How to measure CMB power spectra without losing information

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 arbitrary regions masked out and arbitrary noise behavior. (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 future high-resolution CMB experiments with $n\sim {10}^6$–${10}^7$. It is 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.