On the effect of the cosmological expansion on the gravitational lensing by a point mass

TL;DR

This paper investigates how the universe's expansion influences gravitational lensing by a point mass, finding that cosmological effects are minimal but quantifiable, with corrections around 10^-11 for constant Hubble parameter scenarios.

Contribution

The study derives approximate null geodesic solutions in a McVittie spacetime, showing cosmological corrections to lensing are negligible at leading order but calculable at sub-dominant orders.

Findings

Leading order deflection matches Schwarzschild prediction.

Cosmological corrections are negligible at leading order.

Explicit correction estimate of 10^-11 for constant Hubble parameter.

Abstract

We analyze the effect of the cosmological expansion on the deflection of light caused by a point mass, adopting the McVittie metric as the geometrical description of a pointlike lens embedded in an expanding universe. In the case of a generic, non-constant Hubble parameter $H$ we derive and approximately solve the null geodesic equations, finding an expression for the bending angle $\delta$, which we expand in powers of the mass-to-closest approach distance ratio and of the impact parameter-to-lens distance ratio. It turns out that the leading order of the aforementioned expansion is the same as the one calculated for the Schawarzschild metric and that cosmological corrections contribute to $\delta$ only at sub-dominant orders. We explicitly calculate these cosmological corrections for the case of $H$ constant and find that they provide a correction of order $10^{-11}$ on the lens mass estimate.