The estimate above source spectra can be translated into an expected diffuse signal from certain astrophysical sources by folding the spectrum with the spatial distribution of the sources. In this ansatz, Meli et al. (2005) used Active Galactic Nuclei (AGN) and Gamma Ray Bursts (GRBs) as potential sources, since these are the sources with the highest observed boost factors. They assume that both source types follow Star Formation Rate (SFR) with a red-shift behavior as suggested in Hasinger et al. (2005) For AGN, r = 10 is assumed. For GRBs, the r dependence is considered by taking into account a range of boost factors of 100 < r < 1000 with a maximum in the distribution at r = 300 (Guetta et al., 2004). As a simple model, it is assumed that 10% of all GRBs have r = 100, and further 10% are as powerful as r = 1000. For the remaining 80%, the average expected boost factor of r = 300 is assumed. The normalization of the expected signal is done using the most restrictive upper limit on the neutrino signal from extraterrestrial sources given by the AMANDA experiment (Munich and IceCube Collaboration, 2005):
With an E spectrum for both neutrinos and protons, the spectra are connected by assuming that the expected neutrino energy fluency is a fraction x of the proton spectrum, rdN,, dNr
With x = 1/40, since only 20% of the proton flux goes into pion production via the delta resonance, 1/2 of the remaining flux goes into the charged pion component of which 1/4 goes into neutrinos. The resulting spectrum is shown in Fig. 4.30.3. It can be seen from Fig. 4.30.3 that the only possible contribution to the CR spectrum from the super-luminal shock sources as predicted in the paper of Meli et al. (2005) is around the knee of the measured CR spectrum. It is expected however, that the effective flux is actually even lower, since the normalization is based upon the assumption that the contribution cannot be more than the current neutrino flux limits omit.
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