**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 to answering these open questions in physics.**

The universe has a number of bizarre properties that are difficult to understand with everyday experience. For example, the matter we know, made up of elementary and composite particles that make up molecules and materials, appears to account for only a small portion of the energy of the universe. The largest contribution, about two-thirds, comes from “dark energy“ – a hypothetical form of energy, the background to which physicists are still puzzling. In addition, the universe is not only expanding steadily, but also ever faster.

Both properties seem to be connected, because dark energy is also seen as a driver of the accelerated expansion. In addition, it could reunite two powerful schools of physics: quantum field theory and the general theory of relativity developed by Albert Einstein. But there is a catch: calculations and observations are far from consistent. Now, in an article published by the journal, two researchers from Luxembourg have revealed a new way to solve this 100-year-old mystery *Physical Verification Letters*.

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

“Vacuum has energy. This is a fundamental result of quantum field theory,” explains Prof. Alexandre Tkatchenko, Professor of Theoretical Physics at the Institute for Physics and Materials Science University of Luxembourg. This theory was developed to bring together quantum mechanics and special relativity, but quantum field theory seems to be incompatible with general relativity. Its essential feature: In contrast to quantum mechanics, the theory considers not only particles, but also matter-free fields as quantum objects.

“In this context, many researchers consider dark energy to be an expression of what is known as vacuum energy,” says Tkatchenko: a physical quantity that, in a vivid picture, is caused by a constant formation and interaction of pairs of particles and their antiparticles – such as electrons and positrons – in what is actually empty space Space.

Physicists speak of this coming and going of virtual particles and their quantum fields as vacuum or zero point fluctuations. As the particle pairs quickly disappear back into nothingness, their existence leaves behind a certain energy.

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

#### A colossal mismatch

In contrast to vacuum energy, which can only be derived from the formulas of quantum field theory, the cosmological constant can be determined directly from astrophysical experiments. Measurements with the Hubble Space Telescope and the Planck space mission have given close and reliable values for the fundamental physical quantity. Calculations of dark energy based on quantum field theory, on the other hand, deliver results that correspond to a value of up to 10 for the cosmological constant^{120} times larger – a colossal discrepancy, although in the prevailing world view of physicists today, both values should be the same. The discrepancy found instead is known as the “riddle of the cosmological constant”.

“This is undoubtedly one of the greatest inconsistencies in modern science,” says Alexandre Tkatchenko.

#### Unconventional way of interpretation

Together with his Luxembourg research colleague Dr. Dmitry Fedorov, he has now taken a crucial step closer to solving this mystery that has been unsolved for decades. In a theoretical work, the results of which she recently published in *Physical Verification Letters*the two Luxembourg researchers propose a new interpretation of dark energy. She assumes that the zero-point fluctuations lead to a polarizability of the vacuum that can be both measured and calculated.

“For pairs of virtual particles with opposite electrical charges, it arises from electrodynamic forces that these particles exert on each other during their extremely short existence,” explains Tkatchenko. The physicists speak of a vacuum self-interaction. “This 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 basic model for atoms a few years ago and presented it for the first time in 2018. The model originally served to describe atomic properties, in particular the relationship between the polarizability of atoms and the equilibrium properties of certain non-covalently bound molecules and solids. Since the geometric properties are easy to measure experimentally, the polarizability can also be determined using its formula.

“We have transferred this method to processes in a vacuum,” explains Fedorov. To this end, the two researchers examined the behavior of quantum fields, in particular the “coming and going” of electrons and positrons. The fluctuations in these fields can also be characterized by an equilibrium geometry that is already known from experiments. “We built it into the formulas of our model and ultimately obtained the strength of the intrinsic vacuum polarization,” reports Fedorov.

The last step was then the quantum mechanical calculation of the energy density of the self-interaction between fluctuations of electrons and positrons. The result obtained agrees well with the measured values for the cosmological constant. This means: “Dark energy can be traced 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,” the physicist sums up. “It also provides a testable prediction: namely, that quantum fields such as those of electrons and positrons do indeed possess a small but ubiquitous intrinsic polarization.”

This finding points the way for future experiments to also demonstrate this polarization in the laboratory, 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 that.”

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

In conclusion, Tkatchenko is convinced: “Ultimately, this could also shed light on how quantum field theory and general relativity are intertwined as two ways of looking at the universe and its components.”

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

DOI: 10.1103/PhysRevLett.130.041601