Joint Analysis of Constraints on f(R) Parametrization from Recent Cosmological Observations

In this study, we present constraints on the parameters of three well-known $f(R)$ gravity models, viz. (i) Hu-Sawicki, (ii) Starobinsky, and (iii) ArcTanh by using a joint analysis of recent cosmological observations. We perform analytical approximations for the Hubble parameter, $H(z)$, and cosmological distances in terms of the Hubble constant $(H_0)$, matter density $(\Omega_{m0})$, and a deviation parameter $b$ for each model. {Our analysis combines early and late-universe cosmological data from five cosmological observations:} (a) Hubble parameter measurements (Cosmic Chronometers), (b) Type Ia Supernovae (Union 3.0), (c) Baryon Acoustic Oscillations (DESI-2025), (d) Gamma-Ray Bursts (GRBs) and (e) Cosmic Microwave Background (CMB). We first optimize the models using each dataset independently, and subsequently, we perform a comprehensive joint analysis combining all four datasets. Our results show that the Hu-Sawicki and ArcTanh models do not deviate significantly from the $\Lambda$CDM model at 95% confidence level for individual datasets and remain consistent at 99% confidence level in the joint analysis. In contrast, the Starobinsky model shows a strong deviation and appears as a viable alternative to $\Lambda$CDM. We also constrain the transition redshift parameter ($z_t$), and check that the obtained value agrees with the values inferred from both early-time measurement (Planck) and late-time data from Type Ia Supernovae. These results support the potential support of $f(R)$ gravity to explain the late-time cosmic acceleration effectively. Finally, a statistical model comparison using $\chi^2_{\text{min}}$, AIC, and BIC indicates that all three $f(R)$ models are favored over $\Lambda$CDM, with the Starobinsky model receiving very strong support.