Can we distinguish dark energy from vacuum-decaying models?

In this paper, we study the distinguishability of dark energy and vacuum decay models. We start showing formally that dark energy fluids and dynamic vacuum are kinematically equivalent. Consequently, cosmological probes, such as the distances of type Ia supernovae (SNe Ia), cosmic microwave background (CMB) distance priors, baryon acoustic oscillations (BAO) distilled parameters and cosmic chronometers, are unable to allow us to distinguish such models from each other. Then, we show that, at least in principle, these models are dynamically distinguishable. So, dynamical probes, such as the evolution of linear matter perturbations, can be used to distinguish between dark energy models and dynamic vacuum models. To exemplify the kinematic equivalence between dark energy models and vacuum decay models, we study two simple models: the wCDM, commonly associated with dark energy, and the Wang-Meng model, often associated with dynamic vacuum. We test these two models with 22 measurements of f sigma 8. First, we consider both as dark energy then we consider both as vacuum decay. We show that, although compatible with each other, when seen as dynamical vacuum both models are tightly close to the Lambda CDM model.