Axion as a cold dark matter candidate: Analysis to third order perturbation for classical axion

We investigate aspects of axion as a coherently oscillating massive classical scalar field by analyzing third order perturbations in Einstein's gravity in the axion-comoving gauge. The axion fluid has its characteristic pressure term leading to an axion Jeans scale which is cosmologically negligible for a canonical axion mass. Our classically derived axion pressure term in Einstein's gravity is identical to the one derived in the non-relativistic quantum mechanical context in the literature. We show that except for the axion pressure term, the axion fluid equations are exactly the same as the general relativistic continuity and Euler equations of a zero-pressure fluid up to third order perturbation. The general relativistic density and velocity perturbations of the CDM in the CDM-comoving gauge are exactly the same as the Newtonian perturbations to the second order (in all scales), and the pure general relativistic corrections appearing from the third order are numerically negligible (in all scales as well) in the current paradigm of concordance cosmology. Therefore, here we prove that, in the super-Jeans scale, the classical axion can be handled as the Newtonian CDM fluid up to third order perturbation. We also show that the axion fluid supports the vector-type (rotational) perturbation from the third order. Our analysis includes the cosmological constant.