The statistics of cosmic background radiation fluctuations

We present computations of the radiation correlation functions and angular power spectra for microwave background anisotropies expected in Ω = 1 cold dark matter dominated universes with scale-invariant adiabatic or isocurvature initial conditions. The results are valid on all angular scales. We describe the statistical properties of the radiation pattern and develop the theory of two-dimensional Gaussian random fields. A large number of properties of such fields may be derived analytically or semi-analytically, such as the number densities of hotspots and coldspots, the eccentricities of peaks and peak correlation properties. The formulae 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 allow us 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.