Luminosity correlations for gamma-ray bursts and implications for their prompt and afterglow emission mechanisms

We present the relation between the ($z-$ and $k-$corrected) spectral lags, $\tau$, for the standard Swift energy bands 50-100 keV and 100-200 keV and the peak isotropic luminosity, $L_{\mathrm{i}\mathrm{s}\mathrm{o}}$ (a relation reported first by Norris et al.), for a subset of 12 long Swift GRBs taken from a recent study of this relation by Ukwatta et al. The chosen GRBs are also a subset of the Dainotti et al. sample, a set of Swift GRBs of known redshift, employed in establishing a relation between the (GRB frame) luminosity, $L_X$, of the shallow (or constant) flux portion of the typical XRT GRB-afterglow light curve and the (GRB frame) time of transition to the normal decay rate, $T_{\mathrm{b}\mathrm{r}\mathrm{k}}$. We also present the $L_X-T_{\mathrm{b}\mathrm{r}\mathrm{k}}$ relation using only the bursts common in the two samples. The two relations exhibit a significant degree of correlation ($\rho =-0.65$ for the $L_{\mathrm{i}\mathrm{s}\mathrm{o}}-\tau$ and $\rho =-0.88$ for the $L_X-T_{\mathrm{b}\mathrm{r}\mathrm{k}}$ relation) and have surprisingly similar best-fit power law indices ($-1.19 \pm 0.17$for$L_{$\mathrm{i}\mathrm{s}\mathrm{o}$}-\tau$and$-1.10 \pm 0.03$for$L_{X} - T_{$\mathrm{b}\mathrm{r}\mathrm{k}$}$)\mathrm{.}Evenmoresurprisingly,wenotedthatalthough$\tau$and$T_{$\mathrm{b}\mathrm{r}\mathrm{k}$}$ represent different GRB time variables, it appears that the first relation ($L_{\mathrm{i}\mathrm{s}\mathrm{o}}-\tau$) extrapolates into the second one for timescales $\tau \simeq T_{\mathrm{b}\mathrm{r}\mathrm{k}}$. This fact suggests that these two relations have a common origin, which we conjecture to be kinematic. This relation adds to the recently discovered relations between properties of the prompt and afterglow GRB phases, indicating a much more intimate relation between these two phases than hitherto considered.