Geometric Gaussianity and Non-Gaussianity in the Cosmic Microwave Background

In this paper, Gaussianity of eigenmodes and non-Gaussianity in the CosmicMicrowave Background (CMB) temperature fluctuations in two smallest compacthyperbolic (CH) models are investigated. First, it is numerically found thatthe expansion coefficients of low-lying eigenmodes on the two CH manifoldsbehave as if they are Gaussian random numbers at almost all the places. Next,non-Gaussianity of the temperature fluctuations in the (l,m) space in thesemodels is studied. Assuming that the initial fluctuations are Gaussian, thereal expansion coefficients b_{l m} of the temperature fluctuations in the skyare found to be distinctively non-Gaussian. In particular, the cosmic variancesare found to be much larger than that for Gaussian models. On the other hand,the anisotropic structure is vastly erased if one averages the fluctuations ata number of different observing points because of the Gaussianpseudo-randomness of the eigenmodes. Thus the dominant contribution to thetwo-point correlation functions comes from the isotropic terms described by theangular power spectra C_l. Finally, topological quantities: the total lengthand the genus of isotemperature contours are investigated. The variances oftotal length and genus at high and low threshold levels are found to beconsiderably larger than that of Gaussian models while the means almost agreewith them