Impact of the nuclear symmetry energy on the post-merger phase of a binary neutron star coalescence

The nuclear symmetry energy plays a key role in determining the equation of state of dense, neutron-rich matter, which governs the properties of both terrestrial nuclear matter as well as astrophysical neutron stars. A recent measurement of the neutron skin thickness from the PREX Collaboration has lead to new constraints on the slope of the nuclear symmetry energy, $L$, which can be directly compared to inferences from gravitational wave observations of the first binary neutron star merger inspiral, GW170817. In this paper, we explore a new regime for potentially constraining the slope, $L$, of the nuclear symmetry energy with future gravitational wave events: the post-merger phase of a binary neutron star coalescence. In particular, we go beyond the inspiral phase, where imprints of the slope parameter $L$ may be inferred from measurements of the tidal deformability, to consider imprints on the post-merger dynamics, gravitational wave emission, and dynamical mass ejection. To this end, we perform a set of targeted neutron star merger simulations in full general relativity using new finite-temperature equations of state, which systematically vary $L$ while keeping the magnitude of the symmetry energy at the saturation density, $S$, fixed. We find that the post-merger dynamics and gravitational wave emission are mostly insensitive to the slope of the nuclear symmetry energy. In contrast, we find that dynamical mass ejection contains a weak imprint of $L$, with large values of $L$ leading to systematically enhanced ejecta.

Figure 1

Correlations between $R_{1.4}$ and ${\mathrm{\Lambda }}_{1.4}$ for a sample of $>10,000$ piecewise polytropic EoSs. Each EoS has five polytropic segments, spaced log-uniformly in density between 0.5 and $7.4n_{\mathrm{sat}}$, with uniformly drawn pressures. All EoSs support a maximum mass of at least $1.97M_{\odot }$. The symmetry energy slope is extracted according to Eq. (6). We find that, although $R_{1.4}$ and ${\mathrm{\Lambda }}_{1.4}$ are well correlated, there is a significant scatter in the trend, which depends approximately monotonically on the value of $L$.

Figure 2

Equations of state included in our sample. The top panel shows the zero-temperature, $\beta$-equilibrium pressure; the middle panel shows the mass-radius relation; and the bottom panel shows the tidal deformabilities. The dashed lines indicate EoSs with $R_{1.4}\simeq 11\mathrm{km}$, which were constructed to have identical ${\mathrm{\Lambda }}_{1.4}=193$. The solid lines correspond to EoSs with $R_{1.4}=12\mathrm{km}$, and the dotted lines correspond to the $R_{1.4}=13\mathrm{km}$ EoSs.

Figure 3

Two-dimensional spatial distributions of the rest-mass density $\rho$, temperature $T$, electron fraction $Y_e$, and entropy $s$ per baryon. Shown are meridional (top) and equatorial (bottom) views for three different values of the slope $L$ of the nuclear symmetry energy. The results shown are for the $R_{1.4}=12\mathrm{km}$ models with mass ratio $q=0.85$ at time $t=25\mathrm{ms}$ after the merger.

Figure 4

Temperature $T$ in the equatorial plane at $t\simeq 20\mathrm{ms}$ after the merger for unequal-mass ($q=0.85$) mergers with EoSs having a characteristic radius of $R_{1.4}=12\mathrm{km}$. The green lines indicate contours of constant rest-mass density, with values labeled with respect to the nuclear saturation density. The different panels show results for varying slope parameter $L$ from 40 to 120 MeV.

Figure 5

Temperatures $T$, electron fractions $Y_e$, and lepton chemical potential ${\mu }_l$ probed at different densities $n$ in the massive neutron star remnant. The densities are stated relative to saturation density $n_{\mathrm{sat}}$. The models are the same as shown in Fig. 4.

Figure 6

Time-integrated ejected mass $M_{\mathrm{ej}}$ and mass-weighted average electron fraction $Y_e$ projected onto a sphere at a radius $e=295\mathrm{km}$ from the origin. The data is shown using Mollweide projection for the unequal-mass models.

