The dynamics of Lemaître–Tolman cosmologies

Abstract

In this paper we give a coordinate-independent analysis of the dynamics of the spherically symmetric Lemaître–Tolman (LT) cosmologies, emphasizing their relation with the Friedmann–Lemaître (FL) cosmologies. We show that in general, ever-expanding LT cosmologies isotropize at late times, becoming asymptotic to the de Sitter universe or to the Milne universe, depending on whether or not a cosmological constant is present. The rate of isotropization depends significantly on whether Λ > 0 or Λ = 0. We introduce a dimensionless scalar representing the ratio of the Weyl curvature to the Ricci curvature, and show that in both cases it has a finite limit at late times, whose value determines the asymptotic spatial inhomogeneity in various physical quantities. We coin the phrase ‘residual inhomogeneity’ to describe these effects. We also identify the LT cosmologies for which the initial singularity is isotropic, and show that there is a subclass of models that are close-to-FL globally in time and in space.