Cosmological Horizons and Reconstruction of Quantum Field Theories

Abstract

As a starting point, we state some relevant geometrical properties enjoyed by the cosmological horizon of a certain class of Friedmann-Robertson-Walker backgrounds. Those properties are generalised to a larger class of expanding spacetimes M admitting a geodesically complete cosmological horizon ${{\Im^-}}$ common to all co-moving observers. This structure is later exploited in order to recast, in a cosmological background, some recent results for a linear scalar quantum field theory in spacetimes asymptotically flat at null infinity. Under suitable hypotheses on M, encompassing both the cosmological de Sitter background and a large class of other FRW spacetimes, the algebra of observables for a Klein-Gordon field is mapped into a subalgebra of the algebra of observables ${{\mathcal{W}(\Im^-)}}$ constructed on the cosmological horizon. There is exactly one pure quasifree state λ on ${{\mathcal{W}(\Im^-)}}$ which fulfills a suitable energy-positivity condition with respect to a generator related with the cosmological time displacements. Furthermore λ induces a preferred physically meaningful quantum state λ _M for the quantum theory in the bulk. If M admits a timelike Killing generator preserving ${{\Im^-}}$ , then the associated self-adjoint generator in the GNS representation of λ _M has positive spectrum (i.e., energy). Moreover λ _M turns out to be invariant under every symmetry of the bulk metric which preserves the cosmological horizon. In the case of an expanding de Sitter spacetime, λ _M coincides with the Euclidean (Bunch-Davies) vacuum state, hence being Hadamard in this case. Remarks on the validity of the Hadamard property for λ _M in more general spacetimes are presented.