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.
