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

Abstract

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 equation of state. Using a 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 of the growth index in the past $\gamma_{-\infty}^{\varepsilon}$ does not depend on the equation of state of DE, in sharp contrast with the case $\varepsilon=0$ (for which $\gamma_{-\infty}$ is obtained). We show indeed that there is a mathematical discontinuity: one cannot obtain $\gamma_{-\infty}$ by taking $\lim_{\varepsilon \to 0} \gamma^{\varepsilon}_{-\infty}$ (i.e. the limits $\varepsilon\to 0$ and $\Omega^{\rm tot}_m\to 1$ do not commute). We recover in our analysis that the value $\gamma_{-\infty}^{\varepsilon}$ corresponds to tracking DE in the asymptotic past with constant $\gamma=\gamma_{-\infty}^{\varepsilon}$ found earlier.