The Greisen–Zatsepin–Kuzmin limit (GZK limit) is a theoretical upper limit on the energy of cosmic ray protons traveling from other galaxies through the intergalactic medium to our galaxy. The limit is 5×1019 eV, or about 8 joules (the energy of a proton travelling at ~99.99999999999999999998% the speed of light). The limit is set by slowing interactions of the protons with the microwave background radiation over long distances (~160 million light-years). The limit is at the same order of magnitude as the upper limit for energy at which cosmic rays have experimentally been detected. For example, one extreme-energy cosmic ray, the Oh-My-God Particle, which has been found to possess a record-breaking 3.12×1020 eV (50 joules)^[1][2] of energy (about the same as the kinetic energy of a 95 km/h baseball).

The GZK limit is derived under the assumption that ultra-high energy cosmic rays are protons. Measurements by the largest cosmic-ray observatory, the Pierre Auger Observatory, suggest that most ultra-high energy cosmic rays are heavier elements.^[3] In this case, the argument behind the GZK limit does not apply in the originally simple form, and there is no fundamental contradiction in observing cosmic rays with energies that violate the limit.

In the past, the apparent violation of the GZK limit has inspired cosmologists and theoretical physicists to suggest other ways that circumvent the limit. These theories propose that ultra-high energy cosmic rays are produced nearby our galaxy or that Lorentz covariance is violated in such a way that protons do not lose energy on their way to our galaxy.

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Pions produced in this manner proceed to decay in the standard pion channels—ultimately to photons for neutral pions, and photons, positrons, and various neutrinos for positive pions. Neutrons decay also to similar products, so that ultimately the energy of any cosmic ray proton is drained off by production of high-energy photons plus (in some cases) high-energy electron–positron pairs and neutrino pairs.

The pion production process begins at a higher energy than ordinary electron-positron pair production (lepton production) from protons impacting the CMB, which starts at cosmic-ray proton energies of only about 1017 eV. However, pion production events drain 20% of the energy of a cosmic-ray proton as compared with only 0.1% of its energy for electron–positron pair production. This factor of 200 is from two sources: the pion has only about ~130 times the mass of the leptons, but the extra energy appears as different kinetic energies of the pion or leptons and results in relatively more kinetic energy transferred to a heavier product pion, in order to conserve momentum. The much larger total energy losses from pion production result in the pion production process becoming the limiting one to high-energy cosmic-ray travel, rather than the lower-energy light-lepton production process.

The pion production process continues until the cosmic ray energy falls below the pion production threshold. Due to the mean path associated with this interaction, extragalactic cosmic rays traveling over distances larger than 50 Mpc (163 Mly) and with energies greater than this threshold should never be observed on Earth. This distance is also known as GZK horizon.

A number of observations have been made by the largest cosmic-ray experiments Akeno Giant Air Shower Array, High Resolution Fly's Eye Cosmic Ray Detector, the Pierre Auger Observatory and Telescope Array Project that appeared to show cosmic rays with energies above this limit (called extreme-energy cosmic rays, or EECRs). The observation of these particles was the so-called GZK paradox or cosmic-ray paradox.

These observations appear to contradict the predictions of special relativity and particle physics as they are presently understood. However, there are a number of possible explanations for these observations that may resolve this inconsistency.

Another suggestion involves ultra-high-energy weakly interacting particles (for instance, neutrinos), which might be created at great distances and later react locally to give rise to the particles observed. In the proposed Z-burst model, an ultra-high-energy cosmic neutrino collides with a relic anti-neutrino in our galaxy and annihilates to hadrons.^[6] This process proceeds through a (virtual) Z-boson:

The cross-section for this process becomes large if the center-of-mass energy of the neutrino antineutrino pair is equal to the Z-boson mass (such a peak in the cross-section is called "resonance"). Assuming that the relic anti-neutrino is at rest, the energy of the incident cosmic neutrino has to be

A number of exotic theories have been advanced to explain the AGASA observations, including doubly special relativity. However, it is now established that standard doubly special relativity does not predict any GZK suppression (or GZK cutoff), contrary to models of Lorentz symmetry violation involving an absolute rest frame. Other possible theories involve a relation with dark matter, decays of exotic super-heavy particles beyond those known in the Standard Model.

