Harmonic polynomials for expanding the fluctuations of the Cosmic Microwave Background: The Poincare and the 3-sphere model

TL;DR

This paper constructs harmonic polynomials on the Poincare dodecahedral space to analyze CMB fluctuations, revealing topological insights by comparing them to those on the 3-sphere.

Contribution

It introduces a comprehensive method for constructing harmonic polynomials on the Poincare space and compares them to the 3-sphere to identify topological signatures in CMB data.

Findings

Harmonic polynomials on the Poincare space are explicitly constructed.

Selection rules differentiate the Poincare space from the 3-sphere.

Insights into the universe's topology from CMB fluctuations.

Abstract

Fluctuations of the Cosmic Microwave Background CMB are observed by the WMAP. When expanded into the harmonic eigenmodes of the space part of a cosmological model, they provide insight into the large-scale topology of space. All harmonic polynomials on the multiply connected dodecahedral Poincare space are constructed. Strong and specific selection rules are given by comparing the polynomials to those on the 3-sphere, its simply connected cover.