#### 3.1. Effects of Neutrino Propagation

The relevance of neutrinos oscillations to the interpretation of cosmic neutrinos was remarked quite soon, see for example [

43,

44,

45]. For instance, in Reference [

44] we read “if neutrino oscillations occur and cause the transition

${\overline{\nu}}_{e}\rightleftarrows {\overline{\nu}}_{\mu}$, then the flux of

${\overline{\nu}}_{e}$ increases. This effect is particularly important for

$p\gamma $ neutrinos” while in Reference [

45] we read “if there exist more than two neutrino types with mixing of all neutrinos, cosmic neutrino oscillations may result in the appearance of new type neutrinos, the field of which may be present in the weak interaction hamiltonian together with heavy charged lepton fields.” Actually, these observations still delimit the frontier of research in the field of cosmic neutrinos, for the reasons we recall here below and will elaborate in

Section 5.3 and

Section 5.4.

The vacuum oscillation phases that have been probed in terrestrial experiments can be parameterized as:

This means that vacuum oscillations have to affect neutrino propagating over cosmic distances. When we discuss astrophysical neutrinos, in the energy range between hundreds of TeV and multi-PeV, produced by extragalactic sources (i.e., at least at distances of Mpc), the oscillation phase becomes order of

${10}^{10}$ or more. In this case the only observable and meaningful physical quantity is the phase averaged oscillation. The minimal setup to analyze the effect on cosmic neutrinos is just the average value of the vacuum probabilities when the oscillating phases are set to zero:

This limit, when the oscillation probabilities reduce to constant values, is known as

Gribov-Pontecorvo regime [

43]. Using the most recent best fit values of the mixing angles [

46]:

the resulting approximate values of the oscillation probabilities in Equation (

12) correspond to:

Given a flavor fraction of different neutrinos at the source,

${f}_{\ell}^{0}$, such that

$0\le {f}_{\ell}^{0}\le 1$ and

${f}_{e}^{0}+{f}_{\mu}^{0}+{f}_{\tau}^{0}=1$, one calculates the final flavor fraction at Earth as:

(For a thorough investigation of the relevant parameters and an assessment of the uncertainties, see References [

47,

48].)

The major consequence of Equation (

15) for cosmic neutrinos on Earth is that the spectra of the three neutrino flavors are approximately the same, if they have exactly the same power law distribution. This is particularly true in the case of muon and tau neutrinos, due to the parameters of oscillations, which display an approximate mu-tau exchange symmetry.

Assuming in fact that neutrinos originate from

$pp$ or

$p\gamma $ production (

Section 2.2). In astrophysical environments, the pion decay chain is complete (i.e., also the muon decays). According to Equation (

3), one has exactly a (

${\nu}_{\mu}+{\overline{\nu}}_{\mu}$) pair for each

${\nu}_{e}$ or

${\overline{\nu}}_{e}$, leading to flavor fractions at the source

${f}_{e}^{0}=1/3$,

${f}_{\mu}^{0}=2/3$ and

${f}_{\tau}^{0}<{10}^{-5}$. In all large apparata used to study cosmic neutrinos, there is no experimental method able to distinguish reactions induced by a neutrino or an anti-neutrino; thus, no separation between particles and anti-particles can be made.

The flavor fractions can thus be written using a single parameter,

$f={f}_{e}^{0}$, as:

With oscillation parameters given in Equation (

14), we obtain:

that verify the properties mentioned just above. In particular, for

$f=1/3$ the flavor fractions are

$1/3+\{-0.027,+0.019,+0.008\},$ namely, they are very close to each other after oscillations.

