Neutrino and Cosmic-Ray Emission from Multiple Internal Shocks In Gamma-Ray Bursts

# Neutrino and Cosmic-Ray Emission from Multiple Internal Shocks In Gamma-Ray Bursts

Mauricio Bustamante Center for Cosmology and AstroParticle Physics (CCAPP), The Ohio State University, 191 W. Woodruff Ave., Columbus, OH 43210, USA DESY, Platanenallee 6, D-15738 Zeuthen, Germany Institut für Theoretische Physik und Astrophysik, Universität Würzburg, Am Hubland, D-97074 Würzburg, Germany    Philipp Baerwald Department of Astronomy and Astrophysics; Department of Physics; Center for Particle and Gravitational Astrophysics; Institute for Gravitation and the Cosmos; Pennsylvania State University, 525 Davey Lab, University Park, PA 16802, USA    Kohta Murase Department of Astronomy and Astrophysics; Department of Physics; Center for Particle and Gravitational Astrophysics; Institute for Gravitation and the Cosmos; Pennsylvania State University, 525 Davey Lab, University Park, PA 16802, USA Hubble Fellow – Institute for Advanced Study, Princeton, New Jersey 08540, USA    Walter Winter DESY, Platanenallee 6, D-15738 Zeuthen, Germany
July 17, 2019
###### Abstract

Gamma-ray bursts are short-lived, luminous explosions at cosmological distances, thought to originate from relativistic jets launched at the deaths of massive stars. They are among the prime candidates to produce the observed cosmic rays at the highest energies. Recent neutrino data have, however, started to constrain this possibility in the simplest models with only one emission zone. In the classical theory of gamma-ray bursts, it is expected that particles are accelerated at mildly relativistic shocks generated by the collisions of material ejected from a central engine. We consider neutrino and cosmic-ray emission from multiple emission regions since these internal collisions must occur at very different radii, from below the photosphere all the way out to the circumburst medium, as a consequence of the efficient dissipation of kinetic energy. We demonstrate that the different messengers originate from different collision radii, which means that multi-messenger observations open windows for revealing the evolving GRB outflows.

Gamma-ray bursts (GRBs) are violent outbreaks of energy distributed over cosmological distances. Most of the energy is detected as gamma rays during the so-called prompt phase, lasting from a few seconds to several hundred seconds (see Refs. Piran (2004); Meszaros (2006); Zhang (2007) for reviews). The common view is that relativistic jets are ejected from a central engine, triggered by a collapsing star or a neutron star merger, in the direction of the observer. The inhomogeneity in the jets naturally leads to internal shocks, at which charged particles can be accelerated. In the classical GRB scenario, the observed gamma-ray emission is attributed to synchrotron radiation from nonthermal electrons. It is natural to expect that protons are accelerated as well, and GRBs have also been considered as a possible candidate class for the origin of the ultra-high-energy cosmic rays (UHECRs) Milgrom and Usov (1995); Waxman (1995); Vietri (1995). Whereas the charged cosmic rays cannot be traced back to their origin because of their deflection on magnetic fields during propagation, neutrinos from GRBs, which would be generated via proton-gas or proton-radiation interactions, point back to the sources and could provide crucial clues to the UHECR mystery Waxman and Bahcall (1997).

Neutrinos up to PeV energies from presumably extragalactic sources have now been detected in the IceCube neutrino telescope Aartsen et al. (2013a). While even the signal shape seems compatible with a GRB origin Murase and Nagataki (2006); Murase (2008); Baerwald et al. (2011); Zhang and Kumar (2013), stacking searches for prompt GRB neutrinos using the timing and directional information coming from gamma-ray observations have been so far unsuccessful Aartsen et al. (2013b); Abbasi et al. (2012). Because some of the early analytical predictions of the GRB neutrino fluxes Guetta et al. (2004); Abbasi et al. (2012) have shortcomings which are independent of astrophysical uncertainty (although these do not exist in some numerical works such as LABEL:~Murase and Nagataki (2006)), the model used by IceCube in LABEL:~Abbasi et al. (2012) has been revised by about one order of magnitude Hümmer et al. (2012); Li (2012); He et al. (2012). The current data are even pushing into the expected regime of the latest predictions, enabling us to address whether GRBs can be the sources of the UHECRs, and what the neutrinos can tell us about that.

