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High Energy Neutrino Emission and Neutrino Background

from Gamma-Ray Bursts in the Internal Shock Model

Kohta Murase∗ and Shigehiro Nagataki†

Yukawa Institute for Theoretical Physics, Kyoto University,

Oiwake-cho, Kitashirakawa, Sakyo-ku, Kyoto, 606-8502, Japan

(Dated: February 4, 2008)

High energy neutrino emission from GRBs is discussed. In this paper, by using the simulation kitGEANT4, we calculate proton cooling efficiency including pion-multiplicity and proton-inelasticityin photomeson production. First, we estimate the maximum energy of accelerated protons in GRBs.Using the obtained results, neutrino flux from one burst and a diffuse neutrino background areevaluated quantitatively. We also take account of cooling processes of pion and muon, which arecrucial for resulting neutrino spectra. We confirm the validity of analytic approximate treatmentson GRB fiducial parameter sets, but also find that the effects of multiplicity and high-inelasticitycan be important on both proton cooling and resulting spectra in some cases. Finally, assuming thatthe GRB rate traces the star formation rate, we obtain a diffuse neutrino background spectrum fromGRBs for specific parameter sets. We introduce the nonthermal baryon-loading factor, rather thanassume that GRBs are main sources of UHECRs. We find that the obtained neutrino backgroundcan be comparable with the prediction of Waxman & Bahcall, although our ground in estimation isdifferent from theirs. In this paper, we study on various parameters since there are many parametersin the model. The detection of high energy neutrinos from GRBs will be one of the strong evidencesthat protons are accelerated to very high energy in GRBs. Furthermore, the observations of aneutrino background has a possibility not only to test the internal shock model of GRBs but alsoto give us information about parameters in the model and whether GRBs are sources of UHECRsor not.

PACS numbers: 95.85.Ry, 98.70.Rz, 25.20.-x, 14.60.Lm, 96.50.Pw, 98.70.Sa

I. INTRODUCTION

Gamma-ray bursts (GRBs) are the most powerful ex-plosions in the universe. The observed isotropic energycan be estimated to be larger than 1052 ergs [1, 2]. Thehigh luminosity and the rapid time variability lead to thecompactness problem. This problem and the hardnessof observed photon spectra imply that γ-ray emissionshould be results of dissipation of kinetic energy of rela-tivistic expanding shells. In the standard model of GRBs,such a dissipation is caused by internal shocks - internalcollisions among the shells (see reviews e.g., [3, 4, 5]).Internal shock scenario requires a strong magnetic field,typically 104 − 107 G, which can accelerate electrons tohigh energy enough to explain observed γ-ray spectra bysynchrotron radiation. Usually, Fermi acceleration mech-anism is assumed to be working not only for electronsbut also for protons, which can be accelerated up to ahigh energy within the fiducial GRB parameters. Phys-ical conditions allow protons accelerated to greater than1020 eV and energetics can explain the observed flux ofultra-high-energy cosmic rays (UHECRs), assuming thatsimilar energy goes into acceleration of electrons and pro-tons in the shell [6, 7, 8, 9]. Such protons accelerated tothe ultra-high energy cannot avoid interacting with GRBphotons. This photomeson production process can gen-

∗Electronic address: kmurase@yukawa.kyoto-u.ac.jp†Electronic address: nagataki@yukawa.kyoto-u.ac.jp

erate very high energy neutrinos and gamma rays [10].Whether GRBs are sources of UHECRs or not, internalshock models predict the flux of very high energy neutri-nos at the Earth [11, 12, 13]. Many authors have investi-gated neutrino emission from GRBs. Ice Cherenkov de-tectors such as AMANDA at the South Pole have alreadybeen constructed and are taking data [14, 15, 16]. Now,the future 1 km3 detector, IceCube is being constructed[17, 18, 19]. In the Mediterranean Sea, ANTARES andNESTOR are under construction [20]. If the prediction iscorrect, these detectors may detect these neutrinos cor-related with GRBs in the near future.In this paper, we execute the Monte Carlo simulation kitGEANT4 [21] and simulate the photomeson productionthat causes proton energy loss. As a result, we can getmeson spectrum and resulting neutrino spectrum fromGRBs quantitatively. In our calculation, we take intoaccount pion-multiplicity and proton-inelasticity, whichare often neglected or approximated analytically in pre-vious works, although they may be important in somecases [22, 23]. We also take into account the synchrotronloss of mesons and protons. These cooling processes playa crucial role for the resulting neutrino spectrum. Ourmodels and method of calculation are explained in Sec.II. One of our purposes is to seek physical conditionsallowing proton to be accelerated up to ultra-high en-ergy. Similar calculations are carried by Asano [13], inwhich the possibility that a nucleon creates pions mul-tiple times in the dynamical time scale is included butmultiplicity is neglected. Using obtained results, we alsoinvestigate efficiency of neutrino emission for various pa-

2

rameter sets. Since there are many model uncertaintiesin GRBs, such a parameter survey is meaningful. Ourfinal goal is to calculate a diffuse neutrino backgroundfrom GRBs for specific parameter sets. A unified modelfor UHECRs from GRBs is very attractive [6, 7, 10], al-though it requires GRBs being strongly baryon-loaded[24]. Even if observed UHECRs are not produced mainlyby GRBs, high energy neutrino emission from GRBs canbe expected. Hence, we leave the nonthermal baryon-loading factor as a parameter. In this paper, assumingGRBs trace star formation rate, we calculate a neutrinobackground from GRBs and compare our results withthe flux obtained by Waxman & Bahcall [8]. The designcharacters of neutrino detectors are being determined inpart by the best available theoretical models, so numer-ical investigation of high energy neutrino fluxes shouldbe important. Finally, we will consider the implicationsof neutrino observations and discuss a possibility thatneutrino observation gives some information on the non-thermal baryon-loading factor and the inner engine, ifthe internal shock model is true. Our numerical resultsare shown in Sec. III. Our summary and discussions aredescribed in Sec. IV.

II. METHOD OF CALCULATION

A. Physical Conditions

Throughout this paper, we consider the epoch that in-ternal shocks occurs (see reviews e.g., [3, 4, 5]). We focuson long GRBs, whose duration is typically ∼ 30 s andthey are likely to be related with the deaths of short-lived massive stars (see Sec. II E). The internal shockmodel can produce the observed highly variable tem-poral profiles [25, 26]. In this model, the radial timescale is comparable with the angular time scale (see e.g.,[3]), and each time scale is related to the pulse width.Widths of individual pulses vary in a wide range. GRBpulses with ∼ 0.1 − 10 s durations and separations forbright long bursts are typical ones [27, 28]. The shortestspikes have millisecond or even sub-millisecond widthsin some bursts. It will reflect the intermittent nature ofthe fireball central engine. The internal collision radiiare determined by the separation of the subshells, whichis written by d in the comoving frame. Since the ob-served pulse width is given by δt ≈ d/2Γc, the internalcollision radius is expressed by the commonly used rela-tion, r ≈ 2Γ2cδt ≈ 6 × 1014(Γ/300)

2(δt/0.1 s) cm, where

Γ is a bulk Lorentz factor of the shell, which is largerthan 100. We consider collision radii in the range of(1013 − 1016) cm in our calculation. The width of thesubshells is constrained by the observed variability timeand the duration, and it is typically written by l ≈ r/Γ,where r is the radius at which internal shocks begin tooccur. In this paper, we leave l as a parameter becausethe precise width of each subshell is unknown. The dy-namical time of each collision is tdyn ≈ l/c. For sim-

plicity, we assume each collision radiates similar energy,Eiso

γ = Eisoγ,tot/N , where N is the number of collisions and

it is almost the number of subshells [25]. Although wedo not know about N precisely, we set N ∼ (10 − 100)in our calculation since the number of pulses per brightlong burst is typically the order of dozens [29].The geometrically corrected GRB radiation energy is typ-ically Eγ,tot = 1.24×1051h−1

