On the properties of the massive binary black hole merger GW170729

TL;DR

This paper provides a detailed analysis of GW170729, revealing improved mass and spin estimates through advanced waveform models, and confirms the robustness of the results against systematic errors and alternative formation scenarios.

Contribution

The study introduces improved waveform models with higher-order modes and compares different analysis methods, enhancing the understanding of GW170729's properties and formation history.

Findings

Higher-order modes improve mass ratio estimates.

Data is consistent with a smaller effective spin.

Waveform models accurately describe the signal.

Abstract

We present a detailed investigation into the properties of GW170729, the gravitational wave with the most massive and distant source confirmed to date. We employ an extensive set of waveform models, including new improved models that incorporate the effect of higher-order waveform modes which are particularly important for massive systems. We find no indication of spin-precession, but the inclusion of higher-order modes in the models results in an improved estimate for the mass ratio of $(0.3-0.8)$ at the 90\% credible level. Our updated measurement excludes equal masses at that level. We also find that models with higher-order modes lead to the data being more consistent with a smaller effective spin, with the probability that the effective spin is greater than zero being reduced from $99\%$ to $94\%$. The 90\% credible interval for the effective spin parameter is now $(-0.01-0.50)$. Additionally, the recovered signal-to-noise ratio increases by $\sim0.3$ units compared to analyses without higher-order modes. We study the effect of common spin priors on the derived spin and mass measurements, and observe small shifts in the spins, while the masses remain unaffected. We argue that our conclusions are robust against systematic errors in the waveform models. We also compare the above waveform-based analysis which employs compact-binary waveform models to a more flexible wavelet- and chirplet-based analysis. We find consistency between the two, with overlaps of $\sim 0.9$, typical of what is expected from simulations of signals similar to GW170729, confirming that the data are well-described by the existing waveform models. Finally, we study the possibility that the primary component of GW170729 was the remnant of a past merger of two black holes and find this scenario to be indistinguishable from the standard formation scenario.

Figures

Tables

Table 1. Table 1: List of waveform models we use to model the GW signal. The second and third columns indicate the spin dynamics and higher-order content ( ℓ , | m | ) ℓ 𝑚 (\ell,|m|) of each model in the coprecessing frame respectively. The fourth column gives the algorithm we use with each model. The bottom horizontal line separates the CBC-specific models and the morphology-independent models.

Waveform Model			Spin Dynamics			Modes ($\ell ,\left|m\right|$)			Algorithm
IMRPhenomPv2 Hannam et al. (2014)			Precessing			(2,2)			LALInference
IMRPhenomD Husa et al. (2016); Khan et al. (2016)			Aligned			(2,2)			LALInference
IMRPhenomHM London et al. (2018)			Aligned			(2,2),(2,1),(3,3),(3,2),(4,4),(4,3)			LALInference
SEOBNRv3 Taracchini et al. (2014); Pan et al. (2014)			Precessing			(2,2) (2,1)			LALInference
SEOBNRv4 Bohé et al. (2017)			Aligned			(2,2)			LALInference
SEOBNRv4HM Cotesta et al. (2018)			Aligned			(2,2), (2,1), (3,3), (4,4), (5,5)			LALInference
NR HM  Mroué et al. (2013); Jani et al. (2016); Healy et al. (2017)			Aligned			($\ell \le 4$,$\left|m\right|\le \ell$)			RIFT
NR/NRSur7dq2 HM  Blackman et al. (2015, 2017a, 2017b)			Aligned			($\ell \le 4$,$\left|m\right|\le \ell$)			RIFT
NR  Mroué et al. (2013); Jani et al. (2016); Healy et al. (2017)			Aligned			($\ell =2$,$\left|m\right|\le \ell$)			RIFT
NR/NRSur7dq2  Blackman et al. (2015, 2017a, 2017b)			Aligned			($\ell =2$,$\left|m\right|\le \ell$)			RIFT
Wavelets Cornish and Littenberg (2015)			Flexible			Flexible			BayesWave
Chirplets Millhouse et al. (2018)			Flexible			Flexible			BayesWave

Table 2. Table 2: Parameters of GW170729 obtained with various waveform models and two spin priors, uniform-in- χ 𝜒 \chi (labelled χ 𝜒 \chi ) and uniform-in- χ z subscript 𝜒 𝑧 \chi_{z} (labelled χ z subscript 𝜒 𝑧 \chi_{z} ). We quote median values and 90% credible intervals for the source-frame primary mass, the source-frame secondary mass, the source-frame total mass, the effective spin χ eff subscript 𝜒 eff \chi_{\rm{eff}} , and the effective precession parameter χ p subscript 𝜒 p \chi_{\rm{p}}   Schmidt  et al. ( 2015 ) . For the mass ratio we quote the median value and the 90% HPD. All masses are given in the source frame assuming the cosmological parameters of  Ade  et al. ( 2016 ) to convert luminosity distance to redshift. The effective precession parameter χ p subscript 𝜒 p \chi_{\rm{p}} is absent in the spin-aligned models.

Parameter	$m_1\left(M_{\odot }\right)$	$m_2\left(M_{\odot }\right)$	$M\left(M_{\odot }\right)$	$q$	${\chi }_{\mathrm{eff}}$	${\chi }_p$	SNR	$D_L\left(\text{Gpc}\right)$	$\left|\cos {\theta }_{JN}\right|$
IMRPhenomPv2 ($\chi$)	${51.0}_{-12.4}^{+13.9}$	${31.9}_{-9.6}^{+9.3}$	${83.7}_{-12.0}^{+13.0}$	${0.62}_{-0.23}^{+0.36}$	${0.35}_{-0.23}^{+0.22}$	${0.42}_{-0.29}^{+0.34}$	${10.7}_{-0.4}^{+0.4}$	${2.85}_{-1.28}^{+1.31}$	${0.83}_{-0.40}^{+0.17}$
IMRPhenomPv2 (${\chi }_z$)	${52.0}_{-11.6}^{+15.9}$	${33.2}_{-9.7}^{+9.9}$	${85.8}_{-12.4}^{+12.4}$	${0.64}_{-0.25}^{+0.34}$	${0.41}_{-0.21}^{+0.21}$	${0.58}_{-0.29}^{+0.29}$	${10.7}_{-0.4}^{+0.4}$	${2.96}_{-1.38}^{+1.30}$	${0.84}_{-0.42}^{+0.16}$
IMRPhenomD ($\chi$)	${50.5}_{-11.2}^{+13.5}$	${32.5}_{-8.8}^{+10.0}$	${82.8}_{-12.6}^{+13.4}$	${0.65}_{-0.23}^{+0.32}$	${0.34}_{-0.23}^{+0.21}$	-	${10.7}_{-0.4}^{+0.4}$	${2.75}_{-1.41}^{+1.26}$	${0.80}_{-0.46}^{+0.20}$
IMRPhenomD (${\chi }_z$)	${50.8}_{-11.5}^{+12.8}$	${34.5}_{-9.2}^{+9.7}$	${85.5}_{-11.4}^{+13.9}$	${0.68}_{-0.25}^{+0.29}$	${0.42}_{-0.21}^{+0.19}$	-	${10.8}_{-0.3}^{+0.4}$	${2.90}_{-1.48}^{+1.32}$	${0.79}_{-0.46}^{+0.21}$
IMRPhenomHM ($\chi$)	${57.0}_{-12.0}^{+12.6}$	${29.5}_{-9.0}^{+9.2}$	${86.0}_{-12.0}^{+12.2}$	${0.52}_{-0.21}^{+0.26}$	${0.27}_{-0.27}^{+0.23}$	-	${11.1}_{-0.4}^{+0.4}$	${2.15}_{-1.15}^{+1.19}$	${0.70}_{-0.42}^{+0.30}$
IMRPhenomHM (${\chi }_z$)	${58.1}_{-13.4}^{+12.5}$	${32.0}_{-8.9}^{+9.5}$	${89.7}_{-12.9}^{+13.4}$	${0.55}_{-0.21}^{+0.26}$	${0.36}_{-0.23}^{+0.21}$	-	${11.1}_{-0.4}^{+0.4}$	${2.30}_{-1.22}^{+1.24}$	${0.68}_{-0.46}^{+0.32}$
SEOBNRv3 ($\chi$)	${49.5}_{-10.8}^{+13.2}$	${35.3}_{-8.6}^{+8.9}$	${85.0}_{-12.8}^{+13.8}$	${0.72}_{-0.22}^{+0.28}$	${0.38}_{-0.22}^{+0.23}$	${0.45}_{-0.31}^{+0.32}$	${10.8}_{-0.4}^{+0.4}$	${2.86}_{-1.38}^{+1.38}$	${0.78}_{-0.47}^{+0.22}$
SEOBNRv4 ($\chi$)	${50.8}_{-11.8}^{+12.4}$	${33.4}_{-9.8}^{+9.6}$	${83.9}_{-13.3}^{+13.7}$	${0.66}_{-0.25}^{+0.30}$	${0.35}_{-0.24}^{+0.20}$	-	${10.8}_{-0.3}^{+0.4}$	${2.77}_{-1.45}^{+1.30}$	${0.79}_{-0.46}^{+0.21}$
SEOBNRv4 (${\chi }_z$)	${51.7}_{-12.6}^{+12.4}$	${35.2}_{-9.6}^{+9.4}$	${86.4}_{-12.7}^{+14.9}$	${0.68}_{-0.24}^{+0.29}$	${0.42}_{-0.26}^{+0.21}$	-	${10.8}_{-0.3}^{+0.4}$	${2.97}_{-1.43}^{+1.46}$	${0.79}_{-0.47}^{+0.21}$
SEOBNRv4HM ($\chi$)	${55.2}_{-12.2}^{+10.2}$	${29.8}_{-9.5}^{+9.9}$	${84.6}_{-11.3}^{+12.5}$	${0.54}_{-0.20}^{+0.31}$	${0.25}_{-0.26}^{+0.26}$	-	${11.0}_{-0.4}^{+0.4}$	${2.30}_{-1.17}^{+1.36}$	${0.72}_{-0.39}^{+0.28}$
SEOBNRv4HM (${\chi }_z$)	${54.8}_{-12.1}^{+11.1}$	${32.8}_{-9.2}^{+9.8}$	${87.5}_{-11.7}^{+12.2}$	${0.60}_{-0.21}^{+0.32}$	${0.34}_{-0.24}^{+0.26}$	-	${11.0}_{-0.4}^{+0.4}$	${2.66}_{-1.31}^{+1.38}$	${0.74}_{-0.41}^{+0.26}$

