Closure relations during the plateau emission of Swift GRBs and the fundamental plane

The Neil Gehrels Swift observatory observe Gamma-Ray bursts (GRBs) plateaus in X-rays. We test the reliability of the closure relations through the fireball model when dealing with the GRB plateau emission. We analyze 455 X-ray lightcurves (LCs) collected by emph{Swift} from 2005 (January) until 2019 (August) for which the redshift is both known and unknown using the phenomenological Willingale 2007 model. Using these fits, we analyze the emission mechanisms and astrophysical environments of these GRBs through the closure relations within the time interval of the plateau emission. Finally, we test the 3D fundamental plane relation (Dainotti relation) which connects the prompt peak luminosity, the time at the end of the plateau (rest-frame), and the luminosity at that time, on the GRBs with redshift, concerning groups determined by the closure relations. This allows us to check if the intrinsic scatter sigma_{int} of any of these groups is reduced compared to previous literature. The most fulfilled environments for the electron spectral distribution, p>2, are Wind Slow Cooling (SC) and ISM Slow Cooling for cases in which the parameter q, which indicates the flatness of the plateau emission and accounts for the energy injection, is =0 and =0.5, respectively, both in the cases with known and unknown redshifts. We also find that for the sGRBs All ISM Environments with $q=0$ have the smallest sigma_{int}=0.04 pm 0.15 in terms of the fundamental plane relation holding a probability of occurring by chance of p=0.005. We have shown that the majority of GRBs presenting the plateau emission fulfil the closure relations, including the energy injection, with a particular preference for the Wind SC environment. The subsample of GRBs that fulfil given relations can be used as possible standard candles and can suggest a way to reduce the intrinsic scatter of these studied relationships.