# On astrophysical solution to ultra high energy cosmic rays

###### Abstract

We argue that an astrophysical solution to the ultra high energy cosmic ray (UHECR) problem is viable. The detailed study of UHECR energy spectra is performed. The spectral features of extragalactic protons interacting with Cosmic Microwave Background (CMB) are calculated in model-independent way. Using the power-law generation spectrum as the only assumption, we analyze four features of the proton spectrum: the GZK cutoff, dip, bump and the second dip. We found the dip, induced by electron-positron production on CMB, as the most robust feature, existing in energy range eV. Its shape is stable relative to various phenomena included in calculations: discreteness of the source distribution, different modes of UHE proton propagation (from rectilinear to diffusive), local overdensity or deficit of the sources, large-scale inhomogeneities in the universe and interaction fluctuations. The dip is well confirmed by observations of AGASA, HiRes, Fly’s Eye and Yakutsk detectors. With two free parameters ( and flux normalization constant) the dip describes about 20 energy bins with for each experiment. The best fit is reached at , with the allowed range 2.55 - 2.75. The dip is used for energy calibration of the detectors. For each detector independently the energy is shifted by factor to reach the minimum . We found , and for AGASA, HiRes and Yakutsk detectors, respectively. Remarkably, that after this energy shift the fluxes and spectra of all three detectors agree perfectly, with discrepancy between AGASA and HiRes at eV being not statistically significant. The excellent agreement of the dip with observations should be considered as confirmation of UHE proton interaction with CMB. The dip has two flattenings. The high energy flattening at eV automatically explains ankle, the feature observed in all experiments starting from 1980s. The low-energy flattening at eV provides the transition to galactic cosmic rays. This transition is studied quantitatively in this work. Inclusion of primary nuclei with fraction more than upsets the agreement of the dip with observations, which we interpret as an indication to acceleration mechanism. We study in detail the formal problems of spectra calculations: energy losses (the new detailed calculations are presented), analytic method of spectrum calculations and study of fluctuations with help of kinetic equation. The UHECR sources, AGN and GRBs, are studied in a model-dependent way, and acceleration is discussed. Based on the agreement of the dip with existing data, we make the robust prediction for the spectrum at eV to be measured in the nearest future by Auger detector. We also predict the spectral signature of nearby sources, if they are observed by Auger. This paper is long and contains many technical details. For those who are interested only in physical content we recommend to read Introduction and Conclusions, which are written as autonomous parts of the paper.

###### pacs:

98.70.Sa, 96.50.sb, 96.50.sd, 98.54.Cm## I Introduction

The systematic study of Ultra High Energy Cosmic Rays (UHECR)
started in late 1950s after construction of Volcano Ranch (USA)
and Moscow University (USSR) arrays. During the next 50 years of
research the origin of UHE particles, which hit the detectors, was
not well understood. At present due to the data of the last
generation arrays, Haverah Park (UK)M. Ave *et al.* [Haverah
Park collaboration] (2003), Yakutsk (Russia)
V. P. Egorova *et al.* (2004), Akeno and AGASA (Japan) N. Hayashida *et al.* [AGASA collaboration] (1994); K. Shinozaki *et al.* [AGASA collaboration] (2004), Fly’s
Eye D. J. Bird *et al.* [Fly’s
Eye collaboration] (1994) and HiRes R. U. Abbasi *et al.* [HiRes collaboration] (2004) (USA) we are probably very
close to understanding the origin of UHECR. The forthcoming data
of Auger detector (see The Pierre Auger Collaboration for the first results) will
undoubtedly shed more light on this problem.

On the theoretical side we have an important clue to understanding the UHECR origin: the interaction of extragalactic protons, nuclei and photons with CMB, which leaves the imprint on UHE proton spectrum, most notably in the form of the Greisen-Zatsepin-Kuzmin (GZK) Greisen (1966); Zatsepin and Kuzmin (1966) cutoff for the protons.

We shortly summarize the basic experimental results and the
results of the data analysis, important for understanding of UHECR
origin (for a review see Nagano and Watson (2000)).

(i) The spectra of UHECR are measured with good accuracy at
eV, and these data have a power to
reject or confirm some models. The discrepancy between the AGASA and
HiRes data at eV might have the statistical
origin (see D. De Marco, P. Blasi and Olinto (2003) and discussion in Section
IV.3), and the GZK cutoff may exist.

(ii) The mass composition at eV (as
well as below) is not well known (for a review see Watson (2004)).

Different methods result in different mass composition, and the same methods
disagree in different experiments. In principle, the most reliable
method of measuring the mass composition is given by elongation rate
(energy dependence of maximum depth of a shower, )
measured by the fluorescent method. The data of Fly’s Eye in 1994
D. J. Bird *et al.* [Fly’s
Eye collaboration] (1994) favored iron nuclei at eV with a
gradual transition to the protons at eV. The further
development of this method by the HiRes detector, which is the
extension of Fly’s Eye, shows the transition to the proton composition
already at eV Archbold and Sokolsky (2003); R. U. Abbasi *et al.* [HiRes Collaboration] (2005). At present the
data of HiRes Archbold and Sokolsky (2003); R. U. Abbasi *et al.* [HiRes Collaboration] (2005), HiRes-MIA T. Abu-Zaayad *et al.* [HiRes Collaboration] (2000) and
Yakutsk A. V. Glushkov *et al.* [Yakutsk collaboration] (2000) favor the proton-dominant composition at
eV, data of Haverah Park M. Ave *et al.* [Haverah
Park collaboration] (2003) do not
contradict such composition at eV, while
data of Fly’s Eye D. J. Bird *et al.* [Fly’s
Eye collaboration] (1994) and Akeno M. Honda *et al.* [Akeno Collaboration] (1993) agree
with mixed composition dominated by iron.

(iii) The arrival directions of particles with energy eV show the small-angle clustering within the angular
resolution of detectors. AGASA found 3 doublets and one triplet among
47 detected particles M. Takeda
[AGASA collaboration] (1999) (see the discussion in
Finley and Westerhoff (2004)). In the combined data of several arrays Y. Uchihori *et al.* (2000)
there were found 8 doublets and 2 triplets in 92 events. The stereo
HiRes data R. U. Abbasi *et al.* (2004) do not show small-angle clustering for 27
events at eV, maybe due to limited statistics.

Small-angle clustering is most naturally explained in the case of
rectilinear propagation as a random arrival of two (three) particles
from a single source S. L. Dubovsky, P. G. Tinyakov and
Tkachev (2000). This effect has been calculated in
Refs. Fodor and Katz (2001); H. Yoshiguchi, S. Nagataki and Sato (2003); G. Sigl, F. Miniati and
Ensslin (2003a); H. Yoshiguchi, S. Nagataki and
Sato (2004a); P. Blasi and D. De Marco (2004); Kachelriess and Semikoz (2005a); D. De Marco, P. Blasi and Olinto . In the last five works the
calculations have been performed by the Monte Carlo (MC) simulations
and results agree well.
According to Kachelriess and Semikoz (2005a) the density of the sources, needed to explain
the observed number of doublets is Mpc. In P. Blasi and D. De Marco (2004) the best fit is given by Mpc and the large uncertainties (in particular
due to ones in observational
data) are emphasized.

