Constrained evolution of effective equation of state parameter in non-linear $f(R,L_m)$ dark energy model: Insights from Bayesian analysis of cosmic chronometers and Pantheon samples

We conduct a Bayesian analysis of recent observational datasets, specifically the Cosmic Chronometers (CC) dataset and Pantheon samples, to investigate the evolution of the EoS parameter in dark energy models. Our study focused on the effective EoS parameter, which is described by the parametric form ${\omega }_{eff}=-\frac{1}{1+m(1+z{)}^n}$, where $m$ and $n$ are model parameters. This parametric form is applicable within the framework of $f(R,L_m)$ gravity, where $R$ represents the Ricci scalar and $L_m$ is the matter Lagrangian. Here, we examine a non-linear $f(R,L_m)$ model characterized by the functional form $f(R,L_m)=\frac{R}{2}+L_m^{\alpha }$, where $\alpha$ is the free parameter of the model. We examine the evolution of several cosmological parameters, including the effective EoS parameter ${\omega }_{eff}$, the deceleration parameter $q$, the density parameter $\rho$, the pressure $p$, and the statefinder parameters. Our analysis revealed that the constrained current value of the effective EoS parameter, ${\omega }_{eff}^0=-0.68\pm 0.06$ for both the CC and Pantheon datasets, points towards a quintessence phase. Moreover, at redshift $z=0$, the deceleration parameter, $q_0=-0.61_{-0.01}^{+0.01}$, indicates that the present Universe is undergoing accelerated expansion.