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

TL;DR

This work develops a model-agnostic framework to explore twin-star scenarios arising from strong hadron-quark phase transitions by parametrizing the transition density, transition pressure, transition strength, and quark-matter stiffness within a Maxwell-type EoS. By solving the TOV equations and identifying third-branch solutions via Seidov stability criteria, the authors define witch-hat curves that delineate the viable (P_tr, Delta e) space for given e_tr and c_s,QM. They constrain this space using NICER and GW170817 data, finding that observational bounds tighten the allowed TS regions, constrain the maximum transition density, and cap the maximum twin-star mass (approximately 2.05 solar masses for the stiffest quark EoS). The results show that higher transition densities and stiffer quark matter broaden the allowed region, while softer quark matter or stricter observational limits shrink it, with clear implications for the possible nature of hadron-quark PT in neutron-star cores.

Abstract

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.

Figures (6)

• Figure 1: [Left] Diagram indicating the different parts of EoS, that are controlled by different parameters---transition energy density ($e_\mathrm{tr}$), transition pressure ($P_\mathrm{tr}$), the strength of discontinuity in energy density ($\Delta e$) and constant speed of sound for quark matter EoS ($c^2_\mathrm{s,QM}$). [Right] Schematic diagram of $M$--$R$ sequence of an EoS that constructs TSs. The second (purple) and third (orange) stable branches are separated by a sequence of unstable branch (black dashed curve). The maximum mass on the second and third branches ($M_\mathrm{TOV,2}$ and $M_\mathrm{TOV,3}$) is indicated by purple and red stars, respectively. The sequence of unstable stars begins beyond these stellar configurations. The minimum mass on the third branch is indicated by a red circle. The twin region is defined as the mass range in which equal-mass stellar models are present on both stable branches.
• Figure 2: Allowed region of EoS space with fixed $e_\mathrm{tr}$ that allows TS solutions. The grey region indicates the range of hadronic EoS considered in constructing the hybrid EoSs. The solution-less region (red) indicates the range of $P_\mathrm{tr}$ inside which TSs do not form. The left panel shows the $M$--$R$ sequences of two EoSs in this region with different $\Delta e$. The bound region (blue) is separated from the red region using $P^\mathrm{max}_\mathrm{tr}$ at which a $\Delta e$ exists that allows TS solutions. The middle panels show that the EoS with $\Delta e$ within the allowed values of $\Delta e$ constructs TSs. The sequence of the TSs is indicated by cyan in the inset. The unbound region (green) region only contains the minimum value of $\Delta e$ above which TS solutions are possible. The right panel shows that the EoS with large value of $\Delta e$ form TSs. The sequence of TSs is indicated by turquoise in the inset. The loci of minimum and maximum $\Delta e$ are indicated in black, which we identify as the 'witch-hat' curve. The black dashed line is an extrapolation of the maximum $\Delta e$.
• Figure 3: The effects of change in $e_\mathrm{tr}$ and $c^2_\mathrm{s,QM}$ on witch-hat curve. [Left] The vertical fading lines indicate the $e_\mathrm{tr}$. The respective Seidov lines of each $e_\mathrm{tr}$ are indicated in grey solid lines. [Right] The Seidov lines align in the modified axes.
• Figure 4: Allowed regions under the witch-hat curve for three different values of $e_\mathrm{tr}$, constrained from observational measurements ($2\sigma$ confidence). The vertical fading lines indicate the value of $e_\mathrm{tr}$. For all the EoSs, the quark part is described with $c^2_\mathrm{s,QM}=1.0$. The inset shows the $M$--$R$ sequences of the EoSs with ($P_\mathrm{tr}$, $\Delta e$) marked by circles inside the allowed region. Only the stable branches of $M$--$R$ sequences are shown, where the second (third) branch is indicated using light (dark) colour in the respective colour theme used for the $e_\mathrm{tr}$. Black dashed lines are marked at $1.0~M_\odot$ and $2.0~M_\odot$ for the identification of TS categories.
• Figure 5: Emphasis on particular observational measurements ($2\sigma$ confidence) that are responsible for constraining various regions under the witch-hat curve for three different values of $e_\mathrm{tr}$. The vertical fading lines indicate the value of $e_\mathrm{tr}$. The quark EoS description is set with $c^2_\mathrm{s,QM}=1.0$. The inset shows the $M$--$R$ sequences, along with the particular observational constraint that they violate. These $M$--$R$ sequences are constructed with the EoSs with ($P_\mathrm{tr}$, $\Delta e$) marked by circles outside the allowed region. Only the stable branches of $M$--$R$ sequences are shown, where the second (third) branch is indicated using light (dark) colour in the respective colour theme used for the $e_\mathrm{tr}$.