The existing degeneracy between different dark energy and modified gravity cosmologies at the background level may be broken by analysing quantities at the perturbative level. In this work, we apply a non-parametric smoothing (NPS) method to reconstruct the expansion history of the Universe ($H(z)$) from model-independent cosmic chronometers and high-$z$ quasar data. Assuming a homogeneous and isotropic flat universe and general relativity (GR) as the gravity theory, we calculate the non-relativistic matter perturbations in the linear regime using the $H(z)$ reconstruction and realistic values of $Omega_{m0}$ and $sigma_8$ from Planck and WMAP-9 collaborations. We find a good agreement between the measurements of the growth rate and $fsigma_8(z)$ from current large-scale structure observations and the estimates obtained from the reconstruction of the cosmic expansion history. Considering a recently proposed null test for GR using matter perturbations, we also apply the NPS method to reconstruct $fsigma_8(z)$. For this case, we find a $sim 2sigma$ tension (good agreement) with the standard relativistic cosmology when the Planck (WMAP-9) priors are used.
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