In most of the earlier discussions, a simple one-zone model is assumed: this approach considers one representative collision between two relativistic plasma blobs representing the inhomogeneity in the jet, calculates the emission from this collision, and scales the result for the whole burst by assuming many such identical collisions within the jet. In this simple model, the GRB parameters are fixed during its duration. In particular, the internal shock radius , where the representative collision occurs and which is crucial for neutrino and UHECR production Murase and Nagataki (2006), is often estimated from geometric arguments Halzen and Hooper (2002). Taking the blobs as spherical shells, and using the representative value of the Lorentz factor of the plasma blobs, with the average velocity of the blobs, the variability timescale , and the burst redshift , the collision radius can be estimated as

 RC≃2Γ2ctv/(1+z). (1)

The variability timescale can be obtained by inspection of the pulse rising time of the burst’s light curve; the Lorentz factor can only be estimated, using various approaches Piran (1999); Zou and Piran (2010); and the redshift can be estimated via the observation of the host galaxy of the GRB. In the internal shock model, using the typical variability timescale (which is about three times shorter than the pulse width of  Nakar and Piran (2002a); Bhat (2013)), is expected Nakar and Piran (2002b) and neutrino predictions correspondingly vary Murase and Nagataki (2006); Murase et al. (2008); He et al. (2012). Specifically, in dissipative photospheric scenarios Murase (2008); Wang and Dai (2009); Murase et al. (2013); Bartos et al. (2013); Gao et al. (2012), internal shocks may occur under or around the radius known as the photosphere, at which the Thomson optical depth for scattering Rees and Meszaros (2005) is unity. Gamma rays can directly escape above the photosphere, where the optical depth is low. Even beyond it, high-energy gamma rays are attenuated by interactions. Since the photospheric radius is small, neutrino production is expected to be highly efficient around the photosphere.

UHECR production is also sensitive to ; UHECR escape also depends on GRB parameters Baerwald et al. (2013). Although it is often assumed that UHECRs can escape after the dynamical timescale (i.e., the shock crosses the shell), this is not the case if magnetic fields do not decay. Especially strong constraints on the UHECR-neutrino connection can be obtained if cosmic rays escape only as neutrons which are produced in the same interactions as the neutrinos (“neutron-escape model”) Rachen and Meszaros (1998); Mannheim et al. (2001). While this specific model is essentially excluded Baerwald et al. (2015); Ahlers et al. (2011), a hard flux of protons leaking from the sources (hereafter called “direct escape” around the maximum energy and/or “diffusion escape” at lower energies) can dominate the UHECR emission, which is largely allowed by neutrino observations Baerwald et al. (2013, 2015). As demonstrated in LABEL:~Baerwald et al. (2013), the dominant UHECR escape mechanism is in fact a function of the shell parameters.

Since the one-zone model is not realistic in the internal shock picture, and should evolve even within one GRB. The -dependence of neutrino production efficiency has been discussed in the internal shock model Guetta et al. (2001); Murase et al. (2008); He et al. (2012); Zhang and Kumar (2013), but its integrated effects have not been studied in detail. Hence, it is conceivable that not all collisions occur at the same radius, which has significant consequences for the neutrino and cosmic-ray production, as we show in this work. For example, different UHECR escape mechanisms will dominate in different phases of the evolving GRB.

We demonstrate that the different messengers originate from different collision radii. Even in the internal shock model, the neutrino production can be dominated by emission from around the photosphere, i.e., the radius where the ejecta become transparent to gamma-ray emission. Possible subphotospheric contributions enhance the detectability. We predict a minimal neutrino flux per flavor at the level of for the contribution from beyond the photosphere, with a spectral shape similar to the original theoretical prediction. However, in striking contrast to earlier approaches, this prediction turns out to hardly depend on model parameters such as the Lorentz boost, the baryonic loading, or the variability timescale.

