**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 by everyday experience. For example, matter as we know it, made up of elementary and compound particles that make up molecules and materials, apparently accounts for only a small fraction of the energy of the universe. The largest contribution, about two-thirds,”dark energy” – a hypothetical form of energy that background physicists still confuse. What’s more, the universe is not only expanding steadily, but it is doing so faster than ever before.

Both features seem to be linked, because dark energy it is also considered to be the driving force of accelerated expansion. Moreover, it can reunite two powerful physical schools of thought: the quantum field theory and the general theory of relativity developed by Albert Einstein. But there is a problem: until now, calculations and observations have been far from matching. Now two researchers from Luxembourg have demonstrated a new way to solve this 100-year-old puzzle in a paper published by the journal. *Physical Review Letters*.

#### Trace of virtual particles in a vacuum

“The void has energy. This is a fundamental consequence of quantum field theory,” explains Prof. Alexandre Tkatchenko, Professor of Theoretical Physics in the Department of Physics and Materials Science University of Luxembourg. This theory was developed to combine quantum mechanics with special relativity, but quantum field theory doesn’t seem to be compatible with general relativity. Key feature: Unlike quantum mechanics, the theory considers not only particles but also matter-independent fields as quantum objects.

“In this framework, many researchers see dark energy as an expression of so-called vacuum energy,” says Tkatchenko: a physical quantity caused by the constant emergence and interaction of pairs of particles and their antiparticles in a live image. – like electrons and positrons – actually in empty space.

Physicists refer to this coming and going of virtual particles and their quantum fields as vacuum or zero point fluctuations. As particle pairs rapidly vanish back into nothingness, their presence leaves behind a certain amount of energy.

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

#### A colossal mismatch

Unlike 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 with the Hubble space telescope and the Planck space mission gave close and reliable values for the fundamental physical quantity. On the other hand, dark energy calculations on the basis of quantum field theory give results corresponding to a value of the cosmological constant up to 10.^{120} times larger – a huge inconsistency, but according to the worldview of physicists prevailing today, both values should be equal. The inconsistency found instead is known as the “cosmological constant enigma.”

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

#### unconventional interpretation method

Together with his Luxembourg research colleague Dr. Dmitry Fedorov has now brought an important step closer to solving this puzzle that has been open for decades. In a theoretical study in which they recently published their results *Physical Review Letters*Two Luxembourgian researchers propose a new interpretation of dark energy. It assumes that zero point fluctuations result in a polarizability of the vacuum that can be both measured and calculated.

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

Together with his research colleague Fedorov, they developed the fundamental 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 their equilibrium properties. certain non-covalently bonded molecules and solids. Since geometric properties are fairly easy to measure experimentally, they can also be determined by means of polarizability formulas.

“We transferred this procedure to processes in a vacuum,” explains Fedorov. To this end, the two researchers looked specifically at the behavior of quantum fields, which represent the “come and go” of electrons and positrons. The fluctuations of these fields can also be characterized by an equilibrium geometry already known from experiments. “We added this to the formulas of our model and this way we finally got the power of 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 thus obtained is in 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 Alexandre Tkatchenko.

#### Consistent values and verifiable estimates

“Our work therefore offers an elegant and unconventional approach to solving the cosmological constant riddle,” summarizes the physicist. “It also provides a verifiable prediction: that is, that quantum fields such as those of electrons and positrons do indeed have a small but always present intrinsic polarization.”

The two Luxembourgian researchers say this finding points the way for future experiments to detect this polarization in the lab as well. “Our aim is to derive the cosmological constant from a rigorous quantum theoretical approach,” emphasizes Dmitry Fedorov. “And our work includes a recipe for how to achieve that.”

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

Finally, Tkatchenko was convinced: “Ultimately, this may also shed light on the way in which quantum field theory and general relativity are intertwined 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