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 C _l of the CMB anisotropy are computed for nearly flat models with Ω_tot≤0.95. It is shown that the particular compact fundamental cell that is considered in this paper leads to a suppression in C _l for l ≲10 and Ω_tot≲0.9.