On measuring the covariance matrix of the non‐linear power spectrum from simulations

We show how to estimate the covariance of the power spectrum of a statistically homogeneous and isotropic density field from a single periodic simulation, by applying a set of weightings to the density field, and by measuring the scatter in power spectra between different weightings. We recommend a specific set of 52 weightings containing only combinations of fundamental modes, constructed to yield a minimum variance estimate of the covariance of power. Numerical tests reveal that at non‐linear scales the variance of power estimated by the weightings method substantially exceeds that estimated from a simple ensemble method. We argue that the discrepancy is caused by beat‐coupling, in which products of closely spaced Fourier modes couple by non‐linear gravitational growth to the beat mode between them. Beat‐coupling appears whenever non‐linear power is measured from Fourier modes with a finite spread of wavevector, and is therefore present in the weightings method but not in the ensemble method. Beat‐coupling inevitably affects real galaxy surveys, whose Fourier modes have finite width. Surprisingly, the beat‐coupling contribution dominates the covariance of power at non‐linear scales, so that, counter‐intuitively, it is expected that the covariance of non‐linear power in galaxy surveys is dominated not by small‐scale structure, but rather by beat‐coupling to the largest scales of the survey.