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PHYSICAL REVIEW D, VOLUME 63, 023003

Cosmic ray bound for models of extragalactic neutrino production

Karl Mannheim* Universita¨ts-Sternwarte, Geismarlandstr. 11, D-37083 Go¨ttingen, Germany

R. J. Protheroe† Department of Physics and Mathematical Physics, The University of Adelaide, Adelaide, Australia 5005

Jo¨rg P. Rachen‡ Sterrenkundig Instituut, Universiteit Utrecht, NL-3508 TA Utrecht, The Netherlands ͑Received 22 December 1998; published 22 December 2000͒ We obtain the maximum diffuse neutrino intensity predicted by hadronic photoproduction models of the type which have been applied to the jets of active galactic nuclei or gamma ray bursts. For this, we compare the proton and gamma ray fluxes associated with hadronic photoproduction in extragalactic neutrino sources with the present experimental upper limit on cosmic ray protons and the extragalactic gamma ray background, employing a transport calculation of energetic protons traversing cosmic photon backgrounds. We take into account the effects of the photon spectral shape in the sources on the photoproduction process, cosmological source evolution, the optical depth for cosmic ray ejection, and discuss the possible effects of magnetic fields in the vicinity of the sources. For photohadronic neutrino sources which are optically thin to the emission of neutrons we find that the cosmic ray flux imposes a stronger bound than the extragalactic gamma ray back- ground in the energy range between 105 GeV and 1011 GeV, as previously noted by Waxman and Bahcall ͓Phys. Rev. D 59, 023002 ͑1999͔͒. We also determine the maximum contribution from the jets of active galactic nuclei, using constraints set to their neutron opacity by gamma ray observations. This present upper limit is consistent with the jets of active galactic nuclei producing the extragalactic gamma ray background hadronically, but we point out future observations in the GeV-to-TeV regime could lower this limit. We also briefly discuss the contribution of gamma ray bursts to ultrahigh-energy cosmic rays as it can be inferred from possible observations or limits on their correlated neutrino fluxes.

DOI: 10.1103/PhysRevD.63.023003 PACS number͑s͒: 95.85.Ry, 14.60.Pq, 98.54.Cm, 98.70.Rz

I. INTRODUCTION netic field, can escape and convert into cosmic ray protons after ␤ decay, but their flux may be diminished by photoin- The connection between the emission of cosmic rays, duced n→p conversions. The branching ratios which distrib- gamma rays, and neutrinos from astrophysical accelerators is ute the available energy into the different channels are of considerable interest for the solution of the problem of the thereby generally of order unity. This leads to the conclusion origin of cosmic rays ͓1,2͔. The reason why a fundamental that cosmic proton accelerators produce cosmic rays, relation between these components must exist can be under- gamma rays, and neutrinos with comparable luminosities stood as follows. Particle acceleration mechanisms in cosmic ͓4͔. plasmas generally require the presence of a magnetic field The fundamental relation between cosmic ray and gamma which is able to confine the accelerated charged particles, ray production has the obvious consequence that active ga- i.e., electrons and protons ͑or ions͒. The accelerated electrons lactic nuclei ͑AGN͒, which are known to produce a large lose their energy quickly in synchrotron radiation in the mag- fraction of the gamma rays in the Universe, are a prime netic field. These synchrotron photons provide a target for candidate for the sources of ultrahigh-energy cosmic rays accelerated protons to undergo photohadronic interactions, ͑UHECR͓͒4͔. The spectra of the emitted GeV-TeV gamma resulting in the production of mesons, which decay. The par- radiation of AGN also agree with the predictions of a had- ticles which eventually emerge from this process are high- ronic production of these gamma rays ͓5͔. Other prominent energy photons, electrons ͑pairs͒, neutrons, and neutrinos. gamma ray sources, in particular the violent events con- Neutrinos are directly ejected due to their low interaction nected with gamma ray bursts ͑GRB͒, have also been sug- cross section. Gamma rays and secondary electrons initiate gested as UHECR source candidates. Moreover, most of the electromagnetic cascades, shifting the power from ultrahigh extragalactic gamma ray energy is found in a diffuse back- energies to energies below which the absorption of gamma ground rather than in point sources, which allows for the rays by pair production is unimportant ͓3͔. Finally, the neu- possibility that the UHECR sources could be relatively large trons, which, unlike the protons, are not confined in the mag- objects which would have a low gamma ray surface bright- ness, such as radio galaxies ͓6͔, galaxy clusters ͓7,8͔, or even larger structures ͓9͔. Whatever the sources are, the funda- *Email address: [email protected] mental relation between gamma ray and neutrino fluxes im- †Email address: [email protected] plies that, if in fact the extragalactic gamma ray emission is ‡Email address: [email protected] due to hadronic processes, a neutrino flux of a similar bolo-

0556-2821/2000/63͑2͒/023003͑16͒/$15.0063 023003-1 ©2000 The American Physical Society KARL MANNHEIM, R. J. PROTHEROE, AND JO¨ RG P. RACHEN PHYSICAL REVIEW D 63 023003 metric luminosity must exist. This prediction is the major AGN considering the effect of neutron opacity. In Sec. V, we motivation for high-energy neutrino experiments, which are discuss the possible effect on cosmic ray protons from neu- currently operated, under construction, or planned. In fact, tron decay of the magnetic fields known to exist in clusters most model predictions for extragalactic high-energy neu- of galaxies, and radio galaxies which are considered the trino fluxes have been made by using the source model to hosts of gamma ray emitting AGN. We derive critical ener- determine the spectral shape, and then by normalizing the gies below which they could increase the bound. We con- total flux to some fraction of the diffuse extragalactic gamma clude by discussing the combined effect of our results, and to ray background ͑EGRB͓͒10,11͔. what extent the cosmic ray data can indeed constrain models In contrast to the limits set by gamma ray observations, for expected neutrino fluxes, and vice versa. the limits which could arise from the corresponding cosmic ray emission of the neutrino sources have been given little II. COSMIC RAY, GAMMA RAY, AND NEUTRINO attention. A detailed treatment of this problem regarding pre- EMISSION FROM EXTRAGALACTIC PROTON dictions for neutrino fluxes from the decay of topological ACCELERATORS ͑ ͒ ͓ ͔ defects TD has been given by Protheroe and Stanev 12 , In this section, we obtain the form of the spectra of cos- and a brief discussion of the possible relevance for diffuse mic rays, gamma rays, and neutrinos escaping from cosmic neutrino fluxes from AGN by Mannheim ͓10͔. Recently, it ͓ ͔ proton accelerators. Here, we shall assume that protons are was proposed by Waxman and Bahcall 1 that indeed the confined within the acceleration region whereas photopro- measured flux of ultrahigh-energy cosmic rays provides the duced neutrons may escape to become cosmic rays. This is most restrictive limit on extragalactic diffuse neutrino fluxes justified in particular for cosmic ray sources connected to for a broad class of sources. They claim that this cosmic ray relativistic outflows, like AGN jets or GRB, since the life- bound is for neutrinos of all energies two orders of magni- time of protons is here limited by adiabatic energy losses and tude lower than the bound previously used which was based generally is much shorter than the diffusive escape time. For on the EGRB. In addition to the obvious restriction of their example, if protons are accelerated near the beginning of an result to neutrinos from proton accelerators ͑i.e., excluding AGN jet, say where the jet width is ϳ1016 cm and where the TD models͒, their claim is mainly based on three assump- gamma rays are probably produced, and are released near the tions: ͑i͒ neutrons produced in photohadronic interactions end of the jet after its width has expanded to at least a few can escape freely from the source, ͑ii͒ magnetic fields in the parsec, then their energies on release will be down by a least Universe do not affect the observed flux of extragalactic cos- two orders of magnitude. As a consequence, only protons mic rays, and ͑iii͒ the overall injection spectrum of extraga- resulting from the decay of neutrons which have escaped ϰ Ϫ2. A key role is played by their lactic cosmic rays is E from the acceleration region contribute significantly to the assumption ͑iii͒: By assuming a specific cosmic ray input cosmic ray spectrum. Escaping neutrons are produced by ac- spectrum, they can normalize their bound at the ultrahigh celerated protons which interact with ambient soft photons, energies, where they argue that ͑ii͒ also applies. Assumption together with pions which decay into neutrinos. ͑i͒ is justified by showing that some particular sources of We shall first discuss the properties of photohadronic in- specific interest, like the TeV- Mrk 501, or also GRB, teractions. We shall then obtain the ambient proton spectrum are transparent to the emission of neutrons. The authors in the emission region resulting from shock acceleration fol- claim that this new bound set by cosmic ray data essentially lowed by radiative cooling, adiabatic losses, and advection rules out the hypothesis that hadronic processes in AGN jets away from the shocked region where the photon density is can produce the EGRB, and consequently that their neutrino sufficient for efficient photoproduction of neutrons. From fluxes are overestimated. this we obtain the form of the spectra of neutrons and neu- The purpose of the present paper is to reexamine the role trinos on production, and the escaping neutron spectrum cosmic ray observations can play to constrain models of neu- which may be modified by neutron absorption in photohad- trino production. The paper is organized as follows. In Sec. ronic n→p conversions. II, we give a brief review on the properties of photohadronic interactions, and derive the production spectra for cosmic rays and neutrinos for power-law photon target spectra. In A. Photohadronic interactions Sec. III we briefly describe the effect of the propagation of Photohadronic interactions can be divided into two pro- extragalactic cosmic rays. We then follow Waxman and Bah- cesses: photoproduction of pions ͑and other mesons͒, and call in deriving a cosmic ray bound on neutrino fluxes, Bethe-Heitler production of eϮ pairs. Charged pions decay ͑ ͒ ͑ ͒ ␲Ϯ→␮Ϯ␯ ␮Ϯ→ Ϯ␯ ␯ ͑ adopting their assumptions i and ii , but instead of assum- as ␮ , e ␮ e here and in the following we ing a specific cosmic ray injection spectrum, we assume a disregard the difference between neutrinos and antineutri- specific spectrum for the observable extragalactic cosmic ray nos͒, neutral pions decay into gamma rays as ␲0→␥␥. Elec- flux, which is constructed such that it complies with all ex- trons and positrons ͑from pion decay and Bethe-Heitler pro- isting observational limits on the cosmic ray proton intensity. duction͒ cascade in the magnetic field and radiation field, In Sec. IV we turn to AGN, and discuss in particular the and so can be assumed to convert all their energy into syn- photohadronic opacity of blazar jets as can be estimated from chrotron radiation in the magnetic field required for the ac- observations. We shall show that most AGN in fact have celeration of protons. The production of charged pions al- large photohadronic opacities at ultrahigh energies, and we lows the production of secondary neutrons through isospin derive an upper bound for the neutrino contribution from exchange.

