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We determine the relationship between the turnaround radius, $R_{\rm t}$, andmass, $M_{\rm t}$, in $\Lambda$CDM, and in dark energy scenarios, using anextended spherical collapse model taking into account the effects of shear andvorticity. We find a more general formula than that usually described inliterature, 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 intoaccount shear, and rotation, especially at galactic scales, differing $\simeq30\%$ from the result given in literature. This has effects on the constraintof the $w$ parameter of the equation of state. We compare the $R_{\rm t}-M_{\rmt}$ relationship obtained for the $\Lambda$CDM, and different dark energymodels 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 scenariosare tantamount to the prediction of the $f(R)$ theories. Then, the $R_{\rmt}-M_{\rm t}$ relationship is not a good probe to test gravity theories beyondEinstein's general relativity.

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