The growth of matter perturbations in f(R) models

Abstract

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 ${\gamma }_0^{\prime }\equiv \frac{d\gamma }{dz}(z=0)$, ${\gamma }_0^{\prime }\simeq -0.25$ for ${\Omega }_{m,0}=0.32$ and ${\gamma }_0^{\prime }\simeq -0.18$ 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.

14 pages, 7 figures, improved presentation, references added, results unchanged, final version to be published in JCAP

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