Low finesse scattering and spectral drift of gravitational wave echoes

Abstract

Gravitational wave echoes serve as probes for quantum horizon corrections. While steady-state resonances are assumed in the search of gravitational wave echos, realistic barriers are expected to possess intrinsically low reflectivity. In this work, we investigate this low-finesse limit via time-domain simulations and demonstrate that early-time echoes behave as transient scattered wave packets rather than cavity eigenstates. A central finding is the identification of spectral drift, where the central frequency progressively redshifts. This evolution occurs because high-frequency components dissipate significantly faster than the fundamental mode due to the filtering effect of the potential barrier. To distinguish transient interference from genuine resonance, we establish a physical criterion based on cavity lifetime, identifying a critical reflectivity threshold of approximately $0.37$. Since theoretical models typically operate deep within the overdamped regime below this limit, the resulting signals are spectrally non-stationary. We propose that detection strategies should shift towards dynamic time-frequency tracking to capture these drifting signatures.

Figures (6)

• Figure 1: Time-domain waveform and its Fourier transform
• Figure 2: Spectral stability analysis of the first GW echo as a function of cavity length $L \in [110, 210]$. The frequency shifts $\Delta \omega$ are measured relative to the fundamental Schwarzschild QNM. The effective central frequency (red squares) is defined as the amplitude-weighted mean of the real parts of the eigenfrequencies extracted via MPM, $\omega_{\rm c} = \sum_{j} |A_j| \Re(\omega_j) / \sum_{j} |A_j|$, where $A_j$ and $\omega_j$ are the amplitude and complex frequency of the $j$-th component.
• Figure 3: Temporal and spectral evolution of the first three echoes in the enhanced perturbation model, $\epsilon = 4 \times 10^{-4}$.
• Figure 4: Global Fourier spectrum of the full time-domain waveform for the enhanced perturbation model with amplitude $\epsilon = 4 \times 10^{-4}$ and cavity length $L=180$. The spectrum (red solid line) exhibits a distinct periodic "sawtooth" modulation. The vertical blue dashed lines indicate the theoretical cavity eigenfrequencies calculated via the shooting method, nearly demonstrating a precise alignment with the numerical peaks. The grey dot marks the dominant spectral peak located at $\Re(\omega) \simeq 0.35989$.
• Figure 5: The hallmark of genuine resonance in the high-finesse regime. Here we take cavity length $L=180$, barrier height $V=10$, and width $w=3$.