Even though no high energy neutrino source has been observed yet, it is possible to make some reliable assertions concerning the maximum neutrino flux to be expected from such sources. The idea behind deriving such flux limits is illustrated in Fig. 5.53: In a hadronic particle accelerator, protons are accelerated to high energies and subsequently interact with ambient photons
5.5. FLUX LIMITS 115
confined p
+ p/γ
+ p/γ +γ B
n ν
p
β decay
ν
p
Earth
source
Figure 5.53: Schematic illustration of the reasoning used for deriving neutrino flux limits. In a hadronic source, protons are accelerated to high energies and subsequently may interact with photons or other protons, so that neutrons and neutrinos are created. Whereas the protons are confined within the source by a magnetic field, the neutrons may escape from the source, unless they are turned into a proton by a photohadronic reaction. Neutrons leaving the source decay and thus form the cosmic ray protons. The neutrinos can escape from the source unimpeded.
or other protons, yielding neutrons and pions. The latter decay (if charged) into neutrinos.
Due to the magnetic field in the source (which is required for the particle acceleration) the protons are confined within the source. The neutrons, on the other hand, may escape freely, unless they undergo a photohadronic interaction before leaving the source and are thus turned into a proton. Of course, any produced neutrino can get away unhindered as well.
On their way to Earth, the neutrons decay into protons, so that they can be considered as the origin of the cosmic ray protons observed at Earth.
Hence from the cosmic ray spectrum one may infer the original neutron spec-trum, and as the neutrons and neutrinos are created by the same processes in the source, this clearly allows to estimate the number of emitted neutrinos.
A similar argument is valid for photons.
A quantitative analysis [139] shows that the relation between the muon neutrino and neutron production spectrum Q at the source depends on the spectral index α of the target photon spectrum. Forα=−1 it is given by
Qνµ(Eνµ) = 83.3Qn(25Eνµ),
whereas for α= 0 it has the form
Qνµ(Eνµ) = 416Qn(25Eνµ).
Similarly, the bolometric muon neutrino and photon luminosities are related via
Lνµ = (1 +e−5α−5)Lγ. (5.9) In order to obtain the “best possible” flux limit, one now assumes a (single source) neutron spectrum of the form Qn ∝ E−1exp(−E/Emax), and the corresponding muon neutrino spectrumQνµ is calculated as above. Then the resulting cosmic ray flux φcr observed at the Earth is obtained by summing over all sources and including propagation effects, [139]
φcr(E)∝ 1 4π
Z zmax
zmin
M(E, z)(1 +z)2 4πd2L
dNsource
dz Qn((1 +z)E), (5.10) where dL and dNsource/dz denote the luminosity distance [90] and source distribution, respectively, and whereM(E, z) takes care of the modifications during the propagation. An analogous relation (with φνµ and Qνµ instead of φcr and Qn, and with M(E, z)≡1) holds valid for the muon neutrino flux.
Now the constants of proportionality are chosen so that φcr is as large as possible, but consistent with the (observed) cosmic ray flux limitφcr,limit. Thenφcr and φcr,limitwill coincide at some energy Ecoincide.
Using the neutrino analogue of Eq. 5.10, one may obtain the correspond-ing muon neutrino flux φνµ at the Earth. Its value at Ecoincide can be inter-preted as the maximum allowed flux at that energy. Repeating the process for all cutoff energiesEmax, we thus should get the upper neutrino flux limit.
While this is indeed correct for energies & 104 GeV, at lower energies there is the additional constraint that photon and neutrino production are coupled (cf. Eq. 5.9), so that the diffuse gamma radiation background and the neutrino flux must be consistent. If this is taken into account, one obtains the line labeled “MPR,τnγ<1” in Fig. 5.54. Fig. 5.55 shows this flux limit as a function of the energy and nadir angle, and the corresponding muon event rate as a function of the nadir angle and energy offset.
A slightly different approach is used in [190]. Here, the input neutron spectrumQn is taken to be of the form En−2 and is normalized to match the cosmic ray flux. The resulting neutrino flux limit must also be a power law
5.5. FLUX LIMITS 117
-7.8 -7.6 -7.4 -7.2 -7 -6.8 -6.6 -6.4 -6.2 -6
4 5 6 7 8 9 10 11 12
lg(Eν2 φν(Eν))
lgEν MPR,τnγ>>1
MPR,τnγ<1 AMANDA
WB
Figure 5.54: Limits for the flux of neutrinos from hadronic sources, as given by Waxman and Bahcall [190] (WB) and by Mannheim, Protheroe and Rachen (MPR) [139] for optically thin(MPR,τnγ<1)and thick sources(MPR,τnγ≫1). Also shown is the current observational flux limit obtained by AMANDA. [2] The shaded area gives the range covered by the MPR flux limits. Flavor oscillations are taken into account.
with spectral index -2; it is contained in Fig. 5.54 as the line labeled WB.
Fig. 5.56 shows its dependence on the nadir angle and the corresponding event rate.
So far, we have assumed that the sources are optically thin for neutrons.
If instead we take them to be optically thick, most of the neutrons don’t leave their source and the above analysis would understimate the flux of neutrinos, which still escape from the source unhindered. Hence the only remaining constraint is consistency with the diffuse gamma radiation back-ground. The resulting neutrino flux limit is shown in Fig. 5.54 as the line labeled “MPR,τnγ≫1”. Its dependence on the nadir angle and the corre-sponding event rate can be found in Fig. 5.57.
Throughout this section, the fundamental prerequisite has been that the neutrinos are created by interactions of accelerated protons. Hence the flux neutrinos from non-hadronic sources isn’t covered by the given flux limits and thus could, at least in principle, be arbitrarily high. However, the AMANDA
8
Figure 5.55: Left: Neutrino flux limit for optically thin sources, as given by Mannheim, Protheroe and Rachen. [139]Right: Corresponding muon event rate.
8
Figure 5.56: Left: Neutrino flux limit, as given by Waxman and Bahcall. [190]
Right: Corresponding muon event rate.
5.5. FLUX LIMITS 119
Figure 5.57: Left: Neutrino flux limit for optically thick sources, as given by Mannheim, Protheroe and Rachen. [139] Right: Corresponding muon event rate.
telescope has established an observational upper flux limit, which is already below the MPR limit for optically thick sources [2], as can be seen from the line labeled “AMANDA” in Fig. 5.54.
Chapter 6
Tomography of the inner Earth
One of the most obvious geophysical questions is the one what the structure of the inner Earth looks like. As this cannot be answered by direct inspection, one needs some agency which can cross the Earth and – if it is not produced inside the Earth and thus allows inferences concerning the structure – the propagation of which depends on the density or chemical composition.
Seismic waves are commonly used as the agency, and these allow to estab-lish a fairly robust image of the density profile in the inner Earth. However, neutrinos might be suitable for the task as well, and indeed there are three possible techniques how to use them. Firstly, geoneutrinos produced by the decay of radionuclids in the Earth may provide evidence regarding the chem-ical composition. [78] Secondly, flavor oscillations of neutrinos from the Sun, a supernova, or a neutrino factory which cross the Earth allow inferences con-cerning the density profile (cf., e.g., [8, 157, 195]). Finally, absorption and regeneration of an isotropic flux of astrophysical neutrinos inside the Earth might make a tomography possible (cf., e.g., [194, 112, 100, 101, 102, 171]).
In this chapter, we use the last method together with the propagation analysis developed in the preceding chapters in order to outline the prospects and limitations of an inner Earth tomography. Before doing so, however, we first give a brief description of seismic waves and the Preliminary Reference Earth Model, and develop a simple algorithm for calculating an inverse Radon transform.
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