Lower bias, lower noise CMB lensing with foreground-hardened estimators
  Extragalactic foregrounds in temperature maps of the Cosmic Microwave
Background (CMB) severely limit the ability of standard estimators to
reconstruct the weak lensing potential. These foregrounds are not fully
removable by multi-frequency cleaning or masking and can lead to large biases
if not properly accounted for. For foregrounds made of a number of unclustered
point sources, an estimator for the source amplitude can be derived and
deprojected, removing any bias to the lensing reconstruction. We show with
simulations that all of the extragalactic foregrounds in temperature can be
approximated by a collection of sources with identical profiles, and that a
simple bias hardening technique is effective at reducing any bias to lensing,
at a minimal noise cost. We compare the performance and bias to other methods
such as "shear-only" reconstruction, and discuss how to jointly deproject any
arbitrary number of foregrounds, each with an arbitrary profile. In particular,
for a Simons Observatory-like experiment foreground-hardened estimators allow
us to extend the maximum multipole used in the reconstruction, increasing the
overall statistical power by $\sim 50\%$ over the standard quadratic estimator,
both in auto and cross-correlation. We conclude that source hardening
outperforms the standard lensing quadratic estimator both in auto and
cross-correlation, and in terms of lensing signal-to-noise and foreground bias.
