  The Peccei-Quinn mechanism presents a neat solution to the strong CP problem.
As a by-product, it provides an ideal dark matter candidate, "the axion",
albeit with a tiny mass. Axions therefore can act as dark radiation if excited
with large momenta after the end of inflation. Nevertheless, the recent
measurement of relativistic degrees of freedom from cosmic microwave background
radiation strictly constrains the abundance of such extra relativistic species.
We show that ultra-relativistic axions can be abundantly produced if the
Peccei-Quinn field was initially displaced from the minimum of the potential.
This in lieu places an interesting constraint on the axion dark matter window
with large decay constant which is expected to be probed by future experiments.
Moreover, an upper bound on the reheating temperature can be placed, which
further constrains the thermal history of our Universe.
