An Inpainting Approach to Tackle the Kinematic and Thermal SZ Induced
  Biases in CMB-Cluster Lensing Estimators
  A galaxy cluster's own Sunyaev-Zel{'}dovich (SZ) signal is known to be a
major contaminant when reconstructing the cluster's underlying lensing
potential using cosmic microwave background (CMB) temperature maps. In this
work, we develop a modified quadratic estimator (QE) that is designed to
mitigate the lensing biases due to the kinematic and thermal SZ effects. The
idea behind the approach is to use inpainting to eliminate the cluster's own
emission from the large-scale CMB gradient map. In this inpainted gradient map,
we fill the pixel values at the cluster location using a constrained Gaussian
realization based on the information from surrounding regions. We show that the
noise induced due to inpainting process is small compared to other noise
sources for upcoming surveys and has minimal impact on the final lensing
signal-to-noise. Without any foreground cleaning, we find a stacked mass
uncertainty of 6.5% for the CMB-S4 experiment on a cluster sample containing
5000 clusters with $M_{200c} = 2 \times 10^{14}\ M_{\odot}$ at z = 0.7. In
addition to the SZ-induced lensing biases, we also quantify the low mass bias
arising due to the contamination of the CMB gradient by the cluster
convergence. For the fiducial cluster sample considered in this work, we find
that bias is negligible compared to the statistical uncertainties for both the
standard and the modified QE even when modes up to $\sim 2700$ are used for the
gradient estimation. With more gradient modes, we demonstrate that the
sensitivity can be increased by 14% compared to the fiducial result above with
gradient modes up to $2000$
