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

The universe has a number of strange properties that are difficult to understand through everyday experience. For example, matter as we know it, made up of the elementary and composite particles that make up molecules and materials, appears to make up only a small fraction of the energy of the universe. The largest contribution, about two-thirds of “dark energy” – a hypothetical form of energy, the background of which still baffles physicists. Moreover, the universe is not only expanding steadily, but expanding at a much faster rate.

Because both features seem related dark energy is also considered a driver of accelerated expansion. In addition, it can bring together two powerful schools of physical thought: quantum field theory and general relativity, developed by Albert Einstein. But there is one thing: calculations and observations have not matched up to now. Now, two researchers from Luxembourg have shown a new way to solve this 100-year-old puzzle in a paper published in the journal. *Physical review letters*.

#### Virtual particle track in vacuum

“Vacuum has energy. This is the main result of the quantum field theory,” said Prof. Aleksandr Tkatchenko, professor of theoretical physics at the Department of Physics and Materials Science University of Luxembourg. This theory was developed to bring together quantum mechanics and special relativity, but quantum field theory is incompatible with general relativity. Its main feature: unlike quantum mechanics, the theory considers not only particles, but also non-matter fields as quantum objects.

“In this context, many researchers perceive dark energy as an expression of the so-called vacuum energy,” says Tkatchenko: a physical quantity that results from the constant emergence and interaction of pairs of particles and their antiparticles in the living image. – like electrons and positrons – in essentially empty space.

Physicists speak of this coming and going of virtual particles and their quantum fields as vacuum or zero-point fluctuations. Although particle pairs quickly disappear, their existence releases a certain amount of energy.

“This vacuum energy also has meaning in general relativity,” notes the Luxembourg scientist: “It manifests itself in the cosmological constant, which Einstein included in his equations for physical reasons.”

#### huge discrepancy

Unlike the vacuum energy, which can only be deduced from the formulas of quantum field theory, the cosmological constant can be determined directly by astrophysical experiments. Measurements by the Hubble Space Telescope and the Planck space mission have yielded close and reliable values for a key physical quantity. The calculation of dark energy based on the quantum field theory gives results corresponding to the value of the cosmological constant up to 10.^{120} times larger – a huge discrepancy, although in the world view of the prevailing physicists today, both values should be equal. The discrepancy found instead is known as the “cosmological constant enigma.”

“Undoubtedly, this is one of the biggest inconsistencies in modern science,” said Alexander Tkatchenko.

#### An unconventional method of interpretation

Luxembourgish researcher with his colleague, Dr. Dmitry Fedorov, he has now brought a significant step closer to solving this puzzle, which has been open for decades. In a theoretical work they recently published their results *Physical review letters*, two Luxembourg researchers propose a new interpretation of dark energy. It assumes that zero-point fluctuations lead to both measurable and calculable vacuum polarizabilities.

Tkatchenko explains: “In pairs of oppositely charged virtual particles, these particles interact with each other in extremely short periods of time due to electrodynamic forces.” Physicists call this vacuum self-interaction. “This leads to an energy density that can be determined with the help of the new model,” says the Luxembourg scientist.

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

“We transferred this procedure to vacuum processes,” said Fedorov. To that end, the two researchers looked at the behavior of quantum fields, specifically those that represent the “coming and going” of electrons and positrons. Fluctuations of these fields can be characterized by the equilibrium geometry already known from experiments. “We incorporated it into the formulas of our model and as a result we got the strength of the internal vacuum polarization,” Fedorov said.

The final step was to quantum mechanically calculate the energy density of the self-interaction between the fluctuations of electrons and positrons. The result obtained in this way is in good agreement with the measured values for the cosmological constant. This means that “Dark energy can be traced back to the energy density of the self-interaction of quantum fields,” emphasizes Alexander Tkatchenko.

#### Consistent values and verifiable predictions

“Thus, our work offers an elegant and unconventional approach to solving the puzzle of the cosmological constant,” the physicist concludes. “Furthermore, it makes a testable prediction: that is, quantum fields like electrons and positrons have a really small but ever-present intrinsic polarization.”

The two Luxembourg researchers say the finding points the way for future experiments to detect this polarization in the laboratory as well. “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 make it happen.”

He and Alexander Tkatchenko see the new results as a first step toward a better understanding of dark energy and its connection to Albert Einstein’s cosmological constant.

Finally, Tkatchenko is convinced: “Ultimately, it may also illuminate the way in which quantum field theory and general relativity intertwine as two ways of looking at the universe and its components.”

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

DOI: 10.1103/PhysRevLett.130.041601