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

0.18 MB·English

by  Emilio Elizalde

Λ
CDM epoch reconstruction from F(R,G) and
modified Gauss-Bonnet gravities
E Elizalde1, R Myrzakulov2, V V Obukhov3 and D
0
1 S´aez-G´omez1
0
2 1Consejo Superior de Investigaciones Cient´ıficas ICE/CSIC-IEEC,Campus UAB,

n Facultat de Ci`encies, Torre C5-Parell-2apl, E-08193 Bellaterra (Barcelona) Spain
a 2Dept. Gen. Theor. Phys., Eurasian National University, Astana, 010008,
J
Kazakhstan
0 3Tomsk State PedagogicalUniversity, Tomsk, Russia
2
E-mail: [email protected], [email protected]
]
c
q
Abstract. DarkenergycosmologyisconsideredinamodifiedGauss-Bonnetmodelof
-
r gravitywithandwithoutascalarfield. ItisshownthatthesegeneralizationsofGeneral
g
[ Relativity endow it with a very rich cosmological structure: it may naturally lead to

an effective cosmologicalconstant, quintessence or phantom cosmic acceleration, with
1
v the possibility to describe the transition from a decelerating to an accelerating phase
6 explicitly. It is demonstrated here that these modified GB and scalar-GB theories are
3
perfectly viable as cosmological models. They can describe the ΛCDM cosmological
6
3 era without any need for a cosmologicalconstant. Specific properties of these theories
. of gravity in different particular cases, such as the de Sitter one, are studied.
1
0
0
1
PACS numbers: 04.50.Kd, 95.36.+x,98.80.-k
:
v
i
X
r
a
ΛCDM epoch reconstruction from F(R,G) and modified Gauss-Bonnet gravities 2
1. 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 explained 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). 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 developed in Refs. [3]-[8]. As is
known, F(R) gravity can be written in terms of a scalar fieldquintessence or phantom
likeby 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 [8] 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. [9], [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. [9]-[25]. Both possibilities have, in principle, the capability to explain
the accelerated expansion of the Universe and even the primordial inflationary phase
(see [3] and [8]), 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. [5], the corresponding cosmological theory will be reconstructed for
several cosmological solutions.
We will here work with the spatially-flat FRW universe metric
3
ds2 = dt2 +a(t)2 (dxi)2, (1)
−
i=1
X
where a(t) is the scale factor at cosmological time t. In GR, the corresponding system
ΛCDM epoch reconstruction from F(R,G) and modified Gauss-Bonnet gravities 3
of equations are the usual Friedmann equations, namely
3 1
H2 = ρ, (2H˙ +3H2) = p, ρ˙ = 3H(ρ+p). (2)
k2 k2 − −
The last of them is the continuity equation for a perfect fluid. As usually, the Hubble
rate H is defined by H a˙/a. In (2), ρ and p are the matter energy-density and
≡
pressure. The Gauss-Bonnet invariant is given by
G = R2 4R Rµν +R Rµνλσ (3)
µν µνλσ
−
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 conclusions we will provide a summary of the main results obtained in the paper.