Table 3. Table 3: Parameters of GW170729 obtained with various waveform models and the 1g and 2g priors. We quote median values and 90% credible intervals for the primary mass, the secondary mass, the total mass, the effective spin χ eff subscript 𝜒 eff \chi_{\rm{eff}} , and the effective precession parameter χ p subscript 𝜒 p \chi_{\rm{p}} . For the mass ratio we quote the median value and the 90% HPD. All masses are given in the source frame. The effective precession parameter χ p subscript 𝜒 p \chi_{\rm{p}} is absent in the spin-aligned models.

Parameter	$m_1\left(M_{\odot }\right)$	$m_2\left(M_{\odot }\right)$	$M\left(M_{\odot }\right)$	$q$	${\chi }_{\mathrm{eff}}$	${\chi }_p$	SNR	$D_L\left(\text{Gpc}\right)$	$\left|\cos {\theta }_{JN}\right|$
IMRPhenomPv2 (1g)	${43.9}_{-6.7}^{+8.3}$	${38.0}_{-6.3}^{+7.0}$	${82.1}_{-11.6}^{+12.9}$	${0.88}_{-0.14}^{+0.12}$	${0.39}_{-0.22}^{+0.20}$	${0.47}_{-0.31}^{+0.31}$	${10.7}_{-0.4}^{+0.4}$	${3.08}_{-1.47}^{+1.17}$	${0.83}_{-0.42}^{+0.17}$
IMRPhenomPv2 (2g)	${55.1}_{-12.6}^{+11.3}$	${30.2}_{-7.2}^{+8.0}$	${85.5}_{-11.2}^{+12.1}$	${0.55}_{-0.20}^{+0.23}$	${0.37}_{-0.19}^{+0.18}$	${0.48}_{-0.29}^{+0.25}$	${10.7}_{-0.4}^{+0.4}$	${2.83}_{-1.31}^{+1.19}$	${0.83}_{-0.40}^{+0.17}$
IMRPhenomD (1g)	${43.8}_{-7.2}^{+7.9}$	${38.2}_{-6.4}^{+6.9}$	${81.9}_{-11.2}^{+13.9}$	${0.88}_{-0.13}^{+0.12}$	${0.38}_{-0.20}^{+0.18}$	-	${10.8}_{-0.4}^{+0.4}$	${2.99}_{-1.43}^{+1.26}$	${0.80}_{-0.44}^{+0.20}$
IMRPhenomD (2g)	${53.7}_{-11.6}^{+11.0}$	${30.2}_{-7.0}^{+7.9}$	${84.0}_{-11.8}^{+13.3}$	${0.56}_{-0.19}^{+0.21}$	${0.34}_{-0.20}^{+0.17}$	-	${10.8}_{-0.3}^{+0.4}$	${2.69}_{-1.31}^{+1.23}$	${0.79}_{-0.44}^{+0.21}$
IMRPhenomHM (1g)	${45.5}_{-7.9}^{+10.0}$	${37.9}_{-6.4}^{+6.7}$	${83.6}_{-11.7}^{+13.3}$	${0.84}_{-0.16}^{+0.16}$	${0.38}_{-0.20}^{+0.19}$	-	${10.9}_{-0.5}^{+0.4}$	${2.83}_{-1.41}^{+1.17}$	${0.80}_{-0.38}^{+0.20}$
IMRPhenomHM (2g)	${58.2}_{-10.1}^{+10.5}$	${29.7}_{-7.4}^{+7.1}$	${87.6}_{-11.2}^{+12.7}$	${0.51}_{-0.16}^{+0.17}$	${0.30}_{-0.21}^{+0.18}$	-	${11.1}_{-0.4}^{+0.4}$	${2.18}_{-1.05}^{+1.18}$	${0.69}_{-0.43}^{+0.31}$
SEOBNRv4 (1g)	${44.3}_{-7.2}^{+8.5}$	${38.7}_{-6.9}^{+7.0}$	${83.0}_{-11.6}^{+14.2}$	${0.88}_{-0.14}^{+0.12}$	${0.39}_{-0.20}^{+0.21}$	-	${10.8}_{-0.4}^{+0.4}$	${2.98}_{-1.50}^{+1.34}$	${0.79}_{-0.45}^{+0.21}$
SEOBNRv4 (2g)	${53.9}_{-11.1}^{+11.5}$	${31.2}_{-7.5}^{+8.3}$	${85.1}_{-11.4}^{+14.3}$	${0.57}_{-0.20}^{+0.21}$	${0.34}_{-0.20}^{+0.19}$	-	${10.8}_{-0.4}^{+0.4}$	${2.68}_{-1.36}^{+1.28}$	${0.78}_{-0.46}^{+0.22}$
SEOBNRv4HM (1g)	${45.1}_{-7.7}^{+8.4}$	${38.5}_{-6.3}^{+7.8}$	${83.7}_{-12.0}^{+14.3}$	${0.87}_{-0.14}^{+0.13}$	${0.38}_{-0.19}^{+0.20}$	-	${10.9}_{-0.4}^{+0.4}$	${2.94}_{-1.43}^{+1.35}$	${0.78}_{-0.47}^{+0.22}$
SEOBNRv4HM (2g)	${56.6}_{-9.9}^{+9.3}$	${30.4}_{-7.7}^{+6.2}$	${86.8}_{-11.0}^{+11.5}$	${0.54}_{-0.16}^{+0.16}$	${0.29}_{-0.18}^{+0.19}$	-	${11.1}_{-0.4}^{+0.4}$	${2.43}_{-1.04}^{+1.21}$	${0.72}_{-0.42}^{+0.28}$

Table 4. Table 4: Overlaps between the median BayesWave reconstruction and the maximum likelihood CBC waveform with different waveform models.

