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Geometric theory of cosmological structure via entropic curvature in Wasserstein space

Year: 2026
Type: article
Abstract: We formulate a weighted-Wasserstein framework for curvature statistics of cosmological large-scale structure. A coarse-grained density defines a weighted reference measure whose relative-entropy response separates into kinematic and density-induced Bakry-Émery contributions. On a Euclidean background, the latter is governed locally by H R = − ∇ 2 ln ρ R ; finite windows define density-weighted directional responses, while finite synthetic dimension subtracts a non-negative score-alignment deficit. The controlled local limit recovers the eigenvalue-threshold classification of the smoothed log-density Hessian, not a general equivalence with density-Hessian or gravitational-tidal classifiers. Independent smoothing and probe scales define a resolution flow. Controlled tests recover quadratic tensors to relative accuracy 10 − 9 or better and confirm O ( ɛ ) and O ( ɛ 2 ) convergence. In a periodic three-dimensional Zel’dovich mock, the specified six-direction transport tensor matches its direct finite-window reference with subpercent median error and directional correlation above 0.999 995. Rotated and enriched probe designs reveal appreciable finite-window eigenvalue and classification dependence, distinct from estimator error; a converged angular projection still has a nonquadratic directional residual. Under thinning to f samp = 1 / 16 , the six-direction internal error remains below 0.3 % , while density reconstruction dominates the total error. At the fiducial radius, this finite-window observable has approximately 30 % lower normalized sampling error than the pointwise log-density Hessian, each relative to its own full-sampling reference. In the Gaussian-linear limit, curvature spectra are differential transforms of two-point statistics; finite-window and finite-dimensional observables retain additional profile and directional information beyond that limit.
DOI: 10.1103/yrgj-rb72
Language: English
Open Access status: hybrid
OpenAlex ID: W7148469955
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