Mathematicians from University College London and the University of California, Davis, have published a mathematical proof that the Universe’s accelerating expansion can be explained without dark energy, dealing a serious blow to the Lambda-cold dark matter model — the standard cosmological model that has stood for nearly 30 years.
Alexander et al. provide a mathematical proof that instabilities inherent in the Einstein-Euler equations imply that the current model of the expanding Universe is not viable. Image credit: M. Weiss / Harvard-Smithsonian Center for Astrophysics.
Almost 30 years ago, dark energy was proposed as the force responsible for the accelerating expansion of the Universe.
The idea harkens back to Albert Einstein’s original 1915, gravity-describing equations for general relativity.
To produce a static universe, Einstein initially introduced an antigravity factor in his theory. He called this factor the cosmological constant.
After Edwin Hubble discovered the Universe was expanding in 1929, Einstein famously called the cosmological constant his ‘biggest blunder’ because without it he could have predicted the expansion.
However, the cosmological constant, and the idea that it’s interchangeable with dark energy, was reintroduced to explain the Universe’s accelerating expansion in the 1990s.
“The Friedmann family of spacetimes has been the starting point for modern cosmology since Lemaitre and Hubble first formulated the theory of an expanding Universe of galaxies emanating from an initial Big Bang singularity,” said University of California, Davis Professor Blake Temple and colleagues.
“The theory is based on the explicit solutions of Einstein’s field equations discovered by Alexander Friedmann in the early 1920s.”
“As the story goes, in 1922 Friedmann sent his solutions to Einstein, who initially rejected them believing the Universe was static but shortly thereafter accepted them as correct under appeal by Friedmann.”
“By 1931, Einstein rejected the static model as unstable and accepting Hubble’s 1929 measurement of the expanding Universe, is famously quoted as describing Lemaitre’s cosmology based on Friedmann spacetimes as the most beautiful and satisfactory explanation of creation.”
“In our new paper, we give a theorem stating that the Friedmann spacetimes are in fact all unstable to radial perturbation, at every order.”
Professor Temple and co-authors pursued alternative explanations for the accelerating expansion of the Universe.
“Our first idea was that maybe the Universe was expanding because there was a shockwave, and the anomalous acceleration was the expanding wave behind that shockwave,” Professor Temple said.
“Then we realized there’s a family of self-similar solutions during the radiation epoch of the Big Bang, which might model that expanding wave.”
Self-similar equations describe physical phenomena that maintains a pattern or structure, regardless of its scale.
In their paper, the mathematicians use a self-similar version of the Einstein equations, which they derived in prior work, to represent the standard model of cosmology as a rest point of the equations.
This provides the framework for a complete mathematical characterization of the standard model’s stability, and more generally, the stability all Friedmann spacetimes during the matter-dominated epoch of the Big Bang.
“We prove that, like Einstein’s static model, the Friedmann spacetimes are all unstable to radial perturbation at large length scales,” Professor Temple said.
“This appears to rule out the Lambda-cold dark matter model as a viable stable solution of the Einstein equations of general relativity, with or without dark energy.”
“This means that the Big Bang should generically look exactly like a Friedmann spacetime near the center of symmetry, but generically one should observe accelerations away from Friedmann far from the center.”
The researchers found that the accelerating expansion of the Universe is a direct consequence of the Einstein-Euler equations without the insertion of a cosmological constant or dark energy.
The math also calls into question the Copernican principle — the idea that the Earth’s location does not occupy a special place in the Universe.
“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,” Professor Temple said.
“If this principle rules out one, it has to rule out the other.”
The paper was published this week in the Proceedings of the Royal Society A.
_____
C. Alexander et al. 2026. The instability of critical and underdense Friedmann spacetimes at the Big Bang as an alternative to dark energy. Proc. A 482 (2338): 20250912; doi: 10.1098/rspa.2025.0912