	Wavelets	Chirplets
IMRPhenomPv2	$0.88$	$0.90$
IMRPhenomD	$0.87$	$0.89$
IMRPhenomHM	$0.89$	$0.90$
SEOBNRv4	$0.87$	$0.90$
SEOBNRv4HM	$0.88$	$0.89$

Table 5. Table 5: Estimates for the parameters of GW170729 obtained with RIFT using various priors and waveform models. We quote median values and 90% credible intervals for the the effective spin and HPD for the mass ratio. We follow similar notation as Tables  2 and  3 .

Parameter	$q$	${\chi }_{\mathrm{eff}}$
SEOBNRv4HM ($\chi$)	${0.59}_{-0.24}^{+0.34}$	${0.29}_{-0.30}^{+0.25}$
NR/NRSur7dq2 ($\chi$)	${0.64}_{-0.24}^{+0.32}$	${0.33}_{-0.26}^{+0.22}$
NR/NRSur7dq2 (${\chi }_z$)	${0.67}_{-0.24}^{+0.30}$	${0.40}_{-0.24}^{+0.22}$
NR/NRSur7dq2 HM ($\chi$)	${0.58}_{-0.24}^{+0.34}$	${0.29}_{-0.28}^{+0.23}$
NR/NRSur7dq2 HM (${\chi }_z$)	${0.62}_{-0.23}^{+0.32}$	${0.37}_{-0.24}^{+0.25}$
NR/NRSur7dq2 (1g)	${0.91}_{-0.12}^{+0.08}$	${0.38}_{-0.25}^{+0.20}$
NR/NRSur7dq2 (2g)	${0.58}_{-0.20}^{+0.23}$	${0.34}_{-0.22}^{+0.19}$
NR/NRSur7dq2 HM (1g)	${0.91}_{-0.12}^{+0.09}$	${0.38}_{-0.23}^{+0.21}$
NR/NRSur7dq2 HM (2g)	${0.55}_{-0.19}^{+0.21}$	${0.31}_{-0.23}^{+0.21}$

Equations

$$p_{1g}(q)\propto {\left\{1+\exp \left[-k\left(q-q_0\right)\right]\right\}}^{-1},$$

$${\mathcal{O}}_N\equiv \frac{(h_1,h_2{)}_N}{\sqrt{(h_1,h_1{)}_N(h_2,h_2{)}_N}}$$

$$(h_1,h_2{)}_N=\sum _i^n(h_1^i,h_2^i)$$

$$(h_1^i,h_2^i)\equiv 4\Re {\int }_0^{\infty }\frac{{\tilde{h}}_1^i(f){\tilde{h}}_2^{i\ast }(f)}{S_n^i(f)}df.$$

Full text

I Introduction

GW170729 was observed on July 29, 2017 by the Advanced LIGO Aasi et al. (2015) twin detectors. Its detection was announced in Abbott et al. (2018a) as part of GWTC-1, the gravitational-wave (GW) transient catalog of compact binary coalescences (CBCs) LIGO Scientific Collaboration, Virgo Collaboration (2018). As reported in Abbott et al. (2018a), GW170729 was emitted during the coalescence of two stellar-mass black holes (BH). It was observed during the offline analysis of the detection pipelines GstLAL Messick et al. (2017) and PyCBC Usman et al. (2016); Nitz et al. (2017), that search for signals from CBC events, as well as cWB Klimenko et al. (2016), a pipeline tuned to search for transient signals whose frequency increases with time.

A number of reasons make GW170729 unique among the binary BHs (BBHs) presented in GWTC-1. At a measured source-frame total mass of $\sim 85M_{\odot}$ and a distance of $\sim 3$Gpc (median values), it is likely the most massive and distant BBH. Additionally, it is one of only two GW events that show evidence for nonzero spins with an effective spin of $(0.11-0.58)$ at the 90% credible level Abbott et al. (2018a). Finally, it is the only event for which the more flexible, non-CBC-specific cWB search returns a lower false alarm rate than the CBC-specific GstLAL and PyCBC template-based searches. In spite of these, Ref. Abbott et al. (2018b) concludes that GW170729 is consistent with the population of the other BBH detections 111An additional candidate claimed in Zackay et al. (2019), if confirmed, would also correspond to a binary with a non zero effective spin..

The fact that GW170729 is the most massive BBH found so far makes it a good candidate to observe the effects of higher-order waveform modes. The GW emission from BBHs can be described as a superposition of GW modes $h_{\ell,m}$ as $h\equiv h_{+}-ih_{\times}=\sum_{\ell,m}Y_{\ell,m}^{-2}(\iota,\varphi)h_{\ell,m}(t,\vec{\theta})$ Thorne (1980). Here $h_{+}$ and $h_{\times}$ are the two GW polarizations, $Y_{\ell,m}^{-2}$ denote spin-2 weighted spherical harmonics Goldberg et al. (1967), which depend on the location $(\iota,\varphi)$ of the observer around the binary, while the modes $h_{\ell,m}$ depend on the masses and spins of the binary, denoted by $\vec{\theta}$. During most of the inspiral stage, $h$ is dominated by the quadrupole modes, $({\ell,m})=(2,\pm 2)$. The rest, known as higher-order modes, grow in strength during the merger and ringdown stages, their impact being larger for highly asymmetric and nearly edge-on binaries Pan et al. (2011); Varma et al. (2014); Calderón Bustillo et al. (2017); Pekowsky et al. (2013); Varma and Ajith (2016); Calderón Bustillo et al. (2016); Graff et al. (2015). Finally, for more massive BBH systems the inspiral emission moves out of the sensitive band of advanced detectors, while sensitivity to the merger-ringdown increases and so does the impact of higher-order modes Van Den Broeck and Sengupta (2007); Calderón Bustillo et al. (2016); Graff et al. (2015).

Standard detection and parameter estimation of BBH events is usually performed using GW templates without the higher-order mode content of the signals Mukherjee et al. (2018); Dal Canton and Harry (2017); Abbott et al. (2016a, 2018a). Reference Abbott et al. (2018a) studied the fact that the event is recovered with higher significance by the flexible unmodeled, but less sensitive, search than the template-based searches Calderón Bustillo et al. (2018a). By performing injections of signals without higher-order modes or spin-precession, it was argued that the difference in the measured significances is in fact not unlikely. It was shown that $\sim 4\%$ of the injected signals were recovered with a higher significance from cWB than PyCBC.

At the same time, the presence of strong higher-order modes in the GW signal can potentially lead to biased parameter estimation if they are omitted in the waveform templates Littenberg et al. (2013); Varma and Ajith (2016, 2016); Calderón Bustillo et al. (2016); Graff et al. (2015). So far all reported events are consistent with nearly equal-mass, face-on BBHs, a fact that has prevented such biases, as shown in Abbott et al. (2016b, c); Kumar et al. (2018) for the case of events observed during the first observation run of Advanced LIGO. Even in this case, usage of models with higher-order modes can improve the accuracy of parameter estimation Littenberg et al. (2013); Graff et al. (2015); London et al. (2018); Calderón Bustillo et al. (2018b). Consequently, a reanalysis of GW150914 and GW170104 events using models with higher-order modes obtains modestly tighter parameter constraints with respect to previous analyses Kumar et al. (2018).

In this paper we present a detailed investigation into the properties of GW170729. We carry out a parameter estimation analysis similar to the one in Abbott et al. (2018a) in order to study the effect of spin-precession, higher-order modes, and spin priors on inferences drawn about GW170729. We make use of a more extended set of CBC waveform models belonging to three distinct waveform model families: phenomenological Ajith et al. (2007, 2011); Santamaria et al. (2010); Husa et al. (2016); Khan et al. (2016); London et al. (2018); Hannam et al. (2014), effective-one-body Taracchini et al. (2014); Bohé et al. (2017); Pan et al. (2014); Cotesta et al. (2018); Pan et al. (2011); Babak et al. (2017), and numerical relativity Mroué et al. (2013); Jani et al. (2016); Healy et al. (2017); Blackman et al. (2015, 2017a, 2017b). This set includes two new improved spin-aligned waveform models that include the effect of higher-order waveform modes. We gauge the importance of a physical phenomenon, namely spin-precession and higher-order modes, by comparing posterior densities for various source parameters obtained through analyses using waveform models with and without that physical phenomenon included.

