5 - Qualitative analysis of Bianchi cosmologies

Summary

In Section 5.1 we give an overview of the use of qualitative methods in analyzing Bianchi cosmologies, expanding on the brief remarks in the Introduction to the book. Section 5.2 provides an introduction to the use of expansion–normalized variables in conjunction with the orthonormal frame formalism, thereby laying the foundation for the detailed analysis of the Bianchi models with non–tilted perfect fluid source in Chapters 6 and 7. In Section 5.3 we discuss, from a general perspective, the use of dynamical systems methods in analyzing the evolution of Bianchi cosmologies, referring to the background material in Chapter 4.

Overview

As explained in Section 1.4.2 there are two main approaches to formulating the field equations for Bianchi cosmologies:

1. the metric approach,

2. the orthonormal frame approach.

In the metric approach the basic variables are the metric components g_αβ(t) relative to a group–invariant, time–independent frame (see (1.89)). This approach was initiated by Taub (1951) in a major paper. After a number of years researchers became aware that the Bianchi models admitted additional structure, namely the automorphism group, which plays an important role in identifying the physically significant variables (also referred to as gauge–invariant variables, or the true degrees of freedom). This group is defined to be the set of time–dependent linear transformations (1.87) of the spatial frame vectors that preserve the structure equations (1.88).