Could the mathematical ‘shape’ of the universe solve the cosmological constant problem

The "ugly" term

The cosmological constant first appeared as a term in the equations describing Einstein's canonical theory of space, time and gravity, known as general relativity. Einstein was forced to introduce the term to make his mathematical universe stable. It represented a repulsive force, present in the vacuum of space, that counteracted the force of gravity and kept the universe from collapsing on itself.

In 1929, however, the cosmological constant was dealt an existential blow. Astronomer Edwin Hubble discovered that the universe was not as stable as Einstein had assumed. Rather than holding static, it was expanding. That discovery allowed Einstein to remove the stabilizing term from his equations, which he did with some relief. He had long viewed it as "ugly" and is purported to have called it his "biggest blunder."

Following Hubble's discovery, the cosmological constant spent about a half-century on the scientific scrap heap. That changed in 1998, however, when scientists discovered that the universe's expansion is not happening at a constant rate; it's accelerating. That discovery once again made the cosmological constant necessary to describe the increasing speed of the universe's expansion.

Not only was Einstein's ugly term back, it was uglier than ever. During the constant's exile, quantum field theory had become the backbone of the Standard Model of particle physics. According to QFT, empty space is not empty at all. Rather, it's a boiling soup of elementary particles constantly popping in and out of existence. All that activity should cause the vacuum energy of space - the energy described by the cosmological constant - to be practically infinite. Yet its observed value, which is estimated by the rate of cosmic expansion, is most definitely not infinity. An infinite value would cause the universe to expand far too quickly to allow the formation of things like galaxies, planets or physicists.

Experiments with elementary particles have shown QFT to be among the most precise and successful theories in all of science, which makes its seemingly errant predictions about the cosmological constant all the more puzzling.

Topologically protected

Alexander has spent years studying Chern-Simons-Kodama (CSK) theory, a proposed state of quantum gravity that grows out of quantum field theory. Scientists have yet to settle on a quantum theory of gravity - a theory that explains how gravity works at the tiniest scales - but the CSK state is one of the more straightforward candidates, according to Alexander.

"It's a really conservative approach to quantizing gravity," he said. "This is the approach used by people like Dirac, Schrödinger and Wheeler. It's just good, old-fashioned quantization."

Alexander had been aware of some mathematical similarities between CSK and the math behind the quantum Hall effect, but he wasn't entirely sure what to make of them. That's when he turned to Hui, an assistant professor at Brown who specializes in topological systems like those that emerge in the quantum Hall effect.

"This is the beauty of the Brown Theoretical Physics Center," Alexander said. "We want to be a place where there's a mixing of lots of perspectives, and this is us practicing what we preach - a cosmologist working closely with a condensed matter theorist."

Together, the researchers were able to show that the cosmological constant has a similar "topological protection" in the CSK state as electrical conductivity has in the quantum Hall effect. The quantum Hall effect emerges when electricity flows through very thin materials in the presence of a magnetic field. Imagine a flat, two-dimensional piece of metal cut into a rectangular strip with an electric current running longways down the strip. Introducing a magnetic field produces a second voltage that runs perpendicular to the original current. This is known as a Hall voltage (named after Edwin Hall, who discovered it).

At room temperature and under relatively weak magnetic fields, the Hall voltage increases linearly as the strength of the magnetic field increases. But at very cold temperatures, where the rules of quantum mechanics dominate, and under very strong magnetic fields, the phenomenon changes. Rather than increasing linearly with magnetic field strength, the Hall voltage starts to increase in discrete (or quantized) steps and plateaus. The steps and plateaus are incredibly precise and consistent, taking the exact same values regardless of the type of metal used as a conductor or whether there happen to be any imperfections in it.

That precision and consistency arise because of the system's topology. In these extreme conditions, electrons enter a highly correlated state of collective behavior. It's the mathematical structure of that collective state - its topology - that locks the values of the steps and plateaus into place. The system is topologically protected from perturbations from the material and its imperfections, so the steps and plateaus always have the same value.

The researchers show that a very similar topological protection is present in the equations describing the CSK state. Just as the topology of the electron states locks the Hall voltage into place, the topology of space-time itself locks the cosmological constant into place, even in the face of quantum fluctuations in the vacuum of space.

"What we find is that this quantization of the electrical conductance in quantum Hall has an analog with the cosmological constant," Hui said. "It also ends up becoming quantized for topological reasons. There turn out to be constraints in the theory that force the cosmological constant to take certain allowed quantized values."

There's much more work to be done to fully flesh out a topological solution to the cosmological constant problem, Alexander says. But finding a potential solution to the gravitational aspect of the problem is a crucial start. At the very least, he says, the work bolsters the profile of the CSK state as a candidate for a long-sought theory of quantum gravity.

"We took something old, which is this conservative, canonical approach to quantum gravity, and discovered something new that had been there all along," Alexander said. "Now we're working on a bigger picture of how this phenomenon works."