QCD Axion on Hilltop by a Phase Shift of $\pi$
  We show that the initial misalignment angle of the QCD axion (or axion-like
particles) can be set very close to $\pi$, if the QCD axion has a mixing with
another heavy axion which induces the phase shift $\approx \pi$ after
inflation. In the simplest case, the heavy axion plays the role of the
inflaton, and we call such inflation as "$\pi$nflation." The basic idea was
first proposed by Daido and the present authors in Ref. [1702.03284] in 2017
and more recently discussed in Ref. [1903.00462]. We show that the QCD axion
with a decay constant $f_a \gtrsim 3 \times 10^9\,$ GeV can explain dark matter
by the $\pi$nflation mechanism. A large fraction of the parameter region has an
overlap with the projected sensitivity of ORGAN, MADMAX, TOORAD and IAXO. We
also study implications for the effective neutrino species and isocurvature
perturbations. The $\pi$nflation can provide an initial condition for the
hilltop inflation in the axion landscape, and in a certain set-up, a chain of
the hilltop inflation may take place.
