Dynamics of FRW Cosmology with \(q = a - \frac{b}{H}\) in f(R, \({{L}_{m}}\)) Gravity

We examine the unknown nature of the source of accelerating expansion of universe in general relativity which leads many researchers to delve into the evolution of our universe in the framework of modified theory of gravity. Here, specially we investigate on Friedmann–Robertson–Walker (FRW) space time filled with a perfect fluid in \(f(R,{{L}_{m}})\) modified theory framework formulated by Harko and Lobo (2010), where \(R\) is the Ricci scalar curvature, \({{L}_{m}}\) is the ideal fluid’s Lagrangian. To accomplish this, we use a specific form of \(f(R,{{L}_{m}})\) gravity as \(f(R,{{L}_{m}}) = \frac{R}{2} + \alpha L_{m}^{n} - \beta \), where \(\alpha \), \(\beta \), and \(n\) are positive model parameters. Here we employ the parametrization of the deceleration parameter \(q = a - \frac{b}{H}\) to resolve the modified field equations in the framework of \(f(R,{{L}_{m}})\) gravity for Friedmann–Robertson–Walker (FRW) metric. The evolution of different cosmological parameters have been examined by means of their graphs. We have also obtained our model parameters with 46 points observational Hubble dataset. We have analysed Energy conditions, statefinder and \(Om(z)\) diagnostics. Finally we have estimated the present age of the universe. The proposed model behaves as a dark energy model and we come to know that it behaves as the SCDM model during early stage of the universe and \(\Lambda \)CDM model during the late time.