Propagation of ultra high energy cosmic rays

We briefly describe the energy loss processes of ultrahigh energy protons, heavier nuclei and gamma rays in interactions with the universal photon fields of the Universe. We then discuss the modification of the accelerated cosmic ray energy spectrum in propagation by the energy loss processes and the charged cosmic ray scattering in the extragalactic magnetic fields. The energy lost by the ultrahigh energy cosmic rays goes into gamma rays and neutrinos that carry additional information about the sources of highest energy particles. The new experimental results of the HiRes and the Auger collaborations are discussed in view of the predictions from propagation calculations.


Introduction
The high interest of the astrophysical community to the Ultrahigh Energy Cosmic Rays (UHECR) developed after the Agasa experiment showed its energy spectrum that continued above 10 20 eV without a break [1] and with no indication of a GZK [2] structure. This is not the first time cosmic rays of such high energy were observed. John Linsley reported on the first extensive air shower above 10 20 eV in 1963 [3]. Three years later Greisen, and independently Zatsepin & Kuzmin, predicted the end of the cosmic ray spectrum that results from the interactions of the UHECR with the microwave background radiation (MBR). Every giant air shower experiment has reported events exceeding 10 20 eV since that time. Because of the very low flux of such particles and the various energy estimates of different experiment it was impossible to judge if these highest energy nuclei in the Universe obey the GZK predictions -a sharp decline of the spectrum above 4×10 19 eV or not. At the time Agasa was the highest exposure experiment and it claimed a spectrum that is not consistent with the GZK cut-off.
This result was not confirmed by the contemporary leaders in UHECR statistics, the Auger and the HiRes experiments. Both these groups have published UHECR spectra that seems to be consistent with the GZK prediction. There are, however, distinct differences between the results of these two groups that create new challenges for the scientists who are eager to establish in some detail the acceleration and propagation picture of UHECR.
It has been a common suspicion for more than 50 years that the highest energy cosmic rays are of extragalactic origin [4]. The argument is that our Galaxy is not large enough and the galactic magnetic fields are not strong enough to contain particles of such high energy and thus accelerate them. The gyro radius R g of a 10 20 eV proton in 3µG field is higher than 30 kpc, similar to the dimension of the whole Galaxy that does not contain shocks of the same dimension.
If UHECR are indeed of extragalactic origin they should suffer from the interactions with the MBR unless their sources are cosmologically very close to us. For this reason a comparison of the detected UHECR spectrum with theoretical calculations of the particle propagation in extragalactic space should at least point at the distance distribution of their sources and bring us closer to the source identification. From that point of view the small differences between the HiRes [5] and Auger [6] spectra lead us to very different interpretations. Figure 1 shows the published spectra and the rough fits of the observed spectra that are suggested by the two experimental groups. It is obvious that the extremely small statistics around 10 20 eV makes the fit of the strong decline above 10 19.6 eV very difficult. The statistics around 10 18.5 eV, on the other hand, is significant and the different behavior of the spectra is confusing in addition to the different normalizations of the spectra. Another major difference between the two experiments is their estimate of the chemical composition of UHECR studied from the measured depth of shower maximum X max . HiRes derives a pure proton composition [7], while the Auger analysis tends to reveal a more complex mixed nuclear composition [8]. The UHECR composition at their sources affects the interpretation of the spectra because protons and heavier nuclei have different energy loss mechanisms. This paper is organized as follows: Section 2 discusses the energy loss of different possible primary particles. The next section describes the formation of the ultrahigh energy cosmic ray spectrum on propagation and the astrophysical parameters that are important for it. Section 4 is dedicated to the production of secondary particle fluxes in propagation, and mostly to cosmogenic neutrinos. Section 5 contains a brief summary of the current knowledge.

