Parametrizing growth in dark energy and modified gravity models

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. In the time-dependent (but scale-independent) case $\mu=\mu(a)$, we show that $\beta(a)$ has the same time dependence as $\mu(a)$. As an example, explicit formalae are provided in the DGP model. In the general case, for theories with $\mu(a,k)$, we obtain a perturbative expansion for $\beta(\mu)$ around the General Relativity case $\mu=1$ which, for $f(R)$ theories, reaches an accuracy below $1 \%$. Finally, as an example we apply the obtained fitting functions in order to forecast the precision with which future galaxy surveys will be able to measure the $\mu$ parameter.