PROMPT TeV NEUTRINOS FROM THE DISSIPATIVE PHOTOSPHERES OF GAMMA-RAY BURSTS

Although it has generally been accepted that the prompt gamma-ray emission of gamma-ray bursts (GRBs) results from internal dissipation, likely internal shocks, of a relativistic outflow (e.g., Paczyński & Xu 1994; Rees & Mészáros 1994), the dissipation site and the radiation mechanism for the gamma-ray emission are still largely unknown. Synchrotron and/or inverse-Compton scattering emission by shock-accelerated electrons in the optically thin region has been proposed as an efficient mechanism for the gamma-ray emission. However, this model does not satisfactorily account for a few observational facts, such as the low-energy spectral slopes that are steeper than synchrotron lower energy spectral indices (Preece et al. 2000; Lloyd et al. 2000), the clustering of peak energies, or the correlation between the burst's peak energy and luminosity (Amati et al. 2002). It becomes recognized that an additional thermal component may play a key role and could solve these problems (e.g., Pe'er et al. 2006; Ryde et al. 2006). It has also been pointed out that a hybrid model with both a thermal and a non-thermal component can describe the spectrum equally well as the Band function model (Band et al. 1993), but the former has a more physical meaning (Ryde 2005). Recently it was suggested that a strong quasi-thermal component could result from internal dissipation occurring in the optically thick inner parts of relativistic outflows (Rees & Mészáros 2005; Pe'er et al. 2006; Thompson et al. 2007). Sub-photospheric shock dissipation can increase the radiative efficiency of the outflow, significantly boosting the original thermal photospheric component so that it may well dominate the nonthermal component from optically thin shocks occurring outside the photosphere.

Neutrino emission from gamma-ray bursts has been predicted at different stages of the relativistic outflow, such as the precursor phase (e.g., Bahcall & Mészáros 2000; Mészáros & Waxman 2001; Razzaque et al. 2003a, 2003b; Razzaque et al. 2004; Ando & Beacom 2005; Horiuchi & Ando 2008; Koers & Wijers 2008), the prompt emission phase (e.g., Waxman & Bahcall 1997; Dermer & Atoyan 2003; Guetta et al. 2004; Murase & Nagataki 2006; Gupta & Zhang 2007; Murase et al. 2006) and the afterglow phase (e.g., Waxman & Bahcall 2001; Dai & Lu 2001; Dermer 2002; Li et al. 2002; Murase & Nagataki 2006; Murase 2007; Dermer 2007). Based on the broken power-law approximation for the spectrum of the prompt emission, presumably from optically thin internal shocks, a burst of PeV neutrinos, produced by photomeson production, was predicted to accompany the prompt gamma-ray emission if protons are present and also accelerated in the shocks (Waxman & Bahcall 1997). The neutrino emission from proton–proton (pp) collisions was generally thought to be negligible due to the lower collision opacity for optically thin internal shocks. However, as we show below, if some part of the prompt emission arises from internal shocks occurring in the optically thick inner part of the outflow, as indicated by the thermal emission, a lower energy (≲10 TeV) neutrino component may appear as a result of pp collisions.

Photosphere models have been widely discussed in relation to the prompt emission of GRBs (e.g., Thompson 1994; Ghisellini & Celotti 1999; Rees & Mészáros 2005; Thompson et al. 2007; Ioka et al. 2007). The potential advantage of photosphere models is that the peak energy can be stabilized, which is identified as the thermal or Comptonization thermal peak (see Ioka et al. 2007 and references therein). The photosphere radiation may also produce a large number of electron–positron pairs, which may lead to a pair photosphere beyond the baryon-related photosphere (e.g., Rees & Mészáros 2005), and may also enhance the radiative efficiency (Ioka et al. 2007). On the other hand, it is also suggested that the number of pairs produced does not exceed the baryon-related electrons by a factor larger than a few (Pe'er et al. 2006). For simplicity, we here only consider the dissipation below the baryon-related photosphere, which is more favorable for pp neutrino production.