UHECR energy loss
Apart from the adiabatic energy loss due to the expansion of the universe there are two important processes for protons: photoproduction interactions and e + e − pair production (BH) interactions identical to the pair production interactions of γ-rays in the nuclear field. The back of the envelope estimate of the photoproduction energy loss goes like this: The average interaction length λ ph for interactions with the MBR is the inverse of the product of the interaction cross section σ ph and the photon density n. For σ ph = 10 −28 cm 2 and n = 400 cm −3 λ ph = 8.3 Mpc. Since protons lose about 0.2 of their energy in each interaction it takes about ten interaction length to decrease the particle energy by a factor of 10.
The story of the heavier nuclei energy loss is more complicated. In addition to these two processes heavy nuclei lose energy in photo disintegration (spallation) processes, i.e. when the center of mass energy exceeds the giant dipole resonance the nucleus can lose a nucleon. Since less energy is required in the center of mass the cross section is higher but the energy loss depends on the mass of the nucleus that loses only one or two nucleons. The photoproduction energy loss follows the same energy dependence as for protons but in the Lorentz factor space, i.e. in E/A units. The pair production cross section is a quadratic function of the charge of the nucleus Z.
In the case of exotic theoretical top-down models UHECR are gamma rays from the decay of very heavy X-particles. Although the γ-ray fraction was strongly limited by the Auger Collaboration to not more than 2% for UHECR above 10 19 eV [9] this possibility can not be totally excluded. In such a case the energy loss is due to the γγ → e + e − process.

Proton energy loss length
A photoproduction interaction is possible when at least one pion is generated in the process. This requires that the center of mass energy of the interaction √ s is higher that the sum of a proton mass m p and a pion mass m π . In the laboratory system the square of the center of mass energy s is where ǫ is the photon energy and θ is the angle between the proton and the photon. In a head on collision (cos θ = −1) with a photon of the average MBR energy (6.3×10 −4 eV) the minimum proton energy is There are many MBR photons with higher energy and the threshold proton energy is actually lower, about 3×10 19 eV. The cross section for this interaction is very well studied at accelerators where photons interact with stationary protons. The highest cross section is at the mass of the ∆ + resonance (1232 MeV) which decays to either a proton and a neutral pion (pπ 0 ) or to a neutron and a positive pion (nπ + ). At the peak of the resonance the cross section is about 500 µb. At higher energy the cross section first decreases to about 100 µb and then increases logarithmically. The neutron interaction cross section is, if not identical, very similar to the proton one.
The MBR spectrum and density are also very well known, so the proton interaction length can be calculated exactly, as shown in the left hand panel of Fig. 2 with dash line. Since protons lose only a fraction of their energy (K inel ), another quantity -the energy loss length L loss = − 1 E dE dx becomes important. The energy loss length is longer than the interaction length by 1/K inel , by about a factor of 5 at threshold. At higher energy K inel grows and this factor is about 2.
In the case of e + e − pair production [10] the addition of two electron masses to the center of mass energy √ s requires much lower proton energy and the process has lower threshold. The cross section for pair production is higher than σ ph , but the fractional energy loss is small, of order of the ratio of the electron to proton mass m e /m p . The energy loss length has a minimum around 2×10 19 eV and is always longer than 1,000 Mpc.
The last proton energy loss process is the redshift due to the expansion of the Universe. The current energy loss length to redshift is the ratio of the velocity of light to the Hubble constant (c/H 0 ) and is 4,000 Mpc for H 0 = 75 km.s −1 Mpc −1 .
The energy loss of protons in the MBR is shown in the left hand panel of Fig. 2. This figure shows also the photoproduction interaction length and the decay length of neutrons. The neutron photoproduction cross section is almost identical to the proton one. This means that neutrons of energy less than 4×10 20 will most likely decay and only neutrons of higher energy are likely to have photoproduction interactions.

Energy loss length of heavier nuclei
The energy loss length of heavier nuclei is shown in the right hand panel of Fig. 2 as calculated in Ref. [11]. Its minimum value is significantly lower than that of protons but is achieved at higher energy: A × E p . It is, of course, not obvious that such high energy could be achieved at UHECR acceleration, although we always assume that the maximum acceleration energy is proportional to the charge Z of the nucleus. The effect of propagation on the accelerated UHECR can not be calculated directly from the energy loss lengths shown in Fig. 2 because an accelerated nucleus changes its mass after the first photo disintegration interactions. This means that the code that treats propagation of nuclei should be able to calculate the cross sections for all nuclei and isotopes lighter than the injected nucleus. In addition to the cross sections for losing one, two, and more nucleons the codes should evaluate the decay probability for unstable nuclei that may be generated in the propagation.
Many of the cross sections necessary for a correct simulation of the propagation of nuclei have been experimentally studied at accelerators. There is still a need for approximations for some of the numerous processes taking place in the propagation of nuclei. See, e.g. the early work on propagation of nuclei [12] that is based on such approximations.

