Inferring the post-merger gravitational wave emission from binary neutron star coalescences

We present a robust method to characterize the gravitational wave emission from the remnant of a neutron star coalescence. Our approach makes only minimal assumptions about the morphology of the signal and provides a full posterior probability distribution of the underlying waveform. We apply our method on simulated data from a network of advanced ground-based detectors and demonstrate the gravitational wave signal reconstruction. We study the reconstruction quality for different binary configurations and equations of state for the colliding neutron stars. We show how our method can be used to constrain the yet-uncertain equation of state of neutron star matter. The constraints on the equation of state we derive are complementary to measurements of the tidal deformation of the colliding neutron stars during the late inspiral phase. In the case of nondetection of a post-merger signal following a binary neutron star inspiral, we show that we can place upper limits on the energy emitted.

Figure 1

Spectrum for a GW emitted during the coalescence of two nonspinning NSs with masses $1.35M_{\odot }$ at a fiducial distance of 20 Mpc, optimally oriented and with the DD2 EoS. We show the premerger point-particle phase (blue), the full simulation starting at 1 kHz (green), the post-merger phase only (red), and the expected detector sensitivity (black). Both the full simulation and the post-merger spectrum exhibit the characteristic dominant peak at about $f_{\mathrm{peak}}=2586\mathrm{Hz}$.

Figure 2

Injected and reconstructed whitened time-domain data (top) and spectrum (bottom) for a signal produced by two nonspinning, $(1.35,1.35)M_{\odot }$ NSs with the DD2 EoS [46] at a post-merger SNR of 5-corresponding roughly to a distance of 20 Mpc for an optimally oriented source-as observed by the Hanford detector. The shaded region denotes the 90% CI of the reconstruction. The dashed line in the bottom panel is the detector sensitivity. The BayesWave reconstruction is able to capture the main features of the signal including the post-merger spectrum peak.

Figure 3

Histogram of the number of wavelets BayesWave used for the signal reconstructions of Fig. 2. BayesWave uses model selection to determine the most probable number of wavelets.

Figure 4

Overlap posterior density function for NL3 (top), DD2 (middle), and SFHO (bottom) for different post-merger SNR values. As the SNR of the injected signal increases, BayesWave achieves more faithful reconstructions of the signal.

Figure 5

Peak frequency posterior density function for NL3 (top), DD2 (middle), and SFHO (bottom) for different post-merger SNR values. The vertical line denotes the correct (injected) value. At low SNR, the posterior for the peak frequency is uninformative and similar to the prior. As the SNR increases, BayesWave achieves a more accurate reconstruction of the signal, and the posterior peaks at the correct value for $f_{\mathrm{peak}}$.

Figure 6

EoS-independent relation between the frequency of the post-merger spectrum peak and the radius of a $1.6M_{\odot }$ nonrotating NS for different total masses. The symbols are data calculated from numerical simulations of merging NSs with different total masses; each color represents the fit for data of the same total mass, the black line is the median of each fit, and the shaded regions denote the 50% (dark colored) and 90% (light colored) CIs of the fit.

Figure 7

Radius posterior density function for NL3 (top), DD2 (middle), and SFHO (bottom) for different post-merger SNR values. The shaded posteriors are calculated with the maximum likelihood fit to the $f_{\mathrm{peak}}/M-R_{1.6}$ relation shown in Fig. 6. The nonfilled dashed posteriors are obtained by marginalizing over the uncertainty in the $f_{\mathrm{peak}}/M-R_{1.6}$ relation including the total mass uncertainty (red dashed) and fixing the total mass to its injected value (blue dashed).

Figure 8

SED posterior for the same system as Fig. 2 as a function of the frequency. The shaded regions denote 50% and 90% CIs of the posterior.

Figure 9

Energy posterior density function for NL3 (top), DD2 (middle), and SFHO (bottom) for different post-merger SNR values. The vertical black line denotes the true injected energy. When the SNR is low and the signal is not reconstructed by BayesWave, the energy posterior can be used to place upper bounds on the energy emitted.

Figure 10

Energy posterior density function for NL3. We plot the energy posteriors for injections for which the signal was not reconstructed. The solid vertical line is the value of the injected energy, while the dotted vertical lines are the 95% Bayesian UL obtained from each injection. This UL can be used to place astrophysically interesting bounds on the energy emitted in the case of a nondetection of a post-merger signal from a confirmed BNS inspiral.

Figure 11

Median over the population median and 90% CIs for the overlap (left) and the error in the peak frequency (right) as a function of the SNR for 504 injections with DD2. The inset in the right panel demonstrates the peak frequency uncertainty at high SNR values for which the data are informative.

Figure 12

Median 90% CIs for the peak frequency as a function or the SNR. For $\mathrm{SNR}\lesssim 4$, BayesWave does not reconstruct the signal, and the posterior CI is equal to the prior CI. For $\mathrm{SNR}\gtrsim 4$, BayesWave reconstructs the signal and achieves the usual 1/SNR performance of matched-filtering analyses.