Abstract
We make comparison of the dynamics of the diagonal and nondiagonal Bianchi IX models in the evolution towards the cosmological singularity. Apart from the original variables, we use the Hubble normalized ones commonly applied in the examination of the dynamics of homogeneous models. Applying the dynamical systems method leads to the result that in both cases the continuous space of critical points is higher dimensional and they are of the nonhyperbolic type. This is a generic feature of the dynamics of both cases and seems to be independent on the choice of phase space variables. The topologies of the corresponding critical spaces are quite different. We conjecture that the nondiagonal case may carry a new type of chaos different from the one specific to the usually examined diagonal one.
Highlights
According to the singularity theorems of General Relativity (GR), the evolution of an expanding universe is geodesically past-incomplete
Applying the dynamical systems method leads to the result that in both cases the continuous space of critical points is higher dimensional and they are of the nonhyperbolic type
The Belinskii, Khalatnikov and Lifshitz (BKL) [1,2] scenario predicts that on approach to a space-like cosmological singularity the dynamics of gravitaional field simplifies as time derivatives in Einstein equations dominate over spatial derivatives
Summary
According to the singularity theorems of General Relativity (GR), the evolution of an expanding universe is geodesically past-incomplete. The Belinskii, Khalatnikov and Lifshitz (BKL) [1,2] scenario predicts that on approach to a space-like cosmological singularity the dynamics of gravitaional field simplifies as time derivatives in Einstein equations dominate over spatial derivatives (see [3] for numerical support for BKL) In this regime the evolution of the Universe becomes strongly non-linear and chaotic, comprising expanding and contracting oscillatory phases around the singular point. Original BKL variables and Hubble normalized ones cannot be connected by canonical transformation In both cases, applying dynamical systems method enables identification of the spaces of non-isolated critical (equilibrium) points, which are of nonhyperbolic type. Topologies of these spaces are quite different, and making them explicit constitutes one of the main results of this paper. We apply the Poincaré sphere to deal with the space of critical points in finite region of phase space in “Appendix D”
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