E Elizalde
email: elizalde@ieec.uab.es

R Myrzakulov

V V Obukhov

D Sáez-Gómez

ΛCDM epoch reconstruction from F(R,G) and modified Gauss-Bonnet gravities

Keywords: numbers: 04.50.Kd, 95.36.+x, 98.80.-k

Dark energy cosmology is considered in a modified Gauss-Bonnet model of gravity with and without a scalar field. It is shown that these generalizations of General Relativity endow it with a very rich cosmological structure: it may naturally lead to an effective cosmological constant, quintessence or phantom cosmic acceleration, with the possibility to describe the transition from a decelerating to an accelerating phase explicitly. It is demonstrated here that these modified GB and scalar-GB theories are perfectly viable as cosmological models. They can describe the ΛCDM cosmological era without any need for a cosmological constant. Specific properties of these theories of gravity in different particular cases, such as the de Sitter one, are studied.

Introduction

Recent observational data indicate that our universe is accelerating. This acceleration is explained in terms of the so-called dark energy (DE), which may be expressed in modified gravity models (for a general review see Ref. [1]). DE could also result from a cosmological constant, from an ideal fluid with a different form of equation of state and negative pressure, a scalar field with quintessence-like or phantom-like behavior (see [2] and references therein), etc. The choice of possibilities reflects the indisputable fact that the true nature and origin of DE has not been convincingly explained yet. It is not even clear what type of DE is more seemingly to explain the current epoch of the universe. Observational data point towards some kind of DE with an equation of state (EoS) parameter which is very close to -1, maybe even less than -1, the so-called phantom case (see Ref. [START_REF] Cai | Quintom Cosmology: Theoretical implications and observations[END_REF]). A quite appealing possibility is the already mentioned modification of General Relativity (GR). Modifications of the Hilbert-Einstein action by introducing different functions of the Ricci scalar have been systematically explored, the so-called F (R) gravity models, which reconstruction has been carried out in Refs. [START_REF] Nojiri | [END_REF]- [9]. As is known, F (R) gravity can be written in terms of a scalar field (quintessence or phantom like) by redefining the function F (R) with the use of a scalar field, and then performing a conformal transformation. It has been shown that, in general, for any given F (R) the corresponding scalar-tensor theory can, in principle, be obtained, although the solution is going to be very different from one case to another. Also, attention has been paid to the reconstruction of F (R) gravity from a given scalar-tensor theory. It is known, too, that the phantom case in scalar-tensor theory does not exist, in general, when starting from F (R) gravity. In fact, the conformal transformation becomes complex when the phantom barrier is crossed, and therefore the resulting F (R) function becomes complex. These situations where addressed in [9] in detail, where to avoid this hindrance, a dark fluid was used in order to produce the phantom behavior in such a way that the F (R) function reconstructed from the scalar-tensor theory continues to be real.

On the other hand, it has also been suggested in the literature (see Refs. [10], [11]) to consider modified Gauss-Bonnet gravity, that is, a function of the GB invariant. Different cosmological properties of modified gravity models of this kind have been studied in Refs. [10]- [START_REF] Bamba | Finite-time future singularities in modified Gauss-Bonnet and F(R, G) gravity and singularity avoidance[END_REF]. Both possibilities have, in principle, the capability to explain the accelerated expansion of the Universe and even the primordial inflationary phase (see [START_REF] Nojiri | [END_REF] and [9]), with no need to introduce a new form of energy. In this paper we will study some specific modified Gauss-Bonnet theories and, by using a technique developed in Ref. [START_REF] Nojiri | [END_REF], the corresponding cosmological theory will be reconstructed for several cosmological solutions.

We will here work with the spatially-flat FRW universe metric

ds 2 = -dt 2 + a(t) 2 3 i=1 (dx i ) 2 , (1)
where a(t) is the scale factor at cosmological time t. In GR, the corresponding system of equations are the usual Friedmann equations, namely 3

k 2 H 2 = ρ, 1 k 2 (2 Ḣ + 3H 2 ) = -p, ρ = -3H(ρ + p). (2)
The last of them is the continuity equation for a perfect fluid. As usually, the Hubble rate H is defined by H ≡ ȧ/a. In (2), ρ and p are the matter energy-density and pressure. The Gauss-Bonnet invariant is given by

G = R 2 -4R µν R µν + R µνλσ R µνλσ (3)
The cosmological models coming from the different versions of modified GB gravity here considered will be carefully investigated with the help of several particular examples where calculations can be carried out explicitly. The paper is organized as follows. In Sect. II, the general technique which will be used to reconstruct a gravity theory from a given cosmological evolution setup will be explained in detail. Several specific examples will be worked through, for the simple case where one considers a Hilbert-Einstein action plus a function on the GB invariant. In Sect. III, a general model whose action depends on a function of the Ricci scalar and of the GB invariant will be studied. With the help of several explicit examples we will show how the reconstruction of the cosmological solutions is carried out. In Sect. IV the more involved case of a scalar-tensor theory where the GB invariant is coupled to the scalar field will be considered. It will be shown there that any cosmological solution can be reproduced by an specific scalar potential and a convenient coupling term. Finally, in the concluding section we will provide a summary of the main results obtained in the paper.

