Thinking outside the box: effects of modes larger than the survey on matter power spectrum covariance

Abstract

Accurate power spectrum (or correlation function) covariance matrices are a crucial requirement for cosmological parameter estimation from large scale structure surveys. In order to minimize reliance on computationally expensive mock catalogs, it is important to have a solid analytic understanding of the different components that make up a covariance matrix. Considering the matter power spectrum covariance matrix, it has recently been found that there is a potentially dominant effect on mildly non-linear scales due to power in modes of size equal to and larger than the survey volume. This beat coupling effect has been derived analytically in perturbation theory and while it has been tested with simulations, some questions remain unanswered. Moreover, there is an additional effect of these large modes, which has so far not been included in analytic studies, namely the effect on the estimated average density which enters the power spectrum estimate. In this article, we work out analytic, perturbation theory based expressions including both the beat coupling and this local average effect and we show that while, when isolated, beat coupling indeed causes large excess covariance in agreement with the literature, in a realistic scenario this is compensated almost entirely by the local average effect, leaving only ∼ 10% of the excess. We test our analytic expressions by comparison to a suite of large N-body simulations, using both full simulation boxes and subboxes thereof to study cases without beat coupling, with beat coupling and with both beat coupling and the local average effect. For the variances, we find excellent agreement with the analytic expressions for k < 0.2hMpc −1 at z = 0.5, while the correlation coefficients agree to beyond k = 0.4hMpc −1 . As expected, the range of agreement increases towards higher redshift and decreases slightly towards z = 0. We finish by including the large-mode effects in a full covariance matrix description for arbitrary survey geometry and confirming its validity using simulations. This may be useful as a stepping stone towards building an actual galaxy (or other tracer's) power spectrum covariance matrix.

Super-sample covariance in simulations

“…This verifies that the linear dilation effect, omitted in previous studies, must be included as part of the SSC model for accuracy in the quasilinear regime (cf. [11]). Finally in Fig.…”

“…(11) directly, or using Gaussian realizations of the linear density field. Note that we define density fluctuations relative to the global mean density rather than the mean within the survey window [11] …”

“…It needs only to be calibrated once for a given cosmological model since the small scale response is the same regardless of the survey geometry. In this paper, we perform this calibration and test the accuracy with which it describes the SSC effect directly with survey-windowed simulations [10,11,19].…”

Super-sample tidal modes on the celestial sphere

“…] is related to that defined to the "global" mean density through n g (k) =n global g [1 + δ global g (k)] as follows [16],…”

Covariances for cosmic shear and galaxy–galaxy lensing in the response approach

“…where k W = (1 − δ b /3) k is a wavenumber measured in the SU simulation, and n = 2, 1, and 0 when XY = mm, hm and hh, respectively. The extra factor (1+δ b ) n is necessary when converting matter density fluctuation fields, defined with respect to the SU local density, to that with respect to the global background density (de Putter et al 2012…”