On the uniqueness of $\Lambda$CDM-like evolution for homogeneous and isotropic cosmology in General Relativity

We address the question of the uniqueness of spatially flat $\Lambda$CDM-like evolution for FLRW cosmologies in General Relativity, i.e. whether any model other than the spatially flat $\Lambda$CDM can give rise to the same type of scale factor evolution. Firstly, we elaborate on what we exactly imply by a $\Lambda$CDM-like evolution or kinematic/cosmographic degeneracy with the $\Lambda$CDM model, using the lessons from the statefinder diagnostic. Then, we consider two models with interaction in the dark sector: coupled fluid-fluid model and coupled quintessence model. We enforce the \emph{kinematic} degeneracy with the spatially flat $\Lambda$CDM model via the cosmographic condition $j=1$ ($j$ being the jerk parameter), which in turn fixes the function of the interaction term that is a priori unspecified. We argue that in General Relativity this cosmographic condition is consistent only with spatial flatness. Employing a dynamical system approach, we show that the spatially flat coupled fluid-fluid interacting models kinematically degenerate with $\Lambda$CDM must necessarily be based on a phantom fluid, whereas the set of physically viable spatially flat coupled quintessence models with power law or exponential potential kinematically degenerate to $\Lambda$CDM is of measure zero. Our analysis establishes that coupled fluid-fluid models with non-phantom fluids or coupled quintessence models with power law and exponential potential can never reproduce a cosmological evolution similar to that of the $\Lambda$CDM. The astrophysical consequences of our findings are qualitatively discussed in light of observational cosmological tensions.