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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 $γ_0\equiv γ(z=0)$ takes the value $γ_0\simeq 0.4$ for $Ω_{m,0}=0.32$ and $γ_0\simeq 0.43$ for $Ω_{m,0}=0.23$, allowing for a clear distinction from $Λ$CDM. Though a scale-dependence appears in the growth of perturbations on higher redshifts, we find no dispersion for $γ(z)$ on low redshifts up to $z\sim 0.3$, $γ(z)$ is also quasi-linear in this interval. At redshift $z=0.5$, the dispersion is still small with $Δγ\simeq 0.01$. As for some scalar-tensor models, we find here too a large value for $γ'_0\equiv \frac{dγ}{dz}(z=0)$, $γ'_0\simeq -0.25$ for $Ω_{m,0}=0.32$ and $γ'_0\simeq -0.18$ for $Ω_{m,0}=0.23$. 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.