Could cosmic rays look orderly because their sources are diverse?

A regular pattern with revealing bends

An energy spectrum describes how the abundance of particles changes with their energy. For Galactic cosmic rays, it roughly follows a power law: multiplying the energy by a fixed factor reduces the flux by a predictable factor.

But the pattern is not exact. In a preprint submitted on 9 November 2026, Paolo Lipari examines published proton and helium measurements between about 50 gigaelectronvolts and 10 petaelectronvolts. The analysis combines results from space instruments with ground-based detectors that reconstruct incoming particles from atmospheric showers. It reports no new detector measurements.

Below their pronounced high-energy breaks, called knees, the proton and helium spectra remain within roughly 20% of simple power-law descriptions. Nevertheless, their smaller bends matter. A hardening means the flux begins falling less steeply with energy; a softening means it falls more steeply.

Lipari fits these changes using a flexible mathematical description with four breaks for protons and five for helium. To explore disagreements between instruments, he fits two different combinations of measurements, excluding three low-energy LHAASO proton points with large systematic errors.

Why compare protons with helium?

The revealing comparison uses rigidity: momentum divided by electric charge, which determines how strongly a particle's path bends in a magnetic field. Simple acceleration and transport descriptions therefore predict closely related behavior for different nuclei at the same rigidity.

Some features do align. Both spectra have knees near a rigidity of 3 petavolts, corresponding to particle energies of approximately 3 petaelectronvolts for protons and 6 for helium nuclei.

Other features do not. Around 100 teravolts, the proton spectrum hardens while helium softens. Helium's subsequent hardening appears around 500 teravolts. The proton-to-helium flux ratio, which decreases at lower rigidities, turns upward around 100 teravolts.

These differences challenge models in which every source supplies the same spectral shape for both particles, with propagation modifying them in the same rigidity-dependent way.

Order from many different spectra

One established modeling approach combines a few components, each associated with a class of accelerators producing a characteristic spectrum. Some versions also invoke a recently active nearby source. Lipari argues that these models can fit observations without establishing that their assumed source populations actually exist.

His alternative allows even sources of the same class to produce many different spectral shapes. For a worked mathematical example, he uses smoothly curved spectra called log-parabolas, each with a characteristic energy where its energy-weighted output peaks.

The crucial assumption concerns how the population's cosmic-ray power is distributed across those characteristic energies. If power per interval of characteristic energy falls as that energy raised to minus p, the summed injection spectrum can follow a power law with index p + 1 over an appropriate energy range.

This is not a claim that arbitrary curves automatically average into a power law. The result depends on the assumed distribution of source power and spectral shapes. Propagation through the Galaxy must still connect that injected population to the flux measured at Earth.

Composition could change the average

The framework also allows proton and helium spectra to match in rigidity within every individual source, yet differ after contributions from many sources are combined.

That requires a relationship between composition and spectral hardness. If harder sources are more helium-rich, helium gains relative importance as rigidity increases. Explaining the later reversal would require harder contributors in that range to become more proton-rich. These are proposed correlations, not measured properties established by this paper.

An illustration, not a confirmation

Lipari offers an observational analogy: summing fitted gamma-ray spectra for 4,002 blazar-associated sources in a Fermi catalog produces a spectrum close to a power law, despite substantial differences among individual sources. Over the illustrated range above about 0.15 gigaelectronvolts, the reported deviations are below 7%.

Blazars are not evidence that Galactic accelerators obey the required population relationship. Nor does the paper calculate whether statistical fluctuations would reproduce the sizes and locations of the cosmic-ray bends.

The proposal remains a theoretical framework presented as an arXiv preprint. Gamma-ray and neutrino observations could constrain source particle populations, but interpreting them requires accounting for source environments and evolution: an observed snapshot need not represent the cosmic rays released over an accelerator's entire lifetime.