Hyperonic Neutron Star Matter in Light of GW170817

TL;DR

This paper examines how hyperons affect neutron star equations of state in light of recent mass measurements from GW170817 and other pulsars, showing hyperons can be consistent with these constraints.

Contribution

It demonstrates that hyperonic equations of state, with empirically motivated coupling constants, can satisfy recent neutron star mass constraints.

Findings

Hyperons can constitute over 10% of baryons in neutron star cores.

Hyperonic equations of state can meet the 2.16 solar mass constraint.

Empirical data and SU(3) symmetry guide hyperon coupling constants.

Abstract

Since 2013 the mass of pulsar PSR J0348+0432 ($M = 2.01\,M_{\odot}$) has provided a tight constraint on neutron star equation of state. However, a number of different analyses of the recently detected BNS merger (GW170817) point to a maximum neutron star mass around $2.16\,M_{\odot}$. In addition, a recent study determined the mass of the millisecond pulsar PSR J2215+5135 to be $2.27^{+0.17}_{-0.15}\,M_{\odot}$. In this work we investigate the presence of hyperons in neutron star matter in light of these new mass measurements using equations of state calculated in the relativistic mean-field approximation. Particular attention is paid to the use of the available empirical data from the study of hypernuclei and that of the SU(3) symmetry relations in fixing the meson-hyperon coupling constants. We find that hyperonic equations of state with reasonable choices for the meson-hyperon coupling constants can satisfy these new mass constraints, with hyperons potentially accounting for more than 10% of the baryons in the core of a neutron star.

Tables

Table 1. Table 1: Required z 𝑧 z , maximum strangeness fractions f S subscript 𝑓 𝑆 f_{S} , and maximum hyperon fractions f Y subscript 𝑓 𝑌 f_{Y} for 2.01 ​ M ⊙ 2.01 subscript 𝑀 direct-product 2.01\,M_{\odot} maximum mass NSs extracted from the data in Figures  4 and   5 .

Model	$z$	$f_S$	$f_Y$
$\sigma \omega \rho$	0.217	5.50%	12.2%
$\sigma \omega \rho \varphi$	0.382	5.76%	13.0%
$\sigma \omega \rho \varphi {\sigma }^{\ast }$	0.300	5.69%	11.8%
$\sigma \omega \rho \varphi {\sigma }^{\ast },\Lambda \mathrm{only}$	0.413	5.17%	15.5%

Table 2. Table 2: Same as Table 1 but for 2.16 ​ M ⊙ 2.16 subscript 𝑀 direct-product 2.16\,M_{\odot} maximum mass NSs.

Model	$z$	$f_S$	$f_Y$
$\sigma \omega \rho$	0.105	4.30%	9.58%
$\sigma \omega \rho \varphi$	0.212	4.24%	9.67%
$\sigma \omega \rho \varphi {\sigma }^{\ast }$	0.177	4.46%	9.58%
$\sigma \omega \rho \varphi {\sigma }^{\ast },\Lambda \mathrm{only}$	0.257	3.76%	11.2%

Equations

$$\mathcal{L}$$

$$g_{\rho B}(n)=g_{\rho B}(n_0)\exp \left[-a_{\rho }\left(n/n_0-1\right)\right],$$

$$U_Y^{(N)}(n_0)=g_{\omega Y}{\bar{\omega }}_0+g_{\varphi Y}{\bar{\varphi }}_0-g_{\sigma Y}{\bar{\sigma }}_0.$$

$$U_{\Lambda }^{(\Lambda )}(n_0)=g_{\omega \Lambda }{\bar{\omega }}_0+g_{\varphi \Lambda }{\bar{\varphi }}_0-g_{\sigma \Lambda }{\bar{\sigma }}_0-g_{{\sigma }^{\ast }\Lambda }{\bar{\sigma }}_0^{\ast }.$$

$$U_{\Xi }^{(\Xi )}(n_0)=2U_{\Lambda }^{(\Lambda )}(n_0/2),$$

$$g_{{\sigma }^{\ast }\Sigma }=g_{{\sigma }^{\ast }\Lambda }.$$

$$g_{\rho \Lambda }=0,g_{\rho \Sigma }=g_{\rho \Xi }=g_{\rho N},$$

Full text

††articletype: Article Type00footnotetext: Abbreviations: BNS, binary neutron star; EoS, equation of state; NS, neutron star; RMF, relativistic mean-field; SNM, symmetric nuclear matter

