The GZK horizon and constraints on the cosmic ray source spectrum from observations in the GZK regime

We discuss the GZK horizon of protons and present a method to constrain the injection spectrum of ultrahigh energy cosmic rays (UHECRs) from supposedly identified extragalactic sources. This method can be applied even when only one or two events per source are observed and is based on the analysis of the probability for a given source to populate different energy bins, depending on the actual cosmic ray injection spectral index. In particular, we show that for a typical source density of $4\cdot10^{-5}\,$Mpc$^{-3}$, a data set of 100 events above $6\cdot10^{19}$ eV allows one in 97% of all cases to distinguish a source spectrum $dN/dE\propto E^{-1.1}$ from one with $E^{-2.7}$ at 95% confidence level.