Cosmologically allowed regions for the axion decay constant $F_a$

If the Peccei-Quinn symmetry is already broken during inflation, the decay constant $F_a$ of the axion can be in a wide region from $10^{11}$ GeV to $10^{18}$ GeV 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 $\sim M_{p}$ (Planck scale) during inflation. In this letter, we point out that the allowed region of the decay constant $F_a$ is reduced to a rather narrow region for a given tensor-to-scalar ratio $r$ when Peccei-Quinn scalar field takes $\sim M_{p}$ during inflation. For example, if the ratio $r$ is determined as $r \gtrsim 10^{-3}$ in future measurements, we can predict $F_a \simeq (0.1-1.4)\times 10^{12}$ GeV for domain wall number $N_\text{DW}=6$.