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 1, 2, 3 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 non-thermal 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)4,5,6. 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 mystery7.

Neutrinos up to PeV energies from presumably extragalactic sources have now been detected in the IceCube neutrino telescope8. While even the signal shape seems compatible with a GRB origin9,10,11,12, stacking searches for prompt GRB neutrinos using the timing and directional information coming from gamma-ray observations have been so far unsuccessful13,14. Because some of the early analytical predictions of the GRB neutrino fluxes14,15 have shortcomings that are independent of astrophysical uncertainty (although these do not exist in some numerical works such as ref. 9), the model used by IceCube in ref. 14 has been revised by about one order of magnitude16,17,18. 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 RC, where the representative collision occurs and which is crucial for neutrino and UHECR production9, is often estimated from geometric arguments19. Taking the blobs as spherical shells, and using the representative value of the Lorentz factor Γ≡(1−(v/c)2)−1/2 of the plasma blobs, with the average velocity of the blobs, the variability timescale tv and the burst redshift z, the collision radius can be estimated as

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 approaches20,21; 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 1 s22,23), RC108–1010.5 km is expected24 and neutrino predictions correspondingly vary9,18,25. Specifically, in dissipative photospheric scenarios10,26,27,28,29, internal shocks may occur under or around the radius known as the photosphere, at which the Thomson optical depth for eγ scattering30 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 γγ interactions. Since the photospheric radius Rph108–108.5 km is small, neutrino production is expected to be highly efficient around the photosphere.

UHECR production is also sensitive to RC; UHECR escape also depends on GRB parameters31. Although it is often assumed that UHECRs can escape after the dynamical timescale (that is, 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’)32,33. While this specific model is essentially excluded34,35, 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 observations31,34. As demonstrated in ref. 31, 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, RC and Γ should evolve even within one GRB. The RC dependence of neutrino production efficiency has been discussed in the internal shock model12,18,25,36, 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.

In this work, 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, that is, the radius where the ejecta become transparent to gamma-ray emission. Possible subphotospheric contributions enhance the detectability. We predict a minimal neutrino flux per flavour at the level of E2J10−11 GeV cm−2 sr−1 s−1 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.


Dynamical burst model

To demonstrate neutrino and cosmic-ray emission from various RC, we follow the internal collision model of ref. 37; see Fig. 1 for illustration. We set out a number Nsh of shells from a central engine with equal initial kinetic energies but a spread in the bulk Lorentz factor, around Γ0, of

Figure 1: Illustration of the internal collision model of gamma-ray bursts used in this study.
figure 1

A set of Nsh shells with equal energies, widths and separations l=d=cδteng are emitted from a central engine, where δteng is the uptime of the central emitter. The shells have a spread in the bulk Lorentz factor, but initially equal bulk kinetic energies. The shells propagate, collide and merge (marked by the shell coloured purple) as soon as they meet other shells (multiple collisions are allowed), whereupon their masses, widths and speeds change. The energy dissipated in the collision is assumed to be radiated away immediately.

where Γk,0 is the initial Lorentz factor of the k-th shell and x follows a Gaussian distribution . Note that a large dispersion AΓ0.1 is required to achieve high efficiencies38, 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 δteng, followed by an equally long downtime, which is an input of the simulation. The variability timescale tv will be obtained after running the simulation from the light curve as an output, with a value that is typically similar to δteng. For simplicity, we have assumed constant uptime and downtime, but ref. 24 explored a scenario where δteng 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 (that is, their Lorentz factors) are assumed to be constant, and their mass density decreases r−2, with r 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 37, 39 and are maintained in the simulations presented here. Our results match the analytical predictions for the dissipation of modest-amplitude fluctuations from refs 40, 41. 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 r−2/3, 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 radiation38, this means the efficiency issue of the internal shock model may remain unresolved42. Third, observed light curves from real GRBs may have slow variability components as well as fast variability components43, which are not easily explained by a discrete number of shells from a continuous emitter, whereas continuous outflow models give better agreement44,45,46.

In this study, we choose for our base model the parameter values Γ0=500, Nsh=1,000, δteng=0.01 s and AΓ=1, as well as a perfect acceleration efficiency of η=1 (defined by , with the proton energy; see ref. 31). The simulation yields 990 collisions, tv0.06 s from the average obtained rise time of the light curve pulses (see ref. 37), a burst duration TNcolltv≈59 s and an average 〈Γ〉≈370 (average Lorentz factor of the merged shells, corresponding to the observable Γ), that is, 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 Eiso=1053 erg, 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 31, 47 to compute the spectra for each collision individually, choosing equal energies in electrons (that is, photons) and magnetic field, and a baryonic loading of ten (that is, ten times more dissipated energy in protons than in photons). Neutrinos are produced in pγ interactions. The target photon spectrum is assumed to be a broken power law with spectral indices αγ=1 and βγ=2, 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.

