Cosmological Implications of the Gong-Zhang Parameterization in Rastall Gravity: A Deep Learning and Observational Study

In this study, we have explored the cosmological dynamics of an isotropic, homogeneous universe in Rastall gravity. For this purpose, we use the parameterization of the EoS parameter in the form $\omega(z) = \frac{\omega_{0}}{(z+1)}$ to derive the explicit solution of the field equations in Rastall gravity. We constrained the cosmological parameters for the derived model by the Markov Chain Monte Carlo (MCMC) approach utilizing OHD, BAO, and Pantheon plus compilation of SN Ia datasets. We also constrained the model parameters using deep learning techniques and the CoLFI Python package. This paper introduces an innovative deep-learning approach for parameter inference. The deep learning method significantly surpasses the MCMC technique regarding optimal fit values, parameter uncertainties, and relationships among parameters. This conclusion is drawn from a comparative analysis of the two methodologies. Additionally, we determined the transition redshift $z_t = 0.941$, which signifies the shift in the Universe's model from an early deceleration phase to the present acceleration phase. The diagnosis of the model with diagnostic tools like statefinders, jerk parameter, and $O_m$ diagnostics are presented and analyzed. The validation of the model's energy conditions is also examined.