Optimizing foreground mitigation for CMB lensing with combined multifrequency and geometric methods

A key challenge for current and upcoming CMB lensing measurements is their sensitivity to biases from extragalactic foregrounds, such as Sunyaev-Zeldovich (SZ) signals or cosmic infrared background emission. Several methods have been developed to mitigate these lensing foreground biases, dividing broadly into multi-frequency cleaning approaches and modifications to the estimator geometry, but how to optimally combine these methods has not yet been explored in detail. In this paper, we examine which combination of lensing foreground mitigation strategies is best able to reduce the impact of foreground contamination for a Simons-Observatory-like experiment while preserving maximal signal-to-noise. Although the optimal combination obtained depends on whether bias- or variance-reduction are prioritized and on whether polarization data is used, generally, we find that combinations involving both geometric (profile hardening, source hardening or shear) and multifrequency (symmetric cleaning) methods perform best. For lensing power spectrum measurements from temperature (polarization and temperature), our combined estimator methods are able to reduce the bias to the lensing amplitude below $\sigma/4$ or 0.3% (0.1%), a factor of 16 (30) lower than the standard QE bias, at a modest signal-to-noise cost of only 18% (12%). In contrast, single-method foreground-mitigation approaches struggle to reduce the bias to a negligible level below $\sigma/2$ without incurring a large noise penalty. For upcoming and current experiments, our combined methods therefore represent a promising approach for making lensing measurements with negligible foreground bias.