Extracting equation of state parameters from black hole-neutron star mergers: Aligned-spin black holes and a preliminary waveform model

Information about the neutron-star equation of state is encoded in the waveform of a black hole-neutron star system through tidal interactions and the possible tidal disruption of the neutron star. During the inspiral this information depends on the tidal deformability $\mathrm{\Lambda }$ of the neutron star, and we find that the best-measured parameter during the merger and ringdown is consistent with $\mathrm{\Lambda }$ as well. We performed 134 simulations where we systematically varied the equation of state as well as the mass ratio, neutron star mass, and aligned spin of the black hole. Using these simulations we develop an analytic representation of the full inspiral-merger-ringdown waveform calibrated to these numerical waveforms; we use this analytic waveform and a Fisher matrix analysis to estimate the accuracy to which $\mathrm{\Lambda }$ can be measured with gravitational-wave detectors. We find that although the inspiral tidal signal is small, coherently combining this signal with the merger-ringdown matter effect improves the measurability of $\mathrm{\Lambda }$ by a factor of $\sim 3$ over using just the merger-ringdown matter effect alone. However, incorporating correlations between all the waveform parameters then decreases the measurability of $\mathrm{\Lambda }$ by a factor of $\sim 3$. The uncertainty in $\mathrm{\Lambda }$ increases with the mass ratio, but decreases as the black hole spin increases. Overall, a single Advanced LIGO detector can only marginally measure $\mathrm{\Lambda }$ for mass ratios $Q=2–5$, black hole spins $J_{\mathrm{BH}}/M_{\mathrm{BH}}^2=-0.5–0.75$, and neutron star masses $M_{\mathrm{NS}}=1.2M_{\odot }–1.45M_{\odot }$ at an optimally oriented distance of 100 Mpc. For the proposed Einstein Telescope, however, the uncertainty in $\mathrm{\Lambda }$ is an order of magnitude smaller.

Figure 1

The 21 EOSs used in the simulations are represented by blue points in the parameter space. For a NS of mass $1.35M_{\odot }$, NS radius contours are solid blue and tidal deformability contours are dashed red. Also shown are two dotted contours of maximum NS mass. EOS parameters in the shaded region do not allow a $1.35M_{\odot }$ NS.

Figure 2

Polarization $h_{+}$ evaluated on the orbital axis for two waveforms from numerical BHNS simulations that differ only in their EOS. Also shown in black are the waveform amplitudes $|h_{+}-i{h}_{\times }|$.

Figure 3

Left panel: Amplitude $|\tilde{h}|$ of the Fourier transform of BHNS waveforms for ${\chi }_{\mathrm{BH}}=0.5$, $Q=3$, and $M_{\mathrm{NS}}=1.35M_{\odot }$ and 21 EOSs. Colors indicate the value of $\log (p_1)$ while the line style indicates the value of $\mathrm{\Gamma }$. Also shown are two analytic approximations to the BBH waveform with the same values of ${\chi }_{\mathrm{BH}}$ and $Q$: PhenomC and EOB (discussed in Sec. 3). Right panel: Phase of the Fourier-transformed waveform relative to the PhenomC BBH waveform $\mathrm{\Delta }\mathrm{\Phi }=\mathrm{\Phi }-{\mathrm{\Phi }}_{\text{Phen}}$. The curves are truncated when the amplitude $D_{\text{eff}}|\tilde{h}|/M$ drops below 0.1, and numerical error begins to dominate. The phase of the EOB BBH waveform is also shown relative to the PhenomC BBH waveform. In both figures the BHNS and EOB waveforms have been windowed, matched, and spliced to the tidally corrected PhenomC waveform as described in Sec. 3. The windowing width $\mathrm{\Delta }{t}_{\text{win}}=300M$; the start and end frequencies for matching, represented by solid vertical lines, are $M{f}_i=0.018$ and $M{f}_f=0.026$; and the splicing interval, represented by dotted vertical lines, is $M{s}_i=M{f}_i$ (overlapping with the solid line) and $M{s}_f=M{s}_i+0.001$.

