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

We investigate spherically symmetric perfect-fluid spacetimes and discuss the existence and stability of a dividing shell separating expanding and collapsing regions. We perform a $3+1$ splitting and obtain gauge invariant conditions relating the intrinsic spatial curvature of the shells to the Misner-Sharp mass and to a function of the pressure that we introduce and that generalizes the Tolman-Oppenheimer-Volkoff equilibrium condition. We find that surfaces fulfilling those two conditions fit, locally, the requirements of a dividing shell, and we argue that cosmological initial conditions should allow its global validity. We analyze the particular cases of the Lema\^{}tre-Tolman-Bondi dust models with a cosmological constant as an example of a cold dark matter model with a cosmological constant (Λ-CDM model) and its generalization to contain a central perfect-fluid core. These models provide simple but physically interesting illustrations of our results.