Strong gravitational lensing: why no central black holes?

We investigate how central black holes (BHs) inhabited in galactic dark halos could affect strong gravitational lensing. The distribution of integral lensing probability with image separations are calculated for quasars of redshift 1.5 by foreground dark matter halos. The mass density of dark halos is taken to be the Navarro-Frenk-White (NFW) profile such that, when the mass of a halo is less than $10^{14} M_{\sun}$, its central black holes or a bulge is included as a point mass. The relationship between the masses $M_{\bullet}$ of supermassive black holes and the total gravitational mass $M_{\mathrm{DM}}$ of their host galaxy is adopted from the most recent literature. Only a flat $\Lambda$CDM model is considered here. It is shown that, while a single black hole for each galaxy contributes considerable but not sufficient lensing probabilities at small image separations compared with those without black holes, the bulges (which are about 100--1000 times larger in mass than a typical black hole) would definitely contribute enough probabilities at small image separations, although it gives too high probabilities at large separation angles compared with lensing observations.