Statistical Properties of Cosmological Billiards

Abstract

Belinski, Khalatnikov and Lifshitz (BKL) pioneered the study of the statistical properties of the never-ending oscillatory behavior (among successive Kasner epochs) of the geometry near a space-like singularity. We show how the use of a "cosmological billiard" description allows one to refine and deepen the understanding of these statistical properties. Contrary to previous treatments, we do not quotient the dynamics by its discrete symmetry group (of order 6), thereby uncovering new phenomena, such as correlations between the successive billiard corners in which the oscillations take place. Starting from the general integral invariants of Hamiltonian systems, we show how to construct invariant measures for various projections of the cosmological-billiard dynamics. In particular, we exhibit, for the first time, a (non-normalizable) invariant measure on the "Kasner circle" which parametrizes the exponents of successive Kasner epochs. Finally, we discuss the relation between: (i) the unquotiented dynamics of the Bianchi IX (a, b, c or mixmaster) model; (ii) its quotienting by the group of permutations of (a, b, c); and (iii) the billiard dynamics that arose in recent studies suggesting the hidden presence of Kac-Moody symmetries in cosmological billiards.

51 pages, 9 figures, 9 tables; version 2: part of previous Section IV moved to Appendix, minor restyling, some typos corrected, improved quality of figures, published version

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• The mixmaster universe is chaotic
• The mixmaster universe: A chaotic Farey tale
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Cited by in corpus (9)

• Spike Oscillations
• Recent developments concerning generic spacelike singularities
• Timelike BKL singularities and chaos in AdS/CFT
• Spacetime Singularities: Recent Developments
• Semiclassical and quantum behavior of the Mixmaster model in the polymer approach
• Chaos Removal in the $R+q{R}^2$ gravity: the Mixmaster model
• BKL maps and Poincaré sections
• Spike statistics
• Reflections on the hyperbolic plane