Neutrino signal dependence on gamma-ray burst emission mechanism
PPrepared for submission to JCAP
Neutrino signal dependence ongamma-ray burst emission mechanism
Tetyana Pitik, a Irene Tamborra, a and Maria Petropoulou b a Niels Bohr International Academy and DARK, Niels Bohr Institute, University of Copen-hagen, Blegdamsvej 17, 2100, Copenhagen b Department of Physics, National and Kapodistrian University of Athens, Panepistimiopolis,15783 Zografos, GreeceE-mail: [email protected], [email protected], [email protected]
Abstract.
Long duration gamma-ray bursts (GRBs) are among the least understood astro-physical transients powering the high-energy universe. To date, various mechanisms havebeen proposed to explain the observed electromagnetic GRB emission. In this work, weshow that, although different jet models may be equally successful in fitting the observedelectromagnetic spectral energy distributions, the neutrino production strongly depends onthe adopted emission and dissipation model. To this purpose, we compute the neutrinoproduction for a benchmark high-luminosity GRB in the internal shock model, includinga dissipative photosphere as well as three emission components, in the jet model invokinginternal-collision-induced magnetic reconnection and turbulence (ICMART), in the case of amagnetic jet with gradual dissipation, and in a jet with dominant proton synchrotron radi-ation. We find that the expected neutrino fluence can vary up to 1–1 . to 10 GeV. For our benchmark inputparameters, none of the explored GRB models is excluded by the targeted searches carriedout by the IceCube and ANTARES Collaborations. However, our work highlights the po-tential of high-energy neutrinos of pinpointing the underlying GRB emission mechanism andthe importance of relying on different jet models for unbiased stacking searches. a r X i v : . [ a s t r o - ph . H E ] F e b ontents A.1 Band function 32A.2 Cut-off power-law 32A.3 Power law 33A.4 Double broken power law 33
B Magnetized jet model with gradual dissipation: dependence of the neutrinoemission on the input parameters 33 – 1 –
Introduction
Gamma-ray bursts (GRBs) are irregular pulses of gamma-rays that have puzzled astronomersfor a long time [1]. Exhibiting a non-thermal spectrum, typically peaking in 10–10 keV en-ergy band [2], bursts lasting for more than 2 s are named long-duration GRBs and are thoughtto be harbored within collapsing massive stars [3–5]. They are the brightest explosions inour universe and can release isotropic energies as high as 10 erg in gamma-rays over fewtens of seconds [6].The central engine of a long-duration GRB jet can either be a hyper-accreting black holeor a rapidly spinning magnetar. Because the central engine cannot be directly observed, itsnature can be inferred only indirectly through its impact on the electromagnetic properties ofGRBs (see, e.g., Ref. [7] and references therein). A bipolar outflow is continuously poweredfor a certain time interval, during which gravitational energy [8, 9] (for accreting systems)or spin energy [10, 11] (for spinning-down systems) is released in the form of thermal energyor Poynting flux energy, respectively. Subsequently, the outflow propagates through the starand it is strongly collimated by the stellar envelope. Once it succeeds to break out of thestellar surface, it manifests itself as the jet responsible for the GRBs that we observe atEarth. The dynamical evolution of the jet strongly depends on the initial conditions of thecentral engine. If the magnetic field is negligible, the evolution of the outflow can be welldescribed by the fireball model [12]. If instead the central engine harbours a strong magneticfield, the jet dynamics is significantly different [13].Gamma-ray bursts are candidate sources of ultra-high energy cosmic rays and highenergy neutrinos [14]. In the prompt phase, if the jet contains baryons, protons and nucleiare expected to be accelerated [15]. If a photon field is also present, photo-hadronic ( pγ )interactions can lead to a significant flux of neutrinos [16–18]. Another copious source ofneutrinos comes from hadronic collisions ( pp or pn ) which, however, are most efficient insidethe progenitor star where the baryon density is large [15, 19–21]. Given the typical GRBparameters, neutrinos produced in the optically thin region are expected to be emitted inthe TeV-PeV energy range [16, 22–24].The IceCube Neutrino Observatory routinely detects neutrinos of astrophysical origin inthe TeV–PeV energy range [25–28]. However, despite the fact that several sources have beenproposed as possible candidates to explain the neutrino flux that we observe [22–24, 29, 30],we are still lacking clear evidence on the sources producing the observed neutrinos. Amongthe candidate sources, high-luminosity GRBs are deemed to be responsible for less than 10%of the observed diffuse emission in the TeV energy range [27, 31]. On the other hand, over theyears, the IceCube and ANTARES Collaborations have searched for high-energy neutrinosemitted in coincidence with GRBs observed by the Fermi satellite [31–33], gradually placingmore stringent upper limits on somewhat optimistic GRB emission models. Recent worksuggests that current limits are still not stringent enough to rule out more realistic estimationsproposed in the literature [15, 34–38].Intriguingly, besides the need for increased detection sensitivity, one of the reasons forthe non-detection of GRB neutrinos could be connected to the theoretical modeling of theneutrino emission, which is strictly linked to the electromagnetic modeling of the jet. Infact, a comprehensive explanation of the GRB emission and dissipation mechanism is stilllacking due to the failure of existing models in addressing all observations in the spectral andtemporal domains.On the other hand, the scarce amount of data on high energy photons and the related– 2 –tatistical challenges allow for a certain flexibility in fitting the same set of data with differentinput models for GRBs–see, e.g., Refs. [39–43].Different GRB models may lead to very different predictions for the neutrino emission.The latter depends on the target photon spectrum and the properties of the acceleratedproton distribution (i.e., energy density, power-law slope, and maximum energy), both de-pending on the emission and dissipation mechanisms as well as the location of the protonacceleration region.In this work, we compute the neutrino emission for a benchmark high-luminosity GRBin various jet emission and dissipation scenarios. In particular, we consider an internalshock (IS) model [44], a dissipative photosphere model in the presence of ISs (PH-IS) [45], athree-component model (3-COMP) with emission arising from the photosphere, the IS, andexternal shock [46], and the internal-collision-induced magnetic reconnection and turbulencemodel (ICMART) [47]. We also compute, for the first time, the neutrino signal expected intwo models where the jet is assumed to be magnetically dominated, namely a magnetizedjet model with gradual dissipation (MAG-DISS) [48, 49], and a proton synchrotron emissionmodel (p-SYNCH) [50]. Our goal is to make a fair comparison among the proposed models fordissipation and electromagnetic emission in GRBs for what concerns the expected neutrinosignal.This paper is organized as follows. In Sec. 2, we outline the basics of the dynamicalevolution of the GRB jets considered in this paper. The main model ingredients as well asthe proton energy distributions are reported in Sec. 3. The neutrino production mechanismis discussed in Sec. 4. The neutrino emission is presented in Sec. 5, first in various scenariosinvolving ISs, then in the case of magnetized jets, and lastly for the proton synchrotronmechanism. A discussion on our findings, also in the context of detection perspectives,and conclusions are reported in Secs. 6 and 7, respectively. The fitting functions adoptedfor the photon spectral energy distributions are listed in Appendix A. A discussion on thedependence of the neutrino emission on the input parameters for the magnetic model withgradual dissipation is reported in Appendix B. In this section, we introduce the main physics describing the jet models considered in thiswork. We present the models in the context of kinetic dominated jets, then focus on twocases of Poynting flux dominated jets, and the proton synchrotron model. Note that, despitethe fact that the proton synchrotron model has a Poynting luminosity larger than the kineticone (see Ref. [51] for a dedicated discussion), we treat it separately from the Poynting fluxdominated jets because it does not require knowledge of the jet dynamics.The general GRB model envisages a relativistic jet propagating with Lorenz factor Γ,with respect to the central engine frame, and half opening angle θ j . As long as Γ − < θ j ,which is expected to hold during the prompt phase [52], the radiating region can be consideredspherically symmetric. We therefore use isotropic equivalent quantities throughout the paper.The reference frames used in our calculation are the observer frame (on Earth), theframe of the central engine (laboratory frame), and the jet comoving frame. A quantitycharacteristic of the jet is labeled as X , ˜ X , and X (cid:48) , in each of these frames, respectively. Forexample, energy is transformed through the following relation: ˜ E = (1 + z ) E = (1 + z ) D E (cid:48) ;time instead transforms as t = (1 + z )˜ t = (1 + z ) D − t (cid:48) , with D = [Γ(1 − β cos θ )] − beingthe Doppler factor, β = v/c , Γ = 1 / (cid:112) − β the Lorentz boost factor and θ the angle of– 3 –ropagation of an ejecta element with respect to the line of sight. A characteristic quantityof the jet is the isotropic-equivalent energy, ˜ E iso , which represents the energetic content ofthe outflow and it is related to the bolometric energy ˜ E bol through the opening angle by thefollowing relation: ˜ E bol = (1 − cos θ j ) ˜ E iso ≈ ( θ j /
2) ˜ E iso , where the approximation holds forsmall opening angles.The dominant source of energy in a GRB jet is related to the initial conditions. Thejet is powered by accretion onto a newly formed black hole [3] or a rapidly spinning massiveneutron star [11]. Two mechanisms are invoked to extract energy from the central compactobject and power the GRB jet: neutrino annihilation [8, 53, 54] or the extraction of magneticenergy at the expenses of the central object spin energy [10, 55]. We start with the case of a generic fireball composed of photons, electron/positron pairs,and a small fraction of baryons (primarily protons and neutrons), with negligible magneticfields [56, 57]. The dynamical evolution of the fireball is sketched in Fig. 1 and consists ofthree phases, namely acceleration, coasting, and deceleration:1.
