tokens: like a galaxy cluster creates a small-scale dipole pattern in the direction of the large-scale CMB temperature gradient. The QE (Hu et al. 2007 ) is based on extracting the correlation between the dipole and background gradient to estimate the lensing convergence . Specifically, the lensing convergence can be estimated from a weighted product of filtered versions of a gradient map and small-scale lensing map :Here, is the pointing unit vector, is angular multipole, and A l is a normalization factor. While designed to pull out the lensing-induced correlations between large-scale and small-scale CMB anisotropy, the QE is also sensitive to correlations due to foreground emission. Of particular concern is the cluster’s own SZ emission, which is typically an order of magnitude larger than lensing signal. As pointed out by Hu et al. ( 2007 ), the magnitude of the SZ bias can be lowered by decreasing the characteristic scale of the low-pass filter on the gradient map, which is set to 2000 in the current work. However, a stronger low-pass filter obviously reduces the number of modes used to measure the gradient, and thus decreases the signal-to-noise. The modified QE instead eliminates this bias by using an SZ-cleaned map for the large-scale gradient map. The large-scale gradient map is chosen because the CMB has much more power on large scales, so the noise penalty from SZ removal has minimal impact. Note that while multiple foregrounds can be removed in principle, in practice the focus has been on removing the SZ signal. This is simply because the SZ signal introduces the largest bias. With the SZ signal present in only one of the two maps, there is no SZ-induced correlation between the two maps and no net bias on the reconstruction of the lensing convergence. However the SZ emission in the small-scale lensing map does add noise to the lensing reconstruction. Since under self-similarity the SZ flux (Kaiser 1986 ), y , is expected to scale with cluster mass M as while the lensing signal is linear in mass (in 1-halo regime), the additional SZ variance will generally be more important for high-mass clusters. The SZ variance will also be more important in low-noise surveys, i.e., when it is larger than the instrumental noise in the convergence map. 3.2. Template Fitting to Reduce the SZ Variance An obvious way to eliminate this SZ variance is by using SZ-free maps for both the small-scale and gradient maps. Of course this would undo the advantages of the modified QE for the instrumental noise in the small-scale map. One can also reduce this extra variance by projecting out a model template for the SZ signal, and thereby reducing the total amount of SZ power in the small-scale map. Given the impossibility in creating a “perfect” SZ template, the template fitting will not completely eliminate the SZ signal in the small-scale map. However, template fitting can significantly reduce the SZ power in the small-scale map, and thus reduce the SZ noise penalty
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