(iv) Recently there have been found the statistically
significant correlations between directions of particles with energies
eV and directions to AGN of the special type -
BL Lacs Tinyakov and Tkachev (2001) (see also the criticism N. W. Evans, F. Ferrer and Sarkar (2004) and the reply
Tinyakov and Tkachev (2004)).

The items (iii) and (iv) favor rectilinear propagation of primaries from the point-like extragalactic sources, presumably AGN. However, the propagation in magnetic fields also exhibits clustering M. Lemoine, G. Sigl and Biermann ; G. Sigl, F. Miniati and Ensslin (2003a); H. Yoshiguchi, S. Nagataki and Sato (2004b).

The quasi-rectilinear propagation of ultra-high energy protons is found possible in MHD simulations K. Dolag, D. Grasso, V. Springel and Tkachev (2004) of magnetic fields in large-scale structures of the universe (see however the simulations in G. Sigl, F. Miniati and Ensslin (2003b); G. Sigl, F. Miniati and Ensslin (2004) with quite different results).

There are many unresolved problems in the field of Ultra High Energy Cosmic Rays, such as nature of primaries (protons? nuclei? or the other particles?), transition from galactic to extragalactic cosmic rays, sources and acceleration, but most intriguing problem remains existence of superGZK particles with energies higher than eV. “The AGASA excess”, namely 11 events with energy higher than eV, is still difficult to explain, though there are indications that it may have the statistical origin combined with systematic errors in energy determination (see section IV.3). The AGASA excess, if it is real, should be described by another component of UHECR, most probably connected with the new physics: superheavy dark matter, new signal carriers, like e.g. light stable hadron and strongly interacting neutrino, the Lorentz invariance violation etc.

The problem with superGZK particles is seen in other detectors, too. Apart from the AGASA events, there are five others: the golden Fly’s Eye event with eV, one HiRes event with eV and three Yakutsk events with eV. No sources are observed in the direction of these particles at the distance of order of attenuation length. The most severe problem is for the golden Fly’s Eye event: with attenuation length Mpc and the homogeneous magnetic field 1 nG on this scale, the deflection of particle is only . Within this angle there are no remarkable sources at distance Mpc Elbert and Sommers (1995).

In this paper we analyze the status of most conservative astrophysical solution to ultra-high energy cosmic ray problem, assuming that primary particles are protons or nuclei accelerated in extragalactic sources. In the first part of the paper (Sections II - V) we analyze the signatures of ultra-high energy protons propagating through CMB. We found that the dip, a spectral feature in energy range eV, is well confirmed by observational data of AGASA, HiRes, Yakutsk and Fly’s Eye detectors. In Sections VI -VII we discuss in the model-dependent way the transition from galactic to extragalactic cosmic rays and extragalactic sources: AGN and GRBs.

## Ii Energy losses and the universal spectrum of UHE protons

In this Section we present our recent calculations of energy losses for UHE protons interacting with CMB, and calculate the spectrum of protons, assuming the homogeneous distribution of sources in the space and continuous energy losses. The spectrum is calculated, using the conservation of the number of protons, interacting with CMB. Formally it does not depend on the mode of proton propagation (e.g. rectilinear or diffusive), and we shall discuss when this approximation is valid. The proton spectrum calculated in this way we call universal.

### ii.1 Energy losses

We present here the accurate calculations of energy losses due to pair production, , and pion production, , where is a microwave photon.

The energy losses of UHE proton per unit time due to its interaction with low energy photons is given by

(1) |

where is the Lorentz factor of the proton, is the energy of background photon in the reference system of the proton at rest, is the threshold of the considered reaction in the rest system of the proton, is the cross-section, is the mean fraction of energy lost by the proton in one collision in the laboratory system, (see Fig. 22 in Appendix A), is the energy of the background photon in the laboratory system, and is the density of background photons.

For CMB with temperature Eq. (1) is simplified

(2) |

Further on we shall use the notation

(3) |

for energy losses on CMB at present cosmological epoch, and . For the epochs with red-shift one has:

(4) | |||||

(5) |

Another important physical quantity needed for calculations of spectra is the derivative , which can be calculated as

(6) |

As one can see from Fig. 1, is numerically very similar to , and for approximate calculations one can use values for both functions.

From Eqs. (1) and (2) one can see that apart from cross-section the mean fraction of energy lost by the proton in laboratory system in one collision, (see Eq. (28)), is the basic quantity needed for calculations of energy losses. The threshold values of these quantities are well known:

(7) |

where and are the threshold fractions for and , respectively.

Pair production loss has been previously discussed in many papers. All authors directly or indirectly followed the standard approach of Ref. Blumenthal (1970) where the first Born approximation of the Bethe-Heitler cross-section with proton mass was used. In contrast to Ref. Blumenthal (1970), we use here the first Born approximation approach of Ref. Berg and Linder (1961) accounting for the finite proton mass. This allows us to calculate the average fraction of energy lost by the proton in laboratory system by performing a fourfold integration over invariant mass of electron-positron pair, , over an angle between incident and scattered proton, and polar and azimuthal angles of an electron in the c.m.s. of the -pair (see Appendix A for further details).

Calculating
photopion energy loss we follow methods developed in papers
Berezinsky and Gazizov (1993); Gazizov (1996). Total cross-sections are taken according to
Ref. Gabathuler (1974).
At low c.m.s. energy, , we consider the binary reactions
, (they
include the resonance ). Differential
cross-sections of binary processes at small energies are taken from
D. Menze, W. Pfeil and Wilke (1977); Wolf and Söding (1978); Fisher (1974). At GeV we assume the
scaling behavior of differential cross-sections, the latter being taken
from Ref. Meyer (1974); Brasse (1974); K. C. Moffeit *et al.* (1972). In the intermediate energy range we
interpolate between the angular distribution of these two regimes with
the cross-section being the difference between total cross-section and
cross-section of the all binary processes. Angular distributions for
this part of cross-section vary from isotropic at threshold to those
imposed by inclusive pion photoproduction data at high energies. The
overall differential cross-sections coincide with low-energy binary
description and high-energy scaling distributions and join smoothly
these two regimes in the intermediate region.