Fireball acceleration : A hot relativistic fireball of isotropic energy ˜ E iso = ˜ L iso ˜ t dur iscreated and launched at the radius R by the central engine emitting energy withluminosity ˜ L iso for a time ˜ t dur . Since after the propagation through the envelope of Figure 1 . Schematic representation of a GRB jet (not in scale) where energy dissipation takes placethrough relativistic shocks. The Lorentz factor Γ is shown as a function of the fireball radius for thecase in which the photosphere occurs in the coasting phase, so that the photospheric radius ( R PH )lies above the saturation radius ( R sat ). The photosphere is assumed to produce thermal γ -rays, theISs forming at R IS are thought to produce non-thermal γ -rays, and the external shock, which startsto decelerate at R dec , is responsible for the afterglow. When energy dissipation takes place below thephotosphere, non-thermal radiation is also expected from R PH . – 4 –he progenitor star, the fireball can be re-born [58], we adopt as size of the jet base R = R (cid:63) θ j , with R (cid:63) (cid:39) cm being the progenitor star radius. The width of theemitted shell is ˜∆ = c ˜ t dur . As the fireball shell undergoes adiabatic expansion, andwhile the pair plasma retains relativistic temperatures, baryons are accelerated byradiation pressure and the bulk Lorenz factor increases linearly with radius (Γ ∝ R ),until it reaches its maximum value. We assume that the latter coincides with thedimensionless entropy per baryon η = ˜ E iso /M c , where M is the baryonic mass injectedinto the outflow. The maximum Lorenz factor is achieved at the saturation radius R sat = ηR .2. Fireball coasting : Beyond R sat , the flow coasts with Γ = Γ sat ∼ η = const. As thefireball shell keeps on expanding, the baryon density, obtained by the mass continuityequation ˙ M = 4 πR Γ ρ (cid:48) c = const [59] for a relativistic flow with spherical symmetry,drops as n (cid:48) b = ρ (cid:48) m p = ˙ M πm p R c Γ (cid:39) ˜ L iso πR m p c η Γ , (2.1)where ρ (cid:48) is the baryon density in the comoving frame, R is the distance from the centralengine, and ˜ L iso = η ˙ M c . At a certain point, photons become optically thin to both pairproduction and Compton scattering off free leptons associated with baryons entrainedin the fireball. Once the Thomson optical depth ( τ T = n (cid:48) l σ T R/ Γ) drops below 1, theenergy that has not been converted into kinetic energy is released at the photosphericradius R PH . Let R denote the number of leptons per baryon ( n (cid:48) l = R n (cid:48) p ), we can definethe critical dimensionless entropy [60]: η ∗ = (cid:32) σ T R ˜ L iso Γ πR m p c (cid:33) / , (2.2)where σ T = 6 . × − cm is the Thomson cross section, and Γ = Γ( R (cid:63) ) (cid:39) η ∗ represents the limiting value of the Lorenz factor whichseparates two scenarios: η > η ∗ (the photosphere occurs in the acceleration phase) and η < η ∗ (the photosphere occurs in the coasting phase). For our choice of parameters,we will always be in the second case, thus we can introduce the photospheric radius asthe distance such that τ T = 1 [62]: R PH = σ T ˜ L iso R πη m p c . (2.3)The radiation coming from the photosphere is the first electromagnetic signal detectablefrom the fireball. It emerges peaking at [62] k B ˜ T PH = k B (cid:32) ˜ L iso πR σ B (cid:33) (cid:18) R PH R sat (cid:19) − , (2.4)where σ B is the Stefan-Boltzmann constant and k B the Boltzmann constant. Theenergy ˜ E PH emerging from the photosphere is parametrized through ε PH = ˜ E PH / ˜ E iso .Since the central engine responsible for the launch of the relativistic jet is expected tohave an erratic activity, the produced outflow is unsteady and radially inhomogeneous.– 5 –his causes internal collisions between shells of matter emitted with time lag t v to occurat a distance [44] R IS (cid:39) ct v Γ z ; (2.5)this is the IS radius, where a fraction ε IS of the total outflow energy ( ˜ E iso ) is dissipated,and particles are accelerated.3. Fireball deceleration : The fireball shell is eventually decelerated [57, 63, 64] by thecircumburst medium that can either be the interstellar medium or the pre-ejectedstellar wind from the progenitor before the collapse. Let us consider an external densityprofile [65]: n b ( R ) = AR − s , (2.6)with s = 0 for a homogeneous medium and s = 2 for a wind ejected at constant speed.For a thin shell [66], the deceleration radius is defined as the distance where the sweptmass from the circumburst medium is m CMB = M/η [or Γ( R dec ) = η/
2] [67]: R dec = (cid:32) − s π ˜ E K , iso m p c Aη (cid:33) / (3 − s ) ; (2.7)in alternative, R dec can be obtained from the observed deceleration time t dec [68]: t dec (cid:39) . z ) R dec η c , (2.8)where ˜ E K , iso = ˜ E iso − ˜ E γ, iso is the isotropic equivalent kinetic energy of the ouflow after˜ E γ, iso has been radiated during the prompt phase. At R dec , an external shock formsand propagates into the medium, hence the deceleration radius is essentially the initialexternal shock radius. For long time, the IS model [44, 69, 70] has been considered as the standard model for theprompt emission in the literature. Among the merits of this model there is its ability tonaturally explain the variability of the lightcurves, to provide natural sites for the dissipationof the kinetic energy of the baryonic fireball, as well as sites for particle acceleration andnon-thermal radiation.The erratic activity of the central engine is responsible for the creation of an outflowthat can be visualized as being composed of several shells. Collisions of such shells withdifferent masses and/or Lorenz factors cause the dissipation of the kinetic energy of the jetat R IS1 (see Fig. 1).Part of the dissipated energy, ε IS ˜ E iso , is used for particle acceleration. Non-thermalelectrons (protons) receive a fraction ε e ( ε p ), while the magnetic field receives a fraction ε B of that energy. In this scenario, electrons emit synchrotron radiation in the fast coolingregime. The radiated energy can thus be expressed as E (cid:48) γ, iso = ε e ε IS E (cid:48) iso and the magneticfield as B (cid:48) = (cid:115) π ε B ε e E (cid:48) γ, iso V (cid:48) iso . (2.9) If there is a large spread in the Γ values of the shells, then R IS can also spread a lot [71–73]. – 6 –he protons co-accelerated with electrons interact with the prompt photons through photo-hadronic interactions and produce neutrinos, as discussed in Sec. 4. Here V (cid:48) iso = 4 πR γ Γ c ˜ t dur represents the isotropic volume of the jet in the comoving frame.Within a more realistic setup, various collisions between plasma shells occur along thejet. Scenarios involving collisions of multiple shells have been considered [71–73] and can leadto lower neutrino fluxes. Yet, in this work, since we aim to compare different jet models, weadopt one representative shell with average parameters and spectral properties for simplicity. In the class of photospheric models, it is assumed that the dominant radiation observedin the prompt phase is produced in the optically thick region below the photosphere [74].Depending on the presence of dissipative processes acting in the optically thick parts of theoutflow, photospheric models can be classified in non-dissipative or dissipative ones.In the presence of a non-dissipative photosphere, according to the standard fireballmodel, the thermal radiation advected with the flow and unaffected by the propagation isreleased at R PH , see Fig. 1. Depending on the dimensionless entropy of the outflow (seeEq. 2.2), this component can be very bright or highly inefficient and is characterized by thefraction ε PH = ( η/η ∗ ) / [62].For a dissipative photosphere, strong subphotospheric dissipation is required in theoptically thick inner parts of relativistic outflows in order to account for the detected non-thermal spectra [75–81]. In this scenario, the spectral peak and the low-energy spectrumbelow the peak are formed by quasi-thermal Comptonization of seed photons by mildlyrelativistic electrons when the Thomson optical depth of the flow is 1 (cid:46) τ T (cid:46)
100 [82,83]. In the literature, several sub-photopsheric dissipative mechanisms have been proposed,including ISs at small-radii [78], collisional nuclear processes [84], or dissipation of magneticenergy [82]. One of the most attractive features of these models is their ability to naturallyexplain the observed small dispersion of the sub-MeV peak and the high prompt emissionefficiency [58, 85, 86], that the standard version of the IS model cannot easily explain.The scenario explored in this work considers the main prompt emission as being releasedat R PH with a non-thermal spectrum. These photons cross the IS region and interact withenergetic protons accelerated at the IS to produce neutrinos. We do not consider neutrinoproduction below and at the photosphere, as this would result in neutrino energies wellbelow the PeV range that we are interested in (i.e., GeV neutrinos produced in proton-neutron collisions in the ejecta [87, 88] or TeV neutrinos produced via pp interactions ofprotons accelerated at sub-photospheric ISs [20, 89–92]). Indeed, photo-hadronic interactionsin the opaque region do not lead to efficient production of high-energy neutrinos because ofinefficient Fermi acceleration that limits the maximum proton energy to low values [93]. The three-component GRB model was introduced in Ref. [46], where the authors found that athermal component described by a black body (BB) spectrum, a Band spectrum (sometimesstatistically equivalent to a cut-off power-law, CPL) and a non-thermal power-law (PL)spectrum at high energies (with or without cut-off) represent a globally better description ofthe data than the Band spectral fit for a number of bursts. We refer the reader to Appendix Afor details on the spectral energy distributions of photons.As argued in Ref. [46], the physical interpretation proposed for the three components isthe following. The BB component, given its weakness, is interpreted as thermal photospheric– 7 –mission of a magnetized jet not strongly affected by sub-photospheric dissipation. The non-thermal emission fitted by the Band (or CPL) component, given the observed variability,is assumed to be produced in the optically thin region of the jet from relativistic electrons.The third PL (or CPL) component, which extends over at least 5 decades in energy andsometimes emerges with a slight temporal delay with respect to the trigger of the burst, isthe one with the least clear origin. Because of its initial temporal variability, it is assumed tobe of internal origin; e.g., it might be due to inverse Compton processes, even if this scenariois not able to explain the extension of such a component to lower energies or the temporaldelay. Finally, the fact that in some cases the PL component becomes dominant at the endof the bursts and lasts longer than the prompt emission led to identify it with the emergenceof an early afterglow, which corresponds to the start of deceleration of the outflow.
When the central compact object is a rapidly rotating black hole threaded by open magneticfield lines, it is possible to tap the black hole spin energy to produce Poynting-flux dominatedjets [10]. The electromagnetic luminosity of this jet is much larger than the kinetic luminosityassociated to matter.A characteristic parameter is the magnetization σ , defined as the ratio of the Poyntingluminosity and the kinetic luminosity: σ ( R ) ≡ L B L K = B ( R )4 π Γ ρ ( R ) c = B (cid:48) ( R )4 πρ (cid:48) ( R ) c , (2.10)where B (cid:48) ( R ) and ρ (cid:48) ( R ) are the magnetic field strength and matter density in the comovingframe at a certain distance R from the central engine. Hence, the total jet luminosity at anyradius is L ( R ) = [1 + σ ( R )] L K ( R ). In this work we consider two models for magnetized jets:the ICMART model and the gradual magnetic dissipation model, which we briefly introducebelow. The ICMART model [47] considers Poynting flux dominated jets, whose energy is dissipatedand radiated away at very large radii from the central engine, as shown in Fig. 2. Themain motivation behind this model relies on the non-detection (or detection of a very weak)photospheric component in the spectra of some GRBs, hinting that the jet compositioncannot be largely Poynting flux dominated at the photosphere.The GRB central engine intermittently ejects an unsteady jet with variable Lorentzfactor and with a nearly constant degree of magnetization σ ≡ σ ( R ). Such a jet is composedby many discrete magnetized shells which collide at R IS (see Eq. 2.5 and Fig. 2). Yet, thekinetic energy dissipated at the ISs is smaller by a factor [1 + σ ( R IS )] with respect to theenergy available in the traditional IS model. Hence, the total energy emitted in radiationcould be completely negligible at this stage.In the optically thin region, the early internal collisions have the role of altering, andeventually destroying, the ordered magnetic field configuration, triggering the first reconnec-tion event. The ejection of plasma from the reconnection layer would disturb the nearbyambient plasma and produce turbulence, facilitating more reconnection events which wouldlead to a runaway catastrophic release of the stored magnetic field energy at the radius de-fined as R ICMART . This would correspond to one ICMART event, which would compose one– 8 – igure 2 . Schematic representation of a Poynting flux dominated jet (not in scale) in the ICMARTmodel. The Lorentz factor Γ is shown as a function of jet radius. The radiation from the photosphere( R PH ) and ISs ( R IS ) is strongly suppressed and can be at most 1 / ( σ + 1) of the total jet energy(see Eq. 2.10); typical values for the magnetization parameter σ are shown. The emitting region islocated at R ICMART , where magnetic reconnection causes a strong discharge of magnetic energy andthe emission of gamma-rays. The magnetization at R ICMART is σ in and σ end in the beginning and atthe end of an ICMART event, respectively. GRB pulse. Other collisions that trigger other reconnection-turbulence avalanches wouldgive rise to other pulses.This model successfully reproduces the observed GRB lightcurves with both fast andslow components [94]. The slow component, related to the central engine activity, would becaused by the superposition of emission from all the mini-jets due to multiple reconnectionsites, while the erratic fast component would be related to the mini-jets pointing towards theobserver.
