Modeling the frequency-domain ringdown amplitude of comparable-mass mergers with greybody factors

TL;DR

We address ringdown modeling for comparable-mass BBH mergers by formulating a frequency-domain amplitude model based on the remnant Kerr greybody reflectivity $\\mathcal{R}_{\\ell m}(\\bar{\\omega},\\chi)$, enabling direct inferences of the remnant mass $M$ and dimensionless spin $\\chi$ without relying solely on QNM fits. The core model is a four-parameter spectrum $H^{(2)}_{\\ell m}(\\omega)=M\\,A_{\\ell m}\\,\\sqrt{\\mathcal{R}_{\\ell m}(\\bar{\\omega},\\chi)}/\\bar{\\omega}^{p_{\\ell m}}$ (with the simpler $H^{(1)}_{\\ell m}(\\omega)=M\\,A_{\\ell m}\\,\\sqrt{\\mathcal{R}_{\\ell m}(\\bar{\\omega},\\chi)}$ as a baseline), which captures the frequency-domain amplitude remarkably well across the SXS catalog and improves previous formulations by about two orders of magnitude in mismatch, i.e., achieving $\\mathcal{M}\\sim O(10^{-5})$. Fitting on the SXS NR catalog yields mismatches of order $\\mathcal{M}\\sim O(10^{-5})$, and identifies an optimal frequency window for application in which $A_{\\ell m}$ and $p_{\\ell m}$ stabilize. The coefficients $A_{\\ell m}$ and $p_{\\ell m}$ are modeled as low-degree polynomials in the progenitor parameters $(\\delta,\\chi_+,\\chi_-)$, with cross-validation selecting the degrees and monomials; results from an independent Random Forest Regressor corroborate the polynomial fits and highlight the dominant role of $\\chi_+$ (and symmetry under exchange of the binary components). The work demonstrates that greybody-based ringdown modeling can provide complementary tests of GR to BH spectroscopy and sets the stage for applying the model to real data, including extensions to precession, eccentricity, and phase modeling.

Abstract

It was recently shown that, in a binary coalescence, the greybody factor of the remnant black hole modulates the post-merger ringdown signal. In this work, we demonstrate that a simple four-parameter model based on the greybody factor accurately reproduces the frequency-domain amplitude of a large set of comparable-mass, aligned-spin numerical relativity waveforms from the SXS catalog, achieving mismatches of order ${\cal O}(10^{-5})$ and improving existing models by roughly two orders of magnitude. We also identify the optimal initial frequency for applying the model in the frequency domain and provide analytical fits of the model parameters in terms of the progenitor masses and aligned spins. Our results pave the way for new consistency tests of the ringdown phase, complementary to traditional black hole spectroscopy.

Figures (10)