Gamma ray energy loss length
The process has γγ → e + e − has a resonant character and the cross section peaks at E γ ǫ = 2m 2 e , where ǫ is the energy of the seed photons. For the average energy of MBR this corresponds to E γ of 8×10 14 eV and the mean free path decreases with increasing E γ . For gamma rays of energy 10 20 eV the relevant seed photon frequency is about 1 MHz -in the radio band. This creates a big uncertainty in the estimates of the UHE γ-ray energy loss length because the density of the radio background at such frequencies is not known. One can relate the radio emission of various astrophysical objects to the much better known infrared emission and generate models to calculate the energy loss length. The results of such modeling are energy loss lengths of order that of protons.
A different source of uncertainty in the γ-ray propagation is the strength of the extragalactic magnetic fields. If they are negligible the electrons have inverse Compton interactions, whose interaction length is similar to that of the pair production, and generate a second generation of very high energy γ-rays. This cascading can continue for a significant distance without downgrading very much the gamma ray energy. If, however, the magnetic fields are significant electrons lose energy very fast on synchrotron radiation. The γ-ray energy is rapidly transferred to the MeV-GeV energy range. The range of top-down cosmic ray generation models has been restricted because of overproduction of GeV γ-rays. The energy loss distance on synchrotron radiation is 2.6E −1 18 B −2 −9 Mpc, where E 18 is the electron energy in unites of 10 18 eV and B −9 is the strength of magnetic field in nGauss.

Formation of the cosmic ray energy spectrum after propagation
Predictions of the shape of the cosmic ray spectrum requires much more than the energy loss in propagation. The necessary astrophysical input, currently unknown, includes et least the following four items: • UHECR source distribution • cosmic ray source luminosity • cosmic ray injection (acceleration) spectrum • maximum acceleration energy E max • cosmic ray chemical composition • cosmic ray source cosmological evolution.
The UHECR source distribution was the least known one. After Auger found a correlation with nearby (redshift z less than 0.018) active galactic nuclei (AGN)of their highest energy events [13,14] the situation changed. Although there is no certainty that these AGN are indeed the sources of UHECR, this is the first time we have a suggestion what the source distribution could be.
The other five parameters are not independent of each other. The UHECR source luminosity can in principle be determined by the detected UHECR flux above 10 19 eV. In view of the low current statistics, the derived luminosity depends strongly on the assumed injection spectrum and composition and partially on the assumed cosmological evolution of the sources [15]. The source cosmological evolution may be the best known parameter since it should resemble these of other astrophysical phenomena such as the star formation rate in the Universe.

Formation of the proton spectrum in propagation
