CMB lensing bispectrum: Assessing analytical predictions against full-sky lensing simulations

Cosmic microwave background (CMB) lensing is an integrated effect whose kernel is greater than half the peak value in the range 1 < z < 5. Measuring this effect offers a powerful tool to probe the large-scale structure of the Universe at high redshifts. With the increasing precision of ongoing CMB surveys, other statistics than the lensing power spectrum, in particular the lensing bispectrum, will be measured at high statistical significance. This will provide ways to improve the constraints on cosmological models and lift degeneracies. Following up on an earlier paper, we test analytical predictions of the CMB lensing bispectrum against full-sky lensing simulations, and we discuss their validity and limitation in detail. The tree-level prediction of perturbation theory agrees with the simulation only up to l similar to 200, but the one-loop order allows capturing the simulation results up to l similar to 600. We also show that analytical predictions based on fitting formulas for the matter bispectrum agree reasonably well with simulation results, although the precision of the agreement depends on the configurations and scales considered. For instance, the agreement is at the 10% level for the equilateral configuration at multipoles up to l similar to 2000, but the difference in the squeezed limit increases to more than a factor of 2 at l similar to 2000. This discrepancy appears to come from limitations in the fitting formula of the matter bispectrum. We also find that the analytical prediction for the post-Born correction to the bispectrum is in good agreement with the simulation. We conclude by discussing the bispectrum prediction in some theories of modified gravity.