The Cosmological Tree Theorem

TL;DR

This paper introduces a new cosmological tree theorem derived from causality conditions, enabling the systematic expansion of loop diagrams into tree diagrams, simplifying calculations of cosmological correlators.

Contribution

It presents a novel set of cutting rules for Feynman-Witten diagrams that generalize Feynman's tree theorem to cosmology, reducing complex loop calculations to simpler tree-level diagrams.

Findings

Certain singularities are canceled in equal-time correlators.

The new rules eliminate the need for nested time integrals.

Tree-level exchange diagrams are fully determined by contact diagrams.

Abstract

A number of diagrammatic "cutting rules" have recently been developed for the wavefunction of the Universe which determines cosmological correlation functions. These leverage perturbative unitarity to relate particular "discontinuities" in Feynman-Witten diagrams (with cosmological boundary conditions) to simpler diagrams, in much the same way that the Cutkosky rules relate different scattering amplitudes. In this work, we make use of a further causality condition to derive new cutting rules for Feynman-Witten diagrams. These lead to the cosmological analogue of Feynman's tree theorem for amplitudes, which can be used to systematically expand any loop diagram in terms of (momentum integrals of) tree-level diagrams. As an application of these new rules, we show that certain singularities in the wavefunction cannot appear in equal-time correlators due to a cancellation between "real" and "virtual" contributions that closely parallels the KLN theorem. Finally, when combined with the Bunch-Davies condition that certain unphysical singularities are absent, these cutting rules completely determine any tree-level exchange diagram in terms of simpler contact diagrams. Altogether, these results remove the need to ever perform nested time integrals when computing cosmological correlators.