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.