Dense baryonic matter: Constraints from recent neutron star observations

Updated constraints from neutron star masses and radii impose stronger restrictions on the equation of state for baryonic matter at high densities and low temperatures. The existence of $2M_{\odot }$ neutron stars rules out many soft equations of state with prominent “exotic” compositions. The present work reviews the conditions required for the pressure as a function of baryon density to satisfy these constraints. Several scenarios for sufficiently stiff equations of state are evaluated. The common starting point is a realistic description of both nuclear and neutron matter based on a chiral effective field theory approach to the nuclear many-body problem. Possible forms of hybrid matter featuring a quark core in the center of the star are discussed using a three-flavor Polyakov–Nambu–Jona-Lasinio model. It is found that a conventional equation of state based on nuclear chiral dynamics meets the astrophysical constraints. Hybrid matter generally turns out to be too soft unless additional strongly repulsive correlations, e.g., through vector current interactions between quarks, are introduced. The extent to which strangeness can accumulate in the equation of state is also discussed.

Figure 1

Allowed regions for the equation of state $P\left(\epsilon \right)$ as dictated by neutron star observables. The upper (dark gray) area takes into account the limitations as given by Trümper [28] and constraints from causality. The lower (light gray) band uses, in addition to the $2M_{\odot }$ constraint, a permitted radius window $11.0$–$12.5\mathrm{km}$ from Refs. [21, 22, 23]. For energy densities smaller than ${\epsilon }_1$ and ${\epsilon }_0$ the FKW and SLy EoS, respectively, are used. The matching points ${\epsilon }_1,{\epsilon }_2,{\epsilon }_3$ of the polytropes in Eq. (4) are also shown in the figure.

Figure 2

Energy per particle, $E/N=\epsilon /{\varrho }_n-M_n$, for pure neutron matter as a function of density ${\varrho }_n$. Solid curve, ChEFT result [43, 45, 51] used in the present work; blue shaded area, results deduced from quantum Monte Carlo (QMC) computations reviewed in Ref. [11], using different models of the three-neutron force ($3N$).

Figure 3

Equations of state at $T=0$ derived from the three-flavor PNJL model with inclusion of charge neutrality and $\beta$ equilibrium conditions. The blue dashed and solid lines show results for different vector-coupling strengths $G_v$, as indicated in the figure. The black solid line displays the EoS derived from in-medium ChEFT as described in Sec. 3a and discussed further in the next section. The gray bands show the constraints from neutron star observables (see Fig. 1).

Figure 4

Particle ratios in the three-flavor (P)NJL model subject to $\beta$ equilibrium and charge neutrality. The ratios ${\varrho }_i/{\varrho }_{\mathrm{tot}}$ with ${\varrho }_{\mathrm{tot}}={\varrho }_u+{\varrho }_d+{\varrho }_s$, for the species indicated in the figure, are given as a function of the baryon density (normalized to nuclear saturation density ${\varrho }_0=0.16{\mathrm{fm}}^{-3}$).

Figure 5

Proton fraction, $x_p=\frac{{\varrho }_p}{\varrho }$, shown for the ChEFT EoS including $\beta$ equilibrium.

Figure 6

Mass-radius relation computed with the ChEFT equation of state for neutron stars including $\beta$ equilibrium. Stable neutron stars can exist up to the maximum of this curve. The hardly distinguishable EoS for pure neutron matter (PNM) is also shown for reference. The horizontal band indicates the masses of the pulsars J1614–2230 and J0348+0432. The lighter gray band corresponds to the radius range deduced in Ref. [21].

Figure 7

Density profile in the interior of a neutron star with mass $M=2M_{\odot }$ and resulting radius of about $R=11\mathrm{km}$. Note that the central density does not exceed ${\varrho }_c\sim 4.8{\varrho }_0$.

Figure 8

Equations of state representing the quark-hadron continuity scenario using different quark vector couplings. Quark matter (PNJL) and nuclear matter (ChEFT) EoS are matched continuously at $\epsilon =\overline{\epsilon }=800\mathrm{MeV}/{\mathrm{fm}}^3$. Solid curve, $G_v=1.5G$; dashed curve, $G_v=0$. The gray areas are those of Fig. 1 representing constraints from neutron star observables.

Figure 9

Solutions of the TOV equations (1) and (2) (mass-radius relation) for neutron stars using the EoS given in Ref. (39). The lines correspond to different vector-coupling strengths, as indicated in the figure. The shaded areas are as in Fig. 6.

Figure 10

Equations of state including a first-order phase transition between hadronic and quark matter. The transition region itself is characterized by the flat parts of the curves. The (upper) solid curve includes a vector repulsion of $G_v=0.5G$ between quarks, while the (lower) dashed curve is found using $G_v=0$. The gray areas are as in Fig. 1.

Figure 11

Particle ratios as a function of the (normalized) baryon density for the particles as indicated in the figure. The first-order coexistence region is marked by the rapid decrease of neutrons and the steep rise of quarks. The case without vector interaction ($G_v=0$) is shown.

Figure 12

Solutions of the TOV equations using the EoS incorporating a first-order hadron-quark phase transition (see Fig. 10). Mass-radius trajectory lines correspond to different vector-coupling strengths $G_v$, as indicated in the figure. The shaded areas are as in Fig. 6.

Figure 13

Density profile a neutron star with mass $M=1.95M_{\odot }$ and radius $R\simeq 11.4\mathrm{km}$. The EoS includes the quark-hadron first-order phase transition and no vector interaction, $G_v=0$. The central density is ${\varrho }_c\approx 5{\varrho }_0$. The shaded area shows the onset of the coexistence region in the inner core within a radius $r\lesssim 2\mathrm{km}$.

Figure 14

Density profiles $\varrho \left(r\right)$ multiplied by $r^2$ and scaled with $R^2{\varrho }_0$, where $R$ is the radius of the neutron star and ${\varrho }_0=0.16$ ${\mathrm{fm}}^{-3}$ is the density of normal nuclear matter. Results are shown for a typical $2M_{\odot }$ neutron star. The upper horizontal scale shows the local baryon density. Solid curve, equation of state from nuclear ChEFT ($M=2M_{\odot }$); dashed curve, including hadron-quark coexistence in the center of the star ($M=1.95M_{\odot }$, calculated using $G_v=0$).

Figure 15

Particle ratios as a function of baryon density $\varrho$ (in units of ${\varrho }_0=0.16$ ${\mathrm{fm}}^{-3}$) for the particles indicated, as in Fig. 11 but with inclusion of $\Lambda$ hyperons.