# Bianchi Type-I Universe with Cosmological Constant and Quadratic Equation of State inf

Authors: G. P. Singh; Binaya K. Bishi
Journal: Advances in High Energy Physics
 (2015)

---

## Abstract

This paper deals with the study of Bianchi type-I universe in the context off
(
R
,
T
) gravity. Einstein’s field equations in f
(
R
,
T
) gravity have been solved in the presence of cosmological constant Λ and quadratic equation of state (EoS) p
=
α
ρ
2
-
ρ, where α
≠
0 is a constant. Here, we have discussed two classes of f
(
R
,
T
) gravity; that is, f
(
R
,
T
)
=
R
+
2
f
(
T
) and f
(
R
,
T
)
=
f
1
(
R
)
+
f
2
(
T
). A set of models has been taken into consideration based on the plausible relation. Also, we have studied some physical and kinematical properties of the models.


---

## Body

## 1. Introduction

It is known that [1–5] in the present scenario, our universe is accelerating. However, final satisfactory explanation about physical mechanism and driving force of accelerated expansion of the universe is yet to achieve as human mind has not achieved perfection. From the modern cosmology, it is known that a point of universe is filled with dark energy. It has been addressed by various slow rolling scalar fields. It is supposed that the dark energy is responsible for producing sufficient acceleration in the late time of evolution of the universe. Thus, it is much more essential to study the fundamental nature of the dark energy and several approaches have been made to understand it. The cosmological constant is assumed to be the simplest candidate of dark energy. It is the classical correction made to Einstein’s field equation by adding cosmological constant to the field equations. The introduction of cosmological constant to Einstein’s field equation is the most efficient way of generating accelerated expansion, but it faces serious problems like fine-tuning and cosmic coincidence problem in cosmology [6, 7]. Quintessence [8], phantom [9], k-essence [10], tachyons [11], and Chaplygin gas [12] are the other representative of dark energy. However, there is no direct detection of such exotic fluids. Researchers are taking an interest in exploring dark energy due to the lack of strong evidence of existence of dark energy. Several authors (Pimentel and Diaz-Rivera [13], Singh et al. [14], Singh et al. [15], and Jamil and Debnath [16]) have discussed cosmological model with cosmological constant in different contexts.Dark energy can be explored in several ways, and modifying the geometric part of the Einstein-Hilbert action [17] is treated as the most efficient possible way. Based on its modifications, several alternative theories of gravity came into existence. Some of the modified theories of gravity are f
(
T
), f
(
R
), f
(
G
), and f
(
R
,
T
) gravity. These models are proposed to explore the dark energy and other cosmological problems. Sharif and Azeem [18] discussed the Cosmological evolution for dark energy models in f
(
T
) gravity. Jamil et al. [19] have studied the stability of the interactive models of the dark energy, matter, and radiation for a FRW model in f
(
T
) gravity. Generalized second law of thermodynamics in f
(
T
) gravity with entropy corrections has been studied by Bamba et al. [20]. In this work, they have used the power law and logarithmic corrected form of entropy for cosmological horizon and analysed the validity of the generalized second law of thermodynamics in specific scenarios of the quintessence and the phantom energy dominated eras. The f
(
R
) modified theory produces both cosmic inflation and mimic behavior of dark energy, including present cosmic acceleration [21–23]. Amendola et al. [24] have discussed the cosmologically viable conditions in f
(
R
) theory, which describe the dark energy models. Jamil et al. [25] have analysed the f
(
R
) tachyon cosmology by the Noether symmetry approach. Azadi et al. [26] have discussed the static cylindrically symmetric vacuum solutions in Weyl coordinates in the context of the metric f
(
R
) theories of gravity. This article is devoted to construct the family of solutions with constant Ricci scalar (
R
=
R
0
) explicitly and its possible relation to the Linet-Tian solution in general relativity. Momeni and Gholizade [27] discussed the constant curvature solutions in cylindrically symmetric metric f
(
R
) gravity. In this paper, they have proved that, in f
(
R
) gravity, the constant curvature solution in cylindrically symmetric cases is only one member of the most generalized Tian family in general relativity and further shown that constant curvature exact solution is applicable to the exterior of a string.The basic paper onf
(
R
,
T
) modified gravity was investigated by Harko et al. [28]. From the literature, it is found that Barrientos and Rubilar have pointed out that Harko et al. have missed an essential term, which has consequences in the equation of motion of test particles. Thus, the corrected derivation of this equation of motion is presented by Barrientos and Rubilar [29], who also discussed some of its consequences. Jamil et al. [30] have studied the reconstruction of some cosmological models in f
