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 ra diation, 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(eta) is given for the four-component models in terms of the Weierstrass P-function. The lower parts of the angular power spectra C-l of the CMB anisotropy are computed for nearly flat models with Omega (tot) less than or equal to 0.95. It is shown that the particular com pact fundamental cell that is considered in this paper leads to a suppressi on in C-l for l less than or similar to 10 and Omega (tot) less than or sim ilar to 0.9.