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Kantowski-Sachs Cosmological Model in f(R,T) Theory of Gravity
1
of
V. U. M. Rao1,* and G. Suryanarayana2 Department of Applied Mathematics, Andhra University, Visakhapatnam, India 2 Department of Mathematics, ANITS, Visakhapatnam, India
A new class of spatially homogeneous Kantowski-Sachs cosmological models filled with perfect fluid in the framework f ( R , T ) gravity proposed by Harko et al. (2011) have been studied with an appropriate choice of a function
f ( R , T ) = f1 ( R ) + f 2 (T ) . The models obtained and presented here are anisotropic, expanding and free from singularities for n > 0 and it is established that the additional condition, special law of variation of Hubble parameter proposed by Bermann (1983), taken by Samanta (2013) is superfluous. Also some important features of the models, thus obtained, have been discussed.
1.
Introduction
Several theories of gravitation have been proposed as different alternative to Einstein’s theory of gravitation since there have been many criticisms of general relativity due to the lack of certain desirable features in the theory. For example, Mach’s principle is not fully incorporated in general relativity and appearance of singularities is another problem. Recent observational data suggest that our universe is accelerating. This acceleration is explained in terms of late time acceleration of the universe and the existence of the dark matter and dark energy. Earlier, Harko et al. [1] developed a generalized f(R,T) gravity where the gravitational Lagrangian is given by an arbitrary function of the Ricci scalar R and of the trace T of the stress energy tensor. They have obtained field equations in metric formalism. The equations of motion for test particles, which follow from covariant divergence of the stress energy tensor, are also presented. They have obtained several models in this theory corresponding to some explicit forms of the function f (R,T). In f(R,T) gravity, the field equations are obtained from the Hilbert-Einstein type variational principle. Using gravitational units (by taking G and c as unity) the corresponding field equations of f(R,T) gravity are obtained by varying the action principle with respect to g ij as
___________________ *
[email protected]
∂f ( R, T ) 1 Rij − f ( R, T ) g ij ∂R 2 + ( g ij ∇ µ ∇ µ − ∇i ∇ j ) = 8πTij − Where, Θ = gαβ ij
(
∂f ( R, T ) ∂R
∂f ( R, T ) Tij + Θij ∂T
δTαβ
(1)
)
.
δg ij
Here ∇ i is the covariant derivative and Tij is usual matter energy-momentum tensor derived from the Lagrangian Lm . Reddy et al. [2,3] have obtained Kaluza-Klein cosmological model in the presence of perfect fluid source and Bianchi type-III cosmological model in this theory assuming the law of variation of Hubble parameter that was proposed by Bermann [4]. Houndjo [5] has developed the cosmological reconstruction of f(R,T) gravity as f(R,T) = f1 ( R ) + f 2 (T ) and discussed the transition of matter dominated phase to an accelerated phase. Reddy and Santhi Kumar [6] have discussed some anisotropic cosmological models in a modified theory of gravity by taking different relations between pressure and energy density. Sharif and Zubair [7] have discussed energy conditions constraints and stability of power law solutions in f ( R , T ) gravity. Samanta and Dhal [8] have discussed higher dimensional cosmological models filled with perfect fluid in f(R,T) theory. Sharma and Singh [9] have obtained Bianchi type-II string
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cosmological model with magnetic field, Reddy et al. [10] have obtained Kantowski-Sachs bulk viscous string cosmological models, Sharif and Zubair [11] have discussed Bianchi-I anisotropic model, Rao and Neelima [12,13] have discussed perfect fluid Einstein-Rosen, Bianchi type-VI0 and non-static plane symmetric Universes, and Rao et al. [14] have obtained Bianchi type III, V and VI0 bulk viscous string cosmological models in f(R,T) gravity. Recently, Rao and Suryanarayana [15] have discussed higher dimensional perfect fluid cosmological models in this theory, respectively. Samanta [16] has discussed Kantowski-Sachs universe filled with perfect fluid in this theory by using a special law of variation for the Hubble parameter that was proposed by Bermann [4]. Tiwari and Dwivedi [17], Adhav et al. [18], Katore and Rane [19], Chaubey [20] are some of the authors who have studied Kantowski-Sachs cosmological models in various theories. In this paper, we will investigate spatially homogeneous and anisotropic Kantowski-Sachs cosmological model filled with perfect fluid in the framework of f(R,T) gravity proposed by Harko et al. [1] with an appropriate general choice of a function f(R,T) = f1 ( R ) + f 2 (T ) .
2.
