The bispectrum covariance beyond Gaussianity

A log-normal approach

Argelander-Institut für Astronomie (AIfA), Universität Bonn, Auf dem Hügel 71, 53121 Bonn, Germany
e-mail: martin@astro.uni-bonn.de

Received: 5 September 2011
Accepted: 7 February 2012

Abstract

Context. To investigate and specify the statistical properties of cosmological fields with particular attention to possible non-Gaussian features, accurate formulae for the bispectrum and the bispectrum covariance are required. The bispectrum is the lowest-order statistic providing an estimate for non-Gaussianities of a distribution, and the bispectrum covariance depicts the errors of the bispectrum measurement and their correlation on different scales. Currently, there do exist fitting formulae for the bispectrum and an analytical expression for the bispectrum covariance, but the former is not very accurate and the latter contains several intricate terms and only one of them can be readily evaluated from the power spectrum of the studied field. Neglecting all higher-order terms results in the Gaussian approximation of the bispectrum covariance.

Aims. We study the range of validity of this Gaussian approximation for two-dimensional non-Gaussian random fields.

Methods. For this purpose, we simulate Gaussian and non-Gaussian random fields, the latter represented by log-normal fields and obtained directly from the former by a simple transformation. From the simulated fields, we calculate the power spectra, the bispectra, and the covariance from the sample variance of the bispectra, for different degrees of non-Gaussianity α, which is equivalent to the skewness on a given angular scale θ_g. In doing so, we minimize sample variance by selecting the same “phases” of the Gaussian and non-Gaussian fields.

Results. We find that the Gaussian approximation of the bispectrum covariance provides a good approximation for degrees of non-Gaussianity α ≤ 0.5 and a reasonably accurate approximation for α ≤ 1, both on scales  ≳ 8θ_g. For larger values of α, the Gaussian approximation clearly breaks down. Using results from cosmic shear simulations, we estimate that the cosmic shear convergence fields are described by α ≲ 0.7 at θ_g ~ 4′′.

Conclusions. We therefore conclude that the Gaussian approximation for the bispectrum covariance is likely to be applicable in ongoing and future cosmic shear studies.