70 ergs [1, 2], and we fix Eγ,tot

throughout this paper. The isotropic energy is a few or-ders of magnitude larger than this energy, and we takeEiso

γ,tot = f−1

b Eγ,tot ∼ (1052 − 1054) ergs. Here, fb ≡ θ2j/2is a beaming factor of GRB jet. In the prompt phase1/Γ <

∼ θj has to be satisfied. Once we determine Eγ andN , fb is also determined.The photon energy density in the comoving frame is givenby,

Uγ =Eiso

γ

4πΓr2l(1)

In the standard model, the prompt spectrum is explainedby the synchrotron radiation from electrons. For simplic-ity, we assume that the GRB photon spectrum obeys apower-law spectrum. That is,

dn

dε∝

ε−α (for εmin < ε < εb)ε−β (for εb < ε < εmax)

(2)

The observed break energy is ∼ 250 keV, which cor-responds to the break energy in the comoving frame,εb ∼ a few keV. So we set εb to 1 keV for Eiso

γ =

2 × 1051 ergs, 3 keV for Eisoγ = 2 × 1052 ergs, 6 keV for

Eisoγ = 2 × 1053 ergs. In addition, we set spectral indices

to α = 1, β = 2.2 as fiducial values and α = 0.5, β = 1.5as a flatter case. We take the minimum energy as 1 eVand the maximum energy as 10 MeV. Below 1 eV, thesynchrotron self-absorption will be crucial, while above10 MeV, the pair creation will be important [30].To explain the observed emission, the internal shockmodel requires the strong magnetic energy density, whichis expressed by UB = ǫBUγ . We take ǫB = 0.1, 1, 10 as aparameter.

B. Acceleration and Cooling of Proton

The radiating particles are considered to gain their en-ergies by stochastic acceleration. The most promising ac-celeration process is Fermi acceleration mechanism. Weassume that electrons and protons will be accelerated inGRBs by first order Fermi-acceleration by diffusive scat-tering of particles across storng shock waves. The accel-eration time scale is given by tacc ≈ (3crL/βA

2)(B/δB)2,

where rL is Larmor radius of the particle, βA = vA/c, vA

is Alfven velocity, and δB is the strength of the turbu-lence in the magnetic field [31, 32]. Assuming that thediffusion coefficient is proportional to the Bohm diffusioncoefficient, we can write the acceleration time of proton

3

as follows,

tacc ≡ ηrLc

= ηεp

eBc(3)

In usual cases η >∼ 10 would be more realistic, and η ∼ 1

will give a reasonable lower limit for any kind of Fermiacceleration time scale. As for the ultrarelativistic shockacceleration, the acceleration time scale can be writtentacc ∼ εp/

√2ΓreleBc, where Γrel is the relative Lorentz

factor of the upstream relative to the downstream [33].For the case of mildly relativistic shocks, η ∼ 1 is prob-ably possible [34]. Hence, we set η = 1 optimistically inour calculation.Corresponding to tacc < tdyn, proton energy is restrictedby εp

<∼ eBl/η, which satisfies the requirement that pro-

ton’s Larmor radius has to be smaller than the size ofthe acceleration sites. Even when GRBs are jet-like, thetransverse size will be larger than the radial size r/Γbecause the jet opening angle θj is larger than 1/Γ, sotacc < tdyn holds. Proton’s maximal energy is also con-strained by various cooling processes, for example, protonsynchrotron cooling, inverse Compton (IC) cooling, andphotohadronic cooling and so on. First, the synchrotronloss time scale for relativistic protons is,

tsyn =3m4

pc3

4σTm2e

1

εp

1

UB(4)

Second, we have to consider IC cooling, but we ignoreIC process in this paper for simplicity. This contributionis considered to be small in the region where proton en-ergy is above around 1017eV [13]. Although IC will bemore important than synchrotron loss when proton en-ergy is below that energy, we can ignore this fact becausephotohadronic loss will dominate IC for our parameters.Furthermore, in our cases IC does not matter for the pur-pose to determine the proton maximum energy. Third,the photohadronic cooling process is important, and pi-ons and resulting neutrinos appear through pion decays.When the internal shock occurs at small radii, photo-hadronic cooling may dominate synchrotron cooling. Wewill discuss this process in the next subsection. Finally,we take into account the adiabatic cooling process, whichhas a time scale tad independent of the proton energy, anddirect ejection of protons from the emission region. Thelatter may be dependent on the proton energy if diffu-sive losses are relevant. For simplicity, we neglect diffu-sive losses and assume that protons are confined over thetime scale set by adiabatic expansion. From above, thetotal proton loss time scale is expressed as follows,

t−1p ≡ t−1

pγ + t−1syn + t−1

ad ≤ t−1acc (5)

The maximum energy of proton is determined by thisinequality, but we do not know the minimum energy. Weset the minimum energy of proton to 10 GeV. Protonenergy density can be expressed by Up = ǫaccUγ , whereǫacc is a nonthermal baryon-loading factor. However, we

have few knowledge about how many protons can be ac-celerated for now. So, we adopt ǫacc = 10 as a fiducialparameter, assuming that a significant fraction of rela-tivistic protons can be accelerated. It is also assumedthat proton distribution has dnp/dεp ∝ ε−2

p according tofirst order Fermi acceleration mechanism.

C. Photomeson Production

GRB photons produced by synchrotron radiation andrelativistic protons accelerated in the internal shocks canmake mesons such as π± and K±, and secondary elec-trons, positrons, gamma rays, and neutrinos through thephotomeson production process. Some authors used ∆-approximation for simplicity, but this approximation un-derestimates proton energy loss in the high energy re-gion. This is because the theoretical model predicts theincrease of cross section in the high energy regions. Weuse the Monte Carlo simulation toolkit GEANT4 [21],which includes cross sections up to 40 TeV [35]. Above40 TeV, we extrapolate cross sections up to the maximumenergy, which is around 10 PeV. We checked the accu-racy of GEANT4 photomeson cross section within a fewpercent, comparing with PDG data [36]. Moreover, wetake into account inelasticity. Around the ∆-resonance,inelasticity is about 0.2. But above the resonance, in-elasticity increases with energy, and the value is about0.5− 0.7, that is consistent with Mucke et al. [22]. How-ever, it seems that the current version of GEANT4 tendsto overestimate the amount of produced neutral pions,leading to underestimation of the amount of charged pi-ons by a factor of ∼ 3. So we approximate π+ : π0 = 1 : 1for single-pion production, and π+π− : π+π0 = 7 : 4 fordouble-pion production [32, 37].Photomeson cooling time scale of relativistic protons isgiven by the following equation for isotropic photon dis-tribution [10].

t−1pγ (εp) =

c

2γ2p

∫ ∞

εth

dε σpγ(ε)κp(ε)ε

∫ ∞

ε/2γp

dε ε−2dn

dε(6)

where ε is the photon energy in the rest frame of proton,γp is the proton’s Lorentz factor, κp is the inelasticityof proton, and εth is the threshold photon energy forphotomeson production in the rest frame of the incidentproton, which is εth ≈ 145 MeV. Using this equation, wecan obtain photomeson production efficiency. Protonswith 1 PeV energy will effectively interact with soft X-ray photons with energy >

∼ 0.16keV.Pion spectrum can be obtained by executing GEANT4through the following equation,

dnπ

dεπ=

∫ εmax

p

εminp

dεpdnp

dεp

∫ εmax

εmin

dεdn

dε

∫

dΩ

4π

dσpγ(ε,Ω, εp)ξ

dεπc tp

(7)where dnp/dεp and dn/dε is proton and photon distri-bution in the comoving frame, ξ is the pion-multiplicity,

4

and tp is the proton loss time scale, which is defined bythe equation (5).

