On the non-existence of a sharp cooling break in gamma-ray burst afterglow spectra

Although the widely used analytical afterglow model of gamma-ray bursts (GRBs) predicts a sharp cooling break ν _c in its afterglow spectrum, the GRB observations so far rarely show clear evidence for a cooling break in their spectra or a corresponding temporal break in their light curves. Employing a Lagrangian description of the blast wave, we conduct a sophisticated calculation of the afterglow emission. We precisely follow the cooling history of non-thermal electrons accelerated into each Lagrangian shell. We show that a detailed calculation of afterglow spectra does not in fact give rise to a sharp cooling break at ν _c. Instead, it displays a very mild and smooth transition, which occurs gradually over a few orders of magnitude in energy or frequency. The main source of this slow transition is that different mini shells have different evolutionary histories of the comoving magnetic field strength B, so that deriving the current value of ν _c of each mini shell requires an integration of its cooling rate over the time elapsed since its creation. We present the time evolution of optical and X-ray spectral indices to demonstrate the slow transition of spectral regimes and discuss the implications of our result in interpreting GRB afterglow data.