Astronomy and Astrophysics – Astrophysics – General Relativity and Quantum Cosmology
Scientific paper
2004-02-06
Class.Quant.Grav. 21 (2004) 2455-2464
Astronomy and Astrophysics
Astrophysics
General Relativity and Quantum Cosmology
11 pages, no figure
Scientific paper
Cosmological models where spatial sections are the Poincar\'e dodecahedral space D have been recently invoked to give an account of the lower modes of the angular anisotropies of the cosmic microwave background. Further explorations of this possibility require the 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 term of eigenvectors of a small rank matrix, very easy to calculate numerically. As an illustration we give 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 to express the properties of any eigenfunction of the Laplacian of the three sphere under an arbitrary rotation of SO(4).
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