Gravitational lensing frequencies: Galaxy cross-sections and selection effects

In order to allow more realistic predictions of multiple image galaxy-quasar lensing frequencies, we consider in detail four issues: the best currently available data on the galaxy velocity dispersion distribution, the effects of finite core radii potential ellipticity on lensing cross-sections, the predicted distribution of lens image separations compared to observational angular resolutions, and the preferential inclusion ('amplification bias') of lens systems in flux limited samples. We find (1) that the singular isothermal sphere (SIS) lensing parameter F = 0.047 ± 0.019 with almost 90 per cent contributed by E and SO galaxies, (2) that observed E and SO core radii are remarkably small giving a factor of ≲ 2 reduction in total lensing cross-sections, (3) that 50 per cent of galaxy-quasar lenses have image separations bigger than ∼ 1.3 arcsec (25 per cent, > 2 arcsec) and should be relatively easily detected, and (4) that amplification bias factors are large (few to tens) and must be carefully taken into account. Including all of the above considerations, we conclude that flat Universe models excessively dominated by the cosmological constant are not favoured by the small observed galaxy-quasar lensing rate. Models with a dimensionless cosmological constant λ₀ =0.9 may be marginally consistent with the available data; all λ₀ =0 models give a good fit.