Calculation of the critical overdensity in the spherical-collapse approximation

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 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 $N_{\mathrm{bin}}$, 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 ${\mathrm{\Omega }}_{m0}$ for any $F$.

Figure 1

(a, left panel) Critical density contrast ${\delta }_c$ as a function of ${\mathrm{\Omega }}_{m0}$ for $F=0$ (solid blue line) and $F=1/3$ (dotted red line) when the collapse occurs at redshift $z_c=0$, following our approach. (b, right panel) Critical density contrast ${\delta }_c$ as function of redshift collapse $z_c$ for $F=0$ (solid blue line) and $F=1/3$ (dotted red line) with ${\mathrm{\Omega }}_{m0}=0.3$, following our approach.

Figure 2

Relative errors for ${\delta }_c$ (see text for definition) between our method and (a) the differential-radius one (dotted black line), the constant-infinity one, with (b) $\widetilde{\mathrm{Inf}}=10^5$ (dashed red line) and (c) $\widetilde{\mathrm{Inf}}=10^8$ (solid blue line) for $F=0$ (left panel) and $F=1/3$ (right panel). For all models, we assumed ${\mathrm{\Omega }}_{m0}=0.3$.

Figure 3

Relative errors for $n_{\ln M}$ (see text for definition) between our method and the constant-infinity one, with (a) $\widetilde{\mathrm{Inf}}=10^8$ (solid blue line) and (b) $\widetilde{\mathrm{Inf}}=10^5$ (dashed red line) for redshift $z=0$ in the cases $F=0$ (left panel) and $F=1/3$ (right panel). For all models, we assumed ${\mathrm{\Omega }}_{m0}=0.3$ and $h=0.6774$.

Figure 4

Relative errors for $n_{\ln M}$ (see text for definition) between our method and the constant-infinity one, with (a) $\widetilde{\mathrm{Inf}}=10^8$ (solid blue line) and (b) $\widetilde{\mathrm{Inf}}=10^5$ (dashed red line) as a function of redshift for $F=0$, $F=1/3$ and virial mass of $10^{13}{h}^{-1}{M}_{\odot }$ (upper panels). The same plots, are shown for $10^{15}{h}^{-1}{M}_{\odot }$ (lower panels). For all models, we assumed ${\mathrm{\Omega }}_{m0}=0.3$ and $h=0.6774$.

Figure 5

$N_{\mathrm{bin}}$ (see text for definition) as a function of redshift for $F=0$ and virial masses between $10^{13}$ and $10^{14}{h}^{-1}{M}_{\odot }$ (left panel) and $10^{14}$ and $10^{15}{h}^{-1}{M}_{\odot }$ (right panel). As always, we assumed ${\mathrm{\Omega }}_{m0}=0.3$ and $h=0.6774$.

Figure 6

Relative errors for $N_{\mathrm{bin}}$ (see text for definition) between our method and the constant-infinity one, with (a) $\widetilde{\mathrm{Inf}}=10^5$ (dashed red line) and (b) $\widetilde{\mathrm{Inf}}=10^8$ (solid blue line) as a function of redshift for virial masses between $10^{13}$ and $10^{14}{h}^{-1}{M}_{\odot }$ and $F=0$ (left panel) and $F=1/3$ (right panel). For all models, we assumed ${\mathrm{\Omega }}_{m0}=0.3$ and $h=0.6774$.