Singularities in Cosmological Loop Correlators II : Non Local Interactions and Flat Space limits

TL;DR

This paper investigates the impact of non-local interactions arising from the ADM formalism on 1-loop cosmological correlators during inflation, establishing a connection between their singularities and flat-space scattering amplitudes.

Contribution

It extends previous analysis to include non-local operators in loop corrections, demonstrating the validity of diagrammatic rules and deriving the flat-space limit of inflationary correlators.

Findings

Non-local vertices introduce complex angular dependencies in loop integrals.

Diagrammatic rules for poles and branch cuts remain valid with non-local interactions.

Flat-space limit of 1-loop correlator matches scattering amplitude singularities.

Abstract

Non-local interactions naturally arise in the ADM formalism after solving the constraint equations and substituting their solutions back into the action. However, the effects of these non-local operators on loop corrections to cosmological correlators remain largely unexplored. Extending the analysis of arXiv:2503.21880 , in this work we compute the $1$-loop bispectrum during slow-roll inflation, including the inverse-Laplacian operators present at cubic and quartic order in the ADM action (in the spatially flat gauge). These non-local vertices introduce non-trivial angular dependencies that significantly complicate the evaluation of loop integrals. We find that the diagrammatic rules of arXiv:2503.21880 for identifying poles and branch cuts in correlators without explicit integration remain valid even in the presence of such non-local interactions. Finally, we derive the flat-space limit of the $1$-loop inflationary correlator and verify it with explicit examples, thereby establishing a direct correspondence between its leading total-energy singularities ($\omega_T\rightarrow0$) and the flat-space scattering amplitude.