Comparison of the ultrahigh energy cosmic ray flux observed by AGASA, HiRes, and Auger

The current measurements of the cosmic ray energy spectrum at ultra-high energies ($\mathrm{E}>{10}^{19}\mathrm{eV}$) are characterized by large systematic errors and poor statistics. In addition, the experimental results of the two experiments with the largest published data sets, AGASA and HiRes, appear to be inconsistent with each other, with AGASA seeing an unabated continuation of the energy spectrum even at energies beyond the Greisen-Zatsepin-Kuzmin cutoff energy at ${10}^{19.6}\mathrm{eV}$. Given the importance of the related astrophysical questions regarding the unknown origin of these highly energetic particles, it is crucial that the extent to which these measurements disagree be well understood. Here we evaluate the consistency of the two measurements for the first time with a model-independent method that accounts for the large statistical and systematic errors of current measurements. We further compare the AGASA and HiRes spectra with the recently presented Auger spectrum. The method directly compares two measurements, bypassing the introduction of theoretical models for the shape of the energy spectrum. The inconsistency between the observations is expressed in terms of a Bayes factor, a standard statistic defined as the ratio of a separate parent source hypothesis to a single parent source hypothesis. Application to the data shows that the two-parent hypothesis is disfavored. We expand the method to allow comparisons between an experimental flux and that predicted by any model.

Figure 1

The AGASA [4], Auger [10] and HiRes monocular spectra [5, 6].

Figure 2

BF as a function of $ϵ$ (the difference between the two-parent distributions) and $\beta$ where the BF is evaluated over two spectra with 200 and $200\times \beta$ events.

Figure 3

The Bayes factor (BF) as a function of the percent energy scale uncertainties for the compared spectra.

Figure 4

The distribution of the BF calculated for an averaged spectrum from two experiments. The red lines denote the BF’s measured for data.

Figure 5

The projected BF between the current Auger spectrum and the AGASA and HiRes spectra as a function of the fraction of the current Auger exposure. The calculations assume 30% energy uncertainties in AGASA and HiRes and the projected 15% energy uncertainties in the Auger spectrum.