On the Equivalence between Euclidean and In-In Formalisms in de Sitter QFT

We study the relation between two sets of correlators in interact-ing quantum field theory on de Sitter space. The first are correlators computed using in-in perturbation theory in the expanding cosmo-logical patch of de Sitter space (also known as the conformal patch, or the Poincare ́ patch), and for which the free propagators are taken to be those of the free Euclidean vacuum. The second are correla-tors obtained by analytic continuation from Euclidean de Sitter; i.e., they are correlators in the fully interacting Hartle-Hawking state. We give an analytic argument that these correlators coincide for inter-acting massive scalar fields with any m2> 0. We also verify this result via direct calculation in simple examples. The correspondence holds diagram by diagram, and at any finite value of an appropriate Pauli-Villars regulator mass M. Along the way, we note interesting connections between various prescriptions for perturbation theory in general static spacetimes with bifurcate Killing horizons. 1 ar X i