Relativistic Quantum Chaos in de Sitter Cosmologies

We consider classical and quantal motion in open de Sitter cosmologies of multiple spatial connectivity. The topological structure of the spacelike slices creates on the one hand chaotic trajectories, on the other hand bound states whose wave fields and energies are intimately connected with the Hausdorff dimension and measure of limit sets of chaotic trajectories in the covering space of the manifold. We discuss the time evolution of the energy of wave fields coupled to the curvature scalar of the 4-manifold in the early and late stages of the cosmic expansion, and its dependence on the spectral variable of the Laplace-Beltrami operator on the space sections. This spectrum is in turn entirely determined by the topological and metrical structure of the sections.