Gravitational-Wave Echo Time

• Gravitational-wave echo time is the interval between successive echoes after a compact-object merger, defined by the round-trip travel time across the effective potential cavity.
• It provides detailed insights into remnant mass, spin, and interior structure, with measurements obtained via frequency-domain comb searches and time-domain waveform analyses.
• The echo interval varies across models—from nearly constant in horizonless exotic objects to dynamically evolving in wormhole scenarios—offering a probe into near-horizon quantum effects.

Gravitational-wave echo time is a characteristic timescale associated with the delayed, repeating bursts of gravitational radiation—“echoes”—following the primary ringdown of a compact-object merger. Echoes are predicted in scenarios where the post-merger remnant contains a reflecting surface or altered near-horizon structure, as in horizonless exotic compact objects (ECOs), physical black holes just prior to horizon formation, or certain ultra-compact stars. The echo time is determined primarily by the round-trip travel time of gravitational-wave perturbations in the effective potential cavity bounded externally by the photon-sphere (angular-momentum barrier) and internally by the surface or interior structure. This timescale encodes detailed information about the near-horizon physics, mass, spin, and internal composition of the remnant, and is directly measurable in gravitational-wave data as the interval between successive, quasi-periodic post-merger bursts.

1. Theoretical Origin of Echo Time

The canonical mechanism for gravitational-wave echo generation involves a multi-bounce process in a cavity formed by the exterior curvature potential barrier and a (possibly quantum or model-dependent) reflective inner boundary. In Schwarzschild geometry, the echo delay is governed by twice the separation in tortoise coordinates between the photon sphere and the reflecting surface: $$\Delta t_{\mathrm{echo}} = 2\bigl[ r_*(r_{\text{ph}}) - r_*(r_0) \bigr].$$ Here, $r_*(r) = r + 2M\ln(r/2M - 1)$ is the Schwarzschild tortoise coordinate, $r_{\text{ph}} \approx 3M$ denotes the photon-sphere (light ring), and $r_0 = 2M + \epsilon$ locates the surface or wall, with $\epsilon \ll M$ for Planckian corrections (Price et al., 2017, Cao et al., 13 Oct 2025, Conklin et al., 2017, Ma et al., 2022).

For nearly horizon-sitting surfaces, the echo delay is dominated by a logarithmic term: $$\Delta t_{\mathrm{echo}} \simeq 4M\ln\left( \epsilon^{-1} \right),$$ where $\epsilon = (r_0 - 2M)/M$. In rotating (Kerr) backgrounds, the relevant expression generalizes to

$$\Delta t_{\mathrm{echo}} \approx \frac{4M r_+}{r_+ - r_-} \ln\left( \epsilon^{-1} \right),$$

with $r_{\pm} = M \pm \sqrt{M^2 - a^2}$ and $a$ the dimensionless spin parameter (Xin et al., 2021, Conklin et al., 2017). For physical or dynamically evolving surfaces, the timescale may increase gradually as the surface approaches the horizon (Cao et al., 13 Oct 2025, Wang et al., 2018).

2. Frequency–Time Duality and Spectral Resonances

A defining feature of echo phenomenology is the duality between the time-domain echo interval and the frequency-domain resonance comb. Any sequence of equally spaced echoes separated by $\Delta t$ yields, under Fourier transformation, a comb of narrow resonances or poles in the spectrum with frequency spacing

$\Delta f = \frac{1}{\Delta t}.$

Conversely, the Green’s function formalism for the transfer function $K(\omega)$ of the system reveals poles at

$$\omega_n \approx \omega_0 + \frac{n\pi}{\Delta x},\quad \Delta x = r_{\text{ph}}^* - r_0^*,$$

with spacing $\Delta\omega = \pi/\Delta x$. Thus,

$$T_{\mathrm{echo}} = 2\Delta x = \frac{1}{\Delta f}.$$

Source-integral modulations, two-component resonance structures, and initial condition effects generally only alter the amplitude envelope and relative heights of the comb but not the spacing—the echo time remains robust to these details (Conklin et al., 2019, Ren et al., 2021, Conklin et al., 2017).

3. Model Dependence: Exotic Compact Objects, Physical Black Holes, Wormholes, and Strange Stars

The functional form of the echo time depends sensitively on the internal structure of the remnant:

• Horizonless ECOs, quantum-corrected black holes: The echo interval is set by the near-horizon cavity as above, scaling logarithmically with the distance to the would-be horizon. Pinpointing the echo delay can constrain the location of the reflecting surface to Planckian precision (Cao et al., 13 Oct 2025, Conklin et al., 2017, Ma et al., 2022).
• Wormholes or Dynamical Surfaces: If the post-merger throat pinches off in time, the cavity length grows and the echo intervals are non-stationary, increasing with each successive echo. For example,

$\Delta t_n = 2\left[ L_0 + n\,\Delta L \right],$

where $L_0$ is the initial cavity length and $\Delta L$ encodes its change per cycle (Wang et al., 2018).

• Ultracompact Stars (strange stars, gravastars): For ultracompact matter configurations, the echo time is governed by the sum of interior and exterior light-crossing times between center and photon-sphere,

$$\Delta t_{\mathrm{echo}} = 2\left[ \int_0^R \frac{dr}{\sqrt{e^{2\Phi(r)}(1-2m(r)/r)}} + \int_R^{R_{\text{ph}}} \frac{dr}{\sqrt{1-2M/r}} \right],$$

with the dominant contribution set by the interior compactness (Mannarelli et al., 2018). For maximally stiff strange stars, echo intervals are extremely short ($\sim 10^{-4}$ s), setting frequencies far above most LIGO/Virgo bandwidths.

