Revealing the high-density equation of state through binary neutron star mergers

We present a novel method for revealing the equation of state of high-density neutron star matter through gravitational waves emitted during the postmerger phase of a binary neutron star system. The method relies on a small number of detections of the peak frequency in the postmerger phase for binaries of different (relatively low) masses, in the most likely range of expected detections. From such observations, one can construct the derivative of the peak frequency vs the binary mass, in this mass range. Through a detailed study of binary neutron star mergers for a large sample of equations of state, we show that one can extrapolate the above information to the highest possible mass (the threshold mass for black hole formation in a binary neutron star merger). In turn, this allows for an empirical determination of the maximum mass of cold, nonrotating neutron stars to within $0.1M_{\odot }$, while the corresponding radius is determined to within a few percent. Combining this with the determination of the radius of cold, nonrotating neutron stars of $1.6M_{\odot }$ [to within a few percent, as was demonstrated in Bauswein et al. Phys. Rev. D 86, 063001 (2012)], allows for a clear distinction of a particular candidate equation of state among a large set of other candidates. Our method is particularly appealing because it reveals simultaneously the moderate and very high-density parts of the equation of state, enabling the distinction of mass-radius relations even if they are similar at typical neutron star masses. Furthermore, our method also allows us to deduce the maximum central energy density and maximum central rest-mass density of cold, nonrotating neutron stars with an accuracy of a few percent.

Figure 1

Dominant GW frequency $f_{\text{peak}}$ of the postmerger phase as a function of the total binary mass $M_{\mathrm{tot}}$ for all EoSs. Different EoSs are distinguished by different solid lines. The highest frequency $f_{\text{peak}}^{\text{thres}}$ for a given EoS is highlighted by a thick cross. The dashed line approximates the dependence of $f_{\text{peak}}^{\text{thres}}$ on the maximum binary mass $M_{\text{thres}}$, which still produces an (at least transiently) stable merger remnant.

Figure 2

Dominant GW frequency $f_{\text{peak}}^{\text{thres}}$ of the most massive NS merger remnant as a function of the corresponding total binary mass $M_{\text{thres}}$ for different EoSs (crosses). Nearby circles with the same shade denote the estimated values for $f_{\text{peak}}^{\text{thres}}$ and $M_{\text{thres}}$ extrapolated entirely from GW information from low-mass NS binary mergers using Eqs. (2) and (3). The solid line represents the fit to the stability limit [Eq. (1)].

Figure 3

Derivative of $f_{\text{peak}}(M_{\mathrm{tot}})$ with respect to $M_{\mathrm{tot}}$ at $M_{\mathrm{tot}}=2.7M_{\odot }$ as a function of $f_{\text{peak}}(2.7M_{\odot })$ for different EoSs. The thin solid line separates models with a normal/steep relative slope (above) from models with flat relative slopes (below). The vertical dotted line distinguishes low frequencies and high frequencies (see the main text).

Figure 4

Relation between the radii $R_{1.6}$ of cold, nonrotating NSs with a gravitational mass of $1.6M_{\odot }$ and the dominant postmerger GW frequency $f_{\text{peak}}$ for a total binary mass of $2.7M_{\odot }$ for different EoSs. The line displays a quadratic fit to the data.

Figure 5

Mass-radius relations for the EoSs considered in this study with the circumferential radius $R$ and the gravitational mass $M$. Crosses at higher masses mark maximum masses of nonrotating NSs. Nearby circles with the same shade indicate the values estimated by the extrapolation of GW data of low-mass NS mergers and use of Eq. (4) to obtain $M_{\max }$ from the $M_{\text{thres}}$ estimate. The crosses at $1.6M_{\odot }$ mark the radii of $1.6M_{\odot }$ NSs. Circles at $1.6M_{\odot }$ indicate the radius estimate via the dominant postmerger GW frequency of a $2.7M_{\odot }$ NS merger.

