The Hidden Order of Cosmic Rays: Fractal Scaling and Temporal Complexity

This article explores the temporal dynamics and fractal characteristics of cosmic ray (CR) intensity by conducting a comprehensive analysis of their intrinsic scaling properties. The study utilizes sophisticated methodologies, including standard Detrended Fluctuation Analysis (DFA) and the Multifractal DFA (MF-DFA) approach to robustly evaluate long-memory, self-similarity, and singularity spectra within extensive CR time series. By systematically investigating measurements from two neutron monitor stations with long data archives, the analysis demonstrates the prevalence of multifractal behavior with persistent long-range correlations. Building on the fractal regime revealed in CR time series, this work utilizes the Natural Time Analysis (NTA) tool that is based in the order of occurrence of the extreme cosmic ray events (ECREs). The operational utility of this tool is demonstrated through a case analysis of CR fluctuations during the severe geomagnetic disturbances observed from 9 to 15 May 2024, capturing early-warning signatures and complex temporal responses. Furthermore, the Modified NTA (M-NTA) is used to estimate the occurrence rate of future ECREs. Our findings contribute to a deeper understanding of the scaling laws governing CR intensity and their potential for improving the ECRE modeling, with direct implications for space weather risk mitigation and solar–terrestrial interactions.