In this scenario, significant magnetic dissipation already occurs below the jet photosphere,as schematically shown in Fig. 3. Following Refs. [48, 49], we consider magnetized outflowswith a striped-wind magnetic field structure, where energy is gradually dissipated throughmagnetic reconnection until the saturation radius. This model can naturally explain thedouble-hump electromagnetic spectra sometimes observed [49].The jet is injected at R ∼ cm with magnetization σ (cid:29) = √ σ + 1 ≈ σ . As it propagates, the magnetic field lines of opposite polarity reconnect,causing the magnetic energy to be dissipated at a rate:˙ E diss = − dL B dR = − ddR (cid:18) σσ + 1 L (cid:19) ∝ R / , (2.11)– 9 – igure 3 . Schematic representation of a Poynting flux dominated jet (not in scale) in the gradualenergy dissipation model. The Lorentz factor Γ is shown as a function of jet radius. The radiationfrom the photosphere ( R PH ) can be very bright, depending on the initial magnetization σ of theoutflow. Typical values for the magnetization parameter σ are shown. The emitting region is locatedbetween R PH and R sat , where magnetic reconnection causes the dissipation of magnetic field energy,the emission of thermal gamma-rays at R PH and synchrotron radiation from accelerated electrons inthe optically thin region up to R sat . where σ ( R ) is obtained from the conservation of the total specific energy Γ( R ) σ ( R ) = Γ σ .The Lorentz factor of the flow evolves asΓ( R ) = Γ sat (cid:18) RR sat (cid:19) / , (2.12)until the saturation radius R sat = λ Γ (see Fig. 3), where λ is connected to the characteristiclength scale over which the magnetic field lines reverse polarity. This length scale can berelated to the angular frequency of the central engine (e.g., of millisecond magnetars) or withthe size of the magnetic loops threading the accretion disk [95].Motivated by results of particle-in-cell (PIC) simulations of magnetic reconnection inmagnetically dominated electron-proton plasmas [96–98], we assume that half of the dissi-pated energy in Eq. 2.11 is converted in kinetic energy of the jet, while the other half goesinto particle acceleration and is redistributed among electrons and protons . In particular, Rough energy equipartition between magnetic field, protons and electron-positron pairs is also found inkinetic simulations of reconnection in pair-proton plasmas [99]. – 10 –he fraction of energy which goes into electrons is [98] ε e ≈ (cid:18) (cid:114) σ
10 + σ (cid:19) , (2.13)while the one that goes into protons has been extracted from Fig. 20 of Ref. [98] and is ε p ∼ − ε e . A fraction ξ of electrons injected into the dissipation region are acceleratedinto a power-law distribution n (cid:48) e ( γ (cid:48) e ) ∝ γ (cid:48)− k e e in the interval [ γ (cid:48) e, min , γ (cid:48) e, max ] with the minimumelectron Lorentz factor being γ (cid:48) e, min ( R ) = k e − k e − ε e ξ σ ( R ) m p m e , (2.14)and γ (cid:48) e, max is the maximum electron energy obtained by equating the acceleration time and thetotal cooling time. The power-law slope of the accelerated particles in relativistic reconnectiondepends on the plasma magnetization in a way that harder spectra ( k e <
2) are obtained for σ (cid:29) k e ( σ ) ≈ . . √ σ . (2.15)The proton spectrum will be discussed in detail in Sec. 5.2.2. Recently, Refs. [39, 103, 104] have analyzed the spectra of a sample of GRBs for which datadown to the soft X-ray band and, in some cases, in the optical are available. This extensivework has established the common presence of a spectral break in the low energy tail of theprompt spectra and led to realize that the spectra could be fitted by three power-laws. Thespectral indices below and above the break are found to be α (cid:39) − / α (cid:39) − / β <
2. The values of all photonindexes are consistent with the predicted values for the synchrotron emission in a marginallyfast cooling regime [105]. However, if electrons are responsible for the prompt emission, thenthe parameters of the jet have to change drastically with respect to the standard scenario, inwhich the emission takes place at relatively small radii and with strong magnetic fields in situ.One possible way out to this has been discussed in Ref. [50], where protons are consideredto be the particles which radiate synchrotron emission in the marginally fast cooling regime;in this way, it is possible to recover the typical emitting region size at R γ (cid:39) cm. In this section, we outline some of the quantities characterizing the energetics and geometryof the jet for all models. We also introduce the target particle distributions.
The gamma-ray emission is assumed to originate from an isotropic volume V (cid:48) iso , s = 4 πR γ ∆ (cid:48) s ,where ∆ (cid:48) s = R γ /
2Γ is the comoving thickness of the emitting shell and R γ is the distance fromthe central engine where the electromagnetic radiation is produced. Dissipation–whether itoccurs in the photosphere, in the optically thin region (e.g., ISs) or external shocks–causesthe conversion of a fraction ε d of the total jet energy ˜ E iso into thermal energy, bulk kineticenergy, non-thermal particle energy, and magnetic energy. The energy stored in relativistic– 11 – able 1 . Characteristic parameters assumed for our benchmark GRB jet for the scenarios consideredin this paper: internal shock (IS) model, dissipative photosphere model with internal shocks (PH-IS),three components model (3-COMP), ICMART model, magnetized jet model with gradual dissipation(MAG-DISS), and proton synchrotron model (p-SYNCH). In the case of quantities varying along thejet, the variability range is reported. The dissipation efficiency ε IS has been taken from Ref. [69]. Forthe magnetic model with gradual dissipation, the electron fraction, the electron power-law index, andthe proton power-law index are defined in Eqs. 2.13, and 2.15, respectively. Parameter Symbol Model
IS PH-IS 3-COMP ICMART MAG-DISS p-SYNCHTotal jet energy ˜ E iso . × erg n/aJet opening angle θ j ◦ Lorentz boost factor Γ 300Redshift z t dur
100 sVariability time scale t v . ε d ε IS = 0 . ε IS = 0 .
03 n/a ε d = 0 .
35 0 .
24 n/aElectron energy fraction ε e /
12 0 . . − .
36 n/aProton energy fraction ε p /
12 0 . . − .
65 n/aElectron power-law index k e n/a 2 . . − . k p . − . . R γ σ n/a 45 1 . − .
81 n/a electrons, protons, and magnetic fields in the emitting region can be parameterized throughthe fractions ε p , ε e and ε B , respectively.These parameters ignore the detailed microphysics at the plasma level, but allow toestablish a direct connection with the observables. We assume that energy equipartitionbetween electrons and magnetic fields occurs at the ISs, and 10 times more energy goesinto protons, namely: ε e = ε B = 1 /
12 and ε p /ε e = 10. For the magnetized jet models,instead, these parameters may depend on the magnetization of the jet, as we will see later.At the external shock, in the deceleration phase, we adopt ε e = 4 × − , ε B = 10 − and ε p = 1 − ε e − ε B [106–108]For what concerns the energetics of our reference jet, motivated by recent observationsof GRB afterglows [109], we choose ˜ E iso = 3 . × ergs, where a typical opening angle of θ j = 3 degrees is adopted. Our benchmark Lorenz factor is Γ = Γ sat = 300 [110]. Theduration of the burst is taken to be ˜ t dur = 100 s / (1 + z ), where z = 2 is the redshift weadopt for our reference GRB. Finally we use t v = 0 . η γ = ˜ E γ, iso ˜ E iso . (3.1)For example η γ = ε PH when the dominant radiation is of photospheric origin or η γ = ε IS ε e when the radiation is produced at the IS, assuming a fast cooling regime for electrons.– 12 – .2 Spectral energy distribution of protons For the purposes of this work, it is sufficient to assume that protons and electrons in thedissipation site are accelerated via Fermi-like mechanisms . The accelerated particles acquirea non-thermal energy distribution that can be phenomenologically described as [115]: n (cid:48) p ( E (cid:48) p ) = AE (cid:48)− kp exp (cid:20) − (cid:18) E (cid:48) p E (cid:48) p, max (cid:19) α p (cid:21) Θ( E (cid:48) p − E (cid:48) p, min ) , (3.2)where A = U (cid:48) p (cid:104)(cid:82) E (cid:48) p, max E (cid:48) p, min n (cid:48) p ( E (cid:48) p ) E (cid:48) p dE (cid:48) p (cid:105) − is the normalization of the spectrum (in units ofGeV − cm − ) and Θ is the Heaviside function, with U (cid:48) p = ε p ε d E (cid:48) iso being the fraction ofthe dissipated jet energy that goes into acceleration of protons. The power-law index isfound to be k ≈ . k = 2 for a non-relativistic shock [117], while it depends on the jetmagnetization for magnetically dominated jets, as we will see later. The exponential cut-offwith α p is due to energy losses of protons and we adopt α p = 2 following Ref. [118], E (cid:48) p, min is the minimum energy of the protons that are injected within the acceleration region, and E (cid:48) p, max is the maximum proton energy. The latter is constrained by the Larmor radius beingsmaller than the size of the acceleration region, or imposing that the acceleration timescale, t (cid:48)− p, acc = ζceB (cid:48) E (cid:48) p , (3.3)is shorter than the total cooling timescale for protons. Here ζ = 1 is the acceleration efficiencyadopted throughout this work. The total cooling timescale is given by t (cid:48)− p, cool = t (cid:48)− + t (cid:48)− p, IC + t (cid:48)− p, BH + t (cid:48)− pγ + t (cid:48)− p, hc + t (cid:48)− p, ad ; (3.4)where t (cid:48) sync , t (cid:48) p, IC , t (cid:48) p, BH , t (cid:48) pγ , t (cid:48) p, hc , t (cid:48) p, ad are the proton synchrotron (sync), inverse Compton(IC), Bethe-Heitler ( pγ → pe + e − , BH), hadronic (hc) and adiabatic (ad) cooling times,respectively. They are defined as follows [59, 119, 120]: t (cid:48)− p, sync = 4 σ T m e E (cid:48) p B (cid:48) m p c π , (3.5) t (cid:48)− p, IC = 3( m e c ) σ T c γ (cid:48) p ( γ (cid:48) p − β (cid:48) p (cid:90) E (cid:48) γ, max E (cid:48) γ, min dE (cid:48) γ E (cid:48) γ F ( E (cid:48) γ , γ (cid:48) p ) n (cid:48) γ ( E (cid:48) γ ) , (3.6) t (cid:48)− p, BH = 7 m e ασ T c √ πm p γ (cid:48) p (cid:90) E (cid:48) γ, max mec γ (cid:48)− p d(cid:15) (cid:48) n (cid:48) γ ( (cid:15) (cid:48) ) (cid:15) (cid:48) (cid:40) (2 γ (cid:48) p (cid:15) (cid:48) ) / (cid:20) ln( γ (cid:48) p (cid:15) (cid:48) ) − (cid:21) + 2 / (cid:41) , (3.7) t (cid:48)− pγ = c γ (cid:48) p (cid:90) ∞ E th2 γ (cid:48) p dε (cid:48) γ n (cid:48) γ ( E (cid:48) γ ) E (cid:48) γ (cid:90) γ (cid:48) p E (cid:48) γ E th dE r E r σ pγ ( E r ) K pγ ( E r ) , (3.8) t (cid:48)− = cn (cid:48) p σ pp K pp , (3.9) t (cid:48)− p, ad = 2 c Γ R . (3.10) In the reconnection region there are various particle acceleration sites, see e.g. Ref. [112]. It remains amatter of active research what is the dominant process responsible for the formation of the power-law, seee.g. Refs. [100, 101, 112–114] – 13 –n the definitions above, (cid:15) (cid:48) = E (cid:48) γ /m e c , γ (cid:48) p = E (cid:48) p /m p c , and α = 1 /
137 is the fine structureconstant. The cross sections σ pγ and σ pp , for pγ and pp interactions respectively, are takenfrom Ref. [121]. The function F ( E (cid:48) γ , γ (cid:48) p ) is provided in Ref. [122], while K pγ is the inelasticityof pγ collisions [59]: K pγ ( E r ) = (cid:40) . E th < E r < . E r > , (3.11)where E r = γ (cid:48) p E (cid:48) γ (1 − β (cid:48) p cos θ (cid:48) ) is the relative energy between a proton with gamma factor γ (cid:48) p and a photon of energy E (cid:48) γ , whose directions form an angle θ (cid:48) in the comoving system, E th = 0 .
