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

Cosmic-ray energy spectrum and mass composition are measured by two complementary instrument classes: space- and balloon-borne detectors that resolve individual elements up to sub-PeV energies, and ground-based air-shower observatories that extend reach beyond 10^11 GeV but resolve only broad mass groups. The Global Spline Fit (GSF) combines both into one self-consistent description.

The GSF is a data-driven model describing 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. Energy-scale offsets of participating experiments are cross-calibrated within their quoted systematic uncertainties as part of the fit, and the model is defined at the local interstellar spectrum with solar modulation at the lowest energies accounted for in the fit.

The fit uses the most precise recent data, including elemental spectra from AMS-02, CALET, and DAMPE; knee-region measurements from LHAASO; and the fluorescence-based composition from the Pierre Auger Observatory. It describes about one thousand data points with χ²/ndf ≈ 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. It further provides 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.