A parametrization of the growth index of matter perturbations in various Dark Energy models and observational prospects using a Euclid-like survey

TL;DR

This paper derives exact and approximate solutions for the growth index of matter perturbations in various dark energy models, assessing observational prospects with a Euclid-like survey to distinguish these models.

Contribution

It introduces a parametrization of the growth index in different dark energy models and evaluates the potential of Euclid-like surveys to differentiate them.

Findings

Euclid-like surveys can constrain dark energy parameters with high precision.

The growth index can be effectively parametrized as b3(a)=b3_0+b3_a(1-a).

Different dark energy models can be distinguished using galaxy power spectrum and weak lensing data.

Abstract

We provide exact solutions to the cosmological matter perturbation equation in a homogeneous FLRW universe with a vacuum energy that can be parametrized by a constant equation of state parameter $w$ and a very accurate approximation for the Ansatz $w(a)=w_0+w_a(1-a)$. We compute the growth index $\gamma=\log f(a)/\log\Om_m(a)$, and its redshift dependence, using the exact and approximate solutions in terms of Legendre polynomials and show that it can be parametrized as $\gamma(a)=\gamma_0+\gamma_a(1-a)$ in most cases. We then compare four different types of dark energy (DE) models: $w\Lambda$CDM, DGP, $f(R)$ and a LTB-large-void model, which have very different behaviors at $z\gsim1$. This allows us to study the possibility to differentiate between different DE alternatives using wide and deep surveys like Euclid, which will measure both photometric and spectroscopic redshifts for several hundreds of millions of galaxies up to redshift $z\simeq 2$. We do a Fisher matrix analysis for the prospects of differentiating among the different DE models in terms of the growth index, taken as a given function of redshift or with a principal component analysis, with a value for each redshift bin for a Euclid-like survey. We use as observables the complete and marginalized power spectrum of galaxies $P(k)$ and the Weak Lensing (WL) power spectrum. We find that, using $P(k)$, one can reach (2%, 5%) errors in $(w_0, w_a)$, and (4%, 12%) errors in $(\gamma_0, \gamma_a)$, while using WL we get errors at least twice as large. These estimates allow us to differentiate easily between DGP, $f(R)$ models and $\Lambda$CDM, while it would be more difficult to distinguish the latter from a variable equation of state parameter or LTB models using only the growth index.}