Probing higher curvature gravity via ringdown with overtones

TL;DR

This work investigates linear perturbations of Schwarzschild black holes in a higher-curvature EFT that adds a near-horizon deformation to the odd-parity master equation. Using a parametrized QNM formalism, the authors show overtone frequencies deviate from Schwarzschild values more strongly as the deformation localizes toward the horizon, while the fundamental mode remains comparatively stable. Time-domain waveforms are computed and fitted with EFT- and GR-based QNM templates, revealing that including overtones yields significantly better fits for the EFT case, with the early-time ringdown most sensitive to near-horizon physics. The results imply that ringdown observations, especially with overtone information, can serve as a powerful probe of near-horizon modifications predicted by higher-curvature gravity EFTs and motivate extending the analysis to rotating black holes.

Abstract

We investigate metric perturbations of a spherically symmetric black hole in higher curvature gravity. We show that higher curvature corrections deform the near-horizon region of the effective potential, and that the deviations of the quasinormal mode (QNM) frequencies from their general relativity (GR) values become more pronounced for overtone modes. We find that, as the order of the higher curvature term increases, the deformations approach the horizon and the deviations of the overtone QNM frequencies grow progressively larger. We also analyze the ringdown waveforms in the higher curvature gravity model. We consider setups in which the deviations from the vacuum-GR QNMs remain mild for the fundamental mode and the first few overtones, and show that these shifted QNMs can be identified in the ringdown signal through waveform fitting.

Figures (6)

• Figure 1: The black line shows the Regge-Wheeler potential $V_{\rm RW}$, while the blue and orange lines show $j\,\delta v_j$ for $j=10$ and $j=1000$, respectively. As the value of $j$ increases, the peak location of $\delta v_j$ approaches to the event horizon.
• Figure 2: The blue solid lines are the time-domain waveforms in the EFTs with $\alpha_{10}=0.01$ (left), $\alpha_{100}=0.1$ (center), and $\alpha_{1000}=0.5$ (right), respectively, while the orange dashed lines are those in GR. Although the EFT and the GR waveforms appear nearly overlapping, the difference between those waveforms (green solid line) are larger than the numerical error (gray solid line). The numerical error is estimated by computing the discretization error from the difference between results obtained with different grid resolutions.
• Figure 3: QNM frequencies for the three representative cases, $\alpha_{10}=0.01$, $\alpha_{100}=0.1$, and $\alpha_{1000}=0.5$, together with the GR values. The vacuum GR QNMs are shown as blue dots, while the QNMs for $\alpha_{10}=0.01$, $\alpha_{100}=0.1$, and $\alpha_{1000}=0.5$ are indicated by orange crosses, green squares, and red diamonds, respectively. The numbers attached to the data points denote the overtone number $n$. We show only those QNMs whose deviations from the GR values remain perturbative.
• Figure 4: The mismatch $\mathcal{M}$, the amplitudes $A_{n}$, and the phases $\phi_{n}$ as functions of $(t_{0} - t_{\rm peak})/r_{\rm H}$ for the three representative cases, $(j, \alpha_{j}) = (10,0.01)$, $(100, 0.1)$, and $(1000,0.5)$. In the mismatch plots, the colors correspond to the fitting model \ref{['eq:EFTtemplate']} with different values of $N$. We use up to $N=4$ for $\alpha_{10}=0.01$, up to $N=3$ for $\alpha_{100}=0.1$, and up to $N=1$ for $\alpha_{1000}=0.5$. In the plots of $A_n$ and $\phi_n$, the value of $N$ is fixed to its maximal value for each case, and the colors correspond to different values of $n$. The solid lines are the results for the EFT fit, while the dashed lines are those for the GR fit.
• Figure 5: The orange solid line is the time-domain waveform in GR, while the gray solid line is the numerical error. The dominant contribution to the numerical error comes from the discretization error, which is estimated by comparing results obtained with different resolutions.