**What’s behind dark energy—and what connects it to the cosmological constant proposed by Albert Einstein? Two physicists from the University of Luxembourg point the way to answering this open question in physics.**

The universe has several strange properties that are difficult to understand with everyday experience. For example, matter as we know it, composed of elementary and composite particles that make up molecules and materials, apparently makes up only a small fraction of the energy in the universe. The largest contribution, about two-thirds, comes from “dark power” – a fictitious form of energy whose background still puzzles physicists. Furthermore, the universe is not only expanding steadily, but is also doing so at an accelerating rate.

Because both features seem to be connected dark power This is considered a driver of accelerated expansion. Moreover, it can combine two powerful physical schools of thought: quantum field theory and the general theory of relativity developed by Albert Einstein. But there is a catch: calculations and observations have so far been far from matching. Now two researchers from Luxembourg have shown a new way to solve this 100-year-old puzzle in a paper published by the Journal. *Physical review letter*.

#### Paths of virtual particles in vacuum

“Vacuum has energy. This is a fundamental result of quantum field theory,” explains Professor Alexandre Takatchenko, professor of theoretical physics in the Department of Physics and Materials Science. University of Luxembourg. This theory was developed to unify quantum mechanics and special relativity, but quantum field theory appears to be incompatible with general relativity. Its essential feature: Unlike quantum mechanics, the theory treats not only particles but also matter-free fields as quantum objects.

“In this framework, many researchers consider dark energy as an expression of the so-called vacuum energy,” said Tkatchenko: a physical quantity that, in a vivid picture, is caused by a constant emergence and interaction of pairs of particles and their antiparticles. — like electrons and positrons — actually in empty space.

Physicists refer to this coming and going of virtual particles and their quantum field as vacuum or zero-point fluctuations. Although the particle pair quickly dissipates back into the void, their existence leaves behind a certain amount of energy.

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

#### A huge mismatch

Unlike the vacuum energy, which can only be estimated from quantum field theory formulas, the cosmological constant can be determined directly by astrophysical experiments. Measurements with the Hubble Space Telescope and the Planck space mission have given close and reliable values for fundamental physical quantities. Calculations of dark energy based on quantum field theory, on the other hand, yield results that go up to values of 10 for the cosmological constant.^{120} Many times larger – a huge discrepancy, although in the world view of physicists currently prevailing, both values should be equal. The discrepancy found instead is known as the “cosmic constant mystery”.

“This is undoubtedly one of the greatest paradoxes of modern science,” says Alexandre Kotchenko.

#### Unconventional ways of interpretation

His Luxembourg research colleagues. Together with Dmitry Fedorov, he now brings the solution to this puzzle, which has been open for decades, a significant step closer. In a theoretical work, the results of which they recently published *Physical review letter*, two Luxembourg researchers proposed a new explanation for dark energy. It assumes that zero-point fluctuations lead to vacuum polarization, which can be both measured and calculated.

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

Together with his research colleague Fedorov, they developed the basic model for the atom a few 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. Some non-covalently bonded molecules and solids. Since geometrical properties are quite easy to measure experimentally, polarizabilities can also be determined by their formula.

“We transferred this method to vacuum processes,” Fedorov explained. To this end, the two researchers looked at the behavior of quantum fields, specifically representing the “comings and goings” of electrons and positrons. Fluctuations in these fields can be characterized by an equilibrium geometry already known from experiments. “We inserted this into the formulation of our model and thus finally obtained the intrinsic vacuum polarization strength,” reports Fedorov.

The last step was to quantum mechanically calculate the self-interaction energy density between the electron and positron fluctuations. The result thus obtained agrees well with the measured value for the cosmological constant. This means: “Dark energy can be traced back to the energy density of self-interactions of quantum fields,” emphasizes Alexandre Takatchenko.

#### Consistent values and verifiable forecasts

“Our work thus offers an elegant and unconventional approach to solving the puzzle of the cosmological constant,” added the physicist. “Furthermore, it provides a testable prediction: namely, that quantum fields such as electrons and positrons indeed possess a small but ever-present internal polarization.”

The discovery points the way for future experiments to detect this polarization in the laboratory, the two Luxembourg researchers said. “Our goal is to derive cosmological constants from a rigorous quantum theoretical approach,” emphasized Dmitry Fedorov. “And our work has a recipe for how to accomplish that.”

He sees the new results, obtained together with Aleksandr Tkachenko, as a first step towards dark energy – and its connection to Albert Einstein’s cosmological constant.

Finally, Tkatchenko is convinced: “Ultimately, it may also shed light on how quantum field theory and general relativity are intertwined as two ways of looking at the universe and its elements.”

Reference: Alexandre Kotchenko and Dmitry V. Fedorov, 24 January 2023, “Casimir Self-Interaction Energy Density in Quantum Electrodynamic Fields” *Physical review letter*.

DOI: 10.1103/PhysRevLett.130.041601