| Summary: | <p>Galactic nuclei (GNs) are dense stellar environments abundant in gravitational-wave (GW) sources for the Laser Interferometer Gravitational-Wave Observatory (LIGO), Virgo, and Kamioka Gravitational Wave Detector (KAGRA). The GWs may be generated by stellar-mass black hole (BH) or neutron star mergers following gravitational bremsstrahlung, dynamical scattering encounters, Kozai–Lidov-type oscillations driven by the central supermassive black hole (SMBH), or gas-assisted mergers if present. In this paper, we examine a smoking gun signature to identify sources in GNs: the GWs scattered by the central SMBH. This produces a secondary signal, an astrophysical GW echo, which has a very similar time–frequency evolution as the primary signal but arrives after a time delay. We determine the amplitude and time-delay distribution of the GW echo as a function of source distance from the SMBH. Between ∼10 per cent and 90 per cent of the detectable echoes arrive within ∼(1--100)<em>M</em><sub>6</sub>s after the primary GW for sources between 10 and 10<sup>4</sup> Schwarzschild radius, where <em>M</em><sub>6</sub>=M<sub>SMBH,z/</sub> (10<sup>6</sup><em>M</em><sub>⊙</sub>), and M<sub>SMBH, z</sub> is the observer-frame SMBH mass. The echo arrival times are systematically longer for high signal-to-noise ratio (SNR) primary GWs, where the GW echo rays are scattered at large deflection angles. In particular, ∼10 per cent--90 per cent of the distribution is shifted to ∼(5--1800)<em>M</em><sub>6</sub>s for sources, where the lower limit of echo detection is 0.02 of the primary signal amplitude. We find that ∼5 per cent--30 per cent(∼1 per cent--7 per cent) of GW sources have an echo amplitude larger than 0.2–0.05 times the amplitude of the primary signal if the source distance from the SMBH is 50 (200) Schwarzschild radius. Non-detections can rule out that a GW source is near an SMBH.</p>
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