Generalizing the CPL Parametrization through Dark Sector Interaction

We investigate a hierarchy of interacting dark energy (IDE) models featuring a non-gravitational coupling between dark matter and dark energy. Specifically, we examine scenarios where the background interaction kernel, $Q = 3H(\delta + \eta a)\rho_\mathrm{de}$, allows for both constant and dynamical coupling parameters. Adopting the Chevallier-Polarski-Linder parametrization for the dark energy equation of state, $w_\mathrm{de} = w_0 + w_a(1-a)$, we derive closed analytical expressions for the energy densities of dark matter and dark energy. Afterwards, we obtain observational constraints using joint combinations of DESI DR2 baryon acoustic oscillations, Pantheon$+$ Type Ia supernovae, and Planck$+$ACT compressed cosmic microwave background likelihoods. For constant coupling models, we find parametric deviations from $\Lambda$ ranging from $2.7\sigma$ to $2.9\sigma$; however, for interactions with dynamical couplings, these significances are reduced to $1.3\sigma$--$1.5\sigma$. Ultimately, our Bayesian model comparison reveals that no investigated IDE scenario is statistically preferred over the concordance $\Lambda$CDM model. These results highlight the necessity of reporting Bayesian evidence alongside conventional frequentist maximum-likelihood analyses to ensure robust cosmological claims concerning dark energy evolution and interaction.