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 Ωm appears to be too small. An infinite number of solutions, with Ωm and γ both constant, are found with wDE = 0 corresponding to Einstein-de Sitter universes. For all modified gravity DE models satisfying Geff ≥ G, among them the f(R) DE models suggested in the literature, the condition to have a constant wDE is strongly violated at the present epoch. In contrast, DE tracking dust-like matter deep in the matter era, but with Ωm <1, requires Geff > G and an example is given using scalar-tensor gravity for a range of admissible values of γ. For constant wDE inside GR, departure from a quasi-constant value is limited until today. Even a large variation of wDE 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 Ωm (and with respect to z) diverges and tends to −∞.