The results of our calculations are presented in
Fig. 1 in terms of
as function of energy (curve 1). Also plotted is the
derivative (Fig. 1b). This quantity is needed for
calculation of differential energy spectrum.
In Fig. 1 we plot for comparison the
energy losses as calculated by Berezinsky and Grigorieva 1988
Berezinsky and Grigorieva (1988) (dashed curve 2). The difference in energy losses due
to pion production is very small, not exceeding 5% in the energy
region relevant for comparison with experimental data (eV). The difference with energy losses due to pair production
is larger and reaches maximal value 15%. The results of calculations
by Stanev *et al.* T. Stanev, R. Engel, A. Muecke, R. J. Protheroe and
Rachen (2000) are shown by black squares. These
authors have performed the detailed calculations for both
aforementioned processes, though their approach is different from ours,
especially for photopion process. Our new energy losses are practically
indistinguishable from Stanev *et al.* T. Stanev, R. Engel, A. Muecke, R. J. Protheroe and
Rachen (2000) for pair production
and pion production at low energies, and differ by 15-20% for pion
production at highest energies (see Fig. 1). Stanev *et al.* claimed that energy losses due to pair production is underestimated by
Berezinsky and Grigorieva Berezinsky and Grigorieva (1988) by 30-40%. Comparison of
data files of Stanev *et al.* and Berezinsky and Grigorieva (see
also Fig. 1) shows that this difference is significantly less.
Most probably, Stanev *et al.* scanned inaccurately the data from the
journal version of the paper Berezinsky and Grigorieva (1988).

### ii.2 Universal spectrum

To calculate the spectrum one should first of all evolve the proton energy from the time of observation (or ) to the cosmological epoch of generation (or red-shift ), using the the adiabatic energy losses and given by Eq. (4):

(8) |

where is the Hubble parameter at cosmological epoch ,
with km/s Mpc, and
D. N. Spergel *et al.* [WMAP Collaboration] (2003).

We calculate the spectrum from conservation of number of particles in the comoving volume (protons change their energy but do not disappear). For the number of UHE protons per unit comoving volume, , one has:

(9) |

where is an age of the universe, is a generation energy at age , calculated according to Eq. (8) and is the generation rate per unit comoving volume, which can be expressed through emissivity , the energy release per unit time and unit of comoving volume, at , as

(10) |

where describes the possible cosmological evolution of the sources. In the case of the power-law generation, , with normalization constant for and for . We recall that in these formulae and everywhere below energies are given in GeV, emissivity in GeV cms and source luminosity in GeV s.

From Eq. (9) one obtains the diffuse flux as

(11) |

where and analytic expression for is given by Eq. (36) in Appendix B.

The spectrum (11) is referred to as universal spectrum. Formally it is derived from conservation of number of particles and does not depend on propagation mode (see Eq. (9)). But in fact, the homogeneity of the particles, tacitly assumed in this derivation, implies the homogeneity of the sources, and thus the condition of validity of universal spectrum is a small separation of sources. The homogeneous distribution of particles in case of homogeneous distribution of sources and inhomogeneous magnetic fields follows from the Liouville theorem (see Ref. Aloisio and Berezinsky (2004)).

Several effects could in principle modify the shape of universal spectrum. They include propagation in magnetic fields, discreteness in distribution of the sources, large-scale inhomogeneous distribution of sources, local source overdensity or deficit, and fluctuations in interaction. These effects will be studied in the next sections. With exception of energies beyond GZK cutoff and energies below eV, the universal spectrum is only weakly modified by aforementioned effects. Here we shortly comment on role of magnetic fields.

As numerical simulations (see e.g. G. Sigl, M. Lemoine and Biermann (1999); H. Yoshiguchi *et al.* (2003)) show, the
propagation of UHE protons in strong magnetic fields changes the energy
spectrum (for physical explanation of this effect see Aloisio and Berezinsky (2004)). The
influence of magnetic field on spectrum depends on the separation of
the sources, . For uniform distribution of sources with separation
much less than characteristic lengths of propagation, such as
attenuation length and the diffusion length , the diffuse spectrum of UHECR is the universal one independent
of mode of propagation Aloisio and Berezinsky (2004). This statement has a status of the
theorem. For the wide range of magnetic fields 0.1 - 10 nG and
distances between sources Mpc the spectrum at eV is close to the universal one Aloisio and Berezinsky (2005).

### ii.3 Modification factor

The analysis of spectra is very convenient to
perform in terms of
the modification factor.

Modification factor is defined as a ratio of the spectrum (see
Eq. (11)) with all energy losses taken into account, to
the unmodified spectrum , where only adiabatic energy losses
(red shift) are included.

(12) |

For the power-law generation spectrum for without evolution one has

(13) |

The modification factor is a less model-dependent quantity than the spectrum. In particular, it should depend weakly on , because both numerator and denominator in Eq. (12) include . In the next Section we consider the non-evolutionary case (see Section IV.5 for discussion of evolution). In Fig. 2 the modification factor is shown as a function of energy for two spectrum indices and . As expected above, they do not differ much from each other. Note that by definition .

## Iii Signatures of UHE protons interacting with CMB

The extragalactic protons propagating through CMB have signatures in the form of three spectrum features: GZK cutoff, dip and bump. The dip is produced due to -production and bump – by pile-up protons accumulated near beginning of the GZK cutoff. We add here the fourth signature: the second dip.

The analysis of these features, especially dip and bump, is convenient
to perform in terms of modification factor Berezinsky and Grigorieva (1988); T. Stanev *et al.* (2000).
For the GZK cutoff we shall use the traditional spectra.

### iii.1 GZK cutoff

The GZK cutoff Greisen (1966); Zatsepin and Kuzmin (1966) is most remarkable phenomenon, which describes the sharp steepening of the spectrum due to pion production. The GZK cutoff is a model–dependent feature of the spectrum, e.g. the GZK cutoff for a single source depends on the distance to the source.

A common convention is that the GZK cutoff is defined for diffuse flux from the sources uniformly distributed over the universe. In this case one can give two definitions of the GZK-cutoff position. In the first one it is determined as the energy, , where the steep increase in the energy losses starts (see Fig. 1). The GZK cutoff starts at this energy. The corresponding path length of a proton is Mpc. The advantage of this definition of the cutoff energy is independence of a spectrum index, but this energy is too low to judge about presence or absence of the cutoff in the measured spectrum. More practical definition is , where the flux with cutoff becomes lower by factor 2 than power-law extrapolation. This definition is convenient to use for the integral spectrum, which is better approximated by power-law function, than the differential one.

In Fig. 3 the function
, where , the calculated integral diffuse
spectrum, is plotted as function of energy. Note, that
is an effective index of the power-law approximation
of the spectrum modified by energy losses. For wide range of generation
indices the cutoff energy is the same,
eV. The corresponding proton path
length is Mpc.
In panel a) of Fig. 4 is found from the
integral spectrum of the Yakutsk array in the reasonable agreement with
theoretical prediction. The HiRes data are shown in panel b). These
data have large uncertainties, which prevent the accurate determination
of . However, they agree with the predicted value eV. In the recent paper D. R. Bergman *et al.* [HiRes Collaboration] the
Hires collaboration found eV
in better agreement with the predicted value.

In Fig. 5 the calculated
universal spectra with the GZK cutoff are compared with AGASA and HiRes data
N. Hayashida *et al.* [AGASA collaboration] (1994); K. Shinozaki *et al.* [AGASA collaboration] (2004); R. U. Abbasi *et al.* [HiRes collaboration] (2004). While the HiRes data agree with the
predicted GZK cutoff, the AGASA data show significant excess over this
prediction at eV.

