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.
