Relativistic effects in extreme-mass-ratio inspirals within scalar clouds: Eccentric and inclined orbits

We study extreme-mass-ratio inspirals (EMRIs) evolving in a scalar cloud environment that may form through superradiant instabilities, using a fully relativistic perturbative framework that allows for eccentric and inclined orbits. EMRIs, consisting of a stellar-mass compact object inspiraling into a supermassive black hole, are key sources for space-based gravitational-wave detectors such as LISA. Previous relativistic studies of EMRIs in scalar clouds have been restricted to circular, equatorial motion. Here, instead, we focus on a Schwarzschild black hole background to incorporate eccentricity and orbital inclination. By computing the scalar energy and angular momentum scattered off to spatial infinity and absorbed at the event horizon, we show that orbital eccentricity can induce a dense spectrum of resonances near the last stable orbit, associated with strong relativistic apsidal precession. We further find that orbital inclination can significantly modify the orbital energy and angular momentum losses. In particular, we identify a critical inclination angle below which, at sufficiently small orbital radii, there is a net transfer of energy from the scalar cloud to the orbit. Moreover, for sufficiently large eccentricities, resonances associated with relativistic apsidal precession persist across the full range of inclinations, although their structure changes significantly between prograde and retrograde orbits. These results provide a foundation for future studies of EMRIs in scalar cloud environments on fully generic orbits around spinning black holes.