Anisotropic Cosmological Model Bianchi Type-VI with Viscous Fluid Containing a Varying ?

Abstract

This paper explores the anisotropic nature of the Cosmological Model Bianchi Type -VI with Viscous Fluid in the presence of cosmological constant ?. Cosmological Model Bianchi Type -VI helps in understanding anisotropic behaviour in the early universe and its implications. To find an exact solution for this model, a supplementary condition between the metric potential is used. Also, it is used the coefficient of shear viscosity is proportional to the scale of expansion, i.e., ???. In this study, I assumed that the viscosity coefficient of a bulk viscosity fluid is a simple power function of energy density, ?(t)=?_0 ?^r and it is solved for r=0 and r=1. The behaviour of cosmological parameters like pressure, energy density will be analysed, and it is also found that the cosmological constant ? it is positive and it is decreasing with respect to time. Lastly, some physical aspects of the models are studied.

Introduction

Cosmology studies the universe as a whole, including its origin, structure, evolution, and fate. Modern observations (e.g., supernovae, CMB, LSS) confirm that:

• The universe has been expanding and accelerating since the Big Bang.

• It is homogeneous and isotropic on large scales, but anisotropies may have existed in the early universe.

To understand early-universe anisotropy, researchers use Bianchi cosmological models—general relativistic models that describe spatially homogeneous but anisotropic spacetimes.

II. Bianchi Models and Viscosity in Cosmology

• Bianchi Types V and VI are particularly useful in modeling early-universe anisotropy.

• Most traditional models assume perfect fluids, but including viscous effects offers a more realistic picture.

• Viscous fluids:

  • Help dissipate anisotropies

  • Can lead to non-singular or inflationary cosmologies

Several researchers (e.g., Misner, Belinski, Banerjee, Sadeghi) have studied Bianchi models with viscous fluids, showing that:

• Viscosity can't remove singularities but modifies their nature.

• It can lead to matter generation by gravitational fields.

III. Role of Time-Dependent Cosmological Constant (Λ)

• Observations (e.g., Type Ia supernovae) suggest Λ is not constant but evolves over time.

• A time-varying Λ(t) helps address the cosmological constant problem.

• Many works (e.g., Zeldovich, Carroll, Pradhan, Vishwakarma) explore cosmologies with variable Λ and show they can lead to:

  • Accelerated expansion

  • Crossing of the phantom divide in the equation of state

Some models also incorporate time-varying gravitational constant G and modified geometries (e.g., Lyra geometry).

IV. Present Work: Bianchi Type VI with Viscosity and Λ(t)

This study focuses on:

• A Bianchi Type VI cosmological model

• A viscous fluid as the matter source

• A time-dependent cosmological constant Λ(t)

Objective:

• Derive exact solutions of the Einstein field equations

• Analyze physical and kinematical properties of the universe model

V. Metric and Field Equations

The Bianchi Type VI metric used is:

ds2=−dt2+A(t)2dx2+B(t)2e−2qxdy2+C(t)2e2qxdz2ds^2 = -dt^2 + A(t)^2 dx^2 + B(t)^2 e^{-2qx} dy^2 + C(t)^2 e^{2qx} dz^2ds2=−dt2+A(t)2dx2+B(t)2e−2qxdy2+C(t)2e2qxdz2

Key components:

• A, B, C: scale factors (functions of cosmic time ttt)

• q: constant defining the anisotropy

The Einstein field equations are derived using this metric and a viscous fluid energy-momentum tensor, including both bulk (ξ) and shear (η) viscosity.

Other important definitions:

• Directional Hubble parameters: H1,H2,H3H_1, H_2, H_3H1?,H2?,H3?

• Mean Hubble parameter: HHH

• Anisotropy parameter: Δ\DeltaΔ

• Expansion scalar: θ=3H\theta = 3Hθ=3H

• Shear scalar: σ2=32ΔH2\sigma^2 = \frac{3}{2} \Delta H^2σ2=23?ΔH2

VI. Constraints and Solution Strategy

Since the Einstein field equations provide 5 equations for 8 unknowns, three assumptions are introduced:

1. Relation between B and C:

   B=kC(k=constant)B = kC \quad (k = \text{constant})B=kC(k=constant)

2. Relation between A and B:

   A=Bn(n=constant)A = B^n \quad (n = \text{constant})A=Bn(n=constant)

3. Proportional shear viscosity:

   η∝θ⇒η=l(A?A+2B?B)\eta \propto \theta \quad \Rightarrow \eta = l\left(\frac{\dot{A}}{A} + 2\frac{\dot{B}}{B} \right)η∝θ⇒η=l(AA??+2BB??)

With these assumptions, the model becomes solvable.

VII. Goals of the Study

• Construct and analyze exact solutions of the Einstein field equations with the above constraints.

• Study the cosmic evolution (e.g., scale factor, anisotropy, expansion, viscosity effects).

• Investigate how a time-dependent Λ influences cosmic acceleration, anisotropy decay, and phantom transitions in the equation of state.

Conclusion

I have studied the Bianchi type-VI an isotropic cosmological model in the presence of viscous fluid with varying cosmological constant ?. My work aims to obtain solutions by using cosmological model Bianchi type-VI. Here I have assumed that the fluid obeys an equation of state of the form p=?? and also assumed bulk viscosity is a simple power function of energy density which is given by ?(t)=?_0 ?^r. The models for the values r=0,1 are studied. From this study, I observed from equation (38) the pressure p continuously decreasing for late time and approaching to negative times cosmological constant ? as t??. From equations (44) and (46), it is observed that the positive cosmological constant is a decreasing function of time and approaches to small value in the present epoch. From equations (43) and (45), it is observed that the energy density is a decreasing function of time and approaches to 0 as t??. From equation (37), it is observed that the scalar expansion factor ? is a decreasing function of ‘t’ and approach to zero as t tends to infinity. Since lim?(t??)???/??=constant, the model is not isotropic for the future large value of t. From equation (32) it is shown that the given model starts with a big-bang at t=0 and the expansion in the model increases as time increases. For this model the spatial volume V?? as t??. The Anisotropic parameter of the expansion is found to be constant. The evolution of Hubble parameter (H) with respect to cosmic time (t) indicates that the value of Hubble parameter (H) is very high at the early time of universe and is decreases very rapidly at late time, and it is constant to zero when the time is increases. Thus, I conclude that the cosmological constant in this model is decreasing function with respect to time and it approaches to small positive value as time increases. From all this, it is clear that the universe is experiencing accelerated expansion in the later phase of evolution.

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