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 Nsh=100 and δteng=0.1 s, in Fig. 2b), we will present a more detailed parameter space study in a future work (Bustamante et al., manuscript in preparation). We do not investigate effects of the spectral evolution during the dynamical time for one collision48, as we imply that taking into account contributions from multiple shells is more relevant, like in the case of gamma rays39. 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.

Figure 2: Two simulated light curves.
figure 2

The curves for the energy flux of gamma rays (solid, black) and neutrinos (dotted, red) are built from the collisions of shells output by an engine emitting shells with equal kinetic energies, with tobs the time in the observer’s frame. The light curves in a (b) correspond to a simulation with Nsh=1,000 (100), Ncoll=990 (91), δteng=0.01 s (0.1 s) and tv=0.06 s (0.66 s). A redshift of z=2 was assumed to produce these light curves.

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 RC. 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 106 km and our choice of 5.5 × 1011 km for the deceleration radius49, where outflows terminate by the external shock into the circumburst medium. Most collisions occur around 1010 km—slightly above the estimate from the geometry equation (1), RC≈1.6 × 109 km. Red dots mark collisions in the neutron escape model regime (optically thick to pγ interactions) and blue empty circles, collisions in the direct proton escape regime.

Figure 3: Neutrino fluence and maximum proton and gamma-ray energies for each internal collision.
figure 3

(a) Muon-neutrino fluence (, in the observer’s frame), (b) maximal proton energy (in the source frame, for ideal (η=1) acceleration) and (c) maximal allowed gamma-ray energy (in the source frame, where τγγ(Eγ,max)=1) as a function of the collision radius. Each dot represents one collision: red filled dots represent collisions where cosmic rays mainly escape as neutrons (optical thickness to pγ interactions larger than unity), blue empty circles represent collisions where cosmic-ray leakage dominates over the neutron escape model, and black squares denote subphotospheric collisions or collisions where this picture cannot be maintained (that is, where the Thomson optical depth is large). In b, the ultra-high energy range for cosmic rays, above 1010 GeV, is shown as a green band; the downward-pointing arrow marks the approximate energy above which adiabatic energy losses dominate. In c, the energy ranges that can be reached by the Fermi-GBM (pink), Fermi-LAT (blue) and Cherenkov Telescope Array (CTA) (green) instruments are illustrated as coloured bands. Collisions in which photons with energies above 106 GeV are able to escape are marked as upward-pointing arrows.

Black squares mark subphotospheric collisions, that is, 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 Rph≈2 × 108 km is somewhat larger than the conventional expectation calculated using the dissipated energy in gamma rays (Rph≈3 × 107 km). 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 (1053 erg), 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 observations25,34 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 RC especially when the spread of the Lorentz factor AΓ is large40,41. Note that 12% 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 >1010 km. 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 photosphere10 and particle acceleration becomes inefficient for radiation-mediated shocks50. 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 RC109 km, close to the photospheric radius Rph≈2 × 108 km. 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 as9,10

Here Γm is the Lorentz factor of the merged shell, σpγ is the photohadronic cross section, and κp0.2 is the fraction of proton energy going into the pion per interaction. Since the internal shock model predicts40 for 0q2/3, we expect . Hence, since AΓ has to be sufficiently large for efficient energy dissipation (AΓ=1 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 108.5 kmRC1010 km; 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 RC. 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, for example, in refs 9, 31. Note that the UHECRs come from two different components dominating at different collision radii: for RC108.5 km, neutron escape dominates and for RC108.5 km, 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 Cherenkov Telescope Array (CTA) or other next-generation imaging atmospheric Cherenkov telescopes, come from large collision radii RC109 km, since for lower radii the optical depth for γγ interactions is too high. As a consequence, neutrinos, cosmic rays and Fermi Large Area Telescope (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 useful25.

