Franz Elsner, Benjamin D. Wandelt
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. In
this letter, we show how to compute the Fisher matrix in only O(N^2 log N)
operations for a very large class of CMB experiments without special
symmetries, 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 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 class of experiments. We
discuss the relevance of our findings to other areas of cosmology where optimal
power spectrum estimation plays a role.
View original:
http://arxiv.org/abs/1202.4898
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