023003-2 COSMIC RAY BOUND FOR MODELS OF... PHYSICAL REVIEW D 63 023003

The physics of photohadronic interactions in ambient for pion photoproduction to take place. From the ambient photon spectra has been extensively studied in Monte Carlo proton spectrum we can then obtain the spectra of neutrinos ͓ ͔ Ϫ1 Ϫ1 simulations 13,14 . The properties of the production of sec- and neutrons produced, Qn(En) and Q␯(E␯) ͓s GeV ͔, ondary particles can be expressed in the fraction ␰ of the respectively. Then taking account of the optical depth of the proton energy given to a specific particle component per in- emission region for neutrons escaping photoinduced n→p teraction. For neutrinos, gamma rays, and neutrons, the val- conversions, one can obtain from the neutron production ues spectrum, the spectrum of cosmic ray protons resulting from the ␤ decay of the neutrons escaping from the jet, Q (E ). ␰ Ϸ␰ Ϸ ͑ ͒ cr p ␯ ␥ 0.1, 1 The spectrum of protons on acceleration and the ambient proton spectrum are related by the proton loss time scale ␰ Ϸ ͑ ͒ n 0.5, 2 t p(E p), respectively, have been found for power-law target spectra N ͑E ͒ϳQ ͑E ͒t ͑E ͒. ͑5͒ typical in AGN jets, while for GRB target spectra the values p p p p p p ␰ Ϸ␰ Ϸ␰ Ϸ ͓ ͔ are ␯ ␥ n 0.2 15 . The energy per particle in units of the proton energy have been found for neutrinos and neu- The processes mainly contributing to losses of protons are Ϸ Ϸ interactions with radiation, advection away from the shock trons as ͗E␯͘/E p 0.033 and ͗En͘/E p 0.83, respectively, ϳ ϳ region of dimension R, and adiabatic energy losses if the while for GRB they are ͗E␯͘/E p 0.02 and ͗En͘/E p 0.5, respectively ͓15͔. From this we can immediately define the emission region expands. relative energy of escaping neutrinos and neutrons as The advection time scale is expected to be tadv ϳ ␤ ␤ R/( shc) where sh is the shock velocity and R is the ␩ ϭ Ϸ ͑ ͒ ␯n ͗E␯͘/͗En͘ 0.04 3 dimension of the jet in the shocked region. In relativistic flows streaming away from a central source, this energy in- for both AGN and GRB target spectra. The fractional energy dependent time scale is usually in competition with adiabatic ␬ Ϸ loss of the proton per interaction is p 0.2 in the AGN case, energy losses of the protons due to the expansion of the flow ␬ Ϸ ͓ ͔ and p 0.5 for GRB 15 . Note that the quantities given ͓16͔. In a relativistic outflow, characterized by a bulk Lor- here as typical for GRB apply only at ultrahigh proton ener- entz factor ⌫ and an opening angle ⌰, the expansion velocity ͑ ͒ ␤ Ϸ⌫⌰ ⌰ gies, at lower energies they approach the values found for in the comoving frame in units of c is ex for ͓ ͔ Ͻ⌫Ϫ1 ␤ Ϸ AGN 15 . , and ex 1 otherwise. This leads to adiabatic cool- Ϸ ␤ Electromagnetic radiation initiated by the Bethe-Heitler ing on a time scale tad R/( exc). For example, in GRB one ␤ Ӎ pair production, and by photons and electrons from neutral can generally assume that ex 1, and also observations of and charged pion decay, are reprocessed in synchrotron pair superluminal motions in AGN jets are consistent with ⌰ ϳ⌫Ϫ1 ͓ ͔ ␤ ϳ cascades. This energy will emerge as a component of the 17 , thus ex 1. In the following, we shall assume gamma ray background radiation, for AGN mainly in an en- Շ␤ Շ that adiabatic losses are relevant with 0.3 ex 1. This has ergy range 10 MeV–1 TeV. The contribution of the Bethe- the important consequence that the lifetime of protons in the Heitler process to the production of gamma rays depends on jet ͑or outflow͒ is limited to about one crossing time. The the target energy spectrum index, ␣, since its cross section time scale for diffusive escape of protons is usually much peaks at energies about two orders of magnitude lower than longer ͑except, maybe, near the maximum proton energy͒, that of photopion production. Assuming that the power law Ӷ thus protons with E p Emax can be assumed to be confined in extends over this range without change, one can find the the emitter and do not contribute to the cosmic ray emission. relation The photon target spectrum will be assumed to have a power-law shape n(⑀)ϰ⑀Ϫ␣Ϫ1 extending to energies suffi- L␥ϭ͓1ϩexp͑5␣Ϫ5͔͒L␯ , ͑4͒ ciently above the threshold for photopion production by pro- tons of energy E . Then, for ␣Ͼ0 the time scale for energy where L␥ and L␯ are the bolometric photohadronic luminosi- p ties in gamma rays and neutrinos ͓16͔, and we note that for loss by photohadronic interactions is asymptotically of the form ␣ϭ1, L␥ϭ2L␯ . We also note that in general, L␥уL␯ holds as a direct consequence of the isospin symmetry of charged ͑ ͒ϰ Ϫ␣ ͑ ͒ and neutral pions—hence, for any kind of neutrino produc- t p␥ E p E p , 6 tion involving pion decay, the bolometric flux in correlated photons sets a robust upper limit on the possible bolometric where t p␥ is understood as including Bethe-Heitler and pion neutrino flux. production losses, the cooling time for pion production will Ͼ be called t p,␲ t p␥ . For very flat target spectra, as, for ex- B. Ambient proton spectrum ample, in GRB at ultrahigh proton energies, the photopro- ͓ ͔ We assume a spectrum of protons on acceleration of the duction time scale is approximately constant 16 , thus, Eq. Ϫ Ϫ Ϫ ͑ ͒ ␣ϭ form, Q(E )ϰE 2 exp(ϪE /E ) ͓s 1 GeV 1͔. In order to 6 applies with 0. We shall confine the discussion to the p p p max ␣ϭ ␣ϭ calculate the spectra of cosmic ray protons and neutrinos values 1, relevant in AGN, and 0 hereafter. Hence, ␣ϭ escaping from AGN jets we first need to obtain the ambient for 1 we obtain Ϫ1 spectrum of protons, N p(E p) ͓GeV ͔, in the shocked re- ͒ϭ ͒ ͒ ͑ ͒ gions where the soft photon target density is sufficiently high t p␥͑E p ͑E1 /E p ͑R/c , 7

023003-3 KARL MANNHEIM, R. J. PROTHEROE, AND JO¨ RG P. RACHEN PHYSICAL REVIEW D 63 023003

Ϸ with t p,␲(E p) 1.5t p␥(E p). E1 is the energy corresponding However, because the two break energies aE1 and bE1 are ␶ ϳ to unit optical depth for photoproduction losses, p␥(E1) probably very close, a b, we shall adopt a single break ϭ ͓ ͔ϭ Ϫ1ϭ Ϫ1 ϩ Ϫ1 ϭ R/ ctp␥(E1) 1. Setting t p t p,ad t p␥ we obtain energy Eb bE1 , and use the following approximations: ͑ ͒ϭ ͑ ͒͑ ͓͒͑ ͒ϩ ͔Ϫ1 ͑ ͒ Ϫ Ϫ1 Ϫ1 ͑ Ͻ ͒ N p E p Q p E p R/c E p /E1 a 8 En En Eb En Eb , Q ͑E ,L ͒ϰL expͫ ͬͭ Ϫ ͑13͒ n n p p E 2 ͑E ϽE ͒, ϭ ␤ ␤ ϰ Ϫ2 Emax n b n with a max( ex , sh), and for Q p(E p) E we have Ϫ Ϫ Ϫ Ϫ Ϫ 1 1 ͑ Ͻ ͒ 1 2 ͑ Ͻ ͒ E p E p Eb E p Eb , a E p E p aE1 , Q ͑E ,L ͒ϰL expͫ ͬͭ Ϫ ͑14͒ N ͑E ͒ϰͭ Ϫ ͑9͒ cr p p p 3 ͑ Ͻ ͒ p p 3 ͑ Ͼ ͒ Emax E p Eb Eb E p . E1E p E p aE1 . ͑Note that for clarity, here and in the next section we omit Note that we have now put in explicitly the cutoff in the ͒ accelerated proton spectrum, and the proportionality with the the exponential cutoff in the spectrum at Emax . Obviously, for ␣ϭ0 the optical depth for photoproduction is constant, proton luminosity of the source, L p . In Sec. IV, we shall and N (E )ϰQ (E ). One can show that for typical photon relate L p to the observed photon luminosities for specific p p p p models of neutrino emission by AGN jets. Obviously, in densities in GRB fireballs ␶ ␥Ͻ1, and that Bethe-Heitler p GRB we simply have Q (E ,L )ϷQ (E ,L ) losses are unimportant, viz., t ␲Ϸt ␥ ͓18,16͔. cr n p n n p p, p ϰ Ϫ2 Ϫ L pE p exp( Ep /Emax). The production spectrum of muon neutrinos will have the C. Generic cosmic ray proton and neutrino production same broken power-law form as the neutron production spectra spectrum, and is related to it by The time scale for photohadronic production of neutrons Ϸ ␬ ␣ϭ Ϸ ͗␰ ͘ is t p␥→n t p,␲ p /͗Nn͘. For 1, this is t p␥→n 0.5t p␥ 2 ␯ Q␯ ͑E͒ϭ Q ͑E/␩␯ ͒, ͑15͒ ϰ Ϫ1 ␣ϭ Ϸ ␮ ͗␰ ͘␩2 n n E p , while for 0 we have t p␥→n 2.5t p␥ . This imme- 3 n ␯n diately gives the production spectrum of neutrons, where we count ␯␮ and ¯␯␮ together, and the corresponding ͒Ϸ ͒ ͒ Qn͑En N p͑En /t p␥→n͑En spectrum of electron neutrinos at the source would be Ϸ 1 Q␯ (E) 2 Q␯␮(E). Putting in the numbers given in Sec. Ϫ1 Ϫ1 ͑ Ͻ ͒ e a En En aE1 , II A, we find ϰͭ Ϫ ͑10͒ E E 2 ͑E ϾaE ͒. 1 n n 1 ͑ ͒Ϸ ͑ ͒ ␣ϭ ͑ ͒ Q␯␮ E 83.3Qn 25E for 1, 16 Neutrons may escape to become cosmic ray protons. How- ever, because neutrons themselves suffer pion photoproduc- ͑ ͒Ϸ ͑ ͒ ␣ϭ ͑ ͒ Q␯␮ E 416.Qn 25E for 0. 17 tion losses, the cosmic ray production spectrum will differ from Qn(E p) above the energy bE1 at which the optical We shall refer to Eqs. ͑14͒ and ͑15͒ as the generic cosmic ␶ depth for neutron escape, n␥ , is one. Neutrons can be con- ray and neutrino production spectra. We emphasize the sidered as ‘‘absorbed’’ after they are converted into a proton, strong dependence of the number of produced neutrinos per or after they have lost most of their energy in n␥ interac- produced neutron on the assumed target photon spectral in- tions, whichever time scale is shorter. For ␣ϭ1 this means dex: at ultrahigh energies, GRB produce about 5 times more ␶ Ϸ ␶ Ϸ ␣ϭ n␥ 2 p␥ , giving b 0.5 for AGN jets, while for 0 and neutrinos per neutron than AGN. We shall return to the im- ␶ Ϸ␶ Ͻ typical GRB photon densities, n␥ p␥ 1, which means plications of this result at the end of the paper. that neutron absorption is unimportant in GRB. We note that in a homogeneous spherical medium of ra- III. PROPAGATION OF NEUTRINOS, PHOTONS, dius R the optical depth decreases radially ϰ(1Ϫr/R) from AND PROTONS OVER COSMOLOGICAL DISTANCES its central value ␶ to zero at rϭR giving rise to the geometri- cal escape probability of an interacting particle propagating In this section, we discuss propagation of cosmic rays and in a straight line neutrinos in an expanding Universe filled with the cosmic microwave background radiation. To illustrate the problem, P ͑␶͒Ϸ͑ Ϫ Ϫ␶͒ ␶ esc 1 e / we compare the energy-loss horizons of protons and neutri- nos. We shall then briefly discuss the physical problems con- 1 ␶Ͻ1, Ϸ ͑ ͒ nected to several approaches to cosmic ray propagation cal- ͭ ␶Ϫ1 ␶Ͼ 11 1, culations. Using the numerical propagation code described by Protheroe and Johnson ͓19͔, we then calculate the observ- resulting in the cosmic ray proton production spectrum being able neutrino and cosmic ray spectra from a cosmological steepened above bE , compared with Q (E ). For ␣ϭ1, the 1 n p distribution of generic photohadronic sources, as described cosmic ray proton production spectrum is therefore in the last section. Here we assume that the sources are trans- Ϫ1 Ϫ1 Ϫ1 ͑ Ͻ ͒ parent to neutrons, while protons are confined, and that a E p E1 E p aE1 , Ϫ gamma rays are reprocessed in synchrotron-pair cascades un- ͑ ͒ϰ 2 ͑ Ͻ Ͻ ͒ ͑ ͒ Qcr E p ͭ E p aE1 E p bE1 , 12 til emitted in the energy range of 10 MeV–30 GeV. Using an Ϫ3 ͑ Ͻ ͒ bE1E p bE1 E p . extrapolated cosmic ray spectrum which is consistent with

