From Λ-CDM to extended models: Via equation of state parametrization in f(R) gravity

In this work, we reconstruct the gravitational action f ( R ) corresponding to various cosmological epochs within the framework of modified gravity. Utilizing the field equations of f ( R ) gravity, we develop analytical expressions for the gravitational Lagrangian under different equation-of-state (EoS) scenarios, including dust ( w = 0 ), radiation ( w = 1 3 ), stiff fluid ( w = 1 ), curvature-like matter ( w = − 1 3 ), and dark energy ( w = − 1 ). The theoretical formulations are calibrated using two independent datasets: the 55-point Observational Hubble Data (OHD) and the 1048-point Pantheon Supernovae Type Ia (SN Ia) compilation, leading to precise constraints on the cosmological parameters H 0 and Ω m . Our reconstructed f ( R ) functions reveal distinct redshift-dependent behaviours corresponding to each EoS, with the stiff fluid case exhibiting a particularly steep increase—reaching magnitudes on the order of 10 10 —and dominant for z ≳ 0, thus indicating its significance during the early Universe. In contrast, the w = − 1 scenario mimics the late-time acceleration consistent with the ΛCDM model but differs notably in amplitude. Our results demonstrate the capacity of f ( R ) models to replicate key features of cosmic expansion, including the transition from deceleration to acceleration, and suggest a geometric origin for dark energy effects, potentially obviating the need for additional scalar fields or exotic matter. Furthermore, all reconstructed models satisfy the standard viability conditions ( f ′( R ) > 0, f ″( R ) > 0) and naturally recover the Hu–Sawicki limit at low curvature and the Starobinsky-like R 2 behaviour at high curvature, reinforcing their theoretical consistency across epochs.