Anharmonic evolution of the cosmic axion density spectrum

We present analytic solutions to the spatially homogeneous axion field equation, using a model potential which strongly resembles the standard anharmonic (1-cosN{θ}) potential, but contains only a piecewise second order term. Our model for {θ}(x,t) spans the entire range [-{π}/N,{π}/N] and is exactly soluble for homogeneous initial conditions. For nearly homogeneous initial conditions, as would naturally occur after inflation, we are able to confirm semianalytically (i) Turner's numeric correction factors to the adiabatic and harmonic analytic treatments of homogeneous axion oscillations, and (ii) Lyth's estimate valid near the metastable misalignment angle {π}/N at the peak at the potential. We compute the enhancement of axion density fluctuations that occurs when the axion mass becomes significant at T{~}1 GeV. We find that the anharmonicity amplifies density fluctuations, but only significantly for relatively large initial misalignment angles. The enhancement factor is {~} (2,3,4,13) for θ_in~(0.85,0.90,0.95,0.99)×{}{π}.