Measuring the Hubble constant with neutron star black hole mergers
The detection of GW170817 and the identification of its host galaxy have allowed for the first standard-siren measurement of the Hubble constant, with an uncertainty of . As more detections of binary neutron stars with redshift measurement are made, the uncertainty will shrink. The dominating factors will be the number of joint detections and the uncertainty on the luminosity distance of each event. Neutron star black hole mergers are also promising sources for advanced LIGO and Virgo. If the black hole spin induces precession of the orbital plane, the degeneracy between luminosity distance and the orbital inclination is broken, leading to a much better distance measurement. In addition neutron star black hole sources are observable to larger distances, owing to their higher mass. Neutron star black holes could also emit electromagnetic radiation: depending on the black hole spin and on the mass ratio, the neutron star can be tidally disrupted resulting in electromagnetic emission. We quantify the distance uncertainty for a wide range of black hole mass, spin and orientations and find that the 1-sigma statistical uncertainty can be up to a factor of better than for a non-spinning binary neutron star merger with the same signal-to-noise ratio. The better distance measurement, the larger gravitational-wave detectable volume, and the potentially bright electromagnetic emission, imply that spinning black hole neutron star binaries can be the optimal standard siren sources as long as their astrophysical rate is larger than Gpcyr, a value not excluded by current constraints.
The discovery of the binary neutron star (BNS) merger GW170817 and the kilonova AT 2017gfo has led to the first gravitational-wave standard siren measurement of the Hubble constant (Abbott et al., 2017a). In case of a positive redshift measurement with electromagnetic (EM) facilities, the uncertainty in the measurement of the Hubble constant is typically dominated by the precision with which the luminosity distance can be measured in the gravitational-wave (GW) sector. For example, for GW170817 Ref. (Abbott et al., 2017a) obtained Mpc (68.3% confidence interval), i.e. a relative 1- uncertainty of . That corresponded to a measurement of the Hubble constant of km s Mpc, that is a relative 1- uncertainty of . The rest of the error budget includes the uncertainty in the estimation of the peculiar velocity of the BNS host w.r.t. the Hubble flow.
One way to improve the measurement of the Hubble constant, is to build joint posteriors given many BNS mergers with host identification (Nissanke et al., 2013; Abbott et al., 2016a; Chen et al., 2017a). Other methods have been proposed, which do not necessary rely on the identification of the host. However, these methods typically require a larger number of detections to achieve a precision comparable to what can be done with a redshift measurement (Schutz, 1986; Holz and Hughes, 2005; Messenger and Read, 2012; Del Pozzo, 2012; Del Pozzo et al., 2017; Taylor et al., 2012).
The precision with which luminosity distance can be measured in BNS mergers is usually limited by the well-known degeneracy between orbital inclination and luminosity distance Abbott et al. (2016b). Previous work by many authors has shown how the luminosity distance uncertainty for a typical BNS event is of the order of a few tens of % (standard deviation) Vitale and Zanolin (2011); Rodriguez et al. (2014); Vitale and Del Pozzo (2014); Farr et al. (2016).
However, other potentially EM-bright type of mergers exist. For example, neutron star black hole (NSBH) mergers. Electromagnetic emission from NSBH mergers would be powered by tidal disruption of the neutron star. Whether tidal disruption happens is strongly dependent on the mass ratio of the system, the spin of black hole, and the neutron star equation-of-state. If the neutron star tidal disruption radius is larger than the innermost stable circular orbit of the system, the neutron star is tidally disrupted. Specifically, one would expect tidal disks when the mass ratio is smaller than a few to one and/or the black hole spin is large (Kyutoku et al., 2011; Foucart, 2012; Just et al., 2015; Fernández et al., 2015).
Black hole spins can break the degeneracy between luminosity distance and inclination, resulting in more precise constraints on the parameters of the source Vecchio (2004). For a canonical 10-1.4M NSBH the luminosity distance uncertainty can be a factor of few smaller than what achievable with a typical BNS (Vitale et al., 2014) .
