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We study the fundamentals of quantum field theory on a rigid de Sitter space.We show that the perturbative expansion of late-time correlation functions toall orders can be equivalently generated by a non-unitary Lagrangian on aEuclidean AdS geometry. This finding simplifies dramatically perturbativecomputations, as well as allows us to establish basic properties of thesecorrelators, which comprise a Euclidean CFT. We use this to infer the analyticstructure of the spectral density that captures the conformal partial waveexpansion of a late-time four-point function, to derive an OPE expansion, andto constrain the operator spectrum. Generically, dimensions and OPEcoefficients do not obey the usual CFT notion of unitarity. Instead, unitarityof the de Sitter theory manifests itself as the positivity of the spectraldensity. This statement does not rely on the use of Euclidean AdS Lagrangiansand holds non-perturbatively. We illustrate and check these properties byexplicit calculations in a scalar theory by computing first tree-level, andthen full one-loop-resummed exchange diagrams. An exchanged particle appears asa resonant feature in the spectral density which can be potentially useful inexperimental searches.

Analyticity and Unitarity for Cosmological Correlators