Brownian Axion-like particles

We study the non-equilibrium dynamics of a pseudoscalar axion-like particle (ALP) weakly coupled to degrees of freedom in thermal equilibrium by obtaining its reduced density matrix. Its time evolution is determined by the in-in effective action which we obtain to leading order in the (ALP) coupling but to \emph{all orders} in the couplings of the bath to other fields within or beyond the standard model. The effective equation of motion for the (ALP) is a Langevin equation with noise and friction kernels obeying the fluctuation dissipation relation. A ``misaligned'' initial condition yields damped coherent oscillations, however, the (ALP) population increases towards thermalization with the bath. As a result, the energy density features a mixture of a cold component from misalignment and a hot component from thermalization with proportions that vary in time $(cold)\,e^{-\Gamma t}+(hot)\,(1-e^{-\Gamma t})$, providing a scenario wherein the ``warmth'' of the dark matter evolves in time from colder to hotter. As a specific example we consider the (ALP)-photon coupling $g a \vec{E}\cdot \vec{B}$ to lowest order, valid from recombination onwards. For $T \gg m_a$ the long-wavelength relaxation rate is substantially enhanced $\Gamma_T = \frac{g^2\,m^2_a\,T}{16\pi} $. The ultraviolet divergences of the (ALP) self-energy require higher order derivative terms in the effective action. We find that at high temperature, the finite temperature effective mass of the (ALP) is $m^2_a(T) = m^2_a(0)\Big[ 1-(T/T_c)^4\Big]$, with $T_c \propto \sqrt{m_a(0)/g}$, \emph{suggesting} the possibility of an inverted phase transition, which when combined with higher derivatives may possibly indicate exotic new phases. We discuss possible cosmological consequences on structure formation, the effective number of relativistic species and birefringence of the cosmic microwave background.