Figure 4

Same as Fig. 3, but ${\chi }_{\mathrm{BH}}=-0.5$, $Q=2$, and $M_{\mathrm{NS}}=1.35M_{\odot }$.

Figure 5

Same as Fig. 3, but ${\chi }_{\mathrm{BH}}=0.5$, $Q=4$, and $M_{\mathrm{NS}}=1.35M_{\odot }$. During the ringdown, the PhenomC waveform amplitude is noticeably less than the BHNS waveform with the softest EOS ($p.3\mathrm{\Gamma }3.0$) around $Mf=0.1$. The EOB amplitude, however, is always greater than the BHNS amplitude.

Figure 6

BHNS waveforms with mass ratios of $Q=\{2,3,4,5\}$ with all other parameters fixed at ${\chi }_{\mathrm{BH}}=0$, $M_{\mathrm{NS}}=1.35M_{\odot }$, and the stiffest $\mathrm{EOS}=p.9\mathrm{\Gamma }3.0$. Left panel: Amplitude of BHNS waveform as well as PhenomC BBH waveforms (black dots), and EOB BBH waveforms (solid but same color) with the same values of $Q$ and ${\chi }_{\mathrm{BH}}$. The difference in amplitude between the BBH and BHNS waveforms decreases with the mass ratio $Q$. Right panel: The difference in phase between BHNS and PhenomC BBH waveforms also decreases with $Q$. The disagreement between the PhenomC and EOB waveforms generally increases with $Q$. Windowing, matching, and splicing are identical to Fig. 3.

Figure 7

Same as Fig. 6 except BH spin is varied instead of the mass ratio. Waveforms with spins of ${\chi }_{\mathrm{BH}}=\{-0.5,0,0.5,0.75\}$ are shown, and $Q=2$, $M_{\mathrm{NS}}=1.35M_{\odot }$, $\mathrm{EOS}=p.9\mathrm{\Gamma }3.0$ for all waveforms. In contrast with the mass ratio (Fig. 6), the departure from the BBH waveform increases as the spin increases. However, the effect is more moderate. EOB waveforms are also shown except for ${\chi }_{\mathrm{BH}}=0.75$ which is not available.

Figure 8

Amplitude $D_{\text{eff}}|\tilde{h}(Mf)|/M$ (left) and phase $\mathrm{\Phi }(Mf)$ (right) for a numerical BHNS waveform matched to the PhenomC BBH waveform with and without the tidal correction ${\psi }_T$ [Eq. (7)]. The waveform parameters are (${\chi }_{\mathrm{BH}}=0$, $Q=2$, $M_{\mathrm{NS}}=1.35M_{\odot }$, $\mathrm{EOS}=p.5\mathrm{\Gamma }3.0$). The matching window $f_i<f<f_f$ is bounded by solid vertical lines, and the splicing window $s_i<f<s_f$, which begins at $s_i=f_i$, is bounded by dotted vertical lines. Note that matching the numerical BHNS waveform to a BBH waveform without tidal corrections, as was done in Paper I, results in ignoring a phase term that accumulates linearly even after the matching region, and underestimates the effect of matter. We truncate the waveform when the amplitude drops below 0.1 denoted by a black dot.

Figure 9

Dependence of the hybrid waveform phase at $Mf=0.05$ on window width $\mathrm{\Delta }{t}_{\text{win}}/M$ and matching interval with width $\mathrm{\Delta }M{f}_{\text{match}}$ and midpoint $M{f}_{\text{mid}}$. Top panel: An EOB waveform, truncated to only include the last $800M$ before merger, is Fourier transformed then matched to the same EOB waveform that has not been truncated. The oscillations in the hybrid phase result from the Gibbs phenomenon. This can be partially suppressed by increasing the window width $\mathrm{\Delta }{t}_{\text{win}}/M$ as well as the width of the matching window $\mathrm{\Delta }M{f}_{\text{match}}$. Bottom panel: The effect of eccentricity can be evaluated by matching an eccentric EOB waveform to a long, zero-eccentricity EOB waveform. Here, an eccentric EOB waveform is generated by using the quasicircular initial conditions $M{\mathrm{\Omega }}_0=0.028$. The dependence of the hybrid phase on the matching interval can be reduced by using a larger matching window $\mathrm{\Delta }M{f}_{\text{match}}$. For both panels, ${\chi }_{\mathrm{BH}}=0$ and $Q=2$. Because the overall phase of the hybrid waveform is arbitrary, we have set it to 0 in this figure when $Mf=0.05$.

