Nonspinning black hole-neutron star mergers: A model for the amplitude of gravitational waveforms

Black hole-neutron star binary mergers display a much richer phenomenology than black hole-black hole mergers, even in the relatively simple case—considered in this paper—in which both the black hole and the neutron star are nonspinning. When the neutron star is tidally disrupted, the gravitational wave emission is radically different from the black hole-black hole case and it can be broadly classified in two groups, depending on the spatial extent of the disrupted material. We present a phenomenological model for the gravitational waveform amplitude in the frequency domain that encompasses the three possible outcomes of the merger: no tidal disruption, “mild,” and “strong” tidal disruption. The model is calibrated to general relativistic numerical simulations using piecewise polytropic neutron star equations of state. It should prove useful to extract information on the nuclear equation of state from future gravitational-wave observations, and also to obtain more accurate estimates of black hole-neutron star merger event rates in second- and third-generation interferometric gravitational-wave detectors. We plan to extend and improve the model as longer and more accurate gravitational waveforms become available, and we will make it publicly available online as a Mathematica package. We also present in the Appendix analytical fits of the projected KAGRA noise spectral density, which should be useful in data analysis applications.

Figure 1

GW amplitude of a BH-NS binary merger with a $1.35M_{\odot }$ NS with EOS B, a nonspinning BH, and a mass ratio $Q=5$ (Table , B-M135-Q5). The GW strain of the NR simulation (continuous grey) is compared to the BH-BH phenomenological model of 27 (dot-dashed blue), to the BH-NS model of 17 (dotted green), and to our phenomenological model (dashed red). The location of $m_0{\tilde{f}}_{\mathrm{RD}}$ is shown by a short, blue, vertical line in the top of the graph; the mass-shedding frequency $m_0{f}_{\mathrm{tide}}$ is higher than 0.1.

Figure 2

BH-NS merger case H-M135-Q5 (see Table ). The GW strain of the NR simulation (continuous grey) is compared to the BH-BH phenomenological model of 27 (dot-dashed blue), to the BH-NS model of 17 (dotted green), and to our BH-NS phenomenological model (dashed red). The location of the QNM frequency of the BH remnant is shown by the short, vertical, blue line in the top of the graph; the frequency at the onset of disruption frequency $m_0{f}_{\mathrm{tide}}$ is indicated by the short, vertical, dashed, red line.

Figure 3

BH-NS merger 2H-M135-Q5 (see Table ). In this plot (and in the remainder of this paper) we use the same color and linestyle conventions as in Fig. 2.

Figure 4

Values of $\alpha$ (the factor correcting ${\delta }_2$) versus $\nu$. The fitting function is of the form $w_{{\nu }_0,d_0}^{-}(\nu )+1$, with ${\nu }_0=0.146872$ and $d_0=0.0749456$, for $\nu \ge 5/36$, i.e. for $Q\le 5$. For $\nu <5/36$ we assume $\alpha$ to be constant and equal to $w_{{\nu }_0,d_0}^{-}(5/36)+1\simeq 1.35$.

