Records from the S-Matrix Marathon: Observables in Expanding Universes

TL;DR

This paper reviews recent advances in understanding observables in expanding universes, focusing on their mathematical structure, the Bunch--Davies wavefunctional, and the combinatorics of cosmological polytopes and nestohedra, with implications for infrared divergences.

Contribution

It introduces a new integral representation for the Bunch--Davies wavefunctional and explores the combinatorial structures underlying cosmological observables.

Findings

Integral representation free of spurious poles

Connection between cosmological polytopes and nestohedra

Insights into infrared divergences in expanding backgrounds

Abstract

Observables in expanding universes are crucial to understand the physics of the early universe. In these lectures, we review some recent progress in understanding their mathematical structure and extract the physics encoded in them. After discussing the most salient features of an expanding background and their consequences for defining an observable, we focus on the so-called Bunch--Davies wavefunctional. We analyze its analytic properties on general grounds and introduce an integral representation for it in perturbation theory for a special class of scalar toy models. We discuss both the diagrammatics associated to the usual Feynman rules and combinatorial rules on the graphs, which generate a representation free of spurious poles. Such combinatorial rules find their origin in the combinatorics of the cosmological polytopes of which we provide a gentle introduction to its definition and its main features. Finally, the combinatorics of the cosmological polytopes turns out to determine the combinatorics of a special class of nestohedra that encode the asymptotic behaviour of the cosmological integrals. We provide a general description of such structures and behaviour, which is of crucial importance to understand the infrared divergences which plague observables in an expanding background. These notes are based on a series of lectures held during the S-Matrix Marathon workshop at the Institute for Advanced Study on 11--22 March 2024.