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The Astrophysics of Ultrahigh Energy Cosmic Rays

The origin of the highest energy cosmic rays is still unknown. The discovery of their sources will reveal the workings of the most energetic astrophysical accelerators in the universe. Current observations show a spectrum consistent with an origin in extragalactic astrophysical sources. Candidate sources range from the birth of compact objects to explosions related to gamma-ray bursts or to events in active galaxies. We discuss the main effects of propagation from cosmologically distant sources including interactions with cosmic background radiation and magnetic fields. We examine possible acceleration mechanisms leading to a survey of candidate sources and their signatures. New questions arise from an observed hint of sky anisotropies and an unexpected evolution of composition indicators. Future observations may reach the necessary sensitivity to achieve charged particle astronomy and to observe ultrahigh energy photons and neutrinos, which will further illuminate the workings of the universe at these extreme energies. In addition to fostering a new understanding of high-energy astrophysical phenomena, the study of ultrahigh energy cosmic rays can constrain the structure of the Galactic and extragalactic magnetic fields as well as probe particle interactions at energies orders of magnitude higher than achieved in terrestrial accelerators.


Introduction
The observation that cosmic rays can exceed 10 20 eV poses some interesting and challenging questions: Where do they come from? How can they be accelerated to such high energies? What kind of particles are they? What is the spatial distribution of their sources? What do they tell us about these extreme cosmic accelerators? How strong are the magnetic fields that they traverse on their way to Earth? How do they interact with the cosmic background All particle cosmic ray flux multiplied by E 2 observed by ATIC (Ahn et al. 2008), Proton (Grigorov et al. 1971), RUNJOB (Apanasenko et al. 2001), Tibet AS-γ (Chen 2008), KASCADE (Kampert et al. 2004), KASCADE-Grande (Apel et al. 2009), HiRes-I (Abbasi et al. 2009), HiRes-II (Abbasi et al. 2008b), and Auger (Abraham et al. 2010b). LHC energy reach of p − p collisions (in the frame of a proton) is indicated for comparison. radiation? What secondary particles are produced from these interactions? What can we learn about particle interactions at these otherwise inaccessible energies? Here we review recent progress towards answering these questions.
The dominant component of cosmic rays observed on Earth originate in the Galaxy. As shown in Figure 1, the study of this striking non-thermal spectrum requires a large number of instruments to cover over 8 orders of magnitude in energy and 24 in flux. Galactic cosmic rays are likely to originate in supernova remnants (see, e.g., Hillas 2006, for a recent update on the origin of Galactic cosmic rays). A transition from Galactic to extragalactic cosmic rays should occur somewhere between 1 PeV (≡ 10 15 eV) and 1 EeV (≡ 10 18 eV). Progress on determining this transition relies both on the study of the highest energies reached in Galactic accelerators as well as the search for extragalactic accelerators that produce ultrahigh energy cosmic rays (UHECRs).
We begin with a brief summary of recent observations (Section 2), which reveal a spec- Spectrum of UHECRs multiplied by E 3 observed by HiRes I (Abbasi et al. 2009) and Auger (Abraham et al. 2010b). Overlaid are simulated spectra obtained for different models of the Galactic to extragalactic transition and different injected chemical compositions and spectral indices, s, described in Section 2.1 and 4.
trum whose shape supports the long-held notion that sources of UHECRs are extragalactic. As shown in Figure 2, the crucial spectral feature recently established at the highest energies is a steep decline in flux above about 30 EeV. This feature is reminiscent of the effect of interactions between extragalactic cosmic rays and the cosmic background radiation, named the Greisen-Zatsepin-Kuzmin (GZK) cutoff (Greisen 1966;Zatsepin & Kuzmin 1966). Another important feature shown in Figure 2 is the hardening of the spectrum at a few EeV, called the ankle, which may be caused by the transition from Galactic to extragalactic cosmic rays or by propagation losses if UHECRs are mostly protons. As discussed in Section 2, recent reports of a trend toward a heavier composition from a few EeV up to 40 EeV together with hints of anisotropies in the sky distribution above 60 EeV raise new and unexpected puzzles. An anisotropic sky distribution is expected for trans-GZK energies (i.e., energies above 60 EeV), if UHECRs are mainly protons, due to a combination of the GZK effect (which limits trans-GZK observed sources to lie within a few 100 Mpc), the anisotropic distribution of source bearing galaxies on 100 Mpc scales, and the low magnetic deflection of light trans-GZK nuclei by the Galactic and extragalactic magnetic fields. Therefore, the report of correlations between UHECRs above 55 EeV and the distribution of nearby active galaxies (Abraham et al. 2007) can be simply interpreted as protons from nearby sources within the so-called GZK sphere. However, composition indicators from shower development observations argue for a transition to a heavier component from a few EeV up to 40 EeV (Abraham et al. 2010a). Heavy nuclei dominated injection models are quite rare in the astrophysical literature of candidate sources (see Section 6) and if iron is the main component at the highest energies, Galactic magnetic fields should wash out most anisotropic patterns around 60 EeV. Another possible interpretation of the observed shower development properties is a change in hadronic interactions above 100 TeV center of mass (TeV ≡ 10 12 eV), an order of magnitude higher energy than will be reached by the Large Hadron Collider (LHC) at CERN. A new puzzle is born: an injection at the source dominated by heavy nuclei is astrophysically unexpected, while significant changes in hadronic interactions represent novel particle physics.
To help discriminate between possible interpretations of recent results, we review in Section 3 the well-known physics of the propagation of ultrahigh energy cosmic rays: their interaction with the cosmic background radiation and the effect of cosmic magnetic fields. The effect of propagation on the observed spectrum, sky distribution, and composition depends on the source redshift evolution, the injected spectrum and composition, and the evolution of cosmic backgrounds and magnetic fields. The spectrum is cut-off due to photopion production of protons and photo-dissociation of nuclei off cosmic backgrounds. The composition simplifies to either proton or iron (or a mixture of the two) at trans-GZK energies. Anisotropies in the sky distribution of sources are blurred by magnetic fields for heavier primaries while protons keep most of the original anisotropies at trans-GZK energies.
Different scenarios for the transition between cosmic rays created in the Galaxy and those from extragalactic sources are discussed in Section 4. Specific acceleration mechanisms envisioned for reaching these extremely high energies are the topic of Section 5, including shock acceleration, unipolar inductors, and other proposals. In Section 6, we survey known astrophysical sites that are reasonable candidates for UHECR sources, from compact objects such as, neutron stars (or magnetars), to gamma-ray bursts and active galaxies. Possible signatures of different candidate sources are discussed in light of future observations of UHECRs and other messengers of the extreme universe.
With a significant increase in the integrated exposure to cosmic rays above 60 EeV, next generation observatories may reach the sensitivity necessary to achieve charged particle astronomy and to observe ultrahigh energy photons and neutrinos, which will further illuminate the workings of the universe at the most extreme energies. We end with the ongoing and future search plans for the cosmic sources of ultrahigh energy particles.

Cosmic Ray Observations at Ultrahigh Energies
After many decades of efforts to discover the origin of cosmic rays, current observatories are now reaching the necessary exposure to begin unveiling this longstanding mystery (see Figure 3 for a the history of exposures for the largest observatories). The first detection of UHECRs dates back to Linsley (1963), but it was only during the 1990s that an international effort began to address these questions with the necessary large-scale observatories. The largest detectors operating during the 1990s were the Akeno Giant Air Shower Array (AGASA), a 100 km 2 ground array of scintillators in Japan (Chiba et al. 1992), and the High Resolution Fly's Eye (HiRes) a pair of fluorescence telescopes that operated in Utah until 2006 (Boyer et al. 2002). During their lifetimes, AGASA reached an exposure of 1.6 × 10 3 km 2 sr yr (or 1,600 L 1 ) while HiRes reached twice that. To date, the highest energy recorded event was a 320 EeV fluorescence detection (Bird et al. 1994) by the pioneer fluorescence experiment Fly's Eye (Baltrusaitis et al. 1985).
Completed in 2008, the Pierre Auger Observatory is the largest observatory at present (Abraham et al. 2004). Constructed in the province of Mendoza, Argentina, by a collaboration of 18 countries, it consists of a 3,000 km 2 array of water Cherenkov stations with 1.5 km spacing in a triangular grid overlooked by four fluorescence telescopes. The combination of the two techniques into a hybrid observatory maximizes the precision in the reconstruction of air showers, allowing for large statistics with good control of systematics. The largest observatory in the northern hemisphere, the Telescope Array (TA), is also hybrid (Nonaka et al. 2009;Tokuno et al. 2009). Situated in Utah, it covers 762 km 2 with scintillators spaced every 1.2 km overlooked by three fluorescence telescopes.

Spectrum
The observed cosmic ray spectrum (Figure 1) can be described by a broken power law, E −s , with spectral index s = 2.7 below the knee at ∼ 1 PeV (= 10 15 eV) and s 3 between the knee and the ankle around 3 EeV (Apel et al. 2009). Above the ankle, s 2.6 followed by the recently established flux suppression above about 30 EeV. With exposures around 10 3 L, the measured spectra at energies where the GZK effect was anticipated had conflicting results: AGASA reported no flux supression at trans-GZK energies (Takeda et al. 1998), while early results from HiRes were consistent with the GZK prediction (Abbasi et al. 2004). By 2006, HiRes accumulated enough statistics for the first significant observation of the GZK suppression (Abbasi et al. 2008b), as displayed in Figure 2. This was confirmed by the Auger Observatory (Abraham et al. 2008b) with a recent update starting at 1 EeV (Abraham et al. 2010b) and based on 1.3×10 4 L exposure (shown in Figure 2). The displayed error bars are statistical errors while the reported systematic error on the absolute energy scale is about 22%. This systematic error allows for overall energy shifts that make the two observations consistent within the estimated errors. The highest energy event reported by Auger thus far is of 142 EeV (Abreu et al. 2010). Figure 2 also shows the observed spectrum fit by different models of UHECR sources (taken from Kotera et al. 2010b and references therein). In the mixed composition and iron dominated models (Allard et al. 2007), the ankle indicates a transition from Galactic to extragalactic cosmic rays (see Section 4), the source evolution is similar to the star formation rate (SFR), and the injection spectra are relatively hard (s ∼ 2 − 2.1). In the proton dominated models in the figure, the ankle is due to pair production propagation losses (Berezinsky & Grigorieva 1988), named "dip models" , and the injection spectra are softer for a wide range of evolution models. Models with proton primaries can also fit the spectrum with harder injection with a transition from Galactic to extragalactic at the ankle (Wibig & Wolfendale 2004).
The confirmed presence of a spectral feature similar to the predicted GZK cutoff, settles the question of whether acceleration in extragalactic sources can explain the high-energy Fraction of cosmic rays that survives propagation over a distance > D, for protons above 40, 60, and 100 EeV and for He, CNO, and Fe above 60 EeV. Black solid line shows where 50% of a given species can originate for a given atomic mass and energy. At trans-GZK energies (E 60 EeV), only protons and iron survive the propagation over D 50 Mpc. Adapted from Allard et al. (2007). spectrum, ending the need for exotic alternatives designed to avoid the GZK feature. However, the possibility that the observed softening of the spectrum is mainly due to the maximum energy of acceleration at the source, Emax, is not as easily dismissed. A confirmation that the observed softening is the GZK feature, awaits supporting evidence from the spectral shape, anisotropies, and composition at trans-GZK energies and the observation of produced secondaries such as neutrinos and photons.

