Cosmological billiards

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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 a hyperboloid in the space of logarithmic scale factors). This is done within the Hamiltonian formalism, and for an arbitrary number of spacetime dimensions D ≥ 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 nonlinear σ-model formally invariant under the corresponding Kac–Moody group.