Researchers have developed a new second-order diagnostic tool to study the dynamic nature of dark energy. Unlike first-order analyses, which rely on the instantaneous value of the dark-energy equation-of-state parameter (ωDE), this new formulation explicitly incorporates the time derivative of ωDE (ω'DE), providing a more direct probe of its evolution.

The first-order formulation of the continuity equations, which describes the evolution of energy densities, does not explicitly reveal how the dark energy equation of state changes over time. By differentiating these equations with respect to e-fold time, ω'DE is introduced, allowing for a complementary description where the local evolution of the equation of state appears directly through the curvature of the dark energy density trajectory. This is particularly relevant in models where dark energy interacts with other components of the dark sector.

For a two-fluid interacting dark-sector model with linear coupling, the second-order equation defines a curvature diagnostic, C = ρ''DE / ρDE. In the cosmological-constant limit, the leading contribution to C is α², where α is the interaction strength. Departures from ωDE = -1 generate corrections through δω = 1 + ωDE and, crucially, through the distinctive term -3ω'DE. This latter term is independent of the interaction strength and directly identifies dynamical dark energy, unlike first-order analyses. Applying this diagnostic to a CPL model, consistent with DESI constraints, allows for the recovery of ω'DE across the full redshift range.

Noise propagation calculations indicate that this diagnostic is detectable with a signal-to-noise ratio exceeding three for σH/H ≲ 1.5%. Furthermore, the degeneracy between α and ω'DE remains negligible for α ≲ 0.1. In the non-interacting limit, the formalism naturally recovers the Caldwell-Linder thawing/freezing classification and extends it to interacting dark-energy models, opening new avenues for understanding the mysterious nature of dark energy.