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Describing matter under extreme conditions, such as those found inside neutron stars, remains an unsolved problem. The density of such matter is equivalent to compressing around 100,000 Eiffel Towers into a single cubic centimeter. In particular, the properties of so-called quark matter – which consists of the universe’s fundamental building blocks, the quarks, and may exist in extremely dense regions—play a central role. Researchers from TU Darmstadt and Goethe University Frankfurt have studied this matter and its thermodynamic properties. Their results have now been published in the renowned journal “Physical Review Letters”.
Theoretical studies suggest that quarks at very low temperatures enter a so-called color-superconducting state, which fundamentally alters the nature of matter. This state is analogous to the transition of an electron gas into an electrical superconductor – except that, instead of electrons, quarks pair up and create an energy gap in their excitation spectrum. Unlike conventional superconductors, however, a color superconductor generally does not conduct electric current without resistance, but rather the color charge of the quarks. This charge ultimately determines the strength of the interaction between these elementary building blocks of matter.
The formation of quark pairs and the resulting energy gap fundamentally change the behavior of matter. Even relatively weak pairing effects have a significant impact on the matter and its thermodynamic properties.
Andreas Geißel, Tyler Gorda, and Jens Braun analyze these effects in detail. They calculate correction terms arising from quark pairing and interactions, while accounting for the specific conditions inside neutron stars. This allows the research team to determine both the thermodynamic pressure and the speed of sound in color-superconducting quark matter.
The results show that the color-superconducting state is thermodynamically favored at high densities. Moreover, this state leads to a significant increase in the speed of sound – a direct measure of the mechanical stability of matter. According to the calculations, the speed of sound in the interior of neutron stars could exceed 60 percent of the speed of light, i.e., more than 180,000 kilometers per second. This figure becomes even more impressive when compared to the speed of sound in the hardest terrestrial "everyday material", diamond, which is 10,000 times lower.
Numerical simulations also suggest that such high speeds of sound are necessary to explain the stability of the most massive known neutron stars. The work by Geißel, Gorda, and Braun now suggests that color-superconducting matter may be a crucial ingredient in explaining massive neutron stars – and that observations of these stars could, in turn, help to better constrain the energy gap in the quark spectrum.
About TU Darmstadt
TU Darmstadt is one of Germany’s leading technical universities and a synonym for excellent, relevant research. We are crucially shaping global transformations – from the energy transition via Industry 4.0 to artificial intelligence – with outstanding insights and forward-looking study opportunities. TU Darmstadt pools its cutting-edge research in three fields: Energy and Environment (E+E), Information and Intelligence (I+I), Matter and Materials (M+M). Our problem-based interdisciplinarity as well as our productive interaction with society, business and politics generate progress towards sustainable development worldwide.
Since we were founded in 1877, we have been one of Germany’s most international universities; as a European technical university, we are developing a trans-European campus in the network, “Unite!”. With our partners in the alliance of Rhine-Main universities – Goethe University Frankfurt and Johannes Gutenberg University Mainz – we further the development of the metropolitan region Frankfurt-Rhine-Main as a globally attractive science location.
Andreas Geißel – Postdoctoral researcher at the Institute for Nuclear Physics, Department of Physics, TU Darmstadt; member of the Collaborative Research Centre (CRC) 1245.
Tyler Gorda – At the time of the study: postdoctoral researcher at Goethe University Frankfurt; member of CRC Transregio 211. Since August 2025: Assistant Professor at the Department of Physics at The Ohio State University (USA) and member of the Center for Cosmology and AstroParticle Physics.
Jens Braun – Professor of Theoretical Physics at the Institute for Nuclear Physics, Department of Physics, TU Darmstadt since 2012; project leader in CRC 1245 as well as board member and project leader in CRC Transregio 211.
Andreas Geißel, Tyler Gorda, and Jens Braun: Color Superconductivity under Neutron-Star Conditions at Next-to-Leading Order; Phys. Rev. Lett. 135 (2025) 21, 211901
https://journals.aps.org/prl/abstract/10.1103/54g5-43nk
DOI: https://doi.org/10.1103/54g5-43nk
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