Constraints to dark energy using PADE parameterisations
  We put constraints on dark energy properties using the PADE parameterisation,
and compare it to the same constraints using Chevalier-Polarski-Linder (CPL)
and $\Lambda$CDM, at both the background and the perturbation levels. The dark
energy equation of state parameter of the models is derived following the
mathematical treatment of PADE expansion. Unlike CPL parameterisation, the PADE
approximation provides different forms of the equation of state parameter which
avoid the divergence in the far future. Initially, we perform a likelihood
analysis in order to put constraints on the model parameters using solely
background expansion data and we find that all parameterisations are consistent
with each other. Then, combining the expansion and the growth rate data we test
the viability of PADE parameterisations and compare them with CPL and
$\Lambda$CDM models respectively. Specifically, we find that the growth rate of
the current PADE parameterisations is lower than $\Lambda$CDM model at low
redshifts, while the differences among the models are negligible at high
redshifts. In this context, we provide for the first time growth index of
linear matter perturbations in PADE cosmologies. Considering that dark energy
is homogeneous we recover the well known asymptotic value of the growth index,
namely $\gamma_{\infty}=\frac{3(w_{\infty}-1)}{6w_{\infty}-5}$, while in the
case of clustered dark energy we obtain $\gamma_{\infty}\simeq
\frac{3w_{\infty}(3w_{\infty}-5)}{(6w_{\infty}-5)(3w_{\infty}-1)}$. Finally, we
generalize the growth index analysis in the case where $\gamma$ is allowed to
vary with redshift and we find that the form of $\gamma(z)$ in PADE
parameterisation extends that of the CPL and $\Lambda$CDM cosmologies
respectively.
