Beyond the equation of state: a second-order diagnostic for dynamical dark energy

The first-order continuity equations determine the evolution of the energy densities but depend only on the instantaneous value of the dark-energy equation-of-state parameter. Differentiating these equations with respect to e-fold time introduces the term ${\omega }_{\mathrm{D}\mathrm{E}}^{\mathrm{\prime }}$ explicitly, providing a second-order probe of dark-energy dynamics. Consequently, while information about the evolution of the equation of state is encoded in the full dynamical solution, it is not explicit in the first-order continuity equations evaluated at a given epoch. The second-order formulation, therefore, provides a complementary description in which the local evolution of the equation of state appears directly through the curvature of the density trajectory. For a two-fluid interacting dark-sector model with linear coupling $Q_{AB}=\alpha {\rho }_{A}H$, the resulting second-order equation defines a curvature diagnostic, $\mathcal{C}={\rho }_{DE}^{\mathrm{\prime }\mathrm{\prime }}\mathrm{/}{\rho }_{DE}$, whose leading contribution, in the cosmological-constant limit, is ${\alpha }^2$, while departures from ${\omega }_{DE}=-1$ generate corrections through both $\delta \omega =1+{\omega }_{DE}$ and the distinctive term $-3{\omega }_{DE}^{\mathrm{\prime }}$. Unlike first-order analyses, this contribution is independent of the interaction strength and directly identifies dynamical dark energy. Applying the diagnostic to a CPL model with parameters consistent with DESI constraints, we recover ${\omega }_{DE}^{\mathrm{\prime }}$ across the full redshift range for both weak and strong interactions. Noise propagation shows that the diagnostic is detectable with signal-to-noise ratio exceeding three for ${\sigma }_H\mathrm{/}H\lesssim 1.5\mathrm{\%}$, while the degeneracy between $\alpha$ and ${\omega }_{DE}^{\mathrm{\prime }}$ remains negligible for $\alpha \lesssim 0.1$. In the non-interacting limit, the formalism naturally recovers the Caldwell–Linder thawing/freezing classification and extends it to interacting dark-energy models.