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 $\mathrm{\Lambda }$CDM ${\gamma }_0^{\mathrm{\prime }}\equiv {\frac{d\gamma }{dz}}_0$ lies in a very narrow interval $-0.0195\le {\gamma }_0^{\mathrm{\prime }}\le -0.0157$ for $0.2 \le \Omega_{m,0}\le 0.35$. Models with a constant equation of state inside General Relativity (GR) are characterized by a quasi-constant ${\gamma }_0^{\mathrm{\prime }}$, for ${\mathrm{\Omega }}_{m,0}=0.3$ for example we have ${\gamma }_0^{\mathrm{\prime }}\approx -0.02$ while ${\gamma }_0$ can have a nonnegligible variation. A smoothly varying equation of state inside GR does not produce either $\mathrm{\mid }{\gamma }_0^{\mathrm{\prime }}\mathrm{\mid }>0.02$. A measurement of $\gamma (z)$ on small redshifts could help discriminate between various DE models even if their ${\gamma }_0$ is close, a possibility interesting for DE models outside GR for which a significant ${\gamma }_0^{\mathrm{\prime }}$ can be obtained.