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 ( \Lambda CDM ) 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 .