From Linear ω-γ Correlation to Redshift-Dependent Dynamics: A Complete Phenomenological Framework and Testing Roadmap for Dark Energy

This perspective article proposes and systematically develops a phenomenological framework centered on the correlation between dark energy dynamics and cosmic structure growth. Building upon the foundational linear relation w(a) = -1 + η(γ(a) - 0.55), where w is the dark energy equation of state and γ is the structure growth index, we extend it to allow for redshift-dependent couplings and provide a complete roadmap for empirical testing. We establish its theoretical basis as an interacting dark energy-dark matter model that respects energy-momentum conservation. A key advancement is our complete parameterization w(z)=−1+η(z)[γ(z)− 0.55]+Δw_bg(z), which separates structure-dependent coupling from possible background evolution and allows for redshift-dependent interactions. This work introduces a novel, phenomenologically motivated piecewise parameterization for the coupling strength η(z), designed to capture potential variations across distinct cosmic epochs (z < 0.5, 0.5 ≤ z ≤ 1.5, z > 1.5) based on the history of structure formation.​ We provide a comprehensive testing roadmap using hierarchical Bayesian model comparison, detailing the specific observational data, analysis methods, and systematic error treatments required. Using Fisher matrix forecasts based on detailed survey specifications, we demonstrate that upcoming surveys (DESI, Euclid, Roman Space Telescope) will provide decisive tests, capable of detecting coupling strengths |η| ≳0.05 with strong evidence. This framework offers a unified approach to addressing both the Hubble and S8​ tensions while making distinctive, testable predictions that differentiate it from other proposed solutions. We conclude with specific recommendations for observational teams and theoretical directions for further development.