Putting the precision in precision cosmology: How accurate should your data covariance matrix be?

Abstract

Cosmological parameter estimation requires that the likelihood function of the data is accurately known. Assuming that cosmological large-scale structure power spectra data are multivariate Gaussian distributed, we show that the accuracy of parameter estimation is limited by the accuracy of the inverse data covariance matrix – the precision matrix. If the data covariance and precision matrices are estimated by sampling independent realizations of the data, their statistical properties are described by the Wishart and inverse-Wishart distributions, respectively. Independent of any details of the survey, we show that the fractional error on a parameter variance, or a figure of merit, is equal to the fractional variance of the precision matrix. In addition, for the only unbiased estimator of the precision matrix, we find that the fractional accuracy of the parameter error depends only on the difference between the number of independent realizations and the number of data points, and so can easily diverge. For a 5 per cent error on a parameter error and ND ≪ 102 data points, a minimum of 200 realizations of the survey are needed, with 10 per cent accuracy in the data covariance. If the number of data points ND ≫ 102, we need NS > ND realizations and a fractional accuracy of |$<\sqrt{2/N_{\rm D}}$| in the data covariance. As the number of power spectra data points grows to ND > 104–106, this approach will be problematic. We discuss possible ways to relax these conditions: improved theoretical modelling, shrinkage methods, data compression, simulation and data resampling methods.