Bosonic dark matter halos: excited states and relaxation in the potential of the ground state

An ultra-light axion field with mass $\sim 10^{-22}\ {\rm eV}$, also known as wave or fuzzy dark matter, has been proposed as a component of the dark matter in the Universe. We study the evolution of the axion dark matter distribution in the central region of a halo, assuming the mass is dominated by this axion field, and that gravity is the only important interaction. We calculate the excited axion states in the spherical gravitational potential generated by the self-gravitating ground-state, also known as soliton. These excited states are similar to the states of the hydrogen atom with quantum numbers $(n,l,m)$, here designating oscillation modes of a classical wave. At fixed $n$, the modes with highest $l$ have the lowest energy because of the extended mass distribution generating the potential. We use an approximate analytical treatment to derive the distribution of mass in these states when a steady-state is reached by dynamical relaxation, and find that a corona with a mass density profile $\rho \propto r^{-5/3}$ should be set up around the central soliton, analogous to the Bahcall-Wolf cusp predicted for the stellar distribution around a central black hole. The central soliton accretes dark matter from the corona as dynamical relaxation proceeds and negative orbital energy flows out. This density profile should remain valid out to the radius where the mass in the corona is comparable to the mass of the central soliton; further than that, the gravitational potential depends on the initial distribution of dark matter and the relaxation time increases rapidly with radius.