Anisotropies in the Sky Distribution
The landmark measurement of a flux suppression at the highest energies encourages the search for sources in the nearby extragalactic universe using the arrival directions of trans-GZK cosmic rays. Above GZK energies, observable sources must lie within about 100 Mpc, the so-called GZK horizon or GZK sphere (Harari et al. 2006;Allard et al. 2007  Arrival directions of cosmic rays with energy E ≥ 55 EeV detected by Auger (black dots) in an Aitoff-Hammer projection of the sky in Galactic coordinates restricted to |b| > 10 • (Abreu et al. 2010). Shaded areas represent a smoothed density map (5 • smoothing angle) of the 2MRS galaxies within 200 Mpc over the Auger Observatory field of view. and references therein). This effect is shown in Figure 4 where the fraction of cosmic rays that arrive on Earth from a given distance is plotted for different energy protons (> 40, 60, and 100 EeV) and for different nuclei (He, CNO, and Fe) arriving with energies above 60 EeV. At these trans-GZK energies, light composite nuclei are promptly dissociated by cosmic background photons (see Section 3), while protons and iron nuclei may reach us from sources at distances up to about 100 Mpc. Since matter is known to be distributed inhomogeneously within this distance scale, the cosmic ray arrival directions should exhibit an anisotropic distribution above the GZK energy threshold, provided intervening magnetic fields are not too strong. At the highest energies, the isotropic diffuse flux from sources far beyond this GZK horizon should be strongly suppressed.
Attempts to detect anisotropies at ultrahigh energies date back to the mid 1990s when hints of correlations with the local large scale structure and with distant BL Lacs objects were claimed and debated (see Section 6.2). With the increase in the number of observed ultrahigh energy events, these early claims have not been substantiated while different correlations have been recently reported.
The most recent discussion of anisotropies in the sky distribution of ultrahigh energy events began with the report that the arrival directions of the 27 cosmic rays observed by Auger with energies above 57 EeV exhibited a statistically significant correlation with the anisotropically distributed galaxies in the 12th VCV (Véron-Cetty & Véron 2006) catalog of active galactic nuclei (AGN) (Abraham et al. 2007(Abraham et al. , 2008a. The correlation was most significant for AGN with redshifts z < 0.018 (distances < 75 Mpc) and within 3.1 • separation angles. An independent dataset confirmed the anisotropy at a confidence level of over 99% (Abraham et al. 2007(Abraham et al. , 2008a. The prescription established by the Auger collaboration tested the departure from isotropy given the VCV AGN coverage of the sky, not the hypothesis that the VCV AGN were the actual UHECR sources. In particular, a lack of events from the Virgo region showed that assuming the VCV AGN to be the sources gives a bad match to the observed event distribution (Gorbunov et al. 2008). No corresponding correlation was observed in the northern hemisphere by HiRes (Abbasi et al. 2008c) where the distribution of their 13 trans-GZK events is consistent with isotropy. Figure 5 shows a map of the 69 events used in the recently published Auger update which includes another 42 trans-GZK events (Abreu et al. 2010). With the new events, the correlation with the VCV catalog is not as strong for the same parameters as the original period (20 events correlate out of the original 27 while only 12 correlate out of the new 42). The data after the prescription period shows a departure from isotropy at the 3σ level. With the currently estimated correlation fraction of 38%, a 5σ significance will require at least another four years of Auger observations (Abreu et al. 2010).
The VCV catalog is not a catalog produced by an instrument or survey strategy, but an extensive compilation of known AGN in the literature. A better set of catalogs which give a more homogeneous and statistically complete survey of the nearby universe over the large field of view of Auger has become recently available. In particular, the Swift-BAT catalog of AGN (Tueller et al. 2010) and the 2MASS Redshift Survey (2MRS) catalog of galaxies (Huchra et al. 2005) are two catalogs where correlations may become more meaningful (George et al. 2008;Abreu et al. 2010). Figure 5 shows the example of Auger data superimposed on a density map generated with the 22,000 galaxies within 200 Mpc of the 2MRS catalog (Abreu et al. 2010) with Galactic latitude |b| > 10 • . The Auger trans-GZK events tend to align better with the distribution of galaxies in 2MRS (and with Swift-BAT AGN) than with the isotropic scenario, however a significance of the anisotropy or a source class identification is hard to access with the current limited statistics (Abreu et al. 2010).
Finally, the anisotropy reported by the test with the VCV catalog may indicate the effect of the large scale structure in the distribution of source harboring galaxies or it may be due to a nearby source. An interesting possibility is the cluster of Auger events around the direction of Centaurus A, the closest AGN (at ∼ 3.8 Mpc). The clustering around the Cen A region may also be due to the Centaurus cluster (as shown by the dark red region of Figure 5) which is much further away but on the general direction of Cen A (Kashti & Waxman 2008;Kotera & Lemoine 2008b). Only much higher statistics will tell if Cen A is the first UHECR source to be identified.

Composition
The third key measurement that can help resolve the mystery behind the origin of UHECRs is their composition as a function of energy observed on Earth. Composition measurements can be made directly up to energies of ∼ 100 TeV with space-based experiments (see, e.g., Ahn et al. 2010). For higher energies, composition is derived from the observed development and particle content of the extensive airshower created by the primary cosmic ray when it interacts with the atmosphere.
Presently, the best indicator of the composition of the primary particle is the depth in the Xmax and RMS(Xmax) as a function of primary energy, as measured by Auger fluorescence detectors (Abraham et al. 2010a). MC simulations from different hadronic interaction models are displayed for primary protons (blue) and primary iron nuclei (red).
atmosphere of the shower maximum, Xmax, given in g/cm 2 . The average shower maximum, Xmax , scales approximately as ln(E/A), where E is the energy and A is the atomic mass of the primary cosmic ray which generated the shower (see, e.g., Letessier-Selvon & Stanev 2011 and references therein). On average the shower maximum for protons occurs deeper in the atmosphere than that for the same energy iron nucleus, X p max > X F e max . In addition, proton showers fluctuate more about Xmax providing another measure of composition, for example, the root mean square fluctuations about Xmax . Another useful measure of composition is the particle content of the shower such as the number of muons: proton showers have fewer muons than showers caused by heavier nuclei with the same energy. In practice, observed shower maxima and particle numbers are compared with Monte Carlo airshower simulations which involve an extrapolation to higher energies of hadronic interactions known at energies of laboratory accelerators ( TeV).
Observations of shower properties from the knee to just below the ankle indicate a general trend from light primaries dominating at the knee to heavier primaries dominating up to ∼ 0.1 EeV (see, e.g., Bluemer et al. 2009). These observations follow expectations that the knee is created by a rigidity 2 dependent end of Galactic cosmic rays which may be due to maximum acceleration at the sources and/or containment in the Galactic magnetic field. Just before the ankle, the trend seems to reverse back toward a lighter composition, being consistent with light primaries at 1 EeV as shown in Figure 6. Figure 6 shows Auger data on Xmax and RMS(Xmax) of 3754 events above 1 EeV together with a range covered by simulations for protons and iron nuclei using different hadronic models (Abraham et al. 2010b). The dominance of light nuclei around a few EeV is also observed by HiRes who reported a final reconstruction of 815 events in Abbasi et al. (2010).
A surprising trend occurs in Auger data above 10 EeV, a change toward heavy primaries is seen both in Xmax as well as in RMS(Xmax) up to 40 EeV. As a mixture of different nuclei would increase the RMS(Xmax), the observed narrow distribution argues for a change toward a composition dominated by heavy nuclei. Due to different reconstruction methods, the HiRes measurement of fluctuations is not easily displayed in Figure 6, but their data trend remains closer to light primaries up to around 50 EeV. The two results are consistent within quoted errors, so the situation is currently unclear.
Shower properties observed by Auger up to 40 EeV are quite challenging to candidate acceleration models because they conflict with the prevailing view that the primaries are proton dominated. The observed hint of anisotropies also points to light primaries (Abreu et al. 2010). Studies of ultrahigh energy nuclei propagation require unusual choices in attempts to fit the observed composition indicators, such as a hard injection spectrum (s ∼ 1.6) with primaries dominated by nitrogen or silicon (Hooper & Taylor 2010). Given the measurement uncertainties, some reasonable but "disappointing" options are also possible such as models where protons have a low maximum energy and the observed steepening of the spectrum, above 30 EeV, is due to the maximum energy of iron nuclei (Aloisio et al. 2009;Allard et al. 2008;Allard 2009). In this case, the feature in the observed spectrum is mainly due to the maximum accelerator energy which is coincidently close to the expected GZK cutoff. As discussed in Section 6, most known astrophysical accelerators have Emax close to GZK energies, so the coincidence may actually be real. The challenge to extragalactic candidate sources is to explain the origin of such high metallicities in the accelerated material. These observations have also motivated a return to the possibility that Galactic cosmic ray sources significantly contribute at ultrahigh energies (Calvez et al. 2010).
As recently highlighted in Schwarzschild (2010), changes to hadronic interactions from current extrapolations provide a plausible alternative interpretation to the observed shower development behavior. Auger probes interactions above 100 TeV center of mass, while hadronic interactions are only known around a TeV. The observation of anisotropies and secondary particles (neutrinos and gamma-rays) can lead to astrophysical constraints on the composition of UHECRs, opening the possibility for the study of hadronic interaction cross sections, multiplicities, and other interaction parameters at hundreds of TeV.
The detailed composition of UHECRs is still to be understood, but it is clear that primaries are not dominated by photons (Abraham et al. 2008;Abraham et al. 2009c) or neutrinos (Abraham et al. 2009a;Abbasi et al. 2008a). Limits on the photon fraction place stringent limits on models where UHECRs are generated by the decay of super heavy dark matter and topological defects. Unfortunately, the uncertainties on the UHECR source Proton energy loss lengths: black solid line for photo-pion production on CMB and IR-UV photons; red solid line for pair production on CMB photons. Dashed lines represent the interaction length (or mean free path to interaction) for photo-pion production on CMB photons (thick) and IR-UV photons (thin), assuming the background of Stecker et al. (2006). The dotted line indicates the losses due to cosmological expansion. composition, spectrum, and redshift evolution translates to many orders of magnitude uncertainty in the expected cosmogenic neutrino flux as discussed in Section 6.