(
R
,
T
) gravity, in which they have shown that dust fluid reproduces ΛCDM, phantom-non-phantom era and the phantom cosmology. Jamil et al. [31] have proved that the first law of black hole thermodynamics is violated for f
(
R
,
T
) gravity in general, but there might be some special case exit in which the first law of black hole thermodynamics is recovered. Momeni et al. [32] have investigated Noether symmetry issue for nonminimally f
(
R
,
T
) model and mimetic f
(
R
). They have pointed out that Noether symmetry is able to provide a very excellent way to study cosmological implications of extended f
(
R
) theories. We have observed from the literature that Bianchi type-I model is one of the important anisotropic cosmological models and hence it is widely studied in general relativity and alternative theories of gravitation. The Bianchi type-I model is discussed by Jamil et al. [33, 34] in different contexts. Recently, authors like Sahoo and Sivakumar [35], Ahmed and Pradhan [36], and Pradhan et al. [37] have investigated the cosmological models with cosmological constant in f
(
R
,
T
) gravity for different Bianchi type space-time.Quadratic equation of state is needed to explore in cosmological models due to its importance in brane world model and the study of dark energy and general relativistic dynamics for different models. The general form of the quadratic equation of state is given by(1)
p
=
p
0
+
α
ρ
+
β
ρ
2
,where p
0, α, and β are parameters. Equation (1) is nothing but the first term of Taylor expansion of any equation of state of the form p
=
p
(
ρ
) about ρ
=
0.Nojiri and Odintsov [38] have studied the final state and thermodynamics of a dark energy universe, in which they discussed the model by considering the equation state of the form p
=
f
(
ρ
). Ananda and Bruni discussed the cosmological models by considering different form of nonlinear quadratic equation of state. Ananda and Bruni [39] have investigated the general relativistic dynamics of RW models with a nonlinear quadratic equation of state and analysed that the behaviour of the anisotropy at the singularity found in the brane scenario can be recreated in the general relativistic context by considering an equation of state of form (1). Also they have discussed the anisotropic homogeneous and inhomogeneous cosmological models in general relativity with the equation of state of the form(2)
p
=
α
ρ
+
ρ
2
ρ
c
,and they tried to isotropize the universe at early times when the initial singularity is approached. Astashenok et al. [40] have analysed phantom cosmology without big rip singularity, in which they have considered the equation of state of the form p
=
-
ρ
-
f
(
ρ
). In our present study, we have considered the quadratic equation of state of the form(3)
p
=
α
ρ
2
-
ρ
,where α
≠
0 is a constant quantity and such type of consideration does not affect the quadratic nature of equation of state.Nojiri and Odintsov [41] studied the effect of modification of general equation of state of dark energy ideal fluid by the insertion of inhomogeneous, Hubble parameter dependent term in the late-time universe. The quadratic equation of state may describe the dark energy or unified dark energy [41, 42]. Rahaman et al. [43] investigated the construction of an electromagnetic mass model using quadratic equation of state in the context of general theory of relativity. Feroze and Siddiqui [44] studied the general situation of a compact relativistic body by taking a quadratic equation of state for the matter distribution. Maharaj and Mafa Takisa [45] have investigated the regular models with quadratic equation of state. They have considered static and spherically symmetric space-time with a charged matter distribution and found new exact solutions to the Einstein-Maxwell system of equations which are physically reasonable.A cosmological model based on a quadratic equation of state unifying vacuum energy, radiation, and dark energy has been discussed by Chavanis [46] and also a cosmological model describing the early inflation, the intermediate decelerating expansion, and the late accelerating expansion by a quadratic equation of state has been investigated by the same author [47]. Strange quark star model with quadratic equation of state has been investigated by Malaver [48] and they have obtained a class of models with quadratic equation of state for the radial pressure that correspond to anisotropic compact sphere, where the gravitational potential Z depends on an adjustable parameter n. Recently, Reddy et al. [49] have studied the Bianchi type-I cosmological model with quadratic equation of state in the context of general theory of relativity.Motivated by the aforesaid research, we have investigated the Bianchi type-I cosmological model inf
(
R
,
T
) gravity with quadratic equation of state and cosmological constant. Here, we have discussed two classes of f
(
R
,
T
) gravity.


