Constraining nuclear matter parameters with GW170817

The tidal measurement of gravitational waves from the binary neutron star merger event GW170817 allows us to probe nuclear physics that suffers less from astrophysical systematics compared to neutron star radius measurements with electromagnetic wave observations. A recent work found strong correlation among neutron-star tidal deformabilities and certain combinations of nuclear parameters associated with the equation of state. These relations were then used to derive bounds on such parameters from GW170817 assuming that the relations and neutron star masses are known exactly. Here, we expand on this important work by taking into account a few new considerations: (1) a broader class of equations of state; (2) correlations with the mass-weighted tidal deformability that was directly measured with GW170817; (3) how the relations depend on the binary mass ratio; (4) the uncertainty from equation of state variation in the correlation relations; (5) adopting the updated posterior distribution of the tidal deformability measurement from GW170817. Upon these new considerations, we find GW170817 90% confidence intervals on nuclear parameters (the incompressibility $K_0$, its slope $M_0$, and the curvature of symmetry energy $K_{\mathrm{sym},0}$ at nuclear saturation density) to be $81\mathrm{MeV}\le K_0\le 362\mathrm{MeV}$, $1556\mathrm{MeV}\le M_0\le 4971\mathrm{MeV}$, and $-259\mathrm{MeV}\le K_{\mathrm{sym},0}\le 32\mathrm{MeV}$, which are more conservative than previously found with systematic errors more properly taken into account.

Figure 1

(Top) Correlations between mass-weighted average tidal deformability $\tilde{\mathrm{\Lambda }}$ and linear combination of nuclear parameters (the slope of the incompressibility $M_0$ and the symmetry energy’s slope $L_0$) for a chirp mass of $\mathcal{M}=1.188M_{\odot }$ corresponding to GW170817, using Skyrme EoSs (green square), relativistic mean field (RMF) EoSs (blue diamond), and phenomenologically varied EoSs (red circle). The first two classes were also considered in Ref. [11], while the last class is considered here for the first time. The mass ratio is chosen to be $q=0.87$, consistent with GW170817, though such correlations are insensitive to $q$. The shaded cyan and magenta regions represent the measurement constraints on $\tilde{\mathrm{\Lambda }}$ from GW170817 [7, 14]. The solid black line represents the best fit line through the data, while the dashed lines correspond to the lines drawn with 90% error bars on the $y$ intercept and slope. The Pearson correlation coefficient $C$ measures the amount of correlation ($C=1$ being the absolute correlation and $C=0$ being no correlation). The constant $\beta$ for the linear combination $M_0+\beta L_0$ is chosen to be $\beta =24.28$ such that the correlation between observables becomes 50%. (Bottom) Similar to the top panel but for the curvature of symmetry energy $K_{\mathrm{sym},0}$.

Figure 2

Posterior probability distribution on the curvature of symmetry energy $K_{\mathrm{sym},0}$ as derived in Sec. 5b. These results utilized a distribution on $\tilde{\mathrm{\Lambda }}$ given by Ref. [33] as prior information. The shaded region represents the 90% confidence interval on the data. Also shown by dashed vertical lines are the constraints found in Ref. [11] for priors on the mass-weighted tidal deformability $70\le \tilde{\mathrm{\Lambda }}\le 720$ and the symmetry energy’s slope $30\mathrm{MeV}\le L_0\le 86\mathrm{MeV}$. Notice how the inclusion of EoS variation uncertainty from a larger class of EoSs weakens the bounds found in Ref. [11].

Figure 3

Neutron star mass as a function of radius (left) and tidal deformability $\mathrm{\Lambda }$ (right) for a representative set of the EoSs used in our analysis, separated into groups of phenomenological (red dashed), Skyrme-type (green dotted-dashed), and RMF (blue dotted). Observe how Skyrme and RMF EoSs follow self-consistent behavior, while PEs see a wide variance in properties such as maximum mass and radius, due to the nature of the random sampling in nuclear parameters.

Figure 4

Correlations between nuclear parameters $L_0$, $K_0$, $M_0$, $K_{\mathrm{sym},0}$ and the mass-weighted average tidal deformability ${\tilde{\mathrm{\Lambda }}}_q$ for a chirp mass of $\mathcal{M}=1.188M_{\odot }$ corresponding to GW170817, using Skyrme EoSs (green square), RMF EoSs (blue diamond), and PEs (red circle). Mass ratios are chosen as $q=0.73$ (left), 0.875 (middle), and 1.00 (right) consistent with GW170817. The shaded cyan and magenta regions represent the measurement constraints on $\tilde{\mathrm{\Lambda }}$ from GW170817 [7, 14]. The solid black line in each panel represents the best fit line through the data, and the Pearson correlation coefficient $C$ measures the amount of correlation ($C=1$ being the absolute correlation and $C=0$ being no correlation).

