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We focus on the class of cosmological models with a time-evolving vacuumenergy density of the form $\rho_\Lambda=C_0+C_1 H+C_2 H^2$, where $H$ is theHubble rate. Higher powers of $H$ could be important for the early inflationaryepoch, but are irrelevant afterwards. We study these models at the backgroundlevel and at the perturbations level, both at the linear and at the nonlinearregime. We find that those with $C_0=0$ are seriously hampered, as they areunable to fit simultaneously the current observational data on Hubble expansionand the linear growth rate of clustering. This is in contrast to the $C_0\neq0$ models, including the concordance $\Lambda$CDM model. We also compute theredshift distribution of clusters predicted by all these models, in which theanalysis of the nonlinear perturbations becomes crucial. The outcome is thatthe models with $C_0=0$ predict a number of counts with respect to theconcordance model which is much larger, or much smaller, than the $\Lambda$CDMand the dynamical models with $C_0\neq 0$. The particular case$\rho_\Lambda\propto H$ (the pure lineal model), which in the past wasrepeatedly motivated by several authors from QCD arguments applied tocosmology, is also addressed and we assess in detail its phenomenologicalstatus. We conclude that the most favored models are those with $C_0\neq 0$,and we show how to discriminate them from the $\Lambda$CDM.

Vacuum models with a linear and a quadratic term in H: structure formation and number counts analysis