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 (Ω_m) 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 (Ω_m) to be monotonically decreasing (or increasing). This condition can be violated inside GR for varying w_DE though generically γ (Ω_m) will be monotonically decreasing (like Λ CDM ), except in the far future and past. A bump or a dip in G_eff can also lead to a significant and rapid change in the slope d/γ d Ω_m . On a Λ CDM background, a γ substantially lower (higher) than 0.55 with a negative (positive) slope reflects the opposite evolution of G_eff. In Dvali-Gabadadze-Porrati (DGP) models, γ (Ω_m) is monotonically increasing except in the far future. While DGP gravity becomes weaker than GR in the future and w^DGP→-1 , we still get γ_∞^DGP=γ_∞^{Λ CDM}=2/3 . In contrast, despite G_eff^DGP→G in the past, γ does not tend to its value in GR because d/G_eff^DGP d Ω_m |_-∞ ≠0 .