Shadows and Observational Images of a Schwarzschild-like Black Hole Surrounded by a Dehnen-type Dark Matter Halo

TL;DR

The work addresses how a Dehnen-type DM halo around a Schwarzschild-like BH alters shadows and photon rings, constraining DM parameters using EHT observations via the relation $r_{sh}=b_{ph}$. It derives the horizon radius $r_h$ and photon geodesics in the metric with $f(r)=1-\dfrac{2M}{r}-32\pi\rho_s r_s^2 \sqrt{\dfrac{r+r_s}{r}}$, and analyzes $V_{eff}(r)$, $r_{ph}$, and $b_{ph}$, showing that increasing $\rho_s$ or $r_s$ raises these characteristic scales. Through ray-tracing, the paper investigates optical appearances under three static disk models and two spherical accretion scenarios, finding that larger DM parameters push the bright features to higher impact parameters $b$ and generally reduce overall brightness. The results demonstrate observable signatures of DM halos in BH imaging, suggesting that future high-resolution observations could help distinguish DM profiles around BHs and test gravitational models in DM-rich environments.

Abstract

This paper investigates the optical appearance of a Schwarzschild-like black hole (BH) surrounded by a Dehnen-(1, 4, 5/2) type dark matter (DM) halo, with a focus on how the DM halo's density $ρ_{s}$ and radius $r_{s}$ influence the BH's shadow and photon ring. First, the radius $r_h$ of the BH's event horizon and the equation of motion for photons were derived, and observational data from the Event Horizon Telescope (EHT) for M87* were used to constrain the parameters $ρ_{s}$ and $r_{s}$ of the DM halo. Afterward, by varying the values of $ρ_{s}$ and $r_{s}$, key parameters such as the effective potential $V_{eff}$ of photons, the critical impact parameter $b_{ph}$, the radius $r_{isco}$ of the innermost stable circular orbit, and the radius $r_{ph}$ of the photon sphere were calculated for each case. It was found that as $ρ_{s}$ and $r_{s}$ increase, the above mentioned parameters all show an increasing trend. Subsequently, we investigated the optical appearance of the BH illuminated by two types of accretion models: optically and geometrically thin disk models and spherical accretion models. The findings indicate that as $ρ_{s}$ and $r_{s}$ increase, the peak of the received intensity shifts toward a higher impact parameter $b$, resulting in a distinct optical appearance.

Figures (14)

• Figure 1: The metric function $f(r)$ versus $r$ at fixed $\rho_{s}=0.01$ (left panel) and versus $\rho_{s}$ at fixed $r_{s}=0.2$ (right panel).
• Figure 2: The variation of the event horizon radius with $\rho_{s}$ and $r_{s}$: left panel shows fixed $\rho_{s} = 0.01$; right panel shows fixed $r_{s} = 0.06$.
• Figure 3: Effective potential curves for different values of $\rho_{s}$ and $r_{s}$. Left: fixed $\rho_{s} = 0.01$; right: fixed $r_{s} = 0.06$. Red, green, blue, and black lines correspond to Schwarzschild, and $r_{s}$ or $\rho_{s} = 0.1$, $0.15$, $0.2$, respectively.
• Figure 4: The constraints imposed by the M87* shadow radius on the DM halo density $\rho_{s}$ and radius $r_{s}$. The dashed line indicates the upper limit of the shadow radius (6.25), with the parameter space below this line representing the allowed region for $\rho_{s}$ and $r_{s}$.
• Figure 5: The intersection number curve for the Schwarzschild BH (left panel) and the corresponding photon trajectories (right panel). The black, orange, and red solid lines represent the direct emission, lensing ring, and photon ring, respectively. In the right panel, the blue dashed line denotes the equatorial plane of the BH, while the black disk and the black dashed circle indicate the BH and the photon sphere, respectively.