Improved parametrization of the growth index for dark energy and DGP models

Understand

We propose two improved parameterized form for the growth index of the linear matter perturbations: (I) $\gamma(z)=\gamma_0+(\gamma_{\infty}-\gamma_0){z\over z+1}$ and (II) $\gamma(z)=\gamma_0+\gamma_1 \frac{z}{z+1}+(\gamma_{\infty}-\gamma_1-\gamma_0)(\frac{z}{z+1})^{\alpha}$.

• With these forms of $\gamma(z)$, we analyze the accuracy of the approximation the growth factor $f$ by $\Omega^{\gamma(z)}_m$ for both the $\omega$CDM model and the DGP model.
• For the first improved parameterized form, we find that the approximation accuracy is enhanced at the high redshifts for both kinds of models, but it is not at the low redshifts.
• For the second improved parameterized form, it is found that $\Omega^{\gamma(z)}_m$ approximates the growth factor $f$ very well for all redshifts.