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Although the widely-used analytical afterglow model of gamma-ray bursts(GRBs) predicts a sharp cooling break $\nu_c$ in its afterglow spectrum, theGRB observations so far rarely show clear evidence for a cooling break in theirspectra or its corresponding temporal break in their light curves. Employing aLagrangian description of the blast wave, we conduct a sophisticatedcalculation of the afterglow emission. We precisely follow the cooling historyof non-thermal electrons accelerated into each Lagrangian shell. We show that adetailed calculation of afterglow spectra does not in fact give rise to a sharpcooling break at $\nu_c$. Instead, it displays a very mild and smoothtransition, which occurs gradually over a few orders of magnitude in energy orfrequency. The main source of this slow transition is that differentmini-shells have different evolution histories of the comoving magnetic fieldstrength $B$, so that deriving the current value of $\nu_c$ of each mini-shellrequires an integration of its cooling rate over the time elapsed since itscreation. We present the time evolution of optical and X-ray spectral indicesto demonstrate the slow transition of spectral regimes, and discuss theimplications of our result in interpreting GRB afterglow data.

On the Non-existence of a Sharp Cooling Break in GRB Afterglow Spectra