Optimal Estimation of Non-Gaussianity

TL;DR

This paper uses estimation theory to evaluate whether the WMAP bispectrum-based estimator for primordial non-Gaussianity is information-optimal. It demonstrates that, for weak non-Gaussianity, the three-point (bispectrum) estimator saturates the Cramer-Rao bound and thus preserves all accessible information about $f_{NL}$. The analysis extends from Poisson toy models to scale-invariant curvature perturbations with Sachs-Wolfe projection and full radiative transfer, connecting to Wiener-filter implementations and showing the WMAP estimator is effectively optimal in practice. It further argues, via Edgeworth expansion, that this optimality extends to arbitrary small-amplitude bispectra, making the bispectrum-based approach the right target for constraining primordial non-Gaussianity across models, while acknowledging practical limitations and directions for improved estimators.

Abstract

We systematically analyze the primordial non-Gaussianity estimator used by the Wilkinson Microwave Anisotropy Probe (WMAP) science team with the basic ideas of estimation theory in order to see if the limited Cosmic Microwave Background (CMB) data is being optimally utilized. The WMAP estimator is based on the implicit assumption that the CMB bispectrum, the harmonic transform of the three-point correlation function, contains all of the primordial non-Gaussianity information in a CMB map. We first demonstrate that the Signal-to-Noise (S/N) of an estimator based on CMB three-point correlation functions is significantly larger than the S/N of any estimator based on higher-order correlation functions; justifying our choice to focus on the three-point correlation function. We then conclude that the estimator based on the three-point correlation function, which was used by WMAP, is optimal, meaning it saturates the Cramer-Rao Inequality when the underlying CMB map is nearly Gaussian. We quantify this restriction by demonstrating that the suboptimal character of our estimator is proportonal to the square of the fiducial non-Gaussianity, which is already constrained to be extremely small, so we can consider the WMAP estimator to be optimal in practice. Our conclusions do not depend on the form of the primordial bispectrum, only on the observationally established weak levels of primordial non-Gaussianity.