Ultrahigh energy cosmic rays and the GeV-TeV diffuse gamma-ray flux

Oleg E. Kalashev,1 Dmitry V. Semikoz,1,2 and Gunter Sigl2,3

1INR RAS, 60th October Anniversary pr. 7a, 117312 Moscow, Russia2APC, 10, rue Alice Domon et Leonie Duquet, Paris 75205, France

3II. Institut fur theoretische Physik, Universitat Hamburg, Luruper Chaussee 149, D-22761 Hamburg, Germany(Received 19 April 2007; revised manuscript received 15 December 2008; published 18 March 2009)

Ultra-high energy cosmic rays accelerated in astrophysical objects produce secondary electromagnetic

cascades during propagation in the cosmic microwave and infrared backgrounds. We show that if the

primary cosmic rays are dominated by protons, such cascades can contribute between ’ 1% and ’ 50% of

the GeV-TeV diffuse photon flux measured by the EGRET experiment. The Fermi Gamma-ray Space

Telescope (GLAST) should have a good chance to discover this flux. If most ultra-high energy cosmic

rays consist of heavy nuclei, the secondary GeV-TeV diffuse photon flux can be lower by factors of several

compared to pure proton primaries from the same sources. The diffuse photon flux then depends on the

unknown composition of accelerated particles, their maximum energy, and the source distribution.

DOI: 10.1103/PhysRevD.79.063005 PACS numbers: 98.70.Sa, 98.70.Vc

I. INTRODUCTION

Recently the HiRes collaboration established [1] theexistence of the Greizen-Zatsepin-Kuzmin (GZK)cutoff[2] which was independently confirmed by the PierreAuger Observatory experiment [3]. This result suggestsan astrophysical origin of ultra-high energy cosmic rays(UHECRs). UHECR interactions with the cosmic micro-wave background (CMB) and other low-energy photonbackgrounds produce secondary photons and pairs. Aninterplay of pair production, ��b ! eþe�, and inverseCompton scattering, e� þ �b ! e� þ �, with low-energyphotons �b then leads to electromagnetic (EM) cascades.These cascades effectively shift the flux of high-energyphotons to below the pair production threshold on the CMBat ’ 1015 eV. Cascading on the infrared/optical back-ground tends to shift the photon flux further down toenergies in the GeV-TeV region [4,5]. At multi-GeV en-ergies this gives rise to a diffuse photon background thatcan be detected with satellite based detectors such as theFermi Gamma-ray Space Telescope (GLAST).

There are two important contributions to secondary EMcascades from UHECRs. One comes from the GZK pro-cess of pion production in interactions of UHECR nucleiðA; ZÞ with CMB photons, ðA; ZÞ þ � ! ðA; Z� 1Þ þ�0ð��Þ. Most of the energy transferred to photons, elec-trons and positrons in the subsequent pion decays wouldcascade down to GeV-TeVenergies, at which the Universeis transparent to photons. If the spectrum of primaryUHECRs is a power law / E��, pion production energylosses and thus the energy deposited into EM cascadesincreases with decreasing power law index �.

The second source of EM cascades is pair production byUHECRs on low-energy photons, ðA; ZÞ þ � ! ðA; ZÞ þeþ þ e�. This process is more efficient for steeper injec-tion spectra with larger �. Both processes together imply aminimal secondary photon flux at GeV-TeV energies.

Apart from pion and pair production, nuclei primaries inthe UHECR flux can produce photons also by the photo-disintegration of a nucleus followed by deexcitation of thedaughter. This process contributes less than 10�3 to thediffuse flux measured by EGRET [6] and can thus beneglected in this context [7].The goal of the present work is to study the contribution

of EM cascades from UHECR interactions with back-ground photons to the diffuse �-ray background in astro-physical scenarios. We study the parameter space ofUHECR models which fit the HiRes energy spectrumwith the GZK cutoff. We show that the UHECR contribu-tion to the �-ray flux is in the range ’ 0:5%–50% of thediffuse flux measured by EGRET [6]. Relatively highvalues of this flux should enable the GLAST satellite [8]to disentangle it from other contributions such as fromstarforming galaxies [9], starbursts [10], large scale struc-ture formation shocks [11], active galactic nuclei (AGNs)[12], blazars [13], and �-ray bursts [14].