Tau neutrinos, in this context of discussion, have a central role as they are not produced in charged mesons decay, either in atmospheric or astrophysical environments. The conventional component of atmospheric neutrinos from pion and kaon decays, as described in

Section 6.1, is

mostly composed by

${\nu}_{\mu}+{\overline{\nu}}_{\mu}$; at few 10–100 TeV one expects the onset of a new component due to charmed meson decays, that should instead be almost equally composed by electron and muon neutrinos (more discussion below). Because at energy scales larger than 1 TeV atmospheric neutrinos are not significantly subject to oscillations in a path of length of the Earth diameter, we do not expect the presence of any atmospheric

${\nu}_{\tau}$. Instead it is for sure that cosmic neutrinos are subject to neutrino oscillations, and therefore tau neutrinos have to be present. The interpretation of tau neutrino events should be regarded with high confidence as due to high-energy neutrinos that have traveled upon cosmic distances. See Reference [

49] for a quantitative and updated discussion of the expectations.

Note that, generally, neutrino and antineutrino events are not distinguishable and therefore are added together - the Glashow resonance events due to

${\overline{\nu}}_{e}$’s, discussed in

Section 5.4, are an exception to this rule.

#### 3.2. Neutrino Interactions in the Earth

The distance traveled by a neutrino in the Earth volume, before reaching the detector, is:

where

${\theta}_{\mathrm{N}}$ is the nadir angle,

${R}_{\oplus}=6371$ km the average Earth radius. The quantity

d represents the depth of the detector (of the order of a few km at most). The formula is derived from basic trigonometric considerations.

The Earth is not fully transparent to the highest energy neutrinos, due to the increase of the absorption cross section of the matter,

${\sigma}_{\mathrm{abs}}$, with the increase of the neutrino energy

${E}_{\nu}$. The main contribution is due to the charged current (CC) interaction with nucleons

${\sigma}_{\mathrm{abs}}\simeq {\sigma}_{\mathrm{CC}}$ that can be modeled as deep inelastic scattering. The neutral current interaction produces a neutrino with a reduced energy, and this does not significantly affect the absorption cross section. Detailed calculations (performed also through Monte Carlo methods) can account for the deformation of the arrival energy spectrum due to neutral current interactions and due to the

${\nu}_{\tau}$ regeneration effect [

50]. The propagation of the

${\overline{\nu}}_{e}$ flavor at 6.3 PeV is also affected by the resonant formation of the

W boson in

${\overline{\nu}}_{e}+{e}^{-}$ collisions, described in

Section 5.4.

The absorption coefficient of the neutrinos, according to the usual formula

${e}^{-\tau}$, depends on the

opacity factor $\tau $, defined as:

Here, the

column density $z\left({\theta}_{\mathrm{N}}\right)$, which corresponds to the number of nucleons per cm

${}^{2}$ that are crossed from a certain nadir angle, can be estimated by the formula

where

${N}_{A}$ is the Avogadro number,

x is the coordinate along the neutrino path inside the Earth. Here, we neglected

d (and

h) in the formula (

18) for

$\ell ({\theta}_{\mathrm{N}};h,d)$, which is sufficient for our purposes. The terrestrial density as a function of the distance from the center

$\rho \left(R\right)$, estimated using the PREM model [

51] with an accuracy adequate to the current need, is shown in the left panel of

Figure 3. One can define the inverse of the column density as the

critical value of the cross section
In fact, when the absorption cross section satisfies the condition

${\sigma}_{\mathrm{abs}}\left({E}_{\nu}\right)={\sigma}_{\star}\left({\theta}_{\mathrm{N}}\right)$, the opacity is

$\tau =1$ and the absorption coefficients is

$1/e\sim 0.37$. The right panel of

Figure 3 compares

${\sigma}_{\star}\left({\theta}_{\mathrm{N}}\right)$, depicted in blue and shown as a function of the nadir angle, with the cross section given for three values of the energy, indicated in the black lines. The angles at which the absorption coefficients of the neutrinos is

$1/e$ are,

${32}^{\xb0}$ for

${E}_{\nu}=100$ TeV,

${59}^{\xb0}$ for

${E}_{\nu}=1$ PeV,

${81}^{\xb0}$ for

${E}_{\nu}=10$ PeV. Therefore, at very-high energies, we receive neutrinos arriving only from a limited patch of the sky, close to the local horizon. Note that the rotation of the Earth changes the region of the sky seen in the course of the day, except for the case of a detector located at the poles (such as IceCube) when this does not change.