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

## TL;DR

This paper introduces a precision matrix expansion method that efficiently estimates errors on cosmological parameters using fewer simulations by leveraging analytical models and simulation data, improving accuracy for large-scale structure analyses.

## Contribution

The paper presents a novel expansion technique for the precision matrix that reduces the number of simulations needed to accurately estimate cosmological parameter errors.

## Key findings

- 400 simulations suffice for DES to achieve negligible uncertainties
- 2400 simulations are enough for LSST to reach similar precision
- Standard covariance estimation requires over 100,000 simulations for comparable accuracy

## Abstract

Computing the inverse covariance matrix (or precision matrix) of large data vectors is crucial in weak lensing (and multi-probe) 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 fit 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 true 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, $\smash{\mathbf{C} = \mathbf{A}+\mathbf{B}}$, where $\smash{\mathbf{A}}$ is well understood analytically and can be turned off in simulations (e.g. shape-noise for cosmic shear) to yield a direct estimate of $\smash{\mathbf{B}}$. We test our method in mock experiments resembling tomographic weak lensing data vectors from the Dark Energy Survey (DES) and the Large Synoptic Survey Telecope (LSST). For DES we find that $400$ N-body simulations are sufficient to achive negligible statistical uncertainties on parameter constraints. For LSST this is achieved with $2400$ simulations. The standard covariance estimator would require >$10^5$ simulations to reach a similar precision. We extend our analysis to a DES multi-probe case finding a similar performance.