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.