Symmetry energy determines the internal bulk viscosity of neutron stars

Neutron-star mergers are not just spectacular astrophysical events; they are natural experimental labs for fundamental nuclear physics. The gravitational waves (ripples of spacetime measured by detectors such as LIGO/Virgo) emitted during an inspiral carry information about how neutron-star matter reacts to being periodically squeezed and released by its companion.

Our recent work shows that this reaction is controlled, to a surprising degree, by a familiar nuclear-structure quantity: the nuclear symmetry energy, which is the energy cost of making nuclear matter neutron-rich rather than an equal mixture of neutrons and protons. Concretely, we find that the symmetry energy and its density dependence largely determine both how neutron-star matter dissipates tidal energy into heat and how it can store tidal energy elastically during inspiral. This connects terrestrial constraints on nuclear physics to new, independent constraints from gravitational-wave astronomy.

The key idea is that gravitational-wave observations can probe more than the star's static compressibility. They can also probe an internal, time-dependent stress associated with weak interactions. In fluid dynamics, this stress is quantified by bulk viscosity ζ (the resistance to uniform compression/expansion, distinct from shear viscosity, which resists shape changes at nearly fixed volume). In this framework the out-of-equilibrium isotropic stress Π---sometimes called the bulk-viscous pressure---is treated as a dynamical variable, and the relaxation time τ_Π sets how quickly Π relaxes back toward equilibrium.

In the cold interior, neutron-star matter is mostly neutrons with a small fraction of protons and electrons, often idealized as neutron--proton--electron (npe) matter. The composition is set by beta equilibrium, meaning that weak interactions (processes that turn neutrons into protons and electrons, and vice versa) have adjusted the chemical composition so that there is no net drive to convert one species into another. Tidal deformations during inspiral perturb the density and composition away from beta equilibrium, generating a chemical imbalance (a mismatch of chemical potentials that quantifies the drive for weak reactions). Our approach starts from nonequilibrium thermodynamics: demanding non-negative entropy production yields an Israel--Stewart relaxation equation for the bulk stress Π, from which we extract ζ and τ_Π. Schematically, 

τ_Π (dΠ/dt) + Π = -ζ θ

where θ measures the local compression rate (in nonrelativistic language, θ ≈ ∇·v). For a periodically driven system such as an inspiral, this single equation automatically yields a frequency-dependent, viscoelastic response: part of Π can be in phase with the compression (an elastic restoring component) and part out of phase (dissipation into heat). Importantly, ζ and τ_Π remain meaningful even in the "elastic" or frozen-composition limit---they parameterize how the stress builds up and relaxes, not only how much heat is produced.

To compute these effects, we need an equation of state (EoS), which is the relation between thermodynamic quantities such as pressure, energy density, and composition. The bulk viscosity and relaxation time depend on many detailed derivatives of the EoS, and therefore on the underlying nuclear microphysics. Our main result is that, in the cold, neutrino-transparent regime (meaning neutrinos escape rather than being trapped, so they do not act as an additional conserved component), this complexity collapses near nuclear saturation density, which is the characteristic density of matter inside heavy nuclei and provides a natural reference point for neutron-star physics. Near this density, we show that almost all of the EoS dependence relevant for weak-interaction-driven bulk viscosity can be expressed in terms of the symmetry energy and its slope. 

A standard parametrization near saturation density characterizes it by two numbers: S, the symmetry energy at saturation, and L, its slope with respect to density. These parameters are constrained by a combination of nuclear experiments and theory, but L in particular still carries significant uncertainty. We derive a compact mapping: the bulk-viscous transport properties near saturation density are determined almost entirely by S and L, plus well-understood leptonic contributions. 

This clean mapping turns uncertainties in nuclear physics directly into uncertainties in the dynamical tidal response. When we vary L within its currently allowed range, the bulk-viscous transport coefficients near saturation density can change by orders of magnitude. An intuitive way to understand this sensitivity is to focus on what sets the size of the out-of-equilibrium stress Π: tidal compression pushes the system away from beta equilibrium and builds up a chemical imbalance. The slope L controls how rapidly the pressure of neutron-rich matter rises with density and how sensitively the beta-equilibrium composition shifts under compression, so it controls how strongly a given tidal compression generates this imbalance and therefore Π. In this sense, the bulk response provides a new "lever arm" on the symmetry energy, complementary to traditional constraints from neutron-star radii and nuclear-structure data.

A particularly interesting limit arises at low temperatures, where weak interactions can become so slow that the relaxation time becomes very long compared to the orbital time scale. In this frozen-composition regime, the composition is effectively stuck during each tidal cycle. This stiffness is quantified by an effective bulk modulus ζ/τ_Π (a measure of resistance to compression, analogous to how stiff a spring is). We show that this effective bulk modulus near saturation density is controlled almost entirely by S and L: increasing L makes neutron-rich matter significantly harder to compress, enhancing the elastic component of the response.

These results have direct implications for gravitational-wave modeling. The inspiral waveform is affected not only by the star's familiar tidal deformability but also by frequency-dependent effects known as dynamical tides. The elastic response corresponds to the tidal bulge, while the dissipative process corresponds to the tidal lag, as Fig.1 shows. Gravitational-wave observations have already begun to place constraints on average viscosities inside neutron stars. Our results indicate that waveform models should also account for the symmetry-energy-controlled elastic contribution that emerges in the frozen-composition regime, especially as detector sensitivity improves.

Figure 1. Cartoon of the tidal response of neutron stars in a binary system (not to scale). Fig. 1 from “Dynamical tidal response of neutron stars as a probe of dense-matter properties”.

In this work, we focused on cold, neutrino-transparent npe matter near saturation density, which represents a substantial fraction of a typical neutron star's interior. At higher densities, additional degrees of freedom (such as muons, hyperons, or deconfined quarks) may appear, and they can modify both the composition dynamics and the transport coefficients. Extending the same symmetry-energy-based strategy to these regimes is an important direction for future work. Nevertheless, the central message is simple and robust: the symmetry energy and its slope, familiar from nuclear structure and effective field theory, also control how neutron stars dissipate and store tidal energy during inspiral, opening a new path to constrain nuclear physics with gravitational waves.

This research is described in the article "Symmetry energy dependence of the bulk viscosity of nuclear matter," Yumu Yang, Mauricio Hippert, Enrico Speranza, Jorge Noronha, Phys. Rev. Lett. 135 (2025) 22, 222702.