Global properties of the growth index: Mathematical aspects and physical relevance

We analyze the global behavior of the growth index of cosmic inhomogeneities in an isotropic homogeneous universe filled by cold nonrelativistic matter and dark energy (DE) with an arbitrary equation of state. Using a dynamical system approach, we find the critical points of the system. That unique trajectory for which the growth index γ is finite from the asymptotic past to the asymptotic future is identified as the so-called heteroclinic orbit connecting the critical points (Ωm = 0 , γ∞) in the future and (Ωm = 1 , γ−∞) in the past. The first is an attractor while the second is a saddle point, confirming our earlier results. Further, in the case when a fraction of matter (or DE tracking matter) ϵΩtotm remains unclustered, we find that the limit of the growth index in the past γϵ−∞ does not depend on the equation of state of DE, in sharp contrast with the case ϵ = 0 (for which γ−∞ is obtained). We show indeed that there is a mathematical discontinuity: one cannot obtain γ−∞ by taking limϵ→0 γϵ−∞ (i.e., the limits ϵ → 0 and Ωtotm → 1 do not commute). We recover in our analysis that the value γϵ−∞ corresponds to tracking DE in the asymptotic past with constant γ = γϵ−∞ found earlier.