Perturbative approach to covariance matrix of the matter power spectrum

TL;DR

This paper develops a perturbative method to accurately estimate the covariance matrix of the matter power spectrum, compares it with simulations, and offers practical approaches for small-volume simulations.

Contribution

It introduces a perturbation theory-based framework for covariance estimation, decomposes covariance components, and provides a method to evaluate covariance from small simulations.

Findings

Agreement with simulations is within 10% up to k ~ 1 h/Mpc

Connected components are dominated by large-scale modes

Full covariance can be approximated by the disconnected part

Abstract

We evaluate the covariance matrix of the matter power spectrum using perturbation theory up to dominant terms at 1-loop order and compare it to numerical simulations. We decompose the covariance matrix into the disconnected (Gaussian) part, trispectrum from the modes outside the survey (beat coupling or super-sample variance), and trispectrum from the modes inside the survey, and show how the different components contribute to the overall covariance matrix. We find the agreement with the simulations is at a 10\% level up to $k \sim 1 h {\rm Mpc^{-1}}$. We show that all the connected components are dominated by the large-scale modes ($k<0.1 h {\rm Mpc^{-1}}$), regardless of the value of the wavevectors $k,\, k'$ of the covariance matrix, suggesting that one must be careful in applying the jackknife or bootstrap methods to the covariance matrix. We perform an eigenmode decomposition of the connected part of the covariance matrix, showing that at higher $k$ it is dominated by a single eigenmode. The full covariance matrix can be approximated as the disconnected part only, with the connected part being treated as an external nuisance parameter with a known scale dependence, and a known prior on its variance for a given survey volume. Finally, we provide a prescription for how to evaluate the covariance matrix from small box simulations without the need to simulate large volumes.