Global properties of the growth index of matter inhomogeneities in the
  universe
  We perform here a global analysis of the growth index $\gamma$ 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 $\gamma(\Omega_{m})$ is unique when the
decaying mode of scalar perturbations is negligible. However,
$\gamma_{\infty}$, the value of $\gamma$ in the asymptotic future, is unique
even in the presence of a nonnegligible decaying mode today. Moreover $\gamma$
becomes arbitrarily large deep in the matter era. Only in the limit of a
vanishing decaying mode do we get a finite $\gamma$, from the past to the
future in this case. We find further a condition for $\gamma(\Omega_{m})$ to be
monotonically decreasing (or increasing). This condition can be violated inside
general relativity (GR) for varying $w_{DE}$ though generically
$\gamma(\Omega_{m})$ will be monotonically decreasing (like $\Lambda$CDM),
except in the far future and past. A bump or a dip in $G_{\rm eff}$ can also
lead to a significant and rapid change in the slope
$\frac{d\gamma}{d\Omega_{m}}$. On a $\Lambda$CDM background, a $\gamma$
substantially lower (higher) than $0.55$ with a negative (positive) slope
reflects the opposite evolution of $G_{\rm eff}$. In DGP models,
$\gamma(\Omega_{m})$ 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 $\gamma_{\infty}^{DGP}= \gamma_{\infty}^{\Lambda CDM}=\frac{2}{3}$.
In contrast, despite $G^{DGP}_{\rm eff}\to G$ in the past, $\gamma$ does not
tend to its value in GR because $\frac{dG^{DGP}_{\rm
eff}}{d\Omega_{m}}\Big|_{-\infty}\ne 0$.
