Maximal temperature of strongly-coupled dark sectors

Taking axion inflation as an example, we estimate the maximal temperature (T-max) that can be reached in the post-inflationary universe, as a function of the confinement scale of a non-Abelian dark sector (Lambda(IR)). Below a certain threshold Lambda(IR)< Lambda(0) similar to 2 x 10(-8)m(pl), the system heats up to T-max similar to Lambda(0) > T-c, and a first-order thermal phase transition takes place. On the other hand, if Lambda(IR) > Lambda(0), then T-max similar to Lambda(IR)< T-c: 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 T-max is lowered towards Lambda(0). The heating-up dynamics generates a gravitational wave background. Its contribution to N-eff at GHz frequencies, the presence of a monotonic similar to f03 shape at (10(-4) - 10(2)) Hz frequencies, and the frequency domain of peaked features that may originate via first-order phase transitions, are discussed.