Sub-Horizon and Quasi-Static Approximations in Interacting Dark Sector Models Involving Scalar Field Dark Energy

We investigate the validity of the sub-horizon approximation (SHA) and quasi-static approximation (QSA) for linear matter perturbations in interacting scalar-field dark-energy models. We consider both quintessence and phantom scalar fields and derive their full relativistic perturbation evolution to compare with the corresponding SHA and QSA solutions. We find that, while the SHA reproduces the full evolution with percent-level accuracy for non-interacting and weakly interacting models on sufficiently sub-horizon scales, its accuracy progressively deteriorates with increasing interaction strength and towards larger scales. Moreover, unlike in Λ CDM and non-interacting scalar-field models, applying the SHA to interacting models does not fully recover the standard Newtonian fluid equations. The QSA generally provides a closer description of the full relativistic evolution than the SHA over the scales considered. By comparing the scalar-field model with Λ CDM using the full relativistic equations and, separately, using the SHA system, we evaluated whether the SHA is capable of capturing the full extent of the physical differences introduced by the alternative model. We find that the difference between these two comparisons becomes increasingly important for stronger interactions and smaller wavenumbers. Finally, we demonstrate that the resulting differences can reach several percent on scales accessible to large-volume hydrodynamical simulations, implying that the validity of the SHA cannot be assumed solely from the sub-horizon condition. Our results highlight the need to benchmark the approximation over the relevant interaction parameter and wavenumber ranges before employing interacting-dark-energy models in large-volume structure-formation simulations and precision comparisons with large-scale-structure observations.