Temporal Structure of the Cosmic Web: A Cosmological Formulation of the Cosmic Time Map and Temporal Entropy

This paper develops a cosmological formulation of the cosmic time map, a spatial field describing variations in local proper-time rates across the universe arising from gravitational potential fluctuations. Building on an earlier conceptual proposal (Catrambone 2024), the work introduces temporal entropy, defined as the Shannon entropy of the probability distribution of dimensionless proper-time fluctuations across cosmic volume. Using cosmological perturbation theory in conformal Newtonian gauge, the paper shows how spatial variations in the Bardeen potential produce a statistical distribution of clock-rate deviations across the universe. As density perturbations grow during structure formation, these variations broaden, leading to an increase in temporal entropy. In the linear regime of matter domination, an approximate scaling relation is derived linking temporal entropy to the cosmic scale factor. The framework connects relativistic time dilation, gravitational potential fluctuations, and large-scale structure formation into a thermodynamic description of cosmic temporal structure. The paper also discusses the relationship between temporal entropy and existing measures of gravitational entropy, including approaches based on density-field information entropy and Weyl curvature. A numerical strategy is proposed for measuring temporal entropy in standard ΛCDM N-body simulations by computing gravitational potentials and evaluating the distribution of clock-rate fluctuations across simulation volumes. This work presents temporal entropy as a complementary statistical measure of cosmic structure, emphasizing the distribution of relativistic clock rates rather than density fluctuations alone.