Figure 6

Mass-radius relations for the EoSs considered in this study with the radius $R$ and the gravitational mass $M$. Boxes illustrate the maximum deviation of the estimated properties of the maximum-mass configuration, which are inferred from GW detections of low-mass binary NS mergers. The size of the boxes is chosen to be the largest deviation found in the sample of EoSs with low maximum masses ($M_{\max }<2.2M_{\odot }$) and the sample of EoSs with high maximum masses ($M_{\max }>2.2M_{\odot }$). Bars at $1.6M_{\odot }$ indicate the maximum deviation of the estimated radius inferred from a single GW detection of a low-mass binary NS merger. The size of the bars is chosen to be the largest deviation from the actual value found in the whole sample of EoSs.

Figure 7

Same as Fig. 6 for two EoSs with similar stellar properties in the intermediate mass range around $1.6M_{\odot }$, where the two mass-radius relations cross. Using the extrapolation procedure described in the main text (Sec. 4), the two EoSs can clearly be distinguished.

Figure 8

Dominant GW frequency $f_{\text{peak}}^{\text{thres}}$ of the most massive NS merger remnant as a function of the radius $R_{\max }$ of the maximum-mass configuration of cold, nonrotating NSs for different EoSs (crosses). The diagonal solid line is a fit to $R_{\max }(f_{\text{peak}}^{\text{thres}})$. Circles denote the estimated values for $f_{\text{peak}}^{\text{thres}}$, estimated entirely from GW information from low-mass NS binary mergers. The estimated values for $R_{\max }$ can be inferred by projecting horizontally, i.e., following the short lines to the diagonal line representing the fit to the numerical data (crosses).

Figure 9

Dominant GW frequency $f_{\text{peak}}^{\text{thres}}$ of the most massive NS merger remnant as a function of the maximum central energy density $e_{\mathrm{c},\max }$ of the maximum-mass configuration of nonrotating NSs for different EoSs (crosses). The diagonal solid line is a fit to $e_{\mathrm{c},\max }(f_{\text{peak}}^{\text{thres}})$. Circles denote the estimated values for $f_{\text{peak}}^{\text{thres}}$, extrapolated entirely from GW information from low-mass NS binary mergers. The estimated values for $e_{\mathrm{c},\max }$ can be inferred by projecting horizontally, i.e., following the short lines to the diagonal line representing the fit to the numerical data (crosses).

Figure 10

Dominant GW frequency $f_{\text{peak}}^{\text{thres}}$ of the most massive NS merger remnant as a function of the maximum central rest-mass density ${\rho }_{\mathrm{c},\max }$ of the maximum-mass configuration of nonrotating NSs for different EoSs (crosses). The diagonal solid line is a fit to ${\rho }_{\mathrm{c},\max }(f_{\text{peak}}^{\text{thres}})$. Circles denote the estimated values for $f_{\text{peak}}^{\text{thres}}$, extrapolated entirely from GW information from low-mass NS binary mergers. The estimated values for ${\rho }_{\mathrm{c},\max }$ can be inferred by projecting horizontally, i.e., following the short line to the diagonal line representing the fit to the numerical data (crosses).

Figure 11

Threshold NS binary mass distinguishing the prompt collapse to a BH from the formation of a (at least transiently) stable merger remnant as a function of the maximum central energy density $e_{\mathrm{c},\max }$ of the maximum-mass configuration of cold, nonrotating NSs, for different EoSs.

Figure 12

Approximate dominant GW frequency $f_{\text{peak}}^{\text{thres}}$ of the most massive NS merger remnant as a function of the corresponding estimated total binary mass $M_{\text{thres}}$ for a large sample of EoSs. The estimates are entirely based on the stellar properties of cold, nonrotating NSs (see the text). The plus signs represent the estimated properties for the set of EoSs employed in this work. The squares display the estimated threshold properties for a larger set of EoSs, which covers also the extreme cases.