Black Hole Shadow of Sgr A$^{*}$ in Dark Matter Halo

We study, for the first time, the shadow of the supermassive black hole Sgr A$^{*}$ at the center of the Milky Way in dark matter halos. For the Cold Dark Matter and Scalar Field Dark Matter models considered in this work, the apparent shape of the shadow depends upon the black hole spin $a$ and the dark matter parameter $k$. We find that both dark matter models influence the shadow in a similar way. The shadow is a perfect circle in the non-rotating case ($a=0$) and a deformed one in the rotating case ($a\neq0$). The size of the shadow increases with increasing $k$ in both non-rotating and rotating cases, while the shadow gets more and more distorted with increasing $a$ in the rotating case. We further investigate the black hole emission rate in both dark matter halos. We find that the emission rate decreases with increasing $k$ and the peak of the emission shifts to lower frequency. Finally, by calculating the angular radius of the shadow, we estimate that the dark matter halo could influence the shadow of Sgr A$^{*}$ at a level of order of magnitude of $10^{-3}$ $\mu$as and $10^{-5}$ $\mu$as, for CDM and SFDM, respectively. Future astronomical instruments with high angular resolution would be able to observe this effect and shed light on the nature of Sgr A$^{*}$. More interestingly, it may be possible to distinguish between CDM and SFDM models given the resolutions required differing by two orders of magnitude from each other.