Unitarity limits on thermal dark matter in (non-)standard cosmologies

Using the upper bound on the inelastic reaction cross-section implied by S-matrix unitarity, we derive the thermally averaged maximum dark matter (DM) annihilation rate for general $k \rightarrow 2$ number-changing reactions, with $k \geq 2$, taking place either entirely within the dark sector, or involving standard model fields. This translates to a maximum mass of the particle saturating the observed DM abundance, which, for dominantly $s$-wave annihilations, is obtained to be around $130$ TeV, $1$ GeV, $7$ MeV and $110$ keV, for $k=2,3,4$ and $5$, respectively, in a radiation dominated Universe, for a real or complex scalar DM stabilized by a minimal symmetry. For modified thermal histories in the pre-big bang nucleosynthesis era, with an intermediate period of matter domination, values of reheating temperature higher than $\mathcal{O}(200)$ GeV for $k \geq 4$, $\mathcal{O}(1)$ TeV for $k=3$ and $\mathcal{O}(50)$ TeV for $k=2$ are strongly disfavoured by the combined requirements of unitarity and DM relic abundance, for DM freeze-out before reheating.