Probing Modified Gravity with the Structures Maximum Dimensions

Abstract

By using an extended spherical collapse model taking into account the effects of shear, vorticity, and dynamical friction we found the relationship between the turnaround radius, $${{R}_{t}}$$ , and mass, $${{M}_{t}}$$ , in $$\Lambda $$ CDM and dark energy scenarios. We get a more general formula than that usually described in literature, showing a dependence of $${{R}_{t}}$$ from shear, vorticity and dynamical friction. The $${{R}_{t}}{-} {{M}_{t}}$$ relation differs from that obtained not taking into account shear, and rotation, and dynamical friction especially at galactic scales, differing for $$ \simeq {\kern 1pt} 40$$ % from the result given in literature. The theoretical turnaround–mass relation and the turnaround radius obtained from stable structures, can be used to put some constraints on the equation of state, $$w$$ , parameter. We compare the $${{R}_{t}}{-}{{M}_{t}}$$ relationship obtained for different dark energy models to that obtained in the $$f(R)$$ modified gravity (MG) scenario. The $${{R}_{t}}{-}{{M}_{t}}$$ relationship in dark energy scenarios are tantamount to the prediction of the $$f(R)$$ theories. Consequently, we must conclude that the $${{R}_{t}}{-}{{M}_{t}}$$ relationship is not a good probe to test gravity theories beyond Einstein’s general relativity.