The [R + f (G)]-model

Consider the following action, which describes General Relativity plus a function of the Gauss-Bonnet term (see Refs. [10] and [11]),

S = d 4 x √ -g 1 2κ 2 R + f (G) + L m , (4)
where κ 2 = 8πG N , G N being the Newton constant. By varying the action over g µν , the following field equations are obtained

0 = 1 2κ 2 (-R µν + 1 2 g µν R) + T µν + 1 2 g µν f (G) -2f G RR µν + 4f G R µ ρ R νρ -2f G R µρστ R ν ρστ -4f G R µρσν R ρσ + 2(∇ µ ∇ ν f G )R -2g µν (∇ 2 f G )R -4(∇ ρ ∇ µ f G )R νρ -4(∇ ρ ∇ ν f G )R µρ +4(∇ 2 f G )R µν + 4g µν (∇ ρ ∇ σ f G )R ρσ -4(∇ ρ ∇ σ f G )R µρνσ . (5)
For the metric (1), these equations give the first FRW equation which has the form

0 = - 3 κ 2 H 2 + Gf G -f -24 ĠH 3 f GG + ρ m . (6)
The Hubble rate H is here defined by H = ȧ/a, while G and R are given by

G = 24( ḢH 2 + H 4 ), R = 6( Ḣ + 2H 2 ). (7)
The matter energy density ρ m satisfies the standard continuity equation

ρ m + 3H(1 + w)ρ m = 0. (8)
Let us now rewrite Eq. ( 6) by using a new variable N = ln a a 0 =ln(1 + z), that is, the number of e-foldings, instead of the cosmological time t, where z is the redshift (this method has been implemented in Ref. [START_REF] Nojiri | [END_REF] for f (R) gravity). The following expressions are then easily obtained

a = a 0 e N , H = Ṅ = dN dt , d dt = H d dN , d 2 dt 2 = H 2 d 2 dN 2 + HH d dN , H = dH dN . (9)
Eq. ( 6) can thus be expressed as follows

0 = - 3 κ 2 H 2 + 24H 3 (H + H)f G -f -576H 6 HH + 3H 2 + 4HH f GG + ρ m , (10)
where G and R are now

G = 24(H 3 H + H 4 ), Ġ = 24(H 4 H + 3H 3 H 2 + 4H 4 H ), R = 6(HH + 2H 2 ). (11)
By introducing a new function g as g = H 2 , we have

H = √ g, H = 1 2 g -1/2 g , H = - 1 4 g -3/2 g 2 + 1 2 g -1/2 g . (12)
Hence, Eq. ( 10) takes the form

0 = - 3 κ 2 g+12g(g +2g)f G -f -24 2 g 1 2 g 2 g + 1 2 gg 2 + 2g 2 g f GG +ρ m ,( 13
)
where we have used the expressions

G = 12gg + 24g 2 , Ġ = 12g -1/2 [g 2 g + gg 2 + 4g 2 g ] , R = 3g + 12g. (14)
Finally, we can write the FRW equations in a slightly different form [13] 3

κ 2 H 2 = ρ ef f , 1 k 2 (2 Ḣ + 3H 2 ) = -p ef f , (15)
where the effective energy and pressure densities are

p ef f = w ef f ρ ef f , ρeff = -3H(ρ ef f + p ef f ), (16)
with

ρ ef f = ρ G + ρ m , p ef f = p G + p m . (17)
Here

ρ G = Gf G -f -24H 3 Ġf GG , p G = -ρ G + 8H 2 Ġ2 f GGG -192f GG (4H 6 Ḣ -8H 3 Ḣ Ḧ -6H 2 Ḣ3 -H 4 ... H -3H 5 Ḧ -18H 4 Ḣ2 ). ( 18
)
As it was pointed out in Ref. [13], for de Sitter space

p G = w G ρ G = -ρ G , that is w G = -1.

Example 1

As a first example we consider the ΛCDM-model. As it will be shown below, for the gravity theory described by the action (4), it is possible to reconstruct such evolution with no need of a cosmological constant term, and the transition from a decelerated to an accelerated epoch is achieved (note that this can be also performed in the spinor field scenario, see [START_REF] Cai | [END_REF]). For the ΛCDM case, we have

g = H 2 = H 2 0 + κ 2 ρ 0 3 a -3 = H 2 0 + κ 2 ρ 0 3 a -3 0 e -3N = H 2 0 + lz, (19)
where l =

κ 2 ρ 0 a -3 0 3 , z = e -3N
. Hence, we get

z = -3z, g = -3lz, g = 9lz. ( 20
)
Finally, for G we obtain the expressions 21) where W = 12H 2 0 l -24l 2 z. We can reverse Eq. ( 21) and the number of e-foldings can be written as a function of the Gauss-Bonnet term. This yields

G = 24H 4 0 +12H 2 0 lz-12l 2 z 2 , Ġ = -3zHW = -3Hz(12H 2 0 l-24l 2 z),(
z = 3H 2 0 -81H 4 0 -3G 6l = e -3N , (22)
and, from here

N = 1 3 ln 6l 3H 2 0 -81H 4 0 -3G . (23)
Hence, it follows that

g = H 2 = H 2 0 + 3H 2 0 -81H 4 0 -3G 6 = 9H 2 0 -81H 4 0 -3G 6 . (24)
Finally, for the function f (z) we get the equation

a 2 f zz + a 1 f z + a 0 f + b = 0, (25)
where

a 0 = -w 2 = -24 2 l 4 z 2 + 24 2 H 2 0 l 3 z -144H 4 0 l 2 , a 1 = 12(H 2 0 + lz)[144l 2 z(H 2 0 + lz) + (2H 2 0 -lz)w], a 2 = 72zw(H 2 0 + lz) 2 , b = [ρ m - 3 κ 2 (H 2 0 + lz)]w 2 . ( 26
)
The energy density can be expressed as

ρ m = -Dz -E, (27)
being

D = 144HH 4 0 l 3 83 5κ 2 - 16 9 + 20δ 9 , E = 144HH 4 0 l 2 4 9 - 13 5κ 2 - 5δ 9 .( 28
)
Here δ = const. As a consequence, Eq. ( 25) has the following particular solution

f (z) = θz 2 + ϑz + H 2 0 δ, (29)
where

θ = l 2 H 2 0 [κ -2 - 2 9 (δ + 1)], ϑ = l[ 1 5 κ -2 + 2 9 (δ + 1)]. (30)
Thus, we have

f (G) = θ   3H 2 0 -81H 4 0 -3G 6l   2 + ϑ   3H 2 0 -81H 4 0 -3G 6l   + H 2 0 δ.( 31
)
We observe that this function reproduces exactly the same behavior as the ΛCDM model in the context of Gauss-Bonnet gravity.

