Microphysical Parameter Variation in GRB Stratified Afterglows and Closure Relations: from sub-GeV to TeV Observations

Gamma-ray bursts (GRBs) are one of the most exciting sources that offer valuable opportunities for investigating the evolution of energy fraction given to magnetic fields and particles through microphysical parameters during relativistic shocks. The delayed onset of GeV-TeV radiation from bursts detected by the \textit{Fermi} Large Area Telescope (\textit{Fermi}-LAT) and Cherenkov Telescopes provide crucial information in favor of the external-shock model. Derivation of the closure relations (CRs) and the light curves in external shocks requires knowledge of GRB afterglow physics. In this manuscript, we derive the CRs and light curves in a stratified medium with variations of microphysical parameters of the synchrotron and SSC afterglow model radiated by an electron distribution with a hard and soft spectral index. Using Markov Chain Monte Carlo simulations, we apply the current model to investigate the evolution of the spectral and temporal indexes of those GRBs reported in the Second Gamma-ray Burst Catalog (2FLGC), which comprises 29 bursts with photon energies above 10 GeV and of those bursts (GRB 180720B, 190114C, 190829A and 221009A) with energetic photons above 100 GeV, which can hardly be modeled with the CRs of the standard synchrotron scenario. The analysis shows that i) the most likely afterglow model using synchrotron and SSC emission on the 2FLGC corresponds to the constant-density scenario, and ii) variations of spectral (temporal) index keeping the temporal (spectral) index constant could be associated with the evolution of microphysical parameters, as exhibited in GRB 190829A and GRB 221009A.