Canonical Differential Equations for Cosmology from Positive Geometries

TL;DR

This work develops canonical differential equations for cosmological wavefunction coefficients in power-law FRW spacetimes by marrying positive-geometry methods with the combinatorics of graph tubings. It constructs master integrals as twisted logarithmic forms in the graph variables, yielding a closed, $\epsilon$-form system whose coefficients are governed by region variables associated to graph regions. The framework unifies flat-space and FRW results through twisted integrals and provides explicit diagonalisation in terms of tubings, with concrete examples for tree and star graphs. The approach promises general applicability to any graph, linking cosmological correlators to cosmological polytopes and graph-associahedra via a transparent, combinatorial, and geometric structure with potential computational advantages. Overall, the paper presents a principled, geometry-driven method to generate and solve differential equations for FRW wavefunctions, leveraging canonical forms and region-variable alphabets for scalable cosmological computations.

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.

Figures (7)

• Figure 1: Examples of Feynman graphs for wavefunction coefficients at tree level (left) and one loop (right).
• Figure 2: An example of $b$-tubes that are (a) compatible and (b) not compatible on the star graph with four vertices. An example of (c) a $b$-tubing and (d) a maximal $b$-tubing.
• Figure 3: An example of (a) a $u$-tubing and (b) a maximal $u$-tubing on the star graph with four vertices.
• Figure 4: Two examples of $c$-tubings on the star graph with four vertices. The tubing in figure (b) is maximal, while the one in figure (a) is not.
• Figure 5: Two regions on the star graph with four vertices. The shaded areas are bounded by the parent tube of each region, and they do not include areas bounded by the children tubes.