  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.
