Gravitational Wave Echoes Overview

• Gravitational wave echoes are late-time signals generated by partial trapping of radiation between an angular-momentum barrier and a reflective inner structure.
• The echo delay times and spectral features depend strongly on the object's compactness, spin, and near-horizon modifications.
• Advanced detection methods, including time-domain reconstruction and resonance comb searches, analyze these model-dependent signals.

Gravitational wave echoes (GWEs) are hypothesized late-time signals that follow the prompt post-merger or ringdown emission when gravitational radiation is partially trapped between an exterior angular-momentum barrier and an inner reflection structure. In the compact-star context, the relevant outer structure is the photon sphere at $R=3M$, while the inner boundary can be the stellar surface if the remnant is an ultracompact star; in black-hole scenarios, many models instead postulate a reflective surface or filter close to the would-be horizon. Across the literature, GWEs are treated as probes of ultra-compact stellar structure, near-horizon quantum modifications, and theories beyond general relativity, but their delay times, spectral content, amplitudes, and observability are strongly model dependent (Mannarelli et al., 2018, Bora et al., 2022, Cardoso et al., 2019).

1. Geometric conditions and characteristic timescales

A common geometric criterion for echo production is the existence of a photon sphere. For compact stars, this requires the radius to satisfy $R<3M$, equivalently a compactness $C=M/R>1/3$. A second upper bound is set by Buchdahl’s limit, $R_B=9M/4$, so the compactness range usually quoted for stellar GWE production is

$\frac{1}{3}<\frac{M}{R}<\frac{4}{9},$

or $0.33Bora et al., 2022). In this regime, gravitational waves can be partially trapped between the photon sphere and the stellar surface. If the post-merger remnant is instead a black hole, the standard argument is that an additional internal reflection surface associated with quantum effects near the horizon is needed, otherwise the waves are captured (Mannarelli et al., 2018).

The echo timescale is typically identified with a light-crossing or round-trip time through the cavity. In ultracompact-star calculations, one form used for the echo time is

$$\tau_{\mathrm{echo}}=\int_0^{3M}\frac{dr}{\sqrt{e^{2\Phi(r)}\left(1-\frac{2m(r)}{r}\right)}},$$

with the corresponding echo frequency approximated as

$$\omega_{\mathrm{echo}}=\frac{\pi}{\tau_{\mathrm{echo}}},$$

while some literature instead quotes a repetition rate $1/(2\tau_{\mathrm{echo}})$ (Mannarelli et al., 2018). In searches motivated by near-horizon structure, the time delay between successive pulses is treated as nearly constant, and its inverse sets the average spacing of frequency-domain resonances, $\Delta f \approx 1/t_d$ (Conklin et al., 2017, Ren et al., 2021).

For spinning remnants, one explicit relation employed in echo phenomenology is

$R<3M$0

where $R<3M$1 and $R<3M$2 measures the distance of the inner boundary from the horizon (Conklin et al., 2017). This relation is used to connect an observed delay to remnant mass, spin, and the scale of near-horizon deviation.

2. Ultracompact stars and strange-star realizations

A major branch of the GWE literature asks whether horizonless compact stars can be sufficiently compact to produce echoes without invoking any exotic reflective surface beyond the stellar surface itself. In the strange-star context, one study adopted maximally stiff quark-matter equations of state at the causal limit $R<3M$3, including the MIT bag-model form

$R<3M$4

and found that strange stars can cross the photon-sphere line only marginally and do not approach the Buchdahl limit. For bag constants $R<3M$5 and $R<3M$6, the predicted echo frequencies were $R<3M$7 kHz and $R<3M$8 kHz, respectively, which are not compatible with a claimed $R<3M$9 Hz signal after GW170817 (Mannarelli et al., 2018).

A later comparative study examined MIT bag, linear, and polytropic equations of state for static, spherically symmetric, non-rotating, cold strange stars by integrating the Tolman-Oppenheimer-Volkoff equations. It reported that the MIT bag model and the linear equation of state can emit GWEs in the range of tens of kilohertz, whereas the polytropic equation of state does not reach sufficient compactness to develop a photon sphere and therefore does not emit GWEs. The reported examples were $C=M/R>1/3$0 kHz to $C=M/R>1/3$1 kHz for MIT bag-model choices $C=M/R>1/3$2, and $C=M/R>1/3$3 kHz to $C=M/R>1/3$4 kHz for linear-model choices $C=M/R>1/3$5; the study also emphasized that the GWE frequency increases with bag constant $C=M/R>1/3$6 and decreases with linear constant $C=M/R>1/3$7, exhibiting a model-dependent nature (Bora et al., 2022).

