Closure Relations of Gamma-Ray Bursts in High Energy Emission

The synchrotron external shock model predicts the evolution of the spectral ($\beta$) and temporal ($\alpha$) indices during the gamma-ray burst (GRB) afterglow for different environmental density profiles, electron spectral indices, electron cooling regimes, and regions of the spectrum. We study the relationship between $\alpha$ and $\beta$, the so-called "closure relations" with GRBs detected by \textit{Fermi} Large Area Telescope (\textit{Fermi}-LAT) from 2008 August to 2018 August. The spectral and temporal indices for the > 100 MeV emission from the \textit{Fermi}-LAT as determined in the Second Fermi-LAT Gamma-ray Burst Catalog (2FLGC) are used in this work. We select GRBs whose spectral and temporal indices are well constrained (58 long-duration GRBs and 1 short-duration GRBs) and classify each GRB into the best-matched relation. As a result, we found that a number of GRBs require a very small fraction of the total energy density contained in the magnetic field ($\epsilon_{B}$ $\lesssim$ 10$^{-7}$). The estimated mean and standard deviation of electron spectral index $\mathit{p}$ are 2.40 and 0.44, respectively. The GRBs satisfying a closure relation of the slow cooling tend to have a softer $\mathit{p}$ value compared to those of the fast cooling. Moreover, the Kolmogorov--Smirnov test of the two $\mathit{p}$ distributions from the fast and slow coolings rejects a hypothesis that the two distributions are drawn from the single reference distribution with a significance of 3.2 $\sigma$. Lastly, the uniform density medium is preferred over the medium that decreases like the inverse of distance squared for long-duration GRBs.