$E_{\mathrm{peak}}$-$\hat{I}\pm$ Correlation in Time Resolved GRB Spectra: A Bottom-Up Approach with Optically Thin Inverse Compton Scattering Model

Abstract

Gamma-ray bursts (GRBs) are the brightest explosions in the Universe, yet the origin of their emission remains uncertain. Time-resolved spectral analysis offers key insights into the evolution of spectral shapes, constraining both radiation mechanisms and emission-site microphysics. Observationally, GRB spectra are well described by the empirical Band function, characterized by the peak energy ($E_{\mathrm{peak}}$) and low-energy spectral index ($\hat{I}\pm$). We investigate the temporal evolution of spectra produced by optically thin inverse-Compton scattering (ICS) within a standard fireball jet framework, focusing on the scenarios that can produce the two commonly observed spectral evolution patterns: hard-to-soft evolution and intensity tracking, within a single emission pulse. The evolution is analysed using both Bayesian block and constant-fluence binning, with the observed spectrum modeled consistently using the Band function. Using this bottom-up approach, we find that optically thin ICS yields a positive $E_{\mathrm{peak}}$-$\hat{I}\pm$ correlation, with $\hat{I}\pm$ evolving from hard (Planck-like, $>+0.5$) to softer ($<-0.67$) values. Such hard $\hat{I}\pm$ values are inconsistent with standard synchrotron emission. This characteristic evolution in the $E_{\mathrm{peak}}$-$\hat{I}\pm$ plane, therefore, provides a diagnostic signature of optically thin ICS as the dominant radiation mechanism during the prompt phase of GRBs. Furthermore, this type of smooth evolution of $\hat{I}\pm$ within a single pulse does not require invoking a transition between different radiation mechanisms, unless additional observational evidence supports such a change.

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