Fast calculation of the Fisher matrix for cosmic microwave background experiments

F. Elsner¹ and B. D. Wandelt^1,2

¹ Institut d’Astrophysique de Paris, UMR 7095, CNRS – Université Pierre et Marie Curie (Univ. Paris 06), 98bis Bd Arago, 75014 Paris, France
e-mail: elsner@iap.fr
² Departments of Physics and Astronomy, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA

Received: 8 February 2012
Accepted: 21 February 2012

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(N² log N) operations as long as the inverse noise covariance matrix can be applied to a data vector in time 𝒪(ℓ³_max 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(N²log 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.