In general, the luminosity distance of NSBH sources can be measured significantly better than for BNS. There are two main reasons why. On one side, the degeneracy between luminosity distance and inclination is only present in the inspiral phase of the GW signal, whereas accessing the merger and ringdown can help resolving it (McWilliams et al., 2011; Klein et al., 2016). As the merger frequency decreases for increasing total mass, the luminosity distance of NSBH sources can be measured better than that of BNS coalescences. On the other side, NSBHs can have significant spin precession, as long as the black hole spin is not negligible and is not aligned with the orbital angular momentum. Spin precession gives the waveform a characteristic amplitude (and phase, although not relevant for this work) modulation (Apostolatos et al., 1994), which significantly reduces the degeneracy with the inclination angle Vitale et al. (2014, 2017).
The fact that each NSBH can have its luminosity distance measured more precisely, yielding a better measurement, is compensated for by the fact that NSBH are expected to merge less often (Abbott et al., 2016c). In fact, no NSBH has been discovered to date, either in the EM or in the GW band. Non-detection of NSBH during LIGO’s first science run allowed estimation of their merger rate to be smaller than Gpcyr (Abbott et al., 2016c).
In this letter we explore the potential of NSBH as standard sirens for measurement. We simulate NSBH detections made by ground-based gravitational-wave detectors, considering various mass ratios, spins and orbital inclination angles, and systematically compare the resulting uncertainty in the inferred luminosity distance and hence Hubble constant with what could be obtained with a BNS of identical signal-to-noise ratio (SNR) at the same sky position. We find that NSBHs can potentially serve as a competitive standard sirens to BNS, yielding a more precise measurement if their merger rate is larger than Gpcyr, a value allowed by current constraints.
To verify how much better the luminosity distance can be measured for NSBH than for BNS we simulate sources of both classes and add them into “zero-noise” (which yields the same results that would be obtained averaging over many noise realizations (Vallisneri, 2008; Rodriguez et al., 2014)). We work with a network made by the two LIGO detectors and the Virgo detector. For all instruments, we used the design sensitivity (Abbott et al., 2013). This choice should not significantly affect our results, since we are mostly interested in the ratio of uncertainties for NSBH and BNS, which is not a strong function of the exact noise being used.
The details of EM emission from NSBH are not known. While it is highly likely that whether a system will be electromagnetic bright or not depends on the NSBH mass ratio, spin magnitude and spin tilt angle, the exact dependence is not known. We therefore do not restrict our analysis to a particular combination of mass and spins, but rather cover a large range of possibilities. We consider three different NSBH masses: M, M and M. We do not assign spin to the neutron star (consistent with the fact that known neutron stars have very small spins). If neutron stars turn out to be significantly spinning, that would actually improve the measurement of luminosity distance, by adding extra precession. For each system, we consider three possible orientations of the black hole spins. We will refer to the angle between the black hole spin and the orbital angular momentum as the tilt angle. In case of precession, both the spin vector and the orbital angular momentum precess around the total angular momentum (whose direction is nearly fixed in space (Zhao et al., 2017)). We quote tilt angles at a reference frequency of Hz, corresponding to our choice for the lower frequency of the gravitational-wave analysis. The three values we use are . A tilt angle of means that the spin vector is aligned with the orbital angular momentum. In this case no precession happens, and one would expect the degeneracy between distance and inclination to still be present. Conversely, implies maximum precession. The results we obtain for those two extreme cases will thus bracket what one can expect. For each of the tilt angles, we consider two possible values of the dimensionless BH spin magnitude (Abbott et al., 2016b): moderate () and large ().
