Constraining f(R) gravity model through Hubble Parametrization

In this work, we explore a modified theory of gravity by transitioning from standard General Relativity(GR) to an f(R) gravity framework wherein the Ricci scalar $R$ is replaced by a general function $f\left(R\right)=R+\alpha R^2$. By adopting a specific Hubble parameterization $H\left(z\right)=\frac{H_0}{\sqrt{2}}{\left(1+{(1+z)}^{2(1+\zeta )}\right)}^{\frac{1}{2}}$, where $H_0$ is the present value of Hubble parameter and $\zeta$ be the free model parameter. We investigate the dynamical evolution of the universe under this modified gravity scenario with quadratic equation of state(EoS), $p=\mu {\rho }^2-\rho$. The Raychaudhuri Equation is employed to analyze the focus of geodesics and provide insights into the expansion behavior of the model universe, allowing us to track deviations from the standard cosmological model. To assess the viability of our f(R) gravity model, we analyze 46 Hubble parameter observations using the Markov Chain Monte Carlo(MCMC) technique to constrain cosmological parameters. We further use the 1048 Pantheon dataset of Type Ia supernovae to enhance the statistical robustness and tighten constraints. The combined observational analysis supports the model as a viable alternative to the standard $\Lambda$CDM framework, particularly in explaining late-time cosmic acceleration. Notably the model exhibits deviations at higher redshifts that suggest new insights into cosmic evolution. The study also develops a neural network-based machine learning model to predict the Hubble parameter H(z) across various redshifts, facilitating data-driven insights into cosmic expansion.