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

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, $R_{\rm TA, max}$, of a cosmic structure. We take the first order metric derived recently assuming a static and ideal dark energy fluid with equation of state $P(r)=\alpha\rho(r)$ as a source in Ref. [1], which reproduced the expression for $R_{\rm TA, max}$ 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 non-relativistic regime. We extend this metric up to the third order and compute the bending of light using the Rindler-Ishak method. For $ \alpha\neq -1$, a dark energy dependent term appears in the bending equation, unlike the case of the cosmological constant, $\alpha=-1$. Due to this new term in particular, existing data for the light bending at galactic scales yields, $(1+\alpha)\lesssim {\cal O}(10^{-14})$, 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.