Abstract
We carry out the harmonic analysis on four Platonic spherical three-manifolds with different topologies. Starting out from the homotopies (Everitt 2004), we convert them into deck operations, acting on the simply connected three-sphere as the cover, and obtain the corresponding variety of deck groups. For each topology, the three-sphere is tiled into copies of a fundamental domain under the corresponding deck group. We employ the point symmetry of each Platonic manifold to construct its fundamental domain as a spherical orbifold. While the three-sphere supports an orthonormal complete basis for harmonic analysis formed by Wigner polynomials, a given spherical orbifold leads to a selection of a specific subbasis. The resulting selection rules find applications in cosmic topology, probed by the cosmic microwave background.
Highlights
In 1917, Einstein replaced the Euclidean three-space by the three-sphere and so introduced the first spherical manifold to describe the spatial part of the Universe [1]
Three-manifolds with non-Euclidean topology have in recent years found applications in cosmology, in particular in relation to the multipole analysis of the cosmic microwave background (CMB) radiation
The connection of multipole-resolved CMB measurements to cosmic topology is provided by selection rules for the harmonic analysis
Summary
In 1917, Einstein replaced the Euclidean three-space by the three-sphere and so introduced the first spherical manifold to describe the spatial part of the Universe [1]. There is no systematic account that links the homotopy of Platonic manifolds to deck groups and to their harmonic analysis and selection rules. We construct from the homotopies of [4] the isomorphic deck actions on the three-sphere S 3 and, from them determine, the deck groups for the Platonic manifolds. Our harmonic analysis on orbifolds yields for each orbifold specific multipole selection rules and, predicts topological correlations between different multipole orders (l, l0 ) of the CMB
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