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Extragalactic foregrounds are known to generate significant biases in temperature-based CMB lensing reconstruction. Several techniques, which include ``source hardening'' and ``shear-only estimators'' have been proposed to mitigate contamination and have been shown to be very effective at reducing foreground-induced biases. Here we extend both techniques to polarization, which will be an essential component of CMB lensing reconstruction for future experiments, and investigate the ``large-lens'' limit analytically to gain insight on the origin and scaling of foreground biases, as well as the sensitivity to their profiles.Using simulations of polarized point sources, we estimate the expected bias to both Simons Observatory and CMB-S4 like (polarization-based) lensing reconstruction, finding that biases to the former are minuscule while those to the latter are potentially non-negligible at small scales ($L\sim1000-2000$). In particular, we show that for a CMB-S4 like experiment, an optimal linear combination of point-source hardened estimators can reduce the (point-source induced) bias to the CMB lensing power spectrum by up to two orders of magnitude, at a $\sim4\%$ noise cost relative to the global minimum variance estimator., published in PRD; 18 pages, 7 figures

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.

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 $\ell_{\text{max,}T}$., published in PRD

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., Comment: published in PRD

An observational program focused on the high redshift ($2, Comment: published in JCAP