Ultra-high energy cosmic rays, cascade gamma rays, and high-energy neutrinos from gamma-ray bursts

C D Dermer^1,3 and A Atoyan²

¹ Space Science Division, Code 7653, US Naval Research Laboratory, 4555 Overlook Ave., SW, Washington, DC 20375, USA
² Centre de Recherches Mathématiques, Université de Montréal, Montréal H3C 3J7, Canada

³ Author to whom any correspondence should be addressed.

Email: dermer@gamma.nrl.navy.mil and atoyan@crm.umontreal.ca

Received 25 April 2006
Published 31 July 2006

Abstract. Gamma-ray bursts (GRBs) are sources of energetic, highly variable fluxes of γ rays, which demonstrates that they are powerful particle accelerators. Besides relativistic electrons, GRBs should also accelerate high-energy hadrons, some of which could escape cooling to produce ultra-high energy cosmic rays (UHECRs). Acceleration of high-energy hadrons in GRB blast waves will be established if high-energy neutrinos produced through photopion interactions in the blast wave are detected from GRBs. Limitations on the energy in non-thermal hadrons and the number of expected neutrinos are imposed by the fluxes from pair-photon cascades initiated in the same processes that produce neutrinos. Only the most powerful bursts at fluence levels 3× 10 - 4 erg cm - 2 offer a realistic prospect for detection of   TeV neutrinos. Detection of high-energy neutrinos is likely if GRB blast waves have large baryon loads and Doppler factors    200. Cascade γ rays will accompany neutrino production and might already have been detected as anomalous emission components in the spectra of some GRBs. Prospects for detection of GRBs in the Milky Way are also considered.

Contents

• 1. Introduction
• 2. Photohadronic processes
• 3. Neutrinos and cascade gamma rays
• 4. Cosmic rays from GRBs
• 5. GRBs in the local universe
• 6. Summary and conclusions
• Acknowledgments
• References

1. Introduction

There is general consensus that acceleration of cosmic rays by supernova remnants (SNRs) is the main source of galactic cosmic rays at energies below ~100 TeV (e.g. [1]). It is also generally thought that all cosmic rays with energies up to at least the second knee in the cosmic-ray spectrum at E₂~ 3× 10¹⁷ eV (e.g. [2]) or even up to the ankle at E_ank≊ 3× 10¹⁸ eV are produced in our Galaxy (see, e.g. [3]). Meanwhile, cosmic-ray acceleration to energies significantly exceeding ≈ 10¹⁴ eV with the conventional mechanism of nonrelativistic first-order shock acceleration by SNRs from typical (Type Ia and II) supernovae (SNe) is problematic [4]. The origin of the knee in the cosmic-ray spectrum in the form of a spectral-index break in the power-law all-particle spectrum by ≈ 0.3 units at E₁ ≊ 3× 10¹⁵ eV, accompanied by a change in the cosmic-ray composition, seems to suggest a new contribution to cosmic rays at these energies.

Gamma-ray bursts (GRBs) have been proposed as effective accelerators of cosmic rays in the universe [5] and as probable sources of cosmic rays up to ultra-high energies (UHE) in our Galaxy [6]. Cosmic rays with energies below ~ 100 TeV are produced in the conventional scenario of continuous injection due to nonrelativistic shock acceleration by SNRs formed in all types of SNe, with subsequent modification of the source spectrum through energy-dependent propagation (see [1, 7]). GRBs in our own Galaxy are very rare and probably take place at a rate    10 ^- 4 year ^- 1. Our best opportunity to establish cosmic-ray acceleration in GRB blast waves is by looking for hadronic emission signatures in the spectra of GRBs, or characteristic signatures from remnants of past GRBs. Conclusive evidence for cosmic-ray acceleration in GRBs will come from the detection of high-energy neutrinos.

