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We investigate spherically symmetric perfect-fluid spacetimes and discuss theexistence and stability of a dividing shell separating expanding and collapsingregions. We perform a 3+1 splitting and obtain gauge invariant conditionsrelating the intrinsic spatial curvature of the shells to the Misner-Sharp massand to a function of the pressure that we introduce and that generalizes theTolman-Oppenheimer-Volkoff equilibrium condition. We find that surfacesfulfilling those two conditions fit, locally, the requirements of a dividingshell and we argue that cosmological initial conditions should allow its globalvalidity. We analyze the particular cases of the Lema\^itre-Tolman-Bondi dustmodels with a cosmological constant as an example of a cold dark matter modelwith a cosmological constant (\Lambda-CDM) and its generalization to contain acentral perfect-fluid core. These models provide simple, but physicallyinteresting illustrations of our results.

physical review. d. particles, fields, gravitation, and cosmology/physical review. d, particles, fields, gravitation, and cosmology

Separating expansion from contraction in spherically symmetric models  with a perfect-fluid: Generalization of the Tolman-Oppenheimer-Volkoff  condition and application to models with a cosmological constant