On one-loop corrections to the Bunch-Davies wavefunction of the universe

TL;DR

This paper investigates one-loop quantum corrections to the Bunch-Davies wavefunction in cosmology, revealing polylogarithmic and elliptic structures and analyzing their mathematical properties using cosmological polytopes.

Contribution

It introduces a systematic analysis of one-loop corrections in scalar toy models using cosmological polytopes and explores their complex mathematical structures, including elliptic functions.

Findings

Two-site corrections are expressible with multiple-polylogarithms.

Three-site corrections involve elliptic functions, indicating more complex structures.

The scattering amplitude limit confirms known polylogarithmic results.

Abstract

Understanding the loop corrections to cosmological observables is of paramount importance for having control on the quantum consistency of a theory in an expanding universe as well as for phenomenological reasons. In the present work, we begin with a systematic study of such corrections in the context scalar toy models whose perturbative Bunch-Davies wavefunction enjoys an intrinsic definition in terms of cosmological polytopes, focusing on one-loop graphs. Owing to the underlying twisted period integral representation they admit, their combinatorial structure along with their vector space structure, emerging from polynomial ideals algebra and intersection theory, are exploited to set-up and analyse the differential equations that the two- and three-site one-loop corrections have to satisfy upon variation of the external kinematic variables. We find that, while the two-site contribution can be written in terms of multiple-polylogarithms, this is no longer true for the three-site case, for which elliptic structures appear. As a non-trivial check, we consider the scattering amplitude limit, recovering the known result in terms of polylogarithms only.