  Extending scattering to states with unphysical mass values (particles ``off
their mass shell'') has been instrumental in developing modern amplitude
technology for Minkowski spacetime. Here, we study the off-shell correlators
which underpin the recently proposed S-matrix for scattering on de Sitter
spacetime. By labelling each particle with both a spatial momentum and an
independent ``energy'' variable (the de Sitter analogue of a 4-momentum), we
find that the practical computation of these correlators is greatly simplified.
This allows us to derive compact expressions for all 3- and 4-particle
S-matrices at tree-level for scalar fields coupled through any derivative
interactions. As on Minkowski, we find that the 3-particle and exchange part of
the 4-particle S-matrices are unique (up to crossing). The remaining contact
part of the 4-particle S-matrix is an analytic function of just two
differential operators, which become the usual Mandelstam variables in the
Minkowski limit. Finally, we introduce a spectral decomposition for the
tree-level exchange of a heavy field responsible for a cosmological collider
signal. Once projected onto physical mass eigenstates, these S-matrix elements
encode the statistical properties of the early inflationary perturbations.
