In this work, we examine the possibility that the dark energy (DE) density, ρ de , can be dynamical and _ ...Þ. For the appear as a power series expansion of the Hubble rate (and its time derivatives), i.e., ρ de ðH; H; 2 _ present universe, however, only the terms H, H, and H can be relevant, together with an additive constant term. We fit these models to the current cosmological data on the main observables SnIa þ HðzÞþ BAO þ LSS þ CMB þ BBN. Our analysis involves both the background as well as the cosmic perturbation equations. The latter include, apart from the matter density perturbations, also the DE density perturbations. We assume that matter and dynamical DE are separately self-conserved. As a result the equation of state of the DE becomes a nontrivial function of the cosmological redshift, w D 1⁄4 w D ðzÞ. The particular subset of DE models of this type having no additive constant term in ρ de include the so-called “entropic-force” and “QCD-ghost” DE models, as well as the pure linear model ρ de ∼ H, all of which are strongly disfavored in our fitting analysis. In contrast, the models that include the additive term plus one or _ and H 2 appear more favored than the ΛCDM. In particular, the both of the dynamical components H dynamical DE models provide a value of σ 8 ≃ 0.74–0.77 which is substantially lower than that of the ΛCDM and hence more in accordance with the observations. This helps to significantly reduce the σ 8 tension in the structure formation data. At the same time the predicted value for H 0 is in between the local and Planck measurements, thus helping to alleviate this tension as well.