As an example we will discuss the formation of the cosmic ray spectrum in the case all UHECR are protons. The left hand panel of Fig. 3 shows the evolution of a monochromatic protons of injection energy 10 21.5 eV after propagation from different redshifts. After propagation on z=0.0005 (appr. 2 Mpc) a large fraction (appr. 60%) of the injected protons have not interacted. Some of them, however, have interacted a couple of times and their energy has decreased as much as a factor of 10 as such high energy protons lose much more than 20% of their energy in photoproduction interactions. Almost all protons have interacted at z=0.0078 and the average proton energy at this distance is 2×10 20 eV. In further propagation the energy distribution at arrival becomes narrower as the highly stochastic photoproduction energy loss becomes lower than the almost continuous pair production, and later the adiabatic loss from the expansion of the Universe. The width of the arrival energy decreases until the pair production energy loss length is smaller than that of the adiabatic energy loss length. One can see in the  right hand panel how the sums of the contributions of low redshifts exceeds the value of the injected spectrum after a propagation to z=0.1. In models with the cosmological evolution the effect is stronger and proportional to the strength of the source evolution.
A natural assumption for the source distribution, the result of which is shown in Fig. 3 is that sources are isotropically and homogeneously distributed in the Universe because we do not inhabit a special part of it, and the contribution of all sources are identical. In such a case the cosmic ray flux at Earth could be determined by an integration of the fluxes from different redshifts shown in the right hand panel of Fig. 3. In the case of cosmological evolution of the sources the integral is where L(z) is the cosmic ray source luminosity as a function of redshift and N 0 (E 0 ) reflects the injection spectrum, P (E 0 , E ′ , z) is the probability for a proton injected with energy E 0 at redshift z to reach us with energy E ′ . The derivative dt/dz depends on the cosmological model and is dt dz and is simplified to (1 + z) −5/2 /(H o (1 + z)) for the Einstein-deSitter Universe.
It is important to note that contribution of different redshifts depends not only on the cosmological evolution but also on the injection spectral index as the photoproduction energy loss is a strong function of the injection energy. Since in steep injection spectra a larger fraction of the observed flux comes from lower primary energy (that do not change as much on propagation) the contribution of higher redshifts is larger.
One can see in the right hand panel of Fig. 3 that even z=0.05 does contribute to UHECR above 6×10 19 eV which was observed by Auger to be correlated to AGN within distances of only 71 Mpc (z ≤ 0.017). There are several ways to deal with this seeming controversy. One would be to claim that the energy scale of Auger is not correct and it should be higher by about 25-30%. This would make the contributions of higher redshifts smaller and bring the GZK sphere closer to the expectations for protons.
Another is to claim that the feature observed by Auger and HiRes is not a result of the GZK process, as the experimental groups claim, and is just the end of the acceleration power of the sources that does not much exceed 10 20 eV. The shape of the spectra observed by both experiments (that are shown in Fig. 1) is, however, very close to what we expect from the GZK cutoff. If one attempts to fit these spectra with the same model some problems start showing up. The model of Berezinsky et al [16] that has pure proton composition with steep injection spectrum (E −2.7 ) fits perfectly the HiRes spectrum and does not require other contributions (such as galactic cosmic rays) above 10 18 eV. The same model, however, does not appear to fit well the current Auger data.
One can also involve the extragalactic magnetic fields in the explanation. If it is high the UHECR would scatter often and their real pathlength would be considerably larger than the distance to the sources.

