Evidence for subdominant multipole moments and precession in mergingblack-hole-binaries from GWTC-2.1

Charlie Hoy,1 Cameron Mills,1, 2, 3 and Stephen Fairhurst1

1School of Physics and Astronomy, Cardiff University, Cardiff, CF24 3AA, United Kingdom2Albert-Einstein-Institut, Max-Planck-Institut for Gravitationsphysik, D-30167 Hannover, Germany

3Leibniz Universitat Hannover, D-30167 Hannover, Germany(Dated: November 23, 2021)

The LIGO–Virgo–KAGRA collaborations (LVK) recently produced a catalogue containinggravitational-wave (GW) observations from the first half of the third GW observing run (O3a). Thiscatalogue, GWTC-2.1, includes for the first time a number of exceptional GW candidates producedfrom merging black-hole-binaries with unequivocally unequal component masses. Since subdominantmultipole moments and spin-induced orbital precession are more likely to leave measurable imprintson the emitted GW from unequal component mass binaries, these general relativistic phenomenamay now be measurable. Indeed, both GW190412 and GW190814 have already shown conclusiveevidence for subdominant multipole moments. This provides valuable insights into the dynamicsof the binary. We calculate the evidence for subdominant multipole moments and spin-inducedorbital precession for all merging black-hole-binaries in GWTC-2.1 that were observed during O3aand show that (a) no gravitational-wave candidate has measurable higher order multipole contentbeyond ` = 3, (b) in addition to the already known GW190412 and GW190814, GW190519 153544shows significant evidence for the (`, |m|) = (3, 3) subdominant multipole, (c) GW190521 may havemeasurable subdominant multipole content and (d) GW190412 may show evidence for spin-inducedorbital precession.

I. Introduction

Between 2015 and 2017, the Advanced LIGO [1](aLIGO) and Advanced Virgo [2] (AdV) gravitational-wave (GW) detectors performed their first and sec-ond GW observing runs (O1 and O2). During thistime, the LIGO Scientific and Virgo collaborations(LVC) announced GWs originating from 10 binary-black-hole (BBH) mergers [3–9] and a single binaryneutron star [10]. Additional compact binary candi-dates have also been reported by other groups [11–14].

Two important general relativistic effects that werenot clearly observed during O1 and O2 were higher or-der multipoles [15] (though see Refs. [16, 17] for a dis-cussion regarding marginal evidence for higher ordermultipoles in GW170729 [9]) and spin-induced orbitalprecession [9, 18]. Higher order multipoles are termsbeyond the dominant quadrupole when a GW is ex-panded into multipole moments with spherical polarcoordinates defined in the source frame [19]. Thesehigher order multipoles are present for all binary sys-tems, but they are typically at a much lower amplitudethan the quadrupole [see e.g. 20]. Spin-induced orbitalprecession arises when there is a misalignment betweenthe orbital angular momentum and the spins of eachcompact object leading to characteristic modulationsin the amplitude and phase of the observed GW sig-nal [21]. Including both higher order multipoles andprecession in waveform models that are used to infer

source properties through Bayesian inference [see e.g.22–24] can improve parameter measurement accuracyand provide additional constraints on the in-plane spincomponents of the binary [see e.g. 25–29]. The impor-tance of both of these effects increase as the binary’smass ratio (q = m1/m2 ≥ 1) increases [20, 29–36].Clear evidence for asymmetric masses was absent inthe binaries detected during O1 and O2 [9], makingthe observation of either precession and higher ordermultipoles challenging.

An initial analysis of the first 6 months of data fromthe first half of the third GW observing run (03a) bythe LIGO-Scientific, Virgo and KAGRA collaborations(LVK) revealed a further 39 GW candidates describedin the second GW catalogue (GWTC-2) [37]. Subse-quent searches over the same period revealed a num-ber of subthreshold candidates [38, 39], with most re-cently the extended second GW catalogue, (GWTC-2.1), increasing the number of high-significance GWcandidates observed during O3a to a total of 44 [39].

In contrast to O1 and O2, several events in O3ahad unequivocally unequal masses. First among theseis GW190412 [27], with a mass ratio of ∼4:1. Theunequal mass ratio resulted in more significant higherorder multipoles, and for the first time, imprints of sub-dominant multipole radiation oscillating at three times

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the orbital frequency – the (`,m) = (3, 3) multipole1

– were visible. Similarly, it was the first time that theamount of precession in the system was constrainedaway from the prior [27, 38, 40–42]. Several monthslater GW190814 was detected with highly asymmet-ric component masses (∼9:1) and a secondary compo-nent with a mass larger than any previously discov-ered neutron star and lighter than any black hole [28].GW190814 had significant evidence of the (3,3) mul-tipole [28, 42] and the most restrictive measurementof the in-plane spin components of any event observedto date. A combination of the higher order multipolesand the lack of evidence for precession reduced the un-certainty on the mass of the smaller object.

By comparing parameter estimates that were ob-tained with waveform models that a) included higherorder multipoles and b) excluded higher order multi-poles beyond ` = 2, it has previously been hinted thatseveral GW signals in O3a show possible evidence forhigher order multipoles [37]. Similarly, by comparingthe posterior and prior distributions for parameterscharacterising spin-induced orbital precession, it haspreviously been shown that no event in O3a unam-biguously exhibits spin-induced orbital precession [37],although there is evidence that the observed popula-tion of BBHs have spins which are misaligned with theorbital angular momentum [43]. Other studies have in-vestigated the evidence for higher order multipoles andprecession for GW190412 and GW190814 [27, 28, 40–42], but there has not yet been a study which identi-fies the evidence for precession and higher order multi-poles for the latest gravitational-wave observations onan event by event basis.

In this paper, we take advantage of the multipoledecomposition for identifying the presence of higher or-der multipoles [20] and the two-harmonic formalism foridentifying the presence of precession [18, 29, 44] to es-tablish if any BBH candidates in GWTC-2.1, that wereobserved during O3a, exhibit evidence for higher ordermultipoles and precession. We calculate the signal-to-noise ratio (SNR) in the (`, |m|) ∈ {(2, 1), (3, 3), (4, 4)}multipoles and from precession for the latest BBH ob-servations and compare it to the expected distributionfrom noise. We include GW190814 since it is mostlikely (71%) the result of a BBH merger [28]. We showthat,

1. There is minimal evidence for GW emission in thesubdominant multipole moments beyond ` = 3,

1 (`,m) should everywhere be read as (`, |m|) unless otherwiseindicated.

2. GW190814 has the largest evidence for the(`,m) = (3, 3) multipole for all events in O3awith SNR in the (3, 3) multipole ρ33 = 6.2+1.3

−1.5,

3. GW190814, GW190412 and GW190519 153544show significant evidence for the (3, 3) subdomi-nant multipole moment,

4. GW190521 may show evidence for subdomi-nant multipole moments. The reanalysis ofGW190521 by Nitz et al. [38] suggests evidencefor the (`,m) = (3, 3) multipole while the initialanalysis by the LVK shows minimal evidence,

5. GW190412 may show evidence for spin-inducedorbital precession: two independent analyses in-fer marginal evidence while two other indepen-dent analyses infer significant evidence,

6. GW190929 012149 and GW190915 235702 showmarginal evidence for subdominant multipolemoments and precession respectively.

