Skewing the CMB$\times$LSS: a Fast Method for Bispectrum Analysis
  Upcoming cosmic microwave background (CMB) lensing measurements and
tomographic galaxy surveys are expected to provide us with high-precision data
sets in the coming years, thus paving the way for fruitful cross-correlation
analyses. In this paper we study the information content of the weighted
skew-spectrum, a nearly-optimal estimator of the angular bispectrum amplitude,
as a means to extract non-Gaussian information on both bias and cosmological
parameters from the bispectra of galaxies cross-correlated with CMB lensing,
while gaining significantly on speed. Our results suggest that for the
combination of Planck satellite and the Dark Energy Spectroscopic Instrument
(DESI), the skew-spectrum achieves almost equivalent information for both bias
and cosmological parameters as the bispectrum, where the difference in the
constraints is at most $17\%$. We further compare and find agreement between
our theoretical skew-spectra and those estimated from N-body simulations, and
show that it is crucial to include gravitational non-linearities beyond
perturbation theory and the post-Born effect for CMB lensing. We define an
algorithm to apply the skew-spectrum estimator to the data and, as a
preliminary step, we use the skew-spectra to constrain bias parameters and the
amplitude of shot noise from the simulations through a Markov chain Monte Carlo
likelihood analysis, finding that it may be possible to reach percent-level
estimates for the linear bias parameter $b_1$.
