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

Abstract

The paper examines the cosmological dynamics of a $f\left(R\right)$ 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 $Q=3\alpha H\rho$ 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 $f\left(R\right)$ 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.