Laplacian eigenmodes for spherical spaces

Abstract

The possibility that our space is multi–rather than singly–connected has gained renewed interest after the discovery of the low power for the first multipoles of the CMB by WMAP. To test the possibility that our space is a multi-connected spherical space, it is necessary to know the eigenmodes of such spaces. Except for lens and prism space, and to some extent for dodecahedral space, this remains an open problem. Here we derive the eigenmodes of all spherical spaces. For dodecahedral space, the demonstration is much shorter, and the calculation method much simpler than before. We also apply our method to tetrahedric, octahedric and icosahedric spaces. This completes the knowledge of eigenmodes for spherical spaces, and opens the door to new observational tests of the cosmic topology. The vector space of the eigenfunctions of the Laplacian on the 3-sphere , corresponding to the same eigenvalue λk = −k(k + 2), has dimension (k + 1)2. We show that the Wigner functions provide a basis for such a space. Using the properties of the latter, we express the behaviour of a general function of under an arbitrary rotation G of SO(4). This offers the possibility of selecting those functions of which remain invariant under G. Specifying G to be a generator of the holonomy group of a spherical space X, we give the expression of the vector space of the eigenfunctions of X. We provide a method to calculate the eigenmodes up to an arbitrary order. As an illustration, we give the first modes for the spherical spaces mentioned.