Figure 7

Histograms of the electron fraction, the average entropy per baryon, and the velocity of the dynamical mass ejecta for simulations with mass ratios $q=0.85$. The histograms refer to time-integrated quantities and have been normalized to the respective amount of total mass ejection $M_{\mathrm{ej}}$. The different colors refer to simulations with different slopes $L$ of the nuclear symmetry energy, while $R_{1.4}$ denotes the radius a $1.4M_{\odot }$ star for each EoS.

Figure 8

Gravitational wave strain for the $q=0.85$ binaries, viewed face-on at 40 Mpc. The different radii are plotted in each column, while the rows show different values of $L$. All waveforms are aligned at the time of merger. We find significant differences in the post-merger GW signals, even for EoSs with the same $R_{1.4}$.

Figure 9

Same as Fig. 8, but for the equal-mass binaries.

Figure 10

Characteristic strain, including all $\ell =2$, 3 modes, for a face-on merger at 40 Mpc. The top two and bottom left panels are for the $q=0.85$ binaries, while the bottom right panel corresponds to the equal-mass binary. The vertical solid line marks the dominant $f_2$ spectral peak, while the dotted lines indicate the location of the secondary peaks $f_1$ and $f_3$. The markers indicate the approximate location of spectral peaks associated with the $m=1$ mode, which are expected to occur at $\simeq f_2/2$ where present. Finally, the $R_{1.4}=12\mathrm{km}$, $L=40\mathrm{MeV}$ EoS is also plotted in the lower left panel with the faded, dashed pink line, to illustrate the similarity of this spectrum with the $R_{1.4}=13\mathrm{km}$ spectra. The gray dash-dot and dotted lines indicate the design sensitivity curves for advanced LIGO [132] and Einstein Telescope [133], respectively.

Figure 11

Correlations between the peak frequency $f_2$ and the radii $R_{1.4}$ and $R_{1.8}$ of $1.4M_{\odot }$ and $1.8M_{\odot }$ neutron stars, respectively. We also show correlations with the pressure $P$ at density $n=3n_{\mathrm{sat}}$. The different colors correspond to the value of $L$, while the different symbol shapes indicate $R_{1.4}$ for our chosen EoSs. The hatched symbols correspond to the results from the equal-mass binaries, while the solid-fill symbols correspond to the $q=0.85$ binaries. We find a significant scatter in the correlation between $f_2$ and $R_{1.4}$, and that $f_2$ instead correlates more closely with the high-density part of the EoS, parametrized here either with $R_{1.8}$ or $P_{3n_{\mathrm{sat}}}$.

Figure 12

Inferred radius as a function of the true (inputted) radius, for each of the binaries in this work. The color scheme and symbols are as in Fig. 11. We calculate $R_{1.6}(f_2)$ using the universal relationship from Eq. (21) of [137]. We find that using this standard fitting formula with these more extremal EoSs leads to inferred errors of up to 0.86 km in the radius for the EoSs included in this work.

Figure 13

Gravitational wave energy ($\mathrm{\Delta }{E}_{\mathrm{GW}}$) and angular momentum losses ($\mathrm{\Delta }J$) for the unequal-mass mergers $q=0.85$. Different colors correspond to different slope parameters $L$.

Figure 14

Amplitude of the spherical harmonic components of the GW signal for an edge-on merger at 40 Mpc, for the $R_{1.4}=12\mathrm{km}$ EoSs. The solid and dotted lines correspond to the $(\ell ,m)=(2,1)$ and (2,2) modes, respectively. For the $q=0.85$ binary (top panel), we find that the $m=1$ mode saturates within a few milliseconds of the merger for the values of $L>100\mathrm{MeV}$, but that the $m=1$ mode quickly decays for the $L=40\mathrm{MeV}$ EoS, indicating that this binary is rapidly becoming axisymmetric. In contrast, for the equal-mass binaries (bottom panel), the $m=1$ mode is in general weaker, and decays quickly for all values of $L$.