Following Rees & Mészáros (2005) and Pe'er et al. (2006), we assume that during the early stage of prompt emission, internal shocks of the outflow occur at radii below the baryonic photosphere. Initially, the internal energy is released at the base of the outflow, r₀ ∼ αr_g = 2αGM/c², where α ≳ 1 and r_g is the Schwarzschild radius of a central object of mass M. The internal energy is then converted to the kinetic energy of the flow, whose bulk Lorentz factor grows as γ ∼ r up to a saturation radius at r_s ∼ r₀η, where $\eta =L_0/(\dot{M}c^2)$ is the initial dimensionless entropy, with L₀ and $\dot{M}$ being the total energy and mass outflow rates. Above the saturation radius, the observer-frame photospherical luminosity decreases as L_γ(r) = L₀(r/r_s)^−2/3 and the greater part of energy is in kinetic form, L_k ∼ L₀. If the dissipation is maintained all the way to the photosphere, it will lead to an effective luminosity L_γ ∼ _dL₀ and a temperature (Rees & Mészáros 2005)

$$\begin{equation} T_\gamma =\epsilon _d^{1/4}(r/r_s)^{-1/2}T_0=200 \epsilon _d^{1/4} r_{11}^{-1/2} \Gamma _2^{1/2} L_{0,52}^{1/4}\,{\rm keV}, \end{equation} \tag{ 1 }$$

where _d is the dissipation efficiency, T₀ is the initial temperature of the fireball outflow, and Γ is the bulk Lorentz factor of the outflow. The internal shock occurs at R ≃ 2Γr_s = 6 × 10¹⁰αΓ²₂(M/10 M_☉) cm, for which the optical depth to Thomson scattering by the baryon-related electrons is τ_T = σ_TL_k/(4πRΓ³m_pc³) = 120 L_k,52R⁻¹₁₁Γ⁻³₂, where σ_T is the Thomson scattering cross section and L_k is the kinetic energy luminosity. The photosphere is further out, at radius R_ph = 1.2 × 10¹³L_k,52Γ⁻³₂ cm. A detailed calculation taking into account electron/positron cooling and the Comptonization effect leads to a quasi-thermal emission which peaks at energy ∼300–500 keV for dissipation at a Thomson optical depth of τ_T ∼ 10–100 (Pe'er et al. 2006). This temperature is consistent with the observed peak energies of prompt gamma-ray emission of a majority of GRBs.

Assuming that a fraction of _B ≃ 0.1 of the shock internal energy is converted into magnetic fields, we have a magnetic field B' = 2.5 × 10^71/2_B,−1L^1/2_k,52R⁻¹₁₁Γ⁻¹₂ G. Protons accelerated by internal shocks are assumed to have a spectrum dn/dε_p ∼ ε^−p_p with p ≃ 2, as often assumed for non-relativistic or mildly relativistic shock acceleration. The maximum proton energy is set by comparing the acceleration timescale $t^{\prime }_{\rm acc}=\alpha \varepsilon ^{\prime }_p\big/(e B^{\prime } c)=4.4\times 10^{-12} \alpha \big(\frac{\varepsilon ^{\prime }_p}{1 \,{\rm GeV}}\big) \epsilon _{B,-1}^{-1/2}L_{k,52}^{-1/2} R_{11}\Gamma _2 \,{\rm s}$ with the energy-loss timescales. The synchrotron loss time is $t^{\prime }_{\rm syn}=6\pi m_p^4 c^3\big/\big(\sigma _{\rm T}m_e^2 {\varepsilon ^{\prime }_p} B^{\prime 2}\big)=10^{-4} \epsilon _{B,-1}^{-1}L_{k,52}^{-1} R_{11}^2\Gamma _2^2 \big(\frac{\varepsilon ^{\prime }_p}{10^{8} \,{\rm GeV}}\big)^{-1} \,{\rm s}$. Assuming that the sub-photosphere emission at the dissipation site peaks at ε_γ = 300 keV with a thermal-like spectrum, the number density of photons in the comoving frame is n'_γ = L_γ/(4πR²Γ²cε'_γ) = 5 × 10²¹L_γ,51R⁻²₁₁Γ⁻¹₂(ε_γ/300 keV)⁻¹ cm^-3. The pγ cooling time is approximately $t^{\prime }_{p\gamma }={1}/\big({\sigma _{p\gamma } n^{\prime }_{\gamma } c K_{p\gamma }}\big)=10^{-4}L_{\gamma,51}^{-1} R_{11}^2 \Gamma _2 \big(\frac{\varepsilon _\gamma }{300\, {\rm keV}}\big)\, {\rm s}$, where K_pγ ≃ 0.2 is the inelasticity and σ_pγ = 5 × 10⁻²⁸ cm² is the peak cross section at the Δ resonance. By comparison with the synchrotron loss time, it is found that the most effective cooling mechanism for protons is the pγ process for protons with energies above the pγ threshold, but below ∼10⁸ GeV. Equating t'_acc = t'_pγ, we obtain the maximum proton energy in the shock comoving frame

