**3. VHE Properties of GRBs**

So far, the direct detection of VHE radiation from GRBs has only been possible for a limited number of cases, thanks to ground-based follow-up observations. The monitoring campaign carried out by the *Fermi* Large Area Telescope (*Fermi*-LAT) [44] led to the identification of a high-energy spectral component, coming in the form of power-law emission, which appears to be a common feature of bright GRBs and may even arise very early in the event [45]. Identifying the origin of this component and its relationship with the low energy portion of the spectrum has important implications on the interpretation of GRBs. If it arises as a high-energy extension of the synchrotron spectrum, we would expect a strong degree of correlation between different spectral bands, with direct implications on the energy of the radiating particles. If, on the contrary, it represents an independent contribution, its distribution among GRBs and its spectral characteristics may prove fundamental to understand its origin. The data collected so far, however, do not yet allow the drawing of a conclusive picture.

Due to its limited collecting area, the LAT cannot place strong constraints on the spectral features of short transients at *E* > 100 GeV. In the assumption of a composite synchrotron and inverse Compton spectrum, the upper limits placed above ∼30 GeV for the VHE detected GRB 190114C represented an invaluable reference to estimate the transition between the two regimes, though the possibility of alternative interpretations and the limited temporal information still leave room for open questions. In any case, the results of *Fermi*-LAT observations can be used as a starting point to estimate the possible extension of the GRB properties to the VHE domain and therefore evaluate their detection possibility with other instruments.

Assuming, for the sake of simplicity, that the temporal evolution of the spectrum is only limited to a scaling factor, without relevant spectral changes, we can express the high-energy spectrum of a GRB as a function of energy in the form of:

$$\frac{\mathrm{d}N(t)}{\mathrm{d}E} = N\_0(t) \left(\frac{E}{E\_0}\right)^{-a} \exp[-\tau(E, z)] \quad [\mathrm{photons} \,\mathrm{cm}^{-2} \,\mathrm{s}^{-1} \,\mathrm{GeV}^{-1}],\tag{4}$$

where *N*0(*t*) is the flux of photons per unit energy observed at time *t* and pivot energy *E*0, *α* is the spectral index, which is often within the range 1.5 6 *α* 6 3, with an average value close to 2, and *τ*(*E*, *z*) is the opacity due to pair production on EBL, given as a function of energy and redshift. The temporal evolution of the flux is typically well represented by a power-law, or a broken power-law, which can be written as:

$$N\_0(t) = \begin{cases} N\_{peak} \left( \frac{t - T\_0}{T\_{peak} - T\_0} \right) & \text{for} \quad T\_0 \le t < T\_{peak} \\\ N\_{peak} \left( \frac{t}{T\_{peak}} \right)^{-\gamma} & \text{for} \quad t > T\_{peak} \end{cases} \tag{5}$$

where we denoted with *T*<sup>0</sup> the trigger time, with *Tpeak* the time taken to achieve peak emission, with *Npeak* the maximum flux, and *γ* the temporal evolution index, which is often found to be 1 6 *γ* 6 2.

Using the second catalog of *Fermi*-LAT detected GRBs (2FLGC) [45], which provides measurements of the observed photon fluxes in the energy range between 100 MeV and 10 GeV, together with information on the spectral index and on the light-curve shape, for a sample of GRBs observed during 10 years of regular monitoring operations, we are able to apply Equation (4), with the inclusion of Equation (5), to estimate the expected

high-energy fluxes as a function of time, as illustrated for instance in Figure 2. In principle, we can extend this type of spectra to the VHE domain and, thus, obtain an estimate for the expected fluxes. In practice, this operation is not directly possible, due to the lack of a redshift measurement for most of the LAT detected GRBs, which implies an unknown EBL opacity in Equation (4). Although the effects of EBL are generally negligible for the observed LAT band, they become quickly very important at higher energies, with a typical EBL opacity horizon set by *τ* = 1 for *z* ≈ 1 already at *E* = 100 GeV [41]. For this reason, we combined the spectral and temporal fits, which we obtained from the LAT data, with a set of simulations, aiming at estimating the effects of EBL opacity on the VHE extension of the GRBs that resulted in the observed LAT fluxes.

**Figure 2.** Comparison between the *Fermi*-LAT light curve of GRB 130427A and the model based on Equations (4) and (5). The vertical blue dashed lines mark the temporal window of the LAT signal, the red horizontal line is the average energy flux collected during the emission, the green continuous line is the 2FLGC broken power-law fit to the data, while the blue continuous line is a model using the light curve of Equation (5).

The approach that we adopted in our simulations was to extract the fluences of all the LAT detected GRBs, reported in 2FLGC, and to assign a set of 1000 random redshift values to each GRB without an available redshift measurement. The result of this process is the production of 1000 random GRB redshift distributions, corresponding to an equal number of random luminosity distributions, all of which yield the observed 2FLGC population. As shown in Figure 3, the simulation set provides a distribution of GRB luminosities which is in good agreement with the one followed by the 2FLGC GRBs with a measured redshift. The combination of all the different simulations, therefore, can be used to estimate the likelihood that a GRB with measured spectral and temporal characteristics is associated with a specific redshift range (see [46] for more details on the simulation process).

**Figure 3.** Comparison between the luminosity estimated in the LAT band 0.1 GeV 6 *E* 6 10 GeV for the 2FLGC GRBs with a redshift measurement (large red dots) and the luminosity of the 2FLGC GRBs with 1000 randomly distributed redshifts (small blue points).