Both BNS and NSBH signals are generated using the IMRPhenomPv2 waveform family (Schmidt et al., 2015; Hannam et al., 2014) and have a network SNR of 20. While GW170817 was much louder, with a network SNR of 32.4 Abbott et al. (2017b), an SNR of 20 is more representative of what will be typical (see Sec. IV for a discussion on how this choice might affect the results). We put all sources at the same sky position, near the maximum of LIGO’s antenna patterns, where we would expect the typical detection to be made Chen et al. (2017b). To check that the results we obtain are solid, we also considered a second sky position (near the north pole direction) and verified the main conclusions are the same. In what follows, we will thus only show plots obtained with sources near the maximum of LIGO’s antenna patterns.
The effects of spin precession in the detector frame are not only dependent on the actual degree of precession, but also on the inclination angle, defined in this work as the angle between the line of sight vector and the total angular momentum, Apostolatos et al. (1994). The effects of precession are more visible if the system is observed at inclinations close to (edge-on) Vitale et al. (2014, 2017).
To capture that dependence, we repeat all simulations at several values of inclination angle, uniformly spaced in . Critically, the SNR is always kept at the same value of 20. Therefore, when the inclination angle of a simulated source is varied, its true luminosity distance is also changed to yield the same SNR. This implies that all observable differences are uniquely due to the different morphology of the signals.
After the simulated signals are added into synthetic noise, we measure their parameters using stochastic sampling. Specifically, we use the nested sampling flavor of the LALInference suite Veitch et al. (2015). We use the Reduced Order Quadrature (ROQ) approximation to the likelihood Canizares et al. (2015) to speed up the computation. We notice that the ROQ method has only been tuned up to spin magnitude of 0.89 Smith et al. (2016), which explains our choice for the maximum spin. We also stress that the IMRPhenomPv2 waveform family does not contain higher order harmonics, which might play a role for large mass ratios. This choice is forced on us by the lack of fast-to-compute inspiral-merger-ringdown waveforms with precessing spins. While we do not expect the results to significantly change, this study should be repeated once more sophisticated waveform models are available. Similarly, we assume that the compact objects are in quasi-circular orbits, i.e. we neglect eventual eccentricity. This is a reasonable assumption since eccentricity is expected to be radiated away very quickly, circularizing the binary’s orbit Abbott et al. (2016d).
For the NSBH analysis we use the same prior shapes described by the LVC in Ref. Abbott et al. (2016b). The prior ranges on the component masses are given by Ref. Canizares et al. (2015). For the BNS events, we assume that the neutron stars have no spins, which significantly reduces the computational time. Similarly, we do not account the tidal deformability of the NS. Neither of this effect is expected to impact the measurability of the luminosity distance. In all analyses, we assume the sky position of the source is known, since we work under the assumption that an electromagnetic counterpart to the GW event is found, which provides the necessary redshift measurement. We marginalized over instrumental calibration errors with the same method used by the LIGO-Virgo collaborations (e.g. in Ref. Abbott et al. (2017b)) assuming gaussian priors on the calibration spline points with standard deviations of 3% for the amplitude and 1.5 degrees in phase, for all instruments. These are realistic estimates of what can be achieved by advanced detectors. LALInference provides high-dimensional posterior distribution for all the unknown parameters of the compact binary, including mass, spins, luminosity distance and inclination. The marginalized luminosity distance distribution can be used, together with the redshift inferred from the EM side to estimate (Schutz, 1986)
iii.1 Uncertainty for luminosity distance and inclination
In Figure 1 we plot the fractional 1- luminosity distance uncertainty relative to the true distance against the true value of the inclination angle for all the systems we simulated. Solid lines refer to systems without spins precession, and hence without amplitude modulation. Dashed lines are systems with tilt angle, whereas dotted lines are systems with . The color allows to distinguish the BNS (black) from the NSBHs with spin magnitude (green) or spin magnitude (blue).
