Cosmological Scattering Equations

TL;DR

The paper introduces a worldsheet formulation for cosmological correlators in de Sitter space by defining cosmological scattering equations in momentum space from conformal generators and constructing a Pfaffian-based integrand, yielding a compact expression for even $n$. In the flat-space limit, the construction reproduces the CHY $\phi^4$ amplitudes, establishing a bridge between cosmological correlators and flat-space techniques. The authors validate the framework with explicit four- and six-point checks via the global residue theorem and derive a ladder-recursion for general $n$, showing how tree-level Witten diagrams factorize through bulk-to-bulk propagators. This work opens a path toward applying amplitude-inspired methods to cosmology, with potential extensions to spinning correlators, broader interactions, and UV-complete formulations.

Abstract

We propose a worldsheet formula for tree-level correlation functions describing a scalar field with arbitrary mass and quartic self-interaction in de Sitter space, which is a simple model for inflationary cosmology. The correlation functions are located on the future boundary of the spacetime and are Fourier-transformed to momentum space. Our formula is supported on mass-deformed scattering equations involving conformal generators in momentum space and reduces to the CHY formula for $φ^4$ amplitudes in the flat space limit. Using the global residue theorem, we verify that it reproduces the Witten diagram expansion at four and six points, and sketch the extension to $n$ points.

Figures (4)

• Figure 1: Factorization of ${\cal A}(\mathbb{1}_4)$ with (a) ${\rm Pf} A^{14}_{14}$, (b) ${\rm Pf} A^{12}_{12}$.
• Figure 2: Diagrammatic representation of ${\cal A}(\mathbb{1}_6)$.
• Figure 3: Factorization contribution.
• Figure 4: Factorization of (a) a ladder diagram, (b) a non-ladder diagram.