Greisen–Zatsepin–Kuzmin Limit on Cosmic Rays
Ultra‑high energy cosmic rays (UHECRs) with energies exceeding 1019 eV traverse cosmological distances before reaching Earth. In the mid‑1960s, Kenneth Greisen, Georgiy Zatsepin, and Vadim Kuzmin independently realized that interactions between these energetic protons and the cosmic microwave background would dramatically suppress the observable flux above roughly 5×1019 eV. This energy scale, now known as the GZK cutoff, arises from photopion production: p + γCMB → Δ+ → p π0 or n π+. Once above threshold, each interaction robs the proton of a substantial fraction of its energy, limiting the distance such particles can travel without attenuation.
The kinematic threshold for photopion production follows from conservation of the Mandelstam variable s. For a head‑on collision between a proton of energy E and a CMB photon of energy ε, the threshold condition is
Solving for the proton energy and expressing the Δ resonance mass in terms of the proton and pion masses yields a convenient approximation:
For CMB photons, a characteristic energy ε ≈ 6.3×10⁻⁴ eV (corresponding to a blackbody temperature of 2.725 K) leads to a threshold Ethr ≈ 5×1019 eV. Protons with lower energy can traverse gigaparsecs essentially unimpeded, while those above the threshold interact frequently enough to lose energy over tens of megaparsecs. This effect imprints a spectral suppression in the observed UHECR flux that has been confirmed by experiments such as HiRes, the Pierre Auger Observatory, and the Telescope Array.
The calculator evaluates Ethr using the formula above and compares the user‑supplied proton energy to this threshold. If the proton exceeds the threshold, an approximate attenuation length is returned using the empirical relation
This expression captures the rapid decrease in energy‑loss length with rising proton energy: at 1020 eV, λ is around 25 Mpc, while at 5×1019 eV it is roughly 50 Mpc. Although simplistic, it offers a useful order‑of‑magnitude estimate for how far UHECRs can propagate before losing most of their energy to photopion production.
The existence of the GZK cutoff has profound implications. It limits the observable horizon for the highest energy cosmic rays, restricting their sources to relatively nearby extragalactic objects such as active galactic nuclei, radio lobes, or gamma‑ray bursts. It also implies that the flux of trans‑GZK particles is sensitive to the distribution of nearby sources and to magnetic deflections along their paths. The detection of a few events above 1020 eV continues to spark debate about exotic origins—topological defects, decays of superheavy dark matter, or Lorentz invariance violation—although mainstream data are consistent with the GZK expectation.
Beyond protons, nuclei also suffer photodisintegration when interacting with the CMB and the infrared background. For heavier nuclei the effective threshold occurs at lower energies, further shaping the observed composition at the highest energies. Understanding these interactions is crucial for interpreting UHECR spectra and for designing next-generation observatories seeking to pinpoint their sources.
Our calculator deliberately adopts a simplified framework: a single photon energy representing the CMB and an approximate attenuation length formula. More sophisticated treatments integrate over the full blackbody spectrum, include the inelasticity and energy dependence of the photopion cross section, and account for cosmological redshift. Nevertheless, this tool offers quick intuition about the basic kinematics of the GZK process.
The table below illustrates the computed classification for several proton energies. Energies are given in exa-electronvolts (EeV; 1 EeV = 1018 eV).
| E (EeV) | Classification | Attenuation Length (Mpc) |
|---|---|---|
| 30 | Below threshold | ∞ |
| 60 | Above threshold | 41 |
| 100 | Above threshold | 25 |
Note that the infinite attenuation length for sub-threshold energies reflects the negligible energy loss over cosmological distances, whereas above threshold the finite λ values highlight the limited propagation range. These effects make the observed UHECR spectrum a powerful probe of both astrophysical sources and fundamental interactions.
Beyond providing a quick numerical answer, this calculator encourages exploration. By varying the input energy, users can see how rapidly the attenuation length drops once the GZK threshold is crossed. Researchers planning UHECR observatories can estimate the reachable volume of space for a given energy band, while students can connect textbook derivations to tangible numbers. The simplicity of the interface belies the richness of the underlying physics, which ties together special relativity, particle interactions, and the cosmological microwave background.
Historically, the confirmation of the GZK suppression required decades of effort. Early experiments in the 1970s and 1980s reported events seemingly above the limit, leading to speculation about exotic physics. Only with the advent of large aperture detectors like Auger and Telescope Array did statistics become sufficient to reveal a clear flux suppression consistent with the GZK prediction. Today the cutoff is a cornerstone of astroparticle physics, guiding theoretical models of cosmic-ray acceleration and propagation.
The GZK effect also underscores the interconnectedness of cosmic phenomena. A relic radiation field dating back to the early universe constrains the energy of particles accelerated in extreme astrophysical environments billions of years later. This interplay exemplifies the holistic nature of modern astrophysics, where processes across vast energy and time scales combine to shape observations.
Finally, it is worth considering potential deviations from the standard GZK picture. If Lorentz invariance were violated at ultra-high energies, the photopion threshold could shift, altering the cutoff. Similarly, the existence of new light particles or modifications to photon dispersion could influence the attenuation length. Thus, precision measurements of the UHECR spectrum continue to serve as tests of fundamental physics. Although our calculator assumes conventional physics, it provides a baseline against which such exotic scenarios can be compared.
By converting abstract cross-section calculations into an accessible numerical tool, this calculator demystifies the GZK cutoff. It invites users to appreciate how a seemingly innocuous background of microwave photons imposes a hard limit on the energy of cosmic messengers reaching Earth. Whether used for outreach, education, or quick research estimates, it highlights the power of simple kinematic reasoning in unveiling deep truths about our universe.