The Contribution of The Cosmological Constant/Dark Energy to The Bending of Light and its Applications.*

We report new results on the effect of the cosmological constant Lambda on the bending of light by a spherically symmetric mass. We show that when the Schwarzschild-de Sitter geometry is taken into account, Lambda does indeed contribute to the bending of light. We use this result to discuss an amended lens equation and new time delay expressions. Although the newly derived Lambda-terms are perhaps small for current observations, they do not cancel out. We apply the result to observations of Einstein radii around several systems where the lens is a cluster of galaxies. We find that the contribution of the Lambda-term to the bending angle can be as large as the second-order term in the Einstein deflection angle for many lens systems and even larger in some cases. We show that our results derive from various methods including the exact matching of solutions to Einstein's equations, approximations used in gravitational lensing, and Fermat's Principle.