## I Results

### i.1 Dynamical burst model

In order to demonstrate neutrino and cosmic-ray emission from various , we follow the internal collision model of LABEL:~Kobayashi et al. (1997); see Fig. 1 for illustration. We set out a number of shells from a central engine with equal initial kinetic energies but a spread in the bulk Lorentz factor, around , of

 ln(Γk,0−1Γ0−1)=AΓx, (2)

where is the initial Lorentz factor of the -th shell and follows a Gaussian distribution . Note that a large dispersion is required in order to achieve high efficiencies Kobayashi and Sari (2001), as we have explicitly tested, since the energy dissipation is proportional to the difference between the Lorentz factors of the colliding shells. The shells are assumed to be emitted with an uptime of the emitter , followed by an equally-long downtime, which is an input of the simulation. The variability timescale will be obtained after running the simulation from the light curve as an output, with a value which is typically similar to . For simplicity, we have assumed constant uptime and downtime, but LABEL:~Nakar and Piran (2002b) explored a scenario where is different for each emitted shell and follows a log-normal distribution; post-simulation, it is possible also in this case to infer a variability timescale for the whole burst. While the shells evolve, their widths, masses, and speeds (i.e., their Lorentz factors) are assumed to be constant, and their mass density decreases , with the radial distance to the emitter. Because of the different speeds of the shells, a shell will collide with another and merge into a new one; see Fig. 1. During the burst evolution, shells may collide several times. We assume that after a collision the new shell immediately cools by prompt radiation of the internal energy into gamma rays, cosmic rays, and neutrinos. Derivations of the properties of the newly formed shell are given in Refs. Kobayashi et al. (1997); Aoi et al. (2010) and are maintained in the simulations presented here. Our results match the analytical predictions for the dissipation of modest-amplitude fluctuations from Refs. Beloborodov (2000); Li (2010). Note that we simplify the evolution of the internal shocks in several points, although our approach is enough for the purpose of this work. First, since we focus on the classical internal shock scenario where optically-thin synchrotron emission is the most relevant mechanism, we assume situations where most of the dissipation occurs in the optically-thin regime. If significant dissipation occurs in the optically-thick regime, the internal energy scales adiabatically , which is spent to accelerate the outflow. Second, since we do not consider cases where only a fraction of the internal energy made available after a collision is released as radiation Kobayashi and Sari (2001), this means the efficiency issue of the internal shock model may remain unresolved Kumar (1999). Third, observed light curves from real GRBs may have slow variability components as well as fast variability components Zhang and Yan (2011), which are not easily explained by a discrete number of shells from a continuous emitter, whereas continuous outflow models give better agreement Daigne and Mochkovitch (1998, 2003); Bosnjak et al. (2008).

In this study, we choose for our base model the parameter values , , , and , as well as a perfect acceleration efficiency of (defined by , with the proton energy; see LABEL:~Baerwald et al. (2013)). The simulation yields 990 collisions, from the average obtained rise time of the light curve pulses (see LABEL:~Kobayashi et al. (1997)), a burst duration , and an average (average Lorentz factor of the merged shells, corresponding to the observable ), i.e., the GRB is sufficiently close to conventional assumptions in neutrino production models. Our study focuses on long-duration GRBs, which typically last tens of seconds, and our chosen parameter sets indeed yield burst durations of that order. We normalize the total isotropic photon energy of all collisions in the source frame to , consistent with GRB observations. Note that the fraction of photon energy dissipated in subphotospheric collisions is only about 9%, which means that a renormalization of the gamma-ray energy output to only collisions above the photosphere would hardly affects our result. For the cosmic-ray and neutrino production, we follow Refs. Baerwald et al. (2012, 2013) to compute the spectra for each collision individually, choosing equal energies in electrons (i.e., photons) and magnetic field, and a baryonic loading of ten (i.e., ten times more dissipated energy in protons than in photons). Neutrinos are produced in interactions. The target photon spectrum is assumed to be a broken power law with spectral indices and , respectively, with a fixed break energy of in the merged-shell rest frame (primed quantities are in the merged-shell rest frame). That is, it is implied that the target photon spectrum corresponds to conventional GRB observations regardless of the underlying radiation processes leading to this spectral shape.

### i.2 Simulation results

The light curve of the simulated burst is shown in Fig. 2a, as a black curve. Although we show the light curves for only two representative simulations in this study (the aforementioned one and another one with and s, in Fig. 2b), we will present a more detailed parameter space study in a future work Bustamante et al. (in preparation). We do not investigate effects of the spectral evolution during the dynamical time for one collision Asano and Mészáros (2014), as we imply that taking into account contributions from multiple shells is more relevant, like in the case of gamma rays Aoi et al. (2010). Note that, although we do not calculate hadronic cascades, their feedback on neutrino spectra is unimportant given the value of the baryonic loading factor used in this work.

