On the maximum volume of collapsing structures

In many cosmological models, including the ΛCDM concordance model, there exist theoretical upper bounds on the size of collapsing structures. The most common formulations in the literature refer to a turnaround radius in spherical symmetry or a turnaround surface, defined as the zero-expansion boundary separating the outer Hubble flow from the inner flow of a collapsing fluid. In order to access a generic scenario, we propose an improvement of this cosmological test in terms of the maximum volume of the cosmological structures, which is equivalent to a zero-averaged expansion - instead of the zero-local expansion. By combining the Lagrangian perturbations method and the scalar averaging of Einstein's equations, we obtain a maximum volume for a collapse model without any restricting symmetries. We compare this result with some exact, inhomogeneous solutions and discuss further potential developments.