The dip is a quite faint feature that arises in the spectrum of UHE protons in the energy range 2 × 1018 eV - 5 × 1019 eV due to the pair production process; it is a unique imprint of the interaction of protons with the CMB background. The dip behavior is more pronounced when analyzed in terms of the modification factor (E) = Jp(E) / jpumm(E) , defined as the ratio of proton spectrum divided by the so-called unmodified spectrum, where only adiabatic energy losses (expansion of the Universe) are taken into account.
In this paper we will always assume a power law injection spectrum for UHECR, namely Q(Eg) ∝ Eg-g being Eg the UHECR generation energy eventually bounded from above by the maximum attainable energy at the source Egmax. A useful characteristic of the modification factor is its universality, it depends very weekly on the injection power law index and on various physical phenomena that can be included in the calculation . In figure 1 we plot (E) for different values of g (as labelled); it can be seen that if one includes in the calculation of Jp only adiabatic energy losses then, by definition, (E) = 1 (dash dotted line).
Figure 1. Modification factor for different values of g (see text).
When e+e- production is additionally included one obtains (E) as labelled in figure 1, this behavior corresponds to the pair-production dip. Finally, if all channels of energy losses are taken into account (E) shows also the GZK suppression, curve labelled 'total'. The only "critical" assumption at the very base of the dip behavior is the proton dominated spectrum, already a nuclei pollution at the level of 15% will spoil the behavior of figure 1.
Modification factor is a very useful tool to study the effect of the CMB radiation field on the propagation of UHE protons and can be compared with observations simply dividing the observed spectra by the unmodified spectrum Jpumm(E) ∝ E-g. In figures 2, 3 we compare the modification factor with the observations of HiRes and TA . This comparison involves only two free parameters: the power law index at injection g and the total emissivity of UHE protons sources.
Figure 2. The pair production dip compared with the HiRes experimental data , this experiment confirms the pair-production dip with good accuracy (χ2 / d.o.f. ≃ 1.0 - 1.2).
The fit of HiRes and TA data to the pair-production dip is quite good with a χ2 / d.o.f. ≃ 1 and a best fit value of g ≃ 2.7 for HiRes and g = 2.6 for TA. From figures 2, 3 one can notice that the modification factor observed by HiRes exceeds the theoretical one around EeV energies; since by definition (E) ≤ 1 this excess signals the appearance of a new component of CR at E < 1 EeV. This component can be nothing else but the galactic cosmic rays. Therefore the dip model for UHECR implies a transition from galactic to extra-galactic CR at E ≲ 1 EeV, we will come back to this in section 4.
The experimental observations of HiRes and TA show a strong evidence of the GZK cut-off in the spectra, while Auger data, that we will discuss in the next session, seem less compatible with the expected GZK behavior of the spectra. In general all experiments show a flux suppression at the highest energies, nevertheless to ascribe it to the process of photo-pion production of protons on the CMB field (i.e. GZK suppression) one must prove that: (i) the energy scale of the cut-off and its shape correspond to the theoretical predictions, (ii) the observed chemical composition is strongly dominated by protons.
Figure 3. The pair production dip compared with the TA experimental data .
The integral spectrum of protons Jp(> E) has a power law behavior at low energy. Increasing the energy, because of the photo-pion production process, Jp(> E) shows an abrupt suppression (i.e. GZK cut-off). The energy scale at which this suppression arises can be evaluated as the energy at which the integral flux is reduced by half of its low energy value. This energy scale E1/2 can be computed with extreme accuracy and it is found to be practically model independent , its theoretical prediction is E1/2 = 1019.72 eV ≃ 52.5 EeV .
The HiRes and TA observations show a proton dominated spectrum at all energies E > 1018 eV , moreover the HiRes collaboration measured E1/2 obtaining a value in a pretty good agreement with theoretical expectations, namely E1/2HiRes = 1019.73 ± 0.07 eV. These experimental evidences show a coherent picture of a proton dominated flux that exhibits the expected features: pair-production dip and photo-pion production suppression (GZK).