Figure 10

Dependence of the hybrid waveform phase at $Mf=0.05$ on window width $\mathrm{\Delta }{t}_{\text{win}}/M$ and matching interval as in Fig. 9. Top panel: The BHNS simulation with parameters $\{{\chi }_{\mathrm{BH}}=0,Q=2,M_{\mathrm{NS}}=1.35M_{\odot },\mathrm{EOS}=p.3\mathrm{\Gamma }2.4\}$ is matched to the PhenomC BBH waveform with tidal phase corrections. Middle panel: The BHNS simulation with parameters $\{{\chi }_{\mathrm{BH}}=0,Q=2,M_{\mathrm{NS}}=1.35M_{\odot },\mathrm{EOS}=p.9\mathrm{\Gamma }3.0\}$. Bottom panel: The relative contribution of the 1PN tidal phase correction to the leading-order tidal phase correction provides a crude estimate of the error in the tidal phase correction. For reference, dashed vertical lines give the ISCO frequency $1/(6^{3/2}\pi )$ as well as the innermost circular orbit for nonrotating black holes, defined by the minimum of the third post-Newtonian energy [Eq. (194) of Ref. 56], for mass ratios of $Q=2$ and 5.

Figure 11

Noise PSD for broadband aLIGO, ET-B, and ET-D. Also shown are the weighted amplitudes $2f^{1/2}|\tilde{h}|$ of phenomenological waveforms for four parameter values. For all waveforms, $M_{\mathrm{NS}}=1.35M_{\odot }$ and $D_{\text{eff}}=100\mathrm{Mpc}$. Black circles represent the start of the waveform fit at $Mf=0.01$.

Figure 12

An example of the coordinates $u$ and $v$ used to compute derivatives in Eqs. (29) and (30). At the origin of the $u-\nu$ coordinate system shown here, a $\log (p_1)=\text{const}$ contour is approximately aligned with a ${\mathrm{\Lambda }}^{1/5}=\text{const}$ contour. Fisher matrix components in $u-v$ coordinates involve derivatives that are approximated well by finite differences; in contrast, a component associated with the coordinates $\log (p_1)$ and $\mathrm{\Gamma }$ involves an inaccurate approximation of $\partial \tilde{h}/\partial \mathrm{\Gamma }$ by a difference of two nearly identical waveforms with the same $\log (p_1)$ but different $\mathrm{\Gamma }$.

Figure 13

$1\sigma$ error ellipses for the two-parameter Fisher matrix using the ET-D noise curve. We use hybrid waveforms where the numerical BHNS waveform is matched to a PhenomC BBH inspiral waveform with no tidal correction. The binary is optimally oriented and at a distance of 100 Mpc. Values of ${\chi }_{\mathrm{BH}}$, $Q$, and $M_{\mathrm{NS}}$ are listed in each panel. Evenly spaced contours of constant ${\mathrm{\Lambda }}^{1/5}$ are also shown. Each ellipse is centered on the estimated parameter ${\mathrm{\theta ⃗}}_A$ denoted by a $\times$. Top: The matching window has width $\mathrm{\Delta }M{f}_{\text{match}}=0.008$ and is centered on $M{f}_{\text{mid}}=0.016$. Middle: $\mathrm{\Delta }M{f}_{\text{match}}=0.008$ and $M{f}_{\text{mid}}=0.016$. Bottom: $\mathrm{\Delta }M{f}_{\text{match}}=0.008$ and $M{f}_{\text{mid}}=0.020$.

