  We prove that the axion as a coherently oscillating scalar field acts as a
cold dark matter (CDM) to the second-order perturbations in all cosmological
scales including the super-horizon scale. The proof is made in the
axion-comoving gauge. For a canonical mass, the axion pressure term causes
deviation from the CDM only on scales smaller than the Solar System size.
Beyond such a small scale the equations of the axion fluid are the same as the
ones of the CDM based on the CDM-comoving gauge which are exactly identical to
the Newtonian equations to the second order. We also show that the axion fluid
does not generate the rotational (vector-type) perturbation even to the second
order. Thus, in the case of axion fluid, we have the relativistic/Newtonian
correspondence to the second order, even considering the rotational
perturbation. Our analysis is made in the presence of the cosmological
constant, and can be easily extended to the realistic situation including other
components of fluids and fields.
