Quasinormal ringing of a regular black hole sourced by the Dehnen-type distribution of matter

TL;DR

The paper investigates how a regular black hole sourced by a Dehnen-type matter distribution affects the quasinormal-mode spectrum of massless test fields. It analyzes the Konoplya–Zhidenko metric with $f(r)=1-\frac{2 M r^{2}}{(r+a)^{3}}$ using a sixth- and ninth-order WKB method with Padé corrections, cross-validated by time-domain integration. It finds that increasing the halo-scale parameter $a$ elevates the real part of the frequencies $\mathrm{Re}(\omega)$ while leaving the damping rates $\mathrm{Im}(\omega)$ largely unchanged, implying longer-lived perturbations but only small deviations from Schwarzschild for astrophysically realistic halos. These results confirm linear stability, provide precise benchmarks for perturbations and grey-body factors in halo-embedded black holes, and suggest limited observational impact of ordinary galactic environments on QNM spectra.

Abstract

We study quasinormal modes of test scalar, electromagnetic, and Dirac fields in the background of a new analytic regular black-hole solution obtained as an exact solution of the Einstein equations sourced by a Dehnen-type matter distribution in [R. A. Konoplya, A. Zhidenko, arXiv:2511.03066]. The metric is asymptotically flat and characterized by a simple lapse function $f(r)=1-2 M r^{2}/(r+a)^{3}$, where $M$ is the ADM mass and $a$ represents the characteristic scale of the surrounding dark-matter halo that regularizes the central region. The effective potentials for all perturbing fields possess the standard single-barrier form, ensuring linear stability and the applicability of the WKB formalism. The quasinormal frequencies are computed using the sixth- and ninth-order WKB methods with Padé corrections and verified by time-domain integration, both approaches showing excellent agreement. The parameter $a$ leads to a moderate increase in the real oscillation frequency, while the damping rate remains almost unchanged, producing only small corrections to the Schwarzschild spectrum. Since such deviations become appreciable only for unrealistically dense halos, our results confirm that the quasinormal spectrum of astrophysical black holes is safely unaffected by ordinary galactic dark-matter environments.

Figures (8)

• Figure 1: Effective potential as a function of the tortoise coordinate $r_{*}$ for $\ell=0$ scalar field perturbations: $M=1$; $a=0$ (blue), $a=0.05$ (red) and $a=0.15$ (purple).
• Figure 2: Effective potential as a function of the tortoise coordinate $r_{*}$ for $\ell=1$ scalar field perturbations: $M=1$; $a=0$ (blue), $a=0.05$ (red) and $a=0.15$ (purple).
• Figure 3: Effective potential as a function of the tortoise coordinate $r_{*}$ for $\ell=1$ electromagnetic field perturbations: $M=1$; $a=0$ (blue), $a=0.05$ (red) and $a=0.15$ (purple).
• Figure 4: Effective potential as a function of the tortoise coordinate $r_{*}$ for $\ell=1/2$ Dirac field perturbations: $M=1$; $a=0$ (blue), $a=0.05$ (red) and $a=0.15$ (purple).
• Figure 5: Time-domain profile for the scalar perturbations at $\ell=0$ and $a=0.1$, $M=1$. The WKB data for the fundamental frequency is $\omega = 0.124786 - 0.106453 i$ and the time-domain method gives $\omega = 0.123736 - 0.106996 i$. The difference of about one percent should be attributed to the error of the Prony method, because the period of quasinormal ringing is very short for $\ell=0$ case.