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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

We investigate the cosmological constraints on the Variable Chaplygin gas model from the latest observational data: SCP Union 2.1 compilation dataset of Type Ia supernovae (SNe Ia), Pantheon sample of SNe Ia, Platinum Sample of Gamma Ray Bursts (GRB) and GWTC-3 of gravitational wave merger events. Variable Chaplygin gas is a model of interacting dark matter and dark energy, which interpolates from a dust-dominated era to a quintessence-dominated era. The Variable Chaplygin gas model is shown to be compatible with Type Ia Supernovae and gravitational merger data. We have obtained tighter constraints on cosmological parameters $B_s$ and $n$, using the Pantheon sample. By using the Markov chain Monte Carlo (MCMC) method on the Pantheon sample, we obtain $B_s$=0.108 $\pm$ 0.034, n=1.157 $\pm$ 0.513 and $H_0$=70.020 $\pm$ 0.407, for GRBs, we obtain $B_s$=0.20 $\pm$ 0.11, n=1.45 $\pm$ 1.40 and $H_0$=70.41 $\pm$ 0.67} and on GWTC-3, we obtain $B_s$=0.130 $\pm$ 0.076, n=0.897 $\pm$ 1.182 and $H_0$=69.838 $\pm$ 3.007. The combined constraints from the above data sets are $B_s$=0.11 $\pm$ 0.03, n=1.14 $\pm$ 0.36 and $H_0$=70.34 $\pm$ 0.61, Comment: Accepted for publication by Astrophysics and Space Science. 22 pages, 11 figures and 4 tables