## 2. Gravitational Field Equations off
(
R
,
T
) Modified Gravity Theory

Let us consider the action for the modified gravity as(4)
S
=
∫
f
R
,
T
16
π
G
+
L
m
-
g
d
4
x
,where f
(
R
,
T
) is the arbitrary function of R and T. R is the Ricci scalar and T is the trace of the stress energy tensor of the matter T
i
j. L
m is the matter Lagrangian density. For the choice of f
(
R
,
T
), we will get the action for the different theories. If f
(
R
,
T
)
≡
f
(
R
) and f
(
R
,
T
)
≡
R, then (4) represents the action for f
(
R
) gravity and general relativity, respectively. The stress energy tensor of matter is defined as(5)
T
i
j
=
-
2
-
g
δ
-
g
L
m
δ
g
i
j
, and its stress by T
=
g
i
j
T
i
j. If we consider that the matter Lagrangian density L
m of matter depends only on g
i
j and not on its derivatives, then it will lead us to(6)
T
i
j
=
g
i
j
L
m
-
2
∂
L
m
∂
g
i
j
.By varying action (4) with respect to the metric tensor component g
i
j, we have(7)
f
R
R
,
T
R
i
j
-
1
2
f
R
,
T
g
i
j
+
g
i
j
□
-
∇
i
∇
j
f
R
R
,
T
=
8
π
-
f
T
R
,
T
T
i
j
-
f
T
R
,
T
Θ
i
j
,where(8)
Θ
i
j
=
-
2
T
i
j
+
g
i
j
L
m
-
2
g
l
k
∂
2
L
m
∂
g
i
j
∂
g
l
k
. Here, f
T
(
R
,
T
)
=
∂
f
(
R
,
T
)
/
∂
T,  f
R
(
R
,
T
)
=
∂
f
(
R
,
T
)
/
∂
R, □
≡
∇
i
∇
i is the De Alembert’s operator, and T
i
j is the standard matter energy momentum tensor derived from the Lagrangian L
m. By contracting (7), we obtained the relation between R and T as(9)
f
R
R
,
T
R
+
3
□
f
R
R
,
T
-
2
f
R
,
T
=
8
π
T
-
f
T
R
,
T
T
-
f
T
R
,
T
Θ
,where Θ
=
g
i
j
Θ
i
j. From (7) and (9), the gravitational field equations can be written as(10)
f
R
R
,
T
R
i
j
-
1
3
R
g
i
j
+
1
6
f
R
,
T
g
i
j
=
8
π
-
f
T
R
,
T
T
i
j
-
1
3
T
g
i
j
-
f
T
R
,
T
Θ
i
j
-
1
3
Θ
g
i
j
+
∇
i
∇
j
f
R
R
,
T
.The perfect fluid form of the stress energy tensor of the matter Lagrangian is given by(11)
T
i
j
=
ρ
+
p
u
i
u
j
-
p
g
i
j
,where u
i
=
(
1,0
,
0,0
) is the four-velocity vector and satisfies the relation u
i
u
i
=
1 and u
i
∇
j
u
i
=
0. ρ and p are the energy density and pressure of the fluid, respectively. From (8), we have(12)
Θ
i
j
=
-
2
T
i
j
-
p
g
i
j
.It is to note that the functional f
(
R
,
T
) depends on the physical nature of the matter field through tensor Θ
i
j. Thus, each choice of f
(
R
,
T
) leads us to different cosmological models. Harko et al. [28] presented three classes of f
(
R
,
T
) as follows: (13)
f
R
,
T
=
R
+
2
f
T
f
1
R
+
f
2
T
f
1
R
+
f
2
R
f
3
T
.In this present work, we have discussed two classes of f
(
R
,
T
); that is, f
(
R
,
T
)
=
R
+
2
f
(
T
) and f
(
R
,
T
)
=
f
1
(
R
)
+
f
2
(
T
).For the choice off
(
R
,
T
)
=
R
+
2
f
(
T
) and with the help of (11) and (12), (7) takes the form(14)
G
i
j
=
8
π
+
2
f
′
T
T
i
j
+
2
p
f
′
T
+
f
T
g
i
j
,which is the gravitational field equation in f
(
R
,
T
) modified gravity for the class f
(
R
,
T
)
=
R
+
2
f
(
T
). For the choice of f
(
R
,
T
)
=
f
1
(
R
)
+
f
2
(
T
) and with the help of (11) and (12), (7) takes the form(15)
f
1
′
R
R
i
j
-
1
2
f
1
R
g
i
j
+
g
i
j
□
-
∇
i
∇
j
f
1
′
R
=
8
π
+
f
2
′
T
T
i
j
+
f
2
′
T
p
+
1
2
f
2
T
g
i
jwhich is regarded as the gravitational field equation in f
(
R
,
T
) modified gravity for the class f
(
R
,
T
)
=
f
1
(
R
)
+
f
2
(
T
).


## 3. Field Equations and Cosmological Model forf
(
R
,
T
)
=
R
+
2
f
(
T
)

Inf
(
R
,
T
) theory, the gravitational field equation (14) in the presence of cosmological constant Λ is given as(16)
G
i
j
=
8
π
+
2
f
′
T
T
i
j
+
2
p
f
′
T
+
f
T
+
Λ
g
i
j
, where prime denotes differentiation with respect to the argument. For the choice of f
(
T
)
=
λ
T, (16) takes the form(17)
G
i
j
=
8
π
+
2
λ
T
i
j
+
λ
ρ
-
p
λ
+
Λ
g
i
j
.Let us consider the Bianchi type-I space-time in the form(18)
d
s
2
=
d
t
2
-
X
1
2
d
x
2
-
X
2
2
d
y
2
-
X
3
2
d
z
2
,where X
1, X
2, and X
3 are function of t only. The field equation (17) for the line element (18) takes the form(19)
X
˙
1
X
˙
2
X
1
X
2
+
X
˙
1
X
˙
3
X
1
X
3
+
X
˙
2
X
˙
3
X
2
X
3
=
-
8
π
+
3
λ
ρ
+
p
λ
-
Λ
(20)
X
¨
2
X
2
+
X
¨
3
X
3
+
X
˙
2
X
˙
3
X
2
X
3
=
8
π
+
3
λ
p
-
λ
ρ
-
Λ
(21)
X
¨
1
X
1
+
X
¨
3
X
3
+
X
˙
1
X
˙
3
X
1
X
3
=
8
π
+
3
λ
p
-
λ
ρ
-
Λ
(22)
X
¨
1
X
1
+
X
¨
2
X
2
+
X
˙
1
X
˙
2
X
1
X
2
=
8
π
+
3
λ
p
-
λ
ρ
-
Λ
.


## 4. Solution Procedure

Now, our problem is to solve Einstein’s modified field equations (19)–(22). Here, the system has four equations and six unknowns (X
1, X
2, X
3, p, ρ, and Λ). To obtain the complete solution, we need two more physically plausible relations. The considered two physically plausible relations are(1)
quadratic equation of state;
(2)
expansion law:
(a)
power law:(23)
V
=
V
0
t
3
n
,
(b)
exponential law:(24)
V
=
V
0
e
β
t
, where n and β are the positive constant quantity. According to the choice of expansion law, we have obtained two different models of the Bianchi type-I universe.


