Gravitational Lensing Corrections in Flat LambdaCDM Cosmology

We compute the deflection angle to order (m/r_0)^2 and m/r_0*Lambda r_0^2 for a light ray traveling in a flat LambdaCDM cosmology which 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 non-linear 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\approx 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 Lambda appears is proportional to m/r_0*\sqrt{Lambda r_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 non-linear 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.