Relativistic signatures of scalar dark matter in extreme-mass-ratio inspirals

We study gravitational wave emission by circular extreme-mass-ratio systems in a spherically symmetric scalar environment. Previous studies have focused on the impact of scalar radiation channels, revealing a rich structure of resonances, sharp features and floating orbits. Through the backreaction of the cloud on the metric, corrections to the gravitational sector come in at the same order. We develop the computational methods, and provide a characterization of this new, fully relativistic cloud signature. Remarkably, corrections to the polar sector can dominate all other dissipative corrections. We identify scalar field masses $M\mu\lesssim 0.12$ as the regime where polar can overtake axial and scalar channels at small separation. For small $M\mu$, vacuum dephasing is dominated mostly by conservative and polar cloud corrections, with scalar radiation acting as only a minor correction. At large $M\mu$, both terms terms are shown to be highly non-negligible. Our results therefore motivate including these relativistic signatures in beyond-vacuum EMRI templates.