A Dynamical Systems Approach to the Oscillatory Singularity in Bianchi Cosmologies

Abstract

To understand the observational properties of cosmological models, in particular, the temperature of the cosmic microwave background radiation, it is necessary to study their null geodesics. Dynamical systems theory, in conjunction with the orthonormal frame approach, has proved to be an invaluable tool for analyzing spatially homogeneous cosmologies. It is thus natural to use such techniques to study the geodesics of these models. We therefore augment the Einstein field equations with the geodesic equations, all written in dimensionless form, obtaining an extended system of first-order ordinary differential equations that simultaneously describes the evolution of the grav-itational field and the behavior of the associated geodesics. It is shown that the extended system is a powerful tool for investigating the effect of spacetime anisotropies on the temperature of the cosmic microwave background radiation, and that it can also be used for studying geodesic chaos.

The cosmological billiard attractor

“…is a Kasner sequence with period 3, see [37] or [12, p. 236]. Note that this Kasner sequence is associated with a heteroclinic cycle (or rather, a set of entangled heteroclinic cycles) in the state space description of sequences.…”

Numerical Approaches to Spacetime Singularities

“…LeBlanc et al [178] have shown (analytically and numerically) that MD can arise in Bianchi VI 0 models with magnetic fields (see also [181]). In essence, the magnetic field provides the wall needed to close the potential in a way that yields the BKL map for u [29].…”

Numerical Approaches to Spacetime Singularities

“…LeBlanc et al [131] have shown (analytically and numerically) that MD can arise in Bianchi VI 0 models with magnetic fields (see also [133]). In essence, the magnetic field provides the wall needed to close the potential in a way that yields the BKL map for u [20].…”