II. MODELING THE PRIMARY PROTON FLUX

We first consider proton primaries. Heavier nuclei pri-maries will be discussed in Sec. IV. We assume spatiallyuniform proton injection spectra which we parametrize asdN=dE / E���ðEmax � EÞ, where Emax is the maximalproton energy and � is the power law index for whichwe consider the ranges 2� 1020eV � Emax � 1021 eVand 2 � � � 2:7, respectively. Note that this effectiveinjection spectrum can be the result of averaging oversources with variable maximal energies and injectionpowers [15].We assume the injection rate per energy and comoving

volume to scale as �ðE; zÞ / ð1þ zÞm�ðzmax � zÞ�ðz�zminÞ, where m parametrizes the luminosity evolution forwhich we consider the range�2 � m � 4. The parameterszmin and zmax are the redshifts of the closest and most

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distant sources, respectively. We choose zmin � 0:01 toavoid a GZK cutoff more pronounced than observed[1,3]. We fixed zmax ¼ 3 which is sufficiently large totake into account cosmologically distant sources. Wenote that both the luminosity evolution index m and themaximum redshift zmax have similar qualitative influenceon both proton spectrum and the resulting GeV-TeV cas-cade fluxes.

We furthermore considered two special cases for sourceevolution, motivated by the observed redshift evolution ofactive galactic nuclei (AGN) and of star formation rates,respectively. For AGN sources, we used as an example theevolution of Ref. [16], where the injection rate scales as/ ð1þ zÞ3 up to z ¼ 1:7, stays constant up to z ¼ 3 andexponentially decays at larger redshifts. For sources fol-lowing the star formation rate we adopted the recent modelof Ref. [17], in which evolution approximately scales as/ ð1þ zÞ4:6 up to z ¼ 1, followed by / ð1þ zÞ1:7 up toz ¼ 3, / ð1þ zÞ�3:1 up to z ¼ 6, and ending with a fastdropping tail at z > 6. Both these source evolution scenar-ios give final GeV-TeV flux within the range of evolutionparameters considered in this paper (see Fig. 5).

For the rectilinear propagation of protons and cascadesof secondary electrons, positrons and photons we used twoindependent codes [18,19]. Both codes perfectly agree onthe proton flux and practically agree on the neutrino flux.Photon fluxes agree within 50% which is within signifi-cantly larger uncertainties from unknown astrophysicalparameters such as redshift evolution. An additional diffi-culty comes from the fact that an unknown fraction of thephoton flux is shifted below the lowest energy bin due tointeractions with the infrared/optical background. For theresults presented in this paper we used the code [19], forwhich global numerical energy nonconservation is smallerthen 1% if the CMB background is used.

For pion production by protons and neutrons both codesuse the SOPHIA generator [20]. All decay products of pionand neutron decay are taken into account. In addition,protons lose energy due to production of eþe� pairs, whileneutrons decay. Secondary photons produce single anddouble pairs on the low-energy photon background.Electrons and positrons interact via inverse Compton scat-tering and triplet pair production, and undergo synchrotronenergy losses in extragalactic magnetic fields. Because ofall those reactions secondary photons, electrons, and posi-trons cascade down to TeV energies, where they are notaffected by interactions with the CMB anymore. However,even at these energies they can still interact with infrared(IR) and optical photon backgrounds. We use the recentmodel of Ref. [21] for these backgrounds.

