**What is behind dark energy and what connects it to the cosmological constant introduced by Albert Einstein? Two physicists from the University of Luxembourg point the way in answering these open questions in physics.**

The universe has a number of strange properties that are difficult to understand with everyday experience. For example, the matter we know of, which consists of elementary and compound particles that make up molecules and materials, apparently makes up only a small part of the energy in the universe. The largest contribution, around two-thirds, comes from “dark energy” – a hypothetical form of energy whose background physicists are still puzzling over. Furthermore, the universe is not only constantly expanding, but also at an ever-increasing rate.

Both features seem to be connected, because dark energy it is also considered a driver of accelerated expansion. Furthermore, it could bring together two powerful physical schools of thought: quantum field theory and the general theory of relativity developed by Albert Einstein. But there’s a catch: the calculations and observations so far have been far from matching. Now, two researchers from Luxembourg have shown a new way to solve this 100-year-old riddle in an article published by the journal *Physical Review Letters*.

#### The trail of virtual particles in a vacuum

“The vacuum has energy. This is a fundamental result of quantum field theory,” explains Prof. Alexandre Tkatchenko, Professor of Theoretical Physics at the Department of Physics and Materials Science at University of Luxembourg. This theory was developed to unite quantum mechanics and special relativity, but quantum field theory appears to be incompatible with general relativity. Its essential feature: unlike quantum mechanics, the theory considers not only particles but also free fields of matter as quantum objects.

“In this framework, dark energy is considered by many researchers to be an expression of so-called vacuum energy,” says Tkatchenko: a physical quantity that, in a vivid picture, is caused by a constant appearance and interaction of pairs of particles and their antiparticles. – as electrons and positrons – in what is actually empty space.

Physicists speak of this coming and going of virtual particles and their quantum fields as zero-point or vacuum fluctuations. While the particle pairs quickly fade into nothing again, their existence leaves behind a certain amount of energy.

“This vacuum energy also has a meaning in general relativity”, points out the Luxembourgish scientist: “It manifests itself in the cosmological constant that Einstein included in his equations for physical reasons”.

#### A colossal mismatch

Unlike the energy of a vacuum, which can only be deduced from the formulas of quantum field theory, the cosmological constant can be determined directly by astrophysical experiments. Measurements with the Hubble Space Telescope and the Planck space mission have returned close and reliable values for the fundamental physical quantity. Dark energy calculations on the basis of quantum field theory, on the other hand, yield results that correspond to a value of the cosmological constant that is up to 10^{120} times greater: a colossal discrepancy, although in the world view of physicists that prevails today, both values should be equal. The discrepancy found instead is known as the “cosmological constant puzzle”.

“It is, without a doubt, one of the biggest inconsistencies in modern science,” says Alexandre Tkatchenko.

#### Unconventional way of interpreting.

Together with his research colleague from Luxembourg Dmitry Fedorov, he has now brought the solution to this puzzle, which has been open for decades, one significant step closer. In a theoretical paper, the results of which he recently published in *Physical Review Letters*, the two Luxembourg researchers propose a new interpretation of dark energy. He assumes that zero point fluctuations lead to a polarizability of the vacuum, which can be measured and calculated.

“In pairs of virtual particles with an opposite electrical charge, it arises from the electrodynamic forces that these particles exert on each other during their extremely short existence,” explains Tkatchenko. Physicists refer to this as a self-interaction of the vacuum. “It leads to an energy density that can be determined with the help of a new model,” says the Luxembourgish scientist.

Together with their research colleague Fedorov, they developed the basic model for atoms a few years ago and presented it for the first time in 2018. The model was originally used to describe atomic properties, in particular the relationship between the polarizability of atoms and the balance properties. of certain non-covalently bonded molecules and solids. Since the geometric characteristics are quite easy to measure experimentally, the polarizability can also be determined through its formula.

“We transferred this procedure to processes in a vacuum,” explains Fedorov. To this end, the two researchers observed the behavior of quantum fields, in particular the representation of the “come and go” of electrons and positrons. The fluctuations of these fields can also be characterized by an equilibrium geometry that is already known from experiments. “We inserted it into our model formulas and in this way we finally obtained the strength of the intrinsic polarization of the vacuum,” reports Fedorov.

The last step was then to calculate quantum mechanically the energy density of the self-interaction between the fluctuations of electrons and positrons. The result thus obtained agrees well with the measured values for the cosmological constant. This means: “Dark energy goes back to the energy density of the self-interaction of quantum fields,” emphasizes Alexandre Tkatchenko.

#### Consistent values and verifiable forecasts

“Our work thus offers an elegant and unconventional approach to solving the riddle of the cosmological constant,” summarizes the physicist. “Furthermore, it provides a verifiable prediction: namely, that quantum fields such as those of electrons and positrons have a small but ever-present intrinsic polarization.”

This finding points the way for future experiments to detect this polarization in the laboratory as well, say the two Luxembourg researchers. “Our goal is to derive the cosmological constant from a rigorous quantum theoretical approach,” emphasizes Dmitry Fedorov. “And our work contains a recipe for how to do this.”

He sees the new results obtained together with Alexandre Tkatchenko as the first step towards a better understanding of dark energy and its connection to Albert Einstein’s cosmological constant.

Finally, Tkatchenko is convinced: “In the end, this could also shed light on the way in which quantum field theory and general relativity theory intertwine as two ways of looking at the universe and its components.”

Reference: “Casimir Self-Interaction Energy Density of Quantum Electrodynamic Fields” by Alexandre Tkatchenko and Dmitry V. Fedorov, January 24, 2023, *Physical Review Letters*.

DOI: 10.1103/PhysRevLett.130.041601