Parametrizing growth in dark energy and modified gravity models

Understand

It is well-known that an extremely accurate parametrization of the growth function of matter density perturbations in $\Lambda$CDM cosmology, with errors below $0.25 \%$, is given by $f(a)=\Omega_{m}^{\gamma} \,(a)$ with $\gamma \simeq 0.55$.

• In this work, we show that a simple modification of this expression also provides a good description of growth in modified gravity theories.
• We consider the model-independent approach to modified gravity in terms of an effective Newton constant written as $\mu(a,k)=G_{eff}/G$ and show that $f(a)=\beta(a)\Omega_{m}^{\gamma} \,(a)$ provides fits to the numerical solutions with similar accuracy to that of $\Lambda$CDM.
• In the time-independent case with $\mu=\mu(k)$, simple analytic expressions for $\beta(\mu)$ and $\gamma(\mu)$ are presented.