There has been some evidence that the composition of UHECRs is heavy51. Initial studies such as refs 52, 53, 54 concluded that heavy nuclei cannot survive inside GRBs: photodisintegration on fireball photons would break them up into lighter nuclei and protons. Anchordoqui et al.55 calculated the neutrino emission from the injection of both protons and nuclei and found that the latter cannot survive in internal shocks; however, only collisions at very small radii, around 108 km, were considered. It has been argued that the typical collision radius is much larger (see, for example, ref. 43 and references therein) and that heavy nuclei can be largely loaded in GRB jets56,57. Therefore, acceleration of nuclei to ultra-high energies and their survival against photodisintegration are possible, provided RC is large enough25,58,59. In Fig. 4, we show that this is indeed the case for our simulations. The figure shows the maximum energy to which iron nuclei (A=56, Z=26) can be accelerated at each of the collisions. The energy is a factor of 26 higher than for protons (compare with Fig. 3b), where its absolute magnitude is a consequence of the assumed acceleration efficiency. Here the photodisintegration timescale has been calculated using the approximation in ref. 25. Triangles (blue) and circles (red) represent collisions in which the maximum energy is limited, respectively, by the break-up of the nucleus due to photodisintegration and by adiabatic losses. Even though photodisintegration losses dominate up to 109 km, after which adiabatic losses take over, maximum energies well within the UHE band can be achieved at the turning point, where most of the UHECR emission would come from. Note that this turning point is about a factor of five higher in RC than for protons (compare arrows in Figs 3b and 4), which means that UHECR nuclei on average reach their peak energy at higher RC than UHECR protons. UHECR nuclei may also escape directly at the highest energies, but there is no such thing as neutron escape. It is therefore expected that nuclei come from somewhat larger collision radii than protons at the highest energies, where the radiation densities are too low to break up the nuclei. Since the actual energy output of heavy nuclei depends on the nuclear loading (that is, an additional assumption is required), we do not show their energy output explicitly in the following.

Figure 4: Maximum energy to which iron nuclei can be accelerated in each collision of a simulated GRB.
figure 4

For the iron nuclei, A=56, Z=26. Our standard parameter set is assumed for the simulation. Energy is shown in the source frame and is calculated for ideal (η=1) acceleration. Triangles (blue) and circles (red) represent collisions where the energy is limited by break-up due to photodisintegration and by adiabatic losses, respectively. The photodisintegration timescale has been computed using the approximation in ref. 25. The ultra-high energy range for cosmic rays, above 1010 GeV, is shown as a green band. The downward-pointing arrow marks the approximate energy above which adiabatic energy losses dominate.

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. Figure 5a shows the energy output per bin, while Fig. 5b shows the fraction of energy in each bin compared with 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 RC-bin; especially the latter number is important to get the proper weighing of RC. The result confirms the above observations: the neutrino production is dominated by small values of RC just beyond the photosphere from within a relatively narrow region RC≈108.5–109 km, the cosmic-ray production by intermediate RC≈109–1010 km and the prompt gamma-ray emission is, in fact, dominated by large RC, at around 1010–1011.5 km—compatible with what is typically expected in the literature24. These results have significant implications: our knowledge of the prompt phase of GRBs is obtained from gamma rays, of course, and, consequently, RC 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, for example, ref. 34.

Figure 5: Energy dissipated beyond the photosphere.
figure 5

We consider energy dissipated in (prompt) gamma rays, neutrinos (all flavours) and CR protons (UHECRs from 1010 to 1012 GeV). Energies are binned as a function of the collision radius. (a) Absolute energy values; (b) the fraction of energy output normalized to one for each messenger. Neutron escape dominates the cosmic-ray emission below RC≈108.5 km, while proton escape dominates above this radius. The rough value of the photospheric radius and the assumed radius of the circumburst medium are indicated as dashed lines.

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 Γ0, 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 10−8 GeV cm−2 s−1 sr−1 flux60. 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 Ncoll, tv, 〈Γ〉 and T obtained from the simulation assuming identical shells with a collision radius obtained from equation (1) (RC≈109.2 km in Fig. 6b). The reference flux in Fig. 6b is significantly lower than the prediction in ref. 16. In that reference, the same parameters as in the IceCube analysis61 were used for comparison, implying that RC≈1.9 × 108 km. That is about one order of magnitude smaller than the RC used here; cf. equation (3) for its impact on the neutrino flux. The reference flux in Fig. 6a is comparable to ref. 16.

