Cosmological consequences of the Peccei-Quinn field ψ=${\mathit{re}}^{\mathit{i}\mathrm{\theta }}$/ √2 are explored. It has a Mexican-hat potential W=1/4λ(${\mathit{r}}^2$-${\mathit{f}}_{\mathit{a}}^2$${)}^2$. During inflation the potential may be modified so that ${\mathit{f}}_{\mathit{a}}$ has a different effective value ${\mathit{f}}_{\mathit{a}1}$; it is assumed that r sits in the vacuum at r=${\mathit{f}}_{\mathit{a}1}$. After inflation the temperature is supposed to be less than ${\mathit{f}}_{\mathit{a}}$ so that r=${\mathit{f}}_{\mathit{a}}$, and the only degree of freedom is the axion field ${\mathit{f}}_{\mathit{a}}$θ. It has a Gaussian inhomogeneity coming from the vacuum fluctuation of θ during inflation. When the axion mass ${\mathit{m}}_{\mathit{a}}$(T) becomes significant at T∼1 GeV, θ has dispersion ${\mathrm{\sigma }}_{\mathrm{\theta }}$≃(4/2π)(${\mathit{H}}_1$/${\mathit{f}}_{\mathit{a}1}$) and some mean θ¯ (in the observable Universe). The axion potential is U(θ)=(79 MeV${)}^4$(1-cosNθ), and the ensuing cosmology is determined by the three parameters ${\mathit{f}}_{\mathit{a}}$/N, Nθ¯, and N${\mathrm{\sigma }}_{\mathrm{\theta }}$. The entire domain of parameter space is considered, including the regime where the axion density perturbation is non-Gaussian and the regime where axionic domain walls are produced. Observational constraints on the parameters are established. At the end of the paper the additional assumption is made that during inflation the vacuum is at r=${\mathit{f}}_{\mathit{a}}$. Unless ${\mathit{f}}_{\mathit{a}}$/N is near the Planck scale and axions make up only a small fraction of the dark matter, this leads to the bound ${\mathit{V}}_1^{1/4}$<2×${10}^{15}$ GeV, where ${\mathit{V}}_1$ is the energy density during inflation, at the epoch when the observable Universe leaves the horizon.

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