Searching for patchy reionization from cosmic microwave background with hybrid quadratic estimators

We propose a hybrid quadratic estimator to measure cross correlations between gravitational lensing of the cosmic microwave background (CMB) and differential screening effects arising from fluctuations in the electron column density, such as could arise from patchy reionization. The hybrid quadratic estimators are validated by simulated data sets with both Planck and CMB-Stage 4 (CMB-S4) instrumental properties and found to be able to recover the cross-power spectra with almost no biases. We apply this technique to Planck 2015 temperature data and obtain cross-power spectra between gravitational lensing and differential screening effects. Planck data alone cannot detect the patchy-reionization-induced cross-power spectrum but future experiment like CMB-S4 will be able to robustly measure the expected signal and deliver new insights on reionization.