We present a general framework for analyzing spatially inhomogeneous cosmological dynamics. It employs Hubble-normalized scale-invariant variables which are defined within the orthonormal frame formalism, and leads to the formulation of Einstein's field equations with a perfect fluid matter source as an autonomous system of evolution equations and constraints. This framework incorporates spatially homogeneous dynamics in a natural way as a special case, thereby placing earlier work on spatially homogeneous cosmology in a broader context, and allows us to draw on experience gained in that field using dynamical systems methods. One of our goals is to provide a precise formulation of the approach to the spacelike initial singularity in cosmological models, described heuristically by Belinskiǐ, Khalatnikov and Lifshitz. Specifically, we construct an invariant set which we conjecture forms the local past attractor for the evolution equations. We anticipate that this new formulation will provide the basis for proving rigorous theorems concerning the asymptotic behavior of spatially inhomogeneous cosmological models.

This article is devoted to a study of the asymptotic dynamics of generic solutions of the Einstein vacuum equations toward a generic spacelike singularity. Starting from fundamental assumptions about the nature of generic spacelike singularities, we derive in a step-by-step manner the cosmological billiard conjecture: we show that the generic asymptotic dynamics of solutions is represented by (randomized) sequences of heteroclinic orbits on the "billiard attractor". Our analysis rests on two pillars: (i) a dynamical systems formulation based on the conformal Hubblenormalized orthonormal frame approach expressed in an Iwasawa frame; (ii) stochastic methods and the interplay between genericity and stochasticity. Our work generalizes and improves the level of rigor of previous work by Belinskii, Khalatnikov, and Lifshitz; furthermore, we establish that our approach and the Hamiltonian approach to "cosmological billiards", as elaborated by Damour, Hennaux, and Nicolai, can be viewed as yielding "dual" representations of the asymptotic dynamics.e-print archive: http://lanl.arXiv.org/abs/gr-qc/0702141 J. MARK HEINZLE, CLAES UGGLA, AND NIKLAS RÖHR

We study the late time evolution of a class of exact anisotropic cosmological solutions of Einstein's equations, namely spatially homogeneous cosmologies of Bianchi type VII 0 with a perfect fluid source. We show that, in contrast to models of Bianchi type VII h which are asymptotically self-similar at late times, Bianchi VII 0 models undergo a complicated type of self-similarity breaking. This symmetry breaking affects the late time isotropization that occurs in these models in a significant way: if the equation of state parameter γ satisfies γ ≤ 4 3 the models isotropize as regards the shear but not as regards the Weyl curvature. Indeed these models exhibit a new dynamical feature that we refer to as Weyl curvature dominance: the Weyl curvature dominates the dynamics at late times. By viewing the evolution from a dynamical systems perspective we show that, despite the special nature of the class of models under consideration, this behaviour has implications for more general models.

A canonical transformation is performed on the phase space of a number of homogeneous cosmologies to simplify the form of the scalar (or, Hamiltonian) constraint. Using the new canonical coordinates, it is then easy to obtain explicit expressions of Dirac observables, i.e. phase space functions which commute weakly with the constraint. This, in turn, enables us to carry out a general quantization program to completion. We are also able to address the issue of time through "deparametrization" and discuss physical questions such as the fate of initial singularities in the quantum theory. We find that they persist in the quantum theory inspite of the fact that the evolution is implemented by a 1-parameter family of unitary transformations. Finally, certain of these models admit conditional symmetries which are explicit already prior to the canonical transformation. These can be used to pass to quantum theory following an independent avenue. The two quantum theories -based, respectively, on Dirac observables in the new canonical variables and conditional symmetries in the original ADM variables-are compared and shown to be equivalent.