Bootstrapping Inflationary Correlators in Mellin Space

TL;DR

This work develops a comprehensive Mellin-space framework for boundary correlators in AdS and dS, revealing that AdS-to-dS analytic continuation is encoded by simple phase factors in the Mellin-Barnes representation. It yields analytic, general expressions for late-time three-point functions with spinning exchanges in dS and, at four points, shows that CPWs fix exchange diagrams up to boundary-condition determined factors, enabling an exact EFT expansion and controlled treatment of (partially-)massless and massive fields across general dimensions. The authors construct explicit results for external scalars with arbitrary integer-spin exchange, provide recursion relations that relate different dimensions and spins, and extract inflationary slow-roll corrections from these de Sitter correlators, including non-analytic, oscillatory signatures in the squeezed limit. The framework unifies AdS and dS scattering data, clarifies the role of boundary conditions, and offers a powerful tool for systematic exploration of de Sitter observables and inflationary cosmology via Mellin-space methods.

Abstract

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.

Figures (8)

• Figure 1: Comparison between a scattering process in AdS and dS. In AdS the boundary condition is imposed at the conformal boundary at $z\to 0$, while in dS it is imposed at the infinite past $\eta \to -\infty$.
• Figure 2: Analytic continuation from Euclidean anti-de Sitter space to de Sitter space. Here $z$ is the radial coordinate in EAdS while $\eta$ is conformal time in dS and we display the two possible analytic continuations from complexified dS.
• Figure 3: A pictorial representation of the dS in-in contours, of bulk-to-boundary propagators on the left and of the split representation for bulk-to-bulk propagators on the right. The horizontal direction parameterise the momentum space coordinates while $\Gamma_{\pm}$ represent the (anti-)time ordered contours which we unfolded above the late time de Sitter horizon to distinguish the path ordering along the contour from the actual time-ordering relations.
• Figure 4: Pole structure of the Mellin-Barnes representation \ref{['exchamp']} for the scalar exchange four-point function, focusing w.l.o.g. on that of the Mellin variable $s_1$. The different coloured "$\bullet$" (red, yellow and blue) denote the different sets of Gamma function poles in $s_1$, while the green line represents the integration contour which w.l.o.g. we take to be indented along the imaginary axis with $\mathfrak{Re}(s_j)=0$. Notice that the poles of the $\csc$ factor, which are in blue, run from $-\infty$ to $+\infty$ and are split by the integration contour according to \ref{['cscident']} with $q=0$. For Principal Series representations, where $\nu_1, \nu \in \mathbb{R}$, the different sets of Gamma function poles do not collide as the red and yellow poles can only move vertically along the imaginary axis as one varies the scaling dimensions. Away from the Principal Series these poles can move horizontally, which for certain scaling dimensions pinches the integration contour, generating singularities. Such cases can be treated by regulating the contour pinching to obtain the analytic continuation of the exchange four-point function for these values of the scaling dimensions, as we shall see in section \ref{['subsec::someparticcases']}.
• Figure 5: OPE limit in momentum space vs OPE limit in position space.