# General modal estimation for cross-bispectra

## TL;DR

This paper introduces a fast, optimal estimator for measuring angular bispectra between two weakly non-Gaussian fields, applicable to various shapes and domains, demonstrated on Planck CMB data.

## Contribution

It presents a versatile, separable modal bispectrum estimator capable of analyzing multiple bispectrum shapes and domains, with implementation for Planck CMB data analysis.

## Key findings

- Effective measurement of angular bispectra from observational data.
- Applicable to both factorizable and non-factorizable bispectrum templates.
- Used in the official Planck satellite analysis of CMB maps.

## Abstract

We describe a fast, optimal estimator for measuring angular bispectra between two correlated weakly non-Gaussian fields ($Y$ and $Z$) from observational datasets, based on a separable modal bispectrum expansion. Our methodology is applicable to (1) any shape of the input theoretical bispectrum templates (factorizable or not), (2) both even and odd $\ell_1 + \ell_2 + \ell_3$ multipole domains and (3) both amplitude ($f_{\rm NL})$ bispectrum estimation and full bispectrum reconstruction, considering either joint estimation of ($YYY$, $ZZZ$, $YYZ$ and $ZZY$) shapes, or independent estimation of auto-bispectra ($YYY$ or $ZZZ$) and cross-bispectra ($YYZ$ or $ZZY$); hence, it has quite high versatility. The methodology described here was implemented and used for the official analysis of temperature and polarization cosmic microwave background maps from the $Planck$ satellite.