The statistics of cosmic background radiation fluctuations

Abstract

Computations of the radiation correlation functions and angular power spectra are presented for microwave-background anisotropies expected in Omega = 1 cold-dark-matter-dominated universes with scale-invariant adiabatic or isocurvature initial conditions. The results are valid on all angular scales. The statistical properties of the radiation pattern are described, and the theory of two-dimensional Gaussian random fields is developed. A large number of properties of such fields may be derived analytically or semianalytically, such as the number densities of hot and cold spots, the eccentricities of peaks, and peak correlation properties. The formulas presented here provide valuable insight into the textural characteristics of the microwave-background anisotropies and must be satisfied if the primordial fluctuations are Gaussian. The assumption of Gaussian initial conditions makes it possible to make highly specific predictions for the pattern of the temperature anisotropies. This is demonstrated by the construction of maps of the fluctuations predicted for the total intensity and the polarization.