$$\begin{equation} \varepsilon ^{\prime }_{p, {\rm max}}=10^7 \alpha _1^{-1} \epsilon _{B,-1}^{1/2}L_{k,52}^{1/2}L_{\gamma,51}^{-1}R_{11}\left(\frac{\varepsilon _\gamma }{300 \,{\rm keV}}\right) \, {\rm GeV}. \end{equation} \tag{ 2 }$$

The shock-accelerated protons produce mesons via pp and pγ interactions. Since the meson multiplicity in pp interactions is about 1 for pions and 0.1 for kaons, neutrinos contributed by pion decay are dominant when the cooling effect of pions is not important, which is applicable to the low-energy pp neutrinos. Therefore we here consider only pion production in pp interactions. Pion production by pp interaction in the sub-photosphere dissipation is efficient since the cooling time in the shock comoving frame,

$$\begin{equation} t^{\prime }_{pp}=1/(\sigma _{pp} n^{\prime }_p c K_{pp})=0.008\, L_{k,52}^{-1}R_{11}^2\Gamma _2^2 \, {\rm s}, \end{equation} \tag{ 3 }$$

can be shorter than the shock dynamic time, t'_dyn = R/Γc = 0.03R₁₁Γ⁻¹₂ s, where σ_pp = 4 × 10⁻²⁶ cm² is the cross section for pp interactions, n'_p = L_k/(4πR²Γ²m_pc³) = 2 × 10¹⁷L_k,52R⁻²₁₁Γ⁻²₂ cm^-3 is the proton number density, and K_pp ≃ 0.5 is the inelasticity. Protons also cool through Bethe–Heitler interactions (pγ → pe⁺e⁻) and pγ interactions when the target photon energy seen by the protons is above the threshold energy for each interaction. Denoting by n(_γ)d_γ the number density of photons in the energy range _γ to _γ +  d_γ, the cooling time in the shock comoving frame for pγ and Bethe–Heitler cooling processes is given by

$$\begin{equation} t^{\prime }_{\lbrace p\gamma, BH\rbrace }=\frac{c}{2\Gamma _p^2} \int _{\epsilon _{th}}^\infty d\epsilon \sigma (\epsilon) K(\epsilon)\epsilon \int _{\epsilon /2\Gamma _p}^{\infty } dx x^{-2} n(x), \end{equation} \tag{ 4 }$$

where Γ_p = ε'_p/m_pc², σ and K are, respectively, the cross section and the inelasticity for the pγ (or Bethe–Heitler) process. As a rough estimate, the Bethe–Heitler cooling time is

$$\begin{equation} \frac{t^{\prime }_{\rm BH}}{t^{\prime }_{pp}}=0.5 \left(\frac{(28/9) \,{\rm ln}40-218/27}{(28/9) \,{\rm ln}2k-218/27 }\right)\left(\frac{L_{k,52}}{L_{\gamma, 51}}\right)\left(\frac{\varepsilon ^{\prime }_\gamma }{3 \,{\rm keV}}\right) \end{equation} \tag{ 5 }$$

when k ≡ ε'_pε'_γ/(m_pm_ec⁴) is a large value (a good approximation when k ≳ 10; Chodorowski et al. 1992), where ε'_γ is the thermal peak energy of photons in the comoving frame. So when the proton energy is larger than $\varepsilon ^{\prime (1)}_{p,b}=1500 \big(\frac{\varepsilon ^{\prime }_\gamma }{3 \,{\rm keV}}\big)^{-1}\,{\rm GeV}$, the Bethe–Heitler cooling dominates over the pp cooling. At even higher energies near the threshold for pγ interactions at $\varepsilon ^{\prime (2)}_{p,b}=6\times 10^4 \big(\frac{\varepsilon ^{\prime }_\gamma }{3\, {\rm keV}}\big)^{-1}\,{\rm GeV}$, pγ cooling becomes increasingly dominant.

