Hubble expansion and structure formation in time varying vacuum models

We investigate the properties of the FLRW flat cosmological models in which the vacuum energy density evolves with time, $\Lambda (t)$. Using different versions of the $\Lambda (t)$ model, namely, quantum field vacuum, power series vacuum and power law vacuum, we find that the main cosmological functions such as the scale factor of the Universe, the Hubble expansion rate $H$, and the energy densities are defined analytically. Performing a joint likelihood analysis of the recent supernovae type Ia data, the cosmic microwave background shift parameter and the baryonic acoustic oscillations traced by the Sloan Digital Sky Survey galaxies, we put tight constraints on the main cosmological parameters of the $\Lambda (t)$ scenarios. Furthermore, we study the linear matter fluctuation field of the above vacuum models. We find that the patterns of the power series vacuum $\Lambda =n_{1}H+n_2{H}^2$ predict stronger small scale dynamics, which implies a faster growth rate of perturbations with respect to the other two vacuum cases (quantum field and power law), despite the fact that all the cosmological models share the same equation of state parameter. In the case of the quantum field vacuum $\Lambda =n_0+n_2{H}^2$, the corresponding matter fluctuation field resembles that of the traditional $\Lambda$ cosmology. The power law vacuum ($\Lambda \propto a^{-n}$) mimics the classical quintessence cosmology, the best fit being tilted in the phantom phase. In this framework, we compare the observed growth rate of clustering measured from the optical galaxies with those predicted by the current $\Lambda (t)$ models. Performing a Kolmogorov-Smirnov statistical test we show that the cosmological models which contain a constant vacuum ($\Lambda \mathrm{CDM}$), quantum field vacuum, and power law vacuum provide growth rates that match well with the observed growth rate. However, this is not the case for the power series vacuum models (in particular, the frequently adduced $\Lambda \propto H$ model) in which clusters form at significantly earlier times ($z\ge 4$) with respect to all other models ($z\sim 2$). Finally, we derived the theoretically predicted dark matter halo mass function and the corresponding distribution of cluster-size halos for all the models studied. Their expected redshift distribution indicates that it will be difficult to distinguish the closely resembling models (constant vacuum, quantum field, and power law vacuum), using realistic future x-ray surveys of cluster abundances. However, cluster surveys based on the Sunayev-Zeldovich detection method give some hope to distinguish the closely resembling models at high redshifts.

Figure 1

Upper Panel shows the evolution of the growth factor. The lines correspond to ${\Lambda }_{{\mathrm{PS}}_1}$ (solid), ${\Lambda }_{\mathrm{RG}}$ (dashed), ${\Lambda }_n$ (dot-dashed), and ${\Lambda }_{{\mathrm{PS}}_2}$ (dot). Note, that open triangles correspond to the traditional $\Lambda$ cosmology. Bottom Panel: shows a comparison of the observed (solid circles) [64], (see Table ) and theoretical evolution of the growth rate of clustering $f(z)$. We do not plot the growth rate of the $\Lambda$ cosmology in order to avoid confusion.

Figure 2

The predicted fractional rate of cluster formation as a function of redshift for the current cosmological models (using ${\sigma }_8=0.80$). The meaning of the various lines is as in Fig. 1.

Figure 3

The halo mass function at two different redshifts. The different models are represented by the same line-types as in the previous figures (which however fall on each other and thus are indistinguishable), but only the ${\Lambda }_{\mathrm{PS}2}$ model (upper curve) provides a significantly different $n(M,z)$ than the other models.

Figure 4

The expected cluster redshift distribution (left panel) and the corresponding fractional difference (right panel) of the ${\Lambda }_{\mathrm{RG}}$ (upper blue curve) and ${\Lambda }_n$ (lower magenta curve) models with respect to the standard $\Lambda$ model.

Figure 5

The expected cluster redshift distribution (left panel) and the corresponding fractional difference (right panel) of the ${\Lambda }_{\mathrm{RG}}$ and ${\Lambda }_n$ models with respect to the standard $\Lambda$ model for the case of a realistic (future) x-ray survey with a flux limit of ${10}^{-14}\mathrm{erg}{\mathrm{s}}^{-1}{\mathrm{cm}}^{-2}$.

Figure 6

The expected cluster redshift distribution (left panel) and the corresponding fractional difference (right panel) of the ${\Lambda }_{\mathrm{RG}}$ (upper blue curve) and ${\Lambda }_n$ (lower magenta curve) models with respect to the standard $\Lambda$ model for the case of a realistic (future) SZ survey with a flux limit of 5 mJy.