We show in Fig. 3 the neutrino fluence (a), maximal proton energy (b), and maximal gamma-ray escape energy (c) for each collision (dot) as a function of . The maximal proton energies are obtained from comparing acceleration, dynamical, synchrotron loss, and photohadronic (for protons) timescales. As a result, we find that the collisions are spread between about and our choice of for the deceleration radius Rees and Meszaros (1992), where outflows terminate by the external shock into the circumburst medium. Most collisions occur around — slightly above the estimate from the geometry Eq. (1), . Red dots mark collisions in the neutron-escape model regime (optically thick to interactions) and blue empty circles, collisions in the direct proton escape regime.

Black squares mark subphotospheric collisions, i.e., those for which the Thomson optical depth is larger than unity. The optical depth is obtained by calculating the proton number density from the masses of the shells and assuming that the electron number density is as high as the proton density, which is expected for an electrically neutral plasma. In reality, however, the electron and positron densities may be somewhat higher if there is a significant non-thermal contribution from electron-positron pair production. The obtained photospheric radius is somewhat larger than the conventional expectation calculated using the dissipated energy in gamma rays (). This estimate is affected by the efficiency of the conversion from kinetic energy into dissipated energy, which is roughly 25% in our cases. However, the more important reason is the significant baryonic loading: since most of the energy is dissipated into protons, the masses of the shells have to be upscaled to match the required energy output in gamma rays (), which leads to larger radii of the photosphere because of higher electron densities. It can therefore be expected that the large baryonic loadings that are needed to describe the UHECR observations Murase et al. (2008); Baerwald et al. (2015) will lead to larger fractions of subphotospheric collisions.

We find that the obtained range of collision radii is large, from under the photosphere out into the deceleration radius, since dissipation occurs for a wide range of especially when the spread of the Lorentz factor is large Beloborodov (2000); Li (2010). Note that % of collisions occurs under the photosphere for the chosen parameter set, altogether 118 out of the total 990 collisions, but most of the energy dissipation occurs at large radii . In the internal shock model, gamma-ray emission should be produced beyond the photosphere, so we only consider collisions beyond the photosphere in the following, unless noted otherwise. This is conservative, since the baryonic loading may be smaller under the photosphere Murase (2008) and particle acceleration becomes inefficient for radiation-mediated shocks Murase and Ioka (2013). The ratio of total energy emitted as neutrinos via optically-thin internal shocks to the total energy emitted by these collisions as gamma rays is 4.8% for this representative parameter set.

Most importantly, Fig. 3a demonstrates that neutrinos are dominantly produced at small collision radii , close to the photospheric radius . This result can be understood as follows. In each collision, the emitted gamma-ray energy, , is a fraction of the total dissipated energy. The pion production efficiency (fraction of proton energy going into produced pions) at the photon spectral break , which is neglecting spectral effects, can be approximated as Murase and Nagataki (2006); Murase (2008)

 fpγ∝κpσpγEisoγ−sh4πR2CΓmϵ′γ,break. (3)

Here is the Lorentz factor of the merged shell, is the photohadronic cross section, and is the fraction of proton energy going into the pion per interaction. Since the internal shock model predicts Beloborodov (2000) for , we expect . Hence, since has to be sufficiently large for efficient energy dissipation ( in the simulations in the present study), neutrino production is typically dominated by collisions at radii around the photosphere. UHECR protons come from collisions in the range ; see Fig. 3b. First the maximal proton energy increases with collision radius as (close to the peak) synchrotron losses limit the maximal proton energy, and the magnetic field drops with . The peak occurs where adiabatic losses take over, and the decline comes from a decrease of the acceleration timescale for dropping magnetic fields; the expressions for the different energy-gain and energy-loss timescales can be found, e.g., in Refs. Murase and Nagataki (2006); Baerwald et al. (2013). Note that the UHECRs come from two different components dominating at different collision radii: for , neutron escape dominates, and for , protons directly escaping from the source dominate – which are obviously not related to strong neutrino production; see Fig. 3a. In the chosen example, the main contribution to cosmic rays actually comes from direct escape. Finally, Fig. 3c illustrates that high gamma-ray energies, which can only be observed in CTA or other next-generation imaging atmospheric Cherenkov telescopes, come from large collision radii , since for lower radii the optical depth for interactions is too high. As a consequence, neutrinos, cosmic rays, and Fermi-LAT/CTA gamma rays probe different emission radii. Neutrinos are useful to probe dissipation at small radii, including subphotospheric dissipation. For dissipation at large radii, where heavy nuclei survive, the TeV gamma-ray diagnostics of a GRB would be useful Murase et al. (2008).

