High-z Universe probed via Lensing by QSOs (HULQ): How many QSO lenses are there?

Aims. The evolution of scaling relations between SMBHs and their host galaxies becomes uncertain at high redshifts. The HULQ project proposes to use gravitational lensing to measure the masses of QSO host galaxies, an otherwise difficult goal. SMBH masses of QSOs are relatively easy to determine using either reverberation mapping or the single-epoch method. These measurements, if made for a substantial number of QSOs at various redshifts, will allow us to study the co-evolution of SMBHs and their host galaxies. 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 deflector QSO host galaxies (hereafter QSO lenses). Method and results. Using SMBH masses measured from SDSS DR14 spectra, and the M_BH - Sigma relation, the Einstein radii are calculated as a function of source redshift, assuming singular isothermal sphere mass distributions. Using QSOs and galaxies as sources, the probability of a QSO host galaxy being a QSO lens is calculated, depending on the limiting magnitude. The expected numbers of QSO lenses are estimated for ongoing and future wide-imaging surveys, and additional factors that may affect these numbers are discussed.