In this paper, we study how an extended spherical collapse model (ESCM) taking into account the interaction with external protostructures through shear and vorticity and screening modifies typical parameters of the collapse, such as the collapse threshold, ${\delta }_c(z)$ and ${\delta }_c(M)$, the turnaround density, ${\delta }_t$, the turnaround radius, $R_{\mathrm{t}}$, and halo mass function. Unlike the standard spherical collapse model (SSCM) which excludes shear and vorticity (quantities depending on mass), the collapse threshold ${\delta }_c$ depends on mass, redshift, and the parameter $|f_{\mathrm{R}0}|$ of the Hu-Sawicki $f_{\mathrm{R}}$ theory. We used this result to obtain the halo mass function by means of the excursion set approach. The main goal of the paper is to determine whether the quoted quantities and the mass function can disentangle between general relativity (GR) and the Hu-Sawicki $f(R)$ model. When considering the ESCM prediction for GR and the large field (LF) limit of the $f(R)$ theory, the percentage difference in ${\delta }_c$ is of a few percent and is around 20% if we consider the difference between the GR in the ESCM and the LF limit in the SSCM. In the case of ${\delta }_t$, the values are around 27% and 29% and for $R_t$ is around 6.5%. In the case of ${\delta }_c$ and for $|f_{R0}|={10}^{-4}$ (F4), the difference between the GR value of ${\delta }_c$ and the screened one is 0.25% and goes down to 0 for $|f_{R0}|={10}^{-6}$ (F6). In the case of ${\delta }_t$, they are larger than 20% for F4, $\simeq 10\%$ for F5, and 0% for F6 and in the case of the turnaround radius $\simeq 6.5\%$ (F4), $\simeq 3\%$ (F5), and $\simeq 0\%$ (F6). The observational constraints on $|f_{R0}|$ give values smaller than F4 ($|f_{R0}|\lesssim {10}^{-4.5}$, at large scales); thus, only F5 could be used to disentangle between GR and $f(R)$. Since F4 is excluded, $F6$ is practically equal to GR, and the LF is excluded because it is larger than F4. Unfortunately, as shown by some studies, the error on $R_t$, is of the order of 20%, excluding the possibility to use the turnaround radius, and related quantities, to disentangle between GR and $f(R)$. Concerning the mass function, the haloes number predicted by $f(R)$, especially at high redshift, can disentangle between $f(R)$ and GR.

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• Cosmology
• Dark energy
• Dark matter