Estimating the Power Spectrum of the Cosmic Microwave Background

TL;DR

This paper develops and unifies two approaches to estimating the CMB power spectrum: direct maximum-likelihood evaluation and a fast minimum-variance quadratic estimator. It shows that, when iterated, the quadratic estimator converges to the same peak of the likelihood as direct evaluation, providing a scalable method for obtaining ${\cal C}_\ell$ across datasets with complex noise and sky cuts. The authors apply these methods to COBE/DMR and Saskatoon data, quantify the quadrupole and higher multipoles, discuss the non-Gaussian nature of the posterior for $C_\ell$, and introduce robust binning, radical compression, and forecasting frameworks for future experiments. Their work delivers practical algorithms for power-spectrum extraction, enables data compression without meaningful information loss in the Gaussian limit, and offers guidance for analyzing upcoming megapixel CMB datasets.

Abstract

We develop two methods for estimating the power spectrum, C_l, of the cosmic microwave background (CMB) from data and apply them to the COBE/DMR 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_ls. The quadratic estimator can also be used to directly determine 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.

Figures (16)

• Figure 1: A one-dimensional example of quadratic estimation.
• Figure 2: Maximum-likelihood power spectra from iterative direct evaluation of the likelihood function. The curve is the zeroth iteration: COBE-normalized standard CDM. The points with error bars are, from left to right, the results of the first to third iterations. Here, we define the error bars by a likelihood ratio of $e^{-1/2}$ from the peak.
• Figure 3: Iterative quadratic estimation. The curve is the zeroth iteration: COBE-normalized standard CDM. The points with error bars are, from left to right, the results of the first to third iterations.
• Figure 4: We compare the results of the quadratic and direct evaluation iteration schemes. At each $\ell$, the left error bar (square symbol) is for the quadratic, the right (triangle) is for the direct evaluation.
• Figure 5: Comparison of different groups' power spectrum estimates, as marked. Gorski computes power spectra in both ecliptic and galactic pixelizations of the sky.