Breaking the Dark Sector Degeneracy with Nonparametric Expansion--Growth Reconstruction

The dark energy equation of state (EoS) and a possible dark-sector interaction are degenerate at the level of background expansion: the same expansion history may be interpreted as time-varying dark energy, energy exchange with dark matter, or a mixture of both. We introduce a data-driven, nonparametric expansion--growth framework that breaks this degeneracy by simultaneously reconstructing the dark energy EoS and the dark-sector coupling. Allowing the interacting matter density to evolve freely, we incorporate the coupling into the linear matter perturbation equation and reconstruct the expansion and growth histories using Gaussian processes, with hyperparameters marginalized in a Bayesian treatment. Applying this method to Pantheon+, cosmic chronometers, BAO measurements including DESI DR2, and redshift-space-distortion data, we infer both $w_{\rm de}(z)$ and the interaction history over $0\lesssim z\lesssim 2$, without assuming either a parametric EoS or a prescribed interaction form. We find no statistically significant evidence for either a nonzero interaction or dark energy dynamics: the reconstructed coupling and dark energy EoS remain consistent with the $\Lambda$CDM limit. Our results establish an expansion--growth consistency test for coupled dark-sector physics and provide a model-independent route to distinguish genuine dark energy dynamics from effective dark-sector energy exchange.