1 Introduction

Published mass measurements of PSR J1614-2230 in 2010 and PSR J0348+0432 in 2013 indicate that neutron stars (NSs) can have masses as large as twice the mass of our sun. Including hyperons in the modeled composition of NS matter increases the baryonic degrees of freedom softening the NS equation of state (EoS), the degree of which depends on the choice of meson-hyperon coupling constants in Relativistic Mean-Field (RMF) models. This softening often results in a calculated maximum NS mass that is inconsistent with the aforementioned observations, giving rise to the so-called “hyperon puzzle." Common resolutions of the hyperon puzzle involve stiffening the EoS by introducing repulsive hyperon-nucleon and hyperon-hyperon interactions and adjusting the meson-hyperon coupling constants in various EoS models. However, analyses of the recent multi-messenger observation of a binary neutron star (BNS) merger suggest a maximum NS mass significantly higher than two solar masses, as does the recently published measurement of the mass of PSR J2215+5135, reinvigorating the hyperon puzzle. Therefore, we seek to determine whether or not a reasonably constrained hyperonic NS matter EoS can produce a maximum mass consistent with these new constraints.

In this work we compute the EoS of NS matter including the full baryon octet, using a RMF model consistent with constraints on the properties of the isospin-symmetric nuclear matter (SNM) EoS including the nuclear incompressibility and the isospin asymmetry energy and slope. EoS stiffening due to the inclusion of hyperon-hyperon interactions, mediated via strange-scalar ($\sigma^{*}$) and strange-vector ($\phi$) mesons, is examined by calculating NS properties for different combinations of mesons. Scalar meson-hyperon coupling constants are fit to empirical data on hypernuclei where available, while the vector meson-hyperon couplings are fixed by the SU(3) symmetry. By varying SU(3) coupling parameters we traverse the entire vector meson-hyperon coupling space and calculate NS properties including the maximum mass, strangeness fraction, and hyperon fraction. We determine the couplings that are consistent with the mass constraints provided by PSR J0348+0432 and analyses of the detected gravitational waves (GW170817) and gamma-ray burst (GRB 170817A) produced by the observed BNS merger.

This work is organized as follows. In Sec. 2 we introduce the relativistic mean-field model and parameterization of the EoS, as well as the methods used for determining the meson-hyperon coupling constants. In Sec. 3 we discuss additional constraints on the EoS provided by analyses of GW170817 and GRB 170817A. Our results, including calculations of NS properties for the vector meson-hyperon coupling space, are presented in Sec. 4. Finally, we provide a summary of this work in Sec. 5.

2 Equation of State

In this work we calculate EoSs of cold, nonrotating NSs using the nonlinear relativistic mean-field (RMF) approximation, modeling baryon-baryon interactions in terms of scalar ($\sigma,\sigma^{*}$), vector ($\omega,\phi$), and isovector ($\rho$) meson fields, and represented by the following Lagrangian,

[TABLE]

The EoS is parameterized to reproduce the following properties of SNM at a saturation density of $n_{0}=0.150$ fm*-3*: the energy per nucleon $E_{0}/N=-16.0$ MeV, nuclear incompressibility $K_{0}=250.0$ MeV, effective nucleon mass $m^{*}/m_{N}=0.70$, isospin asymmetry energy $S_{0}=30.3$ MeV, and slope of the asymmetry energy $L_{0}=46.5$ MeV. The parameterization is specifically tailored for consistency with a number of simultaneous constraints on the asymmetry energy ($S_{0}$) and slope of the asymmetry energy ($L_{0}$) at $n_{0}$ as shown in Figure 1.

In order to satisfy the constraints on $L_{0}$ the isovector-vector meson-baryon coupling constants are taken to be density-dependent with a functional density-dependence given by [Typel \BBA Wolter (\APACyear1999), Drago \BOthers. (\APACyear2014)]

[TABLE]

with the parameter $a_{\rho}$ fit to $L_{0}$ at $n_{0}$.

The EoS of SNM and mass-radius relation for the purely nucleonic EoS ($npe\mu$) are given in Figure 2.

The low-density EoS is sufficiently soft to satisfy the constraint from kaon production and produce a relatively small canonical radius, while the high-density EoS is too stiff to fully satisfy the elliptic flow constraint but produces a high maximum mass ($M_{\mathrm{max}}=2.33\,M_{\odot}$) consistent with the masses of PSR J0348+0432 ($M=2.01\pm 0.04\,M_{\odot}$) and PSR J2215+5135 ($M=2.27^{+0.17}_{-0.15}\,M_{\odot}$) [Fuchs (\APACyear2006), Lynch \BOthers. (\APACyear2009), Antoniadis \BOthers. (\APACyear2013), Linares \BOthers. (\APACyear2018), Danielewicz \BOthers. (\APACyear2002)]. Note that outer and inner crust EoSs are appended to the low density EoS [Douchin \BBA Haensel (\APACyear2001), Haensel \BBA Pichon (\APACyear1994)].

