Bispectrum covariance in the flat-sky limit

TL;DR

This paper derives the bispectrum covariance in the flat-sky limit using a Fourier-plane approach, compares it to spherical harmonic methods, and clarifies differences and similarities in their information content.

Contribution

It provides a detailed formal derivation of the bispectrum covariance in the flat-sky approximation and introduces a geometrical interpretation of the estimator.

Findings

A factor of two difference in covariance between Fourier-plane and spherical harmonic approaches.

High agreement in Fisher information between the two formalisms.

A geometrical link between estimator normalization and triangle area/Wigner symbols.

Abstract

To probe cosmological fields beyond the Gaussian level, three-point statistics can be used, all of which are related to the bispectrum. Hence, measurements of CMB anisotropies, galaxy clustering, and weak gravitational lensing alike have to rely upon an accurate theoretical background concerning the bispectrum and its noise properties. If only small portions of the sky are considered, it is often desirable to perform the analysis in the flat-sky limit. We aim at a formal, detailed derivation of the bispectrum covariance in the flat-sky approximation, focusing on a pure two-dimensional Fourier-plane approach. We define an unbiased estimator of the bispectrum, which takes the average over the overlap of annuli in Fourier space, and compute its full covariance. The outcome of our formalism is compared to the flat-sky spherical harmonic approximation in terms of the covariance, the behavior under parity transformations, and the information content. We introduce a geometrical interpretation of the averaging process in the estimator, thus providing an intuitive understanding. Contrary to foregoing work, we find a difference by a factor of two between the covariances of the Fourier-plane and the spherical harmonic approach. We argue that this discrepancy can be explained by the differing behavior with respect to parity. However, in an exemplary analysis it is demonstrated that the Fisher information of both formalisms agrees to high accuracy. Via the geometrical interpretation we are able to link the normalization in the bispectrum estimator to the area enclosed by the triangle configuration at consideration as well as to the Wigner symbol, which leads to convenient approximation formulae for the covariances of both approaches.