Metric and Energy Momentum Tensor
We consider a spatially homogeneous KantowskiSachs metric of the form
(
ds 2 = dt 2 − A2 dr 2 − B 2 dθ 2 + Sin2θ dφ 2
)
(2)
Where, A and B are the functions of time t only. Beside Bianchi type metrics, the KantowskiSachs [21] models are also describing spatially homogeneous Universes. For a review of Kantowski-Sachs metrics one can refer to MacCallum [22]. These metrics represent homogeneous but anisotropically expanding (or contracting) cosmologies and provide models where the effects of anisotropy can be estimated and compared with all well known FriedmannRoberston-Walker class of cosmologies. The field equations in f (R, T) gravity for the function f (R,T)= f1 ( R ) + f 2 (T ) This is so when the matter source is perfect fluid as given by Harko et al. [2] are
∂f1( R ) ∂R
1 Rij − f1( R) gij = 8πTij 2 ∂f (T ) ∂f (T ) Tij + 2 p + 1 f (T ) gij (3) + 2 ∂T 2 2 ∂T
The matter tensor for perfect fluid is
Θij ≡ −2T ij − δ ij p where
(4)
T ij = ( ρ + p)uiu j − δ ij We consider a particular form of the function f ( R) = λ R and f (T ) = λ T where λ1 and λ 2 1
2
1
2
are any parameters, so that f ( R, T ) = λ1R + λ 2T . Then the field equations (3) will reduce to
Rij −
(
)
1 λ 8π + λ2 Rgij − p + T 2 g ij = Tij 2 λ 2 λ1 1
(5)
The field equations (5) in mixed form can be taken as
8π + λ 2 1 G ij ≡ R ij − δ ij R ≡ 2 λ1 3.
i λ2 T i T j + p + δ j 2 λ1 (6) Solutions of Field Equations
Now with the help of Eqns. (3)-(5), the field equations (6) for the metric in Eqn. (2) can be written as 16π + 3λ λ && & 2 2 p− 2 ρ 2 B + B + 1 = B B 2 B 2 2λ 2λ 1 1
(7)
&& B && A& B& 16π + 3λ λ A 2 p − 2 ρ + + = A B AB 2λ1 2λ1
(8)
&& B && A& B& 16π + 3λ λ A 2 p − 2 ρ + + = A B AB 2λ1 2λ1
(9)
16π + 3λ2 λ B& 2 A& B& 1 ρ + 2 p +2 + 2 = − 2 AB B 2λ1 B 2λ1 (10)
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Here the over head dot denotes differentiation with respect to t . Eqns. (9) to (10) is a system of three independent equations in four unknowns A, B, p and ρ . In order to get a deterministic solution, we take the following plausible physical condition: the shear scalar σ is proportional to scalar expansion θ leading to a linear relationship between the metric potentials A and B , i.e.,
B=A
n
(11)
Where, n ≠ 0 is a constant. From Eqns. (7), (8) and (11), we get
Thus the metric in Eqn. (15) together with Eqns. (16) and (17) constitutes Kantowski-Sachs perfect fluid cosmological model in f (R,T) gravity, where f ( R, T ) = λ 1R + λ 2T . For particular values of
f ( R, T ) = R + 2λ T , then the energy density and the total pressure will become
ρ=
p=
&2 && + 2n A = 1 1 A A (1 − n ) A2 n−1
1
A = [n(k1t + k 2 )] n
(13)
1
1 2 and k2 is k1 = , n ≠ ±1 2 1 − n constant of integration. From Eqns. (11) and (13), we get
Where,
B = [n(k1t + k 2 )]
a
(14)
The metric can now be written as 2
)
(15)
From Eqns. (13), (14) and Eqns. (7) to (10), we get the energy density as
ρ=
λ1
(
)
2n n − 1 (k1t + k 2 ) 2
2
2
(
2
4π (2n + 1) + λ (3n + 1) (2π + λ )(4π + λ ) (18)
−1
2
)
2
4π + λ (1 − n ) (2π + λ )(4π + λ ) (19)
λ>
(
)
2
4n n − 1 (k1t + k 2 ) 2
For
− [n(k1t + k 2 )] 2 dθ 2 + Sin 2θ dφ 2
(
1
4n n − 1 (k1t + k 2 ) 2
(12)
From Eqn. (12), we get
ds 2 = dt 2 − [n(k1t + k 2 )] n dr 2
λ1 = 1 and λ 2 = 2λ ,
λ2 (3n + 1) + 8π (2n + 1) (4π + λ2 )(8π + λ2 ) (16)
-1
and
for
1< n < ∞
4π , p > 0 and ρ > 0 . Hence, Eqns. (18) n −1
and (19) satisfy the energy conditions. Also, when p<0 for −∞ < n < 1 and 1 < n < ∞ with
λ<
4π show that the perfect fluid behaves like n −1
a phantom-type dark energy. So, we can conclude that the perfect fluid may be a source of early dark energy due to the negative pressure, since energy conditions are violated. For the above particular values of λ1 and λ 2 , the metric in Eqn. (15) together with Eqns. (18) and (19) represents Kantowski-Sachs perfect fluid cosmological model in f(R,T) gravity with f ( R , T ) = R + 2λ T , which is more general. The additional condition, special law of variation of Hubble parameter proposed by Bermann [4], taken by Samanta [16] is superfluous.