D. Neutrinos from Pion and Muon Decay

By using GEANT4, we can get pion spectra. Hence,neutrino spectra follow from the spectra of pions andmuons. Neutrinos are produced by the decay of π± →µ±+νµ(νµ) → e±+νe(νe)+νµ+ νµ. The life times of pi-ons and muons are tπ = γπτπ and tµ = γµτµ respectively.Here, τπ = 2.6033 × 10−8s and τµ = 2.1970 × 10−6s arethe mean life times of each particle. When pions decaywith the spectrum, dnπ/dεπ by π± → µ± + νµ(νµ), thespectrum of neutrinos are given by [38],

dnν

dεν=mπc

2ε∗ν

∫ ∞

εminπ

dεπ1

pπ

dnπ

dεπ(8)

Here, ε∗ν = (m2π − m2

µ)c2/2mπ and εminπ = (ε∗ν/εν +

εν/ε∗ν)/2. Similarly, we can get muon spectrum from

pion spectrum. Muon decay is the three-body-decay pro-cess, which is slightly more complicated than the case oftwo-body-decay. Given the spectrum of muon, it can becalculated by the following equation [39],

dnν

dεν=

∫ ∞

mµc2

dεµ1

cpµ

dnµ

dεµ

∫ ε∗

µ2

ε∗

µ1

dε∗ν1

ε∗ν(f0(ε

∗ν)∓cosθ∗νf1(ε

∗ν))

(9)

where ε∗µ1 = γµεν − (γ2µ − 1)

1/2εν , ε∗µ2 = min(γµεν +

(γ2µ − 1)

1/2εν , (m2

µ −m2e)c

2/2mµ), and ε∗ν is the energyof muon-neutrino in the muon-rest frame. f0(x) andf1(x) can be calculated by the quantum field theory,which are, f0(x) = 2x2(3 − 2x), f1(x) = 2x2(1 − 2x),and x = 2ε∗ν/mµc

2. However, because of cooling pro-cesses of π± and µ±, these equations have to be appliedat each time step. The pion or muon decay probabilityis (1 − exp(−∆t/tπ,µ)), and meanwhile cooled by syn-chrotron cooling and adiabatic cooling. The synchrotronradiating time scale is given by replacing proton masswith pion or muon mass in the equation (4). The adia-batic cooling time scale is still comparable to dynamicaltime scale. We neglect IC process of pions and muons inour calculation, because Klein-Nishina suppression willwork in our cases [13]. We also neglect neutrinos due toneutron decay n→ p+e−+ νe, whose time scale is muchlarger than the dynamical time scale tdyn.Roughly speaking, in the ∆-resonance picture, neutrinospectrum follows proton spectrum above the break en-ergy which is given by, εb

pεb ∼ 0.3 GeV, and becomes

harder below the break by εβ−1p due to fpγ ≡ tdyn/tpγ ∝

εβ−1p [10]. In addition, if tπ,µ > tad, neutrino spectrum

will be suppressed by tad/tπ,µ due to adiabatic cooling.If tπ,µ > tsyn, neutrino spectrum will be suppressed bytsyn/tπ,µ due to synchrotron cooling. These statementswill be confirmed numerically by above methods.

E. GRB Diffuse Neutrino Background

UHECRs may come from GRBs [6, 7]. This hypothesisrequires that the conversion of the initial energy of a fire-ball into UHECRs must be very efficient. In other words,this needs a very large value of ǫacc. On the other hand,Waxman & Bahcall [8] have derived a model independentupper bound for high energy neutrino background fromsources optically thin to photomeson production. Thisbound is conservative and robust [9] and it is called asthe Waxman and Bahcall limit (WB limit). They havealso estimated the contribution of GRBs to the high en-ergy neutrino background, assuming that UHECRs comefrom GRBs, and showed this contribution is consistentlybelow the WB limit.Whether UHECRs come from GRBs or not, numeri-cal calculations on neutrino spectrum obtained by ourmethod shown above can provide a diffuse neutrino back-ground quantitatively, assuming the GRB rate followssome distribution. The demonstration that long-durationGRBs are associated with core-collapse supernovae [40]implies that GRB traces the deaths of short-lived massivestars. Furthermore, GRBs can be detected to very highredshifts, unhindered by intervening dust and the currentrecord is recently observed GRB 050904 [41]. This holdsthe promise of being useful tracers of star formation rate(SFR) in the universe [42, 43, 44]. However, there re-main many problems such as observational bias and theeffect of dust-enshrouded infrared starbursts, althoughthese are crucial for to which extend GRBs follow theSFR and to which extend they can be used to determinethe SFR at high redshifts.In this paper, we estimate neutrino background for sev-eral models, following Nagataki et al. [45, 46]. First,we adopt the hypothesis that GRB rate traces the SFR,RGRB ∝ RSF. This ansatz that GRBs are likely to tracethe observed SFR in a globally averaged sense, is notprecluded for now although there are some uncertaintiesas described above. Second, we use parameterization ofPorciani & Madau [47], especially employing their mod-els SF1, SF2 and SF3, which are expressed by followingequation in a flat Friedmann-Robertson-Walker universe,

ψSF1∗ (z) = 0.32f∗h70

exp(3.4z)

exp(3.8z) + 45

×F (z,Ωm,ΩΛ)M⊙ yr−1Mpc−3 (10a)

ψSF2∗ (z) = 0.16f∗h70

exp(3.4z)

exp(3.4z) + 22

×F (z,Ωm,ΩΛ)M⊙ yr−1Mpc−3 (10b)

ψSF3∗ (z) = 0.21f∗h70

exp(3.05z − 0.4)

exp(2.93z) + 15

×F (z,Ωm,ΩΛ)M⊙ yr−1Mpc−3 (10c)

5

where F (z,Ωm,ΩΛ) = (Ωm(1 + z)3+ ΩΛ)

1/2/(1 + z)

3/2,

h70 = H0/70 kms−1 Mpc−1, and we adopt the standardΛCDM cosmology (Ωm = 0.3,ΩΛ = 0.7). The correctionfactor f∗ is introduced for uncertainties of SFR. We setf∗ = 1, which is consistent with mildly dust-correctedUV data at low redshift; on the other hand, it may un-derestimate the results of other wave band observationsand we only have to correct f∗ in such cases [48]. Manyworks have modeled the expected evolution of the cosmicSFR with redshift, but there are some uncertainties (inparticular at high redshift z >

∼ 6). In addition, even ob-servational estimates at modest high redshift have beenplagued by uncertainties arising from a result of correc-tion for dust extinction. Due to a few constraints at highredshift, we cannot avoid extrapolating SFRs. For thesereasons, we employ three models.The inner engine of GRBs is still unknown, althoughthere are several plausible candidates. One of the plausi-ble models of GRB progenitors is a collapsar model [49],because GRBs likely have a link with the explosion ofa massive (M >

∼ (35 − 40)M⊙) rotating star whose corecollapses to form a black hole [49, 50, 51]. Assuming thatmassive stars with masses larger than ∼ 35M⊙ explodeas GRBs, GRB rate can be estimated for a selected SFRby multiplying the coefficient,

fcl ×∫ 125

35dmφ(m)

∫ 125

0.4 dmmφ(m)= 1.5 × 10−3fclM

−1⊙ (11)

where φ(m) is the initial mass function (IMF) and massis the stellar mass in solar units. Here we adopt theSalpeter’s IMF (φ(m) ∝ m−2.35), assuming that theIMF does not change with time, which may be a goodapproximation if there are no significant correlationsbetween the IMF and the environment in which starsare born. For now, extant evidences seem to argueagainst such correlations at z <

