Does spatial flatness forbid the turnaround epoch of collapsing structures?

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 D R ∼ −5. We show analytically that a special case gives Ω D R = −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.

A few numbers from the turnaround epoch of collapse

“…In this work we extended the analysis in [3] by relaxing the II i D = III i D = 0 condition. Our results show that the density threshold needed for the collapsing domain to reach the turnaround is lower ( ρ D /ρ H = 4) than the one obtained from Eulerian perturbation theory ( ρ D /ρ H ≈ 5.55) and is scale-independent.…”

“…In our examination of gravitational collapse we will restrict ourselves to the turnaround epoch i.e. a moment when initially expanding domain starts collapsing and departing from Hubble expansion, extending the derivation in [3].…”

Theoretical cosmology

“…The perturbations are allowed to have non-zero spatial curvature. Initially, the negative curvature of underdense regions is compensated by the positive curvature of overdense regions [231,106]. But once the evolution enters the non-linear regime, this symmetry is broken and the mean spatial curvature of the universe slowly drifts from zero towards negative curvature induced by cosmic voids (which occupy more volume than other regions).…”

The role of the elaphrocentre in void galaxy formation

“…What is effectively antigravity in voids -in comparison to the surroundings -is unlikely to be well modelled by a Newtonian approximation. Indeed, relativistically, to reach turnaround, an overdensity has to pass through a strongly positive spatial curvature phase (Roukema & Ostrowski 2019;Ostrowski 2019;Vigneron & Buchert 2019), after which it virialises at an overdensity of a few hundred times the mean density (e.g. Lacey & Cole 1993).…”