One of the biggest unsolved problems in physics revolves around a number known as the cosmological constant. This value describes the energy responsible for the accelerating expansion of the universe. It is also at the heart of a major conflict between two of scientific theories’ most successful theories.
According to quantum field theory (QFT), the framework that describes elementary particles and their interactions, empty space must be filled with quantum fluctuations that contribute a huge amount of energy. In fact, calculations indicate that the cosmological constant must be very large, actually approaching infinity.
However, the observations show something completely different. The actual value of the cosmological constant is incredibly small compared to what theory predicts.
Now, researchers at Brown University have proposed a possible explanation.
Their work suggests that a mathematical property of spacetime itself may prevent the cosmological constant from inflating to the huge values expected from quantum physics. The idea is based on an unexpected connection between quantum gravity and the quantum Hall effect, an observed phenomenon in condensed matter physics.
A surprising link between quantum gravity and the quantum Hall effect
The team found that the mathematics behind the simple approach to quantum gravity is very similar to the mathematics that describes the quantum Hall effect, an unusual state of matter in which electrical conductivity takes on very precise values.
In the quantum Hall effect, these values remain constant even when the conducting material contains defects. Stability comes from topology, a branch of mathematics concerned with the basic “shape” or structure of a system.
The researchers see a similar type of topology appearing in the Chern-Simons-Kodama state, a proposed ground state for quantum gravity.
“What we have shown is that if spacetime has this non-trivial topology, it solves one of the most serious problems of the cosmological constant,” said study co-author Stephon Alexander, a professor of physics at Brown University. “All quantum perturbations that should explode the value of the cosmological constant become inert due to this topology, which keeps the value of the constant stable.”
The study, co-authored by Alexander and Brown Center for Theoretical Physics’ Aaron Hoy and Heliodson 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, and his theory of space, time, and gravity.
At that time, Einstein believed that the universe was static. To prevent his equations from predicting the collapse of the universe, he introduced the cosmological constant as a kind of repulsive effect in empty space that counterbalances gravity.
This idea seemed unnecessary after Edwin Hubble discovered in 1929 that the universe was expanding. Since the universe was not static anyway, Einstein removed this term from his equations. Al Thabet reportedly disliked it and later referred to it as his “biggest blunder”.
For decades, the cosmological constant has largely faded from prominence.
Then, in 1998, astronomers discovered something surprising: the expansion of the universe was accelerating. Instead of disappearing from the story, the cosmological constant suddenly became fundamental again because it could explain this accelerating expansion.
The problem of the cosmological constant
The revival of the cosmological constant created a serious problem.
During the years when the constant was no longer valid, quantum field theory became one of the most successful theories in science and a cornerstone of the Standard Model of particle physics.
QFT describes empty space as anything that is not empty. Instead, it is full of particles that constantly appear and disappear through quantum fluctuations.
All this activity must contribute a huge amount of vacuum energy. This vacuum energy is related to the cosmological constant, which means that the constant must be very large.
But observations show that this is not the case.
If the cosmological constant were as large as QFT predicts, the universe would have expanded so rapidly that galaxies, stars, planets, and life could never have eventually formed.
The incompatibility between theory and observation remains one of the most puzzling problems in modern physics. The puzzle has become more surprising because experiments have repeatedly confirmed the extraordinary accuracy of quantum field theory in other contexts.
Topological solution
Alexander has spent years studying Chern-Simons-Kodama (CSK) theory, a proposed state of quantum gravity 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, CSK’s approach is among the most obvious possibilities.
“It’s a really conservative approach to measuring gravity,” he said. “This is the approach that people like Dirac, Schrödinger, and Wheeler use. It’s just good, old-fashioned quantization.”
Alexander has long noted the similarities between CSK theory and the mathematics of the quantum Hall effect. To better understand these connections, he collaborated with Hui, an assistant professor at Brown University who studies topological systems.
“That’s the beauty of the Brown Center for Theoretical Physics,” Alexander said. “We want to be a place where there’s a mixture of a lot of viewpoints, and that’s where we practice what we preach — a cosmology that works closely with a condensed matter view.”
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 the Hall potential (named after Edwin Hall, who discovered it).
Under normal conditions, the Hall potential changes smoothly as the magnetic field increases.
However, under very cold temperatures and very strong magnetic fields, the behavior changes dramatically. Instead of changing smoothly, Hall’s effort increases in distinct steps and plateaus. It is striking that these values remain the same regardless of the material used or any defects it contains.
This reliability comes from the topology.
In these extreme conditions, electrons behave collectively and enter a tightly coupled quantum state. The topology of this state stabilizes the values of steps and plateaus, making it resistant to disturbances and faults.
The Brown researchers see a similar process occurring in the CSK description of quantum gravity.
Just as the topology locks the Hall potential into fixed values, the topology of space-time can lock the cosmological constant into stable values, protecting it from quantum fluctuations that might push it much higher.
“What we found is that this quantization of electrical conduction in the quantum hall has an analogue to the cosmological constant,” Hui said. “It also ends up being quantized for topological reasons. It turns out that there are constraints in the theory that force the cosmological constant to accept some permissible quantized values.”
A new direction for quantum gravity
Alexander emphasizes that more work is needed before a topological explanation of the cosmological constant can be fully established.
However, he believes the results represent an important step toward solving the gravitational side of the problem. This work also strengthens the case for CSK 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. “We are now working on a bigger picture of how this phenomenon works.”