The propagation of Ultrahigh Energy Cosmic Rays
While propagating from their sources to the observer, UHECRs experience two types of processes: (i) interactions with cosmic backgrounds that affect their energy and their composition, but not their direction; and (ii) interactions with cosmic magnetic fields that affect their direction and travel time, but not their energy and composition. Both leave a variety of signatures on the observables of UHECRs and generate secondary neutrinos and gamma rays (see Section 6.3).

Interaction processes on cosmic backgrounds
In the intergalactic medium, cosmic rays primarily interact with the Cosmic Microwave Background (CMB) photons at the highest energies, and with infrared, optical, and ultraviolet background (IR-UV) photons at slightly lower energy (see, e.g., Kneiske et al. 2004;Stecker et al. 2006 for detailed background models).
Photohadronic interactions between protons and background photons mainly lead to pion production: p γ −→ N + nπ (here N is a nucleon and n is the number of pions produced), or to electron-positron pair production, also called Bethe-Heitler process: p γ −→ p e + e − . The energy threshold of these interactions for a photon of energy reads: Ep,π ∼ 200 EeV ( CMB/ ) for pion production, and for pair production: Ep,ee ∼ 0.8 EeV ( CMB/ ), with CMB 2.7 kBTCMB ∼ 6 × 10 −4 eV, the mean energy of a CMB photon.
We plot in Figure 7 the energy loss lengths x loss ≡ |E −1 dE/c dt| −1 for these two processes, using analytical formulae by Stecker (1968) (also see Maximon 1968;Genzel et al. 1973;Begelman et al. 1990;Mücke et al. 1999). Above E ∼ 60 EeV, the distance that particles can travel without losing their energy shortens considerably. If cosmic rays originate from cosmological distances, their flux above this energy should be consequently suppressed, producing the well-known GZK feature in the UHECR spectrum (see Section 2.1). This property further imposes that the sources of the observed UHECRs at a given energy should be located in our local Universe, within a distance l x loss (E). Losses due to the cosmological expansion are also represented in Figure 7.
For primary cosmic rays with mass number A > 1, different interaction processes come into play. At ultrahigh energies, nuclei photo-disintegrate on CMB and IR-UV photons through three main types of processes that contribute at increasing energy ranges: the Giant Dipolar Resonance (for ∼ 8 − 30 MeV), the Quasi Deuteron process (for ∼ 20 − 150 MeV), and the Baryonic Resonance (for ∼ 150 MeV). In a first approximation, one can consider that the Lorentz factor of the primary nucleus remains unchanged through these interactions. Nuclei also experience photo-pair production that decreases the Lorentz factor without affecting the number of nucleons.
After pioneering work by Puget et al. (1976), energy losses for nuclei have been examined by several groups (Stecker & Salamon 1999;Epele & Roulet 1998a,b;Bertone et al. 2002;Khan et al. 2005;Allard et al. 2005Allard et al. , 2008Hooper et al. 2005Hooper et al. , 2008Aloisio et al. 2008). One remarkable effect of the propagation of nuclei is that nuclei with mass number A < 20 cannot travel farther than few tens of megaparsecs without disintegrating (see Figure 4). In particular, one can conclude that heavy nuclei could be found in abundance at trans-GZK energies only if the composition were essentially dominated by iron group nuclei. Such a composition can arise when the proton Emax is smaller than Ep,π, so that only heavy nuclei are present at greater energies Aloisio et al. 2009).
The effect of photo-hadronic interactions on the cosmic ray spectrum can be calculated analytically for protons (Berezinsky & Grigorieva 1988;Berezinsky et al. 2006). Numerical codes such as SOPHIA (Mücke et al. 1999) enable the precise evaluation of the cross-sections for photo-hadronic interactions taking into account various channels, and of the produced  flux of secondary particles (pioneered by Berezinsky & Gazizov 1993). Numerical Monte-Carlo methods are best suited to model inhomogeneous distribution of sources, calculate secondary emissions, and treat the complex processes intervening in the propagation of nuclei in the intergalactic medium. Among the existing propagation codes that have been developed for this purpose, one might refer to the public code CRPropa  and to the complete nuclei propagation tool by Allard et al. (2006).
The calculated spectra are in very good agreement with the observed spectra for a variety of chemical compositions, Galactic to extragalactic transition models, source evolution histories, and injection spectrum indices between 1.6 − 2.7, for a fixed maximum acceleration energy, Emax (see, e.g., Figure 2). Kachelriess & Semikoz (2006) demonstrate that relaxing the assumption of a single maximum acceleration energy and introducing a power-law distribution of Emax leads to a change in the overall propagated spectrum slope. A key region for models to fit is the ankle around a few EeV where the spectral slope changes (see Section 4). The precise shape of the GZK feature depends on the local source density and on the transient or continuously emitting natures of the sources (see, e.g., Aloisio & Boncioli 2010;Berezinsky et al. 2006;Medina Tanco 1998;. For instance, if Emax 100 EeV a recovery of the spectrum at high energies can be observed by future detectors.

The effects of Magnetic fields
The absence of powerful astrophysical counterparts in the arrival directions of UHECRs is probably related to the effect of cosmic magnetic fields that deflect and delay particles during their propagation. Charged particles are subject to the influence of magnetic fields in the source environment, in the intergalactic medium, and in the Galaxy, as depicted in Figure 8. Since very little is known about cosmic magnetic fields (for recent reviews see, e.g., Beck 2008;Kulsrud & Zweibel 2008;Vallée 2004;Widrow 2002;Kronberg 1994), the parameter space for an accurate description is quite large. Different propagation regimes apply to different cosmic ray rigidities, from weak angular deflection at the highest energies or in weak magnetic fields, to the diffusive regime in strong fields or at low enough energies. Figure 8 shows schematic representations of magnetized regions intervening in UHECR propagation.
Much progress has been made in recent years on Galactic magnetic field observations (see Han et al. 1999Han et al. , 2006Han 2008;Jansson et al. 2009) and their effect on the propagation of UHECRs (Harari et al. 1999;Alvarez-Muñiz et al. 2002;. These studies conclude that the deflection for particles of charge Z and energy E should not exceed ∼ 10 • Z (40 EeV/E). In particular, the regular component of the Galactic magnetic field can distort the angular images of cosmic ray sources: the flux may appear dispersed around the source or globally translated in the sky with a small dispersion (Harari et al. 1999). Source image distortions are stronger for heavier nuclei (see, e.g., Giacinti et al. 2010). Since Galactic magnetic fields are not uniform in the sky, angular deflections also depend on the observed direction (Harari et al. 1999;Kachelrieß et al. 2007;Takami & Sato 2008).
The extragalactic magnetic field (EGMF) 3 is much less known. Measurements of Faraday rotation provide estimates of the magnetic fields in the core of clusters of galaxies, with typical strengths ∼ 1 − 40 µG (see above reviews). Outside clusters upper limits on the integrated strength of the magnetic field parallel to the line of sight, B || , have been obtained using rotation measures (Ryu et al. 1998;): B 2 || λB 1/2 10 −8 G Mpc 1/2 , where the field reversal scale, λB 1 Mpc, given the typical turbulent velocities (e.g., Waxman & Bahcall 1999). The future SKA project (Beck et al. 2007) will enlarge considerably the observations of the EGMF in our local Universe. Meanwhile, Neronov & Semikoz (2009) demonstrated that the observation of extended gamma-ray emission around point sources together with time delays in gamma-ray flares should provide robust measurements of the EGMF, and help constrain scenarios of its origin. Based on recent Fermi (Fermi Gammaray Space Telescope) observations, the existence of an EGMF of order 10 ?16 -10 ?15 G has been suggested using upper limits on the secondary emission of a few blazars Neronov & Vovk (2010) and by the apparent (and debated, see e.g., Neronov et al. 2011) detection of pair halos in stacked images of a large number of sources Ando & Kusenko (2010). These new developments may lead to stronger constraints or even detections of the EGMF, which at present can range from 10 −16 − 10 −9 G.
The origins of extragalactic magnetic fields are not well understood (e.g., Kulsrud & Zweibel 2008). Some models set their origins in the primordial Universe (see Widrow 2002 and references therein) other models through magnetic pollution from astrophysical sources such as galactic winds or jets from radio-galaxies (Kronberg et al. 1999;Birk et al. 2000;Aguirre et al. 2001;Cen et al. 2005;Bertone et al. 2005Bertone et al. , 2006Scannapieco et al. 2006). If magnetic fields have a primordial origin, they should be all-pervading and amplified in dense regions by dynamical effects induced by large-scale structure formation. If galaxies are the primary generators of magnetic enrichment, the field should be rather concentrated in high peaked density regions and nearly suppressed in voids, depending on the efficiency of the winds (Bertone et al. 2006). A combination of the two scenarios may occur, as well as intermediate models in which magnetic pollution happens during the reionization epoch, generating slightly inhomogeneous fields already at high redshift. Whatever the origin of extragalactic fields, dynamical amplification in dense regions during structure collapse should play a key role in setting up their present configuration (Bruni et al. 2003;Siemieniec-Oziȩb lo & Golda 2004;King & Coles 2006;Ryu et al. 2008).
More recently, various groups have developed simulations of the formation of large scale structures including magnetic fields, in order to model more realistic inhomogeneous configurations (Ryu et al. 1998;Sigl et al. 2004;Dolag et al. 2004Dolag et al. , 2005Brüggen et al. 2005;Ryu et al. 2008;Dubois & Teyssier 2008;Das et al. 2008;Donnert et al. 2009). Most of these authors set an initial magnetic seed at high redshift which is then evolved in time. The overall amplitude of the field is rescaled at the end of the simulation so as to reproduce the observed strength of magnetic fields in the core of clusters of galaxies. Ryu et al. (1998) and Das et al. (2008) estimate directly the intensity of the magnetic field using the vorticity and the energy density calculated from the kinetic properties of the gas. With these methods, magnetic fields generated by astrophysical feedback can be hard to implement (see, however Donnert et al. 2009 who have incorporated some of these effects).
These simulations lead to very discrepant results concerning the configuration of the fields (see Figure 9). These differences probably stem from the different methods and assumptions made by each group, and illustrate the inherent complexity of these simulations. Indeed, Ryu et al. (1998) and Sigl et al. (2004) assume that magnetic seeds are generated by Biermann battery effects around accretion shocks, whereas Dolag et al. (2005) implement a homogeneous seed around redshift z ∼ 20, and Donnert et al. (2009) add to the latter method astrophysical magnetic pollution in localized spots at lower redshifts. The difference in configuration probably also results from the Eulerian and Lagrangian treatments used by the different groups and limitations on dynamical range and resolution.
Such discrepancies on the magnetic fields impact the range of deflections induced on UHECRs. For instance, Sigl et al. (2004) find that protons with energy E > 100 EeV should be deflected by 10 − 20 • , while Dolag et al. (2004) find deflections of less than a degree at the same energy.
In the framework of UHECR propagation, a simpler and faster approach to model the inhomogeneous extragalactic magnetic field is to perform a scaling of the field strength to the underlying density field (Medina Tanco 1997;Takami et al. 2006;Takami & Sato 2008;Kotera & Lemoine 2008a). The assumed scaling law enables one to account for various types of magnetic field amplifications due to structure formation and for the high contrast fields that should be produced by astrophysical pollution. In addition, an analytic stochastic approach can be effective in describing the propagation of UHECRs in the extragalactic fields because of the high energy of the particles and the low magnetization of voids (Kotera & Lemoine 2008b). Indeed, the deflection of UHECRs by magnetic fields of strength B < 10 −12 G is lower than typical instrument resolutions which are ∼ 1 • . Particle transport can then be viewed as a succession of rectilinear portions interrupted by deflections on localized magnetized regions (such as filaments, halos of radio-galaxies and galactic winds). This model can be applied to the coherent field amplified in numerical simulations as well as for the local enrichment processes due to astrophysical sources, and provides an effective framework to calculate the influence of magnetic fields on observable quantities of UHECRs (see, for example, the expected angular deflection skymap for protons calculated Expected angular deflection skymap (in degrees) for protons with energy E 6 × 10 19 eV, calculated using the method developed in Kotera & Lemoine (2008b). During their propagation, particles are assumed to be deflected on magnetized scattering centers that are distributed according to the galaxy density estimated using the PSCz catalog (Saunders et al. 2000). The gray mask indicates the regions in the sky that are not covered by PSCz. Here, the parameters of the scattering centers (that sets a particular configuration of the EGMF) are chosen so as to reproduce the typical parameters found in the literature for magnetized filaments and radio halos: we assume that protons encounter ∼ 3 scattering centers in average over a propagation distance of ∼ 100 Mpc and experience a deflection of order 1.7 • after each encountering. White dots indicate the arrival direction of the cosmic rays of energy E 6 × 10 19 eV detected by Auger (Abreu et al. 2010).
for particular assumptions on EGMF presented in Figure 10).
If the magnetic field is sufficiently strong, or the particle energy "not too high", UHECRs may enter a diffusive regime. One can define the scattering time of a particle, tscatt, as the timescale beyond which its quadratic deflection angle becomes of order unity (δθ 2 ∼ 1). This quantity depends on the properties of the turbulence of the magnetic field. Diffusion occurs when the travel time exceeds tscatt, and the diffusion coefficients, which govern the transport on timescales t tscatt, also depend on the properties of the turbulence. The propagated spectrum of UHECRs in the diffusive regime can be calculated analytically for a given injection spectrum, density and distribution of sources, for homogeneous magnetic fields (Aloisio & Berezinsky 2004;Lemoine 2005;Aloisio & Berezinsky 2004). Interestingly, Aloisio & Berezinsky (2004) have demonstrated that the analytical propagated spectra for both rectilinear (without magnetic field) and diffusive propagations converge when the typical distance between two sources is shorter than the characteristic propagation lengths (i.e. the scattering length and the energy loss lengths). This is particularly the case of continuously distributed sources for which the diffusion of particles in the magnetic field does affect the final spectrum.
When the strong magnetic field is not all pervading but limited to a particular geometry, the diffusive regime can lead to interesting confinement effects. For instance, the properties of UHECR transport in a strongly magnetized local supercluster or in clusters of galaxies have been discussed (e.g., Berezinsky et al. 1990aBerezinsky et al. , 1997Lemoine et al. 1999;Ide et al. 2001;Medina Tanco 2001, and see Section 6.3 for more references on clusters of galaxies).
Another interesting aspect of diffusion, from the point of view of phenomenology, is that of magnetic horizons that will be discussed in the next section. If the length of the path travelled by the particle, from source to observer, becomes larger than the Hubble length, cH −1 0 , the source cannot be seen in cosmic rays as it lies beyond the magnetic horizon. In a Hubble time, particles travel a linear distance d ∼ c H −1 0 tscatt 1/2 65 Mpc (ctscatt/1 Mpc) 1/2 . Since tscatt, and hence d, increases with increasing energy, this produces a low-energy cut-off in the propagated spectrum.

