Oscillons of Axion-Like Particle: Mass distribution and power spectrum

In string theory, the simultaneous existence of many Axion-Like Particles (ALPs) are suggested over a vast mass range, and a variety of potentials have been developed in the context of inflation. In such potentials shallower than quadratic, the prominent instability can produce localized dense objects, oscillons. Because of the approximate conservation of their adiabatic invariant, oscillons generally survive quite long, maybe up to the current age of the universe in the case of ultra-light ALPs with $m \sim 10^{-22}\ {\rm eV}$. Such oscillons can have significant effects on the evolution of the recent universe. In this paper, we investigate the oscillons of the pure-natural type potential by classical lattice simulation to explore the key quantities necessary for phenomenological application: the number density of oscillons, the oscillon mass distribution, the energy ratio of oscillons to the ALP field, and the power spectrum. Then, we evolve these values in consideration of the analytic decay rate.