The GZK cutoff as calculated above has many uncertainties.

The energy shape of the GZK feature is model dependent. The local excess of sources makes it flatter, and the deficit – steeper. The shape is affected by (see Fig. 5) and by fluctuations of source luminosities and distances between the sources. The cutoff, if discovered, can be produced as the acceleration cutoff (steepening below the maximum energy of acceleration in the generation spectrum). Since the shape of both, the GZK cutoff and acceleration cutoff, is model-dependent, it will be difficult to argue in favor of any of them, in case a cutoff is discovered.

To illustrate the effect of local overdensity/deficit of the UHECR sources we calculate here the UHECR spectra with different local ratios , where is the local density of the UHECR sources and is mean extragalactic source density. We use the various sizes of overdensity/deficit regions , equal to 10, 20 and 30 Mpc. The results of our calculations for the overdensity case are presented in Fig. 6 for , and for four values of overdensity equal to 1, 2, 3 and 10, assuming the size of overdensity region 30 Mpc (the results for Mpc are not much different). The spectra for the case of the deficit are shown by curves 5 and 6 in the both panels. The theoretical spectra shown in Fig. 6 illustrate uncertainties in the prediction of the shape of the GZK feature.

An interesting question is whether the local overdensity of the sources can explain the AGASA excess. This problem has been already addressed in M. Blanton, P. Blasi and Olinto (2001) for the realistic distribution of visible galaxies, considering them as the UHECR sources. Our calculations show (see left panel of Fig. 6) that unrealistic overdensity is needed to explain the AGASA excess.

Both effects, and local source overdensity (deficit) affect weakly the shape of the GZK cutoff at eV. Thus, the precise measurements of the spectrum in this energy region, as well as measurement of , give the best test of the GZK cutoff. At higher energies theoretical predictions have large model-dependent uncertainties.

### iii.2 Bump in the diffuse spectrum

Protons do not disappear in the photopion interactions, they only loose energy and are accumulated near beginning of the GZK cutoff in the form of a bump.

We see no indication of the bump in the modification-factor energy dependence in Fig. 2. As explained above, it should have been located at merging of and curves. The absence of the bump in the diffuse spectrum can be easily understood. The bumps are clearly seen in the spectra of the single remote sources (see left panel in Fig. 7). These bumps, located at different energies, produce a flat feature, when they are summed up in the diffuse spectrum. This effect is illustrated by Fig. 7 (right panel). The diffuse flux there is calculated in the model where sources are distributed uniformly in the sphere of radius (or ). When are small (between 0.01 and 0.1) the bumps are seen in the diffuse spectra. When radius of the sphere becomes larger, the bumps merge producing the flat feature in the spectrum. If the diffuse spectrum is plotted as this flat feature looks like a pseudo-bump.

### iii.3 The dip

The dip is more reliable signature of interaction of protons with CMB than GZK feature. The shape of the GZK feature is strongly model-dependent (see Section III.1), while the shape of the dip is fixed and has a specific form which is difficult to mimic by other mechanisms, unless they have many free parameters. The protons in the dip are collected from the large volume with the radius about 1000 Mpc and therefore the assumption of the uniform distribution of sources within this volume is well justified, in contrast to the GZK cutoff, which strongly depends on local overdensity or deficit of the sources. The GZK cutoff can be mimicked by acceleration cutoff, and since the shape of the GZK cutoff is not reliably predicted, these two cases could be difficult to distinguish.

The problem of identification of the dip depends on the accuracy of observational data, which should confirm the specific (and well predicted) shape of this feature. Do the present data have the needed accuracy?

The comparison of the calculated modification factor with that obtained from the Akeno-AGASA data, using , is given in Fig. 8. It shows the excellent agreement between predicted and observed modification factors for the dip.

In Fig. 8 one observes that at eV the agreement between calculated and observed modification factors becomes worse and the observational modification factor becomes larger than 1. Since by definition , it signals the appearance of another component of cosmic rays, which is almost undoubtedly given by galactic cosmic rays. The condition implies the dominance of the new (galactic) component, the transition occurs at eV.

To calculate for the confirmation of the dip by Akeno-AGASA data, we choose the energy interval between eV and eV (the energy of intersection of and ). In calculations we used the Gaussian statistics for low-energy bins, and the Poisson statistics for the high energy bins of AGASA. It results in . The number of Akeno-AGASA bins is 19. We use in calculations two free parameters: and the total normalization of spectrum. In effect, the confirmation of the dip is characterized by for d.o.f.=17, or /d.o.f.=1.12, very close to the ideal value 1.0 for the Poisson statistics.

In the right-upper panel of Fig. 8 the comparison of modification factor with the HiRes data is shown. The agreement is also very good: for for the Poisson statistics. The Yakutsk and Fly’s eye data (not shown here) agree with dip as well. The Auger spectrum The Pierre Auger Collaboration at this preliminary stage does not contradict the dip.

The good agreement of the shape of the dip with
observations is a strong evidence for extragalactic protons interacting
with CMB. This evidence is confirmed by the HiRes data on the mass
composition Archbold and Sokolsky (2003); R. U. Abbasi *et al.* [HiRes Collaboration] (2005). While the data of the Yakutsk array
A. V. Glushkov *et al.* [Yakutsk collaboration] (2000) and HiRes-MIA T. Abu-Zaayad *et al.* [HiRes Collaboration] (2000) support this mass
composition, and Haverah Park data M. Ave *et al.* [Haverah
Park collaboration] (2003) do not contradict it at eV, the data of Akeno M. Honda *et al.* [Akeno Collaboration] (1993)
and Fly’s Eye D. J. Bird *et al.* [Fly’s
Eye collaboration] (1994) favor the mixed composition dominated by heavy
nuclei.

The observation of the dip should be considered as independent evidence in favor of proton-dominated primary composition in the energy range eV.

### iii.4 The second dip

The second dip in the spectrum of extragalactic UHE protons appears at energy due to interplay between pair production and photopion production. It is the direct consequence of energy eV, where energy losses due to -production and pion-production become equal (see Fig. 1). This spectrum feature is explained as follows.

The pion-production energy loss increases with energy very fast, and at energy slightly below -production dominates and spectrum can be with high accuracy described in continuous energy-loss approximation. At energy slightly higher than the pion-production dominates and the precise calculation of spectrum should be performed in the kinetic-equation approach. In this method the evolution of number of particles in interval is given by two compensating terms, describing the particle exit and regeneration due to collisions. The small continuous energy losses affect only the exit term and break this compensation, diminishing the flux. The exact calculations are given in Appendix D. The second dip is very narrow and its amplitude at maximum reaches (see Fig. 24). This feature can be observed by detectors with very good energy resolution, and it gives the precise mark for energy calibration of a detector. It can be observed only marginally by the Auger detector.