We note that highly structured sources and large scalemagnetic fields can lead to enhanced synchrotron fluxesfrom discrete sources up to TeVenergies [22], provided theelectrons in the EM cascades are not significantly deflectedin extragalactic magnetic fields [23]. Also, such enhance-

ments are important only for discrete sources becauseinhomogeneities are not expected to have a strong effecton diffuse fluxes at GeV-TeVenergies which are dominatedby sources at large distances, up to redshifts z > 3. On theother hand, extragalactic magnetic fields can have consid-erable influence on the ultra-high energy photon flux atE> 1018 eV which is created over distances & 50 Mpc,see Refs. [24,25]. Since in the present paper we are mostlyinterested in the diffuse photon flux below TeV energies,we neglect effects of structured extragalactic magneticfields in the following.Throughout this paper we will assume a flat, �CDM

universe for which the Hubble rate HðzÞ at redshift z inthe matter dominated regime, z & 103, is given by

HðzÞ ¼ H0½�mð1þ zÞ3 þ���1=2. We use the standardvalues being �m ¼ 0:3, �� ¼ 0:7, and H0 ¼h0100 km s�1 Mpc�1 with h0 ¼ 0:7. The transport equa-tions with the relevant source terms are then integratedover redshift taking into account the geometrical factorsfrom cosmology. When energy losses other than redshiftare negligible, the differential flux jðEÞ is related to theinjection flux �ðE; zÞ by

jðE; zÞ ¼Z 1

0dz

4��½ð1þ zÞE; z�HðzÞ : (1)

We fit the HiRes spectrum with the method described inRef. [26]. Among all models characterized by m, �, Emax,zmin, and zmax we choose those which fit the latest HiResspectrum [1] at the 95% confidence level, taking intoaccount empty bins above the highest energy events ob-served, as well as an energy uncertainty �E=E ¼ 17%,which influences the shape of the spectrum around theGZK cutoff.We consider the two main scenarios currently discussed

in the literature for the transition from a cosmic ray fluxdominated by galactic sources to a flux dominated byextragalactic sources: In the ‘‘dip scenario’’ extragalacticprotons dominate over the galactic contribution down to afew 1017 eV and the dip observed in the energy spectrumbetween ’ 1 and ’ 10 EeV is caused by pair production bythese protons [27]. Primary protons with energies E �1018 eV from cosmological distances would not arrive atEarth due to magnetic horizon effects. These effects wouldalso change the shape of the UHECR spectrum [28].Therefore, we require the predicted fluxes in this scenarioto fit the HiRes spectrum only for E � 2 EeV.In the second or ‘‘ankle’’ scenario, the ankle in the

spectrum around ’ 5� 1018 eV is due to a crossoverfrom low-energy galactic to high-energy extragalactic cos-mic rays. Recent versions of this model include a mixedUHECR composition at the highest energies [29]. To beconservative, in this case we fit the HiRes spectrum onlyfor E � 40 EeV.

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III. THE DIFFUSE GEV-TEV �-RAY FLUX

We now discuss the possible range of contributions ofUHECRs to the diffuse �-ray flux in the EGRET band. InFig. 1 we show a scenario where the dip is due to pairproduction by extragalactic protons. The lower red (solid)line was fitted to the HiRes spectrum [1] at energies E �2 EeV and the corresponding EM cascade flux is shown asblue (dash-dotted) line. Figure 1 shows that practically allEM energy ends up in the GeV-TeV region. Up to 20% ofthe energy is shifted to even lower energies. By alsoshowing the proton spectrum (green, dashed line) and thecorresponding cascade flux (magenta, dotted line) whichresult when pair production by protons is neglected, Fig. 1demonstrates that the GeV-TeV �-ray flux is dominated bypair production losses of protons in this scenario.