This paper is structured as follows: Section II de-tails the method used to infer the presence of higherorder multipoles and precession in the observed GWdata. We provide a brief overview of the two-harmonicapproximation and a summary of how the SNR fromprecession and each subdominant multipole is calcu-lated. We then explain how we construct the ex-pected noise distribution for both measurements. InSection III we present our results and indicate whichGW events show evidence for subdominant multipolesand precession. We then conclude with a summary anddiscussion of future directions.

II. Method

a. Calculating the SNR in precession and highermultipoles

In general relativity, GWs are fully described by twopolarizations. These polarizations can be decomposedinto multipole moments using the −2 spin-weightedspherical harmonic orthonormal basis [45]. Coalescingcompact binaries (CBCs) predominantly emit radia-tion at twice the orbital frequency in the leading or-der (2,2) multipole. The most important subdominantmultipole for most CBCs is the (3,3) multipole, thoughthe (4,4) multipole can be more significant for binarieswhose components have comparable masses [20].

Previous studies have identified evidence for sub-dominant multipole moments by either a) calculatingBayes factors, the difference in Bayesian evidences,through multiple parameter estimation studies [e.g.

3

16, 40, 41] or via likelihood re-weighting [15], b) statis-tically comparing posteriors obtained with waveformmodels which included higher order multipoles andthose which excluded higher order multipoles beyond` = 2 [46], c) analysing time–frequency tracks in theGW strain data [27, 47], d) identifying if there is a lossin the observed coherent signal energy when compar-ing the output from the cWB detection pipeline [48]with predictions from a waveform model which ex-cludes subdominant multipole moments [28], or e) di-rectly calculating the SNR of each (`,m) multipoleρ`m [20, 27, 28, 49]. Here, we identify whether mul-tipoles other than the dominant (2,2) multipole havebeen observed by calculating the orthogonal SNR ofeach (`,m) multipole. This is achieved by decompos-ing a GW signal into each subdominant multipole, ex-tracting the component that is orthogonal to the (2, 2)quadrupole and calculating the SNR associated withthe resulting waveform.

A binary with spin angular momenta S1 and S2

undergoes spin-induced orbital precession when the to-tal spin S = S1 + S2 of the binary is misaligned withthe Newtonian orbital angular momentum L. In mostcases, precession of the orbital plane leads to L pre-cessing around the approximately constant J = L+S,leading to characteristic modulations in the emittedGW signal [21, 50].

Previous studies have identified evidence for spin-induced orbital precession by either a) calculatingBayes factors [e.g. 27, 28, 40, 51], b) statistically com-paring posterior distributions for parameters charac-terizing precession to their prior distributions [e.g.9, 46] or c) directly calculating the precession SNR,described as the contribution to the total SNR of thesystem that can be attributed to precession [18, 27–29, 36, 44, 46]. In this paper, we identify if spin-inducedorbital precession has been observed from a GW signalby calculating the precession SNR. The precession SNRρp is calculated by decomposing the (2,2) quadrupoleinto non-precessing harmonics and isolating the SNRcontained in the second most significant harmonic. Ifρp is small, the amplitude of the second harmonic is in-significant and any beating of the harmonics is negligi-ble. For this case, we would observe a GW signal whichlooks like the dominant non-precessing harmonic. Theprecession SNR has been shown previously to accu-rately identify whether spin-induced orbital precessionhas been observed in simulated GW signals [29, 36].

We follow the method introduced in Refs. [20, 29]and calculate ρ`m and ρp from the inferred propertiesof each compact binary merger in O3a. To do thiswe use posterior samples from the GWTC-2.1 datarelease [52]. For GW candidates not included in the

GWTC-2.1 data release, samples from the GWTC-2data release [53] are used. Further details about thespecific posterior samples used are in Appendix A. Weuse the parameters of each sample to generate the lead-ing order precession and higher multipole contributionsto the waveform and calculate ρ`m and ρp for the net-work at the time of the event. This calculation usesthe conversion module publicly available in PESum-mary [54].2

The formalism for ρ`m and ρp were initially de-veloped for waveform models containing higher ordermultipoles [20] or precession [18, 44] respectively, butnot both. In this paper, we apply them to posteriorsamples obtained with a gravitational waveform modelcontaining both higher multipoles and precession. Weheuristically justify this by noting that, in most cases,both precession and higher multipoles are small cor-rections to the leading order gravitational waveform.It is therefore reasonable to expect that the preces-sion correction to the higher multipoles will be an evensmaller effect which can be safely ignored. We demon-strated that this is indeed the case in Ref. [44]. Briefly,a GW signal containing only the (`,m) = (2, 2) mul-tipole can be written as a sum of 5 non-precessingharmonics, h22 = A22,0h

22,0 + A22,1h22,1 + . . ., where

A22,n is the (complex) amplitude of the nth harmonich22,n which depends upon the orientation of the sig-nal. The amplitude of each harmonic scales with bn

where b = tanβ/2 and β is the opening angle (thepolar angle between L and J). The characteristic am-plitude and phase modulations associated with preces-sion are therefore caused by the beating of these 5 non-precessing harmonics. Since for the majority of casesb � 1, a GW signal containing only the ` = 2 mul-tipoles can approximately be written as a sum of thetwo leading harmonics, h22 ≈ A22,0h

22,0 + A22,1h22,1.

When the GW signal includes other multipoles, theycan be decomposed similarly. For example, we canexpress h33 ≈ A33,0h

33,0 + A33,1h3,1 where A33,n is

the amplitude of the nth harmonic h33,n. As before,the amplitude of the precession harmonics scale as bn.Furthermore, the overall amplitude of the (3,3) multi-

2 When computing ρ`m we generate a non-precessing waveform(i.e. set the in-plane spin components to zero) and calculatethe higher multipole SNR for that waveform, which is notexactly the identical to the prescription above but, providedthat the b � 1, any differences will be small. In particular,precession adds a secular drift in the phase evolution of thewaveform. However, as we only use the waveform to calculatethe expected SNR in the higher multipoles (i.e. we do notmatched filter against the data), this small phase differencewill not impact the value of ρ33.

4

pole is typically much lower than the (2,2) multipole.Therefore, to a good approximation, we can write thewaveform as h ≈ A22,0h

22,0 + A22,1h22,1 + A33,0h

33,0.The (`,m) = (4, 4) precession multipoles can be addedin a similar fashion, although their amplitude is gener-ally smaller than the (3,3) and can often be neglected.When computing ρp throughout this paper, we onlyconsider the precession power in the leading (2, 2) mul-tipole. Similarly, when calculating the power in highermultipoles, we consider only the contribution from,e.g., h33,0 and neglect the precession corrections.

b. Calculating the expected distribution of ρpand ρ`m in the absence of a signal

In order to assess the significance of precession andhigher order multipoles, we compare the inferred ρp

and ρlm distributions to the expected distribution fromnoise. Since the statistical properties of ρp and ρlm aresimilar, the expected noise distribution has the same

functional form for both measurements. Below we sum-marize the derivation of the common noise distribution(parameterized by ρ which denotes either ρp or ρlm)and we refer the reader to Refs. [20, 29] for furtherdetails.

