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. In addition, we examine the validity of the energy conditions (EC) to assess the physical plausibility of the model across cosmic evolution. Our findings show that for [Formula: see text], the model remains stable and aligns with physical expectations. However, for other 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.