Cosmological evolution of interacting dark energy with a CPL equation of state

This paper examines interacting dark energy models within the Chevallier-Polarski-Linder (CPL) parametrization, emphasizing both theoretical structure and observational viability. Two commonly adopted interaction terms are considered: $Q = \beta H \rho_{de}$ and $Q = \beta H \rho_c$. We derive exact analytic solutions that describe how the dark sector evolves. These solutions involve incomplete gamma functions and reveal a non-trivial mathematical structure that is often missed in numerical analyses. We perform a Bayesian analysis using current cosmological observations, including the Hubble parameter (OHD), Type Ia supernovae (SNIa), baryon acoustic oscillations (BAO), and cosmic microwave background (CMB) data. Relative to the non-interacting CPL scenario, the interacting model with $Q = \beta H \rho_{de}$ yields a modestly improved fit, as indicated by the Akaike Information Criterion (AIC). However, the Bayesian Information Criterion (BIC) penalizes increased model complexity, leading to a continued preference for $\Lambda$CDM. In contrast, the interaction model that depends on dark matter density does not provide observational support. The preferred interacting scenario indicates that the dark energy equation of state evolves dynamically, transitioning from an effective phantom regime at high redshift to quintessence-like behavior at late times. Further analysis indicates the potential for a transient phase of cosmic acceleration in the future. These findings suggest that interacting dark energy models within the CPL framework enrich the standard cosmological model by introducing more diverse phenomenology while maintaining consistency with current observations.