Axions and inflation: String formation during inflation

The Peccei-Quinn field $\psi =\frac{r{e}^{i\theta }}{\sqrt{2}}$ has a Mexican hat potential $\frac{1}{4}\lambda {(r^2-f_a^2)}^2$, and the axion, with the field $f_a\theta$, is its Nambu-Goldstone boson. During inflation there is a quantum fluctuation in both the radial component $r$ and in the angle $\theta$. If the temperature after inflation is less than $f_a$ the fluctuation survives, and may be cosmologically significant. Previous investigations of this "inflationary scenario" assumed that $r$ sits in the vacuum, leaving only the angular fluctuation. Here both fluctuations are kept, using the Langevin and Fokker-Planck equations. It is found that for $f_a$ in the usually accepted range $f_a\lesssim {10}^{13}$ GeV, the radial fluctuation is typically big enough to take $\psi$ over the top of the potential. In that case, strings are produced, which invalidate previous treatments of the inflationary scenario.