Turnaround radius in $f(R)$ model

We investigate the turnaround radius in the spherical collapse model, both in General Relativity and in modified gravity, in particular $f(R)$ scenarios. The phases of spherical collapse are marked by the density contrast in the instant of turnaround ${\delta }_t$, and by the linear density contrast in the moment of collapse, ${\delta }_c$. We find that the effective mass of the extra scalar degree of freedom which arises in modified gravity models has an impact on ${\delta }_t$ of up to $\sim 10\mathrm{\%}$, and that ${\delta }_c$ can increase by $\sim 1.0\mathrm{\%}$. We also compute the turnaround radius, $R_t$, which in modified gravity models can increase by up to $\sim 6\mathrm{\%}$ at $z\simeq 0$.