The CMB Bispectrum, Trispectrum, non-Gaussianity, and the Cramer-Rao Bound

TL;DR

This paper investigates the relationship between bispectrum and trispectrum estimators for non-Gaussianity in the CMB, demonstrating that both provide independent information about the parameter fnl, consistent with the Cramer-Rao bound.

Contribution

It clarifies that bispectrum and trispectrum estimators become statistically independent at large scales, showing the trispectrum adds information beyond the bispectrum for estimating fnl.

Findings

Bispectrum and trispectrum estimators are statistically independent at large pixel counts.

The trispectrum provides additional information on fnl beyond the bispectrum.

The analysis aligns with the Cramer-Rao bound, confirming the potential for combined estimators.

Abstract

Minimum-variance estimators for the parameter fnl that quantifies local-model non-Gaussianity can be constructed from the cosmic microwave background (CMB) bispectrum (three-point function) and also from the trispectrum (four-point function). Some have suggested that a comparison between the estimates for the values of fnl from the bispectrum and trispectrum allow a consistency test for the model. But others argue that the saturation of the Cramer-Rao bound by the bispectrum estimator implies that no further information on fnl can be obtained from the trispectrum. Here we elaborate the nature of the correlation between the bispectrum and trispectrum estimators for fnl. We show that the two estimators become statistically independent in the limit of large number of CMB pixels and thus that the trispectrum estimator does indeed provide additional information on fnl beyond that obtained from the bispectrum. We explain how this conclusion is consistent with the Cramer-Rao bound. Our discussion of the Cramer-Rao bound may be of interest to those doing Fisher-matrix parameter-estimation forecasts or data analysis in other areas of physics as well.