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 Bac kground Explorer Differential Microwave Radiometer and Saskatoon datas ets. One method involves a direct evaluation of the likelihood functio n, and the other is an estimator that is a minimum-variance weighted q uadratic function of the data. Applied iteratively, the quadratic esti mator is not distinct from likelihood analysis, but is rather a rapid means of finding the power spectrum that maximizes the likelihood func tion. Our results bear this out: direct evaluation and quadratic estim ation converge to the same C(l)s. The quadratic estimator can also be used to determine directly cosmological parameters and their uncertain ties. While the two methods both require O(N-3) operations, the quadra tic is much faster, and both are applicable to datasets with arbitrary chopping patterns and noise correlations. We also discuss approximati ons that may reduce it to O(N-2) thus making it practical for forthcom ing megapixel datasets. [S0556-2821(98)04404-X].