The Galactic to extragalactic transition
The highest energy cosmic rays are likely to originate in extragalactic sources, given the strength of Galactic magnetic fields and the lack of correlations with the Galactic plane. Low energy cosmic rays are easily created and contained in the Galaxy, so a transition region should occur in some intermediate energy. "A hypothesis blessed by long tradition is that" Galactic cosmic rays end below 10 EeV "and above that a different source is active (most plausibly in the nearby supercluster of galaxies)" quoting Hillas (1984). Modern measurements of the spectrum place a plausible transition region around the ankle at a few EeV (Figure 1 and 2). However, the ankle can also be interpreted as the product of propagation losses due to pair production (Berezinsky & Grigorieva 1988;Berezinsky et al. 2006) in proton dominated scenarios allowing for a transition at lower energies.
The knee in the cosmic ray spectrum is likely to signal the Emax for light nuclei of dominant Galactic sources and/or the maximum containment energy for light nuclei in the Galactic magnetic field. The same effect for heavier nuclei may cause the softer spectrum above the knee (see, e.g., Lemoine 2005;Hillas 2006). Extragalactic sources producing spectra harder than s = 3 can overtake the decaying Galactic flux around the ankle. Recent studies of a transition at the ankle which fit the observed spectrum and the composition trends in this energy region are discussed in Allard et al. (2005) where different models are contrasted. Models based on proton primaries with a hard spectrum (Wibig & Wolfendale 2004), on a mixed composition with proportions similar to the Galactic mix, or even on a composition dominated by heavy nuclei (Allard et al. 2007) fit well the UHECR spectrum and composition data around the ankle. In Figure 2, we show two examples of the so-called "ankle transition models": one with source injection s = 2.1, source composition similar to the Galactic mixture, and source evolution that follows the SFR; and a second model with similar source evolution and s = 2, but a pure iron composition injected. Both models fit well the UHECR spectrum but predict different compositions throughout this energy range.
Ankle transition models work well for UHECR scenarios, but they were thought to chal-lenge models for the origin of Galactic cosmic rays. The requirement that Galactic sources reach energies close to the ankle strained traditional models where acceleration in supernova remnants (SNRs) was expected to fade around 1 PeV (Lagage & Cesarsky 1983). A modification to the traditional SNR scenario, such as magnetic field amplification in SN shocks (Bell & Lucek 2001), or a different progenitors such as Wolf-Rayet star winds (Biermann & Cassinelli 1993), and trans-relativistic supernovae (Budnik et al. 2008) may explain the energy gap from PeV to EeV. Taking into account magnetic field amplification and Alfvenic drift in shocks of Type IIb SNRs, Ptuskin et al. (2010) find that Galactic cosmic ray iron can reach Emax ∼ 5 EeV, allowing extragalactic cosmic rays to begin to dominate above the ankle. The possibility that the ankle is due to pair-production losses during the propagation of extragalactic protons (Berezinsky & Grigorieva 1988) has motivated an alternative model for the Galactic to extragalactic transition, called "dip models" ). The energy of the predicted dip is close to the observed ankle and a good fit to the spectrum over a large energy range is reached with a softer injection index as the dip proton models shown in Figure 2. This option relaxes the need for Galactic cosmic rays to reach close to EeV energies, however it needs to be tuned to avoid strong (unobserved) spectral features between the knee and the ankle. Detailed models where the lower energy behavior of the extragalactic component blends smoothly with the Galactic cosmic rays have been developed using minimum energy and magnetic effects (Lemoine 2005;Aloisio & Berezinsky 2005;Hillas 2006;Kotera & Lemoine 2008a;Globus et al. 2008). In some of these models a feature is produced around the "second knee" which may be observed around 0.5 EeV. The dip model can fit the observed spectrum if the injection is proton dominated Allard et al. 2007) or with at most a primordial proton to helium mix (Hillas 2006), which gives a clear path for distinguishing it from mixed composition models. A proton dominated flux below the ankle region is a necessary condition for this model to be verified.
Clarifying the structure of the transition region is crucial for reaching a coherent picture of the origin of Galactic and extragalactic cosmic rays. This will require accurate spectrum and composition measurements from the knee to the ankle and beyond. KASCADE-Grande (Apel et al. 2009) has made great progress above the knee, while UHECR projects have started to lower their energy threshold such as the Auger Observatory enhancements (Abraham et al. 2009b): HEAT (High Elevation Auger Telescopes) and AMIGA (Auger Muons and Inll for the Ground Array); and the Telescope Array Low Energy Extension (TALE) proposal. Having the same system covering a large range in energy will help control systematic offsets that degrade the accuracy of the needed precision. In addition, a strong multi-wavelength program has shown that magnetic field amplification occurs in SNRs and Galactic sources can reach further than previously believed. Finally, models of hadronic interactions will benefit from the energy reach of the LHC which can probe hadronic interactions at energies higher than the knee (Figure 1) and help constrain composition indicators between the knee and the ankle.

Acceleration Mechanisms
The acceleration of charged particles is easily achieved in the presence of electric fields. However, ubiquitous astrophysical plasmas destroy large scales electric fields throughout the universe. Occasionally, high voltage drops that may lead to particle acceleration can be found in some regions of the magnetosphere or the wind of neutron stars, or near black holes and their accretion disks. In contrast, magnetic fields are omnipresent in astrophysical objects. Their variations in space and time imply the existence of transient electric fields which can supply a consequent amount of energy to charged particles.
In our framework, acceleration mechanisms must fulfill the following criteria: they should enable charged particles to reach ultrahigh energies (from EeV to above 200 EeV) and the accelerated population should bear an injection spectrum (usually power-law) that would fit the observed UHECR spectrum after propagation. Below we summarize two of the most commonly cited acceleration mechanisms, Fermi acceleration and unipolar inductors, and briefly discuss other processes that have been proposed in the literature.

