Axion Perturbations: A General Analytical Treatment

TL;DR

This work provides a general analytic framework for axion dark matter perturbations that remains valid for any cosmological background and temperature-dependent potential. It shows that super-horizon perturbations are fully specified by a family of background solutions parameterized by the initial axion value θ_ini and its initial velocity, decomposing into an adiabatic Weinberg mode and a potentially enhanced isocurvature component. The adiabatic part obeys the universal relation δs ∝ dot{\bar s}, with Φ and Ψ tied to a constant curvature ζ on large scales, while the isocurvature part is governed by derivatives of the background energy density with respect to θ_ini and can exhibit significant non-Gaussianity through a local bispectrum with amplitude f_NL^(S). The results generalize prior analyses to arbitrary backgrounds and potentials, and apply to generic ALP or non-thermal DM scenarios, offering a robust tool for predicting observational signatures in the CMB and large-scale structure.

Abstract

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 $θ_{\rm ini}$. 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, $\partial Ω_a / \partial θ_{\rm ini}$, 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.

Figures (1)

• Figure 1: The density perturbation for the QCD axion $\delta_\theta = \delta\rho_\theta/\overline\rho_\theta$, time-averaged by filtering out high-frequency oscillations. A full numerical solution is shown in the blue solid curve, with the analytical solution based on the adiabatic field fluctuation $\delta\theta = t\dot{\overline \theta} \Psi_{\boldsymbol{k}}$ shown in dashed orange. The black dotted line indicates the initial constant value during the vacuum-dominated phase, while the black dot-dashed line gives the constant about which $\delta_\theta$ oscillates when the QCD axion potential is temperature dependent, and the black dashed line indicates the constant solution after the temperature dependence is gone, identical to the behavior of thermal CDM (when averaging over oscillations). The solutions in this figure assume radiation domination, superhorizon conditions, and the QCD axion potential in \ref{['eq: axion m(T)']} with $n=4$.