We consider a gravitational waveform comprisinga dominant contribution h0, the leading precessionharmonic of the (2,2) multipole, and a single, addi-tional, subdominant contribution h1 arising either froma higher multipole or from precession. The gravita-tional waveform can be written as

h = A0(λ)h0(λ) +A1(λ)h1(λ) (1)

where Ai(λ) are overall amplitudes which depend uponthe orientation of the binary, and λ denotes the param-eters of the system [20, 44]. The GW likelihood maythen be factorized into two components: one describingthe contribution from the dominant harmonic, Λ0(λ),and another describing the contribution from the sub-dominant harmonic, Λ1(λ).

p(d|λ) ∝ exp

(−1

2〈d− [A0(λ)h0(λ) +A1(λ)h1(λ)] |d− [A0(λ)h0(λ) +A1(λ)h1(λ)]〉

)(2)

∝ exp

(A0(λ)〈d|h0(λ)〉 − |A0(λ)|2

2〈h0(λ)|h0(λ)〉

)× exp

(A1(λ)〈d|h1(λ)〉 − |A1(λ)|2

2〈h1(λ)|h1(λ)〉

)=: Λ0(λ)× Λ1(λ) .

In the second line, we have absorbed the (constant)〈d|d〉 term into the proportionality, and we have as-sumed that the dominant and subdominant harmonicsare orthogonal 〈h0|h1〉 = 0.3

The phase evolution of the gravitational waveform,encoded in h0,1(λ), is determined by the masses andaligned spin components of the binary. Since the dom-inant harmonic has the largest SNR, its measurementwill primarily be used to determine the evolution ofthe waveform. An observation of the sub-dominantharmonic will provide a small improvement to the evo-lution of the waveform. However, for simplicity, weneglect it in the following discussion. In this case, thesubdominant harmonic h1 is known and only the over-all amplitude and phase, encoded in A1 remain to be

3 The calculation can be fairly simply generalized to the casewhere the harmonics are not orthogonal by simply replacingh1 by h⊥1 , the component of h1 orthogonal to h0 [20].

determined. Furthermore, the value of A1 will typi-cally be unconstrained by the observation of A0 — inthe case of precession, both the amplitude and phaseof A1 are free as they depend upon the in-plane spins,while for higher multipoles the amplitude and phaseof A1 will depend upon the orientation and mass ratioof the binary which are generally not precisely mea-sured. Therefore, in the simplest approximation, wecan simply maximize the likelihood Λ1(λ) over A1,

Λ1(λ)max = exp

((ρMF

1 )2

2

)(3)

where the matched filter SNR, ρMF1 is defined as

(ρMF1 )2 =

[(s|h1)2 + (s|ih1)2

]|h1|2

. (4)

(ρMF1 )2 will be chi-squared distributed with two de-

grees of freedom in the absence of signal, and non-centrally chi-squared distributed in the presence of a

5

signal. Using the maximum likelihood, we argue, seeRefs. [20, 44], that a threshold of ρ1 ≥ 2.1 is a require-ment for observation of precession or higher multipoles.

Alternatively, we can marginalize the likelihood,Λ1(λ) over the unknown phase to obtain a likelihoodas a function of ρ1 as

Λ1(ρ1) ∝ I0(ρMF1 ρ1) exp

(− (ρMF

1 )2 + ρ21

2

). (5)

Here, I0 is the Bessel function of the first kind.The expected posterior distribution for ρ is, there-

fore,

p(ρ1|d) ∝ p(ρ1) Λ1(ρ1) , (6)

where p(ρ) is the prior distribution for ρ. For the caseof uniform priors on the complex amplitude A1, p(ρ|d)takes the form of a non-central χ distribution with 2degrees of freedom with non-centrality parameter equalto ρMF

1 .To obtain a better prediction for the posterior dis-

tribution p(ρ1|d), we can use the measurement of thedominant harmonic, h0, to place a more informativeprior on the strength of the sub-dominant harmonic.For instance, an observation of a close to equal massor face-on binary reduces the expectation of observinghigher harmonics or precession. We construct an in-formed prior for p(ρ1) using results, where available4,from a parameter estimation analysis that includesonly the dominant multipole. The informed prior iswhat results from calculating p(ρ1|d) in Eq.6 while tak-ing Λ1(ρ1) = 1.5

We estimate p(ρ1|d) for a specific noise realizationby combining the informed prior with the likelihoodfrom Eq. 5, where the value of ρMF

1 is randomly drawnfrom the χ distribution with 2 degrees of freedom. Thisprocedure is repeated 100 times to obtain different re-alizations of the noise. We take the median of thesedistributions as our estimate of the expected noise dis-tribution. Refs. [20, 29] demonstrated that this proce-dure can result in predicted posterior distributions thatmore accurately resemble the inferred distribution. In

4 See Appendix A for details.5 For precession there are additional parameters that must be

marginalized over which are not inferred with dominant multi-pole models: the precession parameter χp [55] and the preces-sion phase. However, the inference of aligned spin and massratio does provide additional constraints on these parameters,and so rather than assuming the default prior on these pa-rameters, we condition on the measured aligned spin and massratio.

particular, for the majority of events, the predictedamplitude of the subdominant waveform is small and,therefore, the predicted noise distribution is shifted to-wards lower values than predicted by using a uniformprior on A1.

c. Assessing the significance of precession andhigher multipoles

To assess the significance of precession and highermultipoles, we compare the distributions for ρ`m andρp obtained from the posterior samples (discussed inSection II a), with the expected noise distribution (dis-cussed in Section II b), by calculating the Jensen-Shannon divergence DJS [56] between the inferred pos-terior and expected noise distribution6; D`m

JS and DpJS

compare the calculated ρ`m and ρp distributions to theexpected noise distributions respectively.The Jensen-Shannon divergence is designed to quantify the differ-ence between probability distributions. The Jensen–Shannon divergence has the property that it is sym-metric under interchange of the two distributions andalways has a finite value: DJS ∈ [0, ln(2)]7. WhenDJS = 0 (DJS = ln(2)), the distributions A and B areidentical (significantly different).

We also calculate the probability that the observed,or lack of, evidence for higher multipoles and precessionis consistent with noise by finding the value of ρMF

1 ,which results in a noise distribution that resembles theinferred posterior for ρ`m or ρp. We can then calculatethe probability of obtaining that value of ρMF

1 or larger,due to noise alone. In practice, we can find the valueof ρMF

1 by finding the specific noise distribution whichminimizes the Jensen-Shannon divergence between thenoise and inferred posterior distribution.

Based on a fiducial noise and ρp distribution, wefound that a Jensen-Shannon divergence of 0.15 (0.1)maps to 5% (10%) probability that the observed ev-idence for precession is consistent with noise. Wetherefore isolate events with a DJS > 0.15 as show-ing significant evidence for subdominant multipole mo-ments and precession, since the null hypothesis thatthe distribution is caused by noise can be rejected ata 5% confidence interval. We argue that events with

6 The Jensen-Shannon Divergence has been used in previousGW analyses, see e.g. Ref. [37].

7 In this work, the Jensen-Shannon divergence is defined interms of the natural logarithm. If instead the base 2 loga-rithm is used, the Jensen-Shannon divergence will be boundedby DJS ∈ [0, 1].

6

a 0.1 < DJS < 0.15 show marginal evidence, since thenull hypothesis can only be rejected at a 10% confi-dence interval. Other works, see e.g. Ref. [37], usea Jensen-Shannon divergence of 0.05 as a thresholdfor showing a difference between distributions. Ourrequirements for significant evidence of subdominantmultipole moments and precession is therefore stricterthan those used in previous papers.

III. Results

Table I presents a summary of the main results.For all of the events in GWTC-2.1, we give the SNRin the (3,3) multipole and the sub-dominant precessionharmonic. Where possible, we also provide the Jenson-Shannon divergence between the observed distributionand the predicted noise distribution. Figure 1 showsthe inferred posteriors for ρ33 and ρp for all events con-sidered in this study. In both the Table and Figures,we also include the re-analysis of GW190521 by Nitzet al. [38].