While we find no indication of spin-precession, higher-order modes have a distinct impact on the posterior density for various source parameters. We find that CBC waveform models that include higher-order modes result in posterior distributions for the mass ratio of the binary that are shifted away from unity, resulting in more support for unequal masses than originally concluded in  Abbott et al. (2018a). In particular, we find a highest probability density (HPD) interval of the mass ratio of $(0.3-0.8)$ at the 90% credible level, while the corresponding upper limit of the HPD interval on the mass ratio without higher-order modes is $\sim 0.96$. This improved measurement, obtained using waveform models that include more physical effects, shows that GW170729 is not consistent with the merger of two equal-mass BHs at the $90\%$ level. At the same time, models with higher-order modes lead to marginally less support for positive effective spin $\chi_{\rm{eff}}$ 222The effective spin parameter is defined as the sum of the mass-weighted projections of the component spins along the orbital angular momentum and it is conserved to at least the second post-Newtonian order Racine (2008). and binary orientations where the orbital angular momentum points along or away from the line of sight. In particular, we find that the probability that the effective spin is positive is reduced from $99\%$ when higher-order modes are omitted to $94\%$ when they are included in the waveform models. We obtain consistent results when we use various CBC waveform models from different waveform families to describe the data. We thus argue that our conclusions are robust against systematic errors in the waveform models.

As the source-frame mass of the more massive BH is close to the proposed mass upper limit due to pair instability333It has been suggested that pulsational pair instability supernovae will result in no BH remnants with masses above $\sim 50M_{\odot}$ as the remnant is disrupted during the explosion, e.g. Woosley (2017); Marchant et al. (2018). Heger and Woosley (2002) and the posterior for the binary mass ratio favors unequal masses, we further investigate the possibility of second generation (2g) merger Gerosa and Berti (2017); Fishbach et al. (2017). In a 2g merger scenario, the primary BH is the remnant of an earlier BBH merger. As such, it is expected to be more massive than its companion in GW170729 resulting in unequal binary masses, and to have a relatively large spin magnitude. We contrast this scenario to a first generation (1g) merger scenario which favors comparable component masses. We reanalyze the data using two different priors tailored to 1g and 2g mergers and calculate the Bayes Factor (BF) of the 2g versus the 1g hypotheses. We find a BF of 4.7:1(1.4:1) in favor of the 2g scenario when using waveforms with (without) higher-order modes. This value favors the 2g model, but not decisively so, in agreement with the results of Abbott et al. (2018b). This question has been earlier and independently addressed by Kimball et al. Kimball et al. (2019) and our results are in agreement with their results.

Finally, we compare the signal reconstruction obtained with CBC waveform models to a morphology-independent reconstruction Cornish and Littenberg (2015). We quantify the consistency between the CBC and the generic reconstruction by computing the noise-weighted overlap between the two. We find broad consistency between the two with overlap values typical of what is expected for this mass range and signal strength Abbott et al. (2016d). This result is only minimally affected by the inclusion of higher-order modes.

Posterior samples from all our analyses are available in Chatziioannou et al. (2019). The rest of the paper presents our analysis and conclusions in detail. Section II describes the analysis we carry out including the CBC waveform models, the priors, and the generic analysis. Section III presents results derived under the CBC waveform models and posterior densities for the source parameters. Section IV gives the results of the generic analysis and how they compare to the CBC-specific analysis. Finally, Sec. V presents our main conclusions.

II Analysis

In this section we describe the details of the analysis we perform including the data, waveform models, and inference approaches we use. Our analysis follows closely and builds off of the work originally presented in Abbott et al. (2018a). We employ two complementary approaches: one is based on waveform models constructed specifically to describe compact binary coalescences, while the other uses a more flexible waveform model that can capture unexpected signal morphology. We describe both analyses in the following.

II.1 Data and setup

We use the publicly available LIGO and VIRGO strain data for GW170729 from the Gravitational Wave Open Science Center Gravitational Wave Open Science Center () (GWOSC); Vallisneri et al. (2015). The LIGO strain data have been post-processed to subtract several sources of instrumental noise Driggers et al. (2018); Davis et al. (2018) and calibrated as described in Abbott et al. (2018a). In particular we analyze $4$s of strain data centered at the GW170729 trigger time. The analysis covers a bandwidth from $f_{\rm low}=20$ Hz with the upper frequency cutoff set to $f_{\rm high}=1024$ Hz for waveform models without higher-order modes and $f_{\rm high}=2048$ Hz when higher-order modes are included. For masses typical of GW170729 (a detector-frame total mass of 120$M_{\odot}$ and mass ratio of 0.5) this upper frequency cutoff ensures that the analysis includes up to at least the $\ell=5$ ringdown harmonic of a compact binary merger, the highest frequency mode available in the waveform models we use.

We assume that the noise in the three detectors is Gaussian and stationary. The power spectral density (PSD) of the noise is obtained from the same $4$s of on-source data with the technique described in Littenberg and Cornish (2015). Specifically, a model consisting of a cubic spline and a number of Lorenzians is used to obtain posterior samples for the PSD from which a median PSD value is computed separately for each frequency bin. This median PSD is used in the estimation of the likelihood function in LALInference, RIFT, and BayesWave and we use the same PSD as Abbott et al. (2018a) that is publicly available in Power Spectral Densities release for GWTC-1() (PSD).

II.2 CBC waveform models

The top half of Table 1 lists the CBC waveform models we use; these models describe the inspiral, merger, and ringdown signal from the coalescence of two BHs as predicted by General Relativity (GR). All CBC waveforms we use can be divided into three main families: (i) phenomenological models (IMRPhenom), effective-one-body models (SEOBNR), and numerical relativity (NR). The first family is based on results of post-Newtonian theory Blanchet (2014) to compute the inspiral phase and a phenomenological approach to describe the merger, aided by calibration to EOB-NR hybrid waveforms Hannam et al. (2014); Husa et al. (2016); Khan et al. (2016). The second family uses the effective-one-body approach Buonanno and Damour (1999, 2000), which is based on a resummation of post-Newtonian results to describe the inspiral, and uses calibration to NR simulations for the late-inspiral and merger Zilhao and Loffler (2013); Loffler et al. (2012); Lovelace et al. (2011); Mroué et al. (2013). Both families describe the ringdown employing results of BH perturbation theory Barausse et al. (2012); Taracchini et al. (2014). The NR waveforms are obtained by solving the full non-linear Einstein equations and are subject only to numerical errors Centrella et al. (2010).

Besides the waveform family, models also differ on whether they include the effect of spin-precession Apostolatos et al. (1994); Kidder (1995) and higher-order modes, as indicated in Table 1. From the IMRPhenom and SEOBNR families we use one aligned-spin and one spin-precessing model without higher-order modes as well as a spin-aligned model with higher-order modes. None of the models from these two waveform families currently include both precessing spins and higher-order modes. In particular, from the IMRPhenom family, we use: the spin-precessing IMRPhenomPv2 Hannam et al. (2014) and the spin-aligned IMRPhenomD Husa et al. (2016); Khan et al. (2016) models. From the SEOBNR family we use: the spin-precessing SEOBNRv3444SEOBNRv3 includes the modes $(\ell,|m|)=(2,2),(2,1)$ in the coprecessing frame, the coordinate system for which the $z$-axis is instantaneously aligned with the Newtonian angular momentum, see Buonanno et al. (2003). Taracchini et al. (2014); Pan et al. (2014) and the spin-aligned SEOBNRv4555In particular, we use the reduced order model implementation of SEOBNRv4 which is computationally less expensive Pürrer (2014, 2016). Bohé et al. (2017) models. As far as higher-order modes are concerned, we use IMRPhenomHM London et al. (2018), a spin-aligned model of the IMRPhenom family that includes the $(\ell,|m|)=[(2,2),(2,1),(3,3),(3,2),(4,4),(4,3)]$ higher-order modes and SEOBNRv4HM Cotesta et al. (2018), a spin-aligned model of the SEOBNR family that includes the $(\ell,|m|)=[(2,1),(3,3),(4,4),(5,5)]$ higher-order modes. Posterior samples obtained with IMRPhenomPv2 and SEOBNRv3 have already been made publicly available by the LIGO-Virgo Collaborations and we use them directly Parameter estimation sample release for GWTC-1 .

