Scale-dependent perturbations finally detectable by future galaxy surveys and their contribution to cosmological model selection

By means of the present geometrical and dynamical observational data, it is very hard to establish, from a statistical perspective, a clear preference among the vast majority of the proposed models for the dynamical dark energy and/or modified gravity theories alternative with respect to the $\Lambda$CDM scenario. On the other hand, on scales much smaller than present Hubble scale, there are possibly detectable differences in the growth of the matter perturbations for different modes of the perturbations, even in the context of the $\Lambda$CDM model. Here, we analyze the evolution of the dark matter perturbations in the context of $\Lambda$CDM and some dynamical dark energy models involving future cosmological singularities, such as the sudden future singularity and the finite scale factor singularity. We employ the Newtonian gauge formulation for the derivation of the perturbation equations for the growth function, and we abandon both the sub-Hubble approximation and the slowly varying potential assumption. We apply the Fisher Matrix approach to three future planned galaxy surveys e.g., DESI, Euclid, and WFirst-2.4. With the mentioned surveys on hand, only with the dynamical probes, we will achieve multiple goals: $1.$ the improvement in the accuracy of the determination of the $f\sigma_{8}$ will give the possibility to discriminate between the $\Lambda$CDM and the alternative dark energy models even in the scale-independent approach; $2.$ it will be possible to test the goodness of the scale-independence finally, and also to quantify the necessity of a scale dependent approach to the growth of the perturbations, in particular using surveys which encompass redshift bins with scales $k<0.005\,h$ Mpc$^{-1}$; $3.$ the scale-dependence itself might add much more discriminating power in general, but further advanced surveys will be needed.