Reconstruction of some cosmological models in f(R,T) gravity

Abstract

In this paper, we reconstruct cosmological models in the framework of
$f(R,T)$ gravity, where $R$ is the Ricci scalar and $T$ is the trace of the
stress-energy tensor. We show that the dust fluid reproduces $\Lambda $CDM,
phantom-non-phantom era and the phantom cosmology. Further, we reconstruct
different cosmological models including, Chaplygin gas, scalar field with some
specific forms of $f(R,T)$. Our numerical simulation for Hubble parameter shows
good agreement with the BAO observational data for low redshifts $z<2$.

Full text

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arXiv:1107.5807v1 [physics.gen-ph] 28 Jul 2011
,
Dust f(R,T) Models
D. Momeni,1, ∗Mubasher Jamil,2, †and Ratbay Myrzakulov3, ‡
1Department of Physics, Faculty of sciences,
Tarbiat Moa’llem university, Tehran, Iran
2Center for Advanced Mathematics and Physics,
National University of Sciences and Technology, H-12, Islamabad, Pakistan
3Eurasian International Center for Theoretical Physics,
Eurasian National University, Astana 010008, Kazakhstan
Abstract
In this Letter, we reconstruct cosmological models in the frame of the new model
f(R,T), where R is the Ricci scalar and T is the trace of the stress-energy ten-
sor. This model was recently proposed by [T. Harko, F.S.N. Lobo, S. Nojiri, S.D.
Odintsov, Phys. Rev.D 84, 024020 (2011)]. We show that the dust fluid re-
produce ΛCDM, phantom-non phantom era and also the phantom cosmology with
some specified forms of the action f(R,T). We also considered the important case
f(R,T) = f1(R) + f2(T).
∗Electronic address: d.momeni@tmu.ac.ir,d.momeni@yahoo.com
†Electronic address: mjamil@camp.nust.edu.pk
‡Electronic address: rmyrzakulov@gmail.com; rmyrzakulov@csufresno.edu

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I.INTRODUCTION
From the cosmological observational data [1, 2], it is now well established that the present
observable Universe is undergoing an accelerating period. While the source driving this
acceleration acceleration is known as ‘dark energy’ its origin has not been identified yet. This
acceleration is driven by the negative pressure of the dark energy. Although a ‘cosmological
constant’ is the most simple candidate for explaining cosmic acceleration, there has been
significant development in the construction of dark energy models by the modification of the
geometrical part of the Einstein equations. This method is called as the Modified Gravity
can successfully explain the rotation curves of galaxies, the motion of galaxy clusters, the
Bullet Cluster, and cosmological observations without the use of dark matter or Einstein’s
cosmological constant [3–5]. The f(R) theories where R is the Ricci scalar, can produce
inflation, mimic behavior of dark matter and current cosmic acceleration, being compatible
with the observational data [6–9] (also see a recent review [10] on f(R) gravity and its
cosmological implications ).
In a recent paper [11], the authors considered a generalized gravity model f(R,T), T
being the trace of stress-energy tensor, with an arbitrary coupling between matter and
geometry. By choosing different functional forms of f, they solved the dynamical equations
to find solutions of astrophysical and cosmological interest. In the present work, we study
the previous model by taking the energy source as purely dust having zero pressure. By
constructing the governing field equation, we solve it to reconstruct the Lambda-Cold Dark
Matter (ΛCDM) model and a universe filled with phantom and non-phantom matter. Finally
we conclude.
II. GENERAL EQUATION OF MOTION FOR f(R,T)
The action of the f(R,T) has the form [11]
S =
?
d4x√−g[f(R,T) + Lm], (1)
where f(R,T) is an arbitrary function of the scalar curvature R and the trace T = Tµ
µof
the energy-momentum tensor Tµν. We define the matter Lagrangian density Lmby
Tµν= −
2
√−g
δ(√−gLm)
δgµν
. (2)

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Note that we can rewrite the action (1) as [4]
S =
?
d4√−g[P1(φ) + P2(φ)T + Q(φ) + Lm], (3)
where P1, P2and Q are proper functions of the auxiliary scalar field φ. By varying this
action with respect to φ, we obtain
P
′
1R + P
′
2T + Q
′= 0. (4)
It has the solution
φ = φ(R,T).(5)
Substituting (5) in (3), we obtain
f(R,T) = P1(φ(R,T)) + P2(φ(R,T))T + Q(φ(R,T)). (6)
Following [11], the equation of motion (EOM) for a dust fluid (p = 0,T = ρ,Tµν= Tuµuν)
is
fR(R,T)
?
Rµν−1
3Rgµν
?
+1
6f(R,T)gµν= T(8π + fT(R,T))
?
uµuν−1
3gµν
?
. (7)
We consider the metric of a flat Friedmann-Robertson-Walker (FRW) spacetime gµν =
diag(1,−a2(t),−a2(t),−a2(t)). The 00−component of the (7) reads now
− 2H2fR(R,T) +1
6f(R,T) =2
3T(8π + fT(R,T)). (8)
In (8) if we know the functional form of the Hubble parameter H as a function of the two
independent variables R,T, then construction of the model f(R,T) is straightforward. But
always, we have a physical intuition on the form of the H(R,T). We clarify it here more1
We will following the same procedure in the next sections.
1In a FRW model, the form of the Ricci scalar is R = 6(˙H +H2). It is convent to use the new independent
variable N is called e-folding by N = log(a
a0). It can be represented in terms of the redshift z as
N = −log(1 + z). The time derivative of any quantity as
used in f(R) gravity [12]. Any cosmological era contains some specified matter fields as matter, radiation,
phantom field or a mixed of them. For example in a universe containing the matter field which is called
ΛCDM we have H2= H2
3
. It is easy to see that we can write the R in terms of the H as
H2=R
functional form f(R,T).
d
dt= H
d
dN. This useful coordinate has been
0+κ2ρ0a−3
3− 3H2
0. Thus usually, we can write H2= H2(R). Thus (8) can be integrated easily to give the

