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Cosmic microwave background (CMB) lensing is an integrated effect whosekernel is greater than half the peak value in the range $1<z<5$. Measuring thiseffect offers a powerful tool to probe the large-scale structure of theUniverse at high redshifts. With the increasing precision of ongoing CMBsurveys, other statistics than the lensing power spectrum, in particular thelensing bi-spectrum, will be measured at high statistical significance. Thiswill provide ways to improve the constraints on cosmological models and liftdegeneracies. Following on an earlier paper, we test analytical predictions ofthe CMB lensing bi-spectrum against full-sky lensing simulations, and discusstheir validity and limitation in detail. The tree-level prediction ofperturbation theory agrees with the simulation only up to $\ell\sim 200$, butthe one-loop order allows capturing the simulation results up to $\ell\sim600$. We also show that analytical predictions based on fitting formulas forthe matter bi-spectrum agree reasonably well with simulation results, althoughthe precision of the agreement depends on the configurations and scalesconsidered. For instance, the agreement is at the $10\%$-level for theequilateral configuration at multipoles up to $\ell\sim2000$, but thedifference in the squeezed limit raises to more than a factor of two at$\ell\sim2000$. This discrepancy appears to come from limitations in thefitting formula of the matter bi-spectrum. We also find that the analyticalprediction for the post-Born correction to the bi-spectrum is in good agreementwith the simulation. We conclude by discussing the bi-spectrum prediction insome theories of modified gravity.

CMB lensing bi-spectrum: assessing analytical predictions against full-sky lensing simulations