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Upcoming cosmic microwave background (CMB) lensing measurements andtomographic galaxy surveys are expected to provide us with high-precision datasets in the coming years, thus paving the way for fruitful cross-correlationanalyses. In this paper we study the information content of the weightedskew-spectrum, a nearly-optimal estimator of the angular bispectrum amplitude,as a means to extract non-Gaussian information on both bias and cosmologicalparameters from the bispectra of galaxies cross-correlated with CMB lensing,while gaining significantly on speed. Our results show that for the combinationof the Planck satellite and the Dark Energy Spectroscopic Instrument (DESI),the difference in the constraints on bias and cosmological parameters from theskew-spectrum and the bispectrum is at most $17\%$. We further compare and findagreement between our theoretical skew-spectra and those estimated from N-bodysimulations, for which it is important to include gravitational non-linearitiesbeyond perturbation theory and the post-Born effect for CMB lensing. We definean algorithm to apply the skew-spectrum estimator to the data and, as apreliminary step, we use the skew-spectra to constrain bias parameters and theamplitude of shot noise from the simulations through a Markov chain Monte Carlolikelihood analysis, finding that it may be possible to reach percent-levelestimates for the linear bias parameter $b_1$.

Skewing the CMB$\times$LSS: a Fast Method for Bispectrum Analysis