In the relativistic blast-wave model for GRBs, photohadronic production is the most important mechanism for producing high-energy neutrinos. In section 2, the photohadron process is described and an approximation used for computing the spectra of secondaries formed in photohadronic processes, which is also useful for analytical studies, is given. Calculations of secondary neutrino and cascade gamma-ray production are presented in section 3, and the requirements for detection of high-energy neutrinos from GRBs are derived. Models for UHECRs from GRBs are described in section 4. A discussion of signatures of GRBs in the Galaxy is found in section 5, where we also briefly consider recent reports that GRBs favour sites of low metallicity. Summary and conclusions are given in section 6.

2. Photohadronic processes

If high-energy hadrons are accelerated by GRB blast waves, then photohadronic processes, which require the presence of target photons, are the most important hadronic energy-loss mechanisms. The target photon field is simply the highly variable radiation formed in the GRB blast wave that is detected as the GRB. Secondary nuclear production, by contrast, requires a large target particle density that would make the GRB energetics untenable [8].

The most important photohadronic process for hadronic energy losses in GRB blast waves is the photopion reaction, which can be written as p + γ → N  +  π, where N stands for a proton or neutron. Another important photohadronic process is photopair production, which can be written as p + γ → p  +  e ⁺   +  e ^- . A large fraction of the initial proton energy is lost in a photopion reaction, so only a few scatterings are required for the proton to lose most of its initial energy. By contrast, only a small fraction of the initial proton energy is lost in photopair production, so hundreds of scatterings are needed for a proton to lose most of its initial energy through this process. Although the photopair reaction is not so important for hadronic energy losses in GRB blast waves, it can play an important role in the evolution of the UHECR spectrum throughout intergalactic space [9].

The photopion process p  +  γ → N  +  π has a threshold photon energy ε_th  =  m_π  +  m_π²/2m_p 150 MeV, and its cross-section is shown in figure 1. Four separate contributions to the total photopion cross-section [10] are shown, namely resonance production involving, for example, the Δ ⁺ (1232) resonance; direct production, which refers to residual, non-resonant contributions to direct two-body channels; multi-pion production; and diffractive scattering.

A useful approximation [8] to the photopion process is to treat it as the sum of two channels, namely the single-pion resonance channel, where the proton loses 20% of its energy on average through the reaction

and a multi-pion channel, where the proton loses on average 60% of its initial energy. The latter channel is assumed to be equally divided into secondary π⁰,π ⁺  and π ^-  particles. In the p + γ → Δ ⁺  → p + π⁰ channel, the neutral pion decays into two γ rays with ≈10% of the energy of the initial proton. Following production, the γ rays will usually convert into e ⁺ -e ^-  pairs through the γγ → e ⁺ -e ^-  absorption processes, initiating an electromagnetic synchrotron/Compton/pair-production cascade.

In the p + γ → Δ ⁺  → n + π ⁺  channel, the decay of the charged pion produces three neutrinos and a positron in the reaction chain

Unless the neutron first interacts with a photon through the photohadronic process and is converted into a proton, it decays with a mean lifetime t_n 886 s [12] in the rest frame through the β-decay reaction

The β-decay electron and neutrino have energies ≈ 1 MeV in the neutron's rest frame. In a single pγ interaction leading to the production of a single π ⁺ , four ν and two leptons are therefore formed, with one of the neutrinos and one of the leptons having ≈ 50× less energy than the others.

The two-step function approximation [8, 13] for the photopion cross-section is given by

and inelasticity

as shown in figure 1. Hence

In this approximation, the lower-energy step function approximates the Δ(1232) resonance production process, and the higher-energy step function approximates the multi-photon production process. For the Δ resonance, isospin statistics imply [14] a charge-changing ratio of 1/3 for single resonance production. Thus the probability of the processes pγ → nπ ⁺  and pγ → pπ⁰ is in the ratio 1:2. In the multi-pion limit, the production of π ⁺ , π ^-  and π⁰ approaches 1:1:1.

