X iv : a st ro - p h / 98 10 03 4 v 3 2 3 Ju n 19 99 Computation of eigenmodes on a compact hyperbolic 3 - space

Measurements of cosmic microwave background (CMB) anisotropy are ideal experiments for discovering the non-trivial global topology of the universe. To evaluate the CMB anisotropy in multiply-connected compact cosmological models, one needs to compute the eigenmodes of the Laplace-Beltrami operator. Using the direct boundary element method, we numerically obtain the low-lying eigenmodes on a compact hyperbolic 3-space called the Thurston manifold which is the second smallest in the known compact hyperbolic 3-manifolds. The computed eigenmodes are expanded in terms of eigenmodes on the unit three-dimensional pseudosphere. We numerically find that the expansion coefficients behave as Gaussian pseudorandom numbers for low-lying eigenmodes. The observed gaussianity in the CMB fluctuations can partially be attributed to the Gaussian pseudo-randomness of the expansion coefficients assuming that the Gaussian pseudo-randomness is the universal property of the compact hyperbolic spaces.