  We investigate the wave propagation on a compact 3-manifold of constant
positive curvature with a non trivial topology, the Poincar\'e dodecahedral
space, when the scale factor is exponentially increasing. We prove the
existence of a limit state as t tends to infinity 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.
