The two-dimensional and three-dimensional relations in the plateau emission in multi-wavelengths

Gamma-Ray Bursts (GRBs) are interesting objects for testing the emission models in highly energetic regimes and are very promising standardizable candles, given their observability at high redshift (up to $z=9.4$) that allows the extension of the Hubble diagram much further the limit of Supernovae Ia (SNe Ia), the most distant one being at $z=2.26$. In this study, we demonstrate that the fundamental plane relation involving the prompt peak luminosity in X-rays, the X-rays plateau-end luminosity, and the plateau-end rest-frame time is not only a robust benchmark for testing GRB emission models like the magnetar but also a promising avenue for high-$z$ cosmology exploration. First, we discuss the connection between the magnetar model and the GRB afterglow correlations. Second, through the simulation of GRBs, we count how many years are needed to achieve the same precision of modern SNe Ia samples in the estimation of $\Omega_{M}$.