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The energy production rate & the generation spectrum of UHECRs

We derive simple analytic expressions for the flux and spectrum of ultra-high energy cosmic-rays (UHECRs) predicted in models where the CRs are protons produced by extra-Galactic sources. For a power-law scaling of the CR production rate with redshift and energy, dṅ/dε∝ε−α(1 + z)m, our results are accurate at high energy, ε > 1018.7 eV, to better than 15%, providing a simple and straightforward method for inferring dṅ/dε(ε) from the observed flux at ε. We show that current measurements of the UHECR spectrum, including the latest Auger data, imply ε2dṅ/dε(z = 0) = (0.45 ± 0.15)(α−1) × 1044 erg Mpc−3 yr−1 at ε > 1019.5 eV with α roughly confined to 2 ≲ α < 2.7. The uncertainty is dominated by the systematic and statistic errors in the experimental determination of individual CR event energy, (Δε/ε)sys ∼ (Δε/ε)stat ∼ 20%. At lower energy, dṅ/dε is uncertain due to the unknown Galactic contribution. Simple models in which α ≃ 2 and the transition from Galactic to extra-Galactic sources takes place at the "ankle", ε ∼ 1019 eV, are consistent with the data. Models in which the transition occurs at lower energies require a high degree of fine tuning and a steep spectrum, α ≃ 2.7, which is disfavored by the data. We point out that in the absence of accurate composition measurements, the (all particle) energy spectrum alone cannot be used to infer the detailed spectral shapes of the Galactic and extra-Galactic contributions.