From AdS to dS exchanges: Spectral representation, Mellin amplitudes, and crossing

In this paper we take the first steps towards bridging the gap between the conformal and cosmological bootstrap programs. We present a simple general relation between tree-level exchanges in anti-de Sitter (AdS) and de Sitter (dS) space with Bunch-Davies intial conditions, providing a complete AdS to dS dictionary at tree-level. This allows us to directly import techniques and results for conformal field theory correlators dual to AdS-Witten diagrams to boundary correlation functions in dS. We apply this relation to extend various indispensable tools from the conformal bootstrap to boundary correlation functions in dS, including Mellin amplitudes and the spectral representation for dS exchanges. We also derive their conformal block decomposition, 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 factorization, conformal symmetry and boundary conditions.