3. Neutrinos and cascade gamma rays

Much effort has been devoted to understanding radiative signatures of leptons accelerated in GRB blast waves (e.g. [15]), and this approach can also be applied to hadronic acceleration [16]. Consider a GRB blast wave with Lorentz factor Γ. Relativistic protons or hadrons are assumed to be injected in the comoving frame of the blast wave with a number spectrum E_p ^- s at comoving proton energies E_p  >  Γ GeV up to a maximum proton energy determined by the condition that the particle Larmor radius is smaller than both the size scale of the emitting region and the photomeson energy-loss length. The injection index s ≈ 2.2, as suggested by relativistic shock acceleration and particle injection determined by analyses of GRB afterglows [17].

The total photon fluence Φ  =  ∫ _- ∞^∞dt ∫₀^∞dE E(E,t) of a GRB, where (E,t) is the differential photon number flux, must be at the level of Φ 10 ^- 4 ergs cm ^- 2 to be detected with a kilometre-scale neutrino telescope such as IceCube, as we now show. For the detection of muon neutrinos, the best sensitivity of a neutrino telescope for detecting a spectrum of neutrinos that is falling ε_ν ^- p with p >  1 is near 100 TeV≈ 160 ergs. This is because of the linear increase of the detection probability 100 TeV) at ε_ν 1 TeV [18], and the increasingly large cosmic-ray induced neutrino background at ε_ν    100 TeV. The detection of neutrinos therefore requires that the neutrino fluence . Because the differential neutrino fluence E²_ν(E)  =  E²∫ _- ∞^∞dt  _ν (E,t), where _ν (E) is the differential neutrino number flux, will be spread over several orders of magnitude, it is necessary that Φ 10 ^- 4 ergs cm ^- 2 in order to produce a sufficient neutrino fluence for detection, given that the ν_μ fluence will generally be smaller than the photon fluence even in the optimistic case that the photon radiation originates substantially from hadronic processes.

In calculations presented here to illustrate the astrophysical importance of photomeson production and subsequent electromagnetic cascades, the observed synchrotron spectral flux in the prompt phase of the burst is parameterized, as shown in figure 2, by the expression F(ν) ν ^- 1 (ν/ν_br)^α, where hν_br = 300 keV, α  =   - 0.5 above ν_br and an exponential cutoff at 10 MeV, and α =  0.5 when 10 keV ≤ hν ≤ h ν_br. At lower energies, α =  4/3. The observed total hard x-ray (keV-MeV) photon fluence Φ_tot t_dur∫₀^∞dν F(ν), where t_dur is the characteristic duration of the GRB. We consider a source at redshift z  =  1 and assume the hard x-ray fluence Φ_tot 3× 10 ^- 4 erg cm ^- 2. One or two GRBs should occur each year above this fluence level. Here we take s  =  2.

A total amount of energy E '  =  4π d_L²f_CR Φ_tot δ ^- 3 (1 + z) ^- 1 is injected in the form of accelerated proton energy into the comoving frame of the GRB blast wave. Here z is the redshift, d_L is the luminosity distance, and

is the Doppler factor. The factor f_CR is the baryon-loading factor, which gives the ratio of energy deposited in non-thermal hadrons compared to the energy detected as electromagnetic radiation, which is assumed to be provided by non-thermal electrons. The energy deposited into each of N_sp light-curve pulses (or spikes) is therefore E_sp '  =  E '/N_sp ergs. We assume that all the energy E_sp ' is injected in the first half of the time interval of the pulse (to ensure variability in the GRB light curve), which effectively corresponds to a characteristic variability time scale t_var  =  t_dur/2N_sp. The proper width of the radiating region forming the pulse is Δ R ' t_varcδ/(1 + z), from which the energy density of the synchrotron radiation can be determined [19]. We set the GRB prompt duration t_dur  =  100 s, and let N_sp  =  50, corresponding to t_var  =  1 s.

