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  • Open Access

Geometric theory of cosmological structure via entropic curvature in Wasserstein space

Tsutomu T. Takeuchi*

  • *Contact author: tsutomu.takeuchi.ttt@gmail.com
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APS Open Sci. 1, 000201 – Published 3 November, 2026

DOI: https://doi.org/10.1103/yrgj-rb72

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 HR=−∇2lnρR; 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(ɛ2) 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 fsamp=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.

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APS Open Sci. 1, 000201– Published 3 November, 2026
Vol. 1
  • Received 1 April 2026
  • Revised 15 August 2026
  • Accepted 27 October 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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