The NR simulations we use include a total of 763 spin-aligned and 625 spin-precessing simulations Mroué et al. (2013); Jani et al. (2016); Healy et al. (2017). We optionally augment the list of NR simulations with waveforms computed using NRSur7dq2 Blackman et al. (2015, 2017a, 2017b), a surrogate model directly based on NR. The surrogate model we use is valid for mass ratio $0.5\leq q\leq 1$ and dimensionless spin magnitude $\chi\leq 0.8$; however, the analysis performed results in a full posterior due to the inclusion of the NR waveforms that cover the remaining region. For both NR-related analyses, we include results with higher-order modes $(\ell\leq 4,|m|\leq\ell)$ and with only the $(\ell=2,|m|\leq\ell)$ modes. When using the NR simulations we also assume that the spins remain aligned to the orbital angular momentum (no spin-precession).

II.3 Priors

Our analysis employs the following priors for the source parameters. The detector-frame component masses $m_{1},m_{2}$ are assumed to be uniform between $10M_{\odot}$ and $200M_{\odot}$ with $m_{1}>m_{2}$, while the mass ratio $q\equiv m_{2}/m_{1}$ is restricted to be above 0.125. The sky location and orientation of the binary, as well as the directions of the component spins are uniform on the unit sphere. The distance is uniform in volume with a maximum cut off of $7$Gpc, while the time and phase of arrival are uniform. We have verified that the mass and distance prior ranges encompass the entire region where the posterior distribution has non-negligible support.

For the magnitude of the dimensionless component spins $\chi_{i},\,i\in\{1,2\},$ we make different choices in order to investigate how this affects the posterior. The first prior is uniform-in-$\chi$ up to 0.99 for both spin-aligned and spin-precessing waveform models. The second prior is uniform-in-$\chi_{z}$, where $\chi_{z}$ is the spin projection along the axis perpendicular to the orbital plane, with the restriction that the spin magnitude is below 1. For spin-precessing model, the in-plane $\chi_{x}$ and $\chi_{y}$ components are also uniform; in that case this prior is sometimes referred to as ‘volumetric’ prior as it corresponds to the spin vector being uniformly distributed within the unit sphere.

Finally, we also use priors targeted toward CBCs where the primary component is the product of a past merger. We study two cases. The 1g case uses a spin prior that is uniform-in-$\chi$ for both component spins and a mass ratio prior that favors equal masses

[TABLE]

where $k=20$ and $q_{0}=0.8$. This prior choice was motivated in Vitale et al. (2017). In the case of a 2g merger, the primary is expected to be more massive than the secondary binary component. We use a Gaussian mass ratio prior with a mean of $0.5$ and a standard deviation of $0.2$, motivated by Fig. 2 of Gerosa and Berti (2017). In the 2g case, the primary is also expected to be spinning more rapidly. We therefore use a prior where $\chi_{1}$ is distributed according to a Gaussian centered at $0.7$ with a width of $0.1$ Gerosa and Berti (2017); Fishbach et al. (2017). The priors of secondary spin magnitude and both spin directions are uniform.

II.4 LALInference and RIFT

Given a waveform model and a set of prior choices, we compute the joint multidimensional posterior distribution of the source parameters. For fast-to-evaluate waveform models we use the publicly available software library LALInference Veitch et al. (2015); LIGO Scientific Collaboration and Virgo Collaboration (2017) to directly sample the posterior distribution. This approach computes the likelihood exactly at various points of the parameter space, but in order to obtain enough independent samples, millions of likelihood evaluation are required. This is prohibitive for models that are slow to evaluate, such as NR.

In these cases we use RIFT Pankow et al. (2015); Lange et al. (2018). RIFT’s three-stage algorithm first evaluates the likelihood on a dense grid; then approximates it via interpolation; and then uses Monte Carlo integration to produce the full posterior distribution. The number of grid points used for this particular NR-only analysis is 63,000, and the number of added surrogate points for the NR/NRSur7dq2 grid was 40,000; this brought the total number of points for the NR/NRSur7dq2 to 103,000. For context, RIFT in general calculates the marginalized likelihood on thousands grid points in parallel. For each marginalized likelihood, which has fixed intrinsic parameters, we evaluate the likelihood at $\approx 10^{6}$ different extrinsic parameters. Even though the number of evaluations are orders of magnitude larger than for LALInference, the overall wallclock time is considerably lower because the likelihood evaluations are faster and done in parallel, see Wysocki et al. (2019) for details. However, due to grid limitations (discreteness and limited range), RIFT does not sample both extrinsic and intrinsic parameters jointly from the full posterior distribution. Instead it marginalizes over all extrinsic parameters to calculate the likelihood and posterior for just the intrinsic parameters.

All results obtained using LALInference marginalize over the same detector calibration amplitude and phase uncertainty as in Abbott et al. (2018a) and publicly available in Calibration uncertainty envelope release for GWTC-1 using the method described in Abbott et al. (2016e); Farr et al. (2015). All RIFT results assume perfect calibration; this choice was shown to not affect the intrinsic binary parameters Vitale et al. (2012).

II.5 BayesWave

Finally, we also use a minimal-assumptions analysis that does not make use of CBC-specific waveform models. We use BayesWave Cornish and Littenberg (2015), a publicly-available algorithm LIGO Scientific Collaboration and Virgo Collaboration (2018) that does not explicitly assume that the signal is a CBC666While BayesWave does not assume an explicit signal morphology, it does assume that the signal is elliptically polarized, that it propagates at the speed of light, and that there is no phase decoherence during the propagation., and instead models it through a linear combination of basis functions, either sine Gaussian (known as Morlet Gabor) wavelets, or “chirplets” Millhouse et al. (2018), as listed in the bottom half of Table 1. The latter are sine Gaussians modified with a linearly evolving frequency. BayesWave relies on a transdimensional sampler GREEN (1995) to explore the multidimensional posterior of the parameters of the wavelets/chirplets (frequency, time, phase, amplitude, quality factor, and possibly the frequency derivative) as well as the number of wavelets/chirplets in the linear combination.

We then compare the signal reconstruction obtained with the morphology-independent models of BayesWave and with CBC waveform models. Broad agreement between the wavelet reconstruction and IMRPhenomPv2 was established in Abbott et al. (2018a) and we here perform the same test for waveform models that include higher-order modes. We also quantify the level of consistency through the detector network overlap Apostolatos (1995), defined as

[TABLE]

where $(h_{1},h_{2})_{N}$ denotes the inner product over the network defined by

[TABLE]

where $i$ sums over all the detectors in the network, and $(h_{1}^{i},h_{2}^{i})$ is the inner product in an individual detector defined by

[TABLE]

In the above, $\tilde{h}_{1}^{i}(f)$ denotes a signal reconstruction sample computed with CBC models and $\tilde{h}_{2}^{i}(f)$ is a reconstruction sample computed with BayesWave. Finally, $S_{n}^{i}(f)$ denotes the PSD of the detector. The superscipt $i$ denotes the quantities as they appear in the the $i^{\rm{th}}$ detector.

III CBC-model-based analysis

In this section we present results on inference with CBC waveform models. We study how the posteriors for the various source parameters are affected by the inclusion of spin-precession and higher-order modes in the waveforms. We also study the effect of spin priors and waveform systematics on the validity of our conclusions.

III.1 Higher-order modes

The importance of higher-order modes on a GW signal observed in the detectors depends on both extrinsic parameters, such as the inclination of the binary, and intrinsic parameters, such as the mass ratio. In general, signals from edge-on and asymmetric binaries include more power in higher-order modes. To study the effect of higher-order modes on GW170729 we analyze the data with waveform models both with and without higher-order modes. Table 2 gives the median and 90% symmetric credible interval and/or HPD (highest probability density interval) for various source parameters from multiple waveform models and the two spin priors.