Let us start by analyzing non-precessing systems. Figure 1 shows that the uncertainty steadily increases from face-on to inclinations quite close to edge-on. It is important to realize that this is the relative uncertainty. Since all the sources are kept at the same SNR, binaries at higher inclination angles have to be closer to yield an SNR of 20. This is why the uncertainty in Figure 1 goes up for non-precessing systems. The actual uncertainty is roughly constant, but the true luminosity distance gets smaller and smaller. For the BNS systems, the 1- uncertainty is roughly Mpc for inclinations in the range . When the true inclination is very close to perfectly edge on, both luminosity distance and inclination measurement get better (that is because the cross polarization of GW goes to zero when the system is edge-on, which breaks the degeneracy with the inclination Vitale et al. (2014)), and both true and relative distance uncertainty reach a minimum.
In case of precession, the relative 1- uncertainty can be a factor of smaller than what achievable with a BNS at the same position. The smallest uncertainties are obtained for the largest spins and tilt we considered (blue dotted). That is unsurprising: a large and misaligned BH spins results in a significant waveform amplitude modulation, which entirely breaks the degeneracy.
Similar conclusions apply to the measurement of the inclination angle itself, Figure 2, which could provide precious information to study the EM emission. For example, precise inclination measurement can resolve the BNS EM emission models (Mooley et al., 2018; Lazzati et al., 2017). NSBH EM emissions are expected to have stronger inclination angle dependency, and therefore better inclination measurement can be used to compare to the EM observation and lead to better understanding of the emission mechanism.
Once again, we start with the BNS and non-precessing NSBH. Here again we see that the 1- uncertainty is flat for inclination angles below , and is comparable for all systems at a value of . As the inclination gets closer to the uncertainty goes up at first, before significantly decreasing at angles extremely close to . This behavior can again be explained by a combination of priors and degeneracy between luminosity distance and inclination. For small to moderate () inclination angles, the posteriors on does not significantly change. For large inclination angles, a secondary posterior mode will start appearing at the true value, which result in the uncertainty to increase significantly. When the inclination is nearly , the posterior is unimodal, narrow, and centered at the true value. When precession is present, the uncertainty in goes monotonically down as the inclination approaches . The main difference between precessing and non-precessing signals is the uncertainty for systems close to face-on/off. For example, for systems with an inclination of , the BNS sources have uncertainty while NSBH can yield uncertainties as small as .
iii.2 Uncertainty in
In this work we assume that an EM counterpart is found to the GW event, which can provide a measurement of the redshift to the source. Furthermore, we assume that the Hubble velocity is perfectly measured from the redshift, in which case the uncertainty in the measurement of the luminosity distance can directly converted to the same relative uncertainty in the measurement of the Hubble constant. In practice, the Hubble velocity measurement is affected by uncertainties due to redshift calibration and peculiar velocity of the host galaxy, this will be discussed in Section IV. The uncertainty after combining N detections can be written as
where is the expected uncertainty for a single event, which we estimate from the simulations described in Section II. This means we are taking sources of SNR 20 as representative of the events used to calculate . While in reality sources with different SNRs will contribute to the measurement, our approach is appropriate to assess the relative precision achievable with NSBH and BNS.
As we have shown in Fig. 1, the uncertainty in the luminosity distance depends significantly on the inclination angle of the source, which cannot be directly averaged out, since GWs from face-on binaries are easier to detect than for edge-on binaries. This happens trivially because more GW energy is emitted along the direction of the final object’s angular momentum Maggiore, M. (2007). Once one folds in this selection effect, the resulting distribution for the inclination angle of detectable sources can be shown to follow a bimodal curve, with maxima at and and a local minima at . An analytical form for the expected distribution, which we use to weight events based on their probability of detection, is provided elsewhere Schutz (2011).
We can now check if and to which extent NSBH can contribute significantly to the measurement of . The answer will obviously depend on the number of NSBH detections, which in turns depends on the very poorly known astrophysical merger rates of NSBH Abbott et al. (2016c).
More specifically, the number of detections for each class of source can be written as , where is the astrophysical rate, is the redshifted volume Chen et al. (2017c), and is the observing time (factoring the duty cycle of the detectors).
