Cosmic strings and ultrahigh-energy cosmic rays

We calculate the flux of ultrahigh-energy protons due to the process of ‘‘cusp evaporation’’ from cosmic-string loops. For the ‘‘standard’’ value of the dimensionless cosmic-string parameter ε==Gμ≊${10}^{\mathrm{-}6}$, the flux is several orders of magnitude below the observed cosmic-ray flux of ultrahigh-energy protons. However, the flux at any energy initially increases as the value of ε is decreased. This at first suggests that there may be a lower limit on the value of ε, which would imply a lower limit on the temperature of a cosmic-string-forming phase transition in the early Universe. However, our calculation shows that this is not the case—the particle flux at any energy reaches its highest value at ε≊${10}^{\mathrm{-}15}$ and it then decreases for further decrease of the value of ε. This is due to the fact that for too small values of ε (<${10}^{\mathrm{-}15}$) the energy loss of the loops through the cusp evaporation process itself (rather than gravitational energy loss of the loops) becomes the dominant factor that controls the behavior of the number density of the loops at the relevant times of emission of the particles. The highest flux at any energy remains at least 4 orders of magnitude below the observed flux. There is thus no lower limit on ε.