High-z Universe probed via Lensing by QSOs (HULQ): Expected Number of QSOs acting as Gravitational Lenses

The HULQ project proposes to use gravitational lensing to determine the masses of QSO host galaxies, an otherwise difficult goal. If these host galaxy masses, along with their SMBH masses from single-epoch measurements, are estimated for a substantial number of QSOs at various redshifts, the co-evolution of SMBHs and their host galaxies can be studied for a large portion of the history of the universe. To determine the feasibility of this study, we present how to estimate the number of sources lensed by QSO hosts, i.e. the number of lensing QSO host galaxies (hereafter QSO lenses). SMBH masses in the literature are transformed into the velocity dispersions of their host galaxies using the M_BH -sigma relation, and in turn the Einstein radii for each QSO -source redshift combination is calculated, assuming singular isothermal spherical mass distributions. Using QSOs and galaxies as potential sources, the probability of a QSO host galaxy being a QSO lens is calculated, as a function of limiting magnitude. The expected numbers of QSO lenses are estimated for ongoing and future wide-imaging surveys, and the Hyper Suprime-Cam Wide survey is illustrated as an example.