Axion energy density, bottle neck period, and $\bar{\theta}$ ratios between early and late times

Abstract

The possibility of the "invisible" axion being cold dark matter relies on the acceptable estimates of the current axion energy density. The estimate depends on the nature of QCD phase transition at a few hundred MeV and the evolution of the misalignment angle $\bar{\theta}$. The onset of $\bar{\theta}$ oscillation undergoes a bottleneck period which occurred during the QCD phase transition. In addition, the anharmonic coupling of order $a^4$ affects the $\bar{\theta}$ evolution. From the time that the anharmonic effect is negligible, it is rather simple to calculate the ratio of $\bar{\theta}$'s between early and late times. For multi GHz oscillations, the current age of the Universe needs at least $10^{27}$ oscillations which limits an exact calculation of $\bar{\theta}$ . We establish a stepwise approximation for numerical solutions of the differential equation and obtain $\bar{\theta}_{\rm now}/\bar{\theta}_f\approx 3\times 10^{-17}$ for $m_a\simeq 10^{-4}\,$eV, where $t_f$ is the first time that the full hadronic phase (after the QCD phase transition) was established.