A Solvable Model for Anharmonic Evolution of Cosmic Axion Overdensitites
  We present analytic solutions to the spatially homogeneous axion field equation, using a model potential which strongly resembles the $(1-\cos \theta)$ standard anharmonic potential. We find that the anharmonicity amplifies early universe axion density fluctuations, but only significantly so for relatively large initial misalignment angles. Enhancements of $\sim$ (2,3,4,13) result for $\theta_{\rm in}\sim (0.85,0.90,0.95,0.99)\times\pi$. The large $\theta$ result agrees with Lyth, and so validates his approximations. 
Karl Strobl and Thomas J. Weiler