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 $\mathrm{\Lambda }$. Working in Boyer-Lindquist coordinates, we consider the effects of the perfect dark matter fluid 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 $\mathrm{\Lambda }$.

Figure 1

Shadows cast by a rotating black hole in PFDM background for different values of $\alpha$; all quantities are in units of $M$. The observer is positioned at $r_0=50$ and ${\theta }_0=\pi /2$.

Figure 2

Variation in shadow of a rotating black hole in the PFDM background with respect to the cosmological constant, when the observer is at position $r_0=50$ and ${\theta }_0=\pi /2$. All quantities are in units of $M$.

Figure 3

The figure shows the quadrilateral ${{}_R{}^{\infty }\square }_S^{\infty }$ domain of integration.

Figure 4

On the left side, we plot the deflection angle $\widehat{\alpha }$ as a function of the impact factor $b$ and the PFDM parameter $\alpha \in (0,2M)$. On the right side, we plot $\widehat{\alpha }$ as a function of the impact factor $b$ and the PFDM parameter $\alpha \in (-7.18M,0)$. We have chosen $M=1$ and $a=0.8$ in both plots.