Effect of the cosmological constant on the bending of light and the cosmological lens equation

Abstract

We revisit the effect of cosmological constant $\Lambda$ on the light deflection and its role in the cosmological lens equation. First, we re-examine the motion of photon in the Schwarzschild spacetime, and explicitly describe the trajectory of photon and deflection angle $\alpha$ up to the second-order in $G$. Then the discussion is extended to the contribution of the cosmological constant $\Lambda$ in the Schwarzschild-de Sitter or Kottler spacetime. Contrary to the previous arguments, we emphasize the following points: (a) the cosmological constant $\Lambda$ does appear in the orbital equation of light, (b) nevertheless the bending angle of light $\alpha$ does not change its form even if $\Lambda \neq 0$ since the contribution of $\Lambda$ is thoroughly absorbed into the definition of the impact parameter, and (c) the effect of $\Lambda$ is completely involved in the angular diameter distance $D_A$.