Estimating the power spectrum of the cosmic microwave background

We develop two methods for estimating the power spectrum, C-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(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. [S0556-2821(98)04404-X].