Consistently constraining fNL with the squeezed lensing bispectrum using consistency relations

We introduce a nonperturbative method to constrain the amplitude of local-type primordial non-Gaussianity (fNL) using squeezed configurations of the CMB lensing convergence and cosmic shear bispectra. First, we use cosmological consistency relations to derive a model for the squeezed limit of angular auto- and cross-bispectra of lensing convergence fields in the presence of fNL. Using this model, we perform a Fisher forecast with specifications expected for upcoming CMB lensing measurements from the Simons Observatory and CMB-S4, as well as cosmic shear measurements from a Rubin LSST/Euclid-like experiment. Assuming a minimum multipole \ell_min = 10 and maximum multipole \ell_max = 1400, we forecast \sigma_fNL = 175 (95) for Simons Observatory (CMB-S4). Our forecasts improve considerably for an LSST/Euclid-like cosmic shear experiment with three tomographic bins and \ell_min = 10 and \ell_max = 1400 (5000) with \sigma_fNL = 31 (16). A joint analysis of CMB-S4 lensing and LSST/Euclid-like shear yields little gain over the shear-only forecasts; however, we show that a joint analysis could be useful if the CMB lensing convergence can be reliably reconstructed at larger angular scales than the shear field. The method presented in this work is a novel and robust technique to constrain local primordial non-Gaussianity from upcoming large-scale structure surveys that is completely independent of the galaxy field (and therefore any nuisance parameters such as b\phi), thus complementing existing techniques to constrain fNL using the scale-dependent halo bias.