023003-4 COSMIC RAY BOUND FOR MODELS OF... PHYSICAL REVIEW D 63 023003

present observational limits on the light component of cos- production reduces the horizon for protons with E p ͑ ͒ տ 4 Ϸ ϫ 11 ␭ ϳ mic rays i.e., protons as an upper limit on the possible m pm␲c /kTmbr 5 10 GeV to p 10 Mpc. Again, in extragalactic proton contribution, and the diffuse extragalac- units of the radius of the Einstein–de Sitter Universe, the tic gamma ray background ͑EGRB͒ observed by the Ener- energy-dependent horizon for protons can be written getic Gamma Ray Experiment Telescope ͑EGRET͒ as an upper limit on the hadronic extragalactic gamma ray flux, we ␭ˆ ͒ϭ ␭ˆ ϩ ␭ˆ ͒ϩ ␭ˆ ͒ Ϫ1 ͑ ͒ p͑E ͓1/ z 1/ p,BH͑E 1/ p,␲͑E ͔ , 22 determine an energy dependent upper bound on the neutrino flux from cosmic ray sources with the assumed properties. where the components expressing redshift, Bethe-Heitler, Our result is compared with the energy independent bound and pion production losses can be written as on extragalactic neutrino fluxes recently proposed by Wax- man and Bahcall ͓1͔. ␭ˆ Ӎ z 0.78,

A. Comparison of energy-loss horizons ␭ˆ ͒Ϸ ͒ ͑ ͒ p,BH͑E 0.27h50 exp͑0.31/E10 , 23 We wish to compare the distances that neutrinos, photons with energies below threshold for cascading in background Ϫ4 ␭ˆ ␲͑E͒Ϸ5ϫ10 h exp͑26.7/E ͒, radiation fields, and protons will travel through the Universe p, 50 10 without significant energy losses. We define the energy loss ϭ 10 ␭ˆ horizon by with E10 E p/10 GeV. The approximations for p,BH and ␭ˆ ͓ ͔ p,␲ fit the exact functions determined numerically in 19 ␭ϭ ͉ ͉ ͑ ͒ ϳ cE/ dE/dt , 18 and the exact interaction kinematics within 10% up to E p ϳ1012 GeV. such that for linear processes the energy is reduced to 1/e of The different energy-loss horizons for gamma rays and ␭ its initial value on traversing a distance . neutrinos, and protons strongly affect the relative intensities ϳ For gamma rays below 30 GeV and neutrinos, the of their diffuse isotropic background fluxes. This is true in ͓ ͔ energy-loss process is due to expansion of the Universe 20 . particular for evolving source populations such as quasars, For simplicity, we adopt an Einstein–de Sitter cosmology galaxies, or GRBs ͑if they trace star formation activity͒, ͑ ⌳ϭ ⍀ϭ ͒ ␭ i.e., 0 and 1 , so that the horizon can be related to since here most of the energy is released at large redshifts. a redshift z by the redshift distance relation ϳ ϫ 9 Cosmic rays above the ankle (Ecr 3 10 GeV) originate only from sources with redshifts zՇ0.27, while neutrinos 2 c Ϫ ␭͑z͒ϭ ͓1Ϫ͑1ϩz͒ 3/2͔ ͑19͒ and gamma rays originate from sources within z␯ϭz␥ϳ1.7. 3 Ho This will give rise to the neutrino intensity being enhanced relative to the protons because of their larger horizon, and ϭ Ϫ1 Ϫ1 where Ho 50h50 km s Mpc is the Hubble constant. because of the evolution of the sources ͑e.g., quasars͒ with Since we require the distance for which the energy is reduced cosmic time ͑redshift͒. ϩ ϭ by a factor e during propagation, we have (1 z) e giving We may illustrate the problem as follows. The basic the horizon for redshift losses: method of calculating the approximate present-day diffuse fluxes of neutrinos, gamma rays below ϳ30 GeV, and cos- 2c ␭ ϭ Ϫ Ϫ3/2͒ ͑ ͒ mic rays of photoproduction origin, would be to integrate the z ͑1 e . 20 3Ho contributions from sources at redshifts up to those corre- sponding to the respective energy-loss horizons. Assuming a This is also the horizon for neutrinos and gamma rays below ϳ constant source number per comoving volume element for 30 GeV. Normalizing the neutrino horizon to the radius of simplicity, the resulting fluxes are proportional to the Einstein–de Sitter universe, ␭ˆ 2 ␭ˆ ϳ␭ˆ ␭ˆ ␭ˆ Vc( )/dL( ) , where Vc( ) and dL( ) are the cosmo- logical comoving volume and the luminosity distance, re- 3Ho ␭ˆ ϵ ␭ , ͑21͒ ␭ˆ z 2c z spectively, corresponding to the horizon . Assuming pho- tohadronic production of neutrinos and cosmic rays in, e.g., a cosmological ͑nonevolving͒ distribution of AGN, the relative we obtain ␭ˆ ␥ϭ␭ˆ ␯ϭ␭ˆ Ϸ0.78. z ͑ ͒ In addition to redshift losses, extragalactic cosmic rays flux of neutrons assuming no absorption with energy En , ϭ suffer energy losses from photohadronic interactions with and corresponding neutrinos with energy E␯ 0.04En ,isat ͓ 2 ͔ ͓ 2 ͔ϭ␰ ␰ Ϸ cosmic backgrounds, mainly the microwave background, and the source given by E␯N␯(E␯) / EnNn(En) ␯ / n 0.2. this is the reason for the Greisen-Zatsepin-Kuzmin ͑GZK͒ The same ratio will be observed in the integrated fluxes, as Ͻ ϫ 9 cutoff expected for a cosmic ray spectrum originating from a long En 4 10 GeV. For higher cosmic ray energies, the ͓ ͔ ␭ˆ ␭ˆ cosmologically homogeneous source distribution 21,22 . flux ratio must be multiplied by a factor ␯ / p , which yields ϳ ϳ 10 ϳ Photopion production and Bethe-Heitler pair production gov- a flux ratio of 0.6 for En 10 GeV, and 30 for En ern the energy loss in different energy regimes due to their ϳ1011 GeV. Obviously, the differences would be much very different threshold energies. The Bethe-Heitler process larger if we had assumed strong source evolution which en- limits the propagation of protons with energies E p hances the contribution from large distances. If we want to Ͼ 4 Ϸ ϫ 9 ␭ ϳ 2m pmec /kTmbr 4 10 GeV to p 1 Gpc, while pion determine a neutrino spectrum from an observed, correlated