We compare these three cooling timescales for protons in Figure 1 for representative parameters, using more accurate cross sections for the pγ and Bethe–Heitler processes in Equation (4). For the photopion production cross section, we take the Lorentzian form for the resonance peak (Mücke et al. 2000) plus a component contributed by multi-pion production at higher energies, while for the Bethe–Heitler process we use the cross section given by Chodorowski et al. (1992). The number density of photons used in the calculation has been assumed to have a blackbody distribution. The numerical result confirms that pp cooling and pγ cooling are dominant, respectively, at lowest and highest energies, while at intermediate energies, Bethe–Heitler cooling is dominant.

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Defining the total cooling time for protons as t'_p = 1/(t'⁻¹_pp + t'⁻¹_BH + t'⁻¹_pγ), the total energy loss fraction of protons is η_p = Min{t'_dyn/t'_p, 1}. The fractions of energy loss by pp and pγ processes are, respectively,

$$\begin{equation} \left\lbrace \begin{array}{@{}ll} \zeta _{pp}={t^{\prime }}^{-1}_{pp}\big/\big({t^{\prime }}^{-1}_{pp}+{t^{\prime }}^{-1}_{\rm BH}+{t^{\prime }}^{-1}_{p\gamma }\big)\\\ms \zeta _{p\gamma }={t^{\prime }}^{-1}_{p\gamma }\big/\big({t^{\prime }}^{-1}_{pp}+{t^{\prime }}^{-1}_{\rm BH}+{t^{\prime }}^{-1}_{p\gamma }\big). \end{array} \right. \end{equation} \tag{ 6 }$$

The cooling of secondary pions may also affect the neutrino production efficiency if they suffer from cooling before decaying to secondary products. The pions suffer from radiative cooling due to both synchrotron emission and inverse-Compton emission. The total radiative cooling time is t'_π,rad = 3m⁴_πc³/[4σ_Tm²_e'_πU'_B(1 + f_IC)] ≃ 0.002('_π/1 TeV)⁻¹⁻¹_B,−1L⁻¹_k,52R²₁₁Γ²₂ s, where U'_B is the energy density of the magnetic field in the shock region and f_IC ≲ 1 is the correction factor accounting for the inverse-Compton loss. The pions also cool due to collisions with protons (Ando & Beacom 2005). The cooling for this hadronic process is t'_π,had = '_π/(cσ_πpn'_pΔ'_π) = 0.006L⁻¹_k,52R²₁₁Γ²₂ s, where σ_πp = 5 × 10⁻²⁶ cm² is the cross section for meson–proton collisions and Δ'_π = 0.5'_π is the energy lost by the meson per collision. The suppression of neutrino emission due to cooling of pions can be obtained by comparing the cooling time t'_π,rad or t'_π,had with the lifetime of pions τ'_π = γ_πτ = 1.9 × 10⁻⁴('_π/1 TeV) s in the shock comoving frame, where γ_π and τ are the pion Lorentz factor and proper lifetime. This defines two critical energies for pions, above which the effect of radiative cooling or hadronic cooling starts to suppress the neutrino flux, i.e., '_π,rad = 3^−1/2_BL^−1/2_k,52R₁₁Γ₂ TeV and '_π,had = 30L⁻¹_k,52R²₁₁Γ²₂ TeV. The total cooling time of pions is t'_π,c = 1/(t'⁻¹_π,rad + t'⁻¹_π,had) and the total suppression factor on the neutrino flux due to pion cooling is (Razzaque et al. 2004)

$$\begin{equation} \zeta _{\pi }={\rm Min}\lbrace t^{\prime }_{\pi, c}/\tau ^{\prime }_{\pi }, 1\rbrace. \end{equation} \tag{ 7 }$$

