  Cosmological correlators, the natural observables of the primordial universe,
have been extensively studied in the past two decades using the in-in formalism
pioneered by Schwinger and Keldysh for the study of dissipative open systems.
Ironically, most applications in cosmology have focused on non-dissipative
closed systems. We show that, for non-dissipative systems, correlators can be
equivalently computed using the in-out formalism with the familiar Feynman
rules. In particular, the myriad of in-in propagators is reduced to a single
(Feynman) time-ordered propagator and no sum over the labelling of vertices is
required. In de Sitter spacetime, this requires extending the expanding
Poincar\'e patch with a contracting patch, which prepares the bra from the
future. Our results are valid for fields of any mass and spin but assuming the
absence of infrared divergences. We present three applications of the in-out
formalism: a representation of correlators in terms of a sum over residues of
Feynman propagators in the energy-momentum domain; an algebraic recursion
relation that computes Minkowski correlators in terms of lower order ones; and
the derivation of cutting rules from Veltman's largest time equation, which we
explicitly develop and exemplify for two-vertex diagrams to all loop orders.
The in-out formalism leads to a natural definition of a de Sitter scattering
matrix, which we discuss in simple examples. Remarkably, we show that our
scattering matrix satisfies the standard optical theorem and the positivity
that follows from it in the forward limit.
