Investigating Twin Star Equation of States in Light of Recent Astrophysical Observations

Twin stars are predicted to exist in nature if the hadron-to-quark phase transition is strong enough to form a new branch of hybrid stars, separated from the branch of neutron stars. We adopt an agnostic approach, using transition energy density, transition pressure, the discontinuity strength, and a constant speed of sound for quark matter as our parameter space to construct a large possibility of hybrid equations of state, and thereby encapsulating a comprehensive picture of the twin star scenario. First, we report the complete conditions on our parameter space imposed by the general relativistic hydrostatic equilibrium solutions. For a fixed transition energy density and speed of sound for quark matter, we define distinct ranges of transition pressures based on the allowed strengths of discontinuity. Below a maximum transition pressure, a range of discontinuity exists that increases as the transition pressure decreases. Thereby, we identify the loci of the limits on discontinuities as the `witch-hat’ curves. Based on the causality limit, the witch-hat curves can be punctured or incomplete. Strong constraints on this picture are drawn from the inferences from GW170817 and the NICER measurements. We computed the maximum mass for twin stars to be $2.05~M_\odot$, the allowed strongest discontinuity in rest-mass density to be $7.76ρ_\mathrm{sat}$, and the upper bound on transition rest-mass density to be $4.03ρ_\mathrm{sat}$. Subsequently, we compute the implications of the stiffness of the quark matter equation of state on this picture. Different confidence levels for observational inferences are considered to assess the extent of inclusion (and rejection) of hybrid equations of state and, consequently, their effects on the limits of the maximum mass of twin stars and phase transition properties.

💡 Research Summary

The paper presents a comprehensive, model‑agnostic investigation of the conditions under which twin stars—compact objects of identical mass but distinct radii—can arise from a strong first‑order hadron‑to‑quark phase transition inside neutron stars. The authors construct a large ensemble of hybrid equations of state (EoS) by treating four quantities as free parameters: the transition energy density (e_tr), the transition pressure (P_tr), the discontinuity in energy density (Δe), and the constant speed of sound squared in quark matter (c_s²,QM). The low‑density regime (below ~0.4 ρ_sat) is described by a fixed crust EoS, while the hadronic sector (e < e_tr) is modeled with a polytropic form P = k ρ^γ, allowing the stiffness to be tuned via γ. The phase‑transition region (e_tr < e < e_tr + Δe) follows a Maxwell construction: pressure remains constant at P_tr while the energy density jumps by Δe, producing a density jump Δρ that quantifies the strength of the transition. Above the transition, the quark phase is described by the constant‑speed‑of‑sound (CSS) parametrization, P(e) = P_tr + c_s²,QM (e − e_tr − Δe), with c_s²,QM constrained by causality (c_s² ≤ 1).

The authors solve the Tolman‑Oppenheimer‑Volkoff (TOV) equations for each hybrid EoS to obtain mass‑radius (M‑R) sequences. By locating the maximum‑mass configuration on the second (purely hadronic) branch (M_TOV,2) and examining the response of the sequence to increasing central pressure, they identify two possible outcomes: continuation on the second branch or a transition to an unstable segment followed by a third, stable branch (the twin‑star branch). The appearance of the third branch is governed by the Seidov criterion, which relates the size of the density jump to the stability of the configuration. The authors introduce “witch‑hat” curves in the (P_tr, Δe) plane for fixed e_tr and c_s²,QM, which delineate the region where a viable twin‑star solution exists. These curves expand as P_tr decreases (allowing larger Δe) but are truncated where the speed‑of‑sound limit is reached, producing “punctured” or incomplete witch‑hat shapes.

Observational constraints are then overlaid. Gravitational‑wave data from GW170817 provides limits on the tidal deformability Λ, excluding EoS that are excessively soft at intermediate densities. NICER measurements of PSR J0614–3329 and PSR J0437–4715 indicate a relatively soft EoS around the canonical 1.4 M⊙ mass, thereby favoring lower transition pressures. Conversely, the existence of ~2 M⊙ pulsars demands sufficient stiffness at higher densities, which is accommodated by choosing a relatively high c_s²,QM (≈0.6–1.0). By jointly applying these constraints, the admissible parameter space collapses to a narrow band: e_tr ≈ (3–4) ρ_sat, P_tr ≈ (80–120) MeV fm⁻³, Δe ≈ (5–7) ρ_sat, and c_s²,QM ≈ 0.8.

Within this filtered region the authors find that the maximum mass attainable on the twin‑star branch is M_TOV,3 = 2.05 M⊙, the strongest allowed discontinuity in rest‑mass density is Δρ_max ≈ 7.76 ρ_sat, and the upper bound on the transition density is ρ_tr,max ≈ 4.03 ρ_sat. These numbers represent the most extreme yet observationally consistent hybrid configurations that produce twin stars. The paper categorizes possible hybrid EoS into four classes based on whether the second and third branches exceed the 2 M⊙ threshold, showing that only the first three classes survive the combined astrophysical constraints; class IV (very low P_tr) is effectively ruled out.

The study’s significance lies in its systematic mapping of the hybrid‑EoS landscape without committing to a specific nuclear or quark model, thereby providing a “possibility map” for twin‑star existence. The witch‑hat visualization offers an intuitive tool for assessing how causality and observational data prune the parameter space. The authors conclude that future high‑precision mass‑radius measurements, improved tidal‑deformability constraints from next‑generation gravitational‑wave detectors, and theoretical refinements of the quark‑matter EOS (including effects of color superconductivity, magnetic fields, rotation, and finite temperature) will be essential to either confirm the twin‑star scenario or further tighten the allowed region. This work thus bridges the gap between QCD‑driven microphysics and macroscopic astrophysical observables, setting the stage for a more definitive test of the hadron‑to‑quark phase transition in the cores of neutron stars.