Reconciling the ultra-high energy cosmic ray spectrum with Fermi shock acceleration

The energy spectrum of ultra-high energy cosmic rays (UHECR) is usually calculated for sources with identical properties. Assuming that all sources can accelerate UHECR protons to the same extremely high maximal energy Emax>1020eV and have the steeply falling injection spectrum 1/E2.7, one can reproduce the measured cosmic ray flux above E≳1018eV. We show that relaxing the assumption of identical sources and using a power-law distribution of their maximal energy allows one to explain the observed UHECR spectrum with the injection 1/E2.2 typically predicted by Fermi shock acceleration.