023003-5 KARL MANNHEIM, R. J. PROTHEROE, AND JO¨ RG P. RACHEN PHYSICAL REVIEW D 63 023003 cosmic ray spectrum from the same sources, the result must lated neutrino spectrum, under the assumption that all ob- therefore approximately reflect the changes in the ratio of the served extragalactic cosmic rays are due to neutrons ejected energy-loss horizons. from the same sources as the neutrinos. A difficulty here is that the contribution of extragalactic cosmic rays to the total B. Exact calculation of present-day neutrino observed cosmic ray flux is unknown. Since neutrons convert and cosmic ray spectra into cosmic ray protons, we can clearly consider only the proton component at all energies. Above the ankle in the The example above, of simply integrating the cosmic ray ϳ ϫ 9 and neutrino contribution of cosmologically distributed cosmic ray spectrum at Ecr 3 10 GeV, observations are sources up to the energy-loss horizon, disregards several im- generally consistent with a ‘‘light’’ chemical composition, portant aspects of particle propagation. First, the particle i.e., a 100% proton composition is possible. Since protons at number is not conserved in this method. Second, the energy these energies cannot be confined in the magnetic field of the evolution of the particles is neglected, which removes the Galaxy, they are also likely to be of extragalactic origin. At dependence on spectral properties of the source. Both cave- extremely high energies, however, there is the problem that ats are removed in a method known as the continuous-loss the event statistics is very low, and different experiments disagree on the mean cosmic ray flux at ϳ1011 GeV by one approximation, which follows the particle energy along fixed ͑ ͓ ͔ trajectories as a function of cosmological distance ͑or red- order of magnitude see Bird et al. 28 for a comparison of shift͓͒23,6͔. For example, the trajectories for neutrino ener- the results until 1994 of the four major experiments Akeno, ϭ ϩ Havarah Park, Fly’s Eye, and Yakutsk͒. This energy region gies would be simply E(z) E0(1 z). This method is exact for adiabatic losses due to the expansion of the Universe is very important, since we expect here the existence of the ͑i.e., particle redshift͒, and is still a very good approximation GZK cutoff due to photoproduction losses in the microwave for Bethe-Heitler losses, but it gives only poor results for background. Currently, no clear evidence for the existence of photopion losses. The reason for the latter is the large mean this cutoff has been found, and the results of at least two major experiments Havarah Park ͓29͔, and Akeno Giant Air free path, and the large inelasticity of this process, which ͑ „ ͒ ͓ ͔ results in strong fluctuations of the particle energy around its Shower Array AGASA 1998 30 … are consistent with the mean trajectory ͓24,25͔. Cosmic ray transport in the regime result obtained from a superposition of all experiments using a maximum likelihood technique ͓31,32͔, that is, a continu- where photopion losses are relevant is therefore best de- ϰ 2.75 scribed by numerical approaches, either solving the exact ation of the cosmic ray spectrum as a power law E up to տ ϫ 11 ͑ ͒ transport equation ͓24͔, or by Monte Carlo simulations ͓26͔. 3 10 GeV see also Fig. 1 . Another important aspect concerning relative fluxes of cos- The situation is even more difficult at lower energies: cos- mic rays, gamma rays, and neutrinos is the fact that the in- mic rays are here assumed to be mainly of Galactic origin, teraction of the cosmic rays with the cosmic background ra- and there is evidence that a considerable, maybe dominant diation themselves produces secondary particles, which have fraction consists of heavy nuclei rather than protons. Around ϳ 6 7 to be considered as an additional contribution to the primary the knee or the cosmic ray spectrum at Ecr 10 –10 GeV, neutrinos and gamma rays. The full problem is treated by the recent results from the KASCADE air shower experiment suggest that the fraction of heavy nuclei in the cosmic ray cascade propagation code which has been described in detail ϳ by Protheroe and Johnson ͓19͔, and which we shall use also flux is at least 30%, and further increasing with energy ͓33͔. Also below the ankle, in the energy range here. The code is based on the matrix-doubling technique for 8 9 cascade propagation developed by Protheroe and Stanev 10 –10 GeV, the analysis of air shower data has produced ͓27͔. tentative evidence of a composition change from heavy to ͑ ͒ For an input spectrum (dP /dV )͗Q(E,z)͘ per unit co- light with increasing energy , supporting a dominantly gal c heavy composition of cosmic rays between the knee and the moving volume per unit energy per unit time, the intensity at ͓ ͔ ͑ Earth at energy E is given by ankle 34 . Note that this result is under dispute, and it has been shown that it depends on the Monte Carlo simulation 2 1 zmax ͑1ϩz͒ dV dP codes used to construct the air shower properties in depen- ͑ ͒ϰ ͵ ͑ ͒ c gal I E M E,z 2 dence of the primary particle mass ͓35͔. These simulation 4␲ 4␲d dz dV zmin L c codes involve particle interaction models based on extrapo- ϫ͗Q͓͑1ϩz͒E,z͔͘dz, ͑24͒ lations many orders of magnitude above the energy range currently accessible with particle accelerators ͓36͔.͒ where dL and Vc are luminosity distance and comoving vol- Using all the available data, we find that an extragalactic ume, and M(E,z) are ‘‘modification factors’’ for injection of cosmic ray spectrum of the form protons at redshift z as defined by Rachen and Biermann ͓6͔; ͑ ͒ϭ ϫ͑ ͒Ϫ2.75 Ϫ2 Ϫ1 Ϫ1 Ϫ1 for neutrinos, M(E,z)ϭ1. The modification factors for pro- N p,obs E 0.8 E/1 GeV cm s sr GeV tons depend on the input spectra, and are calculated numeri- ͑3ϫ106 GeVϽEϽ1012 GeV͒ ͑25͒ cally using the matrix method ͓19͔. is consistent with all data and limits on the cosmic ray proton C. An abstract bound on extragalactic neutrino fluxes flux ͑Fig. 1͒. It represents the current experimental upper from neutron-transparent sources limit on the extragalactic cosmic ray proton flux, which we From the above considerations, it is obvious that one can shall use to construct an upper limit on the possible, diffuse use the observed cosmic ray spectrum to construct a corre- extragalactic neutrino flux. If it can be shown that the inten-

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cosmic ray spectrum above 1011 GeV. The upper bound on the neutrino flux according to this model is given by the minimum of the cosmic ray and the ␶ Ͻ gamma ray bound, shown in Fig. 3 by the curve ‘‘ n␥ 1.’’ We see that the cosmic ray limit starts to dominate the bound for E␯տ100 TeV, which then decreases to a minimum at 9 E␯Ϸ10 GeV, after which it rises again. This rise is a con- sequence of the strongly increasing ratio of neutrino and pro- ton horizons above this energy, in conjunction with the as- sumptions that the distribution of sources is homogeneous in space. A more realistic scenario might be that the lack of evidence for the GZK cutoff in the cosmic ray data is due to the dominant contribution of a local cosmic ray source. In ͓ 2 ͔ ͓ 2 ͔ this case, our estimate that E␯N␯(E␯) / EnNn(En) ϰ␭ˆ ␭ˆ ␯ / p would not apply. Obviously, if an extended data set would confirm the existence of the GZK cutoff in the UHECR spectrum, the neutrino bound would remain at the FIG. 1. The observed all-particle cosmic ray spectrum taken ͑ from the article by T. K. Gaisser and T. Stanev in the 1998 Review level of the Waxman-Bahcall result also shown in Fig. 3, ͒ Ͼ 9 of Particle Properties ͓81͔, and supplemented by Fly’s Eye monocu- see discussion below for E␯ 10 GeV. lar data ͓28͔͑open circles at high energy͒, and recent AGASA data By assumption, the neutrino bound constructed this way ͓30͔͑filled circles at high energy͒. Also shown are estimates of the applies only to sources which are transparent to neutrons. cosmic ray proton component: based on the proton fraction esti- For the opposite extreme, i.e., sources with a very high neu- ␶ ӷ mates by ͓34͔͑thick lines with thick error bars, extended by thin tron opacity, n␥ 1, it is still possible to set an upper limit lines which indicate the systematic error due to normalization to the using the observed EGRB, assuming that the dominant part all-particle data͒; Norikura data ͓82͔͑filled circles with large error of the emitted gamma radiation is in the EGRET range. This ϫ 6 7 ͒ ␶ ӷ bars at 3 10 –10 GeV ; proton fraction estimated from is shown by the line labeled ‘‘ n␥ 1’’ in Fig. 3. The range KASCADE data ͓33͔ normalized to all-particle data ͑hatched band in between can be regarded as the ‘‘allowed range’’ for the 6 7 ͒ ␶ Ͼ from 10 –10 GeV . The spectrum we adopt for the proton compo- neutrino emission from sources with n␥ 1. In the next sec- nent which forms the upper bound to any extragalactic cosmic ray tion, we will estimate specific neutrino spectra for AGN proton spectrum ͓cf. Eq. ͑25͔͒ is shown by the dotted line. ␶ Ͼ models which imply that n␥(En) 1 for high neutron ener- gies. sity of protons at 109 GeV and lower energies is below that Our result may be compared with the cosmic ray bound to assumed in this paper ͑dotted line in Fig. 1͒, then the neu- extragalactic neutrino fluxes recently proposed by Waxman trino bound we have constructed below 107 GeV would need and Bahcall ͓1͔. Their bound was constructed using a differ- to be reduced. ent approach: Waxman and Bahcall assume an EϪ2 input To construct the neutrino bound, we assume test spectra spectrum of extragalactic cosmic rays, and normalize the ϭ ϰ Ϫ1 Ϫ ϭ 10 of the form Qcr(E) Qn(E) E exp( E/Emax) with propagated spectrum to the observed flux at Ecr 10 GeV. 6 Ͻ Ͻ 12 10 GeV Emax 10 GeV. The corresponding neutrino Consequently, their bound ͓determined for a source evolu- spectra are determined using Eq. ͑16͒. We assume a source tion ϰ(1ϩz)3.5͔ agrees with the one derived in this work at 8 distribution following the cosmological evolution function E␯ϳ5ϫ10 GeV, where the cosmic ray limit is most restric- found for galaxies and AGN ͓͑37͔, see next section͒. For a tive. Ϫ1 given Emax the total contribution of cosmic rays and neutri- We have chosen E trial spectra with variable exponen- nos is calculated using Eq. ͑24͒. The resulting spectrum is tial cutoffs in order to be able to mimic the effect of the then normalized so that its maximum reaches the cosmic ray superposition of spectra from various source classes. The ͑ ͒ 6 flux given by Eq. 25 . By varying Emax between 10 and pronounced peak in the energy flux associated with the trial 1012 GeV, we then obtain the desired maximum flux of neu- spectra allows one to normalize the neutrino flux consistent trinos consistent with the present cosmic ray data ͑see Fig. with the experimental upper limit on extragalactic protons 2͒. We also consider the correlated gamma ray output, as- (ϰEϪ2.75) at any chosen energy. Other hard spectra or delta sumed to be twice the neutrino energy flux, and check it does function distributions as trial spectra would have yielded not exceed the observed power-law component of the diffuse practically the same result. Although canonical AGN jets, gamma ray background ͑we estimate the background be- which we discuss in the next section, are an example for tween 3 MeV and 30 GeV to be Ϸ1.5ϫ10Ϫ5 GeV sources with EϪ1 ͑or similar͒ spectra, our result does not cmϪ2 sϪ1 srϪ1 ͓38͔͒. Note that the cosmic ray curve for imply that we assume AGN jets to actually saturate the upper ϭ 6 Emax 10 GeV does not reach the estimated cosmic ray pro- limit in general. As a matter of fact, at neutrino energies ton spectrum in Fig. 2͑a͒ in order to avoid over-producing տ109 GeV, a class of photohadronic sources saturating our diffuse gamma rays. Note also that the propagated cosmic bound has neither been suggested nor does its existence seem 11 ray spectra cut off at or below 10 GeV for all Emax values, likely on the basis of current knowledge. By restricting their and so our bound is insensitive to the assumed extragalactic source spectra to EϪ2, Waxman and Bahcall have con-

023003-7 KARL MANNHEIM, R. J. PROTHEROE, AND JO¨ RG P. RACHEN PHYSICAL REVIEW D 63 023003

FIG. 2. Spectra of ͑a͒ cosmic rays and ͑b͒ neutrinos after propagation through the Universe of input spectra for optically thin pion ϭ 6 ϫ 7 9 ϫ 10 12 ϫ 13 photoproduction sources for Emax 10 ,3 10 ,10,3 10 ,10 , and 3 10 GeV, assuming galaxy evolution as described in the text. Spectra are normalized such that the cosmic ray intensity does not exceed the cosmic ray proton spectrum estimated from observations ͓dotted line in part ͑a͔͒ and such that the neutrino energy flux does not exceed 0.5 of the observed photon energy density above 3 MeV. The dotted curve in part ͑b͒ joins the peaks in the neutrino spectra and forms our neutrino upper bound for optically thin pion photoproduction sources. The dashed line is the bound obtained by Waxman and Bahcall ͓1͔. structed a bound for neutron-transparent sources which is generic source spectra, Eqs. ͑14͒ and ͑15͒, varying the break probably closer to current models for cosmic ray and neu- energy Eb over the range allowed by the models. trino production than our general upper limit above 1019 eV. Examples for such models are the model for diffuse neutrino A. Cosmic ray proton and neutrino production spectra fluxes from GRBs proposed by the same authors ͓18,1͔͑note from that this prediction assumes no cosmological evolution for As our starting point, we assume a target photon spectrum GRBs͒, and model A in Mannheim ͓10͔ which was con- with index ␣ϭ1 which we have already seen leads to structed using the cosmic ray limit with the assumption that the emerging neutrons at ϳ1010 GeV contribute to the extra- galactic cosmic ray spectrum ͑both shown in Fig. 3͒.