Example 2

Let us now consider a second example

g = H 2 = le -3N = lz. ( 32
)
This case describes a time evolution given by

H(t) = 2 3(t -t 0 ) , (33)
which is equivalent to the cosmological evolution of a pressureless fluid in GR, what gives a decelerated expansion. As in the example above, we find that z = ±i

√ 3G
6l . In the present case, we have the following differential equation,

z 2 f zz - 7 6 zf z + 1 3 f - c 3 = 0, (34)
where c = ρ m -3 κ 2 lz. Let us assume that c = 0, then Eq. (34) takes the form

z 2 f zz - 7 6 zf z + 1 3 f = 0. ( 35
)
This equation admits the following exact solution

f 1 (z) = C 1 z 2 + C 2 z 1 6 . ( 36
)
As a consequence, for this second example we obtain the following model

f ( G) = -C 1 G 12 + C 2 ( ±i √ 3 6l ) 1 6 G 1 12 = C 1 G + C 2 G 1 12 . ( 37
)
It is now easy to see from here that the GB terms could actually contribute in the matter dominated epoch of the universe evolution, provided the f (G) function is taken properly, what is certainly an interesting result.

Example 3

As third and last example we will now consider the case when the Hubble parameter and the matter energy density behave as

H = e mN , ρ m = b 1 + 3 κ 2 e 2mN + 96(m + 1) 5 b 2 e 5mN , (38)
where b j =const. This case corresponds to the so-called phantom behavior, what means that we have an effective EoS parameter w ef f < -1, and produces a superaccelerated phase that could end in some kind of future singularity (see Refs. [START_REF] Bamba | Finite-time future singularities in modified Gauss-Bonnet and F(R, G) gravity and singularity avoidance[END_REF] and [28]). We first write the Hubble parameter as a function of time,

H(t) = H 0 t s -t , ( 39
)
where t s , usually called the Rip time, defines the moment when the Big Rip singularity occurs. We can find the f (G) function that reproduces this model by solving the equation ( 10). This yields the solution

F (G) = F (N ) = b 1 + 96(m + 1) 5 b 2 e 5mN = = b 1 + 96(m + 1) 5 b 2 [ G 24(m + 1) ] 5 4 .
(40)

The F (R, G) model

Let us now consider a more general model for a class of modified Gauss-Bonnet gravity. This can be described by the following action

S = d 4 x √ -g[ 1 2κ 2 F (R, G) + L m ]. (41)
Varying over g µν the gravity field equations are obtained [13],

0 = κ 2 T µν + 1 2 g µν F (G) -2F G RR µν + 4F G R µ ρ R νρ -2F G R µρστ R ν ρστ -4F G R µρσν R ρσ + 2(∇ µ ∇ ν F G )R -2g µν (∇ 2 F G )R -4(∇ ρ ∇ µ F G )R νρ -4(∇ ρ ∇ ν F G )R µρ + 4(∇ 2 F G )R µν + 4g µν (∇ ρ ∇ σ F G )R ρσ -4(∇ ρ ∇ σ F G )R µρνσ -F G R µν + ∇ µ ∇ ν F R -g µν ∇ 2 F R . (42)
In the case of a flat FRW Universe, described by the metric (1), the first FRW equation yields

0 = 1 2 (GF G -F -24H 3 F Gt ) + 3( Ḣ + H 2 )F R -3HF Rt + κ 2 ρ m . (43)
And from here, using the techniques developed in the previous section, it is plain that explicit F (R, G) functions can be reconstructed for given cosmological solutions.

De Sitter Solutions

As well known, the de Sitter solution is one of the most important cosmological solutions nowadays, since the current epoch has been observed to have an expansion that behaves approximately as de Sitter. This solution is described by an exponential expansion of the scale factor, which gives a constant Hubble parameter H(t) = H 0 . By inserting it in the Friedmann equation ( 43), one finds that any kind of F (R, G) function can possibly admit de Sitter solutions, with the proviso that the following algebraic equation has positive roots for

H 0 0 = 1 2 (G 0 F G (G 0 ) -F (G 0 , R 0 )) + 3H 2 0 F R (R 0 ) , (44)
being R 0 = 12H 2 0 and G 0 = 24H 4 0 ; we have here neglected the contribution of matter for simplicity. As it was pointed out in Refs. [START_REF] Cognola | [END_REF] and [9] for the case of modified F (R) gravity, the de Sitter points are critical points for the Friedmann equations, what could explain the current acceleration phase as well as the inflationary epoch. This explanation can be extended to the action (41), so that any kind of function F (R, G) with positive real roots for the equation ( 44) could in fact explain the acceleration epochs of the Universe in exactly the same way a cosmological constant does.

