Fast Cosmic Microwave Background Analyses via Correlation Functions

We propose and implement a fast, universally applicable method for extracting the angular power spectrum C_l from cosmic microwave background temperature maps by first estimating the correlation function ξ(θ). Our procedure recovers the C_l using N² (but potentially NlogN) operations, where N is the number of pixels. This is in contrast with standard maximum likelihood techniques that require N³ operations. Our method makes no special assumptions about the map, unlike present-day fast techniques that rely on symmetries of the underlying noise matrix, sky coverage, scanning strategy, and geometry. This makes analysis of megapixel maps without symmetries possible for the first time. The key element of our technique is the accurate multipole decomposition of ξ(θ). The C_l error bars and cross-correlations are found by using a Monte Carlo approach. We applied our technique to a large number of simulated maps with BOOMERanG (Balloon Observations Of Millimetric Radiation and Geophysics) sky coverage in 81,000 pixels. We used a diagonal noise matrix, with approximately the same amplitude as the BOOMERanG experiment. These studies demonstrate that our technique provides an unbiased estimator of the C_l. Even though our method is approximate, the error bars obtained are nearly optimal, and they converged only after a few tens of Monte Carlo realizations. Our method is directly applicable for the nondiagonal noise matrix. This and other generalizations, such as minimum variance weighting schemes, polarization, and higher order statistics, are also discussed.