Cosmological Cutting Rules

TL;DR

This work formalizes a universal set of Cosmological Cutting Rules that arise from unitarity and apply to the wavefunction of the universe in FLRW spacetimes. The rules express the discontinuity of an n-loop wavefunction coefficient in terms of lower-loop (or lower-point) data by summing over all possible internal-line cuts and incorporating boundary power spectra, dramatically simplifying loop computations. By demonstrating explicit Minkowski, de Sitter, and EFT-inflation examples, the approach shows how tree-level calculations can fix one-loop corrections and yield perturbative unitarity bounds, enhancing the bootstrap program for cosmological observables. The framework opens new paths to constrain EFTs in curved spacetime and to connect bulk unitarity with boundary correlators, enabling precise, non-perturbative insights in cosmology.

Abstract

Primordial perturbations in our universe are believed to have a quantum origin, and can be described by the wavefunction of the universe (or equivalently, cosmological correlators). It follows that these observables must carry the imprint of the founding principle of quantum mechanics: unitary time evolution. Indeed, it was recently discovered that unitarity implies an infinite set of relations among tree-level wavefunction coefficients, dubbed the Cosmological Optical Theorem. Here, we show that unitarity leads to a systematic set of "Cosmological Cutting Rules" which constrain wavefunction coefficients for any number of fields and to any loop order. These rules fix the discontinuity of an n-loop diagram in terms of lower-loop diagrams and the discontinuity of tree-level diagrams in terms of tree-level diagrams with fewer external fields. Our results apply with remarkable generality, namely for arbitrary interactions of fields of any mass and any spin with a Bunch-Davies vacuum around a very general class of FLRW spacetimes. As an application, we show how one-loop corrections in the Effective Field Theory of inflation are fixed by tree-level calculations and discuss related perturbative unitarity bounds. These findings greatly extend the potential of using unitarity to bootstrap cosmological observables and to restrict the space of consistent effective field theories on curved spacetimes.

Figures (13)

• Figure 1: An example of the Cosmological Cutting Rules, applied to a particular diagram that contributes to the wavefunction of the universe.
• Figure 2: A graphical representation of the Feynman rules to compute the wavefunction of the universe in perturbation theory.
• Figure 3: An internal line between the largest time vertex $\bar{t}$ and another time $t_j$ is either (i) connecting two otherwise disconnected components, or (ii) forming part of a loop (such that the graph remains connected once it is removed). The pairwise additions shown correspond to \ref{['eqn:PairwiseSum(i)']} and \ref{['eqn:PairwiseSum(i)']} respectively.
• Figure 4: There are four ways to cut two propagators attached to the largest time vertex, $\bar{t}$. They can be paired together as shown in the first and second lines, which amputates all of the $t_1$ dependence. The constants of proportionality are the same, and so adding the two diagrams as on the third line shows that this sum vanishes (see \ref{['eqn:tree_eg_1']} and \ref{['eqn:tree_eg_2']}).
• Figure 5: Diagrammatic representation of \ref{['eqn:LoopCuttingRuleProofEg1']} and \ref{['eqn:LoopCuttingRuleProofEg']}, showing the pairwise sum of two loop diagrams that differ only by the cut of a single line. The two terms on the right-hand-sides exactly cancel.