CMB lensing bi-spectrum: 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 bi-spectrum, will be measured at high statistical significance. This
will provide ways to improve the constraints on cosmological models and lift
degeneracies. Following on an earlier paper, we test analytical predictions of
the CMB lensing bi-spectrum against full-sky lensing simulations, and discuss
their validity and limitation in detail. The tree-level prediction of
perturbation theory agrees with the simulation only up to $\ell\sim 200$, but
the one-loop order allows capturing the simulation results up to $\ell\sim
600$. We also show that analytical predictions based on fitting formulas for
the matter bi-spectrum 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 $\ell\sim2000$, but the
difference in the squeezed limit raises to more than a factor of two at
$\ell\sim2000$. This discrepancy appears to come from limitations in the
fitting formula of the matter bi-spectrum. We also find that the analytical
prediction for the post-Born correction to the bi-spectrum is in good agreement
with the simulation. We conclude by discussing the bi-spectrum prediction in
some theories of modified gravity.
