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.
