  If the Peccei-Quinn symmetry associated to an axion has ever been restored
after inflation, axion strings inevitably produce a contribution to the
stochastic gravitational wave background. Combining effective field theory
analysis with numerical simulations, we show that the resulting gravitational
wave spectrum has logarithmic deviations from a scale invariant form with an
amplitude that is significantly enhanced at low frequencies. As a result, a
single ultralight axion-like particle with a decay constant larger than
$10^{14}~{\rm GeV}$ and any mass between $10^{-18}~{\rm eV}$ and $10^{-28}~{\rm
eV}$ leads to an observable gravitational wave spectrum and is compatible with
constraints on the post-inflationary scenario from dark matter overproduction,
isocurvature and dark radiation. Since the spectrum extends over a wide range
of frequencies, the resulting signal could be detected by multiple experiments.
We describe straightforward ways in which the Peccei-Quinn symmetry can be
restored after inflation for such decay constants. We also comment on the
recent possible NANOgrav signal in light of our results.
