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We determine the Hubble expansion and the general cosmic perturbationsequations for a general system consisting of self-conserved matter andself-conserved dark energy (DE). While at the background level the twocomponents are non-interacting, they do interact at the perturbations level. Weshow that the coupled system of matter and DE perturbations can be transformedinto 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 cosmologicalconstant. As a nontrivial application we analyze a class of dynamical modelswhose DE density $\rho_D$ consists of a constant term, $C_0$, and a series ofpowers of the Hubble rate. These models were previously analyzed from the pointof view of dynamical vacuum models, but here we treat them as self-conserved DEmodels with a dynamical equation of state. We fit them to the wealth ofexpansion history and linear structure formation data and compare the obtainedfit 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 asthe 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 DEcandidates whose phenomenological performance is highly competitive with therigid $\Lambda$-term inherent to the $\Lambda$CDM.

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