Across the majority of the parameter space, theexpected SNR in the (3,3) multipole is largest, withthe only exception being for binaries with a) large to-tal mass – where the power in the (4,4) is larger –and b) close to equal mass components – where the(3,3) multipole vanishes [20]. Consequently, it is unsur-prising that the (3,3) multipole is the most significantsubdominant multipole for every event except one inO3a. GW190910 112807 is the sole exception, havinginferred ρ33 = 0.6+1.3

−0.6 and ρ44 = 1.0+0.5−1.0, both of which

are consistent with noise with Jenson–Shannon diver-gences D33

JS = 0.01 and D44JS = 0.06 respectively. As ex-

pected, GW190910 112807’s source has close to equalmass components, q = m1/m2 = 1.22+0.48

−0.20, and has

relatively large total mass M = m1 +m2 = 79.6+9.3−9.1. It

also has significant support for an edge-on orientation,where the relative amplitude of the (4,4) multipole islargest.

Several events have an SNR in the (3,3) multi-pole which is clearly above the expectation for noisealone. Indeed, the observed distribution for the pop-ulation, shown in Figure 1, shows a clear high-SNRtail that indicates the observation of the (3,3) mul-tipole. Higher multipoles have previously been iden-tified in both GW190814 and GW190412, with theirobservability and their impact on parameter estimatesdiscussed at length in previous works [27, 28, 40–42].Unsurprisingly, we see that among all events in O3a,GW190412 and GW190814 have the largest SNR inthe (3, 3) multipole, with ρ33 = 6.2+1.3

−1.5 for GW190814

and ρ33 = 3.5+0.8−1.2 for GW190412. Both events show

Event ρ33 D33JS ρp Dp

JS

GW190403 051519 1.4+1.3−1.1 - 0.3+0.9

−0.2 -GW190408 181802 0.5+1.1

−0.5 0.02 1.0+1.8−0.9 0.03

GW190412 3.5+0.8−1.2 0.34 3.0+1.6

−1.5 0.26GW190413 134308 0.7+1.2

−0.6 0.0 0.7+1.5−0.6 0.04

GW190413 052954 0.5+1.1−0.5 0.02 0.6+1.4

−0.5 0.01GW190421 213856 0.4+0.9

−0.4 0.01 0.7+1.4−0.6 0.03

GW190426 190642 0.9+1.9−0.8 - 0.2+0.5

−0.2 -GW190503 185404 0.8+1.3

−0.7 0.01 0.8+1.8−0.7 0.03

GW190512 180714 1.1+1.1−0.9 0.02 0.8+1.6

−0.7 0.01GW190513 205428 1.1+1.4

−1.0 0.02 0.8+1.6−0.6 0.01

GW190514 065416 0.4+1.0−0.4 0.0 0.5+1.2

−0.4 0.03GW190517 055101 0.7+1.3

−0.7 0.01 1.0+2.0−0.8 0.02

GW190519 153544 2.3+1.5−1.8 0.20 1.0+1.9

−0.7 0.07GW190521 1.2+2.4

−1.1 - 0.7+1.4−0.6 -

GW190521 074359 1.0+1.5−0.9 0.01 1.6+2.5

−1.2 0.09GW190527 092055 0.6+1.1

−0.5 0.0 0.7+1.7−0.6 0.01

GW190602 175927 0.8+1.4−0.8 0.0 0.5+1.0

−0.4 0.01GW190620 030421 1.1+1.5

−1.0 0.01 0.8+1.7−0.6 0.01

GW190630 185205 1.0+1.2−0.9 0.01 1.0+1.8

−0.8 0.02GW190701 203306 0.5+1.0

−0.4 0.0 0.5+1.0−0.4 0.0

GW190706 222641 1.5+1.5−1.3 0.07 0.5+1.1

−0.4 0.02GW190707 093326 0.4+0.8

−0.3 0.0 0.7+1.5−0.6 0.0

GW190708 232457 0.4+0.9−0.3 0.0 0.7+1.5

−0.6 0.0GW190719 215514 0.7+1.2

−0.6 0.0 0.6+1.5−0.5 0.01

GW190720 000836 0.5+0.9−0.4 0.0 0.6+1.2

−0.5 0.01GW190725 174728 0.5+1.0

−0.5 - 1.0+1.9−0.8 -

GW190727 060333 0.5+1.2−0.4 0.0 0.7+1.6

−0.6 0.02GW190728 064510 0.5+1.2

−0.4 0.0 0.7+1.3−0.6 0.01

GW190731 140936 0.5+1.1−0.4 0.0 0.5+1.3

−0.4 0.01GW190803 022701 0.4+0.9

−0.4 0.0 0.6+1.4−0.5 0.01

GW190805 211137 0.6+1.1−0.6 - 0.5+1.2

−0.4 -GW190814 6.2+1.3

−1.5 0.68 1.8+1.6−1.2 0.02

GW190828 063405 0.5+1.0−0.4 0.0 0.9+1.6

−0.8 0.01GW190828 065509 1.2+1.0

−1.0 0.04 1.0+1.9−0.8 0.03

GW190910 112807 0.6+1.3−0.6 0.01 0.8+1.6

−0.7 0.02GW190915 235702 0.5+1.0

−0.5 0.01 1.5+2.4−1.2 0.13

GW190916 200658 0.8+1.3−0.7 - 0.5+1.2

−0.4 -GW190924 021846 0.5+1.0

−0.5 0.02 0.5+1.1−0.5 0.01

GW190925 232845 0.4+1.2−0.4 - 0.7+1.3

−0.6 -GW190926 050336 0.8+1.3

−0.7 - 0.7+1.7−0.6 -

GW190929 012149 2.0+1.6−1.5 0.10 0.9+2.0

−0.7 0.06GW190930 133541 0.4+1.0

−0.3 0.0 0.6+1.2−0.5 0.01

GW190521 Nitz 2.4+2.2−2.0 - 1.1+2.8

−0.9 -

TABLE I. Table showing the SNR in the (`,m) = (3, 3)multipole moment ρ33 and the SNR from precession ρp forall BBH candidates observed in GWTC-2.1 [39] plus thereanalysis of GW190521 by Nitz et al. [38]. For each eventwe show two Jensen Shannon Divergences (JSDs); D33

JS andDp

JS compare the calculated ρ33 and ρp distributions to theexpected noise distributions respectively. Events with alarger JSD show greater evidence for higher order multi-poles and/or precession. For events where the JSD andρ33/ρp could not be calculated we add a hyphen. Whereapplicable we report the median values along with the 90%symmetric credible intervals.

7

0 2 4 6 8 100.00

0.25

0.50

0.75

1.00

1.25

1.50

Prob

abilit

y De

nsity

GW190412GW190519_153544GW190521GW190814GW190915_235702GW190929_012149GW190521 Nitz

0 1 2 3 4 5 60.0

0.5

1.0

1.5

2.0

2.5

Prob

abilit

y De

nsity

GW190412GW190519_153544GW190521GW190814GW190915_235702GW190929_012149

FIG. 1. Plot showing the Left : ρ33 and Right : ρp distributions for all observations in the second GW catalogue (grey).In red we show the ρ33 and ρp distribution averaged across events. In black we show the average of the median expectednoise distribution for Left: higher multipoles and Right: precession. Events which are discussed in the text are colored.

significant differences (as characterized by the Jenson–Shannon divergence) between the observed ρ33 dis-tribution and that predicted from noise. The nexttwo most significant events are GW190519 153544 andGW190929 012149. Based upon the (3,3) SNR andthe divergence of the distributions, GW190519 153544shows significant evidence for the existence of the (3,3)multipole, but GW190929 01214 only shows marginalevidence. The inferred SNR in the (3, 3) multipolefor GW190521 is dependent on the parameter estima-tion samples used: ρ33 = 1.2+2.4

−1.1 and ρ33 = 2.4+2.2−2.0 for

the LVK and Nitz et al. samples respectively. Con-sequently, GW190521 may have an observable (3, 3)multipole. We discuss these five events in detail in thefollowing Sections. For the other events, there is mini-mal evidence for power in the higher multipoles.

