{'fraction_non_alphanumeric': 0.03882762831794835, 'fraction_numerical': 0.03498437556122266, 'mean_word_length': 4.236411510250141, 'pattern_counts': {'":': 0, '<': 4, '<?xml version=': 0, '>': 3, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Diffuse PeV neutrinos from gamma-ray bursts

The IceCube collaboration recently reported the potential detection of two cascade neutrino events in the energy range 1-10 PeV. We study the possibility that these PeV neutrinos are produced by gamma-ray bursts (GRBs), paying special attention to the contribution by untriggered GRBs that elude detection due to their low photon flux. Based on the luminosity function, rate distribution with redshift and spectral properties of GRBs, we generate, using Monte-Carlo simulation, a GRB sample that reproduce the observed fluence distribution of Fermi/GBM GRBs and an accompanying sample of untriggered GRBs simultaneously. The neutrino flux of every individual GRBs is calculated in the standard internal shock scenario, so that the accumulative flux of the whole samples can be obtained. We find that the neutrino flux in PeV energies produced by untriggered GRBs is about 2 times higher than that produced by the triggered ones. Considering the existing IceCube limit on the neutrino flux of triggered GRBs, we find that the total flux of triggered and untriggered GRBs can reach at most a level of ~10^-9 GeV cm^-2 s^-1 sr^-1, which is insufficient to account for the reported two PeV neutrinos. Possible contributions to diffuse neutrinos by low-luminosity GRBs and the earliest population of GRBs are also discussed.


Introduction
Detection of high-energy neutrinos would be an important step towards identifying the origin of cosmic rays. Interestingly, through analysis designed to search for ultra high-energy ( PeV) and high quality shower events over a period of 672.7 live days between 2010 and 2012 with IceCube, two PeV neutrino-induced cascade events have been found (Ishihara 2012). The two events were observed on August 9, 2011 and January 3, 2012, and each have an energy of approximately 1-10 PeV. The observation of two events with an expected background of 0.14 events from the atmospheric neutrinos corresponds to a significance of 2.36σ. Given this modest statistical significance, we cannot at this time be confident that they are of astrophysical origin.
Gamma-ray bursts (GRBs) have been proposed as a potential source of ultra-high energy cosmic rays (UHECRs) (e.g. Waxman 1995;Vietri 1995;Dermer 2002), and high energy neutrino emission has been predicted to be produced in the dissipative fireballs, where cosmic ray protons interact with fireball photons (e.g. Waxman& Bahcall 1997; Dermer & Atoyan 2006;Guetta et al. 2004;Wang & Dai 2009). Gamma-ray bursts are an attractive possibility for the reported two PeV neutrinos because the predicted neutrino emission in the standard internal shock model peaks at PeV energies. Notably, previous searches of GRB neutrinos with IceCube via stacking analysis of many triggered GRBs have yield none detection (Abbasi et al. 2010(Abbasi et al. , 2011(Abbasi et al. , 2012. The approach to search for neutrinos from triggered GRBs is through analysis of upgoing neutrinos that are correlated in time and direction with known GRBs, which is different from that of detecting diffuse neutrinos. The most stringent constraint is given by the combined analysis of the IceCube 40-and 59-string (IC40+59) data of upgoing neutrinos from 215 triggered GRBs, which leads to an upper limit that is claimed to be below the expected flux (Abbasi et al. 2012) 1 . Beside the contribution by triggered GRBs for which the stacking analysis has been performed, GRBs that do not trigger detectors due to their low flux may also contribute to the diffuse neutrino flux. The purpose of this paper is to study whether the sum contribution of triggered GRBs and untriggered GRBs can explain the reported two PeV neutrinos under the constraint that the stacked neutrino flux of 215 triggered GRBs is below the IC40+59 sensitivity. The untriggered GRBs defined in our paper are normal GRBs (i.e. belonging to the normal GRB population) that do not trigger the current detectors. They are normal GRBs in the low-luminosity end of the luminosity function (for normal GRBs) and/or locating at high redshifts and hence they can elude the detection by current GRB detectors (e.g. Fermi/GBM) due to the limited sensitivity.
