A Note on Kinematic Flow and Differential Equations for Two-Site One-Loop Graph in FRW Spacetime

Abstract: In this work, we investigate the canonical differential equations dictate the behavior of cosmological wavefunction coefficients for two-site loop-level configurations in conformally-coupled scalar field theory within a general power-law FRW cosmology. Employing the tools of relative twisted cohomology and integration-by-parts techniques, we systematically derive these equations, thereby establishing a comprehensive framework for loop-level correlators for the first time. A novel contribution of this work is the extension of kinematic flow framework to the loop-level scenarios. Using a graphical approach based on family trees generated by marked tubing graphs, we efficiently construct the differential equations while capturing the singularity structure and hierarchical relationships within the associated integral families. Furthermore, we present new insights into the tadpole systems, uncovering that their integral families can exhibit multiple parent functions due to distinct, non-overlapping relative hyperplanes, in contrast with the single parent function of bubble and tree-level systems. Additionally, we demonstrate that tadpole wavefunction coefficients selectively probe only a subset of the cohomology space, despite the hyperplane arrangement suggesting a higher-dimensional structure. Beyond these findings, we provide a preliminary discussion on generalization to two-site higher-loop configurations, laying the groundwork for future exploration of more intricate scenarios. This work introduces several first-time methodologies and results, advancing our understanding of the geometric, topological, and dynamical properties of loop-level wavefunction coefficients and offering new insights for probing the fundamental mechanisms of the early universe.