Effects of Ellipticity and Shear on Gravitational Lens Statistics

We study the effects of ellipticity in lens galaxies and external tidal shear from neighboring objects on the statistics of strong gravitational lenses. For isothermal lens galaxies normalized so that the Einstein radius is independent of ellipticity and shear, ellipticity reduces the lensing cross section slightly, and shear leaves it unchanged. Ellipticity and shear can significantly enhance the magnification bias, but only if the luminosity function of background sources is steep. Realistic distributions of ellipticity and shear lower the total optical depth by a few percent for most source luminosity functions, and increase the optical depth only for steep luminosity functions. The boost in the optical depth is noticeable (&5%) only for surveys limited to the brightest quasars (L/L∗ & 10). Ellipticity and shear broaden the distribution of lens image separations but do not affect the mean. Ellipticity and shear naturally increase the abundance of quadruple lenses relative to double lenses, especially for steep source luminosity functions, but the effect is not enough (by itself) to explain the observed quadruple-to-double ratio. With such small changes to the optical depth and image separation distribution, ellipticity and shear have a small effect on cosmological constraints from lens statistics: neglecting the two leads to biases of just ∆ΩM = 0.00± 0.01 and ∆ΩΛ = −0.02± 0.01 (where the errorbars represent statistical uncertainties in our calculations). This is the most direct demonstration to date that ellipticity and shear have small effects on the lensing optical depth and image separation distribution. Subject headings: cosmology: theory — gravitational lensing