We study transition rates and cross sections from first principles in a spatially flat radiation dominated cosmology. We consider a model of scalar particles to study scattering and heavy particle production from pair annihilation, drawing more general conclusions. The S-matrix formulation is ill suited to study these ubiquitous processes in a rapidly expanding cosmology. We introduce a physically motivated adiabatic expansion that relies on wavelengths much smaller than the particle horizon at a given time. The leading order in this expansion dominates the transition rates and cross sections. Several important and general results are direct consequences of the cosmological redshift and a finite particle horizon: (i) a violation of local Lorentz invariance, (ii) freeze-out of the production cross section at a finite time, (iii) sub-threshold production of heavier particles as a consequence of the uncertainty in the local energy from a finite particle horizon, a manifestation of the anti-Zeno effect. If heavy dark matter is produced via annihilation of a lighter species, sub-threshold production yields an enhanced abundance. We discuss several possible cosmological consequences of these effects.