Spherical bispectrum expansion and quadratic estimators

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.