Abstract
Cosmological observational analysis frequently assumes that the Universe is spatially flat. We aim to non-perturbatively check the conditions under which a flat or nearly flat expanding dust universe, including the Λ-cold-dark-matter (ΛCDM) model if interpreted as strictly flat, forbids the gravitational collapse of structure. We quantify spatial curvature at turnaround. We use the Hamiltonian constraint to determine the pointwise conditions required for an overdensity to reach its turnaround epoch in an exactly flat spatial domain. We illustrate this with a plane-symmetric, exact, cosmological solution of the Einstein equation, extending earlier work. More generally, for a standard initial power spectrum, we use the relativistic Zel'dovich approximation implemented in INHOMOG to numerically estimate how much positive spatial curvature is required to allow turnaround at typical epochs/length scales in almost-Einstein-de Sitter (EdS)/ΛCDM models with inhomogeneous curvature. We find that gravitational collapse in a spatially exactly flat, irrotational, expanding, dust universe is relativistically forbidden pointwise. In the spatially flat plane-symmetric model considered here, pancake collapse is excluded both pointwise and in averaged domains. In an almost-EdS/ΛCDM model, the per-domain average curvature in collapsing domains almost always becomes strongly positive prior to turnaround, with the expansion-normalised curvature functional reaching Ω
∼−5. We show analytically that a special case gives Ω=−5 exactly (if normalised using the EdS expansion rate) at turnaround. An interpretation of ΛCDM as literally 3-Ricci flat would forbid structure formation. The difference between relativistic cosmology and a strictly flat ΛCDM model is fundamental in principle, but we find that the geometrical effect is weak.