Algebraic Approaches to Cosmological Integrals

Cosmological correlators encode the statistical properties of the very early universe. They can be written as multi-dimensional Mellin transforms of specific rational functions. Recent progress has shown how systems of differential equations, that are used to efficiently solve these integrals, can be derived diagrammatically -- the so-called `kinematic flow'. The simplicity of these rules suggests an underlying mathematical structure. In this talk, I will present a few algebraic approaches to begin the exploration of this structure. I will discuss systematic ways of deriving shift-relations and how we can think of the cosmological integrals as restrictions of GKZ systems. Furthermore, I will explain a diagrammatic derivation of the partial fraction decomposition for the integrands and show why it can be useful when studying the integrals. Finally, I will highlight a few open questions about the cosmological integrals. This talk is based on the joint work arXiv:2410.14757 with Claudia Fevola, Anna-Laura Sattelberger and Guilherme L. Pimentel.