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.
