Strong gravitational lensing and galactic bulges

Lensing probabilities of quasars with image separations greater than \Delta\theta and flux density ratios less than q_{\mathrm{r}} are calculated by foreground dark matter halos in a flat, cosmological constant dominated (\LambdaCDM) universe . The mass density of the lenses is taken to be the Navarro-Frenk-White (NFW) profile on all mass scales, plus a central point mass for low-mass halos with M<M_c=5\times 10^{13}h^{-1}M_{\odot}. We introduce a quantity M_{\mathrm{eff}}, which is a point mass ranging from 1 to 1000 times the mass M_{\bullet} of a supermassive black hole (SMBH) inhabiting the center of each galaxy, to describe the contributions of galactic central SMBHs and galactic bulges to lensing probabilities. The lensing cross section and thus the lensing probability are quite sensitive to the flux density ratio q_{\mathrm{r}} of multiple images in our calculations. It is shown that, to reproduce the lensing survey results of JVAS/CLASS for q_{\mathrm{r}}<10, about 20% of the bulge mass is needed as a point mass for each galaxy. Since there is still considerable uncertainty regarding the value of the spectrum normalization parameter \sigma_8, we investigate the effect of varying this parameter within its entire observational range (from 0.7 to 1.1), and find that low \sigma_8 values (\leq 0.7) are ruled out, and the best fit value is \sigma_8\simeq 1.0.