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 Omega(gamma)(m) and then suggest a new form, f(a)=Omega(gamma)(m)+(gamma-4/7)Omega(k), as an approximation for the growth factor when the curvature Omega(k) is not negligible, and where the growth index gamma is usually model dependent. The expression recovers the standard results for the curved and flat Lambda CDM and Dvali-Gabadadze-Porrati models. Using the best fit values of Omega(m0) and Omega(k0) 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 gamma(Lambda)(Omega(k)not equal 0)=0.65(-0.15)(+0.17) and gamma(DGP)(Omega(k)not equal 0)=0.53(-0.12)(+0.14). For the Lambda CDM model, the 1-sigma 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.