Waves on accelerating dodecahedral universes

TL;DR

This paper studies wave behavior in a universe with dodecahedral topology undergoing exponential expansion, deriving its long-term state and linking it to cosmic microwave background observations.

Contribution

It provides the first analytical description of wave propagation and asymptotic states in an exponentially expanding dodecahedral universe model.

Findings

Existence of a limit state as time approaches infinity.

Analytic expression for the asymptotic wave profile.

Numerical scheme using invariant finite elements for accurate computation.

Abstract

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.