Cosmological Billiards

TL;DR

This work demonstrates that near spacelike singularities, Einstein-dilaton-p-form systems admit an ultralocal, billiard-like description in hyperbolic beta-space, where the dynamics reduces to free motion interrupted by reflections against a finite set of dominant walls. The analysis leverages Iwasawa decomposition to isolate diagonal scale factors and shows asymptotic freezing of off-diagonal and p-form degrees of freedom, yielding a geodesic billiard in γ-space with walls determined by symmetry, gravitational, and p-form contributions. Depending on the wall volume, the billiard is either finite (chaotic oscillations) or infinite (Kasner-like behavior); the work also connects these dynamics to Kac-Moody algebras, formulating a KM sigma-model description that reproduces the asymptotics and offers a group-theoretical perspective on the underlying symmetries. These findings illuminate the role of p-forms, dilatons, and higher-dimensional gravity in shaping the approach to cosmological singularities and suggest deep links to infinite-dimensional symmetries in fundamental theories.

Abstract

It is shown in detail that the dynamics of the Einstein-dilaton-p-form system in the vicinity of a spacelike singularity can be asymptotically described, at a generic spatial point, as a billiard motion in a region of Lobachevskii space (realized as an hyperboloid in the space of logarithmic scale factors). This is done within the Hamiltonian formalism, and for an arbitrary number of spacetime dimensions $D \geq 4$. A key role in the derivation is played by the Iwasawa decomposition of the spatial metric, and by the fact that the off-diagonal degrees of freedom, as well as the p-form degrees of freedom, get ``asymptotically frozen'' in this description. For those models admitting a Kac-Moody theoretic interpretation of the billiard dynamics we outline how to set up an asymptotically equivalent description in terms of a one-dimensional non-linear sigma-model formally invariant under the corresponding Kac-Moody group.