We can thus write the uncertainty after combining all BNS detections made in the time period , and compare it to what doable with the NSBH detected in the same time:
Where the observing time cancels out. For both NSBH and BNS, we can calculate the redshifted volume using method described in Chen et al. (2017c). At this point, we can plot the ratio of uncertainty achievable with BNS and NSBH as a function of the relative astrophysical merger rate.
This is shown for the 10-1.4 M NSBH in Fig. 3. The different diagonal lines refers to various values of BH spin magnitude and orientation we have considered.
For example, if the merger rates of NSBH with BH with spin magnitude 0.5 and tilt are more than 1/50 of BNS astrophysical rate, then NSBH alone would yield a better constraint than what doable with BNS. If the NSBH population happens to have larger spins, or tilts, or both, fewer NSBH are required to achieve a precision comparable to BNS. In the best cases among those we considered, even if there is a single NSBH merger for every 100 BNS is enough. Conversely, in absence of spin precession the luminosity distance estimate of each NSBH source is only marginally better than for BNS, and an higher relative ratio is required to achieve equal precision. In this case the actual value of the spin magnitude is not very important, and for both the 0.5 and the 0.9 spin magnitude we obtain that more than 1 NSBH for every 10 BNS should exist to yield the same precision.
The vertical shaded area in Fig. 3 represents a likely range of relative merger rates. Those are obtained by taking the minimum, median and maximum projected NSBH rates from Ref. Abbott et al. (2016c) and the median BNS rate measured after the discovery of GW170817 Abbott et al. (2017b). The uncertainties are large for both class of sources, thus these lines should just be taken as an indication of what is possible. In particular, we see relative rates higher than 1 NSBH per 10 BNS are not excluded. Those rates would imply that measurement with NSBH only (no matter of their spin) is better than what doable with BNS.
Lower mass NSBH would require higher rates to achieve equal uncertainty. For example, for the M sources, at least 1 NSBH for every 7 BNS is required, independently on the spin magnitude and orientation.
The precision achievable in standard-siren cosmology is limited by the number of joint GW-EM detection, and by how well one can measure the luminosity distance of a the gravitational-wave sources. Whereas virtually all of the literature has considered binary neutron stars, in this paper we focussed on neutron star black hole mergers and on their potential as standard sirens.
We generated simulated NSBH sources with different mass ratio, spin magnitude and orientations, as well as a canonical 1.4-1.4 BNS for comparison. All sources are placed at distances which result in a network signal-to-noise ratio of 20. This allows for an apple-to-apple comparison, where all the differences in the quality of the parameters’ measurement depends on the morphology of the GW signal, including correlations among parameters. We do not expect our results to depend significantly on the choice of 20 as a “representative” SNR, since our main figure of merit is a ratio of uncertainties. In the limit where the uncertainty decreases with the SNR in the same way for BNS and NSBH, the results we presented in Fig. 3 will hold true. We assumed that the sources positions were known (through identification of the EM counterpart) and estimated the other parameters (including luminosity distance and inclination) using stochastic sampling. The uncertainties in the luminosity distance was then converted into an uncertainty in by averaging over the possible inclination angles, and taking into account that high (i.e. close to edge-on) systems are harder to detect in the GW band.
The main result we found is that inference of with NSBH can be better than with BNS systems, as long as the relative merger rate of NSBH and BNS is larger that 1/10 if all NSBH have aligned spins or 1/50 if significant spin precession is present. Both these values are still allowed by current estimate of the merger rates of BNS and NSBH (Fig. 3).
In what follows we list a few caveats and possible developments of this analysis.
The results presented in Fig. 3 assume that for all detectable NSBH an electromagnetic counterpart can be found, and hence that the probability of finding a counterpart does not significantly depend on the orbital orientation. In reality, since EM emission in NSBH is expected to be produced by equatorial tidal disks Kyutoku et al. (2015), the probability of detecting the EM counterpart could strongly depend on the orientation angle. As models are made available to calculate how the EM detectability depends on the inclination angle and spins, they can be folded in while weighting how systems at different inclination angles contributed to the measurement.
