Gravitational wave spectroscopy of binary neutron star merger remnants with mode stacking

A binary neutron star coalescence event has recently been observed for the first time in gravitational waves, and many more detections are expected once current ground-based detectors begin operating at design sensitivity. As in the case of binary black holes, gravitational waves generated by binary neutron stars consist of inspiral, merger, and postmerger components. Detecting the latter is important because it encodes information about the nuclear equation of state in a regime that cannot be probed prior to merger. The postmerger signal, however, can only be expected to be measurable by current detectors for events closer than roughly ten megaparsecs, which given merger rate estimates implies a low probability of observation within the expected lifetime of these detectors. We carry out Monte Carlo simulations showing that the dominant postmerger signal (the $\ell =m=2$ mode) from individual binary neutron star mergers may not have a good chance of observation even with the most sensitive future ground-based gravitational wave detectors proposed so far (the Einstein Telescope and Cosmic Explorer, for certain equations of state, assuming a full year of operation, the latest merger rates, and a detection threshold corresponding to a signal-to-noise ratio of 5). For this reason, we propose two methods that stack the postmerger signal from multiple binary neutron star observations to boost the postmerger detection probability. The first method follows a commonly used practice of multiplying the Bayes factors of individual events. The second method relies on an assumption that the mode phase can be determined from the inspiral waveform, so that coherent mode stacking of the data from different events becomes possible. We find that both methods significantly improve the chances of detecting the dominant postmerger signal, making a detection very likely after a year of observation with Cosmic Explorer for certain equations of state. We also show that in terms of detection, coherent stacking is more efficient in accumulating confidence for the presence of postmerger oscillations in a signal than the first method. Moreover, assuming the postmerger signal is detected with Cosmic Explorer via stacking, we estimate through a Fisher analysis that the peak frequency can be measured to a statistical error of $\sim 4–20\mathrm{Hz}$ for certain equations of state. Such an error corresponds to a neutron star radius measurement to within $\sim 15–56\mathrm{m}$, a fractional relative error $\sim 4\%$, suggesting that systematic errors from theoretical modeling $(\gtrsim 100\mathrm{m}$) may dominate the error budget.

Figure 1

Histogram for single-event SNR for 100 realizations in the MC simulation. Orange bins represent the SNR with respect to the sensitivity of the ET (single detector). Blue bins are associated with the triple-detector, triangle design of the ET [56]. Green bins represent the SNR with respect to the sensitivity of the CE (wide-band configuration). The detection threshold ($\rho =5$) is indicated by the dashed red line. The TM1 EOS and 1-year observation is assumed, and the binary merger rate is taken to be $R_{\mathrm{BNS}}=1.54{\mathrm{Mpc}}^{-3}{\mathrm{Myr}}^{-1}$.

Figure 2

Histograms of the second and third loudest events with the CE sensitivity, from the same MC runs as in Fig. 1. Orange bins represent the SNR of the second loudest event, while blue bins represent the SNR of the third loudest event.

Figure 3

The same setting as in Fig. 1 but with different EOSs and with respect to the CE sensitivity alone. Orange bins represent the SNR for the SFHo EOS, blue bins for the LS220 EOS, green bins for the DD2 EOS and red bins for the Shen EOS.

Figure 4

Histogram for power stacked, coherently stacked and single-event $\alpha$ (a proxy for SNR, with $\alpha =1$ being the detection threshold), in each realization for the TM1 EOS. Orange bins represent the top event in each MC realization without stacking, with $79/100$ passing the detection threshold. Blue bins represent power stacking using the loudest 5 events, and demonstrate that all realizations pass the detection threshold. Green bins represent power stacking using the loudest 30 events, showing an improvement compared to power stacking 5 events. Pink bins represent coherently stacking the top 5 events with $C=2\pi$ in Eq. (42); all cases pass the detection threshold, and the skew of the distribution toward larger values of $\alpha$ indicates that coherently stacking the top 5 events is more efficient than power stacking the top 30 events.

Figure 5

Vertical axis: The improvement factor of effective SNR for the power stacked signal ($N=30$) over the best single event in each MC realization with the TM1 EOS. Horizontal axis: The ratio of $\rho$ between the second best event and the best event in each MC realization.

Figure 6

Number of identical events needed to satisfy the detection threshold, as a function of single-event SNR $\rho$, with the false alarm rate $P_f=0.01$ and the detection probability $P_d=0.982$. The solid (dashed) blue line represents the requirement for power stacking with (without) the Gaussian approximation. The solid (dashed) red line represents the requirement for coherent mode stacking with (without) phase alignment uncertainty considered. Note that coherent stacking is always more efficient than power stacking (fewer events are needed to cross above the detection threshold).

Figure 7

Average (log of) Bayes factor for different MC realizations, testing EOS TM1 against DD2. Blue bins represent the stacked signals with the loudest 30 events according to Eq. (55). There are 79 realizations where their best events individually pass the detection threshold, and their corresponding Bayes factors are shown in orange bins.

Figure 8

Histogram of $\delta f_{\mathrm{peak}}$ for different MC realizations, assuming the TM1 EOS and a $1.35+1.35M_{\odot }$ BNS merger remnant with corresponding $f_{\mathrm{peak}}=2.3\mathrm{kHz}$. The blue bins correspond to the power stacked signal using the loudest 30 events from each of the 100 realizations that pass the detection threshold, while the orange bins correspond to the 79 individual events that pass the detection threshold. A $\delta f_{\mathrm{peak}}\sim 8\mathrm{Hz}$ measurement roughly corresponds to a $\sim 26\mathrm{m}$ statistical error in the determination of the radius of a NS with mass $1.6M_{\odot }$. However, systematic errors in the universal $f_{\mathrm{peak}}-R_{1.6M_{\odot }}$ relation used in the mapping could be larger than $\sim 100\mathrm{m}$, so these errors would dominate over the statistical measurement error.

Figure 9

Characteristic strain $h_c$ vs frequency for the dominant peak in BNS postmerger gravitational wave spectra and for various equations of state considered in this work. The luminosity distance to the source is set to 50 Mpc. Solid lines correspond to Eq. (1) with the values for the amplitude and quality factor listed in Table 1. Dashed lines correspond to the postmerger spectra obtained from numerical relativity simulations of [13, 70, 71].