  Lensing of the CMB generates a significant bispectrum, which should be
detected by the Planck satellite at the 5-sigma level and is potentially a
non-negligible source of bias for f_NL estimators of local non-Gaussianity. We
extend current understanding of the lensing bispectrum in several directions:
(1) we perform a non-perturbative calculation of the lensing bispectrum which
is ~10% more accurate than previous, first-order calculations; (2) we
demonstrate how to incorporate the signal variance of the lensing bispectrum
into estimates of its amplitude, providing a good analytical explanation for
previous Monte-Carlo results; and (3) we discover the existence of a
significant lensing bispectrum in polarization, due to a previously-unnoticed
correlation between the lensing potential and E-polarization as large as 30% at
low multipoles. We use this improved understanding of the lensing bispectra to
re-evaluate Fisher-matrix predictions, both for Planck and cosmic variance
limited data. We confirm that the non-negligible lensing-induced bias for
estimation of local non-Gaussianity should be robustly treatable, and will only
inflate f_NL error bars by a few percent over predictions where lensing effects
are completely ignored (but note that lensing must still be accounted for to
obtain unbiased constraints). We also show that the detection significance for
the lensing bispectrum itself is ultimately limited to 9 sigma by cosmic
variance. The tools that we develop for non-perturbative calculation of the
lensing bispectrum are directly relevant to other calculations, and we give an
explicit construction of a simple non-perturbative quadratic estimator for the
lensing potential and relate its cross-correlation power spectrum to the
bispectrum. Our numerical codes are publicly available as part of CAMB and
LensPix.