Phantom dark energy

Let us now explore the cosmic evolution described by (38) in the context of the action (41). This solution reproduces a phantom behavior, i.e. a superaccelerated expansion that, according to recent observations, our Universe could be in-or either close to cross the phantom barrier. We can now proceed with the reconstruction method, as explicitly shown in the section above, and a F (R, G) function for the Hubble parameter (38) will be obtained. For simplicity, we consider the following subfamily of functions

F (R, G) = f 1 (G) + f 2 (R) . (45)
Correspondingly, the Friedmann equation ( 43) can be split into two equations, as

0 = -24H 3 Ġf 1GG + Gf 1G -f 1 , 0 = -3H Ṙf 2RR + 3( Ḣ + H 2 )f 2R - 1 2 f 2 + κ 2 ρ m ( 46
)
For the example (38), that is H = e mN , the Ricci scalar and the Gauss-Bonnet terms take the following form,

G = 24(m + 1)e 4mN = 24(m + 1)H 4 , R = 6(m + 2)e 2mN = 6(m + 2)H 2 . ( 47
)
Hence, the first equation in (46) can be written in terms of G, as

G 2 f 1GG - m + 1 4m Gf 1G + m + 1 4m f 1 = 0 . ( 48
)
This is an Euler equation, easy to solve, and yields

f 1 (G) = C 1 G 1+ m+1 4m + C 2 , ( 49
)
where C 1,2 are integration constants. In the same way, for the case being considered here, the second equation in (46), for R, takes the form

R 2 f 2RR - m + 1 2m Rf 2R + m + 2 2m f 2 - κ 2 (m + 2) m ρ m = 0 . (50)
In absence of matter (ρ m = 0) this is also an Euler equation, with solution

f 2 (R) = k 1 R µ + + k 2 R µ - where µ ± = 1 + m+1 2m ± 1 + (m+1) 2 4m 2 -m+3 m 2 , (51)
and k 1,2 are integration constants. Then, the complete function F(R,G), given in (45), is reconstructed (in absence of matter) yielding the solutions (49) and ( 51). The theory (3.12) belongs to the class of models with positive and negative powers of the curvature introduced in [29].

Let us now consider the case where matter is included. From the energy conservation equation ( 8) we have that, for a perfect fluid with constant EoS, p m = w m ρ m , the solution is given by ρ m = ρ 0 e -3(1+wm)N . By inserting the expression for R (47) into this solution, we get

ρ m = ρ m0 = R 6(m + 2) - 3(1+wm ) 2m . (52)
In such case the general solution for f 2 is given by

f 2 (R) = k 1 R µ + + k 2 R µ -+ kR A , where, k = κ 2 ρ m0 (6(m + 1)) -A A(A -1) -1/2m and A = - 3(1 + w m ) 2m . (53)
Hence, we see that the solution for the Hubble parameter (38) can be easily recovered in the context of modified Gauss-Bonnet gravity. Nevertheless, it seems clear that, for more complex examples, one may not be able to solve the corresponding equations analytically and numerical analysis could be required..

Reconstruction of scalar-GB gravity

In this section we consider a four-dimensional action containing the Einstein-Hilbert part, a massless scalar field, and the Gauss-Bonnet term coupled to the scalar field. The corresponding action is

S = d 4 x √ -g[ 1 2κ 2 R - 1 2 ∂ µ φ∂ µ φ -V (φ) -ξ(φ)G + L m ]. (54)
An action of this kind has been proposed as a model for dark energy in Ref. [12]. It can actually be related with modified GB gravity, as explained in Refs. [14] and [30]. The variation of this action over the metric g µν gives [START_REF] Nojiri | Reconstruction and deceleration-acceleration transitions in modified gravity[END_REF]

0 = 1 κ 2 (-R µν + 1 2 g µν R) + T µν + 1 2 ∂ µ φ∂ ν φ - 1 4 g µν ∂ ρ φ∂ ρ φ + 1 2 g µν (-V + ξG) -2ξRR µν -4ξR µ ρ R νρ -2ξR µρστ R ν ρστ +4ξR µρνσ R ρσ + 2(∇ µ ∇ ν ξ)R -2g µν (∇ 2 ξ)R -4(∇ ρ ∇ µ ξ)R νρ -4(∇ ρ ∇ ν ξ)R µρ + 4(∇ 2 ξ)R µν + 4g µν (∇ ρ ∇ σ ξ)R ρσ + 4(∇ ρ ∇ σ ξ)R µρνσ . (55)
In the FRW universe case, these equations and the equation of motion for the scalar become [14] 0 = -

3 κ 2 H 2 + 1 2 φ2 + V (φ) + 24H 3 ξ + ρ m , (56) 0
= 1 κ 2 (2 Ḣ + 3H 2 ) + 1 2 φ2 -V (φ) -8H 2 ξ -16H( Ḣ + H 2 ) ξ + p m , ( 57
) 0 = φ + 3H φ + V φ + ξ φ G. (58)
Note that, as a consequence of the Bianchi identity, Eq. ( 58) is satisfied automatically. From Eqs. (56-58), we get [14]

0 = 2 κ 2 Ḣ + φ2 -8a( H 2 a ξ) t . (59)
For the variables φ and ξ, (56-58) constitute a system of second order differential equations. It is useful to reduce this system to a first order set of differential equations. This can be immediately achieved by introducing new variables, as

u = φ2 , v = ξ, (60)
Then, we obtain the following first order system 0 = -

3 κ 2 H 2 + 1 2 u + V + 24H 3 v, (61) 0
= 1 κ 2 (2 Ḣ + 3H 2 ) + 1 2 u -V -8H 2 v -16H( Ḣ + H 2 )v, (62) 0
= u + 6Hu + 2 V + 2Gv. ( 63
)
From here it is easy to explore the stability and number of attractors of the physical model. To do so we rewrite the system (61-63) as

Ḣ = 1 24 GH -2 -H 2 , (64) u
= -(6Hu + 2 V + 2Gv), (65)
v = 1 8H 2 [ 2 κ 2 ( 1 24 GH -2 -H 2 ) + u -(16H Ḣ -8H 3 )v]. (66)
Let q = u + 2V . Then, this can be put as

0 = - 3 κ 2 H 2 + 1 2 q + 24H 3 v, (67) 0
= 1 κ 2 (2 Ḣ + 3H 2 ) + 1 2 q -2V -8H 2 v -16H( Ḣ + H 2 )v, ( 68
) 0 = q + 6Hq -12HV + 2Gv, (69)
and for (64-66)

