We discuss equilibration process in expanding universes as compared to the thermalization process in Minkowski space--time. The final goal is to answer the following question: Is the equilibrium reached before the rapid expansion stops and quantum effects have a negligible effect on the background geometry or stress--energy fluxes in a highly curved early Universe have strong effects on the expansion rate and the equilibrium is reached only after the drastic decrease of the space--time curvature? We argue that consideration of more generic non--invariant states in theories with invariant actions is a necessary ingredient to understand quantum field dynamics in strongly curved backgrounds. We are talking about such states in which correlation functions are not functions of such isometry invariants as geodesic distances, while having correct UV behaviour. The reason to consider such states is the presence of IR secular memory effects for generic time dependent backgrounds, which are totally absent in equilibrium. These effects strongly affect the destiny of observables in highly curved space--times.
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