IV. DIFFUSE NEUTRINO SPECTRA FROM AGN JETS In this section, we shall consider the spectra of cosmic ray protons and neutrinos emerging from jets of two classes of gamma ray emitting AGN, i.e., BL Lac objects and radio quasars, which are usually combined as the class of blazars. We shall use the estimated optical depths of gamma rays to photon-photon pair production in a typical AGN of each type, to infer the corresponding neutron-photon optical depths in these objects. We shall show that high luminosity AGN ͑like 3C279͒ can be expected to be opaque to neutrons at energies above about 108 –109 GeV, while low luminosity BL Lacs ͑like Mrk501͒ must be transparent to neutrons at all energies. Then assuming a model for the luminosity depen- FIG. 3. Muon neutrino upper bounds for optically thin pion dence of the optical depths, and the local luminosity func- ͑ ␶ Ͻ ͒ photoproduction sources curve labeled n␥ 1 and optically thick tions of BL Lacs and quasars, we shall estimate the form of ͑ ␶ ӷ ͒ pion photoproduction sources curve labeled n␥ 1 ; the hatched the production spectra of cosmic ray protons and neutrinos range between the two curves can be considered the allowed region ␶ Ͼ per unit volume of the local Universe. Applying the source for upper bounds for sources with n␥ 1. For comparison we show evolution functions found for BL Lacs and radio quasars, the bound obtained by Waxman and Bahcall ͓1͔͑for an evolving respectively, we shall derive model estimates for the diffuse source distribution͒. Predictions for optically thin photoproduction neutrino contribution from these sources which are compat- sources are also shown: proton-blazar ͑Mannheim 1995 ͓10͔, model ible with cosmic ray limits. We shall also construct an upper A͒—dotted curve, and GRB sources ͑Waxman and Bahcall 1997, bound for the contribution of AGN jets, using the same ͓18͔͒—dashed curve. Also shown is an observational upper limit method as in the previous section, but for the appropriate from Fre´jus ͓83͔ and the atmospheric background ͓84͔.

023003-8 COSMIC RAY BOUND FOR MODELS OF... PHYSICAL REVIEW D 63 023003

␶ ϰ p␥(E p) E p . A similar energy dependence applies to gamma rays interacting with the same photons by photon- photon pair production (␥␥→eϩeϪ), and so for ␣ϭ1we have

2 ͗␬ ␴ ␥͘ 2m c E ␶ ͑ ͒ϭ p p ␶ ͩ e p ͪ p␥ E p ␴ ␥␥ ϩ ͗ ␥␥͘ m␲͓m p m␲/2͔ Ϸ ϫ Ϫ4␶ ϫ Ϫ6 ͒ ͑ ͒ 5 10 ␥␥͓͑4 10 ͔E p . 26 ␬ ␴ ␴ Ϸ ␮ ␴ Here we have used ͗ p p␥͘/͗ ␥␥͘ 300 barn/ T from av- ␣ϭ ␬ ␴ eraging over a photon spectrum with 1, where ͗ p p␥͘ includes Bethe-Heitler pair production, ␴␥␥ is the total cross ␥␥→ ϩ Ϫ ͓ ͔ ␴ section for the process e e 39 , and T is the Thom- ␶ Ϸ ␶ son cross section. Using n␥(En) 2 p␥(En), and assuming ␶ ϰ 5 Շ р that n␥(E) E holds for a range 10 E␥ En Emax ,weob- tain the relation

Ϫ9 FIG. 4. From of the spectra of neutrinos and cosmic rays escap- ␶ ␥͑E ͒/␶␥␥͑E␥͒Ϸ4ϫ10 E /E␥ . ͑27͒ n n n ing from optically thick photoproduction sources, Eqs. ͑14͒ and ͑15͒. The example shown here has a break energy of 108 GeV, and We apply this relation to two reference AGN: the BL Lac 11 object Mrk501, and the quasar 3C279. The combination of an exponential cutoff above 10 GeV. the observed TeV spectrum and the EGRET flux limits for Mrk501 ͓40,41͔ gives rise to our assumption of a break en- creased relative to ␶␥␥ by a factor of 10—consequently, the ergy at about 0.3–1 TeV in the context of photohadronic possible break in the gamma ray spectrum of 3C279 would ϳ 8 models, while for 3C279 the EGRET spectrum allows for a correspond to Eb 10 GeV. A principal lower limit to the ϳ ͓ ͔ ϳ 7 break at 3–10 GeV 42 . Note that internal opacity due to break energy in EGRET sources is set at Eb 10 GeV, since the presence of low-energy synchrotron photons with spec- otherwise EGRET photons ͑տ1 GeV͒ could not be emitted. tral index ␣, which plays a crucial role in photohadronic Using cosmic ray proton production spectra with a break models for the gamma ray emission, gives rise to a spectral at ϳ108 GeV ͑see Fig. 4͒ will, of course, have a strong effect steepening by EϪ␣, and not to an exponential cutoff, above on the neutrino bound implied by cosmic ray data. Since the the energy where ␶␥␥ϭ1, see Eq. ͑11͒. A TeV power law cosmic ray proton spectrum of the source above the break being steeper than the gamma ray spectrum at EGRET ener- drops faster than the upper limit spectrum on cosmic ray gies could thus imply optical thickness in spite of reaching protons, Eq. ͑25͒, the bound at a neutrino energy E␯ ͓ ͔ Ͼ some 25 TeV as seen by the HEGRA telescopes 43 . Eb/25 is essentially set by the cosmic ray flux at the break Clearly, lower values for the optical depth are obtained if the energy Eb rather than by the more restrictive flux at 25E␯ .If data are interpreted with models which assume that the ob- all sources had the same Eb , the bound would be increased ͑ 0.75 served steepening is due to the radiation process itself e.g., roughly by a factor (25E␯ /Eb) . Clearly, the assumption synchrotron–self-Compton emission͓͒40,41͔. Applying Eq. of optical depths allowing break energies of 108 GeV or be- ͑27͒ we find that the break energy in the cosmic ray produc- low in luminous quasars is not directly supported, but only ϳ 11 tion spectra should be Eb 10 GeV in Mrk501, and Eb consistent with current observations. The maximal neutrino ϳ109 GeV in 3C279. Thus, UHE cosmic rays can escape intensity for AGN jets, which we shall derive below on the from Mrk501 under optically thin conditions, and our upper basis of this assumption, is therefore at the currently allowed bound is not affected by the actual value of the gamma ray maximum for the adopted source model, and likely to be optical depth in this source. lowered when more detailed gamma ray data become avail- We note that the break energies derived above depend on able. the assumption of an undistorted power-law target spectrum As a general limitation of this approach it must be noted of photons. This assumption might be reasonable for BL that most AGN jets have not been detected by EGRET. If Lacs, which show power-law photon spectra extending from their lack of high-energy emission is due to a large gamma the infrared ͑relevant for p␥ interactions͒ into the x-ray re- ray opacity, this would also imply very low break energies gime ͑relevant for ␥␥ absorption of ϳ10–100 GeV photons, for their ejected cosmic ray spectrum. One such possible assuming a Doppler boosting of the emission by a factor ␦ class of sources are the gigahertz-peaked sources ͑GPS͒ and ϳ10͒. In high luminosity radio quasars, however, this is not compact steep-spectrum ͑CSS͒ quasars ͓47͔, which make up the case ͑see, e.g., ͓44͔͒, and relation ͑27͒ does not necessar- about 40% of the bright radio source population. Objects ily hold. Moreover, in these sources the dominant target such as these could thus produce a diffuse neutrino intensity spectrum for p␥ and ␥␥ interactions could be given by the at the level of the EGRB or maybe even above, without external accretion disk photons, forming roughly an ͑un- violating any constraint. Also low-opacity sources for which boosted͒ power law spectrum with ␣ϭ1uptoϳ10 eV ͓45͔, the maximum proton energy remains much less than its theo- where it drops by about one order of magnitude ͑e.g., ͓46͔͒. retically allowed value and, in particular, below the value of ͑ ͒ ␶ This still implies a relation like Eq. 27 , but with n␥ in- the break energy, could lead to a higher neutrino intensity at

