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Cosmological models and their corresponding parameters are widely debated because of the current discrepancy between the results of the Hubble constant, $H_{0}$, obtained by SNe Ia, and the Planck data from the Cosmic Microwave Background Radiation. Thus, considering high redshift probes like Gamma-Ray Bursts (GRBs) is a necessary step. However, using GRB correlations between their physical features to infer cosmological parameters is difficult because GRB luminosities span several orders of magnitude. In our work, we use a 3-dimensional relation between the peak prompt luminosity, the rest-frame time at the end of the X-ray plateau, and its corresponding luminosity in X-rays: the so-called 3D Dainotti fundamental plane relation. We correct this relation by considering the selection and evolutionary effects with a reliable statistical method, obtaining a lower central value for the intrinsic scatter, $\sigma_{int}=0.18 \pm 0.07$ (47.1 \%) compared to previous results, when we adopt a particular set of GRBs with well-defined morphological features, called the platinum sample. We have used the GRB fundamental plane relation alone with both Gaussian and uniform priors on cosmological parameters and in combination with SNe Ia and BAO measurements to infer cosmological parameters like $H_{0}$, the matter density in the universe ($\Omega_{M}$), and the dark energy parameter $w$ for a $w$CDM model. Our results are consistent with the parameters given by the $\Lambda$CDM model but with the advantage of using cosmological probes detected up to $z=5$, much larger than the one observed for the furthest SNe Ia., Comment: 43 pages, 29 figures, accepted for publication in Monthly Notices of the Royal Astronomical Society

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., Comment: 44 pages, 23 figures; Accepted to the PASJ to be published soon