Cosmological model with linear equation of state parameter in f(R, Lm) gravity

In this paper, we examine the universe's expansion in $f(R, L_{m})$ gravity for a particular form of $f(R, L_{m})=\frac{R}{2}+L_{m}^{n}$. The field equations for flat FLRW metric with matter Lagrangian $L_{m}=\rho$ are derive. Hubble parameter in terms of red-shift$(z)$ are derived using the linear form of Equation of State (EoS) parameter $\omega=w_{0}+w_{1}z$. By using Bayesian statistical techniques based on $\chi^{2}$-minimization technique, we have obtained the best fit values of the model parameters of this model for cosmic chronometer and supernovae Pantheon datasets. The evolution of the equation of state parameter$(\omega)$, energy density $(\rho)$, pressure $\mathit{(p)}$ cosmographic parameters, and the impact of the energy conditions with best-fit values of the model parameters are all thoroughly examined. We have also analyze Om diagnostic's behavior and determine the present age of universe for this model.