The growth of matter perturbations in f(R) models

We consider the linear growth of matter perturbations on low redshifts in some $f(R)$ dark energy (DE) models. We discuss the definition of dark energy (DE) in these models and show the differences with scalar-tensor DE models. For the $f(R)$ model recently proposed by Starobinsky we show that the growth parameter $\gamma_0\equiv \gamma(z=0)$ takes the value $\gamma_0\simeq 0.4$ for $\Omega_{m,0}=0.32$ and $\gamma_0\simeq 0.43$ for $\Omega_{m,0}=0.23$, allowing for a clear distinction from $\Lambda$CDM. Though a scale-dependence appears in the growth of perturbations on higher redshifts, we find no dispersion for $\gamma(z)$ on low redshifts up to $z\sim 0.3$, $\gamma(z)$ is also quasi-linear in this interval. At redshift $z=0.5$, the dispersion is still small with $\Delta \gamma\simeq 0.01$. As for some scalar-tensor models, we find here too a large value for $\gamma'_0\equiv \frac{d\gamma}{dz}(z=0)$, $\gamma'_0\simeq -0.25$ for $\Omega_{m,0}=0.32$ and $\gamma'_0\simeq -0.18$ for $\Omega_{m,0}=0.23$. These values are largely outside the range found for DE models in General Relativity (GR). This clear signature provides a powerful constraint on these models.