Studying strong phase transitions in neutron stars with gravitational waves

The composition of neutron stars at the extreme densities reached in their cores is currently unknown. Besides nuclear matter of normal neutrons and protons, the cores of neutron stars might harbor exotic matter such as deconfined quarks. In this paper we study strong hadron-quark phase transitions in the context of gravitational wave observations of inspiraling neutron stars. We consider upcoming detections of neutron star coalescences and model the neutron star equations of state with phase transitions through the Constant-Speed-of-Sound parametrization. We use the fact that neutron star binaries with one or more hadron-quark hybrid stars can exhibit qualitatively different tidal properties than binaries with hadronic stars of the same mass, and hierarchically model the masses and tidal properties of simulated populations of binary neutron star inspiral signals. We explore the parameter space of phase transitions and discuss under which conditions future observations of binary neutron star inspirals can identify this effect and constrain its properties, in particular the threshold density at which the transition happens and the strength of the transition. We find that if the detected population of binary neutron stars contains both hadronic and hybrid stars, the onset mass and strength of a sufficiently strong phase transition can be constrained with 50–100 detections. If the detected neutron stars are exclusively hadronic or hybrid, then it is possible to place lower or upper limits on the transition density and strength.

Figure 1

Mass-radius relation for the EoSs considered in this study. The EoSs are based on the hadronic baseline EoSs DBHF (left) and SFHo (right). They are constructed by choosing different values of the CSS parameters ($n_{\text{trans}}$, $\mathrm{\Delta }ϵ/{ϵ}_{\text{trans}}$) and setting maximal stiffness in quark matter ($c_{\mathrm{QM}}^2=1$). Solid lines (dotted lines) correspond to the stable (unstable) part of each EoS. Grey bands denote the measured masses of heavy pulsars from [58] (light grey) and [60] (dark grey) at the 68% level.

Figure 2

Absolute value of the difference between the true radius and the inferred radius for different EoSs with and without phase transitions when analyzed with EoS-insensitive relations that assume hadronic EoSs. Different color maps correspond to different binary combinations of hadronic and hybrid stars. In all cases here and in Fig. 12, the radius error is less than $\sim 2.5\mathrm{km}$.

Figure 3

Chirp mass and chirp radius for binary systems with the EoSs from Fig. 1 with the DBHF (left) and the SFHo (right) hadronic EoS baseline. Each dot represents a BNS system with masses randomly selected in $[1{\mathrm{M}}_{\odot },M_{\text{max}}]$; the color scheme follows Fig. 1. The grey dashed lines are the $\mathcal{M}(\mathcal{R};R)$ fit for purely hadronic EoSs for increments of $R$ of 1 km. The thick black dashed line is the fit for $R=R_{1.4}$, the radius of a $1.4{\mathrm{M}}_{\odot }$ NS with each hadronic baseline model. The inset in each plot shows the hadronic EoS (blue dots) and the fits. The $\mathcal{M}(\mathcal{R};R)$ fit is based on EoS-insensitive relations and it can model the purely hadronic systems up to a chirp mass of $\sim 1.6{\mathrm{M}}_{\odot }$.

Figure 4

Case A: Marginalized 1-dimensional posterior density for the transition mass $M_{\mathrm{t}}$ (left) and the hadronic radius $R$ (right) for a random population of 100 BNSs with the purely hadronic EoS SFHo. Solid vertical lines denote the true parameters, where applicable. Dashed vertical lines denote the 90% credible lower limit (left) and symmetric interval (right).

Figure 5

Case B: Marginalized 1-dimensional posterior density for the transition mass $M_{\mathrm{t}}$ (left) and the hadronic radius $R$ (right) for a random population of 100 BNSs with EoSs whose transition mass is lower than the detected population masses, $M_{\mathrm{t}}<1{\mathrm{M}}_{\odot }$. Solid vertical lines denote the true parameters, where applicable. The dashed vertical line denotes the 90% credible upper or lower limit as appropriate (left). We show results with DBHF_1032 (top row), SFHo_2009 with a hadronic radius prior lower limit of 10 km (second row), SFHo_2009 with a hadronic radius prior lower limit of 11 km (third row), and SFHo_2506 (bottom row).

Figure 6

Case C: Marginalized 1-dimensional posterior density for the transition mass $M_{\mathrm{t}}$ (left) and the hadronic radius $R$ (right) for a random population of 100 BNSs with EoSs whose transition mass is comparable to the typical NS masses in the detected population. Solid vertical lines denote the true parameters. The dashed vertical lines denote the 90% credible upper limit or symmetric interval as appropriate. We show results with DBHF_2507 (top row), and SFHo_3003 (bottom row).

Figure 7

Case D: Marginalized 1-dimensional posterior density for the transition mass $M_{\mathrm{t}}$ (left) and the hadronic radius $R$ (right) for a random population of 100 BNSs with EoSs whose transition mass is at relatively high masses and the phase transition is not detectable. Solid vertical lines denote the true parameters. The dashed vertical lines denote the 90% credible upper limit (left) or symmetric interval (right). We show results with DBHF_3504 (top row), and DBHF_2507 with a BNS mass distribution that is tightly centered around $\sim 1.32{\mathrm{M}}_{\odot }$ (second row).

