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.
