Fast calculation of the Fisher matrix for cosmic microwave background
  experiments
  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.
