The isotropy of compact universes (Q2724807)

The observational fact that our universe appears to be highly isotropic gives some relevance to the stability properties of Robertson-Walker models (\(=\) locally homogeneous and isotropic models) within the larger class of Bianchi models (\(=\) locally homogeneous models). In the traditional literature, this question is discussed for spatially compact topologies (\(\simeq S^3\)) in the case of Robertson-Walker models with positive spatial curvature (\(k=1\)) and for spatially non-compact topologies (\(\simeq R^3\)) in the case of Robertson-Walker models with negative or vanishing spatial curvature (\(k=-1,0\)).NEWLINENEWLINENEWLINEIn the paper under review the authors demonstrate that the results drastically change if for \(k=-1\) and \(k=0\) a compactified topology is considered. Robertson-Walker models with \(k=-1\) are special cases of the Bianchi types V and \(\text{VII}_h\); however, anisotropic universes of these Bianchi types with compact spatial sections do not exist at all. The case \(k=0\) is more involved. Here Robertson-Walker models are special cases of the Bianchi types I and \(\text{VII}_0\), and anisotropic models are possible with various different compact spatial topologies. The authors discuss, among other things, the restrictions that are imposed by the assumption that Einstein's field equation for a perfect fluid is satisfied. A certain parameter-counting method is used for investigating the stability question.