We start by discussing the binary intrinsic parameters, in particular masses and spins. In Fig. 1 we show multi-dimensional corner plots for the posterior densities of the source-frame total mass $M$, the mass ratio $q$ and the effective spin parameter $\chi_{\rm{eff}}$ with waveform models of the IMRPhenom (left) and the SEOBNR family (right). For these figures and all the figures of this subsection we show results with the uniform-in-$\chi$ spin prior. Within each family, the spin-precessing model is given with a solid line, the spin-aligned model with a dashed line and the spin-aligned model with higher-order modes with a dotted line.

In all cases we find that the inclusion of higher-order modes does not have a large effect on the total mass measurement. The mass ratio of the system and the effective spin posteriors are both shifted. In particular we find that waveforms with higher-order modes consistently provide more support for unequal-mass systems and smaller effective spins. For the mass ratio we find that the 90% HPD interval is $(0.31-0.78)[(0.34-0.85)]$ with higher-order modes and $(0.42-0.97)[(0.41-0.96)]$ without them when using the spin-aligned IMRPhenomHM [SEOBNRv4HM] and IMRPhenomD [SEOBNRv4] models respectively. We conclude that GW170729 is not consistent with an equal-mass merger at the 90% level at least.

The combination of unchanged total mass but lower mass ratio means that the primary mass of GW170729 is inferred to be larger than previously measured. Reference Abbott et al. (2018b) studied the population of the 10 detected BBHs and concluded that no more than 1% of BHs in BBHs are expected to be above $45M_{\odot}$. We find that our updated primary mass measurement is not at odds with this conclusion. In particular, we find that the probability that $m_{1}$ is lower than $45M_{\odot}$ is 17% using IMRPhenomPv2 and reduced to 6% with IMRPhenomHM. Given that we have detected 10 BBHs, it is not unlikely that the true $m_{1}$ of one of them is at the sixth posterior percentile. A more detailed population analysis in needed to quantify this statement, but this is beyond the scope of this paper.

We also find that higher-order modes result in less support for a positive effective spin in GW170729. Reference Abbott et al. (2018a) reported that for GW170729 $\chi_{\rm{eff}}\sim(0.11-0.58)$ at the 90% credible level using combined posterior samples between IMRPhenomPv2 and SEOBNRv3. Interestingly, this credible interval does not include zero, suggesting that at the 90% level GW170729 has a nonzero effective spin. The inclusion of higher-order modes slightly changes this picture as we now find that the corresponding 90% credible intervals no longer exclude zero: $\chi_{\rm{eff}}\sim(-0.01-0.49)$ with IMRPhenomHM and $\chi_{\rm{eff}}\sim(-0.02-0.50)$ with SEOBNRv4HM. The effective spin parameter is still probably positive with the probability of $\chi_{\rm{eff}}>0$ being $94\%$ with higher-order modes, which is slightly reduced from the corresponding probability of $99\%$ when higher-order modes are not taken into account. Overall higher-order modes cause the 95% lower limit for $\chi_{\rm{eff}}$ to shift by $-0.10$ for the IMRPhenom family, $-0.13$ for the SEOBNR family.

Despite the broad consistency between results obtained with different waveform families, it is still possible that our results are partly affected by systematic uncertainties in the waveform models. To address this in Fig. 2 we plot again the mass ratio (left) and effective spin (right) posterior densities for the IMRPhenom and SEOBNR waveform models with (dashed lines) and without (dotted lines) higher-order modes. We find very small differences between both all the dotted and all the dashed lines and in particular between the new waveform models IMRPhenomHM and SEOBNRv4HM that include higher-order modes. More importantly, we find a clear separation between the dotted lines, i.e. the posteriors that include higher-order modes, and the dashed lines, i.e. the posteriors that do not include higher-order modes.

As a further test, in Fig. 3 we compare posteriors for $q$ and $\chi_{\rm{eff}}$ computed with SEOBNRv4HM (red line) with NR waveforms (magenta lines), as well as with NR/NRSur7dq2 (green lines). Compared to Fig. 1 we omit the total mass posterior as RIFT did not compute source-frame quantities. In order to perform a fair comparison, all posteriors have been computed with RIFT, while for technical reasons we cannot use IMRPhenomHM with RIFT. We find excellent agreement between NR with higher-order modes, NR/NRSur7dq2 with higher-order modes, and SEOBNRv4HM. This shows that SEOBNRv4HM is as accurate as NR waveforms in describing GW170729. Moreover, the agreement between SEOBNRv4HM and IMRPhenomHM in Fig. 2 suggests that the latter IMRPhenom waveform is also highly accurate for the event studied here. While these posteriors are broadly consistent with those obtained in Fig. 2, we find a disagreement in results obtained with LALInference and RIFT with the same waveform model for the mass ratio at the $7\%$ level. The nature of this difference and further numerical estimates are described in Appendix A. Overall, Figures 2 and 3 suggest that despite minor differences between the waveform models considered here, our main conclusions are robust.

We now turn to the binary extrinsic parameters. Figure 4 shows the matched-filter network signal-to-noise ratio (SNR) posterior density on the top and the two-dimensional posterior density for the luminosity distance and the inclination on the bottom. The network SNR is defined as the square root of the squared sums of the matched-filter SNR in each interferometer, calculated as $\rho=(h,d)/\sqrt{(h,h)}$, where $d$ is the data and $h$ the signal model. The inclination is defined as the angle between the total angular momentum vector of the binary, whose direction we treat as fixed, and the line of sight. We present results for waveform models of the IMRPhenom (blue) and the SEOBNR (red) families.

The SNR depends on both the intrinsic loudness of the data and the agreement between the signal and the template $h$. Since the data are common for all analyses, a larger SNR indicates a better agreement/overlap between the data and the template. While we find that within both families spin-precession has a minimal impact on the SNR, waveforms with higher-order modes report slightly larger SNR values, see the top panel of Fig. 4. This suggests that their inclusion leads to a marginally better fit of the data. The data, though, include both the GW170729 signal and a random realization of Gaussian noise, so a better fit of the data does not necessarily imply that waveforms with higher-order modes recover a larger fraction of the GW signal.

We quantify the impact of higher-order modes on GW170729 by computing the Bayes Factor in favor of IMRPhenomHM compared to IMRPhenomD. We find a BF of 5.1:1. While it favors the model with HM, this BF is consistent with the fact that the HM waveforms are able to extract marginally more SNR from the data777The BF is related to the SNR through $\log\rm{BF}\propto 1/2\,\rm{SNR}^{2}$. Therefore an SNR increase of $\sim 0.2$ compared to $\sim 10.8$ (see Table 2) would result in a BF of $\sim 8$. This suggests that the measured BF of $5$ is consistent with the SNR increase due to HM.. Moreover, we emphasize that the BF is not the same as the odds ratio in favor of higher-order modes, which quantifies our degree of belief that higher-order modes are present in the signal. The odds ratio is the BF times the prior odds in favor of the presence of higher-order modes. The latter is formally infinite within GR, as the theory of gravity unequivocally predicts that higher-order modes are present in all CBC signals. The BF presented here only quantifies if higher-order modes are a necessary feature of the models in order to describe the data, and not whether we believe that they exist in general.

Regarding the bottom panel of Fig. 4, we find that waveforms with higher-order modes result in less support for face-on/off binary orientations. This observation, coupled to the fact that we see more support for unequal masses and lower spins, see Fig. 1, suggests that higher-order modes lead to more support for sources that are intrinsically of lower amplitude888We have also verified this by computing the posterior of the intrinsic loudness (defined as the product of the SNR and the distance) with and without higher-order modes. We find that higher-order modes lead to larger probability for intrinsically quieter sources than the quadrupole templates.. This in turn leads to a posterior distribution for the luminosity distance that is shifted to lower values, as compared to analyses without higher-order modes, as also seen in the bottom panel of Fig 4.

Overall, we find that the inclusion of higher-order modes induces small but noticeable shifts in the parameter posteriors. Specifically, the matched-filter SNR increases, the mass ratio posterior obtains more support for unequal masses, and the effective spin parameter is more consistent with lower values; parameter measurements are given in Table 2. The general consistency between the waveform model families we study here shows that our conclusions are robust against waveform systematics.