The flux in energy carried by a differential spectrumjðEÞ is given by

RE2jðEÞd lnE. One can then see from

Fig. 1 that the UHECR energy lost to pair production byprotons (energy flux difference between the red, solid andthe green, dashed line) appears as the energy flux differ-ence between the blue, dash-dotted and the magenta, dot-ted line, up to the flux of electron-positron pairs which isnot shown. Furthermore, the UHECR energy lost to pionproduction by protons (energy flux difference between thetwo red, solid lines) appears as the energy flux in the

magenta, dotted line, up to the neutrino flux which is notshown.We also note that the power law index � of the injection

spectrum / E�� and the source evolution parameter m aredegenerate in this model (see, for example, Ref. [26]). Thefit to HiRes data of the proton flux with parameters � ¼2:6 and m ¼ 0, presented in Fig. 1 would be as good as thefit with � ¼ 2:45 and m ¼ 3. However, in the latter casethe diffuse photon flux would be much larger, see Fig. 3below.In Fig. 2 we show a scenario where the UHECR flux is

extragalactic only above the ankle. In the particular case� ¼ 2 one can see the relation between the energy depos-

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m ¼ 0, between zmin ¼ 0 and zmax ¼ 3. In this case the UHECRflux is dominated by extragalactic protons down to 1018 eV [27].The lower red (solid) line is the proton flux, the blue (dash-dotted) line is the corresponding secondary �-ray flux. The green(dashed) line is the proton flux without e� production and themagenta (dotted) line is the corresponding �-ray flux. TheUHECR flux observed by HiRes [1] (magenta error bars) andtwo estimates of the extragalactic diffuse �-ray backgrounddeduced from EGRET data are also shown as blue (higher)[41] and red (lower) crosses [6].

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FIG. 3 (color online). Dependence on redshift evolution indexm of the minimal and maximal fractional contribution ofUHECR proton interactions to the EGRET flux between 1 and2 GeV. Blue, dashed lines are for fitting above 2 EeV (dipscenario of the type shown in Fig. 1) and red, solid lines arefor fitting the UHECR spectrum above 40 EeV (ankle scenario ofthe type shown in Fig. 2).

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ited in GeV-TeV photons and the energy lost by protonsmore directly. For this purpose we artificially continue theproton flux to low energies where it has the same order ofmagnitude as the secondary photon flux. Since protons losesimilar amounts of energy to pion and to pair production inthis scenario, the contribution of these processes to theGeV-TeV �-ray flux is also comparable, contrary to thedip scenario of Fig. 1.

Note that at energies* 1019 eV, our simulations predictphoton fractions of the UHECR flux at the per mille level,especially for relatively hard proton injection spectra. Suchphoton fractions are consistent with current upper limitsand are expected to be detectable within a few years[30,31].

In Fig. 3 we show the range of possible contributions ofUHECR interactions to the EGRET flux, as a function ofredshift evolution index m. We express this as the fractionof integral fluxes between 1 and 2 GeV, the energy band inwhich the EGRET energy flux E2jðEÞ is minimal. Figure 3shows that the diffuse �-ray flux in the EGRET bandstrongly depends on the source luminosity evolution indexm. In the ankle scenario, for a given value of m, it stilldepends on other parameters within a factor 3, whereas inthe dip scenario the scatter is much smaller due to partialdegeneracy between m and �. All realistic astrophysicalsource distributions have m � 0, which implies that thecontribution of secondary photons from UHECRs will beat least ’ 1% if UHECRs are proton dominated. Forstronger evolution, m ¼ 3� 4, this fraction increases tomore than 50% of the EGRET flux. Finally, for the case ofzmax ¼ 4, � ¼ 2:35, and m ¼ 4 one can completely satu-rate the EGRET flux with secondary photons produced byUHECRs. However, we are not aware of any astrophysicalobjects with such strong evolution.