Figure 5

Correlations with $\tilde{\mathrm{\Lambda }}$ as a function of mass ratio $q$ for $K_0+\alpha L_0$, $M_0+\beta L_0$, and $K_{\mathrm{sym},0}+\gamma L_0$ for $\mathcal{M}=1.188M_{\odot }$. These are much stronger than those involving single nuclear parameters, which is also shown for reference. Here we choose $\alpha =2.27$ and $\beta =24.28$ giving 50% correlations in the universal relations, while we choose $\gamma =2.63$ such that the correlation is maximized (see Sec. 5 for more details). Observe that correlations do not change significantly with $q$ across a wide range of mass ratios.

Figure 6

Similar to Fig. 1 but for a linear combination of nuclear parameters $K_0+\alpha L_0$. As discussed in Sec. 5, the linear coefficient $\alpha$ is chosen to be $\alpha =2.27$ such that 50% correlation is achieved in the universal relations. Observe that the correlations here are much stronger than those involving single nuclear parameters, as in Fig. 4.

Figure 7

Comparisons between nuclear parameter constraints and correlations with $\tilde{\mathrm{\Lambda }}$, evaluated at $q=0.87$, as functions of $\alpha$, $\beta$, and $\gamma$ for $K_0$ (left), $M_0$ (middle), and $K_{\mathrm{sym},0}$ (right). (Top) Estimated nuclear parameter constraints for different combinations of priors: (i) $70\le \tilde{\mathrm{\Lambda }}\le 720$, $40\le L_0\le 62$ (magenta); (ii) $279\le \tilde{\mathrm{\Lambda }}\le 822$, $40\le L_0\le 62$ (red); (iii) $70\le \tilde{\mathrm{\Lambda }}\le 720$, $30\le L_0\le 86$ (green); (iv) $279\le \tilde{\mathrm{\Lambda }}\le 822$, $30\le L_0\le 86$ (blue). Dashed horizontal lines correspond to bounds derived by [11] under similar prior assumptions. (Middle) Constraint ranges given as the difference between upper and lower limits. (Bottom) Correlations between $\tilde{\mathrm{\Lambda }}$ and linear combinations of nuclear parameters. Dotted vertical lines represent chosen values of $\alpha$, $\beta$, and $\gamma$ for deriving final bounds on the nuclear parameters. These values for $\alpha$ and $\beta$ are chosen to give a 50% correlation, while that for $\gamma$ gives a 80% correlation.

Figure 8

Two-dimensional normalized probability distributions between $\tilde{\mathrm{\Lambda }}$ and nuclear parameters $K_0+\alpha L_0$ (left), $M_0+\beta L_0$ (center), and $K_{\mathrm{sym},0}$ (right) generated via Eq. (14). Overlaid on the distributions is the set of 120 data points corresponding to each EoS used in this investigation for comparison. Observe how the multivariate Gaussian distributions indicate high levels of covariance between the variables, indicating the importance of estimating bounds using this method.

Figure 9

Posterior probability distribution on $\tilde{\mathrm{\Lambda }}$ as derived by the LIGO Collaboration in Ref. [33]. We take this as a prior distribution when computing the posteriors on nuclear parameters. Additionally shown are the GW and EM counterpart bounds of $70\le \tilde{\mathrm{\Lambda }}\le 720$ [7] (dashed maroon) and $279\le \tilde{\mathrm{\Lambda }}\le 822$ [14] (dotted orange) for comparison.

