CMB anisotropy of spherical spaces

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.