The total energy emitted in neutrinos from pp or pγ processes per GRB is, respectively,

$$\begin{equation} {\varepsilon _\nu ^2}J_{\lbrace pp,p\gamma \rbrace }(\varepsilon _\nu)=\frac{1}{8} \frac{E_p \eta _{p}(\varepsilon _\nu) {\zeta _{\lbrace pp,p\gamma \rbrace }(\varepsilon _\nu)\zeta _{\pi }(\varepsilon _\nu)}}{{\rm ln}(\varepsilon ^{\prime }_{p,{\rm max}}/\varepsilon ^{\prime }_{p,{\rm min}})}, \end{equation} \tag{ 8 }$$

where E_p is the energy in accelerated protons in one burst during the dissipative photosphere phase, ε'_p,max and ε'_p,min are the maximum and minimum energies of acceleration protons. In the absence of pion cooling loss, the neutrinos produced by pion decay carry one-eighth of the energy lost by protons to pion production, since charged and neutral pions are produced with roughly equal probability and muon neutrinos carry roughly one-fourth of the pion energy in pion decay.¹ The mean pion energy is about 20% of the energy of the proton producing the pion, so the mean energy of neutrinos is ε_ν ≃ 0.05ε_p. Assuming that protons are efficiently accelerated in shocks with an energy density of U'_p = 10U'_γ, the number of TeV neutrinos from one GRB is about N_ν = 0.1(Φ_γ/10⁻⁴ erg cm^-2) for η_p ≃ 1, according to Equation (8). So only from very strong bursts with gamma-ray fluence Φ_γ ≳ 10⁻³ erg cm^-2, which are very rare events, can neutrinos from a single GRB be detected.

The aggregated muon neutrino flux from all GRBs is approximately given by

$$\begin{eqnarray} {\varepsilon _\nu ^2}\Phi _{\lbrace pp,p\gamma \rbrace }(\varepsilon _\nu)&\simeq& \left(\frac{c}{4\pi H_0}\right) {\varepsilon _\nu ^2}J_{\lbrace pp,p\gamma \rbrace }(\varepsilon _\nu) R_{\rm GRB}(0) f_z \nonumber\\ &=&1.5\times 10^{-9} E_{p, 53} \left(\frac{R_{\rm GRB}(0)}{1 {\rm Gpc^{-3} yr^{-1}}}\right)\left(\frac{f_z}{3}\right)\nonumber\\ &&\times \eta _{p} {\zeta _{\lbrace pp,p\gamma \rbrace }}\zeta _{\pi } \,{\rm GeV \,cm^{-2}\, s^{-1} sr^{-1}}, \end{eqnarray} \tag{ 9 }$$

