Development of the spectrum of gamma-ray burst pulses influenced by the intrinsic spectral evolution and the curvature effect

The spectral evolution of gamma-ray burst pulses assumed to arise from the emission of fireballs is explored. It is found that due to the curvature effect, the integrated flux is well related to peak energy by a power law in the decaying phase of pulses, where the index is about 3, which does not depend on intrinsic emission and the Lorentz factor. The spectra of pulses in the decaying phase are slightly different from each other when different intrinsic spectral evolution patterns are considered, indicating that it is dominated by the curvature effect. In the rising phase, the integrated flux keeps increasing whilst the peak energy remains unchanged when the intrinsic emission bears an unchanged spectrum. Within this phase, the flux decreases with the increase of the peak energy for a hard-to-soft intrinsic spectrum, and for a soft-to-hard-to-soft intrinsic spectrum, the flux generally increases with the increase of the peak energy. An intrinsic soft-to-hard-to-soft spectral evolution within a co-moving pulse would give rise to a pulse-like evolutionary curve for the peak energy.