  We describe in more detail the general relation uncovered in our previous
work between boundary correlators in de Sitter (dS) and in Euclidean anti-de
Sitter (EAdS) space, at any order in perturbation theory. Assuming the
Bunch-Davies vacuum at early times, any given diagram contributing to a
boundary correlator in dS can be expressed as a linear combination of Witten
diagrams for the corresponding process in EAdS, where the relative coefficients
are fixed by consistent on-shell factorisation in dS. These coefficients are
given by certain sinusoidal factors which account for the change in coefficient
of the contact sub-diagrams from EAdS to dS, which we argue encode
(perturbative) unitary time evolution in dS. dS boundary correlators with
Bunch-Davies initial conditions thus perturbatively have the same singularity
structure as their Euclidean AdS counterparts and the identities between them
allow to directly import the wealth of techniques, results and understanding
from AdS to dS. This includes the Conformal Partial Wave expansion and, by
going from single-valued Witten diagrams in EAdS to Lorentzian AdS, the
Froissart-Gribov inversion formula. We give a few (among the many possible)
applications both at tree and loop level. Such identities between boundary
correlators in dS and EAdS are made manifest by the Mellin-Barnes
representation of boundary correlators, which we point out is a useful tool in
its own right as the analogue of the Fourier transform for the dilatation
group. The Mellin-Barnes representation in particular makes manifest
factorisation and dispersion formulas for bulk-to-bulk propagators in (EA)dS,
which imply Cutkosky cutting rules and dispersion formulas for boundary
correlators in (EA)dS. Our results are completely general and in particular
apply to any interaction of (integer) spinning fields.

High Energy Physics - Theory;General Relativity and Quantum Cosmology