Friedmann spacetimes are unstable solutions — and that may rule out the standard model of the universe

A new mathematical proof published in Proceedings of the Royal Society A argues that the equations governing cosmic expansion are fundamentally unstable — and that dark energy may never have been needed at all.

A model standing on one end

For nearly three decades, dark energy has been the cosmological community’s answer to one of the most disquieting observations in modern astronomy: the universe is not just expanding, it is accelerating. Something had to be pushing space apart at an increasing rate, and dark energy — a mysterious repulsive force encoded in Einstein’s field equations as the cosmological constant — became the standard explanation.

Now, mathematicians from the University of California, Davis, are challenging that assumption from a strictly formal angle. In a paper published in Proceedings of the Royal Society A, Blake Temple, distinguished professor emeritus of mathematics at UC Davis, and his co-authors present mathematical proof that the Friedmann spacetimes underpinning the standard cosmological model are inherently unstable solutions of the Einstein-Euler equations — the mathematical union of general relativity and fluid dynamics used to describe the structure and evolution of the cosmos.

The implication is significant: if the solution is unstable, it cannot be physical. Nature does not sustain unstable equilibria.

Temple illustrated the point with a memorable image. A pencil balanced on its tip satisfies all the equations governing its forces. It is technically a solution. But it is not a solution you will ever encounter in reality, because the slightest perturbation sends it falling. The Friedmann spacetimes, Temple argues, are cosmological pencils.

What the Einstein-Euler equations actually say

The Einstein-Euler equations combine the curvature of spacetime — as described by general relativity — with the behavior of matter as a fluid. They are the standard tool for modeling large-scale cosmic phenomena, from galaxies to black holes to the global expansion of space.

The Lambda-cold dark matter model, or ΛCDM, is the prevailing framework built on top of these equations. It treats the universe’s matter as uniformly distributed at any given moment in time and expanding outward from the Big Bang. The cosmological constant Λ — which Einstein introduced to force a static universe before Hubble’s 1929 discovery of expansion, and which he later dismissed as his «biggest blunder» — was reintroduced in the 1990s to account for the observed acceleration.

Temple and his colleagues used a self-similar reformulation of the Einstein equations — one they had developed in prior work — to represent ΛCDM as a fixed point, or rest point, of those equations. This approach allowed them to perform a complete stability analysis of the standard model during the matter-dominated epoch of the Big Bang.

The result: all Friedmann spacetimes are unstable to radial perturbations at both small and large length scales. The model that has guided cosmology for thirty years may be, in a precise mathematical sense, not a valid physical solution.

Acceleration without dark energy

The team’s original motivation was different. Temple recounted that their starting point was the idea that the universe’s accelerating expansion might be a consequence of an outward-propagating shockwave from the Big Bang, with the anomalous acceleration appearing in the expanding wave behind it. That line of investigation led them to a family of self-similar solutions during the radiation epoch, which they then used to probe the stability of the standard model.

What they found is that the instability of Friedmann spacetimes to accelerated expansion suggests that the acceleration itself could be a natural geometric consequence of the Einstein equations — with no need to invoke dark energy or the cosmological constant. The acceleration, in other words, may already be embedded in the structure of general relativity, emerging once the unstable Friedmann background is perturbed in physically realistic ways.

«The instability of all Friedmann spacetimes to accelerated expansion suggests a simpler, more natural explanation for the acceleration of the universe than dark energy,» Temple said.

The Copernican principle under pressure

The paper carries an additional, philosophically provocative consequence. The Copernican principle — one of the foundational assumptions of modern cosmology — holds that Earth does not occupy a special position in the universe. Cosmological models are generally required to be consistent with this principle; a model that only works if we happen to be near the center of a spherically symmetric structure would be considered physically implausible.

Temple notes, however, that both the ΛCDM model and a spherically symmetric spacetime require a special point — a center of symmetry — for the model to be physically viable. If the Copernican principle disqualifies the latter, it must apply equally to the former.

«Both the Lambda-cold dark matter model and a spherically symmetric spacetime produce a special place where we must lie for the model to be physically plausible,» Temple said. «If this principle rules out one, it has to rule out the other.»

This does not mean the Copernican principle is wrong. It means that the standard model, as currently formulated, may rest on a tension with its own assumptions — a tension that the mathematical instability proof makes unavoidable to confront.

What comes next

This paper does not offer a finished alternative cosmology. It is a proof of instability, not a replacement model. What Temple and his co-authors have established is a formal mathematical result: ΛCDM, expressed as a rest point of the self-similar Einstein-Euler equations, is not a stable physical solution. The standard framework must be reconsidered.

Whether the replacement involves modified gravity, different initial conditions, or a geometric reinterpretation of the Big Bang’s shockwave structure remains an open problem. But the result places the burden of proof squarely on those who would defend dark energy as a necessary ingredient of cosmology — and opens the door to simpler explanations grounded in general relativity alone.

The work was published as: C. Alexander et al, «The instability of critical and underdense Friedmann spacetimes at the Big Bang as an alternative to dark energy,» Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences (2026). DOI: 10.1098/rspa.2025.0912

© 2026 SKYCR.ORG | Homer Dávila Gutiérrez, FRAS. All rights reserved. Total or partial reproduction prohibited without express authorization. More information: University of California, Davis / Proceedings of the Royal Society A (2026). DOI: 10.1098/rspa.2025.0912