Rotating Black Hole Shadow in Perfect Fluid Dark Matter

Abstract: We study analytically the shadow cast by the rotating black hole in the perfect fluid dark matter. The apparent shape of the shadow depends upon the black hole spin $a$ and the perfect fluid dark matter intensity parameter $k$ ($k>0$). In general, the shadow is a perfect circle in the non-rotating case ($a=0$) and a deformed one in the rotating case ($a\neq 0$). The deformation gets more and more significant with the increasing $a$, similar to the Schwarzschild and Kerr black holes. In addition, there exists a reflection point $k_0$. When $k<k_0$, the size of the shadow decreases with the increasing $k$ and the distortion increases with the increasing $k$. When $k>k_0$, the size of the shadow increases with the increasing $k$ and the distortion decreases with the increasing $k$. Furthermore, the energy emission rate of the black hole in the perfect fluid dark matter increases with the increasing $k$ and the peak of the emission shifts to higher frequencies. Finally, we propose that to observe the effect of the black hole spin $a$ and the perfect fluid dark matter intensity $k$ on the shadow of the black hole Sgr A$^{*}$ at the center of the Milky Way, highly improved techniques would be necessary for the development of future astronomical instruments.