Growth of perturbations in dark energy parametrization scenarios

In this paper, we study the evolution of dark matter perturbations in the linear regime by considering the possibility of dark energy perturbations. To do this, two popular parameterizations, CPL and BA with same number of free parameters and different redshift dependency have been considered. We integrate the full relativistic equations to obtain the growth of matter fluctuations for both clustering and smooth versions of CPL and BA dark energy. The growth rate is larger (smaller) than the $\Lambda$CDM in the smooth cases when $w<-1$ ($w>-1$) but the dark energy clustering gives a larger (smaller) growth index when $w>-1$ ($w<-1$). We measure the relative difference of the growth rate with respect to concordance $\Lambda$CDM and study how it changes depending on the free parameters. Furthermore, it is found that the difference of growth rates between smooth CPL and BA is negligible, less than $0.5\%$, while for clustering case, the difference is considerable and might be as large as 2$\%$. Eventually, using the latest geometrical and growth rate observational data, we perform an overall likelihood analysis and show that both smooth and clustering cases of CPL and BA parameterizations are consistent with observations. In particular, we find the dark energy FoM $\sim70$ for the BA and $\sim30$ for the CPL which indicates BA model constraints relatively better than CPL one.