Optimized Large-Scale CMB Likelihood And Quadratic Maximum Likelihood Power Spectrum Estimation

TL;DR

This paper develops an efficient computational framework for large-scale CMB likelihood and power spectrum estimation using linear compression, significantly reducing computational costs and improving robustness, demonstrated with WMAP data.

Contribution

It introduces a fully working, optimized linear compression method for CMB analysis, identifying the signal-to-noise eigenvector basis as most efficient, and presents a stable, low-memory QML implementation.

Findings

Compression reduces likelihood evaluation time by a factor of 5.

Signal-to-noise eigenvector basis is most efficient for data compression.

The method enables low-$mbda$ QML analysis on standard laptops.

Abstract

We revisit the problem of exact CMB likelihood and power spectrum estimation with the goal of minimizing computational cost through linear compression. This idea was originally proposed for CMB purposes by Tegmark et al.\ (1997), and here we develop it into a fully working computational framework for large-scale polarization analysis, adopting \WMAP\ as a worked example. We compare five different linear bases (pixel space, harmonic space, noise covariance eigenvectors, signal-to-noise covariance eigenvectors and signal-plus-noise covariance eigenvectors) in terms of compression efficiency, and find that the computationally most efficient basis is the signal-to-noise eigenvector basis, which is closely related to the Karhunen-Loeve and Principal Component transforms, in agreement with previous suggestions. For this basis, the information in 6836 unmasked \WMAP\ sky map pixels can be compressed into a smaller set of 3102 modes, with a maximum error increase of any single multipole of 3.8\% at $\ell\le32$, and a maximum shift in the mean values of a joint distribution of an amplitude--tilt model of 0.006$\sigma$. This compression reduces the computational cost of a single likelihood evaluation by a factor of 5, from 38 to 7.5 CPU seconds, and it also results in a more robust likelihood by implicitly regularizing nearly degenerate modes. Finally, we use the same compression framework to formulate a numerically stable and computationally efficient variation of the Quadratic Maximum Likelihood implementation that requires less than 3 GB of memory and 2 CPU minutes per iteration for $\ell \le 32$, rendering low-$\ell$ QML CMB power spectrum analysis fully tractable on a standard laptop.