If the temperature after inflation is below the Peccei-Quinn energy scale f(a), the microwave background constraint on the spectrum P(a) of the primeval isocurvature axion density inhomogeneity can provide an upper bound on the inflationary potential V1 (the subscript 1 denotes the epoch when the observable universe leaves the horizon ). Recent work evaluates this bound for the entire regime of parameter space and the essential results are reported here. The crucial new finding is that during inflation the PQ field may be kicked out of the vacuum by its quantum fluctuation. leading to the production of strings which remove the bound. This typically happens if H-1 greater than or similar to f(a) where H-1 is the Hubble parameter. In the opposite case. the only degree of freedom during inflation is the axion field, whose quantum fluctuation is Gaussian. In that case the bound on P(a) is evaluated, taking into account the non-gaussian part of the density inhomogeneity (quadratic in the field) as well as the gaussian part (linear in the field).