Perturbative quantities, such as the growth rate ($f$) and index ($gamma$), are powerful tools to distinguish different dark energy models or modified gravity theories even if they produce the same cosmic expansion history. In this work, without any assumption about the dynamics of the Universe, we apply a non-parametric method to current measurements of the expansion rate $H(z)$ from cosmic chronometers and high-$z$ quasar data and reconstruct the growth factor and rate of linearised density perturbations in the non-relativistic matter component. Assuming realistic values for the matter density parameter $Omega_{m0}$, as provided by current CMB experiments, we also reconstruct the evolution of the growth index $gamma$ with redshift. We show that the reconstruction of current $H(z)$ data constrains the growth index to $gamma=0.56 pm 0.12$ (2$sigma$) at $z = 0.09$, which is in full agreement with the prediction of the $Lambda$CDM model and some of its extensions.
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