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

Previous works have argued that future gravitational-wave detectors will be able to probe the properties of astrophysical environments where binary coalesce, including accretion disks, but also dark matter structures. Most analyses have resorted to a Newtonian modelling of the environmental effects, which are not suited to study extreme-mass-ratio inspirals immersed in structures of ultra-light bosons. In this letter, we use relativistic perturbation 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 depletion of scalars, i.e. dynamical friction. Our results suggest that the Laser Inteferometer Space Antenna will be able to probe ultra-light dark matter structures in the Galaxy by tracking the phase of extreme-mass-ratio inspirals.