Measuring the neutron star equation of state with gravitational wave observations

We report the results of a first study that uses numerical simulations to estimate the accuracy with which one can use gravitational wave observations of double neutron-star inspiral to measure parameters of the neutron-star equation of state. The simulations use the evolution and initial-data codes of Shibata and Uryū to compute the last several orbits and the merger of neutron stars, with matter described by a parametrized equation of state. Previous work suggested the use of an effective cutoff frequency to place constraints on the equation of state. We find, however, that greater accuracy is obtained by measuring departures from the point-particle limit of the gravitational waveform produced during the late inspiral. As the stars approach their final plunge and merger, the gravitational wave phase accumulates more rapidly for smaller values of the neutron-star compactness (the ratio of the mass of the neutron-star to its radius). We estimate that realistic equations of state will lead to gravitational waveforms that are distinguishable from point-particle inspirals at an effective distance (the distance to an optimally oriented and located system that would produce an equivalent waveform amplitude) of 100 Mpc or less. As Lattimer and Prakash observed, neutron-star radius is closely tied to the pressure at density not far above nuclear. Our results suggest that broadband gravitational wave observations at frequencies between 500 and 1000 Hz will constrain this pressure, and we estimate the accuracy with which it can be measured. Related first estimates of radius measurability show that the radius can be determined to an accuracy of $\delta R\sim 1\mathrm{km}$ at 100 Mpc.

Figure 1

Initial choices of EOS for numerical evolution compared to the set of tabled EOSs considered in 27. Candidates are labeled in order of increasing softness: 2H, H, HB, B, 2B.

Figure 2

Solid lines show numerical waveforms, scaled by $c^{2}D/G{M}_{\mathrm{tot}}$, and aligned in time and phase to the same point-particle post-Newtonian inspiral (dashed line), using the method described in Sec. 4b. The two dashed vertical bars indicate the portion of the waveform used for matching; the last vertical bar indicates the end of inspiral time $t_{\mathrm{M}}$ for the numerical waveform. The top four simulations, 2H, H, HB, and B, show the start of post-merger oscillations from a hyper-massive NS remnant in the simulation. 2B shows quasinormal ringdown from a prompt collapse to a black hole following merger.

Figure 3

Phase-optimized, time-limited match between numerical inspiral waveforms and point-particle post-Newtonian waveforms. Contours are shown at 0.95, 0.98, 0.99, and 0.997, as a function of match region truncation at some time before numerical merger, $T_{\mathrm{F}}-t_{\mathrm{M}}$, and relative shift between numerical and point-particle waveforms, $t_{\mathrm{c}}^{\mathrm{PP}}-t_{\mathrm{M}}$. The start of the match is fixed to 1.5 ms after the start of the numerical waveform. Subsequent analysis in this paper is done using the best match at a fixed $T_{\mathrm{F}}-t_{\mathrm{M}}$ of 1.8 ms for each waveform.

Figure 4

Time-frequency behavior, vertical line markings as previous figure. The departure from the point-particle time-frequency relations, shown using a long-dashed line, occurs between 700–1000 Hz depending on EOS.

Figure 5

DFT of full numerical waveforms, at an effective distance $D_{\mathrm{eff}}=100\mathrm{Mpc}$, compared to noise spectra for Advanced LIGO ( labeled “AdvLIGO” for the standard configuration and “Broadband” for the broadband configuration) and the Einstein Telescope ( labeled “ET”) shown by thick grey lines. The DFT of the numerical waveforms turned off after $t_{\mathrm{M}}$ is shown by dot-dashed lines, the stationary-phase point particle is shown by a dashed line for reference. The lower right figure shows a combined plot of inspiral-truncated waveforms, smoothly joined on to best-match PP inspiral time series before the DFT is taken.