Figure 10

Resulting posterior distributions on the nuclear incompressibility $K_0$ and its slope $M_0$, and the curvature of symmetry energy $K_{\mathrm{sym},0}$ derived by integrating over the product of probability distributions [$P(\tilde{\mathrm{\Lambda }},K_0+\alpha L_0)$, $P(\tilde{\mathrm{\Lambda }},M_0+\beta L_0)$, and $P(\tilde{\mathrm{\Lambda }},K_{\mathrm{sym},0})$] in Fig. 8 and $P_{\mathrm{LIGO}}(\tilde{\mathrm{\Lambda }})$ in Fig. 9. Further, for the linear combinations of $K_0+\alpha L_0$ and $M_0+\beta L_0$, one more integration over the probability distribution of $30\mathrm{MeV}\le L_0\le 86\mathrm{MeV}$ was required to directly find the posterior distributions on $K_0$ and $M_0$. Overlaid are the resulting 68% and 90% confidence intervals in orange and maroon, respectively, as well as the corresponding bounds calculated in Sec. 5a shown by dashed maroon vertical lines. Additionally shown in dotted blue are the corresponding bounds on $M_0$ and $K_{\mathrm{sym},0}$ computed by Ref. [11], using priors of $\tilde{\mathrm{\Lambda }}\in [70,720]$ and $L_0\in [30,86]\mathrm{MeV}$. Observe how the results for the 90% confidence intervals obtained in this section are slightly smaller than those found in Sec. 5a, indicating that previously the error was slightly overestimated (as the probability distribution on $\tilde{\mathrm{\Lambda }}$ and the covariances between $\tilde{\mathrm{\Lambda }}$ and the nuclear parameters were not properly taken into account).

Figure 11

(Top) Correlation between mass-weighted average tidal deformability $\tilde{\mathrm{\Lambda }}$ and ${\mathrm{\Lambda }}_{1.4}$ (individual tidal deformability at the mass of $1.4{\mathrm{M}}_{\odot }$) for various EoSs, each evaluated at mass ratios $q=0.73$, 0.80, 0.87, 0.93, and 1.00. The chirp mass is fixed to be the measured value of $\mathcal{M}=1.188M_{\odot }$. (Bottom) Fractional difference from the fit for each EoS. Notice how the HS EoSs interrupt the universality between the two parameters by up to 60% (5% maximal percent difference in the absence of hybrid EoSs).

Figure 12

$\tilde{\mathrm{\Lambda }}$ for a representative set of EoSs as a function of mass ratio $q$ in GW170817’s observed range of $q\in [0.73,1.00]$ with the chirp mass fixed to $\mathcal{M}=1.188M_{\odot }$. Notice how $\tilde{\mathrm{\Lambda }}$ only varies slightly in this region of interest for Skyrme, RMF, and PEs. Hybrid EoSs, on the other hand, admit two different configurations for GW170817, HS/NS (solid maroon) and HS/HS (dashed maroon), with the former giving a significant variation in $\tilde{\mathrm{\Lambda }}$. For demonstration purposes, the black vertical line corresponds to mass ratio $q=0.995$, where it can be seen that two different binary configurations emerge, discussed in more detail in Appendix pp3.

Figure 13

Similar to Fig. 5 upon the removal of PEs. Observe that the correlations for linear combinations involving lower-order parameters improve by up to 55%, while linear combinations with high-order parameter $K_{\mathrm{sym},0}$ show slightly diminished, yet comparable, correlations. Observe also that the correlations are insensitive to $q$.

Figure 14

Scatter plots demonstrating variation in correlation at $q=0.995$ (with $\mathcal{M}=1.188M_{\odot }$) when introducing HS EoSs based on three different fiducial nuclear matter EoSs with $\tilde{\mathrm{\Lambda }}\approx 465$, $\tilde{\mathrm{\Lambda }}\approx 800$, $\tilde{\mathrm{\Lambda }}\approx 1045$, represented by filled red circles. These correspond to soft, intermediate, and stiff fiducial EoSs, respectively. These are followed by two different star configurations of hybrid HS/NS values (purple triangle) and HS/HS (purple diamond) with a reduction in $\tilde{\mathrm{\Lambda }}$, as is shown in Fig. 12 for two different HS EoSs, corresponding to the $j=0.8$ and $j=1.0$ configurations. As demonstrated in Fig. 12, $q=0.995$ clearly admits both HS/NS and HS/HS binary configurations. Shown in gray as reference are the PE, Skyrme, and RMF EoSs, irrelevant to this investigation. Displayed in the bottom right corner is the correlation between nuclear parameter combinations and $\tilde{\mathrm{\Lambda }}$ when imposing soft, intermediate, and stiff fiducial EoSs in the generation of HS structure. Notice how the choice of fiducial EoS alters correlations between $\tilde{\mathrm{\Lambda }}$ and combinations of nuclear parameters differently. This indicates that potential HS EoSs could impact nuclear bounds significantly.