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

TL;DR

This paper reviews and improves the numerical implementation of the spherical collapse model for dark energy, highlighting potential errors and providing better methods for accurate estimation of key parameters across various dark energy models.

Contribution

It offers a refined numerical approach for the spherical collapse model, addressing common errors and extending applicability to diverse dark energy scenarios.

Findings

Identification of errors in existing numerical implementations.

Development of a better recipe for code design in dark energy models.

Comparison of numerical results with analytic predictions for standard cosmologies.

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 Herrera et al. 2017, 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 $\delta_{\rm c}$, 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 $\Lambda$CDM models. We further discuss the evolution of $\delta_{\rm c}$ 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.