The favoured twin: on the dynamical response of twin stars to perturbations

TL;DR

This work addresses which twin-star configuration is likely to be realized in nature when a hadron–quark first-order phase transition yields HB and TB equilibria for the same mass. Using a large suite of general-relativistic hydrodynamic simulations, the authors identify a mass-dependent critical perturbation strength $\lambda_{\rm crit}$ that determines whether a star remains on its original branch or migrates to the neighboring one while conserving rest-mass $M_{\rm b}$. They show that the favoured branch is the one with the larger $\lambda_{\rm crit}$, a result that can be predicted from the binding-energy difference $E_{\rm b}$ and a neutral mass $M_{\rm neut}$ where both branches are equally favoured. The findings refine the nonlinear stability picture of twin stars and provide a practical criterion to assess which configuration is more likely in nature, with implications for interpreting potential observations of compact-star systems.

Abstract

If a strong first-order phase transition takes place at sufficiently high rest-mass densities in the equation of state (EOS) modelling compact stars, a new branch will appear in the mass-radius sequence of stable equilibria. This branch will be populated by stars comprising a quark-matter core and a hadronic-matter envelope, i.e., hybrid stars, which represent ``twin-star'' solutions to equilibria having the same mass but a fully hadronic EOS. While both branches are stable to linear perturbations, it is unclear which of the twin solutions is the ``favoured'' one, that is, which of the two configurations is expected to be found in nature. We assess this point by performing a large campaign of general-relativistic simulations aimed at assessing the response of compact stars on the two branches to perturbations of various strength. In this way, we find that, independently of whether the stars populate the hadronic or the twin branch, their response is characterised by a critical-perturbation strength such that the star will oscillate on the original branch for subcritical perturbations and migrate to the neighbouring branch for supercritical perturbations while conserving rest-mass. Because the critical values are different for stars with the same rest-mass but sitting on either branch, it is possible to define as favoured the part of the branch that has the largest critical perturbation, thus correcting the common wisdom that stellar models on the twin branch are the favoured ones. Interestingly, we show that the binding energies on the two branches can be used to deduce without simulations which of the stellar configurations is more likely to be found in nature.

Figures (5)

• Figure 1: Representative example of a mass-radius sequence for an EOS leading to the coexistence of a hadronic branch (${\rm HB}$) and of a twin branch (${\rm TB}$). Solid and dashed lines are used to indicate branches that are stable or unstable to linear perturbations, respectively. Shown with a shaded area is the range in gravitational masses where the twin solutions exist, i.e., the twin region. The yellow star on ${\rm TB}$ and the yellow box on ${\rm HB}$ indicate the models $\mathtt{TB.1.4}$ and $\mathtt{HB.1.4}$ which are selected to discuss the migration process in Sec. \ref{['sec:1.4Mb_evolution']}. The properties of these configurations are listed in Tab. \ref{['tab:model']}.
• Figure 2: Top panel: Evolution of the central rest-mass density $\rho_\mathrm{c}$ normalised to the nuclear-saturation density for the representative stellar configurations $\mathtt{TB.1.4}$ and $\mathtt{HB.1.4}$ that are shown with yellow symbols in Fig. \ref{['fig:fig1']} (see also Tab. \ref{['tab:model']}). Because no perturbations are introduced in this case, the inset is needed to show the minute relative oscillations and their decay with time. The shaded area reports the range of rest-mass densities across which the PT takes place. Clearly, both $\mathtt{TB.1.4}$ and $\mathtt{HB.1.4}$ are stable to linear perturbations. Bottom panel: the same as in the top but when the stars experience critical perturbations. In both cases, the stars migrate to the neighbouring branch of static solutions.
• Figure 3: Numerical values of the critical perturbation velocities $\lambda_{\rm T, crit}$ for models on the ${\rm TB}$ (filled blue circles) and the corresponding values $\lambda_{\rm H, crit}$ for models on the ${\rm HB}$ (filled red circles) for the range of masses in the twin region. The different colours in the two shaded regions show which of the two branches is favoured when comparing $\lambda_{\rm T, crit}$ and $\lambda_{\rm H, crit}$; shown with a horizontal dashed line is the value of the neutral-favouritism mass $M_\mathrm{neut} = 1.2988 \, M_\odot$, where the two branches are equally favoured. Shown in the inset is a representation of the favourite branches in an $(M,R)$ diagram.
• Figure 4: The same as in Fig. \ref{['fig:fig3']} but in terms of the binding energy $E_\mathrm{b}:= M_{\rm b} - M$; the inset shows a magnified view of the binding energies near the neutral mass $M_{\rm neut}=1.2988 \, M_\odot$. Clearly, the representations in terms of $E_\mathrm{b}$ and $\lambda_\mathrm{crit}$ are very similar, so that the binding energies can be used to deduce the favoured twin without resorting to numerical simulations.
• Figure 5: Left panel: the same as in Fig. \ref{['fig:fig1']} but for three EOSs that lead to twin stars of category II. The stable ${\rm HB}$ is marked with a black solid line, while the stable ${\rm TBs}$ are marked with solid lines of different colours; dashed and dotted lines mark the (linearly) unstable branches and the neutral masses $M_{\rm neut}$, respectively. Right panel: the same as in Fig. \ref{['fig:fig4']} but for the three EOSs on the left panel; also here, the coloured dotted lines mark the neutral masses. Note that no neutral mass exists for the EOS with the smallest maximum mass on the ${\rm TB}$; hence, no stellar model on the ${\rm TB}$ is favoured for this EOS.