Sweeping Horndeski Canvas: New Growth-Rate Parameterization for Modified-Gravity Theories

We propose and numerically validate a new fitting formula that is sufficiently accurate to model the growth of structure in Horndeski theories of modified gravity for upcoming Stage IV and V large-scale structure surveys. Based on an analysis of more than 18,000 Horndeski models and adopting the popular parameterization of the growth rate $f(z)={\mathrm{\Omega }}_M(z{)}^{\gamma }$, we generalize the constant growth index $\gamma$ to a two-parameter redshift-dependent quantity, $\gamma (z)$, that more accurately fits these models. We demonstrate that the functional form $\gamma (z)={\gamma }_0+{\gamma }_1{z}^2\mathrm{/}(1+z)$ improves the median ${\chi }^2$ of the fit to viable Horndeski models by a factor of $\sim 40$ relative to that of a constant $\gamma$, and is sufficient to obtain unbiased results even for precise measurements expected in Stage IV and V surveys. Finally, we constrain the parameters of the new fitting formula using current cosmological data.