This paper is devoted to studying the generalized Chaplygin gas models in Bianchi type III space- time geometry with time varying bulk viscosity, cosmological and gravitational constants. We are considering the condition on metric potential . Also to obtain deterministic models we have considered physically reasonable relations like , and the equation of state for generalized Chaplygin gas given by . A new set of exact solutions of Einstein’s field equations has been obtained in Eckart theory, truncated theory and full causal theory. Physical behaviour of the models has been discussed.
The motivation behind the stimulated interest in anisotropic cosmological models is experimental study of isotropy of the cosmic microwave background radiation and speculation about the amount of the helium formed at the early stages of the evolution of the universe. The existence of anisotropic stage of the universe is supported by experimental data and numbers of scientific arguments in the literature which is supposed to be phased out during evolution. The present day universe is isotropic and homogeneous. In understanding the behavior of universe at early stages, anisotropic cosmological models have played a significant role. Singh and Singh [
The astronomical observations of type Ia supernovae [
The idea of variability of G originated with the work of Dirac [
In the literature it has been discussed that during the early stages of evolution of the universe, bulk viscosity could arise in many circumstances and could lead to an effective mechanism of galaxy formation [
It has been observed that the universe has entered an acceleration phase and some exotic dark energy must presently dominate [
We consider the Bianchi type III metric in the form
For perfect fluid distribution Einstein’s field equations with gravitation and cosmological constant may be written as
where G is gravitational constant,
The energy momentum tensor
where
where p is equilibrium pressure,
Einstein’s filed Equation (2) for the metric (1) leads to
where the over head dot denote differentiation with respect to time t. An additional equation for the time changes of G and
Equation (10) splits into two equations as
For the full causal non-equilibrium thermodynamics the causal evolution equation for bulk viscosity is given by [
Since there are five basic Equations (5)-(9) and eight unknowns viz.
Case I: Non-Causal Cosmological Solution
For non causal solution
To find the complete solution of the system of equations, following relations are taken into consideration.
The power law relation for bulk viscosity is taken as
The equation of state is
We assume the solution of the system in the form
where n is constant. On integrating Equation (17), we get
where a and b are constants of integration.
Using Equations (16) and (17) in (11), we obtain
which on solving yields
where C is constant of integration.
From Equation (20) and
On differentiating Equation (20), one can get
Now with the help of Equations (17) and (18), Equation (8) becomes
which on differentiation leads to
Substituting Equations (12), (14) and (17) into Equation (23), we have
By use of Equations (15) and (21), Equation (24) yields
where
From Equation (25) and
Using Equations (20) and (25), Equation (22) gives
From Equation (26) and
Now from Equations (15) and (20), we have
From Equation (27) and
From Equations (14) and (17), the expression for bulk viscous stress is given by
Thus the metric (1) reduces to the form
The shear scalar [
For this model the Shear scalar is
From Equation (31) it is clear that as
The expansion scalar is defined by
For this model expansion scalar is given by
The deceleration parameter is related to the expansion scalar as
For this model
Foe accelerating expansion of the universe the deceleration parameter q < 0 for
Case II: Causal Cosmological Solution
In addition to physically plausible relations (16), (17), in this case we assume
where H is Hubble parameter, given by
From Equations (17) and (39), the Hubble parameter is given by
Using Equations (17)-(18), (38) and (40) in Equation (8), we get
where
From Equations (20) and (41),
From Equation (42) and
Substitute the values from Equations (17), (20), (38) and (42) in Equation (5), we get
where
By use of Equation (20), Equation (43) gives
(i) Evaluation of Bulk viscosity in Truncated Causal Theory
Now we study variation of bulk viscosity coefficient
In order to have exact solution of the system of equations one more physically plausible relation is required.
Thus, we consider the well known relation
Using Equations (17), (20), (44) and (46) in Equation (45) one can obtain
(47)
where
(ii) Evaluation of Bulk Viscosity in Full Causal Theory
It has already been mentioned that for full causal theory
On the basis of Gibb’s inerrability condition, Maartens [
which with the help of Equation (16) gives
using Equations (20), (40), (46) and (50) in Equation (48) one can obtain
which on simplification yields the expression for bulk viscosity
(51)
In this paper we have studied bulk viscous Bianchi type III space-time geometry with generalized Chaplygin gas and time-varying gravitational and cosmological constants. We have obtained a new set of Einstein’s equations
by considering
constant are decreasing as gravitational constant G(t) is increasing with time. Shear dies out with evolution of the universe for large value of t. For accelerating model of the universe, the deceleration parameter q < 0
for
which is considered to be fundamental and match with the observations. In order to have clear idea of variation in behavior of cosmological parameters, relevant graphs have been plotted. All graphs of cosmological parameters go with cosmological observations.
S. K. would like to thank U. G. C. New Delhi for providing financial support under the scheme of major research project F. No. 41-765/2012 (SR). S. K. and R. K. would like to thank Inter University Centre for Astronomy and Astrophysics for providing facilities.