2. The [R+f(G)]-model
Consider the following action, which describes General Relativity plus a function of the
Gauss-Bonnet term (see Refs. [9] and [11]),
1
S = d4x√ g R+f(G)+L , (4)
− 2κ2 m
Z (cid:20) (cid:21)
where κ2 = 8πG , G being the Newton constant. By varying the action over g , the
N N µν
following field equations are obtained
1 1 1
0 = ( Rµν + gµνR)+Tµν + gµνf(G) 2f RRµν +4f RµRνρ 2f RµρστRν
2k2 − 2 2 − G G ρ − G ρστ
4f RµρσνR +2( µ νf )R 2gµν( 2f )R 4( µf )Rνρ 4( νf )Rµρ
G ρσ G G ρ G ρ G
− ∇ ∇ − ∇ − ∇ ∇ − ∇ ∇
+4( 2f )Rµν +4gµν( f )Rρσ 4( f )Rµρνσ. (5)
G ρ σ G ρ σ G
∇ ∇ ∇ − ∇ ∇
For the metric (1), these equations give the first FRW equation which has the form
3
0 = H2 +Gf f 24G˙H3f +ρ . (6)
−k2 G − − GG m
The Hubble rate H is here defined by H = a˙/a, while G and R are given by
G = 24(H˙ H2 +H4), R = 6(H˙ +2H2). (7)
ΛCDM epoch reconstruction from F(R,G) and modified Gauss-Bonnet gravities 4
The matter energy density ρ satisfies the standard continuity equation
m
ρ˙ +3H(1+w)ρ = 0. (8)
m m
Let us now rewrite Eq. (6) by using a new variable N = ln a = ln(1+z), that is, the
a0 −
number of e-foldings, instead of the cosmological time t, where z is the redshift (this
method has been implemented in Ref. [5] for f(R) gravity). The following expressions
are then easily obtained
dN d d
a = a eN, H = N˙ = , = H ,
0
dt dt dN
d2 d2 d dH
= H2 +HH , H = . (9)
′ ′
dt2 dN2 dN dN
Eq. (6) can thus be expressed as follows
3
0 = H2 +24H3(H +H)f f
−k2 ′ G −
576H6 HH +3H 2 +4HH f +ρ , (10)
′′ ′ ′ GG m
−
where G and R are now(cid:16) (cid:17)
G = 24(H3H +H4), G˙ = 24(H4H +3H3H 2 +4H4H ),
′ ′′ ′ ′
R = 6(HH +2H2). (11)
′
By introducing a new function g as g = H2, we have
1 1 1
H = √g, H = g 1/2g , H = g 3/2g2 + g 1/2g . (12)
′ − ′ ′′ − ′ − ′′
2 −4 2
Hence, Eq. (10) takes the form
3 1 1
0 = g+12g(g+2g)f f 242g g2g + gg2+2g2g f +ρ ,(13)
−k2 ′ G− − 2 ′′ 2 ′ ′ GG m
(cid:20) (cid:21)
where we have used the expressions
G = 12gg +24g2, G˙ = 12g 1/2[g2g +gg2+4g2g ] ,
′ − ′′ ′ ′
R = 3g +12g. (14)
′
Finally, we can write the FRW equations in a slightly different form [12]
3 1
H2 = ρ , (2H˙ +3H2) = p , (15)
k2 eff k2 − eff
where the effective energy and pressure densities are
p = w ρ , ρ˙ = 3H(ρ +p ), (16)
eff eff eff eff eff eff
−
with
ρ = ρ +ρ , p = p +p . (17)
eff G m eff G m
Here
ρ = Gf f 24H3G˙f ,
G G GG
− −
p = ρ +8H2G˙2f
G G GGG
−
...
192f (4H6H˙ 8H3H˙ H¨ 6H2H˙ 3 H4 H 3H5H¨ 18H4H˙ 2). (18)
GG
− − − − − −
As it was pointed out in Ref. [12], for de Sitter space p = w ρ = ρ , that is
G G G G
−
w = 1.
