The cosmological tree theorem

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 on any time-dependent spacetime background. 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.

On the IR divergences in de Sitter space: loops, resummation and the semi-classical wavefunction

“…For cosmology, the distinction between classical and quantum effects is somehow obscured in the in-in formalism computation for correlation functions. One explanation is that, as shown by the definition (2.1), the wavefunction is an in-out object like the S-matrix in Minkowski spacetime but with a fixed-time future boundary (see [65,72,73] for elaboration of this perspective). Therefore it is more natural to discuss quantum and classical origins in the wavefunction coefficients.…”

Nonanalyticity and on-shell factorization of inflation correlators at all loop orders

“…The analytic property of dS amplitudes was much less explored than their flat-space counterparts, but this direction has attracted lots of attentions recently. New results include various bootstrap methods using a variety of basic physical principles as input [68][69][70][71][72][73][74][75][76][77][78], the study of unitarity, causality, symmetries, and their implications to analytical structures [79][80][81][82][83][84][85][86], the Mellin-space approach to dS amplitudes [87][88][89][90][91][92], the study of analytical structure and explicit results using techniques of partial Mellin-Barnes (MB) representation [51,[93][94][95], spectral decomposition [96,97], and new results for spinning fields [98][99][100][101] and parity violations [60,[102][103][104]. There are other approaches to explore the analytical structure of dS amplitudes, including the cosmological polytopes [105,106], the scattering equation [107,108].…”