## Iv Robustness of the dip prediction

We calculated the dip for the universal spectrum, i.e. for the case when distances between sources are small enough, and the spectrum does not depend on the propagation mode. In this Section we shall study stability of the dip relative to other possible phenomena, namely, discreteness in the source distribution, propagation in magnetic fields etc. We shall consider also some phenomena related to existence of the dip.

### iv.1 Discreteness in the source distribution

As it follows from analysis of the small-scale anisotropy (see S. L. Dubovsky, P. G. Tinyakov and Tkachev (2000); Fodor and Katz (2001); H. Yoshiguchi, S. Nagataki and Sato (2003); G. Sigl, F. Miniati and Ensslin (2003a); H. Yoshiguchi, S. Nagataki and Sato (2004a)), the average distance between UHECR sources is Mpc P. Blasi and D. De Marco (2004); Kachelriess and Semikoz (2005a). Such discreteness affects the spectrum, especially at highest energies, when energy attenuation length is comparable with .

In this subsection we demonstrate the stability of the dip relative to discreteness of the sources. We illustrate the effect of discreteness by an example of UHE protons propagating rectilinearly from sources located in the vertices of a 3D cubic lattice with spacing . Positions of sources are given by coordinates , and , where . The observer is assumed at with no source there. The diffuse flux for the power-law generation spectrum is given by summation over all vertices. The maximum distance is defined by the maximum red-shift . Then the observed flux is given by

(14) |

where is emissivity, is the red-shift for a source with coordinates , and factor takes into account the time dilation.

The calculated spectra for and Mpc are shown in Fig. 9 in comparison with the AGASA-Akeno data. In calculations we used eV, (no evolution), and . Emissivity is chosen to fit the AGASA data. One can see that discreteness in the source distribution affects weakly the dip, but the effect is more noticeable for the shape of the GZK cutoff.

### iv.2 Dip in the case of diffusive propagation

The dip, seen in the universal spectrum, is also present in the case of diffusive propagation in magnetic field Aloisio and Berezinsky (2005). The calculations are performed for diffusion in random magnetic fields with the coherent magnetic field nG

and up to nG on the basic scale Mpc. The calculated spectrum is shown in Fig. 10 for the case and distance between sources Mpc. The critical energy, where diffusion changes its regime is eV. The spectra are shown for three different regimes at as indicated in Fig. 10. The universal spectrum is also presented. One can see that the dip agrees well with universal spectrum and observational data, while the shape of the GZK cutoff differs considerably from the universal spectrum.

These calculations demonstrate stability of the dip relative to changing of the propagation mode, and sensitivity of the GZK cutoff to the way of propagation.

### iv.3 Energy calibration of the detectors using the dip

Since the position and shape of the dip is robustly fixed by proton interaction with CMB, it can be used as energy calibrator for the detectors. We use the following procedure for the calibration. Assuming the energy-independent systematic error, we shift the energies in each given experiment by factor to reach minimum for comparison with the calculated dip. The systematic errors in energy determination of existing detectors exceed 20%, and it determines the expected value of . The described procedure results in , and for AGASA, Yakutsk and HiRes detectors, respectively. After this energy calibration the fluxes in all experiments agree with each other.

First we consider the AGASA and HiRes data. There are two discrepancies between these data (see the upper-left panel of Fig. 11): one is described by factor 1.5 - 2.0 in energy region eV, and the second – at eV. In Fig. 11 the spectra of Akeno-AGASA and HiRes are shown before and after the energy calibration. One can see the good agreement of the calibrated data at eV and their consistency at eV. This result should be considered together with calculations D. De Marco, P. Blasi and Olinto (2006), where it was demonstrated that 11 superGZK AGASA events can be simulated by the spectrum with GZK cutoff in case of 30% error in energy determination. We may tentatively conclude that existing discrepancy between AGASA and HiRes spectra at all energies are due to systematic energy errors and statistics.

In Fig. 11 the energy spectra are shown for AGASA and Yakutsk spectra before and after energy calibration. Again, the best fit to the dip shape results in excellent agreement in the absolute values of fluxes.

The agreement between spectra of all three detectors after energy calibration by the dip confirms the dip as the spectrum feature produced by interaction of the protons with CMB, and demonstrates compatibility of fluxes measured by AGASA, HiRes and Yakutsk detectors.

### iv.4 Dip and extragalactic UHE nuclei

The proton dip has very good agreement with
observations (see Fig. 8). However, in all astrophysical
sources the nuclei must be also accelerated to the energies, naively, Z
times higher than that for protons. Do UHE nuclei in primary flux
upset the good agreement seen in Fig. 8? This problem has been
recently considered in V. Berezinsky, A. Gazizov and
Grigorieva (2005); D. Allard, E. Parizot and
Olinto (a); Sigl and Armengaud (2005) (for study of
propagation UHE nuclei through CMB see
e.g. F. W. Stecker and Salamon (1992); Sigl (2004); T. Yamamoto *et al.* (2004)).

The presence of nuclei in primary extragalactic spectrum modifies the dip V. Berezinsky, A. Gazizov and Grigorieva (2005). In Fig. 12 the dips for helium and iron nuclei are shown in comparison with that for protons. From this figure one can see that presence of 15 - 20 % of nuclei in the primary flux breaks the good agreement of proton deep with observations. The modification factor for cosmic rays composed of protons and nuclei with the fraction , where is generation function for nuclei (A) and protons (p), can be easily calculated as

(15) |

The fraction is a model-dependent value, which depends on ratio of number densities of gas components in media, where acceleration operates, on ionization of the gas and on the injection mechanism of acceleration V. Berezinsky . Besides, the UHE nuclei can be destroyed inside the source or in its vicinity Sigl and Armengaud (2005).

The strongest distortion of proton modification factor is given by helium nuclei, for which corresponding to the helium mass fraction . In Fig. 13 the modification factors are shown for the mixed composition of protons and helium with the mixing parameter (left panel) and (right panel). One can judge from these graphs about allowed values of mixing parameter . If agreement of the proton dip with observations is not incidental (the probability of this is small according to small /d.o.f.), Fig. 13 should be interpreted as indication to possible acceleration mechanism V. Berezinsky .

### iv.5 Dip and cosmological evolution of the sources

The cosmological evolution of the sources, i.e. increase of the luminosities and/or space densities with red-shift , is observed for many astronomical populations. The evolution is reliably observed for star formation rate in the normal galaxies, but this case is irrelevant for UHECR, because neither stars nor normal galaxies can be the UHECR sources due to low cosmic-ray luminosities and maximum energy of acceleration .

AGN, which satisfy these
requirements, also exhibit the evolution seen in radio, optical and X-ray
observations. The X-ray radiation is probably most relevant tracer for
evolution of UHECR because both radiations are feed by the energy release
provided by accretion to a massive black hole:
X-rays – through radiation of accretion disk, and UHECR – through
acceleration in the jets. According to
recent detailed analysis in Ueda et al. (2003) and A. J. Barger *et al.* (2005) the
evolution of AGN seen in X ray radiation can be described by factor
up to and is saturated at larger
. In Ueda et al. (2003) the pure luminosity evolution and pure density
evolution are allowed with and , respectively, and
with for both cases. In A. J. Barger *et al.* (2005) the
pure luminosity evolution is considered as preferable with
and . These authors do not distinguish between
different morphological types of AGN. It is possible that some AGN
undergo weak cosmological evolution, or no evolution at all.
BL Lacs Tinyakov and Tkachev (2001), which are important as potential UHECR sources,
show no signs of positive cosmological evolution S. L. Morris *et al.* (2003).