In order to obtain an even more quantitative statement of how much energy is released as a function of collision radius, we show in Fig. 5 binned distributions for the prompt gamma rays, neutrinos, and cosmic-ray protons, which are all directly calculable within our model. Fig. 5a shows the energy output per bin, while Fig. 5b shows the fraction of energy in each bin compared to the total, for each particle species. We note that the energy per messenger per bin is obtained as a product of energy released per collision, and the number of collisions occurring per -bin; especially the latter number is important to get the proper weighing of . The result confirms the above observations: the neutrino production is dominated by small values of just beyond the photosphere from within a relatively narrow region to , the cosmic-ray production by intermediate to , and the prompt gamma-ray emission is, in fact, dominated by large , at around to – compatible with what is typically expected in the literature Nakar and Piran (2002b). These results have significant implications: our knowledge of the prompt phase of GRBs is obtained from gamma rays, of course, and, consequently, is derived from gamma-ray observations. This collision radius is, however, not the one to be used for neutrino or cosmic-ray calculations. It is therefore conceivable that multi-zone predictions are different from the naive one-zone expectation based on the gamma-ray emission radius. One can also read off from Fig. 5 that a significant amount of energy in UHECRs is transported away by direct escape, unrelated to neutrino production, which may affect the predicted neutrino flux if normalized to the observed UHECRs, as in, e.g., LABEL:~Baerwald et al. (2015).

We show in Fig. 6 the predicted quasi-diffuse neutrino spectra from collisions beyond the photosphere as thick orange curves for three different values of , where Fig. 6b corresponds to our standard assumptions. Note that the neutrino fluence per burst has been rescaled to a quasi-diffuse flux prediction by assuming 667 (identical) bursts per year and is significantly below the current diffuse neutrino signal reported by IceCube at the level of flux Aartsen et al. (2014). The dashed curves correspond to the standard assumption that all collisions occur at the same radius, derived from gamma-ray observations. To generate these curves, we use the parameters , , , and obtained from the simulation assuming identical shells with a collision radius obtained from Eq. (1) ( in Fig. 6b). The reference flux in Fig. 6b is significantly lower than the prediction in LABEL:~Hümmer et al. (2012). In that reference, the same parameters as in the IceCube analysis Abbasi et al. (2011) were used for comparison, implying that . That is about one order of magnitude smaller than the used here; cf., Eq. (3) for its impact on the neutrino flux. The reference flux in Fig. 6a is comparable to LABEL:~Hümmer et al. (2012).

We first of all find that the neutrino spectra from collisions beyond the photosphere (thick orange curves) all exhibit the same flux level quite independently of (and even of , as we have explicitly tested). The expected neutrino flux per flavor is at the level of , peaking between and . This contribution can be regarded as a minimal prediction for the neutrino flux, as it can be inferred from gamma-ray observations and hardly depends on the parameters. Note that this flux is probably outside the sensitivity of the existing IceCube experiment, but it will provide a target for the optimization of the planned high-energy volume upgrade. There is a significant qualitative difference to conventional models such as Refs. Waxman and Bahcall (1997); Guetta et al. (2004), for which the pion production efficiency contains a factor coming from the collision radius estimate in Eq. (1) applied to Eq. (3). However, the optical thicknesses to Thomson scattering and photohadronic interactions both scale , which leads to the following estimate for the pion production efficiency at the photosphere independent of  Murase (2008):

 fphpγ∼5×ε0.25×ϵe0.1×1keVϵ′γ,break. (4)

Here is the fraction of the dissipated energy going into photons, and is the dissipation efficiency (ratio between dissipated and kinetic energies). Notably, drops out of – unless the break energy is fixed in the observer’s frame, in which case there is a single factor of retained.

When the innermost collisions give the dominant contributions, the time-integrated neutrino fluence roughly scales as

 Fν∝Ncoll(fpγ≳1)Ncoll×min[1,fphpγ]×ϵpϵe×Eiso, (5)

where is the number of collisions with efficient neutrino production close to the photosphere, is the total number of collisions, and is the fraction of energy going into protons. Since the number of dominant collisions contributing to the neutrino flux is of order ten almost independently of the model parameters (see thin solid curves in Fig. 6), the neutrino flux prediction is relatively robust. The neutrino prediction above the photosphere hardly depends on the baryonic loading () as well, as long as most of the energy is dissipated into protons. Increasing the baryonic loading in Eq. (5) is compensated by a correspondingly smaller in Eq. (4). As a result, the neutrino flux is roughly independent of – which we have explicitly tested numerically.

