Relative spectral lag: an new redshift indicator to measure the cosmos with gamma-ray bursts

Using 64 ms count data for long gamma-ray bursts ($T_{90}>$2.6s), we analyze the quantity, relative spectral lag (RSL), which is defined as $\tau_{31}/FWHM_{(1)}$, where $\tau_{31}$ is the spectral lag between energy bands 1 and 3, and $FWHM_{(1)}$ denotes full width at half maximum of the pulse in energy channel 1. To get insights into features of the RSLs, we investigate in detail all the correlations between them and other parameters with a sub-sample including nine long bursts. The general cross-correlation technique is adopted to measure the lags between two different energy bands. We can derive the below conclusions. Firstly, the distribution of RSLs is normal and oncentrates on around the value of 0.1. Secondly, the RSLs are weakly correlated with $FWHM$, asymmetry, peak flux ($F_{p}$), peak energy ($E_{p}$) and spectral indexes ($\alpha$ and $\beta$), while they are uncorrelated with $\tau_{31}$, hardness-ratio ($HR_{31}$) and peak time ($t_m$). The final but important discovery is that redshift ($z$) and peak luminosity ($L_{p}$) are strongly correlated with the RSLs which can be measured easily and directly. We find that the RSL is a good redshift and peak luminosity estimator