Nonparametric inference of neutron star composition, equation of state, and maximum mass with GW170817

The detection of GW170817 in gravitational waves provides unprecedented constraints on the equation of state (EOS) of the ultradense matter within the cores of neutron stars (NSs). We extend the nonparametric analysis first introduced by Landry and Essick (2019), and confirm that GW170817 favors soft EOSs. We infer macroscopic observables for a canonical $1.4M_{\odot }$ NS, including the tidal deformability ${\mathrm{\Lambda }}_{1.4}={211}_{-137}^{+312}$ $({491}_{-181}^{+216})$ and radius $R_{1.4}=10.86_{-1.42}^{+2.04}$ $({12.51}_{-0.88}^{+1.00})\mathrm{km}$, as well as the maximum mass for nonrotating NSs $M_{\max }=2.064_{-0.134}^{+0.260}$ $({2.017}_{-0.087}^{+0.238})M_{\odot }$, with nonparametric priors loosely (tightly) constrained to resemble candidate EOSs from the literature. Furthermore, we find weak evidence that GW170817 involved at least one NS based on gravitational-wave data alone ($B_{\mathrm{BBH}}^{\mathrm{NS}}=3.3\pm 1.4$), consistent with the observation of electromagnetic counterparts. We also investigate GW170817’s implications for the maximum spin frequency of millisecond pulsars, and find that the fastest known pulsar is spinning at more than 50% of its breakup frequency at 90% confidence. We additionally find modest evidence in favor of quark matter within NSs, and GW170817 favors the presence of at least one disconnected hybrid star branch in the mass-radius relation over a single stable branch by a factor of 2. Assuming there are multiple stable branches, we find a suggestive posterior preference for a sharp softening around nuclear density followed by stiffening around twice nuclear density, consistent with a strong first-order phase transition. While the statistical evidence in favor of new physics within NS cores remains tenuous with GW170817 alone, these tantalizing hints reemphasize the promise of gravitational waves for constraining the supranuclear EOS.

Figure 1

Example synthetic EOSs drawn from our agnostic (left) and informed (right) nonparametric priors, constructed as mixture models with equal prior odds for hadronic, hyperonic, and quark compositions. Draws from the prior are colored according to the maximum nonrotating NS mass they support: blue for $M_{\max }\ge 1.93M_{\odot }$, and black otherwise. Candidate EOSs from the literature, used as input for our GPs, are shown in red (see Table 7). Vertical lines indicate once, twice, and six times nuclear saturation density.

Figure 2

Distributions for $M_1$, ${\mathrm{\Lambda }}_1$, and ${\mathrm{\Lambda }}_2$ after marginalizing over NS composition. (Left) Model-agnostic prior (cyan), posterior (magenta), and low-spin marginal likelihood (green). (Right) Model-informed prior (blue), posterior (red), and low-spin marginal likelihood (green). Contours in the joint distributions denote highest-posterior-density 50% and 90% credible regions.

Figure 3

EOS processes after marginalizing over composition. (Left) Model-agnostic prior (cyan) and posterior (magenta) processes. (Right) Model-informed prior (blue) and posterior (red) processes. Shaded regions correspond to 50% and 90% symmetric marginal credible regions for the pressure at each density. Solid lines denote the median and vertical lines denote ${\rho }_{\mathrm{nuc}}$, $2{\rho }_{\mathrm{nuc}}$, and $6{\rho }_{\mathrm{nuc}}$.

Figure 4

Processes relating a few macroscopic observables after marginalizing over EOS composition with the agnostic prior. Prior (cyan) and posterior (magenta) processes for (left) tidal deformability ($\mathrm{\Lambda }$; top), radius ($R$; middle), and moment of inertia ($I$; bottom) as functions of mass as well as (right) the maximum spin frequency ($f_{\max }$; top), maximum dimensionless spin parameter (${\chi }_{\max }$; middle), and the dimensionless spin parameter divided by the spin frequency (bottom), useful when estimating $\chi$ for a pulsar with unknown mass. Shaded regions denote the 50% and 90% symmetric credible regions for the marginal distribution of each observable at each mass. Solid lines denote the median and vertical lines denote canonical $1.4M_{\odot }$ stars. Horizontal grey lines in the top-right panel denote the measured spin frequencies of $\mathrm{J}1748-2446\mathrm{ad}$ (716 Hz [36]) and $\mathrm{B}1937+241$ (641 Hz [106]), which lie below the 90% lower limits for $f_{\max }$ only when $M\gtrsim M_{\odot }$.

Figure 5

Agnostic EOS prior (cyan) and posterior (magenta) processes for EOSs that support multiple stable branches in the $M\text{−}R$ relation above ${\rho }_c=0.8{\rho }_{\mathrm{nuc}}$ after marginalizing over composition. The equivalent processes for EOSs that support a single stable branch are indistinguishable from Fig. 3. The general preference for softer EOSs a posteriori manifests as a dramatic softening at or below ${\rho }_{\mathrm{nuc}}$ before stiffening at approximately $2{\rho }_{\mathrm{nuc}}$. Gray lines denote ${\rho }_{\mathrm{nuc}}$, $2{\rho }_{\mathrm{nuc}}$, and $6{\rho }_{\mathrm{nuc}}$, respectively. Bayes factors between multiple and single stable branches for the agnostic priors are $B_{n=1}^{n>1}=2.0513\pm (9.5\times {10}^{-3})$, $1.5274\pm (6.8\times {10}^{-3})$, $4.084\pm (1.9\times {10}^{-2})$, and $1.684\pm (1.7\times {10}^{-2})$ for the marginalized, hadronic, hyperonic, and quark compositions, respectively.

Figure 6

Processes relating a few macroscopic observables after marginalizing over EOS composition with the informed prior. Prior (blue) and posterior (red) processes for (left) tidal deformability ($\mathrm{\Lambda }$; top), radius ($R$; middle), and moment-of-inertia ($I$; bottom) as functions of mass as well as (right) the maximum spin frequency ($f_{\max }$; top), maximum dimensionless spin parameter (${\chi }_{\max }$; middle), and $\chi /f$ (bottom). Shaded regions denote the 50% and 90% symmetric credible regions for the marginal distribution of each observable at each mass. Solid lines denote the median and vertical lines denote canonical $1.4M_{\odot }$ stars. Horizontal grey lines in the top-right panel denote the measured spin frequencies of J1748–2446ad (716 Hz [36]) and $\mathrm{B}1937+241$ (641 Hz [106]), which lie below but near the lower limits for $f_{\max }$.

Figure 7

Distributions for $M_{b,1.4}$, $M_{\max }$, ${\mathrm{\Lambda }}_{1.4}$, and $R_{1.4}$ after marginalizing over EOS composition. (Left) Model-agnostic prior (cyan) and posterior (magenta). (Right) Model-informed prior (blue) and posterior (red). Contours in the joint distributions denote minimal 50% and 90% credible regions. The bimodal behavior seen with the agnostic posterior is a combination of the inherited multimodal likelihood from GW170817 (see Fig. 11 of Ref. [8]) as well as the preference for multiple stable branches. EOSs with multiple stable branches in the $M\text{−}R$ relation only inhabit the low-$R_{1.4}$ mode, for example, while EOSs with a single stable branch inhabit both. The multimodal behavior seen with the informed prior is mostly due to the tight constraints imposed for each composition separately, which result in the separate peaks observed in these canonical observables.