The growth factor parametrization versus numerical solutions in flat and non-flat dark energy models

In the present investigation we use observational data of $f \sigma_ {8}$ to determine observational constraints in the plane $(\Omega_{m0},\sigma_{8})$ using two different methods: the growth factor parametrization and the numerical solutions method for density contrast, $\delta_{m}$. We verified the correspondence between both methods for three models of accelerated expansion: the $\Lambda CDM$ model, the $w_{0}w_{a} CDM$ model and the running cosmological constant $RCC$ model. In all case we consider also curvature as free parameter. The study of this correspondence is important because the growth factor parametrization method is frequently used to discriminate between competitive models. Our results we allow us to determine that there is a good correspondence between the observational constrains using both methods. We also test the power of the $f\sigma_ {8}$ data to constraints the curvature parameter within the $\Lambda CDM$ model. For this we use a non-parametric reconstruction using Gaussian processes. Our results show that the $f\sigma_ {8}$ data with the current precision level does not allow to distinguish between a flat and non-flat universe.