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The non-linear hydrodynamic equations for axion/scalar field dark matter (DM)in the non-relativistic Madelung-Shcr\"{o}dinger form are derived in a simplemanner, including the effects of universal expansion and Hubble drag. Thehydrodynamic equations are used to investigate the relative velocity betweenaxion DM and baryons, and the moving-background perturbation theory (MBPT)derived. Axions massive enough to be all of the DM do not affect the coherencelength of the relative velocity, but the MBPT equations are modified by theinclusion of the axion effective sound speed. These MBPT equations arenecessary for accurately modelling the effects of axion DM on the formation ofthe first cosmic structures, and suggest that the 21cm power spectrum couldimprove constraints on axion mass by up to four orders of magnitude withrespect to the current best constraints. A further application of these resultsuses the "quantum force" analogy to model scalar field gradient energy in asmoothed-particle hydrodynamics model of axion DM. Such a model can treat axionDM in the non-linear regime and could be incorporated into existing N-bodycodes.

Non-linear hydrodynamics of axion dark matter: relative velocity effects and "quantum forces"