QCD Axion on Hilltop by 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.