Previous estimates of diffuse neutrino flux from GRBs are made through integrating the neutrino flux over the whole luminosity range and the redshift range of GRBs, assuming that the their number density follows a given luminosity function and redshift distribution function (e.g. Gupta & Zhang 2007;He et al. 2012;Cholis & Hooper 2012). The estimates show that the total flux from all the GRBs (including both the triggered ones and the untriggered ones) reach a level of ∼ 10 −9 − 10 −8 GeVcm −2 s −1 sr −1 (Gupta & Zhang 2007;He et al. 2012;Cholis & Hooper 2012), which depends on the luminosity function and redshift distribution that are used. These estimates made simplified assumptions about some properties of GRBs, e.g. assuming the same photon spectrum for every GRBs, and do not consider the existing IceCube limit on triggered GRBs. To overcome this simplification, we here present a new approach to calculate the diffuse neutrino flux.
We generate a GRB sample with Monte-Carlo simulation 2 , requiring that the triggered population of GRBs reproduce the observed properties of Fermi/GBM GRBs. A population of dim, untriggered GRBs will be generated simultaneously in the simulation. Then, the neutrino flux of every individual GRBs are calculated, so that the accumulative flux of the whole sample can be obtained.
The rest part of this paper is arranged as follows. In §2, we describe the properties of GRBs and some empirical relations used in our simulation generating the GRB sample. We calculate the accumulative neutrino flux from both triggered GRBs and untriggered GRBs in the generated sample and compare them with the reported two PeV events in §3. We give our discussion and conclusion in §4 and 5. Throughout the paper, we use eV for particle energy, c.g.s units for other quantities and denote by Q x the value of the quantity Q in units of 10 x unless specified.

Simulation of a GRB sample
The distributions of GRBs in luminosity and in redshift are assumed to be independent of each other in this work, i.e., n(L, z) = φ(L)ρ(z), where φ(L) and ρ(z) are luminosity function and redshift distribution function of GRBs respectively. Wanderman & Piran (2010, hereafter, W10) obtained the luminosity function and cosmic rate by inverting directly the redshift -luminosity distribution of observed long Swift GRBs, which are, respectively, given by where a 1 = 1.2, a 2 = 2.4, L * = 10 52.5 ergs −1 , n 1 = 2.1, n 2 = −1.4, z * = 3.1, and ρ 0 = 1.3Gpc −3 yr −1 are the best fit parameters. Note that L is the isotropic, peak bolometric luminosity of GRBs in energy range 1 keV-10 MeV. The low-and high-luminosity ends of the luminosity function are set as 10 50 ergs −1 and 10 54 ergs −1 respectively, and the maximum redshift is set as z max = 8. We generate a sample of GRBs that follow the luminosity distribution and redshift distribution given by Eq. 1 and Eq. 2 with Monte-Carlo simulation. The isotropic energy of each GRBs is obtained through the relation log E iso,52 = 1.07log L iso,52 + (0.66 ± 0.54) which is obtained from the data in Ghirlanda et al. (2012). The photon spectra of GRBs can be described by an empirical function known as the Band function (Band et al. 1993), which can be written as where N 0 is the normalization constant and α, β are the photon indices for low-and high-energy spectra respectively. Generally, α > −2 and β < −2, and the energy spectrum peaks at E peak = (2 + α)E 0 . The values of α and β are generated following the observed distribution of BATSE long GRBs in the simulation and only bursts with β < −2 are adopted (Kaneko et al. 2006). The rest frame peak energy of GRBs is found to be correlated with their isotropic energy and luminosity, which are known as the Amati relation and the Yonetoku relation, respectively (Amati et al. 2002;Yonetoku et al. 2004). In this work, to generate the peak energies of GRBs in the simulation, we adopt the E peak − E iso relation recently found by Ghirlanda et al. (2012), i.e.
where E ob peak is the observer frame peak energy (hereafter the subscript "ob" represents the quantity in the observer frame). Since the neutrino emission is produced dominantly by long duration GRBs, we only include bursts with T 90 > 2s in our sample. Following Kakuwa et al. (2012), we determine T 90 roughly by T 90 = (1 + z)(E iso /L ave ) where L ave = 0.3L iso is the average luminosity of a GRB.
