Shadow and deflection angle of rotating black holes in perfect fluid dark matter with a cosmological constant

The presence of dark matter around a black hole remarkably affects its spacetime. We consider the effects of dark matter on the shadow of a new solution to the Einstein equations that describes a rotating black hole in the background of perfect dark matter fluid, along with its extension to nonzero cosmological constant. Working in Boyer-Lindquist coordinates, we consider the effects of the perfect dark matter fluid parameter on the shadow cast by a black hole with respect to an observer at position (r₎, ₎). By applying the Gauss-Bonnet theorem to the optical geometry, we find that notable distortions from a Kerr black hole can occur. We describe their dependence on and.