When is the growth index constant?

Abstract

The growth index $γ$ is an interesting tool to assess the phenomenology of dark energy (DE) models, in particular of those beyond general relativity (GR). We investigate the possibility for DE models to allow for a constant $γ$ during the entire matter and DE dominated stages. It is shown that if DE is described by quintessence (a scalar field minimally coupled to gravity), this behaviour of $γ$ is excluded either because it would require a transition to a phantom behaviour at some finite moment of time, or, in the case of tracking DE at the matter dominated stage, because the relative matter density $Ω_m$ appears to be too small. An infinite number of solutions, with $Ω_m$ and $γ$ both constant, are found with $w_{DE}=0$ corresponding to Einstein-de Sitter universes. For all modified gravity DE models satisfying $G_{\rm eff}\ge G$, among them the $f(R)$ DE models suggested in the literature, the condition to have a constant $w_{DE}$ is strongly violated at the present epoch. In contrast, DE tracking dust-like matter deep in the matter era, but with $Ω_m <1$, requires $G_{\rm eff} > G$ and an example is given using scalar-tensor gravity for a range of admissible values of $γ$. For constant $w_{DE}$ inside GR, departure from a quasi-constant value is limited until today. Even a large variation of $w_{DE}$ may not result in a clear signature in the change of $γ$. The change however is substantial in the future and the asymptotic value of $γ$ is found while its slope with respect to $Ω_m$ (and with respect to $z$) diverges and tends to $-\infty$.