Cooling Regimes of Nonthermal Electrons: The Slow, the Fast, and the Very Fast

In many astrophysical systems, nonthermal radiation is emitted by accelerated charged particles, whose cooling rates play a crucial role in determining their luminosity and spectral properties. We consider synchrotron radiation from a power-law energy distribution of relativistic electrons and provide a comprehensive framework for understanding particle cooling regimes, focusing on the fast cooling (FC) and very fast cooling (VFC) regimes. In FC, all electrons cool significantly on the dynamical time but remain relativistic. In VFC, all electrons cool to nonrelativistic speeds much faster than the dynamical time. The VFC regime is realized only in highly relativistic sources, with no Newtonian counterpart. It is highly relevant for the prompt phase of gamma-ray bursts (GRBs), where it can naturally occur regardless of the dissipation mechanism and with weak dependence on the degree of magnetization. We find that the FC and VFC regimes are satisfied in a broad range of parameters, under which bright optical emission can be observed during the prompt GRB phase. The brightest optical emission corresponds to an unbroken power-law spectrum extending from the sub-MeV peak of the bolometric luminosity to optical/UV frequencies. Additionally, we propose that VFC in the Thomson-thick regime is necessary for producing a bright and narrow electron-positron annihilation line, as was observed in the B.O.A.T. event, GRB 221009A.