We have also tested that this prediction does not depend on the variability timescale of the burst: Fig. 7 shows predictions for two different values of the emitter uptime s (a) and s (b), where the fixed leads to a longer burst duration . Clearly, the quasi-diffuse flux coming from the simulations is independent of the value of , as expected from Eq. (4). This markedly contrasts with the standard numerical prediction, in which larger variability timescales unavoidably imply lower particle densities at the source and, therefore, a reduced neutrino production. In Fig. 7c, with a reduced number of shells is chosen, corresponding to the light curve in Fig. 2b, that has a similar duration as the light curve for our standard simulation, but fewer pulses. In this case, the obtained result depends on the actual realization of the -distribution, as only a few collisions dominate the neutrino flux and lead to strong variations – six different realizations of the -distribution are shown in Fig. 7c. As expected, our conclusions hold for a sufficiently large sample of GRBs centered around our average prediction. We expect that these fluctuations become more severe for even fewer pulses from fewer collisions, as it has been studied for the contributions from different bursts in Section 4 of LABEL:~Baerwald et al. (2012). The independence on the model parameters implies that the predicted flux is very robust. The only exception may be increasing (see Eq. (5)) or the baryonic loading, which may in fact be required to match the injected energy needed to describe UHECR observations; see Section 2 in LABEL:~Baerwald et al. (2015).

The photon spectra can still be approximated by the Band function up to a Thomson optical thickness of ten or so Pe’er (2008); Beloborodov (2011), which occurs under the photosphere. This means that we can extrapolate our assumptions to below the photosphere to some degree. In the most extreme case, all energy may be dissipated into neutrinos, whereas the energy of neutrons, protons, and gamma rays is reconverted into kinetic energy – this is, however, very speculative, as nonthermal particle acceleration may not occur efficiently Murase (2008). We show the corresponding subphotospheric extrapolations for the neutrino spectra as highly uncertain shaded regions in Fig. 6, corresponding to the contribution to the black squares in Fig. 3. Since the photospheric radius increases with decreasing , the number of subphotospheric collisions increases with it, and their contribution in Fig. 6a can be much higher than in Fig. 6b (and in Fig. 6c much lower). As a consequence, the subphotospheric extrapolation may even reach the current sensitivity limit, and can be already constrained with current data. However, note again that this extrapolation is highly uncertain, as gamma-ray data cannot be used to obtain information about below the photosphere.

Finally, we show the “neutrino light curve” for our standard parameter set in Fig. 2a as a dotted (red) curve; Fig. 2b shows it for a simulation with fewer collisions and longer emitter uptime, corresponding to the neutrino spectra in Fig. 7c. It can be clearly seen that the neutrino flux is typically much lower than the gamma-ray flux except in some rare cases where the collision occurs close to the photosphere. Furthermore, the variation of the neutrino flux is larger due to the strong dependence of the pion production efficiency on . One qualitative prediction that could help neutrino searches is that neutrinos are more likely to be associated with gamma-ray spikes that are pulses with very short variability timescales.

## Ii Discussion

In summary, we have studied neutrino, gamma-ray (at different energies), and cosmic-ray production in an evolving GRB outflow. We have demonstrated that they are produced at different collision radii. Consequently, the typical emission radius derived from prompt gamma rays cannot be directly applied to neutrino and UHECR production, and the GRB will look very different from the point of view of different messengers. This concept is well known from conventional astronomical observations, where astrophysical objects look very different in different wavelength bands.

The neutrino spectra derived from gamma-ray observations are dominated by the emission close to the photosphere at to , as the pion production efficiency depends on the collision radius in a non-linear way. UHECR protons have been shown to be produced at intermediate collision radii to , where the magnetic fields are high enough for efficient acceleration, but not so high that synchrotron losses limit the maximal proton energies. We have taken into account two possibilities for UHECR escape: emission as neutrons, which are not magnetically confined, and emission as protons from the edges of the shells – the dominant mechanism in each collision depends on the parameters of the colliding shells. Since the neutrons come from photohadronic interactions, their production dominates at smaller collision radii, where the optical depth is higher, whereas protons tend to be directly emitted at large radii. Heavier nuclei can also survive for sufficiently large collision radii; their actual contribution depends on the nuclear loading. The main energy in gamma rays is deposited between around to , compatible with earlier estimates. In particular, gamma rays at the highest energies, such as in the energy range only accessible to CTA, cannot come from collision radii as the photon densities are too high there to let them escape.