III.2 Spin prior

Besides the CBC waveform models, posterior measurements are also affected by prior choices, in particular the spin prior Vitale et al. (2017). To test the effect of the spin prior, we reanalyze the data this time assuming a uniform-in-$\chi_{z}$ prior, where $\chi_{z}$ is the spin component perpendicular to the orbital plane. The results are presented in Fig. 5 for the mass ratio (left) and the effective spin (right) and for waveforms of the IMRPhenom family. We have verified that we obtain qualitatively similar results when using SEOBNR and NR waveforms models. Due to computational constraints we have only checked results with SEOBNRv3 and the uniform-in-$\chi$ prior, as computed in Abbott et al. (2018a).

We find that the spin prior has a minimal effect on the mass ratio posterior. This is expected as the correlation between mass ratio and effective spin is mostly present in the inspiral phase of a CBC. High-mass systems, such as GW170729 are instead dominated by the merger and ringdown in the LIGO sensitive frequency band. In this case little correlation exists between mass ratio and effective spin Ng et al. (2018), and changing the spin prior doesnt affect the mass ratio posterior. The effective spin parameter, on the contrary, is directly affected by the choice of the spin prior, and clear differences are visible. The uniform-in-$\chi_{z}$ prior favors larger spin magnitudes than the uniform-in-$\chi$ prior. As a result, the effective spin posterior is shifted to larger values. The median and 90% credible interval for the effective spin is $0.41^{+0.21}_{-0.21}$ under the uniform-in-$\chi_{z}$ prior and $0.35^{+0.22}_{-0.23}$ under the uniform-in-$\chi$ prior using the IMRPhenomPv2 waveform model. Additional spin measurements for other waveform models are presented in Table 2.

III.3 Second generation merger

Finally, we study the possibility of a 2g merger. In that scenario, GW170729 is created in a dense environment such as a nuclear or a globular cluster and its primary mass is the product of a previous merger of two BHs Gerosa and Berti (2017); Fishbach et al. (2017); Di Carlo et al. (2019). In that case, the mass ratio of the system is expected to be closer to 2:1 (as we find when we use waveform models with HM) and the spin magnitude of the primary should be close to 0.7 (the typical spin of the remnant BH after the merger of two equal-mass, nonspinning BHs) Gonzalez et al. (2007); Berti et al. (2007); Buonanno et al. (2008); Campanelli et al. (2006); Baker et al. (2004) 999Though a 2g merger scenario provides a simple way to produce a highly spinning BH, such systems could also be produced in certain astrophysical scenarios, e.g Bouffanais et al. (2019)..

We repeat our analysis with two more priors tailored to the cases of a 1g and a 2g merger scenario. Table 3 gives the median and 90% symmetric credible interval and/or HPD for various source parameters obtained with the 1g and 2g priors. Figure 6 shows the effect of this 2g prior on the mass ratio (left) and the effective spin (right) of the binary when using IMRPhenomD and IMRPhenomHM. We have verified that we obtain similar results with SEOBNR and NR waveform models. As expected from the priors we have selected, 2g runs show strong support for unequal masses. This support is even more evident for waveforms with higher-order modes, as anticipated from the results of the previous section. The effective spin parameter is similar, with 1g runs showing more support for nonzero binary components.

To further quantify the prior effect, we calculate the Kullback-Leibler (KL) divergence of posterior against prior Kullback and Leibler (1951). We find a KL divergence for the mass ratio in the 1g case that is $\sim 10$ times smaller than the KL divergence in 2g or default prior scenarios, implying that we do not extract much information by applying the 1g mass ratio prior. For the effective spin, we find that the KL divergence in both priors are comparable as expected because the prior on the effective spin does not change as drastically as the prior on the mass ratio.

We also compute the 2g-vs-1g BF. We find 4.7:1 (1.4:1) for waveforms with (without) higher-order modes. We conclude that there is not enough support for the hypothesis that the GW170729 primary needs to be the product of a past merger in order to explain the data’s properties. Note that both 2g and 1g models have the same number of parameters, but different distributions in mass ratio and primary spin. The resulting BFs are therefore only affected by how well each model fits the data.

IV Morphology-Independent analysis

The studies presented in the previous section relied on specific waveform models for the signal emitted during a CBC as predicted by GR. We here follow a more generic approach and analyze GW170729 in a morphology-independent way that does not explicitly assume it is a CBC described by the currently available waveform models. We use BayesWave to reconstruct the signal and then compare this reconstruction to the one obtained with CBC models. The comparison is shown in Fig. 7, where we plot the whitened strain as a function of time. At each time, the shaded region denotes the 90% credible interval of the reconstruction using CBC waveform models (blue) and BayesWave (orange). The left panel is obtained with the wavelets, while the right panel is obtained with chirplets. The top row is made with the CBC waveform model IMRPhenomPv2 which includes the effect of spin precession, but not higher-order modes. The bottom row uses IMRPhenomHM which assumes that the spins remain aligned with the orbital angular momentum, but includes higher-order modes.

As discussed in Abbott et al. (2018a), BayesWave can sometimes reconstruct features that are not present in the CBC reconstructions as can be seen in the right hand side panels of Fig. 7, for example around t=0.27s on the right panel. Unlike CBC model waveforms, wavelet-based models are not limited to a physically motivated waveform morphology. As a result, the BayesWave sampler can sometimes pick up random coherence between nearby noise samples. However, these outlying wavelets do not point to any potential additional features in the waveform. In fact, they are absent in the 50% credible intervals of the reconstruction, implying that they have a low significance. Similar outliers were observed in BayesWave analyses applied to simulated signals added to real data.

We find broad agreement between the CBC reconstruction and the BayesWave reconstruction in all cases. In particular, the 90% credible intervals obtained with the two methods overlap for all waveform models and BayesWave basis functions. The agreement suggests that the omission of higher-order modes does not degrade the reconstruction enough to leave a coherent residual detectable by BayesWave. This conclusion is in agreement with the results of the previous section, as well as the corresponding reconstruction plot in Abbott et al. (2018a).

In order to make this statement quantitative, we draw 1000 samples from the BayesWave posterior and compute their overlap with the maximum likelihood (ML) CBC waveform from the analyses using IMRPhenomPv2 and IMRPhenomHM. The posterior distribution for the overlap is given in Fig. 8 for wavelets (left) and chirplets (right). We find overlaps typically between $0.6-0.9$. This large spread in the overlaps is a result of the inherent flexibility in wavelet-based analyses causing a large variance in the reconstructed signal morphology. Therefore, unlike CBC waveform samples which are motivated by a physical theory, a single BayesWave sample is not constrained by physical reconstruction considerations, other than propagation at the speed of light. Instead, the median BayesWave waveform (MBW), defined as being the median across the sample waveforms at each time or frequency step, represents a collective estimate across samples, assumed to represent “the wisdom of the crowd”. The MBW is a stable estimate of the reconstruction as it is relatively immune to the stochastic fluctuations in the variable dimensional sampler. Each of the four vertical lines in Fig. 8 represent the overlap values between the MBW and the ML CBC waveform. They are summarized in Table 4 where we also show results obtained using SEOBNR waveform models. The ML CBC waveform has in general a higher agreement with the MBW waveform than with each of the individual samples.

We find that waveforms both with and without higher-order modes achieve large overlaps with the MBW reconstruction, in the range of $0.87-0.9$, consistent with expectations. In fact, Ref. Abbott et al. (2016d) showed that for masses and SNRs typical of GW170729 the expected overlap between simulated signals and their median reconstructions is in the $0.85-0.9$ range at the 1-$\sigma$ level, similar to what we obtain here. The small remaining disagreement between the ML CBC reconstruction and the MBW reconstruction is due to the fact that, unlike modeled analyses, BayesWave is only sensitive to excess signal that stands out and above the detector noise. This means that it is less sensitive than CBC analyses in the lower frequencies, $<40$ Hz. We find that if we increase the low frequency cut off in the overlap calculation to $40$ Hz, the overlaps improve by $\sim 0.07$ for each pair of BayesWave and CBC waveforms.

Besides the good reconstruction, we find that waveforms with higher-order modes lead to similar overlaps with the BayesWave reconstruction to waveforms without higher-order modes. We perform the Kolmogorov–Smirnov test for the two overlap distributions for each panel of Fig. 8 and find only 0.048 and 0.017, which implies that both IMRPhenomPv2 and IMRPhenomHM reconstruct the data comparably well. This confirms that GW170729 is consistent with a CBC and that the higher-order modes are not strong enough to lead to a degradation of the signal reconstruction if neglected.