Ḣ = 1 24 GH -2 -H 2 , (70) q
= 12HV -6Hq -2Gv, (71)
v = 1 8H 2 [ 2 k 2 ( 1 24 GH -2 -H 2 ) + q -2V -(16H Ḣ -8H 3 )v]. (72)
Let us now rewrite Eqs. (56-58) in terms of the number of e-foldings N . We have

0 = - 3 κ 2 H 2 + 1 2 H 2 φ 2 N + V (φ) + 24H 4 ξ 1N , (73) 0
= 1 κ 2 (2HH N + 3H 2 ) + 1 2 H 2 φ 2 N -V (φ) -8H 4 ξ 1NN -8H(2HH N + 3H 2 )Hξ 1N , (74) 0
= H 2 φ N φ NN + (H N + 3H 2 )φ 2 N + V N + ξ 1N G. (75)
If we introduce new variables

h = φ 2 N = H -2 u, p = ξ 1N = H -1 v, (76)
the system (73-75) takes the form

0 = - 3 κ 2 H 2 + 1 2 H 2 h + V (φ) + 24H 4 p, (77) 0
= 1 κ 2 (2HH N + 3H 2 ) + 1 2 H 2 h -V -8H 4 p N -8H(2HH N + 3H 2 )Hp,(78) 0 = H 2 h N + 2(H N + 3H 2 )h + 2V N + 2Gp. ( 79
)
To summarize, the original system of equations has been written in several ways that can be used to construct solutions. Needless to say, all the corresponding solutions are equally valid in every form of the system. Here we will find solutions for the one given by Eqs. (61-63). To do tat, we rewrite them in terms of the e-folding variable N :

0 = - 3 κ 2 H 2 + 1 2 u + V + 24H 3 v, (80) 0
= 1 κ 2 (2HH N + 3H 2 ) + 1 2 u -V -8H 3 v N -16H 2 (H N + H)v, (81) 0
= Hu N + 6Hu + 2HV N + 2Gv. ( 82
)
This has the following solution

u = 8H 4 W NN + 8H 3 (3H N -H)W N - 2 k 2 HH N , (83)
v = HW N , (84)
V = -4H 4 W NN -(12H 2 H N + 20H 3 )HW N + 3 κ 2 H 2 + 1 k 2 HH N , (85)
which has the solution

y(t) = y(φ) = y i = δ ± √ δ 2 -4 σ 2 . (99)
Hence, we have

N = N i (t) = N i (φ) = 1 η ln y i β (100)
and

H = H(t) = H(φ) = √ α + y i . (101)
We have thus seen that, from the action (54), any of the usual cosmologies can be obtained. The system of equations (83-85) provides a quite simple setup to reproduce any kind of cosmological solution, as it has been clearly illustrated with the three examples above, where for a given Hubble parameter the scalar Gauss-Bonnet theory has been constructed.

Conclusions

In the present paper, several types of DE cosmologies in modified GB gravity-which can be viewed as being inspired by string considerations, Ref. [14]-have been investigated. We have studied different kinds of theories, in all of which the GB invariant plays an important role in the corresponding equations. First, we have shown that GR plus some function of the GB term provides a very powerful theory, where no sort of dark energy is actually needed to reproduce the standard ΛCDM cosmology. More general theories have been considered too, as in Sect. III, where an action depending on a function of the Ricci scalar and GB invariant has been studied. In this case the dark energy behavior can also be reproduced, while its extra degrees of freedom provide us with a powerful method to constraint the theory, in order to avoid violation of the local gravity tests.

In the same way, the scalar GB theory studied in the last section adequately reproduces any kind of cosmological solution. As a follow up, we have shown, with the help of several particular examples corresponding to explicit choices of the functions f (G), F (R, G) or the scalar field that, in principle, any cosmic evolution can be obtained from these models, what includes the unification of early-time inflation with the late-time acceleration coming from astronomical observations. To finish, it has been indicated that DE cosmologies in a more general (and complicated) F (R, G) framework-generically requiring numerical analysis-can be reconstructed in a similar fashion, too. also supported by MEC (Spain) grant PR2009-0314, and by AGAUR (Generalitat de Catalunya), contract 2009SGR-994.

Acknowledgments

DSG acknowledges a grant from MICINN (Spain), project FIS2006-02842. Part of EE's research was performed while on leave at Department of Physics and Astronomy, Dartmouth College, 6127 Wilder Laboratory, Hanover, NH 03755, USA. This work was

where W = W (N ) is some function of N . At the same time, φ, ξ 1 are given by

Let us now consider several examples.

(i) As the first one consider the case when W = 0, which corresponds to the Einsteinscalar gravity. In this case, we have

(ii) A second example is the model

where ν =const. Here the solution of the system (83)-( 85) is

It corresponds to the so-called Einstein-Gauss-Bonnet gravity.