15 GeV is the threshold for the photo-hadronic interaction, n (cid:48) γ ( E (cid:48) γ ) is the targetphoton density field (in units of GeV − cm − ), K pp = 0 .
8, and n (cid:48) p is the comoving protondensity defined as n (cid:48) p = n (cid:48) b /
2, where n (cid:48) b is the baryonic density defined in Eq. 2.1. Aswe will see in Sec. 5.3, the proton synchrotron scenario is such that the properties of theproton distribution (e.g., minimum energy, power-law slope), as well as the shape of energydistribution itself, can be directly inferred from the observed GRB prompt spectra. The simultaneous presence of a high density target photon field in the site of proton acceleration–that can be radiated by co-accelerated electrons, by protons themselves or have an externalorigin–leads to an efficient production of high-energy neutrinos through photo-hadronic inter-actions. Since the number of target photons is always much larger than the number densityof non-relativistic (cold) protons in all cases of study, we neglect the pp contribution.Photo-hadronic interactions lead to charged pion and kaon (as well as neutron) produc-tion, which subsequently cool and decay in muons and neutrinos. According to the standardpicture, pion production occurs through the ∆(1232) resonance channel: p + γ −→ ∆ + −→ (cid:40) n + π + / p + π / π + → µ + + ν µ (4.2) µ + → ¯ ν µ + ν e + e + . (4.3)In order to accurately estimate the neutrino spectral energy distribution and the relatedneutrino flavor ratio, we rely on the photo-hadronic interaction model of Ref. [118] (modelSim-B and Sim-C) based on SOPHIA [123]. The latter includes higher resonances, directand multi-pion production contributions. Note that, although we compute the neutrino andantineutrino spectral distributions separately, in the following we do not distinguish betweenthem unless otherwise specified.Given the photon and proton energy distributions in the comoving frame, n (cid:48) γ ( E (cid:48) γ ) and n (cid:48) p ( E (cid:48) p ), the production rate of secondary particles is given by (in units of GeV − cm − s − ) Q (cid:48) l ( E (cid:48) l ) = (cid:90) ∞ E (cid:48) l dE (cid:48) p E (cid:48) p n (cid:48) p ( E (cid:48) p ) (cid:90) ∞ E th / γ (cid:48) p dE (cid:48) γ n (cid:48) γ ( E (cid:48) γ ) R α ( x, y ) , (4.4)where x = E (cid:48) l /E (cid:48) p is the fraction of proton energy going into daughter particles, y = γ (cid:48) p E (cid:48) γ , and l stands for π + , π − , π , and K + . Since kaons suffer less from radiative cooling than charged– 14 –ions due to their larger mass and shorter lifetime, their contribution to the neutrino fluxbecomes important at high energies [36, 124, 125], whilst it is sub-leading at lower energies,given the low branching ratio for their production. The “response function” R l ( x, y ) containsall the information about the interaction type (cross section and multiplicity of the products);we refer the interested reader to Ref. [118] for more details.Once produced, the charged mesons undergo different energy losses before decaying intoneutrinos. Their energy distribution at decay is Q (cid:48) dec l ( E (cid:48) l ) = Q (cid:48) l ( E (cid:48) l ) (cid:34) − exp (cid:32) − t (cid:48) l, cool E (cid:48) l τ (cid:48) l (cid:33)(cid:35) , (4.5)with t (cid:48) l, cool being the cooling time scale and τ (cid:48) l the lifetime of the meson l . The neutrinoenergy distribution originating from the decay processes like the one in Eq. 4.2 is Q (cid:48) ν α ( E (cid:48) ν α ) = (cid:90) ∞ E (cid:48) να Q (cid:48) dec l ( E (cid:48) l ) 1 E (cid:48) l F l → ν α (cid:18) E (cid:48) ν α E (cid:48) l (cid:19) , (4.6)where F l → ν α is defined in Ref. [115] for ultra-relativistic parent particles. The same procedureis followed for antineutrinos.The steps above also allow to compute the spectra of charged muons, with an additionaldependence on the muon helicity. Again, the cooled muon spectra are derived as in Eq. 4.5and the neutrinos generated by the muon decay are computed following Ref. [115].The total neutrino injection rate Q (cid:48) ν α ( E (cid:48) ν α ) at the source is obtained by summing overthe contributions from all channels. Finally, the fluence for the flavor ν α at Earth from asource at redshift z is (in units of GeV − cm − )Φ ν α ( E ν α , z ) = ˆ N (1 + z ) πd L ( z ) (cid:88) β P ν β → ν α ( E ν α ) Q (cid:48) ν β (cid:20) E ν α (1 + z )Γ (cid:21) , (4.7)where [29] P ν e → ν µ = P ν µ → ν e = P ν e → ν τ = 14 sin θ , (4.8) P ν µ → ν µ = P ν µ → ν τ = 18 (4 − sin θ ) , (4.9) P ν e → ν e = 1 −
12 sin θ , (4.10)with θ (cid:39) . P ν β → ν α = P ¯ ν β → ¯ ν α , and ˆ N = V (cid:48) iso , s t dur [127] being the normal-ization factor depending on the volume of the interaction region. The luminosity distance d L ( z ) is defined in a flat ΛCDM cosmology as d L ( z ) = (1 + z ) cH (cid:90) z dz (cid:48) (cid:112) Ω Λ + Ω M (1 + z (cid:48) ) (4.11)with Ω M = 0 . Λ = 0 .
685 and the Hubble constant H = 67 . − Mpc − [128].– 15 – Results: Gamma-ray burst neutrino emission
Each of the dissipation mechanisms introduced in Sec. 2, according to the radius at whichit takes place, leads to different photon energy distributions. In Appendix A we report theempirical functions usually adopted to fit the observed photon spectra. For each of the GRBmodels considered in this section, we assume that the spectral energy distribution of photonsis either given by one of the fitting functions or a combination of them. In this section, weinvestigate the neutrino production in the prompt phase for each scenario.
We focus on the IS model introduced in Sec. 2.1.1 with the photon spectrum produced atthe IS radius and described by the Band function in Eq. A.2. The radiative efficiency is η γ = ε IS ε e (cid:39) .
02 (see Table 1).In order to establish the relative importance of the various energy loss processes inthis scenario, we compute the proton and the secondary particle ( K ± , π ± and µ ± ) coolingtimes as illustrated in Sec. 3.2. The cooling times are shown in Fig. 4 as functions of theparticle energy in the comoving frame. With the parameters adopted for our benchmarkGRB, protons are mainly cooled by adiabatic expansion up to E (cid:48) p, max (left panel of Fig. 4),with the second dominant energy loss mechanism being photo-hadronic interactions. Formesons and muons (right panel of Fig. 4) adiabatic and synchrotron cooling at low and highenergies, respectively, are the two dominant cooling processes. E p [GeV] − − − − − − − − t − [ s − ] t − p,ad t − p,cool t − pγ t − p,BH t − p,IC t − p,sync t − hc t − p,acc E µ,π,K [GeV] − − − − t − [ s − ] τ − π τ − µ τ − K t − ad t − π,sync t − µ,sync t − K,sync t − π,tot t − µ π ,tot t − K,tot
Figure 4 . Left:
Inverse cooling timescales for protons at the IS radius as functions of the protonenergy in the comoving frame for our benchmark GRB, see Table 1. The thin solid lines mark theindividual cooling processes introduced in Sec. 3.2; the thick black and red solid lines represent thetotal cooling timescale and the acceleration timescale, respectively. The red star marks the maximumcomoving proton energy such that t (cid:48)− p, cool = t (cid:48)− p, acc . Protons are mainly cooled by adiabatic expansionand pγ interactions. Right:
Analogous to the left panel, but for the inverse cooling timescales forpions, muons, and kaons. The dominant energy losses in this case are adiabatic cooling at low energiesand synchrotron cooling at higher energies. – 16 – − − − − − − − − E γ [GeV] − − − − − E γ Φ γ ( E γ ) [ G e V c m − ] BAND 10 E ν [GeV] − − − − − − E ν Φ ν ( E ν ) [ G e V c m − ] ν µ + ¯ ν µ ν e + ¯ ν e Figure 5 . Left:
Band photon fluence observed at Earth for our benchmark GRB in the IS model, seeTable 1.
Right:
Correspondent ν α + ¯ ν α fluence in the observer frame (in red for the electron flavorand in blue for the muon flavor) in the presence of flavor conversions. The fluence for the muon flavorpeaks at E peak ν = 3 . × GeV.
Following Sec. 4, we compute the neutrino production rate in the comoving frame at R IS and the correspondent fluence at Earth including flavor conversions. The results are shownin the right panel of Fig. 5, while the photon spectrum described by the Band function isshown in the left panel.Our results are in good agreement with analogous estimations reported in the literaturefor comparable input parameters, see e.g. Ref. [129]. In this scenario, the fluence for themuon flavor peaks at E peak ν = 3 . × GeV and rapidly declines at higher energies. Theeffects due to the cooling of kaons are not visible because, as shown in Fig. 4, the maximumproton energy is more than one order of magnitude lower than the one at which kaons coolby synchrotron radiation.
We now explore the model introduced in Sec. 2.1.2 and consider a jet with an efficient pho-tospheric emission, described by a Band spectrum peaking at the energy given by Eq. A.4,and undergoing further IS dissipation. At R IS , protons and electrons are efficiently accel-erated and turbulent magnetic fields may build up. In this scenario, electrons cool, otherthan by emitting synchrotron radiation, also by Compton up-scattering of the non-thermalphotospheric photons. As we are interested in investigating the case where the photosphericemission is dominant in the MeV energy range, we consider Case (I) of Table 1 of Ref. [45],corresponding to the luminosity hierarchy L PH (cid:29) L UP (cid:29) L SYNC , where L PH , L UP , and L SYNC stand for the photospheric luminosity, up-scattered photospheric luminosity of theaccelerated electrons at R IS , and synchrotron luminosity radiated by the electrons at R IS ,respectively.Following Ref. [45], we define x = ε IS ε B ε PH and Y = U (cid:48) SYNC U (cid:48) B = 43 ( k e − k e − τ T γ (cid:48) e, min γ (cid:48) e, cool h , (5.1)– 17 –here Y is the Compton parameter, k e is the slope of the electron energy distribution, h is a function of γ (cid:48) e, min and γ (cid:48) e, cool and depends on the cooling regime, γ (cid:48) e, min is the minimumLorentz factor of the electrons injected in the acceleration region γ (cid:48) e, min = m p m e k e − k e − R − ξ − ε IS ε e , (5.2)with ξ being the fraction of electrons accelerated at the shock and R being the numberof leptons per baryon. Finally, γ (cid:48) e, cool is the electron cooling Lorentz factor obtained from γ (cid:48) e, cool m e c = P ( γ (cid:48) e, cool ) t (cid:48) ad and given by γ (cid:48) e, cool ( R ) (cid:39) m e R m p τ T ε PH x (1 + Y ) + 1 ; (5.3) t (cid:48) ad being the adiabatic cooling timescale and P ( γ (cid:48) e, cool ) = 4 / σ T cγ (cid:48) e, cool ( U (cid:48) B + U (cid:48) SYNC + U (cid:48) PH )the cooling rate for electrons. The conditions we need to fulfill in order to satisfy L PH (cid:29) L UP (cid:29) L SYNC are η < η ∗ , x (cid:28) , xY (cid:28) , Y = ε IS ε e hε PH (cid:28) . (5.4)In this way, it is possible to estimate L UP = Y L PH and L SYNC = xY L PH .We adopt the electron slope k e = 2 . ε IS by relying on the observations inthe optical band; by assuming that the synchrotron extended emission in this range shouldnot be brighter than what is typically observed, the following constraint on the flux holds: F sync ν ( E γ, opt ) <
100 mJy with E γ, opt = 2 eV [130]. In our case, ε IS ≤ .
