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

We focus on the class of cosmological models with a time-evolving vacuum energy density of the form $ρ_Λ=C_0+C_1 H+C_2 H^2$, where $H$ is the Hubble rate. Higher powers of $H$ could be important for the early inflationary epoch, but are irrelevant afterwards. We study these models at the background level and at the perturbations level, both at the linear and at the nonlinear regime. We find that those with $C_0=0$ are seriously hampered, as they are unable to fit simultaneously the current observational data on Hubble expansion and the linear growth rate of clustering. This is in contrast to the $C_0\neq 0$ models, including the concordance $Λ$CDM model. We also compute the redshift distribution of clusters predicted by all these models, in which the analysis of the nonlinear perturbations becomes crucial. The outcome is that the models with $C_0=0$ predict a number of counts with respect to the concordance model which is much larger, or much smaller, than the $Λ$CDM and the dynamical models with $C_0\neq 0$. The particular case $ρ_Λ\propto H$ (the pure lineal model), which in the past was repeatedly motivated by several authors from QCD arguments applied to cosmology, is also addressed and we assess in detail its phenomenological status. We conclude that the most favored models are those with $C_0\neq 0$, and we show how to discriminate them from the $Λ$CDM.

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