The magnetic field is determined by assuming equipartition between the energy densities of the magnetic field and the non-thermal electron energies inferred from the synchrotron radiation spectrum. For the parameters considered here, the equipartition magnetic field range from several 100 G to a few kG (depending on δ). For fields B    kG, energy losses of the pions and muons can introduce a break in the ν_μ spectrum at multi-PeV energies [20], but we neglect that effect in our calculations. The constraint on maximum possible proton energy imposed by the condition that the gyroradius is less than the source size is also imposed. In addition, relativistic expansion of the blast wave causes adiabatic cooling of the particles, which limits neutrino production for a duration set by the blast-wave shell light-crossing time.

In figure 3 we show the neutrino fluence expected in the collapsar GRB scenario from a model burst with photon fluence Φ_rad  =  3× 10 ^- 4 erg cm ^- 2 at redshift z  =  1. In order to demonstrate the dependence of the neutrino fluxes on δ, we consider 3 values of δ. The value of f_CR is set equal to 20 in this calculation. Here we take s  =  2.2. The numbers of muon neutrinos that would be detected from a single GRB with IceCube for these parameters and with δ  =  100,  200 and 300 are N_ν  =  1.32, 0.105 and 0.016, respectively. We should note, however, that for the assumed value of f_CR, the calculated total fluence of neutrinos (both ν_μ and ν_e) produced when δ  =  100 is Φ_ν,tot  =  7.2 × 10 ^- 4 erg cm ^- 2, i.e., by a factor 7.2/3  =  2.4larger than the assumed radiation fluence. This means that the maximum value of the baryon loading that could be allowed if the high-energy radiation fluence is less than the X/γ fluence for this particular case should be about 8-10, instead of 20, in order that the hadronic cascade γ-ray flux is guaranteed not to exceed the measured photon flux. Consequently, the expected number of neutrinos for δ  =  100 should be reduced to ≊ 0.6. On the other hand, the neutrino fluence for the case δ  =  200 (300) is equal to Φ_ν,tot  =  1.4 × 10 ^- 4 (3 × 10 ^- 5) erg cm ^- 2, so this accommodates an increased baryon-loading from    20 up to f_CR≊ 45 (200), with the expected number of neutrinos observed by IceCube being N_ν,corr ≊ 0.23 (0.16). If the radiation fluence at MeV-GeV energies is allowed to exceed the X/γ fluence by an order of magnitude, a possibility that GLAST will resolve, then the expected number of detected neutrinos could be increased correspondingly.

For the large baryon load f_CR 20, which is required in the hypothesis that GRBs are the sources of UHECRs, as discussed in the next section, calculations show that 100 TeV-100 PeV neutrinos could be detected several times per year from all GRBs with kilometre-scale neutrino detectors such as IceCube [13, 21]. It is important that at these energies the number of atmospheric background neutrinos expected for a kilometre-scale neutrino detector in the time window of a typical long GRB is negligible. Detection of even 1 or 2 neutrinos from GRBs with IceCube or a northern hemisphere neutrino detector will provide compelling support for this scenario for the origin of high-energy and UHE cosmic rays.

A pair-photon cascade initiated by photohadronic processes between high-energy hadrons accelerated in the GRB blast wave and the internal synchrotron radiation field produces a γ-ray emission component that appears during the prompt phase of a GRB, as shown in figure 4. The various generations of synchrotron and Compton radiation initiated by the cascade are shown, along with the total radiation spectrum. As can be seen, the cascade radiation approaches the spectrum of an electron distribution cooling by synchrotron losses, that is, a spectrum with photon number index between  - 1.5 and  - 2.

Photomeson interactions in the relativistic blast wave also produce a beam of UHE neutrons, as proposed for blazar jets [8], which may escape from the site of the GRB to deposit UHECRs at distances ranging from the close exterior neighbourhood of the blast wave to multi-parsec scales. Subsequent photopion production of these neutrons with photons outside the blast wave can also produce a directed hyper-relativistic electron-positron beam in the process of charged pion decay and the conversion of high-energy photons formed in π⁰ decay [22]. These energetic leptons produce a synchrotron spectrum in the radiation reaction-limited regime extending to GeV energies, with properties in the 1-200 MeV range similar to that measured from GRB 941017 [23]. Large fluence GRBs displaying these anomalous γ-ray components are most likely to be detected as sources of high-energy neutrinos [24].

