Classical equipartition dynamics between axions and non-Abelian gauge fields

TL;DR

The paper investigates how an axion-like inflaton transfers energy to a non-Abelian SU(2) gauge sector using non-linear real-time lattice simulations in Minkowski space. It demonstrates initial exponential growth of low-momentum gauge modes, delayed damping of the axion condensate, and eventual energy equipartition between axion and gauge degrees of freedom, with distinct dynamics compared to the Abelian case. The results quantify growth and damping rates, reveal smooth SU(2) spectra due to self-interactions, and indicate a gauge-dominated final state consistent with rapid equilibration, while flagging the limitations of classical lattices for true thermalization. The work provides a foundation for understanding non-Abelian thermalization and its implications for warm inflation and sphaleron friction, and outlines clear paths for incorporating expansion and fermions in future studies.

Abstract

Motivated by axion-like inflation and its warm embedding within the Standard Model, we study the early stages of the energy transfer between an axion condensate and an SU(2) gauge ensemble, by employing non-linear classical real-time lattice simulations. The discretized equations of motion are worked out, elaborating on Gauss constraints. A numerical solution is implemented on the CosmoLattice platform. Adopting a quadratic potential, and omitting universe expansion for the moment, we establish initial exponential growth of the low-momentum gauge modes; damping of axion oscillations after some delay; and subsequent energy equipartition between axion and gauge ensembles. A clear difference between the SU(2) and U(1) dynamics is observed, likely associated with non-Abelian self-interactions. We elaborate on what this implies for the possible thermalization of the SU(2) ensemble.

Figures (9)

• Figure 1: Left: A numerical solution of eq. (\ref{['B_modes']}), with initial conditions from eq. (\ref{['B_initial']}), background modelling from eq. (\ref{['ansatz']}), and the result plotted in terms of the power spectrum from eq. (\ref{['B_P']}). The curves correspond to snapshots at equal time intervals, $\Delta \tilde{t} = 0.50$, and they display exponential growth for as long as $\dot{\bar{\varphi}}\neq 0$ in eq. (\ref{['B_modes']}). As a function of $\tilde{k}$, they also show a band structure, characteristic of systems with periodically varying coefficients. The overall time period is the same as in fig. \ref{['fig:power_spectra']}, however the linearized solution lacks the transfer of power to higher momenta that is seen there. Right: The time evolution of the positive and negative helicity modes, at fixed $\tilde{k} = 2.01$. The blobs correspond to the snapshots shown in the left panel (though there both helicities are summed together).
• Figure 2: Left: Evolution of gauge-field energy densities as a function of time, $\tilde{t}$, for several $\widetilde{\kappa}$, with the vector notation denoting a sum over both spatial and colour indices. The grey bands indicate domains of exponential growth. Right: The coefficient characterizing the exponential growth, $\widetilde{\Gamma}$ (cf. eq. (\ref{['def_Gamma']})), as a function of $\widetilde{\kappa}$.
• Figure 3: Illustration of axion oscillations and their subsequent damping, for SU(2) (left) and U(1) (right), for three representative values of $\widetilde{\kappa}$. The dotted lines show a fit to the envelope of eq. \ref{['ansatz']}, whereas the dashed lines represent a fit to the full eq. \ref{['ansatz']}.
• Figure 4: Illustration of the coefficient $\tilde{t}^{ }_0$, characterizing the delay of when damping starts, as extracted from a fit to eq. (\ref{['ansatz']}). With a lighter shade we show points in the overdamped regime, where less than a full oscillation occurs before the damping sets in. In the latter regime, the error of the fitting procedure is significant.
• Figure 5: The axion damping coefficient $\widetilde{\Upsilon}$, as extracted from a fit to eq. (\ref{['ansatz']}), for both SU(2) and U(1). We also illustrate possible quadratic and constant representations of the data. The points in the overdamped regime are shown with a lighter shade. Here less than a full oscillation takes place before the damping sets in, which significantly adds to the error of the fitting procedure.