Dark energy reconstruction based on the PADE approximation; an expansion around the $\Lambda$CDM

We study the dynamical properties of dark energy based on a large family of PADE parameterizations for which the dark energy density evolves as a ratio between two polynomials in the scale factor of the universe. Using the latest cosmological data we perform a standard likelihood analysis in order to place constraints on the main cosmological parameters of different PADE models. We find that the basic cosmological parameters, namely $(\Omega_{m0},h,\sigma_{8})$ are practically the same for all PADE parametrizations explored here. Concerning the free parameters which are related to dark energy we show that the best fit values indicate that the equation of state parameter at the present time is in the phantom regime ($w<-1$), however we can not exclude the possibility of $w>-1$ at $1\sigma$ level. Finally, for the current family of PADE parametrizations we test their ability, via AIC and Jeffreys' scale, to deviate from $\Lambda$CDM cosmology. Among the current PADE parametrizations, the model which contains two dark energy parameters is the one for which a small but non-zero deviation from $\Lambda$CDM cosmology is slightly allowed by AIC test. Moreover, based on Jeffreys' scale we show that a deviation from $\Lambda$CDM cosmology is also allowed and thus the possibility of having a dynamical dark energy in the form of PADE parametrization cannot be excluded.