Does spatial flatness forbid the turnaround epoch of collapsing
  structures?
  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 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 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 difference between relativistic
cosmology and a strictly flat LCDM model is fundamental in principle, but we
find that the geometrical effect is weak.
