High-$z$ Universe probed via Lensing by QSOs (HULQ) I. Number Estimates of QSO-QSO and QSO-Galaxy Lenses.

It is unclear how galaxies and their central supermassive black holes (SMBHs) co-evolve across cosmic time, especially for the non-local universe ($z \gtrsim 0.5$). The High-$z$ Universe probed via Lensing by QSOs (HULQ) project proposes to utilize quasi-stellar object (QSO) host galaxies acting as gravitational lenses (QSO lenses) to investigate this topic. This paper focuses on the feasibility of this project, i.e., whether sufficiently large numbers of QSO lenses are expected to be found in various concurrent and future imaging surveys. We find that $\sim 440$ QSO lenses will reside in the Hyper Suprime-Cam Wide survey (HSC/Wide), which is expected to be the most prolific concurrent survey, with this number being boosted by one to two orders of magnitude (to $\sim 10000$) with upcoming surveys such as that conducted with the Large Synoptic Survey Telescope (LSST). We discuss several methods of how to study the redshift evolution of the $M_{\rm BH} - \sigma_*$ relation, which is a stand-out illustration of the co-evolution. In addition, we demonstrate how the intimacy of lensed images to the bright deflector QSO for most systems will affect the \textit{detectability} of QSO lenses. We estimate that only $\sim 82$ and $900$ will be detectable for HSC/Wide and LSST, respectively; the decrease is significant yet still yields an acceptable sample for the main objective. This decrease will be less of a problem for space-based imaging surveys, for their small point spread function FWHMs will allow detections of lensed images lying relatively close to the deflector QSO, and thus unveil the less massive yet more numerous QSO hosts.