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 (PFDM), along with its extension to nonzero cosmological constant $\mathrm{\Lambda }$. Working in Boyer-Lindquist coordinates, we consider the effects of the PFDM parameter $\alpha$ on the shadow cast by a black hole with respect to an observer at position $(r_o,{\theta }_o)$. 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 $\alpha$ and $\mathrm{\Lambda }$.