A Measure-Theoretic Transport Formulation of Galaxy Evolution on the Galaxy Manifold: Geometric Constraints

Abstract

We develop a measure-theoretic framework for galaxy evolution in which galaxy populations are described as probability measures on a state space. Galaxy evolution is represented as the time evolution of a measure $ν_t$, governed by the sum of a continuous transport term and a jump operator. The transport term describes internal galaxy evolution, while the jump operator captures discrete events such as mergers and interactions, yielding a unified reaction--transport system on the space of measures. We further equip the space of probability measures with the Wasserstein distance and impose a curvature--dimension condition CD$(K,\infty)$ to reveal the geometric structure of the dynamics. In this setting, the transport term is interpreted as a gradient flow of a free-energy functional, whereas the jump operator generates nonlinear rearrangements induced by many-body interactions. In the low-density limit, these interactions reduce effectively to two-body processes, leading to a closed dynamical system. A central consequence is that galaxy evolution is not arbitrary, but constrained by a variational structure, curvature bounds, and an interaction hierarchy. Admissible trajectories are restricted by energy dissipation, geometric contractivity, and effective interaction closure. The framework also separates intrinsic galaxy dynamics from observational projection, treating observables as pushforwards of measures. It thus provides a unified foundation for structure formation and galaxy evolution as a geometrically constrained reaction--transport process.