Authors:Tsutomu T. Takeuchi (Nagoya University and Institute of Statistical Mathematics)

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Abstract:We develop a transport-geometric theory of cosmological peak statistics based on optimal transport and entropy geometry. The density field is treated as a probability measure in Wasserstein space, and its local structure is characterized by a logarithmic curvature tensor obtained as the localized response of an entropy functional. Peaks are thereby defined as positive-curvature stationary points, and their abundance is formulated as a curvatureconditioned measure on local tensor space. In the Gaussian-linear limit, this measure admits a finite-dimensional closure in terms of the spectral moments of the density field. The resulting peak abundance reduces exactly to the classical BBKS formula, identifying BBKS as a solvable Gaussian closure of a more general geometric structure. This formulation separates peak statistics into three fundamental ingredients: the probability distribution of local variables, the positive-curvature constraint, and the induced geometric measure. The theory extends naturally beyond the Gaussian approximation. Nonlinear evolution appears as a deformation of the logarithmic curvature geometry, while primordial non-Gaussianity is interpreted as a deformation of the curvature-conditioned measure itself. We further formulate two- and three-point peak statistics as higher-order curvature-conditioned measures and show that the resulting hierarchy can be organized as response functions to long-wavelength background modes, with conventional peak bias emerging as the lowest-order response coefficient. These results provide a unified description linking optimal transport, curvature geometry, peak statistics, and cosmological observables, and establish a systematic framework for studying nonlinearity, scale dependence, primordial non-Gaussianity, and higher-order peak correlations.

Submission history

From: Tsutomu Takeuchi T. [view email]
[v1] Wed, 1 Apr 2026 13:19:11 UTC (78 KB)
[v2] Fri, 19 Jun 2026 13:45:38 UTC (1,824 KB)
[v3] Wed, 9 Sep 2026 00:58:30 UTC (1,811 KB)