Figure 2 also shows calculations of photomeson neutrino production for a GRB with Φ_tot  =  3× 10 ^- 4 erg cm ^- 2 and δ  =  100, as well as the accompanying electromagnetic radiation induced by pair-photon cascades from the secondary electrons and γ rays from the same photomeson interactions. The total number of ν_μ expected with IceCube is 0.1. The total fluence of cascade photons shown here is contributed by lepton synchrotron (dot-dashed) and Compton (dashed) emissions. For comparison, the dotted curve shows the primary lepton synchrotron radiation spectrum assumed for the calculations. The level of the fluence of the cascade photons is ≈ 10% of the primary synchrotron radiation. This means that the maximum allowed baryon loading for these parameters cannot exceed a factor of ≈ 30 in order not to overproduce the primary synchrotron radiation fluence, as in the case of GRB 941017 [23]. This limits the maximum number of ν_μ to ≈ 3 even in the case of large baryon loading for rare, powerful GRBs, unless a very strong hadronic emission component accompanies the GRB. This component, whose existence is indicated by joint analysis of BATSE and EGRET/TASC data in GRB 941017 and at least two other GRBs [23], will be strongly detected by the GRB Monitor and Large Area Telescope on GLAST.

4. Cosmic rays from GRBs

The original argument [5] that GRBs are the sources of UHECRs was based on the coincidence between the energy density of UHECRs and the amount of power necessary to supply cosmic rays with energies 10²⁰ eV. These GZK cosmic rays, named after the discoverers of the effect [25], are subject to strong photomeson energy losses on cosmic microwave background photons. The effective distance for 10²⁰ eV protons to lose 50% of their energy is ≈140 Mpc [26], so that the photopion energy-loss timescale of 10²⁰ eV cosmic-ray protons is t_GZK 140 Mpc/c 1.5× 10¹⁶ s.

The energy density u_uhecr of GZK cosmic rays observed near Earth, as measured with the AGASA air shower array and the High Resolution air fluorescence detector [27], is ≈ 10 ^- 21 ergs cm ^- 3. If these cosmic rays are powered by GRBs with luminosity L_GRB throughout the universe, then

where V_prod is the production volume of the universe. Here UHECRs are assumed to be produced locally with an efficiency ζ compared with the mean γ-ray power of GRBs.

The mean γ-ray fluence of BATSE GRBs is F_γ ≈ 3× 10 ^- 6 ergs cm ^- 3 and their rate over the full sky is . If most GRBs are at redshift z ~ 1, as implied by redshift measurements of GRBs detected with Beppo-SAX, which had similar triggering criteria as BATSE, then their mean distance is d ≈ 2× 10²⁸ cm. Thus the average isotropic energy release of a typical GRB source is E_γ ≈ 4π d ²F_γ/(1 + z) 8× 10⁵¹ ergs, implying a mean GRB power into the universe of L_GRB ≈ 2× 10⁴⁷ ergs s ^- 1. (This estimate is independent of the beaming fraction, because a smaller beaming fraction implies a proportionately larger number of sources.) This implies that the energy density observed locally is u_uhecr ≈ 10 ^- 22 ζ ergs cm ^- 3.

Thus super-GZK particles could in principle be powered by GRBs if roughly equal energies are deposited in super-GZK particles with energies exceeding 10²⁰ eV as is radiated by GRBs. Other effects, for example, bolometric corrections to the total photon fluence, the lower production rate of GRBs at z    1 than at z ≈ 1, the enhancement due to GRB-type sources which do not trigger GRB detectors, complicate the estimate.

In the model of Waxman and Bahcall [28], sources deposit UHECRs with energies between ≈ 10¹⁹ and 10²¹ eV with a  - 2 injection number spectrum. In this way, the coincidence is preserved, but the model requires a separate origin for UHECRs with energies below the cosmic-ray ankle energy of    5× 10¹⁸ eV. In their model, the ankle is interpreted as the energy of the transition between the galactic and extragalactic cosmic rays.

