Long-lived quasinormal modes and grey-body factors of supermassive black holes with a dark matter halo

TL;DR

The paper investigates how a galactic dark-matter halo affects quasinormal modes and grey-body factors of a massive scalar field around a Schwarzschild black hole. It uses a combination of sixth- and seventh-order WKB methods with Padé approximants and time-domain integration with Prony analysis to compute the QNM spectrum and transmission coefficients. The results show that increasing the field mass $\mu$ raises the oscillation frequency $\mathrm{Re}(\omega)$ while decreasing the damping rate $\mathrm{Im}(\omega)$, yielding longer-lived modes, and that halo parameters $V_c$ and $a$ have negligible effects for astrophysically realistic halos. Grey-body factors decline with $\mu$ and angular momentum $\ell$, with halo corrections well below percent level, reinforcing the Schwarzschild baseline and the robustness of ringdown signatures in galactic environments.

Abstract

We study quasinormal modes and grey-body factors of a massive scalar field in the background of a Schwarzschild black hole surrounded by a spherically symmetric galactic dark matter halo. The background metric, recently obtained as an analytic generalization of the Schwarzschild geometry, depends on the halo velocity parameter $V_{c}$ and the core radius $a$. Using the sixth- and seventh-order WKB methods with Pade approximants, supported by time-domain integration and Prony analysis, we compute the fundamental quasinormal frequencies and transmission coefficients. The results show that the real part of the frequency slightly increases while the damping rate decreases with growing field mass $μ$, leading to longer-lived oscillations. The influence of the dark matter halo parameters is found to be negligible for astrophysically realistic values, confirming the robustness of Schwarzschild-like ringdown signatures. Grey-body factors decrease with increasing field mass and multipole number, while the effect of the halo parameters remains small.

Figures (7)

• Figure 1: Effective potentials for $\ell=0$ massive scalar field perturbations at $M=1$, $V_{c}=0.1$, $a=10$, $\mu=0$ (black), $\mu=0.2$ (blue) and $\mu=0.3$ (red).
• Figure 2: Effective potentials for $\ell=1$ massive scalar field perturbations at $M=1$, $V_{c}=0.1$, $a=10$, $\mu=0$ (black), $\mu=0.3$ (blue) and $\mu=0.5$ (red).
• Figure 3: Semi-logarithmic time-domain profile for $\ell=0$ perturbations. Here $\mu=0.1$, $V_{c}=0.1$, $a=10$. The WKB data is $0.113638-0.096752 i$, and the extraction the dominant frequency from the time-domain profile via the Prony method gives $\omega = 0.121193 - 0.0955343 i$. This does not mean immediately insufficient accuracy of the WKB data, because the period of quasinormal oscillations is quickly changed by the late-time tail at $t \approx 80$.
• Figure 4: Semi-logarithmic time-domain profile for $\ell=1$ perturbations. Here $\mu=0.1$, $V_{c}=0.1$, $a=10$. The WKB data is $0.296515 - 0.094775 i$, and the extraction the dominant frequency from the time-domain profile via the Prony method gives $\omega = 0.296517 - 0.0947754 i$.
• Figure 5: Grey-body factors for $\ell=1$ massive scalar field perturbations at $M=1$, $V_{c}=0.1$, $a=10$, $\mu=0$ (blue), $\mu=0.1$ (red), $\mu=0.2$ (green), and $\mu=0.3$ (black).