Minimizing gravitational lensing contributions to the primordial
  bispectrum covariance
  The next generation of ground-based CMB experiments aim to measure
temperature and polarization fluctuations up to $\ell_{\rm max} \approx 5000$
over half of the sky. Combined with Planck data on large scales, this will
provide improved constraints on primordial non-Gaussianity. However, the
impressive resolution of these experiments will come at a price. Besides signal
confusion from galactic foregrounds, extra-galactic foregrounds and late-time
gravitational effects, gravitational lensing will introduce large
non-Gaussianity that can become the leading contribution to the bispectrum
covariance through the connected 4-point function. Here, we compute this effect
analytically for the first time on the full sky for both temperature and
polarization. We compare our analytical results with those obtained directly
from map-based simulations of the CMB sky for several levels of instrumental
noise. Of the standard shapes considered in the literature, the local shape is
most affected, resulting in a 35\% increase of the estimator standard deviation
for an experiment like the Simons Observatory (SO) and a 110\% increase for a
cosmic-variance limited experiment, including both temperature and polarization
modes up to $\ell_{\rm max} = 3800$. Because of the nature of the lensing
4-point function, the impact on other shapes is reduced while still non
negligible for the orthogonal shape. Two possible avenues to reduce the
non-Gaussian contribution to the covariance are proposed. First by
marginalizing over lensing contributions, such as the ISW-lensing 3pt function
in temperature, and second by delensing the CMB. We show the latter method can
remove almost all extra covariance, reducing the effect to below $<$5\% for
local bispectra. At the same time, delensing would remove signal biases from
secondaries induced by lensing, such as ISW-lensing.