023003-9 KARL MANNHEIM, R. J. PROTHEROE, AND JO¨ RG P. RACHEN PHYSICAL REVIEW D 63 023003

͑ ͒ ϰ ϰ Ϫ1 low E␯ limited only by the EGRB . We did not consider this and use the relations L p LX , and Eb LX with Eb ϭ 11 ϭ ϫ 44 Ϫ1 ͑ ͒ possibility by keeping E p,max fixed. 10 GeV for LX 3 10 erg s Mrk501 . Here we ϰ have assumed that Lo LX . B. Blazar luminosity functions and generic models for the For quasars, we use the EGRET luminosity function ͓ ͔ neutrino contribution from BL Lacs and radio quasars given by Chiang and Mukherjee 51 , N ϰ Ϫ2.2 46 Ͻ Ͻ 48 Ϫ1͒ In order to obtain a parameterization of blazar neutrino d q /dL␥ L␥ ͑10 erg s L␥ 10 erg s , ␶ spectra, we need to express n␥ as a function of the blazar ͑29͒ luminosity L, given at some frequency. Here we take into ϰ ϰ Ϫ1/2 consideration that blazars are assumed to be beamed emit- and the relations L p L␥ , and Eb L␥ where we consider 48 Ϫ1 ters, with a Doppler factor ␦ϳ10. The optical depth intrinsic two possible normalizations for L␥ϭ10 erg s ͑3C279͒, ϭ 9 to the emission region is proportional to Lt /R, where Lt is which are Eb 10 GeV in case that jet-intrinsic photons ϭ 8 the intrinsic target photon luminosity and R is the size of the dominate the target field, and Eb 10 GeV for the assump- emitter. Since blazars are strongly variable objects ͓48͔,we tion that external photons are the dominant target. For illus- can use the variability time scale Tvar to estimate the intrinsic tration, we show in Fig. 4 cosmic ray and neutrino spectra on ϳ ␦ ϭ 8 ϭ 11 size by R Tvarc . Using also the relation between intrinsic emission for Eb 10 GeV and Emax 10 GeV. ϭ ␦4 ␶ and observed luminosity, L Lt , we obtain n␥ Then we obtain the form of the production spectra of ϰ Ϫ1␦Ϫ5 cosmic ray protons and neutrinos averaged over the local LTvar . Although there is no detailed study of a possible universe due to these two classes of AGN, systematic dependence of Tvar on L, the observations are compatible with no such correlation existing—for example, ͐Q ␯͑E,L͒͑dN/dL͒dL variability time scales of order 1 day are common in both, ͒ ϭ cr, ͑ ͒ ͗Qcr,␯͑E ͘ , 30 moderately bright BL Lacs like Mrk 501 or Mrk 421, and ͐L͑dN/dL͒dL ϩ powerful quasars like 3C279, PKS0528 134, or PKS1622- ͑ ͒ 297, whose optical luminosities differ by at least three orders where the input spectra Qcr , Q␯ are given by Eqs. 14 and ͑ ͒ ϭ 11 of magnitude ͓48͔. For the Doppler factors, there is no evi- 15 , with Emax 10 GeV, and Eb , L p given as functions of dence for a systematic dependence on L either ͓17͔, although L as discussed above. unification models for blazars and radio galaxies ͑see next To integrate properly over redshift, we note that while ͑ section͒ suggest that BL Lacs are on average slightly less quasars show strong evolution similar to galaxies see be- ͒ ͓ ͔ beamed (͗␦͘ϳ7) than radio quasars (͗␦͘ϳ11) ͓49͔. There- low , BL Lacs show little or no evolution 52 , and we shall fore, we may assume that for both BL Lacs and quasars take this into account when propagating these spectra ␶ ϰ ␶ through the Universe from large redshifts. For quasars, their n␥(E,L) L holds on average, and that n␥(E,L,BL Lac) ϳ ␶ ͑ ͒ luminosity per comoving volume has a pronounced peak at 10 n␥ E, L, quasar . It is interesting to note that this rela- ϳ tion would imply a relation of the break energies of Mrk501 redshifts of z 2, and declines or levels off at higher red- ϳ 44 Ϫ1 ϳ 47 shifts ͓53͔. We shall assume that this effect is due to evolu- (Lopt 10 erg s ) and 3C279 (Lopt 10 erg/s) as E (3C279)ϳ10Ϫ2E (Mrk501), consistent with our estimate tion of the number of quasars with z, rather than evolution of b b the luminosity of individual sources, which keeps the pro- obtained for intrinsic absorption from the gamma ray spec- ͗ ͘ tral break. Of course, this does not rule out the possibility duction spectra Qcr,␯(E) independent of z. A particular pa- rameterization of the redshift-dependence of the ͑comoving that Eb might be systematically lower in 3C279 due to ex- ͒ ternal photons, as argued above. frame UV luminosity density of AGNs as inferred by Boyle and Terlevich ͓37͔, assuming an Einstein–de Sitter cosmol- To determine the contribution of all blazars in the Uni- ϭ ogy and h50 1, is given by verse, we have to relate the proton luminosity L p to the blazar luminosity L in some frequency range, and then inte- ͓͑1ϩz͒/2.9͔3.4 ͑zϽ1.9͒, grate over the luminosity function dN/dL, determined for dP gal ϭ ͑ р Ͻ ͒ ͑ ͒ the same frequency range. Here we have to distinguish be- P0 ͭ 1.0 1.9 z 3 , 31 dVc tween the luminosities in the energy range where the target exp͓Ϫ͑zϪ3͒/1.099͔͑zу3͒, photons are, Lo , and the luminosity of the gamma rays L␥ , ϭ Ϯ ϫ 44 Ϫ1 Ϫ3 which are here assumed to be produced by hadronic interac- where P0 (3.0 0.3) 10 erg s Mpc . Clearly, the ϰ tions. Obviously, the latter implies L p L␥ , but on the other normalization plays no role here since L p is adapted to match ␶ ϰ hand n␥ Lo . We also have to distinguish between the two the cosmic ray flux at earth. So for BL Lacs ͑no evolution͒, ϰ ϭ classes of blazars: while for BL Lacs L␥ Lo , observations we simply use dPgal /dVc 1. ϰ 2 ͓ ͔ rather suggest that for quasars L␥ Lo 44 . This leads to the The result is shown in Fig. 5. As expected, the diffuse following simple models: neutrino fluxes from AGN jet models exceed the bound for For BL Lacs, we use the x-ray luminosity functions given optically thin sources, but fall into the allowed region for by Wolter et al. ͓50͔ for x-ray selected BL Lacs, sources optically thick for neutron emission. The BL Lac contribution falls below the bound, because it was derived N ϰ Ϫ1.6 d BL /dLX LX for a nonevolving source distribution—we note that it still exceeds the corresponding Waxman-Bahcall bound for the ͑3ϫ1043 erg sϪ1ϽL Ͻ3ϫ1046 erg sϪ1͒, X case of no evolution. In addition to the models discussed ͑28͒ above, we have also constructed and estimated an upper

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FIG. 5. ͑a͒ Comparison of neutrino spectra for optically thick pion photoproduction sources with neutrino upper bounds obtained in the present work and by Waxman and Bahcall ͓1͔. Neutrino intensities obtained in the present work plotted are maximal superposition of spectra 7 11 ͑ ͒ having Eb in the range 10 –10 GeV thick solid curve ; maximum source spectra averaged over the quasar luminosity function assuming ␶ ϰ 1/2 ϭ 48 Ϫ1 Ϸ 8 ͑ ͒ Ϸ 9 ͑ n␥ L with break energies corresponding to L 10 erg s at Eb 10 GeV thick dashed curve and at Eb 10 GeV thick dotted ͒ ␶ ϰ curve ; maximum source spectra averaged over the BL Lac luminosity function assuming n␥ L with the break energy corresponding to ϭ ϫ 44 Ϫ1 Ϸ 11 ͑ ͒ ͓ ͔ L 3 10 erg s at Eb 10 GeV thin dot-dashed curve . Also shown is the prediction by Protheroe 45 for an external photon optically thick proton blazar model normalized down in accordance with the recent estimates of the blazar contribution to the diffuse gamma ray background ͓53͔͑thick chain curve͒, and the bounds obtained by Waxman and Bahcall ͓1͔ with and without source evolution ͑note that the maximum BL Lac intensity was calculated assuming no evolution͒. Other symbols shown correspond to Fig. 3. ͑b͒ Cosmic ray intensities for the neutrino intensities obtained in the present work shown in part ͑a͓͒key to these curves is as in part ͑a͔͒. The upper bound to any extragalactic cosmic ray proton spectrum is shown by the thin line, which corresponds to the dotted line in Fig. 1. bound on the diffuse neutrino contribution on AGN jets. photoproduced neutrons are trapped, or suffer severe adia- Here we used the same method as for the construction of Fig. batic deceleration in the large-scale magnetized environment 3, but implying input spectra of the form of Eqs. ͑14͒ and such as the host galaxy and its halo, a , or a ͑ ͒ ϭ 11 15 , with variable Eb and fixed Emax 10 GeV. The break supercluster. We emphasize that although there is now useful 7 Ͻ energy Eb was then varied in the range 10 GeV Eb information available about magnetic fields in galaxies and Ͻ ϭ 11 Emax 10 GeV, and the normalization chosen such that clusters, the magnetic field structure and topology is not suf- the superposed spectra approximately represented the upper ficiently well known for us to predict reliably magnetic trap- limit on the extragalactic cosmic ray spectrum ͑Fig. 5, left ping and adiabatic losses in host galaxies and clusters. How- panel͒. We note that this upper bound corresponds within a ever, we shall discuss in this section the fate of the cosmic factor of 2 with the prediction of a previously published rays resulting from photoproduced neutrons, and show that model by Protheroe ͓45͔. Other published models, for ex- in some plausible scenarios, our bound for optically thin ample by Halzen and Zas ͓11͔, or Mannheim ͓͑10͔, model photoproduction sources can be exceeded for neutrinos from B͒, exceed our bound by about one order of magnitude at optically thin photoproduction sources. 8 energies E␯տ10 GeV, but their predictions for the impor- X-ray observations and measurements of extragalactic 7 tant energy range below ϳ10 GeV are compatible with this Faraday rotation suggest that structures surrounding compact bound. We return to the discussion of these models later, AGN jets carry magnetic fields of the order of 0.1–10 ␮G after we discuss the effect of magnetic fields in the AGN ͓54͔. These can influence the propagation of cosmic ray pro- environment in the next section. tons in essentially two ways: ͑a͒ particles may be physically у ϭ ͑ ͒ confined in the structure for a time t tH 1/Ho ,or b the diffusive escape of the particles can lead to adiabatic energy V. MAGNETIC FIELDS AND THEIR IMPACT losses. Obviously, magnetic fields on scales larger than the ON COSMIC RAY PROPAGATION ␤ Ϸ mean free path of a neutron for decay, ln 10 kpc 9 As discussed in Sec. II, protons accelerated inside AGN (En/10 GeV), also affect cosmic rays which are ejected as jets are likely to be trapped in the jet to be released later near neutrons from the source. Here we shall discuss what influ- the end of the jet, and will consequently suffer severe adia- ence these effects can have on the strength of the measured batic losses as a result of jet expansion. The bound that we cosmic ray flux relative to the corresponding neutrino flux. calculated for optically thin photoproduction sources shown We shall estimate the critical energy E* below which our in Fig. 3 may of course be exceeded for optically thick bound could plausibly be exceeded for optically thin photo- sources. However, it may even be exceeded for optically thin production sources for a number of scenarios, in particular photoproduction sources if the cosmic rays resulting from for clusters of galaxies and radio galaxies hosting AGN.