Fermi acceleration at shock waves
The principle of Fermi acceleration is the transfer of energy from macroscopic motion to microscopic particles through their interaction with magnetic inhomogeneities. In the version elaborated by Fermi himself (Fermi 1949), magnetic scattering centers had random velocities which led to an energy gain of order ∆E/E ∝ β 2 , where β is the average velocity of the scattering centers in units of c. This process is now called 2nd order Fermi acceleration, while the 1st order Fermi process is the case where the macroscopic motion is coherent, such as a shock wave where particles can gain energy as they bounce back and forth, making ∆E/E ∝ β (Axford et al. 1977;Bell 1978;Blandford & Ostriker 1978). Shock waves are quite frequent in the Universe, for instance where an ejecta encounters the interstellar medium. SNR shocks are believed to be the sites where Galactic cosmic rays are accelerated via 1st order Fermi processes. Popular shock regions for UHECR acceleration are GRB shocks, jets and hot spots of AGN, and gravitational accretion shocks.
Second order Fermi acceleration depends on the scattering time of particles in the magnetic turbulence and thus on the characteristics of the latter, which are poorly known in astrophysical objects. It is less efficient than 1st order Fermi in the non relativistic limit (β 1). In the relativistic case (Pelletier 1999), it has been applied to acceleration studies inside GRBs (Pelletier & Kersalé 2000;Gialis & Pelletier 2003. First order Fermi processes under the 'test particle' approximation 4 , lead to simple powerlaw predictions for the spectrum of the accelerated population (see, e.g., Gaisser 1991). Several recent studies show that 1st order Fermi acceleration at relativistic shock waves is more intricate, even under the test particle approximation Lemoine & Pelletier 2003;Lemoine et al. 2006;Pelletier et al. 2009). Lemoine et al. (2006) have pointed out that Fermi acceleration cannot happen if the Larmor radius of the accelerated particle is much smaller than the coherence length of the magnetic field. Indeed, the particle is then captured on a field line and advected far away downstream since the magnetic field is shock-compressed to a perpendicular configuration. Fermi acceleration should thus stop after a first back and forth cycle around the shock, unless the magnetic field is strongly amplified on spatial scales much smaller than the Larmor radius rL of the particle. It was further demonstrated by Pelletier et al. (2009) that the noise associated with the motion in the small scale turbulent magnetic field needs to overcome the unperturbed trajectory in the large scale coherent field, which sets an upper bound to rL (see also, Niemiec et al. 2006).
Such requirements being hardly reached for ultra-relativistic shocks, as discussed by Pelletier et al. (2009), one might advocate that the most efficient Fermi acceleration occurs around mildly relativistic shocks. The characteristics of the accelerated population strongly depend on shock parameters (e.g., the shock Lorentz factor Γ sh , temperature and pressure values). No general tendency is easily derived, as can be concluded from the large span of spectral properties obtained by the groups who have performed detailed studies of particle acceleration around mildly relativistic shocks in various situations (e.g., Bednarz & Ostrowski 1998;Kirk et al. 2000).
Accelerated particles can act as precursors of the shock and induce significant modifications of the gas flow upstream and downstream. The current of energetic particles can initiate plasma instabilities that tend to increase the level of magnetohydrodynamic (MHD) turbulence which scatter the particles. The possibility that streaming instabilities amplify the magnetic field, and that the acceleration efficiency could increase accordingly, was first discussed in the framework of non relativistic shocks in SNRs (Bell 1978;Lagage & Cesarsky 1983). More detailed work was conducted by, e.g. The estimated amplification of the upstream field is at most of two orders of magnitude. The generalization of the instability to the relativistic regime (Milosavljević & Nakar 2006;Reville et al. 2007;Pelletier et al. 2009) reveals that this mechanism is not sufficient to allow efficient 1st order Fermi acceleration around ultra-relativistic shocks. Other mechanisms outside the MHD range, such as Weibel like instabilities (Medvedev & Zakutnyaya 2009) or resonant Cerenkov effects with plasma modes (Pelletier et al. 2009) might yet provide enough amplification to allow successful Fermi acceleration. Downstream of the shock, the relativistic two stream Weibel instability could amplify the magnetic field on small scales, to the level required by afterglow modeling of gamma-ray bursts (Gruzinov & Waxman 1999;Medvedev & Loeb 1999). In order to make progress on these issues, many groups are performing Particle-In-Cell simulations that endeavor to solve self-consistently the field-particle interactions (e.g., Silva et al. 2003;Hededal & Nishikawa 2005;Dieckmann et al. 2008;Spitkovsky 2008;Riquelme & Spitkovsky 2010). It is promising that the latest simulations see evidences of Fermi acceleration and particle-wave interactions.
Finally, when modeling Fermi acceleration around shocks, mechanisms of "shear acceleration" have to be taken into account. Particles traversing a velocity gradient perpendicularly to a jet axis (instead of going along the axis as in the case of pure shock acceleration) can experience acceleration (e.g., Rieger et al. 2007, see Rieger & Duffy 2005 for an application to acceleration of UHECRs in GRBs, Lyutikov & Ouyed 2007 for a variant). This mechanism however depends on the characteristics of the plasma velocity gradients, which are not fixed, unlike the shock characteristics.

Unipolar Inductors
Unipolar inductors have been suggested as alternative ways to accelerate particles to ultrahigh energies (see, e.g., Berezinsky et al. 1990b). Below we focus specifically on neutron stars, but acceleration may also be caused by unipolar inductors in other relativistic magnetic rotators, such as black holes with magnetized disks that lose rotational energy in jets.
Rapidly rotating neutron stars generally create relativistic outflows ("winds"), where the combination of the rotational energy and the strong magnetic field induces an electric field E = v × B/c (where v and B are the velocity and the magnetic field of the outflowing plasma). The wind thus presents voltage drops where charged particles can be accelerated to high energy. This model was first developed in the framework of ordinary pulsars (see, e.g., Shapiro & Teukolsky 1983 and references therein), but the latter do not supply enough energy to reach the highest energies (E > 10 20 eV). Blasi et al. (2000) pointed out that young neutron stars with millisecond rotation periods Ω and very high surface magnetic fields B * (i.e., magnetars) could easily accelerate particles to: E(Ω) ∼ 3 × 10 21 eV Zη1(B * /2 × 10 15 G)(R * /10 km) 3 (Ω/10 4 s −1 ) 2 , where η1 = 0.1 is the fraction of the voltage drop experienced by a particle and R * is the radius of the star. The spin down of the magnetar due to energy losses and the dependency of the particle acceleration energy on Ω lead to a power-law spectrum for the population of cosmic rays produced by a magnetar (Blasi et al. 2000;Arons 2003): dN/dE = 9c 2 I(1+E/Eg) −1 (2ZeB * R 3 * E) −1 . I is the principal inertial momentum of the magnetar and Eg is the energy corresponding to its angular velocity, at which gravity wave losses and electromagnetic losses are equal. Arons (2003) calculated that UHECRs should be produced at the very early stages of the lives of magnetars (after a few days to get down to E ∼ 6 EeV), hence their emission can be considered as an impulsive burst. Note that this model was introduced in the AGASA era, in order to explain the absence of the GZK cut-off; with our current data however, such a hard injection spectrum (s = 1) is problematic as it does not fit the observed slope. An adequate distribution of initial voltages among magnetars can be found to soften the spectrum while leading to gravitational waves (Kotera 2011).
The toy model described above needs to be further investigated on several issues, for instance on the nature of the ions injected in the wind, on the mechanism through which the current in the wind taps the available voltage, and on the escape of the accelerated particles from the wind and the wind nebula. A detailed discussion of such issues can be found in Arons (2003).

Other models
Among other models that have been proposed in the literature, one might consider magnetic reconnection acceleration, wake-field acceleration (often related to ponderomotive acceleration), and re-acceleration in sheared jets. In a plasma, a local reconfiguration of the magnetic field topology (or reconnection) happens when the plasma conductivity is not high enough to support the current associated with a magnetic field structure (see Zweibel & Yamada 2009 for a review). The field reaches a lower energy level configuration, and the liberated energy can be devoted to particle acceleration. This mechanism, responsible for the generation of high energy particles in solar flares, has been applied to UHECR acceleration in pulsar winds (Coroniti 1990;Lyubarsky & Kirk 2001), in newborn millisecond pulsars (de Gouveia Dal Pino & Lazarian 2000), in termination shocks of pulsar winds (Lyubarsky 2003), and in GRB outflows (Thompson 2006). Some authors also proposed that, in Poynting-flux dominated flows (e.g., flows with Poynting to kinetic flux ratio 1), 1st order Fermi acceleration could aliment this mechanism, as particles reflected in the magnetized plasma could converge in the reconnection region (de Gouveia dal Pino & Lazarian 2005; Giannios 2010).
A wake-field is created in a plasma when waves with high charge separation travel through the plasma. It leads to the formation of ponderomotive forces (longitudinal Lorentz invariant nonlinear forces that a charged particle experiences in an inhomogeneous oscillating electromagnetic field) that can accelerate particles if they are trapped in the wave: particles surf-ride the waves (Tajima & Dawson 1979;Chen et al. 2002). A simple form of surf-riding acceleration was developed in the case of a pulsar wind by Buckley (1977) and Contopoulos & Kazanas (2002). A detailed discussion for magnetar winds can also be found in Arons (2003).
Other acceleration mechanisms have been proposed and may contribute to the acceleration of cosmic rays in the Galaxy. These include a variety of second order processes and many of them can be observed to operate in solar physics. However, they are believed to be too slow to be relevant to the acceleration of UHECRs.

Candidate Sources and their signatures
The requirements for astrophysical objects to be sources of UHECRs are quite stringent. After reviewing some of the basic requirements, we briefly discuss plausible sources such as accretion shocks in large scale structures, active galactic nuclei, gamma-ray bursts, and neutron stars or magnetars. For these different classes of candidate sources, we discuss the possibility of locating the sources with UHECR observations and review possible ways of discovering the sources with secondary photons and neutrinos.