In case of UHECR there is no need to distinguish between luminosity and density evolution, because the diffuse flux is determined by the emissivity, which includes both luminosity and space evolution, as it follows from Eq. (10).

In Fig. 14 we present the calculated dip spectrum in
evolutionary models, inspired by the data cited above. For
comparison we show also the case of absence of evolution , as
can be valid for BL Lacs.
From Fig. 14 one can see that the spectra with evolution up
to can explain the observational data down to
(a few) eV and even below, in accordance with early
calculations V. Berezinsky, A. Z. Gazizov and
Grigorieva (a); Scully and Stecker (2002); Z. Fodor, S. D. Katz, A. Ringwald and
Tu (2003) (see D. R. Bergman *et al.* for recent
analysis). However, for any
reasonable magnetic fields protons with these energies have small
diffusion lengths and the spectrum
acquires the diffusion ’cutoff’ at energy eV
(see Section IV.2).

We conclude that for many reasonable evolution regimes the dip agrees with observational data as well as the non-evolutionary case .

## V Role of interaction fluctuations

UHE proton spectrum is affected by fluctuations in the photopion production. These fluctuations may change the proton spectrum only at energy substantially higher than eV. At this energy the half of energy losses is caused by production which does not fluctuate. Up to energy eV the photoproduction of pions occurs at the threshold in collisions with photons from high-energy tail of the Planck distribution, and fraction of energy lost does not fluctuate, being fixed by the threshold value. Indeed, for eV the minimal energy of CMB photon needed for pion production is eV to be compared with energy of photon in the Planck distribution maximum eV. The only fluctuating value is the interaction length.

The noticeable effect of fluctuations is expected for protons with energies eV.

As it is well known Lifshitz and Pitaevskii (1981); Ginzburg and Syrovatskii (1964), the kinetic equations give an adequate method to account for the fluctuations in interaction. Neglecting the conversion of proton to neutron (neutron decays back to proton with small energy loss) the kinetic equation for UHE protons with adiabatic energy losses and with and scattering in collisions with CMB photons can be written down as follows:

(16) | |||

where is the number density of UHE protons per unit energy, is the generation rate, given by Eq. (10) with and is the Hubble parameter. The first term in the r.h.s. of Eq. (16) describes expansion of the universe. The energy loss due to -pair production is treated as continuous energy loss. The photopion collisions are described with help of probability for proton exit from energy interval due to -collisions and with help of their regeneration in the same energy interval described by probability . These two probabilities describe fluctuations in the interaction length and in fraction of energy lost in the interaction; the interaction length is equivalent in this picture to time of proton exit from energy interval .

The exit probability due to collisions with CMB photons with temperature can be written as:

(17) |

where is the total c.m.s. energy of colliding proton and photon, , is the photopion cross-section and is the CMB temperature at cosmological epoch .

Similarly, in regeneration term of Eq. (16) the probability for a proton with energy to produce a proton with energy is given by

(18) |

where and are the energies of primary and secondary protons, respectively, and . The minimum value of the allowed c.m.s. energy in this case is given by

(19) |

This bound corresponds to the process with minimum invariant mass, namely to .

We have solved Eq. (16) numerically. The calculated spectrum , where is the age of the universe at red-shift , is presented in Fig. 15 as modification factor for generation spectrum with and eV (left panel) and eV (right panel). For comparison the modification factors for universal spectrum with continuous energy losses is also shown. The difference in these two spectra at highest energies must be due to fluctuations in energy losses, though formally we have to say that this is the difference between solution to kinetic equation (16) and the continuous energy loss approximation. For eV one can see the difference in the spectra about 25% at highest energies and tiny difference above intersection of and curves. For eV the difference is small.

Note, that modification factors do not vanish at even when generation function goes abruptly to zero, since both solutions vanish keeping the same value of ratios . It is easy to demonstrate analytically that ratio of flux in continuous loss approximation to unmodified flux , given by Eq. (13) tends to , when . From Fig. 15 one can see that this ratio coincides exactly with our numerical calculations (e.g. the analytical value is for eV), and this gives a proof that our numerical calculations are correct.

The effect of interaction fluctuations is usually taken into account with help of Monte-Carlo simulations. The method of kinetic equation corresponds to averaging over large number of Monte-Carlo simulations, and if all other assumptions are the same, the results must coincide exactly. These assumptions include and parameters of -interaction. However, the existing Monte-Carlo simulations in most cases include some other assumptions in comparison with kinetic equations, which modify spectrum stronger than interaction fluctuations. One of them is discreteness in the source distribution (in kinetic equations the homogeneous distribution is assumed), the other is fluctuations of distances to the nearby sources.

It is possible however to make the comparison with Monte-Carlo simulations for homogeneously distributed sources and using identical interaction model. Such a comparison is discussed in Appendix C.

As to results presented here, it is necessary to emphasize that the difference of kinetic-equation solution and continuous-energy-loss approximation presented in Fig. 15 includes fluctuations, but not only fluctuations. The transition of kinetic equation to continuous-energy-loss equation depends on some other conditions which can fail. What the presented calculations demonstrate (see Fig. 15) is that continuous-energy-loss approximation describes with very good accuracy more reliable kinetic-equation solution, which in particular includes interaction fluctuations.

## Vi Transition from extragalactic to galactic cosmic rays

In the analysis above we obtained several indications that transition from extragalactic to galactic cosmic rays occurs at eV. These evidences are summarized in Fig. 16.

The predicted spectrum above eV describes perfectly well the observed spectra: see the modification factor in the upper-left panel of Fig. 16 compared with AGASA data and in Fig. 8 with HiRes. However, at experimental modification factor becomes , in contrast to definition . It signals the appearance of a new component, which can be nothing but the galactic cosmic rays.

In the right panel the spectrum for rectilinear propagation from the sources with different separation and with is compared with AGASA data. One can see that at eV the calculated extragalactic spectrum becomes less than that observed.

In the lower panel the similar comparison is shown for diffusive propagation with different diffusion regimes at lower energies Aloisio and Berezinsky (2005); Lemoine (2005). The random magnetic field with basic scale Mpc and magnetic field on this scale nG is assumed. The dash-dotted curve (universal spectrum) corresponds to the case when the separation between sources .

In all cases the transition from extragalactic to galactic component begins at eV, with index for “beginning”.

What is the reason of this universality? We study the transition, moving from high towards low energies. is the beginning of transition (or its end, if one moves from low energies). is determined by energy eV, where adiabatic and pair-production energy losses become equal. The quantitative analysis of this connection is given in Aloisio and Berezinsky (2005). We shall give here the semi-quantitative explanation.

