Expected Diffuse Neutrino Fluxes

Galactic sources of charged cosmic rays and γ-rays are likely to produce neutrinos with energies belowO(100 TeV) and a diffuse flux might be difficult to detect [72]. They are not further discussed here, for a review of galactic neutrinos sources see [35]. The extra-galactic sources mentioned above are

the prime candidates to produce neutrinos, whether observed as a diffuse flux above the atmospheric background or as point-sources. Especially, the observation of high energy γ-rays which might be produced in neutral pion decays motivates these candidates. The neutrino flux depends on the optical thickness of the sources and the energy transfer to the secondary particles in equation (2.2) and (2.3), in particular the energy transfer is different for pp and pγ interactions.

The measured cosmic ray spectrum in the energy range from 10 PeV to 100 EeV can be used to estimate the diffuse flux from extra-galactic optically thin sources under the assumption of hadronic acceleration. An upper bound is derived by calculating the flux normalization for a generic E⁻² injection spectrum which generates the same energy density as the charged cosmic ray flux. This assumes that the entire energy of the primary proton is transferred to the pions. The resulting bound is known as the Waxman-Bahcall bound and it is loosely confined to the range [72, 134]:

E²dE/dN = 1∼5·10⁻⁸GeV cm⁻²s⁻¹sr⁻¹.

Mannheim, Protheroe and Rachen (MPR) do not assume a fixed E⁻² in-jection spectrum and take into account source characteristics, like the opac-ity to neutrons which determines the rate of the cosmic flux observed at the Earth [107]. The less restrictive bound is derived from sources being optically thick for nuclear interactions, where high energy neutrons are shifted towards lower energies before escaping the acceleration region, and the observed high energy cosmic ray flux is lower than at the acceleration site. Thus, the de-rived neutrino flux is higher than for an optically thin source, where the cosmic rays at the highest energies are not attenuated at the sources.

The neutrino spectrum from GRBs is modelled following the observed photon spectrum as a combination of two power laws. Again, this spectrum can be normalized to the observed flux of ultra-high energy cosmic rays, which has been done by Waxman and Bahcall in [135] for the prompt phase of the GRB. Also in the afterglow and hours before the GRB different models predict high energy neutrinos. A review of several extra-galactic neutrino flux models can be found in [34].

Cosmological Neutrinos

The Greisen-Zatseptin-Kuzmin (GZK) cutoff mentioned above describes the process of protons losing energy by interactions with the CMB radiation:

p+γ_CMB →

( ∆⁺

p+e⁺+e⁻ , (2.8)

with threshold energiesE_th^∆≈5·10¹⁹eV andE_th^e+e− ≈5·10¹⁸eV, respectively.

The attenuation length λ = E/(−dE/dx) for resonant production of ∆⁺ is only ∼10 Mpc, while it is ∼1000 Mpc for the pair creation process [34]. In particular the resonant ∆⁺ production leads to a rapid decrease of the cos-mic particle spectrum. Two experiments, namely Hires and Agasa, mea-sured incompatible fluxes. While Hires observed a decline of the spectrum [34], Agasa detected two events above 50 EeV [130]. However, taking large systematic uncertainties into account, it is possible to interpret the two re-sults to be consistent with the GZK cutoff [47]. Recent observations from the Auger experiment exclude the continuation of the power law behavior above ∼40 EeV with a significance of 6σ [145].

An important consequence of the GZK cutoff is the existence of ultra high energy neutrinos, which are produced in the decay chain of the ∆⁺-resonance similar to the pγ interactions described in equation (2.3). The resulting neutrinos are commonly referred to as cosmogenic or GZK neutrinos. The energy ranges from approximately 10 PeV to 1 ZeV [34]. The strength and shape of the spectrum depends on several parameters. In particular the evolution of the source population with redshift has a large impact on the modelling of the spectrum [41], as well as the maximum energy of the primary particle flux. Also the composition of the cosmic ray flux is crucial for the calculation. If heavier elements dominate at the highest energies, photo-disintegration processes need to be taken into account, which suppress the cosmogenic neutrino flux [83].

Figure 2.4 shows a compilation of electron neutrino flux expectations and experimental flux limits for extraterrestrial neutrinos in the energy range from 10 TeV to 10 EeV. Up to ∼50 TeV the spectrum is dominated by at-mospheric neutrinos which is shown as a shaded area, to account for the variations due to the zenith angle dependence and the unknown contribu-tions from prompt neutrinos. At these energies cosmic neutrinos could be detected as an excess of events in the energy spectrum of atmospheric neu-trinos. Above 100-500 TeV the contribution from extraterrestrial neutrino fluxes exceeds that of the atmospheric neutrinos due to the steeply falling spectrum. The fluxes of cosmic neutrinos shown are the theoretical bounds discussed above, namely the Waxman-Bahcall bound and the MPR bound.

The MPR bound is given by a shaded area, where the upper bound is valid for optically thick and the lower bound for optically thin sources. Also shown are the MPR model for neutrino production in AGN jets [107] and the model by Stecker, Done, Salamon and Sommers (SDSS) [126, 127]. Additionally, the Waxman-Bahcall GRB flux prediction is depicted, which has been mentioned above. A range of cosmogenic neutrino fluxes with different evolution param-eters from [147] is indicated as a shaded area. Limits for fluxes ∝E⁻² from

different experiments are shown, all-flavor limits are scaled with 1/3 to get comparable one-flavor limit: Amanda-IIanalyses using 4 years ν_µ data (1), 3 year all neutrino flavors (3), 5 years cascade signatures (4), and 1 year all flavors with an enhanced data acquisition system (5) [3, 10, 13, 125];Baikal all neutrino flavors (2) [30]; Rice all neutrino flavors (6) [101]; Auger ν_τ (7) [38];Anita-lite all neutrino flavors (8) [33]; an estimatedν_µ sensitivity of the completeIceCube detector within one year of operation (9) [21].

Figure 2.4: Flux predictions and theoretical bounds for electron neutrino fluxes.

Models based on muon neutrino fluxes are scaled down by a factor of 2 to obtain the electron neutrino flux taking into account the production ratio and oscillation effects.

The SDSS model is not scaled, because it does not account for anti-neutrinos and is therefore a factor 2 lower. Experimental limits forE⁻²fluxes are shown from different experiments, all-flavor limits are scaled with 1/3: Amanda-IIanalyses using 4 years νµ data (1), 3 year all neutrino flavors (3), 5 years cascade signatures (4), and 1 year all flavors with an enhanced data acquisition system (5) [3, 10, 13, 125]; Baikal all neutrino flavors (2) [30]; Rice all neutrino flavors (6) [101]; Auger ντ (7) [38];

Anita-liteall neutrino flavors (8) [33]; an estimatedνµ sensitivity of the complete IceCubedetector within one year of operation (9) [21].