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.

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Inflationary cosmology has been in vogue for over a decade. However, it does not appear to provide satisfactory answers to all of the cosmological questions which provided the motivation for the theory. It is argued that quiescent cosmology and the related ideas of Penrose (1989) regarding gravitational entropy may well provide a viable alternative when formulated within the framework of isotropic singularities.

We use Bardeen's gauge-invariant formalism to analyze the behavior of, and relationship between, various geometric and physical quantities of cosmological interest at the linear level. This leads to a cosmologically oriented gauge-invariant characterization of the different perturbation modes that can arise. In particular a link is made between the existence of gravitational-wave modes and the conformal curvature of hypersurfaces in spacetime. We indicate how these results can be useful in the analysis of exact solutions of the Einstein field equations.