Extragalactic foregrounds in Cosmic Microwave Background (CMB) temperature maps lead to significant biases in CMB lensing reconstruction if not properly accounted for. Combinations of multi-frequency data have been used to minimize the overall map variance (internal linear combination, or ILC), or specifically null a given foreground, but these are not tailored to CMB lensing. In this paper, we derive an optimal multi-frequency combination to jointly minimize CMB lensing noise and bias. We focus on the standard lensing quadratic estimator, as well as the "shear-only" and source-hardened estimators, whose responses to foregrounds differ. We show that an optimal multi-frequency combination is a compromise between the ILC and joint deprojection, which nulls the thermal Sunyaev-Zel'dovich (tSZ) and Cosmic Infrared Background (CIB) contributions. In particular, for a Simons Observatory-like experiment with $\ell_{\text{max},T}=3000$, we find that profile hardening alone (with the standard ILC) reduces the bias to the lensing power amplitude by $40\%$, at a $20\%$ cost in noise, while the bias to the cross-correlation with a LSST-like sample is reduced by nearly an order of magnitude at a $10\%$ noise cost, relative to the standard quadratic estimator. With a small amount of joint deprojection the bias to the profile hardened estimator can be further reduced to less than half the statistical uncertainty on the respective amplitudes, at a $20\%$ and $5\%$ noise cost for the auto- and cross-correlation respectively, relative to the profile hardened estimator with the standard ILC weights. Finally, we explore possible improvements with more aggressive masking and varying $\ell_{\text{max,}T}$.
Recovering unbiased CMB polarization maps using modern ground-based experiments with minimal assumptions about atmospheric emission
Université Paris Cité, CNRS, Astroparticule et Cosmologie（巴黎西岱大学，法国国家科学研究中心，亚原子粒子与宇宙学）；CNRS-UCB International Research Laboratory, Centre Pierre Binétruy（法国国家科学研究中心-加州伯克利国际研究实验室，皮埃尔·比内屈中心）；Jodrell Bank Centre for Astrophysics, University of Manchester（乔德雷尔银行天体物理学中心，曼彻斯特大学）
