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 $\mathrm{\Lambda }$CDM cosmology, with errors below $0.25\mathrm{\%}$, is given by $f(a)={\mathrm{\Omega }}_m^{\gamma }(a)$ 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 $\mu (a,k)=G_{eff}\mathrm{/}G$ and show that $f(a)=\beta (a){\mathrm{\Omega }}_m^{\gamma }(a)$ provides fits to the numerical solutions with similar accuracy to that of $\mathrm{\Lambda }$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 $1\mathrm{\%}$. 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.