### 4.1. Power Law Model

With the help of (20)–(22), we have obtained the metric potentials as(25)
X
i
t
=
X
i
0
V
1
/
3
exp
⁡
∫
X
0
i
V
,
i
=
1,2
,
3
, where X
i
0 and X
0
i are constant of integration (
i
=
1,2
,
3
) which satisfies the relation ∏
i
=
1
3
X
i
0
=
1 and ∑
i
=
1
3
X
0
i
=
0. From (19)-(20) and along with (3), we have got(26)
ρ
2
=
1
α
8
π
+
2
λ
X
¨
2
X
2
+
X
¨
3
X
3
-
X
˙
1
X
˙
2
X
1
X
2
-
X
˙
1
X
˙
3
X
1
X
3
.Using (23) in (25), we have the metric potential as(27)
X
i
t
=
X
i
0
V
1
/
3
exp
⁡
-
X
0
i
t
-
3
n
+
1
3
n
-
1
V
0
,
i
=
1,2
,
3
.The directional Hubble parameters are obtained as H
i
=
n
/
t
+
X
0
i
/
V
0
t
3
n,  i
=
1,2
,
3. The Hubble parameter (
H
), deceleration parameter (
q
), expansion scalar (
Θ
), and Shear scalar (
σ
2
) are as follows: (28)
H
=
n
t
,
q
=
-
1
+
1
n
,
Θ
=
3
n
t
,
σ
2
=
X
02
2
+
X
03
2
+
X
02
X
03
V
0
2
t
6
n
. Using the observational value for q
=
-
0.33
±
0.17 [50], we have restricted n as n
∈
(
1.19,2
) in case of power law model. Here, we noticed that H,  Θ, and σ
2 die out for larger values of t. With the help of (27) from (26), the energy density is obtained as (29)
ρ
2
=
1
4
π
+
λ
α
X
02
2
+
X
03
2
+
X
02
X
03
V
0
2
1
t
6
n
-
n
t
2
.Using (29) in (3), we have the pressure as follows: (30)
p
=
X
02
2
+
X
03
2
+
X
02
X
03
t
-
6
n
+
2
-
V
0
2
n
4
π
+
λ
V
0
2
t
2
-
X
02
2
+
X
03
2
+
X
02
X
03
t
-
6
n
+
2
-
V
0
2
n
4
π
+
λ
V
0
2
α
t
2
.With the help of (27)–(30) from (19), the cosmological constant Λ is obtained as(31)
Λ
=
-
4
4
π
+
λ
V
0
2
t
6
n
+
2
2
π
+
λ
4
π
+
λ
V
0
2
t
6
n
+
2
-
X
02
2
+
X
03
2
+
X
02
X
03
t
-
6
n
+
2
-
V
0
2
n
4
π
+
λ
V
0
2
α
t
2
-
π
+
λ
2
X
02
2
+
X
03
2
+
X
02
X
03
t
2
+
1
4
3
λ
n
+
λ
+
12
n
π
V
0
2
n
t
6
n
.Figures1 and 2 show the variation of energy density ρ and pressure p against time t for different values as in the figures. Here, we noticed that ρ,  p
→
0 when t
→
∞. In the increase of n, energy density and pressure increase and decrease, respectively. Figure 3 represents the variation of cosmological constant Λ against time t for different values as in the figures. It is observed that cosmological constant Λ is decreasing with the increase of n and with the evolution of time it approaches towards zero. Variation of different energy conditions against time for different n is presented in Figure 4. We observed from the figure that DEC (dominant energy condition, ρ
-
p
≥
0) is satisfied, but NEC (null energy condition, ρ
+
p
≥
0) and SEC (strong energy condition, ρ
+
3
p
≥
0) are violated in this case. This violation may be responsible for the accelerated expansion of the universe.Figure 1
Variation of energy densityρ against time t for λ
=
1, α
=
-
0.1, V
0
=
1, X
02
=
0.01, X
03
=
0.01, and different n
(
1.2,1.4,1.6,1.8
).Figure 2
Variation of pressurep against time t for λ
=
1, α
=
-
0.1, V
0
=
1, X
02
=
0.01, X
03
=
0.01, and different n
(
1.2,1.4,1.6,1.8
).Figure 3
Variation of cosmological constantΛ against time t for λ
=
1, α
=
-
0.1, V
0
=
1, X
02
=
0.01, X
03
=
0.01, and different n
(
1.2,1.4,1.6,1.8
).Figure 4
Variation of energy conditions (ρ
+
p
≥
0, ρ
-
p
≥
0, ρ
+
3
p
≥
0) against time t for power law model.
(a)
(b)
(c)


