How to measure CMB power spectra without losing information

TL;DR

The paper introduces a lossless, quadratic estimator for the CMB angular power spectrum $C_\ell$ that is faster than traditional maximum-likelihood methods, scales as $O(n^2)$, and remains applicable to arbitrary survey geometries. It proves that the estimator can achieve the Fisher information bound, making it optimal in the Cramer-Rao sense, and provides an intuitive interpretation via high-pass filtering, edge tapering, and a truncated spherical-harmonics expansion. A concrete COBE/DMR demonstration shows the method yields a precise, information-preserving compression from megapixel maps to a manageable set of bandpowers with well-characterized windows. The paper also outlines practical downstream steps for visualization and parameter estimation, arguing that simple chi-squared analyses can be nearly as informative as full likelihood approaches when using the lossless power-spectrum representation. Overall, it presents a computationally feasible, transparent data-analysis pipeline capable of delivering accurate cosmological constraints from future large CMB maps, provided systematic effects are controlled.

Abstract

A new method for estimating the angular power spectrum C_l from cosmic microwave background (CMB) maps is presented, which has the following desirable properties: (1) It is unbeatable in the sense that no other method can measure C_l with smaller error bars. (2) It is quadratic, which makes the statistical properties of the measurements easy to compute and use for estimation of cosmological parameters. (3) It is computationally faster than rival high-precision methods such as the nonlinear maximum-likelihood technique, with the crucial steps scaling as n^2 rather than n^3, where n is the number of map pixels. (4) It is applicable to any survey geometry whatsoever, with arbitraty regions masked out and arbitrary noise behaviour. (5) It is not a "black-box" method, but quite simple to understand intuitively: it corresponds to a high-pass filtering and edge softening of the original map followed by a straight expansion in truncated spherical-harmonics. It is argued that this method is computationally feasible even for futute high-resolution CMB experiments with n=10^6-10^7. It is also shown that C_l computed with this method is useful not merely for graphical presentation purposes, but also as an intermediate (and arguably necessary) step in the data analysis pipeline, reducing the data set to a more manageable size before the final step of constraining Gaussian cosmological models and parameters - while retaining all the cosmological information that was present in the original map.

Figures (8)

• Figure 1: Power spectrum estimation as an intermediate data-compression step in converting the raw data from a future megapixel experiment to measurements of cosmological parameters. If all three data compression steps are lossless, then this data analysis pipeline will measure the parameters with just as small error bars as a (computationally unfeasible) likelihood analysis measuring the parameters directly from the time-ordered data.
• Figure 2: The power spectrum estimated from the COBE/DMR 4 year data.
• Figure 3: How the window functions depend on the target multipole.
• Figure 4: How the window functions depend on the sky coverage.
• Figure 5: Sky maps before and after power tilting.