Late-time echoes from planar wave-packet scattering by an ultracompact star

We simulate the propagation and scattering of an incident plane scalar wave in the spacetime of an ultracompact, horizonless constant-density star using a time-domain multipole mode-sum method. By reconstructing the coherently superposed time-domain waveforms, we compare the star's dynamical response with that of a Schwarzschild black hole. We find that the early-time response closely mimics a black-hole ringdown due to wave scattering near the exterior unstable photon sphere. Conversely, the late-time signal develops a sequence of delayed echoes originating from wave trapping near the interior stable light ring and subsequent leakage through the exterior potential barrier. While individual multipoles exhibit regular echoes, their coherent superposition induces interference, producing complex waveforms with irregular profiles that nonetheless retain an underlying temporal organization with alternating peak separations. Finally, as a proof of concept, we perform a signal-to-noise ratio analysis using an assumed strain normalization, demonstrating that these weak, late-time signatures can be resolved by third-generation gravitational-wave detectors.