Exact Cosmological Models in Modified \(\boldsymbol{f(R,L_{m})}\) Gravity with Observational Constraints

This study is an investigation of exact cosmological models in modified \(f(R,L_{m})\) gravity with observational constraints, where \(R\) is the Ricci scalar, and \(L_{m}\) is the matter Lagrangian for a perfect fluid. We have obtained the field equations using a flat FLRW metric with matter Lagrangian \(L_{m}=-p\) and \(f(R,L_{m})=R/2+\alpha L_{m}^{n}-\beta\), where \(\alpha\), \(\beta\), \(n\) are positive parameters. We have solved the field equations for the scale factor \(a(t)\) with the equation of state (EoS) \(p=\omega\rho\), where \(p\) is the isotropic pressure and \(\rho\) is the energy density. We have obtained the scale factor \(a(t)=k_{0}[\sinh(k_{1}t+k_{2})]^{[2(n+\omega-n\omega]/[3n(1+\omega)]}\), where \(k_{1}=\frac{\sqrt{3\beta}}{2}\frac{n(1+\omega)}{n+\omega-n\omega}\), and \(k_{0}\), \(k_{2}\) are integration constants. Using this scale factor, we have analyzed various cosmological parameters \(\{H_{0},q_{0},j_{0},s_{0},t_{0}\}\) with observational constraints by applying the \(\chi^{2}\) test with four observational datasets \(H(z)\), Union 2.1, JLA and Bined datasets. Also, we have analyzed the Om diagnostic parameter.