Listening to the Long Ring: How Gravitational Waves Reveal the Neutron Star Equation of State ✛
The collision of two neutron stars is a strike heard across the universe. Gravitational wave detectors pick up the ripples in spacetime, but the most interesting sound begins after the impact: the merger remnant, a rapidly spinning hypermassive neutron star, continues to vibrate, emitting waves for a few more milliseconds. Physicists call this short but information-rich signal the long ring. By listening to how the bell fades after the strike, you can learn what metal it was cast from. The same applies here: in the damped oscillations of the remnant, the equation of state is encoded, the relation between pressure and density of matter compressed to unimaginable values.
In the cores of neutron stars, matter is compressed to densities several times that of an atomic nucleus. The equation of state of such matter remains one of the main unsolved problems in the physics of strong interactions. Existing constraints come from observations of neutron star mergers (e.g., GW170817), mass and radius measurements, and theoretical calculations based on quantum chromodynamics. However, all of them apply to densities up to about three times nuclear saturation, while in the centers of the most massive neutron stars, the density can reach five to six times saturation. This work joins the line of research using post-merger gravitational waves to probe these extreme conditions, but it proposes a new observable parameter that had not been considered before.
The authors constructed an ensemble of equations of state using Gaussian processes constrained by data from chiral effective field theory and perturbative quantum chromodynamics. Then, using principal component analysis, they selected six "golden" equations of state that cover the 68% confidence region. For each of them, numerical simulations of neutron star mergers were performed with parameters consistent with GW170817: a fixed chirp mass of 1.18 solar masses and three mass ratios of the components (0.7, 0.85, 1.0). From the resulting gravitational wave signals, the energy and angular momentum carried away by the waves, as well as the instantaneous frequency, were extracted. In the long ring phase (from 1 to 15 ms after merger), the slope of the energy versus angular momentum relation was calculated, which turned out to be constant.
The key result: the slope dE_GW/dJ_GW in the long ring phase correlates with the pressure and density at the center of the most massive neutron star (the TOV point). For all six "golden" equations of state and all three mass ratios, this correlation is described by a bilinear model. For example, when the pressure at the TOV point increases from 0.2 to 0.6 GeV/fm³, the slope changes from about 3.0 to 3.5 (in dimensionless units). Modeling measurements with an uncertainty of ±4% showed that using the slope together with the peak frequency f2 can reduce the uncertainty of the equation of state in the high-density region by a factor of several compared to using only f2.
The work proposes a new observable parameter of the gravitational wave signal after a neutron star merger that is directly related to the equation of state at maximum densities. Unlike previous studies that focused on spectral peaks (e.g., f2), here the ratio of energy to angular momentum is used, which is easier to extract from the signal and is less sensitive to modeling details. This opens the way to direct constraints on the equation of state in the density range of 4–6 times saturation, where currently there are no data.
With the advent of third-generation detectors such as the Einstein Telescope and Cosmic Explorer, many neutron star merger events with high signal-to-noise ratio are expected to be detected. This will allow the proposed method to be applied to real data and to obtain increasingly precise constraints on the equation of state. Furthermore, the correlation can be refined with more detailed simulations that include magnetic fields, neutrino emission, and other effects. In this way, we will learn to read from the fading ring what material the most extreme objects in the universe are made of.
The long ring lasts only a few milliseconds, but during that time the merger remnant loses energy equivalent to the mass of several Jupiters.
In Carl Sagan's novel "Contact," gravitational waves from a neutron star merger serve as a key to deciphering an alien message; here, too, they carry information about the most extreme states of matter.