Estimating the power spectrum of the cosmic microwave background

Abstract

We develop two methods for estimating the power spectrum, $C_{\mathcal{l}},$ of the cosmic microwave background from data and apply them to the Cosmic Background Explorer Differential Microwave Radiometer and Saskatoon datasets. One method involves a direct evaluation of the likelihood function, and the other is an estimator that is a minimum-variance weighted quadratic function of the data. Applied iteratively, the quadratic estimator is not distinct from likelihood analysis, but is rather a rapid means of finding the power spectrum that maximizes the likelihood function. Our results bear this out: direct evaluation and quadratic estimation converge to the same $C_{\mathcal{l}}$s. The quadratic estimator can also be used to determine directly cosmological parameters and their uncertainties. While the two methods both require ${O(N}^3)$ operations, the quadratic is much faster, and both are applicable to datasets with arbitrary chopping patterns and noise correlations. We also discuss approximations that may reduce it to ${O(N}^2)$ thus making it practical for forthcoming megapixel datasets.