Kilohertz gravitational waves from binary neutron star remnants: Time-domain model and constraints on extreme matter

The remnant star of a neutron star merger is an anticipated loud source of kilohertz gravitational waves that conveys unique information on the equation of state of hot matter at extreme densities. Observations of such signals are hampered by the photon shot noise of ground-based interferometers and pose a challenge for gravitational-wave astronomy. We develop an analytical time-domain waveform model for postmerger signals informed by numerical relativity simulations. The model completes effective-one-body waveforms for quasicircular nonspinning binaries in the kilohertz regime. We show that a template-based analysis can detect postmerger signals with a minimal signal-to-noise ratio (SNR) of 8.5, corresponding to GW170817-like events for third-generation interferometers. Using Bayesian model selection and the complete inspiral-merger-postmerger waveform model it is possible to infer whether the merger outcome is a prompt collapse to a black hole or a remnant star. In the latter case, the radius of the maximum mass (most compact) nonrotating neutron star can be determined to kilometer precision. We demonstrate the feasibility of inferring the stiffness of the equation of state at extreme densities using the quasiuniversal relations deduced from numerical-relativity simulations.

Figure 1

Gravitational-wave merger $f_{\mathrm{mrg}}$ and postmerger peak $f_2$ frequency for GW170817. The distributions are estimated from the LIGO-Virgo posterior distributions [3] for the $\tilde{\mathrm{\Lambda }}$ parameters using (i) the quasiuniversal relation proposed in [4] for the merger frequency; (ii) the relation proposed in [16] and further refined in this work for the postmerger peak frequency. The distribution of $f_2$ is cut at ${\kappa }_2^{\mathrm{T}}<70$ to exclude binaries that undergo prompt collapse at merger.

Figure 2

Merger and postmerger waveform from two very different BNS with mass $M=(1.35+1.35)M_{\odot }$. The MS1b BNS is an example of a long-lived remnant, and the SLy BNS is an example of a short-lived remnant collapsing at $\widehat{t}\sim 1200$ after merger time, $\widehat{t}={\widehat{t}}_{\mathrm{mrg}}$. In both cases the postmerger waveform amplitude has characteristic maxima and minima ${\widehat{A}}_i$ at times ${\widehat{t}}_i$ with $i=0,\dots ,3$. Note the jump in the phase at ${\widehat{t}}_0$, where the instantaneous frequency is not defined.

Figure 3

Characteristic frequency information from NR simulations. Markers represent the frequencies extracted from the NR data and the uncertainties are estimated using simulations at different resolutions; the black lines are the fits and the gray bands are the 90% credible regions. Left and right panels show the same data: the colors on the left panel correspond to the EOS variation, those on the right panel to the mass ratio.

Figure 4

Mismatches between NRPM model and the CoRe NR waveforms. The validation set is indicated with cyan triangle markers. Vertical bars indicate the range of mismatches among NR waveforms at different grid resolutions (when available); a single marker indicates the mismatch between waveforms from two grid resolutions or the average from many resolutions. LIGO design sensitivity [83, 84, 85] is used in the calculation of $\overline{F}$ and the frequency ranges start from $f_{\mathrm{mrg}}$ (computed with relation extracted above) and reach 4 kHz.

Figure 5

Complete TEOBResumS_NRPM (2, 2) waveforms and corresponding spectra. Left panel: Time-domain TEOBResumS_NRPM (2, 2) waveforms compared with selected NR hybrids around merger. From top to bottom, $\mathrm{BHB}\mathrm{\Lambda }\varphi$ $M=(1.25+1.25)M_{\odot }$ is the best mismatch case, DD2 $M=(1.50+1.50)M_{\odot }$ represents an intermediate case and SLy4 $M=(1.364+1.364)M_{\odot }$ is the worst mismatch case. Right panel: Corresponding spectra from 400 Hz to 4 kHz with sources located at 40 Mpc and analytical power spectral densities of LIGO design [83, 84, 85] and Einstein Telescope [91, 92].

Figure 6

Mismatches between hybrid waveforms (TEOBResumS+NR) and the complete model TEOBResumS_NRPM as a function of lower cutoff frequency $f_{\min }\in [50\mathrm{Hz},f_{\mathrm{mrg}}]$. The latter quantity is taken from the NR fits.

Figure 7

Marginalized posterior distributions of $f_2$ for three injected cases at different SNRs: the first case, $\mathrm{BHB}\mathrm{\Lambda }\varphi$ $M=(1.25+1.25)M_{\odot }$, is a case where the peak frequency is well recovered and this is also supported by the low mismatch between the NRPM model and the injected signal. In the second case, DD2 $M=(1.50+1.50)M_{\odot }$, we can see that for high SNRs biases appear systematically and the recovered peak is below the injected one. The third case, SLy4 $M=(1.364+1.364)M_{\odot }$, shows a bimodal distribution: a dominant peak appears at frequency $\sim 5.2\mathrm{kHz}$ (beyond the Nyquist limit, not in the plot) while the secondary peak is close to the injected value. The primary peak is compatible with the frequency $f_{2–0}$ aliased at high frequencies.

Figure 8

Characteristic postmerger frequency ${\widehat{f}}_2$ against ${\widehat{R}}_{\max }$ extracted from NR data for different EOS. The black solid line represents the fit with its 90% credible region. The right panel shows the marginal posterior distributions of ${\widehat{f}}_2$ for three selected injections while the top panel shows the respective ${\widehat{R}}_{\max }$ marginal distributions.

Figure 9

Binary neutron stars described by the $\mathrm{BHB}\mathrm{\Lambda }\varphi$ and the DD2 EOS and simulated signals [58]. Top: Mass of individual spherical equilibrium NS as a function of the central density. Markers refer to simulated BNS. Bottom: Real part of the (2, 2) waveforms for BNSs with mass $M=(1.50+1.50)M_{\odot }$ and $M=(1.25+1.25)M_{\odot }$.

Figure 10

Inference of EOS properties at extreme densities. Left panel: Marginalized posterior distributions of $f_2$ and ${\kappa }_2^{\mathrm{T}}$ for the low-mass cases (SNR 11 and 16). The postmerger posteriors agree with the value predicted by the fit and with the measurement from the inspiral. Right panel: Marginalized posterior distributions of $f_2$ and ${\kappa }_2^{\mathrm{T}}$ for the high-mass cases (SNR 11 and higher). The panels also shows $f_2({\kappa }_2^{\mathrm{T}})$ fits related to the injected values with the associated 90% credible regions. The uncertainties associated to the injected $f_2$ are the widths of the relative peaks in the frequency domain.

Figure 11

Characteristic amplitude and time information from NR simulations. Markers represent the quantities extracted from the NR data; the black lines are the fits with their 90% credible regions. All upper panels show the same data; the colors on the left panel correspond to the EOS variation, and on the right panel to the mass ratio. Note that we impose a lower bound for ${\widehat{A}}_0$ equal to zero for all those values of $\xi$ that lead to negative results in the fits.

Figure 12

Postmerger frequencies $f_2$ from the CoRe database (gray crosses) and the SACRA catalog [106, 107] (colored dots), averaged on different resolutions. The black solid line is the quasiuniversal relation for ${\widehat{f}}_2$ extracted from CoRe data with its 90% credible region.

Figure 13

Dependence of NR waveform on the grid resolution for the simulation SLy4 $M=(1.30+1.30)M_{\odot }$. VLR, LR, SR, HR stand respectively for maximal resolutions $h=[0.415,0.246,0.185,0.136]\mathrm{km}$ in each direction.