Figure 8

Relation between the strength of the phase transition $\mathrm{\Delta }ϵ/{ϵ}_{\text{trans}}$ and the onset mass for quarks $M_{\mathrm{t}}$ at fixed maximum mass of hybrid stars $M_{\text{max}}=2.0{\mathrm{M}}_{\odot }$ (solid) and $M_{\text{max}}=2.2{\mathrm{M}}_{\odot }$ (dashed for selected EoSs). Points and dashed curves correspond to numerical solutions to the TOV equations, while solid curves are fits with the coefficients from Table 2. The gray-shaded regions mark the area excluded by the corresponding heavy pulsar measurements $M_{\text{max}}\ge 2.0{\mathrm{M}}_{\odot }$ for DBHF. The detection of even heavier pulsars will further decrease the allowed parameter space.

Figure 9

Dependence of the chirp radius ${\mathcal{R}}^{\mathrm{QQ}}(\mathcal{M}=1.2{\mathrm{M}}_{\odot })$ on the strength of the phase transition $\mathrm{\Delta }ϵ/{ϵ}_{\text{trans}}$ for a small transition mass $M_{\mathrm{t}}\lesssim 1{\mathrm{M}}_{\odot }$ (case B); equal-mass binaries are assumed. We again consider two hadronic baseline EoSs and vary the onset density of the transition $n_{\text{trans}}$ (indicated next to each line). At the lowest transition density $n_{\text{trans}}=1.0n_0$, the hadronic baseline EoS has little effect though its influence increases with increasing $n_{\text{trans}}$. The horizontal line denotes ${\mathcal{R}}^{\mathrm{HH}}(\mathcal{M}=1.2{\mathrm{M}}_{\odot },R=10\mathrm{km})\approx 9\mathrm{km}$. For fixed $n_{\text{trans}}$, the maximum value of $\mathrm{\Delta }ϵ/{ϵ}_{\text{trans}}$ is determined by $M_{\text{max}}=2.0{\mathrm{M}}_{\odot }$ (dashed curves), which also corresponds to the leftmost part of the boundary contours in Fig. 8 when $M_{\mathrm{t}}\lesssim 1{\mathrm{M}}_{\odot }$. Note that for a given $n_{\text{trans}}$, different hadronic base EoSs reflect different values of $M_{\mathrm{t}}$.

Figure 10

Contours of constant reduction in chirp radius $\mathrm{\Delta }\mathcal{R}\equiv {\mathcal{R}}^{\mathrm{HH}}(\mathcal{M}=1.2{\mathrm{M}}_{\odot })-{\mathcal{R}}^{\mathrm{HQ}}(\mathcal{M}=1.2{\mathrm{M}}_{\odot })$ for “HQ” binaries with mass ratio $q=0.7$ (solid) or $q=0.9$ (dashed) on the ($M_{\mathrm{t}}$, $\mathrm{\Delta }ϵ/{ϵ}_{\text{trans}}$) plane. We use the DBHF baseline for which the chirp radius is ${\mathcal{R}}^{\mathrm{HH}}(\mathcal{M}=1.2{\mathrm{M}}_{\odot })\approx 13.5\mathrm{km}$ for purely hadronic binaries. The dash-dotted horizontal line denotes $\mathrm{\Delta }ϵ/{ϵ}_{\text{trans}}=0.6$, the approximate threshold for a separate hybrid branch to form, while the shaded region is excluded by $M_{\text{max}}\ge 2.0{\mathrm{M}}_{\odot }$.

Figure 11

Marginalized posterior distributions for $M_{\mathrm{t}}$ (left subplot) and median 90% (light shading) and 50% (dark shading) credible interval for $R$ (right subplot) for different numbers of detected BNSs $N$ for various EoSs (see subplot caption). In the left subplots solid curves show the posterior averaged over 100 random population realizations, while thin dashed lines show the posterior for 5 random population realizations for each $N$ (same color as the averaged posterior). Solid vertical or horizontal black lines denote the injected $M_{\mathrm{t}}$ and $R_{1.4}$, where applicable. Dashed vertical lines in the $M_{\mathrm{t}}$ posteriors denote upper or lower 90% credible intervals, where appropriate (same color as the averaged posterior).

Figure 12

Same as Fig. 2 for the remaining EoSs of Fig. 1.

Figure 13

$M_{\text{max}}=2.0{\mathrm{M}}_{\odot }$ (solid) and $M_{\text{max}}=2.2{\mathrm{M}}_{\odot }$ (dotted) contours for the DBHF and HLPS hadronic EoSs with soft ($c_{\mathrm{QM}}^2=0.33$; left panel) and relatively stiff ($c_{\mathrm{QM}}^2=0.5$; right panel) quark matter. For the softest quark EoS, the allowed values of $\mathrm{\Delta }ϵ/{ϵ}_{\text{trans}}$ are so small that only very weak transitions can occur, leading to indistinguishable features compared to purely hadronic stars. By increasing the stiffness in quark matter (right panel), with the stiff DBHF hadronic baseline EoS it is now possible to have strong transitions ($\mathrm{\Delta }ϵ/{ϵ}_{\text{trans}}\gtrsim 0.5$).