Greisen–Zatsepin–Kuzmin limit

The Greisen-Zatsepin-Kuzmin limit (GZK limit) is a theoretical upper limit on the energy of cosmic rays from distant sources.

How to compute GZK limit?
This extra limit was independently computed in 1966 by Kenneth Greisen^[1], and Vadim Kuz'min and Georgiy Zatsepin^[2], based on interactions between the cosmic ray and the photons of the cosmic microwave background radiation (CMB). They predicted that cosmic rays with energies over the threshold energy of 5×10¹⁹ eV would interact with cosmic microwave background photons γ_CMB to produce pions via the Δ resonance,

or

This 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.