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

Synchrotron radiation from accelerated electrons above the photosphere of a relativistic ejecta is a natural candidate for the dominant process for the prompt GRB emission. There is however a tension between the predicted low-energy spectral index $\alpha=-3/2$ 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. To study the impact on the synchrotron spectrum, we compute the radiation from electrons in a decaying magnetic field, including adiabatic cooling, synchrotron radiation, IC scatterings and pair production. We explore the physical conditions in the comoving frame of the emission region and focus on the fast cooling regime where the radiative timescale of electrons with a Lorentz factor $\Gamma_{m}$ responsible for the peak of the emission, $t_{syn}(\Gamma_{m})$, is much shorter than the dynamical timescale $t_{dyn}$. We find that the effect depends on the decay characteristic timescale $t_{B}$: (i) for a slow decay with $t_{B}\gtrsim 10 t_{syn}(\Gamma_{m})$, the effect is very weak and the spectral shape is mostly determined by the impact of the IC scatterings on the electron cooling, leading to $-3/2\le \alpha\le -1$; (ii) for a fast decay with $0.1 t_{syn}(\Gamma_{m})\lesssim t_{B}\lesssim 10 t_{syn}(\Gamma_{m})$, 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 IC component is enhanced in this regime ; (iii) for an even faster decay, the whole electron population is slow cooling. We conclude that efficient synchrotron radiation in a rapidly decaying magnetic field can reproduce low-energy photon indices ranging from $\alpha=-3/2$ to $-2/3$, in agreement with the measured value in the majority of GRB spectra.