Hardness ratio evolutionary curves of gamma-ray bursts expected by the curvature effect

We have investigated the gamma-ray bursts pulses assumed to arise from relativistically expending fireballs. We find, due to the curvature effect, the evolutionary curve of the raw hardness ratio (when the background count is included) varies significantly, depending on the hardness of a burst. For a soft burst, an upside-down pulse of its raw hardness ratio curve (HRC) would be observed; for a hard burst, its raw HRC shows a pulse-like profile with a sinkage in its decaying phase; for a very hard burst, the raw HRC possesses a pulse-like profile without a sinkage in its decaying phase. For a pulse-like raw HRC as shown in the case of the hard and very hard bursts, its peak would appear in advance of that of the corresponding light curve. We have studied here the HRC of GRB 920216, GRB 920830 and GRB 990816 in detail. The features of the raw HRC expected in the hard burst are observed in these bursts. A fit to the three bursts shows that the curvature effect alone could indeed account for the predicted characteristics of HRCs. In addition, we find that the observed hardness ratio tends to be harder at the beginning of the pulses than what the curvature effect could predict and be softer at the late time of the pulses. We believe this is an evidence showing the existence of intrinsic hard-to-soft radiation