Candidate Source Requirements
The Larmor radius, rL = E/ZeB ∼ 110 kpc Z −1 (µG/B)(E/100 EeV), of UHECRs in Galactic magnetic fields is much larger than the thickness of the Galactic disk. Thus, confinement in the Galaxy is not maintained at the highest energies, motivating the search  for extragalactic sources. Requiring that candidate sources be capable of confining particles up to Emax, translates into a simple selection criterium for candidate sources with magnetic field strength B and extension R (Hillas 1984): rL ≤ R, i.e., E ≤ Emax ∼ 1 EeV Z (B/1 µG)(R/1 kpc). Figure 11 presents the so-called Hillas diagram where candidate sources are placed in a B − R phase-space, taking into account the uncertainties on these parameters (see also Ptitsyna & Troitsky 2010 for an updated discussion on the Hillas diagram). Most astrophysical objects do not even reach the iron confinement line up to 10 20 eV, leaving the best candidates for UHECR acceleration to be: neutron stars, Active Galactic Nuclei (AGN), Gamma Ray Bursts (GRBs), and accretion shocks in the intergalactic medium. The Hillas criterion is a necessary condition, but not sufficient. In particular, most UHECR acceleration models rely on time dependent environments and relativistic outflows where the Lorentz factor Γ 1. In the rest frame of the magnetized plasma, particles can only be accelerated over a transverse distance R/Γ, which changes subsequently the Hillas criterion.
The maximum accessible energy further depends on many details of the acceleration region but can be estimated by comparing the acceleration time, tacc, the escape time of particles from the acceleration region tesc, the lifetime of the source, tage, and the energy loss time due to expansion and to interactions with the ambient medium, t loss (see, e.g., Norman et al. 1995a;Lemoine & Waxman 2009). The condition for successful acceleration can then be written tacc tesc, tage, t loss . The escape timescale tesc = R 2 /(2D), where D is the diffusion coefficient, depends on the characteristics of the transport of particles in the ambient medium, i.e., on the magnetic field and on the turbulence features. Detailed studies of this subject can be found in, e.g., Jokipii (1966); Giacalone & Jokipii (1999); Casse et al. (2002); Yan & Lazarian (2002); Candia & Roulet (2004); Marcowith et al. (2006). Energy losses during acceleration are generally due to synchrotron radiation, to interactions with the radiative backgrounds, or to hadronic interactions, the latter process being mostly inefficient in diluted astrophysical media. The timescale for energy losses through synchrotron emission and pion production can be expressed in a generic way (Biermann & Strittmatter 1987) Uγ/UB corresponds to the ratio of the energy density of radiation leading to pion production Uγ, to the magnetic energy density UB = B 2 /8π. In the central region of an AGN for example, assuming equipartition with the magnetic field (corresponding to the Eddington luminosity), for E20 = E/10 20 eV and BG = B/1G, t rad ∼ 10 5 s E −1 20 B −2 G . This timescale has to be compared to the acceleration timescale which reads (Lemoine & Waxman 2009): tacc = A tL, where tL is the Larmor timescale and A 1 for all types of Fermi acceleration (non, mildly, or ultra-relativistic, 1st and 2nd order Fermi accelerations). For a non relativistic 1st order Fermi acceleration for instance, A ∼ g/β 2 sh and tacc ∼ 10 7 s g E20B −1 G β −2 sh , where the shock velocity β sh 1 and g ≡ D/(rLc) 1. Majoring this timescale with the radiative loss timescale leads to a maximum acceleration energy in the central region of AGN of order: Emax ∼ 10 19 eV g −1/2 B −1/2 G β sh . In the generic case of acceleration in an outflow, Lemoine & Waxman (2009) compare this acceleration time and the dynamical time t dyn ∼ R/βWΓWc of the outflow, to set a robust lower bound on the luminosity that a source must possess in order to be able to accelerate particles up to E = 10 20 eVE20: L > LB ≡ ΓWR 2 B 2 /2 > 10 45 Z −2 E 2 20 erg s −1 . The magnetic luminosity LB of the source is written as a function of the size of the acceleration region R in the observer frame, in motion with Lorentz factor ΓW (and velocity βW) and imparted with a magnetic field of characteristic strength B. This quantity is not straightforward to derive: the classical determination of the field strength using the synchrotron emission (assuming equipartition between the total energy density of non thermal particles and of the magnetic field for example), depends notably on the hardly known ratio between the leptonic and hadronic accelerated particles (e.g., Beck & Krause 2005). In the case of blazars for example, Celotti & Ghisellini (2008) discuss that their jets are not magnetically dominated and that Faranoff-Riley I (FRI) radio galaxies, TeV blazars, and BL Lac objects only possess magnetic luminosities of order 10 42−44 erg s −1 .
It should be noted that the escape of particles from acceleration regions is an intricate issue that has been scarcely discussed in detail in the literature (note however the works of Norman et al. 1995a;Mannheim et al. 2001;Rachen 2008;Allard & Protheroe 2009). Mannheim et al. (2001) and Rachen (2008) argue that one promising way to have high energy particles leave the magnetized acceleration site would be that they be transformed into neutrons. However, it is not obvious that such a scenario could produce a power-law spectrum; the shape of the spectrum depends on the evolution of the optical depth to photo-pion production (see, e.g., Waxman 2001).
In addition to being able to accelerate up to Emax > 200 EeV, candidate UHECR accelerators should have luminosities that can account for the observed fluxes. A simple estimate of the required luminosity can be done assuming that all sources have the same injection spectral index s, a L19 steady luminosity in cosmic rays above E19 ≡ 10 19 eV, and that they are distributed homogeneously in the Universe with a density ns. To account for the observed flux of UHECRs at E19, the main quantity at play, nsL19, scales as: (E 3 dN/dE)E=E 19 ∼ 10 24 eV 2 m −2 s −1 sr −1 (ns/10 −5 Mpc −3 )(L19/10 42 erg s −1 ), for the case of s = 2.3 and Emax = 10 20.5 eV. For reference, the number density of normal galaxies in the Universe today is of order 10 −2 Mpc −3 (Blanton et al. 2003), and it drops to 10 −9 −10 −8 Mpc −3 for Faranoff-Riley II(FRII) type galaxies (Wall et al. 2005). For transient sources, this scaling can be translated into: (ṅs/10 −9 Mpc −3 yr −1 ) (Etot,19/3 × 10 53 ergs), whereṅs is the birth rate of the source and Etot,19 the total injected energy in cosmic rays above E19.
Below we discuss the main astrophysical sites where UHECRs may originate.

Gravitational accretion shocks
The accretion of dark matter and gas produce shocks around the largest structures of the Universe (clusters of galaxies, filaments, walls), where diffusive shock acceleration can happen. For clusters of galaxies, one can estimate the linear extension of the magnetized shock to ∼ 1 − 10 Mpc and the magnetic field intensity on both sides of the shock to ∼ 1 µG (see, e.g., the recent radio detection of synchrotron radiation from bow shocks in a merging cluster by van Weeren et al. 2010, that indicates the presence of micro-gauss level magnetic fields far from the cluster center), which would allow particles to be confined up to E ∼ 10 20 eV. Note however that the strength of the magnetic field upstream of the shock could actually be 1 µG, as it was produced out of the weakly magnetized void; thus shock acceleration can occur only if the magnetic field upstream can be strongly amplified (M. Lemoine and E. Zweibel, private comm., and see tentative amplification mechanisms by Schlickeiser & Shukla 2003;Zweibel & Everett 2010). The detection of very high energy gamma rays from these shocks would enable us to constrain these parameters. Clusters of galaxies have been considered as promising accelerators by various authors: Norman et al. (1995b); Kang et al. (1996Kang et al. ( , 1997; Miniati et al. (2000); Ryu & Kang (2003); Inoue et al. (2005Inoue et al. ( , 2007; Murase et al. (2008a). Recently, Vannoni et al. (2009) performed a detailed time-dependent numerical calculation, including energy losses due to interactions of protons with radiative backgrounds and demonstrated that for realistic shock speeds of a few thousand km/s and a background magnetic field close to 1 µG, the maximum energy achievable by protons cannot exceed a few times 10 19 eV, due to radiative losses.

Active Galactic Nuclei
AGN are composed of an accretion disk around a central super-massive black hole and are sometimes associated with jets terminating in lobes (or hot spots) which can be detected in radio. One can classify these objects into two categories: radio quiet AGN with no prominent radio emission or jets and radio loud objects presenting jets. Both categories could in principle accelerate particles in their nuclei: for a black hole of mass M bh ∼ 10 9 M , the equipartition magnetic field in the central region yields B ∼ 300 G. Assuming the central region to be of order R ∼ 100 A.U., particles could be confined up to Emax ∼ 150 EeV and accelerated to E 10 20 eV by electrostatic acceleration in the black hole magnetosphere (e.g., Boldt & Ghosh 1999). This energy is hardly reached by particles in practice due to energy losses that they experience in this dense region (see above and Norman et al. 1995a;Henri et al. 1999;Rieger & Mannheim 2000). Radio loud galaxies could also accelerate particles in their inner jets, where one possible mechanism at play could be shear acceleration (see end of Section 5.1 and e.g. Rieger et al. 2007). The quantity B R ∼ 0.3 G pc for the jets of M bh ∼ 10 9 M , leading to Emax ∼ 300 EeV, but the acceleration is limited by photo-interactions and adiabatic losses making the escape of UHE particles non trivial (Mannheim 1993). The most powerful radio-galaxies (Faranoff-Riley II galaxies and their associated Flat Spectrum Radio Quasars, noted FSRQ) present hot spots and bow shocks, formed at the termination of the jets by interaction with the intergalactic medium. For these regions, the same estimates for particle confinement energies as in jets can be found. For hot spots, shock acceleration and escape should be easier than in the jet (see, e.g., Rachen & Biermann 1993), but the acceleration in the bow shock is non trivial (Berezhko 2008). UHECR acceleration in AGN should lead to particular signatures in the gamma-ray spectrum of these sources, through various emissions such as proton synchrotron, photo-hadronic interactions that induce pair cascades, muon synchrotron etc. With more gamma-ray data on each source over a wide energy range (with, e.g., the future Cherenkov Telescope Array, or CTA), one could distinguish these hadronic signatures from leptonic acceleration ones and probe the UHECR acceleration in AGN (Hinton & Hofmann 2009). As discussed above, Lemoine & Waxman (2009) point out that only FRII/FSRQ radio-galaxies with magnetic luminosity LB 10 45 erg s −1 (following the modeling of Celotti & Ghisellini 2008) meet the energetic requirements to accelerate particles to the highest energies. Under this assumption, one may argue that the local FRII galaxies are well known and do not seem to correlate with the arrival direction of the highest energy events. Possible reasons for this could be that the extragalactic magnetic fields are stronger than expected, and/or that cosmic rays are heavy nuclei. AGN are often suggested as continuous emitters of UHECRs, however transient events such as AGN flares more easily meet the UHECR acceleration requirements (Farrar & Gruzinov 2009). Other transient models for UHECR acceleration are discussed next.

