Saving the invisible axion

Recently it has been shown that the energy density contributed today by coherent oscillations of the invisible axion field corresponds to Ω {reversed tilde equals} 0.1 ( f{hook}_a 10¹² GeV) ^{11 9} (f{hook}_a is the scale of peccei-Quinn SSB), apparently ruling out f{hook}_a ≳ O(10¹³ GeV). We propose a simple way to avoid this bound: entropy production when the temperature of the universe is between about 1 GeV and 1 MeV. The entropy might be produced by a first-order phase transition, or the out-of-equilibrium decays of a relic particle species. For f{hook}_a {reversed tilde equals} 10¹⁵ GeV, the entropy of the universe must be increased by a factor of O(10²-10³) to have Ω_a {reversed tilde equals} O(1). If inflation also occurs at a temperature {reversed tilde equals}10¹⁵ GeV, then the de Sitter space produced fluctuations in the axion field lead to a Zel'dovich-like spectrum of (isothermal) perturbations in the axion energy density, with amplitude about 10^-4. It has recently been argued that in an axion-dominated universe such a spectrum could lead to the formation of of the structure observed in the universe today.