# On the implementation of the spherical collapse model for dark energy models

## Abstract

In this work we review the theory of the spherical collapse model and critically analyse the aspects of the numerical implementation of its fundamental equations. By extending a recent work by [1], we show how different aspects, such as the initial integration time, the definition of constant infinity and the criterion for the extrapolation method (how close the inverse of the overdensity has to be to zero at the collapse time) can lead to an erroneous estimation (a few per mill error which translates to a few percent in the mass function) of the key quantity in the spherical collapse model: the linear critical overdensity δ{sub c}, which plays a crucial role for the mass function of halos. We provide a better recipe to adopt in designing a code suitable to a generic smooth dark energy model and we compare our numerical results with analytic predictions for the EdS and the ΛCDM models. We further discuss the evolution of δ{sub c} for selected classes of dark energy models as a general test of the robustness of our implementation. We finally outline which modifications need to be taken into account to extend the code to more general classes of models, suchmore »

- Authors:

- Jodrell Bank Centre for Astrophysics, School of Physics and Astronomy, The University of Manchester, Manchester, M13 9PL (United Kingdom)
- Zentrum für Astronomie der Universität Heidelberg, Institut für theoretische Astrophysik, Philosophenweg 12, D-69120, Heidelberg (Germany)

- Publication Date:

- OSTI Identifier:
- 22667628

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Cosmology and Astroparticle Physics; Journal Volume: 2017; Journal Issue: 10; Other Information: Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 79 ASTROPHYSICS, COSMOLOGY AND ASTRONOMY; COMPARATIVE EVALUATIONS; DESIGN; EQUATIONS; EXTRAPOLATION; FORECASTING; GRAVITATIONAL COLLAPSE; MASS; MODIFICATIONS; NONLUMINOUS MATTER; SPHERICAL CONFIGURATION

### Citation Formats

```
Pace, Francesco, Meyer, Sven, and Bartelmann, Matthias, E-mail: francesco.pace@manchester.ac.uk, E-mail: sven.meyer@uni-heidelberg.de, E-mail: bartelmann@uni-heidelberg.de.
```*On the implementation of the spherical collapse model for dark energy models*. United States: N. p., 2017.
Web. doi:10.1088/1475-7516/2017/10/040.

```
Pace, Francesco, Meyer, Sven, & Bartelmann, Matthias, E-mail: francesco.pace@manchester.ac.uk, E-mail: sven.meyer@uni-heidelberg.de, E-mail: bartelmann@uni-heidelberg.de.
```*On the implementation of the spherical collapse model for dark energy models*. United States. doi:10.1088/1475-7516/2017/10/040.

```
Pace, Francesco, Meyer, Sven, and Bartelmann, Matthias, E-mail: francesco.pace@manchester.ac.uk, E-mail: sven.meyer@uni-heidelberg.de, E-mail: bartelmann@uni-heidelberg.de. Sun .
"On the implementation of the spherical collapse model for dark energy models". United States.
doi:10.1088/1475-7516/2017/10/040.
```

```
@article{osti_22667628,
```

title = {On the implementation of the spherical collapse model for dark energy models},

author = {Pace, Francesco and Meyer, Sven and Bartelmann, Matthias, E-mail: francesco.pace@manchester.ac.uk, E-mail: sven.meyer@uni-heidelberg.de, E-mail: bartelmann@uni-heidelberg.de},

abstractNote = {In this work we review the theory of the spherical collapse model and critically analyse the aspects of the numerical implementation of its fundamental equations. By extending a recent work by [1], we show how different aspects, such as the initial integration time, the definition of constant infinity and the criterion for the extrapolation method (how close the inverse of the overdensity has to be to zero at the collapse time) can lead to an erroneous estimation (a few per mill error which translates to a few percent in the mass function) of the key quantity in the spherical collapse model: the linear critical overdensity δ{sub c}, which plays a crucial role for the mass function of halos. We provide a better recipe to adopt in designing a code suitable to a generic smooth dark energy model and we compare our numerical results with analytic predictions for the EdS and the ΛCDM models. We further discuss the evolution of δ{sub c} for selected classes of dark energy models as a general test of the robustness of our implementation. We finally outline which modifications need to be taken into account to extend the code to more general classes of models, such as clustering dark energy models and non-minimally coupled models.},

doi = {10.1088/1475-7516/2017/10/040},

journal = {Journal of Cosmology and Astroparticle Physics},

number = 10,

volume = 2017,

place = {United States},

year = {Sun Oct 01 00:00:00 EDT 2017},

month = {Sun Oct 01 00:00:00 EDT 2017}

}