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 $\Omega_{m}^\gamma$ and then suggest a new form,
$f_a=\Omega_m^\gamma+(\gamma-4/7)\Omega_k$, 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 supernovae, baryon acoustic
oscillation, WMAP5, and $H(z)$ data, we fit the growth index parameter to
current growth factor data and obtain $\gamma_{\Lambda}(\Omega_{k} \not= 0) =
0.65^{+0.17}_{-0.15}$ and $\gamma_{DGP}(\Omega_{k} \not= 0) =
0.53^{+0.14}_{-0.12}$. 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.
