Statistical Isotropy of CMB and Cosmic Topology

The breakdown of statistical homogeneity and isotropy of cosmic perturbations is a generic feature of ultra large scale structure of the cosmos, in particular, of non trivial cosmic topology. The statistical isotropy (SI) of the Cosmic Microwave Background temperature fluctuations (CMB anisotropy) is sensitive to this breakdown on largest scales comparable to, and even beyond the cosmic horizon. We study a set of measures, $\kappa_\ell$ ($\ell=1,2,3, >...$) which for non-zero values indicate and quantify statistical isotropy violations in a CMB map. The main goal here is to interpret the $\kappa_\ell$ spectrum and relate it to characteristic patterns in the correlation function of CMB anisotropy arising from cosmic topology. We numerically compute the predicted $\kappa_\ell$ spectrum for CMB anisotropy in flat torus universe models. The essential features are captured in the leading order approximation to the correlation function where $\kappa_\ell$ can be calculated analytically. The $\kappa_\ell$ spectrum is shown to reflect the number, importance and relative orientation of principal directions in the CMB correlation dictated by the shape of the Dirichlet domain (DD) of the compact space and its size relative to cosmic horizon. Hence, besides detecting cosmic topology, $\kappa_\ell$ can discriminate between different topology of the universe complementing ongoing search for cosmic topology in CMB anisotropy data.