Freeze-in and ultra-relativistic freeze-out during general reheating scenarios

The dark-matter relic abundance can depend sensitively on the thermal history before radiation domination. We derive a general analytic framework for dark-matter production from the Standard Model bath during a non-instantaneous reheating era, unifying freeze-in, ultra-relativistic freeze-out and the approach to ordinary non-relativistic freeze-out. The reheating background is described by an effective equation-of-state parameter omega and a cooling index alpha, while the dark-matter interaction rate is parametrised by an effective scale and a leading temperature power n. We show that the production history is organised by two critical temperature exponents: one controls whether a thermalised relativistic species decouples during reheating or after radiation domination begins, and the other controls whether post-decoupling production is infrared dominated, ultraviolet dominated or logarithmic. We derive analytic relic yields in the main regimes, including both the entropy-diluted freeze-out contribution and the post-freeze-out production term. These results explain the scaling of relic-density contours and are checked against numerical Boltzmann solutions. For matter-like reheating our framework reproduces the known IR/UV ultra-relativistic freeze-out structure, while more general reheating histories can shift the same microscopic interaction between freeze-in, ultra-relativistic freeze-out and ordinary freeze-out regimes.