Challenges for heavy QCD axion inflation

We examine the theoretical possibility for the heavy QCD axion to induce slow-roll inflation while solving the strong CP problem through the Peccei-Quinn mechanism. If the cancellation between contributions from a small-size instanton and another Peccei-Quinn symmetry breaking occurs with high accuracy, the potential can be flattened enough near its maximum to achieve hilltop inflation. This comes at the cost of severe fine-tuning of their relative size and phase, but it also allows us to relate the high quality problem of the Peccei-Quinn symmetry to the fine-tuning problem of the hilltop inflation. There are two classes of such axion hilltop inflation, each giving a different relation between the axion mass at the minimum and the decay constant. The first class predicts the relation $m_\phi \sim 10^{-6}f_\phi$, and the axion can decay via the gluon coupling and reheat the universe. Most of the predicted parameter region will be covered by various experiments such as CODEX, DUNE, FASER, LHC, MATHUSLA, and NA62 where the production and decay proceed through the same coupling that induced reheating. The second class predicts the relation $m_\phi \sim 10^{-6} f^2_\phi/ M_{\rm pl}$. In this case, the axion mass is much lighter than in the previous case, and one needs another mechanism for successful reheating. The viable decay constant is restricted to be $10^8{\rm GeV} \lesssim f_\phi \lesssim 10^{10}\,$GeV, which will be probed by future experiments on the electric dipole moment of nucleons. In both cases, requiring the axion hilltop inflation results in the strong CP phase that is close to zero.