Anisotropic Behavior of Cosmological Models with Exponential and Hyperbolic Scale Factors

In this paper, the cosmological models of the universe are constructed in ( ) , f R T gravity with choice of the functional ( ) , f R T in the form ( ) 1 f R R λ = and ( ) 2 f T T λ = . The space-time considered here is Bianchi Type I and the energy momentum tensor is in the form of perfect fluid. Two cosmological models are presented using a power form of exponential function and a hyperbolic form. The energy conditions along with the state finder diagnostic pair have been obtained and analyzed.


Introduction
Modified theory of gravity is one of the prominent alternative theories to Einstein's general theory of relativity (GR).Though GR is very much known for its excellent success, it is silent on certain specific topics on the study of the universe.One of such topics is the late time cosmic acceleration.It has been learnt that modified theory of gravity has been successful in describing the late time cosmic acceleration and accelerated expansion of the universe.Thus, modified gravity nowadays is a topic of great interest.Out of all prospective variants of modified gravity, the ( ) , f R T gravity theory proposed by Harko et al. [1] is one of the most fascinating theories.In this theory, they have generalized the basic ( ) f R theory by taking the gravitational Lagrangian as a random function of Ricci scalar R and trace of the stress-energy tensor T. In this theory, the action for a polynomial function of Ricci scalar is defined as the Einstein-Hilbert action that leads to obtain the Einstein field equation according to principle of least action.Second, different types of operation such as variation of action were placed.Considering the universe was filled by perfect fluid that the stress energy-momentum tensor of the matter is shown as T µν , the action for ( ) , f R T can be described as After some manipulation on variation of action and assuming the relation 2T pg ) where the functional ( ) , f R T is the function of Ricci scalar R and stress-energy momentum tensor T, R f and T f respectively denote the partial derivative of ( ) , f R T with respect to R and T respectively.
Several aspects of ( ) , f R T gravity have been investigated so far: Yousaf et al. [2] have investigated causes of irregular energy density in ( ) Alves et al. [3] have studied the gravitational waves in ( ) , f R T gravity.It can also be noted that, the recent starting of the gravitational wave astronomy through various detections by LIGO (Laser Interferometer Gravitational-Wave Observatory) will be useful in discriminating between Einstein's General Theory of Relativity and the extended theories of gravity [4] [5].Sharif and Zubair [6] have discussed the equilibrium picture of thermodynamics at the apparent horizon of FRW universe whereas Zaregonbadi et al. [7] obtained the dark matter solution from ( ) , f R T gravity.Mishra et al. [8] have studied the dynamical behaviour of the ( ) , f R T gravity.Also, Mishra and Vadrevu [9] have obtained the exact solution in cylindrically symmetric space-time.Agrawal and Pawar [10] obtained the magnetized domain wall in ( )

T gravity in a
Bianchi type V space-time.Chakraborty [11] has suggested that a part of the arbitrary function ( ) , f R T can be determined if the conservation of stress-energy tensor is taken into consideration in the original version [1].Several authors have studied different aspects of viscous fluid in modified theory of gravity [12] [13] [14] [15] [16].Shamir and Kanwal [17] studied the anisotropic universe using Noether symmetries in locally rotationally symmetric Bianchi type I whereas Mishra et al. [18] reconstructed the anisotropic universe in Bianchi type V space-time.The anisotropic behaviour of the cosmological models has been investigated by many authors [19] [20] [21] [22] [23].
We intend to assume the functional form of ( ) , f R T in such a way that it can be reduced to the usual field equations in GR.A popular choice is ( ) , where λ is constant.So, the field equation in modified theory of gravity can be reduced to where ( ) is the Effective Cosmological Constant henceforth Journal of High Energy Physics, Gravitation and Cosmology abbreviated as ECC.

Basic Formalism of the Model
We have considered Bianchi type I space-time in the form ( ) The metric potentials A and B are function of cosmic time only.The energy-momentum tensor for the perfect fluid can be expressed as ( ) whereas u µ is the four velocity vector of the fluid.Now, the field Equation ( 3), for the space-time (4) with the energy momentum tensor ( 5) can be obtained as where 8π 3 2 α λ = + and dot represents the derivative with respect to the cosmic time.Equations ( 6)-( 8) are having four knowns viz.: A, B, p, ρ.In view of this with an intention to obtain a solution to the field equation, we have transformed all the field Equations ( 6)-( 8) in terms of Hubble rate.The directional Hubble rates can be considered as . Now, the set of field Equations ( 6)-( 8) can be reduced to Now, with suitable algebraic manipulations the pressure and energy density with respect to Hubble term can be obtained as Then We can study the dynamical features of the model, which would be decided by the physical quantities given in ( 12)- (15).However all these quantities depend on the Hubble parameter which is subsequently depends on the mean scale factor.So, if we understand the behaviour of mean scale factor, the background cosmology and the associated dynamics can be studied.