∼ 2, although this va-lidity at high redshift is uncertain [52]. For compar-ison, we can obtain RSN(z) = 0.0122M−1

⊙ ψSF∗ (z) as-

suming that all stars with M > 8M⊙ explode as core-collapse supernovae. This result combined with f∗ = 1agrees with the observed value of local supernova rate,RSN(0) = (1.2 ± 0.4) × 10−4h3

70 yr−1Mpc−3 [53]. Inthe equation (11), we introduce an unknown parameter,fcl, which expresses the fraction of the collapsars whosemass range is in (35 − 125)M⊙ are accompanied withGRBs. We normalize fcl using the value of GRB rate,RGRB(0) = 17h3

70 yr−1Gpc−3, obtained by recent analy-sis using BATSE peak flux distribution [54, 55]. Fromequations (10a), (10b), and (10c), combined with (11),we obtain following expressions in units of yr−1Gpc−3,

RGRB1(z) = 17

(

fcl

1.6 × 10−3

)

46 exp(3.4z)

exp(3.8z) + 45(12a)

RGRB2(z) = 17

(

fcl

1.6 × 10−3

)

23 exp(3.4z)

exp(3.4z) + 22(12b)

RGRB3(z) = 21

(

fcl

1.6 × 10−3

)

24 exp(3.05z − 0.4)

exp(2.93z) + 15(12c)

We also consider the Rowan-Robinson SFR [56] that canbe fitted with the expression in units of yr−1Gpc−3,

RRR(z) = 41

(

fcl

1.6 × 10−3

)

100.75z (for z < 1)100.75 (for z > 1)

(13)

The parameter, fcl is unknown for now, but ∼ 2 × 10−3

will give the reasonable upper limit for the probability forone collapsar to generate a GRB, which corresponds tothe value if all the GRBs come from collapsars. On theother hand, direct estimates from the sample of GRBswith determined redshifts are contaminated by observa-tional biases and are insufficient to determine the pre-cise rate and luminosity function. In addition, the ob-served peak luminosity also depends on intrinsic spec-trum. More precise data on GRBs will give us informa-tion on the accurate GRB rate. For these reasons, weleave the value of fcl as a parameter here.In order to get the differential number flux of backgroundneutrinos, first we compute the present number densityof the background neutrinos per unit energy from GRBs.The contribution of neutrinos emitted in the interval ofthe redshift z ∼ z + dz is given as,

dnobν (Eν) = RGRB(z)(1 + z)

3 dt

dzdzdNν(E

′

ν)

dE′

ν

dE′

ν(1 + z)−3

(14)

where E′

ν = (1 + z)Eν is the energy of neutrinos at red-

shift z, which is now observed as Eν and dNν(E′

ν)/dE′

ν

is the number spectrum of neutrinos emitted by oneGRB explosion. The differential number flux of GRBbackground neutrinos, dFν/dEν , using the relationdFν/dEν = c dnob

ν /dEν ,

dFν

dEνdΩ=

c

4πH0

∫ zmax

zmin

dz RGRB(z)dNν((1 + z)Eν)

dE′

ν

×1

√

(1 + Ωmz)(1 + z)2 − ΩΛ(2z + z2)

(15)

where we assume zmin = 0, and zmax = 7 or zmax = 20.This is because the high redshift GRB event GRB 050904is recently reported by Swift observation and the GRBdistribution likely extends beyond z = 6 [41]. In addition,some GRBs are expected to exist at much higher redshiftsthan z = 7. First, preliminary polarization data on thecosmic microwave background collected by WMAP indi-cate a high electron scattering optical depth, hinting thatthe first stellar objects in the universe should have formedas early as z ∼ 20 [57]. Second, theoretical simulationsof the formation of the first stars similarly conclude thatthese should have formed at redshifts z ∼ (15 − 40) [58].Because there are convincing evidences that at least longGRBs are associated with the deaths of massive stars,it is conceivable that high-z GRBs (z >

∼ 15 − 20 or evenhigher) exist.

6

-8

-6

-4

-2

0

2

1 2 3 4 5 6 7 8 9

log(

f pγ)

log(εp [GeV])

FIG. 1: Proton cooling efficiencies, fpγ , for r = 2 × 1013 cmand Eiso

γ = 2 × 1051 ergs, by GEANT4 (solid line). For com-parison, we also show the case with cross section having acutoff at 2 GeV (dashed line) and rough analytic approxima-tion (dotted line).

-8

-6

-4

-2

0

2

1 2 3 4 5 6 7 8 9

log(

f pγ)

log(εp [GeV])

FIG. 2: Proton cooling efficiencies, fpγ ; case A for r = 2 ×

1013 cm and Eisoγ = 2 × 1051 ergs (solid line), case B for r =

5.4 × 1014 cm and Eisoγ = 2 × 1052 ergs (dotted line), case C

for r = 2 × 1013 cm and Eisoγ = 2 × 1053 ergs (dashed line).

Each shell width is given by l = r/Γ.

-8

-6

-4

-2

0

2

4

6

8

10

1 2 3 4 5 6 7 8 9

log(

t-1 [s

ec-1

])

log(εp [GeV])

tad-1

tacc-1

tsyn-1

tpγ-1

FIG. 3: Various cooling time scales and acceleration timescales for case A, r = 2×1013 cm, and ǫB = 0.1. The hatchedlines show an uncertainty of the acceleration time, which cor-responds to η = 1 − 10.

-8

-6

-4

-2

0

2

4

6

8

10

1 2 3 4 5 6 7 8 9

log(

t-1 [s

ec-1

])

log(εp [GeV])

tad-1

tacc-1

tsyn-1

tpγ-1

FIG. 4: Various cooling time scales and acceleration timescales for case A, r = 2 × 1013 cm, and ǫB = 1. The hatchedlines show an uncertainty of the acceleration time, which cor-responds to η = 1 − 10.

III. RESULTS

A. Photomeson Production Efficiency

We calculate the proton cooling efficiency through pho-tomeson production by the method explained above. Theobtained results on fpγ ≡ tdyn/tpγ by using GEANT4are shown in Fig. 1. For comparison, we also showthe case where the cross section has a cutoff at 2 GeVin the proton-rest frame. In this case, we checked thatour result agrees with Asano [13] within a few percent.Analytic rough approximation by ∆-resonance is alsoshown in Fig. 1 [10]. In this approximation we setσpγ ≈ 5 × 10−28 cm2, ε ≈ 0.3 GeV, ∆ε ≈ 0.3 GeV,and κp ≈ 0.2 in the equation (6). At γp

<∼ 107GeV,

∆-resonance is a good approximation and the break en-ergy is determined by εb

pεb ∼ 0.3 GeV2. Protons be-

low the break energy mainly produce pions with photonswhose energies are above the break energy, so it leads tofπ ∝ εβ−1

p . On the other hand, protons above the breakenergy mainly interact with harder photons, which leadsto fπ ∝ εα−1

p . The photomeson production efficiency ob-tained by GEANT4 is larger than other two cases whichhave the cutoff and monotonically increasing in the highenergy region of the order of γp

>∼ 107GeV. This is be-

cause κp ≈ 0.5−0.7 rather than κp ≈ 0.2 at ∆-resonanceare satisfied and multi-pion production occurs in this re-gion.In this paper we adopt three parameter sets for GRB

isotropic energy, photon break energy, and photon spec-

7

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-6

-4

-2

0

2

4

6

8

1 2 3 4 5 6 7 8 9

log(

t-1 [s

ec-1

])

log(εp [GeV])

tpγ-1

tsyn-1

tad-1

tacc-1

FIG. 5: The same as Fig. 3. But a result for case B, r =5.4 × 1014 cm, and ǫB = 1.

