A Comparative Study of the Power-law Relationship between the Pulse width and Energy of Precursor and Main Burst

In gamma-ray burst prompt emission, there is still no consistent conclusion if the precursor and main burst share the same origin. In this paper, we try to study this issue based on the relationship between pulse width and energy of the precursor and main burst. We systematically search the light curve data observed by Swift/BAT and Fermi/GBM, and find 13 long bursts with well-structured precursors and main bursts. After fitting the precursor light curve of each different energy channel with the Norris function, we find that there is not only a power-law relationship between precursor width and energy, but also a power-law relationship between the ratio of the rising width to the decaying width and energy. By comparing the relationship between the precursors and the main burst pulses, we find that the distribution of the precursors and the relationship between the power-law indices are roughly the same as those of the main burst. In addition, it is found that the precursor width distribution as well as the upper limit of the pulse width ratio does not exceed 1 and both are asymmetric, which are also consistent with the main burst. These indicate that the precursor and the main burst are indistinguishable, and the precursor and the main burst may have the same physical origin.