  Here we generally prove that the axion as a coherently oscillating scalar
field acts as a cold dark matter in nearly all cosmologically relevant scales.
The proof is made in the linear perturbation order. Compared with our previous
proof based on solutions, here we compare the equations in the axion with the
ones in the cold dark matter, thus expanding the valid range of the proof.
Deviation from purely pressureless medium appears in very small scale where
axion reveals a peculiar equation of state. Our analysis is made in the
presence of the cosmological constant, and our conclusions are valid in the
presence of other fluid and field components.