To compare our simulated sample with observed GRBs by Fermi /GBM, the trigger threshold in our simulation is set to be the same as that of Fermi /GBM, i.e., 0.74 photons cm −2 s −1 (Meegan et al. 2009). Taking into account the effect of Earth occultations and the South Atlantic Anomaly (SAA) passages, about 65% GRBs above the Fermi /GBM trigger threshold can be observed. To reproduce the GBM detection rate of ∼ 250 GRBs per year, the value of ρ 0 is required to be 1.01 Gpc −3 yr −1 , which is within the 1σ error of the best-fit value obtained by W10. For this value of ρ 0 , the total event rate of GRBs in the luminosity range from 10 50 ergs −1 to 10 54 ergs −1 and redshift range from z = 0 to z = 8 is ∼ 8000 GRBs per year in the generated sample so the vast majority of them are untriggered GRBs. The total number of GRBs occurred from z = 0 to z = 8 per year is obtained 3 Ω M + Ω Λ being the Hubble constant at redshift z in a flat universe.
In Fig. 1, we compare the 10-1000keV fluence distributions dP/d(logF γ ) of GRBs above the trigger threshold in our simulated sample with that of 482 Fermi /GBM bursts 3 . The K-S test of two distributions returns a high accepting probability of 89.5%. The total gamma-ray fluence of GRBs in 10-1000keV above the trigger threshold in our simulated sample is 0.0048 erg cm −2 when the number of bursts is normalized to be 482, which also agree well with the total gamma-ray fluence, 0.0044 erg cm −2 , of the observed 482 GBM bursts.
In Fig. 2 we show the distribution of the cumulative bolometric fluence i F γ,i (> F γ,th ) above a certain flux F γ,th for the subsample of both triggered and untriggered GRBs. As is shown, the total gamma ray fluence of the untriggered GRBs (F γ,unt = 0.0136 erg cm −2 ) is 2-3 times larger than that of the triggered ones (F γ,tri = 0.0053 erg cm −2 ). This is mainly due to two aspects. First, the number of untriggered GRBs is much larger than that of triggered GRBs, although the luminosity of untriggered GRBs is on average much lower than that of triggered GRBs; Second, a fraction (about 35%) of bright GRBs occulted by the Earth also contribute to the gamma-ray fluence in the untriggered GRBs. This demonstrates that the total contribution of untriggered GRBs can be important.
There is significant variation in the luminosity function and the redshift distribution of GRBs in the literature, see e. g. Liang et al. (2007, hereafter, L07) 4 and Guetta & Piran (2007, hereafter, G07) 5 . We find that the GRB sample generated using the above approach with the luminosity function and rate redshift distribution given in L07 contain much more triggered GRBs than the observed ones, while the G07 luminosity function in combination with the relevant redshift distribution gives rise to too few triggered GRBs. In other words, to maintain the rate of triggered GRBs of 250/yr by Fermi/GBM, ρ 0 is required to be ∼ 0.2 Gpc −3 yr −1 and ∼ 5 Gpc −3 yr −1 for L07 and G07 respectively, which are beyond the 3σ error of the best-fit values (1.12 Gpc −3 yr −1 and 0.27 Gpc −3 yr −1 for L07 and G07 correspondingly). So here we adopt the luminosity function and redshift distribution of W10 in generating the GRB sample in our simulation. 4 The luminosity function obtained by L07 is where ρ0 = 1.12Gpc −3 yr −1 is the local rate of GRBs, and Φ0 is a normalization constant to assure the integral over the luminosity function being equal to the local rate ρ0. This luminosity function breaks at L γb = 2.25 × 10 52 erg s −1 , with indices α1 = 0.65 and α2 = 2.3 below and above the break. The rate distribution with redshift used by L07 in obtaining this luminosity function is S(z) = 23e 3.4z /(e 3.4z + 22.0) (Porciani & Madau 2001).

