When is the growth index constant?

TL;DR

This paper explores conditions under which the growth index gamma remains constant in dark energy models, analyzing scalar field and modified gravity scenarios, and finds that constant gamma behavior is generally excluded except in specific Einstein-de Sitter cases.

Contribution

It provides a comprehensive analysis of the conditions for constant growth index gamma in dark energy models, including scalar field and modified gravity theories, highlighting when such behavior is possible or excluded.

Findings

Constant gamma is excluded in quintessence models unless transitioning to phantom behavior.

An infinite set of solutions with constant gamma exist in Einstein-de Sitter universes.

Modified gravity models generally violate the conditions for constant gamma at the present epoch.

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 $G_{\rm eff}\ge G$, 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 $\Omega_m <1$, requires $G_{\rm eff} > G$ 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 $-\infty$.