• Figure 1: Mismatch $\mathcal{M}$, defined in Eq. \ref{['eq:mismatch']}, between the numerical Fourier transform $H_{\ell m}^{\rm data}(\omega) = |h_{\ell m}(\omega)|$ from the simulation SXS:BBH:3922 and two analytical models. The mismatch is shown over a portion of the $(M,\chi)$ plane of the remnant. The upper panel corresponds to the model $H^{(1)}_{\ell m}(\omega) = MA_{\ell m}\,\sqrt{\mathcal{R}_{\ell m}(\bar{\omega},\chi)}$ proposed in Ref. Okabayashi:2024qbz, for which the fit is performed with a single free parameter $A_{\ell m}$, while the lower panel shows results for the modified model proposed in this work, $H^{(2)}_{\ell m}(\omega) = MA_{\ell m}\,\sqrt{\mathcal{R}_{\ell m}(\bar{\omega},\chi)}/\bar{\omega}^{p_{\ell m}}$ (see Eq. \ref{['eq:model']}), where both $A_{\ell m}$ and $p_{\ell m}$ are fitted. The fit is performed over the range $\bar{\omega} \in [0.35,\,0.75]$, determined for this specific simulation as discussed in Appendix \ref{['subsec:fittingprocedure']}. For the model $H^{(1)}_{\ell m}$, the region of minimal mismatch does not exactly coincide with the true remnant parameters, whereas for the modified model $H^{(2)}_{\ell m}$ the true values lie within the minimum-mismatch region. Overall, the agreement improves by about two orders of magnitude, with the mismatch $\mathcal{M}$ dropping from $\sim\!10^{-3}$ to $\sim\!10^{-5}$ around the true values of the remnant parameters.
• Figure 2: Illustration of the fitting procedure for the simulation SXS:BBH:3982, considering the dominant $(\ell,m)=(2,2)$ mode. The goal of the fit is both to extract the model parameters $A_{\ell m}$ and $p_{\ell m}$ and to determine the frequency range over which the model remains valid. For this purpose, we perform fits over intervals $\bar{\omega} \in [\bar{\omega}_i,\,\bar{\omega}_{\rm cut}]$ for several starting frequencies $\bar{\omega}_i$. The top-left panel shows the numerical frequency-domain spectrum $H_{22}(\omega)=|h_{22}(\omega)|$ (solid black) and the best-fit model $A/\bar{\omega}^p$ (dashed pink), evaluated for $\bar{\omega} < \bar{\omega}_{\rm cut}$ as described in Appendix \ref{['app:fft']}. The bottom-left panel displays the mismatch $\mathcal{M}$, computed according to Eq. \ref{['eq:mismatch']}, as a function of the initial fitting frequency $\bar{\omega}_i$. The top-right and bottom-right panels show the fitted parameters $p_{\ell m}$ and $A_{\ell m}$ as functions of $\bar{\omega}_i$. Both parameters become stable beyond a certain frequency $\bar{\omega}_x$, indicated by the shaded region, where their variations remain below $0.1\%$. The frequency $\bar{\omega}_x$ therefore marks the lower bound of validity of the model. The mismatch reaches its minimum in this stability region, with $\mathcal{M}\simeq2\cdot10^{-5}$, demonstrating the excellent agreement between the model and the numerical data.
• Figure 3: Equal-mass slice for the $(\ell,m)=(2,2)$ coefficients. We select simulations with $q\in[0.999,\,1.001]$, corresponding to $\delta\simeq 0$, and show the variation of $A_{22}$ (top) and $p_{22}$ (bottom) over the $(\chi_{+},\chi_{-})$ plane. Each point represents a numerical relativity simulation. The color scale encodes the value of the coefficients. The dependence on $(\chi_{+},\chi_{-})$ at fixed $\delta$ appears smooth. Moreover, the plot clearly shows that the dominant dependence is on $\chi_+$: variations in the values of $A_{22}$ and $p_{22}$ along $\chi_-$ (i.e., along vertical slices) are comparatively small.
• Figure 4: Cross-validation of the polynomial model for the fitted coefficients $A_{22}$ and $p_{22}$ for the $\delta=0$ slice of the $(2,2)$ mode, with $\delta$ defined in Eq. \ref{['eq:delta']}. The upper (lower) panel shows the cross-validation mean squared error (CV MSE), defined in Eq. \ref{['eq:CVMSE']}, as a function of the polynomial degree $\mathcal{N}$ for $A_{22}$ ($p_{22}$). Results are displayed for both the training set (solid black) and the validation set (dashed pink). The training error decreases monotonically with $\mathcal{N}$, whereas the validation error reaches a minimum and then increases due to overfitting. The optimal degree $\mathcal{N}_{\rm opt}$, indicated by the vertical dashed line, is chosen as the value that minimizes the validation CV MSE, thus providing the best compromise between accuracy and generality.
• Figure 5: Comparison between the values predicted by the polynomial model of Eq. \ref{['eq:poly_model']}, $X_{\mathrm{fit}}$, and the corresponding values directly extracted from the simulations, $X_{\mathrm{data}}$, through the fitting procedure described in Sec. \ref{['subsec:fittingprocedure']}. The top panel refers to $A_{22}$, and the bottom panel to $p_{22}$. Black (pink) points correspond to the training (validation) set. The dashed line shows $X_{\mathrm{fit}} = X_{\mathrm{data}}$, along which both datasets lie, indicating a good fit with no evidence of overfitting.