Geometric Gaussianity and non-Gaussianity in the cosmic microwave background

In this paper, the Gaussianity of eigenmodes and non-Gaussianity in the cosmic microwave background (CMB) temperature fluctuations in the two smallest compact hyperbolic (CH) models are investigated. First, it is numerically found that the expansion coefficients of low-lying eigenmodes on the two CH manifolds behave as if they are Gaussian random numbers in almost all places. Next, the non-Gaussianity of the temperature fluctuations in the $\mathrm{}(l,m)\mathrm{}$ space in these models is studied. Assuming that the initial fluctuations are Gaussian, the real expansion coefficients $b_{\mathrm{lm}}$ of the temperature fluctuations in the sky are found to be distinctively non-Gaussian. In particular, the cosmic variances are found to be much larger than for Gaussian models. On the other hand, the anisotropic structure is vastly erased if one averages the fluctuations at a number of different observation points because of the Gaussian pseudorandomness of the eigenmodes. Thus the dominant contribution to the two-point correlation functions comes from the isotropic terms described by the angular power spectra $C_l.$ Finally, topological quantities, the total length and the genus of isotemperature contours are investigated. The variances of total length and genus at high and low threshold levels are found to be considerably larger than that of Gaussian models while the means almost agree with them.