Axion Perturbations: A General Analytical Treatment

Cosmological data provides us two key constraints on dark matter (DM): it must have a particular abundance, and it must have an adiabatic spectrum of density perturbations in the early universe. Many different cosmological scenarios have been proposed that establish the abundance of axion DM in qualitatively different ways. In this paper we emphasize that, despite this variety of backgrounds, the perturbations in axion DM can be understood from universal principles. How does a feebly interacting axion field acquire perturbations proportional to those of photons? How do the isocurvature power spectrum and non-Gaussianity depend on the background evolution of the universe? We answer these questions for a completely general choice of cosmological background and temperature-dependent axion potential. We show that the most general solution to the axion field equation on super-horizon scales is entirely determined by the family of background solutions for different initial field values $\theta_{\rm ini}$. This holds for both the component in the field perturbation solution contributing to the DM isocurvature perturbation (enhanced at late times by the sensitivity of the DM abundance to the initial condition, $\partial \Omega_a / \partial \theta_{\rm ini}$, which can be large for initial conditions near the hilltop), and the other component that contributes to the DM curvature perturbation. In particular, we explain that an unperturbed axion field in the early universe evolving into one with nontrivial adiabatic perturbations is guaranteed by Weinberg's theorem on adiabatic modes. These results have been derived before with various assumptions, such as a radiation dominated background or a quadratic potential. Our aim is to give a clear, simple derivation that is manifestly independent of those assumptions, and thus can be applied to any cosmological axion scenario.