CMB Anisotropy in Compact Hyperbolic Universes

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Measurements of 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 eigenmodes of the Laplace-Beltrami operator. We numerically obtain the eigenmodes on a compact 3-hyperbolic space cataloged as m003(-2,3) in SnapPea (computer program by Jeff Weeks) using the direct boundary element method, which enables one to simulate the CMB in multiply-connected compact models with high precision. The angular power spectra Cₗ's (2 ≤ l≤18) are calculated using computed eigenmodes for 5.4<10 and Gaussian random approximation for the expansion coefficients for 10<20. Assuming that the initial power spectrum is the Harrison-Zeldovich spectrum, the computed Cₗ's are consistent with the COBE data for 0.1 ≤Ωₒ ≤ 0.6. In low Ωₒ models, the large-angular fluctuations can be produced at periods after the last scattering as the curvature perturbations decay in the curvature dominant era.