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

Abstract

We focus on the class of cosmological models with a time-evolving vacuum energy density of the form |$\rho _\Lambda (H)=C_0+C_1 H+C_2 H^2$|⁠, 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 non-linear regime. We find that those with C0 = 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 C0 ≠ 0 models, including the concordance Λ cold dark matter (ΛCDM) model. We also compute the redshift distribution of clusters predicted by all these models, in which the analysis of the non-linear perturbations becomes crucial. The outcome is that the models with C0 = 0 predict a number of counts with respect to the concordance model which is much larger, or much smaller, than the ΛCDM and the dynamical models with C0 ≠ 0. The particular case |$\rho _\Lambda (H)\propto H$| (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 favoured models are those with C0 ≠ 0, and we show how to discriminate them from the ΛCDM.