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 ${\hat{I}}^3$ is finite from the asymptotic past to the asymptotic future is identified as the so-called heteroclinic orbit connecting the critical points $(\hat{I}{\text{c◯}}_m=0,{\hat{I}}_{\infty }^3)$ in the future and $(\hat{I}{\text{c◯}}_m=1,{\hat{I}}_{-\infty }^3)$ 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) $\epsilon \hat{I}{\text{c◯}}_m^{\mathrm{tot}}$ remains unclustered, we find that the limit of the growth index in the past $γ_{-\infty}^{\varepsilon}$ does not depend on the equation of state of DE, in sharp contrast with the case $\epsilon =0$ (for which ${\hat{I}}_{-\infty }^3$ is obtained). We show indeed that there is a mathematical discontinuity: one cannot obtain ${\hat{I}}_{-\infty }^3$ by taking $\lim_{\varepsilon \to 0} γ^{\varepsilon}_{-\infty}$ (i.e. the limits $\epsilon \to 0$ and $\hat{I}{\text{c◯}}_m^{\mathrm{tot}}\to 1$ do not commute). We recover in our analysis that the value $γ_{-\infty}^{\varepsilon}$ corresponds to tracking DE in the asymptotic past with constant $γ=γ_{-\infty}^{\varepsilon}$ found earlier.

18 pages, 3 figures; v2: improved presentation, matches published version