CMB anisotropy in compact hyperbolic universes. II. COBE maps and limits

Abstract

The measurements of CMB anisotropy have opened up a window for probing the global topology of the universe on length scales comparable to, and even beyond, the Hubble radius. For compact topologies, the two main effects on the CMB are: (1) the breaking of statistical isotropy in characteristic patterns determined by the photon geodesic structure of the manifold and (2) an infrared cutoff in the power spectrum of perturbations imposed by the finite spatial extent. We calculate the CMB anisotropy in compact hyperbolic universe models using the regularized method of images described in detail in the preceding paper, including the line-of-sight “integrated Sachs-Wolfe” effect, as well as the last-scattering surface terms. We calculate the Bayesian probabilities for a selection of models by confronting our theoretical pixel–pixel temperature correlation functions with the COBE-DMR data. Our results demonstrate that strong constraints on compactness arise: if the universe is small compared to the horizon size, correlations appear in the maps that are irreconcilable with the observations. This conclusion is qualitatively insensitive to the matter content of the universe, in particular, the presence of cosmological constant. If the universe is of comparable size to the “horizon,” the likelihood function is very dependent upon orientation of the manifold w.r.t. the sky. While most orientations may be strongly ruled out, it sometimes happens that for a specific orientation the predicted correlation patterns are preferred over those for the conventional infinite models. The full Bayesian analysis we use is the most complete statistical test that can be done on the cosmic background explorer maps, taking into account all possible signals and their variances in the theoretical skies, in particular the high degree of anisotropic correlation that can exist. We also show that standard visual measures for comparing theoretical predictions with the data such as the isotropized power spectrum C_l are not so useful in small compact spaces because of enhanced cosmic variance associated with the breakdown of statistical isotropy.