Nonlinear hydrodynamics of axion dark matter: Relative velocity effects and quantum forces

The nonlinear hydrodynamic equations for axion/scalar field dark matter (DM) in the nonrelativistic Madelung-Shcr\"odinger form are derived in a simple manner, including the effects of universal expansion and Hubble drag. The hydrodynamic equations are used to investigate the relative velocity between axion DM and baryons, and the moving-background perturbation theory (MBPT) derived. Axions massive enough to be all of the DM do not affect the coherence length of the relative velocity, but the MBPT equations are modified by the inclusion of the axion effective sound speed. These MBPT equations are necessary for accurately modeling the effects of axion DM on the formation of the first cosmic structures, and suggest that the 21-cm power spectrum could improve constraints on axion mass by up to 4 orders of magnitude with respect to the current best constraints. A further application of these results uses the ``quantum force'' analogy to model scalar field gradient energy in a smoothed-particle hydrodynamics model of axion DM. Such a model can treat axion DM in the nonlinear regime and could be incorporated into existing N-body codes.