Einstein's equations may explain the universe without dark energy

Mathematicians are raising a major challenge to one of modern cosmology's most widely accepted explanations: the idea that dark energy is driving the accelerating expansion of the universe.

A study published in Proceedings of the Royal Society A presents mathematical findings that researchers at the University of California, Davis, say reveal a fundamental instability in the standard model of the expanding universe. Their calculations suggest that cosmic acceleration could emerge naturally from Einstein's equations, potentially eliminating the need to invoke mysterious dark energy.

The work challenges the Lambda-cold dark matter model, the leading framework scientists use to describe the universe's evolution since the Big Bang. The researchers reached their conclusions by studying the Einstein-Euler equations, which combine Einstein's theory of general relativity with equations describing how fluids behave. Together, these mathematical tools help scientists investigate everything from galaxies and black holes to the expansion of the cosmos.

To explain the problem, Blake Temple, the study's corresponding author and a distinguished professor emeritus of mathematics at UC Davis, offered a simple comparison: a pencil balanced vertically on its tip.

"All the forces are in balance when a pencil is standing on end, so it is a 'solution of the equations,'" he said. "But it's unstable. Any breath of air and it falls away."

According to Temple, the team's mathematical analysis shows that Friedmann spacetimes, which form the foundation of conventional models of cosmic expansion, are unstable at both small and large scales in the context of the Big Bang. He described the standard solution as exceptionally unstable.

This instability could have far-reaching consequences. If the universe naturally evolves away from the behavior predicted by these conventional models, its accelerating expansion might not require a separate form of energy to explain it. Instead, Temple argues, the answer could already be contained within Einstein's original theory of gravity.

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

For nearly three decades, dark energy has been the leading explanation for one of astronomy's most puzzling discoveries: the universe is not simply expanding, but its expansion is accelerating.

The concept has roots in Einstein's groundbreaking 1915 theory of general relativity, which describes gravity as the curvature of space and time. Einstein originally believed the universe was static, neither expanding nor contracting. To make his equations compatible with that assumption, he introduced a mathematical term that counteracted gravity, calling it the "cosmological constant."

That assumption changed after Edwin Hubble discovered evidence in 1929 that the universe was expanding. Einstein famously referred to his introduction of the cosmological constant as his "biggest blunder," since his original equations without the additional term could have predicted an expanding universe.

The cosmological constant returned to prominence in the 1990s, when astronomers discovered that cosmic expansion was speeding up. It became closely associated with dark energy, the name given to the unknown component thought to be responsible for this acceleration.

Today's standard cosmological framework builds on the Friedmann universe, a mathematical description in which space expands while matter remains evenly distributed across the cosmos at any given moment. This assumption of large-scale uniformity plays a central role in how scientists reconstruct the universe's history.

But Temple and his collaborators began questioning whether the underlying mathematics could remain stable, prompting them to investigate other possible explanations for the observed acceleration.

The researchers initially considered whether a massive shockwave might help explain the universe's expansion and the unexpected acceleration associated with it.

"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," 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."

In mathematics and physics, self-similar solutions describe patterns that retain their underlying structure when viewed at different scales. A familiar example is a fractal, in which similar shapes can appear at increasingly larger or smaller sizes.

The researchers identified a family of these solutions associated with the radiation-dominated period of the early universe, when radiation played a major role in determining how the cosmos evolved.

Building on their earlier work, they used a self-similar form of Einstein's equations to represent the standard cosmological model as a rest point, a mathematical state in which the system's defining variables remain unchanged. This approach allowed them to examine whether small departures from that state would disappear or grow over time.

Their analysis provided a mathematical framework for evaluating the stability of the standard model and, more broadly, Friedmann spacetimes during the matter-dominated era of the early universe, when matter became more influential than radiation.

The results suggest that the conventional mathematical descriptions of the expanding universe are vulnerable to disturbances that grow rather than fade.

"We prove that, like Einstein's static model, the Friedmann spacetimes are all unstable to radial perturbation at large length scales," 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."

Radial perturbations are disturbances that change with distance from a central point. In the researchers' mathematical framework, such changes can cause the behavior of an expanding universe to depart from the predictions of the standard Friedmann model.

"This means," he added, "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 argue that this instability may offer a natural route to accelerated cosmic expansion without adding a cosmological constant or invoking dark energy. Their calculations indicate that the acceleration can emerge directly from the Einstein-Euler equations themselves.

The findings represent a theoretical challenge to standard cosmology rather than observational proof that dark energy is unnecessary. Establishing whether this alternative can describe the actual universe would require determining how well its predictions match astronomical observations.

Named after astronomer Nicolaus Copernicus, who helped overturn the belief that Earth sits at the center of the solar system, the principle holds that our location in the universe is not fundamentally special or privileged.

This assumption has become an important part of modern cosmology, supporting the view that the universe looks broadly similar when observed from different locations on sufficiently large scales.

The researchers' alternative mathematical framework introduces a center of spherical symmetry, raising questions about where an observer would need to be located for the model to agree with observations.

"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."

The work opens another avenue for investigating why the universe's expansion is accelerating. Although dark energy remains a central component of the prevailing cosmological model, the researchers contend that the mathematics of general relativity may offer a different explanation for the same phenomenon.

The research received funding from the United Kingdom's Engineering and Physical Sciences Research Council and the American Institute of Mathematics SQuaREs Program.