The flattening of the spectrum occurs at energies , where and should be estimated as red-shift up to which the main contribution to unmodified spectrum occurs. The simplified analytic estimate for gives and hence eV. In fact, the right and lower panels of Fig. 16 present the exact calculations of this kind.

In experimental data the transition is searched for as a feature started at some low energy - the second knee. Its determination depends on experimental procedure, and all we can predict is . Determined in different experiments eV.

The transition at the second knee appears also in the study of
propagation of cosmic rays in the Galaxy (see e.g. P. L. Biermann *et al.* ; Hoerandel (2003); S. D. Wick, C. D. Dermer and Atoyan (2004)).

Being thought of as purely galactic feature, the position of the second knee in our analysis appears as direct consequence of extragalactic proton energy losses.

The transition at the second knee is illustrated by
Fig. 17. The clue to understanding of this transition is
given by the KASCADE data K.-H. Kampert *et al.* [KASCADE collaboration] (2001); J. R. Hoerandel *et al.* [KASCADE collaboration] (2002). They confirm the
rigidity model, according to which position of a knee for nuclei with
charge Z is connected with the position of the proton knee as
. There are two versions of this model. One is the
confinement-rigidity model (bending above the knee is due to
insufficient confinement in galactic magnetic field), and the other is
acceleration-rigidity model ( is determined by rigidity).
In both models the heaviest nuclei (iron) start to disappear at eV, if the proton knee is located at
eV. The shape of the spectrum above the
iron knee () is model dependent, with two reliably
predicted features: it must be steeper than the spectrum below the iron
knee (), shown by the dash-dot curve, and iron nuclei
must be the dominant
component there (see Fig. 17). The high energy part of
the spectrum has a characteristic energy , below which the
spectrum becomes more flat, i.e. drops down when multiplied to
(see Fig. 17). This part of the spectrum is
shown for the diffusive propagation described in section VII.2.
These two falling parts of the spectrum inevitably intersect at some
energy , which can be defined as transition energy from
galactic to extragalactic cosmic rays. The ’end’ of galactic cosmic rays
eV and the beginning of full dominance
of extragalactic component eV differ by
an order of magnitude. Note, that power-law extrapolation of the total
galactic spectrum, shown by dot-dash line, beyond the iron knee has no physical meaning in the rigidity models and must not be
discussed.

The second-knee transition gives an alternative possibility in comparison with ankle-transition hypothesis known from end of 1970s. It is inspired by flattening of the spectrum at eV seen in the AGASA and Yakutsk data (left panels in Fig. 8) and possibly at eV in the Hires data (the right panel in Fig. 8). Being multiplied to factor , as in Fig. 17, the ankle transition looks very similar to that at the second knee. Note that in the latter case the ankle is just an intrinsic part of the dip.

The ankle transition has been recently discussed in Refs. Hillas (2004); Wibig and Wolfendale (2005); D. Allard, E. Parizot and Olinto (a, b); D. De Marco and Stanev (2005); Hillas .

In the ankle model it is assumed that galactic cosmic ray spectrum has a power-law shape from the proton knee eV to about eV where it becomes steeper and crosses the more flat extragalactic spectrum (see the right panel of Fig. 17). The ankle transition in Fig. 17 is shown for extragalactic proton spectrum with generation index , while galactic spectrum, given by curve “gal.CR” is calculated as difference of the observed total spectrum and calculated extragalactic proton spectrum.

The ankle model has the problems with galactic component of cosmic rays. The spectrum at eV is taken ad hoc to fit the observations, while in the second knee model this part of the spectrum is calculated with excellent agreement with the data. In the rigidity models the heaviest nuclei (iron) start to disappear at eV. How the gap between eV and eV is filled?

Galactic protons start to disappear at eV. Where they came from at eV to be seen e.g. in the Akeno detector with fraction ?

The ankle model needs acceleration by galactic sources up to eV (at least for iron nuclei), which is difficult to afford. The second knee model ameliorates this requirement by one order of magnitude.

The second-knee model predicts the spectrum shape down to eV with extremely good accuracy (/d.o.f.= 1.12 for Akeno-AGASA and /d.o.f.= 1.03 for HiRes). In the ankle model one has to consider this agreement as accidental, though such hypothesis has very low probability, determined by cited above. As an alternative the ankle model-builders can suggest only hopes for future development of galactic propagation models to be as precisely calculated as the dip.

## Vii Astrophysical sources of UHECR

In the sections above we have performed the model-independent analysis of spectra of extragalactic protons interacting with CMB. We have calculated the features of the proton spectrum assuming the power-law generation spectrum valid at eV, and compared predicted features with observations. We found that proton dip, a model-independent feature at energy between eV and eV, is well confirmed by observations. Only two free parameters are involved in fitting of observational data: (the allowed range is 2.55 - 2.75) and the flux normalization constant. The various physical phenomena included in calculations, such as discreteness in the source distribution, the different modes of propagation (rectilinear and diffusive), cosmological evolution with parameters similar to AGN evolution, fluctuations in interaction etc, do not upset this agreement.

The transition of extragalactic to galactic cosmic rays is also discussed basically in model-independent manner.

In this Section we shall discuss the models: realistic energy spectra, the sources and the models for transition from extragalactic to galactic cosmic rays.

The UHECR sources have to satisfy two conditions: they must be very powerful and must accelerate particles to large eV. There is one more restriction S. L. Dubovsky, P. G. Tinyakov and Tkachev (2000); Fodor and Katz (2001); H. Yoshiguchi, S. Nagataki and Sato (2003); G. Sigl, F. Miniati and Ensslin (2003a); H. Yoshiguchi, S. Nagataki and Sato (2004a); P. Blasi and D. De Marco (2004); Kachelriess and Semikoz (2005a), coming from observation of small-scale clustering: the space density of the sources should be Mpc probably with noticeable uncertainty in this value. Thus, these sources are more rare, than typical representatives of AGN, e.g. the Seyfert galaxies, whose space density is Mpc. The sources could be the rare types of AGN, and indeed the analysis of Tinyakov and Tkachev (2001) show statistically significant correlation between directions of particles with energies eV and directions to AGN of the particular type – BL Lacs (see also criticism N. W. Evans, F. Ferrer and Sarkar (2004) and reply Tinyakov and Tkachev (2004)). The acceleration in AGN can provide the maximum energy of acceleration up to eV for non-relativistic shock acceleration (see e.g. Biermann and Strittmatter (1987)).

The relativistic shock acceleration can occur in AGN jets. Acceleration to eV in the AGN relativistic shocks is questionable (see discussion below).

Gamma Ray Bursts (GRBs) are another potentially possible sources of UHECR Milgrom and Usov (1995); Vietri (1995); Waxman (1995). They have very large energy output and can accelerate particles up to eV Waxman (1995); Vietri (1995). These sources have, however, the problems with explaining small-angle anisotropy and with energetics (see discussion below).

