Fast and precise way to calculate the posterior for the local non-Gaussianity parameter $f_\text{nl}$ from cosmic microwave background observations

TL;DR

This paper introduces an efficient approximate Bayesian method to determine the posterior distribution of the local non-Gaussianity parameter $f_{nl}$ from CMB data, validated with novel tests and applicable to flat-sky and spherical models.

Contribution

It presents a new approximation technique for calculating the $f_{nl}$ posterior that avoids expensive sampling, along with a validation method and extension to higher-order non-Gaussianity.

Findings

Accurate posterior calculation under flat-sky approximation.

Inaccuracies in spherical case likely due to numerical transform issues.

Deviations from Gaussian posterior increase with larger true $f_{nl}$.

Abstract

We present an approximate calculation of the full Bayesian posterior probability distribution for the local non-Gaussianity parameter $f_{\text{nl}}$ from observations of cosmic microwave background anisotropies within the framework of information field theory. The approximation that we introduce allows us to dispense with numerically expensive sampling techniques. We use a novel posterior validation method (DIP test) in cosmology to test the precision of our method. It transfers inaccuracies of the calculated posterior into deviations from a uniform distribution for a specially constructed test quantity. For this procedure we study toy cases that use one- and two-dimensional flat skies, as well as the full spherical sky. We find that we are able to calculate the posterior precisely under a flat-sky approximation, albeit not in the spherical case. We argue that this is most likely due to an insufficient precision of the used numerical implementation of the spherical harmonic transform, which might affect other non-Gaussianity estimators as well. Furthermore, we present how a nonlinear reconstruction of the primordial gravitational potential on the full spherical sky can be obtained in principle. Using the flat-sky approximation, we find deviations for the posterior of $f_{\text{nl}}$ from a Gaussian shape that become more significant for larger values of the underlying true $f_{\text{nl}}$. We also perform a comparison to the well-known estimator of Komatsu et al. [Astrophys. J. 634, 14 (2005)] and finally derive the posterior for the local non-Gaussianity parameter $g_{\text{nl}}$ as an example of how to extend the introduced formalism to higher orders of non-Gaussianity.