Light bending by the cosmological constant

TL;DR

This study demonstrates that the cosmological constant has negligible impact on gravitational lensing of light, confirming that standard lensing equations remain valid without modifications for $ ext{Lambda}$ effects.

Contribution

The paper provides a numerical analysis showing that the cosmological constant does not significantly influence light bending beyond standard distance-based effects.

Findings

Negligible $ ext{Lambda}$ dependence in light bending, at the level of 1 part in 10^7.

Standard lensing equations are sufficient without modifications for $ ext{Lambda}$ effects.

Small residual effects can be explained by finite region considerations.

Abstract

We revisit the question of whether the cosmological constant $\Lambda$ affects the cosmological gravitational bending of light, by numerical integration of the geodesic equations for a Swiss cheese model consisting of a point mass and a compensated vacuole, in a Friedmann-Robertson-Walker background. We find that there is virtually no dependence of the light bending on the cosmological constant that is not already accounted for in the angular diameter distances of the standard lensing equations, plus small modifications that arise because the bending is restricted to a finite region covered by the hole. The residual $\Lambda$ dependence for a $10^{13}\,M_{\odot}$ lens is at the level of 1 part in $10^7$, and even this might be accounted for by small changes in the hole size evolution as the photon crosses. We therefore conclude that there is no need for modification of the standard cosmological lensing equations in the presence of a cosmological constant.