# Global properties of the growth index of matter inhomogeneities in the universe

## TL;DR

This paper analyzes the evolution of the growth index of matter inhomogeneities across cosmic history, revealing conditions for its behavior in different gravity models and implications for understanding cosmic structure growth.

## Contribution

It provides a comprehensive analysis of the growth index's behavior from the matter era to the far future in GR and modified gravity, identifying conditions for monotonicity and effects of varying effective gravity.

## Key findings

- $eta$ becomes arbitrarily large deep in the matter era.
- $eta$ tends to a finite value only with negligible decaying modes.
- DGP models show a monotonic increase in $eta$ except in the far future.

## Abstract

We perform here a global analysis of the growth index $\gamma$ behaviour from deep in the matter era till the far future. For a given cosmological model in GR or in modified gravity, the value of $\gamma(\Omega_{m})$ is unique when the decaying mode of scalar perturbations is negligible. However, $\gamma_{\infty}$, the value of $\gamma$ in the asymptotic future, is unique even in the presence of a nonnegligible decaying mode today. Moreover $\gamma$ becomes arbitrarily large deep in the matter era. Only in the limit of a vanishing decaying mode do we get a finite $\gamma$, from the past to the future in this case. We find further a condition for $\gamma(\Omega_{m})$ to be monotonically decreasing (or increasing). This condition can be violated inside general relativity (GR) for varying $w_{DE}$ though generically $\gamma(\Omega_{m})$ will be monotonically decreasing (like $\Lambda$CDM), except in the far future and past. A bump or a dip in $G_{\rm eff}$ can also lead to a significant and rapid change in the slope $\frac{d\gamma}{d\Omega_{m}}$. On a $\Lambda$CDM background, a $\gamma$ substantially lower (higher) than $0.55$ with a negative (positive) slope reflects the opposite evolution of $G_{\rm eff}$. In DGP models, $\gamma(\Omega_{m})$ is monotonically increasing except in the far future. While DGP gravity becomes weaker than GR in the future and $w^{DGP}\to -1$, we still get $\gamma_{\infty}^{DGP}= \gamma_{\infty}^{\Lambda CDM}=\frac{2}{3}$. In contrast, despite $G^{DGP}_{\rm eff}\to G$ in the past, $\gamma$ does not tend to its value in GR because $\frac{dG^{DGP}_{\rm eff}}{d\Omega_{m}}\Big|_{-\infty}\ne 0$.