Opening the 1$\,$Hz axion window
  An axion-like particle (ALP) with mass $m_\phi \sim 10^{-15}$eV oscillates
with frequency $\sim$1 Hz. This mass scale lies in an open window of
astrophysical constraints, and appears naturally as a consequence of grand
unification (GUT) in string/M-theory. However, with a GUT-scale decay constant
such an ALP overcloses the Universe, and cannot solve the strong CP problem. In
this paper, we present a two axion model in which the 1 Hz ALP constitutes the
entirety of the dark matter (DM) while the QCD axion solves the strong CP
problem but contributes negligibly to the DM relic density. The mechanism to
achieve the correct relic densities relies on low-scale inflation ($m_\phi
\lesssim H_{\rm inf}\lesssim 1$ MeV), and we present explicit realisations of
such a model. The scale in the axion potential leading to the 1 Hz axion
generates a value for the strong CP phase which oscillates around
$\bar{\theta}_{\rm QCD}\sim 10^{-12}$, within reach of the proton storage ring
electric dipole moment experiment. The 1 Hz axion is also in reach of near
future laboratory and astrophysical searches.
