Long-lived PeV–EeV neutrinos from gamma-ray burst blastwave

Long duration gamma-ray bursts are powerful sources that can accelerate particles to ultrahigh energies. Acceleration of protons in the forward shock of the highly relativistic gamma-ray burst (GRB) blastwave allows PeV-EeV neutrino production by photopion interactions of ultrahigh energy protons with x-ray to optical photons of the GRB afterglow emission. Four different blastwave evolution scenarios are considered: adiabatic and fully radiative blastwaves in a constant density circumburst medium and in a wind environment with the particle density in the wind decreasing inversely proportional to the square of the radius from the center of the burst. The duration of the neutrino flux depends on the evolution of the blastwave and can last up to a day in the case of an adiabatic blastwave in a constant density medium. Neutrino fluxes from the three other blastwave evolution scenarios are also calculated. Diffuse neutrino fluxes calculated using the observed rate of long-duration GRBs are consistent with the recent IceCube upper limit on the prompt GRB neutrino flux below PeV. The diffuse neutrino flux needed to explain the two neutrino events at PeV energies recently detected by IceCube can partially come from the presented GRB blastwave diffuse fluxes. Future observations by IceCube and upcoming huge radio Askaryan experiments will be able to probe the flux models presented here or constrain the GRB blastwave properties.

Figure 1

Opacity of $p\gamma$ interactions in the GRB blastwave for cosmic-ray protons that are accelerated by the forward shock. The solid and dashed curves correspond to the opacity calculated by the $\Delta (1232)$ resonance $p\gamma$ cross section and by the full $p\gamma$ cross section, respectively. Each panel corresponds to a particular blastwave evolution scenario. The pairs of solid and dashed curves are calculated, from left to right, at times $t=t_{\mathrm{dec}}$, $10t_{\mathrm{dec}}$, ${10}^2{t}_{\mathrm{dec}}$, ${10}^3{t}_{\mathrm{dec}}$, and ${10}^4{t}_{\mathrm{dec}}$ for the adiabatic cases (upper panels), and at times $t=t_{\mathrm{dec}}$, $3t_{\mathrm{dec}}$, $10t_{\mathrm{dec}}$, $30t_{\mathrm{dec}}$, and ${10}^2{t}_{\mathrm{dec}}$ for the radiative cases (lower panels). Temporal behavior of the break energies [Eq. (14)] and ${\tau }_{p\gamma }$ at the lower break energy [Eq. (15)] are indicated with labeled arrows. Approximate power-law behavior of different segments of the curves are also indicated. Here we have used $k=2.5$ in Eq. (11).

Figure 2

Neutrino fluxes from a GRB at redshift $z=1$ for four different blastwave models. The model parameters are the same as in Fig. 1 for each case: ${\Gamma }_0={10}^{2.8}$ and $n_0=1$ for the ISM ($t_{\mathrm{dec}}=10.5\mathrm{s}$); ${\Gamma }_0={10}^{2.6}$ and $A_{\star }=0.1$ for the wind ($t_{\mathrm{dec}}=11.6\mathrm{s}$); the common parameters are $E_k={10}^{55}\mathrm{erg}$, ${ϵ}_p=1$, ${ϵ}_e={ϵ}_B=0.1$, $\varphi =10$, $k=2.5$, and $d_L=2.047\times {10}^{28}\mathrm{cm}$. The dot-dashed lines are for ${\nu }_{\mu }$ fluxes from the ${\pi }^{+}$ decay, the solid lines are for ${\overline{\nu }}_{\mu }$ fluxes from the ${\mu }^{+}$ decay, and the dashed lines are for ${\nu }_e$ fluxes from the ${\mu }^{+}$ decay. Neutrino oscillation has not been taken into account for the plotted fluxes.

Figure 3

Diffuse ${\nu }_{\mu }+{\overline{\nu }}_{\mu }$ fluxes from GRB blastwave in four different evolution scenarios. The top and bottom lines of the same type, with shaded region in between, correspond to 100% and 10% of the GRBs having high bulk Lorentz factors. Also shown is the recent upper limit on the GRB internal shock $\nu$ flux model [8] from IceCube [9]. Neutrino oscillation in vacuum has been taken into account for the diffuse flux models, which leads to equal fluxes of ${\nu }_e+{\overline{\nu }}_e$ and ${\nu }_{\tau }+{\overline{\nu }}_{\tau }$ as the ${\nu }_{\mu }+{\overline{\nu }}_{\mu }$ flux plotted here.