Quasinormal modes from numerical relativity with Bayesian inference

TL;DR

This work presents a Bayesian framework for extracting quasinormal modes from numerical-relativity waveforms by modeling NR numerical uncertainties with a physically motivated Gaussian-process kernel. The QNM model is linearized around a reference, yielding analytically sampleable Gaussian posteriors for mode amplitudes and remnant parameters, and is trained on a public CCE NR waveform catalog to define a robust likelihood. Compared with simpler noise models, the GP-based approach produces tighter posteriors and enables a principled significance measure for including specific QNMs in a model, along with posterior predictive checks to assess fit quality. The method demonstrates efficient, scalable QNM inference across multiple modes and domains (strain, news, and curvature) and offers a practical tool for probing subdominant ringdown content in gravitational-wave data.

Abstract

Numerical relativity (NR) enables the study of physics in strong and dynamical gravitational fields and provides predictions for the gravitational-wave signals produced by merging black holes. Despite the impressive accuracy of modern codes, the resulting waveforms inevitably contain numerical uncertainties. Quantifying these uncertainties is important, especially for studies probing subdominant or nonlinear effects around the merger and ringdown. This paper describes a flexible Gaussian-process model for the numerical uncertainties in all the spherical-harmonic waveform modes across a state-of-the-art catalog of NR waveforms and a highly efficient procedure for sampling the posteriors of quasinormal mode models without the need for expensive Markov chain Monte Carlo. The Gaussian-process model is used to define a likelihood function which allows many Bayesian data analysis techniques - already widely used in the analysis of experimental gravitational wave data - to be applied to NR waveforms as well. The efficacy of this approach is demonstrated by applying it to the analysis of quasinormal modes in Cauchy-characteristic evolved waveforms.

Figures (20)

• Figure 1: These plots illustrate the standard GP model (Eq. \ref{['eq:kernel']}) for the uncertainty in selected waveform modes ($\beta=$(2,2), (3,2), and (4,4) in the top middle and bottom panels respectively) of simulation $i=0001$. The colored curves show the real and imaginary parts of the residuals in the Bondi news, $\mathfrak{r}_i^\beta(t)$. The shaded regions show the amplitude of GP model for the waveform uncertainties. What cannot be seen from the shaded regions is the timescale over which the GP model is correlated; this is illustrated with the horizontal bars on each panel, scaled to the squared-exponential timescale as described in the text.
• Figure 2: PGM of the GP model for the NR waveform uncertainty described in the main text. The observed variables (shaded circles) are the real and imaginary parts of the waveform residuals in each simulation in the catalog (indexed by $i$) and in each spherical harmonic model (indexed by $\beta$). The residuals are the differences between the two highest resolution simulation available in the catalog. The parameters of the model are shown in empty circles. The values of the two pooled parameters in the top row are ultimately to be inferred from the NR catalog. The other latent parameters are determined from the pooled parameters and from the various fixed quantities which are shown without circles.
• Figure 3: The symmetrized KLD, $D$, (in bits) between the Gaussian distributions defined using the three kernels across the full catalog and all spherical modes used in training. The difference between the standard kernel (GP) or complicated kernel (GPc) and the white noise (WN) kernel are significant, however those between the GP kernel and the GPc are localized at smaller values, indicating that they are more similar.
• Figure 4: The mismatch of the MAP QNM model compared to the NR data as a function of the ringdown start time for Model 1. The top panel gives the mismatch, calculated with respect to an inverse covariance matrix computed using the standard kernel function ($\mathcal{M}_{\rm GP}$) while the lower panel shows the typical white-noise mismatch $\mathcal{M}_{\rm WN}$. The vertical black line indicates $t_0=17M$ which is used as the reference start time in later plots. In both panels the gray shaded regions gives an indication of the noise floor of the simulation and show the mismatch between the NR waveforms at the two highest resolution levels.
• Figure 5: The QNM decay-corrected amplitudes $|\hat{C}_\alpha|$ as a function of the ringdown start time for Model 1. Solid (dashed) lines show the results obtained using the GP (WN) covariance and the colors distinguish the QNM overtones. The median value is plotted and the shaded regions show the 50% confidence intervals for the Prior 1 posterior on the WN kernel. The interval is too small to be seen for the GP kernel, so has been omitted. The logarithmic scale used on this plot to show all the overtones gives the misleading impression that the width of the amplitude posteriors is constant at late times; this is related to the behavior of Prior 1 at small amplitudes and is discussed in the main text.