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 Einsteins general relativity.
تحميل البحث