The research, carried out by Khushbu Zala and Sreemoyee Sarkar of SVKM’s NMIMS University in Mumbai, focuses on a phenomenon known as the r-mode instability. R-modes are non-radial oscillation modes in a rotating star that couple to gravity: as the stellar fluid sloshes back and forth, it emits gravitational radiation that carries away angular momentum. In the absence of any counteracting effect, these modes would grow without bound, spinning the star down dramatically in a runaway process. Yet astronomers observe that many millisecond pulsars—stars rotating hundreds of times per second—remain remarkably stable. Something inside them must be damping the oscillations, and the leading candidates are the two forms of viscosity that govern how dense matter dissipates energy: shear viscosity, which resists the sliding of adjacent fluid layers, and bulk viscosity, which dissipates energy when the fluid is periodically compressed and expanded.

What makes the new work distinctive is its treatment of a star whose core contains a mixed phase, a region where hadronic and quark matter coexist in thermodynamic equilibrium. Below a transition density of about 2.23 times nuclear saturation density, the outer layer is modeled as ordinary baryonic matter described by the relativistic mean-field equation of state known as DDME2. Above that threshold, the researchers assume a hybrid interior in which quarks and hadrons mingle. The problem, as the authors emphasize, is inherently ill-posed: the equation of state at supranuclear densities is poorly constrained, quantum chromodynamics cannot yet be solved reliably in the non-perturbative regime relevant to stellar cores, and the transport coefficients—the viscosities that control damping—are even harder to pin down from first principles.

To cut through these uncertainties, the team turned to Bayesian inference, a statistical framework that updates prior physical knowledge with observational data to produce posterior distributions for unknown parameters. The prior distributions for the key dimensionless coefficients—labeled S-tilde, V-tilde, W-tilde and J-tilde, which encode the shear and bulk viscous response of the mixed phase along with the star’s equilibrium structure—were drawn from existing calculations for neutron stars, strange stars and hybrid stars. The likelihood function combined two independent observational constraints: mass-radius measurements from NASA’s Neutron Star Interior Composition Explorer, or NICER, mission, and the spin-frequency and temperature observations of neutron stars in low-mass X-ray binaries. Using the UltraNest nested sampling algorithm, the researchers explored the parameter space efficiently, focusing on regions of high likelihood and estimating the Bayesian evidence for their model.

The formalism at the heart of the analysis describes how the amplitude of an r-mode evolves in time as an exponential whose decay constant is set by the competition between three timescales: gravitational radiation, which drives the instability, and shear and bulk viscosity, which suppress it. At low temperatures, shear viscosity dominates and stabilizes the star; at high temperatures, bulk viscosity takes over. Between these regimes lies a window where damping is least effective and the star is most vulnerable. The minimum of the instability curve—the lowest spin frequency at which the mode can grow—is therefore an exquisitely sensitive probe of the microphysics inside the star, including the equation of state and the weak-interaction processes, such as the direct Urca reaction, that generate bulk viscosity.

Applying this framework to two hybrid star configurations of 1.5 and 1.75 solar masses, the team obtained concrete estimates for the dissipation timescales. The shear viscous damping time came out as approximately 4.99 times ten to the eight, multiplied by the temperature to the five-thirds power, in seconds, while the bulk viscous timescale follows a more complex dependence on both temperature and spin, scaling inversely with the square of the angular velocity. From the inferred coefficients, the researchers calculated the minima of the instability curves: the critical angular velocity reaches its lowest value of about 451.87 hertz at a temperature of 0.259 megaelectronvolts for the 1.5 solar mass star, and 517.47 hertz at 0.234 megaelectronvolts for the 1.75 solar mass star. Normalized to the Kepler frequency, the maximum spin rate a star can sustain before mass shedding, these minima correspond to ratios of roughly 0.069 and 0.071.

Crucially, the resulting instability window does more than produce elegant numbers—it matches what astronomers actually see. When the team compared their inferred instability curves with the observed spin frequencies and temperatures of real millisecond pulsars, they found that the enhanced viscous dissipation from the mixed hadron-quark phase provides sufficient damping to explain the stability of several well-known objects. Among them are the accreting low-mass X-ray binary sources XTE J0929-314 and XTE J1807-294, and the radio millisecond pulsars J0437-4715 and J2124-3358. These stars all spin faster than 100 hertz, placing them squarely in the frequency range where r-mode physics matters, and all of them sit safely outside the region where the instability would grow—precisely as the two-layer hybrid model predicts.

The posterior distributions themselves carried informative structure. The corner plots of the inferred parameters showed moderate correlations among the shear and bulk viscous coefficients, reflecting the coupled role of the two viscosities in setting the instability boundary, while correlations involving the equilibrium parameter remained comparatively weak. Notably, the posterior contours for the more massive 1.74 solar mass configuration were narrower and more tightly localized than those for the lighter star, indicating that frequency-temperature observations constrain the dense-core dissipation properties more stringently in heavier hybrid stars. The team also found that the position of the instability minimum is remarkably robust: it barely shifts when the equilibrium and viscous parameters vary across their full credible intervals, suggesting that the result is not an artifact of statistical noise.

The broader significance of the work lies in its demonstration that r-mode phenomenology, combined with modern statistical inference, can serve as a practical observational tool for probing phase transitions in ultra-dense matter. Because pulsar rotational frequencies and their time derivatives are among the most precisely measured quantities in all of astrophysics, and because NICER continues to deliver mass-radius constraints, the approach links macroscopic observables directly to microscopic transport physics. If the inferred viscous properties of the mixed phase continue to align with observations, it would strengthen the case that some neutron stars genuinely harbor deconfined quark matter in their cores—a question that has remained open since the earliest theoretical speculations about quark stars.

The authors outline several directions for extending the framework. A more realistic three-layer stellar model could better capture the stratification of a hybrid star’s interior, and generalizing the formalism to derive quantitative constraints on the shear and bulk viscosities of each individual layer would sharpen the physical picture. Perhaps most ambitiously, they aim to characterize the nature of the hadron-quark phase transition itself—determining whether it is first order, second order, or a smooth crossover—by performing statistical inference on transport coefficients constrained by the gravitational-wave signatures that r-mode oscillations generate. As gravitational-wave detectors grow more sensitive, the faint hum of a wobbling hybrid star may one day confirm what this Bayesian analysis already hints at: that the universe’s most extreme matter hides its secrets in the way it dissipates motion.

Subject of Research: Bayesian inference of viscous dissipation timescales governing r-mode instability in hybrid stars with hadron-quark mixed phases

Article Title: Modelling dissipative dynamics of r-mode instability in hybrid stars

Article References: Zala, K., & Sarkar, S. (2026). Modelling dissipative dynamics of r-mode instability in hybrid stars. The European Physical Journal C, 86(9), Article 1064. https://doi.org/10.1140/epjc/s10052-026-16199-6

Image Credits: AI Generated

DOI: 10.1140/epjc/s10052-026-16199-6

Keywords: hybrid stars, r-mode instability, neutron stars, bulk viscosity, shear viscosity, Bayesian inference, quark matter, millisecond pulsars, NICER, low-mass X-ray binaries, gravitational waves, dense matter