Abstract
We revisit the question of whether the cosmological constant Λ 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 Λ dependence for a 1013 M ☉ lens is at the level of 1 part in 107, 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.
Highlights
IntroductionAs the observer and source are both in the FRW region, we can use standard angular diameter distance formulae to calculate observed angles and distances as we would in conventional lensing, and bending only happens in a finite region inside the hole
In discussing the results of this numerical integration, let us take a step back to look at the specific parts of ray-tracing that have a Λ-dependence
Λ affects the bending through its well-known effect on the angular diameter distances, but beyond this we find no effects that could genuinely be attributed directly to Λ
Summary
As the observer and source are both in the FRW region, we can use standard angular diameter distance formulae to calculate observed angles and distances as we would in conventional lensing, and bending only happens in a finite region inside the hole. The expected corrections to the standard gravitational lensing formalism in the Swiss-cheese solution arising from the finite size of the hole were calculated in [15]. Schucker [6, 16] has done some similar work in numerical simulations of the Swiss-cheese, and concluded that he agrees with Rindler and Ishak. We reproduce his results and expand upon it, but come to the opposite conclusion, and explain some of the apparent discrepancies. The rate of expansion of the Kottler hole in static coordinates changes with Λ, as a result of the matching conditions between the two metrics at the boundary
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