Relaxation in a Fuzzy Dark Matter Halo. II. Self-consistent Kinetic Equations

Abstract

Fuzzy dark matter (FDM) is composed of ultra-light bosons having a de Broglie wavelength that is comparable to the size of the stellar component of galaxies at typical galactic velocities. FDM behaves like cold dark matter on large scales. However, on the scale of the de Broglie wavelength, an FDM halo exhibits density fluctuations that lead to relaxation, a process similar to the two-body relaxation that occurs in classical gravitational N -body systems and is described by the Fokker–Planck equation. We derive the FDM analog of that kinetic equation, which describes the evolution of the velocity distribution in a spatially homogeneous FDM halo. We show that the evolution of the velocity distribution predicted by our kinetic equation matches numerical solutions of the coupled Schrödinger–Poisson equations. We also determine the dielectric function and the dispersion relation for linear waves in an FDM halo.