Splitting Rules from Kinematic Flow: Local Evolution and Emergent Time

Abstract

The differential equations satisfied by the wavefunction coefficients of conformally coupled scalars in a power-law cosmology can be recast as an iterative differential system of basis functions. These functions can be encoded within graph tubings and are governed by a set of rules describing how they flow in kinematic space. In this paper, we formulate a set of splitting rules equivalent to the kinematic flow at tree level by reversing the flow direction of graph tubings. Specifically, for any basis function, these rules identify all other basis functions whose total differentials contain it. From the splitting perspective, we uncover deeper physical information captured by graph tubings, such as the formation of singularity structures and the realization of local evolution. In an alternative basis based on time ordering, the splitting processes correspond to the emergence of time integrals. These rules can also be generalized naturally to the $\mathrm{tr}\,\phi^3$ theory beyond individual graphs, providing a physical interpretation of the structure of the associahedron. This suggests that graph tubings and the kinematic flow may be more fundamental objects than the differential equations, and may have a life of their own.