Impact of Perfect Fluid Dark Matter on the Appearance of Rotating Black Hole

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 parameterk and the spin parametera. Within the resulting physically allowed parameter regime, we perform high-resolution, general-relativistic ray-tracing simulations of thin accretion disks at 87GHz and 230GHz, 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.

💡 Research Summary

The paper investigates how a surrounding perfect‑fluid dark matter (PFDM) halo modifies the observable appearance of a rotating black hole (BH). Starting from Kiselev’s PFDM model, the authors construct a Kerr‑PFDM spacetime by applying the Newman‑Janis algorithm, which introduces a new mass function m(r)=M−(k/2)ln|r/k|. The PFDM intensity parameter k (positive for attractive, negative for repulsive effects) directly controls the dark‑matter density around the BH, while the spin parameter a governs rotation. By solving the horizon condition Δ=0, they map the (a,k) parameter space, identifying a green region where a true event horizon exists and a white region corresponding to naked singularities.

A key physical constraint comes from the innermost stable circular orbit (ISCO). Using the effective potential V(r,E,L) for equatorial timelike geodesics, the authors impose V=0, ∂rV=0, and ∂r²V=0 to locate ISCO radii for given (a,k). They find a non‑monotonic dependence: as k grows, the ISCO radius first shrinks then expands, and for sufficiently large k two distinct ISCO solutions appear, which would break the thin‑disk assumption. Consequently, the study restricts k to values that yield a single, well‑defined ISCO (roughly k≲0.5 M for the spins considered).

Photon trajectories are derived from the Hamilton‑Jacobi equation with Carter’s constant, leading to critical impact parameters ξ and η. These are transformed into celestial coordinates (α,β) for an observer at infinity, producing shadow contours. The shadow area contracts with increasing k up to a critical value, then expands again, reflecting the competing influences of spin‑induced frame dragging and PFDM‑induced curvature modifications.

To translate these geometric effects into observable signatures, the authors perform high‑resolution general‑relativistic ray‑tracing simulations of an optically thin, geometrically thin accretion disk at 87 GHz and 230 GHz. They explore both prograde and retrograde disk orientations and a range of inclination angles. The synthetic images reveal that PFDM subtly shifts the shadow centre, enhances brightness asymmetry, and modifies the photon ring thickness. Notably, when k≈0.1–0.3 M and a≈0.8–0.94, the simulated 230 GHz images reproduce the compact shadow size and east‑west brightness contrast observed in the Event Horizon Telescope (EHT) image of M87*.

Finally, the measured angular diameters of M87* (≈42 μas) and Sgr A* (≈50 μas) are used to further constrain (a,k). The combined ISCO and EHT constraints narrow the viable parameter space to a narrow band where the Kerr‑PFDM model outperforms the pure Kerr solution in fitting the data. The authors conclude that PFDM provides a physically motivated extension to the Kerr geometry that can leave detectable imprints on horizon‑scale images. This work opens a new avenue for probing dark‑matter distributions in the immediate vicinity of supermassive black holes using current and future very‑long‑baseline interferometry campaigns, and suggests that more sophisticated GRMHD simulations incorporating PFDM could refine these constraints further.