No abstract

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.

The authors consider the problem of describing the asymptotic states of orthogonal spatially homogeneous cosmologies, near the big bang and late times. They assume perfect fluid source with a linear equation of state and zero cosmological constant. The Einstein field equations are written as an autonomous system of first-order differential equations, so that results from the theory of dynamical systems can be used. The emphasis is on describing certain general properties of the orbits of the differential equations and the local stability of the equilibrium points.

Motivated by the ideas of quiescent cosmology and Penrose's Weyl tensor hypothesis concerning the 'big bang', the authors give a geometric (and hence coordinate-independent) definition of the concept of 'isotropic singularity' in a spacetime. The definition generalises previous work on 'quasi-isotropic' and 'Friedman-like' singularities. They discuss simple consequences of the definition. In particular it is shown that an isotropic singularity is a scalar polynomial curvature singularity at which the Weyl tensor is dominated by the Ricci tensor. Finally they impose the Einstein field equations with irrotational perfect fluid source. This enables them to give a detailed description of the geometric structure of an isotropic singularity.

We present a new formulation of the two classes of Szekeres solutions of the Einstein field equations, which unifies the solutions as regards their dynamics, and relates them to the Friedmann-Robertson-Walker (FRW) cosmological models in a particularly transparent way. This reformulation enables us to give a general analysis of the scalar polynomial curvature singularities of the solutions, and of their evolution in time. In particular, the solutions which are close to an FRW model near the initial singularity, or in the late stages of evolution, are identified.