### Abstract

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.

Original language | English (US) |
---|---|

Pages (from-to) | 1188-1204 |

Number of pages | 17 |

Journal | Monthly Notices of the Royal Astronomical Society |

Volume | 371 |

Issue number | 3 |

DOIs | |

State | Published - Sep 2006 |

### Fingerprint

### Keywords

- Large-scale structure of Universe
- Methods: data analysis

### ASJC Scopus subject areas

- Space and Planetary Science

### Cite this

*Monthly Notices of the Royal Astronomical Society*,

*371*(3), 1188-1204. https://doi.org/10.1111/j.1365-2966.2006.10709.x

**On measuring the covariance matrix of the non-linear power spectrum from simulations.** / Hamilton, Andrew J S; Rimes, Christopher D.; Scoccimarro, Román.

Research output: Contribution to journal › Article

*Monthly Notices of the Royal Astronomical Society*, vol. 371, no. 3, pp. 1188-1204. https://doi.org/10.1111/j.1365-2966.2006.10709.x

}

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

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

UR - http://www.scopus.com/inward/record.url?scp=33748324610&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33748324610&partnerID=8YFLogxK

U2 - 10.1111/j.1365-2966.2006.10709.x

DO - 10.1111/j.1365-2966.2006.10709.x

M3 - Article

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 -