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

Abstract

Since the Schwarzschild-de Sitter spacetime is static inside the cosmological event horizon, if the dark energy state parameter is sufficiently close to −1, apparently one could still expect an effectively static geometry, in the attraction dominated region inside the maximum turn around radius, RTA,max, of a cosmic structure. We take the first order metric derived recently assuming a static and ideal dark energy fluid with equation of state P(r) = αρ(r) as a source in Ref. [1], which reproduced the expression for RTA,max found earlier in the cosmological McVittie spacetime. Here we show that the equality originates from the equivalence of geodesic motion in these two backgrounds, in the nonrelativistic regime. We extend this metric up to the third order and compute the bending of light using the Rindler-Ishak method. For α 6= −1, a dark energy dependent term appears in the bending equation, unlike the case of the cosmological constant, α = −1. Due to this new term in particular, existing data for the light bending at galactic scales yields, (1 + α) . O(10−14), thereby practically ruling out any such static and inhomogeneous dark energy fluid we started with. Implication of this result pertaining the uniqueness of the Schwarzschild-de Sitter spacetime in such inhomogeneous dark energy background is discussed.