Parametrizing growth in dark energy and modified gravity models

Abstract

It is well known that an extremely accurate parametrization of the growth function of matter density perturbations in $\mathrm{\ensuremath{\Lambda}}\mathrm{CDM}$ cosmology, with errors below 0.25%, is given by $f(a)={\mathrm{\ensuremath{\Omega}}}_{m}^{\ensuremath{\gamma}}(a)$ with $\ensuremath{\gamma}\ensuremath{\simeq}0.55$. 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 $\ensuremath{\mu}(a,k)={G}_{\mathrm{eff}}/G$ and show that $f(a)=\ensuremath{\beta}(a){\mathrm{\ensuremath{\Omega}}}_{m}^{\ensuremath{\gamma}}(a)$ provides fits to the numerical solutions with similar accuracy to that of $\mathrm{\ensuremath{\Lambda}}\mathrm{CDM}$. In the time-independent case with $\ensuremath{\mu}=\ensuremath{\mu}(k)$, simple analytic expressions for $\ensuremath{\beta}(\ensuremath{\mu})$ and $\ensuremath{\gamma}(\ensuremath{\mu})$ are presented. In the time-dependent (but scale-independent) case $\ensuremath{\mu}=\ensuremath{\mu}(a)$, we show that $\ensuremath{\beta}(a)$ has the same time dependence as $\ensuremath{\mu}(a)$. As an example, explicit formulas are provided in the Dvali-Gabadadze-Porrati (DGP) model. In the general case, for theories with $\ensuremath{\mu}(a,k)$, we obtain a perturbative expansion for $\ensuremath{\beta}(\ensuremath{\mu})$ around the general relativity case $\ensuremath{\mu}=1$ 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 $\ensuremath{\mu}$ parameter.