Strong Gravitational Lensing and Galactic Bulges

The lensing probabilities of quasars with image separations greater than Δθ and flux density ratios less than q r are calculated by foreground dark matter halos in a flat, cosmological constant-dominated (Λ cold dark matter) universe. The mass density of the lenses is taken to be the Navarro-Frenk-White profile on all mass scales plus a central point mass for low-mass halos with M < M c = 5 × 10 13 h -1 M ☉ . We introduce a quantity M eff , which is a point mass ranging from 1 to 1000 times the mass M • 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 r of multiple images in our calculations. It is shown that in order to reproduce the lensing survey results of the Jodrell-Bank VLA Astrometric Survey and the Cosmic Lens All-Sky Survey for q 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 σ 8 , we investigate the effect of varying this parameter within its entire observational range (from 0.7 to 1.1), and we find that low-σ 8 values (≤0.7) are ruled out and that the best-fit value is σ 8 ≃ 1.0.