Gamma-ray Bursts
The explosion of a GRB leads to the formation of multiple shock regions which are potential acceleration zones for UHECRs. The value of the magnetic field at these shocks is estimated to be of order B ∼ 10 6 G at a distance R ∼ 10 12 cm from the center. These values are derived for internal shocks that happen before the ejected plasma reaches the interstellar medium, assuming B ∼ 10 12 G near the central engine (of size R ∼ 10 km) and an evolution B ∝ R −1 . The wide green region presented in Figure 11 stems from this dependency: parameters can take different values at different times of the GRB explosion. These objects have been examined by various authors as possible sources of UHECRs (Waxman 1995(Waxman , 2001Vietri 1995;Gialis & Pelletier 2003;Murase et al. 2006Murase et al. , 2008b. These authors invoke Fermi processes at external shocks (Vietri 1995), at mildly relativistic internal or reverse shocks (Waxman 1995(Waxman , 2001Murase et al. 2008b), or 2nd order Fermi processes through multiple interactions with mildly relativistic internal shocks (Gialis & Pelletier 2003). Overall, models allow acceleration up to ∼ 10 20 eV provided that some assumptions on the source parameters are verified (e.g., magnetic field strength, turbulence, etc.). The flux of gamma-rays reaching the Earth from GRBs is generally comparable to the observed flux of UHECRs, implying a tight energetic requirement for GRBs to be the sources of UHECRs. With a GRB rate of ∼ 0.3 Gpc −3 yr −1 at z = 0, it can be calculated that the energy injected isotropically (regardless of beaming) in UHECRs is of order EUHECR 10 53 erg, (Guetta & Piran 2007;Zitouni et al. 2008;Budnik et al. 2008). Note also that the transient nature of these objects could possibly explain the lack of powerful counterparts correlating with the arrival direction of the highest energy cosmic rays.

Neutron Stars
Neutrons stars -young millisecond magnetars to be precise, easily fulfill the Hillas criterion, and might prove to be very good candidate sources, though they are scarcely discussed in the literature. Magnetars are neutron stars with extremely strong surface dipole fields of order 10 15 G (see Wood & Thompson 2004, Harding & Lai 2006 and Merghetti 2008 for reviews). Blasi et al. (2000) studied the possibility that UHE-CRs are accelerated through unipolar induction in the relativistic winds for rapidly rotating magnetars, building up on previous constraints by Venkatesan et al. (1997). The maximum energy reached by particles injected by these objects is very promising (see Section 5.2). Arons (2003) further developed the model and found that only 5% of the extragalactic magnetar population need to be fast-rotators to account for the observed UHECR energetics. Magnetars, as GRBs, are transient sources and should not be observed in coincidence with UHECR arrival directions. The possibility of injecting large proportions of heavy nuclei into an acceleration region may be more easily met by young neutron stars than alternative models, due to their iron rich surface and early environment.

Cosmic Ray Astronomy at Ultrahigh Energies
One of the most puzzling facts concerning UHECRs is the absence of clear sources in the arrival directions of the highest energy events. Indeed, if sources are powerful astrophysical objects, one would expect to see a counterpart in the arrival direction at the highest observed energies. At trans-GZK energies, current upper limits on the strength of cosmic magnetic fields suggest that particles should not be deflected by more than a few degrees (unless they are heavy nuclei), thus some correlation should exist with the underlying baryonic matter.
As a result, many authors have searched for correlations between existing data and astrophysical object catalogs. For example, Tinyakov & Tkachev (2001; Gorbunov et al. (2002); Gorbunov & Troitsky (2005) found a correlation between HiRes event directions and BL Lac catalogs, that has been much debated (Evans et al. 2002(Evans et al. , 2004Tinyakov & Tkachev 2004). Stanev et al. (1995) noted an association between arrival directions of UHECRs and the supergalactic plane that was not confirmed by AGASA results (Takeda et al. 1998) and that seems to have reappeared in the Auger results (Stanev 2009). The latest correlation result concerns the highest energy events (E > 55 EeV) detected by the Auger Observatory and AGN within distance < 75 Mpc (Abraham et al. 2007(Abraham et al. , 2008aAbreu et al. 2010).
It must be underlined that the AGN correlating with the Auger events are mostly not very powerful Seyfert type galaxies, and are thus not favored as accelerators of UHECRs. Hence, what should be retained from that correlation is that it brings to light the hint of an anisotropic distribution of events at the highest energy with 99% CL, its most reasonable interpretation then being that events trace the large scale structures along which AGN are distributed. Another possible interpretation is that Auger may be observing in part the last scattering surface of UHECRs rather than their source population (Kotera & Lemoine 2008b). The possibility that this 'fake correlation' effect could play a non negligible role was shown numerically by Ryu et al. (2010).
Another explanation to the absence of counterparts in the arrival direction of UHECRs could also reside in the very nature of the sources. The delay induced by extragalactic magnetic fields of mean strength B and coherence length λB on particles of charge Z and energy E with respect to photons over a distance D reads (Alcock & Hatchett 1978): (1) For intergalactic magnetic fields of lower overall strength (B 10 −12 G), this formula indicates that the time delay be shorter than a year over 100 Mpc. However, the crossing of one single magnetized filament (sizeri and field strength B) will lead to a slight deflection that induces a time delay with respect to a straight line of order (Alcock & Hatchett 1978;Harari et al. 2002b): δti 0.93 × 10 3 yr (ri/2 Mpc) 2 (Bi/10 −8 G) 2 (λi/0.1 Mpc)(E/10 20 eV) −2 .
For transient sources like GRBs, neutron stars, or AGN flares which have an activity timescale δt, this delay is sufficient to erase any temporal coincidence between UHECRs and their progenitors (Vietri 1995;Waxman 1995). Because these bursts are fairly rare in our local Universe (where observed UHECRs originate), one only expects to see particles from a particular bursting source if their arrival times are spread out over σt 10 3 yrs at 10 20 eV. This spread should be particularly increased in the presence of magnetized scattering centers between the source and the observer. Kotera & Lemoine (2008b) and Kalli et al. (2011) discuss that this effect could induce an artificially enhanced correlation of UHECR arrival directions with the foreground matter density, which could be measured to identify the transient nature of the sources.
As discussed in Section 2.3, UHECR sky anisotropies and their composition are tightly connected. In particular, if an anisotropy signal is measured above an energy E thr assuming that it is produced by heavy nuclei of charge Z, one expects an anisotropy signal to be also present at energy > E thr /Z due to the proton component, depending on the proton to heavy nuclei ratio qp/qZ injected at the source (Lemoine & Waxman 2009). The evaluation of the anisotropy signals at both high and low energies should thus place constraints on the ratio qp/qZ .