### 4.2. Exponential Law Model

In this case, with the help of (24) in (25), we have found the metric potential as(32)
X
i
t
=
X
i
0
V
1
/
3
exp
⁡
-
-
β
2
V
0
t
+
3
X
0
i
e
-
β
t
3
β
V
0
,
i
=
1,2
,
3
.The directional Hubble parameters are obtained as H
i
=
β
/
3
+
X
0
i
/
V
0
e
β
t,  i
=
1,2
,
3. The Hubble parameter (
H
), deceleration parameter (
q
), expansion scalar (
Θ
), and Shear scalar (
σ
2
) are as follows: (33)
H
=
β
3
,
q
=
-
1
,
Θ
=
β
,
σ
2
=
X
02
2
+
X
03
2
+
X
02
X
03
V
0
2
e
2
β
t
. Here, we noticed that σ
2 die out for larger values of t. From (32) and (26), the energy density is expressed as(34)
ρ
2
=
X
02
2
+
X
03
2
+
X
02
X
03
e
-
2
β
t
V
0
2
α
4
π
+
λ
.Using (34) in (3), the pressure is expressed as(35)
p
=
1
4
π
+
λ
V
0
2
-
4
π
+
λ
V
0
2
X
02
2
+
X
03
2
+
X
02
X
03
e
-
2
β
t
V
0
2
α
4
π
+
λ
+
X
02
2
+
X
03
2
+
X
02
X
03
e
-
2
β
t
.With the help of (32)–(35) from (19), the cosmological constant Λ is obtained as(36)
Λ
=
-
8
π
+
4
λ
X
02
2
+
X
03
2
+
X
02
X
03
e
-
2
β
t
V
0
2
α
4
π
+
λ
+
2
2
π
+
λ
X
02
2
+
X
03
2
+
X
02
X
03
e
2
β
t
V
0
2
4
π
+
λ
-
β
2
3
.Figures5 and 7 show the variation of energy density ρ and pressure p against time t for different values as in the figures. Here, we noticed that ρ,  p
→
0 when t
→
∞. In the increase of β, energy density and pressure decrease, respectively. Variation of different energy conditions against time for different n is presented in Figure 6. We observed from the figure that NEC and DEC are satisfied, but in this case SEC is violated. This violation may be responsible for the accelerated expansion of the universe. Figure 8 represents the variation of cosmological constant Λ against time t for different values as in the figures. It is observed that cosmological constant Λ is not approaching towards zero with the evolution of time and also it takes negative values.Figure 5
Variation of energy densityρ against time t for λ
=
1, α
=
0.1, V
0
=
1, X
02
=
0.01, X
03
=
0.01, and different β
(
0.2,0.4,0.6,0.8,1
).Figure 6
Variation of energy conditions (ρ
+
p
≥
0, ρ
-
p
≥
0, ρ
+
3
p
≥
0) against time t for exponential law model.
(a)
(b)
(c)Figure 7
Variation of pressurep against time t for λ
=
1, α
=
0.1, V
0
=
1, X
02
=
0.01, X
03
=
0.01, and different β
(
0.2,0.4,0.6,0.8,1
).Figure 8
Variation of cosmological constantΛ against time t for λ
=
1, α
=
0.1, V
0
=
1, X
02
=
0.01, X
03
=
0.01, and different β
(
0.4,0.6,0.8,1
,
1.2
).


## 4.1. Power Law Model

With the help of (20)–(22), we have obtained the metric potentials as(25)
X
i
t
=
X
i
0
V
1
/
3
exp
⁡
∫
X
0
i
V
,
i
=
1,2
,
3
, where X
i
0 and X
0
i are constant of integration (
i
=
1,2
,
3
) which satisfies the relation ∏
i
=
1
3
X
i
0
=
1 and ∑
i
=
1
3
X
0
i
=
0. From (19)-(20) and along with (3), we have got(26)
ρ
2
=
1
α
8
π
+
2
λ
X
¨
2
X
2
+
X
¨
3
X
3
-
X
˙
1
X
˙
2
X
1
X
2
-
X
˙
1
X
˙
3
X
1
X
3
.Using (23) in (25), we have the metric potential as(27)
X
i
t
=
X
i
0
V
1
/
3
exp
⁡
-
X
0
i
t
-
3
n
+
1
3
n
-
1
V
0
,
i
=
1,2
,
3
.The directional Hubble parameters are obtained as H
i
=
n
/
t
+
X
0
i
/
V
0
t
3
n,  i
=
1,2
,
3. The Hubble parameter (
H
), deceleration parameter (
q
), expansion scalar (
Θ
), and Shear scalar (
σ
2
) are as follows: (28)
H
=
n
t
,
q
=
-
1
+
1
n
,
Θ
=
3
n
t
,
σ
2
=
X
02
2
+
X
03
2
+
X
02
X
03
V
0
2
t
6
n
. Using the observational value for q
=
-
0.33
±
0.17 [50], we have restricted n as n
∈
(
1.19,2
) in case of power law model. Here, we noticed that H,  Θ, and σ
2 die out for larger values of t. With the help of (27) from (26), the energy density is obtained as (29)
ρ
2
=
1
4
π
+
λ
α
X
02
2
+
X
03
2
+
X
02
X
03
V
0
2
1
t
6
n
-
n
t
2
.Using (29) in (3), we have the pressure as follows: (30)
p
=
X
02
2
+
X
03
2
+
X
02
X
03
t
-
6
n
+
2
-
V
0
2
n
4
π
+
λ
V
0
2
t
2
-
X
02
2
+
X
03
2
+
X
02
X
03
t
-
6
n
+
2
-
V
0
2
n
4
π
+
λ
V
0
2
α
t
2
.With the help of (27)–(30) from (19), the cosmological constant Λ is obtained as(31)
Λ
=
-
4
4
π
+
λ
V
0
2
t
6
n
+
2
2
π
+
λ
4
π
+
λ
V
0
2
t
6
n
+
2
-
X
02
2
+
X
03
2
+
X
02
X
03
t
-
6
n
+
2
-
V
0
2
n
4
π
+
λ
V
0
2
α
t
2
-
π
+
λ
2
X
02
2
+
X
03
2
+
X
02
X
03
t
2
+
1
4
3
λ
n
+
λ
+
12
n
π
V
0
2
n
t
6
n
.Figures1 and 2 show the variation of energy density ρ and pressure p against time t for different values as in the figures. Here, we noticed that ρ,  p
→
0 when t
→
∞. In the increase of n, energy density and pressure increase and decrease, respectively. Figure 3 represents the variation of cosmological constant Λ against time t for different values as in the figures. It is observed that cosmological constant Λ is decreasing with the increase of n and with the evolution of time it approaches towards zero. Variation of different energy conditions against time for different n is presented in Figure 4. We observed from the figure that DEC (dominant energy condition, ρ
-
p
≥
0) is satisfied, but NEC (null energy condition, ρ
+
p
≥
0) and SEC (strong energy condition, ρ
+
3
p
≥
0) are violated in this case. This violation may be responsible for the accelerated expansion of the universe.Figure 1
Variation of energy densityρ against time t for λ
=
1, α
=
-
0.1, V
0
=
1, X
02
=
0.01, X
03
=
0.01, and different n
(
1.2,1.4,1.6,1.8
).Figure 2
Variation of pressurep against time t for λ
=
1, α
=
-
0.1, V
0
=
1, X
02
=
0.01, X
03
=
0.01, and different n
(
1.2,1.4,1.6,1.8
).Figure 3
Variation of cosmological constantΛ against time t for λ
=
1, α
=
-
0.1, V
0
=
1, X
02
=
0.01, X
03
=
0.01, and different n
(
1.2,1.4,1.6,1.8
).Figure 4
Variation of energy conditions (ρ
+
p
≥
0, ρ
-
p
≥
0, ρ
+
3
p
≥
0) against time t for power law model.
(a)
(b)
(c)


