  We consider late-time correlators in de Sitter (dS) space for initial states
related to the Bunch-Davies vacuum by a Bogoliubov transformation. We propose
to study such late-time correlators by reformulating them in the familiar
language of Witten diagrams in Euclidean anti-de Sitter space (EAdS), showing
that they can be perturbatively re-cast in terms of corresponding dS boundary
correlators in the Bunch-Davies vacuum and in turn, Witten diagrams in EAdS.
Unlike the standard relationship between late-time correlators in the
Bunch-Davies vacuum and EAdS Witten diagrams, this involves points on the upper
and lower sheet of the EAdS hyperboloid which account for antipodal
singularities of the two-point functions. Such Bogoliubov states include an
infinite one parameter family of de Sitter invariant vacua as a special case,
where the late-time correlators are constrained by Conformal Ward identities.
In momentum space, it is well known that their late-time correlators exhibit
singularities in collinear ("folded") momentum configurations. We give a
position space interpretation of such solutions to the conformal Ward
identities, where in embedding space they can be generated from the solution
without collinear singularities by application of the antipodal map. We also
discuss the operator product expansion (OPE) limit of late-time correlators in
a generic dS invariant vacuum. Many results are derived using the Mellin space
representation of late-time correlators, which in this work we extend to
accommodate generic dS invariant vacua.
