Light bending by the cosmological constant

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.

Gravitational Light Bending in Weyl Gravity and Schwarzschild–de Sitter Spacetime

“…Considering the small magnitude of the cosmological constant, one would expect that the effects of the first-order terms in Λ in Equations ( 13) and ( 18) would be much smaller than the first-order Schwarzschild contribution ∆ϕ sch ∼4m/R, so that in reality the presence of these terms in the bending angle formulae would not effect gravitational lensing [14,15]. However, by considering examples of galaxies and galaxy clusters and obtaining the magnitudes of these cosmological contributions (as was done in Table 1 of Ref.…”

“…The reactions to this proposal were mixed [6][7][8][9][10][11][12][13] with some against and some in favor of Rindler and Ishak's method, while a few others still questioned whether Λ contributes effectively to lensing. This debate has since calmed down, with the recent general consensus being that the cosmological constant does indeed play a role in gravitational lensing, but that the effect is way too tiny to be significant and can thus be ignored for practical purposes, given that its magnitude is much smaller when compared with other lensing affects such as aberration and uncertainties in cosmological distances [14,15]. In the time that the debate surrounding the SdS spacetime has been going on, another spacetime that has also attracted a lot of interest with regards to light bending is the static and spherically symmetric vacuum solution to conformal Weyl gravity, which was first derived by Riegert [16] but is more commonly know as the Mannheim-Kazanas (MK) solution [17,18].…”

Total light bending in non-asymptotically flat black hole spacetimes

“…At this point, it is necessary to carefully choose the branch of the solution of the tangent function, see [77] for a discussion of this. Using the trigonometric property sin 2 (x) = tan 2 (x)/(1 − tan 2 (x)) we rewrite (41) in a more compact and usual form…”

“…See, for example, the [39,40]. Although other works such as [41,42] indicate that the effect would be too small to be observed. Nevertheless, using a different approach, [43] concludes that Λ does not produce changes in the bending of light.…”

On the Physical Nature of Einstein’s Gravitational Lensing Effect

“…Several years ago, a review article proposed some questions about gravity such as the phenomenon of gravitational lensing as well as observed rotational velocities of stars orbiting the galactic center that are deviating from Newton's law [18]. Recently, another question asked whether the cosmological constant affects the cosmological gravitational bending of light [19]. To clarify these questions, this paper re-examines the physical nature of Einstein's gravitational lensing effect from three aspects.…”