Platonic topology and CMB fluctuations : Homotopy , anisotropy , and multipole selection rules

The Cosmic Microwave Background CMB originates from an early stage in the history of the universe. Observed low multipole contributions of CMB fluctuations have motivated the search for selection rules from the underlying topology of 3-space. Everitt (2004) has generated all homotopies for Platonic spherical 3-manifolds by face gluings. We transform the glue generators into isomorphic deck transformations. The deck transformations act on a spherical Platonic 3-manifold as prototile and tile the 3-sphere by its images. A complete set of orthonormal functions on the 3-sphere is spanned by the Wigner harmonic polynomials. For a tetrahedral, two cubic and three octahedral manifolds we construct algebraically linear combinations of Wigner polynomials, invariant under deck transformations and with domain the manifold. We prove boundary conditions on polyhedral faces from homotopy. By algebraic means we pass to a multipole expansion. Assuming random models of the CMB radiation, we derive multipole selection rules, depending on the point symmetry of the manifold.