  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 to
all orders can be equivalently generated by a non-unitary Lagrangian on a
Euclidean AdS geometry. This finding simplifies dramatically perturbative
computations, as well as allows us to establish basic properties of these
correlators, which comprise a Euclidean CFT. We use this to infer the analytic
structure of the spectral density that captures the conformal partial wave
expansion of a late-time four-point function, to derive an OPE expansion, and
to constrain the operator spectrum. Generically, dimensions and OPE
coefficients do not obey the usual CFT notion of unitarity. Instead, unitarity
of the de Sitter theory manifests itself as the positivity of the spectral
density. This statement does not rely on the use of Euclidean AdS Lagrangians
and holds non-perturbatively. We illustrate and check these properties by
explicit calculations in a scalar theory by computing first tree-level, and
then full one-loop-resummed exchange diagrams. An exchanged particle appears as
a resonant feature in the spectral density which can be potentially useful in
experimental searches.
