Evolution of the universe with quintessence model in Rastall gravity

We investigate the universe's evolution within the framework of Rastall gravity, which is an extension of the standard $\Lambda$CDM model. Utilizing a linear parametrization of the Equation of State (EoS) in a Friedmann-Lema\^{\i}tre-Robertson-Walker (FLRW) background, we constrain the model parameters through analysis of cosmic chronometers (CC), Pantheon, Gold, Gamma Ray Burst (GRB), and Baryon Acoustic Oscillations (BAO) datasets, as well as their joint analysis, under $1\sigma$ and $2\sigma$ confidence levels, considering the Rastall parameter $\lambda$. The constrained parameters are then used to compare our model with the standard $\Lambda$CDM model. Our findings include a detailed examination of the model's physical interpretations and demonstrate the potential for an accelerating universe expansion in later times, aligning with the observed behavior of dark energy.