High-overtone ringdown fits: start time, no-hair tests, and correlations

TL;DR

This paper assesses the practical value of including many overtones in black-hole ringdown analyses by applying NR-based fits (both least-squares and Bayesian) to evaluate how the ringdown start time, and tests of the no-hair theorem, depend on the number of overtones $N$. It shows that while adding overtones can push the usable start time earlier and improve remnant-property recovery, there is no unique highest overtone; the gains diminish with increasing $N$ and the overtones become highly correlated, complicating individual amplitude measurements. Through perturbing overtone frequencies and mapping the posterior correlations, the work reveals that joint measurements of amplitudes (via the correlation structure) retain sensitivity to the frequencies and decay times of even high-$n$ overtones, offering a potential path for GR-consistency tests. The findings inform pragmatic ringdown modeling and BH spectroscopy, suggesting that future work should pursue full Bayesian parameter estimation with high-overtone models and additional QNMs, along with accessible code for reproducibility.

Abstract

Overtones are known to improve the performance of fits to the ringdown, both in numerical-relativity simulations and gravitational-wave observations. Although the overtone frequencies are a concrete prediction of general relativity, it remains an open question whether they are excited to the extent that fits would suggest. In this work, we take a pragmatic approach and investigate the practical utility of each additional overtone in extracting information from the ringdown. We look at the dependence of the ringdown start time on the number of overtones, and the feasibility of detecting deviations from general relativity in the ringdown frequencies. We suggest that there is no clear "maximum" overtone, but rather the utility of each additional overtone decreases compared to the one before. Finally, we perform Bayesian parameter estimation (as opposed to least-squares fits) to obtain posterior distributions on the overtone amplitudes and phases, allowing us to investigate their correlation structure. Due to strong correlations it becomes increasingly hard to measure individual amplitudes and phases for the highest overtones. However, we find that the joint measurement of overtone amplitudes (i.e., the correlation structure itself) is sensitive to the frequencies and decay times of even the highest overtones, possibly offering an avenue to perform consistency tests with general relativity.

Figures (9)

• Figure 1: Top: The remnant mass-spin error ($\epsilon$) from an overtone model fitted to CCE:01, for different numbers of overtones in the model ($N$) and for three different ringdown start times (line colors). For each start time there is a choice of $N$ which gives a minimum in the curve (indicated with a black circle --- we take the first minimum that occurs, not the global). Middle row: The amplitude of each best-fit QNM from a fit at the minimum of the curve from the top panel (indicated by the connecting lines). We report the amplitudes rescaled to what they would be at the time $h_\mathrm{peak}^{h_{22}}$. For the left and right panels, where the fit is performed $10\,M$ after and before that time respectively, the unscaled amplitudes are shown in light gray. Bottom row: The same as the middle panel, but for the phase of each best-fit QNM. We only show the phases reported at the time $h_\mathrm{peak}^{h_{22}}$, and the axis limits are chosen to aid comparison of the phases across the three fits.
• Figure 2: Overtone morphology. Top: The frequency evolution of CCE:01 around the time of peak strain, with the overtone frequencies plotted as horizontal lines. The length of each line to the right of the vertical dashed line is proportional to the decay time of the mode (the line length is given by $3\tau_{22n}$). Bottom: A 17-overtone fit to CCE:01, deconstructed into the individual QNMs. The start time of the fit, indicated by the vertical dashed line, is determined by minimizing the mismatch (see Sec. \ref{['sec:start_time']}). Due to the large number of QNMs in the model we highlight only a subset of them. As $n$ increases, the overtones grow to extremely large amplitudes (note the change in $y$-axis scale below $-2$ and above $2$) and barely oscillate before decaying away.
• Figure 3: Top left: The mismatch vs ringdown start time for fits to the $h_{22}$ mode of CCE:01 with a model consisting of $N$ overtones of the fundamental $(2,2,0)$ QNM. Crosses mark the location of $t_0^{N,\mathcal{M}}$ for each $N$. The gray region indicates the NR error, obtained from the mismatch between the two highest available levels. Top right: As in the top left panel, but for the error on the remnant mass and spin. Crosses mark the location of $t_0^{N,\epsilon}$ for each $N$. Bottom left: The ringdown start times, $t_0^{N,\mathcal{M}}$ and $t_0^{N,\epsilon}$, plotted against each other for CCE:01 (crosses) and the other nine CCE waveforms considered in this work (faint circles). We also indicate the approximate SNR (in the high-SNR limit) as a function of start time, in units of the SNR from a fundamental-mode-only fit. Bottom right: As in the bottom left panel, but for the change in ringdown start time, Eq. \ref{['eq:delta_t0']}. This is not defined for $N=0$, and we only plot up to $N=17$ since $\Delta t_0^{18,\mathcal{M}}$ and $\Delta t_0^{18,\epsilon}$ are negative.
• Figure 4: Top: The induced error on the remnant mass and spin as a function of the number of overtones included in the fit, when perturbing all the overtone frequencies simultaneously (dashed lines) and when perturbing only the $n=1$ mode (solid lines), for different values of the perturbation $\delta$ (line colors). We fit to CCE:01 at a start time $t_0 = t_0^{N=10,\epsilon}$ (as determined in Sec. \ref{['sec:start_time']}), but now we include the $(3,2,0)$ mode in the fit. Middle: The remnant error for a $N=13$ overtone fit as a function of $\delta$, where now we modify each overtone frequency individually in turn (line colors). Bottom: A slice through the middle panel at $\delta = 0.05$. Alongside CCE:01 we plot the result from each of the CCE waveforms considered in this work.
• Figure 5: How we pick the noise level in this work. Given the $(2,2)$ mode of an NR waveform (here we show CCE:01, thin gray line), we perform a fit of the $(3,2,0)$ QNM over a stable window to reconstruct $h_{320} = A_{320}e^{i\phi_{320}}e^{-i\omega_{320}t}$ thin blue line). We then choose the noise level, $\sigma$ (gray shaded region) to be equal to $A_{320}$ (thick blue line) at the chosen $t_0$.