Maximal Coupling of Bianchi Type- II, VIII and IX Space-Time with HDE and DM in Gravity

Received: 23/Mar/2018, Accepted: 19/Apr/2018, Online: 31/Dec/2018 AbstractThe present investigation represents the study the role of Non-Viscous and Viscous Holographic Dark Energy (HDE) and Dark Matter (DM) for a homogeneous anisotropic Bianchi type-II, type-VIII and type-IX cosmological models within the frame work of a maximal coupling between geometry and matter represented by ) , ( T R f theory of gravity proposed by Harko


I. INTRODUCTION
The HDE models have been emerged as a viable candidate to explain the problems of modern cosmology such as the recent accelerated expansion as well as the coincidence problem of the Universe in the recent years. The concept of HDE is based on the holographic principle proposed by Hooft [1] and found its roots in the quantum field theory. Cohan et al. [2] have shown that in the quantum field theory, the formation of black hole set a limit which relates cut-off to IR cut-off. In the formalism of HDE, the Hubble's horizon is a most natural choice for the IR cut-off, but it leads to a wrong Equation of State of dark energy [3]. However, Pavon and Zimdhal [4], Banerjee and Pavon [5] have shown that the viable EoS of dark energy could be achieved by taking the interaction between HDE and (DM). It is strongly believed that the Universe has entered a phase of the accelerated expansion which has been confirmed by the recent observations like Supernovae-Ia, Cosmic Microwave Background Radiation, Baryon Acoustic Oscillation and Planck data. The existence of this accelerated expansion is due to of two types, one is within the framework of General Relativity (GR), the cause of the acceleration can be attributed to the existence of a mysterious component of the Universe dubbed as Dark Energy, which makes up to 70% of the total cosmic energy in the Universe and second is to modification in an action of General Relativity called Modified gravity.
Recently, a new modified gravity theory known as ) , ( T R f gravity have proposed by Harko et al. [6], where R as usual stands for the Ricci scalar and T denotes the trace of energymomentum tensor. This modified theory presents a maximal coupling between geometry and matter. A number of authors have discussed the modified ) , ( T R f gravity in different context to explain the early and late time acceleration of the universe. The general form of the Einstein-Hilbert action for the modified ) , ( T R f gravity in the unit where g stands for the determinant of the metric tensor ij g , R is the Ricci scalar and T represents the trace of the energy-momentum tensor while m L denotes the matter Lagrangian density. The speed of light is taken to be unity. The energy-momentum tensor of matter is defined as Basically this gravity is the generalization of ) (R f gravity based on the coupling between geometry and matter. The .
Assuming that the Lagrangian density m L of matter depends only on the metric tensor components ij g and not on its derivatives, in this case, we obtain ij m m ij ij ( Varying the action S with respect to the metric tensor components ij g , the field equation of The contraction of equation (4) yields The variation of stress energy of perfect fluid has the following expression On the physical nature of the matter field, the field equations also depend through the tensor ij  . Several theoretical models corresponding to different matter contributions for In this paper, we have focused on the first class is an arbitrary function of tress energy tensor of the form where  is coupling constant. For this choice the gravitational field equations of where the prime denotes differentiation with respect to the argument. If the matter source is a perfect fluid then the field equations (in view of Eq. (7)) becomes Many authors had described the recent accelerated expansion by assuming the interaction between HDE and DM in the different theories of gravity. In this paper, instead of taking the interaction between HDE and DM to describe the recent acceleration, we consider that the HDE interacts with the geometry of gravity. This is due to the fact that this modified gravity theory has the interaction between matter and geometry. Therefore, we consider To be more realistic, the prefect fluid Universe is just an approximation of the viscous Universe. The dissipative processes in the relativistic fluid may be modeled as bulk viscosity. The phenomenon of the bulk viscosity arises in the cosmological fluid when the fluid expands (contracts) to fast due to which the system is out of thermal equilibrium. Then, © 2018, IJSRPAS All Rights Reserved 141 the effective pressure becomes negative to restore the thermal equilibrium [7]. Therefore, it is natural to consider the bulk viscosity in an accelerating Universe. It has been shown that inflation and recent acceleration can be explained using the viscous behavior of the Universe, and plays an important role in the phase transition of the Universe [8][9][10].
The concept of viscous DE has been discussed extensively in the literature [11]. Feng and Li [12] show that the age problem of the Ricci dark energy can be alleviated using the bulk viscosity. Motivated by the above works, we extend our analysis to viscous HDE with the same IR cut-off which gives the recent phase transition of the universe.
The paper is organized as follows, Section I contains the introduction of related work and basics of modified ) , ( T R f gravity, Section II contain Metric and field equations, Section III contain the solutions of the field equations to obtained non-viscous holographic dark energy cosmological models, Section IV contain the solutions of the field equations to obtained Viscous holographic dark energy cosmological models, Section V concludes summery of research work with future directions.

