We estimate the range of possible neutrino signals in three ways: first on the basis of an energetics argument, second by referring to some particular models, and third by comparing to known sources of TeV photons.
Models for the origin of the highest energy cosmic-rays typically predict associated neutrino fluxes. A requirement on the sources is that they must provide sufficient power to supply the observed energy in the galactic or extragalactic component of the cosmic radiation. The assumption that comparable amounts of energy go into high-energy neutrinos allows an estimate of the corresponding neutrino signal that is independent of the specific nature of the sources, but which depends only on their distribution in the universe.
In the case of galactic cosmic-rays, the energy flux carried by neutrinos is much lower than that carried by the parent cosmic-rays because the charged particles are trapped locally as they propagate diffusively in the turbulent galactic magnetic fields. Thus an observer inside the galaxy has several chances to see a given cosmic ray particle, but only one chance to see a neutrino that passes directly out of the galaxy. Nevertheless, there is a chance that some nearby galactic sources could be visible in neutrinos in a km-scale detector. Examples will be discussed in the next section.
In contrast, the neutrino energy flux for a cosmological distribution of sources can be comparable to or larger than the observed cosmic-ray energy flux. The flux depends on how the sources have evolved over cosmological time scales{provided that the fraction of accelerated protons that interact near the source is large. One natural scenario that gives comparable energy in neutrinos and cosmic rays occurs when protons remain trapped in the acceleration region until they suffer inelastic collisions, while secondary neutrons escape and decay to become cosmic-ray protons. It is generally believed that sources of the ultra-high energy cosmic-rays are indeed extragalactic, or at least not confined to the plane of the galaxy. There is some evidence for a transition from one particle population to another somewhere above 1018 eV as well as for a trend from heavy toward lighter composition. Measurements of the cosmic-ray spectrum above 1017 eV are summarized in fig. 2.
We now wish to estimate the neutrino signal expected if the energy in neutrinos is comparable to the energy in the extra-galactic component of the cosmic radiation. The first step is to determine what fraction of the observed spectrum is the extra-galactic component. It is generally assumed that the acceleration processes produce a power-law spectrum α E-α with differential index α = 2 or slightly higher [8, 10]. But the measured spectrum has an index close to α = 3, so it is not clear just how to normalize with an extra-galactic component with a much harder spectrum.
As an illustration, the lower heavy line in fig. 2 shows a spectrum with α = 2 and an exponential cutoff at 5 × 1019 eV to represent the Greisen-Zatsepin-Kuzmin (GZK) effect [11]. Particles with energies above the GZK cutoff have interaction lengths in the microwave background of order 50 Mpc or less and must therefore be from a local or exotic component (indicated by Super-GZK in the figure), which we do not consider for the moment. If the excess of data above the curve for E < 1019 eV in fig. 2 is attributed to the tail of the galactic cosmic-ray spectrum, then the energy in a universal component of the cosmic radiation may be estimated. Integrating the energy content under the α = 2 curve gives for the energy density in cosmic rays of extragalactic origin, ρEG ∼ 2 × 10-19 eV. The estimated power calculated from ρEG is
consistent with that observed from AGN and GRBs.
Shifting the normalization point lower (or higher) by half a decade in energy would increase (decrease) this estimate by roughly a factor of two. This is comparable to the systematic differences among the different measurements of the spectrum. If the spectrum is steeper (indicated by the upper solid curve in fig. 2, as expected for acceleration by relativistic shocks [12, 13, 14]) then the energy content will be somewhat [15] larger.
Using our estimate of the energy in the extragalactic component of the cosmic-ray spectrum and assuming a spectral index α ∼ 2:0 for the neutrinos as well as the cosmic-rays, one predicts a neutrino event rate of f × 30 events/km2/yr [17], where f is the efficiency for production of neutrinos relative to cosmic rays. For f = 0:3 this estimate gives a diffuse flux at the level of E2νdN/dEν ∼ 10-8 GeVcm-2sr-1, which is comparable to the "upper bound" estimate of Waxman & Bahcall [8] before accounting for the likely effect of evolution of sources over cosmological times.
In a sense, this estimate is conservative, and there are several ways in which the neutrino flux could be larger:
The above discussion is summarized in Fig. 3.
