HIGHER DIMENSIONAL PERFECT FLUID COSMOLOGICAL MODEL IN GENERAL RELATIVITY WITH QUADRATIC EQUATION OF STATE (EoS)

Higher dimensional Bianchi type-V cosmological model in the general theory of relativity with quadratic equation of state interacting with perfect fluid has been studied. For higher dimensional Bianchi type-V space time, the general solutions of the Einstein's field equations have been obtained under the assumption of quadratic equation of state (EoS) p = αρ − ρ, where α ≠ 0 is arbitrary constant. The physical and geometrical aspects of the model are discussed.


INTRODUCTION
It is now almost proved from theoretical and astronomical observations such as type Ia supernovae (Riess et al. [1], Perlmutter et al. [2], Riess et al. [3]) that the universe is expanding with acceleration from the big-bang till today. However, no one can guarantee for forever expansion because there is no final conclusion about the expansion or contraction of universe till today and represents an open question for theoretical physicists which inspired numerous cosmologists and physicists for similarly research within the area of studies on this field to resolve this problem including the modified theory of gravity and possible existence of dark energy (DE). From various literatures and opinions it can be belief that the acceleration of present universe may be accompanied by way of deceleration. In the recent past years, several models in cosmology has been proposed by different authors in order to explain the hidden reasons of expansion of the existing universe with the acceleration in the framework of general theory of relativity and modified theories. Spatially homogeneous and anisotropic Bianchi type cosmological model plays a great role to describe the large-scale behavior of the universe.
Modern cosmology revolves around the study about the past and present state of the universe and how it will evolve in future.
In the study of universe, among different models of Bianchi types, Bianchi type-V cosmological models is the natural generalization of the open Friedmann-Robertson-Walker(FRW) model which plays an important role in the description of universe and more interesting is that these models contain isotropic special cases and it may permit small anisotropy levels at some of time.
There are theoretical arguments from the recent experimental data which support the existence of an anisotropic phase approaching to isotropic phase leading to the models of the universe with anisotropic background. Cosmological models which are spatially homogeneous and anisotropic play significant roles in the description of the universe at its early stages of evolution. Some authors such as Maartens and Nel [4], Meena and Bali [5] have studied Bianchi type-V models in different contexts. Coley [6] have investigated Bianchi type-V spatially homogeneous universe model with perfect fluid cosmological model which contains both viscosity and heat flow. In recent years, the solution of Einstein's field equation for homogeneous and anisotropic Bianchi type models have been studied by several authors such as Hajj-Boutros [7], Shri Ram [8,9], Pradhan and Kumar [10] by using different generating techniques. Ananda and Bruni [11] 3157 HIGHER DIMENSIONAL PERFECT FLUID COSMOLOGICAL MODEL discussed the cosmological models by considering different form and non-linear quadratic equation of state. A few models that describe an anisotropic space time and generate particular interest are among Lorenz and Pestzold [12], Singh and Agrawal [13], Marsha [14], Socorro and Medina [15]. Furthermore, from several kinds of literature and findings one can actually locate that the anisotropic model had been taken as possible models to initiate the expansion of the universe. Banerjee and Sanyal [16] have considered Bianchi type-V cosmological models with bulk viscosity and heat flow. Conformally flat tilted Bianchi type-V cosmological models in the presence of a bulk viscous fluid are investigated by Pradhan and Raj [17]. Kandalkar et al. [18] discussed the variation law of Hubble's parameter, average scale factor in spatially homogeneous anisotropic Bianchi type-V space time filled with viscous fluid where the universe exhibits power law and exponential expansion.
In relativity and cosmology, the equation of state is nothing but the relationship among combined matter, pressure, temperature, energy and energy density for any region of space, plays an important role in the study about universe. For the study of dark energy and general relativistic dynamics in different cosmological models, the quadratic EoS plays an important role. Many researchers like Ivanov [19], Sharma and Maharaj [20], Thirukkanesh and Maharaj [21] A cosmological model in higher-dimensions performs a crucial role in different aspects of the early phases of the cosmological evolution of the universe and are not observable in real universe.
It is not possible to unify the gravitational forces in nature in typical four-dimensional space-times. So the theory in higher dimensions may be applicable in the early evolution. The study on higher-dimensional space-time gives us an important idea about the universe that our universe was much smaller at initial epoch than the universe observed in these days. Due to these reasons studies in higher dimensions inspired and motivated many researchers to enter into such a field of study to explore the hidden knowledge of the present universe. Subsequently, many researchers have already investigated various cosmological models in five dimensional space-time with various Bianchi type models in different aspects [35,38,41,42,43,44]. gives the solutions of the cosmological problems. In sec.4, some of the important physical and geometrical parameters are derived. The results found are discussed in Sec.5. Finally, in Sec.6, concluding points are provided.

Where a, b, c and D are functions of time only and m is the extra dimension (Fifth dimension)
which is space-like.
The energy momentum tensor for the perfect fluid is given by Where ρ is the energy density, p is the pressure and = (0,0,0,0,1) is the five velocity vector of particles.

Also
(3) = 1 We have assumed an equation of state in the general form = ( ) for the matter distribution.
In this case, we consider the quadratic form as Where ≠ 0 is arbitrary constant.
The Einstein's field equations in general relativity is given by If R(t) be the average scale factor, then the spatial volume is given by Where R(t) is given by The expansion scalar is defined as The Hubble's parameter is defined as Deceleration parameter q is defined as The mean anisotropy parameter is defined as The shear scalar is defined as

(19) 3̇=̇+̇+̇
The overhead dots here denote the order of differentiation with respect to time `t'.

COSMOLOGICAL SOLUTION
From Equation (19) we get Here, is the constant of integration. Without loss of generality we can choose = 1 and so we get Subtracting (14) from (15) The solution of above equation is Where ≠ 0, are constants and + 1 ≠ 0 Using (7), (22), (31) and (33)  The Hubble's Parameter is given by (37) = ( +1)( + ) The expansion scalar ( ) is given by The shear scalar is The average anisotropy parameter is (42) ∆= 0

DISCUSSION
The variation of some of the important cosmological parameters (Physical and Geometrical) with respect to time for the model are shown below by taking, = −0.5, = = 1, 5 = 2 and = 1.
From the equation (36) it is seen that the spatial volume is finite when time, = 0. With the increases of the time the spatial volume V also increases and it becomes infinitely large as → ∞, so spatial volume expands as shown in fig.1. This shows that the universe is expanding with the increases of time. We observe from fig.3 or from equation (39), that the evolution of the energy density is large constant at the time = 0 and with increases of time the energy density decreases and finally when time → ∞ the energy density becomes zero. This shows that the universe is expanding with time. For this model the value of shear scalar is zero throughout the evolution as seen from the equation (41), so the model is shear free. The ratio 2 2 tends to zero as time tends to infinity, showing that the model approaches isotropy at the late time universe, which agrees with Collins and Hawking [45].
From the equation (42) it is seen that average anisotropy parameter, ∆= 0, so the model is isotropic one and this model also shows the late time acceleration of the universe.
It is observed that the spatial volume, all the three scale factors and all other physical and kinematical parameters are constant at initial epoch, = 0. This shows that the present model is free from the initial singularity.

CONCLUSION
Here in this paper we studied a Bianchi type-V cosmological model in the context of general

CONFLICT OF INTERESTS
The author(s) declare that there is no conflict of interests.