Reconstructing the neutron-star equation of state with gravitational-wave detectors from a realistic population of inspiralling binary neutron stars

Gravitational-wave observations of inspiralling binary neutron-star systems can be used to measure the neutron-star equation of state (EOS) through the tidally induced shift in the waveform phase that depends on the tidal deformability parameter $\lambda$. Previous work has shown that $\lambda$, a function of the neutron-star EOS and mass, is measurable by Advanced LIGO for a single event when including tidal information up to the merger frequency. In this work, we describe a method for stacking measurements of $\lambda$ from multiple inspiral events to measure the EOS. We use Markov chain Monte Carlo simulations to estimate the parameters of a four-parameter piecewise-polytrope EOS that matches theoretical EOS models to a few percent. We find that, for “realistic” event rates ($\sim 40$ binary neutron-star inspiral events per year with signal-to-noise ratio $>8$ in a single Advanced LIGO detector), combining a year of gravitational-wave data from a three-detector network with the constraints from causality and recent high-mass neutron-star measurements, the EOS above nuclear density can be measured to better than a factor of 2 in pressure in most cases. We also find that in the mass range $1M_{\odot }–2M_{\odot }$, the neutron-star radius can be measured to better than $\pm 1\mathrm{km}$ and the tidal deformability can be measured to better than $\pm 1\times 10^{36}\mathrm{g}{\mathrm{cm}}^2{\mathrm{s}}^2$ (10%–50% depending on the EOS and mass). The overwhelming majority of this information comes from the loudest $\sim 5$ events. Current uncertainties in the post-Newtonian waveform model, however, lead to systematic errors in the EOS measurement that are as large as the statistical errors, and more accurate waveform models are needed to minimize this error.

Figure 1

The joint constraint imposed by causality and the existence of a $1.93M_{\odot }$ NS. In the dark shaded volume outlined in black, the EOS parameters allow a NS to have $v_s>c$ at some density below ${\rho }_{\mathrm{c},\max }$ and are therefore ruled out. In the light shaded volume outlined in red, the EOS parameters are ruled out because they do not allow a NS to have a maximum mass of at least $1.93M_{\odot }$. Also outlined in blue is the surface where the maximum mass is $2.4M_{\odot }$. A hypothetical $2.4M_{\odot }$ NS observation would rule out EOS parameters below this surface.

Figure 2

Radius and tidal deformability of tabulated EOS models (solid line) and the least-squares piecewise-polytrope fits (dashed line) to those tabulated models given in Table 1. The 20 vertical lines represent the most likely NS masses of the ten known BNS systems [38]. Some of these masses, however, have significant uncertainties. The overlapping vertical bands represent the $1\sigma$ uncertainty in the masses of the pulsars J1614-2230 ($1.97\pm 0.04M_{\odot }$) [1] and $\mathrm{J}0348+0432$ ($2.01\pm 0.04M_{\odot }$) [2], both in neutron-star–white-dwarf binaries.

Figure 3

Uncertainty in the recovered EOS, radius, and tidal deformability for the three-detector aLIGO-aVirgo network. Results are shown for the loudest 20 events with network SNR from 63.7 down to 13.6. The red, green, and blue regions represent the 68% ($1\sigma$), 95% ($2\sigma$), and 99.7% ($3\sigma$) credible regions, respectively. In the bottom left panel, $p_{\text{true}}$ is the pressure of the true injected EOS, which in this case is the fit to the MPA1 EOS. In the right panels, the vertical line at $1.93M_{\odot }$ is the minimum mass, set by the prior, where some accepted EOS parameters do not produce a stable NS.

Figure 4

The same BNS population as Fig. 3. Contours represent the 95% credible regions for the loudest one, five, and 20 inspiral events. Including quieter events from the population does not improve the results. Also shown is the lower limit on the 95% credible region from just the maximum mass ($M_{\max }\ge 1.93M_{\odot }$) and causality priors.

Figure 5

Dependence of credible regions on the highest known NS mass. The 95% credible regions are shown for the same 20 BNS systems as in Fig. 3. The maximum mass prior is set to $M_{\max }\ge 1.93M_{\odot }$ for the red shaded region and $M_{\max }\ge 2.4M_{\odot }$ for the blue shaded region. Also shown are the lower limits on the 95% credible regions from just the maximum mass and causality priors, with $M_{\max }\ge 1.93M_{\odot }$ (red dashed line) and $M_{\max }\ge 2.4M_{\odot }$ (blue dashed line).

Figure 6

Dependence of credible regions on the masses of NSs in the simulated BNS population. Results use the same BNS parameters as in Fig. 3, except the masses are varied to be all $1.0M_{\odot }$ (green dot-dashed line), all $1.4M_{\odot }$ (blue dashed line), and all $1.8M_{\odot }$ (magenta line). Also shown is the lower limit from just the maximum mass ($M_{\max }\ge 1.93M_{\odot }$) and causality priors.

Figure 7

The same BNS population as Fig. 3. The SNR in the Fisher matrix calculation is set to match the network SNR of the LALInferenceMCMC calculation. Contours represent 95% credible regions.

Figure 8

Dependence of the EOS credible region on the true EOS for the 20 loudest BNS systems. Contours represent the 68% and 95% credible regions. The BNS parameters are the same as in Fig. 3 except for the choice of tabulated EOS. Results were calculated using the Fisher matrix approximation to the quasilikelihood, and the SNR in the Fisher matrix calculation was scaled to match the network SNR of the three-detector network as in Fig. 7. From left to right then top to bottom, the true EOSs are SLy, ENG, MPA1, MS1, MS1b, H4, and ALF2. Unlike the results above, the true EOSs used here are the tabulated EOSs, so the recovered EOSs also include the systematic bias from the inexact EOS parameterization. The corresponding best-fit piecewise polytropes are shown as dashed curves. For each panel, the other EOSs are also shown relative to the true tabulated EOS. The quantity $p/p_{\text{true}}$ is not defined above the highest density in the EOS table, and for SLy, MPA1, and H4, the EOS table ends before the ${10}^{15.5}\mathrm{g}/{\mathrm{cm}}^3$ right limit of each panel.

Figure 9

The 95% credible regions for the radius and tidal deformability for the same BNS systems and tabulated EOSs used in Fig. 8. In the left panels the EOSs are SLy, MPA1, H4, MS1 from bottom to top, and in the right panels they are ENG, ALF2 and MS1b from bottom to top. The tabulated EOS models, and not the fits, were used in generating the simulated waveforms. The lower bound from the mass and causality prior is also shown.

Figure 10

The same BNS population and priors as Figs. 3 and 4. Contours represent 95% credible regions for the loudest five events. The different contours correspond to different noise realizations. The dashed contour corresponds to the zero-noise data also shown in Fig. 4.

Figure 11

Five different populations of simulated BNS events for a one-year period using the realistic event rate. Contours represent the 95% credible regions for the loudest five events in each population using zero-noise data. Population 1 is the same as in Figs. 3 and 4.

Figure 12

Systematic errors in the recovered EOS due to uncertainty in the correct GW model. TaylorF2, TaylorT1, and TaylorT4 waveforms were injected into the data, and the parameters were recovered with the TaylorF2 waveform. The 95% credible regions are shown for the loudest five systems.