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The quasinormal mode content of binary black hole ringdown

Richard Dyer, Christopher J. Moore

TL;DR

The paper develops a fully Bayesian, data-driven framework to identify quasinormal modes in binary black hole ringdown waveforms, including overtones, retrograde, and nonlinear (quadratic and cubic) modes, as well as constant offsets and potential power-law tails. It uses Bayes-factor driven model selection and posterior predictive checks to robustly determine mode content at each ringdown start time, applied to the public SpECTRE CCE catalog. The results provide a systematic reference of mode content across simulations and start times, clarifying the behavior of overtones and confirming the absence of late-time power-law tails in these waveforms. This framework delivers a rigorous, scalable approach for theoretical and observational ringdown analyses and can accommodate future catalogs including CCM-based tails.

Abstract

We present a fully Bayesian, data-driven framework for identifying quasinormal modes in high-accuracy Cauchy-Characteristic Evolution (CCE) gravitational waveforms. Applying this to a public catalog, we identify QNM overtones, retrograde modes, and nonlinear modes up to cubic order in the ringdown. The ringdown mode content is tabulated across a wide range of start times for all available simulations, providing a systematic reference for theoretical and observational studies. We also search for late-time power-law tails, which are, as expected, absent from the CCE waveforms.

The quasinormal mode content of binary black hole ringdown

TL;DR

The paper develops a fully Bayesian, data-driven framework to identify quasinormal modes in binary black hole ringdown waveforms, including overtones, retrograde, and nonlinear (quadratic and cubic) modes, as well as constant offsets and potential power-law tails. It uses Bayes-factor driven model selection and posterior predictive checks to robustly determine mode content at each ringdown start time, applied to the public SpECTRE CCE catalog. The results provide a systematic reference of mode content across simulations and start times, clarifying the behavior of overtones and confirming the absence of late-time power-law tails in these waveforms. This framework delivers a rigorous, scalable approach for theoretical and observational ringdown analyses and can accommodate future catalogs including CCM-based tails.

Abstract

We present a fully Bayesian, data-driven framework for identifying quasinormal modes in high-accuracy Cauchy-Characteristic Evolution (CCE) gravitational waveforms. Applying this to a public catalog, we identify QNM overtones, retrograde modes, and nonlinear modes up to cubic order in the ringdown. The ringdown mode content is tabulated across a wide range of start times for all available simulations, providing a systematic reference for theoretical and observational studies. We also search for late-time power-law tails, which are, as expected, absent from the CCE waveforms.
Paper Structure (4 sections, 3 equations, 3 figures, 1 table)

This paper contains 4 sections, 3 equations, 3 figures, 1 table.

Table of Contents

  1. Description of the ringdown model building algorithm
  2. An Example of the Ringdown Model Building Algorithm
  3. Spherical Harmonics
  4. Power-Law Tail Analysis

Figures (3)

  • Figure 1: The ringdown mode content for simulation 0010. Fits are performed independently at each start time, $t_0$. Colored horizontal bands indicate a mode detected at that $t_0$. Prograde QNMs are grouped and colored based on their $(\ell,m)$ indices with the fundamental ($n\!=\!0$) uppermost and overtones ($n\!\geq\!1$) below. If the retrograde mode is also detected then the band is hatched. Nonlinear QNMs are shown with the harmonic that they mix strongly with. The shaded region $t_0\!\leq\!3M$ indicates where the PPC fails and the model fits the data poorly. Similar plots for all simulations in the catalog are available at https://bgp-qnm-fits.github.io/bgp_qnm_content/website.
  • Figure 2: Plots of selected (decay-corrected) amplitudes $\hat{C}_\alpha$methods_paper for the detected modes in simulation 0010 as a function of $t_0$. Horizontal lines indicate modes that decay exponentially at the rate predicted by perturbation theory. The left/center/right plots show the $(\ell,m)\!=\!(2,2)$/$(4,4)$/$(6,6)$ sectors from Fig. \ref{['fig:mode_content']} with matching colors. Thick lines denote fundamental QNMs with thinner lines denoting higher overtones; nonlinear QNMs are indicated in the legends (only prograde modes are shown). The central lines indicate the median posterior amplitudes and the shaded regions indicate the $90\%$ ranges; in most cases the shaded regions are too small see clearly at this scale. Similar plots for all simulations in the catalog are available at https://bgp-qnm-fits.github.io/bgp_qnm_content/website.
  • Figure S1: Bayesian algorithm for determining the QNM content of the ringdown. The bayes_fit (line 8) refers to a full Bayesian inference of a ringdown model with the specified mode content to the NR data while sig (line 9) refers to the calculation of the significance of a particular mode $c$ (related to the Bayesian evidence for that mode) in that fit. In practice, the modifications described in the text speed this up.