The Isotropy of Compact Universes

We discuss the problem of the stability of the isotropy of the universe in the space of ever-expanding spatially homogeneous universes with a compact spatial topology. The anisotropic modes which prevent isotropy being asymptotically stable in Bianchi-type $VII_h$ universes with non-compact topologies are excluded by topological compactness. Bianchi type $V$ and type $VII_h$ universes with compact topologies must be exactly isotropic. In the flat case we calculate the dynamical degrees of freedom of Bianchi-type $I$ and $VII_0$ universes with compact 3-spaces and show that type $VII_0$ solutions are more general than type $I$ solutions for systems with perfect fluid, although the type $I$ models are more general than type $VII_0$ in the vacuum case. For particular topologies the 4-velocity of any perfect fluid is required to be non-tilted. Various consequences for the problems of the isotropy, homogeneity, and flatness of the universe are discussed.