Hill-top inflation from Dai-Freed anomaly in the standard model -- A solution to the iso-curvature problem of the axion dark matter

The discrete symmetry $Z_4$ in the standard model (SM) with three right-handed neutrinos is free from the Dai-Freed anomaly. Motivated by this $Z_4$ symmetry, we constructed a topological inflation model consistent with all known constraints and observations. However, we assumed a specific inflaton potential in the previous work. In this paper we extend the inflaton potential in a more general form allowed by the discrete $Z_4$ gauge symmetry and show that consistent hilltop inflation is realized. We find that the Hubble parameter $H_\mathrm{inf}$ can be smaller than $\simeq 10^{9}$ GeV so that the isocurvature fluctuations of the axion dark matter are sufficiently suppressed. Furthermore, the running of the spectral index can be as large as $dn_s/\ln k \simeq 0.0018$ which will be tested in future CMB observations. Since this discrete $Z_4$ acts on the SM, the inflaton can couple to pairs of the right-handed neutrinos and hence the reheating temperature can be high as $\sim 10^{10}$ GeV, producing the cosmic baryon asymmetry naturally through the thermal leptogenesis.