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

Abstract

The cosmic microwave background (CMB) originates from an early stage in the history of the universe. Observations show low-multipole contributions of CMB fluctuations. A possible explanation is given by a non-trivial topology of the universe and has motivated the search for topological selection rules. Harmonic analysis on a topological manifold must provide basis sets for all functions compatible with a given topology and so is needed to model the CMB fluctuations. We analyze the fundamental groups of Platonic tetrahedral, cubic and octahedral manifolds using deck transformations. From them we construct the appropriate harmonic analysis and boundary conditions. We provide the algebraic means for modeling the multipole expansion of incoming CMB radiation. From the assumption of randomness, we derive selection rules, depending on the point symmetry of the manifold.