By contrast, in the model of [21], cosmic rays with energies between the knee at ≈ 3× 10¹⁵ eV and the second knee at E₂≈ 5× 10¹⁷ eV are mostly due to a single or a few relatively recent Galactic GRB/supernova events that occurred some t₀   10⁶ years ago at distances r₀    1 kpc from us. UHECRs from extragalactic GRBs dominate at E E₂. This model explains the entire CR spectrum from GeV up to UHE with a single population of sources, namely SNe. The coincidence between GRB electromagnetic power and UHECR power is however lost, and a large baryon load, f_CR 20-50, is required in GRBs, as compared to f_CR≈ 1 in [28], due to the softer spectrum and greater injection-energy range. The transition from galactic to extragalactic component takes place at the second knee of the cosmic-ray spectrum, in accord with indications for a change from heavy-to-light composition near the second knee [29].

A `single-source' model was proposed earlier by Erlykin and Wolfendale [30], who suggested that the knee could be due to a single `normal' supernova event that occurred some t~ 10⁴ years ago within r~ 100 pc from us. This differs from the approach of [21], because the latter model can explain acceleration of particles up to UHE by the relativistic shocks formed by GRB outflows, which is very problematic in the case of SNRs formed in the collapse to neutron stars. Moreover, the much larger total energy of cosmic rays injected by a SN/GRB, which permits the source to have occurred at larger (~ 1 kpc) distances and from a significantly older GRB than for a single normal SN source, makes it then easier to explain the likelihood of such an event, as well as the low degree of anisotropy observed in HECRs.

The model of [21] provides a way to explain the origin and the sharpness of the knee at E₁≊ 3 PeV as a consequence of pitch-angle scattering of cosmic rays on plasma waves injected in the interstellar medium (ISM) through dissipation of bulk kinetic energy of SNRs on the pc-scale Sedov length. Furthermore, the origin of the second knee in the cosmic-ray spectrum at E₂ ≊ 4× 10¹⁷ eV is explained as a consequence of diffusive propagation through scattering with turbulence injected on a scale of ~ 100 pc, corresponding to the thickness of the Galactic disc, which perhaps represents the largest natural scale for effective injection of plasma turbulence in the Galaxy. The transition from Galactic to extragalactic CRs occurs around and above the second knee.

The rapid decline of the CR flux from local GRBs above the second knee results in the contribution of the extragalactic component to the all-particle spectrum dominating near and above the second knee. Calculations of the extragalactic component are shown in figure 5 for our model of CRs from GRBs, which includes photomeson interactions, e ⁺  - e ^-  pair production, and adiabatic cooling of UHECRs [7, 31] during transport. We assume that the rate density of GRBs is proportional to the cosmological SFR of the universe. For the two rates shown in figure 5(a) that correspond to minimum and maximum SFRs, calculations in [21] result in the two spectra for the extragalactic component shown in figure 5(b). An interesting result here is that in the framework of this model, the ankle in the spectrum of CRs observed at E≊ 3× 10¹⁸ eV is formed in the process of cooling of UHE protons on cosmological timescales.

Similar spectral behaviour for the extragalactic CR component at E≥ 10¹⁸ eV as shown in figure 5(b), where the ankle is explained as a consequence of photopair losses of UHECRs formed at high redshift, was proposed also by Berezinsky and collaborators [9]. They consider a model where UHECRs are accelerated by active galactic nuclei and assume cosmological evolution of the injection rate of UHECRs (1 + z)⁴ (see figure 5(a)). It remains to be studied if these two principal options (GRBs and AGNs) for the sources of UHECRs in the universe can be distinguished from each other observationally as a result of differences in their evolutionary histories.

The spectra of UHECRs resulting from injection of UHECR protons in the universe on cosmological timescales show a sharp (`GZK') cutoff above the GZK energy E≊ 6× 10¹⁹ eV. The UHECR spectrum in figure 5(b) agrees with the HiRes data, but is in disagreement with the AGASA results at E 10²⁰ eV. If Auger observations show any significant excess over the exponential GZK cutoff at these energies, this would imply that there are recent (   10⁸ years) local source/sources of extragalactic origin in our vicinity at    10 Mpc that produce this flux. One possibility is that the excess would be due to cosmic ray ions [32].

