2023MNRAS.524.1246P

We generalize the result of Rindler-Ishak for the lensing deflection angle in a Schwarzschild-deSitter (SdS) space-time, to the case of a general spherically symmetric fluid beyond the cosmological constant. We thus derive an analytic expression to first post-Newtonian order for the lensing deflection angle in a general static spherically symmetric metric of the form $\mathrm{ d}s^2 = f(r)\mathrm{ d}t^{2} -\frac{\mathrm{ d}r^{2}}{f(r)}-r^{2}(\mathrm{ d}\theta ^2 +\sin ^2 \theta \mathrm{ d}\phi ^2)$ with $f(r) = 1 - \frac{2m}{r}-\sum _{i} b_\mathrm{ i}\,\, r_0^{-q_i}\,\, \left(\frac{r_0}{r}\right)^{q_i}$, where r₀ is the lensing impact parameter, $b_i\ll r_0^{q_i}$, m is the mass of the lens, and q_i are real arbitrary constants related to the properties of the fluid that surrounds the lens or to modified gravity. This is a generalization of the well known Kiselev black hole metric. The approximate analytic expression of the deflection angle is verified by an exact numerical derivation and in special cases it reduces to results of previous studies. The density and pressure of the spherically symmetric fluid that induces this metric is derived in terms of the constants b_i. The Kiselev case of a Schwarzschild metric perturbed by a general spherically symmetric dark fluid (e.g. vacuum energy) is studied in some detail and consistency with the special case of Rindler-Ishak result is found for the case of a cosmological constant background. Observational data of the Einstein radii from distant clusters of galaxies lead to observational constraints on the constants b_i and through them on the density and pressure of dark fluids, field theories, or modified gravity theories that could induce this metric.