# Sound velocity bound and neutron stars

### P. Bedaque and A.W. Steiner (2015)

Figure: The squared speed of sound as a function of density in a neutron star. The sound speed is small at low densities near the surface of the neutron star and higher at high densities near the neutron star core. These three lines represent three different models for how the sound speed behaves inside the neutron star. The quantitiy on the x-axis, $\varepsilon$, is energy density (closely related to mass density in the usual sense).

Summary:

Neutron stars1 are the final stage in the evolution of a star originally between 8 and 20 times the mass of the sun. It has been possible to measure the mass of several neutron stars, and until recently, all accurate mass measurements were near 1.4 times the mass of our sun. However, within the past few years, two neutron stars have been discovered to have a mass around twice that of our sun2.

The speed of sound in air is about 346 meters per second, and the speed of sound tends to increase with either the density or the temperature of the medium in which it travels. Since neutron stars contain the most dense matter in the universe (except for the matter inside a black hole which we cannot observe) one might wonder how fast the speed of sound is inside neutron stars.

Everywhere else in the universe3, the speed of sound seems to be limited to the speed of light ( $c$ ), divided by $\sqrt{3}$ (see for example the figures here). At high enough densities or temperatures, the speed of sound always approaches this value, $c/\sqrt{3}$. This result comes from quantum chromodynamics (QCD)4 - the physical theory which describes how neutrons and protons (made of quarks) interact. At high enough densities and temperatures, QCD exhibits "asymptotic freedom", meaning that the interaction becomes weaker.5 Unfortunately, neutron star densities are not large enough so that quarks are weakly interacting.

Paulo and I showed that the speed of sound in neutron stars must, at some point, exceed this value. The reason is that models where the speed of sound is smaller than $c/\sqrt{3}$ at all densities (those like the black lines in the figure) cannot produce neutron stars with masses twice the mass of the sun. Thus, the only alternative is that the speed of sound must look something like either the blue dotted or red dashed lines. At some density the speed of sound must exceed $c/\sqrt{3}$.

This result is important because it tells us more about how neutrons and protons interact, not only in neutron stars, but also here on earth6. It gives us more insight into how QCD behaves at high densities. Finally, it also helps us understand some of the more extreme neutron star-related processes like core-collapse supernovae, magnetar flares, and neutron star mergers.

Our paper is published in Phys. Rev. Lett. (as an "Editor's suggestion") and is also accessible on arXiv.org/1408.5116.

Notes:

1See a diagram of stellar evolution from the Chandra X-ray observatory, their neutron star page, or the wikpedia entry on neutron stars.
2See Demorest et al. (2010) and Antoniadis et al. (2013).
3The only possible exception is matter inside the event horizon of a black hole, which is not causally connected with the rest of the universe anyway.
4See the Wikipedia article on Quantum Chromodynamics.
5This finding led to a Nobel prize in physics for Gross, Politzer, and Wilczek in 2004.
6Neutrons and protons are the basic building blocks of all atomic nuclei.