03 satisfies sucha condition. The radiative efficiency of this GRB is η γ = ( ˜ E PH + ˜ E SYNC + ˜ E UP ) / ˜ E iso =˜ E PH (1 + Y + xY ) / ˜ E iso (cid:39) .
2. Since the high-energy photopsheric photons are absorbed bythe e ± pair creation at R PH , we use a cut-off for the Band spectrum at R PH , defined inEq. A.1.We define the total photon energy distribution in the comoving frame at R IS as n (cid:48) γ, tot ( E (cid:48) γ ) = (cid:18) R PH R IS (cid:19) n (cid:48) γ, PH ( E (cid:48) γ ) + n (cid:48) γ, SYNC ( E (cid:48) γ ) + n (cid:48) γ, UP ( E (cid:48) γ ) (5.5)and compute the cooling processes of protons at the IS. The results are qualitatively similar tothose shown in Fig. 4, except for the BH cooling being strongly dominant over the pp one, and t pγ being much shorter [ O (40)] than in the IS case; this is due to the larger photon numberdensity at R IS . Synchrotron losses are negligible. Furthermore, the acceleration timescale inthis scenario is much longer due to a weaker magnetic field in the acceleration region. This setsthe proton energy distribution cutoff at a lower energy than before ( E (cid:48) p, max ∼ . × GeV).Figure 6 shows the resultant photon (on the left panel) and neutrino (on the right panel)fluences for the IS shock scenario with a dissipative photosphere. The black curve in the leftpanel represents the overall photon fluence. The total spectrum is consistent with Fermiobservations [131], being the high energy component subdominant with respect to the Bandone. From the right panel, one can see that the muon fluence peaks at E peak ν = 6 . × GeVand it is comparable in normalization to the one of the IS model (see Fig. 5). This is due toa counterbalance between the radiative inefficiency in the IS model and a lower high energyproton number density in this scenario. Similar conclusions are reported in Ref. [132], wherea further reduction in the neutrino production efficiency is envisaged as due to the anisotropy– 18 – − − − − − − − E γ [GeV] − − − − − − − − − E γ Φ γ ( E γ ) [ G e V c m − ] PHSYNCUPTOT E ν [GeV] − − − − − − E ν Φ ν ( E ν ) [ G e V c m − ] ν µ + ¯ ν µ ν e + ¯ ν e Figure 6 . Left:
Photon fluence observed at Earth for the IS model with dissipative photosphere. Thephotospheric emission (PH, violet line), the photospheric up-scattered emission (UP, orange line), andthe synchrotron emission of electrons accelerated at R IS (SYNC, green line) are plotted together withthe total photon fluence (in black). Right: ν α + ¯ ν α fluence in the observer frame produced at R IS with flavor oscillations included (in red for electron and blue for muon flavors). The astrophysicalparameters for this GRB are reported in Table 1, and ˜ E PH = 6 . × erg, ˜ E SYNC (cid:39) . × erg,˜ E UP (cid:39) . × erg, R = 20, ξ =1. The fluence for the muon flavor peaks at E peak ν = 6 . × GeVand its normalization is comparable to the one in Fig. 5. This is due to a counterbalance between theradiative inefficiency in the IS model and the lower high energy proton density in this scenario. of the incoming photon field at R IS , an effect that we do not take into account here. Finally,the high energy tail of the distribution has a decline sharper in this scenario than in theIS case because of the lower proton energy cutoff. The low E (cid:48) p, max is also the reason whyspectral modifications due to kaon decay are not visible ( E (cid:48) π, max , E (cid:48) K, max > E (cid:48) p, max ). We are interested in a representative GRB of the class of bursts introduced in Sec. 2.1.3,hence we adopt average values for the spectral index and intensity of each component. Tothis purpose, we rely on Refs. [46, 133, 134].Once the outflow becomes transparent to radiation, a BB component is emitted at R PH , with spectral index α BB = 0 . α CPL1 = −
1. An additional cut-off power-law (CPL2) begins to appear after a slight delaywith respect to CPL1, with the cut-off shifting to higher energies until its disappearance. Atlater times, this additional component is well described by a simple PL, and we associate itto the beginning of the afterglow. With this choice, we take into account both interpretationsof the additional energetic component, namely the internal or external origin of CPL2.At the deceleration radius R dec (see Eq. 2.7), the external shock starts accelerating pro-tons and electrons of the wind and the magnetic field builds up. Motivated by the afterglowmodeling [135], we use the following values for the energy fractions: ε e = 4 × − , ε B = 10 − and ε p = 1 − ε e − ε B [108], compatible with our choice for the prompt efficiency.– 19 –e consider a wind type circumburst medium with A = 3 × cm − [136] and anadiabatic blastwave, with Γ( t ) = Γ( t dec / t ) / [136] and R ( t ) = 2Γ ( t ) ct/ (1 + z ) describingthe temporal evolution of the Lorenz factor and the radius of the forward shock after t dec ,respectively. The energy of the accelerated particles in the blastwave, at a time t after thedeceleration, is ˜ U p = 4 πε p AR ( t ) m p c [Γ ( t ) − α PL = − . E PL = ε e /ε p ˜ U p . Thephoton field target for pγ interactions at the forward shock is the sum of the PL, BB andCPL1 components; the latter two being Lorentz transformed in the comoving frame of theblastwave.Since we are interested in computing the neutrino fluence emitted at the forward shockduring the prompt phase, we take a representative average radius R ∗ in logarithmic scalebetween R dec and R ( t dur − t dec ). The photon energy distribution is n (cid:48) tot ( E (cid:48) γ , R ∗ ) = n (cid:48) PL ( E (cid:48) γ ) + (cid:18) R PH R ∗ (cid:19) n (cid:48) BB (cid:18) E (cid:48) γ Γ r (cid:19) + (cid:18) R IS R ∗ (cid:19) n (cid:48) CPL1 (cid:18) E (cid:48) γ Γ r (cid:19) , (5.6)where Γ r is the relative Lorenz factor between Γ and Γ ∗ ≡ Γ( R ∗ ).The BB component is always subdominant, while CPL1 and CPL2 are expected to varyin absolute and relative intensity from burst to burst; this is true also in the same GRB, oncethe temporal evolution is considered. In order to investigate to what extent the neutrinospectrum may be affected by these factors, we considered two scenarios of study for theprompt phase: case (I) such that the energetics of the three components is ˜ E BB (cid:39) . E CPL1 and ˜ E CPL1 = 3 ˜ E CPL2 (solid black line in the top left panel of Fig. 7) and case (II) with˜ E CPL1 = 1 / E CPL2 (dotted black line in the top left panel of Fig. 7). The three cut-off power-laws (BB, CPL1, and CPL2) follow Eq. A.5 with peak energies E BB , peak (cid:39) × − GeV, E CPL1 , peak (cid:39) × − GeV, and E CPL2 , peak (cid:39) × − GeV, respectively. These valuesare consistent with the ones in Refs. [46, 137]. With this set of parameters, the hierarchyand intensity of the various cooling processes is analogous to the IS case for the promptphase (Fig. 4), while adiabatic cooling is the dominant cooling process by many orders ofmagnitude at R ∗ . For what concerns the forward shock, we assume a differential numberdensity of protons n (cid:48) p ( E (cid:48) p ) ∝ E (cid:48) − p injected between the minimum energy E (cid:48) p, min = m p c Γ ∗ and the maximum E (cid:48) p, max , derived from the condition that the proton acceleration time t (cid:48) acc is limited by the adiabatic time t (cid:48) ad (see Sec. 3.2).The neutrino fluence at the IS and forward shock is displayed in Fig. 7. In the top leftpanel, the solid line represents the neutrino fluence for case (I), while the dotted line standsfor case (II). The enhancement of the energetic component CPL2 of almost one order ofmagnitude leads to a negligible impact on the neutrino energy distribution, producing onlya slight increase of the fluence at low energies.The bottom panels of Fig. 7 display the photon fluence and the correspondent neutrinofluence when the emission from the forward shock starts during the prompt phase. Thedashed lines represent the neutrino fluence produced at the IS from the interaction of ac-celerated protons and the photon field (BB+CPL1), while the dash-dotted line representsthe neutrino outcome from the forward shock at R ∗ , where we rely on Eq. 5.6 for the pho-ton field. The solid line describes the total neutrino fluence expected during the promptphase. The forward shock contribution is significantly higher than what expected for theafterglow phase [136]. This is mainly due to a much larger photon number density in theacceleration region. Furthermore, given the very low magnetic field and its inefficiency to– 20 – − − − − − − − − E γ [GeV] − − − − − − E γ Φ γ ( E γ ) [ G e V c m − ] BBCPL1CPL2TOTCPL2TOT E ν [GeV] − − − − − − E ν Φ ν ( E ν ) [ G e V c m − ] ν µ + ¯ ν µ ν µ + ¯ ν µ − − − − − − − E γ [GeV] − − − − − E γ Φ γ ( E γ ) [ G e V c m − ] BBCPL1PLTOT E ν [GeV] − − − − − − E ν Φ ν ( E ν ) [ G e V c m − ] ν µ + ¯ ν µ ν e + ¯ ν e Figure 7 . Top left:
Photon fluence in the prompt phase for the model with three components. It iscomposed by a thermal BB component (violet dashed curve), a cut-off power law CPL1 (green dashedcurve), and a second cut-off power law CPL2 (in orange dashed for case (I) such ˜ E CPL1 = 3 ˜ E CPL2 ;in orange dotted for case (II) with ˜ E CPL1 = 1 / E CPL2 ). Furthermore ˜ E BB = 0 . E CPL1 . The totalfluence is plotted in black (solid and dotted lines).
Top right:
Correspondent ν α + ¯ ν α fluence in theobserver frame with flavor oscillations included (in red for electron and blue for muon flavors). Thesolid line represents the total contribution during the prompt for case (I), while the dotted one is forcase (II). The fluence for the muon flavor peaks at E peak ν = 4 × GeV for the case (I). The lowenergy tail is affected by the interaction of protons with CPL2.
Bottom left:
Photon fluence for thescenario such that the emission from the forward shock (PL, blue dash-dotted line) starts during theprompt phase (dashed, BB+CPL1).
Bottom right:
Corresponding ν α + ¯ ν α fluence (dashed curve forthe IS emission, dash-dotted for the forward shock, and solid line for the total). accelerate particles to very high energies, the cutoff in the neutrino spectrum occurs at alower energy compared to the prompt case (see dot-dashed line). The overall intensity atpeak energy of the neutrino emission in this scenario is comparable to the IS case with adissipative photosphere. – 21 – .2 Poynting flux dominated jets5.2.1 ICMART model For the model introduced in Sec. 2.2.1, the typical radius necessary to make sure that runawayreconnection has enough time to grow is R ICMART (cid:39) cm [47], while the typical widthof the reconnection region is ∆ = ct v , where we adopt t v (cid:39) . .
35 in this model [138] and this is the value we adopt for ε d .It has been shown that a Band-like spectrum may be reproduced in this scenario by con-sidering an appropriate time dependent injection rate of particles in the emitting region [139],and this is the spectrum we adopt for this model. If σ is the magnetization parameter at R ICMART , the magnetic field in the bulk comoving frame can be expressed as [47]: B (cid:48) = (cid:32) L iso Γ cR σσ + 1 (cid:33) / . (5.7)We use as initial jet magnetization σ = σ in = 45 (see Fig. 2); this choice, as shown in thefollowing, allows for a consistent comparison with the results of Sec. 5.2.2. By relying on theresults from particle-in-cell simulations, we assume k p ( σ ) (cid:39) ε p = 0 . ε e = 0 . η γ = ε d ε e (cid:39) .