The evidence for precession is weaker than forhigher multipoles. There are only three events whichshow any deviation from the expected noise distribu-tion and, overall, the population distribution for ρp isconsistent with that expected from a non-precessingpopulation. GW190412 has ρp = 3.0+1.6

−1.5 and DpJS =

0.26, both of which indicate evidence for precession inthe system. While GW190814 has the second-largestprecession SNR, ρp = 1.8+1.6

−1.2, the observed distribu-tion is consistent with zero precession (Dp

JS = 0.02).There is marginal evidence that GW190915 235702has measurable precession, with ρp = 1.5+2.4

−1.2 andDp

JS = 0.13. We discuss these three events in moredetail in the following Sections. For the other events,there is no evidence for precession. The lack of observ-able precession does not necessarily mean that mostevents in O3a have aligned-spins but rather it meansthat if the binaries were precessing, the imprint of pre-

cession on the observed signal is not strong enough tobe observed with the current detector sensitivities.

a. GW190814

GW190814 is the most unequal mass ratio binaryobserved in O3a. The component masses were inferredto be 23.2+1.1

−1.0M� and 2.59+0.08−0.09M� which makes the

secondary component mass either the heaviest neutronstar or lightest black hole ever recorded. Previously,GW190814 was found to have significant evidence forsubdominant multipole moments owing to the unequalcomponent masses [28].

Unsurprisingly, we infer that GW190814 has themost significant measurement of ρ33 in O3a. We findthat the inferred ρ33 measurement is inconsistent withnoise since it’s Jensen-Shannon divergence is signifi-cantly larger than 0 and is the largest for any event inO3a, D33

JS = 0.68.Although GW190814 has the second largest ρp,

there is no evidence for precession in the observed GWsignal since Dp

JS = 0.02. The inferred ρp measurementcan be reproduced from noise in 28% of cases, as shownin Figure 2. The lack of evidence for precession isexpected given the near-zero precession measurement,χp = 0.04+0.04

−0.03 [28] (we parameterize precession by theeffective precession spin parameter 0 ≤ χp ≤ 1 [55]).GW190814 is an example where a large ρp does notalways correlate with observable precession since therelatively large SNR can trivially be reproduced fromnoise. GW190814’s noise distribution is shifted to largeSNR because, as explained in Ref. [29], there exists acorrelation between ρp and the binary’s mass ratio: ρp

8

FIG. 2. ρp distributions for First row : GW190412, Sec-ond row : GW190915, Third row : GW190814. The blueline shows the expected distribution of ρp in a stretch ofnoisy data under the assumption that the source is non-precessing, ρNp . The blue shaded region shows the 1σ un-

certainty of ρNp , the grey dotted line shows the expected ρNpdistribution in 10% of cases and the black dashed line showsthe expected ρNp distribution in First row : 1%, Second row :7% and Third row : 28% of cases. The black dashed linewas calculated by finding the value of ρMF

1 which minimizedthe Jensen-Shannon divergence between the ρNp and the in-ferred ρp posterior. The dashed red and blue lines in theFirst row show the ρp and ρNp distributions calculated usingthe samples from Colleoni et al. [40].

is larger for more unequal mass binaries, assuming con-stant χp. This correlation between ρp and χp meansthat GW190814’s informed prior is peaked at largerρp values than average, leading to a noise distributionwhich allows for larger ρp. Consequently, the inferred

Analysis ρ33 D33JS ρp Dp

JS

LVK 3.5+0.8−1.2 0.34 3.0+1.6

−1.5 0.26Colleoni et al. 3.5+1.1

−1.2 - 2.5+1.8−1.4 0.12

Nitz et al. 3.5+1.3−1.3 - 2.4+1.9

−1.3 -Zevin et al. 3.6+0.9

−1.2 - 3.0+1.7−1.6 -

TABLE II. Table as in Table I but showing only the inferredposteriors and Jensen-Shannon divergences for GW190412.We compare analyses from the LVK [27, 53], Colleoni etal. [40, 57], Nitz et al. [38] and Zevin et al. [58, 59]. Wecalculate ρ33 and ρp for Zevin et al. by using posteriorsamples obtained from the “Model A” analysis since thepriors are the same as those used in Ref. [27]. We equallycombined the posterior samples obtained with the SEOB-NRv4PHM [60] and IMRPhenomPv3HM [61] waveformmodels as was done in Ref. [27].

ρp distribution for GW190814 is well contained withinthe 1σ noise uncertainty and a low Jensen-Shannon di-vergence is recovered. GW190814 demonstrates the ef-ficacy of using the Jensen-Shannon divergence to inferthe presence of precession.

The lack of observable precession in GW190814 im-plies that it’s source is either non-precessing or weare unable to observe the precession at current detec-tor sensitivities. This is a similar conclusion to thatstated in Ref. [36] which highlighted that a precessingGW190814-like system with in-plane spin 0 < χp < 0.1is indistinguishable from a non-precessing system basedon the difference in Bayesian evidence.

b. GW190412

GW190412 was the first detection of a BBH withconclusively unequal component masses: 30.1+4.6

−5.3M�and 8.3+1.6

−0.9M� and the first observation where sub-dominant multipole moments were clearly observed.GW190412 was also the first observation where an in-formative precession measurement was inferred, withthe posterior deviating significantly from the prior [27].Several groups later re-analysed GW190412 and foundsimilar results [38, 40, 41, 58].

We infer that GW190412 has the second most sig-nificant measurement of ρ33 in O3a. We find thatthe inferred ρ33 measurement is inconsistent withnoise with D33

JS = 0.34, which suggests that the in-clusion of higher order multipoles will significantlychange the inferred posterior density estimates, a re-sult consistent with parameter estimation studies fromRefs. [27, 40, 41]. The inferred ρ33 measurement isrobust since it remains approximately constant for dif-

9

0.4 0.8

2.5

5.0

2.5 5.0

LVKColleoni et al.Nitz et al.Zevin et al.

FIG. 3. Corner plot comparing the inferred χp and ρp forGW190412 from the LVK [27, 53], Colleoni et al. [40, 57],Nitz et al. [38] and Zevin et al. [58, 59]. Shading shows the1σ, 3σ and 5σ confidence intervals.

ferent independent analyses, see Table II8.The evidence for precession in GW190412 is depen-

dent on which Bayesian analysis is considered. We cal-culate that GW190412 shows significant evidence forprecession when using data from the initial analysisconducted by the LVK [27]. We show that the inferredρp measurement is inconsistent with noise since theJensen-Shannon divergence is Dp

JS = 0.26, which im-plies that the inferred ρp measurement can only bereproduced from noise in 1% of cases, see Figure 2.The low probability of recovering the inferred ρp mea-surement from noise implies that GW190412 may haveoriginated from a binary system with spins misalignedwith the total orbital angular momentum.

