  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 $f\sigma_8(z)$
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 $f\sigma_8(z)$. For this case, we find a $\sim 2\sigma$
tension (good agreement) with the standard relativistic cosmology when the
Planck (WMAP-9) priors are used.
