Spherical bispectrum expansion and quadratic estimators

TL;DR

This paper introduces a comprehensive spherical bispectrum expansion into orthogonal modes, enabling improved analysis of CMB bispectra and construction of quadratic estimators that are robust against foregrounds and noise.

Contribution

It presents a novel full-sky bispectrum expansion using orthogonal polynomials, facilitating the development of new quadratic estimators for CMB lensing that are immune to certain foregrounds.

Findings

Orthogonal basis for spherical bispectra developed

New quadratic estimators constructed for CMB lensing

Estimators are robust against foreground contamination

Abstract

We describe a general expansion of spherical (full-sky) bispectra into a set of orthogonal modes. For squeezed shapes, the basis separates physically-distinct signals and is dominated by the lowest moments. In terms of reduced bispectra, we identify a set of discrete polynomials that are pairwise orthogonal with respect to the relevant Wigner 3j symbol, and reduce to Chebyshev polynomials in the flat-sky (high-momentum) limit for both parity-even and parity-odd cases. For squeezed shapes, the flat-sky limit is equivalent to previous moment expansions used for CMB bispectra and quadratic estimators, but in general reduces to a distinct expansion in the angular dependence of triangles at fixed total side length (momentum). We use the full-sky expansion to construct a tower of orthogonal CMB lensing quadratic estimators and construct estimators that are immune to foregrounds like point sources or noise inhomogeneities. In parity-even combinations (such as the lensing gradient mode from $TT$, or the lensing curl mode from $EB$) the leading two modes can be identified with information from the magnification and shear respectively, whereas the parity-odd combinations are shear-only. Although not directly separable, we show that these estimators can nonetheless be evaluated numerically sufficiently easily.