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

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

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