Reconstruction of cosmological matter perturbations in Modified Gravity

The analysis of perturbative quantities is a powerful tool to distinguish between different Dark Energy models and gravity theories degenerated at the background level. In this work, we generalise the integral solution of the matter density contrast for General Relativity gravity to a wide class of Modified Gravity (MG) theories. To calculate this solution is necessary prior knowledge of the Hubble rate, the density parameter at the present epoch ($\Omega_{m0}$) and the functional form of the effective Newton's constant that characterises the gravity theory. We estimate in a model-independent way the Hubble expansion rate by applying a non-parametric reconstruction method to model-independent cosmic chronometer data and high-$z$ quasar data. In order to compare our generalised solution of the matter density contrast, using the non-parametric reconstruction of $H(z)$ from observational data, with purely theoretical one, we choose a parameterisation of the Screened MG and the $\Omega_{m0}$ from WMAP-9 collaborations. Finally, we calculate the growth index for the analysed cases, finding very good agreement between theoretical values and the obtained ones using the approach presented in this work.