Kasner Cosmological Models with Two-Fluid Source in General Relativity

In this paper we present Bianchi type-I metric of the Kasner form describing two-fluid source of the universe in general relativity. In Kasner cosmological models one fluid is a radiation field modeling the cosmic microwave background, while the other is a matter field, modeling material content of the universe. The radiation and matter content of the universe are in interactive phase. We have also presented anisotropic, homogeneous nature of Kasner cosmological models with two-fluid. The behavior of fluid parameters and kinematical parameters of the models are also discussed.


Introduction
In recent years, there has been considerable interest in the study of Friedman-Robertson-Walker (FRW) spatially homogeneous isotropic models which are widely considered as good approximation of the present and early stages of the universe. Two-fluid FRW models of the universe have been investigated where one fluid is the radiation field corresponding to the observed cosmic background radiation, while a perfect fluid is chosen to represent the matter content of the universe Coley and Tupper [1,2]. Coley and Dunn [3] examined Bianchi type-VI 0 model with two-fluid source. Two-fluid cosmological models using Bianchi type II space-time has been investigated by pant and Oli [4]. The generalization of Kasner model were proposed by many authors [5][6][7][8][9][10][11] has defined an analytic expression for the anisotropy of the Kasner metric. When cosmological constant is zero and the model isotropizes in Einstein theory for a nonzero cosmological constant, Mothanty and Daud [12] have shown that the model reduces to Kasner [13] form, wherein they studied Bianchi type-I cosmological model with gauge function in vacuum. In Einstein general relativity, a viscous cosmological fluid does not permit the Kasner metric to be anisotropic which have been proved by [14][15][16].
The Kasner metric is one of the more widely studied metric. It has played a central role in the elucidation of the existence and structure of anisotropic cosmological models and their singularities in general relativity motivates authors to study this problems. The discovery of 2.73 K isotropic cosmic microwave background radiation (CMBR) motivated many authors to investigate FRW model with a two-fluid source [17,18].
Several prominent results obtained in the development of the super-string theory therefore the study of higher dimensional physics is important. In the latest study of super-strings and super-gravity theories, Weinberg [19] studied the unification of the fundamental forces with gravity which reveals that the space-time should be different from four. The string theories are discussed in 10-dimension or 26-dimensions of space-time, since the concept of higher dimensions is not unphysical. For that reason, many researcher to enter into such field of study to explore the hidden knowledge of the universe to studies in higher dimensions. Several authors [20][21][22][23][24][25][26][27] have studied the multi-dimensional cosmological models in general relativity and in other alternative theories of gravitation.
In this paper, we have investigated two fluid Kasner cosmological models in four and five dimension; an analytic solution of general relativistic field equation is presented and also discussed the physical behavior of the corresponding models.

Field Equations
We consider an anisotropic Bianchi type-I metric of Kasner form as where p 1 , p 2 and p 3 are three parameters that we shall requires to be constant, satisfying the two relation p 1 + p 2 + p 3 = 1 and p 2 1 + p 2 2 + p 2 3 = 1. The Einstein's field equations for a two fluid source in natural unit (gravitational units) are written as The energy momentum tensor for a two fluid source is given by where T (m) ij is the energy momentum tensor for matter field and T (r) ij is the energy momentum tensor for radiation field [3] is given by

4)
Thus the matter field T (m) ij is described by a perfect fluid with density ρ m , pressure p m and the radiation field T (r) ij , has density ρ r , pressure p r = 1 3 ρ r . The off diagonal equations of (2.2) together with energy conditions imply that the matter and radiation are both co-moving, we get, Using (2.1), (2.3), (2.4), (2.5) and (2.6), the field equations (2.2) reduces to: . By comparing Eqs. (2.8), (2.9) and (2.10), we get where p is arbitrary constant. Using above, Bianchi type-I metric of Kasner form can be written as

