The Dynamics of a Two-Fluid Bianchi Type I Universe

TL;DR

This paper analyzes the global dynamics of a two-fluid Bianchi Type I universe with bulk viscosity and vacuum energy, identifying key fixed points and conditions for isotropization relevant to current cosmological observations.

Contribution

It provides a detailed fixed-point and global dynamical analysis of a two-fluid Bianchi Type I model with viscosity and vacuum energy, highlighting conditions for isotropization.

Findings

Identification of fixed points including FLRW, Kasner, and de Sitter universes.

Conditions under which the universe approaches isotropy.

The flat FLRW universe with vacuum and non-vacuum energy acts as a future attractor.

Abstract

We use a dynamical systems approach based on the method of orthonormal frames to study the dynamics of a two-fluid, non-tilted Bianchi Type I cosmological model. In our model, one of the fluids is a fluid with bulk viscosity, while the other fluid assumes the role of a cosmological constant and represents nonnegative vacuum energy. We begin by completing a detailed fixed-points analysis of the system which gives information about the local sinks, sources and saddles. We then proceeded to analyze the global features of the dynamical system by using topological methods by finding the $\alpha$- and $\omega$-limit sets. The fixed points found are a flat FLRW universe, an Einstein-de Sitter universe, a de Sitter universe, a mixed FLRW universe with both vacuum and non-vacuum energy, and a Kasner universe. We then find conditions for which each equilibrium point was a saddle, sink, or source, and attempt to describe the global and past asymptotic behaviour of the model with respect to each fixed point. The flat FLRW universe solution we found with both vacuum and non-vacuum energy is clearly of primary importance with respect to modelling the present-day universe. In particular, we show that this equilibrium point is a local sink and a saddle of the dynamical system, so there are orbits that approach this equilibrium point in the future. Therefore, there exists a time period for which our cosmological model will isotropize and be compatible with present-day observations of a high degree of isotropy of the cosmic microwave background in addition to the existence of both vacuum and non-vacuum energy.