Extragalactic sources could be connected with starburst galaxies in the local group, such as M82 and NGC 253, both at distances r~ 3.5 Mpc. Taking into account that the supernova rate in these galaxies is about 0.3-1 per year, and that the estimated GRB rate in our Galaxy is about (0.3-1)% of the supernova rate, the mean GRB rate in the starburst galaxies is estimated as one per ~ 300-1000 years. If the total energy of CRs accelerated by a typical GRB is indeed about 10⁵² ergs, as for our local Galactic GRB, the characteristic injection power of UHECRs from starburst galaxies averaged over the timescale of ~ 10⁸ years can be ~(1-3)× 10⁴² ergs s ^- 1.

5. GRBs in the local universe

There are two important issues relevant to the question of the frequency and effects of GRBs in the Milky Way and the local universe. These are (1) the rate of GRBs as inferred from the GRB beaming factors; and (2) the dependence of GRBs on the SFR of the universe and the tendency of GRBs to be found in sites of different metallicity. These questions are crucial to determine at what rate GRBs occur in the Milky Way and whether they can be sources of cosmic rays above the knee of the cosmic ray spectrum. At the present time, it is not possible to answer either of these questions satisfactorily.

Regarding issue (1), achromatic beaming breaks in the afterglow light curves of GRBs can be used to infer the opening angle of the GRB jets. Direct observations [33] indicate that the mean measured opening angle is ≈ 0.1 radian, so that the opening angle of the GRB jets is ≈ 1/500th of the full sky, leading to a rate of GRBs in the Milky Way Galaxy of 1 every 3000-30 000 years [6]. A self-consistent calculation [34] for both the redshift and opening angle distribution suggests that the mean GRB opening angle is ≈ 1/75th of the full sky, greatly reducing the rate in the Milky Way. The initial (prompt) opening angle of the GRB jet may, however, be smaller than the opening angles of the lower bulk-Lorentz factor outflows that produce, at later times, beaming breaks observed at optical frequencies, as in structured jet models (e.g. [35]). Hence the actual opening angles of the high Lorentz factor jets making the γ rays that trigger GRB detectors, and consequently define the rate of GRBs in the Milky Way, remains very uncertain.

The second important issue is whether GRBs follow the SFR of the universe, as shown in figure 5(b), or follow a significantly different rate. If GRBs are the population of massive stars that collapse to black holes, then they may sample only a select portion of the high-mass stars that produce the blue/UV luminosity used to determine the SFR. If the dependence of the GRB rate on the SFR changes with cosmic time, for example, due to the lower metallicities in the early universe, then the GRB rate in the recent epoch in the Milky Way may be much lower than inferred from the high-redshift GRBs [36, 37]. The tendency of GRBs to favour low-metallicity hosts is not certain because GRBs are preferentially found in OB associations [36], which may tend to have higher metallicity than the host galaxy; the host galaxy of GRB 980425 associated with SN 1998bw is a spiral galaxy with fairly high metallicity [37]; and high metallicity is found for GRB 060206 at redshift 4.048 from absorption line studies [38]. Moreover, the conclusions of [37] rest on only five GRBs with anomalously low beaming-corrected energies, which may simply indicate that underluminous GRBs have a tendency to be born in low metallicity environments.

Evidence for local GRBs can be established by identifying remnants of GRB in the Milky Way [39]. The identification of a ~1 Myr old GRB remnant (GRBR) that happened at a    1 kpc from us, which explains [21] the origin of the knee(s) in the CR spectrum, is virtually impossible by now. However, there are other hopes to distinguish the radiation characteristics of the Galactic GRBRs from the radiation from the remnants of `ordinary' (nonrelativistic) supernovae in the Galaxy.

