An equation of state for dense nuclear matter such as neutron stars

Neutron stars are some of the densest objects in the universe. They are the core of a collapsed megastar that went supernova, have a typical radius of 10 km—just slightly more than the altitude of Mt. Everest—and their density can be several times that of atomic nuclei.

Physicists love extreme objects like this because they require them to stretch their theories into new realms and see if they are confirmed or if they break, requiring new thinking and new science.

For the first time, researchers have used lattice quantum chromodynamics to study the interior of neutron stars, obtaining a new maximum bound for the speed of sound inside the star and a better understanding of how pressure, temperature and other properties there relate to one another.

Their work is published in Physical Review Letters.

Neutron stars present other challenges besides a humongous density. Their small size makes them impossible to study visually with telescopes, as they appear no more than a point. (The nearest neutron star to Earth is 400 light-years away.)

Laboratories on Earth can't form bulk materials that match their density, about a quadrillion times that of water—or come close to their dimensions. Even studying them theoretically is difficult, as the relevant equations cannot be solved with standard mathematical or computational techniques.

This new approach, utilizing both particle theory and simulations, has determined new, rigorous constraints on the interior of neutron stars. In particular, a maximum speed of sound has been established—exceedingly high but definite—and such stars may be able to grow more massive than was previously thought.

Like any substance, neutron stars have an equation of state, or more precisely a phase diagram, such as that for water.

The properties of a neutron star are determined by quantum chromodynamics (QCD), the theory of the strong force that relates to the interactions of protons and neutrons, quarks and gluons.

But QCD makes calculating particle interactions extremely difficult, because the force-carrying boson, the gluon, itself carries the "color" charge that is the main quantum number of strong force particles. It's as if the photon, the boson that transmits the electromagnetic (EM) force, had an electrical charge. (Instead, the photon is electrically neutral.)

As such, QCD is called a "nonlinear" theory. QCD also has the peculiar property of asymptotic freedom—the force is small and essentially vanishing at small distances, like inside a proton, but it grows larger as distance increases, just the opposite of the other three forces.

When the coupling is large, quantum field theorists are unable to use their standard, well-honed mathematical technique called perturbation theory, which involves breaking the calculation up into an infinite series (such as a Taylor series familiar in basic calculus) and calculating only one or a few of the first terms.

Perturbation theory works well in EM because successive powers of the EM coupling constant, alpha ~ 1/137, get smaller fast. But that does not happen for the entire energy spectrum of QCD.

So, lead author Ryan Abbott of the Center for Theoretical Physics at the Massachusetts Institute of Technology (MIT) and his colleagues turned to an established alternative, lattice QCD.

There, the space and time where particle interactions occur are divided up into a discrete grid, and the dynamics of the interactions are calculated only at those grid points by a computer. Even this technique has problems at neutron star densities.

But another simplification is possible: using isospin, another quantum number which the proton and neutron have in opposite values, +1/2 or -1/2 respectively (the idea being that the proton and neutron can be treated as isospin states of the same particle but with opposite isospins).

The quantum mechanical mathematics of isospin is very similar to that of ordinary particle spin in quantum mechanics and quantum electrodynamics. It's known that nuclear matter at any density has a pressure less than the nuclear matter at nonzero isospin density.

Using this limit on the pressure, the group was able to "drill down" into high-density regions of the neutron star and obtain rigorous results. To do this, the team reduced the full mathematical description of a neutron star, then ran extensive lattice QCD models taking "a few thousand GPU hours," splitting the work over several supercomputers.

Many parts of the calculation had previously been done by other researchers; Abbott estimated that in all the problem required "several million GPU hours" on a supercomputer. To correct for the isospin nuclear matter being simulated on a discrete space-time grid, they were able to obtain the "continuum limit" where the small lattice spacing vanishes, something never before done for isospin nuclear matter.

They obtained an equation of state of isospin-dense matter for any isospin chemical potential (the energy change when adding or reducing the particle number of the system) at zero temperature, a result presented for the first time.

From conformal field theory, it had previously been suggested that the speed of sound in a neutron star, traveling as compressed waves, had a maximum of c/√3 in strongly interacting QCD matter, where c is the speed of light. But Abbott and his group found a speed of sound that exceeded this—though uncertain, it was higher, with the range peaking at ¾ c.

The results by Abbott and colleagues open a window into further computational studies of the matter of neutron stars. More refined calculations may be possible, such as conductivities and viscosity, and perhaps someday be able to interpret astronomical observations, and perhaps even predict them.

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