Constraining gravity theories with the gravitational stability mass

The measurement of the size of gravitationally bounded structures is an important test of gravity theories. For a given radius different theories can in fact predict a different gravitational stability mass (GSM) necessary to ensure the stability of the structure in presence of dark energy. We compute the GSM of gravitationally bounded structures as a function of the radius for different scalar-tensor theories, including $f\left(R\right)$ and generalized Brans-Dicke, and compare the theoretical predictions to observational data. Since the GSM only gives a lower bound, the most stringent constraints come few objects with a mass lower that the one expected in general relativity. The analysis of different observational data sets shows that modified gravity theories (MGT) are compatible with observational data, and in some cases fit the data better than general relativity (GR), but the latter is not in strong tension with the observations. The data presently available does not provide a statistically significant evidence of the need of a modification of GR, with the largest deviation of order $2.6\sigma$ for the galaxy cluster NGC5353/4. Due to the limited number of objects not satisfying the GR bound, for these structures it may be important to take into account non gravitational physics or deviations from spherical symmetry.