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