G
−
ΛCDM epoch reconstruction from F(R,G) and modified Gauss-Bonnet gravities 5
2.1. Example 1
As a first example we consider the ΛCDM-model. As 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. For the ΛCDM case, we have
k2ρ k2ρ
g = H2 = H2 + 0a 3 = H2 + 0a 3e 3N = H2 +lz, (19)
0 3 − 0 3 −0 − 0
where l = k2ρ0a−03, z = e 3N. Hence, we get
3 −
′
z = 3z, g = 3lz, g = 9lz. (20)
′ ′′
− −
Finally, for G we obtain the expressions
G = 24H4+12H2lz 12l2z2 , G˙ = 3zHW = 3Hz(12H2l 24l2z),(21)
0 0 − − − 0 −
where W = 12H2l 24l2z. We can reverse Eq. (21) and the number of e-foldings can
0 −
be written as a function of the Gauss-Bonnet term. This yields
3H2 81H4 3G 1 6l
z = 0 ± 0 − = e 3N N = ln . (22)
q6l − → 3 3H2 81H4 3G
0 ± 0 −
Hence, it follows that q
3H2 81H4 3G 9H2 81H4 3G
g = H2 = H2 + 0 ± 0 − = 0 ± 0 − . (23)
0 q6 q6
Finally, for the function f(z) we get the following equation
a f +a f +a f +b = 0, (24)
2 zz 1 z 0
where
a = w2 = 242l4z2 +242H2l3z 144H4l2,
0 − − 0 − 0
a = 12(H2+lz)[144l2z(H2 +lz)+(2H2 lz)w],
1 0 0 0 −
3
a = 72zw(H2+lz)2, b = [ρ (H2 +lz)]w2. (25)
2 0 m − k2 0
The energy density can be expressed as
ρ = Dz E, (26)
m
− −
being
83 16 20δ 4 13 5δ
D = 144HH4l3[ + ], E = 144HH4l2[ ]. (27)
0 5k2 − 9 9 0 9 − 5k2 − 9
Here δ = const. As a consequence, Eq. (24) has the following particular solution
f(z) = θz2 +ϑz +H2δ, (28)
0
where
l2 2 1 2
θ = [k 2 (δ +1)], ϑ = l[ k 2 + (δ +1)]. (29)
− −
H2 − 9 5 9
0
Thus, we have
3H2 81H4 3G 3H2 81H4 3G
f(G) = θ[ 0 ± 0 − ]2 +ϑ[ 0 ± 0 − ]+H2δ. (30)
q6l q6l 0
Weobserve thatthisfunctionreproducesexactly thesamebehavior astheΛCDMmodel
in the context of Gauss-Bonnet gravity.
ΛCDM epoch reconstruction from F(R,G) and modified Gauss-Bonnet gravities 6
2.2. Example 2
Let us now consider a second example
g = H2 = le 3N = lz. (31)
−
This case describes a time evolution given by
1
H(t) = , (32)
t t
0
−
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. In
± 6l
the present case, we have the following differential equation,
7 1 c
z2f zf + f = 0, (33)
zz z
− 6 3 − 3
where c = ρ 3 lz.
m − k2
Let us assume that c = 0, then Eq. (33) takes the form
7 1
z2f zf + f = 0. (34)
zz z
− 6 3
This equation admits the following exact solution
f1(z) = C1z2 +C2z16. (35)
As a consequence, for this second example we obtain the following model
G i√3
1 1 1
f(G) = −C112 +C2(±6l )6G12 = C1′G+C2′G12. (36)
It is now easy to see from this example 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.
2.3. Example 3
As third and last example we will now consider the case when the Hubble parameter
and the matter energy density behave as
3 96(m+1)
H = emN, ρ = b + e2mN + b e5mN, (37)
m 1 k2 5 2
where b = const. This case corresponds to the so-calledphantom behavior, what means
j
that we have an effective EoS parameter w < 1, and produces a superaccelerated
eff
−
phase that could end in some kind of future singularity (see Refs. [25] and [26]). We
first write the Hubble parameter as a function of time,
H
0
H(t) = , (38)
t t
s
−
where t , usually called the Rip time, defines the moment when the Big Rip singularity
s
occurs. We can find the f(G) function that reproduces this model by solving the
ΛCDM epoch reconstruction from F(R,G) and modified Gauss-Bonnet gravities 7
equation (10). This yields following solution
96(m+1)
F(G) = F(N) = b + b e5mN =
1 2
5
96(m+1) G
5
= b1 + b2[ ]4. (39)
5 24(m+1)
3. The F(R,G) model
Let us now consider a more general model for a kind of modified Gauss-Bonnet gravity.