-12

-10

-8

-6

-4

-2

0

2

4

6

8

1 2 3 4 5 6 7 8 9

log(

t-1 [s

ec-1

])

log(εp [GeV])

tpγ-1

tsyn-1

tad-1

tacc-1

FIG. 6: The same as Fig. 3. But a result for case B, r =1.6 × 1015 cm, and ǫB = 1.

tral indices. That is, case A: Eisoγ = 2 × 1051 ergs,

εb = 1 keV, and α = 1, β = 2.2; case B: Eisoγ =

2 × 1052 ergs, εb = 3 keV, and α = 1, β = 2.2; case C:Eiso

γ = 2 × 1053 ergs, εb = 6 keV, and α = 0.5, β = 1.5.Fig. 2 shows the proton cooling efficiencies for each case.Case A and case B have the fiducial spectral indices ofGRB photons, while case C has a rather flatter photonspectrum. In addition, case C corresponds to the casewhere a more energetic burst will be observed.Fig. 3 - Fig. 6 show various cooling time scales andacceleration time scale. Proton’s maximum energy is de-termined by equation (5). Note that most of protons willbe depleted in the case optically thick to photomesonproduction, even if a fraction of protons are acceleratedup to the maximal energy indicated by the equation (5).For example, Fig. 3 and Fig. 4 corresponds to such cases.Only in the optically thin case, a significant fraction ofprotons with the maximum energy can escape from thesource. In these figures, for comparison, we also show ηin the equation (3) as a parameter having the range of 1 -10 (hatched lines). Fig. 3 shows the case where the pho-tohadronic cooling is crucial for determining the maxi-mum energy. Multiplicity and inelasticity will be impor-tant because the contribution of photohadronic coolingis comparable to synchrotron cooling if cross section hasthe cutoff at 2 GeV, or smaller than synchrotron cool-ing in the case of analytic ∆-resonance approximation.At inner radii, photon density is large, so photohadronicprocess can be important unless ǫB is enough large forsynchrotron cooling to be a dominant process. When theproton’s maximum energy is determined by the photo-hadronic process, a significant fraction of protons withthe very high energy cannot escape from the acceleratingsite in many cases of GRB parameters. It is only pos-sible within the limited range of radii or for the case ofthe moderately smaller radiation energy than our cases.In many cases of possible parameter sets for GRBs, thesynchrotron cooling is the most dominant process to de-

termine the maximum energy and such a case is demon-strated by Fig. 4 and Fig. 5. At outer radii, the dy-namical time scale will be more and more important dueto decreasing of photon energy density and magnetic en-ergy density. Fig. 6 corresponds to such a case and themaximum energy of proton is restricted by tad. In ourcases, significant acceleration of protons is possible onlyat larger radii, r >

∼ 1014 cm. This result is consistentwith Asano [13]. Smaller Eiso

γ and larger r are favorableto generate UHECRs. Fig. 5 and Fig. 6 demonstratesuch cases where sources are optically thin to photomesonproduction and the production of UHECRs is possible.The effects of multiplicity and high-inelasticity appear inthe very high energy region and enhances photohadroniccooling of protons. However, these effects affect coolingtime scale only for the cases of inner collision radii andsmaller magnetic fields, ǫB = 0.1. In the next subsection,we will see these effects can become important only forlimited cases.

B. Neutrino Spectrum and Flux

We can get the spectrum of pions from photomesonproduction efficiencies calculated by GEANT4. Frompion obtained spectra we can also calculate neutrinospectra following from the method explained in Sec. II Dand II E. Fig. 7 shows spectra of νµ + νν as one ofour results for r = 2 × 1013 cm, Γ = 100, and Γ = 300.The high-energy break is mainly determined by tsyn/tπ.

Hence the high-energy break changes satisfying εsν ∝ ǫ

1/2

B(see Fig. 7). Note, from the equation (1), different valuesof Γ give the similar neutrino spectra for the same Eiso

γ

and r as long as l = r/Γ is hold.Fig. 8 - Fig. 10 show spectra from single-pion, double-pion, and multi-pion production origins. As seen in Fig.8 and Fig. 9, for the case of α = 1 and β = 2.2 thatis typical for GRB, the effects of double- and multi-

8

4

6

8

10

12

14

0 1 2 3 4 5 6 7 8

log(

ε ν2 dn

ν/dε

ν [G

eV c

m-3

])

log(εν [GeV])

εB=0.1εB=1

εB=10

FIG. 7: Neutrino spectra in the comoving frame with ǫB

changing for r = 2×1013 cm. The upper lines are for Γ = 100and Eiso

γ = 2×1052 ergs. The lower lines are for Γ = 300 and

Eisoγ = 2 × 1051 ergs. Three cases are shown, ǫB = 0.1 (solid

line), ǫB = 1 (dashed line), ǫB = 10 (dotted line).

2

4

6

8

10

12

0 1 2 3 4 5 6 7 8

ε ν2 dn

ν/dε

ν [G

eV c

m-3

]

log(εν [GeV])

from single πfrom single µ

from double πfrom double µ

from multi πfrom multi µ

FIG. 8: Muon-neutrino (νµ + νµ) spectra in the comovingframe. Muon-neutrinos are produced by the decay of pionand muon which origins are single- or double- or multi-pionphotomeson production. Each case is shown. A result forr = 2 × 1013 cm on case A. ǫB is set to 1.

0

1

2

3

4

5

6

7

8

9

0 1 2 3 4 5 6 7 8 9

log(

ε ν2 dn

ν/dε

ν [G

eV c

m-3

])

log(εν [GeV])

from single πfrom single µ

from double πfrom double µ

from multi πfrom multi µ

FIG. 9: The same as Fig. 8. But a result for r = 5.4×1014 cmon case B. ǫB is set to 1.

2

4

6

8

10

12

0 1 2 3 4 5 6 7 8 9

log(

ε ν2 dn

ν/dε

ν [G

eV c

m-3

])

log(εν [GeV])

from single πfrom single µ

from double πfrom double µ

from multi πfrom multi µ

FIG. 10: The same as Fig. 8. But a result for r = 1.8 ×

1014 cm on case C. ǫB is set to 0.1.

pion are negligible or comparable to that of single-pion,whose contribution can be well described by ∆-resonanceapproximation. Protons of energy εp

<∼ 105 GeV can

interact only with the steep part of the photon spec-trum above the break, dn/dε ∝ ε−2.2. Hence the con-tribution of non-resonance, εpε ≫ 0.16 GeV, is negligi-ble. Protons with the energy εp

>∼ 105 GeV can inter-

act with the flatter part of the photon spectrum belowthe break, dn/dε ∝ ε−1. In this case, the contributionof double- and multi-pion production can be importantbecause the flatter part of photon spectrum cover sig-nificant energy range. Such contribution can be crucialat the very high energy range, εp

>∼ 107 GeV. At in-

ner radii the contributions of double- and multi-pion arenegligible or comparable (see Fig. 8). Even when itis comparable, such a region is above the high-energybreak. Because the nonthermal proton’s maximum en-

ergy is around (108 − 108.5)GeV, there are only a fewvery high energy protons which can produce multi-pions.At outer radii these contributions are comparable to orlarger than single-pion fraction by a factor of ∼ (2 − 3)(see Fig. 9). However, in most of the region where multi-pion production dominates, the resulting neutrino spec-tra are suppressed by synchrotron and adiabatic coolingprocesses because such a region belongs to the high en-ergy region above around the high-energy break. For thecase of α = 0.5 and β = 1.5, which is the flatter photonspectrum, the multi-pion production has the significanteffect for neutrino spectra. In this case there are sufficienthigh energy photons, which can interact with very highenergy protons, εp