### vii.1 Spectra

The assumption of the power-law generation spectrum with extrapolated to GeV results in too large emissivity required for observed fluxes of UHECR. To avoid this problem the broken generation spectrum has been suggested in Refs. V. Berezinsky, A. Z. Gazizov and Grigorieva (a, b):

(20) |

where is the generation function (rate of particle production per unit of comoving volume), defined by Eqs. (9) and (10), and was considered as a free parameter.

Recently it was demonstrated that broken generation spectrum can naturally emerge under most reasonable physical assumption. In Ref. Kachelriess and Semikoz (2005b) it has been argued that while spectrum is universal for non-relativistic shock acceleration, the maximum acceleration energy is not, being dependent on the physical characteristics of a source, such as its size, regular magnetic field etc. Distribution of sources over results in steepening of the generation function, so that the distribution of the sources explains the observational data, if Kachelriess and Semikoz (2005b).

In Appendix E we address the generalized problem: what should be the distribution of spectral emissivity over to provide the generation function with the broken spectrum. We use there the notation and introduce the spectral emissivity , where is particle luminosity of a source and is the space density of the sources. The total emissivity is given by . Distribution of spectral emissivity over maximal energies determines the energy steepening of the generation function at energy in the distribution. This function is calculated analytically for arbitrary , assuming that is confined to the interval (. It is demonstrated that can be the power-law function exactly, only if distribution is power-law, too . For the source generation function at , the generation index in the interval is found to be , including the case . The steepening of the generation spectrum from to occurs approximately at energy . At energy the spectrum is suppressed as or stronger.

### vii.2 Active Galactic Nuclei

The AGN as sources of UHECR meet the necessary requirements: (i) to accelerate particles to eV, (ii) to provide the necessary energy output and (iii) to have the space density Mpc, required by small-scale clustering. We shall discuss below these problems in some details.

#### vii.2.1 Acceleration and spectra

The flow of the gas in AGN jet can be terminated by the non-relativistic shock which accelerates protons or nuclei in the radio lobe up to eV with spectrum Biermann and Strittmatter (1987).

In some cases the observed velocities in AGN jets are ultra-relativistic with Lorentz factor up to . It is natural to assume there the existence of internal and external ultra-relativistic shocks. Acceleration in relativistic shocks relevant for UHECR has been recently studied in Refs. Vietri (1995); Waxman (1995); Gallant and Achterberg (1999); J. G. Kirk, A. W. Guthmann, Y. A. Gallant and Achterberg (2000); Vietri (2003); Blasi and Vietri (2005); Lemoine and Pelletier (2003); Lemoine and Revenu (2006). The acceleration spectrum is and in the case of isotropic scattering of particles upstream and downstream, the spectrum index is J. G. Kirk, A. W. Guthmann, Y. A. Gallant and Achterberg (2000). However, recently it was understood that this result depends on scattering properties of the medium Blasi and Vietri (2005); Lemoine and Revenu (2006), and the spectrum can be steeper. In the regime of large angle scattering in Blasi and Vietri (2005), was found possible for the shock with velocity and compression ratio . In Monte-Carlo simulation Lemoine and Revenu (2006) it is demonstrated that effect of compression of upstream magnetic field results in increasing of up to the limiting value 2.7 in ultra-relativistic case .

The maximal energy of acceleration is a controversial issue. While in most works (very notably Gallant and Achterberg (1999)) it is obtained that cannot reach eV for all realistic cases of relativistic shocks, the authors of Ref. M. Vietri, D. De Marco and Guetta (2003) argue against this conclusion.

We shall divide a problem with into two: the reliably estimated energy gain in relativistic shocks and model-dependent absolute value of . The energy gain in the first full cycle of particle reflection upstream-downstream-upstream () is about . The next reflections are much less effective, as was first observed in Gallant and Achterberg (1999): a particle lives a short time in the upstream region, before it is caught up by the shock. As a result, a particle is deflected by upstream magnetic field only to a small angle, and thus it occurs in the downstream region with approximately the same energy as in the first cycle. Then the energy upstream will be also almost the same as in the first cycle. According to Monte-Carlo simulation Lemoine and Revenu (2006) the average energy gain per each successive cycle is only 1.7 (see Fig. 4 in Lemoine and Revenu (2006) with clear explanation). With these energy gains ( in the first cycle and with in each successive cycle) it is not possible to get eV in the conservative approach for AGN and GRBs, but there are some caveats in this conclusion as indicated in Gallant and Achterberg (1999); M. Vietri, D. De Marco and Guetta (2003). The magnetic field in the upstream region can be large, and then the deflection angle of a particle after the shock crossing is large, too. Relativistic shock acceleration can operate in medium filled by pre-accelerated particles, and thus initial energy can be high.

There are some other mechanisms of acceleration to energies up to eV relevant for AGN: unipolar induction and acceleration in strong electromagnetic waves (see V. S. Berezinsky, S. V. Bulanov, V. A. Dogiel, V. L. Ginzburg and Ptuskin (1990) for description and references). The mechanisms of jet acceleration have very special status.

The observed correlations between arrival directions of particles with energies eV and BL Lacs Tinyakov and Tkachev (2001) imply the jet acceleration. This is because BL Lacs are AGN with jets directed towards us. For this correlation the propagation of particles (most probably protons) with energies above eV must be (quasi)rectilinear, that can be realized in magnetic fields found in MHD simulations in K. Dolag, D. Grasso, V. Springel and Tkachev (2004) (see however the simulations in G. Sigl, F. Miniati and Ensslin (2003b); G. Sigl, F. Miniati and Ensslin (2004) with quite different results).

An interesting mechanism of jet acceleration, called pinch acceleration, was suggested and developed in plasma physics V. V. Vlasov, S. K. Zhdanov and Trubnikov (1990). It is based on pinch instability well known in plasma physics, both theoretically and observationally. The electric current along jet produces the toroidal magnetic field which stabilizes the jet flow. The pinch instability is caused by squeezing the tube of flow by magnetic field of the current. It results in increasing the electric current density and magnetic field. The magnetic field compresses further the tube and thus instability develops. The acceleration of particles in V. V. Vlasov, S. K. Zhdanov and Trubnikov (1990) is caused by hydrodynamical increase of velocity of the flow and by longitudinal electric field produced in the pinch. This process is illustrated by Fig. 18.

The pinch acceleration has been developed for tokamaks and was confirmed there by observations. The generation spectrum is uniquely predicted as

(21) |

Scaling a size of the laboratory tube to the cosmic jet, one obtains exceeding eV. The particle beam undergoes a few pinch occurrences during traveling along cosmic jet (a few kpc in case of AGN), and thus spectrum obtains a low energy cut at high value of . This mechanism needs more careful study.

We finalize this subsection concluding, that there are at least three possibilities for the broken generation spectrum with at : The acceleration by non-relativistic shocks with distribution of sources (more precisely emissivity) over , the acceleration by relativistic shocks, where naturally occurs due to the first cycle as with being particle energy before acceleration, and in the pinch acceleration where