A Geometric Theory of Cosmological Structure via Entropic Curvature in Wasserstein Space

Abstract

We construct a geometric framework for cosmological large-scale structure based on optimal transport theory and Wasserstein geometry. In this framework, Ricci curvature on the probability measure space $\mathcal{P}_2(M)$ is characterized by the geodesic convexity of entropy and is formulated as the response of probability distributions to optimal transport. We introduce effective Ricci curvatures $K_{\mathrm{eff}}^{(\infty)}$ and $K_{\mathrm{eff}}^{(N)}$ associated with Kullback--Leibler-type and Rényi-type entropies, corresponding respectively to the curvature-dimension conditions CD$(K,\infty)$ and CD$(K,N)$. By localizing these curvatures to finite scales using local and reference measures, we construct curvature indicators applicable to observational data. Under a local quadratic approximation, the effective curvature reduces to the Hessian of the log-density, showing that conventional Hessian-based structure classifications arise as a limiting case of the present framework. We further show that effective curvature depends on observational scale and formulate this dependence as a scale flow, distinct from Ricci flow because it describes a change of resolution rather than a time evolution of geometry. Treating curvature as a random field then extends the statistical description of density fields: curvature statistics are given by higher-order weighted integrals of the power spectrum and by spatial derivatives of the correlation function, emphasizing geometric rather than amplitude information. This framework provides a unified connection between optimal transport geometry and cosmological structure analysis, and offers a new perspective on multiscale structure and nonlinear statistics.

Figures (1)

• Figure 1: Schematic illustration of the local effective curvature in large-scale structure. Top: a local region $B_R(x)$ defines a probability measure $\mu_x^{(R)}$, which is compared to a reference measure $\nu_x^{(R)}$ through optimal transport. The curvature is quantified by the entropy variation $\Delta S$ along the Wasserstein geodesic. Void regions correspond to low-density, dark areas, while halo regions correspond to high-density, bright concentrations. Bottom: the corresponding local curvature structure is characterized by the Hessian of the log-density $H_x=-\nabla^2 \ln \rho(x)$. Different eigenvalue configurations distinguish void, filament, and halo structures.

Theorems & Definitions (2)

• Definition 2.1: CD$(K,\infty)$
• Definition 3.1: KL-type effective Ricci curvature