Stacked lensing estimators and their covariance matrices: Excess surface mass density vs. Lensing shear

Stacked lensing is a powerful means of measuring the average mass distribution around large-scale structure tracers. There are two stacked lensing estimators used in the literature, denoted as $\Delta\Sigma$ and $\gamma_+$, which are related as $\Delta\Sigma=\Sigma_{\rm cr}\gamma_+$, where $\Sigma_{\rm cr}(z_l,z_s)$ is the critical surface mass density for each lens-source pair ($z_l$ and $z_s$ are lens and source redshifts, respectively). In this paper we derive a formula for the covariance matrix of $\Delta\Sigma$-estimator focusing on `weight' function to improve the signal-to-noise ($S/N$). We assume that the lensing fields and the distribution of lensing objects obey the Gaussian statistics. With this formula, we show that, if background galaxy shapes are weighted by an amount of $\Sigma_{\rm cr}^{-2}(z_l,z_s)$, the $\Delta\Sigma$-estimator maximizes the $S/N$ in the shot noise limited regime. We also show that the $\Delta\Sigma$-estimator with the weight $\Sigma_{\rm cr}^{-2}$ gives a greater $(S/N)^2$ than that of the $\gamma_+$-estimator by about 5--25\% for lensing objects at redshifts comparable with or higher than the median of source galaxy redshifts for hypothetical Subaru HSC and DES surveys. However, for low-redshift lenses such as $z_l<0.3$, the $\gamma_+$-estimator has higher $(S/N)^2$ than $\Delta\Sigma$. We also discuss that the $(S/N)^2$ for $\Delta\Sigma$ at large separations in the sample variance limited regime can be boosted, by up to a factor of 1.5, if one adopts a weight of $\Sigma_{\rm cr}^{-\alpha}$ with $\alpha>2$. Our formula allows one to explore how the combination of the different estimators can approach an optimal estimator in all regimes of redshifts and separation scales.