Propagation in magnetic fields
Our knowledge of the extragalactic magnetic fields is quite limited. We do know that they exist and are most likely proportional to the mass density in the Universe, i.e. high in regions of high matter concentration ans low in voids. Magnetic fields introduce three main effects. The cosmic ray scattering increases the propagation pathlength and thus restricts the radius of the possible sources. The scattering creates a deviation of the arrival direction of the UHECR from the direction of the source. The gyro radius of a 10 20 eV proton in 10 −9 G (nG) field is 100 Mpc. If the field is random with a correlation length ℓ the deviation angle θ after propagation on distance D is where B −9 is the r.m.s. field strength in nG, D 100 is the distance in units of 100 Mpc, ℓ 1 is the correlation length in units of 1 Mpc and E 20 is the energy in units of 10 20 eV. Protons below 10 20 eV would scatter much more around the direction of the source but the highest energy particles would point at the source with an angle comparable to the experimental resolution. The scattering angles for heavier nuclei are proportional to their charge. The scattering also introduces time delay compared to the rectilinear propagation of light. The time delay δτ has a much stronger dependence on the particle energy, magnetic field strength and the propagation distance. For small angle scattering it is If the source of the observed UHECR were an explosive process, such as a gamma ray burst at a distance of 100 Mpc, all protons would accelerated at once, but because of time delay they would arrive at Earth in a reverse order of their energy. The highest energy particles would reach us first, while the lower energy ones would be delayed with millions of years. It is important to note that the time delay depends on the square of the particle charge. Iron nuclei coming from 10 Mpc will be still delayed seven times more than protons arriving from 100 Mpc. Time delays could prevent some of the extragalactic protons from reaching us, because their travel time could exceed the the age of the Universe. Particles of energy below 5.5×10 17 eV from the gamma ray burst at 100 Mpc, for example, propagating in 1 nG field will not reach us because their time delay will exceed Hubble time, taken here of 10 10 yrs for simplicity.
In a simulation that propagated protons in 1 nG random field [17] we calculated the proton horizon R 50 , which is the distance at which 1/e fraction of protons maintain at least one half of their injection energy. This distance is smaller than the proton energy loss length even for 10 20 eV protons. The ratio R 50 to the energy loss length in 1 nG field is proportional to E −1.2 at the approach to 10 20 eV. Using Eq. 5 one can estimate the additional pathlength in 1 nG field for 6×10 19 eV (60 EeV) protons emitted at distance of 70 Mpc to 66 kpc. This does not make a huge difference but remember that in 10 nG field it would grow to 6.6 Mpc. For iron and 1 nG field the additional pathlength would increase to 45 Mpc -for a total pathlength of 115 Mpc.
The numbers quoted above do not include the cosmic ray energy loss which contributes significantly to the low R 50 values. At 60 EeV R 50 is only 35% of the proton energy loss length and would be less than a Mpc for Fe nuclei.
The magnetic field of our Galaxy is a good example for a combination of organized and random magnetic fields, which most likely exist on different scales in the Universe. The regular field B reg in the Galaxy has a spiral structure of axisymmetric or bisymmetric type resembling the matter distribution. The local strength of the field is about 1.8 µG with direction pointing inwards approximately along the Orion arm. The strength of the random field is not known exactly, with estimates between 1/2 and 2 B reg . The correlation length ℓ of the random galactic fields is of order 50 to 100 pc. More general estimates of the total field strength over the whole Galaxy give 5-6 µG [18], and it is possible that a galactic halo field, that does not contribute much locally, also exists. It is likely that the random field dominates the total field strength within the galactic arms, while the regular field is dominant in the inter arm space.
In a 5 µG galactic field the gyro radius of a 10 20 eV proton would be 20 kpc. The real deflection depends on the distance of the proton trajectory from the galactic center and on the pitch angle of the trajectory to the regular field. The Auger analysis attributes scattering angle of about 3 o to their events above 60 EeV which often pass relatively close to the galactic center. This is consistent with the correlation angle with nearby AGN measured by the collaboration. Events in the HiRes field of view will on the average smaller scattering than those of Auger.
The amount of scattering depends strongly on the exact magnetic field model. Field models with alternating polarity, such as BSS [19] give the smallest deflections. Models with a z component of the field (perpendicular to the galactic plane, that may be due to a dipole field) cause the largest ones.
The possible existence of regular large scale fields makes the consequences of proton propagation even more complicated. The following exercise in Ref. [22] demonstrates the problems in the following geometry: a cosmic ray source at the origin injects isotropically protons above 10 18.5 eV on a power law spectrum with spectral index of 2 and exponential cutoff at 10 21.5 eV. The source is in the central yz plane of a 3 Mpc wide magnetic wall, that is a simplified version of the Supergalactic plane (SGP), which is the plane of weight of galaxies within redshift of 0.04. Magnetic field with strength of B reg = 10 nG fills the SGP, points in z direction and decays exponentially outside the SGP. The regular field is accompanied by random field with strength B rndm = B reg /2.
Under these conditions the protons leaving the 20 Mpc sphere have not only very large deflections, but also modified energy spectra. Protons with trajectories perpendicular to the magnetic field lines exhibit much flatter spectrum since the lower energy protons are caught up in the magnetic walls. Protons with trajectories parallel to the magnetic field lines have softer spectra that include the particles that could not penetrate the supergalactic plane.
In these simple cases one can scale the effects in proton energy as a function of the magnetic field strength. If B reg were 1 µG, for example, all effects would be the same in a 200 kpc sphere. There are lobes of radio galaxies, including Cen A, that have bigger lobes that may have ordered magnetic fields of that strength and may also cause significant deflections and modifications of the injection spectra depending on the position of the observer.

