When is the growth index constant?

Abstract

The growth index $\gamma$ 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 $\gamma$ 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 $\gamma$ 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 ${\Omega }_m$ appears to be too small. An infinite number of solutions, with ${\Omega }_m$ and $\gamma$ both constant, are found with $w_{DE}=0$ corresponding to Einstein-de Sitter universes. For all modified gravity DE models satisfying $G_{\mathrm{eff}}\ge G$, 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 ${\Omega }_m<1$, requires $G_{\mathrm{eff}}>G$ and an example is given using scalar-tensor gravity for a range of admissible values of $\gamma$. 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 $\gamma$. The change however is substantial in the future and the asymptotic value of $\gamma$ is found while its slope with respect to ${\Omega }_m$ (and with respect to $z$) diverges and tends to $-\infty$.

16 pages, 5 figure; incorrect reference corrected; v3 matches published version in JCAP; v4: comment added and expanded references

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Cited by in corpus (1)

• Reconstruction of cosmological matter perturbations in Modified Gravity