Waves on accelerating dodecahedral universes

We investigate the wave propagation on a compact 3-manifold of constant positive curvature with a non trivial topology, the Poincaré dodecahedral space, if the scale factor is exponentially increasing. We prove the existence of a limit state as $t\to +\infty$ and we get its analytic expression. The deep sky is described by this asymptotic profile thanks to the Sachs–Wolfe formula. We transform the Cauchy problem into a mixed problem posed on a fundamental domain determined by the quaternionic calculus. We perform an accurate scheme of computation: we employ a variational method using a space of second order finite elements that is invariant under the action of the binary icosahedral group. (paper)