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

## 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{sub 0}+w{sub a}(1−a). We compute the growth index γ = log f(a)/log Ω{sub 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 γ(a) = γ{sub 0}+γ{sub a}(1−a) in most cases. We then compare four different types of dark energy (DE) models: wΛCDM, DGP, f(R) and a LTB-large-void model, which have very different behaviors at z∼>1. 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 ≅ 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 usemore »

- Authors:

- Instituto de Física Teórica, Universidad Autónoma de Madrid, Cantoblanco 28049 Madrid (Spain)

- Publication Date:

- OSTI Identifier:
- 22280222

- Resource Type:
- Journal Article

- Journal Name:
- Journal of Cosmology and Astroparticle Physics

- Additional Journal Information:
- Journal Volume: 2011; Journal Issue: 10; Other Information: Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 1475-7516

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 79 ASTROPHYSICS, COSMOLOGY AND ASTRONOMY; APPROXIMATIONS; ASTROPHYSICS; COMPARATIVE EVALUATIONS; COSMOLOGICAL CONSTANT; COSMOLOGY; DISTURBANCES; EQUATIONS OF STATE; EXACT SOLUTIONS; GALAXIES; LEGENDRE POLYNOMIALS; NONLUMINOUS MATTER; RED SHIFT; UNIVERSE

### Citation Formats

```
Belloso, Alicia Bueno, García-Bellido, Juan, and Sapone, Domenico, E-mail: alicia.bueno@uam.es, E-mail: juan.garciabellido@uam.es, E-mail: domenico.sapone@uam.es.
```*A parametrization of the growth index of matter perturbations in various Dark Energy models and observational prospects using a Euclid-like survey*. United States: N. p., 2011.
Web. doi:10.1088/1475-7516/2011/10/010.

```
Belloso, Alicia Bueno, García-Bellido, Juan, & Sapone, Domenico, E-mail: alicia.bueno@uam.es, E-mail: juan.garciabellido@uam.es, E-mail: domenico.sapone@uam.es.
```*A parametrization of the growth index of matter perturbations in various Dark Energy models and observational prospects using a Euclid-like survey*. United States. doi:10.1088/1475-7516/2011/10/010.

```
Belloso, Alicia Bueno, García-Bellido, Juan, and Sapone, Domenico, E-mail: alicia.bueno@uam.es, E-mail: juan.garciabellido@uam.es, E-mail: domenico.sapone@uam.es. Sat .
"A parametrization of the growth index of matter perturbations in various Dark Energy models and observational prospects using a Euclid-like survey". United States. doi:10.1088/1475-7516/2011/10/010.
```

```
@article{osti_22280222,
```

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

author = {Belloso, Alicia Bueno and García-Bellido, Juan and Sapone, Domenico, E-mail: alicia.bueno@uam.es, E-mail: juan.garciabellido@uam.es, E-mail: domenico.sapone@uam.es},

abstractNote = {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{sub 0}+w{sub a}(1−a). We compute the growth index γ = log f(a)/log Ω{sub 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 γ(a) = γ{sub 0}+γ{sub a}(1−a) in most cases. We then compare four different types of dark energy (DE) models: wΛCDM, DGP, f(R) and a LTB-large-void model, which have very different behaviors at z∼>1. 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 ≅ 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{sub 0},w{sub a}), and (4%, 12%) errors in (γ{sub 0},γ{sub 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 ΛCDM, 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.},

doi = {10.1088/1475-7516/2011/10/010},

journal = {Journal of Cosmology and Astroparticle Physics},

issn = {1475-7516},

number = 10,

volume = 2011,

place = {United States},

year = {2011},

month = {10}

}