Impact of Perfect Fluid Dark Matter on the Appearance of Rotating Black Hole
Huang Yu-Xiang, Guo Sen, Liang En-Wei, Lin Kai
TL;DR
This work addresses how dark matter in the form of a perfect fluid around a rotating black hole imprints on horizon-scale images. It develops a Kerr–PFDM spacetime with mass function $m(r)=M- rac{k}{2}\ln\left(\frac{r}{|k|}\right)$ and uses Hamilton–Jacobi separability to compute photon geodesics, deriving shadow contours and ISCO properties via $\xi=L/E$ and $\eta=\mathcal{C}/E^{2}$ and the conditions $V=\partial_r V=\partial_r^2 V=0$. By performing high-resolution ray-tracing of optically thin disks at 87 and 230 GHz and comparing with EHT angular-diameter measurements for M87$^{*}$ and Sgr A$^{*}$, the study finds that PFDM can shrink shadows and modify brightness distributions, with quantitative bounds on the PFDM intensity $k$ (e.g., $0.278\le k\le 0.5$ for M87$^{*}$ and $0.419\le k\le 0.5$ for Sgr A$^{*}$ at representative spins). The results support PFDM as a physically plausible extension to Kerr that can be tested with current and upcoming horizon-scale VLBI data, motivating PFDM-inclusive modeling in future GRMHD radiative-transfer analyses. Key equations include $m(r)=M- rac{k}{2}\ln\left(\frac{r}{|k|}\right)$, the shadow relation on the celestial plane, and the emission integral $I_{ u_o}=\,\sum_s f_s\, g^{3}(r_s)\, J_{\rm model}(r_s)$.
Abstract
Understanding how dark matter affects the immediate environment of black holes (BHs) is crucial for interpreting horizon-scale observations. We study rotating BHs surrounded by perfect fluid dark matter (PFDM), exploring their observable features through both analytical and numerical approaches. Using the existence criterion of the innermost stable circular orbit (ISCO), we first derive joint constraints on the PFDM intensity parameter~k and the spin parameter~a. Within the resulting physically allowed parameter regime, we perform high-resolution, general-relativistic ray-tracing simulations of thin accretion disks at 87~GHz and 230~GHz, capturing the detailed brightness morphology and photon ring structure shaped by PFDM. By incorporating angular diameter measurements of M87^{*} and Sgr~A^{*} from the Event Horizon Telescope (EHT), we further narrow down the viable parameter space and directly compare synthetic images with EHT observations of M87^{*}. We find that the inclusion of PFDM improves the agreement with the observed compact shadow and asymmetric brightness distribution, suggesting that dark matter may leave observable imprints on horizon-scale images. Our results position PFDM as a physically motivated extension to the Kerr geometry and highlight a promising astrophysical pathway for probing dark matter near BHs with current and future VLBI campaigns.