# Does spatial flatness forbid the turnaround epoch of collapsing structures?

## TL;DR

This paper investigates whether spatial flatness in cosmological models prevents the gravitational collapse of structures, finding that exact flatness forbids collapse pointwise, but slight deviations allow it, with implications for interpreting LCDM.

## Contribution

It provides a non-perturbative analysis of the conditions under which spatial flatness forbids structure formation, using exact solutions and relativistic approximations.

## Key findings

- Exact spatial flatness forbids collapse pointwise.
- Almost-flat models exhibit strongly positive curvature before turnaround.
- Strictly flat LCDM interpretation would prevent structure formation.

## 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 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.