BIANCHI COSMOLOGIES AS DYNAMICAL SYSTEMS WITHOUT THE CONSTRAINT

Abstract

The Bianchi class A cosmology is treated as a nonlinear dynamical system. In the new variables in which Hamiltonian constraint is solved algebraically, the Bianchi class A model assumes the form of autonomous dynamical system in ℝ4 with polynomial form of vector field. It is proposed that the dimension of minimum reduced phase spaces of unconstrained autonomous systems be treated as a measure of generality of solution. The behavior of these models is studied in terms of qualitative analysis of differential equations. It is shown that the more general Bianchi IX and Bianchi VIII models (called Mixmaster models) can be presented as four-dimensional. We argue that the reduced Mixmaster dynamical systems are chaotic in the same sense as the original ones. The Bianchi I and Bianchi II world models are described by one-dimensional and two-dimensional systems, respectively. We also study dynamics of Bianchi VI0 and Bianchi VII0 models as a three-dimensional dynamical system. For two-dimensional dynamical system, the phase portraits are constructed with the Poincaré sphere which allows the analysis of dynamics both in finite domain and at infinity. For the last class of models we find an invariant submanifold on which systems are analyzed in details.