Axion as a fuzzy-dark-matter candidate: Uniqueness of the comoving gauge

Axion as a coherently oscillating massive scalar field is known to behave as a zero-pressure irrotational fluid with characteristic quantum stress on a small scale. In relativistic perturbation theory, the case was proved in the axion-comoving gauge up to fully nonlinear and exact order. Our basic assumption is that the field is oscillating with Compton frequency and the Compton wavelength is smaller than the horizon scale. Here, we revisit the relativistic proof to the linear order in the other gauge conditions. To our surprise, we show that {\it the same} equation for density perturbation known in the non-relativistic treatment can be consistently derived {\it only} in the comoving gauge. In other gauge conditions either we have inconsistencies or the quantum stress term is missing. We clarify that our relativistic analysis is valid for all scales larger than the Compton wavelength. For comparison, we review the non-relativistic quantum hydrodynamics and present the Schr\"odinger equation to first-order post-Newtonian expansion in the cosmology context.