When using data produced from a re-analysis ofGW190412 using the latest suite of Phenomenologicalwaveform models (PhenomX [63–67]) [40], GW190412shows marginal evidence for precession since the in-ferred ρp is smaller than that reported by the LVKand can be reproduced from noise in ∼ 12% of cases.The smaller ρp is a consequence of inferring a lower

8 Islam et al. [41] also re-analysed GW190412 using the NR-Sur7dq4 waveform model [62] but their samples are not pub-licly available and therefore not included in this work.

in-plane spin, as shown in Figure 3. Colleoni et al. [40]calculated the Bayes factor in favour of precession andfound B ∼ 2, which shows positive but no substantialevidence for precession [68]. The Bayes factor agreeswith the conclusion from comparing ρp to the expectednoise distribution, as expected since it has previouslybeen shown that there exists a strong relationship be-tween the ρp and the Bayes factor [29, 36].

Nitz et al. [38] and Zevin et al. [58] also re-analysedGW190412 but these additional analyses did not pro-duce publicly available posterior samples for aligned-spin higher order multipole waveform models. The lackof aligned-spin posterior samples means that noise dis-tributions, and hence evidences for precession, couldnot be calculated. However, the inferred ρp and χp

distributions from Nitz et al. and Zevin et al. arecomparable to the measurements reported in Colleoniet al. and the LVK respectively, see Figure 3. Thisagreement implies that our analysis would likely in-fer marginal and significant evidence for precession forNitz et al. and Zevin et al. respectively.

Of those considered, Colleoni et al. was the only in-dependent analysis which reported a definitive Bayesfactor in favour of precession. The LVK mentionedthat log10 B was smaller or comparable to the sys-tematic uncertainties of order unity. By utilizing theapproximately linear relationship between lnB andρ2

p/2 [29, 36], we can approximate the Bayes factor infavour of precession from the LVK posterior samples.The relation between lnB and ρ2

p/2 implies that an in-crease in ρp from ∼ 2.5 to ∼ 3.0 leads to the Bayes fac-tor increasing by a factor of ∼ 4. We therefore approxi-mate that the Bayes factor in favour of precession fromthe LVK analysis is B ∼ 8 (log10 B = 0.9). While thisis consistent with the conclusions from the LVK, thisBayes factor may also be interpreted as showing sub-stantial evidence for precession according to Ref. [68].The larger χp inferred from the LVK and Zevin et al.is therefore a tell-tale sign for a larger observability ofprecession and larger Bayes factor [29].

Colleoni et al. and Nitz et al. both used the IMR-PhenomXPHM waveform model [65] for the Bayesianinference while the LVK and Zevin et al. used a combi-nation of the IMRPhenomPv3HM [61] and SEOB-NRv4PHM [60] waveform models. This suggests thatthe differences we see between interpretations could ei-ther be a consequence of waveform systematics, dif-ficulties in sampling the complex parameter space orsampler differences.

Understanding if GW190412 originated from a pre-cessing binary has substantial implications on under-standing the preferred BBH formation channel [see e.g.69, for a review] since a clear observation of a precess-

10

ing binary is considered a “smoking gun” for decipher-ing between the two favoured formation mechanisms:dynamical where the spin orientation is anticipated tobe uniform [70] and hence precession is expected, andisolated where the spin orientation is expected to bepreferentially aligned with a small opening angle [seee.g. 71–73] and thus little to no precession is expected.

c. GW190519 153544 and GW190929 012149

GW190519 153544 originated from a binary withrelatively high total mass: 50% posterior probabil-ity for M > 100M�, and has spins preferentiallyaligned with the orbital angular momentum withχeff = 0.31+0.20

−0.22. Ref. [46] found that the analysis ofGW190519 153544 benefits significantly from the in-clusion of higher order multipoles (see Figure 13 ofRef. [46]), indicating possible evidence for subdominantmultipole moments. GW190929 012149 also has rela-tively large total mass M� = 104.3+34.9

−25.2M� and is con-

sistent with unequal component masses: 80.8+33.0−33.2M�

and 24.1+19.3−10.6M�.

We infer that GW190519 153544 shows evidencefor higher order multipoles with D33

JS = 0.20, thoughless significantly than both GW190814 and GW190412.There is a 1.5% probability of recovering the inferredρ33 posterior from noise, see Figure 4. It is the ex-tra likelihood from the (3,3) multipole that resultsin narrower posteriors for mass ratio and inclination.The mass ratio is constrained more tightly betweenq = 1.17−2.34, compared with q = 1.04−2.80 from the(2,2) multipole alone. With the inclusion of higher har-monics, the inclination peaks at edge-on, rather thanface-on/face-off. What’s more, the tighter constraintsin inclination and mass ratio lead to improved con-straints on other properties of the system that are cor-related, such as the effective spin χeff [74–77], totalmass, distance, redshift, source frame masses and po-larization angle.

GW190929 012149 shows marginal evidence forhigher-order multipoles with D33

JS = 0.10. Ascan be seen in the right-hand panel of Figure 4,GW190929 012149’s ρ33 posterior can be recoveredfrom noise in 10% of cases. Inclusion of higher har-monics does not improve the constraint on inclination,but does result in a slightly tighter constraint on massratio with q = 1.16− 6.77 becoming q = 1.34− 5.28.

Both GW190519 153544 and GW190929 012149show no evidence for precession, as the inferred ρp isconsistent with noise (Dp

JS ∼ 0.07). No evidence forprecession is expected, since both GW190519 153544and GW190929 012149’s precession measurement are

consistent with the prior. GW190519 153544 ob-serves a larger ρp than GW190929 012149 sinceGW190519 153544’s inclination peaks at edge-on,where precession is easier to measure [see e.g. 29], whileGW190929 012149 is consistent with face-on/face-off.

d. GW190521

GW190521 is the most massive detection inGWTC-2.1. An initial analysis conducted by the LVKdemonstrated that GW190521 was the first evidenceof a new population of black holes that resist straight-forward interpretation as supernovae remnants, withat least one black hole lying firmly in the pulsationalpair-instability mass gap (∼ 65 − 120M�) [51, 78]. Itwas found that GW190521 was consistent with compo-nent masses 85+21

−14M� and 66+17−18M�. Nitz et al. [79]

later challenged this view, showing that it is possible toobtain parameter estimates consistent with componentmasses that instead straddle this gap. They found thatwhen using a uniform in mass-ratio prior, GW190521’smass posterior was multi-model with additional modesat larger mass ratio, q ∼ 6 and q ∼ 10. Prior con-straints on the mass ratio imposed by the initial analy-sis [37, 51, 78] ruled out any possibility of sampling thishigh mass ratio region of the parameter space. It waslater discovered that the waveform approximant usedby Nitz et al. did not accurately account for possibilityof transitional precession [21, 38]. Nitz et al.’s alter-native interpretation of GW190521 was therefore laterrevised in Ref. [38] with the high mass ratio q ∼ 10peak no longer significantly supported, while the modeat q ∼ 6 remained. GW190521 may therefore haveoriginated from either a near equal mass system, wherethe SNR in both the (3, 3) multipole [20] and preces-sion [29] are expected to be small, or an unequal massratio system, where it is likely that higher order multi-pole and precession effects could be directly measured.

