Putting the Precision in Precision Cosmology: How accurate should your data covariance matrix be?

TL;DR

This paper analyzes how the accuracy of the data covariance matrix impacts cosmological parameter estimation, providing quantitative requirements for the number of data realizations needed for reliable results.

Contribution

It derives the statistical properties of the covariance and precision matrices, establishing how their estimation errors affect parameter uncertainties in cosmological analyses.

Findings

Fractional error on parameter variance equals fractional variance of the precision matrix.

Minimum of 200 realizations needed for 5% parameter error with fewer than 100 data points.

For large data sets, the number of realizations must exceed data points, with fractional covariance accuracy of less than sqrt(2/N_D).

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 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 realisations 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 realisations and the number of data points, and so can easily diverge. For a 5% error on a parameter error and N_D << 100 data-points, a minimum of 200 realisations of the survey are needed, with 10% accuracy for the data covariance. If the number of data-points N_D >>100 we need N_S > N_D realisations and a fractional accuracy of <sqrt[2/N_D] in the data covariance. As the number of power spectra data points grows to N_D>10^4 -10^6 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.