Simulations of neutron stars provide new bounds on their properties, such as their internal pressure and their maximum mass.
So if the equations are known, why is it so difficult to solve them in the case of neutron stars?
But in the bulk of neutron stars, where the density is between those two extremes, perturbation theory fails.
Finally, by using the rigorous relations between the pressure inside neutron stars and nonzero-isospin nuclear matter [5, 6], the researchers were able to put rigorous bounds on the properties of matter inside neutron stars.
The future of drilling down into neutron stars with computers looks bright, indeed.
Neutron star simulations yield new limits on nuclear star characteristics, including maximum mass and internal pressure.
Neutron star research is challenging. With today’s space-faring technology, sending a probe would probably take half a million years because the closest one is roughly 400 light-years away. Neutron stars are merely points in the sky, about the size of a small city, so telescopes don’t show much detail from our perspective. Furthermore, neutron stars are too dense—many times denser than atomic nuclei—for any laboratory on Earth to replicate their interior. Since the equations for neutron-star matter cannot be solved using conventional computational methods, that high density also presents a theoretical challenge. Nevertheless, attempts to comprehend these enigmatic objects have continued despite these challenges. Through a combination of theory-based techniques and computer simulations, Ryan Abbott and associates at MIT have produced new, exacting constraints for neutron star interior properties [1]. According to their findings, the speed of sound inside these compact objects may be relatively high, which could allow neutron stars to become more massive than previously believed.
Quantum chromodynamics (QCD) describes the strong force acting on protons, neutrons, and their constituent quarks. Its equations govern the internal properties of a neutron star, including pressure and density. Therefore, if the equations are known, why is it so hard to solve them for neutron stars? The reason is that we use perturbation theory as our primary calculation tool, which expands the equations in terms of a small parameter (ignoring higher-order terms). The outer atmosphere and upper crust, where the density is relatively low [2], and the core of the most massive neutron stars, where the QCD coupling parameter is small [3], are two areas where perturbation theory is a feasible approach for neutron-star matter. But perturbation theory doesn’t work in the bulk of neutron stars, where the density lies in the middle. Although interpolation between low and high densities is possible, the results are not precise [4]. ).
Luckily, lattice QCD is another tool available to physicists. QCD can be simulated on a computer thanks to this numerical approach, which treats quark and gluon interactions on a discretized space-time lattice. Lattice QCD breaks down at the neutron star densities of interest, but QCD can be solved directly at small densities using this method. On the other hand, there is a clever way to frame this issue. It makes use of isospin, a kind of nuclear charge that regards neutrons as negative and protons as positive. Since protons and neutrons make up about equal amounts of most nuclear matter, the isospin density is almost zero. However, a large (or “nonzero”) isospin density state of matter where protons greatly outnumber neutrons is conceivable. Nuclear matter must have a lower pressure at any density than it does at nonzero isospin density, according to earlier research [5, 6].
Abbott and his associates have obtained rigorous results by “drilling” down into a neutron star’s high-density regions using this upper pressure limit [1]. On multiple of the most potent supercomputers, the group concurrently carried out comprehensive numerical lattice QCD simulations for nonzero isospin density. Since the “real world” is continuous and lattice QCD assumes a discrete space-time, a direct solution to the equation of state for isospin nuclear matter was not achievable even with that much processing power. This was the first time the group had carefully extended their computer simulations to the “continuum limit” of vanishingly small lattice spacing for nonzero-isospin nuclear matter in order to produce systematically controlled results.
Abbott and colleagues were able to derive a number of important new findings regarding the characteristics of extreme-density matter using the nonzero-isospin calculations available. They first demonstrated that nuclear matter at high isospin density is a kind of superconducting material and calculated its superconducting gap, a measure that describes the system’s potential energy. They calculated the gap by taking the difference between their calculated pressure and the known pressure for nonsuperconducting matter [4], and the result is more accurate than the value that others have obtained using analytic calculations [7], but it agrees with it.
Second, the scientists clearly showed that the speed of sound in nonzero-isospin nuclear matter is higher than a speed limit called the conformal bound [8], but lower than a speed limit that was recently proposed [9]. The maximum mass a neutron star can have before exploding under the weight of its own gravity into a black hole is affected by this finding. The maximal speed of sound in nuclear matter caps this maximum mass, so breaking the conformal bound, as demonstrated by Abbott and colleagues, implies that neutron stars could theoretically grow larger than the 2-solar-mass limit that was previously determined using the conformal bound.
Finally, the researchers were able to place rigorous bounds on the properties of matter inside neutron stars by using the rigorous relations between the pressure inside neutron stars and nonzero-isospin nuclear matter [5, 6]. One cannot overestimate the significance of these boundaries. For nonzero-isospin nuclear matter, the availability of exact and rigorous results offers a very challenging test bed for a wide range of models and approximation techniques. Now that they can compare their models to these bounds, modelers can continue to develop new ideas for trying to approximate the matter inside neutron stars.
This method does not only apply to nuclear matter with nonzero-isospin. There are already plans to delve even deeper into the characteristics of neutron stars using different types of lattice QCD calculations [10]. Therefore, a completely new area of computational research on neutron-star matter has been made possible by the findings of Abbott and associates. In order to better understand the spin down and cooling of neutron stars, additional extensions of this work could provide constraints on more complex nuclear matter properties like conductivities and viscosities. Astrophysical observations can be directly interpreted and potentially predicted by lattice QCD once this more comprehensive picture is available. Computer-assisted neutron star drilling appears to have a bright future.
Citations.
R. Abbott and associates. QCD constraints on isospin-dense matter and the nuclear equation of state (NPLQCD Collaboration), Phys. Rev. Lett. 011903 (2025), 134.
A. The T. M. A and Son. At finite isospin density, Stephanov, “QCD,” Phys. Rev. Lett. (2001), 86, 592.
A. Annala et al. “Proof for massive neutron stars’ quark-matter cores,” Nat. Physical. 16, 907 (2020).
1. Kurkela and associates. “Cold quark matter,” Physics. The Rev. 105021 (2010), D 81.
T. . D. Cohen, “QCD inequalities for the free energy of baryonic matter and the nucleon mass,” Phys. Rev. Let’s. 032002 (2003), 91.
Y. “Restrictions on the equation of state from QCD inequalities and lattice QCD,” by Fujimoto and S. Reddy, Phys. The Rev. D 109, 014020 (2024).
Y. Fujimoto, “Raised pairing gap contribution to QCD equation of state at large isospin chemical potential,” Phys. Rev. D 109, 2024, 054035.
A. . Cherman & Co. “Limited by the speed of sound from holography,” Phys. Rev. . 066003, D 80 (2009).
Ms. Hippert and associates. “Transport-based upper bound on sound speed in nuclear matter,” Phys. Let’s. 139184, B 860 (2025).
G. . Moore and T. Ddot. Gorda, “Using the lattice to bound the QCD equation of state,” J. High Vitality. Physically. 2023, 133 (2023).
About the Writer.
Now a professor in the University of Colorado Boulder’s Department of Physics, Paul Romatschke earned his doctorate from the Technical University of Vienna. He is interested in a wide range of theoretical physics topics, including particle physics, relativistic fluid dynamics, cold quantum gases, and nuclear physics. His book on contemporary relativistic fluid dynamics has been published, and he is now developing a rival mechanism to the Higgs without breaking symmetry.
Subject Areas.