The same overall conclusion persists when additional physics is introduced. With a nonvanishing cosmological constant, strange-star models based on MIT bag and linear equations of state still produce echoes in the tens-of-kHz regime, while increasing $C=M/R>1/3$8 increases the stellar radius, decreases the compactness, increases the echo time, and decreases the echo frequency. The study identified an effective range

$C=M/R>1/3$9

and reported echo-frequency shifts such as $R_B=9M/4$0 for the MIT bag model and $R_B=9M/4$1 for the linear equation of state as $R_B=9M/4$2 increases over the sampled range (Bora et al., 2021). In $R_B=9M/4$3 gravity, compact stars modeled with the MIT bag model and color-flavor-locked phase equations of state were likewise found capable of producing GWEs, but in a lower high-frequency band of $R_B=9M/4$4–$R_B=9M/4$5 kHz; surface redshift and adiabatic-index analysis were used there to confirm stability (Sinha et al., 5 Aug 2025).

Taken together, these results sharply constrain a stellar interpretation of low-frequency echo candidates. Standard nuclear equations of state do not cross the photon-sphere line, and even maximally stiff strange-star models predict kHz rather than Hz echoes (Mannarelli et al., 2018).

3. Black-hole horizon modifications and exotic compact objects

In black-hole-centered echo scenarios, the central assumption is that the horizon is not a perfect absorber. One realization comes from the Bekenstein-Mukhanov proposal of black-hole area quantization, in which the horizon absorbs only at discrete transition frequencies,

$R_B=9M/4$6

At other frequencies, the near-horizon region acts as a partially reflective, frequency-selective surface. The initial ringdown remains essentially the classical one determined by the photon sphere, while the late-time response consists of distorted echoes whose spectra inherit absorption notches from the quantized horizon filter (Cardoso et al., 2019).

Another realization appears in ghost-free massive gravity. There, black holes carry scalar hair associated with Stückelberg fields, and a coupling of gravitational perturbations to that background can generate a second peak in the effective potential outside the horizon. The total potential takes the schematic form

$R_B=9M/4$7

and the resulting double-peak structure traps perturbations between two barriers, producing delayed echoes after the main quasinormal ringing (Dong et al., 2020). This mechanism is structurally different from the reflective-surface picture because the cavity is produced by a modified exterior potential rather than solely by a boundary condition at the horizon.

For exotic compact objects (ECOs), the standard boundary model places a partially reflective surface at

$R_B=9M/4$8

An infalling perturbation is then trapped between the angular-momentum barrier and this surface, and the first-echo amplitude depends strongly on the reflectivity and on the progenitor binary parameters. A perturbative treatment using a physically motivated Boltzmann reflectivity found that binaries with comparable masses have a stronger first echo, and that a GW150914-like event would require a ringdown signal-to-noise ratio in the range $R_B=9M/4$9–$\frac{1}{3}<\frac{M}{R}<\frac{4}{9},$0 for first-echo detection (Micchi et al., 2020).

These black-hole and ECO models share the cavity picture but differ substantially in how the inner reflection is realized: selective absorption, an explicit reflective wall, or an additional potential barrier.

4. Echoes from alternative gravity, lensing, and environmental structure

Not all proposed GWEs are generated by near-horizon reflection alone. In theories beyond general relativity, gravitational-wave propagation eigenstates can differ from the metric polarizations $\frac{1}{3}<\frac{M}{R}<\frac{4}{9},$1 and $\frac{1}{3}<\frac{M}{R}<\frac{4}{9},$2, and kinetic mixing with additional degrees of freedom can produce birefringent propagation around lenses. In that framework, echoes arise when the accumulated delay between propagation eigenstates exceeds the duration of the gravitational-wave signal; shorter delays instead produce waveform scrambling. The formalism is based on identifying the dynamical propagation eigenstates in a short-wave expansion and computing both the speed difference and the geometric time delay along lensed paths (Ezquiaga et al., 2020).

