On lensing by a cosmological constant

Several recent papers have suggested that the cosmological constantdirectly influences the gravitational deflection of light. We place this problem in a cosmological context, deriving an expression for the linear potentials which control the cosmological bending of light, finding that it has no explicit dependence on the cosmological constant. To explore the physical origins of the apparent �-dependent potential that appears in the static Kottler metric, we highlight the two classical effects which lead to the aberration of light. The first relates to the observer's motion relative to the source, and encapsulates the familiar concept of angular-diameter distance. The second term, which has proved to be the source of debate, arises from cosmic acceleration, but is rarely considered since it vanishes for photons with radial motion. This apparent form of light-bending gives the appearance of curved geodesics even within a flat and homogeneous universe. However this cannot be construed as a real lensing effect, since its value depends on the observer's frame of reference. Our conclusion is thus that standard results for gravitational lensing in a universe containingdo not require modification, with any influence ofbeing restricted to negligible high-order terms. do not exclude a contribution ofto the lensing equa- tions at some level, but show that the linear potential is unaffected by �, with the apparentr 2 =3 contribu- tion appearing as a consequence of the choice of a static metric. We verify this with numerical solutions in x III. The remainder of this work aims to clarify the phys- ical interpretation of the apparent light bending. We revisit the analysis of Ishak (5) in x IV, before extending this to evaluate the photon's deflection angle from differ- ent perspectives within the Kottler metric. The source of the extra term is revealed in x V, and its relation to the angular-diameter distance is outlined in x VI. Final discussions are presented in x VIII.