Constraints to dark energy using PADE parameterisations

Abstract

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.