1l2RabRcd +2α3l2DωkabRcd) =κδLMδee, (9)
 α3l2εabcdeRabRcd=κδLMδhe, (10)
 2α3l2εabcdeRcdTe=κδLMδkab, (11)
 2εabcde(α1l2RcdT e+α3l2DωkabTe +α3ecedTe+α3l2RcdDωhe) +2α3εabcdel2Rcdke fef =κδLMδωab. (12)

For simplicity, we will assume that the torsion vanishes () and . In this case the Eqs.(9 - 12) takes the form

 εabcde(2α3Rabeced+α1l2RabRcd) =κδLMδee, (13) α3l2εabcdeRabRcd =κδLMδhe, (14) δLMδkab =0 (15) 2α3l2εabcdeRcdDωhe =κδLMδωab. (16)

This field equations system can be written in the form

 εabcdeRabeced =4κ5(δLMδee+αδLMδhe), (17) l2εabcdeRabRcd =8κ5δLMδhe, (18) l2εabcdeRcdDωhe =4κ5δLMδωab (19)

where we introduce and .

The field equation (9) contains three terms. The first one, proportional to the Einstein tensor. The second one corresponds to a quadratic term in the curvature, and a third one, a term that describes the dynamics of the field . Since we asume the last term in left side of Eq. (9) vanishes.

In order to write this field equation manner analogous to Einstein’s equations, one chooses to leave the term proportional to the Einstein tensor on the left side of Eq. (9)

 ϵabcdeRbcedee=κ2α3δLMδea−α12α3l2ϵabcdeRbcRde

and using the Eq. (14) we obtain Eq. (17).

This result allows us to interpret as the energy momentum tensor for a second type of matter, not ordinary. Henceforth we will say that corresponds to the energy-momentum tensor for the field .

The equation of motion for the -field is given by Eq.(19). The condition (usual in gravity theories), imposed for consistency with the condition , leads to the equation of motion (22) for the -field . This means that -field is governed by the following field equations

 εabcdeRabeced =4κ5(δLMδee+αδLMδhe), (20) l28κ5εabcdeRabRcd =δLMδhe, (21) εabcdeRcdDωhe =0. (22)

This means that the Einstein-Chern-Simons field equations, subject to the conditions and , can be re-written in a way similar to the Einstein-Maxwell field equations.

From (20-22) we can see that if , then in five dimensions there is no solution of Schwarzschild type Izaurieta2009213 (), PhysRevD.85.124026 ().

## Iii Einstein-Chern-Simons Equations in the range of validity of General Relativity

From (20-21) we can see that general relativity is valid when (i) the curvature takes values not excessively large (ii) the parameter takes small values Izaurieta2009213 (); (iii) the constant takes values not excessively large. In fact, in this case we have that (21) takes the form

 δLMδhe≈0. (23)

Introducing (23) into (20) we obtain the Einstein’s field equation

 εabcdeRabeced≈4κ5δLMδee. (24)

If is not large then 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 (20-22) are given by

 εabcdeRabeced =4κ5δLMδee, (25) εabcdeRcdDωhe =0. (26)

On the another hand, if is large enough, so that when it is multiplied by (which is very small) will have a non-negligible results, then we will find that is not negligible.  This means that, in this case, we must consider the entire system of equations (20-22).

## Iv Einstein-Chern-Simons Field Equations for a Friedmann-Robertson-Walker-like spacetime

The shape of the field is obtained from of the application of the cosmological principle to the metric tensor of spacetime: it is considered a splitting of the 5D-manifold in a maximally symmetric four-dimensional manifold and one temporal dimension (). This leads to five dimensional Friedmann-Robertson-Walker (FRW) metric. So that, the vielbein can be chosen like in PhysRevD.84.063506 ():

 e0 =dt, e1 =a(t)√1−kr2dr, e2 =a(t)rdθ2, e3 =a(t)rsinθ2dθ3, e4 =a(t)rsinθ2sinθ3dθ4 (27)

where is the scale factor of the universe and is the sign of the curvature of space (): (i) for a closed space (), (ii) for a flat space () and (iii) for an open space (hyperbolic).

The application of the cosmological principle to the metric tensor of the spacetime also constrains the shape of the field (see for example PhysRevD.84.063506 ()). A detailed discussion can be also found in Ref. weinberg1972gravitation (). The bosonic field is given by

 h0 =h(0)dt=h(0)e0, h1 =h(t)a(t)√1−kr2dr=h(t)e1, h2 =h(t)a(t)rdθ2=h(t)e2, h3 =h(t)a(t)rsinθ2dθ3=h(t)e3, h4 =h(t)a(t)rsinθ2sinθ3dθ4=h(t)e4 (28)

where is a constant and is a function of time that must be determined. Substituting (28) into Eq. (22) we obtain the explicit form of the equations of motion for the -field, which will be displayed in Eq.(39).

