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 , given by the quotient of
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]. Thus, for this restrictive case there would remain only the two homogeneous spherical spaces
and
with Ωtot of about 1.038 and 1.018, respectively, as possible topologies for our Universe.