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

Understand

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.