where f_z is the correction factor for the contribution from high redshift sources and R_GRB(0) is the overall GRB rate at redshift z = 0. Assuming that the GRB rate traces the star-formation rate in the universe, the calculation gives f_z ≃ 3 (Waxman & Bahcall 1999). It is not clear how efficiently the protons are accelerated in GRB shocks. Assuming an optimistic case that protons are efficiently accelerated in shocks and that half of the kinetic energy dissipation occurs below the photosphere,² we take a mean value E_p = 1.5 × 10⁵³ erg for the isotropic equivalent energy in accelerated protons in one GRB during the dissipative photosphere phase, based on a typically used value L_k = 10⁵² erg s^-1 for the isotropic kinetic energy luminosity and a typical long GRB duration of ΔT = 30 s. The GRB rate³ at redshift z = 0 is taken to be R_GRB(0) = 1 Gpc^-3yr⁻¹ (Guetta et al. 2005; Liang et al. 2007). The isotropic luminosity is taken to be L_γ = 10⁵¹ erg.⁴ The energy-dependent neutrino fluxes contributed by pp and pγ interactions are plotted in Figure 2 for a set of representative parameters of the dissipative photosphere model and three different dissipation radii. If the kinetic energy is dissipated at a radius of R = 10¹¹ cm, the calculation (the red solid curves) shows that at energies below tens of TeV, the neutrino flux is dominated by a pp component. Taking the detection probability of P_νμ = 10⁻⁶(ε_ν/1 TeV) for TeV neutrinos (Gaisser et al. 1995), the expected flux of upward moving muons contributed by this pp component is about 8–10 events each year for a km³ neutrino detector, such as Icecube. We can also estimate the atmospheric neutrino background expectation in coincidence with these GRB sources, noting that the search for neutrinos accompanying GRBs requires that the neutrinos be coincident in both direction and time with gamma rays. Taking an average GRB duration of ≃30 s, an angular resolution of Icecube of ≃1° and an atmospheric neutrino background flux of ≃10⁻⁴ GeV cm^-2 s⁻¹ sr⁻¹ at 1 TeV (Ahrens et al. 2004), the atmospheric neutrino background expectation is ≃5 × 10⁻³ events from 500 GRBs (in one year). Such a low background in coincident with GRBs allows the claim of detection of TeV neutrinos from GRB sources. At energies from a few TeV to tens of TeV, the Bethe–Heilter cooling suppresses the pp cooling, resulting in a steepening at several TeV in the neutrino spectrum. At energies above the threshold for pγ interactions, the neutrino from pγ process is heavily suppressed due to the strong radiative cooling of secondary pions. For a larger dissipation radius at R = 10¹² cm (the blue dashed curves), the neutrino emission flux from pγ process is no longer suppressed and in this case both pp and pγ neutrino components could be detected by km³ detectors. We also calculated the neutrino flux, shown by the green dotted lines in Figure 2, for shock at the photosphere radius R_ph, assuming that the radiation spectrum is a broken power-law (due to Comptonization) peaking at ε_γ = 100 keV, with lower-energy and higher-energy photon indexes given by −1 and −2, respectively. In this case, the pp neutrino flux becomes small (may be marginally detectable), while the pγ neutrino flux spectrum is similar to the analytic result obtained by Waxman & Bahcall (1997), as expected for a broken power-law photon spectrum. Note that in one burst the shock dissipation could be continuous from small to large radii, as indicated by the larger variability timescales seen in GRBs. By comparing the three cases of different dissipation radii in Figure 2, one can see that as the shock radius increases, the neutrino emission from the pp component decreases, while the pγ component increases until it reaches the saturation level. The total pp neutrino flux from such continuous dissipation is thus contributed predominantly by the deepest internal shocks below the photosphere. In the whole neutrino spectrum, a "valley" is seen between the pp and pγ components of the spectrum, which may be a potential distinguishing feature of the sub-photosphere dissipation effect.

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Waxman & Bahcall (1997) as well as later works have studied neutrino emission from the photomeson process during the prompt internal shocks of GRBs, assuming that the radiation in the shock region has a broken power-law nonthermal spectrum. It was found that the neutrino emission peaks at energies above 100 TeV. Toward lower energies, the neutrino emission intensity decreases as ε²_νΦ_ν ∼ ε_ν (Waxman & Bahcall 1997). However, if internal shocks, especially at the early stage of the prompt emission, occur below the photosphere, a quasi-thermal spectrum will arise. In this Letter, we have discussed the neutrino emission associated with the dissipative photosphere that produces such prompt thermal emission. We find that pp interaction process becomes important for shock-accelerated protons and provides a new neutrino component, which dominates at energies below tens of TeV. The neutrino emission from photopion processes of protons interacting with sub-photosphere radiation could be significantly suppressed due to radiative cooling of secondary pions, when the dissipation radius is relatively small. Nevertheless, the total contribution by photopion processes will not be suppressed since the shock dissipation could be continuous and occur at large radii as well. Although TeV neutrinos may also be produced during the early precursor stage of a GRB, i.e., before the jet breaking out the progenitor star (e.g., Razzaque et al. 2004; Ando & Beacom 2005), we want to point out that the TeV neutrino component discussed here can be distinguished from these, because, in our case, the neutrino emission is associated in time with the prompt emission.

After this work was completed and later put onto the arXiv Web site (arXiv:0807.0290), we became aware that K. Murase was also working on the sub-photosphere neutrino independently (Murase 2008). X.Y.W. would like to thank P. Mészéros, S. Razzaque, K. Murase, E. Waxman, Z. Li, and K. Ioka for useful comments or discussions. This work is supported by the National Natural Science Foundation of China under grants 10221001, 10403002, and 10873009, and the National Basic Research Program of China (973 program) under grants No. 2007CB815404 and 2009CB824800.