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 _ { 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 _ { 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 .