17. The photon number density is normalizedto η γ E (cid:48) iso . Since R IS (cid:39) R ICMART , we obtain similar trends for the cooling times as in Fig. 4,except for the synchrotron loss that starts to dominate at E (cid:48) p ∼ GeV and a slightlyincreased rate of pγ interactions due to a larger photon number density in the dissipationregion. − − − − − − − − E γ [GeV] − − − − − E γ Φ γ ( E γ ) [ G e V c m − ] BAND 10 E ν [GeV] − − − − − − E ν Φ ν ( E ν ) [ G e V c m − ] ν µ + ¯ ν µ ν e + ¯ ν e Figure 8 . Left:
Band photon fluence observed at Earth and emitted at R ICMART = 10 cm forthe ICMART model. The parameters of this GRB are reported in Table 1 and η γ = 0 . Right:
Correspondent ν α + ¯ ν α fluence in the observer frame in the presence of flavor conversions for theICMART model (in blue and red for the muon and the electron flavors, respectively). The muonneutrino fluence peaks at E peak ν = 1 . × GeV; the high energy tail of the neutrino distributionshows the double bump structure due to kaon decay. – 22 –he neutrino fluence is displayed in the right panel of Fig. 8. The fluence for themuon flavor peaks at E peak ν = 1 . × GeV. Note that the double bump due to kaondecay is clearly visible in the high-energy tail of the energy distribution. This is due to E (cid:48) p, max ∼ E (cid:48) K, max ∼ × GeV (while E (cid:48) π, max ∼ . × GeV). This feature is determinedby the stronger magnetic field in the acceleration region ( B (cid:48) (cid:39) B (cid:48) (cid:39) For the model introduced in Sec. 2.2.2, we follow Ref. [48] and assume that the energy which isdissipated in the optically thick region is reprocessed into quasi-thermal emission, leading to ablack-body-like emission from the photosphere. In the optically thin region, the synchrotronradiation from electrons is the dominant emission mechanism and it represents the non-thermal prompt emission. The energy emitted at the photosphere is obtained by integratingthe energy dissipation rate (Eq. 2.11) up to R PH and considering that only the fraction( R/R PH ) / of the energy dissipated at R remains thermal at R PH . In the optically thinregion, electrons are always in the fast cooling regime and E (cid:48) γ, ssa (cid:29) E (cid:48) γ, cool for R ph < R < R sat with our choice of parameters. Here E (cid:48) γ, ssa is the synchrotron self-absorption energy [49]: E (cid:48) γ, ssa ∼ (cid:32) h πm p ξ ˜ L iso π Γ sat R Γ( R ) (cid:33) / . (5.8)The shape of the synchrotron spectrum follows Eq. A.10, but we replace E (cid:48) cool with E (cid:48) γ, ssa and use α γ = 1 for E (cid:48) γ < E (cid:48) γ, ssa [49]. Furthermore, only a fraction ξ = 0 . λ = 4 × cm [48]. The terminal Lorenz factorof the outflow is Γ sat (cid:39) Γ σ . We choose Γ sat = 300 and the initial jet magnetization is σ = Γ / ∼ γ (cid:48) p, min = max (cid:20) , k p − k p − ε p σ ( R ) (cid:21) (5.9)and extending up to a maximum value determined by balancing the energy gain and lossrates, as described in Sec. 3.2. We also assume that the power of the proton distribution isthe same as the one of electrons, namely k p = k e (see Eq. 2.15). The latter assumption ismotivated by particle-in-cell simulations of magnetic reconnection for σ (cid:29) σ ∼ R = aR PH with a = 3. Being an arbitrary choice for the starting radius, weexplore the effects of a on the neutrino fluence in Appendix B. For illustration purposes,we compute the neutrino production rate at three radii ( R , R , and R ) equally distancedin logarithmic scale. We make this choice in order to establish the qualitative trend of theneutrino production during the evolution of the outflow in the optically thin region. Thephoton and proton distributions are normalized at each radius R i along the jet to the energydissipated between R i − and R i , where R = R PH and R = R sat . At each R i , the photonfield coming from R j is Lorentz transformed through the relative Lorentz factorΓ rel ,ij = 12 (cid:18) Γ i Γ j + Γ j Γ i (cid:19) , (5.10)– 23 – able 2 . Summary table for the input parameters adopted at the radii R , R and R in themagnetic model with gradual dissipation: the radius ( R ), the comoving magnetic field ( B (cid:48) ), theLorentz factor (Γ), the maximum energy of protons ( E (cid:48) p, max ), pions ( E (cid:48) π, max ), muons ( E (cid:48) µ, max ), andkaons ( E (cid:48) K, max ), as well as the power-law slope ( k e = k p ) of electrons and protons. R [cm] B (cid:48) [kG] Γ E (cid:48) p, max [GeV] E (cid:48) π, max [GeV] E (cid:48) µ, max [GeV] E (cid:48) K, max [GeV] k e R . × . ×
176 1 . × . × . × . × . R . × . ×
230 1 . × × . × . × . R . × . ×
300 2 . × . × . × . × . that holds as long as Γ i , Γ j (cid:29)
1. The total photon number density used as input at eachradius R i for producing neutrinos is thus n (cid:48) tot ( E (cid:48) γ , R i ) = i (cid:88) j =0 (cid:18) R j R i (cid:19) n (cid:48) j (cid:18) E (cid:48) γ Γ rel ,ij (cid:19) rel ,ij (5.11)where n (cid:48) j ( E (cid:48) γ ) is the photon energy distribution at R j (in units of GeV − cm − ).Once the photon distributions are set, we evaluate the proton cooling times at eachradius. In all the three cases, dominant losses are due to the adiabatic cooling up to (cid:39) GeV, and pγ interactions for 10 GeV (cid:46) E (cid:48) p (cid:46) E (cid:48) p, max . Synchrotron losses becomerelevant around 10 GeV. Given the very strong magnetic field (see Table 2), the secondariessuffer strong synchrotron losses; this considerably affects the resulting neutrino spectrum,which is damped at energies much lower than in all the other models investigated so far inthis work. A summary of the input parameters at the three R i is reported in Table 2.The (photon) neutrino fluence at Earth is shown in the (left) right panel of Fig. 9. Theslope of the three non-thermal synchrotron components and their distribution peaks decreaseas the distance from the source increases. The high energy cut-off of each spectral componentis given by Eq. A.1. Notably, the dominant component comes from the smallest radius, whilethe contribution coming from larger radii gets lower and lower (67%, 26%, and 7% from R , R and R , respectively). The significant drop in the neutrino flux between R and R is mainly due to the decrease of the proton power slope (see Table 2), which causes a morepronounced drop in proton number density in the energy range of interest. This is a peculiarfeature of this model, which predicts parameters depending on the jet magnetization, andthus changing with the radius.The neutrino fluence for the muon flavor peaks at E peak ν = 7 . × GeV, which is about O (10 − ∼
22 keV. The second bump visible in the spectrum isinstead represented by the kaon contribution. Apart from the ICMART model, this is theonly other case out of the ones studied in this work in which this feature is clearly identified athigher energies. The reason is the very strong magnetic field in these two magnetic models.Another peculiar feature of this model is the low-energy tail of the neutrino distribution,which is higher than in previous cases. This is due to a combination of the larger number– 24 – − − − − − − − − − E γ [GeV] − − − − − E γ Φ γ ( E γ ) [ G e V c m − ] R PH R R R TOT 10 E ν [GeV] − − − − − − E ν Φ ν ( E ν ) [ G e V c m − ] R : ν µ + ¯ ν µ R : ν µ + ¯ ν µ R : ν µ + ¯ ν µ ν µ + ¯ ν µ ν e + ¯ ν e Figure 9 . Left : Photon fluence in the observer frame for the GRB model invoking continuous mag-netic dissipation for the parameters reported in Tables 1 and 2. The total photon energy distributionis shown in black, and its components at R PH , R , R , and R are plotted in violet, orange, green,and coral respectively. Right:
Correspondent ν α + ¯ ν α fluence at Earth after flavor oscillations in theleft panel (in blue for the muon flavor and in red for the electron one). The fluence for the muonflavor peaks at E peak ν = 7 . × GeV. An unusual spectral structure is clearly visible. density [ O (10 − )] of protons at low energies in the acceleration region and the extendedphoton field at higher energies. In order to estimate the neutrino production in the proton synchrotron model (see Sec. 2.3),we need to evaluate the fraction of the proton energy which goes into pγ interactions. Weconsider the photon spectral fit as in Eq. A.10 and follow Ref. [39], which provides thecooling energy E γ, cool , the peak energy (or minimum injection energy) E γ, peak ≡ E γ, min , andthe energy flux at the cooling energy ( F γ, cool ).Another inferred quantity is the cooling timescale of the radiating particles, t cool ∼ t cool is related, after Lorentz transforming, to the comoving magnetic field B (cid:48) and γ (cid:48) cool by means of Eq. A.7. The variability timescale is assumed to be t v = 0 . B (cid:48) ,– 25 – − − − − − − Y p o f G R B s Figure 10 . Histogram of Y p (see Eq. 5.17) for a subset of GRBs analyzed in Ref. [39] for whichredshift information is available; Γ = 300 and t v = 0 . Y p quantifiesthe relative importance between the proton synchrotron emission and pγ interactions. The very lowvalues of Y p for most of GRBs in the sample suggest a negligible neutrino production of this class ofGRBs. γ (cid:48) min , R γ , Γ, and E (cid:48) γ, bol , iso through the following relations [39]: E γ, peak = 32 (cid:126) eB (cid:48) γ (cid:48) m p c Γ1 + z , (5.12) E γ, cool = (cid:18) m p m e (cid:19) π (cid:126) em e cσ T B (cid:48) t z Γ , (5.13) F γ = F γ, cool (cid:18) E γ, cool h (cid:19) (cid:34)
34 + 2 (cid:115) E γ, peak E γ, cool − k p − (cid:115) E γ, peak E γ, cool (cid:35) , (5.14) E (cid:48) γ, bol , iso = 4 πd L ( z ) F γ t dur Γ(1 + z ) , (5.15) R γ = 2 ct v Γ (1 + z ) . (5.16)where F γ = ˜ L γ, bol , iso / πd L ( z ) is the bolometric isotropic radiative flux (in units of GeV cm − s − ),˜ L γ, bol , iso being the bolometric isotropic luminosity of the burst over the whole energy range.Using these relations we can infer B (cid:48) , γ (cid:48) min , R γ and E (cid:48) γ, bol , iso as functions of Γ.In order to figure out the relative importance between proton synchrotron and pγ coolingfor the sample of GRBs studied in Ref. [39], we introduce the following parameter [141]: Y p ≡ L (cid:48) p ,pγ L (cid:48) p , syn ≈ σ pγ σ p,T U (cid:48) p, syn U (cid:48) B = σ pγ σ p,T E (cid:48) γ, tot , iso V (cid:48) iso πB (cid:48) = σ pγ σ p,T πF γ d L ( z )Γ R γ cB (cid:48) , (5.17)– 26 – − − − − − − − E γ [GeV] − − − − − E γ Φ ( E γ ) [ G e V c m − ] p-SYNCH E ν [GeV] − − − − − − E ν Φ ν ( E ν ) [ G e V c m − ] ν µ + ¯ ν µ ν e + ¯ ν e Figure 11 . Left : Photon fluence in the observer frame for the proton synchrotron model; see Table 1for the model parameters, in addition ˜ E γ, iso = 5 × ergs, t cool = 0 . γ min /γ cool = 12, E γ, cool =7 keV. Right:
Correspondent ν α + ¯ ν α fluence (in red and blue for the electron and muon flavors,respectively). The peak in the neutrino distribution ( E peak ν = 3 . × GeV), due to the cooling energybreak E γ, cool , is shifted to lower energies with respect to the other analized models. The damping athigh energies is due to the very strong magnetic field in the emitting region ( B (cid:48) (cid:39) . × G). where L (cid:48) p,pγ and L (cid:48) p, sync are the proton energy loss rates for pγ interactions and synchrotronemission respectively, and σ p,T = σ T ( m e /m p ) . By relying on Eqs. 5.13 and 5.16, Y p can beestimated as a function of the bulk Lorentz factor.For our bencnhmark Γ = 300, we compute Y p for the GRBs studied in Ref. [39] forwhich redshift information is available. The histogram of Y p is shown in Fig. 10. We cansee that Y p spreads over almost three orders of magnitude, with very low typical values.Hence, assuming proton synchrotron radiation as the main emission mechanism, we expectthis class of GRBs to be poor emitters of high energy neutrinos. To show this quantitatively,we compute the neutrino fluence for our representative GRB.We adopt the following GRB parameters: γ min /γ cool = 12, E γ, cool = 7 keV, Γ = 300, z = 2 E γ, peak = ( γ min /γ cool ) E γ, cool , which result in ˜ E γ, bol , iso (cid:39) × erg. These values arecompatible with the average ones inferred from the sample considered in Ref. [39], namely (cid:104) ˜ E γ, bol , iso (cid:105) (cid:39) . × erg, (cid:104) E γ, cool (cid:105) (cid:39) . (cid:104) γ min /γ cool (cid:105) (cid:39) .
