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 Δrs. The Gauss-Bonnet theorem, formulated for asymptotic spacetimes, is found to be ill-behaved in the third-order of 1/Δrs for very large Δrs. 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/Δrs using Ishihara (et al.) method. The result showed that the required dark matter thickness is ∼23mM 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 (∼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.

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