Some Physical and Kinematical Properties
We assume the relation between pressure and energy density of matter field through the "gamma-law" equation of state which is given by We get energy density of matter, energy density of radiation and total energy density as 3) The generalized mean Hubble parameter H is given by where H 1 , H 2 and H 3 are the directional Hubble Parameter in the direction of x, y and z axes respectively.
Case I: Dust model In order to investigate the physical behavior of the fluid parameters we consider the particular case of dust i.e. when γ = 1. The scalar of expansion, shear scalar and deceleration parameter are given by The density parameters are where Ω 0 is total density parameter.
Case IV: Zeldovich Universe (γ = 2) In this case, we get, the scalar of expansion, shear scalar and deceleration parameter as We get, the energy density of matter, energy density of radiation and total energy density as Here ρ m and total density ρ is positive and ρ r is negative.

Field Equations
We consider an anisotropic Bianchi type-I metric of Kasner form as where p 1 , p 2 , p 3 and p 4 are four parameters that we shall require to be constant, satisfying the two relation The Einstein's field equations for a two fluid source in natural unit (gravitational units) are written as The energy momentum tensor for a two fluid source is given by where T (m) ij is the energy momentum tensor for matter field and T (r) ij is the energy momentum tensor for radiation field [3] is given by The off diagonal equations of (4.3) together with energy conditions imply that the matter and radiation are both co-moving, we get, Using (4.1), (4.4), (4.5), (4.6) and (4.7) the field equations (4.3) reduces to: where s = p 1 + p 2 + p 3 + p 4 , φ = p 2 1 + p 2 2 + p 2 3 + p 2 4 . By comparing Eqs. (4.9), (4.10), (4.11) and (4.12) we get where P is arbitrary constant. Using above, Bianchi type-I metric of Kasner form can be written as

Some Physical and Kinematical Properties
We assume the relation between pressure and energy density of matter field through the "gamma-law" equation of state which is given by We get energy density of matter, energy density of radiation and total energy density as The generalized mean Hubble parameter H is given by where H 1 , H 2 , H 3 and H 4 are the directional Hubble Parameter in the direction of x, y, z and u axes respectively.
Case I: Dust model In order to investigate the physical behavior of the fluid parameters we consider the particular case of dust i.e. when γ = 1.
The scalar of expansion, shear scalar and deceleration parameter are given by The density parameters are Case II: Radiation universe (when γ = 4 3 ) On substituting γ = 4 3 , we get, the scalar of expansion, shear scalar and deceleration parameter as And the density parameters using are γ = 4 3 in (5.1) and (5.2) are given by Case III: Hard Universe (γ ∈ ( 4 3 , 2) let γ = 5 3 ) The scalar of expansion, shear scalar and deceleration parameter in Hard Universe are For γ = 5 3 , (5.1) and (5.2) imply density parameters as Here ρ m and total density ρ are positive whereas ρ r is negative.
Case IV: Zeldovich Universe (γ = 2) In this case, we get, the scalar of expansion, shear scalar and deceleration parameter as We get, the energy density of matter, energy density of radiation and total energy density as Here ρ m and total density ρ is positive and ρ r is negative.

Conclusion
In this paper, we have provided a detailed Kasner Cosmologies dominating by two relativistic cosmic fluids. Both the cases can be exactly solved in the framework of Einstein field equations. The positive sign of (q > 1) corresponds to decelerating model whereas the negative sign (−1 < q < 0) indicates acceleration and q = 0 corresponds to expansion with constant velocity.
In Sects. 2 and 3, here, in all cases, we get q = −1 indicating that the universe is accelerating. This observation is consistent with the present day observation. the ratio ( σ θ ) 2 = 1 6 = 0. Therefore, these models do not approach isotropy for large value of t .
In Sects. 4 and 5, here we consider five-dimensional anisotropic Bianchi type-I space time of Kasner from.
Here, in all cases, we get negative deceleration parameter indicating that the universe is accelerating. This observation is consistent with the present day observation.
It is observed that the results in four dimension retained in higher dimension too.