Case-I
In this case, we have considered the hyperbolic scale factor in the form ( ) . Subsequently the Hubble parameter and deceleration parameter can be obtained as linear anisotropic relation among the directional Hubble rates is assumed i.e.
x y

H kH =
, where k is a constant.Hence, from Equations (( 12), ( 13)), the pressure and energy density can be obtained as The graphical behaviour of pressure and energy density has been represented respectively in Figure 1 and Figure 2. It has been observed that the pressure starts from a high negative value and subsequently settled near a small value.At the same time, the energy density starts from a high positive value and at late times reduces to a small value, however in the entire evolution it remains in the positive domain only.It can be noted that the scaling constant λ is also The EoS parameter ECC can be reduced from Equation ( 14) with the hyperbolic scale factor as ( )( ) ( )( ) The graphical behaviour of the EoS parameter and ECC respectively represented in Figure 3 and

Case-II
In order to have a better understanding on the anisotropic universe, in this case, we have considered the scale factor in the power form of as a fraction of exponential function in the form ( ) The pressure increases from some large negative value 0.8 ≈ − at an initial stage and slowly vanishing to small values at late times.It is felt that the choice of scaling parameter affects substantially the magnitude of the pressure (Figure 5).On the other hand, the energy density always remain in the positive domain and decreases to small value at late times (Figure 6).
The EoS parameter and ECC can be obtained as ( ) ( ) It is observed that the EoS parameter as represented in Figure 7

Physical Parameters of the Model
The state finder diagnostic pair shows the geometrical behaviour of the model.
The acceptability of the model depends on the ( ) , r s diagnosis that compares with the Λ CDM model.For Case-I, the pair can be calculated as Similarly for case II, the pair can be obtained as  For Case-I the energy conditions can be calculated as Similarly, for Case-II, the energy conditions can be obtained as The graphical behaviour of the energy conditions for Case-I and Case-II are represented respectively in Figure 11     .It is found that for 1 k = , the rate of anisotropy vanishes and confirms the isotropic nature of the space-time.

Conclusion
In this paper, we have studied the anisotropic behaviour of the cosmological model constructed with a Bianchi type I space-time and two different scale factors.In both the models, the state finder diagnostic pair is in the accepted value.The effect of the anisotropic parameters k is investigated.The ratio of anisotropy reduces to zero when the anisotropic parameters in both the cases are unity.The behaviour of energy conditions remain same irrespective of the value of the scaling constant.The state finder diagnostic pair found to be in accordance with the present observed value.It is also concluded that an increase in cosmic anisotropy value in this work substantially affects the energy conditions.
Different scale factors has different roles in the study of cosmic dynamics of the universe.For instance: power law cosmology emerged as an alternative to ΛCDM model; the cosmic dynamics dominated by the exponential scale factor at late phase; and the hybrid scale factor produces a cosmic transit from early deceleration to late time cosmic acceleration.However not many research has been done on hyperbolic scale factor and fractional power of an exponential functions to understand the cosmic dynamics of the universe.Therefore, we have considered two scale factors such where η and ζ both are constants.

Figure 4 .Figure 3 .Figure 4 .
Figure 3. Graphical behaviour of w versus t ( ) 0.01, 0.45, 0.5 .k η λ = = = initial time and as time increases the value of the parameter increases.However, it remains in the negative domain and stay in the quintessence region as suggested observationally.At the same time, the ECC (Figure 8) remains entirely in the positive domain and gradually decreases and remain idle at the late time of the evolution.

Figure 9 .
Figure 9. Graphical behaviour of r versus t.

Figure 10 .
Figure 10.Graphical behaviour of s versus t.
and Figure 12 for representative values of the parameters.The energy conditions in Figure 11 are calculated for 0(Q), Orange(U), Blue (G) phase in both the graphs respectively represent the behaviour for WEC, SEC and DEC.We have explored the effect of

Figure 11 .
Figure 11.Graphical behaviour of Q, U, G versus t.

Figure 12 .
Figure 12.Graphical behaviour of Q, U, G versus t.