  Axion cosmology is reexamined taking into account effect of kinetic pseudo
Nambu-Goldstone modes, with its importance recently pointed out. When
Peccei-Quinn (PQ) symmetry is broken by a chiral U(1) singlet, it is found that
the effect of kinetic Nambu-Goldstone mode makes the axion dark matter
untenable. When PQ symmetry is extended and is broken by two singlets, we find
axion cosmology to work, but there are several differences from the axion
cosmology studied in the literature. The differences are (1) ordinary type of
dark matter scaling as $1/{\rm cosmic \; scale\; factor}^3$ arising from a
modulus field and not from the usual angular field, (2) mass of the dark matter
quantum in the ultralight range, $(10^{-32} \sim 10^{-14})\,$eV, (3) emergence
of dark energy with the present density of order (a few meV)$^4$ consistent
with observations, (4) presence of a long range spin-dependent force, (5)
slow-roll inflation after PQ symmetry breaking when conformal coupling to
gravity is introduced.
