Spectral densification and macroscopic phase delay of gravitational echoes from exotic compact objects

Abstract

Gravitational-wave echoes from Exotic Compact Objects (ECOs) provide an observable probe for horizon-scale physics. Standard phenomenological models for these signals typically assume a constant Free Spectral Range, relying on the geometric optics approximation. In this work, we demonstrate that wave dispersion at the photon sphere induces a systematic deviation from this assumption, manifesting instead as a hyperbolic spectral densification. By employing an analytical framework based on the Riccati equation and macroscopic impedance mapping, we extract the spectrum of these high-finesse resonances without semi-classical approximations. We characterize the structural transition from the eikonal geometric asymptote ($\ell \gg 1$) down to the wave-tunneling dominated quadrupolar mode ($\ell=2$). In this wave-dominated regime ($\ell \in [2, 10]$), the macroscopic deviation from the semi-classical limit is governed by a phenomenological $\mathcal{L}^{-3/2}$ inverse power law. Finally, we show that this macroscopic densification isolates the structural dispersion of the external spacetime, decoupled from the boundary microphysics, provided the membrane phase shift is frequency-independent.