While the EM emission from NSBH is not yet fully understood, there exist models suggesting that large spin tilts can reduce the amount of ejecta Kawaguchi et al. (2016). This, of course, might reduce the fraction of NSBH sources than can contribute to the measurement. This is why we provide results for different values of spin magnitude and orientation, finding that they only differ by a factor of 5.
In this paper we only considered spin orientations which have positive projection on the orbital angular momentum. Negative projections could results in dimmer EM emission Shibata and Taniguchi (2011); Pannarale (2014), making it hard to find an EM counterpart for those NSBH. Nevertheless, the improved distance uncertainty might still reduce the number of potential host galaxies, which could allow to perform a statistical measurement of by considering all the galaxies inside the GW localization volume Schutz (1986).
One might expect that NSBH hosts are hard to localize given their smaller bandwidth Fairhurst (2011). While it is true that NSBHs will typically be localized to larger areas than BNSs, the main issue is whether the localization is so poor that the area cannot be covered by optical facilities. This will be less of a concern as the network of gravitational-wave detector expands with the inclusion of KAGRA in Japan Aso et al. (2013) and LIGO India Iyer et al. (2011). Even the LIGO-Virgo network can detect heavy sources to within a few tens of square degrees (see e.g. Fig. 5 of Ref. Vitale et al. (2017)) or smaller Chen and Holz (2016). This was shown by the BBH GW170814, which was localized within an area of , despite being sub-threshold in Virgo (SNR of Abbott et al. (2017c)). This is an area that can be comfortably covered by present (and future) optical facilities.
Another possible concern are waveform systematics. In particular, the waveform model we used does not include tidal effects, which are important in the last stages of the orbital evolution. In general, waveform modeling for NSBH sources is extremely complex and uncertain. However, the main result that can make NSBH competitive standard sirens is that their luminosity distance can be estimated precisely, due to spin precession. We expect this result to hold true even as more sophisticated waveform models are developed. This is because spin precession plays a major role at low frequency, when the orbital separation is large. Whereas the waveform models might improve at frequencies above a few hundred Hertz, at lower frequencies the current models are sufficient. Using a very different waveform family, which does not even model merger and ringdown, one can find improvements in the measurement of the luminosity distance similar to what we present here Vitale et al. (2014).
As mentioned above, a non negligible fraction of the total uncertainty for GW170817 came from the uncertainty on the peculiar velocity of the host galaxy relative to the Hubble flow. This will be less of a problem for NSBH sirens. The average redshift of M NSBH detected by advanced detectors at design sensitivity is , where the velocity of the Hubble flow is much larger. At that redshift a representative peculiar velocity uncertainty of km/s would contribute to of the uncertainty, which is significantly smaller than the uncertainty arising from the GW analysis. On the other hand, a milky way like galaxy would have an apparent magnitude of 15.5 at this distance, well within reach of many EM facilities. This would allow for a systematic follow-up of the host galaxy, if the EM counterpart is identified.
In conclusion, while significant uncertainties still exist on the actual merger rate of NSBH, and on their EM emission, NSBH have the potential to significantly contribute to the measurement of the Hubble constant.
Acknowledgements.We acknowledge valuable discussions with Jolien Creighton, Thomas Dent, Daniel Holz, Scott Hughes, Brian Metzger, Francesco Pannarale, Bernard Schutz, Licia Verde and John Veitch. SV acknowledges the support of the National Science Foundation and the LIGO Laboratory. LIGO was constructed by the California Institute of Technology and Massachusetts Institute of Technology with funding from the National Science Foundation and operates under cooperative agreement PHY-0757058. HYC was supported by the Black Hole Initiative at Harvard University, through a grant from the John Templeton Foundation. The authors would like to acknowledge the LIGO Data Grid clusters, without which the simulations could not have been performed. We are grateful for computational resources provided by Cardiff University, and funded by an STFC grant supporting UK Involvement in the Operation of Advanced LIGO.
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