Optimal multi-frequency weighting for CMB lensing

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 $ell_{text{max,}T}$.
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