Overview
Neutron stars provide a natural laboratory for testing matter under densities unreachable on Earth, bridging the limits of nuclear and particle physics. This project explored how quark deconfinement within such stars could leave detectable signatures in gravitational-wave observations from binary mergers. Using a hybrid equation of state (EoS) approach, the study connects microphysical phase transitions to macroscopic observables such as mass–radius relations and tidal deformabilities relevant to the Einstein Telescope’s sensitivity range.
A Python implementation of a TOV solver (converted from C++) was used to generate mass–radius and tidal-deformability sequences for a set of hybrid EoS models. Results are compared against constraints consistent with GW170817, and key transition properties are visualised through thermodynamic and structural diagnostics described below.
Hybrid Equations of State & Phase Transition
The pressure–chemical potential (P–μ) relation provides a clear thermodynamic view of the hadron–quark phase transition in dense stellar matter. When plotted, the intersection of the hadronic and quark branches marks the coexistence point, where both phases share equal pressure and chemical potential—a characteristic of a first-order transition constructed via the Maxwell method. Below this transition, pressure rises smoothly with μ, reflecting stable hadronic behaviour, while above it the quark matter branch exhibits a shallower slope, signifying a reduced stiffness at higher densities. The location of this intersection depends sensitively on the model parameters, particularly the effective gluon mass, and determines the onset density of deconfinement within the stellar core. This relationship therefore bridges microscopic quark-level physics and macroscopic stellar observables.
The pressure–energy density (P–ε) relation defines the macroscopic stiffness of matter and governs the overall structure and stability of compact stars. For hadronic matter, the pressure increases steadily with energy density, reflecting the growing resistance to compression. As the phase transition to quark matter begins, the curve softens— its slope decreases—indicating a temporary loss of stiffness. When the transition is strong, this discontinuity can produce an unstable region in the mass–radius sequence, eventually leading to the formation of a re-stabilised hybrid branch. This so-called twin-star behaviour illustrates how small changes in the microphysics of the equation of state can give rise to two classes of stars with similar masses but markedly different radii.
TOV Modelling, Twin-Star Behaviour & Mass–Radius Sequences
Integration of the Tolman–Oppenheimer–Volkoff (TOV) equations was performed for each EoS to obtain equilibrium sequences of mass and radius. The resulting tracks exhibit the characteristic pattern predicted by strong first-order phase transitions: a stable hadronic branch, an intermediate unstable region, and a secondary, re-stabilised hybrid branch. The existence of this distinct second family—the twin-star configuration— demonstrates how the hybrid core’s stiffness recovery counteracts gravitational collapse once deconfined quark matter dominates the central pressure.
In the generated models, the onset of instability corresponds to the softening visible in the pressure–energy density relation, while re-stabilisation coincides with the quark-matter branch becoming sufficiently stiff. The mass–radius diagram therefore encodes the entire transition sequence, providing direct visual evidence of the hybrid star phenomenon that underlies potential multi-branch populations detectable by next-generation gravitational-wave observatories.
Tidal Deformability
The tidal deformability (Λ) acts as the critical observable linking microscopic matter properties to the gravitational-wave signatures detected during binary neutron star mergers. It quantifies how easily a star’s shape distorts under its companion’s gravitational field, directly encoding the stiffness of the underlying equation of state. As the star becomes more compact, Λ falls sharply—reducing the tidal imprint on the waveform. This behaviour, captured in the Λ–M relation, is particularly sensitive to the presence of a quark-matter core: once deconfinement occurs, the reduced internal stiffness drives a visible kink in the sequence, providing an indirect but measurable indicator of the phase transition.
For hybrid EoS models, this transition region produces a striking signature—an abrupt drop in Λ around the mass range corresponding to the hybrid branch in the M–R diagram. Within the context of GW170817 and the sensitivity of upcoming observatories such as the Einstein Telescope, this feature defines an astrophysical “hook”: a testable prediction that the imprint of quark deconfinement could emerge directly in the tidal phase evolution of binary inspirals. Thus, the tidal deformability is not only a measure of compactness but a probe of the fundamental structure of matter at supranuclear densities.
Summary
The figures above summarise the workflow: hybrid EoS construction with a first-order transition, TOV-based M–R generation including twin-star branches, and tidal-deformability diagnostics used for gravitational-wave comparisons. The qualitative behaviours align with published predictions: the emergence of a re-stabilised hybrid branch under strong phase transitions and the associated decrease in Λ for more compact configurations. These results collectively demonstrate how microphysical assumptions about deconfinement manifest as testable astrophysical signatures.