Stacked lensing estimators and their covariance matrices: excess surface mass density versus 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 ΔΣ and γ₊, which are related as ΔΣ = Σ_crγ₊, where Σ_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 ΔΣ-estimator focusing on `weight' function to improve the signal-to-noise ratio (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 Σ_cr^{-2}(z_ l,z_ s), the ΔΣ-estimator maximizes the S/N in the shot-noise-limited regime. We also show that the ΔΣ-estimator with the weight Σ_cr^{-2} gives a greater (S/N)² than that of the γ₊-estimator by about 5-25 per cent for lensing objects at redshifts comparable with or higher than the median of source galaxy redshifts for hypothetical Subaru Hyper Suprime-Cam and Dark Energy Survey. However, for low-redshift lenses such as z_ l ≲ 0.3, the γ₊-estimator has higher (S/N)² than ΔΣ. We also discuss that the (S/N)² for ΔΣ 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 Σ_cr^{-α } with α > 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.