9. Furthermore,we choose k p = 2 . Q (cid:48) ( γ (cid:48) p ), that reproducesthe typical value of the high energy photon index β ∼ − .
3; note that k p is almost neverconstrained for the sample in Ref. [39].As for the radiated energy, this fiducial GRB is comparable to the ones analyzed inthe previous sections, except for the total energetics. In fact, given the very high magneticfield, B (cid:48) (cid:39) . × G, the total isotropic energy is ˜ E B, iso ∝ R γ Γ B (cid:48) ∼ O (10 ) erg, muchlarger than the typical energy that a GRB jets is able to release (spin down of magnetars orthrough the Blandford-Znajek mechanism [10]). Since the synchrotron radiation dominatesby many orders of magnitude over all the other proton energy loss mechanisms, we assume U (cid:48) p, iso (cid:39) E (cid:48) γ, bol , iso , where U (cid:48) p is the total isotropic proton energy in the comoving frame. Such ajet turns out to be highly inefficient in radiating energy, given that ˜ E iso ∼ ˜ E B, iso (cid:29) ˜ E γ, bol , iso .– 27 –y considering the proton energy distribution as in Eq. A.10 and normalizing it to U (cid:48) p , we compute the neutrino fluence and show it in the right panel of Fig. 11. The left ofthe same figure shows the total synchrotron photon fluence. Analogously to the model inSec. 5.2.2, the peak in the neutrino distribution ( E peak ν = 3 . × GeV) is due to the coolingenergy break E γ, cool and it is shifted to lower energies. The neutrino spectrum is furthermorestrongly damped at high energies due to the synchrotron cooling of mesons in the jet. Ourestimation of the neutrino emission results to be in agreement with the one reported in theindependent work of Ref. [51], for GRBs with similar parameters.The proton synchrotron model, besides requiring unreasonable total jet energies, pre-dicts the smallest neutrino fluence among all models considered in this work. We note thatwith the choice made of parameters, our representative GRB has Y p ∼ O (10 − ); hence, ourestimation may be considered an optimistic one, given the distribution of Y p shown in Fig. 10.We refer the interested reader to Ref. [51] for additional details and discussion on this model. In this work, we have computed the neutrino fluence for a class of models adopted to describethe prompt phase of long GRBs, all having the same ˜ E iso . Because of the diversity of elec-tromagnetic GRB data and the uncertainties inherent to the models (e.g., jet composition,energy dissipation mechanism, particle acceleration, and radiation mechanisms), an exhaus-tive theoretical explanation of the mechanism powering GRBs is still lacking. To comparethe neutrino production across models, we have selected input parameters for a benchmarkGRB motivated by observations. In addition, the modeling of the dissipative and accelerationefficiencies, as well as the properties of the accelerated particle distributions have been guidedby the most recent simulation findings. A summary of our input parameters is reported inTable 1. In this section, we compare the energetics of the GRB models explored in this workas well as the detection prospects of stacked neutrino fluxes. A summary of our findings is reported in Table 3, where the radiative efficiency of thejet (Eq. 3.1) is listed for the six GRB models investigated in this paper together with theisotropic photon and neutrino (for six flavors) energies, as well as the ratio of the lattertwo. As already discussed in Sec. 5, the least efficient model in converting ˜ E iso in ˜ E γ, iso isthe proton synchrotron model, whilst the most efficient one is the model which considers adissipative photosphere as the main source of prompt emission. This is mainly due to thehigh dissipative efficiency suggested by recent three-dimensional simulations [86]. Note thatthe radiative efficiency is an input parameter of each model, since we do not compute theradiation spectra self-consistently.Among the models considered in this work, all with identical ˜ E iso , neutrinos carry thelargest amount of energy in the ICMART model; the jet model with a dissipative photosphereand the jet model with three emission components have very similar ˜ E ν, iso .The reason for such a high neutrino yield in these three cases lies in the large photonnumber density in the acceleration region. Indeed, although the model with three componentspresents a slightly smaller E γ, iso , the high E ν, iso is due to the fact that protons interact witha high-energy photon component comparable in intensity to the one in the γ -ray range (i.e.1 keV −
10 MeV). This leads to an enhancement in the neutrino flux by a factor of ∼ E ν, iso / ˜ E γ, iso reported in Table 3 (note that ˜ E γ, iso in Table 3 is estimated over the energy range 1 keV −
10 MeV; hence, this ratio, when definedwith the bolometric photon energy used for neutrino production, should be slightly smallerthan the one reported for the IS model with a dissipiative photosphere, the model with threecomponents, the magnetic one with gradual dissipation, and the proton synchrotron model).
In order to compare the neutrino detection perspectives for our six models, we compute theall-sky quasi-diffuse flux for neutrinos and antineutrinos. We assume that our benchmarkGRB at z = 2 yields a neutrino emission that is representative of the entire GRB population.For ˙N (cid:39)
667 yr − long GRBs per year [32], the stacking flux for the muon flavor over thewhole sky is defined as F ν µ ( E ν ) = 14 π ˙NΦ ν µ ( E ν , z = 2) . (6.1)Figure 12 shows the resultant all-sky quasi-diffuse fluxes for the muon flavor for the sixGRB models as functions of the neutrino energy (colored curves). For comparison, we alsoshow the GRB staking limits of IceCube [32] and the projected ones for IceCube-Gen2 [142](black curves). In agreement with the non-detection of high-energy neutrinos from targetedGRB searches [32], our forecast for the neutrino fluxes lies below the experimental limitsand is in agreement with the upper limits reported by the ANTARES Collaboration [31]and with the ones expected for KM3NeT [143]. All considered models predict comparableneutrino flux at peak energy, except for the IS model and the proton synchrotron one (seealso Table 3).An important aspect to consider in targeted GRB searches is the large spread in energyand shape of the expected neutrino fluxes for different jet models. It is evident from Fig. 12that the neutrino flux peak energy ranges from O (10 ) GeV for the proton synchrotron model Table 3 . Summary of the derived quantities for the models considered in this work and ourbenchmark parameters value (see Table 1). The radiative efficiency of the jet (Eq. 3.1), the isotropicphoton energy in the 1 keV–10 MeV energy range, the isotropic neutrino energy for neutrinos andantineutrinos of all flavors, the ratio between the isotropic total neutrino and photon energies, theneutrino energy at the fluence peak, and the maximum proton energy are listed. The model withthe smallest radiative efficiency is the proton synchrotron model; this model has also the smallest˜ E ν, iso . The most radiatively efficient model is the one with a dissipative photosphere; while ˜ E ν, iso is comparable among the jet model with a dissipative photosphere, the one with three emissioncomponents, and the ICMART model. Model η γ (%) ˜ E γ, iso [erg] ˜ E ν, iso [erg] ˜ E ν, iso / ˜ E γ, iso E peak ν µ [GeV] E p, max [GeV]IS 1 . . × . × . × − . × . × PH-IS 20 6 . × . × . × − . × . × . . × . × . × − × . × ICMART 17 . × . × × − . × . × MAG-DISS 8 2 . × . × × − . × . × p-SYNCH 2 × − × × . × − . × . × – 29 – E ν [GeV] − − − − − − − − − − − − E ν F ν ( E ν ) [ G e V c m − s − s r − ] ISPH-IS3-COMP ICMARTMAG-DISSp-SYNCH IceCubeIceCube-Gen2
Figure 12 . Model comparison of the expected all-sky quasi-diffuse fluxes for the six GRB modelsconsidered in this work for the benchmark jet parameters listed in Table 1. The quasi-diffuse fluxhas been computed by relying on Eq. 6.1 for ν µ + ¯ ν µ ; all models have identical ˜ E iso . For comparison,the IceCube staking limits (combined analysis for 1172 GRBs) [32] and the expected sensitivity forIceCube-Gen2 (based on a sample of 1000 GRBs) [142] are reported (solid and dashed black lines).By relying on the most up-to-date best-fit GRB parameters, all models predict a quasi-diffuse fluxthat lies below the sensitivity curves; however, a large spread in energy and shape of the expectedneutrino fluxes is expected for different jet models. to O (10 − ) GeV for the IS model with a dissipative photosphere. As such, targetedsearches assuming one specific GRB model, such as the IS one, as benchmark case for theGRB neutrino emission may lead to biased results.It is worth noticing that our predictions rely on best fit values of the input parametersand assume one GRB as representative of the whole population. Variations of the jet param-eters within their uncertainty range could lead to variations of the expected neutrino flux upto a couple of orders of magnitude, see e.g. Refs. [36, 73], possibly hitting the IceCube-Gen2sensitivity. In addition, since none of the considered jet models can account for all obser-vational constraints, a population study [42] may further affect the expected quasi-diffuseemission.Another caveat of our modeling is that the spectral energy distributions of photonsand the ones of the secondary particles produced through pγ interactions are not computedself-consistently; this may affect the overall expected emission, see e.g. Refs. [144–148] fordedicated discussions. – 30 – Conclusions
Long duration gamma-ray bursts (GRBs) are subject of investigation since long time, beingamong the most mysterious transients occurring in our universe. In the attempt of explainingthe observed electromagnetic GRB emission, various models have been proposed. The maingoal of this work is to show that the neutrino emission strongly depends on the chosen jetmodel, despite the fact that different jet models may be equally successful in fitting theobserved electromagnetic spectral energy distributions.To this purpose, we choose a benchmark GRB and compute the neutrino emission forkinetic dominated jets, i.e. in the internal shock model, also including a dissipative pho-tosphere as well as three spectral components. We also consider Poynting flux dominatedjets: a jet model invoking internal-collision-induced magnetic reconnection and turbulence(ICMART) and a magnetic jet model with gradual dissipation. A jet model with dominantproton synchrotron radiation in the keV-MeV energy range is also taken into consideration.In particular, the neutrino production for the latter two models has been investigated for thefirst time in this work.Defining the radiative efficiency as the ratio of isotropic gamma-ray energy to the totalisotropic energy of the jet, we find that the least radiatively efficient model is the protonsynchrotron one, while the most efficient one is the model with a dissipative photosphere.However, the model predicting the largest amount of isotropic-equivalent energy going intoneutrinos is the ICMART one.In the context of targeted searches, it should be noted that the expected quasi-diffuseneutrino flux can vary up to 1–1 . to 10 GeV. The predicted spectral shape of the neutrino distribution isalso strongly dependent on the adopted jet model. A summary of our findings is reported inTable 3 and Fig. 12.This work highlights the great potential of neutrinos in pinpointing the GRB emissionmechanism in the case of successful neutrino detection. In particular, it suggests the need torely on a wide range of jet models in targeted stacking searches.
