Global Spline Fit: A unified data-driven view of the cosmic-ray spectrum and mass composition from GeV to the highest energies

Abstract

The energy spectrum and mass composition of cosmic rays are measured by two complementary classes of instruments: space- and balloon-borne detectors that resolve individual elements up to sub-PeV energies, and ground-based air-shower observatories that extend the reach to beyond $10^{11}$ GeV but resolve only broad mass groups. The Global Spline Fit (GSF) is a data-driven model that combines both into a single, self-consistent description of the flux of all elements from hydrogen to nickel. The flux of four leading mass groups is parametrized by cubic basis splines, imposing smoothness but no astrophysical expectation on the spectral shape, while the energy-scale offsets of the participating experiments are cross-calibrated within their quoted systematic uncertainties as part of the fit. The model is defined at the local interstellar spectrum, with the effect of solar modulation at the lowest energies accounted for in the fit. The fit uses the most precise recent data, among them the elemental spectra from AMS-02, CALET, and DAMPE, the knee-region measurements of LHAASO, and the fluorescence-based composition of the Pierre Auger Observatory, and describes about one thousand data points with $χ^2/\mathrm{ndf}\approx 1.3$, or 0.83 after de-weighting localized disagreements between data sets, demonstrating that the global body of cosmic-ray data is consistent once energy scales are aligned. The model delivers the flux, the mass composition, and their full covariance, along with a compact reduced representation for uncertainty propagation. We further provide the cosmic-ray nucleon flux, which differs from widely used parametrizations by 20-50% over four decades in energy. This exceeds the remaining model uncertainty several times over and has direct consequences for atmospheric neutrino and muon flux predictions.