(iii) The third example will be

where µ, ν are some constants. Then, the solution of the system (83)-( 85) is given by

The next step is finding the explicit forms of φ(t), H(t), ξ 1 (t), V (t). To this end we consider the case when φ has a kink form, that is, it obeys the (0+1)-dimensional Sine-Gordon equation

where γ=const and

In this case W obeys the equation

We will solve this equation for the case (91) and H = α + βe mN . There, y satisfies y 2 +δy+σ = (12µη-8µ)y 2 +(12µη-16αµ-η κ 2 )y-(u+8µα 2 ) = 0,(98)


Brian P Dolan

Chiral fermions and torsion in the early Universe

Keywords: 04.20.-q, 04.20.Cv, 11.30.Rd torsion, chiral fermions, early Universe, Einstein-Cartan theory

come

It is often the case that quantum matter acts as a source for a classical field in situations where quantum aspects of the field itself can be ignored. This approximation has proven extremely useful for Einstein's equations where

G ab = 8πG < T ab >, (1)
works well when the matter source is degenerate fermionic matter, where <> is a quantum expectation value, and for thermal radiation, where <> is a thermal average of photons. There are difficulties with this approach however, not least that the singularities inherent in fully fledged quantum field theory for the sources render (1) ambiguous and some criterion for cutting off the integrals must be introduced. For example it is well known that a naïve calculation of the vacuum energy density of the standard model of particle physics leads to far too high a value of the cosmological constant to be compatible with observations [1]. Nevertheless (1) seems to work well in the early Universe when the dynamics is dominated by radiation, as long as temperatures are well below the Planck temperatures. In the radiation dominated Universe Einstein's equations boil down to the Friedmann-Robertson-Walker (FRW) equation, ignoring spatial curvature this is ȧ2

a 2 = 8πG 3 < T 00 >= N ef f 4π 3 45
T 4 m 2 P l (2)
where a(t) is the cosmological scale factor and N ef f is the effective number of degrees of freedom in the relativistic gas,

N ef f := N B + 7 8 (N + + N -) ( 3 )
with N B the number of bosonic degrees of freedom (2 for photons), N + and N -are the number of positive chirality and negative chirality fermionic degrees of freedom respectively (for a standard model neutrino N + = 2, N -= 0; for a Dirac fermion N + = N -= 2), [START_REF] Weinberg | Gravitation and Cosmology[END_REF]. The Planck mass, m 2 P l = G -1 (we use units with h = c = 1), appears in ( 2) not because we are considering a theory of quantum gravity but because of the quantum nature of the source for classical gravity.

In the Einstein-Cartan formulation of general relativity fermionic matter is expected to induce torsion (recent bounds on the magnitude of torsion have been derived from tests of violation of Lorentz invariance [START_REF] Kostelecký | [END_REF] and from cosmic microwave polarization [START_REF] Das | Constraints on background torsion from birefringence of CMB polarization[END_REF]). When the connection is varied in the Einstein-Cartan action the torsion twoforms τ a = 1 2 τ a bc e b ∧ e c are determined by a spinor field Ψ via the algebraic equation

τ a = 2πG abcd (Ψγ 5 γ d Ψ)e b ∧ e c , (4)
a, b, c, . . . are orthonormal indices (for a review of torsion in Einstein-Cartan formulation in general see [START_REF] Hehl | [END_REF] and [6]).

In the spirit of (1) the equation of motion (4) would be interpreted as

τ a,bc = 4πG abcd < Ψγ 5 γ d Ψ > . (5)
As is well known fermions generate torsion in the antisymmetric class of tensors, according to the classification of [7].

We shall examine the effect of torsion arising from relativistic fermions in the early Universe, assuming isotropy and spatial homogeneity of both the geometry and the matter. It will be assumed that the metric of Robertson-Walker type and that the energy-momentum is of the form

T ab = ρ 0 0 p δ ij (6)
Confidential: not for distribution. Submitted to IOP Publishing for peer review 24 February 2010

where the density ρ and pressure p are homogeneous and depend only on time and i, j = 1, 2, 3 are space-like indices.

The Riemann tensor involves the square of the connection and the net effect of including the torsion (4) into the gravitational connection is that Einstein's equations are modified to

3 ȧ2 a 2 - τ 2 4 = 8πGρ (7) - 2ä a - ȧ2 a 2 + τ 2 4 = 8πG 3 ρ, (8)
where a(t) is the Robertson-Walker scale factor and

τ 2 = -τ a τ a = -16π 2 G 2 (Ψγ 5 γ a Ψ)(Ψγ 5 γ a Ψ) (9)
(the metric signature is (-, +, +, +)). Eliminating τ from ( 7) and [START_REF] Watanabe | Dirac-field model of inflation in Einstein-Cartan theory[END_REF] gives

ä a = - 8πG 3 ρ. ( 10
)
We assume in (8) that the pressure p = ρ 3 for relativistic matter, or equivalently that the energymomentum tensor has zero trace, as befits highly relativistic fermions, and we ignore any possible spatial curvature. The Clifford algebra convention is {γ a , γ b } = -2η ab , with γ 0 hermitian. Equations ( 7) and ( 8) are are not independent and are related by the (first) Bianchi identity. We shall see below that this has important implications for the form of ρ.

Since fermions constitute quantum matter, it seems natural to interpret (4) in the early Universe as meaning a thermal average [START_REF] Hehl | [END_REF]. However there is an ambiguity as to whether (9) should be interpreted using

< (Ψγ 5 γ a Ψ)(Ψγ 5 γ a Ψ) > (11)
or

< Ψγ 5 γ a Ψ >< Ψγ 5 γ a Ψ > . (12)
These are different in general. The former can be Fierz re-arranged to give

< (Ψγ 5 γ a Ψ)(Ψγ 5 γ a Ψ) > = 4 < (Ψ † + Ψ -)(Ψ † -Ψ + ) > (13)
where Ψ + and Ψ -are the positive and negative chirality components of Ψ. This is always positive definite for Dirac spinors and vanishes for Weyl spinors [START_REF] Hehl | [END_REF], hence τ 2 ≤ 0 in [START_REF] Nurgaliev | [END_REF], making the torsion space-like (in the classification of [7] this is denoted As). The cosmological consequences of this formulation in inflationary models are explored in [START_REF] Watanabe | Dirac-field model of inflation in Einstein-Cartan theory[END_REF]. The same philosophy, applied to spin densities rather than the pseudo-vector Ψγ 5 γ a Ψ, is followed in [START_REF] Nurgaliev | [END_REF] and [10].

