Abstract The cosmological history and evolution are examined for gravitational models that include interaction within the dark sector of the Universe. In particular, we consider dark energy to be described by a phantom scalar field, while dark matter ρ m is modeled as a pressureless ideal gas. We introduce an interaction term of the form Q = β ( t ) ρ m , where the function β ( t ) is taken to be proportional to ϕ ̇ , ϕ ̇ 2 H − 1 , H , or a constant parameter with dimensions of H 0 . For each of the four interacting models, we perform a detailed phase-space analysis by calculating the stationary points, which are then used to reconstruct the cosmological evolution. The use of compactified variables is essential for capturing the global structure of the phase space and for assessing the cosmological viability of the interacting models. The full analysis is carried out for the exponential potential V ϕ = V 0 e λ ϕ . In addition, the impact of alternative scalar field potential functions on the cosmological dynamics is also investigated.
Paliathanasis et al. (Thu,) studied this question.