We infer that GW190521 has a measurable (3, 3)multipole if we use the posterior samples obtainedfrom Nitz et al. (ρ33 = 2.4+2.2

−2.0) otherwise the in-ferred ρ33 is consistent with Gaussian noise (ρ33 =1.2+2.4

−1.1). Figure 5 shows that this difference is be-cause Nitz et al. infers a non-equal mass ratio system,q = 1.8+2.8

−0.6 [38], while the analysis from the LVK infers

that q = 1.3+1.2−0.3 [37, 51, 78]. It is the extra likelihood

from the measurement of the (3,3) multipole that iskey to the Nitz et al. reinterpretation of GW190521.Unfortunately, we are unable to construct an informedprior for GW190521 as there are no publicly availablesamples generated from a precessing non-higher ordermultipole waveform model.

11

FIG. 4. ρ33 for Left : GW190519 153544 and Right : GW190929 012149. The blue line shows the expected distribution ofρ33 in a stretch of noisy data, ρN33. The blue shaded region shows the 1σ uncertainty of ρN33, the grey dotted line showsthe expected ρN33 distribution in 10% of cases and the black dashed line shows the expected ρN33 distribution in Left : 1.5%and Right : 10% of cases. The black dashed line was calculated by finding the ρMF

1 which minimized the Jensen-Shannondivergence between the ρN33 and the inferred ρ33 posterior.

5 10

2.5

5.0

7.5

2.5 5.0 7.5

Nitz et al.LVKNitz et al.LVKNitz et al.LVK

FIG. 5. Corner plot showing the inferred mass ratio and ρ33for the reanalysis of GW190521 by Nitz et al. [38] comparedto the results from the LVK [37, 51, 53, 78]. Shading showsthe 1σ, 3σ and 5σ confidence intervals.

GW190521 has the largest inferred precession mea-surement of any event observed so far with χp =

0.68+0.26−0.44 and χp = 0.5+0.31

−0.33 as reported by theLVK [37, 51, 78] and Nitz et al. respectively. De-spite this, GW190521 shows no evidence for preces-sion, with ρp = 0.7+1.4

−0.6 and ρp = 1.1+2.8−0.9 respectively.

The lack of observable precession is unsurprising giventhat GW190521 is the largest mass event observed todate and merges at a lower frequency. Just four in-spiral wave cycles (two orbits) are visible in the detec-tors’ sensitive frequency band. As a result, GW190521is decomposed into two near parallel non-precessingharmonics, meaning that there is very little power or-thogonal to the dominant harmonic and ρp is small asa result. Several explanations for the large χp havebeen suggested, including possible evidence for eccen-tricity [80, 81] and head-on collisions [82].

Since the evidence for higher order multipole con-tent is dependent on which Bayesian analysis is consid-ered, we perform matched filtering [83–85] to directlyextract the SNR in the (3, 3) multipole from the GWstrain data [86, 87]. Although matched filtering is astandard procedure when searching for potential GWsignatures when there is an accurate model of the grav-itational waveform available [see e.g. 88–100], only theSNR in the dominant (2, 2) multipole is typically ex-tracted [see e.g. 9, 11, 37, 38, 101]. In order to extractthe SNR in the (3, 3) multipole we simply calculatethe matched filter SNR in the (3, 3) multipole that isconsistent with the masses and spins of the templatewhich gives the largest SNR in the dominant (2, 2)quadrupole. As with previous studies, we ignore theeffects of spin-induced orbital precession.

We find that GW190521 best matches a domi-nant (2, 2) quadrupole waveform with a detector-framechirp mass M = 114.33M�, mass ratio q = 1.0005and spins aligned with the orbital angular momentumS1z = S2z = 0.01. Both the detector-frame chirp massand aligned spin components match those from a fullBayesian analysis [38, 51, 78]. This template gives a

12

FIG. 6. Left : SNR time series around the time of GW190521 for the preferred template with detector-frame chirp massM = 114.33M�, mass ratio q = 1.0005 and spins aligned with the orbital angular momentum S1z = S2z = 0.01. Thecoloured circles show the position of the peak SNR as seen in each detector. Right : Region of the inclination ι and massratio q parameter space consistent with the inferred matched filter SNR in the (3, 3) multipole. Red contours show the 50%and 90% credible intervals consistent with the initial analysis conducted by the LVK [51, 78] and the black contours andblack dots show the 50% and 90% credible intervals and posterior samples from the Nitz et al. analysis [38] respectively.The red star and black arrow show the maximum likelihood samples from the LIGO–Virgo and Nitz et al. analysesrespectively. The dark green and light green regions show the 1σ and 2σ uncertainty on ρ33.

network matched filter SNR in the (2, 2) multipoleof ρMF

22 = 13.94 and SNR in the (3, 3) multipole ofρMF

33 = 2.81, see the left panel of Figure 6. The ρMF33

implies that GW190521 has the third largest SNR inthe (3, 3) multipole for any event in O3a. The matchedfilter SNR in the (3, 3) multipole is consistent with theinferred matched filter SNR from a full Bayesian anal-ysis using a precessing higher order multipole wave-form [51, 78], ρ = 14.2+0.3

−0.3, since the missing powercan be attributed to the inferred SNR from precessionpresented in Table I,

ρp ≈√ρ2 − (ρMF

22 )2 − (ρMF33 )2 (7)

0.3+2.7−0.3 =

√(14.2+0.3

−0.3)2 − 13.942 − 2.812.

The left panel of Figure 6 shows a peak in the SNRtime series of the (3, 3) multipole ∼ 0.02s after themaximum SNR in the (2, 2) multipole. This is due tothe large overlap between the (3, 3) and (2, 2) multipoleat this time owing to the higher frequency evolutionof the (3, 3) compared to the (2, 2). At the time ofGW190521, the (2, 2) and (3, 3) multipoles are closeto orthogonal with an overlap of 0.1.

Since the SNR in the (3, 3) multipole is stronglydependent on the inclination angle and mass ratio of

the system [see e.g. 20], we bound the region of pa-rameter space consistent with ρMF

33 . The right panel ofFigure 6 shows that the inferred matched filter SNRρMF

33 = 2.81 is consistent with a close-to-equal massratio system viewed edge-on or a high mass ratio sys-tem viewed face-on/face-off. In general, we find thatthe close-to-equal mass ratio peak inferred from Nitzet al. is consistent with the matched filter SNR in the(3, 3) multipole with the majority of posterior sampleslying within the ρMF

33 uncertainty. As a result of theLVK analysis preferring lower mass ratios and inclina-tion angles, where the power in the (3, 3) multipole isreduced, the majority of samples are inconsistent withthe inferred matched filter SNR. This implies that, un-like the LVK, Nitz et al. were able to measure thesubdominant multipole content. We also find that theq ∼ 6 peak inferred from Nitz et al. seems to be in-consistent with the inferred ρ33 since it prefers a muchlarger ρ33 owing to the large inclination angle.

Understanding if GW190521 has a measurable (3,3) multipole is key for understanding the binary’s for-mation mechanism. If GW190521 has a measurable(3,3) multipole, it cannot have originated from an equalmass ratio system. The preferred formation mecha-nism has been investigated previously with some au-thors suggesting that GW190521 may be a result of ahierarchical merger [102] owing to possible evidence for

13

eccentricity of the binaries orbit [80, 81] (although theinitial LVK analysis found no conclusive evidence thatGW190521 resulted from a hierarchical merger [51]).The right panel of Figure 6 implies that the matchedfilter SNR in the (3, 3) multipole is consistent withbinary of q >∼ 1.5. This constraint on the mass ratioimplies that if GW190521 originated from a hierarchi-cal merger, it is more likely a result of merger betweena second generation black hole (2G; the remnant of afirst generation black hole merging with another firstgeneration black hole) and a first generation black hole(1G) as 2G+2G mergers likely have mass ratios closeto 1 while 1G+2G likely have mass ratios close to 2 [seee.g. 103].

e. GW190915 235702

GW190915 235702 is likely to have originatedfrom a BBH with component masses 35.3+9.5

−6.4M�and 24.4+5.6

−6.1M�. The LVK analysis found thatGW190915 235702 had an informative precession mea-surement with the posterior deviating from the prior,see Figure 11 of Ref. [37].

