Excitation factors for horizonless compact objects: long-lived modes, echoes, and greybody factors

TL;DR

The paper analyzes quasinormal excitation factors (QNEFs) for horizonless ultracompact objects (ECOs and wormholes) to understand how long-lived cavity modes contribute to gravitational-wave signals. It shows that these long-lived modes have very small excitation factors, causing their imprint to appear predominantly at late times as echoes, while the prompt ringdown is governed by standard photon-sphere modes. Analytically and numerically, the authors derive scaling laws for QNEFs and demonstrate that high-frequency cavity modes drive the first echoes; they propose a practical ringdown waveform model combining BH QNMs and cavity modes, needing only a few modes per echo. They further establish the robustness of long-lived modes and greybody factors under small perturbations, arguing that ECO signatures are stable observables. This provides a unified framework linking excitation, stability, and echoes, with direct implications for gravitational-wave spectroscopy of ECOs.

Abstract

We present an analytical and numerical investigation of the quasinormal excitation factors of ultracompact horizonless objects. These systems possess long lived quasinormal modes with extremely small imaginary parts, originating from the effective cavity between the photon sphere and the object's interior. We show that the excitation of such modes is strongly suppressed, scaling with the imaginary part of their frequency, and therefore they contribute to the waveform only at very late times. This hierarchy naturally explains the structure of echo signals: the prompt ringdown is dominated by standard light ring modes, the early echoes arise from moderately damped cavity modes, and only the latest echoes are governed by long lived modes. Based on this, we propose a practical ringdown waveform model based on a superposition of ordinary black hole quasinormal modes and cavity modes, which captures the complexity of the ringdown of horizonless ultracompact objects. We further demonstrate that the combination of small excitation factors and weak damping enhances the robustness of long lived modes against localized perturbations, in contrast to the spectral instabilities affecting standard black hole quasinormal modes. Finally, we extend the analysis of greybody factors to exotic compact objects and wormholes, showing that they remain stable under small deformations of the effective potential and thus represent robust observables. Our results provide a unified framework for understanding excitation, stability, and echoes in ultracompact horizonless objects, with direct implications for their spectral properties and gravitational wave signatures.

Figures (7)

• Figure 1: QNMs and excitation factors for axial gravitational perturbations with $l=2$ in the wormhole scenario. Top left panel: imaginary part of the first four long-lived modes, as a function of the throat location (horizontal axis). Top right panel: absolute value of $dA^{\rm in}/d\omega$ for the corresponding modes, using the same color coding. $dA^{\rm in}/d\omega$ becomes very large for long-lived modes, indicating that these modes are extremely suppressed (see Eq. \ref{['excitation']}). Bottom left panel: the product ${\omega^I_n}\, dA^{\rm in}/d\omega$, which approaches unity when the throat location satisfies $\frac{(r_{\rm throat} - 2M )}{2M}\ll 1$. This confirms the analytical predictions of Eq. \ref{['eqproduct']}. Bottom right panel: $|B_n|$ for the considered modes, using the same color coding. The QNEF is smaller for smaller imaginary part of the mode.
• Figure 2: All panels refer to the $l=2$ mode of a gravitational perturbation for different ECO scenarios. Top panel: imaginary part of the fundamental long-lived quasinormal mode for gravitational perturbations as a function of the object's radius $r_0$, for three different ECO reflectivities: constant reflectivity with $R_{\rm ECO} = 0.9$, $R_{\rm ECO} = 0.5$, and a Boltzmann reflectivity profile. Middle panel: corresponding behavior of the absolute value of the $dA^{\rm in}/d\omega$ for the same modes and models. Again, for long-lived modes the derivative $\dv{A^{\rm in}_0}{\omega}$ is large, which implies that the corresponding modes are only weakly excited. Bottom panel: the corresponding values of the QNEFs $B_0$ for the cases under consideration.
• Figure 3: Fourier transform of the reflectivity for a wormhole with throat location $r_{\rm throat} = (2 + 10^{-6})M$. To reconstruct the time-domain signal, we employ Eq. \ref{['QNMdecompositionWORMHOLE']}, with $n^{\rm BH}_{\rm max} = 7$ for BH QNMs and $n^{\rm cavity}_{\rm max} = 12$ for cavity modes. Here the cavity length crresponds to $L \sim 54 M$. The resulting waveform accurately reproduces the oscillatory structure of the signal.
• Figure 4: Mode content and echo reconstruction for a wormhole with throat location at $r_{\rm throat}=(2+10^{-6})M$Left: representative portion of the cavity-mode spectrum. The longest-lived modes do not dominate the echo onset because their excitation factors are highly suppressed. Right: comparison between the numerical echo signal and QNM-based reconstructions with/without high-frequency modes (defined as those with ${\omega^R_n}>\sqrt{V_{\max}}$). Including high-frequency modes yields an accurate reconstruction of the first echoes; excluding them reproduces only very-late-time echoes.
• Figure 5: Reconstruction of a wormhole echo with throat location $r_{\rm throat}=2.00001M$. The inset in the upper left shows a representative set of QNMs. The numerical signal is shown as a black solid line, focusing on the second echoes in the interval $4L<t<6L$. The dashed blue line corresponds to a reconstruction with two modes, specifically the fourth and fifth ones, i.e. the first two modes above the threshold $\sqrt{V_{\rm max}}$, which already provide a good agreement. The dashed orange line shows the case with four modes, yielding an even better match. This demonstrates that only a small number of modes is sufficient to reproduce the echo.