From AdS to dS Exchanges: Spectral Representation, Mellin Amplitudes and Crossing

TL;DR

The paper establishes a simple, general relation between tree-level exchanges in AdS and dS using a Mellin-Barnes representation of boundary correlators, enabling direct import of AdS Witten-diagram techniques to in-in correlators on the dS boundary. It defines Mellin amplitudes and a spectral representation for dS exchanges and derives conformal block decompositions in direct and crossed channels from their AdS counterparts, with explicit BD-vacuum normalisations. The key result expresses any dS exchange as a linear combination of AdS exchanges, weighted by sinusoidal factors, and this holds both in position and momentum space, providing a concrete bridge between AdS/CFT methods and the Cosmological Bootstrap program. This framework enables systematic analysis of dS correlators, clarifies the role of vacuum choice, and offers a pathway to connect with flat-space limits via Mellin amplitudes.

Abstract

We present a simple general relation between tree-level exchanges in AdS and dS. This relation allows to directly import techniques and results for AdS Witten diagrams, both in position and momentum space, to boundary correlation functions in dS. In this work we apply this relation to define Mellin amplitudes and a spectral representation for exchanges in dS. We also derive the conformal block decomposition of a dS exchange, both in the direct and crossed channels, from their AdS counterparts. The relation between AdS and dS exchanges itself is derived using a recently introduced Mellin-Barnes representation for boundary correlators in momentum space, where (A)dS exchanges are straightforwardly fixed by a combination of factorisation, conformal symmetry and boundary conditions.