>∼ 107 GeV. Fig. 10 demonstrates one

of such cases. In this case the contribution of multi-pionorigin dominates single-pion origin by one order of mag-nitude even around the high-energy break. Note that in

9

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4

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8

10

12

14

0 1 2 3 4 5 6 7 8 9

log(

ε ν2 dn

ν/dε

ν [G

eV c

m-3

])

log(εν [GeV])

2*1013 cm6*1013 cm

1.8*1014 cm5.4*1014 cm

FIG. 11: Muon-neutrino (νµ + νµ) spectra in the comovingframe for various collision radii on case A. The subshell widthis l = 6.7 × 1010 cm. ǫB is set to 1

2

4

6

8

10

12

14

0 1 2 3 4 5 6 7 8 9

log(

ε ν2 dn

ν/dε

ν [G

eV c

m-3

])

log(εν [GeV])

2*1013 cm6*1013 cm

1.8*1014 cm5.4*1014 cm

FIG. 12: Muon-neutrino (νµ + νµ) spectra in the comovingframe for various collision radii on case A. The subshell widthis given by l = r/Γ. ǫB is set to 1

the very high energy region, not only inelasticity but alsomultiplicity are also high. As a result, the average pion’senergy which can be estimated by the parent proton’senergy multiplied by inelasticity and divided by multi-plicity cannot be so large [23].The difference between neutrino spectra from pion decayand from muon decay is explained as follows. Neutri-nos from muons dominate those emitted directly frompions in the low energy region. When a pion decays asπ± → µ± + νµ(νµ), the energy fraction a muon obtainsis typically ∼ mµ/mπ ∼ 0.76. Hence a direct neutrinofrom π± has a fraction of ∼ 0.24. When a muon decaysas µ± → e± + νe(νe) + νµ + νµ, each of three specieswill carry similar energy. But a neutrino from µ± has asmaller energy fraction than a direct neutrino from π±,since muon has the longer life time than pion so that itis subject to synchrotron cooling.Fig. 11 shows neutrino spectra that occur at various col-lision radii with the fixed shell width. In this case photonenergy density changes with Uγ ∝ r−2. This result is con-sistent with Asano [13]. Fig. 12 shows neutrino spectrawhich occurs at various collision radii with changing theshell width holding l ≈ r/Γ. In this case photon energydensity changes with Uγ ∝ r−2l−1 ∝ δt−3. Here, δt isthe typical variability time scale. Roughly speaking, thehigh-energy break is determined by tsyn/tπ again. This

implies the high-energy break is proportional to rl1/2.On the other hand the low-energy break is determinedby the minimum energy of pions produced from protons.The dynamical time scale, tdyn is proportional to l, whiletpγ and tsyn are proportional to r2l, so the proton cool-ing efficiencies, which is expressed by fpγ ≡ tdyn/tpγ andfsyn ≡ tdyn/tsyn, are proportional to r−2 independentlyof l [13]. So, this minimum energy will depend on onlyr. Both the low-energy and high-energy break increaseroughly proportionally to r. Fig. 11 and Fig. 12 confirmthese statements.

To get the total neutrino spectrum of a single burstwith a collision radius r and a subshell width l simply,we only have to multiply the number of collisions N .However, here we cumulate the neutrino spectra over var-ious internal collision radii, assuming each collsion emitssimilar energy. It is because internal shocks may occurwithin some range of distances, although the distributionof collision radii is not so clear. In the internal shockmodel, the sequence of collsions takes place. Collisionsof the subshells with the smaller separation will occurat smaller radii. The subsequent collsions will occur atlarger radii. For simplicity, we assume the number ofsuch collisions is proportional to ∆/d, where ∆ is thetotal width of the shell. We also assume the fixed sub-shell width through one burst. We may need to take intoaccount the spreading of subshells since internal shockradii are roughly comparable to the spreading radii ofsubshells [4]. Even so, it does not affect the resultingspectra so much because photon density becomes muchsmaller at larger radii. To improve our calculation, wewill need a detailed calculation on the internal shockmodel [25, 26], which is beyond the scope of this paper.We adopt three parameter sets in this study. That is,set A: Eiso

γ = 2 × 1051 ergs, εb = 1 keV, α = 1, β = 2.2,

and r = (1013 − 8.1 × 1014) cm, and l = 6.7 × 1010 cm;set B: Eiso

γ = 2 × 1052 ergs, εb = 3 keV, α = 1, β = 2.2,

r = (2.7 × 1014 − 7.3 × 1015) cm, and l = 1.8 × 1012 cm;set C: Eiso

γ = 2× 1053 ergs, εb = 6 keV, α = 0.5, β = 1.5,

r = (1013 − 8.1 × 1014) cm, and l = 6.7 × 1010 cm. Set Ademonstrates internal shocks begin at somewhat smallerradii, r ∼ 1013 cm. In this set, the typical variabilitytime scale is δt ∼ 30 ms and if N = 200, the typical jetangle is θj ∼ 0.1 rad. Set B demonstrates internal shocksbegin at r ∼ 1014 cm. The typical variability time of thisset is δt ∼ 0.3 s and if N = 20, the typical jet angle isθj ∼ 0.1 rad. Set C is a very energetic case which hasthe flatter photon spectrum. Three sets are shown in

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3 4 5 6 7 8 9 10 11

log(

Eν2 φ ν

[erg

cm

-2])

log(Eν [GeV])

B

C

A

from π++π-

from µ++µ-

total

FIG. 13: The observed muon-neutrino (νµ + νµ) spectra forone GRB burst at z = 1. Neutrino spectra from pion andmuon are shown respectively on set A with N = 200, set Bwith N = 20, set C with N = 20. Set A and Set B are ǫB = 1,but Set C is ǫB = 0.1

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-8

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3 4 5 6 7 8 9 10 11

log(

Eν2 φ ν

[erg

cm

-2])

log(Eν [GeV])

B

C

A

εB=0.1εB=1

εB=10

FIG. 14: The observed muon-neutrino (νµ + νµ) spectra forone GRB burst at z = 1. Neutrino spectra with various ǫB

values are shown respectively on set A with N = 200, setB with N = 20, set C with N = 20. Set A and Set B areǫB = 1, but Set C is ǫB = 0.1

Fig. 13 and Fig. 14. To evaluate observed flux from oneburst, we set a source at z = 1. Obtained neutrino flux ofset A and set B are comparable with Guetta et al. [12].However, such levels of neutrino flux are hardly detectedby km3 detector such as IceCube. Only the most pow-erful bursts or nearby sources can give a realistic chancefor detection of νµ [11]. In our sets, only set C has theprospect for detection by IceCube. To see this, here weestimate neutrino events in IceCube. We use the follow-ing fitting formula of the probability of detecting muonneutrinos [59, 60].