Analytical calculation of neutrino flux
To illustrate the dependence of neutrino flux on the parameters, we first present an analytical approximation of calculating the neutrino flux.
It has been suggested that protons can be accelerated up to ǫ ob p ∼ 10 20 eV through internal or external shocks (e.g. Waxman 1995;Vietri 1995;Wang et al. 2008;Murase et al. 2008). The accelerated protons collide with the fireball photons in the same region and produce charged pions. The charged pions decay into four final-state leptons via the processes π + (π − ) → ν µ (ν µ )+µ + (µ − ) → ν µ (ν µ ) + e + (e − ) + ν e (ν e ) +ν µ (ν µ ). In the fireball comoving frame, the fractional energy loss rate of a proton with Lorentz factor γ p via photopion production is where c is the speed of light,Ẽ is the photon energy in the rest frame of the proton andẼ 0 is the threshold energy of photonpion production. σ π is the cross section of photopion production and ξ is the inelasticity. x and n(x) are the photon energy and the photon spectrum in the GRB comoving frame respectively. When only the ∆ resonance channel for the photopion production is considered, the fraction of the proton energy converted into the pion can be approximated by where Γ is the bulk Lorentz factor of GRB and R is the internal shock radius which relates with the bulk Lorentz factor and variability time δt through R = 2Γ 2 cδt. The break energy ǫ ob pb in the proton spectrum is caused by the break in the photon spectrum. Correspondingly, a break occurs in the neutrino spectrum at energy ǫ ob νb = 7.5 × 10 14 (1 + z) −2 Γ 2 2.5 E ob,−1 peak,MeV eV.
Additionally, there is another break in the neutrino spectrum caused by the synchrotron cooling of charged pions in the magnetic field of the dissipative fireball, which can be written as The final neutrino spectrum (single flavor) of a GRB is approximately a three-section power law with two breaks at ǫ ob νb and ǫ ob νc respectively, i.e., where η p is the ratio between the energy in the accelerated protons and the radiation energy, ǫ p,max and ǫ p,min are respectively the maximum and minimum energy of accelerated protons, and F γ,bol is the bolometric (1 keV-10 MeV) gamma-ray fluence of the GRB. The factor 1/8 comes from the fact that half of the produced pions are charged pions and each neutrino takes about 1/4 of the pion's energy.

Numerical results
In our numerical calculation, besides the ∆ resonance channel, we also consider the contribution from the direct pion channel and the multi pion channel. The cross-section for photopion production is taken from Mücke et al. (2000) and the branching ratio of each channel is approximated as the same in H12. Muon cooling, pion cooling and neutrino oscillation effects are also included in the calculation. We calculate the GRB neutrino flux within the standard internal shock model, i.e., we use R = 2Γ 2 cδt, similar to the treatment by the IceCube Collaboration (Abbasi et al. 2010(Abbasi et al. , 2011(Abbasi et al. , 2012. We consider two cases of the choice of the bulk Lorentz factor Γ, one is the relation Γ = 29.8E 0.51 iso,52 that is obtained by Ghirlanda et al. (2012), and another is the benchmark choice, i.e. Γ = 10 2.5 for all GRBs. As in H12, we fix δt ob = δt(1 + z) at 0.01s. We assume the accelerated proton spectrum to be a power law with an index of −2. The baryon ratio is taken to be η p = 10.
