Waxman-Bahcall bound

Atmospheric neutrinos are produced in the atmosphere, about 15 km above the Earth's surface.[2] They are the result of particles, usually protons or light atomic nuclei, hitting other particles in the atmosphere and causing a shower or neutrinos into the Earths surface.[2]

Sample proton cosmic shower as a result of atmospheric collisions

Atmospheric neutrinos were successfully detected in the 1960s when experiments were able to successfully find muons that resulted from the neutrinos in the cosmic shower.[4] From that, they were able to find the energy of the neutrinos and the flux associated with them. Currently, neutrinos are able to be detected by many different experiments, such as IceCube Lab, allowing for the higher accurate measurements of their energy and fluxes. Looking back at figure 1 above, the energy of the atmospheric neutrinos ranges from 10⁶ to 10¹⁴ eV while the flux is around 10⁻¹² to 10⁻²⁶ Ev⁻¹ cm⁻² s⁻¹.[5][3]

The GZK limit exists as a limit on the highest possible neutrino that can come into the atmosphere and successfully detected which is expected to be around 5 x 10¹⁹ eV.[6] The limit exists because at these higher energies, and at travel distances further than 50 kpc, interactions with the CMB photons increases.[6] With these interactions, the new product particles have lower and lower energy, and do not reach Earth until the products are now within the expected tolerances and start to blend is as atmospheric or solar neutrinos.[6]

Initially in the derivation for the muon neutrino intensity above, redshift factors were ignored. However, if a correction factor was included, it could also be found that the neutrinos detected above either started out at high energies and were detected a lo energies due to redshift, or started at low energies but were blue-shifted to a higher energy level exceeding the initial limit.[1]

It is known, however, that if red-shift is to be the prime factor in the limit, that the proton would have had to have a redshift z of less than 1. If the particle started from outside this range, as told by the GZK limit, other interactions would take place during the particles travel, and make it so that the neutrinos detected would be far below the threshold discussed.[1]

Deriving a correction factor to multiply by Imax to change the threshold of the system, it was found to be:

${\displaystyle \xi _{z}={\frac {\int _{0}^{z_{max}}dzg(z)(1+z)^{-7/2}f(z)}{\int _{0}^{\inf }dzg(z)(1+z)^{-5/2}}}}$

Working with nearby galaxies and clusters, it was found that there is no significant change on the limit from the redshift correction, and that the reason for the limit and expected values outside of the limit has to come from some other external source.[1]

Another factor to consider was the addition of the magnetic fields at the source of the neutrinos and how it might allow for increased energy of an incoming charged particle from a cosmic ray.[1] If protons can be prevented from leaving the source due to a magnetic field, then only neutrinos would be allowed to go through, meaning we would be able to see higher level neutrinos.[1] Bahcall and Waxman quickly ruled this out as a permanent option, as when there is a proto-meso interaction a charged pion is created but a proton is then also turned into a neutron. The neutron will not be affected by the field in any way, and will travel about 100 kpc when with high energies. This makes it impossible to exceed the upper bound found earlier from Waxman and Bahcall.[1]

Another theory is that the intergalactic magnetic field would be able to change the direction of the protons on their way to Earth, allowing for the neutrinos to come in relatively in a straight line.[7]

To derive this theory, Waxman and Bahcall started with the basic proton traveling with energy E, in a magnetic field B, and with correlation length λ. If the proton travels a distance of λ, the resulting angle of deflection is:[1]

${\displaystyle angle=\lambda /R_{l}}$

Where R_l is the Larmor Radius.[1]

${\displaystyle R_{l}=E/eB}$

If the angle is kept small, and propagating a distance l, the new deflection angle becomes:[1]

${\displaystyle angle={\sqrt {l/\lambda }}*\lambda /R_{l}}$

Plugging in values for time, which would give us a maximum propagation distance that the particle could travel in that time, we find that the existence of a uniformly distributed inter-galactic magnetic field would have no effect on the limit.[7][1]

When looking out into the galaxy, and starting to think about what could have caused such high energy neutrinos to appear, it was thought of that jets from Active Galactic Nuclei (AGN) were the main cause. Looking further into the details, Waxman and Bachall saw that the intensities for jets from AGN's are two times higher in magnitude than the limit discussed above.[8]

Initially, it was thought that the photons and protons were accelerated into the jets thanks to Fermi acceleration with an energy spectrum:[1]

${\displaystyle {\frac {dN}{dE}}\varpropto E^{-2}}$

for both protons and photons (simply plug in the values for photons or protons for either quantity). This implies the optical depth is related to E_p and assuming a small optical depth allows us to have the neutrino spectrum of:[1]

${\displaystyle {\frac {dN}{dE}}\varpropto E^{-1}}$

Later, it was realized that the decay of neutral pions, which are created along with charged pions, cause a high energy gamma ray emission. It was then found that the large energies being seen was not a result of Compton scattering of protons and photon, but of neutral pion decay.[8] Once this emission was fixed, the intensity of the neutrinos found from AGN was under the max limit discussed above, and AGN then became a valid cause for these higher energy neutrinos if the area was optically thin and the energy burst was cause by a single interaction of a decaying neutral pion.[1]

The Gamma-Ray Bursts (GRB) fireball model has also been another candidate for the reasoning behind higher energy neutrinos.[9][1]

Gamma-ray-burst-Mechanism

The high energy neutrino model already took multiple variables into account and was a match for the limit discussed above.[10] Similar to AGN's, the GRB's are optically thin, however, unlike AGN's which needed some more assumptions to be made on how the energy was being expelled and reached to match the flux calculations, the GRB model was able to correctly match this limit.[10][1]

The fireball model works by having the initial burst of the GRB, but then has another shock later on which goes onto explain the afterglow associated with GRB's. This second shock continues to push particles away and allows them to reach detectors on Earth within the limits discussed earlier.[10]