Rotating black holes in the Hernquist galactic halo and its accretion disk luminosity

Technical Summary: Rotating Black Holes in the Hernquist Galactic Halo and Its Accretion Disk Luminosity

Problem Statement
While the existence of dark matter is confirmed by various observational data (rotation curves, CMB, gravitational lensing), the specific influence of dark matter halos on the dynamics and electromagnetic signatures of supermassive black holes (BHs) remains an active area of investigation. Previous studies have derived static, spherically symmetric BH solutions immersed in dark matter halos with Hernquist-type density profiles (Cardoso et al., Ref. [16]). However, realistic astrophysical black holes are expected to be rapidly rotating. The challenge addressed in this work is to extend the static Cardoso solution to a stationary, axially symmetric rotating metric and to determine how the presence of a Hernquist-type dark matter halo affects the observable properties of thin accretion disks, specifically comparing these effects against the standard Kerr black hole model in a vacuum.

Methodology
The authors employ a multi-step theoretical framework:

1. Metric Construction: Starting from the static, spherically symmetric solution for a BH in a Hernquist halo (derived by Cardoso et al. using an anisotropic dark matter fluid model with zero radial pressure), the authors apply the Newman-Janis algorithm. This allows them to generate the rotating, axially symmetric counterpart (the "rotating Cardoso BH") in Boyer-Lindquist coordinates.
2. Geodesic Analysis: The study analyzes the motion of test particles in the equatorial plane of this new metric. The authors derive expressions for the specific energy ($\tilde{E}$), specific angular momentum ($\tilde{L}$), and orbital angular velocity ($\Omega$) using the Killing vectors of the spacetime.
3. Accretion Disk Modeling: The electromagnetic properties of the accretion disk are calculated using the steady-state Novikov-Thorne model. This model assumes a geometrically thin, optically thick disk in local thermal equilibrium. Key quantities derived include:
   • The radius of the Innermost Stable Circular Orbit (ISCO), determined by the condition $V_{\text{eff},rr} = 0$.
   • The radiant energy flux $F(r)$ and disk temperature $T(r)$.
   • The emission spectrum $L(\nu)$ and radiative efficiency $\epsilon$.
4. Numerical Comparison: The authors numerically compute these parameters for various values of the spin parameter ($a$) and the halo compactness parameter ($C = M/a_0$). These results are systematically compared against the standard Kerr BH solution (where dark matter is absent) and the previously studied static Cardoso BH.

Key Contributions

• Derivation of Rotating Metric: The paper successfully constructs the metric for a rotating black hole immersed in a Hernquist-type dark matter halo, generalizing the static solution found in Ref. [16]. This metric covers the full range of spin parameters, unlike previous slow-rotation approximations.
• Quantitative Analysis of Disk Properties: The study provides a detailed numerical analysis of how the dark matter halo parameters (mass $M$ and scale length $a_0$) and the black hole spin ($a$) influence the ISCO radius, energy flux, temperature, and luminosity spectra of the accretion disk.
• Comparative Assessment: The work offers a direct comparison between the rotating Cardoso BH, the static Cardoso BH, and the standard Kerr BH, isolating the specific contributions of rotation versus the dark matter environment.

Results

• ISCO Radius Trends: The ISCO radius decreases as the spin parameter $a$ increases. Similarly, for a fixed spin, the ISCO radius decreases as the halo compactness parameter $C$ increases. A smaller ISCO radius leads to a larger disk surface area, resulting in higher energy flux, temperature, and luminosity.
• Dark Matter Impact: While the presence of dark matter does increase the luminosity and temperature of the disk compared to a vacuum Kerr BH, the magnitude of this effect is relatively small compared to the effect of the spin parameter.
• Indistinguishability at High Spin: The most significant finding is that for astrophysical black holes with large spin parameters, the optical properties (flux, temperature, spectra) of a rotating Cardoso BH become nearly indistinguishable from those of a standard Kerr BH. The deviation caused by the dark matter halo is minimal in the high-spin regime.
• Event Horizon Behavior: The study notes that decreasing the compactness parameter $C$ reduces the event horizon radius, making it smaller than that of a Kerr BH for the same spin, though this geometric difference does not translate into a significant observational difference in the accretion disk's electromagnetic output for high-spin objects.

Significance and Claims
The paper concludes that while dark matter halos do theoretically alter the spacetime geometry and accretion disk properties, these alterations are not significant enough to serve as a primary diagnostic tool for distinguishing between a standard Kerr black hole and a rotating black hole immersed in a Hernquist-type dark matter halo, particularly for rapidly spinning black holes.

The authors modestly claim that their results suggest the optical properties of accretion disks are not a promising tool for distinguishing a Kerr BH from a rotating BH in a Hernquist halo (or potentially other halo types) when the spin parameter is large. This implies that current observational constraints based on accretion disk luminosity and spectra may be insufficient to detect the specific influence of Hernquist-type dark matter distributions on supermassive black holes, reinforcing the robustness of the Kerr metric as a description of astrophysical black holes even in the presence of such halos.