^{1}

^{*}

^{1}

^{*}

^{2}

Bianchi type-IX cosmological models with variable equation of state (EoS) parameter have been investigated in general relativity when universe is filled with dark energy. The field equations have been solved by considering (i)
*q*=
*B* (variable); (ii)
, where k and m are constants; (iii)
, where k is constant and R is average scale factor; (iv)
which gives
. This renders early decelerating and late time accelerating cosmological models. The physical and geometrical properties of the models are also discussed.

Most remarkable observational discoveries in cosmology prevail that the universe is undergoing an accelerated expansion. Analysis of type-Ia supernovae (SN Iae) [

DE is characterized by the equation of state (EoS) parameter defined by

The studies of Bianchi type models are important in achieving better understanding of anisotropy in the universe. Moreover, the anisotropic universes have greater generality than FRW isotropic models. The simplicity of the field equations made Bianchi type space-times useful. Bianchi type I-IX cosmological models are homogeneous and anisotropic. Bianchi type-IX universe is studied by a number of cosmologists because of familiar solutions like Robertson-Walker Universes, the de-sitter universe, the Taub-Nut solutions, etc. Reddy and Naidu [

To study cosmological models one of the important observational quantities is the deceleration parameter q. In any cosmological model, the Hubble constant

cosmological solutions called the models with Constant Deceleration Parameter (CDP) by assuming

(Berman and Gomide [

During 1960s and 1970s, Redshift magnitude test claimed that, the DP lied between 0 and 1 and thus the universe was decelerating. But the observations of CMBR and SNe-Ia experiments concluded that the expansion of the universe was accelerating. Riess et al. [

In 2006, Pradhan et al. [

where R is the average scale factor. Yadav [

In 2011, Akarsu and Dereli [

In 2009, Singha and Debnath [

q is defined as

investigated Bianchi cosmological models by using special form of DP. Recently, Chirde and Shekh [

In 2012, Saha et al. [

time dependent deceleration parameter such that the model generates a transition of the universe from early decelerating phase to the recent accelerating phase. Pradhan and Amirshachi [

[

Motivated by this study about the deceleration parameter from constant to time dependent, an attempt is made to study Bianchi type-IX space-time when universe is filled with DE with time dependent DP in general relativity. This work is organized as follows: In Section 2, the model and field equations have been presented. The field equations have been solved in Section 3 by choosing four different time depending deceleration parameters. The physical and geometrical behaviors of the models have been discussed in Sections 3.1-3.4. In the last Section 4, concluding remarks have been expressed.

Bianchi type-IX metric is considered in the form,

where a, b are scale factors and are functions of cosmic time t.

The energy-momentum tensor for the anisotropic dark energy fluid is

Here

The Einstein field equations in gravitational units (

Here

In the co-moving coordinate system the field Equations (3) for the metric (1) and with the help of energy- momentum tensor (2) can be written as

where the overdot (^{×}) denotes the differentiation with respect to t.

From Equations (6) and (7) we see that, the deviations from

The field Equations (4) to (6) are a system of three highly non-linear differential equations with five unknown parameters

(i) The expansion scalar (

, (8)

where m is proportionality constant.

The motive behind assuming condition is explained with reference to Thorne [

dies place the limit

(ii) Now one extra condition is needed to solve the system completely. Hence different models of deceleration parameters are considered as

The average scale factor as an integrating function of time is (Saha et al. [

where r and l are positive constants.

Using Equation (9), the value of DP becomes

The proposed law yields a time-dependent DP which describes the transition of the universe from the early decelerating phase to current accelerating phase.

The metric (1) is completely characterized by average scale factor R is given by

Solving Equations (8) and (11), Equation (9) reduces to

With the help of Equation (8), Equation (12) leads to

Using Equations (12) and (13), the metric (1) takes the form

Equation (14) represents Bianchi type-IX DE cosmological model in general relativity with time-dependent deceleration parameter.

Some Physical Properties of the Model

For the cosmological model (14), the physical quantities such as spatial volume V, Hubble parameter H, expansion scalar

The spatial volume is in the form

The Hubble parameter is given by

The expansion scalar is

The mean anisotropy parameter is

The shear scalar is given by

Here

The energy density is obtained as

The EoS parameter is

The skewness parameter is given by

For illustrative purposes, evolutionary behaviors of some cosmological parameters are shown graphically (Figures 1-3).

Physical Behavior of the Model

From Equations (15) and (19), we observed that, the spatial volume is zero at

and

also constant, hence the model is anisotropic throughout the evolution of the universe except at

We consider the deceleration parameter to be variable parameter (Pradhan et al. [

where R is the average scale factor.

