# Calculation of the critical overdensity in the spherical-collapse approximation

## TL;DR

This paper compares methods for calculating the critical overdensity in spherical collapse models, emphasizing the importance of precise definitions, especially in modified gravity theories like $f(R)$, and provides analytical and numerical results for these calculations.

## Contribution

It clarifies the accuracy of the constant-infinity method and introduces an analytical expression for density contrast in $f(R)$ gravity models.

## Key findings

- The constant-infinity method requires careful definition of numerical infinities.
- Relative errors in $\, ext{delta}_c$, $n_{ ext{ln} M}$, and $N_{ ext{bin}}$ depend on redshift.
- An analytical formula for density contrast as a function of collapse redshift and matter density parameter is derived.

## Abstract

Critical overdensity $\delta_c$ is a key concept in estimating the number count of halos for different redshift and halo-mass bins, and therefore, it is a powerful tool to compare cosmological models to observations. There are currently two different prescriptions in the literature for its calculation, namely, the differential-radius and the constant-infinity methods. In this work we show that the latter yields precise results {\it only} if we are careful in the definition of the so-called numerical infinities. Although the subtleties we point out are crucial ingredients for an accurate determination of $\delta_c$ both in general relativity and in any other gravity theory, we focus on $f(R)$ modified-gravity models in the metric approach; in particular, we use the so-called large ($F=1/3$) and small-field ($F=0$) limits. For both of them, we calculate the relative errors (between our method and the others) in the critical density $\delta_c$, in the comoving number density of halos per logarithmic mass interval $n_{\ln M}$ and in the number of clusters at a given redshift in a given mass bin $N_{\rm bin}$, as functions of the redshift. We have also derived an analytical expression for the density contrast in the linear regime as a function of the collapse redshift $z_c$ and $\Omega_{m0}$ for any $F$.