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.
