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 = 0 between the background and dark energy parameters and the matter perturbations growth parameters. For ΛCDM γ₀^′ ≡ frac(d γ, d z) |₀ lies in a very narrow interval - 0.0195 ≤ γ₀^′ ≤ - 0.0157 for 0.2 ≤ Ω_{m, 0} ≤ 0.35. Models with a constant equation of state inside General Relativity (GR) are characterized by a quasi-constant γ₀^′, for Ω_{m, 0} = 0.3 for example we have γ₀^′ ≈ - 0.02 while γ₀ 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) on small redshifts could help discriminate between various DE models even if their γ₀ is close, a possibility interesting for DE models outside GR for which a significant γ₀^′ can be obtained.