On the growth of linear perturbations

We consider the linear growth of matter perturbations in various dark energy (DE) models. We show the existence of a constraint valid at z=0z=0 between the background and dark energy parameters and the matter perturbations growth parameters. For ΛCDM γ0′≡dγdz|0 lies in a very narrow interval −0.0195⩽γ0′⩽−0.0157 for 0.2⩽Ωm,0⩽0.350.2⩽Ωm,0⩽0.35. Models with a constant equation of state inside General Relativity (GR) are characterized by a quasi-constant γ0′, for Ωm,0=0.3Ωm,0=0.3 for example we have γ0′≈−0.02 while γ0γ0 can have a nonnegligible variation. A smoothly varying equation of state inside GR does not produce either |γ0′|>0.02. A measurement of γ(z)γ(z) on small redshifts could help discriminate between various DE models even if their γ0γ0 is close, a possibility interesting for DE models outside GR for which a significant γ0′ can be obtained.