The shape of the CMB lensing bispectrum

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 similar to 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.