- qc / 0 70 10 17 v 1 1 J an 2 00 7 Harmonic polynomials for expanding the fluctuations of the Cosmic Microwave Background : The Poincaré and the 3 - sphere model

1 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 Poincaré space are constructed. Strong and specific selection rules are given by comparing the polynomials to those on the 3-sphere, its simply connected cover. The global topology of 3-space is not fixed by Einstein general relativity, since this is formulated in terms of local differential equations. Einsteins first static cosmological models used for the space-part of the universe a simply connected sphere S 3. With present-day cosmological information it becomes possible to test multiply-connected topologies for the space-part of the universe. J-P Luminet et al. [7] 2003 and J Weeks [10] 2004 propose to explore the topology of 3-space from temperature fluctuations of the cosmic microwave background (CMB). These fluctuations are measured by the Wilkinson Microwave Anisotropy Probe (WMAP) with very high precision. A way to test the topology is to expand the temperature fluctuations of the CMB into harmonic polynomials of the chosen topological 3-manifold, for example the Poincaré dodecahedral manifold P. The topology will be verified if the harmonic polynomials of the manifold suffice to expand these fluctuations. The simply connected space parts of cosmological models are the 3-sphere S 3 for positive, and the hyperbolic 1