Indirect Detection of Dark Matter Around a Supermassive Black Hole with High Energy-Resolution Gamma-Ray Telescopes

We explore whether the unprecedented energy resolution of upcoming gamma-ray telescopes can uncover relativistic effects in photon spectra resulting from dark matter (DM) annihilation or decay near the supermassive black hole (SMBH) at the Galactic Center (GC), specifically, gravitational redshift, Doppler broadening due to Lorentz boosts, and kinetic energy enhancements arising from high DM velocities. By modeling DM density and velocity profiles under various SMBH formation scenarios and DM properties, we calculate the corresponding gamma-ray spectra and identify the conditions under which SMBH-induced spectral distortions become observable. We find that, in favorable cases, the observed spectra encode the DM velocity distribution near the SMBH, enabling potential discrimination among annihilation mechanisms with different velocity dependencies. Even when SMBH-induced effects are modest, the upcoming COSI mission, with sub-percent energy resolution surpassing the typical DM velocity dispersion at the GC, $\mathcal{O}(10^{-3})$, may still be able to detect subtle Doppler broadening. These results highlight a promising pathway for determining the origin of gamma-ray signals and probing DM properties through high-resolution spectral measurements.