Contiguous redshift parameterizations of the growth index

The growth rate of matter perturbations can be used to distinguish between different gravity theories and to distinguish between dark energy and modified gravity at cosmological scales as an explanation to the observed cosmic acceleration. We suggest here parameterizations of the growth index as functions of the redshift. The first one is given by $γ(a)=\tildeγ(a) \frac{1}{1+(a_{_{ttc}}/a)}+γ_{_{early}} \frac{1}{1+(a/a_{_{ttc}})}$ that interpolates between a low/intermediate redshift parameterization $\tildeγ(a)=γ_{_{late}}(a)= γ_0 + (1-a) γ_a$ and a high redshift $γ_{_{early}}$ constant value. For example, our interpolated form $γ(a)$ can be used when including the CMB to the rest of the data while the form $γ_{_{late}}(a)$ can be used otherwise. It is found that the parameterizations proposed achieve a fit that is better than 0.004% for the growth rate in a $Λ$CDM model, better than 0.014% for Quintessence-Cold-Dark-Matter (QCDM) models, and better than 0.04% for the flat Dvali-Gabadadze-Porrati (DGP) model (with $Ω_m^0=0.27$) for the entire redshift range up to $z_{_{CMB}}$. We find that the growth index parameters $(γ_0,γ_a)$ take distinctive values for dark energy models and modified gravity models, e.g. $(0.5655,-0.02718)$ for the $Λ$CDM model and $(0.6418,0.06261)$ for the flat DGP model. This provides a means for future observational data to distinguish between the models.

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