Fast calculation of the Fisher matrix for cosmic microwave background experiments

TL;DR

This paper introduces an efficient method to compute the Fisher matrix for CMB experiments, reducing computational complexity from cubic to quadratic logarithmic time, enabling analysis of large datasets with realistic systematics.

Contribution

The paper presents a novel O(N^2 log N) algorithm for Fisher matrix calculation applicable to general CMB experiments, including realistic noise and sky coverage.

Findings

Computational complexity reduced from O(N^3) to O(N^2 log N).

Method accounts for realistic noise correlations and sky coverage.

Enables feasible optimal power spectrum estimation for large datasets.

Abstract

The Fisher information matrix of the cosmic microwave background (CMB) radiation power spectrum coefficients is a fundamental quantity that specifies the information content of a CMB experiment. In the most general case, its exact calculation scales with the third power of the number of data points N and is therefore computationally prohibitive for state-of-the-art surveys. Applicable to a very large class of CMB experiments without special symmetries, we show how to compute the Fisher matrix in only O(N^2 log N) operations as long as the inverse noise covariance matrix can be applied to a data vector in time O(l_max^3 log l_max). This assumption is true to a good approximation for all CMB data sets taken so far. The method takes into account common systematics such as arbitrary sky coverage and realistic noise correlations. As a consequence, optimal quadratic power spectrum estimation also becomes feasible in O(N^2 log N) operations for this large group of experiments. We discuss the relevance of our findings to other areas of cosmology where optimal power spectrum estimation plays a role.