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A. Particle confinement in clusters and superclusters to the plasma flow, we require that in order to escape from Clusters of galaxies have been recently discussed in the the cluster, the average ‘‘speed’’ of particle diffusion, ϳ literature as a possible ‘‘storage room’’ for cosmic rays ͓55– D/Rcl , exceeds the inflow velocity. If we assume the ϳ 57͔, because of their relatively strong magnetic fields (B lower value for the magnetic field at Rcl 3 Mpc, i.e., B0 0 ϳ ␮ ϳ տ ␮ 0.1 G and a0 200 kpc we obtain a critical energy 1 G) extending over large scales, i.e., cluster radii, Rcl տ1 Mpc. The time scale for diffusive escape of cosmic rays B͑R ͒ 6 cl in a turbulent magnetic field of homogeneous strength B0 E* Ϸ2ϫ10 GeVϫͫ ͬ acc 0.1 ␮G within a central radius R0 is given by 3 ͒ Ϫ2 2 2 Rcl vacc a͑Rcl R0 3eB0R0 ϫͫ ͬ ͫ ͬ ͑ ͒ ϳ ϭ ͑ ͒ Ϫ1 . 35 tesc ␭ ͒ , 32 3 Mpc 300 km s 200 kpc D cEcr ͑Ecr Of course, with higher magnetic fields and a shorter field- where we have used for the diffusion coefficient D ϭ 1 ␭ ␭ reversal length a0 , Eacc* would be higher. Similar processes 3 rLc, and (Ecr) is the scattering length in units of the ϭ would occur in the cooling flows observed in some rich clus- Larmor radius, rL Ecr /eB0 of a cosmic ray proton of en- ␭ ters harboring powerful radio galaxies, but the confinement ergy Ecr . The function depends on the turbulence spectrum Ϫ energies for typical cooling flow parameters ͓64,65͔ are even of the magnetic field, ͓␦B(k)͔2ϰk y, where k is the wave lower. Also the effect of drift in the nonhomogeneous cluster number. Cosmic ray scattering is dominated by field fluctua- ͓ ͔ ϳ Ϫ1 field 1 is small compared to diffusion, and cannot lead to tions on the scale of the Larmor radius, i.e., k rL . This larger confinement energies. implies that Ensslin et al. ͓57͔ point out that simple diffusion, as as- sumed above, may not be the best description of cosmic ray eB a y ␭͑ ͒ϭͫ 0 0ͬ Ͻ ͑ ͒ propagation in clusters of galaxies, and suggest that a one- Ecr for Ecr eBa0 , 33 Ecr dimensional random walk along static, but randomly tangled,

Ϫ magnetic field lines may be more realistic for particles with where a ϭk 1 is the largest scale of the turbulence, i.e., the Ӷ 0 min rL a0 . In this case the field line topology may be consid- ‘‘cell size’’ ͑or ‘‘reversal scale’’͒ of the magnetic field. For ered as arising from a three-dimensional random walk with Ͼ Ϫ1 k a0 , the magnetic turbulence in clusters of galaxies steps of size a0 , yielding an effective diffusion length of seems to be well described by the Kolmogorov law for fully ϭ 2 R0 Rcl/a0 . For the escape time we then obtain ϭ 2 developed hydrodynamical turbulence, i.e., y 3 . This can 4 1/3 be easily seen from relating the typical turbulent magnetic Rcl eB0a0 t ϭ ͫ ͬ տt for E ӶeB a , ͑36͒ field and cell size found from Faraday rotation measure- esc 3 E H cr 0 0 ϳ ϳ ␮ ͓ ͔ 2ca0 cr ments, a0 20 kpc and B0 1 G 58,59 , to the diffusion ϳ coefficient found for electrons of energy 1 GeV which means for typical cluster parameters that essentially ͓ ͔ from the synchrotron radio emission spectrum 60 , all cosmic rays with E Շ3ϫ109 GeV are confined to the ϳ ϫ 29 2 Ϫ1 Ͼ Ͼ cr D 2 10 cm s . For Ecr eBa0 , i.e., rL a0 , the par- cluster. Of course, this result neglects cross-field diffusion so ϳ ticle motion is a random walk with scattering angles a0 /rL that the maximum confinement energy is probably lower, but in each step. This can also be approximately described by it could still be higher than the result obtained in Eq. ͑34͒ for ͑ ͒ ͑ ͒ ϭϪ Eqs. 32 and 33 by setting y 1. the case of simple diffusion, in particular in the likely case Confinement of cosmic rays over the cluster radius, R0 that the field strength decreases with the distance from the ϭ ϳ Ͼ Rcl 3 Mpc, is then obtained for tesc tH , corresponding cluster core. to a critical energy Confinement of cosmic rays in clusters would lead to a

6 Ϫ2 decrease of the cosmic ray flux measured at earth relative to B0 Rcl a0 E* Ϸ5ϫ108 GeVϫͫ ͬͫ ͬ ͫ ͬ , ͑34͒ the corresponding neutrino flux, causing an increase of the diff 1 ␮G 3 Mpc 20 kpc neutrino bound. It is important to note, however, that cosmic ray confinement may exist even on larger scales, i.e., super- Ͼ 2 provided a0 1 kpc (Rcl/3 Mpc) . This assumes that the clusters. It has been shown that magnetic fields ϳ0.1 ␮Gin magnetic field strength is homogeneous over the entire clus- superclusters are consistent with observations, and expected ϳ ␮ ϳ ter. However, if it dropped to 0.1 G, and a0 200 kpc at in simulations of structure formation which also predict ac- ϳ 3 ϳ Ϫ1 ͓ ͔ the edge of the cluster then Ediff* would be a factor 10 cretion of gas with speeds vacc 1000 km s 66,67 . The lower. On the other hand, some observations suggest also cosmic ray confinement energies for these larger scales (R larger magnetic fields on smaller reversal scales ͓61,62͔, ϳ10 Mpc, aϳ1 Mpc) could therefore be տ108 GeV, where which would imply higher confinement energies. again topological aspects connected to the detailed structure The scenario above assumes that the background plasma of the field may allow even higher values. Since our Galaxy filling the cluster is at rest. However, simulations of structure itself is located in a supercluster, this latter scenario would formation suggest that clusters of galaxies are accreting ex- tend to increase the cosmic ray flux relative to the corre- tragalactic hot gas ͓63͔, forming inflows of typical speeds sponding neutrino flux, which fills the Universe տ Ϫ1 vacc 300 km s downstream of an accretion shock near the homogeneously—thus it could actually decrease the bound 7 outer radius of the cluster ͓7͔. Since particles diffuse relative below E␯ϳ10 GeV. The net effect of confinement in clus-

023003-12 COSMIC RAY BOUND FOR MODELS OF... PHYSICAL REVIEW D 63 023003 ters and our local supercluster is obviously strongly model The observed synchrotron spectra from radio lobes imply dependent, and therefore difficult to estimate. typical magnetic fields in the range 10–50 ␮G, turbulent An effect of the unknown structure of magnetic fields on with a maximum scale ͑or cell-size͒ of ϳ0.5 kpc ͓72,71͔. galaxy cluster and supercluster scales may therefore most The observed asymmetric depolarization in double radio gal- likely affect the observed extragalactic cosmic ray flux at axies leads to the suggestion that the magnetized plasma 6 energies below ϳ2ϫ10 GeV, corresponding to E␯ around the radio galaxy extends into a halo of radius ϳ300 ϳ105 GeV. On the other hand, we can exclude an effect only kpc, with Bϳ0.3 ␮G and a cell size of ϳ5 kpc at a radius տ ϫ 9 տ 8 ϳ ͓ ͔ for energies Ecr 3 10 GeV (E␯ 10 GeV), but we note R 100 kpc 73 . The properties of the magnetic field in the that cosmic ray confinement at energies higher than central lobe and the halo can be connected by assuming that 8 ϭ Ϫ2 ϳ10 GeV requires extreme assumptions on the strength and magnetic field and cell size scale as B B0(R/R0) and a ϭ ϳ ␮ ϳ topology of the magnetic fields. We therefore agree with a0(R/R0), respectively, with B0 30 G, a0 0.5 kpc, ͓ ͔ ϭ the conclusion of Waxman and Bahcall 1 that at and R0 10 kpc. ϳ ϫ 8 E␯ 5 10 GeV an effect of large scale magnetic fields on Within radius R0 , we assume the properties of the turbu- the relation of cosmic ray and neutrino fluxes cannot be ex- lent magnetic field are constant. This corresponds to the as- pected. sumption of an isotropic magnetic field expanding in a ϭ ϳ plasma outflow with Rmin R0 and Rmax 300 kpc, which B. Adiabatic losses in expanding radio lobes and halos will be used as a working hypothesis in the following. We can consider cosmic ray protons as nearly isotropized in the Although many galaxy clusters have powerful radio gal- Ͻ plasma if rL(R) a(R), corresponding to energies Ecr(R) axies at their centers, it is also a fact that most powerful radio ϽE* (R) with galaxies are not found in such environments ͓68,69͔. For a ad radio galaxy located in the normal extragalactic medium, B a R Ϫ1 evidence has been found that a pressure equilibrium with the ͑ ͒ϭ 10 ϫͫ 0 ͬͫ 0 ͬͫ ͬ ͑ ͒ Ead* R 10 GeV 38 external medium cannot be obtained within the lifetime of 30 ␮G 0.5 kpc 10 kpc the source (Շ108 yr) ͓70͔. Therefore, it can be concluded that the lobes must expand. For a sample of powerful double- for 10 kpcϽRՇ300 kpc. To justify the assumption of near lobe ͑or FR-II͒ radio galaxies, lobe propagation velocities of isotropy below E* (R), we have to show that advection in Ϫ ad order 104 km s 1 have been inferred ͓71͔. Since the aspect the flow indeed dominates over diffusion, i.e., that the cen- ratio of the sources is found to be independent of their size, ters of the diffusive cloud of cosmic ray protons move ap- consistent with a propagation of the lobes along a constant proximately steady with the flow. Indeed, for the parameters opening angle of ϳ10°, expansion velocities of the lobes and adopted above we find for the ‘‘speed’’ of diffusive escape ϳ Ϫ1 ϳ 1 Ͻ р 1 the connecting ‘‘bridges’’ of vex 1000 km s can be in- D/R 3 a(R)vcell /R vex , where vcell 3 c is the average ferred. velocity of the particles to cross the cell, considering that Neutrons produced by pion photoproduction interactions also this motion is diffusive if the magnetic turbulence pro- at an acceleration site near the base of the jet will be beamed ceeds to smaller scales as expressed in Eq. ͑33͒. preferentially along the jet direction decaying farther out Consequently, we can consider the cosmic rays with Ecr along the jet or in the radio lobe, where they decay. The Շ Ead* (R) as are advected with the flow at a distance R from resulting cosmic ray protons will then be advected with the the center of the galaxy, so that they suffer adiabatic losses outflowing plasma on a time scale of the galaxy lifetime following Eq. ͑37͒ leading to E (R)ϰRϪ1. This implies that ϳ 8 Ӷ cr 10 yr tH . Additionally, the protons perform a random Ecr /Ead* is independent of R, i.e., that cosmic rays confined in walk in the magnetic field, which may even decrease their the outflow at some radius R remain confined for larger radii. confinement time. Expanding radio lobes can therefore not If we consider cosmic rays which are ejected as neutrons, the confine cosmic rays indefinitely. However, in plasma out- radius where they couple to the magnetic field is given by the flows all particles which are isotropized in the flow due to ␤ Ϸ 9 -decay mean free path, ln 10 kpc (En/10 GeV). The re- scattering with plasma turbulence will experience adiabatic sulting protons are confined to the outflow and are subject to losses before their release. Kinetic theory implies that the adiabatic losses provided E ϽE* (l ), or particle energy at ejection from the flow is related to the n ad n injected particle energy by B 1/2 a 1/2 Ͻ ϫ 9 ͫ 0 ͬ ͫ 0 ͬ ͑ ͒ En 3 10 GeV . 39 Eej Rinj 30 ␮G 0.5 kpc ϭ . ͑37͒ Einj Rmax Ϸ The energy on ejection from the outflow is then Eej En/30 Ͻ 9 ϭ 2 ϫ 10 9 Here, Rmax is the radius of the outer termination shock of the for En 10 GeV, and Eej En/(3 10 GeV) for 10 GeV Ͻ Ͻ ϫ 9 Ͼ ϫ 9 flow, and Rinj is in general given by some minimum radius En 3 10 GeV. For En 3 10 GeV, cosmic rays ͑ ͒ Rmin where the outflow starts. For the case of neutron ejec- neutrons and protons traverse the lobe-halo along almost ϭ ͑ tion from a central source, as considered here, Rinj ln if ln straight paths and adiabatic losses do not apply. Note that Ͼ ϭ Rmin and Rinj Rmin otherwise. Note that the lobes we are the corresponding energies where these modifications may ϳ considering here are much more extended than the jets in affect the neutrino bound are E␯ En/20, thus E␯ which the particles are accelerated. Ͻ108 GeV.͒

