Quasinormal modes and their excitation beyond general relativity. II: isospectrality loss in gravitational waveforms

TL;DR

The paper examines how isospectrality breaking between polar and axial quasinormal modes, induced by a cubic-in-curvature EFT, manifests in the time-domain ringdown of a nonrotating black hole. By evolving the EFT-modified master equations and mapping the gauge-invariant perturbations to gravitational-wave polarizations, it quantifies the difficulty of extracting both parity-mode frequencies from a single-time-domain signal. The results show that parity mixing and mode doubling generally hinder model-agnostic fits from identifying both fundamental EFT QNMs, but EFT-informed fits can reveal deviations (notably when the ringdown is polar-dominated). These findings have implications for black-hole spectroscopy in beyond-GR theories and guide future work on rotating black holes and self-consistent wave generation.

Abstract

We continue our series of papers where we study the quasinormal modes, and their excitation, of black holes in the simplest beyond general relativity model in which first-principle calculations are tractable: a nonrotating black hole in an effective-field-theory extension of general relativity with cubic-in-curvature terms. In this theory, the equivalence between the quasinormal mode spectra associated with metric perturbations of polar and axial parities ("isospectrality") of the Schwarzschild black hole in general relativity no longer holds. How does this loss of isospectrality translate into the time-domain ringdown of gravitational waves? Given such a ringdown, can we identify the two "fundamental quasinormal modes" associated to the two metric-perturbation parities? We study these questions through a large suite of time-domain numerical simulations, by a prescription on how to relate the gauge-invariant master functions that describe metric perturbations of each parity with the gravitational polarizations. Under the assumptions made in our calculations, we find that it is in general difficult to identify either of the two fundamental modes from the time series, although finding evidence for a non-general-relativistic mode is possible sometimes. We discuss our results in light of our assumptions and speculate about what may occur when they are relaxed.

Figures (25)

• Figure 1: Illustration of isospectrality breaking. Two degenerate quasinormal frequencies (black circle) belonging to metric perturbations of opposite parities move apart in the complex frequency plane when the parameter $\varepsilon$ is tuned from zero ("general relativity") to a nonzero value. Only one pair of frequencies is shown in the figure, however this split also occurs, not necessarily in the same way, for all other frequencies. In our problem, $\varepsilon$ is related to the lengthscale $l$ associated with higher-curvature terms in the EFT. Understanding how isospectrality breaking impacts the black-hole ringdown is the main goal of this paper.
• Figure 2: Deviation from unity of the propagation speed squared of linear metric perturbations. We vary the parameter $\varepsilon = \lambda \, l^4/M^4$ from zero (general relativity) to $0.05$ in increments of $0.01$. The maximum deviation from the speed of light happens at $r = 6 r_\textrm{h} / 5$, near the hole's horizon $r_\textrm{h}$, and vanishes at the horizon and at spatial infinity.
• Figure 3: The effective potentials $V_{2}^{(\pm)}$ for perturbations of polar (left panel) and axial (right panel) parity. We vary the parameter $\varepsilon = \lambda \, l^4/M^4$ from zero (general relativity) to $0.05$ in increments of $0.01$. Deviations from general relativity are mostly bound to the region between the event horizon, pushed to $r_{\ast} \to -\infty$, and the location of the potential peak.
• Figure 4: Quadrupolar axial- and polar-parity waveforms in general relativity obtained by evolving the same initial data. The inset zooms near the peak of the waveforms and shows that they are offset by approximately $M$.
• Figure 5: Snapshots of the incident perturbation (solid line and rescaled by a factor of $5$) as it propagates into the region of variable propagation speed. The filled gray curve represents $1 - c_\textrm{s}^2$, for $\varepsilon = 0.05$, as also shown in Fig. \ref{['fig:speed']}. As the left-moving perturbation reaches the region in which $c_\textrm{s}^2$ deviates from one, part of it is reflected, as shown in the top panels. This reflected pulse propagates rightward (see the bottom panels) and is measured by an observer at $r_{\ast}^{\rm ext}$ as the waveforms in Fig. \ref{['fig:rwz_no_potential']}.