Stability of the Fundamental Quasinormal Mode in Time-Domain Observations: The Elephant and the Flea Redux

Abstract

Black hole spectroscopy with gravitational waves is an important tool to measure the mass and spin of astrophysical black holes and to test their Kerr nature. Next-generation ground-and spacebased detectors will observe binary black hole mergers with large signal-to-noise ratios and perform spectroscopy routinely. It was recently shown that small perturbations due, e.g., to environmental effects (the "flea") to the effective potential governing gravitational-wave generation and propagation in black hole exteriors (the "elephant") can lead to arbitrarily large changes in the black hole's quasinormal spectrum, including the fundamental mode, which is expected to dominate the observed signal. This raises an important question: is the black hole spectroscopy program robust against perturbations? We clarify the physical behavior of time-domain signals under small perturbations in the potential, and we show that changes in the amplitude of the fundamental mode in the prompt ringdown signal are parametrically small. This implies that the fundamental quasinormal mode extracted from the observable time-domain signal is stable against small perturbations. The stability of overtones deserves further investigation.

Quasinormal modes of Schwarzschild black holes on the real axis

“…A similar analysis has recently been made in Ref. [21] for slightly different perturbations, and our results seem consistent with theirs qualitatively. This analysis motivates us to search for the signature of unperturbed quasinormal modes, particularly the fundamental mode, from real-frequency solutions of the scattering problem.…”

“…It is obvious that the ringdown signals, i.e., the time evolution of the field characterized by exponential damping, are essentially unchanged under perturbations until the echo signal arrives after ∼ 2b = 200M from the direct signal. These features agree with those found in the analysis of double rectangular barriers [19] and the Regge-Wheeler potential perturbed by a Gaussian bump [21].…”

“…Thus, the averaging over the frequency is naturally introduced when we focus on a short time scale of 2b (see also Refs. [19][20][21] for relevant discussions).…”

“…However, the spectral instability of quasinormal-mode spectra does not destabilize observable ringdown signals, particularly on the damping time scale of unperturbed quasinormal modes [19][20][21]. Rather, a lot of evidence is accumulating that the ringdown signal is fairly universal even for nonlinear perturbations.…”

Collective filters: a new approach to analyze the gravitational-wave ringdown of binary black-hole mergers

“…Cardoso et al [43] drew a similar conclusion while considering nearhorizon perturbations. In fact, as pointed out by Hui et al [168] and Berti et al [166], despite the nonlocality of QNMs, one needs to appreciate the causal structure of the system while considering the time-domain signals -a time response reflects the nature of each single potential bump that is causally connected to the observer, e.g., the prompt ringdown of a regular BH. The QNMs of the perturbed system do not show up until very late times as "echoes" [169] when the initial Cauchy data travels and experiences the entire potential 8 .…”