2.1 Meson-Hyperon Coupling Constants

Inclusion of the hyperonic degrees of freedom softens the neutron star EoS, and the corresponding reduction of the maximum mass is extremely sensitive to the choice of meson-hyperon coupling constants. It is therefore essential that the coupling constants be fixed using empirical data wherever possible.

The scalar meson-hyperon coupling constants $g_{\sigma Y}$ and $g_{\sigma^{*}Y}$ can be fit to hyperon single-particle potentials and self-potentials derived from the available empirical data on hypernuclei, but first the vector meson-hyperon couplings $g_{\omega Y}$ and $g_{\phi Y}$ must be specified. In SU(3) symmetry the vector couplings can be written in terms of three parameters: the mixing angle $\theta_{V}$, the $F/(F+D)$ ratio $\alpha_{V}$, and the octet-singlet coupling ratio $z$ [Dover \BBA Gal (\APACyear1984)]. The most commonly used values for these parameters are $\theta_{V}=35.26^{\circ}$, $\alpha_{V}=1$, and $z=1/\sqrt{6}$, corresponding to SU(6) symmetry. However, sophisticated baryon-baryon interaction models such as the Nijmegen extended-soft-core model (ESC08) predict much lower values of the coupling ratio $z$ ($z_{\mathrm{ESC08}}\approx 0.195$) that lead to stiffer EoSs and much higher maximum masses than those constructed using SU(6) symmetry [Rijken \BOthers. (\APACyear2010)]. Rather than calculate a single EoS constructed from fixed values of $\alpha_{V}$ and $z$, we will instead calculate NS maximum masses, strangeness fractions, and hyperon fractions for the $z$ parameter space with $\alpha_{V}=1$, and for the entire $\alpha_{V}$-$z$ parameter space. Note that the $\phi$ meson couples to the nucleons if $z\neq 1/\sqrt{6}$, and as a result $g_{\omega N}$ must be recalculated to restore the saturation properties of the RMF parameterization.

With the vector meson-hyperon couplings specified, the scalar couplings are set to reproduce empirical hyperon single-particle potentials at saturation, $U_{Y}^{~{}\!(N)}(n_{0})$, using the following,

[TABLE]

In this work we employ the following hyperon potentials: $U_{\Lambda}^{(N)}(n_{0})=-28$ MeV, $U_{\Sigma}^{(N)}(n_{0})=+30$ MeV, and $U_{\Xi}^{(N)}(n_{0})=-14$ MeV. If the strange-scalar field $\sigma^{*}$ is included, the strange-scalar meson-$\Lambda$ coupling constant $g_{\sigma^{*}\Lambda}$ is set to reproduce a saturation self-potential of $U_{\Lambda}^{(\Lambda)}(n_{0})=-1$ MeV in isospin-symmetric $\Lambda$-matter, a value close to that suggested by the Nagara event [Ahn \BOthers. (\APACyear2013)], using the following,

[TABLE]

The other strange-scalar meson-hyperon couplings are determined relative to that of the $\Lambda$ as follows [Oertel \BOthers. (\APACyear2015)],

[TABLE]

[TABLE]

The isovector-vector meson-hyperon coupling constants $g_{\rho Y}$ are given as follows,

[TABLE]

with the differences in hyperon isospins accounted for by the isospin operator in the Lagrangian (see Equation (1)).

3 Constraints from GW170817

Analyses of gravitational waves (GW170817) and a gamma-ray burst (GRB 170817A) emitted from the recent BNS merger have provided additional constraints on the NS EoS, with particularly tight constraints on hyperonic EoSs coming from estimates of the NS maximum mass [Abbott \BOthers. (\APACyear2017\APACexlab\BCnt1), Abbott \BOthers. (\APACyear2017\APACexlab\BCnt2)]. A range of $2.15\,M_{\odot}\leq M_{\mathrm{max}}\leq 2.25\,M_{\odot}$ was determined from numerical relativity simulations and electromagnetic observations [Shibata \BOthers. (\APACyear2017)]. Kilonova modeling suggests $2.01\pm 0.04\,M_{\odot}\leq M_{\mathrm{max}}\leq 2.16^{+0.17}_{-0.15}\,M_{\odot}$, the lower limit taken to be the measured mass of PSR J0348+0432 [Rezzolla \BOthers. (\APACyear2018)]. Finally, analyses of the gamma-ray burst and kilonova ejecta provide an upper limit of $M_{\mathrm{max}}\leq 2.17\,M_{\odot}$ [Margalit \BBA Metzger (\APACyear2017)]. These maximum mass constraints are also relatively consistent with the measured mass of PSR J2215+5135. From these findings, and for the purposes of constraining the hyperonic EoSs calculated in this work, we determine a likely range for the maximum NS mass to be $2.01\,M_{\odot}\leq M_{\mathrm{max}}\leq 2.16\,M_{\odot}$ [Banik \BBA Bandyopadhyay (\APACyear2017)].

