Analysis of GRB Closure Relationship in Multi-wavelengths

Gamma-ray bursts (GRBs) are intense pulses of high-energy emission associated with massive stars' death or compact objects' coalescence. Their multi-wavelength observations help verify the reliability of the standard fireball model. We analyze 14 GRBs observed contemporaneously in gamma-rays by the \textit{Fermi} Large Area Telescope (LAT), in X-rays by the \textit{Swift} Telescope, and in the optical bands by \textit{Swift} and many ground-based telescopes. We study the correlation between the spectral and temporal indices using closure relations according to the synchrotron forward-shock model in the stratified medium ($n \propto r^{-k}$) with $k$ ranging from 0 to 2.5. We find that the model without energy injection is preferred over the one with energy injection in all the investigated wavelengths. In gamma-rays, we only explored the $\nu >$ max\{$\nu_c,\nu_m$\} (SC/FC) cooling condition (where $\nu_c$ and $\nu_m$ are the cooling and characteristic frequencies, namely the frequencies at the spectral break). In the X-ray and optical bands, we explored all the cooling conditions, including $\nu_m < \nu < \nu_c$ (SC), $\nu_c < \nu < \nu_m$ (FC), and SC/FC, and found a clear preference for SC for X-rays and SC/FC for optical. Within these cooling conditions, X-rays exhibit the highest rate of occurrence for the density profile with $k = 0$, while the optical band has the highest occurrence for $k$ = 2.5 when considering no energy injection. Although we can pinpoint a definite environment for some GRBs, we find degeneracies in other GRBs.