V Conclusions

We analyze the publicly available strain data for GW170729, the highest mass and most distant confirmed GW detection by the LIGO and Virgo detectors. In particular we investigate the effect of higher-order modes and spin priors on the inference of the source parameters. We find that higher-order modes leave small but noticeable effects, while spin priors affect the spin measurements as anticipated.

We find that the inclusion of higher-order modes in the models leads to changes in the estimates for the mass ratio, the effective spin, and the SNR. Our updated parameter measurements imply decreased support for equal binary component masses and nonzero effective spin. In particular we conclude that the mass ratio is $(0.3-0.8)$ at the 90% credible level, a value that excludes equal masses. We also find that the 90% credible interval for the effective spin parameter has changed from $(0.11-0.58)$ as reported in Abbott et al. (2018a) to $(-0.01-0.50)$, which now marginally includes zero. The effective spin parameter still has a 94% probability of being positive.

Consistent with these findings, we compute the BF in favor of the presence of higher-order modes, and find it to be 5.1:1. Moreover, their omission does not dramatically change the measured parameters, which would happen if they were strong Varma and Ajith (2016); Calderón Bustillo et al. (2016); Graff et al. (2015). This conclusion is also consistent with the fact that current matched-filter searches for CBCs have a reduced efficiency toward signals with strong higher harmonics Harry et al. (2018); Capano et al. (2014) and this event was indeed observed in both the GstLAL and PyCBC searches.

We argue that the observed changes in parameter measurements are not due to systematic errors in the CBC waveform models. We compare results obtained with different waveform models with and without higher-order modes, including waveforms computed with NR. That leads us to believe that both the increased support for unequal masses and the decreased support for nonzero spin are robust conclusions.

We emphasize that the fact that the evidence for higher-order modes is weak does not contradict the fact that waveforms with higher-order modes lead to improved parameter measurements. In fact, accurate modeling of relevant physical effects can improve parameter measurements. This can be because said physical effect is present even if it is too weak to unequivocally detect, or because it helps exclude regions of the parameter space for which that effect would be larger than what we observe. Similar shifts in the posteriors (though in the opposite direction for the distance and inclination) where in fact observed in Kumar et al. (2018) when reanalyzing GW150914 with waveforms that include higher-order modes.

We augment the analysis using CBC waveform models with a morphology-independent analysis using BayesWave. We find broad agreement between the CBC analysis and the generic analyses regardless of whether the CBC model uses higher-order modes or not. We quantify this conclusion in terms of the overlap between the CBC and the generic reconstruction, which we find to be $\sim 0.9$, consistent with expectations for signals of this mass and SNR Abbott et al. (2016d).

Posterior samples from all analyses are available at Chatziioannou et al. (2019).

Acknowledgements.

We thank Christopher Berry, Thomas Dent, Tristano DiGirolamo, Bhooshan Gadre, Roland Haas, Nathan Johnson-McDaniel, Riccardo Sturani, and Aaron Zimmerman for helpful comments and suggestions. This research has made use of data, software and/or web tools obtained from the Gravitational Wave Open Science Center (https://www.gw-openscience.org), a service of LIGO Laboratory, the LIGO Scientific Collaboration and the Virgo Collaboration. LIGO is funded by the U.S. National Science Foundation. Virgo is funded by the French Centre National de Recherche Scientifique (CNRS), the Italian Istituto Nazionale della Fisica Nucleare (INFN) and the Dutch Nikhef, with contributions by Polish and Hungarian institutes. The Flatiron Institute is supported by the Simons Foundation. JCB acknowledges support from Australian Research Council Discovery Project DP180103155. CJH, KKYN and 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-1764464. MH was supported by Science and Technology Facilities Council (STFC) grant ST/L000962/1 and European Research Council Consolidator Grant 647839. Parts of this research were conducted by the Australian Research Council Centre of Excellence for Gravitational Wave Discovery (OzGrav), through project number CE170100004. ROS and JL are supported by National Science Foundation (NSF) PHY-1707965 and PHY-1607520. PS acknowledges support from the NWO Veni grant no. 680-47-460 and the Science and Technology Facilities Council grant ST/N000633/1. MH acknowledges support from the Swiss National Science Foundation (SNSF) Grant IZCOZ0_177057. LS was supported by the Young Elite Scientists Sponsorship Program by the China Association for Science and Technology (2018QNRC001), and partially supported by the National Science Foundation of China (11721303), and XDB23010200. The GT authors gratefully acknowledge the NSF for financial support from Grants No. PHY 1806580, PHY 1809572, and TG-PHY120016. Computational resources were provided by XSEDE and the Georgia Tech Cygnus Cluster. The RIT authors gratefully acknowledge the NSF for financial support from Grants No. PHY-1607520, No. PHY-1707946, No. ACI-1550436, No. AST-1516150, No. ACI-1516125, No. PHY-1726215. This work used the Extreme Science and Engineering Discovery Environment (XSEDE) [allocation TG-PHY060027N], which is supported by NSF grant No. ACI-1548562. Computational resources were also provided by the NewHorizons, BlueSky Clusters, and Green Prairies at the Rochester Institute of Technology, which were supported by NSF grants No. PHY-0722703, No. DMS-0820923, No. AST-1028087, No. PHY-1229173, and No. PHY-1726215. The SXS authors at Caltech acknowledge the Sherman Fairchild Foundation, and NSF grants PHY-1708212 and PHY-1708213. GL, NA, and AG are supported by NSF PHY-1606522, PHY-1654359, PHY-1654359. Computations were done using the orca cluster supported in part by PHY-1429873 and by Cal State Fullerton. AG is also supported in part by Nancy Goodhue-McWilliams. The authors are grateful for computational resources provided by the LIGO Laboratory and supported by National Science Foundation Grants PHY-0757058 and PHY-0823459. Some of the computational work for this manuscript was also carried out on the computer cluster Vulcan at the Max Planck Institute for Gravitational Physics in Potsdam. Plots have been made with matplotlib Hunter (2007) and corner Foreman-Mackey (2016).

Appendix A Differences between LALInference and RIFT

In Sec. III.1 we discuss how higher-order modes affect the posterior distributions for the various source properties of GW170729. We argue that waveform systematics are small since results with IMRPhenomHM and SEOBNRv4HM agree well with each other, and the latter agrees well with NR waveform models. The investigation of waveform systematics, though, also reveals that there is a residual small disagreement between results with SEOBNRv4HM obtained with LALInference and RIFT. We have performed extensive investigations into the nature of this disagreement and have been ultimately unable to pinpoint its origin.

LALInference and RIFT are independently-implemented codes with differences in data-handling, likelihood estimation, algorithm, etc. Despite these differences, in this work we have found good agreement between results obtained by the two algorithms for waveform approximants that do not include higher-order modes, see Tables 2 and 5. However, for waveforms with higher-order modes and in particular the direct comparison using SEOBNRv4HM, we find that the two codes produce results that differ for the mass ratio at the $7\%$ level. We also find that the two codes produce consistent results for the effective spin and the detector-frame total mass of GW170729, though we are unable to check the source-frame total mass which RIFT did not compute. See Table 5 for more estimates.

We performed a number of reanalyses of the data in order to test the effects of various differences between the two algorithms. On the RIFT side these tests include: the NR grid, the specific choice of fitting coordinates, the noise PSD calculation, the data handling, the sampling rate, the lower frequency cut-off, the Monte-Carlo integration, the likelihood evaluation, the summation of higher-order modes to get the waveform, and the time window of the analysis. More technical details about these tests are presented in O’Shaughnessy and Lange (2018). We also performed LALInference runs ignoring the detector calibration uncertainty. We found that none of these tests could account for the shift in the mass ratio posteriors.

Given that and the long history of testing and usage of LALInference, in this paper we also follow previous studies by the LIGO/Virgo Collaborations, for example Abbott et al. (2019, 2018a), and use LALInference for our main results. We do note, though, that RIFT results are qualitatively consistent and quantitatively close to LALInference and the discrepancy is only noticeable when higher-order modes are taken into account. The small residual disagreement will be the focus of future investigations.