[Paper Review] Is backreaction in cosmology a relativistic effect? On the need for an extension of Newton's theory to non-Euclidean topologies

Quentin Vigneron|arXiv (Cornell University)|Sep 21, 2021

Cosmology and Gravitation Theories24 references6 citations

TL;DR

This paper challenges the widely held view that cosmological backreaction is purely a relativistic effect by showing that, in a Newtonian theory extended to non-Euclidean topologies, backreaction can be non-zero. Using a heuristic model on a 3-torus with spatial curvature, the authors demonstrate that inhomogeneities can influence global expansion even without general relativity, implying backreaction might be non-relativistic depending on the universe's topology. The key contribution is the need to develop a non-Euclidean Newtonian cosmology compatible with the non-relativistic limit of general relativity.

ABSTRACT

Cosmological backreaction corresponds to the effect of inhomogeneities of structure on the global expansion of the Universe. The main question surrounding this phenomenon is whether or not it is important enough to lead to measurable effects on the scale factor evolution eventually explaining its acceleration or the Hubble tension. One of the most important result on this subject is the Buchert-Ehlers theorem (Buchert \& Ehlers, 1997) stating that backreaction is exactly zero when calculated using Newton's theory of gravitation, which may not be the case in general relativity. It is generally said that this result implies that backreaction is a purely relativistic effect. We will show that this is not necessarily the case, in the sense that this implication does not apply to a universe which is still well described by Newton's theory on small scales but has a non-Euclidean topology. The theorem should therefore be generalised to account for such a scenario. In a heuristic calculation where we construct a theory which is locally Newtonian but defined on a non-Euclidean topology, we show that backreaction is non-zero, meaning that it might be non-relativistic depending on the topological class of our Universe. However, that construction is not unique and remains to be justified from a non-relativistic limit of general relativity.

Motivation & Objective

• To challenge the assumption that cosmological backreaction is exclusively a relativistic phenomenon.
• To identify the limitations of the Buchert-Ehlers theorem in non-Euclidean topologies.
• To argue for the necessity of extending Newton’s theory to non-Euclidean spatial geometries to properly assess backreaction.
• To motivate the development of a non-Euclidean Newtonian cosmology compatible with the non-relativistic limit of general relativity.
• To propose a heuristic framework where backreaction arises in a Newtonian setting on curved spatial manifolds.

Proposed method

• Formulating a Newtonian cosmology on a 3-torus with non-zero spatial Ricci curvature, using a modified Poisson equation with a curvature-dependent term.
• Introducing a Hubble flow vector field on a non-Euclidean manifold via a Galilean structure, ensuring local Newtonian behavior.
• Applying the Buchert averaging formalism to the Raychaudhuri equation in this curved Newtonian framework to compute the backreaction term QΣ.
• Deriving a modified gravitational potential equation: ΔΦ = 4πG(ρ−⟨ρ⟩Σ) − (R/3)(PcPc − ⟨PcPc⟩Σ), which includes a curvature-induced nonlinearity.
• Comparing two distinct non-Euclidean Newtonian models: one with curvature in the kinematical equations and one with curvature in the gravitational equations, both locally Newtonian but globally inequivalent.
• Proposing that a consistent non-Euclidean Newtonian theory must emerge from the Galilean limit of Lorentzian spacetimes, using Newton-Cartan geometry as a foundation.

Experimental results

Research questions

• RQ1 Can cosmological backreaction be non-zero in a Newtonian theory if the universe has a non-Euclidean spatial topology?
• RQ2 Is the Buchert-Ehlers theorem's conclusion—that backreaction vanishes in Newtonian gravity—invalid when applied to non-Euclidean topologies?
• RQ3 What are the implications for cosmology if backreaction arises even in a non-relativistic, locally Newtonian theory on curved spatial manifolds?
• RQ4 How can a consistent non-Euclidean extension of Newton's theory be constructed that reduces to Newtonian gravity on small scales but incorporates spatial curvature?
• RQ5 Can such a theory be derived from general relativity via a non-relativistic limit, specifically the Galilean limit of Lorentzian spacetimes?

Key findings

• In a heuristic model on a 3-torus with non-zero Ricci curvature, the backreaction term QΣ is non-zero, indicating that inhomogeneities can influence global expansion even in a Newtonian framework.
• The standard Buchert-Ehlers theorem assumes a Euclidean topology, which limits its applicability to universes with non-Euclidean spatial geometry.
• A non-Euclidean Newtonian theory with curvature in the kinematical equations leads to a modified Poisson equation with a nonlinear curvature term, making N-body simulations infeasible.
• Two distinct non-Euclidean Newtonian models were found to be locally Newtonian but globally inequivalent, showing the non-uniqueness of such extensions.
• The Newton-Cartan formalism provides a promising framework for constructing a consistent non-Euclidean Newtonian cosmology compatible with general relativity in the c→∞ limit.
• The paper concludes that backreaction may not be inherently relativistic, but depends on the topological class of the universe, necessitating a new non-Euclidean Newtonian theory derived from general relativity.