Figure 6: Quasi-diffuse neutrino spectra from simulations of multiple internal shocks.
figure 6

Muon-neutrino spectra from collisions beyond the photosphere (thick orange curves), reference spectra computed from averaged burst parameters in the conventional approach (dashed curves) and maximal subphotospheric extrapolations (shaded regions) for three different values of Γ0 in the different panels: 300 (a), 500 (b) and 1,000 (c). The individual (dominant) collisions (contributing to the thick orange curves) are shown also as thin red and blue curves corresponding, respectively, to the optically thick to pγ interactions regime, with the neutron escape dominating at the maximum energy, and to the regime dominated by direct proton escape instead. The (thick orange) spectra (‘evolving fireball’) are estimates of the diffuse flux obtained from the single-burst fluence (one GRB at z=2) by assuming =667 per year long bursts per year over the whole sky . The diffuse GRB flux limit from the IC40+59 analysis14 is shown as a thin black curve. The obtained average values 〈Γ〉 from the simulation, corresponding to the observable Γ, are depicted as well.

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 Γ0 (and even of AΓ, as we have explicitly tested). The expected neutrino flux per flavour is at the level of E2J10−11 GeV cm−2 sr−1 s−1, peaking between 105 and 107 GeV. 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 7, 15, for which the pion production efficiency contains a factor Γ−4 coming from the collision radius estimate in equation (1) applied to equation (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 Γ (ref. 10):

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

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

where Ncoll(fpγ1) is the number of collisions with efficient neutrino production close to the photosphere, Ncoll is the total number of collisions and ɛp 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 (ɛp/ɛe) as well, as long as most of the energy is dissipated into protons. Increasing the baryonic loading in equation (5) is compensated by a correspondingly smaller ɛe in equation (4). As a result, the neutrino flux is roughly independent of ɛp/ɛe—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 δteng=0.1 s (a) and 1 s (b), where the fixed Nsh=1,000 leads to a longer burst duration T. Clearly, the quasi-diffuse flux coming from the simulations is independent of the value of tv, as expected from equation (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, δteng=0.1 s with a reduced number of shells Nsh=100 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 Γ-distribution, as only a few collisions dominate the neutrino flux and lead to strong variations—six different realizations of the Γ-distribution are shown in Fig. 7c. As expected, our conclusions hold for a sufficiently large sample of GRBs centred 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 ref. 47. The independence on the model parameters implies that the predicted flux E2J10−11 GeV cm−2 sr−1 s−1 is very robust. The only exception may be increasing Eiso (see equation (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 ref. 34.

Figure 7: Quasi-diffuse muon-neutrino spectra from simulations with alternative parameter sets.
figure 7

The neutrino spectra in this figure should be compared with the ones obtained using our standard parameter set, Fig. 6b (δteng=0.01 s, Nsh=1,000 and Γ0=500). (a,b) Two larger values of δteng for Nsh=1,000, which leads to a longer burst duration T. (c) Using δteng=0.1 s with a reduced number of shells Nsh=100 (corresponding to the light curves in Fig. 2b), that is, T is similar to the original example, but the light curve is less spiky because of fewer collisions. Since in that case the statistical fluctuations from burst to burst increase, we show six different realizations of the predicted flux.

The photon spectra can still be approximated by the Band function up to a Thomson optical thickness of 10 or so62,63, 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 efficiently10. 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 Γ, 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 RC. 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.


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 RC≈108.5–109 km, as the pion production efficiency depends on the collision radius in a nonlinear way. UHECR protons have been shown to be produced at intermediate collision radii RC≈108.5–1010 km, 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 pγ 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 1010 and 1011.5 km, 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 109 km 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 flavour E2J10−11 GeV cm−2 sr−1 s−1 for an assumed isotropic energy of 1053 erg emitted in gamma rays. Especially, there is no strong dependence on the Lorentz boost Γ, 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 that one would expect in stacking analyses based on the actual gamma-ray observations, such as ref. 14. 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 ref. 34 for a detailed discussion, which will need to be addressed in a future study.

The neutrino flux is significantly lower than earlier predictions16 because (a) we have explicitly excluded subphotospheric contributions, (b) large photospheric radii have been obtained as a consequence of significant baryonic loadings (10) 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 extensions64. No gamma-ray information from deep below the photosphere can be directly obtained, and the neutrino production in that regime is more speculative50. 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 with 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 AΓ, 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 RC, 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, ref. 59 appeared, which shares some common aspects.

Additional information

How to cite this article: Bustamante, M. et al. Neutrino and cosmic-ray emission from multiple internal shocks in gamma-ray bursts. Nat. Commun. 6:6783 doi: 10.1038/ncomms7783 (2015).