Exploring Advanced Curvature and Linear Matter Coupling with Its Impact on the Expanding Universe

This study extends the scalar curvature term [Formula: see text] within the framework of modified gravity, incorporating both the Ricci scalar [Formula: see text] and the energy-momentum tensor [Formula: see text]. We explore a specific formulation of [Formula: see text] gravity, defined as [Formula: see text], in the context of a spatially homogeneous and anisotropic Bianchi type-I cosmological model with a perfect fluid. Within this framework, we derive and analyze the modified field equations for both positive and negative values of [Formula: see text]. By utilizing observational datasets, we constrain the model parameters and visualize its cosmological implications. Our findings show that for negative [Formula: see text], the model remains stable and aligns with physical expectations. However, for positive [Formula: see text], anomalous behavior emerges in specific intervals. Furthermore, we validate our model through metric perturbation analysis and cosmological diagnostic tools like [Formula: see text] and [Formula: see text]. The combined analysis of EC and observational diagnostics confirms the physical viability and late-time accelerated expansion behavior of the model within the admissible parameter range.