In Fig. 4 we show the range of possible contributions ofUHECR interactions to the EGRET flux, as a function ofthe UHECR injection power law index �. Contrary to the

case of Fig. 3, the scatter is larger, especially for smallvalues of � ¼ 2� 2:4. This is due to the strong depen-dence of the flux in the EGRET band on the value of m forany given �, see Fig. 3. The lower lines correspond tominimal values of m ¼ �2, while the maximum is occursform ¼ 4 for� & 2:4, and for smaller values ofm for� *2:4. Other parameter combinations would overproduce thecosmic ray flux below ’ 10 EeV.In Fig. 5 we compare the range of EM cascade fluxes

from UHECR protons with other possible astrophysicalcontributions in the EGRET band. The possible range ofsecondary photons from proton UHECRs is presented bythe light blue shaded band in this figure. Note that most ofthe uncertainty of this flux comes from the unknown sourceevolution. In Fig. 5 we also show the photon fluxes for thetwo specific examples of source evolution discussed inSec. II: UHECR sources with evolution similar to AGNevolution result in diffuse photon fluxes in the dark blueshaded region entitled ‘‘UHECR AGN evolution’’. Thecase where UHECR source evolution traces star formationis presented by the solid red line in Fig. 5.The scatter for a given evolution scenario such as for

AGN type sources is only a factor 2–3, as seen from Fig. 5.In the case where UHECR emission traces star formationthe scatter is practically zero, because there is no freedomin the parameters in this case to fit the UHECR flux. Weremark that dark matter annihilations may also contributeto the diffuse flux [32].For comparison we presented in Fig. 5 also contributions

to the diffuse GeV-TeV background from several astro-

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FIG. 5 (color online). The possible range of UHECR protoninduced cascade fluxes (light shaded band) compared to esti-mated �-ray fluxes directly produced by starforming galaxies[9], starbursts [10], large scale structure formation shocks [11],AGNs [12], and �-ray bursts [14]. The dark shaded band showsthe range of EM cascade fluxes from UHECR sources evolvingas AGNs [16]. The thick solid red line shows the secondaryphoton flux from sources evolving as star formation rates [42].The extragalactic diffuse �-ray background from EGRET is as inFig. 1.

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physical sources studied in the literature, which includes�-rays directly produced by starforming galaxies [9], star-bursts [10], large scale structure formation shocks [11],AGNs [12], and �-ray bursts [14]. For the assumptionsunderlying these scenarios we refer the reader to the re-spective references.

The GeV-TeV cascade flux in scenarios where extraga-lactic cosmic rays dominate down to below the ankle at ’5� 1018 eV would be practically diffuse for an instrumentsuch as Fermi/GLASTwhich will be sensitive to anisotro-pies down to �0:1% [33]. This is because the cosmic rayflux below the ankle is dominated by cosmological sourcesand is isotropic at the percent level, and because large scalemagnetic fields should lead to significant additional iso-tropization of primary protons [34] and secondary pairs.

In contrast, UHECR anisotropies at the 10% level arelikely at energies around or above the GZK cutoff, E>40 EeV, because sources of such UHECRs may have asmall density�10�5 Mpc�3 [35]. Furthermore, at energiesE� 200 GeV–10 TeV the �-ray absorption length in theIR background becomes small compared to the Hubbleradius, but still large compared to the �10 Mpc lengthscales of pion production and EM cascade developmentaround the source. This is also reflected in the amount of�-ray flux suppression in Figs. 1 and 2. Independent of thepoorly known size of deflection, this could lead to consid-erable correlation with nearby UHECR sources, as in thecase where discrete sources emit very high energy �-raysdirectly [36]. In scenarios such as the one shown in Fig. 2,where the secondary cascade flux is dominated by pionproduction due to relatively hard UHECR injection spec-tra, this flux could, therefore, exhibit small scale anisotropyat the 10% level around �200 GeV. Note, however, thatthe average energy flux would be comparable to theUHECR energy flux at E> 40 EeV, and the anisotropiccomponent would thus be about 0.1% of the EGRET flux.This would be at the sensitivity limit of the Fermi/GLASTsatellite but could be detected in principle.

