On the shape of the ultrahigh energy cosmic ray spectrum

We fit the ultrahigh energy cosmic ray spectra above ${10}^{19}\mathrm{eV}$ with different injection spectra at cosmic ray sources that are uniformly and homogeneously distributed in the Universe. We conclude that the current ultrahigh energy spectra are consistent with power laws of index $\alpha$ between 2.4 and 2.7. There is a slow dependence of these indices on the cosmological evolution of the cosmic ray sources, which in this model determines the end of the galactic cosmic rays spectrum.

Figure 1

Solid line: loss length for photo-pion and photo-pair production for protons as calculated in Ref. 6. The dashed lines report the separate contribution of the two processes. The dotted line shows the loss length for redshift losses. The squares represent the loss length of Ref. 16.

Figure 2

Upper panel: Best fits to the AGASA $-15\%$ data set in the ${10}^{19}–{10}^{20}\mathrm{eV}$ energy range. Squares with error bars: HiRes data set. Circles with error bars: AGASA&Akeno data set. Filled symbols: points used in the fit. Smaller data points: data from HiRes2 and from Akeno (not shifted) to show the low-energy shape of the spectrum. Solid lines: best fits for different values of $m$. Dotted lines: galactic component needed in order to fit the spectrum at lower energy. The numbers attached to the lines indicate the value of $m$. Lower panel: best fit for the HiRes $+15\%$ data set.

Figure 3

Best fits to the HiRes $+15\%$ data set in the ${10}^{18.5}–{10}^{20}\mathrm{eV}$ energy range. Symbols are the same as in Fig. 2.

Figure 4

Contour plots of ${log}_{10}{\chi }^2$ as a function of $\alpha$ and $m$ for the AGASA $-15\%$ data set (upper panel) and for the HiRes $+15\%$ data set (lower panel) fits above ${10}^{19}\mathrm{eV}$. The white lines are the contours corresponding to $1\sigma$, $2\sigma$, and $3\sigma$.

Figure 5

Required emissivity as a function of $(\alpha ,m)$. Lines: best fit emissivity (above ${10}^{19}\mathrm{eV}$) as a function of $\alpha$. The numbers attached to the lines give the value of $m$. Every line is highlighted with different thicknesses corresponding to the confidence bands for $1\sigma$, $2\sigma$, and $3\sigma$. The black squares are the best fits obtained in other works. W&B corresponds to Refs. 7, 21; BGG corresponds to Ref. 6.

Figure 6

One $\sigma$ error band (shaded) of the AGASA fit with $m=0$ above ${10}^{19}\mathrm{eV}$.