II. METRIC AND FIELD EQUATIONS
We consider a spatially homogeneous Bianchi type spacetime in the combined form as where the scale factors A and B are functions of cosmic time t only and  ,  ,  are the Eulerian angles. The equation (15) represents The corresponding field equations (10) for spatially homogeneous Bianchi type space-time in the combined form (15) can be written as Now we discussed the solutions of the field equations to obtained cosmological Models in case of Non-viscous and viscous holographic dark energy in next sections.

BIANCHI TYPE-II COSMOLOGICAL MODEL:
From equations (4.1), we obtain scale factor as   The Generalized Hubble parameter H is found to be The expansion scalar is found as The energy density is given as Energy density of DM is found to be The Equation of State given by The coincidence parameter is given by The matter density parameter and HDE parameter are respectively given by (34)

BIANCHI TYPE-VIII COSMOLOGICAL MODEL
The metric (15) with the help of equations (37) and (38) can be written as The Generalized mean Hubble parameter H is found to be (40) The expansion scalar  is found to be The energy density is given as (43) Energy density of DM is found to be The Equation of State is given by The coincidence parameter is given by The matter density parameter and HDE parameter are respectively given by (48)

BIANCHI TYPE-IX COSMOLOGICAL MODEL
(52) This model shows the same result as that of Bianchi type-VIII model. The metric (15) with the help of equations (51) and (52) can be written as (53) The Generalized mean Hubble parameter H is found to be (54) The expansion scalar  is found to be The energy density is given as (57) Energy density of DM is found to be (58) The Equation of State given by The coincidence parameter is given by The matter density parameter and HDE parameter are respectively given by

IV. VISCOUS HOLOGRAPHIC DARK ENERGY COSMOLOGICAL MODELS
A viscous HDE with the Hubble's horizon as an IR cut-off could be helpful to find the phase transition; Bulk viscosity can produce an accelerated expansion even without dark energy matter due to the presence of an effective negative pressure. We can assume that the effective pressure of HDE is a sum of the thermo dynamical pressure ( h p ) and the bulk viscous pressure  i.e.
where  is the positive coefficient of the bulk viscosity.

Now, the matter Lagrangian is taken as
To analyze the behavior of the Universe, we assume that the viscous HDE matter interacts with the geometry of the Universe. Using where  is coupling parameter of matter with geometry. Trace T of energy momentum tensor is given by From equation (64), the field equations for viscous HDE for spatially homogeneous Bianchi type metrics in the combined form in the framework of   Energy density of DM is found to be The Equation of State given by The coincidence parameter is given by The matter density parameter and HDE parameter are respectively given by

BIANCHI TYPE-VIII VISCOUS COSMOLOGICAL MODEL
Anisotropic pressure of HDE is found to be Energy density of DM is found to be The Equation of State given by The coincidence parameter is given by The matter density parameter and HDE parameter are respectively given by (78)

BIANCHI TYPE-IX VISCOUS COSMOLOGICAL MODEL
Anisotropic pressure of HDE is found to be Energy density of DM is found to be The coincidence parameter is given by

V. CONCLUSIONS
In this paper, we have studied non-viscous and viscous HDE cosmological models with Hubble horizon as an IR cut-off in the frame work of modified ) , ( T R f gravity. The ) , ( T R f gravity theory presents a maximal coupling between geometry and matter has been studied in this work. The consequences of the coupling of matter with the geometry of the Universe instead of taking the interaction between HDE and DM have been explored. However, we have assumed that only HDE of total matter (HDE+DM) couple with geometry. By assuming the interaction between HDE and geometry, we have investigated the possibility whether the Hubble horizon as an IR cut-off could explain an accelerated expansion in ) , ( T R f gravity. The non-viscous and viscous HDE models with Hubble's horizon as an IR cut-off can explain accelerated expansion in the frame work of this modified theory.