Growth factor parametrization in curved space

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 Ωmγ and then suggest a new form, fa=Ωmγ+(γ-4/7)Ωk, as an approximation for the growth factor when the curvature Ωk is not negligible, and where the growth index γ is usually model dependent. The expression recovers the standard results for the curved and flat ΛCDM and Dvali-Gabadadze-Porrati models. Using the best fit values of Ωm0 and Ωk0 to the expansion/distance measurements from Type Ia SNe, baryon acoustic oscillation, WMAP5, and H(z) data, we fit the growth index parameter to current growth factor data and obtain γΛ(Ωk≠0)=0.65-0.15+0.17 and γDGP(Ωk≠0)=0.53-0.12+0.14. For the ΛCDM 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 fa are fit simultaneously. Importantly, we find that, in the presence of curvature, the analytical expression proposed for fa provides a better fit to the growth factor than other forms and should be useful for future high precision missions and studies.