CMB Lensing Beyond the Power Spectrum: Cosmological Constraints from the One-Point PDF and Peak Counts

Unprecedentedly precise cosmic microwave background (CMB) data are expected from ongoing and near-future CMB Stage-III and IV surveys, which will yield reconstructed CMB lensing maps with effective resolution approaching several arcminutes. The small-scale CMB lensing fluctuations receive non-negligible contributions from nonlinear structure in the late-time density field. These fluctuations are not fully characterized by traditional two-point statistics, such as the power spectrum. Here, we use $N$-body ray-tracing simulations of CMB lensing maps to examine two higher-order statistics: the lensing convergence one-point probability distribution function (PDF) and peak counts. We show that these statistics contain significant information not captured by the two-point function, and provide specific forecasts for the ongoing Stage-III Advanced Atacama Cosmology Telescope (AdvACT) experiment. Considering only the temperature-based reconstruction estimator, we forecast 9$\sigma$ (PDF) and 6$\sigma$ (peaks) detections of these statistics with AdvACT. Our simulation pipeline fully accounts for the non-Gaussianity of the lensing reconstruction noise, which is significant and cannot be neglected. Combining the power spectrum, PDF, and peak counts for AdvACT will tighten cosmological constraints in the $\Omega_m$-$\sigma_8$ plane by $\approx 30\%$, compared to using the power spectrum alone.