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Mathematicians from the University of California, Davis, are challenging the idea that dark energy is responsible for the accelerating expansion of the universe also known as cosmic expansion
In a study published in the Proceedings of the Royal Society A, the researchers provide mathematical proof that instabilities inherent in the Einstein-Euler equations imply that the current model of the expanding universe is not viable.
These equations combine general relativity and fluid dynamics to model astronomical phenomena such as galaxies, black holes, and cosmic expansion.
The research directly challenges the Lambda-cold dark matter model, which operates as the standard cosmological model of the Big Bang.
The instability of Friedmann spacetimes
Study corresponding author Blake Temple, a distinguished professor emeritus of mathematics at UC Davis, compared the standard cosmological model to a pencil standing on its tip.
“All the forces are in balance when a pencil is standing on end, so it is a ‘solution of the equations,’” Temple said. “But it’s unstable. Any breath of air and it falls away.”
The mathematics prove that Friedmann spacetimes, the mathematical models that govern cosmic expansion, are unstable at both small and large length scales at the Big Bang. This makes it the most unstable solution of all.
Temple noted that unstable solutions are considered non-physical in science because they are never observed in nature. This instability suggests a simpler, more natural explanation for the acceleration of the universe based entirely within the original framework of Einstein’s theory, without the need for dark energy.
Explaining cosmic expansion without dark energy
Dark energy was proposed nearly 30 years ago to explain why the expansion of the universe is accelerating. This idea connects back to Albert Einstein’s 1915 equations for general relativity. To produce a model of a static universe, Einstein originally introduced an antigravity factor called the “cosmological constant.” After Edwin Hubble discovered the universe was expanding in 1929, Einstein removed it, calling the constant his “biggest blunder.”
However, the cosmological constant was reintroduced in the 1990s as a proxy for dark energy. Standard cosmological models are based on a “Friedmann universe,” which describes all matter as expanding while remaining evenly distributed throughout space at any fixed time.
The mathematics did not add up for Temple and his colleagues, leading them to pursue alternative explanations. They realised that a family of self-similar solutions during the radiation epoch of the Big Bang might model an expanding wave. Self-similar equations describe physical phenomena that maintain a consistent pattern regardless of scale.
Rethinking the Copernican principle
In the new paper, the mathematicians use a self-similar version of the Einstein equations, derived in their prior work, to represent the standard model of cosmology as a rest point. This provides a framework for a complete mathematical characterisation of stability during the matter-dominated epoch of the Big Bang.
The team proved that, like Einstein’s static model, Friedmann spacetimes are all unstable to radial perturbation at large length scales. 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.
As a result, the Big Bang should look exactly like a Friedmann spacetime near the centre of symmetry, but observers should see accelerations away from Friedmann far from that centre. This means the accelerating expansion of the universe is a direct consequence of the Einstein-Euler equations without inserting a cosmological constant or dark energy.
The mathematical findings also call into question the Copernican principle, which states that Earth’s location does not occupy a special place in the universe. Temple noted that 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. If the principle rules out one model, it must logically rule out the other.
The study was funded by the United Kingdom’s Engineering and Physical Sciences Research Council and the American Institute of Mathematics SQuaREs Program.