Several constraints on the canonical radius have also been deduced from GW170817 and GRB 170817A and are presented in Figure 3. As shown, the EoS utilized in this work is consistent with these constraints with a canonical radius of 12.7 km. However, hyperons are not typically present in large enough quantities in a $1.4\,M_{\odot}$ NS to have an appreciable affect on the radius, so these constraints are not directly relevant to the hyperonic EoSs presented in the remainder of this work.

4 Results

A thorough investigation of hyperonic NS EoSs requires the calculation of NS properties for a large range of possible meson-hyperon coupling constants. In this section we will discuss NS properties calculated for a range of vector meson-hyperon coupling constants specified by varying the $\alpha_{V}$ and $z$ SU(3) coupling parameters, with the scalar meson-hyperon couplings fit to the hyperon saturation potentials as discussed in Section 2.1. To examine the affect of including hyperon-hyperon interactions, we will compute NS properties for EoS models that include the $\sigma$, $\omega$, and $\rho$ mesons ($\sigma\omega\rho$), the $\phi$ meson ($\sigma\omega\rho\phi$), and the $\sigma^{*}$ meson ($\sigma\omega\rho\phi\sigma^{*}$).

First we set $\alpha_{V}=1$ and calculate the maximum mass ($M_{\mathrm{max}}$) and strangess fractions ($f_{S}$) (Figure 4) as well as the maximum hyperon fractions ($f_{Y}$) (Figure 5) of NSs for all possible $z$. Decreasing $z$ increases the values of the vector couplings $g_{\omega Y}$ and $|g_{\phi Y}|$, along with the maximum NS mass. The $\sigma\omega\rho\phi$ model produces a stiff EoS with maximum masses up to $\sim 0.2\,M_{\odot}$ greater than that of the $\sigma\omega\rho$ and $\sigma\omega\rho\phi\sigma^{*}$ models. However, if one considers the possibility that $\Sigma$ and more importantly $\Xi$ hyperons do not appear in NS matter, then the maximum mass of the $\sigma\omega\rho\phi\sigma^{*}$ model increases dramatically, the $\Xi-$ being responsible for considerable softening of the EoS. Total strangeness and hyperon fractions of the $\sigma\omega\rho\phi$ model increase monotonically with $z$, while those of the $\sigma\omega\rho\phi\sigma^{*}$ model reach maximums $f_{S}^{\mathrm{max}}\approx 6\%$ and $f_{Y}^{\mathrm{max}}\approx 12\%$ at $z\approx 0.37$ before decreasing with increasing $z$.

Results for the 2.01 $M_{\odot}$ and 2.16 $M_{\odot}$ mass constraint boundaries extracted from Figures 4 and 5 are summarized in Tables 1 and 2. Only by excluding the $\Sigma$ and $\Xi$ hyperons from the $\sigma\omega\rho\phi\sigma^{*}$ model can any of the calculated EoSs satisfy even the $2.01\,M_{\odot}$ lower limit of the mass constraint, indicated by the bottom of the green shaded region in Figure 4, if SU(6) symmetry ($z_{\mathrm{SU(6)}}\approx 0.408$) is assumed, let alone the $2.16\,M_{\odot}$ upper limit. If instead $z_{\mathrm{ESC08}}\approx 0.195$ is chosen, the $\sigma\omega\rho\phi$ and $\sigma\omega\rho\phi\sigma^{*}$ models easily satisfy the $2.01\,M_{\odot}$ constraint with maximum masses $M_{\mathrm{max}}(z_{\mathrm{ESC08}})\approx 2.17\,M_{\odot}$ and $M_{\mathrm{max}}(z_{\mathrm{ESC08}})\approx 2.14\,M_{\odot}$ respectively. However, the $\sigma\omega\rho\phi\sigma^{*}$ model requires $z\sim 0.18$ (just under $z_{\mathrm{ESC08c}}\approx 0.182$) to satisfy the $2.16\,M_{\odot}$ upper limit unless the $\Sigma$ and $\Xi$ hyperons are neglected, in which case $z\sim 0.26$ is sufficient.

