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

TL;DR

This paper investigates how the cosmological constant influences light bending and the lens equation, showing that while it affects the impact parameter and distances, it does not alter the fundamental bending angle.

Contribution

It clarifies the role of the cosmological constant in light deflection, demonstrating its absorption into impact parameters and distances rather than changing the bending angle itself.

Findings

The cosmological constant appears in the photon orbital equation.

The bending angle remains unchanged by $\Lambda$ when properly defined.

$\Lambda$ influences the angular diameter distance in lensing calculations.

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 $D_A$.