Researchers have developed a graphical formalism to analyze Friedmann-Robertson-Walker (FRW) integrals, which are fundamental in cosmology for describing the evolution of the universe. This method applies to conformally-coupled scalar theories with non-conformal polynomial interactions, allowing these integrals, their discontinuities, and derivatives to be decomposed into basic components. The construction relies on intersection theory within the context of partial/relative twisted (co)homology, which facilitates a purely graphical description of the coaction.

The approach allows the building blocks of FRW integrals to be represented as decorations of the original Feynman diagrams. This provides a comprehensive combinatorial framework for dissecting the analytic properties of cosmological observables. A key result is that the combinatorics of the differential equations governing FRW integrals—their so-called kinematic flow—naturally emerges from this graphical coaction. The ability to visualize and manipulate these integrals graphically simplifies the study of their complex structure.

To facilitate its use, the authors have developed a web application and a Mathematica notebook that compute the graphical coaction for any graph, and when possible, also its differentials and discontinuities. This computational tool is publicly accessible and represents a significant advance for the community, enabling other researchers to explore the analytic properties of cosmological integrals more efficiently and in greater detail, opening new avenues for understanding the dynamics of the early universe and the formation of large-scale structures.