Large-angle cosmic microwave background anisotropies in an open universe

If the universe is open, scales larger than the curvature scale may be probed by observation of 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. Such spectra may have arisen if, for example, the universe underwent a period of "frustrated" inflation. The resulting large-angle anisotropies of the CMB are computed. The amplitude generally increases as Ω is decreased but decreases as h is increased. Interestingly enough, for all three Ansätze, anisotropies on angular scales larger than the curvature scale are suppressed relative to the anisotropies on scales smaller than the curvature scale, but cosmic variance makes discrimination between various models difficult. Models with 0.2 ≲ Ωh ≲ 0.3 appear compatible with CMB fluctuations detected by COBE and the Tenerife experiment and with the amplitude and spectrum of fluctuations of galaxy counts in the APM, CfA, and 1.2 Jy IRAS surveys. COBE normalization for these models yields σ₈ ≃ 0.5-0.7. Models with smaller values of Ωh when normalized to COBE require bias factors in excess of 2 to be compatible with the observed galaxy counts on the 8 h^-1 Mpc scale. Requiring that the age of the universe exceed 10 Gyr implies that Ω ≳ 0.25, while requiring that the age exceed 13 Gyr implies that Ω≳ 0.35. Unlike in the flat universe case, where the anistropy 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 ≲ Ω^-1 Thus, if the universe is open, COBE has been detecting temperature fluctuations produced at moderate redshift rather than at z ∼ 1300.