We calculate the neutrino fluence from each GRB in the generated sample and the aggregated neutrino fluence from all the triggered GRBs in the sample is shown in Fig.3. To compare with the IceCube limit from the combined analysis of IC40 and IC59 for 215 GRBs, we have multiplied the aggregated fluence by a factor of 215/N tri , where N tri = 250 is the number of triggered GRBs in our sample. The aggregated neutrino fluence peaks around 1 PeV and reach a level of ∼ 0.1GeVcm −2 for both choices of Γ. For comparison, we also show the fluence limit of IC40+59 in Fig.3 (dashed lines). The 90% upper limit on the fluence is obtained by requiring that the upgoing muon neutrino event number 6 observed by IC40+59 should be smaller than N limit = 1.9. The non-detection of neutrinos by IC40+59 implies η p 20. We note that the aggregated neutrino fluence of 215 GRBs in our former work H12 (see Fig. 2 in H12) is a factor of two lower and peaks at higher energy. The difference is due to that, in that work the luminosity and bulk Lorentz factor of all the GRBs without measured redshift are fixed as 10 52 ergs −1 and 10 2.5 respectively.
The diffuse neutrino flux can be obtained by multiplying the aggregated neutrino fluence of the all-sky GRBs occurred in one year by a factor of 1/4π yr ≃ 2.5 × 10 −9 s −1 sr −1 . Fig. 4 presents the diffuse neutrino flux from all GRBs in our generated sample (solid lines), which consists of the flux contributed by the triggered GRBs (dashed line) and the flux contributed by untriggered GRBs (dash-dotted line). The top panel corresponds to the case that assumes Γ = 29.8E 0.51 iso,52 (Ghirlanda et al. 2012) and the bottom to the case of Γ = 10 2.5 . In both cases, the total flux around PeV energy are ∼ 10 −9 GeVcm −2 s −1 sr −1 . The flux produced by the untriggered GRBs is about a factor of two higher than that of the triggered ones.
In Fig. 5, we show the contribution to the diffuse neutrino flux by untriggered GRBs with different luminosity separately. As can be seen, GRBs in the intermediate luminosity range (10 51 − 10 53 ergs −1 ) contribute the largest fraction of the diffuse neutrino flux in both cases of the choice of the bulk Lorentz factor. These GRBs do not trigger the detectors either due to occurring at relatively high redshifts, and/or due to having high spectral peaks (e.g., E ob peak ≫ 300 KeV). Besides, the occulted bright GRBs also contribute a significant fraction to the total neutrino flux.

Comparison with two PeV neutrinos
The IceCube Collaboration recently reported the detection of two cascade neutrinos in the range 1-10 PeV (Ishihara 2012), which are likely to be electron (or anti-electron) neutrinos. Several scenarios have been proposed to explain the origin of these two neutrinos (Barger et al. 2012;Baerwald et al. 2012;Bhattacharya et al. 2012;Roulet et al. 2012;Cholis & Hooper 2012). Since the energies of the reported two neutrinos are close to the peak energy of the typical GRB neutrino spectrum, we here study the possibility that they originate from diffuse GRB neutrinos. In studying the possible GRB origin, one must assure that the stacked neutrino flux of those 215 triggered GRBs observed by IceCube do not exceed the IC40+59 limit. The expected number of diffuse neutrinos from GRBs in the energy range 1-10PeV can be obtained by where dnν e dǫν e diff is the number spectrum of diffuse electron neutrinos (in unit of cm −2 s −1 sr −1 eV −1 ) and S(ǫ νe ) is the IceCube exposure of electron neutrinos at ultra-high energies in 2 years (2010-2012) (Ishihara 2012), which includes the contribution by the Glashow resonance. Notably, since the analysis approach for diffuse cascade neutrinos is different from that for upgoing neutrinos from triggered GRBs, the IceCube effective area is significantly different. Based on the total diffuse neutrino flux we obtained above, we get N diff ≈ 0.1 for the case of Γ = 29.8E 0.51 iso,52 and N diff ≈ 0.2 for the case of Γ = 10 2.5 (η p = 10 is assumed). Since the non-detection of GRBs by IC40+59 implies η p < 20 (see §3.2), we have N diff < 0.2 or 0.4 respectively, which is insufficient to account for the reported two neutrinos. There is still a small possibility that these two neutrinos arise from, for instance, some very strong GRBs that happen to be occulted by the Earth and hence do not trigger the detectors. In addition, strong statistical fluctuation can also possibly lead to detection of one or two events.