4.
Some Other Important Features of the Model
The volume element of the model in Eqn. (15) is given by 2n+1
1 V = (−g) 2 = [n(k1t + k2 )] n Sinθ
And, the total pressure as
p=
(
λ1
)
2n 2 n 2 − 1 (k1t + k 2 )2
λ 2 (n − 1) − 8π (4π + λ 2 )(8π + λ 2 ) (17)
with
(20)
We can observe that the spatial volume is increasing with time. The expression for the expansion scalar θ is given by
The African Review of Physics (2015) 10:0019
2n + 1 k1 n k1t + k 2
θ = u i ,i =
142
(21)
a(t ) =V
And, the shear σ is given by
σ 2 = 1 σ ijσij = 7 (2n + 1)k1
2
18 n(k1t + k2 )
2
The average scale factor given as
(22)
1 n −1 q = (−3θ −2 )(θ, iu i + θ 2 ) = 3 2n + 1
θ 2n + 1 k1 = 3 3n k t + k 1 2
ρ 3H 2
=
−3λ1
n +1 n + 2 (2n + 1) 8π + 2λ2 8π + λ2
J =
(24)
(25)
2 2 1 3 H i − H n −1 2 = ∑ 2n + 1 3 i = 1 H
2 n +1 3n
= [n(k1t + k 2 )] −
1
(27)
1
Sin 3θ − 1 (28)
( 4 n − 1 )( n − 1 ) &a&& = 3 a H ( 2 n + 1) 2 1
(29)
From Eqn. (29) it can be observed that for n= -0.087658 the jerk parameter value overlap with the value j ≈ 2.16, which is obtained from the three kinematical data sets: the gold sample of type Ia supernovae, the SNIa data from the SNLS project and the X-ray galaxy cluster distance measurements. Luminosity parameter
The mean anisotropy parameter Am is given by
Am =
1 −1 a
(23)
The overall density parameter Ω is given by Ω=
2 n+1
And the jerk parameter as
The Hubble’s parameter H is given by
H=
= [n(k1t + k 2 )] 3n Sin 3θ
3
The red shift as
Z= The deceleration parameter q is given by
1
where r1 =
∫
t0 t
d
L
= r1 (1 + z )
1 dt a (t )
(26)
Where, ∆H i = H i − H (i = 1,2,3) .
dL =
3(n)
− ( n + 2) 3n 2
n −1 −( 2n+1) −( n + 2) (k1t 0 + k 2 ) 3n (k1t + k 2 ) 3n − (k1t + k 2 ) 3n (n − 1)k1 Sin 3θ
wij = ui , j − u j ,i is identically zero and hence this Universe is non-rotational. 5.
Conclusions
In this paper, we have presented spatially homogeneous and anisotropic Kantowski-Sachs cosmological model filled with perfect fluid in the framework of f ( R, T ) gravity proposed by Harko et al. [5] and which have been obtained with an appropriate choice of a function f ( R , T ) = f1 ( R ) + f 2 (T ) . We observe that at
t = − k 2 1 − n 2 , the spatial volume vanishes and
(30)
increases continuously with time for n > − 1 . This 2
shows that at the initial epoch the universe starts with zero volume and expands continuously with time. Also the model has no singularity for n > 0 . The expansion scalar θ , the shear scalar σ and the Hubble parameter H decreases with the increase in time and diverges at t = − k 2 1 − n 2 . From Eqns. (16) and (17), we can see that energy density and matter pressure will vanish with the increase of time. From Eqn. (23), we can observe that the deceleration parameter is negative for − 1 < n < 1 and 2
hence it represents an accelerating universe. Since
The African Review of Physics (2015) 10:0019
Am ≠ 0 , this indicates that this model is always anisotropic. The model presented here is anisotropic, non-rotating, expanding and also accelerating. Interestingly, for particular values of λ1 and λ 2 , the metric in Eqn. (15) together with Eqns. (18) and (19) represents Kantowski-Sachs perfect fluid cosmological model in f(R,T) gravity with f ( R, T ) = R + 2λ T , which is more general
143
[19] [20] [21] [22]
S. D. Katore and R. S. Rane, Astrophys. Space Sci. 323, 293 (2009). R. Chaubey, Int. J. Theor. Phys. 51, 3933(2012). R. K. Kantowski-Sachs, J. Math. Phys. 7, 443 (1966). MacCallum, Nature 230, 5289 (1971).
than the model investigated by Samanta [20].
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Received: 17 April, 2015 Accepted: 18 August, 2015
