The Cosmological Bootstrap: Inflationary Correlators from Symmetries and Singularities

TL;DR

This work develops a cosmological bootstrap framework that constrains inflationary correlators using boundary conformal symmetry and singularity structure in de Sitter space. By solving for the four-point function of conformally coupled scalars with massive-scalar exchange, the authors generate a complete set of building blocks that, through spin-raising and weight-shifting operators, yield all inflationary three- and four-point functions for weakly broken conformal symmetry. A key result is that the inflationary bispectrum from arbitrary spin exchange is determined by the soft limit of the simplest scalar-exchange four-point function plus a collection of contact terms, enabling a compact and physically motivated template basis for cosmological non-Gaussianity and collider-like signals. The analysis also reveals a deep link between boundary correlators and flat-space scattering amplitudes via analytic continuation, connecting cosmological collider physics to high-energy amplitude structures. The framework offers a principled route to systematically classify inflationary signals and to explore UV completions through how k_t→0 and related limits encode high-energy physics.

Abstract

Scattering amplitudes at weak coupling are highly constrained by Lorentz invariance, locality and unitarity, and depend on model details only through coupling constants and particle content. In this paper, we develop an understanding of inflationary correlators which parallels that of flat-space scattering amplitudes. Specifically, we study slow-roll inflation with weak couplings to extra massive particles, for which all correlators are controlled by an approximate conformal symmetry on the boundary of the spacetime. After classifying all possible contact terms in de Sitter space, we derive an analytic expression for the four-point function of conformally coupled scalars mediated by the tree-level exchange of massive scalars. Conformal symmetry implies that the correlator satisfies a pair of differential equations with respect to spatial momenta, encoding bulk time evolution in purely boundary terms. The absence of unphysical singularities completely fixes this correlator. A spin-raising operator relates it to the correlators associated with the exchange of particles with spin, while weight-shifting operators map it to the four-point function of massless scalars. We explain how these de Sitter four-point functions can be perturbed to obtain inflationary three-point functions. We reproduce many classic results in the literature and provide a complete classification of all inflationary three- and four-point functions arising from weakly broken conformal symmetry. The inflationary bispectrum associated with the exchange of particles with arbitrary spin is completely characterized by the soft limit of the simplest scalar-exchange four-point function of conformally coupled scalars and a series of contact terms. Finally, we demonstrate that the inflationary correlators contain flat-space scattering amplitudes via a suitable analytic continuation of the external momenta.

Figures (13)

• Figure 1: Cosmological observations can be traced back to the end of inflation where they become spatial correlations on the boundary of the approximate de Sitter spacetime.
• Figure 2: Schematic illustration of the logical connections between the different parts of the paper.
• Figure 3: The four-point point function arising from tree exchange in the $s$-channel satisfy a pair of ordinary differential equations (\ref{['equ:exchange']}), that determine the dependence as $u \propto (k_1 + k_2)^{-1}$ is varied, or as $v \propto (k_3 + k_4)^{-1}$ is varied. These correspond to two different "holographic" pictures for time evolution, which are nontrivially mutually consistent.
• Figure 4: Illustration of two important singularities of the solution $\hat{F}(u,v)$. The singularity in the collinear limit should be absent for the adiabatic vacuum. The singularity in the factorization channel is an avatar of the standard factorization of the scattering amplitude and therefore expected to be present.
• Figure 5: Illustration of the analytic structure of the function $\hat{F}(u,v)$.