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 spatially constant dark energy X-component are considered. The general solution of the Friedmann equation for the cosmic scalefactor a(η) is given for the four-component models in terms of the Weierstrass ℘-function. The lower parts of the angular power spectra Cl of the CMB anisotropy are computed for nearly flat models with Ωtot≤0.95. It is shown that the particular compact fundamental cell that is considered in this paper leads to a suppression in Cl for l≲10 and Ωtot≲0.9.