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, $R_{\mathrm{TA},\max }$, 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 $R_{\mathrm{TA},\max }$ 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 $\alpha \ne -1$, 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, $(1+\alpha )\lesssim \mathcal{O}(10^{-14})$, 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.

arXiv commentv2, 18pp, 1 fig; title changed, added references, discussions and clarifications; improved presentation, main results unchanged; accepted in PRD

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