Inference of gravitational lensing and patchy reionization with future CMB data

We develop an optimal Bayesian solution for jointly inferring secondary signals in the Cosmic Microwave Background (CMB) originating from gravitational lensing and from patchy screening during the epoch of reionization. This method is able to extract full information content from the data, improving upon previously considered quadratic estimators for lensing and screening. We forecast constraints using the Marginal Unbiased Score Expansion (MUSE) method, and show that they are largely dominated by CMB polarization, and depend on the exact details of reionization. For models consistent with current data which produce the largest screening signals, a detection (3\,$\sigma$) of the cross-correlation between lensing and screening is possible with SPT-3G, and a detection of the auto-correlation is possible with CMB-S4. Models with the lowest screening signals evade the sensitivity of SPT-3G, but are still possible to detect with CMB-S4 via their lensing cross-correlation.