## 4.2. Exponential Law Model

In this case, with the help of (24) in (25), we have found the metric potential as(32)
X
i
t
=
X
i
0
V
1
/
3
exp
⁡
-
-
β
2
V
0
t
+
3
X
0
i
e
-
β
t
3
β
V
0
,
i
=
1,2
,
3
.The directional Hubble parameters are obtained as H
i
=
β
/
3
+
X
0
i
/
V
0
e
β
t,  i
=
1,2
,
3. The Hubble parameter (
H
), deceleration parameter (
q
), expansion scalar (
Θ
), and Shear scalar (
σ
2
) are as follows: (33)
H
=
β
3
,
q
=
-
1
,
Θ
=
β
,
σ
2
=
X
02
2
+
X
03
2
+
X
02
X
03
V
0
2
e
2
β
t
. Here, we noticed that σ
2 die out for larger values of t. From (32) and (26), the energy density is expressed as(34)
ρ
2
=
X
02
2
+
X
03
2
+
X
02
X
03
e
-
2
β
t
V
0
2
α
4
π
+
λ
.Using (34) in (3), the pressure is expressed as(35)
p
=
1
4
π
+
λ
V
0
2
-
4
π
+
λ
V
0
2
X
02
2
+
X
03
2
+
X
02
X
03
e
-
2
β
t
V
0
2
α
4
π
+
λ
+
X
02
2
+
X
03
2
+
X
02
X
03
e
-
2
β
t
.With the help of (32)–(35) from (19), the cosmological constant Λ is obtained as(36)
Λ
=
-
8
π
+
4
λ
X
02
2
+
X
03
2
+
X
02
X
03
e
-
2
β
t
V
0
2
α
4
π
+
λ
+
2
2
π
+
λ
X
02
2
+
X
03
2
+
X
02
X
03
e
2
β
t
V
0
2
4
π
+
λ
-
β
2
3
.Figures5 and 7 show the variation of energy density ρ and pressure p against time t for different values as in the figures. Here, we noticed that ρ,  p
→
0 when t
→
∞. In the increase of β, energy density and pressure decrease, respectively. Variation of different energy conditions against time for different n is presented in Figure 6. We observed from the figure that NEC and DEC are satisfied, but in this case SEC is violated. This violation may be responsible for the accelerated expansion of the universe. Figure 8 represents the variation of cosmological constant Λ against time t for different values as in the figures. It is observed that cosmological constant Λ is not approaching towards zero with the evolution of time and also it takes negative values.Figure 5
Variation of energy densityρ against time t for λ
=
1, α
=
0.1, V
0
=
1, X
02
=
0.01, X
03
=
0.01, and different β
(
0.2,0.4,0.6,0.8,1
).Figure 6
Variation of energy conditions (ρ
+
p
≥
0, ρ
-
p
≥
0, ρ
+
3
p
≥
0) against time t for exponential law model.
(a)
(b)
(c)Figure 7
Variation of pressurep against time t for λ
=
1, α
=
0.1, V
0
=
1, X
02
=
0.01, X
03
=
0.01, and different β
(
0.2,0.4,0.6,0.8,1
).Figure 8
Variation of cosmological constantΛ against time t for λ
=
1, α
=
0.1, V
0
=
1, X
02
=
0.01, X
03
=
0.01, and different β
(
0.4,0.6,0.8,1
,
1.2
).