Effect of variation in luminosity function and redshift distribution
There is significant variation in the luminosity function φ(L) and the rate distribution with redshift ρ(z) in the literature(e.g. Firmani et al. 2004, G07, L07, W10, Zitouni et al. 2008Cao et al. 2011). The variation may be caused by different approaches in obtaining the luminosity function and different spectral properties as well as trigger thresholds being used in the calculation.
Under the requirement that the total neutrino fluence from the triggered GRBs do not exceed the upper limit placed by IceCube, the possibility that GRBs are responsible for the two events increases if the fraction of neutrino flux from the untriggered GRBs increases. This requires more dim and/or distant GRBs to take place. Therefore, luminosity functions with steep slopes or rate distribution with higher rate at high redshift are favorable for producing more dim, untriggered GRBs. As the slopes in the luminosity function of L07 below and above the break luminosity L b are shallower than that in W10, adopting this luminosity function will lead to a smaller fraction of untriggered GRBs. However, the rate redshift distribution used in L07 suggests higher rate of GRBs at high redshift, which, on the contrary, will lead to a higher fraction of untriggered GRBs. We estimate that the neutrino flux produced by untriggered GRBs is about 90% of that of triggered GRBs for the luminosity function and rate distribution in L07. For the luminosity function and rate distribution in G07, this factor increases to 2.5 because of the steep slope in the luminosity function of G07 and higher GRB rate at high redshift. Even considering the variation in the luminosity function and rate distribution in the literature, the diffuse neutrino flux from GRBs may be still insufficient to account for the two neutrinos.

Discussions
Other possible sources of dim, untriggered GRBs are low-luminosity GRBs (LLGRBs) and Population III (Pop III) GRBs. LLGRBs are GRBs with luminosity 10 49 ergs −1 , which constitute a distinct population from the normal GRBs(e.g. Soderberg et al. 2006;Liang et al. 2007;Dai 2009;Virgili et al. 2009). Although only several cases are observed within ∼ 200Mpc so far, which is mainly due to their low luminosity, the intrinsic local event rate of LLGRBs is suggested to be about two orders of magnitude higher than that of the normal GRBs(e.g. Liang et al. 2007;Dai 2009;Virgili et al. 2009). Thus, the large local population could compensate for the low energy in LLGRBs, and lead to a significant amount of neutrino flux (e.g. Gupta & Zhang 2007;Murase et al. 2008;Liu et al. 2011). In Liu et al. (2011), we calculated the diffuse neutrino flux from LLGRBs for different LLGRB luminosity functions and found that the flux can reach a level of ∼ 10 −9 GeVcm −2 s −1 sr −1 or higher around PeV energies, depending on the parameters of LLGRBs that are used, such as the choice of Γ. We also found that most LLGRBs are untriggered and the energy budget of gamma rays in untriggered LLGRBs can be several times larger than that in the triggered ones. Thus more than 0.2-0.4 electron neutrinos in the range 1-10 PeV could come from LLGRBs in principle. However, since the origin of LLGRBs are not clear (e.g. Wang et al. 2007) and the properties of LLGRBs have large uncertainties, we should be cautious about this estimate.
