The Asymptotic Structure of Cosmological Integrals

TL;DR

This paper analyzes the asymptotic behavior of cosmological integrals in power-law FRW models, revealing a combinatorial structure using polytopes and graph tubings to identify divergences and their degrees.

Contribution

It introduces a novel geometric and combinatorial framework for understanding divergences in cosmological integrals using nestohedra and graph tubings.

Findings

Divergences are characterized by facets of nestohedra linked to graph tubings.

The asymptotic behavior is governed by special classes of polytopes in graph-weight space.

Sector decomposition can be systematically applied using the combinatorial structure.

Abstract

We provide a general analysis of the asymptotic behaviour of perturbative contributions to observables in arbitrary power-law FRW cosmologies, indistinctly the Bunch-Davies wavefunction and cosmological correlators. We consider a large class of scalar toy models, including conformally-coupled and massless scalars in arbitrary dimensions, that admits a first principle definition in terms of (generalised/weighted) cosmological polytopes. The perturbative contributions to an observable can be expressed as an integral of the canonical function associated to such polytopes and to weighted graphs. We show how the asymptotic behaviour of these integrals is governed by a special class of nestohedra living in the graph-weight space, both at tree and loop level. As the singularities of a cosmological process described by a graph can be associated to its subgraphs, we provide a realisation of the nestohedra as a sequential truncation of a top-dimensional simplex based on the underlying graph. This allows us to determine all the possible directions -- both in the infrared and in the ultraviolet --, where the integral can diverge as well as their divergence degree. Both of them are associated to the facets of the nestohedra, which are identified by overlapping tubings of the graph: the specific tubing determines the divergent directions while the number of overlapping tubings its degree of divergence. This combinatorial formulation makes straightforward the application of sector decomposition for extracting both leading and subleading divergences from the integral, as the sectors in which the integration domain can be tiled are identified by the collection of compatible facets of the nestohedra, with the latter that can be determined via the graph tubings. Finally, the leading divergence can be interpreted as a restriction of the canonical function of the relevant polytope onto a special hyperplane.