Note added : The modeling of the neutrino emission for the proton synchrotron model is alsopresented in the independent work of Ref. [51]. Our paper focuses on the comparison of theneutrino production across different GRB models for the prompt emission, while Ref. [51]investigates the plausibility of the proton synchrotron interpretation.
Acknowledgments
We are grateful to Jochen Greiner and Gor Oganesyan for insightful discussions. This projecthas received funding from the Villum Foundation (Project No. 13164), the Carlsberg Foun-dation (CF18-0183), the Knud Højgaard Foundation, the Deutsche Forschungsgemeinschaftthrough Sonderforschungbereich SFB 1258 “Neutrinos and Dark Matter in Astro- and Par-ticle Physics” (NDM), and the MERAC Foundation.
A Spectral energy distributions of photons: fitting functions
In order to describe the electromagnetic emission, in this appendix we introduce the mainspectral functions used to fit the electromagnetic data: the Band function, the cut-off power-– 31 –aw, a simple power-law, and a double broken power-law usually representing the synchrotronemission from a marginally fast cooling particle population. The various spectral functionsintroduced here are then employed to model the GRB emission in Sec. 5.When the energy distribution does not present an intrinsic cut-off at high energies (e.g.,Band function and synchrotron spectrum), we define E (cid:48) γ, cutoff as the energy at which theopacity to photon-photon pair production becomes unity τ γγ ( E (cid:48) γ, cutoff ) (cid:39) . σ T E (cid:48)∗ n (cid:48) γ ( E (cid:48)∗ ) R γ
2Γ = 1 (A.1)where E (cid:48)∗ = m e c /E (cid:48) γ, cutoff and τ γγ ( E (cid:48) γ, cutoff ) is the opacity for the photons with energy E (cid:48) γ, cutoff and number density distribution n (cid:48) γ ( E (cid:48) γ ). A.1 Band function
The Band function [149] is the most used empirical function to fit the time-integrated elec-tromagnetic spectra. Despite fitting well the data, it is still lacking a clear physical meaning.It consists of a smoothly joint broken power-law: n Band γ ( E γ ) = C (cid:16) E γ
100 keV (cid:17) α γ exp (cid:104) − ( α γ +2) E γ E γ,p (cid:105) E γ < E γ,c (cid:16) E γ
100 keV (cid:17) β γ exp( β γ − α γ ) (cid:16) E γ,c
100 KeV (cid:17) α γ − β γ E γ ≥ E γ,c (A.2)where E γ,c = (cid:18) α γ − β γ α γ + 2 (cid:19) E γ, peak , (A.3) C is a normalization constant (in units of GeV − cm − ), α γ and β γ are the low-energy andhigh-energy power-law photon indices, E γ,c represents the energy where the low-energy power-law with an exponential cutoff ends and the pure high energy power-law begins. The peakenergy E γ, peak is chosen to satisfy the Amati relation [150]:˜ E γ, peak = 80 (cid:32) ˜ E γ, iso erg (cid:33) . keV . (A.4)The typical spectral parameters inferred from observations are: α γ (cid:39) − . β γ (cid:39) − .
2, and E γ, peak (cid:39)
300 keV [2].
A.2 Cut-off power-law
Although the Band spectrum is the best fitting function for most GRBs, it has been shownthat in some cases a cut-off power-law (CPL) can represent the preferred model [151–153].The CPL is a power-law model with a high energy exponential cut-off : n CPL γ ( E γ ) = C (cid:18) E γ
100 keV (cid:19) α γ exp (cid:20) − ( α γ + 2) E γ E γ, peak (cid:21) , (A.5)where α γ is the photon index and E γ, peak the peak energy, whose value will be specified later.In an optically thick thermal scenario, α γ = 1 in the Rayleagh-Jeans limit, α γ = 2 in theWien limit, α γ = 0 . α γ < .3 Power law In the cases of faint bursts or narrow detector bandpass, the whole GRB spectrum, or oneof its components, can be fitted with a simple power-law [154] defined as n PL γ ( E γ ) = C (cid:18) E γ
100 keV (cid:19) α γ , (A.6)where C is the normalization and α γ is the power-law photon index. A.4 Double broken power law
This is a spectral model that is commonly adopted to describe the synchrotron emission ofa fast cooling population of particles that are being injected into the emitting region witha power-law distribution at a rate Q ( γ ) ∝ γ − k with γ min < γ < γ max . During an emissionperiod t , the charged particles of mass m loose most of their energy above the characteristicvalue γ cool : γ cool ( t ) = 6 πmcσ T β B t (cid:18) mm e (cid:19) , (A.7)where m e is the electron mass. Considering a constant injection rate of particles in theemitting region which radiate in the fast cooling regime ( γ min > γ cool ) at a rate ∝ γ , aftera time t the emitting particle distribution has the following shape [68]: n ( γ, t ) ∝ γ < γ cool and γ > γ max γ − γ cool < γ < γ min γ − ( k +1) γ min < γ < γ max . (A.8)Given that each particle radiates photons with a characteristic synchrotron energy E γ ( γ ) = 32 (cid:126) emc γ B, (A.9)the particle distribution in Eq. A.8 emits the following synchrotron spectrum: n sync γ ( E γ ) = C (cid:16) E γ E γ, cool (cid:17) − E γ < E γ, cool (cid:16) E γ E γ, cool (cid:17) − E γ, cool < E γ < E γ, min (cid:16) E γ, min E γ, cool (cid:17) − (cid:16) E γ E γ, min (cid:17) − k +22 E γ, min < E γ < E γ, max , (A.10)where E γ, cool , E γ, min , and E γ, max correspond to the characteristic photon energies mainlyemitted by particles with gamma factors γ cool , γ min , and γ max , respectively. B Magnetized jet model with gradual dissipation: dependence of the neu-trino emission on the input parameters
One of the main, but less certain, parameters of the jet model with gradual magnetic dis-sipation is the initial magnetization σ . This, in turn, determines the photospheric radius,the saturation Lorentz factor, the energy dissipation rate, and other parameters. For thisreason, we investigate the impact of σ on the photon and neutrino fluences by considering– 33 – − − − − − − − − E γ [GeV] − − − − − E γ Φ γ ( E γ ) [ G e V c m − ] R PH R R R TOT E ν [GeV] − − − − − − − E ν Φ ν ( E ν ) [ G e V c m − ] R : ν µ + ¯ ν µ R : ν µ + ¯ ν µ R : ν µ + ¯ ν µ ν µ + ¯ ν µ ν e + ¯ ν e Figure 13 . Similar to Fig. 9, but for σ = 100. The fluence for the muon flavor peaks at E peak ν =3 . × GeV. In addition, E ν, iso = 8 . × erg, E γ, iso = 2 . × erg, Γ sat = 1000, σ = 100, R PH = 7 . × cm, R = 2 . × cm, R = 3 × cm, R = 4 × cm, Γ = 180 , Γ =422 , Γ = 1000, ˜ E ν, iso / ˜ E γ, iso = 3 . × − , η γ = 7%. − − − − − − − − E γ [GeV] − − − − − E γ Φ γ ( E γ ) [ G e V c m − ] σ = 100 σ = 45 E ν [GeV] − − − − − − − E ν Φ ν ( E ν ) [ G e V c m − ] σ = 100 : ν µ + ¯ ν µ σ = 100 : ν e + ¯ ν e σ = 45 : ν µ + ¯ ν µ σ = 45 : ν e + ¯ ν e Figure 14 . Left : Total photon fluence in the observer reference frame, obtained as the sum of thecomponents produced at R PH , R , R and R for the σ = 45 (solid line) and σ = 100 (dashed line)cases, respectively. Right : Correspondent ν α + ¯ ν α fluence (in red and in blue for the electron andmuon flavors, respectively). For parameters used, see captions of Figs. 9 and 13. a case with σ = 100. All the other parameters, like ˜ E iso , are identical to the ones adoptedin Sec. 5.2.2. We follow the same procedure to calculate the neutrino flux as outlined inSec. 5.2.2.In Fig. 13, we show snapshots of the photon fluence (left panel) and neutrino fluence(right panel) for σ = 100 at three indicative radii. A comparison between the σ = 45 and σ = 100 cases is shown in Fig. 14. A noticeable difference is appreciable between the photonspectral energy distributions. As the initial magnetization increases, the saturation Lorentz– 34 – − − − − − − − − E γ [GeV] − − − − − E γ Φ γ ( E γ ) [ G e V c m − ] R PH R R R TOT 10 E ν [GeV] − − − − − − E ν Φ ν ( E ν ) [ G e V c m − ] R : ν µ + ¯ ν µ R : ν µ + ¯ ν µ R : ν µ + ¯ ν µ ν µ + ¯ ν µ ν e + ¯ ν e Figure 15 . Left : Total photon fluence in the observer reference frame, obtained as the sum of thecomponents produced at R PH , R , R and R for the σ = 45 and a = 13 case. Right : Correspondent ν α + ¯ ν α fluence (in red and in blue for the electron and muon flavors, respectively). factor increases, namely Γ sat = σ / = 1000 for σ = 100. The energy is dissipated at a rate˙ E ∝ R / , while the photosphere occurs at a smaller distance from the source ( R PH ∝ / Γ sat ).As a result, less energy is dissipated during the optically thick regime (most of the energyis dissipated at R > R PH ) and the photospheric emission becomes dimmer (compare dashedand solid lines at E γ ∼ − –10 − GeV in Fig. 14). The characteristic synchrotron energy E γ, min ∝ Γ γ (cid:48) B (cid:48) (Eq. A.9) decreases with the radius (see, e.g., dashed colored curves in theleft panel of Fig. 13), while the normalization of the synchrotron photon spectra increaseswith respect to the case of σ = 45 (see dashed curve in the left panel of Fig. 14) because ofthe higher dissipation rate (Eq. 2.11). For a higher σ , particle acceleration begins at smallerradii and so does the production of neutrinos. Moreover, the power slopes of the electronand proton distributions (accelerated via reconnection) are harder [100, 101] because of thehigher magnetization in the acceleration region. Because of the larger saturation radius( R sat ∝ Γ ) found for higher σ , the dissipated energy up to R = aR ph ∝ Γ − that isavailable for relativistic particles is less than in the case of lower initial magnetizations. Thecombination of a smaller amount of dissipated energy up to a given radius, smaller volumeand harder proton power slope leads to a neutrino flux at peak (whose main contributioncomes from R ) comparable to the one with σ = 45 (see right panel of Fig. 14).For the case of σ = 100, the second bump in the neutrino spectrum has a fluence thatis comparable to the one of the first bump at ∼ GeV. On the contrary, the second bumpis barely visible in the neutrino energy distribution with σ = 45 (compare solid and dashedlines in the right panel of Fig. 14). This is because in the σ = 100 case, pions suffer strongersynchrotron losses, hence the neutrino intensity resulting from the decay of pions decreasesto the level of the one produced by kaons. This is also the reason for a slight shift in theneutrino flux peak to lower energies. Another noticeable feature is the low energy tail. Thelatter turns out to be higher in the σ = 100 case, given the higher number density of photonsat higher energies.Finally, in order to explore the effects of the arbitrary choice of the parameter a , weconsidered the case with σ = 45 and a = 13, where R ∼ R sat , see Fig. 15. Since most of– 35 –he energy is dissipated within R , the neutrino contribution from R is dominant, althoughlower by a factor O (10) if compared to the case with a = 3. 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