This can be described by the following action
1
S = d4x√ g[ F(R,G)+L ]. (40)
− 2k2 m
Z
Varying over g the gravity field equations are obtained [12],
µν
1
0 = Tµν + gµνF(G) 2F RRµν +4F RµRνρ
2 − G G ρ
2F RµρστRν 4F RµρσνR +2( µ νF )R 2gµν( 2F )R
− G ρστ − G ρσ ∇ ∇ G − ∇ G
4( µF )Rνρ 4( νF )Rµρ +4( 2F )Rµν +4gµν( F )Rρσ
ρ G ρ G G ρ σ G
− ∇ ∇ − ∇ ∇ ∇ ∇ ∇
4( F )Rµρνσ F Rµν + µ νF gµν 2F . (41)
ρ σ G G R R
− ∇ ∇ − ∇ ∇ − ∇
In the case of a flat FRW Universe, described by the metric (1), the first FRW equation
yields
1
0 = (GF F 24H3F )+3(H˙ +H2)F 3HF +k2ρ . (42)
G Gt R Rt m
2 − − −
And from here, using the techniques developed in the above section, it is plain that
explicit F(R,G) functions can be reconstructed for given cosmological solutions.
3.1. De Sitter Solutions
As well known, thede Sitter solutionis oneof themost 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 . By inserting it in
0
the Friedmann equation (42), 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
1
0 = (G F (G ) F(G ,R ))+3H2F (R ) , (43)
2 0 G 0 − 0 0 0 R 0
being R = 12H2 and G = 24H4; we have here neglected the contribution of matter for
0 0 0 0
simplicity. As it was pointed in Refs. [7] and [8] 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 (40), so that any kind of function F(R,G) with positive real
roots for the equation (43) could in fact explain the acceleration epochs of the Universe
in exactly the same way a cosmological constant does.
ΛCDM epoch reconstruction from F(R,G) and modified Gauss-Bonnet gravities 8
3.2. Phantom dark energy
Let us now explore the cosmic evolution described by (37) in the context of the action
(40). This solution reproduces a phantom behavior, i.e. a superaccelerated expansion
that, as pointed by 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
(37) will be obtained. For simplicity, we consider the following subfamily of functions
F(R,G) = f (G)+f (R) . (44)
1 2
Correspondingly, the Friedmann equation (42) can be split into two equations, as
0 = 24H3G˙f +Gf f ,
1GG 1G 1
− −
1
0 = 3HR˙f +3(H˙ +H2)f f +κ2ρ (45)
2RR 2R 2 m
− − 2
For the example (37), that is H = emN, the Ricci scalar and the Gauss-Bonnet
terms take the following form,
G = 24(m+1)e4mN = 24(m+1)H4,
R = 6(m+2)e2mN = 6(m+2)H2 . (46)
Hence, the first equation in (45) can be written in terms of G, and this yields
m+1 m+1
G2f Gf + f = 0 . (47)
1GG 1G 1
− 4m 4m
This is an Euler equation, easy to solve, and yields
f1(G) = C1G1+m4m+1 +C2 , (48)
where C are integration constants. In the same way, for the case being considered
1,2
here, the second equation in (45), for R, takes the form
m+1 m+2 κ2(m+2)
R2f Rf + f ρ = 0 . (49)
2RR 2R 2 m
− 2m 2m − m
In absence of matter (ρ = 0) this is also an Euler equation, whose solution yields
m
f (R) = k Rµ+ +k Rµ−
2 1 2
1+ m+1 1+ (m+1)2 m+3
where µ = 2m ± 4m2 − m , (50)
± q 2
and k are integration constants. Then, the complete function F(R,G), given in (44),
1,2
is reconstructed (in absence of matter) yielding the solutions (48) and (50). The theory
(3.12) belongs to the class of models with positive and negative powers of the curvature
introduced in [27].
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 ρ , the solution is given by ρ = ρ e 3(1+wm)N. By inserting the expression for R
m m m 0 −
(46) into this solution, we get
3(1+wm)
R − 2m
ρ = ρ = . (51)
m m0
6(m+2)!
ΛCDM epoch reconstruction from F(R,G) and modified Gauss-Bonnet gravities 9
In such case the general solution for f is given by
2
f (R) = k Rµ+ +k Rµ− +kRA ,where,
2 1 2
ρ (6(m+1)) A 3(1+w )
k = κ2 m0 − and A = m . (52)
A(A 1) 1/2m − 2m
− −
Hence, we see that the solution for the Hubble parameter (37) 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 it could require numerical analysis.
