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

Abstract

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. For chosen $\alpha$, the relative error is below 0.003% for the $\Lambda$CDM model and 0.028% for the DGP model when $\Omega_{m}=0.27$. Thus, the second improved parameterized form of $\gamma(z)$ should be useful for the high precision constraint on the growth index of different models with the observational data. Moreover, we also show that $\alpha$ depends on the equation of state $\omega$ and the fractional energy density of matter $\Omega_{m0}$, which may help us learn more information about dark energy and DGP models.