Maximal temperature of strongly-coupled dark sectors

Taking axion inflation as an example, we estimate the maximal temperature ($T_{\rm max}^{ }$) that can be reached in post-inflationary universe, a function of confinement scale non-Abelian dark sector ($\Lambda_{\rm IR}^{ }$). Below certain threshold $\Lambda_{\rm } < \Lambda_{\rm 0}^{ \sim 2\times 10^{-8}_{ m_{\rm pl}^{ }$, system heats up to $T_{\rm > T_{\rm c}^{ and first-order thermal phase transition takes place. On other hand, if then }$: very high temperatures reached, but there is no transition. If inflaton thermalizes during heating-up (which find unlikely), or plasma includes light degrees freedom, heat capacity entropy density are larger, }$ lowered towards }$. The dynamics generates gravitational wave background. Its contribution $N^{ }_{\rm eff}$ at GHz frequencies, presence monotonic $\sim f_{\rm 0}^3$ shape $(10^{-4}_{ - 10^2_{ })\,$Hz frequency domain peaked features may originate via transitions, discussed.