Eigenmodes of dodecahedral space

Cosmological models wherein spatial sections are the Poincaré dodecahedral space D have been recently invoked to explain the missing power in the low-order angular modes of the cosmic microwave background observed anisotropies. Further exploration of this possibility requires knowledge of the eigenmodes of the Laplacian of D. Only the first modes have been calculated numerically. Here we give an explicit form for these modes up to arbitrary order, in terms of the eigenvectors of a small rank matrix, which are very easy to calculate numerically. As an illustration we give numerical estimates of the first modes, up to the eigenvalue −k(k + 2) for k = 62. These results are obtained by application of a more general method (presented in a previous work) which allows one to express the properties of any eigenfunction of the Laplacian of the 3-sphere under an arbitrary rotation of SO(4).