Cosmologically allowed regions for the axion decay constant Fa

If the Peccei-Quinn symmetry is already broken during inflation, the decay constant Fa of the axion can be in a wide region from 1011GeV to 1018GeV for the axion being the dominant dark matter. In this case, however, the axion causes the serious cosmological problem, isocurvature perturbation problem, which severely constrains the Hubble parameter during inflation. The constraint is relaxed when Peccei-Quinn scalar field takes a large value ∼Mp (Planck scale) during inflation. In this letter, we point out that the allowed region of the decay constant Fa is reduced to a rather narrow region for a given tensor-to-scalar ratio r when Peccei-Quinn scalar field takes ∼Mp during inflation. For example, if the ratio r is determined as r≳10−3 in future measurements, we can predict Fa≃(0.1-1.4)×1012GeV for domain wall number NDW=6.