TY - JOUR

T1 - On measuring the covariance matrix of the non-linear power spectrum from simulations

AU - Hamilton, Andrew J.S.

AU - Rimes, Christopher D.

AU - Scoccimarro, Román

N1 - Copyright:
Copyright 2006 Elsevier B.V., All rights reserved.

PY - 2006/9

Y1 - 2006/9

N2 - 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.

AB - 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.

KW - Large-scale structure of Universe

KW - Methods: data analysis

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U2 - 10.1111/j.1365-2966.2006.10709.x

DO - 10.1111/j.1365-2966.2006.10709.x

M3 - Article

AN - SCOPUS:33748324610

VL - 371

SP - 1188

EP - 1204

JO - Monthly Notices of the Royal Astronomical Society

JF - Monthly Notices of the Royal Astronomical Society

SN - 0035-8711

IS - 3

ER -