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Two types of quasinormal modes of Casadio-Fabbri-Mazzacurati brane-world black holes

Bekir Can Lütfüoğlu, Sardor Murodov, Mardon Abdullaev, Javlon Rayimbaev, Munisbek Akhmedov, Muhammad Matyoqubov

TL;DR

This work analyzes how a nonzero mass term for a scalar perturbation modifies the quasinormal-mode spectrum of the Casadio–Fabbri–Mazzacurati brane-world black hole. Using the convergent Leaver continued-fraction method, seeded by WKB estimates, the authors compute QNMs across a range of masses $μ$ and tidal parameters $γ$, revealing two qualitatively distinct spectral branches: one where the real part $Re(ω)$ vanishes and another where the imaginary part $Im(ω)$ vanishes as $μ$ grows. Importantly, when either part reaches zero, the corresponding mode disappears and is replaced by the first overtone, indicating a reorganization of the spectrum. The results underscore the role of brane-world deformation, encoded in $γ$, and the multipole structure in determining spectral behavior, with potential observational implications for ringdown in higher-dimensional scenarios.

Abstract

Using the convergent Leaver method, we investigate the quasinormal modes of a massive scalar field propagating in the background of the Casadio--Fabbri--Mazzacurati brane-world black hole. We show that the spectrum exhibits two distinct types of modes, depending on their behavior as the field mass $μ$ increases. In one class, the real oscillation frequency decreases and eventually approaches zero, while in the other the damping rate tends to vanish. When either the real or imaginary part of the frequency reaches zero, the corresponding mode disappears from the spectrum, and the first overtone replaces it.

Two types of quasinormal modes of Casadio-Fabbri-Mazzacurati brane-world black holes

TL;DR

This work analyzes how a nonzero mass term for a scalar perturbation modifies the quasinormal-mode spectrum of the Casadio–Fabbri–Mazzacurati brane-world black hole. Using the convergent Leaver continued-fraction method, seeded by WKB estimates, the authors compute QNMs across a range of masses and tidal parameters , revealing two qualitatively distinct spectral branches: one where the real part vanishes and another where the imaginary part vanishes as grows. Importantly, when either part reaches zero, the corresponding mode disappears and is replaced by the first overtone, indicating a reorganization of the spectrum. The results underscore the role of brane-world deformation, encoded in , and the multipole structure in determining spectral behavior, with potential observational implications for ringdown in higher-dimensional scenarios.

Abstract

Using the convergent Leaver method, we investigate the quasinormal modes of a massive scalar field propagating in the background of the Casadio--Fabbri--Mazzacurati brane-world black hole. We show that the spectrum exhibits two distinct types of modes, depending on their behavior as the field mass increases. In one class, the real oscillation frequency decreases and eventually approaches zero, while in the other the damping rate tends to vanish. When either the real or imaginary part of the frequency reaches zero, the corresponding mode disappears from the spectrum, and the first overtone replaces it.
Paper Structure (7 sections, 15 equations, 3 figures, 1 table)

This paper contains 7 sections, 15 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. The CFM black-hole spacetime and wave-like equation
  3. Numerical and semi-analytical methods for finding quasinormal modes
  4. WKB approach
  5. Leaver method
  6. Quasinormal Modes
  7. Conclusions

Figures (3)

  • Figure 1: Left Panel: Effective potentials for $\ell=0$, $\mu=0$ perturbations ($M=1$): $\gamma=3$, $\gamma=0$, and $\gamma=-3$ from bottom to top. Middle Panel: The same for $\mu=0.2$. Right Panel: The same for $\mu=0.5$.
  • Figure 2: Real (left) and imaginary (right) parts of the fundamental QNM $\ell=n=0$ for $\gamma=3$ (yellow), $\gamma=1$ (green), $\gamma=0$ (red), $\gamma=0.5$ (blue), $\gamma=-1$ (orange), $\gamma=-3$ (magenta).
  • Figure 3: Real (left) and imaginary (right) parts of the first overtone $\ell=0$, $n=1$ for $\gamma=3$ (black), $\gamma=2$ (red), $\gamma=0$ (blue), $\gamma=0.5$ (blue).