IV. NUCLEI PRIMARIES

Recent results of the Pierre Auger Observatory suggestthat the UHECR flux at energies E> 1019 eV could con-sist of a significant amount of heavier nuclei [37]. In orderto generalize our results to the case of nuclei primaries weused the code of Ref. [38]. Heavy nuclei are attenuatedbasically by two processes: creation of e� pairs and photo-disintegration on the diffuse photon backgrounds. Thelatter is dominated by the giant dipole resonance (GDR).These processes where taken into account in the code. TheGDR simulation, in particular, was based on the GDR crosssection parametrization described in Ref. [39]. Deflectionof nuclei in the magnetic fields was not taken into account.It may lead to additional suppression of the spectrum atenergies below few �1018 eV, which is not crucial for ourpurposes.

However, considering UHECR fluxes with a mixed nu-clear composition adds a large number of new unknownparameters which makes a detailed study of the parameterspace impractical. We, therefore, restrict our study of thediffuse GeV-TeV photon flux from UHECR interactions totwo limiting cases.In the first case we assumed that UHECR sources accel-

erate iron nuclei up to 26� 1021 eV in qualitative agree-ment with the usual assumptions on proton acceleration. Inthis case a big fraction of secondary UHECRs with E>1018 eV are protons from photo-disintegration of nuclei.As a result, the secondary photon flux at GeV-TeVenergiescan reach ’ 10% of the EGRET flux in some cases. Inaddition, any additional contribution of protons and lightnuclei to the primary flux of sources would increase thesecondary photon flux in the EGRET region. This wouldhappen, in particular, in so called mixed compositionmodels [29].As an example we show in Fig. 6 a simulation with iron

primaries with spectrum E�2:1 up to a maximum energyEmax ¼ 26 1021 eV, with the sources evolving as AGNs.One can see that the secondary GeV-TeV photon flux incase of primary iron nuclei is a factor 3–5 lower as com-pared to the case where the same sources emit only primaryprotons, shown in Fig. 5. This can be understood as fol-lows: Since the UHECR spectrum is relatively hard in thisexample, the main source of secondary photons is pionproduction by the nucleons having been knocked out of thenuclei by photo-disintegration, whereas pair production bynuclei is subdominant: Pion production can also occur with

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FIG. 6 (color online). Primary iron nuclei with spectrum E�2:1,maximum energy Emax ¼ 26� 1021 eV and source evolutionsimilar to AGNs give rise to secondary nuclei and protons,whose total flux (solid red line) fits the UHECR flux observedby HiRes [1] at energies above the ankle, E> 3� 1018 eV.Contributions of nucleons and iron are shown as magenta dottedand green dashed lines, respectively. The secondary photon fluxis shown by the blue dashed-dotted line. The extragalacticdiffuse �-ray background from EGRET is as in Fig. 1.

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the nucleons bound in the propagating nuclei, but the crosssection per nucleon is reduced by geometric effects and theeffective energy flux of bound nucleons at nucleon energyE is suppressed by a factor A2�� (�> 2) compared to theenergy flux at the same energy E of the primary nuclei ofmass A and injection spectrum / E��. Secondary nucleonsare, therefore, the main source of photons, but they makeup only a fraction of the total observed UHECR flux at agiven energy in case of heavy primaries. This is obviousfrom Fig. 6 which shows the nucleon and iron contributionas magenta dotted and green dashed line, respectively.

Note that the total extragalactic UHECR flux fits theobserved UHECR spectrum only above the ankle, E> 3�1018 eV in Fig. 6 (red solid line). This is an importantfeature of models with nuclei acceleration, similar tomixed composition scenarios [29]. This implies that thecosmic ray spectrum below 3� 1018 eV has to be ex-plained by a galactic component in these scenarios. Onecan see that although all primaries in our example startedas iron nuclei, the observed flux is dominated by primaryprotons at the highest energies E> 1020 eV. On the otherhand, at lower energies the flux is dominated by intermedi-ate nuclei. If the galactic component is dominated by ironnuclei at energies below the ankle, the composition wouldchange in this case from heavy to light above 1019 eV, inaccordance with experimental data [37].