P (Eν) = 7 × 10−5

(

Eν

104.5 GeV

)β

(16)

where β = 1.35 for Eν < 104.5 GeV, while β = 0.55 forEν > 104.5 GeV. Using a geometrical detector area ofAdet = 1km2, the numbers of muon events from muon-neutrinos a burst are given by,

Nµ(> Eν,3) = Adet

∫

TeV

dEν P (Eν)dNν(Eν)

dEνdA(17)

where Eν = 103 GeVEν,3. Hence, the numbers of muon-neutrinos to be expected by IceCube for set C withN = 20 are Nµ = 1.9 particles. In the case of set A withN = 200 and set B with N = 20, we obtain Nµ = 0.05and Nµ = 0.004 respectively. Of course, if ǫacc is morelarger, flux can be enhanced. But too large ǫacc will besuspicious. We will discuss this later.Since we obtain neutrino spectra on various parameters

above, we can calculate a diffuse neutrino backgroundfrom GRBs for our specific parameter sets. The resultsare shown in Fig. 15 - Fig. 17. Fig. 15 is for zmax = 7.Fig. 16 and Fig. 17 are for zmax = 20. Our adoptedSFR models generate similar results. When zmax is 20,

SF3 model gives higher flux by a factor of ∼ (2 − 3) be-low the low-energy break than other SFR models. Thisis because SF3 model predicts higher SFR at high red-shifts. Whether zmax is 7 or 20 does not affect neutrinoflux up to a factor. Neutrino signals from GRBs can bemarginally detected or not by IceCube in both figures.In fact, when ǫB is set to 1 and we use SF3 model, wecan obtain Nµ = 14 particles a year for zmax = 7 andNµ = 17 particles a year for zmax = 20. On the otherhand, in the case of set B, we get Nµ = 1.2 particlesa year for zmax = 7 and Nµ = 1.5 particles a year forzmax = 20.Our result for set A is very similar to the prediction ofWaxman & Bahcall [8], although our ground is differentfrom theirs. In this parameter set, our calculation onneutrino background from GRBs also satisfy WB bound,although this case is optically thick to photomeson pro-duction at inner radii, r <

∼ 1014 cm. Set B expresses neu-trino spectra for the case of larger collision radii thanset A. In this parameter set, GRB sources are opticallythin to photomeson production and can be sources of veryhigh energy cosmic rays. It has to satisfy WB bound andthis implies ǫacc can be constrained from UHECR obser-vations if set B is fiducial for GRBs and other parame-ters are appropriate. If GRBs are sources of UHECRs,this set implies GRB neutrino fluxes with z-evolution,E2

νdFν/dEνdΩ ∼ 10−8 GeV cm2 s−1 str−1. However, thissuggests very large baryon-loading factor, ǫacc ∼ 100 ifcurrent GRB rate estimation is correct and our model isvalid. If set A is more fiducial, similar arguments leadsto even larger baryon-loading factor because UHECRscan be accelerated only at large radii. So GRBs are notmain sources of UHECRs in set A. If we adopt largerisotropic energy by one order in this set with N fixed,the flux level will increase within a factor, because largerisotropic energy implies smaller θj (but for set B, it will

11

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-8

-7

-6

-5

3 4 5 6 7 8 9 10 11

log(

Eν2 Φ

ν [G

eV/c

m2 s

str

])

log(Eν [GeV])

AMANDA-B10

AMANDA-2

IceCube

R-RSF1SF2SF3WB

FIG. 15: Diffuse neutrino background from GRBs for zmax =7 on several SFR models. R-R means the case using Rowan-Robinson SFR. SF1 - SF3 correspond to the models of Por-ciani & Madau. The left lines are for set A, while the right isfor set B. For comparison, we show WB bounds (three dashedlines). The upper WB bound is for z-evolution of QSOs. Thelower is for no z-evolution. ǫacc = 10 and ǫB = 1 are assumed.

-13

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-7

-6

-5

3 4 5 6 7 8 9 10 11

log(

Eν2 Φ

ν [G

eV/c

m2 s

str

])

log(Eν [GeV])

AMANDA-B10

AMANDA-2

IceCube

R-RSF1SF2SF3WB

FIG. 16: Diffuse neutrino background from GRBs for zmax =20 on several SFR models. R-R means the case using Rowan-Robinson SFR. SF1 - SF3 correspond to the models of Por-ciani & Madau. The left lines are for set A, while the right isfor set B. For comparison, we show WB bounds (three dashedlines). The upper WB bound is for z-evolution of QSOs. Thelower is for no z-evolution. ǫacc = 10 and ǫB = 1 are assumed.

-13

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-7

-6

-5

3 4 5 6 7 8 9 10 11

log(

Eν2 Φ

ν [G

eV/c

m2 s

str

])

log(Eν [GeV])

AMANDA-B10

AMANDA-2

IceCube

εB=0.1εB=1

εB=10

FIG. 17: Diffuse neutrino background from GRBs for zmax =20 with ǫB changing. But the large baryon-loading factor,ǫacc = 100, is assumed. The upper lines are for set A, whilethe lower lines are for set B. All lines use the SF3 model.

increase by one order of magnitude, which is easily seenfrom the equations (1), (6), and (7)). Fig. 17 showsǫB dependence of neutrino background. But the nearextreme case, ǫacc = 100 is presumed. If it is possible,neutrino will be surely observed by the detector such asIceCube. For example, when ǫB is set to 1 and we useSF3 model, we obtain Nµ = 170 particles a year for setA with zmax = 20 and Nµ = 15 particles a year for set Bwith zmax = 20. Observations of neutrino background, ifdetected, may give us the evidence of protons being ac-celerated in GRBs, support to the internal shock modelof GRBs, and information about these parameters inde-pendently of X/γ rays from GRBs.

Finally, we summarize the parameter dependence of ourresults as follows. The collision radii, r and the width ofa subshell, l determine the photon energy density. Thelarger radii and width of a subshell make the result-ing neutrino emissivity smaller. The low-energy breakof neutrino spectrum, εb

ν is determined by the photonbreak energy, but it is also roughly proportional to r.The high-energy break is determined by the synchrotron(or adiabatic) cooling and satisfies εs

ν ∝ ǫB1/2. We take

ǫB = 0.1, 1, 10 (in the case of aftergrow, ǫB = 0.1 ispreferred). The high-energy break is also roughly pro-portional to r2l. One of the most important parametersis the nonthermal baryon-loading factor, ǫacc. We setǫacc to 10 except Fig. 17 and the larger ǫacc can raisethe flux level of neutrino, although too large ǫacc is notplausible. We take Eiso

γ,tot ∼ (1052 − 1054) ergs, and moreenergetic bursts can produce more neutrinos when otherparameters are fixed.

IV. SUMMARY AND DISCUSSION

In this paper we calculate proton’s photomeson cool-ing efficiency and resulting neutrino spectra from GRBsquantitatively. We are able to include pion-multiplicityand proton-inelasticity by executing GEANT4. These ef-fects of multi-pion production and high-inelasticity in thehigh energy region enhance the proton cooling efficiencyin this region, so they help prevent protons from acceler-ating up to the ultra-high energy region in several cases.But in many cases, the synchrotron loss time scale andthe dynamical time scale determine proton’s maximumenergy. Furthermore, resulting neutrino spectra are notso sensitive to the nonthermal proton’s maximum energy

12

except the very high-energy region above the high-energybreak.In our cases GRBs are optically thick to photomeson pro-duction at r <

∼ 1014 cm, while at r >∼ 1014 cm GRBs are

optically thin to it, so the production of UHECRs is pos-sible at larger radii. This result is consistent with Asano[13]. Using the obtained proton cooling efficiency, we cancalculate neutrino spectra. The effects from multi-pionproduction on resulting spectra are also calculated. Weshow that the contribution of multi-pion is almost negli-gible at inner radii but can be larger than that of single-pion at outer radii by a factor. In addition, the contri-bution of multi-pion production is somewhat sensitive tothe proton’s maximum energy, which would be actuallydifficult to determine precisely due to uncertainty of η.But such a contribution can be significant for the flatterphoton spectrum, even though mesons lose their energythrough these cooling process. Radiative cooling of pi-ons and muons plays a crucial role in resulting spectra.Neutrino spectra are suppressed above the high-energybreak energy. If the magnetic field is strong, such sup-pression becomes large and vice versa, if the magneticfield is weak, such suppression becomes weak. The ob-servations of neutrino has the possibility that gives ussome information about such a parameter of GRBs inde-pendently of gamma-ray observation. However, as shownin Dermer & Atoyan [11], only the most powerful bursts,which are brighter than ∼ 1053 ergs or bursts at z <

