[Paper Review] Bianchi Type-II Cosmological Model with Viscous Fluid

A. Banerjee, S. B. Duttachoudhury+1 more|arXiv (Cornell University)|May 8, 2021

Cosmology and Gravitation Theories11 references39 citations

TL;DR

The paper derives exact solutions for a locally rotationally symmetric Bianchi type-II cosmology with viscous fluid, under a barotropic equation of state, highlighting the role of shear and bulk viscosity on singularity, entropy production, and evolution. It also analyzes special cases with vanishing bulk viscosity and with a perfect fluid.

ABSTRACT

A spatially homogeneous and locally rotationally symmetric Bianchi type-II cosmological model under the influence of both shear and bulk viscosity has been studied. Exact solutions are obtained with a barotropic equation of state between thermodynamics pressure and the energy density of the fluid, and considering the linear relationships amongst the energy density, the expansion scalar and the shear scalar. Special cases with vanishing bulk viscosity coefficients and with the perfect fluid in the absence of viscosity have also been studied. The formal appearance of the solutions is the same for both the viscous as well as the perfect fluids. The difference is only in choosing a constant parameter which appears in the solutions. In the cases of either a fluid with bulk viscosity alone or a perfect fluid, the barotropic equation of state is no longer an additional assumption to be imposed; rather it follows directly from the field equations.

Motivation & Objective

• Motivate the study by incorporating viscosity into anisotropic cosmological models to understand dissipative effects on evolution and singularities.
• Obtain exact solutions for an LRS Bianchi type-II spacetime under viscosity with a barotropic equation of state.
• Explore how viscosity alters expansion, shear, and density relations, and compare with non-viscous (perfect fluid) cases.
• Investigate special cases where bulk viscosity vanishes or both bulk and shear vanish (perfect fluid) and compare with the general viscous solutions.

Proposed method

• Adopt an LRS Bianchi type-II metric with scale factors R(t) and S(t).
• Use the viscous fluid energy-momentum tensor with effective pressure ar{p}=p-(eta-2/3 eta); v^a_{;a} = ; anisotropic shear B.
• Impose barotropic EOS p= ho and linear relations = ho/ heta^2 and =^2 ^2 between density, expansion R, and shear 2.
• Derive solutions by reducing field equations to a relation = abla(R) with S=R^, leading to a set of coupled ODEs for R(t) and derived quantities.
• Obtain exact time dependences for R(t), S(t), , , , , , and the viscosity coefficients  and , including special cases with =0 (bulk viscosity) and =0 (perfect fluid).
• Analyze energy conditions and entropy production via the Raychaudhuri equation and the entropy relation  R^2 S  with viscosity terms.

Experimental results

Research questions

• RQ1 What are the exact cosmological solutions for an LRS Bianchi type-II model with viscous fluid under p=psilon ho?
• RQ2 How do shear and bulk viscosity affect the expansion, shear, density, and entropy production in the early and late universe within this model?
• RQ3 Do the viscous solutions reduce to known perfect-fluid or bulk-viscosity-only solutions, and what changes in parameter constraints arise?
• RQ4 Under what conditions do the Hawking-Penrose energy conditions hold, and what is the nature of the initial singularity (point-like) in these solutions?
• RQ5 What are the special cases when bulk viscosity vanishes or when both bulk and shear vanish, and how do these compare to the viscous case?

Key findings

• The model yields self-consistent exact solutions with S=R^\lambda, leading to explicit R(t) and S(t) forms.
• In expansion, viscosity diminishes over time; in contraction, viscosity grows toward a final singular state.
• The initial singularity is a zero-volume (point-type) singularity with divergent densities and viscosities, and the expansion rate (R) decreases over time.
• Entropy production due to viscosity is significant, with the entropy scaling as  R^k with k depending on  and psilon, indicating growth in the expanding phase.
• Imposing energy conditions yields parameter constraints (lambda, C1, epsilon) that must be satisfied for physical solutions; stiff fluid (p= ho) is not allowed in the perfect fluid case.
• Special case analyses show that with only bulk viscosity or with a perfect fluid, the solutions formally coincide with the viscous case, differing only by a constant parameter, and the barotropic EOS can emerge directly from field equations in these cases.