Hyperon fractions in $2.16\,M_{\odot}$ maximum mass NSs are substantial at nearly 10% when the entire baryon octet is included, with $f_{\Lambda}=5.6\%$ and $f_{\Xi^{-}}=3.9\%$ for the $\sigma\omega\rho\phi\sigma^{*}$ model. The fraction of $\Lambda$s significantly exceeds that of the $\Xi^{-}$ in the $\sigma\omega\rho$ and $\sigma\omega\rho\phi$ models, but the two are much more comparable when the $\sigma^{*}$ meson is included as shown. Since no experimental data from the study of double-hypernuclei yet exists to constrain the $\Xi$ saturation self-potential, $g_{\sigma^{*}\Xi}$ was instead fixed theoretically using Equation (5). An overestimation of $g_{\sigma^{*}\Xi}$ will likewise overestimate $f_{\Xi^{-}}$, softening the hyperonic EoS. Therefore, it is possible that the $M_{\mathrm{max}}$ vs. $z$ relation in Figure 4 for the $\sigma\omega\rho\phi\sigma^{*}$ model may be closer to that displayed for the $\Lambda$ hyperon alone, making the model more consistent with the stiff mass constraints deduced from GW170817. More experimental data from the study of hypernuclei and double-hypernuclei is sorely needed to sufficiently constrain the hyperon single-particle- and self-potentials.

Finally, we let both the $\alpha_{V}$ and $z$ SU(3) parameters vary and calculate the maximum NS mass and hyperon fraction for the entire space using the $\sigma\omega\rho\phi\sigma^{*}$ model as shown in Figures 6 and 7. Neither $\alpha_{V}$ nor $z$ are necessarily restricted by the $2.01-2.16\,M_{\odot}$ maximum mass constraint described by the bold contours in Figure 6. Increasing $z$ beyond $\sim 0.3$ requires subsequent reduction of $\alpha_{V}$ below 1 to keep $M_{\mathrm{max}}>2.01\,M_{\odot}$. A hyperon fraction hotspot is centered around $\alpha_{V}=0.525$ and $z=0.775$ with $f_{Y}^{\mathrm{max}}\approx 16.5\%$, while hyperon fractions consistent with the mass constraints range from $9.4-16\%$.

5 Summary

In this contribution we introduced a new RMF parameterization, producing NS EoSs consistent with tight constraints on the isospin asymmetry energy and slope, and with constraints on the canonical NS radius deduced from analyses of the gravitational waves (GW170817) and gamma-ray burst (GRB 170817A) emitted by the recent BNS merger. This particular parameterization is of great utility in the study of hyperonization, as it is relatively straightforward to recover the intended saturation properties of symmetric nuclear matter when choosing vector meson-hyperon coupling constants that are inconsistent with SU(6) symmetry ($z\neq 1/\sqrt{6}$).

We calculated the maximum mass, strangeness fraction, and hyperon fraction of NSs for hyperonic EoSs, exploring the full range of possible vector meson-hyperon coupling constants consistent with SU(3) symmetry. Maximum masses were compared to constraints deduced from analyses of GW170817 and GRB 170817A that suggest the maximum NS mass falls in the following range: $2.01\,M_{\odot}\leq M_{\mathrm{max}}\leq 2.16\,M_{\odot}$. Taking the SU(3) parameter $\alpha_{V}=1$, we find this maximum mass range suggests $0.18\lesssim z\lesssim 0.30$ when considering the full baryon octet and including both the $\phi$ and $\sigma^{*}$ mesons. However, if only the $\Lambda$ hyperon is considered, we find that $0.26\lesssim z\lesssim 0.41$, highlighting the importance of acquiring additional empirical data with which to constrain the saturation single-particle- and self-potential of the $\Xi$ hyperon. Finally, the strangeness and hyperon fractions found for the $\sigma\omega\rho\phi\sigma^{*}$ model and the given maximum mass range are $4.5\%\lesssim f_{S}\lesssim 5.7\%$ and $9.6\%\lesssim f_{Y}\lesssim 12\%$ respectively.

Acknowledgements

We thank D. Blaschke for his valuable comments regarding this manuscript. This research was supported by the National Science Foundation (USA), grant number PHY-1714068.