A suppression of the cosmic-ray flux that can be explained with the GZK limit has been confirmed by the latest generation of cosmic-ray observatories. A former claim by the AGASA experiment that there is no suppression was overruled. It remains controversial whether the suppression is due to the GZK effect. The GZK limit only applies if ultra-high-energy cosmic rays are mostly protons.

In July 2007, during the 30th International Cosmic Ray Conference in Mérida, Yucatán, México, the High Resolution Fly's Eye Experiment (HiRes) and the Pierre Auger Observatory (Auger) presented their results on ultra-high-energy cosmic rays. HiRes observed a suppression in the UHECR spectrum at just the right energy, observing only 13 events with an energy above the threshold, while expecting 43 with no suppression. This was interpreted as the first observation of the GZK limit.^[7] Auger confirmed the flux suppression, but did not claim it to be the GZK limit: instead of the 30 events necessary to confirm the AGASA results, Auger saw only two, which are believed to be heavy-nuclei events.^[8] The flux suppression was previously brought into question when the AGASA experiment found no suppression in their spectrum. According to Alan Watson, spokesperson for the Auger Collaboration, AGASA results have been shown to be incorrect, possibly due to the systematic shift in energy assignment.

In 2010 and the following years, both the Pierre Auger Observatory and HiRes confirmed again a flux suppression,^[9][10] in case of the Pierre Auger Observatory the effect is statistically significant at the level of 20 standard deviations.

After the flux suppression was established, a heated debate ensued whether cosmic rays that violate the GZK limit are protons. The Pierre Auger Observatory, the world's largest observatory, found with high statistical significance that ultra-high-energy cosmic rays are not purely protons, but a mixture of elements, which is getting heavier with increasing energy.^[3] The Telescope Array Project, a joint effort from members of the HiRes and AGASA collaborations, agrees with the former HiRes result that these cosmic rays look like protons.^[11] The claim is based on data with lower statistical significance, however. The area covered by Telescope Array is about one third of the area covered by the Pierre Auger Observatory, and the latter has been running for a longer time.

The controversy was partially resolved in 2017, when a joint working group formed by members of both experiments presented a report at the 35th International Cosmic Ray Conference.^[12] According to the report, the raw experimental results are not in contradiction with each other. The different interpretations are mainly based on the use of different theoretical models (Telescope Array uses an outdated model for its interpretation) and the fact that Telescope Array has not collected enough events yet to distinguish the pure-proton hypothesis from the mixed-nuclei hypothesis.

EUSO, which was scheduled to fly on the International Space Station (ISS) in 2009, was designed to use the atmospheric-fluorescence technique to monitor a huge area and boost the statistics of UHECRs considerably. EUSO is to make a deep survey of UHECR-induced extensive air showers (EASs) from space, extending the measured energy spectrum well beyond the GZK cutoff. It is to search for the origin of UHECRs, determine the nature of the origin of UHECRs, make an all-sky survey of the arrival direction of UHECRs, and seek to open the astronomical window on the extreme-energy universe with neutrinos. The fate of the EUSO Observatory is still unclear, since NASA is considering early retirement of the ISS.

Launched in June 2008, the Fermi Gamma-ray Space Telescope (formerly GLAST) will also provide data that will help resolve these inconsistencies.

In November 2007, researchers at the Pierre Auger Observatory announced that they had evidence that UHECRs appear to come from the active galactic nuclei (AGNs) of energetic galaxies powered by matter swirling onto a supermassive black hole. The cosmic rays were detected and traced back to the AGNs using the Véron-Cetty-Véron catalog. These results are reported in the journal Science.^[15] Nevertheless, the strength of the correlation with AGNs from this particular catalog for the Auger data recorded after 2007 has been slowly diminishing.^[16]