Dynamical system analysis of interacting dark energy-matter scenarios at the linearized inhomogeneous level

We carry out the dynamical system analysis of interacting dark energy-matter scenarios by examining the critical points and stability for not just the background level cosmological evolution, but at the level of the linear density perturbations as well. While an analysis at the background level can lead to a stable phase space trajectory implying that the universe eventually transpires to a dark energy dominated (de-Sitter) era, a two-fold degeneracy in the spectrum of the critical points is found to arise in the inhomogeneous picture, due to the possible growth and decay of matter density perturbations. Analyzing the phase space dynamics of the growth factor, we show that it turns out to be greater than unity initially, for one of the critical points, and leads to a stable configuration as the fluctuations in the matter density die out asymptotically. As to the growth index, we show that the only trajectory which is physically plausible is the one that evolves mildly at high redshifts and gets steeper as time progresses. However, such a trajectory amounts to the average value of the growth index, throughout the expansion history of the universe, not much deviated from the value $6/11$, corresponding to the background $\L$CDM cosmology.