Comparing the dynamics of diagonal and general Bianchi IX spacetime

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.

On the dynamics of the general Bianchi IX spacetime near the singularity

“…Our numerical simulations give strong support to the conjecture concerning the asymptotic dynamics of the general Bianchi IX spacetime put forward long ago by Belinski, Khalatnikov, and Ryan [31]. We remark that approximating the diagonal or non-tumbling case by the asymptotic dynamics (43) is invalid (see [32] for more details).…”

“…One only encounters an effectively diagonal case, which is expressed in terms of the directional scale factors {a, b, c}. So there exist serious differences between the purely diagonal and effectively diagonal cases (see [32] for more details).…”

Generic singularity of general relativity and its quantum fate

“…Numerical simulations of the evolution of the non-diagonal BIX model towards the singularity confirm the existence of the asymptotic form of the dynamics [7]. The evolution defined by Equations ( 1) and ( 2) is different from the commonly known Misner's mixmaster dynamics [8] specific to the diagonal (special) Bianchi IX model [9].…”

Quantum chaos of the BKL scenario

“…17 The dynamics of both models were compared within the dynamical systems method, and it was found that the topologies of the corresponding spaces of critical points are quite different. 18 In what follows we focus our attention on the dynamics of the massive model. It is defined by the following system of ordinary differential equations:…”