Thermal axion production at hard and soft momenta

Hot axions, thermally produced in the Early Universe, would contribute to dark radiation and are thus subject to present and future constraints from $N_{\rm eff}$. In this paper we quantify the contribution to $N_{\rm eff}$ and its uncertainty in models with axion-gluon couplings from thermal dynamics above the QCD transition. In more detail, we determine the leading-order thermal axion production rate for axion momenta of the order of the temperature adopting three different schemes for the incorporation of the collective dynamics of soft gluons. We show how these three schemes extrapolate differently into the regime of softer axion production, thus giving us a first quantitative handle on the theory uncertainty of the rate. Upon solving the Boltzmann equation, we find that this theory uncertainty translates to an uncertainty of order 0.002 for the contribution to $N_{\rm eff}$ prior to the QCD crossover. The uncertainty from common momentum-averaged approximations to the Boltzmann equation is smaller. We also discuss how QCD transition dynamics would need to be integrated into our results and we show how existing rate determinations in the literature based on gauge-dependent resummations are problematic.