Ringdown spectroscopy of phenomenologically modified black holes

TL;DR

This work addresses testing beyond-GR modifications to black hole QNM spectra by extending the theory-agnostic parametrized QNM framework to the time domain. It evolves perturbations with modified potentials and fits the ringdown using a damped-sinusoid/price-tail model, validating that the fundamental QNM frequency agrees with perturbative predictions across single and multiple modifications, provided an overtone or Price tail is included. A complementary WKB-inspired analysis links the QNM real part to the potential peak height and the product of real and imaginary parts to the potential curvature, supporting robust inferences about local potential properties from ringdown data. Overall, the study supports the viability of time-domain black hole spectroscopy as a robust probe of phenomenological modifications to GR and informs observational strategies for incorporating tails and overtones.

Abstract

The characteristic oscillations of black holes, as described by their quasinormal mode (QNM) spectrum, play a fundamental role in testing general relativity with gravitational waves. The so-called parametrized QNM framework was introduced to predict perturbative changes in the spectrum due to small deviations from general relativity. In this work, we extend the framework to model time domain signals and study the excitation of quasinormal modes from the time evolution of initial data. Specifically, we quantify whether the perturbative eigenvalue predictions agree with extracting excited quasinormal modes from such simulations. Addressing this issue is particularly important in the context of agnostic ringdown tests and the possible presence of spectral instabilities, which may diminish the promises of black hole spectroscopy. We find that the extracted quasinormal modes agree well with the perturbative predictions, underlining that these types of modifications can, in principle, be robustly tested from real observations. Moreover, we also report the importance of including late-time tails for accurate mode extractions. Finally, we provide a WKB-inspired analysis supporting the importance of the peak of the scattering potential and show robust scaling relations.

Figures (13)

• Figure 1: Exemplary waveforms produced from the same initial data given from Eqs. \ref{['eq:initial_data']} and \ref{['eq:initial_data2']} and from different potentials, GR (black), with $\alpha^{(2)}=1.00$ (blue) and with $\alpha^{(2)}=-1.00$ (red). The gray-shaded area indicates a typical portion included in the fits.
• Figure 2: Modifications to GR potential and respective waveforms. Top row: we show the effective potentials when varying a single $\alpha^{(k)}=\epsilon\alpha^{(k)}_\mathrm{max}$ modification at a time. Middle row: we plot the corresponding deviations from the original GR potential. Bottom row: we provide the corresponding waveforms. In each panel, we fix the order $k$ and show a family of curves corresponding to increasing the strength of the modification (different colors). The GR case is shown for comparison (black line).
• Figure 3: Mismatches between gravitational waveforms and fit results. In the top panels, we consider a model with one QNM; in the central panels, we use two QNMs; and in the bottom panels, we consider one QNM and the power-law tail. From left to right, we consider different powers $k\in[2,3,5,7]$ of the modifications $\alpha^{(k)}$. In each panel, we show mismatches as a function of the starting time $t_0-t_\text{peak}$. Different curves in each panel correspond to different amplitudes of $\alpha^{(k)}$ that are indicated by different colors in the color bar.
• Figure 4: Extracted parameters from gravitational waveforms. We show from top to bottom the extracted real and imaginary parts of the fundamental QNM, its amplitude $A_0$ and phase $\phi_0$, as function of the starting time $t_0-t_\text{peak}$ when using different fitting models, $n=0$ (solid lines), $n=0,1$ (dashed lines) and $n=0,\mathrm{tail}$ (dotted lines). From left to right, we consider different powers $k\in[2,3,5,7]$ of the modifications $\alpha^{(k)}$.
• Figure 5: Extraction of the $n=0$ QNM for $t_0 - t_\mathrm{peak}=50M$ and comparison with the perturbative prediction. In the top panels, we show the real part, and in the bottom panels, the imaginary part. From left to right, we consider different powers $k\in[2,3,5,7]$ of the modifications $\alpha^{(k)}=\epsilon \alpha^{(k)}_\mathrm{max}$ and we also indicate each $\alpha^{(k)}_\mathrm{max}$ value (vertical dashed lines). In each panel, we show the perturbative prediction (red line) obtained from Eq. (\ref{['eq:omega_expansion']}) and an estimate for its accuracy (grey-shaded region) as a function of the coefficient $\alpha^{(k)}$. Different points correspond to the extracted QNM from the time domain fits of different models.