[Paper Review] Ringdown stability: greybody factors as stable gravitational-wave observables

Romeo Felice Rosato, Kyriakos Destounis+1 more|arXiv (Cornell University)|Jun 3, 2024

Geophysics and Sensor Technology4 citations

TL;DR

This paper proposes greybody factors (GFs) as stable gravitational-wave observables during black hole ringdown, offering a robust alternative to quasinormal modes (QNMs). Despite QNMs being spectrally unstable under small perturbations, GFs—derived from the tunneling probability of waves through the black hole potential—remain stable and accurately describe high-frequency ringdown spectral amplitudes, enabling a more reliable test of general relativity and black hole nature.

ABSTRACT

The quasinormal mode spectrum of black holes plays a crucial role in the modelling of post-merger ringdown signals. However, the spectrum is extremely sensitive to small deformations of the system and describes the linear response only in a certain (not precisely defined) timeframe after the merger. We argue here that the greybody factors, recently shown to describe the ringdown spectral amplitude at relatively high frequencies, are instead stable under small perturbations of the system and free of certain ambiguities that plague the quasinormal mode spectrum. Our analysis also unveils a nontrivial interplay: while certain ringdown quantities are dominated by the contribution of spectrally unstable quasinormal modes, these modes conspire to produce stable observables. Thus, we propose a complementary approach to ringdown studies, which circumvents some limitations of the standard quasinormal mode description.

Motivation & Objective

• To address the spectral instability of quasinormal modes (QNMs) under small perturbations in black hole ringdown signals.
• To identify observables that remain stable despite QNM spectral instability, ensuring robustness in gravitational-wave data analysis.
• To establish greybody factors (GFs) as viable, stable alternatives to QNMs for black hole spectroscopy.
• To demonstrate that GFs accurately describe high-frequency ringdown spectral amplitudes, even when QNMs are unstable.
• To reveal a nontrivial interplay where spectrally unstable QNMs conspire to produce stable physical observables.

Proposed method

• Use the Regge-Wheeler-Zerilli formalism to model linear perturbations of spherically symmetric black holes in the frequency domain.
• Define greybody factors (GFs) as the squared ratio of outgoing to ingoing wave amplitudes at infinity, characterizing transmission through the effective potential.
• Compute spectral amplitudes of ringdown signals via the Green's function method, linking GFs to observable wave amplitudes.
• Perform numerical simulations of ringdown signals for point-particle sources, varying parameters like energy $E$ and angular momentum $L$.
• Compare spectral amplitudes $|h_{lm}( u)|$ with a model $\propto \sqrt{1 - \Gamma_{lm}} / \omega^p$ to test universality and stability.
• Assess stability by perturbing the effective potential with a small bump and measuring changes in total emitted energy and spectral amplitudes.

Figure 1: Perturbed effective potential, Eq. ( 4 ), for $l=2$ , $\epsilon=5\times 10^{-4}$ , and two representative choices for the small bump location, $c=-25M$ (left inset) and $c=50M$ (right inset).

Experimental results

Research questions

• RQ1 Can greybody factors serve as stable observables in black hole ringdown despite the spectral instability of quasinormal modes?
• RQ2 How do small perturbations of the black hole potential affect the quasinormal mode spectrum and the resulting ringdown signal?
• RQ3 To what extent do spectral amplitudes at high frequencies follow a universal behavior governed by greybody factors?
• RQ4 Can the total energy radiated in a given mode remain stable under small background deformations, even when QNMs shift?
• RQ5 What is the physical mechanism by which unstable QNMs produce stable observables in the ringdown phase?

Key findings

• Greybody factors remain stable under small perturbations of the black hole potential, such as a bump of height $\epsilon = 10^{-3}$ and location $c = 15M$, while quasinormal modes (QNMs) exhibit ${\cal O}(1)$ shifts in overtones.
• The spectral amplitude $|h_{lm}( u)|$ at high frequencies is well described by a single-parameter model $\propto \sqrt{1 - \Gamma_{lm}} / \omega^p$, with $p$ depending on $E$, $L$, and $l$, indicating universal behavior.
• The total energy emitted in $l=2$ modes remains stable under perturbations, with relative differences in energy less than $10^{-4}$ for $\epsilon = 10^{-4}$, demonstrating robustness.
• Despite significant changes in QNM excitation factors (e.g., $n=1,2$ overtones), the time-domain ringdown signal is reconstructed equally well from both unperturbed and perturbed QNMs, indicating a hidden stability.
• The energy flux $dE_{lm}/d\omega \sim 1 - \Gamma_{lm}$ at high frequencies confirms that GFs directly govern the spectral shape of the ringdown signal.
• The interplay between unstable QNMs and stable observables is analogous to that seen in quantum scattering via Regge poles, suggesting a deeper connection in wave scattering theory.

Figure 2: Comparison between the spectral instability of the fundamental QNM (left panel) and the stability of the (integrated) GF (right panel) of a Schwarzschild BH under a small perturbation centered at $r_{*}=c$ with amplitude $\epsilon\ll 1$ (see Eq. ( 4 )). While the QNM deviations from the un