From dS to AdS and back

TL;DR

This work establishes a comprehensive, perturbative bridge between boundary correlators in de Sitter space and Euclidean anti-de Sitter space by showing that, under Bunch-Davies initial conditions, any dS diagram is a precise linear combination of Euclidean AdS Witten diagrams with coefficients fixed by on-shell factorisation and sinusoidal BD phases. The authors deploy a Mellin-Barnes representation to make factorisation, dispersion, and unitarity transparent, enabling the import of AdS techniques such as conformal partial waves and inversion formulas to dS. They extend these results to arbitrary integer-spin fields, providing a unified set of rules for decomposing dS diagrams into AdS building blocks, including trees and loops, and they analyze unitarity constraints that differentiate dS from AdS correlators. The analytic structure revealed by this approach also opens paths to applying Lorentzian AdS methods to dS diagrams, and to formulating boundary bootstrap-style approaches for cosmological correlators in a controlled, perturbative setting.

Abstract

We describe in more detail the general relation uncovered in our previous work between boundary correlators in de Sitter (dS) and in Euclidean anti-de Sitter (EAdS) space, at any order in perturbation theory. Assuming the Bunch-Davies vacuum at early times, any given diagram contributing to a boundary correlator in dS can be expressed as a linear combination of Witten diagrams for the corresponding process in EAdS, where the relative coefficients are fixed by consistent on-shell factorisation in dS. These coefficients are given by certain sinusoidal factors which account for the change in coefficient of the contact sub-diagrams from EAdS to dS, which we argue encode (perturbative) unitary time evolution in dS. dS boundary correlators with Bunch-Davies initial conditions thus perturbatively have the same singularity structure as their Euclidean AdS counterparts and the identities between them allow to directly import the wealth of techniques, results and understanding from AdS to dS. This includes the Conformal Partial Wave expansion and, by going from single-valued Witten diagrams in EAdS to Lorentzian AdS, the Froissart-Gribov inversion formula. We give a few (among the many possible) applications both at tree and loop level. Such identities between boundary correlators in dS and EAdS are made manifest by the Mellin-Barnes representation of boundary correlators, which we point out is a useful tool in its own right as the analogue of the Fourier transform for the dilatation group. The Mellin-Barnes representation in particular makes manifest factorisation and dispersion formulas for bulk-to-bulk propagators in (EA)dS, which imply Cutkosky cutting rules and dispersion formulas for boundary correlators in (EA)dS. Our results are completely general and in particular apply to any interaction of (integer) spinning fields.

Figures (19)

• Figure 1: Throughout we represent EAdS pictorially as a Poincaré disk with grey circular boundary. Left: Bulk-to-boundary propagator in EAdS with boundary condition $\Delta$, where $\Delta=\Delta^\pm$. Right: Bulk-to-bulk propagator in EAdS with $\Delta^\pm$ boundary condition.
• Figure 2: Left: Diagrammatic representation of the bulk-to-boundary propagator \ref{['adsodsbubo']} in dS where the point in the bulk sits in the $\pm$ branch of the in-in contour. Right: Bulk-to-bulk propagator in dS for a field of mass \ref{['dSmass']}, where one bulk point sits on the $\pm$ branch of the in-in contour and the other on the ${\hat{\pm}}$ branch.
• Figure 3: This figure displays the Wick rotations \ref{['wickinin']} from the perspective of the complex $z$ plane (left) and the complex $\left(-\eta\right)$ plane (right). Left: Starting from the flat slicing \ref{['adsflatslice']} of EAdS, where $z \in \left[0,\infty\right)$, by Wick rotating anti-clockwise one lands on the $+$ branch of the in-in contour in the flat slicing \ref{['dsflatslice']} of dS and by Wick rotating clockwise one lands on the $-$ branch of the in-in contour. Right: From the $-(+)$ branch of the in-in contour one arrives to EAdS by rotating (anti-)clockwise. Similar figures were given in Sleight:2019mgdSleight:2019hfp, where further details can also be found.
• Figure 4: At the level of the Mellin-Barnes representation the relation between bulk-to-boundary propagators and harmonic functions in EAdS and the in-in formalism of dS amounts to a multiplication by a phase, which implements the Wick rotations \ref{['wickinin']}, and by a constant $c^{\text{dS-AdS}}_{\Delta^\pm}$ which accounts for the change in two-point function coefficient from EAdS to dS.
• Figure 5: In both plots the blue dots represent the poles \ref{['polesMBharm']} in the internal Mellin variables $u$ and ${\bar{u}}$. Left: The crosses mark the zeros of the projector $\omega_{\Delta^+}\left(u,{\bar{u}}\right)$ which overlap with the poles of $\Omega^{\text{AdS}}_{\mu,J}$ in the upper-half plane that violate the $\Delta^+$ boundary condition. Right: The crosses mark the zeros of the projector $\omega_{\Delta^-}\left(u,{\bar{u}}\right)$ which overlap with the poles of $\Omega^{\text{AdS}}_{\mu,J}$ in the lower-half plane that violate the $\Delta^-$ boundary condition.