Fast cosmic microwave background analyses via correlation functions

We propose and implement a fast, universally applicable method for extracti ng the angular power spectrum C-l from cosmic microwave background temperat ure maps by first estimating the correlation function xi(theta). Our proced ure recovers the C-l using N-2 (but potentially N log N) operations, where N is the number of pixels. This is in contrast with standard maximum likeli hood techniques that require N-3 operations. Our method makes no special as sumptions about the map, unlike present-day fast techniques that rely on sy mmetries of the underlying noise matrix, sky coverage, scanning strategy, a nd geometry. This makes analysis of megapixel maps without symmetries possi ble for the first time. The key element of our technique is the accurate mu ltipole decomposition of xi(theta). The C-l error bars and cross-correlatio ns are found by using a Monte Carlo approach. We applied our technique to a large number of simulated maps with BOOMERanG (Balloon Observations Of Mil limetric Radiation and Geophysics) sky coverage in 81,000 pixels. We used a diagonal noise matrix, with approximately the same amplitude as the BOOMER anG experiment. These studies demonstrate that our technique provides an un biased estimator of the C-l. Even though our method is approximate, the err or bars obtained are nearly optimal, and they converged only after a few te ns of Monte Carlo realizations. Our method is directly applicable for the n ondiagonal noise matrix. This and other generalizations, such as minimum va riance weighting schemes, polarization, and higher order statistics, are al so discussed.