Growth factor parametrization in curved space

Abstract

The growth rate of matter perturbation and the expansion rate of the Universe can be used to distinguish modified gravity and dark energy models in explaining cosmic acceleration. We explore here the inclusion of spatial curvature into the growth factor. We expand previous results using the approximation $Ω_{m}^γ$ and then suggest a new form, $f_a=Ω_m^γ+(γ-4/7)Ω_k$, as an approximation for the growth factor when the curvature $Ω_k$ is not negligible, and where the growth index $γ$ is usually model dependent. The expression recovers the standard results for the curved and flat $Λ$CDM and Dvali-Gabadadze-Porrati models. Using the best fit values of $Ω_{m0}$ and $Ω_{k0}$ to the expansion/distance measurements from Type Ia supernovae, baryon acoustic oscillation, WMAP5, and $H(z)$ data, we fit the growth index parameter to current growth factor data and obtain $γ_Λ(Ω_{k} \not= 0) = 0.65^{+0.17}_{-0.15}$ and $γ_{DGP}(Ω_{k} \not= 0) = 0.53^{+0.14}_{-0.12}$. For the $Λ$CDM model, the 1-$σ$ observational bounds are found consistent with theoretical value, unlike the case for the Dvali-Gabadadze-Porrati model. We also find that the current data we used is not enough to put significant constraints when the 3 parameters in $f_a$ are fit simultaneously. Importantly, we find that, in the presence of curvature, the analytical expression proposed for $f_a$ provides a better fit to the growth factor than other forms and should be useful for future high precision missions and studies.