Eigenmodes of 3-dimensional spherical spaces and their application to cosmology

TL;DR

This paper computes eigenmodes of the Laplacian in 3D spherical spaces, develops numerical methods, and discusses implications for cosmology, especially CMB anisotropies, highlighting potential topologies consistent with current data.

Contribution

It provides analytical solutions and three numerical methods for eigenmodes in spherical spaces, advancing understanding of their cosmological implications.

Findings

Eigenmodes are computed for lens and prism spaces.

Numerical methods are validated against analytical solutions.

Spherical topologies remain viable in cosmology even near critical density.

Abstract

This article investigates the computation of the eigenmodes of the Laplacian operator in multi-connected three-dimensional spherical spaces. General mathematical results and analytical solutions for lens and prism spaces are presented. Three complementary numerical methods are developed and compared with our analytic results and previous investigations. The cosmological applications of these results are discussed, focusing on the cosmic microwave background (CMB) anisotropies. In particular, whereas in the Euclidean case too small universes are excluded by present CMB data, in the spherical case there will always exist candidate topologies even if the total energy density parameter of the universe is very close to unity.