Turnaround radius in $\Lambda$CDM, and dark matter cosmologies with shear and vorticity

We determine the relationship between the turnaround radius, $R_{\rm t}$, and mass, $M_{\rm t}$, in $\Lambda$CDM, and in dark energy scenarios, using an extended spherical collapse model taking into account the effects of shear and vorticity. We find a more general formula than that usually described in literature, showing a dependence of $R_{\rm t}$ from shear, and vorticity. The $R_{\rm t}-M_{\rm t}$ relation differs from that obtained not taking into account shear, and rotation, especially at galactic scales, differing $\simeq 30\%$ from the result given in literature. This has effects on the constraint of the $w$ parameter of the equation of state. We compare the $R_{\rm t}-M_{\rm t}$ relationship obtained for the $\Lambda$CDM, and different dark energy models to that obtained in the $f(R)$ modified gravity (MG) scenario. The $R_{\rm t}-M_{\rm t}$ relationship in $\Lambda$CDM, and dark energy scenarios are tantamount to the prediction of the $f(R)$ theories. Then, the $R_{\rm t}-M_{\rm t}$ relationship is not a good probe to test gravity theories beyond Einstein's general relativity.