KALUZA - KLEIN COSMOLOGICAL MODEL WITH QUARK AND STRANGE QUARK MATTER INF(R,T)GRAVITY

Samadhan L. Munde Assistant Professor, Department of Mathematics, Shri R. L. T. College of Science, Akola 444001, Maharashtra, [INDIA]. ...................................................................................................................... Manuscript Info Abstract ......................... ........................................................................ Manuscript History Received : 05 February 2021 Final Accepted : 10 March 2021 Published : April 2021

In the present paper, Kaluza-Klein cosmological model with quark matter and strange quark matter in ( , ) theory of gravity has been studied. The general solutions of the field equations of Kaluza-Klein space-time have been obtained under the assumption of constant deceleration parameter in the context of exponential volumetric expansion model. The physical and geometrical aspects of the model are also discussed in detailed.
Gravitational field equations of ( , ) theory of gravity:-In ( , ) theory of gravity, the field equations are obtained from the Hilbert-Einstein type variation principle. The action for this modified theory of gravity is given by where ( , ) is an arbitrary function of the Ricci scalar and of the trace of the stress-energy tensor of the matter and is the matter Lagrangian.
The corresponding field equations of the ( , ) gravity is found by varying the action (1) with respect to the metric : where ( , ) = ( , ) , is the covariant derivative and is the standard matter energy-momentum tensor derived from the Lagrangian .
The stress-energy tensor of matter is The tensor Θ in equation (2) is given by the matter Lagrangian L may be chosen as L = − , where p is the thermodynamical pressure of matter content of the Universe. Now, equation (5) gives the variation of the stress-energy tensor as
Harkoet al. [11] have investigated FRW cosmological models in this theory by choosing appropriate function ( ).
They have also discussed the case of scalar fields since scalar fields play a vital role in cosmology. The equations of motion of test particles and a Brans-Dickey type formulation of the model are also presented.

Metric and Field Equations:-
Consider a five-dimensional Kaluza-Klein metric in the form as where ( ) and ( ) are the scale factors (metric tensors) and functions of cosmic time only and the fifth coordinate is taken to be space-like.
In the present study, we assume that the energy momentum tensor for the quark matter (Aktas et. al. [31] , Yilmaz, et. al. [32]) in the form as where = + is the energy density, = − is pressure of the fluid and = (1,0,0,0,0) is the five-velocity vector in the comoving coordinates system which satisfies the condition = 1. Since quark matter behaves nearly perfect fluid (Adams et al. [33], Adcoxet al. [34], Back et al. [35], Aktas et al. [31], Yilmaz et al. [32]). We will use the following equation of state for quark matter in the form as Also, the linear equation of state for strange quark matter (Sharma et al. [36,37]) in the form as where 0 is the energy density at zero pressure and is a constant. When = 1 3 and 0 = 4 , the above linear equation of state is reduced to the followingequation of state for strange quark matter in the bag model (Aktaset al. [31], Yilmaz et al. [32]) as where is the Bag constant.
In the present paper, we consider Kaluza-Klein cosmological model for the particular choice of ( , ) given by where the ( ) is an arbitrary function of the trace of the stress-energy tensor of matter.
Using equation (6) & (13) in equation (2) then the gravitational field equation in ( , )gravity becomes where prime denotes differentiation with respect to the argument.

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Now, In the present work, we choose the function ( ) of the trace of the stress-energy tensor of the matter as The corresponding field equations (14) for the metric (8) with the help of equations (9) and (15) can be written as where the overhead dot ( . ) denote derivative with respect to the cosmic time t.
The spatial volume ( ) is defined as where is the average scale factor.
The directional Hubble parameters in the directions of , , and axes respectively are defined as The mean Hubble parameter ( ) is given by The volumetric deceleration parameter ( ) is given by The anisotropic parameter ( ) of the expansionis defined as where ( = 1,2,3,4) represent the directional Hubble parameters in the direction of , , and respectively.
The expansion scalar ( ) is defined as The Shear scalar ( 2 ) is defined as

Solutions of the field equations:-
Since there are three highly non-linear equations (16) to (18) with four unknowns A, B, and . In order to solve the system completely, we impose a law of variation for the Hubble parameter which was initially proposed by Berman [38] for RW (Robertson-Walker) space-time and yields the constant value of deceleration parameter. Adhavet al. [39] used this law for LRS Bianchi type-I metric in creation field cosmology. According to this law, the variation of the mean Hubble parameter for the Kaluza-Klein metric given by where > 0 and ≥ 0 are constants.
The sign of q indicates whether the model accelerates or not. The positive sign if ( > 1) corresponds to decelerating models where as the negative sign −1 ≤ < 0 for 0 ≤ < 1 indicates acceleration and = 0 for = 1 corresponds to expansion with constant velocity.

In this paper, we consider the model for = , ( = − ): (Exponential Volumetric Expansion Model )
Subtracting equation (17) from (18) and using mean Hubble parameter from equation (21), we get On integration of equation (33) and considering equation (29), we obtain where 3 is constant of integration.
The spatial volume ( ) is found to be

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The directional Hubble parameters inthe directions of , , and axes respectively are The mean Hubble parameter ( ) is obtained as The anisotropic parameter ( ) of the expansionis found to be The expansion scalar ( ) is found to be The Shear scalar ( 2 ) is found to be Using equations (35) and (36) in equation (16) with the help of linear equation of state (10) for = 1 3 , we obtain the energy density and pressure of the quark matter as ] .
Similarly, using equations (35) and (36) in equation (16) with the help of linear equation of state (12), we obtain the energy density and pressure of the strange quark matter as

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In Exponential Volumetric Expansion Model, it is observed that the spatial volume is finite at = 0, expands exponentially as increases and become infinitely large as → ∞ as shown in figure-1. From equations (38) and (39), it is observed that the directional Hubble parameters , are finite at = 0 and = ∞. The mean Hubble parameter ( ), the expansion scalar ( ) are constant for all values of . Thus, the model represents uniform expansion.
The anisotropy of the expansion ( ) is not promoted by the anisotropy of the fluid. Here the anisotropy of the expansion Δ → as → 0 and then decreases to null exponentially as increases provided that = 3 = 1. The space approaches to isotropy in this model since → 0 as → ∞ as shown in figure-2. From equations (4.20) to (4.23), one can observed that density and pressure of quark matter (including strange quark models) become constants when as → 0 and then decreases exponentially as t increases and remain constant through out the evolution and hence there is no big bang type of singularity.
From equation (31), it is observed that the present model for = 0 ( = −1) with negative deceleration parameter indicating that the universe is accelerating which is consistent with the present-day observations. For this model, we get = −1 which implies the fastest rate of expansion of the universe. Riess et al. [1,40] and Perlmutter et al. [2] have shown that the deceleration parameter of the universe is in the range −1 ≤ ≤ 0 and the present-day universe is undergoing accelerated expansion.