Abstract This paper explores the cosmological implications of a scalar field with a specific potential, crucial for achieving the final equilibrium state of gravitational collapse.We consider a system with two fluids: minimally coupled matter representing dust-like dark matter and a scalar field acting as dark energy.Our model, akin to General the top-hat collapse model, focuses on isolated over-dense regions within a closed FLRW metric, while music box the background follows a flat FLRW metric.We analyze spacetime configurations where these regions undergo initial expansion followed by contraction, deriving the scalar field potential responsible for their equilibrium state.
Our fully relativistic approach offers a comprehensive understanding of stable cosmic over-dense regions, without the need for ad-hoc Newtonian virialization.