Growth Diagnostics for Dark Energy models and EUCLID forecast

In this work we introduce a new set of parameters $(r_g,s_g)$ involving the linear growth of matter perturbation that can distinguish and constrain different dark energy models very efficiently. Interestingly, for $\mathrm{\Lambda }$CDM model these parameters take exact value $(1,1)$ at all red shifts whereas for models different from $\mathrm{\Lambda }$CDM, they follow different trajectories in the $(r_g,s_g)$ phase plane. By considering the parametrization for the dark energy equation of state ($w$) and for the linear growth rate ($f_g$), we show that different dark energy behaviours with similar evolution of the linear density contrast, can produce distinguishable trajectories in the $(r_{g}, s_{g})$ phase plane. Moreover, one can put stringent constraint on these phase plane using future measurements like EUCLID ruling out some of the dark energy behaviours.