A Transport–Information Geometric Formulation of Cosmic Structure Formation: A Unified Dual-Affine Perspective

Cosmic large-scale structure formation is commonly described in terms of the evolution of density fluctuations and correlation statistics. However, such approaches primarily characterize amplitude variations and do not directly capture the spatial rearrangement of mass distributions. Recent developments based on optimal transport theory have introduced a complementary perspective, in which structure formation is understood as a transport process in the space of probability measures equipped with Wasserstein geometry. In this work, we extend this framework by introducing transport–information geometry, which unifies transport geometry with information geometry. Within this formulation, cosmological states are represented as elements of the product space of probability measures and statistical manifolds, allowing gravitational mass transport and generative deformations associated with galaxy formation to be treated in a unified manner. Using entropic optimal transport, we demonstrate that Wasserstein geometry and Kullback–Leibler-based information geometry are connected within a single mathematical structure, leading to a geometric interpretation of cosmological evolution as a coupled transport–information process endowed with a dual-affine structure. In this picture, gravitational evolution corresponds to generative deformation associated with e-geometry, while observational processes, including finite sampling and survey selection, are described as mixing and projection in m-geometry. This dual-affine cosmology provides a unified framework in which gravitational transport, galaxy bias, observational effects, and nonlinear multi-stream structures are consistently incorporated. The resulting formulation offers a systematic basis for cosmological inference, data analysis, and stochastic descriptions of structure formation.