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

## TL;DR

This paper evaluates the accuracy of analytical models predicting the CMB lensing bi-spectrum by comparing them with full-sky simulations, highlighting their strengths and limitations across different scales and configurations.

## Contribution

It provides a detailed comparison of analytical predictions and simulations for the CMB lensing bi-spectrum, including the effects of perturbation theory and fitting formulas, and discusses implications for modified gravity theories.

## Key findings

- Tree-level perturbation theory matches simulations up to 200.
- One-loop order captures results up to 600.
- Fitting formula predictions agree within 10% for equilateral configurations up to 2000.

## Abstract

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.