Canonical Differential Equations for Cosmology from Positive Geometries

TL;DR

This paper introduces a new method to derive canonical differential equations for cosmological wavefunction coefficients using positive geometries and graph combinatorics, offering a systematic approach for analyzing cosmological correlators.

Contribution

It develops a general technique to obtain differential equations from twisted integrals of logarithmic forms, applied to functions labeled by Feynman graph tubings, unifying combinatorial and geometric aspects.

Findings

Derived differential equations for cosmological wavefunctions.

Established a uniform combinatorial framework for these equations.

Provided explicit examples and conjectured general applicability.

Abstract

Cosmological correlation functions are central observables in modern cosmology, as they encode properties of the early universe. In this paper, we derive novel canonical differential equations for wavefunction coefficients in power-law FRW cosmologies by combining positive geometries and the combinatorics of tubings of Feynman graphs. First, we establish a general method to derive differential equations for any function given as a twisted integral of a logarithmic differential form. By using this method on a natural set of functions labelled by tubings of a given Feynman diagram, we derive a closed set of differential equations in the canonical form. The coefficients in these equations are related to region variables with the same notion of tubings, providing a uniform combinatorial description of the system of equations. We provide explicit results for specific examples and conjecture that this approach works for any graph.