Environmental modifications can also create echo-supporting potentials. A study of black holes embedded in an Einasto dark-matter halo for M87 introduced a geometric parameter $\frac{1}{3}<\frac{M}{R}<\frac{4}{9},$3 such that $\frac{1}{3}<\frac{M}{R}<\frac{4}{9},$4 corresponds to a regular black hole and $\frac{1}{3}<\frac{M}{R}<\frac{4}{9},$5 to a wormhole geometry. Under axial perturbations, the effective potential becomes a double barrier when $\frac{1}{3}<\frac{M}{R}<\frac{4}{9},$6, and a series of gravitational-wave echoes appears after the quasinormal-mode phase. The same study reported that the Einasto shape parameter $\frac{1}{3}<\frac{M}{R}<\frac{4}{9},$7 affects both the quasinormal modes and the echoes, and argued for an upper limit $\frac{1}{3}<\frac{M}{R}<\frac{4}{9},$8 after finding zero difference between frequencies computed for $\frac{1}{3}<\frac{M}{R}<\frac{4}{9},$9 and $0.33Liu et al., 2023).

These alternative mechanisms broaden the scope of the subject. In some cases the operative structure is a near-horizon cavity; in others it is a propagation-induced splitting or a double-barrier exterior geometry. This suggests that the term “echo” covers several physically distinct late-time phenomena rather than a single universal waveform class.

5. Waveform morphology, resonance structure, and spectral evolution

Early echo searches often emphasized a simple picture: semi-periodic pulses in the time domain and a comb-like pattern of nearly evenly spaced narrow resonances in the frequency domain. A study of several LIGO/Virgo events explicitly developed time-domain and frequency-domain windowing methods around this expectation and treated the resonance spacing as inversely related to the echo delay (Conklin et al., 2017). That picture remains useful, but later work has shown that it is not exhaustive.

When the near-horizon region is modeled as a multiple-barrier filter rather than a single reflector, the late-time waveform can exhibit echo mixing and superpositions. In this case, the $0.33Li et al., 2019).

A further revision arises in the low-finesse limit. Time-domain simulations of weakly reflective barriers show that early-time echoes behave as transient scattered wave packets rather than cavity eigenstates, and that a central frequency progressively redshifts because high-frequency components dissipate faster than the fundamental mode. That work identified spectral drift as a characteristic feature and proposed a critical reflectivity threshold of approximately $0.33Hu et al., 11 Dec 2025).

On the theoretical side, the Fredholm approach reformulates the perturbation problem as an integral equation and expresses the solution in terms of Fredholm determinants. By splitting the kernel into a baseline part and a reflectivity-dependent part, it constructs an expansion

$0.33

with $0.33Huang et al., 2019).

The collective implication is that GWE morphology can range from nearly uniform pulse trains to mixed, overlapping, and spectrally drifting transients. Search strategies based on a single decay law or a rigid comb therefore probe only part of the modeled signal space.

6. Detection and characterization methodologies

A central methodological challenge is that detailed echo waveforms are uncertain. One response is morphology-independent reconstruction. Extending the BayesWave framework, one study modeled echoes as sums of generalized wavelets, each wavelet being a comb of sine-Gaussians,

$0.33

with parameters for interval $0.33$$\tau_{\mathrm{echo}}=\int_0^{3M}\frac{dr}{\sqrt{e^{2\Phi(r)}\left(1-\frac{2m(r)}{r}\right)}},$$0. Using Bayesian evidence and coherent-vs-incoherent model comparison, that method found both stylized sine-Gaussian trains and an echoing extreme-mass-ratio-inspiral signal detectable for combined signal-to-noise ratios $$\tau_{\mathrm{echo}}=\int_0^{3M}\frac{dr}{\sqrt{e^{2\Phi(r)}\left(1-\frac{2m(r)}{r}\right)}},$$1, and it recovered $$\tau_{\mathrm{echo}}=\int_0^{3M}\frac{dr}{\sqrt{e^{2\Phi(r)}\left(1-\frac{2m(r)}{r}\right)}},$$2, $$\tau_{\mathrm{echo}}=\int_0^{3M}\frac{dr}{\sqrt{e^{2\Phi(r)}\left(1-\frac{2m(r)}{r}\right)}},$$3, and $$\tau_{\mathrm{echo}}=\int_0^{3M}\frac{dr}{\sqrt{e^{2\Phi(r)}\left(1-\frac{2m(r)}{r}\right)}},$$4 from posterior distributions (Tsang et al., 2018).

A complementary route targets frequency-domain resonance structure directly. The “search with combs” framework uses a uniform comb model with a phase-marginalized likelihood, treating the echo signature as a set of narrow, quasiperiodic resonances whose spacing is $$\tau_{\mathrm{echo}}=\int_0^{3M}\frac{dr}{\sqrt{e^{2\Phi(r)}\left(1-\frac{2m(r)}{r}\right)}},$$5. The algorithm was validated with signal injections in Advanced LIGO noise and applied to GW150914 and GW151012, for which it found no clear evidence of a comblike structure (Ren et al., 2021).

