An apparently innocuous property of nature is the universal alignment of the arrow of time for all physical systems, from the subatomic world to the overall cosmological dynamical evolution. This seems to point towards an explanation of irreversibility as the manifestation of a universal intrinsic property, shared by physical systems of arbitrary scales. A major step towards an understanding of this property was taken in quantum mechanics. This required no alteration of the theory itself, but an extension of its mathematical formalism. A natural attitude is therefore to adapt this procedure to cosmological considerations. The present essay is precisely devoted to the description of the peculiarities of this interpretation of the irreversible dynamical behaviour of the cosmological history.