III. Interacting Dark Energy: Summary of Models, Pathologies, and Constraints

We present an overview of the main results from our two companion papers that are relevant for observational constraints on interacting dark energy (IDE) models. We provide analytical solutions for the dark matter and dark energy densities, $\rho_{\rm dm}$ and $\rho_{\rm de}$, as well as the normalized Hubble function $h(z)$, for eight IDE models. These include five linear IDE models, namely $Q=3H(\delta_{\rm dm} \rho_{\rm dm} + \delta_{\rm de} \rho_{\rm de})$ and four special cases: $Q=3H\delta(\rho_{\rm dm}+\rho_{\rm de})$, $Q=3H\delta(\rho_{\rm dm}-\rho_{\rm de})$, $Q=3H\delta \rho_{\rm dm}$, and $Q=3H\delta \rho_{\rm de}$, together with three non-linear IDE models: $Q=3H\delta \left( \tfrac{\rho_{\rm dm} \rho_{\rm de}}{\rho_{\rm dm}+\rho_{\rm de}} \right)$, $Q=3H\delta \left( \tfrac{\rho_{\rm dm}^2}{\rho_{\rm dm}+\rho_{\rm de}} \right)$, and $Q=3H\delta \left( \tfrac{\rho_{\rm de}^2}{\rho_{\rm dm}+\rho_{\rm de}} \right)$. For these eight models, we present conditions to avoid imaginary, undefined, and negative energy densities. In seven of the eight cases, negative densities arise if energy flows from DM to DE, implying a strong theoretical preference for energy transfer from DE to DM. We also provide conditions to avoid future big rip singularities and evaluate how each model addresses the coincidence problem in both the past and the future. Finally, we propose a set of approaches and simplifying assumptions that can be used when constraining IDE models, by defining regimes that restrict the parameter space according to the behavior researchers are willing to tolerate.