Bootstrapping Inflationary Correlators in Mellin Space

We develop a Mellin space approach to boundary correlation functions in anti-de Sitter (AdS) and de Sitter (dS) spaces. Using the Mellin-Barnes representation of correlators in Fourier space, we show that the analytic continuation between AdS$_{d+1}$ and dS$_{d+1}$ is encoded in a collection of simple relative phases. This allows us to determine the late-time tree-level three-point correlators of spinning fields in dS$_{d+1}$ from known results for Witten diagrams in AdS$_{d+1}$ by multiplication with a simple trigonometric factor. At four point level, we show that Conformal symmetry fixes exchange four-point functions both in AdS$_{d+1}$ and dS$_{d+1}$ in terms of the dual Conformal Partial Wave (which in Fourier space is a product of boundary three-point correlators) up to a factor which is determined by the boundary conditions. In this work we focus on late-time four-point correlators with external scalars and an exchanged field of integer spin-$\ell$. The Mellin-Barnes representation makes manifest the analytic structure of boundary correlation functions, providing an analytic expression for the exchange four-point function which is valid for general $d$ and generic scaling dimensions, in particular massive, light and (partially-)massless fields. When $d=3$ we reproduce existing explicit results available in the literature for external conformally coupled and massless scalars. From these results, assuming the weak breaking of the de Sitter isometries, we extract the corresponding correction to the inflationary three-point function of general external scalars induced by a general spin-$\ell$ field at leading order in slow roll. These results provide a step towards a more systematic understanding of de Sitter observables at tree level and beyond using Mellin space methods.