Late-time acceleration and structure formation in interacting $\alpha$-attractor dark energy models

We investigate the cosmological dynamics of interacting dark energy within the framework of $\alpha$-attractor models. Specifically, we analyze the associated autonomous system, focusing on its fixed points that represent dark energy and scaling solutions, along with their stability conditions. We employ center manifold theory to address cases where some fixed points display eigenvalues with zero and negative real parts. The model reveals attractors describing dark energy, enabling a smooth transition from the radiation-dominated era to the matter-dominated era, and ultimately into the dark-energy-dominated phase. Additionally, we identify a scaling matter solution capable of modifying the growth rate of matter perturbations during the matter-dominated epoch. Consequently, we study the evolution of matter perturbations by obtaining both analytical and numerical solutions to the density contrast evolution equation. Based on these results, we compute numerical solutions for the weighted growth rate $f\sigma_{8}$, indicating that interacting $\alpha$-attractor dark energy models may provide a better fit to structure formation data than the standard $\Lambda$CDM scenario.