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III.COSMOLOGICAL IMPLICATIONS
We now discuss the solutions of (8) relevant in cosmological context. In the coming
sections we will show that, any cosmological epoch (radiation, matter, dark energy domi-
nated eras) can be constructed in a model of f(R,T) only with dust fluid as the source.
We mention here that the form of the action f(R,T) is not unique. For this reason we
will continue our investigation of the cosmological reconstruction for another model with
f(R,T) = f1(R) + f2(T) [11]. When we set f2(T) = 0, we recover the f(R) theory which
has been discussed in [12]. Adopting the technique of the reconstruction the f(R,T) models,
we have the following different models.
A.
ΛCDM model
In Einstein gravity, the Hubble parameter for a flat FRW model with real matter field
describes the ΛCDM model by H2= H2
0+κ2
3ρ0a−3, κ2= 8πG in units c = 1. Since
H2=R
3− 3H2
0then solving (8) we obtain
f(R,T) = −32π
3
T +
4
?
R − 9H2
0F1
?
T
R − 9H2
0
?
. (9)
The form of F1 is not unique, since it can be any arbitrary function of its argument. It
shows that that ΛCDM model can be reconstructed by a model of f(R,T) with the dust
fluid only. Try on form F1(x) = x1/4. This surprisingly shows that the LCDM model can
be constructed from a pure f(T) model without any need to a curvature term.
B.
f(R,T) reproducing the system with non-phantom and phantom matter
In Einstein gravity, a universe filled with a mixture of the phantom and non-phantom
components is governed by H2=
κ2
3(ρqa−c1+ ρpac1).Here the set of the variables ρq
(energy density of non-phantom matter) ρp, (energy density of phantom matter) c1are the
parameters of the model. In early universe when the scale factor a was small , the first term
in this equation dominates and it behaves as the universe described by the Einstein gravity
with a matter whose EoS parameter is w = −1 + c1/3 > −1 which means that the matter
field is non-phantom like. But for now era we have w = −1 −c1/3 < −1 which it means we
live in a phantom epoch of the universe.

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As a special case, c1= 4 we have H2=
a
R+ bR which is a special form discussed [12] we
have
f(R,T) = −32π
3
T + (a + bR2)
1
24bF2(T(a + bR2)−1
6b), (10)
where F2 is an arbitrary function of the arguments x = T(a + bR2)−1
24Gpκ2
0
and b =
6b. Here a =
3ρqa−c1
κ2
3ρpac1
24Gp. Again we try on form F2(x) = x1/4. This surprisingly
shows that this model can be constructed from a pure f(T) model without any need to a
0
curvature term.
C. Phantom era
In this case with H = H0e
N
H0 we have
f(R,T) = −32π
3
T + R
1
12γF3(RT−1
3γ) (11)
Here γ =
H0
3(1+4H0)and F3is an arbitrary function of the arguments x = RT−1
choose F3(x) = x−
3γ. Now we
1
12γ. This surprisingly shows that the phantom model can be constructed
from a pure f(T) model without any need to a curvature term. But if we choose F3(x) = x
1
4
the model is a class of the additive models f(R,T) = f1(R)+f2(T) which will be described
in next section.
IV.
f(R,T) = f1(R) + f2(T) MODEL: EXACT ΛCDM RECONSTRUCTIONS
Let us consider the particular case of f(R,T) gravity when
f(R,T) = f1(R) + f2(T),(12)
In this case, the equations of motion of f(R,T) gravity (1) take the form
f1RRµν−1
2f1gµν+ (gµν
− ∇µ∇ν)f1R=
?1
2− f2T
?
Tµν− f2TΘµν+1
2f2gµν.(13)
For a perfect fluid case, this equation becomes
f1RRµν−1
2f1gµν+ (gµν
− ∇µ∇ν)f1R=
?1
2+ f2T
?
Tµν+
?
f2Tp +1
2f2
?
gµν.(14)
Alternatively as
f1RRµν−1
2f1gµν+ (gµν
− ∇µ∇ν)f1R= tµν,(15)