The High Energy Stereoscopic System (HESS) collaboration has recently discovered a population consisting of several γ-ray sources in the Galactic plane [40, 41], which are spatially extended - up to tens of arcmin - and TeV-bright, but are quiet in all other frequency bands and remain unidentified. The suppressed level of low-frequency synchrotron flux from TeV-bright sources is very unexpected for conventional SNRs. It is proposed in [42] through detailed calculations of particle diffusion and radiation processes and modelling of the spatial and spectral energy fluxes detected, that at least one of these sources, HESS J1303-631 [43], can be explained as a ~10-20 kyr old GRBR at a distance ~ 10-15 kpc from us. The hadronic origin of the detected TeV flux, implied by the non-detectable synchrotron flux, requires 3× 10⁵¹ ergs energy in relativistic protons with E > 5 TeV. Furthermore, it is qualitatively argued [42] that the suppressed level of synchrotron flux could be the specific signature of the GRBRs. Unlike in the process of diffusive Fermi-type acceleration by nonrelativistic shocks for normal SNRs, the bulk kinetic energy of the relativistic shocks is converted to relativistic particles (mostly to protons rather than to electrons, because of the large difference in the rest masses) at the very early stages while the shock is relativistic. This can explain the large baryon load in the GRBs. Even at the early non-relativistic stage of evolution of the GRBR, the magnetic fields (~ 0.1 G) are still high and can quickly remove the relativistic leptons from the pool of accelerated particles. The most important observational feature predicted in [42] for the GRBRs, in contrast to SNRs, is the very hard spectrum of γ rays below 100 GeV energies. A weak detection, confirming the hard spectrum, or non-detection by GLAST of HESS J1303-631 would therefore signify detection of the signatures of proton acceleration by relativistic shocks of a GRB.

Confirmation of HESS J1303-631 and perhaps some other extended unidentified TeV source as GRBRs would confirm the high rate, of order 10 ^- 4 year ^- 1, of local GRBs. These questions are important for determining whether GRBs have had an impact upon the evolution of life at Earth [6, 44], as described in a related contribution to this focus issue [45].

6. Summary and conclusions

The possibility that GRBs accelerate relativistic hadrons could solve the problem of the origin of the high-energy cosmic rays. There are several ways to establish whether GRBs accelerate cosmic rays, from: high-energy neutrino detection from GRBs; anomalous radiation signatures associated with hadronic acceleration and energy losses in GRB blast waves; radiations made by neutrons that escape from the blast wave; and evidence for cosmic-ray production from GRBRs in our own and nearby galaxies.

In the case of neutrino and γ-ray production, at most only a few high-energy ν_μ can be detected with kilometre-scale neutrino detectors even from bright GRBs at the fluence level Φ_tot 3× 10 ^- 4 erg cm ^- 2, and only when the baryon loading is high [13, 21]. This is because the detection of a single ν_μ requires a ν_μ fluence 10 ^- 4 erg cm ^- 2 above 1 TeV. Since the energy release in high-energy neutrinos and electromagnetic secondaries is about equal, this energy will be reprocessed in the pair-photon cascade and emerge in the form of observable radiation at γ-ray energies, and this radiation cannot exceed the measured fluence in this regime. Detection of high-energy neutrinos and hadronic γ-ray emission components from GRBs will mean that GRBs have a large load in relativistic baryons, which is required for UHECR production.

The advent of the IceCube neutrino telescope at the South Pole and the launch of GLAST in late 2007 will provide the important instruments to establish whether GRBs accelerate ultra-relativistic hadrons. In the meantime, searches for GRBRs at other wavelengths will indicate the importance of GRBs in the local universes. With these developments, the longstanding puzzle of the origin of the cosmic rays - now nearly 100 years since their discovery - should finally be answered.

Acknowledgments

We thank A Reimer for permission to reproduce figure 1 from [10], and for useful comments on the paper. The work of CDD is supported by the Office of Naval Research. We also acknowledge support of GLAST Science Investigation DPR-S-1563-Y.

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