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

Technical Summary: Investigating Twin Star Equation of States in Light of Recent Astrophysical Observations

Problem Statement
The internal composition of neutron stars (NSs) and the nature of the phase transition (PT) from hadronic matter (HM) to quark matter (QM) remain unresolved. While Quantum Chromodynamics (QCD) predicts a transition at high densities, the specific conditions—such as the onset density, transition pressure, and the strength of the discontinuity—are unknown. A strong first-order PT could lead to the existence of "twin stars" (TSs): a third branch of stable compact objects in the mass-radius ($M-R$) diagram, separated from the standard neutron star branch by an unstable region. These twin stars would possess equal mass but smaller radii than their neutron star counterparts. The challenge lies in identifying the hybrid equations of state (EoS) that satisfy both general relativistic hydrostatic equilibrium and the increasingly precise constraints from multi-messenger astrophysics, including NICER mass-radius measurements and gravitational wave observations (GW170817).

Methodology
The authors adopt a model-agnostic approach to construct a comprehensive ensemble of hybrid EoSs capable of producing twin star solutions. The methodology relies on the Tolman-Oppenheimer-Volkoff (TOV) equations to solve for hydrostatic equilibrium. The EoS construction involves four distinct regions:

1. Crust Region: A fixed piecewise polytropic EoS below $\sim 0.4\rho_{sat}$.
2. Hadronic Region: Modeled using a polytrope with a free adiabatic index $\gamma$, determined by boundary conditions at the transition.
3. Phase Transition Region: Modeled via a Maxwell construction, characterized by a constant transition pressure ($P_{tr}$) and a discontinuous jump in energy density ($\Delta e$).
4. Quark Region: Modeled using a constant speed of sound (CSS) parametrization ($c^2_{s,QM}$).

The parameter space is defined by four independent variables: transition energy density ($e_{tr}$), transition pressure ($P_{tr}$), the strength of the energy density discontinuity ($\Delta e$), and the speed of sound in quark matter ($c^2_{s,QM}$). The study systematically varies these parameters within causality limits to identify conditions that yield a third stable branch in the $M-R$ sequence.

Key Contributions

• Definition of "Witch-Hat" Curves: The paper introduces a geometric representation in the $(P_{tr}, \Delta e)$ parameter space for fixed $e_{tr}$ and $c^2_{s,QM}$. These curves delineate the boundaries of allowed discontinuity strengths that permit twin star formation. The authors identify three distinct regions within this space:
  • Solution-less Region: No twin stars form regardless of $\Delta e$.
  • Bound Region: Twin stars form for a specific range of $\Delta e$ below a maximum transition pressure.
  • Unbound Region: Twin stars form for $\Delta e$ above a minimum limit, with no upper bound on $\Delta e$ (within the numerical setup).
• Analysis of Causality and Completeness: The study reveals that "witch-hat" curves can be "punctured" or incomplete. This occurs when the maximum allowed transition pressure ($P_{tr}^{max}$) required to form a twin star exceeds the causality limit ($c_s = 1$) or the central pressure of the maximum mass hadronic configuration.
• Integration of Recent Observational Constraints: The authors apply 2$\sigma$ (and 1$\sigma$/3$\sigma$) confidence intervals from recent NICER measurements (PSR J0614–3329, PSR J0437–4715, PSR J0740+6620, PSR J0030+0451), the massive pulsar PSR J0348+0432, and the tidal deformability from GW170817 to constrain the parameter space.

Results
The application of observational data significantly restricts the viable parameter space for twin stars:

• Constraints on Transition Density: Recent measurements of PSR J0614–3329 and PSR J0437–4715 suggest a softer EoS around the canonical mass, which tightly constrains the transition energy density. The study establishes an upper bound on the transition energy density of $e_{tr}^{max} \approx 1.40 \times 10^{15}$ g/cm$^3$ ($\approx 4.03\rho_{sat}$) for the stiffest quark EoS ($c^2_{s,QM}=1.0$).
• Constraints on Discontinuity Strength: The maximum allowed discontinuity in rest-mass density is found to be $\Delta\rho_{max} \approx 7.76\rho_{sat}$ ($\Delta e_{max} \approx 2.94 \times 10^{15}$ g/cm$^3$). This limit is governed by EoSs that produce Category II twin stars (where the second branch supports $>2.0 M_\odot$ but the third branch does not).
• Maximum Mass of Twin Stars: The maximum mass achievable by twin stars is calculated to be $M_{TS}^{max} \approx 2.05 M_\odot$ (for $c^2_{s,QM}=1.0$ and 2$\sigma$ confidence). This limit is set by Category I twin stars (where both branches support $>2.0 M_\odot$).
• Impact of Confidence Levels: The results are sensitive to the confidence level of observational inferences. At a strict 1$\sigma$ confidence level, Category I and II twin stars are ruled out entirely, leaving only Category III and IV solutions with significantly lower maximum masses ($\approx 1.37 M_\odot$). Conversely, at 3$\sigma$, the constraints relax, allowing for a wider range of parameters, including cases where the transition pressure lies in the "unbound" region.
• Category Dependence: The study categorizes twin stars based on the maximum masses of their second ($M_{TOV,2}$) and third ($M_{TOV,3}$) branches. It finds that Category I twins (massive on both branches) are only permitted at extremely high transition densities ($\approx 4\rho_{sat}$), whereas softer transition densities favor Categories II, III, and IV.

Significance
The paper asserts that the observation of twin stars, particularly massive ones, would serve as a "smoking gun" signature for a strong first-order hadron-to-quark phase transition in the cores of neutron stars. By mapping the "witch-hat" curves and applying rigorous observational constraints, the study provides specific thermodynamic upper bounds on the transition density ($e_{tr}$), the discontinuity strength ($\Delta e$), and the maximum mass of twin stars. These bounds offer a critical framework for interpreting future astrophysical data and narrowing down the possible nature of dense matter. The authors note that while their analysis is based on hydrostatic equilibrium, the existence of Category II twin stars is particularly fragile; previous dynamical studies suggest they may be unstable against perturbations, migrating to hadronic branches or collapsing into black holes. Thus, the identification of viable twin star EoSs is a necessary, though potentially insufficient, step toward confirming the existence of quark matter in neutron stars.