Background history and cosmic perturbations for a general system of self-conserved dynamical dark energy and matter

We determine the Hubble expansion and the general cosmic perturbations equations for a general system consisting of self-conserved matter and self-conserved dark energy (DE). While at the background level the two components are non-interacting, they do interact at the perturbations level. We show that the coupled system of matter and DE perturbations can be transformed into a single, third order, matter perturbation equation, which reduces to the (derivative of the) standard one in the case that the DE is just a cosmological constant. As a nontrivial application we analyze a class of dynamical models whose DE density $\rho_D$ consists of a constant term, $C_0$, and a series of powers of the Hubble rate. These models were previously analyzed from the point of view of dynamical vacuum models, but here we treat them as self-conserved DE models with a dynamical equation of state. We fit them to the wealth of expansion history and linear structure formation data and compare the obtained fit quality with that of the concordance $\Lambda$CDM model. Those with $C_0=0$ include the so-called "entropic-force" and "QCD-ghost" DE models, as well as the pure linear model $\rho_D\sim H$, all of which appear strongly disfavored. The models with $C_0\neq 0$, in contrast, emerge as promising dynamical DE candidates whose phenomenological performance is highly competitive with the rigid $\Lambda$-term inherent to the $\Lambda$CDM.