Multi-messenger approach
Secondary neutrinos and photons can be produced by UHECRs when they interact with ambient baryonic matter and radiation fields inside the source or during their propagation from source to Earth. These particles travel in geodesics unaffected by magnetic fields and bear valuable information of the birthplace of their progenitors. The quest for sources of UHECRs has thus long been associated with the detection of neutrinos and gamma rays that might pinpoint the position of the accelerators in the sky.
The detection of these particles is not straightforward however: first, the propagation of gamma rays with energy exceeding several TeV is affected by their interaction with CMB and radio photons. These interactions lead to the production of high energy electron and positron pairs which in turn up-scatter CMB or radio photons by inverse Compton processes, initiating electromagnetic cascades. As a consequence, one does not expect to observe gamma rays of energy above ∼ 100 TeV from sources located beyond a horizon of a few Mpc (Wdowczyk et al. 1972;Protheroe 1986;Protheroe & Stanev 1993). Above EeV energies, photons can again propagate over large distances, depending on the radio background, and can reach observable levels around tens of EeV (Lee 1998). Secondary neutrinos are very useful because, unlike cosmic-rays and photons, they are not absorbed by the cosmic backgrounds while propagating through the Universe. In particular, they give a unique access to observing sources at PeV energies. However, their small interaction cross-section makes it difficult to detect them on the Earth requiring the construction of km 3 or larger detectors (see, e.g., Anchordoqui & Montaruli 2010).
Secondary neutrinos and gamma-rays generated at UHECR sources have been investigated by a number of authors (Szabo & Protheroe 1994;Rachen & Mészáros 1998;Waxman & Bahcall 1999;Mücke et al. 1999;Mücke et al. 2000;Anchordoqui et al. 2008;Kachelrieß et al. 2008;Ahlers et al. 2009;Allard & Protheroe 2009;Mannheim et al. 2001) with particular focus on emissions from AGN and transient sources such as GRBs. The case of cluster accretion shocks has been studied by Inoue et al. (2007) and Murase et al. (2008a), and transient sources have been examined in details by Waxman & Bahcall (2000); Dai & Lu (2001); Dermer (2002); Murase et al. (2006Murase et al. ( , 2008b, and Murase & Ioka (2008) for GRBs and by Murase et al. (2009) for magnetars. The normalization and the very existence of these secondaries highly depend on assumptions about the opacity of the acceleration region and on the shape of the injection spectrum as well as on the phenomenological modeling of the acceleration. For instance, Waxman & Bahcall (1999) obtain an estimate for the cosmic neutrino flux, by comparing the neutrino luminosity to the observed cosmic ray luminosity, in the specific case where the proton photo-meson optical depth equals unity. If the source is optically thick, Allard & Protheroe (2009) demonstrate that cosmic rays are not accelerated to the highest energies and neutrinos above E ∼ EeV are sharply suppressed.
The existence of secondaries from interactions during the propagation of cosmic rays is less uncertain, but it is also subject to large variations according to the injected spectral index, chemical composition, maximum acceleration energy, and source evolution history. The magnetic field in the source environment, especially in clusters of galaxies, can play an important role by confining the charged UHECRs and thus leading to increased interaction probabilities (Berezinsky et al. 1997;Colafrancesco & Blasi 1998;Rordorf et al. 2004;de Marco et al. 2006;Armengaud et al. 2006;Murase et al. 2008a;Wolfe et al. 2008;Kotera et al. 2009). A number of authors have estimated the cosmogenic neutrino flux with varying assumptions (e.g., Engel et al. 2001;Ave et al. 2005;Seckel & Stanev 2005;Hooper et al. 2005;Berezinsky 2006;Stanev et al. 2006;Allard et al. 2006;Takami et al. 2009;Kotera et al. 2010b). Figure 12 summarizes the effects of different assumptions about the UHECR source evolution, the Galactic to extragalactic transition, the injected chemical composition, and Emax, on the cosmogenic neutrino flux. It demonstrates that the parameter space is currently poorly constrained with uncertainties of several orders of magnitude in the predicted flux. UHECR models with large proton Emax(> 100 EeV), source evolution corresponding to the star formation history or the GRB rate evolution, dip or ankle transition models, and pure proton or mixed 'Galactic' compositions are shaded in grey in Figure 12 and give detectable fluxes in the EeV range with 0.06 − 0.2 neutrino per year at IceCube and 0.03 − 0.06 neutrino per year for the Auger Observatory. If EeV neutrinos are detected, PeV information can help select between competing models of cosmic ray composition at the highest energy and the Galactic to extragalactic transition at ankle energies. With improved sensitivity, ZeV (=10 21 eV) neutrino observatories, such as ANITA and JEM-EUSO could explore the maximum acceleration energy.
The detectability of photons from the electromagnetic cascade triggered by pion production interactions has been addressed by several groups (e.g., Lee 1998;Ferrigno et al. 2004;Armengaud et al. 2006;Gelmini et al. 2007;Kotera et al. 2010a). The dilution of the cascaded signal -due to the deflection of the electrons and positrons generated during the cascade -depends on the assumptions made regarding the configuration of the EGMF. More specifically, the gamma-ray flux scales as the fraction of the line of sight in which the magnetic field is smaller than the value B θ such that the deflection of the low energy cascade is θ (Kotera et al. 2010a). For reference, B θ 2 × 10 −14 G for θ = 1 • . One can refer to Figure 9 for numerical estimates of the magnetic filling factor to arrive at the appropriate fraction for a specific line of sight and source location. Even under optimistic assumption on the magnetic field configuration, only sources with extremely high luminosities LE,19 3 × 10 44 erg s −1 (d/100 Mpc) −2 and LE,19 10 43 erg s −1 (d/100 Mpc) −2 for E > 10 19 eV could be detected by current instruments such as H.E.S.S. and by the future CTA respectively, with fluxes of order ∼ 10 −10 GeV cm −2 s −1 around 1 − 10 TeV (Ferrigno et al. 2004;Kotera et al. 2010a). Gabici & Aharonian (2005, 2007 argued that one should rather search for the GeV photons emitted by the synchrotron radiation of the secondary electrons, in presence of substantial magnetic fields in the source environment. Again, only the cases of rare powerful sources with cosmic ray luminosity LE,19 > 10 44−46 erg s −1 are promising in terms of detectability with both current and up-coming instruments. A source with a cosmic ray Cosmogenic neutrino flux for all flavors, for different UHECR parameters compared to instrument sensitivities. Pink solid line corresponds to a strong source evolution case (FRII evolution, see Wall et al. 2005) with a pure proton composition, dip transition model, and Emax = 3 ZeV. Blue lines correspond to uniform source evolution with: iron rich (30%) composition and E Z,max < Z 10 EeV (dotted line) and pure iron injection and E Z,max = Z 100 EeV (solid). Grey shaded range brackets dip and ankle transition models, with evolution of star formation history for z < 4, pure proton and mixed 'Galactic' compositions, and large proton Emax(> 100 EeV)). Including the uniform source evolution would broaden the shaded area down to the black solid line. Current experimental limits (solid lines) assume 90% confidence level and full mixing neutrino oscillation. The differential limit and the integral flux limit on a pure E −2 spectrum (straight line) are presented for IceCube 22 lines (pale blue, Abbasi et al. 2010), ANITA-II (green, Gorham et al. 2010) and Auger South (red, Abraham et al. 2009a). For future instruments, we present the projected instrument sensitivities (dashed lines) for IceCube 80 lines (pale blue, acceptances from S. Yoshida, private communication, see also Karle 2010), and for JEM-EUSO (purple, Takahashi et al. 2009). luminosity of LE,19 ∼ 10 44 erg s −1 located at a distance d ∼ 100 Mpc nearly overshoots the observed cosmic ray spectrum and is thus marginally excluded. Farther sources, with higher luminosity (e.g. LE,19 = 10 46 erg s −1 at d = 1 Gpc) would thus be more promising to observe in gamma-rays. Such distant sources would contribute to about 10% of the observed spectrum of UHECRs up to E ∼ 10 19 eV, and the cosmic rays produced with higher energy would not reach the Earth due to energy losses.
Finally, sources located at a distance 10 Mpc accelerating UHECRs should produce ultrahigh energy photons during their propagation, that can reach the Earth before experiencing Compton cascading. Taylor et al. (2009) studied this potential signature in the particular case of our closest radio-galaxy Cen A (3.8 Mpc) and concluded that Auger should be able to detect 0.05 − 0.075 photon per year from Cen A, assuming that it is responsible for 10% of the cosmic ray flux above 60 EeV, and assuming a 25% efficiency for photon discrimination.
One last messenger that is scarcely discussed in relation to UHECRs is gravitational waves. If anisotropy signals reveal that the source is of the transient type, one way to establish if UHECR sources are GRBs or magnetars would be to look for gravitational waves produced by the latter, as the former are believed to produce only faint signals below detectability (e.g., Piran 2004). Kotera (2011) shows that the distribution of magnetar initial voltages required to reconcile the produced spectrum to the observed one, should lead to higher stochastic gravitational wave signals from these objects than previously calculated (e.g., Regimbau & Mandic 2008). The observation of such a gravitational wave signal could be a probe that these objects meet the requirements (in terms of magnetization, rotation velocity, inertial momentum) to accelerate UHECRs to the highest energies.
It should be highlighted that due to the delay induced by EGMF on charged cosmic rays (see Section 6.2), secondary neutrinos, photons, and gravitational waves should not be detected in time coincidence with UHECRs if the sources are not continuously emitting particles, but are transient such as GRBs and young magnetars.

The Search for Ultrahigh Energy Cosmic Ray Sources
The resolution of the long standing mystery of the origin of ultrahigh energy cosmic rays will require a coordinated approach on three complementary fronts: the direct ultrahigh energy cosmic ray frontier, the transition region between the knee and the ankle, and the multi-messenger interface with high-energy photons and neutrinos.
The most direct route to a resolution of this open question would be a precise measurement of the three pillars of UHECR observations: spectrum, anisotropies, and composition. The spectrum is much better measured today than just a few years ago and will certainly improve in the years to come with current observatories. The precise shape and energy scale of the ankle and the cutoff are excellent selectors of models. For instance, a possible recovery at the highest energies will clearly show the GZK nature of the cutoff as opposed to a cutoff caused by the maximum acceleration energy. A precise energy scale for the ankle will test a propagation dip versus a Galactic to extragalactic cosmic ray transition.
More discriminating than the precise shape of the spectrum, but much harder to plan for success, is a clear observation of anisotropies. A nearby source, the first UHECR source, will clearly be a watershed in the field. A clear correlation with the large scale structure within 200 Mpc will also clear the path to zeroing in at the possible accelerators. The most difficult but key observable of the three pillars is a clear composition measurement. The dependence on hadronic models to translate shower properties into composition measurements make it difficult to reach clear conclusions. Great progress in this arena can be done at the transition region between the knee and the ankle, where the LHC has a direct access to the energy scale. Progress on enlarging the range of cosmic ray measurements to higher energies from the knee (Kascade-Grande) and to lower energies from the ankle (Auger HEAT and AMIGA, and TALE) will help construct a unified model of the cosmic ray properties in a region that hadronic models can be tested at the LHC 5 .
A clear anisotropy determination, especially above 60 EeV, will help determine the composition astrophysically. At these energies possible composition mixtures simplify tremendously for sources above tens of Mpc into two options: protons or iron-like. Iron-like nuclei will have much broader spread in arrival directions due to known magnetic fields erasing small scale anisotropies and opening the possibility of a clear signature of proton primaries for some anisotropic patterns. An astrophysically determined primary composition will open a fruitful avenue to compare interaction models with observed shower properties at energies orders of magnitude above the reach of laboratory accelerators.
The spectrum and composition between the knee and the ankle should signal the transition from Galactic to extragalactic cosmic rays. A clear observation of a second knee or composition studies around the ankle can produce clear signatures of the transition. Anisotropies are not expected at these energies unless neutrons manage to escape acceleration sites. Progress in this energy range will also come from direct gamma-ray observations of cosmic accelerators (e.g., by Fermi, CTA, and HAWC) and their deeper understanding. Neutrinos in this energy range can also be crucial if sources can be identified or if cosmogenic neutrinos are observed in the PeV range (e.g., by IceCube or the future KM3NeT).
Observations of photons and neutrinos at ultrahigh energies will be extremely useful in distinguishing proposed scenarios (e.g., by IceCube, Auger, JEM-EUSO, and ANITA). Photons will only reach us from nearby accelerators, so they can be directly connected to acceleration sites. Neutrinos may be observed from nearby sources or from the diffuse background expected from the propagation of UHECRs from cosmologically distant objects. Having both a nearby view of the apparently brightest accelerators as well as the integrated flux from more distant objects will strongly constrain possible candidates.
As an exercise, we end by contrasting two of the many possible outcomes of future observations. In outcome A, the large scale structure correlation is confirmed above the 5σ level showing that UHECR sources exist within the GZK sphere and that their sky position is not smeared by more than a few degrees. These observations will imply that: 1) UHECR sources are more common than clusters of galaxies or powerful blazars, and 2) the composition above 60 EeV should be dominated by protons. If shower properties continue to show a trend toward iron-like behavior, it is likely to be due to changes in hadronic interactions, given that at 60 EeV only iron or protons arrive from sources further than tens of Mpc. This scenario is rich of multi-messengers: ultrahigh energy photons from nearby sources and neutrinos from sources and the diffuse background should be observable. The type of galaxies that UHECR events correlate with may signal an origin in active galaxies, encouraging models based on central supermasive black holes. If starburst galaxies correlate better, magnetars and GRBs would be better candidate sources. Protons above 60 EeV do not prevent a mixed composition at EeV, so measurements of the composition at the ankle will help determine the injected composition and source spectral index.
In case B, above 60 EeV the spectrum does not show a recovery, anisotropies are not observed even after a significant increase in statistics, and shower properties indicate ironlike primaries. In this case, the maximum energy of the accelerators are likely to be below GZK energies for protons and the spectrum cutoff is likely to be a combined effect of the maximum iron energy and the GZK effect. This coincidence is not an elegant solution but it is a clear possibility. A heavy composition at injection is more natural for models based on magnetars, while scenarios based on AGN and GRBs need to be modified to account for the suprisingly heavy composition. In this scenario, cosmogenic neutrinos and photons will not be easily detected leaving only the hope of observing a nearby source or of major technological advances to reach down orders of magnitude in flux.
Great progress in the ultrahigh energy cosmic ray frontier may lead to a completely different outcome than our speculative exercise above, but that will require a significant increase in statistics at trans-GZK energies. Current data suggest that watershed anisotropies will only become clear above 60 EeV and that very large statistics with good angular and energy resolution will be required. Auger will add 7 × 10 3 L per year in the South while TA will add about 2 × 10 3 L per year in the North. Current technologies can reach a goal of another order of magnitude if deployed by bold scientists over very large areas (e.g., Auger North). New technologies may ease the need for large number of detector units to cover similarly large areas. Future space observatories (e.g., JEM-EUSO, OWL, Super-EUSO) promise a new avenue to reach the necessary high statistics especially if improved photon detection technologies are achieved. With a coordinated effort, the next generation observatories can explore more of the ∼ 5 million trans-GZK events the Earth's atmosphere receives per year.