Windowing methods provide a third line of attack. Searches using Hann or square windows in time and combs in frequency have been proposed to isolate generic echo structure while minimizing dependence on any single phenomenological template. In that framework, cross-correlation between detectors is used to suppress noise triggers, and the frequency-domain implementation was reported as the most successful among the three windowing strategies considered (Conklin et al., 2017).

These methods reflect a broader division in the field: some analyses emphasize model-agnostic robustness, while others exploit the resonance structure of specific cavity pictures. The choice is consequential because later theoretical work questions whether stationary resonance combs are generic in low-reflectivity scenarios (Hu et al., 11 Dec 2025).

7. Observational claims, constraints, and controversies

The observational literature contains both positive claims and null results. Several studies reported candidate echoes in LIGO/Virgo data, including a claimed $$\tau_{\mathrm{echo}}=\int_0^{3M}\frac{dr}{\sqrt{e^{2\Phi(r)}\left(1-\frac{2m(r)}{r}\right)}},$$6 feature at about $$\tau_{\mathrm{echo}}=\int_0^{3M}\frac{dr}{\sqrt{e^{2\Phi(r)}\left(1-\frac{2m(r)}{r}\right)}},$$7 Hz in GW170817 and signals with p-values of order $$\tau_{\mathrm{echo}}=\int_0^{3M}\frac{dr}{\sqrt{e^{2\Phi(r)}\left(1-\frac{2m(r)}{r}\right)}},$$8 or significantly less in GW151226, GW170104, GW170608, GW170814, and GW170817 (Mannarelli et al., 2018, Conklin et al., 2017). A status review argued that the various searches can be mutually consistent if echoes are most prominent at lower frequencies and/or in binary mergers of more extreme mass ratio, and noted that the only reported $$\tau_{\mathrm{echo}}=\int_0^{3M}\frac{dr}{\sqrt{e^{2\Phi(r)}\left(1-\frac{2m(r)}{r}\right)}},$$9 detection, at $$\omega_{\mathrm{echo}}=\frac{\pi}{\tau_{\mathrm{echo}}},$$0 s after GW170817, is coincident with the inferred formation time of the black hole from electromagnetic observations (Abedi et al., 2020).

At the same time, dedicated analyses have produced null results. The Bayesian comb search found no clear evidence for a comblike structure in GW150914 and GW151012 (Ren et al., 2021). More generally, the existence of an echo depends not only on data analysis but also on whether the underlying object can survive the backreaction of the incoming radiation.

Two papers formulate the backreaction objection sharply. Using ingoing Vaidya spacetime, one study showed that energy carried by incident gravitational waves can cause the event horizon to form out of a static ECO, leaving no echo signals toward spatial infinity. Avoiding collapse imposes a lower bound on the distance $$\omega_{\mathrm{echo}}=\frac{\pi}{\tau_{\mathrm{echo}}},$$1 of the surface above $$\omega_{\mathrm{echo}}=\frac{\pi}{\tau_{\mathrm{echo}}},$$2, but increasing $$\omega_{\mathrm{echo}}=\frac{\pi}{\tau_{\mathrm{echo}}},$$3 makes the echoes less distinct; the paper therefore emphasized a trade-off between detectability and distinguishability, together with fine tuning for LIGO-scale observations (Chen et al., 2019). A separate analysis argued that, if causality is maintained to leading order in gently curved spacetime, the reflected wave is trapped by a new closed trapped surface produced by its own backreaction, so no detectable echo can emerge to infinity; within that framework, an actual detection would imply a profound change in the understanding of physics (Guo et al., 2022).

Current detectability estimates also remain restrictive. For ECO models with a physically motivated reflectivity, first-echo detection in a GW150914-like event requires ringdown SNR in the range $$\omega_{\mathrm{echo}}=\frac{\pi}{\tau_{\mathrm{echo}}},$$4–$$\omega_{\mathrm{echo}}=\frac{\pi}{\tau_{\mathrm{echo}}},$$5, although the same study estimated that in an optimistic scenario one or two events per year might reach the required SNR during O4 (Micchi et al., 2020). This leaves the field in a deliberately unresolved state: claimed candidates exist, null searches exist, and theoretical arguments dispute whether clean echoes should escape at all.

The resulting picture is not one of a settled observable, but of a technically rich diagnostic whose meaning depends on compactness bounds, reflectivity, backreaction, potential structure, and the assumptions built into the search. Within the literature surveyed here, GWEs remain a proposed probe of ultra-compact objects and near-horizon physics, with their interpretation constrained as much by dynamical consistency as by signal-processing methodology.