Some mathematical aspects of global properties of the growth index

We analyze the global behaviour of the growth index of cosmic inhomogeneities in an isotropic homogeneous universe filled by cold non-relativistic matter and dark energy (DE) with an arbitrary (and not universal necessarily) equation of state. Using the dynamical system approach, we find the critical points of the system. That unique trajectory for which the growth index $\gamma$ is finite from the asymptotic past to the asymptotic future is identified as the so-called heteroclinic orbit connecting the critical points $(\Omega_m=0,~\gamma_{\infty})$ in the future and $(\Omega_m=1,~\gamma_{-\infty})$ 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) $\varepsilon \Omega^{\rm tot}_m$ remains unclustered, we find that the limit in the past $\gamma_{-\infty}^{\varepsilon}$ does not depend on the equation of state of DE, in sharp contrast with the case $\varepsilon=0$. This is possible because the limits $\varepsilon\to 0$ and $\Omega^{\rm tot}_m\to 1$ do not commute. The value $\gamma_{-\infty}^{\varepsilon}$ corresponds to a solution with tracking DE, $\Omega_m=1-\varepsilon,~\Omega_{DE}=\varepsilon$ and $w_{DE}=0$ found earlier