We infer that GW190915 235702 has no evidencefor subdominant multipole moments with D33

JS = 0.01.The lack of evidence for subdominant multipole mo-ments is consistent with the findings from the LVKanalysis which found that the effect of higher modesis either negligible or subdominant to the systematicsbetween precessing non-higher order multipole wave-forms [37].

GW190915 235702 has the second-largest evidenceof precession in O3a with Dp

JS = 0.13. As shown inFigure 2 we find that there is a 7% probability ofrecovering the inferred ρp posterior from noise. Wefind that the extra likelihood from precession results inan inclination measurement constrained away from theprior θJN = 0.9+0.5

−0.6 (folded between [0, π/2]). We notethat an aligned-spin non-higher order multipole anal-ysis was used to calculate the noise distribution in theabsence of precession in the signal, but since there isno evidence for subdominant multipole moments, Dp

JSis likely to be robust.

IV. Discussion

We calculated the inferred SNR in the sub-dominant multipole moments, ρlm (for (`, |m|) ∈{(2, 1), (3, 3), (4, 4)}, and from precession, ρp, for allBBH candidates in GWTC-2.1 that were observed dur-ing O3a. We determined which events show evidence

for subdominant multipole moments and precessionby comparing the inferred SNRs with predicted dis-tributions expected from noise alone. We found thatmost BBHs in O3a show minimal evidence for sub-dominant multipole moments, but there are a few no-table exceptions. GW190412, GW190519 153544 andGW190814 all show significant evidence for a (3,3)multipole and GW190929 012149 shows marginal ev-idence for the (3, 3) multipole. We also found thatno BBH observed in O3a shows significant evidencefor higher order multipole content beyond ` = 3. Wethen discussed GW190521 and indicated that the GWstrain data is consistent with an observable (3, 3) mul-tipole but the inferred SNR from posterior samples isdependent on which Bayesian analysis is considered.We then discussed evidence for spin-induced preces-sion and again found that most BBHs in O3a showno evidence for precession. However, we found thatGW190412 may have originated from a precessing bi-nary system, but the evidence for precession is depen-dent on which Bayesian analysis is considered. We alsodemonstrated that GW190915 235702 shows marginalevidence for precession.

The method we have demonstrated here is straight-forward, and clearly identifies the evidence for subdom-inant multipole moments and precession from the ob-served GW signal. In the future we wish to expandthis method and calculate the power in the second po-larization as this can also be important in breakingparameter degeneracies [104]. As discussed briefly inSection III d, we also wish to calculate the subdominanthigher multipole and precession SNR directly from theGW strain data. In principle, this should enable theconstruction of predicted posteriors including the ef-fects of precession and higher multipoles using pos-teriors computed with a simpler waveform model. Asimilar method to this has been suggested for highermultipoles [15]. There, the authors demonstrated thatre-weighting posteriors inferred with a (2,2) only wave-form model based on the full likelihood could result inposteriors that closely match those inferred with wave-form models including higher order multipoles.

V. Acknowledgements

We are grateful to Duncan Brown, Davide Gerosa,Bernard Schutz and Vivien Raymond for discus-sions during C. Hoy’s and C. Mills’s Ph.D. defenceswhere this work was first presented. We also thankMark Hannam and Jonathan Thompson for usefuldiscussions and N V Krishnendu for comments onthis manuscript. This work was supported by Sci-

14

ence and Technology Facilities Council (STFC) grantST/N005430/1 and European Research Council (ERC)Consolidator Grant 647839 and we are grateful forcomputational resources provided by Cardiff Univer-sity and LIGO Laboratory and supported by STFCgrant ST/N000064/1 and National Science FoundationGrants PHY-0757058 and PHY-0823459.

This research has made use of data, software and/orweb tools obtained from the Gravitational Wave OpenScience Center (https://www.gw-openscience.org), aservice of LIGO Laboratory, the LIGO Scientific Col-laboration and the Virgo Collaboration. LIGO isfunded by the U.S. National Science Foundation. Virgois funded by the French Centre National de RechercheScientifique (CNRS), the Italian Istituto Nazionaledella Fisica Nucleare (INFN) and the Dutch Nikhef,with contributions by Polish and Hungarian institutes.

Plots were prepared with Matplotlib [105], Cor-ner (https://corner.readthedocs.io) [106] and PESum-mary [54]. Functions within PyCBC [96] were used toperform the matched filtering described in Section III dand LALSuite [107], NumPy [108] and Scipy [109]were used during the analysis.

A. Posterior samples used

For all calculations we used posterior samples re-weighed to a flat-in-comoving-volume prior to remainconsistent with the results in Refs [37, 39]. For themajority of events we used the same posterior sam-ples as those published in GWTC-2 and GWTC-2.1 (the “PublicationSamples” and “PrecessingSpin-IMRHM comoving” datasets respectively). In caseswhere these datasets did not correspond to sam-ples obtained with a precessing higher-order multi-pole approximant we used the “C01:SEOBNRv4PHM”dataset which includes posterior samples obtained withthe SEOBNRv4PHM [60] (precessing and higher-

order multipole) approximant for both analyses9.Since we calculate the inferred ρ`m and ρp with sam-

ples obtained from a precessing higher order multipolewaveform model, we calculate the informed prior us-ing samples obtained with a precessing non-higher or-der multipole and aligned-spin higher order multipolewaveform model in order to ensure that the noise distri-bution is not biased by the absence of precession andhigher order multipoles respectively. Although bothRef. [37] and Ref. [39] performed parameter estima-tion using multiple models, Ref. [39] only analysed eachcandidate with precessing higher order multipole wave-form models while Ref. [37] analysed each candidatewith aligned-spin and precessing waveform models, seeTable VIII of [37].

Due to the lack of samples, we are unable to cal-culate an informed prior, and hence noise distribution,for candidates specific to Ref. [39]. For candidates de-scribed in Ref. [37] we were able to use samples ob-tained with a precessing non-higher order multipolewaveform model (the “PrecessingSpinIMR” dataset)to calculate the informed prior for ρ`m but, becausenot every candidate was analysed with an aligned-spinhigher order multipole waveform model, we generallyused samples obtained with an aligned-spin non-higherorder multipole waveform model to calculate the in-formed prior for ρp (the “AlignedSpinIMR” dataset).We expect that using an aligned-spin non-higher ordermultipole waveform model will not cause a significancedifference in the obtained noise distribution for theabsence of precession, since for the majority of casesthe power from higher order multipoles is expected tobe small and therefore parameter estimates compara-ble. For GW190412 and GW190814, both of whichexhibit strong evidence for subdominant multipole mo-ments [27, 28], we were able to use samples obtainedwith an aligned-spin higher order multipole waveformmodel (the “AlignedSpinIMRHM” dataset) to calcu-late the expected noise distribution for the absence ofprecession.

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