A Mellin Space Approach to Cosmological Correlators

TL;DR

This work develops a Mellin-space framework to compute late-time momentum-space cosmological correlators in de Sitter space, unifying contact and exchange diagrams for general scalars across arbitrary spacetime dimension $d$. By leveraging split representations, Fourier-space Mellin-Barnes integrals, and the analytic structure of generalized hypergeometric functions (notably Appell $F_4$), the authors obtain compact, analytically tractable expressions for $n$-point contact diagrams and $4$-point exchange diagrams, with clear reductions in special cases such as conformally coupled or massless scalars. The Mellin approach encodes the Operator Product Expansion (OPE) and Effective Field Theory (EFT) expansions via poles and zeros, enabling systematic asymptotics in soft, collapsed-triangle, and high-energy limits, and revealing the interplay between boundary conditions (BD–type phases) and bulk dynamics. The formalism connects naturally to AdS/CFT (Witten diagrams) and offers a robust toolkit for inflationary cosmology, including slow-roll inflaton corrections and bootstrap-esque constraints on late-time de Sitter correlators. Overall, the paper provides a powerful, general method to compute and analyze cosmological correlators in momentum space with wide applicability to theoretical and observational cosmology.

Abstract

We propose a Mellin space approach to the evaluation of late-time momentum-space correlation functions of quantum fields in $\left(d+1\right)$-dimensional de Sitter space. The Mellin-Barnes representation makes manifest the analytic structure of late-time correlators and, more generally, provides a convenient general $d$ framework for the study of conformal correlators in momentum space. In this work we focus on tree-level correlation functions of general scalars as a prototype, including $n$-point contact diagrams and $4$-point exchanges. For generic scalars, both the contact and exchange diagrams are given by (generalised) Hypergeometric functions, which reduce to existing expressions available in the literature for $d=3$ and external scalars which are either simultaneously conformally coupled or massless. This approach can also be used for the perturbative bulk evaluation of momentum-space boundary correlators in $\left(d+1\right)$-dimensional anti-de Sitter space (Witten diagrams).

Figures (9)

• Figure 1: Analytic continuation from EAdS to dS
• Figure 2: Depiction of the split representation for de Sitter propagators on the $++$ and $--$ branches of the in-in contour, for $\eta_1 > \eta_2$, ${\bar{\eta}}_1 > {\bar{\eta}}_2$ (figure (a)) and $\eta_2 > \eta_1$, ${\bar{\eta}}_2 > {\bar{\eta}}_1$ (figure (b)). The arrows along the vertical axis indicate the path along the in-in contour.
• Figure 3: Contact diagram contributing to the correlator $\langle \phi^{(\nu_1)}\phi^{(\nu_2)}\,...\,\phi^{(\nu_n)}\rangle$ at late times $\eta_0 \sim 0$.
• Figure 4: Integration contour (Green) for the Mellin-Barnes representation \ref{['2varrep3pt']} of the three-point conformal structure. W.l.o.g. we focus on the integral in the Mellin variable $s_2$ and take $\mathfrak{Re}\left(s_1\right)=0$. In this figure all scaling dimensions are taken to lie on the Principal Series \ref{['PS']}, $\nu_j \in \mathbb{R}$. Poles of the $\Gamma$-functions $\Gamma\left(s_2\pm \tfrac{i\nu_2}{2}\right)$ are displayed in red and the poles of $\Gamma\left(\tfrac{d}{4}-s_1-s_2\pm\tfrac{i\nu_3}{2}\right)$ in blue, which the integration contour is prescribed to separate.
• Figure 5: Poles of the Mellin representation for the conformal structure \ref{['ccsclnptconfstr']} when the $n$-th scalar does not lie on the Principal Series, $\nu_n=i\mu$ with $\mu \in \mathbb{R}$. For ease of presentation, we display the $n=3$ case. As $\mu$ varies, the poles of $\Gamma\left(\frac{d}{4}\pm \frac{i\nu_n}{2}-s\right)$ (in blue) move along the real axis while the poles of $\Gamma\left(2s-1\right)$ (in red) remain fixed. When $\mu=\frac{d}{2}-1$ these poles collide and the contour separating them (in Green) becomes pinched. The pinching is regulated by sending $d \to d+\epsilon$, with $\epsilon >0$.