We may summarize the previous paragraphs by saying that an argument based on energetics coupled with a natural association between cosmic particle accelerators and their secondary neutrino beams suggests that a km-scale detector will be needed to see the neutrinos. A similar conclusion is reached by relating the neutrino flux to the source of the highest energy cosmic-rays in the context of specific models.
In a recent review [19] Learned and Mannheim have summarized recent work on models of AGN, GRB and other sources of high energy astrophysical neutrinos. As relevant examples, we quote here just two new papers that appeared since ref. [19]. Schuster, Pohl and Schlickeiser [20] work out the consequences of a model of AGN blazars [21] in which the TeV γ-rays come from decay of neutral pions produced in proton-proton collisions as a relativistic cloud of dense plasma plows into the ambient interstellar medium of the host galaxy. In this picture, there is a similar flux of TeV neutrinos from decay of charged pions. They conclude that the neutrino flux from a typical bright TeV γ-blazar would be detectable at the 3 σ level above atmospheric background in an exposure of a km2-year. During an extended flare, such as occurred for Mrk501 from March to September of 1997 [22, 23, 24], the rate would be correspondingly higher in this model (see below).
The frequently quoted estimates [25, 26] of neutrino production in GRB sources by collisions of extremely energetic protons with MeV photons in the gamma-ray jets typically produces ∼ 100 TeV neutrinos at a level sufficient to produce a few events per km2-year. Now Mészáaros and Waxman [27] argue that in gamma-ray bursts that involve collapse of massive progenitors, there will be an earlier phase of production of multi-TeV neutrinos from collisions of energetic protons with X-rays as the jet pushes through the envelope of the progenitor. They predict that there may also be a class of collapses so massive that the jets do not emerge. These would not appear as visible gamma-ray bursts but would generate neutrino bursts. The event rates estimated range from a few hundred to a thousand per year over the whole sky, depending on the ratio of "choked" to "bursting" fireballs. The northern hemisphere bursts would be detectable by IceCube.
The question whether the intriguing sources of TeV γ-rays such as Markarian 421, 501 are cosmic proton accelerators has not yet been answered. Most experts argue that the photons are produced by radiative processes from accelerated electrons. There are, however, some hints that this may not be the case. For example, the spectrum of Mrk501 during the extended (six month) flare in 1997 [22] entends to above 20 TeV, which strains the electronic models because of the short synchrotron loss time for electrons. However, only neutrinos can provide incontrovertible evidence of proton acceleration.
For the case of the Mrk501 flare, it is possible to estimate the associated neutrino flux and signal that would be expected if the observed photons are produced from decay of neutral pions. The neutrino flux is closely related to the photon flux at the source. If the pion production mechanism is photoproduction by protons, then the energy in muon neutrinos will be about 1/4 the energy in photons. If the pions are produced in proton-gas collisions, then the energy in muon neutrinos and the energy in photons will be comparable. To obtain the source spectrum of photons, it is necessary first to account for the attenuation of photons during propagation through the infrared background radiation. This has been estimated by Konopelko et al. [28]. They find a source photon spectrum that extends from below a TeV to above 20 TeV with a power law differential spectral index of α = 2. Starting from the six-month average flux in Ref. [22] and correcting for the infra-red absorption, one would expect 10–100νμ events per km2 of effective area in four months, depending on the origin of the produced pions (photo-production or proton-gas interactions).
In principle, high-energy neutrino astronomy has the potential to discriminate between hadronic and electromagnetic origin of the TeV emission from objects as diverse as supernova remnants, gamma-ray bursts and active galactic nuclei. The fluxes are likely to be low, however. Estimates for a variety of sources show that reasonable expectations are at the level of a few events per km2-year. Moreover, the estimates depend on how many interactions the photons experience in the source as well as whether all observed photons are hadronic in origin. As an example of the latter point, the hadronic model of Bednarek and Protheroe [29] for the CRAB nebula attributes only the high-energy end of the photon spectrum to decay of neutral pions. For a range of parameters their model predicts a signal from < 1 to as much as 4 events per km2·yr with Eμ > 1 TeV. The corresponding rate from atmospheric background within a 1° cone is ≈ 0:4.