As a second case we consider iron nuclei with spectrumE�2:1 accelerated up to lower maximal energy, Emax ¼1021 eV. Results of this simulation are presented inFig. 7. In this case secondary protons would have energiesbelow ’ 1019 eV, see magenta dotted line in Fig. 7, and thecomposition around 1020 eV would thus be heavier than inFig. 6. Note that there may be a tendency for the experi-mentally determined composition to become heavier forE * 4� 1019 eV. If this is confirmed in the future, itwould disfavor large maximal iron injection energiessuch as Emax ¼ 26� 1021 eV shown in Fig. 6.

In Fig. 7 we assumed no redshift evolution of thesources, which usually gives the smallest secondary fluxes,corresponding to the proton case in Fig. 1. As a result, thesecondary photon flux is close to lowest possible value inthis case. For the example considered this flux is on thelevel of 0.3% of the EGRET flux, which is a factor 10 lowerthan the corresponding photon flux presented in Fig. 1.However, composition data cannot be fitted without anyprotons for E> 1019 eV, which means that the real protonfraction should be larger than the one in Fig. 7. As aconsequence, the secondary photon fraction is likely largerthan the one shown in Fig. 7.To conclude, nuclei primaries produce lower secondary

photon fluxes mostly due to the fact that photons are mostlyproduced as secondaries of protons, whose flux in turn islower in case of nuclei primaries. We note that a similarreduction occurs correspondingly in the neutrino flux [40].

V. CONCLUSIONS

Ultra-high energy cosmic ray interactions with low-energy photons can significantly contribute to the observeddiffuse flux of �-rays at energies between �100 MeV and�TeV. In this paper we studied the dependence of thiscontribution on unknown parameters of astrophysical sce-narios of the ultra-high energy cosmic ray origin. We foundthat if ultra-high energy cosmic rays are mainly protonsthey contribute no less than 1% to the observed EGRETflux, and up to 50% in some cases. If highest energy cosmicrays are dominated by heavy nuclei and the proton flux isnegligible for E * 1019 eV, still 0.3% of the observedEGRET flux can be due to cosmic ray interactions. Thissuggests that the Fermi/GLAST satellite, which at GeVenergies will be ’ 30 times more sensitive to point sourcesthan the EGRET experiment, will likely be sensitive to theUHECR induced contribution. Even ground-based instru-ments such as HESS and the future CTA may be sensitiveto the cascade flux between ’ 0:1 and ’ 10 TeV, althoughsuch experiments are less well suited for diffusebackgrounds.If the sources of extragalactic highest energy cosmic

rays are rare and dominate the flux down to only ’ 5�1018 eV, the cascade background may have significant (upto 10%) anisotropy at energies around 200 GeV.To summarize, future measurements of resolved and

unresolved components of the diffuse EGRET �-ray back-ground or upper limits on such components can giveimportant information on the origin of ultra-high energycosmic rays and the distribution of their sources.

ACKNOWLEDGMENTS

We thank F. Stecker for providing us with tables for theIR/optical backgrounds for the model of Ref. [21]. Thenumerical simulations were performed at the computercluster of the Theory Division of INR RAS. Work of

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FIG. 7 (color online). Same as Fig. 6, but for maximum ironenergy Emax ¼ 1021 eV and no redshift evolution of sources.

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O.K. was supported in part by the grant of the President ofthe Russian Federation MK-1966.2008.2 and RFBR GrantNo. 07-02-00820. O.K. also acknowledges financial sup-

port from in2p3/CNRS for a collaboration visits at APC,Paris. G. S. acknowledges support by the DFG (Germany)under Grant No. SFB-676.

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