For the quasi-diffuse neutrino flux prediction, we have identified two distinctive contributions. Above the photosphere, gamma-ray observations can be used to infer the pion production efficiency, which leads to a neutrino flux per flavor for an assumed isotropic energy of emitted in gamma rays. Especially, there is no strong dependence on the Lorentz boost , in contrast to conventional one-zone models, as both the photosphere and the pion production efficiency scale with the collision radius in the same way. This is the minimal neutrino flux which one would expect in stacking analyses based on the actual gamma-ray observations, such as LABEL:~Abbasi et al. (2012). There is also a significantly milder dependence on the baryonic loading, as this parameter changes the photosphere of the model at the same time that it rescales the neutrino flux. The prediction hardly depends on the time variability or number of pulses in the GRB light curve within a certain time window either. However, if the overall number of pulses is low, these will only come from a very small number of collisions, which means that large statistical fluctuations of the neutrino flux from burst to burst are expected even for the same parameter values. In that case, our observations have to be instead interpreted for a large enough ensemble of bursts. Note that the chosen isotropic energy and baryonic loading may not be sufficient to describe UHECR observations, see Section 2 of LABEL:~Baerwald et al. (2015) for a detailed discussion, which will need to be addressed in a future study.

The neutrino flux is significantly lower than earlier predictions Hümmer et al. (2012) because a) we have explicitly excluded subphotospheric contributions, b) large photospheric radii have been obtained as a consequence of significant baryonic loadings (ten) and the moderate energy dissipation efficiency of the fireball (25%), and c) only a small number of collisions beyond the photosphere occurs at radii where the neutrino production efficiency is high. This expected “minimal” flux is beyond the sensitivity of the current IceCube experiment, but could be reached in future high-energy extensions Aartsen et al. (2014). No gamma-ray information from deep below the photosphere can be directly obtained, and the neutrino production in that regime is more speculative Murase and Ioka (2013). In principle, however, a high-energy extension of the detector could also constrain the subphotospheric neutrino production.

Our results imply that model-dependent studies of the multi-messenger connection, such as a GRB stacking analysis of neutrino fluences, can be improved, and give a stronger case for testing the hypothesis that UHECRs originate from GRBs. Compared to the one-zone model, some additional assumptions need to be made for the distribution of the collision radii. In particular, the width of the initial distribution of the bulk Lorentz factor , with which the shells are set out by the central engine, turns out to be the key additional parameter. It can in principle be obtained from comparing the light curves between simulation and observation. On the other hand, we have the advantage that the uncertainty in , which is the key issue in the standard model, disappears, as a collision radius distribution is now predicted by the theory. While we expect that the bulk Lorentz factor distribution has to be broad in some way to maintain a high dissipation efficiency, it remains to be studied how the results change for qualitatively different distributions. There should also be new opportunities stemming from our results: different messengers can be used to study different regions of an evolving GRB outflow. For instance, direct neutrino and gamma-ray observations, in CTA, of a single GRB would open windows to very different regions of the GRB.

During completion of this work, LABEL:~Globus et al. (2014) appeared, which shares some common aspects.

Acknowledgements. We thank V. Mangano, E. Waxman, and B. Zhang for discussion and comments. This work is supported by NASA through Hubble Fellowship Grant No. 51310.01 awarded by the STScI, which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA, under Contract No. NAS 5-26555 (K.M.). M.B. and W.W. would also like to acknowledge support from DFG grants WI 2639/3-1 and WI 2639/4-1, and the “Helmholtz Alliance for Astroparticle Physics HAP”, funded by the Initiative and Networking Fund of the Helmholtz Association. P.B. acknowledges support from NASA grant NNX13AH50G. M.B., K.M., and W.W. would like to thank the Kavli Institute for Theoretical Physics at UCSB for its hospitality during the development of part of this work. This research was supported in part by the National Science Foundation under Grant No. NSF PHY11-25915.

Author contributions. All authors contributed to all aspects of this work, discussed the results, and commented on the manuscript.

Competing financial interests. The authors declare no competing financial interests.

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