[Paper Review] On the ease of excitation of black hole ringing: Quantifying the importance of overtones by the excitation factors

Naritaka Oshita|arXiv (Cornell University)|Sep 20, 2021

Pulsars and Gravitational Waves Research58 references59 citations

TL;DR

This paper quantifies the excitation ease of Kerr black hole quasinormal modes using excitation factors (QNEFs), showing that the 4th, 5th, and 6th overtones have the highest excitation factors for intermediate to high spin parameters (j ≲ 0.9). This provides a theoretical explanation for why these overtones dominate early ringdown waveforms in numerical relativity, consistent with observed data from GW150914-like mergers.

ABSTRACT

The excitation factors of black hole quasinormal modes quantify the $ extit{ease of excitation}$ of the quasinormal modes and are independent of the source of perturbation. We compute the excitation factors of Kerr black holes up to the 20th overtone and find that the 4th, 5th, and 6th overtones have the first three highest excitation factors for intermediate and high spin parameters. This provides an independent confirmation of the importance of overtones that has been confirmed by the fitting data analysis of numerical relativity waveforms beginning around the strain peak amplitude.

Motivation & Objective

• To quantify the intrinsic ease of excitation of quasinormal modes in Kerr black holes using excitation factors independent of the source.
• To resolve the long-standing puzzle of why higher overtones are significantly excited in early ringdown, despite their exponential damping.
• To provide a theoretical basis for the empirical success of including up to seven overtones in waveform fitting of numerical relativity waveforms.
• To validate the role of QNEFs in predicting the relative amplitude hierarchy of ringdown modes across different spin parameters.

Proposed method

• Computed quasi-normal excitation factors (QNEFs) for l = m = 2 modes up to the 20th overtone using a modified definition that ensures invariance under perturbation variable choice.
• Solved the Sasaki-Nakamura equation numerically to simulate a particle plunging into a black hole and extracted excitation coefficients C22n for comparison.
• Defined a new decay time metric tlmn based on QN frequency and QNEF to estimate when each overtone becomes subdominant relative to the fundamental mode.
• Performed mismatch analysis between full numerical waveforms and QN-mode superpositions to assess the necessity of including higher overtones.
• Extracted source factors T22n from the SXS:BBH:0305 waveform and found them to be nearly constant across overtones, indicating QNEFs dominate amplitude hierarchy.

Experimental results

Research questions

• RQ1 Why are the 4th, 5th, and 6th overtones particularly prominent in early ringdown waveforms from binary black hole mergers?
• RQ2 What determines the relative excitation amplitudes of overtones, independent of the initial perturbation?
• RQ3 How does the excitation factor hierarchy relate to the observed success of including up to seven overtones in waveform fitting?
• RQ4 Why is the 5th overtone suppressed in near-extremal black holes (j ≳ 0.9)?
• RQ5 To what extent do QNEFs alone determine the amplitude hierarchy of ringdown modes, as opposed to source-dependent factors?

Key findings

• The 4th, 5th, and 6th overtones have the three highest excitation factors for spin parameters 0.3 ≤ j ≲ 0.9, providing a universal explanation for their prominence.
• For j ≳ 0.9, the 5th QNEF is suppressed, explaining the observed suppression of the 5th overtone in near-extremal cases.
• The mismatch between numerical waveforms and QN-mode fits decreases significantly with higher overtone inclusion, especially for high-spin cases (e.g., j = 0.99), confirming the need for higher overtones.
• The decay time tlmn, derived from QNEF and QN frequency, estimates when each overtone becomes subdominant and matches the time of mode dominance shift observed in Giesler et al. (2019) for SXS:BBH:0305.
• Source factors T22n extracted from SXS:BBH:0305 show weak dependence on overtone number, indicating that the amplitude hierarchy is primarily governed by QNEFs rather than source-specific effects.
• The QNEF hierarchy explains the empirical success of truncating at n = 7 in waveform fitting, as the top three QNEFs correspond to the 4th–6th overtones.