LRS Bianchi type-I model with polytropic EoS in ) ( R f gravity

In the present work we have focused on investigated dynamical aspects of LRS Bianchi type –I metric within the frame work of ) ( R f theory of gravitation. The solution of the metric is accelerating universe are derived by assuming the negative constant deceleration parameter and utilizing a power law relation. Additionally, we have visually analyzed the metric’s dynamical and geometrical properties through graphical representations.


Introduction
In the beginning stage of evolution of the universe, the nature of the universe remains a mystery. After a Big-bang, there are several phase changes as the temperature dropped down. The recent discovery of the modern cosmos such as Type Ia supernovae, large scale structure gives indirect proofs of the universe undergoing exponential acceleration [1][2][3][4][5]. Adhav [6] shows the expanding universe with respect to cosmic time Researcher did their work to study the actual nature of the universe by using ) (R f theory of gravity. Santhi et al. [7] studied Bianchi type-II, VII and IX bulk viscous string cosmological models in context of ) (R f gravity and shows that how the cosmos changes from early deceleration to the acceleration. LRS Bianchi type-I metric in ) (R f gravity has been studied by Aditya and Reddy [8] and he got the result that at late time the anisotropic effect disappears. Hatkar et al. [9] explore Kasner type of Bianchi type-I model with non-interacting string and HDE in ) (R f theory of gravitation and also found that the string like phase in the cosmos existed early in the history of universe. Also different authors studied different conceptual phenomenon in the context of ) (R f theory of gravity [10,21].
In the General Theory of Relativity the Adhav et al. [22] investigate "higher dimensional spherically symmetric universe with polytropic equation of state" by considering the five dimensional static spherically star. The physical aspect of Kantowski-Sachs universe has been studied by Samdurkar et al. [23]

Basic formation of
(R f theory of gravitation was proposed by Buchdahl [29] in 1970 in which he used  instead of is a general function of the Ricci scalar R , g is the determinant of metric is an D'Alemberts operator and ij T is standard energy momentum tensor obtained by using Lagrangian m L given by For the perfect fluid the energy momentum tensor is,

Metric and dynamical parameter
Here we studied LRS Bianchi type-I cosmological model given by The Dynamical parameters related to cosmological model (5)  The special volume V in the form of average scale factor ) (t a of model is defined as The mean Hubble parameter where respectively.
To discuss the anisotropy of universe the anisotropic parameter  is define as The expansion parameter  is defined as The deceleration parameter   t q is defined as It grants the acceleration of the universe for The Ricci scalar R of model is given by

State finding Parameter
The pair of state finding parameter (r, s) is given by [30], Where r is jerk parameter, H is a mean Hubble parameter given in equation (7), q is deceleration parameter, a is average scale factor and over headed triple dots denotes the derivative with respect to cosmic time t .

Solution of field equation
In the presence of perfect fluid as source given in equation (4), the field equation (2) analogous with metric given in equation (5) gives the set of linearly independent equations as Here the overhead dots represent a differentiation with respect to t .
The plytropic equation of state followed by perfect fluid (4) is given by Adhav et al. [28]    On subtracting equation (12), (11) and solving them, we get The power law relation between F and a presented by Sharif and Shamir [27], is given by where k is constant.
Using equation (7) and (19), we get (20) Using equation (17) and equation (20), we get Using equation (13), (14) and (16), we get . Hence to solve the above system of non-linear independent equations we consider the special law of variation of Hubble parameter given by Berman [31] which gives the constant deceleration parameter of model.
where a is average scale factor.
Here the deceleration parameter is taken negative for purpose of accelerating model of the universe.
On solving equation (23), it gives q t a    is the condition of expansion given by equation (24).
Using equation (21) and (24) (26) From above equations it is seen that product of power and exponential exist in  (27) On taking constant of integration 0 1   the above metric has initial singularity at cosmic time 0  t and approaches to isotropy.

Properties of cosmological model
Using equation (25) and (26) the Ricci scalar R of model is given by Using equation (28) the value of function (29) where, Above equation (29) shows that the ) (R f is function of Ricci scalar of studied line element and is positive and decresing function of cosmic time t . This nature of function is equivalent to the function found by Sharif et al. [32]. The special volume of model by using equation (6) is given by (30) on using equation (7)  On using equation (7) the anisotropic parameter  of model is 0   , …………….. (32) This means that the model shows the completely homogeneous universe. Also this outcome of parameter gives the full support to cosmological principle stated as "on large scales, the universe is both homogeneous and isotropic." Using equation (9) the expansion parameter  is given by (33) Here in the throughout discussion the value of q is taken as negative constant term.
The state finding parameters are given by using equation (12)

Conclusion
In current study the examination of LRS Bianchi type-I line element has been done with constant deceleration parameter q and polytropic EoS in the context of ) (R f gravity. Throughout the work, the deceleration parameter is assumed to be negative having a constant value which is in line with the accelerating expansion of the universe. It can be observed that the function of Ricci scalar R is gradually decreasing with cosmic time t and approaching to zero. The graphical representations show that, the energy density  of universe is decreasing with cosmic time t and also that with cosmic time t the pressure is reducing gradually and as   t , the pressure 0  p . It is seen that the volume of the universe is under exponential expansion and hence universe has cosmic expansion. The observed result is consistent with the present cosmological observations. Hence, we feel that the obtained result is very useful for the researchers to analyze the characteristics of the universe in the context of ) (R f gravity.

Disclosure of conflict of interest
No conflict of interest to be disclosed.