Nonparametric Equation of State Realizes a Generalized Quark-Hadron Crossover ✦
A nonparametric approach is proposed to construct a statistical equation of state (EOS) continuously from the nuclear crust to the asymptotic freedom regime. The joint requirement of supporting neutron stars with two solar masses and relatively small radii at low masses, along with reaching the asymptotically soft limit of perturbative QCD, forces a peak in the squared speed of sound followed by prolonged softening, with c_s^2 returning to ~1/3 near ~30 n_sat. Accordingly, the trace anomaly Δ ≡ 1/3 − p/ε becomes positive beyond neutron star densities and approaches the pQCD limit from above. A quantitative assessment of the degree of this softening in the posterior distribution indicates the presence of a hadron-quark transition in the cores of the most massive neutron stars. The thermodynamic structure of the full statistical EOS naturally realizes a generalized quark-hadron crossover. The quark EOS above neutron star densities is thus soft and nonperturbative, unlike the stiff quark EOS underlying the quark star picture.
Context
Neutron stars are unique laboratories for studying matter at extreme densities unattainable in Earth-based experiments. Back in the 1930s, Fritz Zwicky predicted their existence, and Jocelyn Bell Burnell discovered pulsars in 1967, which became the main probes of their internal structure. Today, gravitational-wave detectors (LIGO/Virgo) and the NICER X-ray telescope provide complementary data, enabling reconstruction of the equation of state (EOS)—the relationship between pressure and energy density. Stellar evolution leads to the formation of compact remnants, and understanding the phase transition from hadrons to quarks in their interiors is a key challenge at the intersection of nuclear physics and QCD.
Methods
We construct a Bayesian ensemble of EOS by representing the squared speed of sound through an auxiliary field φ and imposing a prior distribution using Gaussian processes (GP). The EOS consists of three continuous segments: the low-density nuclear crust (χEFT), a high-density segment of perturbative QCD (anchored at μ_B = 2.6 GeV, where asymptotic freedom, discovered by Frank Wilczek, applies), and an intermediate action-optimized bridge. The key idea is a least-action principle for c_s^2 trajectories, where the action penalizes deviations from thermodynamically consistent evolution. Exact adherence to pQCD bounds is achieved through iterative projection via the GP covariance. The gravitational field of a neutron star is described by the metric of curved spacetime and the Tolman–Oppenheimer–Volkoff equations.
Results
Posterior analysis incorporating NICER data for several millisecond pulsars (PSR J0030+0451, J0740+6620, J0437−4715, and J0614−3329) and the GW170817 merger revealed a nonmonotonic c_s^2 profile. The peak reaches 0.68 (+0.14, -0.13) — well above the conformal limit of 1/3, followed by significant softening, with c_s^2 returning to ~1/3 at a density of about 30 n_sat. The trace anomaly Δ ≡ 1/3 − p/ε becomes positive inside the neutron star and approaches the pQCD limit from above. To quantify the transition, an accumulated softening index ς is introduced; its posterior distribution shows that the softening substantially exceeds the maximum achievable in purely hadronic models (≤ 1.01×10⁻³). The maximum mass M_TOV = 2.11 (+0.12, -0.09) M☉ is weakly correlated with the height of the c_s^2 peak.
Implications
The discovered “peak–softening” structure is not imposed by the model but dictated by thermodynamics: supporting massive neutron stars requires an early stiffening of the EOS, but this also quickly accumulates energy density; to avoid exceeding the pQCD limit, a subsequent softening is necessary. Thus, the nonparametric statistical EOS naturally realizes the thermodynamic profile of a generalized quark-hadron crossover—a smooth transition analogous to the one known at finite temperature. Quark matter above nuclear densities turns out to be soft and nonperturbative, contrasting with hypothetical quark stars that require a stiff EOS.
Future development
Future observations with next-generation gravitational-wave detectors (Einstein Telescope, Cosmic Explorer) and X-ray missions (STROBE-X) will pinpoint the crossover location and probe the behavior of matter at finite temperature. Numerical simulations of neutron star mergers incorporating the quark-hadron transition and the formation of quark-gluon plasma may reveal unique multimessenger signatures—from the gravitational-wave signal to kilonova afterglows.
Impact
The results directly impact the physics of dense matter, the astrophysics of compact objects, and quantum chromodynamics, providing a consistent picture from the nuclear crust to the asymptotic freedom regime.
Next steps
Priority steps: incorporating strange quark mass and color superconductivity effects into the pQCD bounds, and applying the method to new NICER data and LIGO–Virgo–KAGRA merger catalogs.
Key open problems
The study connects fundamental problems: the equation of state of cold dense matter, the nature of deconfinement in QCD, and the origin of the maximum mass of neutron stars, offering a unified framework for their joint analysis.
🎯 Although the speed of sound in a neutron star can exceed 1/3 the speed of light, it does not violate causality, and information from the core reaches the surface in fractions of a millisecond.
🎬 In Robert Forward’s novel “Dragon’s Egg,” intelligent life is described on the surface of a neutron star, based on nuclear interactions, echoing the idea of exotic states of matter in such objects.
Key numbers
- c_s^2 peak: 0.68 +0.14 -0.13
- Maximum mass M_TOV: 2.11 +0.12 -0.09 M☉
- Crossover density: ~30 n_sat
- pQCD boundary energy: \mu_B = 2.6 \text{ GeV}
- Maximum softening index for hadronic models: ≈ 1.01×10⁻³