CMB anisotropy of spherical spaces

Abstract. The first-year WMAP data taken at their face value hint that the Universe might be slightly positively curved and therefore necessarily finite, since all spherical (Clifford-Klein) space forms M 3 = S 3 /Γ, given by the quotient of S 3 by a group Γ of covering transformations, possess this property. We examine the anisotropy of the cosmic microwave background (CMB) for all typical groups Γ corresponding to homogeneous universes. The CMB angular power spectrum and the temperature correlation function are computed for the homogeneous spaces as a function of the total energy density parameter Ωtot in the large range 1.01, 1.20 and are compared with the WMAP data. We find that out of the infinitely many homogeneous spaces only the three corresponding to the binary dihedral group T ⋆ , the binary octahedral group O ⋆ , and the binary icosahedral group I ⋆ are in agreement with the WMAP observations. Furthermore, if Ωtot is restricted to the interval 1.00, 1.04, the space described by T ⋆ is excluded since it requires a value of Ωtot which is probably too large being in the range 1.06, 1.07. We thus conclude that there remain only the two homogeneous spherical spaces S 3 /O ⋆ and S 3 /I ⋆ with Ωtot of about 1.038 and 1.018, respectively, as possible topologies for our Universe. PACS numbers: 98.80.-k, 98.70.Vc, 98.80.Es 1.