Transport-Geometric Formulation of Peak Statistics: Curvature-Conditioned Point Processes and Response Hierarchy

Abstract: We develop a geometric formulation of peak statistics in cosmological density fields based on optimal transport and entropy. In this framework, the density field is treated as a probability measure, and its local structure is characterized by the Hessian of the log-density, which arises as the local response of an entropy functional in Wasserstein space. Peaks are thereby defined as positive-curvature stationary points, and their number density is expressed as a curvature-conditioned point process. In the linear Gaussian limit, the joint distribution of local variables closes in terms of a finite set of spectral moments, recovering the standard theory of peak statistics, known as BBKS. This clarifies that BBKS corresponds to a solvable limit of a more general structure combining probability distributions, curvature constraints, and geometric measure. The framework extends naturally beyond Gaussianity and linearity. Deviations from Gaussianity are incorporated as deformations of the joint distribution of curvature variables, while nonlinear structures are described through the curvature of the log-density. We further derive the two- and three-point peak statistics as curvature-conditioned $n$-point measures, and show that the full hierarchy of peak statistics can be organized as response functions to long-wavelength background modes. In this formulation, the conventional peak bias appears as the lowest-order response coefficient, with higher-order correlations arising as its natural extensions. This work embeds peak theory into a unified geometric framework and provides a systematic basis for incorporating nonlinearity, non-Gaussianity, and higher-order statistics, with direct relevance for observational applications.