Full-sky correlations of peaks in the microwave background

We compute precise predictions for the two-point correlation function of local maxima (or minima) in the temperature of the microwave background, under the assumption that it is a random Gaussian field. For a given power spectrum and peak threshold there are no adjustable parameters, and since this analysis does not make the small-angle approximation of Heavens & Sheth, it is essentially complete. We find oscillatory features which are absent in the temperature autocorrelation function, and we also find that the small-angle approximation to the peak-peak correlation function is accurate to better than 0.01 on all scales. These high-precision predictions can form the basis of a sensitive test of the Gaussian hypothesis with upcoming all-sky microwave background experiments MAP and Planck, affording a thorough test of the inflationary theory of the early Universe. To illustrate the effectiveness of the technique, we apply it to simulated maps of the microwave sky arising from the cosmic string model of structure formation, and compare the two-point correlation function of peaks with the bispectrum as a non-Gaussian discriminant. We also show how peak statistics can be a valuable tool in assessing and statistically removing contamination of the map by foreground point sources.