LAUSR

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 LCDM 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 for turnaround to be allowed at typical epochs/length scales in almost-EdS and almost-LCDM models that allow inhomogeneous curvature. We find that gravitational collapse in a spatially exactly flat, irrotational, expanding, dust universe is relativistically forbidden pointwise. We explain why in the spatially flat plane-symmetric model considered here, pancake collapse is excluded both pointwise and in averaged domains. In an almost-EdS or LCDM model, the per-domain average curvature in collapsing domains almost always becomes strongly positive prior to turnaround, with the expansion-normalised curvature functional reaching $\Omega_{\cal R}^{\cal D} \sim -5$. We show analytically that a special case gives $\Omega_{\cal R}^{\cal D} = -5$ exactly (if normalised using the EdS expansion rate) at turnaround. An interpretation of LCDM as literally 3-Ricci flat would forbid structure formation. The fundamental difference between relativistic cosmology and a strictly flat LCDM model should not be ignored in cosmological N-body simulations and in analysing new surveys.