Weighing dark matter particles with gravitational waves

It is commonly assumed that the mass of dark matter particles cannot be determined from their gravitational effects when dark matter behaves as collisionless matter. Here we show that extreme-mass-ratio inspirals observable by millihertz gravitational-wave observatories such as the Laser Interferometer Space Antenna (LISA) could measure the mass of ultralight dark matter particles in sufficiently dense dark matter spikes. The separation of scales between the two black holes allows ultralight dark matter to form a collisionless density spike around the massive primary while retaining wave effects in its interaction with the stellar-mass companion. We incorporate wave-dependent accretion and dynamical friction into relativistic inspiral waveforms and jointly infer the dark matter particle mass, spike normalisation and binary parameters from simulated LISA signals. For massive black holes at the low-mass end of the LISA population ($M\sim 10^5 M_\odot$), our forecasts show that particle masses spanning $10^{-14}-10^{-11}\,\mathrm{eV}$ could be measured to within an order of magnitude.