Turnaround radius in $\Lambda$CDM, and dark matter cosmologies II: the
  role of dynamical friction
  This paper is an extension of the paper by Del Popolo, Chan, and Mota (2020)
to take account the effect of dynamical friction. We show how dynamical
friction changes the threshold of collapse, $\delta_c$, and the turn-around
radius, $R_t$. We find numerically the relationship between the turnaround
radius, $R_{\rm t}$, and mass, $M_{\rm t}$, in $\Lambda$CDM, in dark energy
scenarios, and in a $f(R)$ modified gravity model. Dynamical friction gives
rise to a $R_{\rm t}-M_{\rm t}$ relation differing from that of the standard
spherical collapse. In particular, dynamical friction amplifies the effect of
shear, and vorticity already studied in Del Popolo, Chan, and Mota (2020). A
comparison of the $R_{\rm t}-M_{\rm t}$ relationship for the $\Lambda$CDM, and
those for the dark energy, and modified gravity models shows, that the $R_{\rm
t}-M_{\rm t}$ relationship of the $\Lambda$CDM is similar to that of the dark
energy models, and small differences are seen when comparing with the $f(R)$
models. The effect of shear, rotation, and dynamical friction is particularly
evident at galactic scales, giving rise to a difference between the $R_{\rm
t}-M_{\rm t}$ relation of the standard spherical collapse of the order of
$\simeq 60\%$. Finally, we show how the new values of the $R_{\rm t}-M_{\rm t}$
influence the constraints to the $w$ parameter of the equation of state.
