Interpretations of gamma-ray burst spectroscopy I. Analytical and numerical study of spectral lags

We describe the strong spectral evolution that occurs during a gamma-ray burst (GRB) pulse and the means by which it can be analyzed. In particular, we discuss the change of the light curve as a function of energy and the spectral lag. Based on observed empirical correlations, an analytical model is constructed which is used to describe the pulse shape and quantize the spectral lags and their dependences on the spectral evolution parameters. Using this model, we find that the spectral lag depends mainly on the pulse-decay time-scale and that hard spectra (with large spectral power-law indices α) give the largest lags. Similarly, large initial peak-energies, E0, lead to large lags, except in the case of very soft spectra. The hardness ratio is found to depend only weakly on α and the hardness-intensity–correlation index, η. In particular, for low E0, it is practically independent, and is determined mainly by E0. The relation between the hardness ratio and the lags, for a certain E0 are described by power-laws, as α varies. These results are the consequences of the empirical description of the spectral evolution in pulses and can be used as a reference in analyses of observed pulses. We also discuss the expected signatures of a sample of hard spectral pulses (e.g. thermal or small pitch-angle synchrotron emission) versus soft spectral pulses (e.g. optically-thin synchrotron emission). Also the expected differences between a sample of low energetic bursts (such as X-ray flashes) and of high energetic bursts (classical bursts) are discussed.