• Braneworld and higher-dimensional scenarios: Echo times depend on the geometric size of the extra-dimensional cavity and source–detector separation. For brane cavity width $d$ and source–observer distance $l$,

$\Delta t \simeq \frac{2\sqrt{(l/2)^2 + d^2}-l}{c}$

and are further constrained by detector operation windows (Zhu et al., 2024).

• Rogue-echo regime (objects with slow interior crossing times): For ECOs whose interiors possess extreme time dilation or slow-crossing regions, the echo lag can scale as $\Delta t \sim M\,\epsilon^{-1/2}$ and reach years to Gyr, rendering echoes essentially uncorrelated with the original merger (“rogue echoes”) (Zimmerman et al., 2023).

4. Empirical Measurement and Search Methodologies

Gravitational-wave echo times are extracted from data in both time and frequency domains:

• Frequency-domain comb search: The most robust technique leverages the resonance comb generated by the quasi-periodic echo train. Peaks in the amplitude spectrum $|\tilde h(f)|$ are identified, and their regular spacing $\Delta f$ measures the echo interval via $T_{\mathrm{echo}} = 1/\Delta f$ (Conklin et al., 2019, Ren et al., 2021, Conklin et al., 2017).
• Time-domain algorithms: Morphology-independent approaches decompose strain data into generalized wavelet sums manifesting as trains of time-delayed pulses. Bayesian model selection is employed to discriminate echoes from noise/glitches, and the marginal posterior $p(\Delta t|\text{data})$ is extracted (Tsang et al., 2018).
• Windowing/comb filtering: Methods based on time–frequency domain windowing, using “comb” templates or synchronized time windows, enhance coherence and visibility of the expected periodic structure (Conklin et al., 2017).

Precision on $\Delta t$ is routinely—under optimal SNR conditions—at the level of $\lesssim 0.1\%$, substantially exceeding that of direct time-domain template fitting (Ren et al., 2021, Tsang et al., 2018).

5. Astrophysical Parameter Dependence and Illustrative Ranges

The echo time is controlled parametrically by the remnant mass $M$, spin $a$, and the location or nature of the inner boundary:

• For standard mass and Planck-scale surface correction $\epsilon \sim 10^{-40}$, a stellar-mass black hole ($M \sim 30\,M_\odot$) yields

$$\Delta t_{\mathrm{echo}} \sim 10\text{–}100\,\text{ms}$$

which falls naturally into LIGO/Virgo frequency bands (Price et al., 2017, Cao et al., 13 Oct 2025, Ma et al., 2022).

• For more moderate corrections ($\epsilon \sim 10^{-10}$), delays decrease to $\sim 10\,\text{ms}$.
• For neutron stars or strange stars near causality bounds, delays are $\sim 10^{-4}\,\text{s}$, outside most detector sensitivity (Mannarelli et al., 2018).
• For certain ECOs or “rogue” interiors, delays can be $\gg 1\,\text{s}$ and may exceed observational windows entirely (Zimmerman et al., 2023).

6. Modifications: Nonstandard Cavity Structure and Dynamical Effects

Several scenarios introduce significant deviations from the canonical echo time:

• Dynamical boundary movement: Gradually shrinking cavities (e.g., pinching wormholes) cause the echo interval to grow monotonically with echo index, degrading constant-interval template search efficiency and demanding adaptive algorithms (Wang et al., 2018).
• Nontrivial interior structure: In objects where interior time-dilation is extreme, delays can be so large that no echo is expected within practical observational timescales (Zimmerman et al., 2023).
• Spin- or frequency-dependent boundary conditions: Phase or amplitude shifts arising from frequency-dependent reflectivity alter the relative timing by $O(M)$ per echo beyond the geometric delay but do not affect leading-order interval (Price et al., 2017).
• Braneworld and multidimensional models: Echo time depends explicitly on extra-dimensional geometry and can be used to constrain model parameters if echoes are found (Zhu et al., 2024).

7. Observational Status and Astrophysical Applications

Measured echo delays in post-merger data from several LIGO/Virgo events (e.g., GW151226, GW170104, GW170608, GW170814, GW170817) are consistent with theoretical predictions for Planck-scale wall placements and inferred remnant masses and spins, with model-experiment agreement within uncertainties (Conklin et al., 2017).

Absence of echoes within analyzed time windows constrains the possible location of inner surfaces (via upper bounds on $\Delta t_{\mathrm{echo}}$) and, in certain models, can be translated into lower limits on the proximity of the remnant to the classical horizon.

Specialized cases, such as echoes from SMBH scattering in galactic nuclei, predict typical delays ranging from seconds to thousands of seconds depending on the source–SMBH separation, and are subject to amplitude selection and geometric lensing enhancement (Gondán et al., 2021).

Ongoing improvements in search sensitivity, network coverage, and advanced windowing/statistical techniques are expected to further constrain or potentially detect echoes, providing a unique probe of strong-field gravity and quantum corrections at the horizon scale.

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References:

(Conklin et al., 2019, Price et al., 2017, Ren et al., 2021, Tsang et al., 2018, Cao et al., 13 Oct 2025, Xin et al., 2021, Conklin et al., 2017, Mannarelli et al., 2018, Wang et al., 2018, Ma et al., 2022, Zhu et al., 2024, Zimmerman et al., 2023, Gondán et al., 2021)