The Cosmic Microwave Background for a Nearly Flat Compact Hyperbolic Universe

The fluctuations of the cosmic microwave background (CMB) are investigated for a hyperbolic universe with finite volume. Four-component models with radiation, matter, vacuum energy, and an extra dark energy component, which is a particular version of quintessence, are considered. The general solution of the Friedmann equation for the cosmic scale factor a(η) is given for the four-component models in terms of the Weierstrass P -function. The angular power spectra Cl of the CMB anisotropy are computed for nearly flat models with Ωtot 6 0.95. It is shown that the particular compact fundamental cell, which is considered in this paper, leads to a suppression in Cl for l . 10 and Ωtot . 0.9.