According to quantum field theory (QFT), the framework that describes elementary particles and their interactions, empty space should be filled with quantum fluctuations that contribute an enormous amount of energy. In fact, calculations suggest the cosmological constant should be extraordinarily large, effectively approaching infinity.
Yet observations show something very different. The actual value of the cosmological constant is incredibly small compared with what theory predicts.
Now, researchers at Brown University have proposed a possible explanation.
Their work suggests that a mathematical feature of space-time itself may prevent the cosmological constant from ballooning to the huge values expected from quantum physics. The idea draws on an unexpected connection between quantum gravity and the quantum Hall effect, a remarkable phenomenon in condensed matter physics.
A Surprising Link Between Quantum Gravity and the Quantum Hall Effect
The team found that the mathematics behind a simple approach to quantum gravity closely resembles the mathematics that describes the quantum Hall effect, an unusual state of matter in which electrical conductance takes on highly precise values.
In the quantum Hall effect, those values remain fixed even when the conducting material contains imperfections. The stability comes from topology, a branch of mathematics concerned with the underlying "shape" or structure of a system.
The researchers argue that a similar type of topology appears in the Chern-Simons-Kodama state, a proposed ground state of quantum gravity.
"What we've shown is that if space-time has this non-trivial topology, then it resolves one of the deadliest problems of the cosmological constant," said study co-author Stephon Alexander, a professor of physics at Brown. "All the quantum perturbations that should blow up the value of the cosmological constant are rendered inert by this topology, which keeps the constant's value stable."
The study, co-authored by Alexander and Brown Theoretical Physics Center colleagues Aaron Hui and Heliudson Bernardo, was published in Physical Review Letters.
Einstein's "Ugly" Cosmological Constant
The cosmological constant first appeared in Albert Einstein's equations of general relativity, his theory of space, time, and gravity.
At the time, Einstein believed the universe was static. To keep his equations from predicting a collapsing universe, he introduced the cosmological constant as a kind of repulsive effect in empty space that counterbalanced gravity.
That idea seemed unnecessary after Edwin Hubble discovered in 1929 that the universe was expanding. Since the cosmos was not static after all, Einstein removed the term from his equations. He reportedly disliked the constant and later referred to it as his "biggest blunder."
For decades, the cosmological constant largely faded from prominence.
Then, in 1998, astronomers discovered something surprising: the expansion of the universe is speeding up. Rather than disappearing from the story, the cosmological constant suddenly became essential again because it could account for this accelerating expansion.
The Cosmological Constant Problem
The revival of the cosmological constant created a serious problem.
During the years when the constant had fallen out of favor, quantum field theory had become one of the most successful theories in science and a cornerstone of the Standard Model of particle physics.
QFT describes empty space as anything but empty. Instead, it is filled with particles constantly appearing and disappearing through quantum fluctuations.
All of this activity should contribute a vast amount of vacuum energy. That vacuum energy is associated with the cosmological constant, which means the constant should be extraordinarily large.
But observations show that it is not.
If the cosmological constant were as large as QFT predicts, the universe would have expanded so rapidly that galaxies, stars, planets, and ultimately life could never have formed.
The mismatch between theory and observation remains one of the most perplexing problems in modern physics. The puzzle is made even more striking because experiments have repeatedly confirmed the extraordinary accuracy of quantum field theory in other contexts.
A Topological Solution
Alexander has spent years studying Chern-Simons-Kodama (CSK) theory, a proposed quantum gravity state that emerges from quantum field theory.
Physicists still lack a complete quantum theory of gravity that describes gravity at the smallest scales. According to Alexander, the CSK approach is among the more straightforward possibilities.
"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 long noticed similarities between CSK theory and the mathematics of the quantum Hall effect. To better understand those connections, he collaborated with Hui, an assistant professor at Brown who studies topological systems.
"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."
How Topology Creates Stability
The researchers found that the cosmological constant in the CSK framework appears to benefit from the same kind of topological protection seen in the quantum Hall effect.
The quantum Hall effect occurs when electricity flows through extremely thin materials exposed to a magnetic field.
Imagine a thin rectangular strip of metal carrying an electric current. When a magnetic field is applied, a second voltage develops at right angles to the current. This effect produces what is known as a Hall voltage (named after Edwin Hall, who discovered it).
Under ordinary conditions, the Hall voltage changes smoothly as the magnetic field increases.
Under extremely cold temperatures and very strong magnetic fields, however, the behavior changes dramatically. Instead of varying smoothly, the Hall voltage increases in distinct steps and plateaus. Remarkably, those values remain identical regardless of the material being used or any imperfections it contains.
That reliability comes from topology.
In these extreme conditions, electrons behave collectively and enter a highly correlated quantum state. The topology of that state fixes the values of the steps and plateaus, making them resistant to disturbances and defects.
The Brown researchers argue that an analogous process occurs in the CSK description of quantum gravity.
Just as topology locks the Hall voltage into specific values, the topology of space-time could lock the cosmological constant into stable values, protecting it from the quantum fluctuations that would otherwise drive it much higher.
"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."
A New Direction for Quantum Gravity
Alexander emphasizes that much more work is needed before a topological explanation of the cosmological constant can be fully established.
Still, he believes the findings represent an important step toward solving the gravitational side of the problem. The work also strengthens the case for the CSK state as a serious candidate for a future 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." %!s()
