Gravitational lensing, finite galaxy cores, and the cosmological constant

We reexamine the sensitivity of the optical depth for gravitational lensing of distant quasars to the role of finite galaxy core radii (including uncertainties), and the nature of the cosmological model used to calculate the redshift-distance relation, in order to examine limits on a possible nonzero cosmological constant. Particular attention is paid to how these factors affect differential and integrated optical depths in a flat universe. We develop new formulas for optical depths in the "filled beam" approximation, make numerical estimates in the empty beam approximation, and derive analytic approximations which help explore the effect of the core radius-luminosity relation on the optical depth. Fitting data for ellipticals, we find that finite core radii reduce the optical depth for lensing by ≈2-3 compared to the isothermal sphere result (filled beam), with very little dependence on the elliptical core radius-luminosity relation. The contribution of low-luminosity galaxies to the optical depth, and thus the mean angular splitting, is sensitive to this relation, however. Also, the inclusion of nonzero core radii exacerbates the change in optical depth as a function of cosmological constant, thus allowing in principle a finer distinction between the different models to be made by comparing predictions and observations. This is further enhanced by the fact that both empty and full beam approximations yield similar results for a cosmological constant dominated universe, while the former reduces optical depths compared to the latter in a matter-dominated universe. Finally, the differential optical depth as a function of redshift has a maximum which depends sensitively on the cosmological constant, and not on core radius, giving a different robust theoretical handle on a cosmological constant.