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 , d \dot { n } / d \varepsilon \propto \varepsilon ^ { - \alpha } ( 1 + z ) ^ { m } , our results are accurate at high energy , \varepsilon > 10 ^ { 18.7 } eV , to better than 15 \% , providing a simple and straightforward method for inferring d \dot { n } / d \varepsilon ( \varepsilon ) from the observed flux at \varepsilon .
We show that current measurements of the UHECR spectrum , including the latest Auger data , imply \varepsilon ^ { 2 } d \dot { n } / d \varepsilon ( z = 0 ) = ( 0.45 \pm 0.15 ) ( \alpha - 1 ) \times 10 ^ { 4 % 4 } \mbox { erg } \mbox { Mpc } ^ { -3 } \mbox { yr } ^ { -1 } at \varepsilon > 10 ^ { 19.5 } \text { eV } with \alpha roughly confined to 2 \lesssim \alpha < 2.7 .
The uncertainty is dominated by the systematic and statistic errors in the experimental determination of individual CR event energy , ( \Delta \varepsilon / \varepsilon ) _ { sys } \sim ( \Delta \varepsilon / \varepsilon ) _ { % stat } \sim 20 \% .
At lower energy , d \dot { n } / d \varepsilon is uncertain due to the unknown Galactic contribution .
Simple models in which \alpha \simeq 2 and the transition from Galactic to extra-Galactic sources takes place at the ” ankle ” , \varepsilon \sim 10 ^ { 19 } \mbox { 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 , \alpha \simeq 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 can not be used to infer the detailed spectral shapes of the Galactic and extra-Galactic contributions .