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

TL;DR

This paper studies the global behavior of the growth index of cosmic inhomogeneities in a universe with matter and dark energy, revealing mathematical properties and physical implications through a dynamical system approach.

Contribution

It identifies the unique heteroclinic orbit of the growth index connecting critical points and explores the effects of unclustered matter on its asymptotic behavior.

Findings

The growth index trajectory is a heteroclinic orbit between critical points.

Unclustered matter fraction affects the past limit of the growth index.

Mathematical discontinuity exists in the limits of unclustered matter fraction.

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.