Precision matrix expansion - efficient use of numerical simulations in estimating errors on cosmological parameters

Computing the inverse covariance matrix (or precision matrix) of large data vectors is crucial in weak lensing (and multiprobe) analyses of the large-scale structure of the Universe. Analytically computed covariances are noise-free and hence straightforward to invert; however, the model approximations might be insufficient for the statistical precision of future cosmological data. Estimating covariances from numerical simulations improves on these approximations, but the sample covariance estimator is inherently noisy, which introduces uncertainties in the error bars on cosmological parameters and also additional scatter in their best-fitting values. For future surveys, reducing both effects to an acceptable level requires an unfeasibly large number of simulations. In this paper we describe a way to expand the precision matrix around a covariance model and show how to estimate the leading order terms of this expansion from simulations. This is especially powerful if the covariance matrix is the sum of two contributions, C = A + B, where A is well understood analytically and can be turned offin simulations (e.g. shape noise for cosmic shear) to yield a direct estimate of B. We test our method in mock experiments resembling tomographic weak lensing data vectors from the Dark Energy Survey (DES) and the Large Synoptic Survey Telescope (LSST). For DES we find that 400 N-body simulations are sufficient to achieve negligible statistical uncertainties on parameter constraints. For LSST this is achieved with 2400 simulations. The standard covariance estimator would require > 10⁵ simulations to reach a similar precision. We extend our analysis to a DES multiprobe case finding a similar performance.