Optimal multifrequency weighting for CMB lensing

Extragalactic foregrounds in cosmic microwave background (CMB) temperature maps lead to significant biases CMB lensing reconstruction if not properly accounted for. Combinations of multifrequency data have been used minimize the overall map variance (internal linear combination, or ILC), specifically null a given foreground, but these are tailored lensing. In this paper, we derive an optimal combination jointly noise and bias. We focus on standard quadratic estimator, as well ``shear-only'' source-hardened estimators, whose responses differ. show that is compromise between ILC joint deprojection, which nulls thermal Sunyaev-Zel'dovich (tSZ) infrared (CIB) contributions. particular, for Simons Observatory-like experiment with ${\ensuremath{\ell}}_{\mathrm{max},T}=3000$, find profile hardening alone (with ILC) reduces bias power amplitude by 40%, at 20% cost noise, while cross-correlation LSST-like sample reduced nearly order magnitude 10% cost, relative estimator. With small amount deprojection hardened estimator can be further less than half statistical uncertainty respective amplitudes, 5% auto- respectively, weights. Finally, explore possible improvements more aggressive masking varying ${\ensuremath{\ell}}_{\mathrm{max},T}$.