Gamma-ray burst prompt emission from the synchrotron radiation of relativistic electrons in a rapidly decaying magnetic field

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Synchrotron radiation from accelerated electrons above the photosphere of a relativistic ejecta is a natural candidate for the dominant radiative process for the prompt gamma-ray burst emission. There is, however, a tension between the predicted low-energy spectral index, $ in the fast cooling regime and observations. Radiating electrons have time to travel away from their acceleration site and may experience an evolving magnetic field. We study the impact of a decaying field on the synchrotron spectrum. We computed the radiation from electrons in a decaying magnetic field, including adiabatic cooling, synchrotron radiation, inverse Compton scatterings, and pair production. We explored the physical conditions in the co-moving frame of the emission region and focused on the fast cooling regime where the radiative timescale of electrons with a Lorentz factor $ m $ responsible for the peak of the emission, $t_ syn m )$, is much shorter than the dynamical timescale $t_ dyn We find that the effect of the magnetic field decay depends on its characteristic timescale $t_ B $: (i) for a slow decay with $t_ B 10\, t_ syn m )$, the effect is very weak and the spectral shape is mostly determined by the impact of the inverse Compton scatterings on the electron cooling, leading to $-3/2 -1$, and (ii) for a fast decay with $0.1 \,t_ syn m t_ B 10 \,t_ syn m )$, the magnetic field decay has a strong impact, leading naturally to the synchrotron marginally fast cooling regime, where alpha tends to $-2/3$, while the radiative efficiency remains high. The high-energy inverse Compton component is enhanced in this regime. (iii) For an even faster decay, the whole electron population is in the slow cooling regime. We conclude that efficient synchrotron radiation in a rapidly decaying magnetic field can reproduce low-energy photon indices ranging from $ to $-2/3$, which is in agreement with the measured value in the majority of gamma-ray burst spectra.