The astrophysical models discussed present difficulties in providing a completely satisfactory explanation of the super-GZK events, if there are any. This may simply reflect our present lack of statistics, our present ignorance of the true conditions of processes in some highly energetic regions of the universe, or, perhaps, may imply that exotic mechanisms are at play. Physics from the most favored theories beyond the standard model (SM) like string/M theory, supersymmetry (SUSY), grand unified theories (GUTs), and TeV-scale gravity have been invoked to explain the possible flux above the GZK energy limit. This review is not mainly concerned with beyond-SM scenarios (for more comprehensive surveys see e.g. [1, 70, 380, 381, 382]), but for the sake of completeness, we provide here a brief account of some of the most relevant exotic explanations.
The most economical among hybrid proposals involves a familiar extension of the SM, namely, neutrino masses. It was noted many years ago that 's arriving at Earth from cosmologically distant sources have an annihilation probability on the relic neutrino background of roughly 3 h65-1% . Inspired on this analysis, Weiler  and Fargion et al.  noted that neutrinos within a few Z widths of the right energy,
to annihilate with the relic neutrinos at the Z-pole with large cross section,
may produce a "local" flux of nucleons and photons. (35) Remarkably, the energy of the neutrino annihilating at the peak of the Z-pole has to be well above the GZK limit. The mean energies of the ~ 2 nucleons and ~ 20 -rays in each process can be estimated by distributing the resonant energy among the mean multiplicity of 30 secondaries. The proton energy is given by
whereas the -ray energy is given by
The latter is a factor of 2 smaller to account for the photon origin in two body 0 decay. This implies that the highly boosted decay products of the Z could be observed as super-GZK primaries. (36) However, to reproduce the observed spectrum, the Z-burst mechanism requires very luminous sources of extremely high energy neutrinos throughout the universe [387, 388, 389]. The present limits on these sources are near the threshold of sensitivity to the required flux .
In 1931, Georges Lemaître  - a forerunner of the Big Bang hypothesis - introduced the idea that the entire material filling the universe, as well as the universe's expansion, originated in the super-radioactive disintegration of a "Primeval Atom", which progressively decayed into atoms of smaller and smaller atomic weight. The CRs were introduced in this picture as the energetic particles emitted in intermediate stages of the decay-chain. Echoing Lemaître, in the so-called "top-down models", charged and neutral primaries arise in the quantum mechanical decay of supermassive elementary X particles [392, 393, 394, 395, 396, 397, 398, 399, 400, 401, 402, 403, 404, 405, 406].
To maintain an appreciable decay rate today, it is necessary to tune the X lifetime to be longer (but not too much longer) than the age of the universe, or else "store" short-lived X particles in topological vestiges of early universe phase transitions (such as magnetic monopoles, superconducting cosmic strings, vortons, cosmic necklaces, etc.). Discrete gauged symmetries [407, 408, 409] or hidden sectors [410, 411] are generally introduced to stabilize the X particles. Higher dimensional operators, wormholes, and instantons are then invoked to break the new symmetry super-soflty to mantain the long lifetime [399, 400] (collissional annihilation has been considered too ). Arguably, these metastable super-heavy relics (MSRs) may constitute (a fraction of) the dark matter in galactic haloes.
Of course, the precise decay modes of the X's and the detailed dynamics of the first generation of secondaries depend on the exact nature of the X particles under consideration. However, in minimal extensions of the SM, where there are no new mass scales between MSUSY ~ 1 TeV and mX, the squark and sleptons would behave like their corresponding supersymmetric partners, enabling one to infer from the "known" evolution of quarks and leptons the gross features of the X particle decay: the strongly interacting quarks would fragment into jets of hadrons containing mainly pions together with a 3% admixture of nucleons [413, 414, 415]. (37) This implies that the injection spectrum is a rather hard fragmentation-type shape (with an upper limit usually fixed by the GUT scale) and dominated by -rays and neutrinos produced via pion decay. Therefore, the photon/proton ratio can be used as a diagnostic tool in determining the CR origin . In light of the mounting evidence that UHECRs are not -rays, one may try to force a proton dominance at ultrahigh energies by postulating efficient absorption of the dominant ultrahigh energy photon flux on the universal and/or galactic radio background. (38) However, the neutrino flux accompanying a normalized proton flux is inevitably increased to a level where it should be within reach of operating experiments .
