Maximal temperature of strongly-coupled dark sectors

Taking axion inflation as an example, we estimate the maximal temperature that can be reached in the post-inflationary universe, as a function of the confinement scale of a non-Abelian dark sector. Below a certain threshold for the confinement scale, the system heats up above the critical temperature, and a first-order thermal phase transition takes place. On the other hand, above the threshold very high temperatures can be reached, but there is no phase transition. If the inflaton thermalizes during heating-up (which we find to be unlikely), or if the plasma includes light degrees of freedom, then heat capacity and entropy density are larger, and the aximal temperature is lowered towards the threshold. The heating-up dynamics generates a gravitational wave background. Its contribution to Neff at GHz frequencies, the presence of a monotonic shape at mHz-MHz frequencies, and the frequency domain of peaked features that may originate via first-order phase transitions, are discussed.