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 derivatives), i.e.$\rho_{de}(H,\dot{H},...)$. For the
present universe, however, only the terms $H$, $\dot{H}$ 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 $\rho_{de} \sim H$ 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 $\dot{H}$ and $H^2$
appear more favored than the $\Lambda$CDM. In particular, the dynamical DE
models provide a value of $\sigma_8\simeq 0.74-0.77$ 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.
