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Previous works have argued that future gravitational-wave detectors will beable to probe the properties of astrophysical environments where binarycoalesce, including accretion disks, but also dark matter structures. Mostanalyses have resorted to a Newtonian modelling of the environmental effects,which are not suited to study extreme-mass-ratio inspirals immersed instructures of ultra-light bosons. In this letter, we use relativisticperturbation theory to consistently study these systems in spherical symmetry.We compute the flux of scalar particles and the rate at which orbital energy(and angular momentum) is dissipated via gravitational radiation and depletionof scalars, i.e. dynamical friction. Our results suggest that the LaserInteferometer Space Antenna will be able to probe ultra-light dark matterstructures in the Galaxy by tracking the phase of extreme-mass-ratio inspirals.

Axion Weak Leaks: extreme mass-ratio inspirals in ultra-light dark matter