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where
tµν=
?1
2+ f2T
?
Tµν+
?
f2Tp +1
2f2
?
gµν. (16)
For ΛCDM model, the Hubble parameter can be written as
H =
?
ρ0
3a3+Λ
3, (17)
where ρ0 ≥ 0 is the matter density (which consists of the observed and the cold dark
matter) and Λ = 3H2
0is the cosmological constant. For dust fluid solutions of (15) we set
p = 0,T = ρ,Tµν= Tuµuν. Thus the 00-component of the (15) is written as
2f1R(˙H + H2) + (
− ∇0∇0)f1R=
?1
2+ f2T
?
T +1
2(f1+ f2). (18)
For ΛCDM system we have
?
−R
3+ 6H2
0
?
f1R+ (
− ∇0∇0)f1R=
?1
2+ f2T
?
T +1
2(f1+ f2). (19)
For a flat FRW ΛCDM system, (
− ∇0∇0)f1R= −3f1RR(R − 9H2
0)(R − 12H2
0), Thus we
have
?
−R
3+ 6H2
0
?
f1R− 3(R − 9H2
0)(R − 12H2
0)f1RR=
?1
2+ f2T
?
T +1
2
?
f1+ f2
?
. (20)
The solution for (20) is
f2(T) =c1
2−T
6+c2
?−4
9
?2
T,
−1
9−1
(21)
f1(R) =
2F1
18
√10,−4
9
+1
18
√10;1
3;1
3;1
3
−9H02+ R
H02
−9H02+ R
H02
?
C2+?−18H02+ 2R?2/3
C1− 2c1.×2F1
18
√10,1
18
√10 +2
9;53
?
(22)
Here c1is a constant and the set of the constants {C1,C2,c2} must be determined from the
possible boundary conditions.
V. REALISTIC MODELS OF f(R,T) = f1(r) + f2(T)
If we want to have a viable model of modified gravity, such as f(R) or now, f(R,T) it must
be in good agreement with cosmological observations and solar system tests. For example
it must have a physically reasonable weak field limit, in comparison to the Einstein gravity.

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These models are such so called the realistic models and have been discussed previously
in the context of the f(R) gravity. If we search for Realistic f(R,T) models [13] which
describes the LCDM models that be additive in the form f(R,T) = f1(R) + f2(T) and it’s
f(R) part be realistic, i.e. be comparable and be in a good agreement with a definitive
accuracy with cosmological observations , we assume that the R dependence part of the
model is f1(R) = a + bRmwhere a,b are constants and m is near 2 for a large class of
models. Now, we replace the general form f(R,T) = a + bRm+ g(T) in field equation
(8) with H2=
R
3− 3H2
0in the regime of LCDM. But this simple assumption leads to a
contradiction with the T dependence part of the function. Indeed only if we take R << R0,
the solution given in (22) converts to f(R,T) = a+bRm+f2(T). Thus we have such realistic
models. Another interesting model may be exponential model of R dependent part, which
has been investigated in context of the f(R) theory [13].
VI.EINSTEIN STATIC UNIVERSE IN f(R,T) MODEL
If we live in a universe filled by dust and the scale factor increases exponentially with time
as a(t) = a0eH0t, then Hubble parameter is constant. Such model which has been proposed
firstly by Einstein is called the Einstein static universe. In this solution, it is assumed that
the only source of matter filling the universe is dust. Thus it is contained in our treatment
on dust formation. For a static Einstein universe with H = H0, equation (8) gives the
following explicit form
f(R,T) = −32
3πT + e
R
12H02Φ
?
Te
−
R
3H02?
. (23)
Here Φ(x) is an arbitrary function. It is very interesting that if we choose the function
Φ(x) = x
1
4 then it is proved that the static Einstein universe is a solution of pure f(T)
gravity. It means that we can perform our analysis on Einstein universe in a model that in
it the curvature is absent. The explicit form of such a model is
f(R,T) = −32
3Tπ + αT
1
4. (24)
VII. CONCLUSION
We investigated the f(R,T) model by considering the source term as purely dust. By
constructing the governing field equation, we solved it to reconstruct the Lambda-Cold Dark

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Matter (ΛCDM) model and a universe filled with phantom and non-phantom matter. In
investigating realistic and viable ΛCDM models of the form f(R,T) = f1(R) + f2(T) [13],
we assumed that the R dependence part of the model is f1(R) = a + bRm. We found that
there is only one possible solution of this equation for g(T) which is valid only in the regime
R << R0, and it is just the second order power series term of (22). Further we showed
that the static Einstein universe can be addressed in a pure f(T) model completely in the
absence of any curvature term. Also, we can write purely T dependent function for all the
cosmological eras. It is easy to show that in all the epoches choose the arbitrary function as
Fi(x) = x
1
4 reduces the function f(R,T) to the form of pure f(T) gravity.
Acknowledgement
The authors would like to thank Sergei Odintsov for giving useful on this work.
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