Abstract
Roger Penrose's Weyl curvature hypothesis states that the Weyl curvature is small at past singularities, but not at future singularities. We review the motivations for this conjecture and present estimates for the entropy of our Universe. We then extend this hypothesis to the quantum regime by demanding that the initial state of primordial quantum fluctuations be the adiabatic vacuum in a (quasi-)de Sitter space. We finally attempt a justification of this quantum version from a fundamental theory of quantum gravity and speculate on its consequences in the case of a classically recollapsing universe.
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