∼ 0.1,produce detectable neutrino bursts with a km3 detectorsuch as IceCube. Set C of our parameter sets would beone of such detectable cases, but only a few neutrinos areexpected even for the brightest bursts. For one neutrinoburst, we adopt Γ = 300 and consider only the on-axisobservations. If we observe at off-axis, we will observelower energy neutrinos.A diffuse neutrino background is also calculated in thispaper. In our specific parameter sets, neutrino back-ground observations by IceCube can expect a few or a fewtens order of neutrinos per year, although it is importantwhich parameter set is fiducial. Extrapolation of SFRto high redshifts may not be valid. Even so, our resultswould not be so much affected as we have seen above. Inaddition, GRBs may trace not SFR, but metallicity. Inthe collapsar model, the presence of a strong stellar wind(a consequence of high metallicity) would hinder the pro-duction of a GRB, therefore metal-poor hosts would befavored sites [49]. There remains large uncertanty at lowmetallicity at present, and as more bursts are followed upand their environments are better studied by Swift, thiscorrelation will be testable. However, our results will notbe changed so much even when we can take into accountthis.Throughout this paper, we set N to the range of ∼(10 − 100). If we change N , a neutrino signal from oneburst will change with N . On the other hand, whenwe calculate a neutrino background, the results are notchanged unless we fix Eiso

γ . This is because θj varies ac-

cording to the change of N since we fix Eisoγ and Eγ,tot.

However, the results will be changed when we fix θj andEγ,tot. For example, on set B, we will get the lower fluxlevel by one order of magnitude if we adopt N = 200fixing Eγ,tot, θj , and ǫacc. This is easily seen from the

Eisoγ,tot = f−1

b Eγ,tot, and the equations (1), (6), and (7).When we fix Eγ,tot, θj , and ǫacc, more shells mean lowerphoton energy density. For set B, which is the case opti-cally thin to photomeson production, it leads to decreasednπ/dεπ roughly by two orders of magnitude. Hence,the total neutrino spectra will be lower by one order ofmagnitude. If we fix θj , our results in which N is set to∼ 10, would give the reasonable flux level in the opti-mistic case.In our parameter sets both one burst emission and a dif-fuse neutrino background give the neutrino flux we canbarely observe by IceCube. To raise flux, GRBs requirethe larger nonthermal baryon-loading factor, ǫacc, whichis difficult to estimate from microphysics at present.However, there are some clues and assuming too largeǫacc will not be plausible. First, as seen in Sec. III B,UHECRs observations can give the upper limit to ǫacc.Furthermore, the large baryon-loading factor suggests asignificant contribution of the accelerated protons in theobserved hard radiation through secondaries produced inphotomeson production. Such emission is expected to beobserved in the multi-GeV energy range by electromag-netic cascades [10, 24, 61]. If the flux level of multi-GeVemission is comparable to the neutrino flux obtained inthis paper, the flux level will be below that of X/γ emis-sion and the EGRET limit. But assumed photon spectrawill be modified by the radiation from secondaries. Inaddtion, such high energy emission may be detected bythe near future GLAST observation [62]. Although itis important both to compare X/γ emission with suchmulti-GeV emission and to investigate whether GLASTcan detect such multi-GeV emission or not, a detailed cal-culation for this purpose is needed and it is beyond thescope of this paper. If such a calculation is done, GLASTobservation in the near future has the possibility to givemore information about ǫacc. This will be the secondclue about ǫacc. Third, ǫacc will be constrained by theGRB total explosion energy, which is still unknown. Thelarge baryon-loading factor leads to the large explosionenergy. For example, if the true total explosion energy is<∼ 1053 ergs, this suggests ǫacc

<∼ 100.

On the other hand, GRBs associated with supernovaemay imply the isotropic kinetic energy, Eiso

kin>∼ 1052 ergs,

which is larger than usual supernovae by one order ofmagnitude [63]. This leads to a collapsar model as a failedsupernova in the sense that a core collapse event failedto form a neutron star and instead produced a black hole[49]. Here, we assume the internal shock scenario is cor-rect and a collapsar model is valid. If the true total explo-sion energy of a collapsar is assumed to be <

∼ 1053 ergs,neutrino spectra will be limited by fclǫacc

<∼ 10−1 be-

cause fcl is constrained by fcl<∼ 10−3 from current GRB

rate estimations. So if observed neutrino spectra is higherthan these values, the collapsar model cannot explain the

13

spectra alone. On the other hand, the supernova modelpredicts higher neutrino flux by a pulsar wind [11, 64]. Ifthe observed neutrino flux is higher than expected in thecollapsar model, the supernova model might be likely toexist.If IceCube can detect neutrinos and confirms they havethe expected level of neutrino flux by our calculation, thiswill be one of the evidences that a significant fraction ofprotons can be accelerated and our employed internalshock model is valid. Of course, to estimate neutrinobackground more precisely and make our discussion jus-tified, we have to choose the most fiducial parameters forGRBs. The results depend on photon density, and thecontribution from a fraction of bursts with large photondensity might be large. So, we should take into accountthe respective distributions of parameters to execute themost refined calculaion. Unfortunately, many parame-ters have large uncertainty at present. For this reason,we calculate for a wide range of these parameters in thispaper. The signature of GRBs may depend on z. For ex-ample, the total isotropic energy of a GRB and photonspectral indices may depend on z. More and more obser-vations in the near future and more refined theoreticalmodels will allow our results to be improved.So far, we have not taken account of neutrino oscilla-tions. Since we have considered many decaying modes,the production ratio of high energy muon and electronneutrinos is not 2:1 exactly. However, the neutrinos willbe almost equally distributed among flavors as a resultof vacuum neutrino oscillations [10]. So there may be apossibility that tau neutrinos are detected through dou-ble bang events [65].In summary, we obtain the neutrino spectrum fromGRBs quantitatively by using GEANT4 simulation kit.We show that photomeson cooling process can constrainthe proton’s maximum energy and the effects of multi-pion production and high-inelasticity can enhance thecooling efficiency. Furthermore, these effects affect theresulting neutrino spectra slightly and can be signifi-cant for the flatter photon spectrum. We quantitatively

checked radiative cooling of pion and muon play a crucialrole, which is controlled by ǫB. We also confirmed thatUHECRs can be accelerated at r >

∼ 1014 cm. We havecalculated not only neutrino spectra from one burst butalso the GRB diffuse neutrino background using severalSFR models. We have found our specific parameter setsgive neutrino spectra comparable with the prediction ofWaxman & Bahcall [10] without supposing the very largenonthermal baryon-loading factor, which is necessary forthe assumption that GRBs are main sources of UHECRs.We have also discussed influences on neutrino spectra bychanging parameters. Such a study is important sincethere are many parameters. If neutrino signals are de-tected by AMANDA, ANTARES, NESTOR, or IceCube,it will be one of the evidences that protons can be acceler-ated to very high energy in GRBs and the internal shockmodel of GRBs are plausible. Furthermore, such neu-trino observations in the near future may give us someinformation about the nonthermal baryon-loading factorand the inner engine of GRBs.

Acknowledgments

We are very grateful to K. Asano for many usefulcomments and advises for calculation. We also thankthe referee and K. Ioka for a lot of profitable sugges-tions. K.M. is grateful to K. Miwa and G. Normannfor using GEANT4 simulation. We also thank M. Mori,H. Kurashige, T. Hyodo, and D. Wright for GEANT4.This work is in part supported by a Grant-in-Aid for the21st Century COE “Center for Diversity and Universal-ity in Physics” from the Ministry of Education, Culture,Sports, Science and Technology of Japan. S.N. is par-tially supported by Grants-in-Aid for Scientific Researchfrom the Ministry of Education, Culture, Sports, Sci-ence and Technology of Japan through No. 14102004,14079202, and 16740134.

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