Figure 14

Same as Fig. 13, except the hybrid waveforms are generated by matching the numerical BHNS waveforms to PhenomC inspiral waveforms with the tidal correction ${\psi }_T$. For all panels, the matching window has width $\mathrm{\Delta }M{f}_{\text{match}}=0.008$ and is centered on $M{f}_{\text{mid}}=0.022$.

Figure 15

Left panel: Amplitude $|\tilde{h}|$ of the Fourier transform of BHNS waveforms for (${\chi }_{\mathrm{BH}}=0$, $Q=2$, $M_{\mathrm{NS}}=1.35M_{\odot }$) and for the four EOSs $p.3\mathrm{\Gamma }3.0$ (blue), $p.5\mathrm{\Gamma }3.0$ (red), $p.7\mathrm{\Gamma }3.0$ (orange), $p.9\mathrm{\Gamma }3.0$ (green). Numerical waveforms are solid, the fit based on the PhenomC BBH approximation is dotted, and the fit based on the EOB BBH approximation is dashed. Also shown are the PhenomC and EOB BBH approximations. Right panel: Phase of the Fourier-transformed waveform relative to the PhenomC BBH waveform. Matching and splicing intervals are the same as those in Fig. 3.

Figure 16

Same as Fig. 15 except for (${\chi }_{\mathrm{BH}}=0.5$, $Q=5$, $M_{\mathrm{NS}}=1.35M_{\odot }$). The three EOSs plotted are $p.3\mathrm{\Gamma }3.0$ (blue), $p.7\mathrm{\Gamma }3.0$ (red), $p.9\mathrm{\Gamma }3.0$ (green). The disagreement between the PhenomC and EOB models as well as the quality of the fits to the numerical waveforms is significantly worse than that in Fig. 15.

Figure 17

$1\sigma$ error ${\sigma }_{{\mathrm{\Lambda }}^{1/5}}$ for various values of the mass ratio, BH spin, and tidal deformability. The NS mass is fixed at $1.35M_{\odot }$. The noise curve is for broadband aLIGO. For ${\chi }_{\mathrm{BH}}=-0.5$, curves are not shown for $Q=4$ and 5 because these values have not been calibrated to numerical BHNS waveforms.

Figure 18

Same as Fig. 17, but with the ET-D noise curve. The uncertainty ${\sigma }_{{\mathrm{\Lambda }}^{1/5}}$ is an order of magnitude smaller.

Figure 19

Same as Fig. 17, but for ${\sigma }_{\mathrm{\Lambda }}$ instead of ${\sigma }_{{\mathrm{\Lambda }}^{1/5}}$.

Figure 20

Same as Fig. 18, but for ${\sigma }_{\mathrm{\Lambda }}$ instead of ${\sigma }_{{\mathrm{\Lambda }}^{1/5}}$.

Figure 21

Top: Waveforms (as in Fig. 2) for the parameters (${\chi }_{\mathrm{BH}}=0.75$, $Q=4$, $M_{\mathrm{NS}}=1.35M_{\odot }$, $\mathrm{EOS}=p.4\mathrm{\Gamma }3.0$). Three different resolution runs were performed with $N=\{36,42,50\}$, where the grid spacing is proportional to $1/N$ as defined in Ref. [35]. Simulations for all other parameters used $N=50$. Middle: Amplitude of the hybrid waveforms for the three resolutions. Also shown is the phenomenological fit based on the PhenomC waveform. Bottom: Phase of the hybrid waveforms as well as the phenomenological fit. There is a significant discrepancy in the phase between the fit and the hybrid waveform for these parameters which is much larger than the difference between the different resolution runs. Matching and splicing intervals are the same as those in Fig. 3.

Figure 22

Same as Fig. 21 but with the parameters (${\chi }_{\mathrm{BH}}=0.75$, $Q=5$, $M_{\mathrm{NS}}=1.35M_{\odot }$, $\mathrm{EOS}=p.9\mathrm{\Gamma }3.0$).