Anisotropic Standard Decelerating Cosmological Model with Quadratic Equation of State

Spatially homogeneous and anisotropic Bianchi type-I cosmological model containing perfect fluid with quadratic equation of state has been diagnosed in general theory of relativity. To obtain a deterministic solution, we have used a relation between metric potentials. The exact solution of Einstein’s field equations thus obtained represents an expanding and decelerating universe. The physical and kinematical parameters of the model have also been analyzed with certain constrained between the parameters of the quadratic equation of state.


Introduction
In a large scale structure, the universe is highly homogeneous and isotropic, shown in [1]. These properties are also supported by the work of Benet et al. [2] and Spergel et al. [3] via the cosmic microwave background radiation. Aside homogeneity property, isotropy throws a challenge that anisotropic models cannot be isotropized appropriately because of their time evolvement in future. However this is to note that inflation is a suitable method to solve this problem.
Ananda and Bruni's [4] works in relativistic dynamics study of Robertson and Bruni [5] to study the effect of quadratic EoS in homogeneous and inhomogeneous cosmological models to isotropize the universe at early age when initial singularity is continued.
In a firm study, Nojiri and Odintsov [6] used inhomogeneous Hubble parameter term to discuss the variation of dark energy universe relative to different equation of state. In 2006, Capozzelio [7] pointed out that the observational constraint impact along with quadratic EoS on dark energy. Nojiri et al. [6] works reveal that quadratic EoS may be nice tool to describe dark energy or unified dark matter in a more generous way. In 2009, Rahaman et al. [8] showed how electron can be modeled precisely instead of spherically symmetric charged perfect fluid distribution of matter characterized by quadratic EoS. Followed by the works of Rahaman et al. [8], Firoze et al. [9] introduced "Charged anisotrop- It reveals from the above discussion that quadratic EoS plays an important and interesting role in many areas including dark energy and dynamics of different models in general relativity. In the study of anisotropic problem, in particular, Bianchi type-I homogeneous cosmological model containing perfect fluid is an interesting idea that lead us to continue research using quadratic EoS to generate an expanding and decelerating universe.
We have organized this paper as follows: In Section 2, we have applied Bianchi type-I metric to obtain field equations. Explicit solution of the equations is presented in Section 3 followed by relevant interpretations of the results in Section 4. We summarize this paper by giving a conclusion in Section 5.
M. M. Alam et al.

Metric and Field Equations
The Bianchi type-I line element is given by where A, B and C are scale factors and are functions of time t only. The Einstein field equations, in natural limits ( 8 1 G π = and 1 c = ), are where ij R is the Ricci tensor, R is the Ricci scalar and ij T is the energy-momentum tensor.
The energy-momentum tensor ij T for the perfect fluid is given by where ρ is the energy density, p is the pressure and i u is the four velocity vector satisfying 1 Here, we have assumed an equation of state (EoS) in the general form ( ) p p ρ = for the matter distribution.
We have considered it in the quadratic form as follows In co-moving coordinate system, the Einstein field Equations (2) for the metric (1) with the help of Equation (3) reduce to following set of equations: where overhead dot ( ⋅ ) denotes differentiation with respect to time t.
The vanishing divergence of Einstein tensor which leads to ; 0 ij j T = , yields the following energy conservation equation We have defined the spatial volume V and average scale factor "a" for Bianchi type-I space-time as The mean Hubble parameter H for Bianchi type-I universe is defined as

Solution of the Field Equations
The field Equation (5) Subtracting (7) from (8), we have From equations (12) and (13), we get which can be reduced to the following form Integrating Equation (15), we obtain ( ) 1 1 Equation (16) can be written as follows which is a first order linear differential equation in where 1 l and 2 l are integrating constants.
It is to be noted that in this case the solution of Einstein's field equations reduces to the integration of (18) if the explicit form of the scale factor C is known.
We obtain particular solution of (18) for a simple choice of the function C.
For this, we choose, From Equation (12), (19) and (21), we get ( ) ( ) The metric of the model can be written in the form Using the directional scale factors for the model (23) in (10), we compute the spatial volume V as follows For the spatial volume to be positive, we must have From (4) and (25), we obtain the pressure as   σ and deceleration parameter q which are defined and found to be

Results and Discussions
It has been observed from Equation (25)    Again, Figure 2, corresponding to Equation (28) for 1 α = and 1 β = − , depicts the evolution of pressure p and shows that p is positive and 0 p → when t → ∞ as in matter dominated universe.
From Equation (24), it has been observed that initially, the spatial volume V vanishes and linearly increases as time t increases and becomes infinitely large for t → ∞ . This shows that the universe starts expanding with initial zero volume and keeps expanding with increase of cosmic time t. From Equation (29), it has been observed that Hubble parameter is a decreasing function of time and 0 H → when t → ∞ . The expansion scalar and shear scalar start with infinitely large value at 0 t = and tend to zero as t → ∞ corresponding to the Equations (30) and (32) which confirms the expansion of the universe. The mean anisotropy parameter, as in Equation (31), is independent of time and is uniform throughout the whole expansion of the universe when 1 m ≠ − , but for 1 m = − , it tends to infinity. Also, since ( ) The remarkable thing is that though the recent observations of SNeIa [19] [20] as well as CMBR [2] [3] are in favour of accelerating models ( 0 q < ), but both do not entirely rule out the decelerating ones which are also realistic with these observations supported by the work of Vishwakarma [21]. Moreover, the study of the decelerating model of the universe is important to make a bridge between early inflation and late time acceleration.

Conclusion
We investigated spatially homogeneous and anisotropic Bianchi type-I cosmo-Journal of Applied Mathematics and Physics logical models with quadratic equation of state (EoS) in the context of general relativity. The important physical and kinematical parameters in the discussion of cosmological model have been determined and studied. The limitation in the choices of the parameters of quadratic equation of state has been explored and discussed. We also showed, the universe will not be shear free and will remain anisotropic throughout the cosmic evolution with standard deceleration.