Despite decades of observations, the behavior of matter inside neutron stars remains one of astrophysics' greatest mysteries. At densities exceeding that of atomic nuclei (about 2.6×$10^{14}$ g/cm³), theoretical predictions diverge, and laboratory experiments are impossible. The speed of sound—the derivative of pressure with respect to energy density—serves as the direct 'voice' of matter: the higher it is, the stiffer the matter resists compression. In cold neutron stars, its profile determines the maximum mass and radius, and in merger dynamics, it decides the outcome: formation of a hypermassive remnant or swift collapse into a black hole. That’s why astrophysical observations become the only testing ground for our ideas about strong interactions. Back in 1916, Karl Schwarzschild found the first exact solution to the Einstein field equations, laying the foundation for compact object theory, and the discovery of pulsars by Jocelyn Bell Burnell in 1967 confirmed the existence of neutron stars.
To isolate the influence of the speed of sound at supranuclear densities, the authors built a one-parameter model in which the square of the sound speed grows linearly with the baryon chemical potential: $c_s$²(μ) = $c_{sm}^2$ + $c_{ref}^2$ (μ−$μ_m$)/$μ_m$. Using the WhiskyTHC code to solve the full Einstein field equations and the equations of relativistic hydrodynamics, they ran a series of simulations of mergers of equal-mass (1.3–1.4 M⊙) binary neutron stars with varying slope parameters. The gravitational-wave signal was extracted using the Newman–Penrose formalism, and for spectral analysis, quadrupole radiation was applied as the primary energy-loss mechanism. Simultaneously, the flux of dynamically ejected material—giving rise to a kilonova—was calculated. Thermal effects were accounted for by adding a Γ-law with an adiabatic index of 1.7. A grid resolution of 245 m kept computational costs manageable.
The simulations revealed a clear correlation between the sound speed slope and the peak frequency of post-merger gravitational waves $f_{pk}$ (Fig. 7b): for moderate parameter values, the growth is nearly linear, and this dependence is only weakly sensitive to the component masses. The total radiated energy $E_{GW}$ and the mass of ejected material showed non-monotonic behavior—peaking at intermediate stiffness (Figs. 8, 9). Soft equations of state led to prompt collapse into a black hole, while stiff ones resulted in a weak bounce and little ejecta. Additional analysis of simulations from the CoRe database with realistic equations of state yielded an approximately EOS-independent relationship between the dimensionless peak frequency M $f_{pk}$ and a new tidal response parameter ˜D, which directly incorporates the star-integrated sound speed (Fig. 11). An exponential fit gave a coefficient of determination R²=0.811.
The detected correlations indicate that post-merger gravitational waves carry information about the behavior of matter at densities significantly exceeding those probed by tidal deformability measurements during the inspiral. The tidal response parameter ˜D partially 'folds away' the scatter, but the remaining non-monotonicity suggests that global stellar properties do not capture all the microphysics of the core. Thus, future observations of gravitational waves in the kiloHertz range can directly constrain the stiffness of ultra-dense matter, and multi-messenger analysis can constrain the mechanisms of nucleosynthesis of heavy elements. The speed of sound remains within the fundamental limit—the speed of light in vacuum, but approaching it implies extreme stiffness that must eventually give way to a phase transition to satisfy QCD asymptotics.
Parameterized sound-speed models are an ideal basis for building surrogate emulators based on neural networks. Replacing direct simulations with fast approximators will enable scanning of the multi-dimensional EOS parameter space at the pace needed to interpret data from next-generation detectors. Future plans include incorporating additional parameters: the location and magnitude of the sound speed maximum, and possible dips associated with phase transitions. Such a reduced approach will drastically cut computational costs and accelerate the extraction of observational constraints.
The results directly impact the physics of dense matter, the astrophysics of neutron star mergers, and nucleosynthesis of elements heavier than iron. They are also important for planning future gravitational-wave missions and interpreting data from kilonovae.
The immediate task is to expand the parameter set to include possible phase transitions and to align the models with the strict constraints from perturbative QCD at large chemical potentials. Simultaneously, simulations with non-zero mass ratios and spins are needed.
The work directly addresses two unsolved problems: the equation of state of strongly interacting matter at densities of 2–10 n₀ and the possible existence of phase transitions (such as quark deconfinement) in the cores of neutron stars. Integral inequalities linking the average pressure to the speed of sound, along with QCD asymptotics, impose global constraints that any realistic model must satisfy.
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What the work does
The authors constructed a one-parameter model of the speed of sound, which grows linearly with baryon chemical potential, and ran a series of numerical simulations of equal-mass binary neutron star mergers using the WhiskyTHC code. They analyzed the gravitational-wave signal and the dynamical ejecta, revealing a clear correlation between the slope of the speed of sound and the peak frequency of post-merger gravitational waves, as well as a non-monotonic dependence of the radiated energy and ejecta mass on the stiffness of the equation of state.
Where its strength is
The work demonstrates that a simple parametrization of the speed of sound is sufficient to capture the key features of merger dynamics, and offers a convenient bridge between the microphysics of the interior and observable gravitational-wave signatures.
Where it could go next
Worth trying
Why this may matter
Such a simple parametrization could become a handy tool for extracting information about the stiffness of matter at supranuclear densities from future gravitational-wave observations, without requiring full knowledge of the equation of state.
Nearby works in our archive
The archive is crowded around this work: 10 reviews nearby, closest at 0.70. This ground is well walked — and there is plenty to compare against.
A teaspoon of neutron star material weighs as much as Mount Everest — about a billion tons. And the speed of sound in such matter can reach 95% of the speed of light: a sound signal would cross a city-sized object faster than the blink of an eye.
In Robert Forward’s novel 'Dragon’s Egg,' life is described on the surface of a neutron star, where monstrous gravity and superfluid matter create a completely alien world. The sound-speed estimates from our paper could serve as a musical backdrop for such a story.