2010ApJ...718..913K

We compute the deflection angle to order (m/r₀)² and m/r₀x Λr ²₀ for a light ray traveling in a flat ΛCDM cosmology that encounters a completely condensed mass region. We use a Swiss cheese model for the inhomogeneities and find that the most significant correction to the Einstein angle occurs not because of the nonlinear terms but instead occurs because the condensed mass is embedded in a background cosmology. The Swiss cheese model predicts a decrease in the deflection angle of ∼2% for weakly lensed galaxies behind the rich cluster A1689 and that the reduction can be as large as ∼5% for similar rich clusters at z {&x2248;} 1. Weak-lensing deflection angles caused by galaxies can likewise be reduced by as much as ∼4%. We show that the lowest order correction in which Λ appears is proportional to {m/r_0×sqrt(A^AΛr_0^2)} and could cause as much as a ∼0.02% increase in the deflection angle for light that passes through a rich cluster. The lowest order nonlinear correction in the mass is proportional to {m/r_0×sqrt(m/r_0)} and can increase the deflection angle by ∼0.005% for weak lensing by galaxies.