Magnification bias corrections to galaxy-lensing cross-correlations

Abstract

Galaxy-galaxy or galaxy-quasar lensing can provide important information on the mass distribution in the Universe. It consists of correlating the lensing signal (either shear or magnification) of a background galaxy/quasar sample with the number density of a foreground galaxy sample. However, the foreground galaxy density is inevitably altered by the magnification bias due to the mass between the foreground and the observer, leading to a correction to the observed galaxy-lensing signal. The aim of this paper is to quantify this correction. The single most important determining factor is the foreground redshift ${z}_{f}$: the correction is small if the foreground galaxies are at low redshifts but can become non-negligible for sufficiently high redshifts. For instance, we find that for the multipole $\ensuremath{\ell}=1000$, the correction is above $1%\ifmmode\times\else\texttimes\fi{}(5{s}_{f}\ensuremath{-}2)/{b}_{f}$ for ${z}_{f}\ensuremath{\gtrsim}0.37$, and above $5%\ifmmode\times\else\texttimes\fi{}(5{s}_{f}\ensuremath{-}2)/{b}_{f}$ for ${z}_{f}\ensuremath{\gtrsim}0.67$, where ${s}_{f}$ is the number count slope of the foreground sample and ${b}_{f}$ its galaxy bias. These considerations are particularly important for geometrical measures, such as the Jain and Taylor ratio or its generalization by Zhang et al. Assuming $(5{s}_{f}\ensuremath{-}2)/{b}_{f}=1$, we find that the foreground redshift should be limited to ${z}_{f}\ensuremath{\lesssim}0.45$ in order to avoid biasing the inferred dark energy equation of state $w$ by more than 5%, and that even for a low foreground redshift ($<0.45$), the background samples must be well separated from the foreground to avoid incurring a bias of similar magnitude. Lastly, we briefly comment on the possibility of obtaining these geometrical measures without using galaxy shapes, using instead magnification bias itself.