Large-Angle Cosmic Microwave Background Anisotropies in an Open Universe

Abstract

If the Universe is open, scales larger than the curvature scale may be probed by large-angle fluctuations in the cosmic microwave background (CMB). We consider primordial adiabatic perturbations and discuss power spectra that are power laws in volume, wavelength, and eigenvalue of the Laplace operator. The resulting large-angle anisotropies of the CMB are computed. The amplitude generally increases as $\Omega$ is decreased, but decreases as $h$ is increased. Interestingly enough, for all three ansatzes, anisotropies on angular scales larger than the curvature scale are suppressed relative to the anisotropies on scales smaller than the curvature scale. Models with $0.2<\Omega h<0.3$ appear compatible with CMB fluctuations detected by COBE and Tenerife and with the amplitude and spectrum of fluctuations of galaxy counts in galaxy surveys. COBE normalization for these models yields ${\sigma }_8\simeq 0.5-0.7$. Models with smaller values of $\Omega h$ when normalized to COBE require bias factors in excess of 2 to be compatible with the observed galaxy counts on the $8/h$ Mpc scale. Requiring that the age of the universe exceed 10 Gyr implies that $\Omega >0.25$, while requiring that the age exceed 13 Gyr implies that $\Omega >0.35$. Unlike in the flat-Universe case where the anisotropy comes only from the last-scattering term in the Sachs-Wolfe formula, large-angle anisotropies come primarily from the decay of potential fluctuations at $z<1/\Omega$. Thus, if the Universe is open, COBE has been detecting fluctuations produced at moderate redshift rather than at $z\sim 1300$.

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