Gamma-ray Burst Spectrum with a Time-dependent Injection Rate of High-energy Electrons

Although the physical origin of prompt emission in gamma-ray bursts (GRBs) remains inconclusive, previous studies have considered the synchrotron radiation of relativistic electrons as a promising mechanism. These works usually adopted a invariable injection rate of electrons ($Q$) which may be discordant with that in a Poynting-flux dominated jet. In a Poynting-flux dominated jet (e.g., ICMART model, Zhang & Yan 2011), the number of magnetic reconnections occurred simultaneously may grow rapidly with time and results in an increase of $Q$ with time. This paper is dedicated to study the synchrotron radiation spectrum in this scenario. It is found that the radiation spectrum would obviously get harder if an increasing $Q$ is adopted and a Band-like radiation spectrum can be obtained if the increase of $Q$ is fast enough. The latter is related to the fact that a bump-shape rather than a power-law spectrum appears in the low-energy regime of the obtained electron spectrum. This effect can strongly harden the low-energy radiation spectrum. It indicates that an increasing $Q$ can help to alleviate the "fast-cooling problem" of synchrotron radiation for GRBs. Our studies also reveal that a Poynting-flux dominated jet with a large emission radius, a small length of the magnetic reconnection region, or a low-minimum energy of injected electron would prefer to form a Band-like radiation spectrum. We suggest that the Band spectrum found in GRBs may be the synchrotron emission of the electrons with a bump-shape distribution in its low-energy regime.