In accordance with the equation (20), 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 (). The energy-momentum tensors in the comoving frame, are given by

 Tμν=diag(ρ,p,p,p,p), (29)
 T(h)μν=diag(ρ(h),p(h),p(h),p(h),p(h)), (30)

where is the matter density and is the pressure of fluid. Then, the energy-momentum tensor for the composed fluid is

 ~Tμν =Tμν+αT(h)μν (31) =diag(ρ+αρ(h),p+αp(h), p+αp(h),p+αp(h),p+αp(h)) (32) =diag(~ρ,~p,~p,~p,~p). (33)

In the torsion-free case, the energy momentum tensor of ordinary matter satisfies a conservation equation and the Einstein tensor has also zero divergence. In this case the energy momentum tensor for the non-ordinary matter must also satisfy a conservation equation. In fact, from Eq. (20) we find

 ∇μTμbν=0,∇μT(h)μbbbbbν=0 (34)

Introducing (27 - 33) into eqs. (20 - 22) we find the following field equations (see Ref. PhysRevD.84.063506 () and Appendix A)

 6(˙a2+ka2) =κ5~ρ, (35) 3[¨aa+(˙a2+ka2)] =−κ5~p, (36) 3l2κ5(˙a2+ka2)2 =ρ(h), (37) 3l2κ5¨aa(˙a2+ka2) =−p(h), (38) (˙a2+ka2)[(h−h(0))˙aa+˙h] =0. (39)

We should note that equation (35) was studied in Ref. 1475-7516-2012-12-005 () in the context of inflationary cosmology .

The Equations (35) and (36) are very similar to the Friedmann equations in five dimensions. However now and are subject to restrictions imposed by the remaining equations.

## V Acelerated Solution for Einstein-Chern-Simons Field Equations

In order to recover the known results of the standard cosmology in the context of accelerated expansion we use the approach

 Tμν≪αT(h)μν

This approach is analogous to the case when, in the era of Dark Energy, the energy momentum tensor is neglected compared to the cosmological constant. This means that the contribution from the ordinary matter is negligible compared to the contribution from the field . In this case, the energy-momentum tensor fluid is given by

 ~Tμν =diag(~ρ,~p,~p,~p,~p) =αT(h)μν =diag(αρ(h),αp(h),αp(h),αp(h),αp(h)) (40)

and the equations (35 - 39) take the form

 6(˙a2+ka2) =κ5αρ(h), (41) 3[¨aa+(˙a2+ka2)] =−κ5αp(h), (42) 3l2κ5(˙a2+ka2)2 =ρ(h), (43) 3l2κ5¨aa(˙a2+ka2) =−p(h), (44) (˙a2+ka2)[(h−h(0))˙aa+˙h] =0. (45)

### v.1 Case Tμν=0 and k=−1

Introducing (43) into (41) we obtain

 6(˙a2+ka2)=3l2α(˙a2+ka2)2 (46)

which can be rewritten

 (˙a2+ka2)(2αl2−˙a2+ka2)=0. (47)

#### v.1.1 Solution ¨a=0

Consider the solution , i.e., a solution without accelerated expansion. For the first term in left side of (47) we have

 ˙a2+ka2=0, (48)

remembering we have

 ˙a=√−k. (49)

The solution is

 a(t)=√−k(t−t0)+a0. (50)

In this case is increase linearly, i.e., there is no accelerated expansion.

Replacing this solutions into equations (41 - 44) we find

 ρ(h)=p(h)=0 (51)

and equation (45) is satisfied for arbitrary.

#### v.1.2 Solution ¨a≠0

From (47) we obtain we obtain

 ˙a2−2αl2a2=−k. (52)

From (52) we can see two options (i) and (ii) .

##### Case α>0:

Consider the case where the constant is positive. Using the following ansatz111This ansatz can be obtained from whose solution is (, ) using an hyperbolic substitution

 a(t)=Asinh(√2αl2(t−t′)) (53)

where is a constant of integration, we obtain

 A=√−αl2k2 (54)

and therefore

 a(t)=√−αl2k2sinh(√2αl2(t−t′)), (55)

 a(t) =√−αl2k2 ×sinh[√2αl2(t−t0)+arsinh(√−2αl2ka0)] (56)

and

 ˙a(t) =√−k ×cosh[√2αl2(t−t0)+arsinh(√−2αl2ka0)]. (57)

This results shows that if , then there is an accelerated expansion (see Fig. 2).

