Global properties of the growth index of matter inhomogeneities in the universe

Abstract

We perform here a global analysis of the growth index ${\hat{I}}^3$ behaviour from deep in the matter era till the far future. For a given cosmological model in GR or in modified gravity, the value of ${\hat{I}}^3(\hat{I}{\text{c◯}}_m)$ is unique when the decaying mode of scalar perturbations is negligible. However, ${\hat{I}}_{\infty }^3$, the value of ${\hat{I}}^3$ in the asymptotic future, is unique even in the presence of a nonnegligible decaying mode today. Moreover ${\hat{I}}^3$ becomes arbitrarily large deep in the matter era. Only in the limit of a vanishing decaying mode do we get a finite ${\hat{I}}^3$, from the past to the future in this case. We find further a condition for ${\hat{I}}^3(\hat{I}{\text{c◯}}_m)$ to be monotonically decreasing (or increasing). This condition can be violated inside general relativity (GR) for varying $w_{DE}$ though generically ${\hat{I}}^3(\hat{I}{\text{c◯}}_m)$ will be monotonically decreasing (like $\hat{I}›$CDM), except in the far future and past. A bump or a dip in $G_{\mathrm{eff}}$ can also lead to a significant and rapid change in the slope $\frac{d{\hat{I}}^3}{d\hat{I}{\text{c◯}}_m}$. On a $\hat{I}›$CDM background, a ${\hat{I}}^3$ substantially lower (higher) than $0.55$ with a negative (positive) slope reflects the opposite evolution of $G_{\mathrm{eff}}$. In DGP models, ${\hat{I}}^3(\hat{I}{\text{c◯}}_m)$ is monotonically increasing except in the far future. While DGP gravity becomes weaker than GR in the future and $w^{DGP}\to -1$, we still get $γ_{\infty}^{DGP}= γ_{\infty}^{ΛCDM}=\frac{2}{3}$. In contrast, despite $G_{\mathrm{eff}}^{DGP}\to G$ in the past, ${\hat{I}}^3$ does not tend to its value in GR because $\frac{d{G}_{\mathrm{eff}}^{DGP}}{d\hat{I}{\text{c◯}}_m}{|}_{-\infty }\ne 0$.

15 pages, 7 figures; v3: improved presentation, to appear in Phys.Rev.D