Can dark energy be expressed as a power series of the Hubble parameter?

In this work, we examine the possibility that the dark energy (DE) density, rho(de), can be dynamical and appear as a power series expansion of the Hubble rate (and its time derivatives), i.e., rho(de)(H, (H) over dot, ...). For the present universe, however, only the terms H, (H) over dot, and H-2 can be relevant, together with an additive constant term. We fit these models to the current cosmological data on the main observables SnIa + H(z) + BAO + LSS + CMB + 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) = w(D)(z). The particular subset of DE models of this type having no additive constant term in rho(de) include the so-called entropic-force and QCD-ghost DE models, as well as the pure linear model rho(de) similar to H, all of which are strongly disfavored in our fitting analysis. In contrast, the models that include the additive term plus one or both of the dynamical components (H) over dot and H-2 appear more favored than the Lambda CDM. In particular, the dynamical DE models provide a value of sigma(8) similar or equal to 0.74-0.77 which is substantially lower than that of the Lambda CDM and hence more in accordance with the observations. This helps to significantly reduce the sigma(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.