On the another hand, from (56) and (57) we can see that

 ¨a(t)=2αl2a(t), (58)

replacing (56), (57) and (58) into (41 - 44) we obtain

 ρ(h)=−p(h)=12κ5αl2, (59)

i.e., we have an accelerated expansion when the energy density is positive and pressure is negative (like a cosmological constant positive).

From equation (45) we find

 −˙hh−h(0)=˙aa. (60)

Integrating, we find

 h(t)=Csinh[√2αl2(t−t0)+arsinh(√−2αl2ka0)]+h(0) (61)

where is a constant of integration. The initial condition leads

 h(t)=(h0−h(0))√2αl2ka0sinh[√2αl2(t−t0)+arsinh(√−2αl2ka0)]+h(0)

from where we can see that when

##### Case α<0:

Consider now the case when the constant is negative. The ansatz

 a(t)=Asin(√−2αl2(t−t′)) (62)

with a contant of integration, leads

 A=√αl2k2, (63)

therefore

 a(t)=√αl2k2sin(√−2αl2(t−t′)). (64)

 a(t) =√αl2k2 ×sin[√−2αl2(t−t0)+arcsin(√2αl2ka0)] (65)

and

 ˙a(t) =√−k ×cos[√−2αl2(t−t0)+arcsin(√2αl2ka0)]. (66)

Therefore if then , which shows that if , then there is a decelerated expansion (see Fig. 3).

On the another hand, replacing (65) and (66) into (41 - 44) we obtain

 ρ(h)=−p(h)=12κ5αl2. (67)

Since the energy momentum tensor is given by

 ~Tμν=αT(h)μν=diag(αρ(h),αp(h),αp(h),αp(h),αp(h)) (68)

we have that the corresponding energy density and pressure are ()

 ~ρ=αρ(h)=12κ5αl2<0, (69)
 ~p=αp(h)=−12κ5αl2>0, (70)

i.e., the energy density is negative and the pressure is positive (like a cosmological constant negative).

From equation (45) we find

 −˙hh−h(0)=˙aa. (71)

Integrating, we find

 h(t)=Csin[√−2αl2(t−t0)+arcsin(√2αl2ka0)]+h(0) (72)

where is a constant of integration. The initial condition , leads

 h(t)=(h0−h(0))√2αl2ka0sin[√−2αl2(t−t0)+arcsin(√2αl2ka0)]+h(0). (73)

### v.2 Case Tμν=0 and k=0

Introducing (43) into (41) and considering , we obtain

 6(˙aa)2=3l2α(˙aa)4 (74)

which can be rewritten as

 (˙aa)2(2αl2−˙a2a2)=0. (75)

#### v.2.1 Static solution ˙a=0

The solution for an static universe is given by

 a(t)=a0 (76)

 ρ(h)=p(h)=0 (77)

and the equation (45) is satisfied for all .

#### v.2.2 Non-static solution ˙a≠0

From (75) we obtain

 ˙a2−2αl2a2=0. (78)

This equation have solution, only if .

##### Case α>0:

In this case we have an expanding universe

 a(t)=Aexp(√2αl2t). (79)

 a(t)=a0exp(√2αl2(t−t0)) (80)

and

 ρ(h)=−p(h)=12κ5αl2.

Replacing (80) into equation (45), solving for and using the initial condition , we find

 h(t)=h0−h(0)exp(√2αl2(t−t0))+h(0). (81)

##### Case α<0:

In this case it is not possible to find a solution.

### v.3 Case Tμν=0 and k=1

Introducing (43) into (41) we obtain

 6(˙a2+ka2)=3l2α(˙a2+ka2)2 (82)

which can be rewritten as

 (˙a2+ka2)(2αl2−˙a2+ka2)=0. (83)

#### v.3.1 Case ¨a=0

In this case it is not possible to find a solution.

#### v.3.2 Case ¨a≠0

From equation (83) we obtain

 2αl2a2−˙a2=k. (84)

From (84) we can see two cases:

##### Case α>0:

If we can postulate a solution given by

 a(t)=Acosh(√2αl2(t−t′)) (85)

where is a constant of integration, which leads

 A=√αl2k2. (86)

 a(t)=√αl2k2cosh[√2αl2(t−t′)+arcosh(√2αl2a0)] (87)

and

 ˙a(t)=√ksinh[√2αl2(t−t0)+% arcosh(√2αl2a0)] (88)

which shows an accelerated expansion (see Fig. 5)

Replacing (87) and (88) into (41 - 44) we obtain

 ρ(h)=−p(h)=12κ5αl2

i.e., we have an accelerated expansion when the energy density is positive and pressure is negative (like a cosmological constant positive)

From equation (45) we find

 −˙hh−h(0)=˙aa, (89)

so that

 h(t)=Ccosh[√2αl2(t−t0)+arcosh(√2αl2ka0)]