## 5. Field Equations and Cosmological Model forf
(
R
,
T
)
=
f
1
(
R
)
+
f
2
(
T
)

Inf
(
R
,
T
) theory, the gravitational field equation (15) for the choice of f
1
(
R
)
=
λ
R and f
2
(
T
)
=
λ
T, along with cosmological constant Λ, is given as(37)
G
i
j
=
8
π
+
λ
λ
T
i
j
+
ρ
-
p
+
2
Λ
2
g
i
j
.In this case, the field equations are given by(38)
X
˙
1
X
˙
2
X
1
X
2
+
X
˙
1
X
˙
3
X
1
X
3
+
X
˙
2
X
˙
3
X
2
X
3
=
-
16
π
+
3
λ
2
λ
ρ
+
p
2
-
Λ
X
¨
2
X
2
+
X
¨
3
X
3
+
X
˙
2
X
˙
3
X
2
X
3
=
16
π
+
3
λ
2
λ
p
-
ρ
2
-
Λ
X
¨
1
X
1
+
X
¨
3
X
3
+
X
˙
1
X
˙
3
X
1
X
3
=
16
π
+
3
λ
2
λ
p
-
ρ
2
-
Λ
X
¨
1
X
1
+
X
¨
2
X
2
+
X
˙
1
X
˙
2
X
1
X
2
=
16
π
+
3
λ
2
λ
p
-
ρ
2
-
Λ
.


### 5.1. Power Law Model

Following the same procedure as in Section4.1, we have obtained the same metric potential as in (27) and the other parameters like energy density ρ, pressure p, and cosmological constant Λ are expressed as follows:(39)
ρ
2
=
2
λ
V
0
2
α
8
π
+
λ
X
03
2
+
X
02
2
+
X
02
X
03
t
6
n
-
V
0
2
n
t
2
p
=
2
λ
V
0
2
8
π
+
λ
X
03
2
+
X
02
2
+
X
02
X
03
t
6
n
-
V
0
2
n
t
2
-
2
λ
V
0
2
α
8
π
+
λ
X
03
2
+
X
02
2
+
X
02
X
03
t
6
n
-
V
0
2
n
t
2
Λ
=
1
2
t
2
λ
8
π
+
λ
α
V
0
2
-
41
λ
4
π
+
λ
8
π
+
λ
α
V
0
t
-
3
n
+
1
2
V
0
2
n
t
6
n
-
2
t
2
X
03
2
+
X
02
2
+
X
02
X
03
+
4
4
π
+
λ
X
03
2
+
X
02
2
+
X
02
X
03
λ
α
t
-
6
n
+
2
-
2
n
3
λ
n
+
λ
+
24
n
π
V
0
2
α
λ
.Here, also we have noticed similar qualitative results as in Section4.1 (see Figures 9–12).Figure 9
Variation of energy densityρ against time t for λ
=
1, α
=
-
0.1, V
0
=
1, X
02
=
0.01, X
03
=
0.01, and different n
(
1.2,1.4,1.6,1.8
).Figure 10
Variation of pressurep against time t for λ
=
1, α
=
-
0.1, V
0
=
1, X
02
=
0.01, X
03
=
0.01, and different n
(
1.2,1.4,1.6,1.8
).Figure 11
Variation of energy conditions (ρ
+
p
≥
0, ρ
-
p
≥
0, ρ
+
3
p
≥
0) against time t for polynomial law model.
(a)
(b)
(c)Figure 12
Variation of cosmological constantΛ against time t for λ
=
1, α
=
-
0.1, V
0
=
1, X
02
=
0.01, X
03
=
0.01, and different n
(
1.2,1.4,1.6,1.8
).


### 5.2. Exponential Law Model

Following the same procedure as in Section4.2, we have obtained the same metric potential as in (34), and the other parameters like energy density ρ, pressure p, and cosmological constant Λ are as follows: (40)
ρ
=
1
V
0
e
β
t
2
X
03
2
+
X
02
2
+
X
02
X
03
λ
α
8
π
+
λ
p
=
2
X
03
2
+
X
02
2
+
X
02
X
03
λ
V
0
2
e
2
β
t
8
π
+
λ
-
1
V
0
e
β
t
2
X
03
2
+
X
02
2
+
X
02
X
03
λ
α
8
π
+
λ
Λ
=
-
2
2
4
π
+
λ
λ
2
X
03
2
+
X
02
2
+
X
02
X
03
λ
e
-
2
β
t
V
0
2
α
8
π
+
λ
+
2
X
03
2
+
X
02
2
+
X
02
X
03
4
π
+
λ
e
-
2
β
t
V
0
2
8
π
+
λ
-
β
2
3
.Here, also we have noticed the similar qualitative results as in Section4.2 (see Figures 13–16).Figure 13
Variation of energy densityρ against time t for λ
=
1, α
=
0.1, V
0
=
1, X
02
=
0.01, X
03
=
0.01, and different β
(
0.2,0.4,0.6,0.8,1
,
1.2
).Figure 14
Variation of pressurep against time t for λ
=
1, α
=
0.1, V
0
=
1, X
02
=
0.01, X
03
=
0.01, and different β
(
0.2,0.4,0.6,0.8,1
,
1.2
).Figure 15
Variation of cosmological constantΛ against time t for λ
=
1, α
=
0.1, V
0
=
1, X
02
=
0.01, X
03
=
0.01, and different β
(
0.4,0.6,0.8,1
,
1.2
).Figure 16
Variation of energy conditions (ρ
+
p
≥
0, ρ
-
p
≥
0, ρ
+
3
p
≥
0) against time t for polynomial law model.
(a)
(b)
(c)


