Algebraic Approaches to Cosmological Integrals

TL;DR

This paper explores algebraic methods to analyze cosmological integrals, revealing their singularity structures and proposing algorithms for their decomposition, which enhances understanding of the universe's initial conditions.

Contribution

It introduces algebraic techniques to study the differential equations and singularities of cosmological correlators, and proposes a graph-based algorithm for multivariate partial fractioning.

Findings

Characterization of singularities via hyperplane arrangements

Derivation of differential and difference equations for integrals

Development of a graph-based algorithm for partial fractioning

Abstract

Cosmological correlators encode statistical properties of the initial conditions of our universe. Mathematically, they can often be written as Mellin integrals of a certain rational function associated to graphs, namely the flat space wavefunction. The singularities of these cosmological integrals are parameterized by binary hyperplane arrangements. Using different algebraic tools, we shed light on the differential and difference equations satisfied by these integrals. Moreover, we study a multivariate version of partial fractioning of the flat space wavefunction, and propose a graph-based algorithm to compute this decomposition.