2+1 is not 3: Angular bispectrum and Super-Sample Covariance on the light cone

Observations of the Large Scale Structure sit on the light cone. Cosmological statistics must then be projected from 3D to angular space. For the bispectrum and higher orders, the polyspectra crucially depend on the angles between Fourier modes in addition to their moduli. This leads to delicate cancellations in the squeezed limit and make Limber's approximation fail. This is problematic for predicting the Super-Sample Covariance (SSC), with no known full-sky computation of the angle-dependent terms. Here we develop a new analytical approach to the angular bispectrum, writing Fourier vectors as derivatives that we project on the sphere and tackle with spin-weighted spherical harmonics. We find bispectrum equations free of cancellations and amenable to beyond-Limber numerical methods. Limber's approximation becomes well-behaved and works in the squeezed limit and any other configuration. We then derive the consequence for the covariance between 1-point (e.g. cluster or peak counts) and 2-point statistics (e.g. galaxy clustering, weak lensing or 21cm), which relates to the squeezed bispectrum. The results split into intra-survey covariance and SSC, where we find significant differences with past literature. Our results are in line with the 2+1 geometry of the light cone, having e.g. temporal derivatives in addition to spatial ones. We also find that the tidal term is degenerate with the simpler growth-only term. The total SSC contains a wealth of new terms including wide-angle effects and responses to the background velocity. However we show numerically that it is well approximated by a single term with a density-only response. This ultimately makes our results simpler than past literature. This full-sky result should be more representative than the past flat-sky limit for large survey areas, thus being the one relevant for coming observations with e.g. Euclid, LSST and Roman.