We reach a radically different conclusion if we use (12), which follows from taking the thermal average of (4) before calculating the Riemann tensor. Applying the usual Robertson-Walker assumptions of spatial homogeneity and isotropy to the connection, and hence the torsion, we would conclude that, in the cosmic frame,

< Ψγ 5 γ i Ψ >= 0, (14)
while

< Ψγ 5 γ 0 Ψ >= n --n + (15)
where n + and n -are the number density of positive and negative chirality fermions respectively. In this interpretation

τ 2 = 16π 2 G 2 (n + -n -) 2 (16)
is positive in any chiral model for matter with n + = n -. We therefore define

τ = 4πG(n + -n -), ( 17
)
in terms of which the non-vanishing components of the torsion are

τ i,jk = ijk τ,
(in the classification of [7] this is time-like, At). Rotational invariance of the thermal average is not incompatible with the conclusion of [11], where classical solutions of the Weyl equation were analyzed in spherical symmetric space-times with torsion -thermal averages do not necessarily have the same symmetries as solutions of the equations of motion. The general form of the torsion compatible with Robertson-Walker symmetries was given in [12]. The fact that chiral fermions can have interesting consequences when torsion is taken into consideration was noted in the context of anomalies for lepton currents in the Standard Model of particle physics in [13]. Both (11) and ( 12) have interesting, though very different, cosmological consequences. The form (11), being the square of a vector, has a dual description as the square of a 3-form and as such is in the class of models described in [START_REF]Inflation and dark energy from three-forms[END_REF]. Indeed a term of this form is present in the Landau-Ginsparg models discussed in [START_REF]Inflation and dark energy from three-forms[END_REF], though the stabilising quartic term is absent and there is no kinetic term here. A kinetic term would require time derivatives of the torsion and so would go beyond Einstein-Cartan theorysuch terms would be expected to appear in an effective action description of gravity involving higher derivatives and powers of the Riemann tensor but we shall focus here on (11). The form (12) was explored in a cosmological context in [START_REF] Watanabe | Dirac-field model of inflation in Einstein-Cartan theory[END_REF].

So which should one use (11) or (12)? Weinberg [START_REF] Weinberg | [END_REF] takes the point of view that there is nothing special about torsion: it is just another tensor and one can always move it to the right hand side of Einstein's equations and consider it to be part of the matter rather than part of the geometry. We see here that, in the context of (1), there is an ambiguity. If the torsion terms are absorbed into the energy momentum tensor before expectation values are taken then it would seem that (11) is appropriate. In the Einstein-Cartan formulation however the torsion is determined by the equation of motion [START_REF] Das | Constraints on background torsion from birefringence of CMB polarization[END_REF], in which the square of the torsion does not appear. If the gravitational field itself is not quantised, it is hard to see any interpretation of the equation of motion (4) other than (11). When the Riemann tensor is calculated it is then (12) that arises and not (11). Much of the literature has focused on (11) however so in this letter the consequences of ( 5) and ( 12) will be explored and developed.

We shall see that, in the context of the early Universe, thermal number densities of chiral fermions can in principle, remove the initial singularity. The mechanism here is different to torsion induced avoidance of the initial singularity due to spin fluids considered previously, [16][17][START_REF] De Berredo-Piexoto | On the torsion effects of a relativistic spin fluid in early cosmology[END_REF][START_REF] Gasperini | Repulsive gravity in the very early Universe[END_REF][START_REF] Garcia De Andrade | Spin-Torsion in Chaotic Inflation[END_REF], in which the spin density necessarily breaks either rotational or translation invariance.

In a thermal state the number density depends on temperature, when the temperature is large enough all fermions are relativistic and

n ± (T ) = 3 4 ζ(3) π 2 N ± T 3 , (18)
where

ζ(3) = ∞ p=1 1 p 3 ≈ 1.
202 is the Riemann ζfunction. For a model with N + positive chirality degrees of freedom and N -negative chirality degrees of freedom

τ = 3ζ(3) π (N + -N -) T 3 m 2 P l := A T 3 m 2 P l , (19)
where A = 3ζ(3) π (N + -N -). When there is torsion the Bianchi identity does not require that G ab be co-variantly constant, in general one has

∇ b G ba = -τ c bc G ba + 1 2 R adbc τ d,bc , (20)
where R adbc := 1 4 ada d R a d b c bcb c . In the case of Friedmann-Robertson-Walker (FRW) Universes under study here only the second term on the right hand side contributes giving

∇ b G b0 = - 3 2 τ a d dt (τa), ∇ b G bi = 0. ( 21
)
One strategy is to demand ∇ b G ba = 0 and use this to determine the torsion, implying that τ ∝ 1/a [START_REF] Minkowski | [END_REF], but this is too restrictive for our purposes. Instead we take thermodynamic averages as above and use [START_REF] Gasperini | Repulsive gravity in the very early Universe[END_REF] for the form of the torsion. Assuming adiabatic expansion of the Universe requires that T is inversely proportional to the cosmological scale factor a, T ∝ 1/a, [START_REF] Weinberg | Gravitation and Cosmology[END_REF]. The scale factor a can then be eliminated from Einstein's equations in favour of T to give

Ṫ 2 T 2 = 8π 3 ρ m 2 P l + A 2 4 T 6 m 4 P l (22)
T

T = 8π ρ m 2 P l + A 2 2 T 6 m 4 P l . ( 23
)
The behaviour τ 2 ∼ 1/a 6 for torsion arising from spin in the early Universe has been studied before, [START_REF] Gasperini | Repulsive gravity in the very early Universe[END_REF][START_REF] Garcia De Andrade | Spin-Torsion in Chaotic Inflation[END_REF] -what is new in the discussion here is that the torsion arising from thermal fermions is only non-zero in chiral theories.