From Equation (24), we obtain

To solve Equation (25), we assume

The general solution of (25) with assumption

To solve (26) we have to choose

From Equations (26) and (27), we get

The choice of

where

Integrating Equation (28) and without loss of generality assuming constants of integration to be zero, we have

Solving Equations (8) and (11), Equation (30) reduces to

With the help of (8), Equation (31) leads to

Using Equations (31) and (32), the metric (1) takes the form

Equation (33) represents Bianchi type-IX DE cosmological model in general relativity with variable deceleration parameter.

Some Physical Properties of the Model

For the cosmological model (33), the physical quantities such as spatial volume V, Hubble parameter H, expansion scalar

The spatial volume is given by

The Hubble parameter is in the form

The expansion scalar is

The mean anisotropy parameter is obtained as

The shear scalar is given by

Here

The energy density is obtained as,

The EoS parameter is,

The skewness parameter is given by,

The deceleration parameter is obtained as,

For illustrative purposes, evolutionary behaviors of some cosmological parameters are shown graphically (Figures 4-6).

Physical Behavior of the Model

From Equations (34) and (38), we observed that, the spatial volume is zero at

epoch

model does not approach isotropy. In

that the model starts with infinite density and as time increases the energy density tends to a finite value. Hence, after some finite time, the model approaches steady state. In

We consider the linearly varying deceleration parameter (Akarsu and Dereli [

where R is the average scale factor,

For

giving constant value of deceleration parameter.

Using this law one can generalize the cosmological solutions that are obtained via constant deceleration parameter.

After solving (45), we obtain the three different forms of the mean scale factors

where

Solving Equations (8) and (11), Equation (49) reduces to

With the help of Equation (8), Equation (50) leads to

With the help of Equations (50) and (51), the metric (1) takes the form

Equation (52) represents Bianchi type-IX DE cosmological model in general relativity with linearly varying deceleration parameter.

Some Physical Properties of the Model

For the cosmological model (52), the physical quantities such as spatial volume V, Hubble parameter H, expansion scalar

The spatial volume is given by

The Hubble parameter is in the form

The expansion scalar is obtained as

The mean anisotropy parameter is

The shear scalar is given by

Here

The energy density is

The EoS parameter is obtained as

The skewness parameter is given by

For illustrative purposes, evolutionary behaviors of some cosmological parameters are shown graphically (Figures 7-9).

Physical Behavior of the Model

From Equations (53), we observed that, the spatial volume is finite i.e. the universe starts evolving with some finite volume at

except at

Following Singha and Debnath [

where R is the average scale factor,

Solving Equation (62), the average scale factor R is given by

where

Solving Equations (8) and (11), Equation (63) reduces to

With the help of Equation (8), Equation (64) leads to

Using Equations (64) and (65), the metric (1) takes the form

Equation (66) represents Bianchi type-IX DE cosmological model in general relativity with special form of deceleration parameter.

Some Physical Properties of the Model

For the cosmological model (66), the physical quantities such as spatial volume V, Hubble parameter H, expansion scalar

The spatial volume is given by

The Hubble parameter is in the form

The expansion scalar is obtained as

The mean anisotropy parameter is

The shear scalar is given by

Here

The energy density is

The EoS parameter is given by

The skewness parameter is in the form,

For illustrative purposes, evolutionary behaviors of some cosmological parameters are shown graphically (Figures 10-12).

Physical Behavior of the Model

From Equation (67), the spatial volume is finite i.e. the universe starts evolving with some finite volume at

and

cept at

A Bianchi type-IX cosmological model has been obtained when universe is filled with DE in general relativity. To find deterministic solution, we have considered five different models of deceleration parameter which yields time-dependent scale factors.

In model 3.1, the solution of the field equations has obtained by choosing the time dependent DP

type singularity. In early phase of universe, the value of deceleration parameter is positive while as

In model 3.2, the solution of the field equations has obtained by choosing the variable DP

In model 3.3, the solution of the field equations has obtained by choosing the linearly varying DP

early phase of universe, the value of deceleration parameter is positive, after some finite time the model changes from positive to negative, while as

In model 3.4, the solution of the field equations has obtained by choosing the special form of DP

non-singular. In early phase of universe, the value of deceleration parameter is positive while as

It is worth mentioning that in all cases, the models obtained are expanding, shearing, non-rotating and do not approach isotropy for large t. Further the models are anisotropic throughout the evolution. Thus, DE models are in good harmony with recent cosmological observations (Perlmutter et al. [

H. R.Ghate,Atish S.Sontakke,Yogendra D.Patil, (2015) Bianchi Type-IX Anisotropic Dark Energy Cosmological Models with Time Dependent Deceleration Parameter. International Journal of Astronomy and Astrophysics,05,302-323. doi: 10.4236/ijaa.2015.54033