## 5.1. Power Law Model

Following the same procedure as in Section4.1, we have obtained the same metric potential as in (27) and the other parameters like energy density ρ, pressure p, and cosmological constant Λ are expressed as follows:(39)
ρ
2
=
2
λ
V
0
2
α
8
π
+
λ
X
03
2
+
X
02
2
+
X
02
X
03
t
6
n
-
V
0
2
n
t
2
p
=
2
λ
V
0
2
8
π
+
λ
X
03
2
+
X
02
2
+
X
02
X
03
t
6
n
-
V
0
2
n
t
2
-
2
λ
V
0
2
α
8
π
+
λ
X
03
2
+
X
02
2
+
X
02
X
03
t
6
n
-
V
0
2
n
t
2
Λ
=
1
2
t
2
λ
8
π
+
λ
α
V
0
2
-
41
λ
4
π
+
λ
8
π
+
λ
α
V
0
t
-
3
n
+
1
2
V
0
2
n
t
6
n
-
2
t
2
X
03
2
+
X
02
2
+
X
02
X
03
+
4
4
π
+
λ
X
03
2
+
X
02
2
+
X
02
X
03
λ
α
t
-
6
n
+
2
-
2
n
3
λ
n
+
λ
+
24
n
π
V
0
2
α
λ
.Here, also we have noticed similar qualitative results as in Section4.1 (see Figures 9–12).Figure 9
Variation of energy densityρ against time t for λ
=
1, α
=
-
0.1, V
0
=
1, X
02
=
0.01, X
03
=
0.01, and different n
(
1.2,1.4,1.6,1.8
).Figure 10
Variation of pressurep against time t for λ
=
1, α
=
-
0.1, V
0
=
1, X
02
=
0.01, X
03
=
0.01, and different n
(
1.2,1.4,1.6,1.8
).Figure 11
Variation of energy conditions (ρ
+
p
≥
0, ρ
-
p
≥
0, ρ
+
3
p
≥
0) against time t for polynomial law model.
(a)
(b)
(c)Figure 12
Variation of cosmological constantΛ against time t for λ
=
1, α
=
-
0.1, V
0
=
1, X
02
=
0.01, X
03
=
0.01, and different n
(
1.2,1.4,1.6,1.8
).


## 5.2. Exponential Law Model

Following the same procedure as in Section4.2, we have obtained the same metric potential as in (34), and the other parameters like energy density ρ, pressure p, and cosmological constant Λ are as follows: (40)
ρ
=
1
V
0
e
β
t
2
X
03
2
+
X
02
2
+
X
02
X
03
λ
α
8
π
+
λ
p
=
2
X
03
2
+
X
02
2
+
X
02
X
03
λ
V
0
2
e
2
β
t
8
π
+
λ
-
1
V
0
e
β
t
2
X
03
2
+
X
02
2
+
X
02
X
03
λ
α
8
π
+
λ
Λ
=
-
2
2
4
π
+
λ
λ
2
X
03
2
+
X
02
2
+
X
02
X
03
λ
e
-
2
β
t
V
0
2
α
8
π
+
λ
+
2
X
03
2
+
X
02
2
+
X
02
X
03
4
π
+
λ
e
-
2
β
t
V
0
2
8
π
+
λ
-
β
2
3
.Here, also we have noticed the similar qualitative results as in Section4.2 (see Figures 13–16).Figure 13
Variation of energy densityρ against time t for λ
=
1, α
=
0.1, V
0
=
1, X
02
=
0.01, X
03
=
0.01, and different β
(
0.2,0.4,0.6,0.8,1
,
1.2
).Figure 14
Variation of pressurep against time t for λ
=
1, α
=
0.1, V
0
=
1, X
02
=
0.01, X
03
=
0.01, and different β
(
0.2,0.4,0.6,0.8,1
,
1.2
).Figure 15
Variation of cosmological constantΛ against time t for λ
=
1, α
=
0.1, V
0
=
1, X
02
=
0.01, X
03
=
0.01, and different β
(
0.4,0.6,0.8,1
,
1.2
).Figure 16
Variation of energy conditions (ρ
+
p
≥
0, ρ
-
p
≥
0, ρ
+
3
p
≥
0) against time t for polynomial law model.
(a)
(b)
(c)


## 6. Concluding Remarks

In this paper, we have the Bianchi type-I cosmological model inf
(
R
,
T
) modified gravity for two different classes of f
(
R
,
T
) in the presence of cosmological constant and quadratic equation of state. Here, we have discussed two models based on the expansion law. From both models, case of f
(
R
,
T
)
=
R
+
2
f
(
T
), we have concluded the following points.(i)
In both the models, energy densityρ is decreasing function of t and ρ approaches towards zero with the evolution of time.
(ii)
In both the models, pressurep is negative and approaches towards zero with the evolution of time.
(iii)
In both the models, cosmological constantΛ is negative, but here we notice that in case of power law Λ approaches towards zero with the evolution of time whereas it does not approach towards zero with the evolution of time in case of exponential law. Similar qualitative observations are also noticed for the case of f
(
R
,
T
)
=
f
1
(
R
)
+
f
2
(
T
). Here, all the observation are in fare agreement with the observational data.



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*Source: 816826-2015-11-26.xml*