It is clear that because of the wide variety of top down models the ratio of the volume density of the X-particle to its decay time is model dependent. However, if a top down scenario is to explain the origin of UHECRs, the injection spectrum should be normalized to account for the super-GZK events without violating any observational flux measurements or limits at higher or lower energies . In particular, neutrino and -ray fluxes depend on the energy released integrated over redshift, and thus on the specific top down model. Note that the electromagnetic energy injected into the Universe above the pair production threshold on the CMB is recycled into a generic cascade spectrum below this threshold on a short time scale compared with the Hubble time. Therefore, it can have several potential observable effects, such as modified light element abundances due to 4He photodisintegration, or induce spectral distortions of universal -ray and neutrino backgrounds [422, 423]. Additionally, measurements of the diffuse GeV -ray flux , to which the generic cascade spectrum would contribute directly, limit significantly the parameter space in which X's can generate the flux of the UHECRs [425, 426, 427], especially if there is already a significant contribution to this background from conventional sources such as unresolved -ray blazars. Recently, a possible lower extragalactic contribution to the diffuse -ray background measured by EGRET has been pointed out [428, 429]. The ~ 50% smaller EGRET flux practically rules out extragalactic top down and Z-burst scenarios .
If MSRs are the progenitors of the observable UHECRs, then the flux will be dominated by the decay or annihilation products of X's in the Galactic halo, i.e., by sources at distances smaller than all relevant interaction lengths. A clean signal of this scenario is the predicted anisotropy due to the non-central position of the Sun in our galaxy [431, 432]. Although it was noted that the predicted anisotropy is consistent with AGASA and Haverah Park data , recent analyses [433, 434] of SUGAR data excludes the MSR hypothesis at the 5 level if all events above 1019.6 eV are due to metastable X clustered in the halo. (For the extreme case where the population of MSRs is responsible only for CRs with energies 1019.8 eV, the annihilation scenario is disfavored at least at the 99% CL, whereas decaying MSRs still have a probability ~ 10% to reproduce SUGAR data.)
A more exotic explanation postulates that the X's themselves constitute the primaries: magnetic monopoles easily pick up energy from the magnetic fields permeating the universe and can traverse unscathed through the primeval radiation, providing an interesting candidate to generate extensive air showers . In particular, a baryonic monopole encountering the atmosphere will diffuse like a proton, producing a composite heavy-particle-like cascade after the first interaction  with a great number of muons among all the charged particles . Although this feature was observed in a poorly understood super-GZK event [438, 439], it seems unlikely that a complete explanation for the UHECR data sample would be in terms of magnetic monopoles alone. Moreover, any confirmed directional pairing of events would appear difficult to achieve with the monopole hypothesis.
A novel beyond-SM-model proposal to break the GZK barrier is to assume that UHECRs are not known particles but rather a new species, generally referred to as the uhecron, U [440, 441, 442]. The meager information we have about super-GZK particles allows a naïve description of the properties of the U. The muonic content in the atmospheric cascades suggests U's should interact strongly. At the same time, if U's are produced at cosmological distances, they must be stable, or at least remarkably long lived, with mean-lifetime 106 (mU / 3 GeV) (d / Gpc) s, where d is the distance to the source and mU, the uhecron's mass. Additionally, since the threshold energy increases linearly with mU, to avoid photopion production on the CMB mU 1.5 GeV. In recent years, direct searches of supersymmetric hadrons [443, 444, 445, 446] have severely eroded the attractiveness of the U scenario. However, adequate fine-tunings leave a small window still open [447, 448].
On a similar track, it was recently put forward  that strangelets (stable lumps of quark matter with roughly equal numbers of up, down, and strange quarks) can circumvent the acceleration problem in a natural way (due to a high mass and charged, but low charge-to-mass ratio) and move the expected cutoff to much higher energies.
Another possibility in which super-GZK CRs can reach us from very distance sources may arise out of photons that mix with light axions in extragalactic magnetic fields . These axions would be sufficiently weakly coupled to travel large distances unhindered through space, and so they can convert back into high energy photons close to the Earth. An even more radical proposal postulates a tiny violation of local Lorentz invariance, such that some processes become kinematically forbidden . In particular, photon-photon pair production and photopion production may be affected by Lorentz invariance violation. Hence, the absence of the GZK-cutoff would result from the fact that the threshold for photopion production "disappears" and the process becomes kinematically not allowed. This implies that future PAO observations of faraway sources could provide constraints on, or even a measurement of, the violation of Lorentz symmetry, yielding essential insights into the nature of gravity-induced wave dispersion in the vacuum [452, 453, 454, 455, 456, 457, 458, 459, 460, 461].
In summary, future UHECR data may not only provide clues to the particle sources, but could enhance our understanding of fundamental particle physics. We are entering this new High Energy Physics era with the Pierre Auger Observatory [462, 463, 464].
35 GF = 1.16639(1) × 10-5 GeV-2 is the Fermi coupling constant. Back.
36 Similarly, gravi-burst fragmentation jets can contribute to the super-GZK spectrum . Back.
37 In light of this, the sensitivity of future CR experiments to test the SUSY parameter space has been estimated [416, 417]. Back.
38 Of course, this is not the case in traditional scenarios (for an exception see ) of decaying massive dark matter in the Galactic halo, which due to the lack of absorption, predict compositions directly given by the fragmentation function, i.e., domination by -rays. Back.