Figure 5

Correction parameters appearing in our phenomenological models for nondisruptive mergers (left panels) and disruptive mergers (right panels). Top left: correction factor ${ϵ}_{\mathrm{tide}}$ to ${\delta }_1$ versus $x_{\mathrm{ND}}$, as defined in Eq. (29), for the six nondisruptive cases with $M_{\mathrm{NS}}=1.35M_{\odot }$ and ${\Gamma }_2=3.0$. The data points are fitted by the function $2w_{x_0,d_0}^{+}(x_{\mathrm{ND}})$, with $x_0=-0.088166$ and $d_0=0.066167$. Top right: correction factor ${ϵ}_{\mathrm{ins}}$ to ${\tilde{f}}_0$ versus $x_{\mathrm{D}}$, as defined in Eq. (35), for the six disruptive cases with $M_{\mathrm{NS}}=1.35M_{\odot }$ and ${\Gamma }_2=3.0$. The data points are fitted by the linear function $a{x}_{\mathrm{D}}+b$, with $a=-6.04599$ and $b=0.490086$. Bottom left: the additive correction ${\sigma }_{\mathrm{tide}}$ to $d$ versus $x_{\mathrm{ND}}$, as defined in Eq. (29), for the six nondisruptive cases with $M_{\mathrm{NS}}=1.35M_{\odot }$ and ${\Gamma }_2=3.0$. The data points are fitted by the function $2w_{x_0,d_0}^{-}(x_{\mathrm{ND}})$, with $x_0=-0.170321$ and $d_0=0.162074$. Bottom right: the additive correction ${\sigma }_{\mathrm{tide}}$ to $d$ versus the $x_{\mathrm{D}}$, as defined in Eq. (35), for the six nondisruptive cases with $M_{\mathrm{NS}}=1.35M_{\odot }$ and ${\Gamma }_2=3.0$. The data points are fitted by the function $A{w}_{x_0,d_0}^{-}(x_{\mathrm{D}})$, with $x_0=-0.0419235$, $d_0=0.0930419$, and $A=0.129459$.

Figure 6

The numerical simulations used to build the model (solid lines), discussed in Sec. 5a, are compared to our phenomenological amplitude model (dashed lines), to the phenomenological amplitude model of Lackey et al. [17] (dotted lines), and to the BH-BH model of Santamaría et al. [27] (dash-dotted line). Top left: Cases 2H-M135-Q4, H-M135-Q4, and B-M135-Q4 (from left to right, so that the rightmost model is the closest to the BH-BH case). Top right: Cases 2H-M135-Q3, H-M135-Q3, HB-M135-Q3, and B-M135-Q3. Bottom: Cases 2H-M135-Q2, H-M135-Q2, HB-M135-Q2, and B-M135-Q2.

Figure 7

Left: First set of numerical simulations used to test our model (cases 2H-M12-Q2, H-M12-Q2, HB-M12-Q2, and B-M12-Q2), as discussed in Sec. 5b. Right: Second set of numerical simulations used to test our model (cases 1.5H-M135-Q2, 1.5H-M135-Q3, and 1.5H-M135-Q5), as discussed in Sec. 5c; we obtain similar results for the two remaining cases (not shown in this plot to avoid cluttering), i.e. 1.5H-M135-Q4 and 1.25H-M135-Q2.

Figure 8

Top left: Cases 1.5Hl-M135-Q2, 1.25Hl-M135-Q2, Hl-M135-Q2, HBl-M135-Q2, and Bl-M135-Q2. Top right: Cases 1.5Hs-M135-Q2, 1.25Hs-M135-Q2, Hs-M135-Q2, HBs-M135-Q2, and Bs-M135-Q2. Bottom: Cases 1.5Hss-M135-Q2, 1.25Hss-M135-Q2, Hss-M135-Q2, HBss-M135-Q2, and Bss-M135-Q2.

Figure 9

The cutoff frequency $f_{\mathrm{Cut}}$, as defined in Eq. (44), computed with our BH-NS GW amplitude model. We consider the EOSs B, HB, H, 2H, and report contour lines in Hz, with a spacing of 250 Hz. The two white lines in each panel divide the plane in three regions: a top-right one in which the BH-NS coalescences are nondisruptive, a bottom-left one in which they are disruptive, and a middle region in which mildly disruptive coalescences occur. This classification is discussed in Sec. 4b.

Figure 10

Total noise data for KAGRA (dashed blue) and its best fit (continuous orange). The top panel refers to the variable KAGRA configuration in broadband mode (varBRSE); the middle panel refers to KAGRA in broadband mode and optimized for binary neutron star inspiral detection (maxBRSE); the bottom panel refers to the variable KAGRA configuration in detuned mode (varDRSE).