4. 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
1 1
S = d4x√ g[ R ∂ φ∂µφ V(φ) ξ(φ)G+L ]. (53)
− 2k2 − 2 µ − − m
Z
An action of this kind has been proposed as a model for dark energy in Ref. [10]. It can
actually be related with modified GB gravity, as explained in Refs. [13] and [28]. The
variation of this action over the metric g gives [4]
µν
1 1 1 1
0 = ( Rµν + gµνR)+Tµν + ∂µφ∂νφ gµν∂ φ∂ρφ
k2 − 2 2 − 4 ρ
1
+ gµν( V +ξG) 2ξRRµν 4ξRµRνρ 2ξRµρστRν
2 − − − ρ − ρστ
+4ξRµρνσR +2( µ νξ)R 2gµν( 2ξ)R 4( µξ)Rνρ
ρσ ρ
∇ ∇ − ∇ − ∇ ∇
4( νξ)Rµρ +4( 2ξ)Rµν +4gµν( ξ)Rρσ +4( ξ)Rµρνσ. (54)
ρ ρ σ ρ σ
− ∇ ∇ ∇ ∇ ∇ ∇ ∇
In the FRW universe case, these equations and the equation of motion for the scalar
become [13]
3 1
0 = H2 + φ˙2 +V(φ)+24H3ξ˙+ρ , (55)
−k2 2 m
1 1
0 = (2H˙ +3H2)+ φ˙2 V(φ) 8H2ξ¨ 16H(H˙ +H2)ξ˙+p , (56)
m
k2 2 − − −
¨ ˙ ′ ′
0 = φ+3Hφ+V +ξ G. (57)
φ φ
Note that, as a consequence of the Bianchi identity, Eq. (57) is satisfied automatically.
From the Eqs. (55-57), we get [13]
2 H2
0 = H˙ +φ˙2 8a( ξ˙) . (58)
k2 − a t
For the variables φ and ξ Eqs. (55-57) 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 = φ˙2, v = ξ˙, (59)
ΛCDM epoch reconstruction from F(R,G) and modified Gauss-Bonnet gravities 10
Then, we obtain the following first order system
3 1
0 = H2 + u+V +24H3v, (60)
−k2 2
1 1
0 = (2H˙ +3H2)+ u V 8H2v˙ 16H(H˙ +H2)v, (61)
k2 2 − − −
˙
0 = u˙ +6Hu+2V +2Gv. (62)
From here it is easy to explore the stability and number of attractors of the physical
model. To do so we rewrite the system (60-62) as
1
H˙ = GH 2 H2, (63)
−
24 −
˙
u˙ = (6Hu+2V +2Gv), (64)
−
1 2 1
v˙ = [ ( GH 2 H2)+u (16HH˙ 8H3)v]. (65)
−
8H2 k2 24 − − −
Let q = u+2V. Then this can be put as
3 1
0 = H2 + q +24H3v, (66)
−k2 2
1 1
0 = (2H˙ +3H2)+ q 2V 8H2v˙ 16H(H˙ +H2)v, (67)
k2 2 − − −
0 = q˙+6Hq 12HV +2Gv, (68)
−
and for (63-65)
1
H˙ = GH 2 H2, (69)
−
24 −
q˙ = 12HV 6Hq 2Gv, (70)
− −
1 2 1
v˙ = [ ( GH 2 H2)+q 2V (16HH˙ 8H3)v]. (71)
−
8H2 k2 24 − − − −
Let us now rewrite Eqs. (55-57) in terms of the number of e-foldings N. We have
3 1
0 = H2 + H2φ2 +V(φ)+24H4ξ , (72)
−k2 2 N 1N
1 1
0 = (2HH +3H2)+ H2φ2 V(φ)
k2 N 2 N −
8H4ξ 8H(2HH +3H2)Hξ , (73)
1NN N 1N
− −
0 = H2φ φ +(H +3H2)φ2 +V +ξ G. (74)
N NN N N N 1N
If we introduce new variables
h = φ2 = H 2u, p = ξ = H 1v, (75)
N − 1N −
the system (72-74) takes the form
3 1
0 = H2 + H2h+V(φ)+24H4p, (76)
−k2 2