No practical lensing by Lambda: Deflection of light in the Schwarzschild–de Sitter spacetime

Abstract

Debate persists as to whether the cosmological constant Λ can directly modify the power of a gravitational lens. With the aim of reestablishing a consensus on this issue, I conduct a comprehensive analysis of gravitational lensing in the Schwarzschild-de Sitter spacetime, wherein the effects of Λ should be most apparent. The effective lensing law is found to be in precise agreement with the Λ = 0 result:, where the effective bending angle α eff and impact parameter b eff are defined by the angles and angular diameter distances measured by a comoving cosmological observer. [These observers follow the timelike geodesic congruence which (i) respects the continuous symmetries of the spacetime and (ii) approaches local isotropy most rapidly at large distance from the lens.] The effective lensing law can be derived using lensed or unlensed angular diameter distances, although the inherent ambiguity of unlensed distances generates an additional uncertainty O(m 5 /Λb 7 eff ). I conclude that the cosmological constant does not interfere with the standard gravitational lensing formalism.

On the Effect of the Cosmological Expansion on the Gravitational Lensing by a Point Mass

“…The reason for this choice is to use a metric which properly takes into account the Hubble flow to which source, lens and observer are subject. We considered the general case in which the Hubble factor is a generic function of time and find that no contribution coming from cosmology enters the bending angle at the leading order (see Equation (47)), thus strengthening the results obtained by [13][14][15][16].…”

“…On the other hand, there are a few works that do not agree with the above-mentioned results (see e.g., [13][14][15][16]). The main criticism is that the Hubble flow is not properly taken into account, i.e., the relative motion among source, lens and observer is neglected.…”

“…[13][14][15][16], the standard approach to gravitational lensing does not need modifications. For the sake of clarity, the standard approach to gravitational lensing consists of using the result on the deflection angle obtained from the Schwarzschild metric (which models the lens) together with the cosmological angular diameter distances calculated from the Friedmann-Lemaître-Robertson-Walker (FLRW) metric.…”

“…Indeed, the author of Ref. [16] constructs "by hand" cosmological observers in the Kottler metric and finds that Λ has no observable effect on the deflection of light.…”

Light bending, static dark energy, and related uniqueness of Schwarzschild–de Sitter spacetime

“…[35,36,37,38], as well. The chief objection raised is that, if one uses the cosmological McVittie description instead of the static geometry, there will be no light bending due to Λ at all, in the leading order.…”