Production of secondary fluxes in propagation
The energy that UHECR lose in propagation ends up in fluxes of gamma rays and neutrinos from π 0 and π ± as well as other charged and neutral mesons decays. Gamma rays have even shorter energy loss length than nuclei and develop pair production/inverse Compton cascades where synchrotron radiation may play important role. In the presence of noticeable (1 nG) extragalactic field 10 18 eV electrons quickly lose energy to GeV synchrotron photons.
Neutrinos, however, have very low interaction cross section and arrive at Earth with just redshift energy loss from almost any distance. For this reason the cosmological evolution of the cosmic ray sources are very important input in the calculation of these cosmogenic [20,21] neutrinos.
The left hand panel of Fig. 4 shows what fraction of the injection energy of E −2 cosmic ray spectrum above 10 19 is contained in nucleons and secondary particles. After propagation on 200 Mpc less than 50% of the primary energy content is in nucleons. The rest is distributed between neutrinos, gamma rays and electron-positron pairs. The largest fraction of the energy loss is in gamma rays from neutral meson decays. The reason for this is that the ∆ + decays to p + π 0 twice as often as it decays to n + π + . The muon neutrino flux consists of approximately equal numbers of muon neutrinos and antineutrinos. The electron neutrino flux is mostly electron neutrinos. The production of cosmogenic particle fluxes depends on the same general astrophysical parameters as the cosmic ray flux. The main difference is that the cosmological evolution of the cosmic ray sources is much more important. As we saw above UHE cosmic rays arrive only from very small redshifts, where even very strong cosmological evolution would make 10% difference while the cosmogenic neutrinos may easily come from z=2 which would correspond to an increase of order of magnitude or more.
The infrared background (IRB) is also a target worth considering when the production of cosmogenic neutrinos is discussed. Its number density is a factor of 400 or more less than that of the MBR, but its energy is significantly higher. This means that lower energy nucleons will occasionally interact on IRB photons and generate lower energy neutrinos. Since even in a flat (γ=1) injection spectrum there are many more lower energy protons the contribution of interactions in IRB to the cosmogenic neutrino flux, although of lower energy, may be significant. The contribution of IRB increases with the steepness of the UHECR spectrum.
Two among the Auger results presented above may have an important connection to the flux of cosmogenic neutrinos -the correlation of the highest Auger events with nearby AGN and the high cosmological evolution required by one of the spectrum fits with relatively flat proton injection spectrum.
The cosmological evolution of different types of astrophysical objects is studied mostly in star forming regions (SFR) by infrared observations or in AGNs from their X-ray emission. The cosmological evolution of SFR is derived as (1 + z) 3 on the average for up to redshifts close to 2.
It is indeed impossible to judge what is the best proxy for the cosmic rays source evolution, infrared radiation or X-rays in the 0.5 -2 KeV range where most of the statistics is. The conclusions from the X-ray observations are that the evolution of AGN is close to (1 + z) 5 [23,24]. Very powerful AGN have even a stronger evolution but the large number of AGN correlating with the Auger UHECR require using the evolution of average ones.
It may be just a coincidence, but the Auger collaboration analysis of the spectrum shown in Fig. 1 can be best described by two proton models [25]: one with γ = 1.55 and no cosmological evolution, and another on with γ=1.3 and cosmological evolution (1 + z) 5 , the same as that of AGN. The right hand panel of Fig. 4 shows the cosmogenic neutrino spectra generated by these two models. The higher energy peak of these energy spectra consists mostly of ν µ ,ν µ , and ν e from charged meson decays. The peak at about 10 15 eV consists only ofν e from neutron decay. The difference between the two neutrino  Cosmogenic neutrinos from the two proton models that fit best the Auger cosmic rays spectrum. Note that this is only the production in interactions on MBR.
models is huge, almost two orders of magnitude at maximum. One can also see the redshift of the maximum in the γ=1.3 model where most of the neutrinos are generated at high redshift. This also shows up in the lower energyν e peak that is enhanced because of the higher number of protons (and respectively secondary neutrons) injected at high redshifts. The γ=1.55 model flux will significantly increase if the production in IRB is included. This would, however, be at lower energy and the detection rate will not change a lot. The two other models that fit the Auger spectrum are mixed composition ones. If UHECR are heavy nuclei they generate a few cosmogenic neutrinos and only the proton fraction is efficient in production of meson decay neutrinos. Since there are many neutrons that are released in the spallation process there is a significant flux of ν e from neutron decay. There are several calculations [26] of the cosmogenic neutrino fluxes from mixed composition models that show this effect.
A possible detection even of a few cosmogenic neutrinos would greatly help the interpretation of the detected UHECR fluxes. Because of the big difference in the neutrino flux only models with strong cosmological evolution may lead to detection. For those without cosmological evolution we need Gigaton neutrino detectors.

Summary
• All known stable particles that are candidates for the ultra high energy cosmic rays lose energy in interactions with the cosmic microwave background and other astrophysical photon fields.
The cosmic rays spectrum measured by the HiRes collaboration agrees very well with a model of pure proton composition while the Auger collaboration spectrum may require mixed primary composition.
• The energy loss sets a horizon for the sources of such particles (a maximum distance from the observer) and modify the injection spectrum of these cosmic rays.
• Charged UHECR, such as protons and heavier nuclei, scatter in the extragalactic magnetic fields. This scattering causes increased pathlength, decreases the particle horizon, and introduces deflection from the source direction and significant time delay.
• The cosmogenic neutrino fluxes generated in cosmic rays propagation are currently close to detection in the km 3 detectors if the models with strong cosmological evolution of UHECR are correct.
• Only protons of energy approaching 10 20 eV reveal the source position and spectrum after an account for the energy loss and scattering in propagation.
• The identification of the UHECR sources will bring very valuable general information about the power and size of the sources and the magnetic fields in them, as well in the intergalactic space.