Dynamics of an interacting matter − f(R) model with expansion histories approximating ΛCDM

The paper examines the cosmological dynamics of a f ( R ) model that integrates interactions between matter and an effective fluid. It employs a dynamical system approach with three cosmographic parameters. The equilibrium state's physical characteristics and linear stability within the cosmological dynamical system , derived from the field equations, are examined, taking into account an interacting term Q = 3 α H ρ and the specific relationships among cosmological parameters as per ΛCDM cosmology. It is found that the solutions of the model, while imposing a global ΛCDM alike evolution, exhibit significant sensitivity to initial conditions when evaluating cosmological parameters, primarily due to the phase-space constraints imposed by the viability conditions for f ( R ) models. The presence of the interacting term alters most equilibrium points in the system and contributes to the cosmological evolution , imposing constraints on the parameter α of interaction strength. Additionally, by using Poincaré compactification, we conduct a global dynamics analysis for the spatially flat scenario, revealing all possible evolutions of the flat universe in the form of a Poincaré disk and a comprehensive dynamic flow chart. Several equilibrium points and the cosmological evolution within this model exhibit slight deviations from the non-interacting case, yet it successfully captures the universe's acceleration phase.