Weak deflection angle of a dirty black hole

In this paper, we present the weak deflection angle in a Schwarzschild black hole of mass m surrounded by the dark matter of mass M and thickness Delta r(s). The Gauss-Bonnet theorem, for-mulated for asymptotic spacetimes, is found to be ill-behaved in the third-order of 1/Delta r(s) for very large Delta r(s). Using the finite-distance for the radial locations of the source and the receiver, we derived the expression for the weak deflection angle up to the third-order of 1/Delta r(s) using Ishihara (et al.) method. The result showed that the required dark matter thickness is similar to 2 root 3mM for the deviations in the weak deflection angle to occur. Such thickness requirement is better by a factor of 2 as compared to the deviations in the shadow radius (similar to root 3mM). It implies that the use of the weak deflection angle in detecting dark matter effects in one's galaxy is better than using any deviations in the shadow radius.