Pop III GRBs are thought to arise from the death of the first stars that locate at high redshift (z 10). It is suggested that the their luminosity is typically a few times 10 52 ergs −1 and their duration is 10 4 s (Mészáros & Rees 2010;Komissarov & Barkov 2010;Suwa & Ioka 2011), so they may release energy up to 10 56−57 erg in one burst. At such large distances, their gamma-ray flux is below ∼ 10 −8 ergcm −2 s −1 , so they may not be able to trigger current GRB detectors. Thus, if Pop III GRBs can produce high energy neutrinos, these neutrinos will contribute to the diffuse background. It has been shown by Iocco et al. (2008); Gao et al. (2011) that the neutrino flux in 1-10 PeV range can reach ∼ 10 −9 GeVcm −2 s −1 sr −1 , or even ∼ 10 −8 GeVcm −2 s −1 sr −1 when some preferred parameters are used, e.g., with large progenitor mass and/or high interstellar medium density. However, since the flux peaks at ∼ 10 17 eV, one would expect even stronger flux at ∼ 10 17 eV, which is inconsistent with the non-detection by IceCube at such energies. Moreover, the rate of Pop III GRBs is quite uncertain. The Pop III GRB rate adopted in Gao et al. (2011) is from Bromm & Loeb (2006), which obtains a much higher rate than that in de Souza et al. (2011) and Campisi et al. (2011). So whether Pop III GRBs are responsible for the two neutrinos remains to be studied.
Finally we note that we only consider the neutrino emission under the standard internal shock model. For other models for the GRB prompt emission, such as the photosphere emission models (e.g. Rees & Mészáros 2005) or magnetically reconnection models (e.g. Zhang & Yan 2011), the efficiency for producing neutrinos f pγ is different (e.g. Wang & Dai 2009;Zhang & Kumar 2012), due to different dissipation radii R involved. However, the ratio between the flux from triggered GRBs and that from untriggered GRBs is not expected to change. Therefore, the diffuse neutrino flux can not be readily increased under the constraint that no neutrinos are detected from from triggered GRBs.

Conclusions
In this paper, we studied the diffuse neutrino emission from GRBs in light of the recent report that two PeV cascade events are detected by IceCube. We first generate a GRB sample that reproduces the realistic properties of the observed GRBs by Fermi/GBM. An accompanying sample of untrigged GRBs is generated simultaneously. Then we calculated the neutrino flux from the triggered and the untriggered GRBs in the sample within the standard internal shock scenario. The advantage of this approach is that the spectral properties (i.e. the spectral peak and indices) of every individual GRBs can be taken into account properly in the calculation. We found that the neutrino flux from the untriggered GRBs is about a factor of two larger than that of the triggered GRBs. Under the constraint of the non-detection of any upgoing muon neutrinos from 215 triggered GRBs in the IC40+59 combined analysis, the total diffuse neutrinos flux (single flavor) can reach a level of ∼ 10 −9 GeVcm −2 s −1 sr −1 at most. Such a flux can only give rise to 0.2 − 0.4 cascade neutrino events in 1-10 PeV for IceCube 2-years exposure. Low-luminosity GRBs and/or Pop III GRBs can also contribute to the diffuse neutrino flux, but large uncertainties in their properties prevents us from drawing a firm conclusion.  Since about 35% GRBs with flux above the trigger threshold are ascribed to untriggered GRBs due to occultation by the Earth and the South Atlantic Anomaly passages, the cumulative fluence of untriggered GRBs is non-zero below the Fermi/GBM trigger threshold 0.74 phcm −2 s −1 . Fig. 3.-The aggregated neutrino fluence spectrum (single flavor) from triggered GRBs in our generated sample. The blue and green solid lines represent the fluence obtained assuming Γ = 29.8E 0.51 iso,52 (Ghirlanda et al. 2012) and the benchmark value Γ = 10 2.5 respectively. The corresponding dashed lines represent the 90% upper limits from IC40+59. In the calculation, we assumed that the proton spectrum is a power law with index of −2 and η p = 10. iso,52 is used, while in the bottom panel Γ is fixed at 10 2.5 . We assumed that the proton spectrum is a power law with index of −2 and η p = 10.