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

We perform here a global analysis of the growth index γ behavior from deep in the matter era till the far future. For a given cosmological model in general relativity (GR) or in modified gravity, the value of γ(ωm) is unique when the decaying mode of scalar perturbations is negligible. However, γ∞, the value of γ in the asymptotic future, is unique even in the presence of a non-negligible decaying mode today. Moreover, γ becomes arbitrarily large deep in the matter era. Only in the limit of a vanishing decaying mode do we get a finite γ, from the past to the future in this case. We find further a condition for γ(ωm) to be monotonically decreasing (or increasing). This condition can be violated inside GR for varying wDE though generically γ(ωm) will be monotonically decreasing (like ΛCDM), except in the far future and past. A bump or a dip in Geff can also lead to a significant and rapid change in the slope dγdωm. On a ΛCDM background, a γ substantially lower (higher) than 0.55 with a negative (positive) slope reflects the opposite evolution of Geff. In Dvali-Gabadadze-Porrati (DGP) models, γ(ωm) is monotonically increasing except in the far future. While DGP gravity becomes weaker than GR in the future and wDGP→-1, we still get γ∞DGP=γ∞ΛCDM=23. In contrast, despite GeffDGP→G in the past, γ does not tend to its value in GR because dGeffDGPdωm|-∞≠0.