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Energy losses of cosmic rays in the lobes and halos of structure, and the distribution of cosmic ray–neutrino radio galaxies are of particular relevance for models of neu- sources. We therefore do not regard the bound based on the trino production in AGN jets, which we discussed in Sec. IV. cosmic ray flux as a firm upper limit on neutrino fluxes be- These models apply to radio loud AGN, which are likely to low about 105 GeV, the main energy region for underwater- be the beamed counterparts of radio galaxies ͓49͔. The two ice Cherenkov experiments such as the Antarctic Muon and classes of AGN discussed in the last section correspond to Neutrino Detector Array ͑AMANDA͓͒75͔. Models which ͑ ͓ ͔͒ the two Fanaroff-Riley FR 74 classes of radio galaxies: predict neutrino emission mainly in this energy range are radio quasars might be associated to the powerful double- therefore not rigorously bounded by cosmic ray data, as for lobed FR-II radio galaxies, while BL Lac objects might cor- example the model of Berezinsky et al. ͓56͔ predicting neu- respond to the less luminous FR-I radio galaxies which gen- trino fluxes from cosmic rays stored in clusters of galaxies. erally have diffuse lobes centered around the AGN. The Such sources could in principle produce neutrino fluxes al- parameters used above for the lobes-halos of radio galaxies most up to the EGRB limit within the AMANDA range. were mainly obtained from observations of FR-II radio gal- However, the neutrino contribution from clusters of galaxies axies or radio quasars. Therefore, the cosmic ray ejection is expected to be much lower, as it is limited by their ex- ͑ ͒ from radio quasar FR-II sources can be expected to be di- pected contribution to the EGRB of ϳ1% ͓76͔. minished by more than an order of magnitude below At neutrino energies above 109 GeV, the upper limit rises ϳ 9 10 GeV. For the less luminous FR-I galaxies, it could be up to the point where the energy flux of secondary gamma that the magnetic fields, turbulence scales, and halo sizes rays increases above the level of the observed EGRB. The used above are overestimated, so that Ead* could be lower by reason for the increasing bound is that while cosmic rays about one or two orders of magnitude for these sources. from evolving extragalactic sources above the nominal GZK cutoff reach us exponentially damped due to interactions VI. DISCUSSION AND CONCLUSIONS with the microwave background, neutrinos reach us essen- tially unattenuated. An observed excess of cosmic ray events We have looked at the problem of to what extent a pos- above the GZK cutoff would therefore correspond to an even sible flux of extragalactic neutrinos in the broad energy range more pronounced excess of neutrinos in the framework of 5 11 10 GeVՇE␯Շ10 GeV is bounded by the observed flux of such models. Cosmic rays from a fiducial class of extraga- cosmic rays and gamma rays. As the minimum contribution lactic photohadronic sources with a sufficiently flat spectrum to the cosmic rays we consider the neutrons produced to- could thus saturate the rising bound at the highest energies gether with other neutrals such as gamma rays and neutrinos and would still remain below the local cosmic ray flux at all in photohadronic interactions. The most restrictive bound energies. Clearly, this is not the most likely scenario to ex- arises for sources which are transparent to the emission of plain the events beyond the GZK cutoff. For example, if neutrons and where the protons from the decaying neutrons these events are due to a single strong nearby source, then we are unaffected by large-scale magnetic fields. The bound for would not expect the extragalactic neutrino flux above this case is approximately in agreement with the bound pre- 109 GeV to be at a level higher than given by the bound viously computed by Waxman and Bahcall ͓1͔ in the neu- computed by Waxman and Bahcall, even if this source 7 9 trino energy range 10 GeVՇE␯Շ10 GeV but is higher at would have the spectral properties which we used to con- lower and at higher energies. The difference is a conse- struct our bound in Sec. III C. On the other hand, we note quence of the different approaches: Waxman and Bahcall that the flux from non-photohadronic sources, as, for ex- assume a particular model spectrum, which could explain ample, from the decay of topological defects, can exceed this the observed UHECR spectrum, but which has a pronounced bound because of their different branching ratios for the pro- GZK cutoff that is not clearly confirmed by the present low duction of baryons relative to mesons ͓12͔. Measuring the statistics data, and which does not reproduce the steep slope neutrino flux in this energy region, as is planned using large of the observed CR spectrum below ϳ1010 GeV. It thus air-shower experiments such as the Pierre Auger Observa- leaves room for additional contributions from extragalactic tory ͓77͔, would therefore be highly relevant for understand- sources ͑with different spectra͒ outside the narrow energy ing the nature and cosmic distribution of UHECR sources. range where it matches with the observed cosmic ray flux. The bound we derive for extragalactic neutrinos is indeed Our approach considers this possibility by using the best an upper limit, since we do not consider the possibility that currently available experimental upper limit on the extraga- ultrarelativistic protons are ejected from their sources with- lactic proton contribution. We have also pointed out in Sec. out preceding isospin flip interactions ͑‘‘prompt protons’’͒. II C, referring to recent Monte Carlo simulations ͓15͔, that Obviously, any additional prompt proton emission would the fundamental properties of photohadronic interactions can have the effect to lower the relative neutrino flux consistent affect the bound by up to a factor of 5. with the upper bound. However, for the most interesting At neutrino energies below 107 GeV, the cosmic ray sources, i.e., GRBs and AGN jets, the ejection of prompt bound computed in this work rises and equals the bound protons can be expected to be strongly suppressed due to the inferred from the EGRB flux at about 105 GeV, below which rapid expansion of the emission region and the implied large the EGRB constraint is tighter. In the same energy region, adiabatic losses. This will allow us in the future to use mea- we have also shown that effects from extragalactic magnetic sured neutrino fluxes, or experimental limits thereof, as a fields come into play; they could either increase or reduce limit to the contribution of these sources to the cosmic ray the bound, depending on the details of the field strength and flux. If, for example, GRBs produce the observed UHECR

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flux and if the global GRB emissivity follows a similar evo- At present, the most general bound to EGRET detected AGN lution as star formation and AGNs, bursts would have to is given by their contribution to the EGRB. The most ex- emit a neutrino flux on the level of their optically thin bound, treme class of neutrino sources opaque to the emission of which is roughly two orders of magnitude above the original UHECRs could be AGN jets which have not been detected prediction ͓18͔. This is independent of the total fractional in gamma rays. Prime candidates are the GPS and CSS qua- burst energy converted by the protons in their interactions. A sars mentioned in Sec. IV A, which together make up about factor of ϳ3–6 is due to evolution, a factor of ϳ10 due to 40% of the bright extragalactic radio sources. the limited efficiency of p→n conversion ͓78͔, and an addi- Finally, we wish to clarify that our result is not in conflict tional factor ϳ5 due to an accurate treatment of photopro- with, but complementary to, the upper limit previously ob- duction yields in the hard photon fields typical for GRBs tained by Waxman and Bahcall ͓1͔. Their result applies to ͑note that the yield factors used in our figures correspond to photohadronic sources with a particular spectral shape which AGN-like target photon spectra͒. In this case, the gamma are transparent to the emission of UHE cosmic rays, such as rays would also account for most of the EGRB, and one may be representative of low-luminosity BL Lacertae objects could observe strong TeV bursts from nearby sources ͑like Mrk501͒ or GRBs. Our result is more general, and ͓79,80͔. Currently, only AGN jets are known to contribute to therefore less restrictive, indicating that there may be other the EGRB in a major way and this has motivated the original classes of sources, such as quasars, with a different spectral model A of Mannheim ͓10͔͑which is exactly on the level of shape, and/or which are opaque to the emission of UHE cos- the optically thin bound as a consequence of the assumption mic rays, which can produce a higher neutrino flux than the that AGN jets produce the observed UHECRs͒. We note that source classes considered by Waxman and Bahcall. We also the neutrino flux due to nucleons escaping the AGN jets and confirm the claim by Waxman and Bahcall that large scale diffusing through the surrounding AGN host galaxies is dif- magnetic fields are unlikely to have an effect for cosmic ray ficult to assess and limited by the EGRB only ͓10͔, although propagation at ϳ1010 GeV, but we additionally point out that this neutrino flux is the one most relevant in the 1–100 TeV magnetic field effects cannot be disregarded at lower cosmic range. ray energies. We show that hadronic processes in AGN Generally, the relevance of a bound on optically thin which are optically thick to the emission of UHE cosmic sources should not be overestimated. A large number of ex- rays could produce the extragalactic gamma ray background tragalactic neutrino sources could be opaque to the emission according to present observational constraints. However, the of UHE cosmic rays producing a neutrino flux well above limiting model for EGRET-detected AGN presented in this this bound. For the particularly important class of AGN jets, paper is mainly determined by the current instrumental limits we have therefore developed an upper bound to their total of gamma ray astronomy—future observations in the GeV- neutrino flux using constraints on their gamma ray and neu- to-TeV energy range may impose stricter bounds, and there- tron opacity set by present gamma ray data. This maximum fore also limit the possible hadronic contribution to the contribution of AGN jets to the extragalactic diffuse neutrino EGRB. Together with an independent estimate on the total flux is up to an order of magnitude higher than our optically contribution of AGN jets to the EGRB, this may allow one to thin bound in the energy range between 107 GeV and constrain the overall cosmic ray content of AGN jets in the 109 GeV. Additionally, we have discussed that for energies near future. below ϳ108 GeV an influence of magnetic fields in the radio lobes of active galaxies or on larger scales cannot be ex- ACKNOWLEDGMENTS cluded. For example, the model of Halzen and Zas ͓11͔ pre- dicts a neutrino flux about one order of magnitude above our R.J.P. is supported by the Australian Research Council. 8 upper limit for AGN at E␯ϳ10 GeV, which is certainly J.P.R. acknowledges support by the EU-TMR network extreme, but still cannot be considered as completely ruled Astro-Plasma Physics, under contract number ERBFMRX- out, because at this energy the flux of neutrons after turning CT98-0168. K.M. acknowledges support by the Deutsche into cosmic ray protons may be diminished by an additional Forschungsgemeinschaft. We thank Torsten Ensslin for help- factor ϳ10 due to adiabatic losses in expanding radio halos. ful comments, and Anita Mu¨cke for reading the manuscript.

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