The final ingredient that we need to derive the FRW equation is the equation of state relating ρ and T . For a relativistic gas consisting of different particle species the thermal energy-density is

ρ 0 = π 2 30 N ef f T 4 . ( 24
)
However ( 24) is incompatible with ( 22) and ( 23): ( 24) is inconsistent with the Bianchi identity and the assumption of adiabaticity. We can keep the assumption of adiabaticity by modifying ρ to

ρ = 1 8π BT 4 + C T 6 m 2 P l , (25)
with B = 4π 3 15 N ef f . Then ( 22) and ( 23) are consistent with (25) if and only if

C = - 3 2 A 2 . ( 26
)
Using this in (22) we derive the FRW equation with torsion

Ṫ 2 T 2 = N ef f 4π 3 45 T 4 m 2 P l - 9ζ(3) 2 4π 2 (N + -N -) 2 T 6 m 4 P l (27)
while ρ in (25) must be

ρ = π 2 30 N ef f T 4 - 27ζ(3) 2 16π 3 (N + -N -) 2 T 6 m 2 P l . (28)
At temperatures much less than the Planck temperature T P l ≈ 10 32 K the torsion term can only be physically relevant if (N + -N -) 2 >> N ef f . When the temperature approaches the Planck temperature we expect quantum gravity effects to become important and the classical FRW-equation will break down, so we must restrict our analysis of (27) to T << T P l . For the standard model of particle physics, N ef f = 106.75 while (N + -N -) 2 = (3 × 2) 2 = 36, so, even at T = m P l the second term on the right hand side of (27) only contributes 4%: the torsion is never relevant in any regime where (27) can be trusted. The effect of torsion is even weaker if right-handed neutrinos exist, as implied by recent neutrino experiments [22].

However little is known about the fundamental particle content of dark matter -if dark matter is highly chiral, with (N + -N -) 2 >> N ef f , then the T 6 term in (27) could become important while the dynamics of the early Universe is still governed by classical gravity.

In the regime where (27) can be trusted the last term in (28) makes a negative contribution to the energy density which ameliorates the singularity of the Big Bang and can avoid it completely. Equation ( 27) implies a maximum temperature given by

T 2 max = 16π 5 405ζ(3) 2 N ef f (N + -N -) 2 m 2 P l , (29)
at which Ṫ = 0. If the co-efficient is such that T max << m P l the classical FRW equation can be used and the Universe could have started expanding from finite temperature at t = 0 with ȧ = 0 and ä > 0, since at this temperature ρ in (10) is negative. Such a boundary condition is most natural in the context of bouncing cosmologies [START_REF] Gasperini | Repulsive gravity in the very early Universe[END_REF]23].

It is unlikely that the effects above would have any relevance at temperatures corresponding to grand unified energies, unless there is a huge imbalance between N ef f and (N + -N -) 2 . If there was a period of inflation when the temperature was around 10 16 GeV then the torsion discussed here would have a negligible contribution unless (N + -N -) ≈ 10 7 N ef f which seems unlikely. The above discussion is likely to be at most only relevant to the period before inflation. If the inflation picture is correct then any physical effects due to torsion arising from chiral fermions are likely to be diluted by inflation to the point where they would not be directly observable at the present day.

At first sight if might seem disconcerting that energymomentum does not appear to be conserved in this formalism -because of ( 21) and the Einstein equations T ab cannot be co-variantly constant unless aτ is constant. However an "improved" energy-momentum tensor, which is conserved, can be defined. We make the co-variant decomposition of the Einstein tensor

G ab = 0 Gab + ∆G ab (30)
where 0 Gab is the Einstein tensor constructed from the torsion-free connection. We similarly decompose the connection one-forms as

An "improved" energy-momentum tensor can be defined

T ab := T ab - 1 8πG ∆G ab (36)
which is conserved using the torsion free connection,

0 ∇b T ba = 0. ( 37
)
In terms of temperature the improved energymomentum tensor for FRW space-time with torsion is

T ab = ρ 0 0 0 ρ0 3 δ ij - 27ζ(3) 2 32π 3 (N + -N -) 2 T 6 m 2 P l δ ab (38)
with ρ 0 given in (24). In fact both terms in (38) are separately conserved with the torsion-free connection. Finally we observe that the geometrical significance of non-zero τ follows from the anti-symmetrised action of two co-variant derivatives on an arbitrary vector field with components U a ,

[∇ a , ∇ b ]U c = -τ d ab ∇ d U c + R c dab U d . (39)
In addition to the algebraic (rotation) term involving the Riemmann tensor there is a derivative term involving the torsion -a deficit displacement implying that parallelograms generated by parallel transport do not close. The deficit displacement in Robertson-Walker space-time described here is compatible with 3-dimensional rotational symmetry -a vector field with Robertson-Walker symmetries must have U i = 0 and U 0 independent of position, in which case

[∇ i , ∇ j ]U 0 = -τ ij k ∇ k U 0 . (40)
Space-like parallelograms do not close in FRW space with the torsion studied here. The author wishes to thank the Perimeter Institute, Waterloo, Canada, and the Dublin Institute for Advanced Studies for financial support.

0

  the torsion-free connection. Expanding ∆ω a b = ∆ω a b,c e c the components ∆ω a b,c , being the difference of two connections, constitute a tensor field so (31) is again a co-variant decomposition. Gab is the zero torsion Einstein tensor for which the first Bianchi identity implies b G ba = 0 ∇b ∆G ba + ∆ω b c,b G ca + ∆ω a c,b G bc . (33) We also have, by definition, ∇ b T ba = 0 ∇b T ba + ∆ω b c,b T ca + ∆ω a c,b T bc (34) for T ab . Einstein equations, G ab = 8πGT ab , now imply 0 ∇b ∆G ba = 8πG 0 ∇b T ba .