Relation between the Turnaround radius and virial mass in $f(R)$ model
  We investigate the relationship between the turnaround radius ($R_t$) and the
virial mass ($M_v$) of cosmic structures in the context of $\Lambda$CDM model
and in an $f(R)$ model of modified gravity -- namely, the Hu-Sawicki model. The
$R_t$ is the distance from the center of the cosmic structure to the shell that
is detaching from the Hubble flow at a given time, while the $M_v$ is defined,
for this work, as the mass enclosed within the volume where the density is
$200$ times the background density. We consider that gravitationally bound
astrophysical systems follow a Navarro-Frenk-White density profile, while
beyond the virial radius ($R_v$) the profile is approximated by the 2-halo term
of the matter correlation function. By combining them together with the
information drawn from solving the spherical collapse for the structures, we
are able to connect two observables: the $R_t$ and the $M_v$. We show that, in
$\Lambda$CDM, the turnaround mass ($M_t$) at $z=0$ is related to the $M_v$ of
that same structure by $M_t \simeq 3.07 \, M_v$, while in terms of the radii we
have that $R_t \simeq 3.7 \, R_v$ (for $M_v$ of $10^{13} \, h^{-1} \,
M_\odot$). In the $f(R)$ model, we have $M_t \simeq 3.43 \, M_v$ and $R_t
\simeq 4.1 \, R_v$, for $|f_{R0}|=10^{-6}$ and the same mass scale. Therefore,
the difference between $\Lambda$CDM and $f(R)$ in terms of these observable
relations is of order $\sim 10-20\%$ even for a relatively mild strength of the
modification of gravity ($|f_{R0}|=10^{-6}$). For the $R_t$ itself we find a
difference of $\sim 9\%$ between the weakly modification in gravity considered
in this work ($|f_{R0}|=10^{-6}$) and $\Lambda$CDM for a mass of $10^{13} \,
h^{-1} \, M_\odot$. Once observations allow precisions of this order or better
in measurements $R_t$, as well as the $M_v$, these quantities will become
powerful tests of modified gravity.
