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In this work we study the distribution of temporal power-law decay indices,$\alpha$, in the Gamma Ray Burst (GRB) afterglow phase, fitted for $176$ GRBs(139 long GRBs, 12 short GRBs {\it with extended emission} and 25 X-Ray Flashes(XRFs)) with known redshifts. These indices are compared with the temporaldecay index, $\alpha_W$, derived with the light curve fitting using the\cite{willingale07} model. This model fitting yields similar distributions of$\alpha_W$ to the fitted $\alpha$, but for individual bursts a difference canbe significant. Analysis of ($\alpha$, $L_a$) distribution, where $L_a$ is thecharacteristic luminosity at the end of the plateau, reveals only a weakcorrelation of these quantities. However, we discovered a significant regulartrend when studying GRB $\alpha$ values along the \cite{dainotti2008}correlation between $L_a$ and the end time of the plateau emission in the restframe, $T_a^*$, hereafter LT correlation. We note a systematic variation of the$\alpha$ parameter distribution with luminosity for any selected $T_a^*$. Weanalyze this systematics with respect to the fitted LT correlation line,expecting that the presented trend may allow to constrain the GRB physicalmodels. We also attempted to use the derived correlation of $\alpha(T_a)$versus $L_a(T_a)$ to diminish the luminosity scatter related to the variationsof $\alpha$ along the LT distribution, a step forward in the effort ofstandardizing GRBs. A proposed toy model accounting for this systematicsapplied to the analyzed GRB distribution results in a slight increase of the LTcorrelation coefficient.

Study of GRB light curve decay indices in the afterglow phase