More on lensing by a cosmological constant

Abstract

The question of whether or not the cosmological constant affects the bending of light around a concentrated mass has been the subject of some recent papers. We present here a simple, specific and transparent example where bending clearly takes place, and where it is clearly neither a coordinate effect nor an aberration effect. We then show that in some recent works using perturbation theory the contribution was missed because of initial too stringent smallness assumptions. Namely, our method has been to insert a Kottler (Schwarzschild with ) vacuole into a Friedmann universe, and to calculate the total bending within the vacuole. We assume that no more bending occurs outside. It is important to observe that while the mass contribution to the bending takes place mainly quite near the lens, the bending continues throughout the vacuole. Thus, if one deliberately restricts one's search for bending to the immediate neighbourhood of the lens, one will not find it. Lastly, we show that the bending also follows from standard Weyl focusing, and so again, it cannot be a coordinate effect.

The cosmological constant and the gravitational light bending

“…The cosmological constant affects the orbit, introducing corrections to the "classical" bending in agreement with some recent investigations [4,8,10,11]. The leading correction terms differ among these investigation, a consequence of the choice of the observer's position, not always consistent with the choice of the orientation of the adopted polar frame.…”

“…(8), ordering the terms of same power in the perturbation parameter α and keeping only up to the first order terms, one obtains the following set of equations…”

“…Using the well known perturbation procedure to obtain an approximate solution of Eq. (8) in terms of the small parameter α, one can write…”

“…A further analysis by [7] reached a similar conclusion and the authors interpreted the correction term obtained by [4] as an "aberration" effect caused by the adopted gauge. Ishak et al [8] criticized these studies and reported new calculations in which the bending was now estimated by considering a model where a Schwarzschild-de Sitter vacuole is embedded in a Friedmann-Robertson-Walker background. According to them, the correction introduced by the cosmological constant is larger than the second-order term of the "classical" solution.…”

Effect of the Cosmological Constant on Light Deflection: Time Transfer Function Approach

“…See [14] for a review and also [15][16][17][18][19][20][21][22][23][24][25]. In addition, for the cosmological constant and cosmological lensing equation, see, e.g., [17,18,26,27]. As we assess the circumstances, the origin of confusion, e.g., the appearance/disappearance of Λ or the coupling/uncoupling with the mass of the central body M, is essentially attributable to the use of the standard geodesic equation of a photon to obtain light deflection due to Λ, because Λ does not appear.…”